User:Sesquihypercerebral/sandbox

= Title and Introduction =

A Theory Of Primordial Inflation.

This theory postulates that:

1. Dark Energy has two components, the zero point energy density of the vacuum and the zero point curvature of space time. Although both these values are fixed they are incomparable without a conversion factor.

If the conversion factor is variable then Dark Energy is variable.

2. The no-boundary condition requires that proper time began at a temporal pole. At a temporal pole $$ \ g_{tt}=0 \ $$.

3. Unruh radiation causes Dark Energy to reduce, otherwise conservation of energy is violated.

4. A limit on the amount of matter and/or radiation that can fit into a given quantity of space would cause inflation to roll slowly.

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This conversion factor is a function of the temporal scale factor, and varies with it.

Introduction:

This is a theory of Cosmological Inflation. It attempts to explain: Dark Energy, fine tuning, and why inflation rolled slowly. It also attempts to clarify the no-boundary condition.

Abstract:

This Theory postulates that:

Inflation happened before the big bang.

Dark energy has two components, one positive and one negative. Its positive component is the zero point energy density of the vacuum. Its negative component is the zero point curvature of space time, subject to a conversion factor. This conversion factor is a function of the temporal scale factor, and varies with it.

Unruh radiation causes dark energy to dump matter and radiation into the vacuum. As it does so dark energy reduces in strength due to conservation of energy.

The amount of matter and/or radiation that can fit into a given quantity of space has a maximum value. The energy density of this maximum is considerably short of the energy density of the vacuum catastrophe.

Dark energy can only reduce by dumping matter and/or radiation into the vacuum. Once full, space would have to expand before more matter and/or radiation could be dumped into the vacuum. This would limit the rate at which dark energy reduced. Full Space would make Inflation roll slowly.

= The Theory =

Re-writing Einstein's Field Equations
$$\text{ }$$

$$\text{A modified Friedmann Lemaitre Robertson Walker Metric, using spherical co-ordinates, can be written. }$$


 * $$ c_{}^2\mathrm{d}\tau_{s}^2 = c_{*}^2(t) \mathrm{d}t^2 - {a}^2(t)

\left( \frac{\mathrm{d}r^2}{1-k r^2} + r^2 \mathrm{d}\theta^2 + r^2 \sin^2 \left( \theta \right) \, \mathrm{d}\phi^2 \right) $$


 * $$\mbox{Where:}$$


 * $$\ r,\theta(theta), \mbox{ and  } \phi(phi)\text{ are the polar co-ordinates of space.}$$


 * $$\ t\mbox{ is the co-ordinate of time.}$$


 * $$\ a\mbox{ is the spatial scale factor and is taken to be a function of time.}$$


 * $$\ c_{*} \text{ is the temporal scale factor and is taken to be a function of time.}$$


 * $$\ c_{} \text{ is a constant and is the limiting value of } c_{*} \text{. It may be equated with Maxwell's constant.}$$


 * $$\ \tau_{s} \text{(tau subscript s) is the proper time of the line element.}$$


 * $$\ \tau_{c} \text{(tau subscript c) is cosmological time, i.e. the proper time for any world line with constant spatial co-ordinates. }

$$


 * $$k \text{ is a real number; positive, negative, or zero, giving the co-ordinate curvature. Values other than -1,0, and +1, are subject to the fine tuning problem.}

$$


 * $$\kappa(kappa) \text{ is the gravitational parameter and is taken to be a function of the temporal scale factor.}$$


 * $$\Lambda(Lambda) \text{ is the cosmological parameter and is taken to be a function of the temporal scale factor.}$$


 * $$\ \overset{{\ }_{\boldsymbol{\circ}}}{\rho} (rho\ zero\ point) \text{ is the zero point energy density of the vacuum.} $$


 * $$\ \overset{{}_{\boldsymbol{\circ}}}{\Lambda} (Lambda\ zero\ point) \text{ is the zero point curvature of space. } $$


 * $$\chi (chi) \text{ is an arbitary parameter that is useful for calculations. } $$

$$ \text{ Note on Cosmological time. }$$

$$ \text{ Since }\tau_{c} \text{ is defined as the proper time for any world line with constant spatial co-ordinates, it follows that } $$

$$ \ \ c_{}d \tau_{c} = c_{*}dt \ \ \therefore \ \ \frac{d\tau_{c}}{dt} = \frac{c_{*}}{c_{}} $$

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The manifold has no boundary. Co-ordinate time had no beginning. Proper time had a beginning. It follows that proper time began at a temporal pole

At a temporal pole the temporal scale factor would have to be zero. We can define the zero point energy density of the vacuum to be the value of dark energy when the temporal scale factor was zero. It follows that the conversion factor would also be zero when the temporal scale factor was zero.

_______________________________________________________________________________________________

Re-writing Einstein's Field Equations
$$\text{ }$$ $$ \text{ }$$

$$ \text{Einstein's field equations may be written }$$

$$ -G_{\mu \nu} - \Lambda g_{\mu \nu} = \kappa T_{\mu \nu}$$

$$ \text{If the cosmological and gravitational constants are replaced with parameters dependant upon } c_{*} \text{ then}$$

$$ -G_{\mu \nu} - \Lambda(c_{*}) g_{\mu \nu} = \kappa(c_{*}) T_{\mu \nu}$$

$$ \text{If Einstein's field equations are then rewritten using zero point values the results are.}$$

$$ -G_{\mu \nu} + \overset{{}_{\boldsymbol{\circ}}}{\Lambda} g_{\mu \nu} = \kappa(c_{*}) \left( T_{\mu \nu} + \overset{{\ }_{\boldsymbol{\circ}}}{\rho} g_{\mu \nu} \right)$$

$$ -G_{\mu \nu} + \overset{{}_{\boldsymbol{\circ}}}{\Lambda} g_{\mu \nu} = \kappa(c_{*}) T_{\mu \nu} + \kappa(c_{*}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} g_{\mu \nu} $$

$$ \text{Subtracting the second equation from the fourth gives}$$

$$ \overset{{}_{\boldsymbol{\circ}}}{\Lambda}g_{\mu \nu} + \Lambda(c_{*}) g_{\mu \nu} = \kappa(c_{*}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho}g_{\mu \nu}  $$

$$\Lambda(c_{*}) g_{\mu \nu} = \kappa(c_{*}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho}g_{\mu \nu} - \overset{{}_{\boldsymbol{\circ}}}{\Lambda}g_{\mu \nu} $$

$$\Lambda(c_{*}) =  \kappa(c_{*})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} - \overset{{}_{\boldsymbol{\circ}}}{\Lambda} $$

$$ \text{If } c_{} \text{ is defined as the limiting value of } c_{*} \text{ such that } \Lambda(c_{})=0 \text{ then }$$

$$ \kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} = \overset{{}_{\boldsymbol{\circ}}}{\Lambda} $$

$$\Lambda(c_{*}) =  \kappa(c_{*})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} - \kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho}  =  \left(  \kappa(c_{*}) - \kappa(c_{})  \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho} $$

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The Temporal Pole.
The surface of a sphere has no boundary. With a system of co-ordinates of latitude and longitude it has two poles. Lines of latitude begin and end at the poles

The simplest case of a curved space-time is a manifold of two spatial dimensions of unit uniform positive cuvature. This is the surface of a Euclidean sphere of unit radius. If $$ \theta $$ (theta) is used to represent latitude and $$ \phi $$ (phi) is used to represent longitude then the line element equation is   $$ 	\ dS^2 = d{\theta}^2 + sin^2({\theta})d{\phi}^2 \ $$ The non-zero values of the metric are $$ 	g_{\theta\theta} =1 \ and \ g_{\phi\phi} = sin^2({\theta}) $$ The poles occur where $$ g_{\phi\phi} = 0 $$. At the poles there is latitude but no longitude.

the south pole is no different from any of the surrounding points. the manifold has no boundary, yet northernness has a beginning.

Likewise a temporal pole occurs when $$ \ g_{tt} = 0 \ $$ $$ $$

two prong ( see Sean Carroll ) the temporal pole is at a mid-range value of t, say, for convenience, t = 0

white hole the temporal pole is at $$ t = -\infty $$

In both cases, proper time can be finite without the manifold having a boundary.

At a temporal pole the temporal scale factor is zero.

Einstein's Field Equations In Terms Of Zero Point Values.

To present this theory it will be necessary to rewrite Einstein's field equations in terms of zero point values.

Presenting The Story Itself:

In the begining, the temporal scale factor was zero. The line element had only spatial components. No events were causally connected. The universe was a point, a quantum point. Extention without structure, duration without activity. This was the primordium. The uncaused cause, the begining of all things. If you want to know the value of entrophy at the begining of time then count the quantum states. The gravitational parameter was infinite, its reciprocal was zero. The zero point energy density of the vacuum was unopposed. The vacuum catastrophe was in full swing. As the effective scaler field dumped matter and radiation into the vacuum the temporal scale factor rose, the gavitational parameter reduced, and the effective scaler field began to go down. Instantly space was full, limiting Unruh radiation thereby causing inflation to roll slowly.

The first Friedmann equation give us the Hubble parameter

The second Friedmann equation gives us the derivative of the Hubble parameter with respect to cosmological time.

By taking the derivative of the first Friedmann equation with respect ot cosmological time we can equate ........ and thereby determine Chi.

We arrive at the surprising result that the parameter Chi is directly proportional to cosmological time.

Because Chi is an angle that can only vary from 0 to pi/2 this shows us that the period of primordial inflation was finite.

Conclutions and questions:

Although this theory is presented with a gavitational parameter that is the ratio of the spacial curvature divided by energy density it would make more sense to have a convertion factor that was the ratio of the energy density divided by the curvature of space time. That way the convertion factor would begin at zero rather than infinity

= End of Theory =

Spoof Advertisement
Does your cosmology violate conservation of energy but you don't really know what to do about it?

Then why not try the new, improved, Friedmann Lemaitre Robertson Walker Metric with Non-uniform Time.

Non-uniform Time allows conservation of energy to be violated in a controlled and predictable way leaving you safe to enjoy your Free Lunch.

Professor X writes: "I was pondering the hypothesis that the end state of the universe is a warm de-sitter space when I realised that this would not conserve energy. Here was a scalar field dumping energy into the vacuum but staying exactly the same.  I just couldn't get my Boltzmann's Brain around it.  Then I discovered Non-uniform Time.  Non-uniform Time produces an extra term in the second Friedmann equation that deals with non-conservation of energy in a simple and straight forward way.  I'd forgotten that there was a second Friedmann equation."

And this from a student: "I was trying to get the zero point energy density of the vacuum to match the zero point curvature of space time to better than one part in a million million million million million million million million million million million million million million million million million million million million without actually being equal. It seemed impossible.  But with Non-uniform Time all I needed to do was make the gravitational parameter dependent upon the temporal scale factor and I could just sit back and let Unruh Radiation do the matching for me, simple.

'''Non-uniform Time. You can't have a Free Lunch without it.'''

A Theory Of Eternal Inflation.

This theory attempts to explain: why inflation happened, why it rolled slowly, and why it stopped.

In the process it: explans Dark Energy, shows us how to live with The Vacuum Catastrophe,

and side-steps the problem of Fine Tuning.

All this using only: General Relativity, Unruh Radiation, Non-uniform Time, and the concept of Full Space.

Preliminary

Noether's theorem tells us that conservation of energy is a consequence of the uniformity of time, therefore A Free Lunch needs Non-uniform Time.

For the purpose of this theory, Full Space is space where the density of matter and/or radiation is at a maximum and that this maximum in independant of the amount of vacuum energy.

It is also assumed that the zero point energy density of the vacuum exceeds the density of Full Space.

A factor too small would fail to cause enough inflation.

A factor too large would be problematic for baryogenisis.

These constraints leave a wide range of possible values. Something like 2 to 63 orders of magnitude.

Assumptions.
Non-uniform Time. The Vacuum Catastrophe. Full Space. Space is full when it has an energy density excluding vacuum energy well short of the Vacuum Catastrophe. Unruh Radiation (or some other such mechanism whereby vacuum energy produces matter and/or radiation).

Energy is not conserved and therefore time is not uniform.

The Zero Point Energy Density Of The Vacuum has a catastrophically high value.

The Zero Point Curvature Of Space has a correspondingly high value.

The Gravitational Parameter is variable and is dependent upon the Temporal Scale Factor.

Conclusions.
we arrive at the seemingly perverse idea that non-uniform time is an inherent property of the manifold while uniform time is no more than an imposed perspective. An important perspective none the less in that this is where energy is conserved. unifrom time is where energy lives. conservation of energy simplifies the mathematics

A variable temporal scale factor produces an extra term in the second Friedmann equation. The first Friedmann equation is unaffected.

A significant restraint on the amount of energy density that can be held in the vacuum in the form of matter and radiation prolongs the period of inflation.

The Big Bang was preceded by (Non)Eternal Inflation which lasted for at most not much more than a million million years in proper time, although it might have been an infinite amount of co-ordinate time.

How much of the zero point energy density of the vacuum is used to oppose the zero point curvature of space time is dependant upon the gravitational parameter. The surplus is Left Over Vacuum Energy.

All the matter and radiation in the universe is made from Left Over Vacuum Energy.

Abstract
The universe began with inflation. Inflation started off as the vacuum catastrophe.

During the period of inflation the gravitational parameter fell from its initial high value and the temporal scale factor increased from its initial low value.

A significant limit on the amount of energy density that can be held in the vacuum in the form of matter and radiation caused the period of inflation to be prolonged and inflation to continue until this limit was no longer a restraint.

The density and pressure of matter and radiation stayed roughly constant at their maximum values. Space was full.

It is possible that from the end of inflation or shortly thereafter the temporal scale factor and the gravitational parameter have had almost fixed values.

Proportionally, the cosmological parameter can be expected to have changed by a more significant amount.

Note: The cosmological parameter is proportional to the Left Over Vacuum Energy.

Energy is extracted from the vacuum through changes in the gravitaltional parameter.

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Theory:

Preliminaries:

The New Metric.

To present this theory it will be necessary to generalise the FLRW metric by introducing a temporal scale factor. The temporal scale factor will be represented by "c subscript *". Because of the temporal scale factor, cosmological time will be different from co-ordinate time. As with the standard metric, cosmological time is taken to be the proper time along any world line having constant spatial co-ordinates. Cosmological time will be represented by "tau subscript c".

Insert the new metric with glossary

Re-writing Einstein's Field Equations
$$\text{ }$$

$$\text{A modified Friedmann Lemaitre Robertson Walker Metric, using spherical co-ordinates, can be written. }$$


 * $$ c_{}^2\mathrm{d}\tau_{s}^2 = c_{*}^2(t) \mathrm{d}t^2 - {a}^2(t)

\left( \frac{\mathrm{d}r^2}{1-k r^2} + r^2 \mathrm{d}\theta^2 + r^2 \sin^2 \left( \theta \right) \, \mathrm{d}\phi^2 \right) $$


 * $$\mbox{Where:}$$


 * $$\ r,\theta(theta), \mbox{ and  } \phi(phi)\text{ are the polar co-ordinates of space.}$$


 * $$\ t\mbox{ is the co-ordinate of time.}$$


 * $$\ a\mbox{ is the spatial scale factor and is taken to be a function of time.}$$


 * $$\ c_{*} \text{ is the temporal scale factor and is taken to be a function of time.}$$


 * $$\ c_{} \text{ is a constant and is the limiting value of } c_{*} \text{. It may be equated with Maxwell's constant.}$$


 * $$\ \tau_{s} \text{(tau subscript s) is the proper time of the line element.}$$


 * $$\ \tau_{c} \text{(tau subscript c) is cosmological time, i.e. the proper time for any world line with constant spatial co-ordinates. }

$$


 * $$k \text{ is a real number; positive, negative, or zero, giving the co-ordinate curvature. Values other than -1,0, and +1, are subject to the fine tuning problem.}

$$


 * $$\kappa(kappa) \text{ is the gravitational parameter and is taken to be a function of the temporal scale factor.}$$


 * $$\Lambda(Lambda) \text{ is the cosmological parameter and is taken to be a function of the temporal scale factor.}$$


 * $$\ \overset{{\ }_{\boldsymbol{\circ}}}{\rho} (rho\ zero\ point) \text{ is the zero point energy density of the vacuum.} $$


 * $$\ \overset{{}_{\boldsymbol{\circ}}}{\Lambda} (Lambda\ zero\ point) \text{ is the zero point curvature of space. } $$


 * $$\chi (chi) \text{ is an arbitary parameter that is useful for calculations. } $$

$$ \text{ Note on Cosmological time. }$$

$$ \text{ Since }\tau_{c} \text{ is defined as the proper time for any world line with constant spatial co-ordinates, it follows that } $$

$$ \ \ c_{}d \tau_{c} = c_{*}dt \ \ \therefore \ \ \frac{d\tau_{c}}{dt} = \frac{c_{*}}{c_{}} $$

_______________________________________________________________________________________________

The cosmological constant and zero point values
$$ \text{ }$$

$$ \text{Einstein's field equations may be written }$$

$$ -G_{\mu \nu} - \Lambda g_{\mu \nu} = \kappa T_{\mu \nu}$$

$$ \text{If the cosmological and gravitational constants are replaced with parameters dependant upon } c_{*} \text{ then}$$

$$ -G_{\mu \nu} - \Lambda(c_{*}) g_{\mu \nu} = \kappa(c_{*}) T_{\mu \nu}$$

$$ \text{If Einstein's field equations are then rewritten using zero point values the results are.}$$

$$ -G_{\mu \nu} + \overset{{}_{\boldsymbol{\circ}}}{\Lambda} g_{\mu \nu} = \kappa(c_{*}) \left( T_{\mu \nu} + \overset{{\ }_{\boldsymbol{\circ}}}{\rho} g_{\mu \nu} \right)$$

$$ -G_{\mu \nu} + \overset{{}_{\boldsymbol{\circ}}}{\Lambda} g_{\mu \nu} = \kappa(c_{*}) T_{\mu \nu} + \kappa(c_{*}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} g_{\mu \nu} $$

$$ \text{Subtracting the second equation from the fourth gives}$$

$$ \overset{{}_{\boldsymbol{\circ}}}{\Lambda}g_{\mu \nu} + \Lambda(c_{*}) g_{\mu \nu} = \kappa(c_{*}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho}g_{\mu \nu}  $$

$$\Lambda(c_{*}) g_{\mu \nu} = \kappa(c_{*}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho}g_{\mu \nu} - \overset{{}_{\boldsymbol{\circ}}}{\Lambda}g_{\mu \nu} $$

$$\Lambda(c_{*}) =  \kappa(c_{*})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} - \overset{{}_{\boldsymbol{\circ}}}{\Lambda} $$

$$ \text{If } c_{} \text{ is defined as the limiting value of } c_{*} \text{ such that } \Lambda(c_{})=0 \text{ then }$$

$$ \kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} = \overset{{}_{\boldsymbol{\circ}}}{\Lambda} $$

$$\Lambda(c_{*}) =  \kappa(c_{*})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} - \kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho}  =  \left(  \kappa(c_{*}) - \kappa(c_{})  \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho} $$

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A Theory Of Inflation
$$\text{ }$$

$$ \text{From appendix B2 The two Friedmann equations are} $$

$$ H^2 = -\frac{kc^2}{a^2} + \frac13 c_{}^2 \Lambda(c_*) + \frac13 c_{}^2\kappa(c_*)\rho$$

$$ \dot{H} = \frac{kc^2}{a^2} - \frac12 c_{}^2\kappa(c_*) \left( p + \rho \right) $$

$$\text{If } \Lambda(c_*) \text{ is replaced by zero point values.}$$

$$ H^2 = -\frac{kc^2}{a^2} + \frac13 \left( c_{}^2 \kappa(c_*) - c_{}^2\kappa(c) \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 c_{}^2\kappa(c_*)\rho $$

$$ \dot{H} = \frac{kc^2}{a^2} - \frac12 c_{}^2\kappa(c_*) \left( p + \rho \right) $$

$$ \text{If the assumtion is made that the co-ordinate curvature is zero, or has a negligible effect, then the equations can be simplified accordingly} $$

$$ H^2 = \frac13 \left( c_{}^2 \kappa(c_*) - c_{}^2\kappa(c) \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 c_{}^2\kappa(c_*)\rho $$

$$ \dot{H} = - \frac12 c_{}^2\kappa(c_*) \left( p + \rho \right) $$

Applying Full Space
$$\text{ }$$

$$ H^2 = \frac13 \left( c_{}^2 \kappa(c_{*}) - c_{}^2\kappa(c_{}) \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 c_{}^2\kappa(c_{*})\rho^{full} $$

$$ \dot{H}= - \frac12 c_{}^2\kappa(c_{*}) \left( p^{full} + \rho^{full} \right) $$

parameterise
$$\text{ }$$

$$ \text{If } \ \ \chi \ \text{ is defined such that} \ \ \kappa(c_{*}) = \kappa(c_{}) \ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}   \  \sin^{-2}{(\chi)} \ \ \text{ then }$$

second friedmann equation
$$\text{ }$$

$$ \dot{H}= - \frac12 c_{}^2\kappa(c_{*}) \left( p^{full} + \rho^{full} \right) $$

$$ \dot{H} = - \frac12 c_{}^2 \left( p^{full} + \rho^{full} \right) \kappa(c_{}) \ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}   \  \sin^{-2}{(\chi)} $$

$$ \dot{H} = - \frac12 c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} \ \frac{ p^{full} + \rho^{full} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}  \  \sin^{-2}{(\chi)} $$

first friedmann equation
$$\text{ }$$

$$ H^2 = \frac13 \left( c_{}^2 \kappa(c_{*}) - c_{}^2\kappa(c_{}) \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 c_{}^2\kappa(c_{*})\rho^{full} $$

$$ H^2 = \frac13 c_{}^2 \left( \kappa(c_{*})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \kappa(c_{*})\rho^{full} - \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho}     \right)$$

$$ H^2 = \frac13 c_{}^2 \left( \kappa(c_{*}) \left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} +\rho^{full} \right) - \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho}     \right)$$

$$ H^2 = \frac13 c_{}^2 \left( \kappa(c_{}) \ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}   \  \sin^{-2}{(\chi)} \left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} +\rho^{full} \right) - \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho}     \right)$$

$$ H^2 = \frac13 c_{}^2 \left( \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} \sin^{-2}{(\chi)}  - \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho}     \right)$$

$$ H^2 = \frac13 c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} \left( \sin^{-2}{(\chi)} - 1 \right)$$

$$ H^2 = \frac13 c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} \cot^{2}{(\chi)} $$

$$ H = \sqrt{ \frac13  c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} } \ \cot{(\chi)} $$

solve
$$\text{ }$$

$$ H = \sqrt{ \frac13  c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} }  \ \cot{(\chi)} $$

$$ \dot{H} = \sqrt{ \frac13 c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} } \ \frac{ \mathrm{d}\cot{(\chi)} } {\mathrm{d}{\tau} } $$

$$ \dot{H} = \sqrt{ \frac13 c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} } \ \frac{ \mathrm{d} \chi } {\mathrm{d}{\tau} } \frac{ \mathrm{d}\cot{(\chi)} } {\mathrm{d}\chi  } $$

$$ \dot{H} = - \sqrt{ \frac13 c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} } \ \frac{ \mathrm{d}\chi } {\mathrm{d}{\tau} } \sin^{-2}{(\chi)} \ $$

$$ \dot{H} = - \frac12 c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} \ \frac{ p^{full} + \rho^{full} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}  \  \sin^{-2}{(\chi)} $$

$$ - \sqrt{ \frac13 c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} } \ \frac{  \mathrm{d}\chi } {\mathrm{d}{\tau} } \sin^{-2}{(\chi)} = - \frac12 c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} \ \frac{ p^{full} + \rho^{full} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}  \  \sin^{-2}{(\chi)} $$

$$ \sqrt{ \frac13 c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} } \ \frac{ \mathrm{d}\chi } {\mathrm{d}{\tau} } = \frac12 c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} \ \frac{ p^{full} + \rho^{full} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}   \ $$

$$ \frac{  \mathrm{d}\chi } {\mathrm{d}{\tau} } = \frac{ \frac12 c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{ \sqrt{ \frac13 c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} } }\ \frac{ p^{full} + \rho^{full} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}   \ $$

$$ \frac{  \mathrm{d}\chi } {\mathrm{d}{\tau} } = \sqrt{ \frac34 c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} } \ \frac{ p^{full} + \rho^{full} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}  \ $$

$$ \int \mathrm{d}\chi = \sqrt{ \frac34 c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} } \ \frac{ p^{full} + \rho^{full} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}  \  \int \mathrm{d}{\tau} $$

$$ \chi = \sqrt{ \frac34 c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} } \ \frac{ p^{full} + \rho^{full} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}  \  {\tau} $$

conclusions
$$\text{ }$$

$$ H = \sqrt{ \frac13  c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} } \ \cot{(\chi)} $$

$$ \int  H \mathrm{d}{\tau} = \sqrt{ \frac13 c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} }  \int \cot{(\chi)} \mathrm{d}{\tau} $$

$$ \int  H \mathrm{d}{\tau} = \sqrt{ \frac13 c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} }  \int \cot{(\chi)} \frac{\mathrm{d}{\tau} } { \mathrm{d}\chi } \mathrm{d}\chi $$

$$ \int  H \mathrm{d}{\tau} = \sqrt{ \frac13 c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} }  \ \frac{\mathrm{d}{\tau} } {  \mathrm{d}\chi } \int \cot{(\chi)} \mathrm{d}\chi $$

$$ \int  H \mathrm{d}{\tau} = \frac{ \sqrt{ \frac13  c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} }  }{ \sqrt{ \frac34  c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} } }\ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}} { p^{full} + \rho^{full} }    \ \int \cot{(\chi)} \mathrm{d}\chi $$

$$ \int \frac{ \dot{a} }{a} \mathrm{d}{\tau} = \frac23\ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}} { p^{full} + \rho^{full} }   \ \int \cot{(\chi)} \mathrm{d}\chi $$

$$ \ln{(a)} = \frac23\ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}} { p^{full} + \rho^{full} }    \ \ln{\left( \sin{(\chi)}  \right)} + C $$

$$ \ln{(a)} = \frac23\ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}} { p^{full} + \rho^{full} }    \ \ln{\left( \sin{\left(\sqrt{ \frac34  c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} } \ \frac{ p^{full} + \rho^{full} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}  \  {\tau}\right)}  \right)} + C $$

The relationship between the grivitational parameter and cosmological time
$$\text{ }$$

$$ \kappa(c_{*}) = \kappa(c_{}) \ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}   \  \sin^{-2}{(\chi)} $$

$$ \chi = \sqrt{ \frac34 c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} }\ \ \frac{ p^{full} + \rho^{full} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}   \  {\tau} $$

$$ \kappa(c_{*}) = \kappa(c_{}) \ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}   \  \sin^{-2}{\left( \sqrt{ \frac34  c_{}^2 \kappa(c_{}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} }\ \frac{ p^{full} + \rho^{full} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}   \  {\tau} \right)} $$

The relationship between the spatial scale factor and the grivitational parameter
$$\text{ }$$

$$ \ln{(a)} = \frac23\ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}} { p^{full} + \rho^{full} }    \ \ln{\left( \sin{(\chi)}  \right)} + C $$

$$ \ln{(a)} = - \frac13\ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}} { p^{full} + \rho^{full} }    \ \ln{\left( \sin^{-2}{(\chi)}  \right)} + C $$

$$ \ln{(a)} = - \frac13\ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}} { p^{full} + \rho^{full} }    \ \ln{\left( \kappa(c_{}) \ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}   \  \sin^{-2}{(\chi)}  \right)} + \frac13\ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}} { p^{full} + \rho^{full} }    \ \ln{\left( \kappa(c_{}) \ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}   \    \right)} + C $$

$$ \ln{(a)} = - \frac13\ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}} { p^{full} + \rho^{full} }    \ \ln{\left( \kappa(c_{*}) \right)} + \frac13\ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}} { p^{full} + \rho^{full} }    \ \ln{\left( \kappa(c_{}) \ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}   \    \right)} + C $$

$$ \text{ The penultimate term on the right hand side is constant and can therefore be rolled into the constant of integration.} $$

$$ \ln{(a)} = - \frac13\ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}} { p^{full} + \rho^{full} }    \ \ln{\left( \kappa(c_{*}) \right)}  + {\tilde{ C } } $$

How long does inflation last?
Let's do the sums first and then see how applicable they are.

$$\frac{d\tau}{d{\chi}} =   \frac{1 } {\sqrt{\frac34 c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \left(\frac{ \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full} }{ p^{full} + \rho^{full} } \right)$$

$$ d\tau =   \frac{1 } {\sqrt{\frac34 c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \left(\frac{ \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full} }{ p^{full} + \rho^{full} } \right) d{\chi} $$

Integrating

$$ \int d\tau = \int \frac{1 } {\sqrt{\frac34 c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \left(\frac{ \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full} }{ p^{full} + \rho^{full} } \right) d{\chi} $$

Moving the constant factors outside the integral

$$ \int d\tau = \frac{1 } {\sqrt{\frac34 c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \left(\frac{ \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full} }{ p^{full} + \rho^{full} } \right) \int d{\chi} $$

The integral for cosmological time covers the period from the begining of time to the end of inflation.

The integral for the parameter chi is over a quarter of a cycle where sin(chi) goes from zero to one.

$$ \int_{Begining\ Of \ Time}^{End\ Of\ Inflation} d\tau = \frac{1 } {\sqrt{\frac34 c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \left(\frac{ \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full} }{ p^{full} + \rho^{full} } \right) \int_{0}^{\pi/2} d{\chi} $$


 * $${Period\ Of\ Inflation} =

\frac{{\pi}/{2} } {\sqrt{\frac34 c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \left(\frac{ \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full} }{ p^{full} + \rho^{full} } \right) $$


 * $${Period\ Of\ Inflation\ } =

\frac{\pi} {\sqrt{3 c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \left(\frac{ \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full} }{ p^{full} + \rho^{full} } \right) $$

Draft
We seem to be in the paradoxical situation that non-uniform time is an inherent property of the manifold and uniform time is no more than an imposed perspective. Further, that this non-uniformaty is largely indeterminate, but never-the-less necessary. On the positive side this avoids the problem of a singularity in the manifold and despite uniform time only being an imposed perspective it is within this perspective that energy is conserved.

$$ \text{The following example shows how it is possible to have a begining to time without having a singularity in the manifold.}$$

$$ \text{The foundations of this theory need three fundamental values that are dimensionally independent and do not change under any circumstances. }$$

$$ \text{ Two obvious candidates are the zero point values. }$$

$$ \text{Because Einstein's field equations can be derived from the principle of least action it seems necessary to have as the third value a fundamental unit of action. }$$

$$ \text{The three values are therefore: the zero point energy density of the vacuum, the zero point curvature of space, and Planck's reduced constant. }$$

$$ \ \ \text{With the three values represented respectively by: }$$ $$ \text{rho zero point, Lambda zero point, and aitch bar. }$$

$$ \ \ \overset{{\ }_{\boldsymbol{\circ}}}{\rho} \overset{{}_{\boldsymbol{\circ}}}{\Lambda} \hbar $$

2 $$ \text{These values together with the Temporal Scale Factor can be aranged to produce a dimensionless quantity }$$ $$ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho}} {\left( \overset{{}_{\boldsymbol{\circ}}}{\Lambda} \right)^2 \hbar c_*} $$

$$ \text{ }$$

4 $$ \text{The arbitary function kappa can be re-writen in terms of an arbitary dimensionless funtion }$$ $$ f: x \in \mathbb{R} \mapsto f(x) \in \mathbb{R}   $$

$$ \kappa(c_{*}) \equiv f\left( \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho}} {\left( \overset{{}_{\boldsymbol{\circ}}}{\Lambda} \right)^2 \hbar c_*}  \right) \frac{\overset{{}_{\boldsymbol{\circ}}}{\Lambda}} {\overset{{\ }_{\boldsymbol{\circ}}}{\rho}} $$

5  $$ \text{Please note that the above implies } f\left( \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho}} {\left( \overset{{}_{\boldsymbol{\circ}}}{\Lambda} \right)^2 \hbar c}  \right) \equiv 1 $$ $$ \text{ It does not imply that }$$ $$ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho}} {\left( \overset{{}_{\boldsymbol{\circ}}}{\Lambda} \right)^2 \hbar c}  \equiv 1 $$

$$ \ \text{Because }\ \   f: x  \mapsto constant \ \  \text{  is ruled out, }$$ $$ \text{the simplest available option for } f \text{ would be }\ \   f: x  \mapsto x \ \ \ \text{i.e.}\ \ \ f(x)\equiv x  $$

9 $$ \text{Taking this as our example. }$$

$$ \kappa(c_{*}) \equiv \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho}} {\left( \overset{{}_{\boldsymbol{\circ}}}{\Lambda} \right)^2 \hbar c_*} \frac{\overset{{}_{\boldsymbol{\circ}}}{\Lambda}} {\overset{{\ }_{\boldsymbol{\circ}}}{\rho}} \equiv \frac{1} {\overset{{}_{\boldsymbol{\circ}}}{\Lambda} \hbar c_*} $$

11 $$ \text{From the definition of } \ \ \chi \ \text{ (chi) } \ \ $$

$$\kappa(c_{*}) = \kappa(c_{}) \ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}  \  \sin^{-2}{(\chi)} \ \ $$

12 $$ \text{Substituting. }$$

$$ \frac{1} {\overset{{}_{\boldsymbol{\circ}}}{\Lambda} \hbar c_*} = \kappa(c_{}) \ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}  \  \sin^{-2}{(\chi)} \ \ $$

13 $$ \text{Integrating with respect to cosmological time. }$$

$$ \int \frac{1} {\overset{{}_{\boldsymbol{\circ}}}{\Lambda} \hbar c_*} \  \mathrm{d}{\tau} = \int \kappa(c_{}) \ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}  \  \sin^{-2}{(\chi)} \ \mathrm{d}{\tau} \ $$

$$ \text{ }$$ $$ \int \frac{1} {\overset{{}_{\boldsymbol{\circ}}}{\Lambda} \hbar c_*} \ \frac{\mathrm{d}{\tau} } { \mathrm{d}t } \ \mathrm{d}t = \int \kappa(c_{}) \ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}  \  \sin^{-2}{(\chi)} \ \frac{\mathrm{d}{\tau} } { \mathrm{d}\chi } \ \mathrm{d}\chi $$

15 $$ \text{from the definition of cosmological time } \ \ c\mathrm{d}\tau = c_*\mathrm{d}t \ \ \text{ we find the substitution }$$

$$ \int \frac{1} {\overset{{}_{\boldsymbol{\circ}}}{\Lambda} \hbar c_*} \ \frac{ c_* } { c } \ \mathrm{d}t = \int \kappa(c_{}) \ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}  \  \sin^{-2}{(\chi)} \ \frac{\mathrm{d}{\tau} } { \mathrm{d}\chi } \ \mathrm{d}\chi $$

$$ \text{ }$$ $$ \int \frac{1} {\overset{{}_{\boldsymbol{\circ}}}{\Lambda} \hbar c} \ \ \mathrm{d}t = \int \kappa(c_{}) \ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}  \  \sin^{-2}{(\chi)} \ \frac{\mathrm{d}{\tau} } { \mathrm{d}\chi } \ \mathrm{d}\chi $$

17 $$ \text{Rearranging to take all of the constants outside the integrals. }$$

$$ \frac{1} {\overset{{}_{\boldsymbol{\circ}}}{\Lambda} \hbar c} \ \ \int \mathrm{d}t = \kappa(c_{}) \ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}  \ \frac{\mathrm{d}{\tau} } { \mathrm{d}\chi } \int \sin^{-2}{(\chi)} \ \ \mathrm{d}\chi $$

18 $$ \text{Then just using proportionality  }$$

$$ \int \mathrm{d}t \propto\int \sin^{-2}{(\chi)} \ \ \mathrm{d}\chi $$

$$ \text{ }$$

$$ \int \mathrm{d}t \propto -\cot{\left( \chi \right)} +C $$

20 $$ \text{We find the result that although cosmological time is finite into the past, co-ordinate time could go back infinitely far.}$$

$$\text{A universe beginning infinitely far back in co-ordinate time but finitely far back in proper time would be a White Hole solution.}$$

$$\kappa(c_{*}) = \frac{8\pi G}{c_*^{4}}$$

$$\kappa(c_{*}) = \frac{8\pi G}{{c_*}^{4}}$$

In order to avoid a fine-tuning problem it seems necessary that Maxwell's Constant be given no other signicance than it be the limiting value of the Temporal Scale Factor.

the gravitational parameter is inversely proportional to the temporal scale factor or in other words the the inverse gravitational parameter is directly proportional to the temporal scale factor.

Rather perversely proper time is an emergent property and co-ordinate time is a an inherent property of the manifold.

The fourth dimension is never a dimension of space. It is time like at all points except the begining of time where it is NULL. It starts off Null and after that is time-like.

Apendix A:       Derivation Of The Einstein Tensors
The following is adapted from "Curvature tensor components for the diagonal metric." from pages 419-421 of "Relativity: Special, General, and Cosmological." second edition by Wolfgang Rindler ISBN 978-0-19-856732-5

$$\text{Given the metric}$$


 * $$ c_{}^2\mathrm{d}\tau^2 = c_{*}^2(t) \mathrm{d}t^2 - {a}^2(t)

\left( \frac{\mathrm{d}r^2}{1-k r^2} + r^2 \mathrm{d}\theta^2 + r^2 \sin^2 \left( \theta \right) \, \mathrm{d}\phi^2 \right) $$

$$ g_{tt} = c_{*}^2(t) \ \ \ \ \ \ \ \ \ \ \ \ g_{rr} = - \frac{{a}^2(t)}{1-k r^2} \ \ \ \ \ \ \ \ \ \ \ \ g_{\theta\theta} = - {a}^2(t) r^2 \ \ \ \ \ \ \ \ \ \ \ \ g_{\phi\phi} = - {a}^2(t) r^2 \sin^2 \left( \theta \right) \ \ \ \ \ \ \ \ \ \ \ \ \mu \neq \nu \implies g_{\mu\nu} = 0 $$

$$ \therefore \ \ \ \ \ \ \ \ \ \ \ \ g^{tt} = \frac{1}{c_{*}^2(t)} \ \ \ \ \ \ \ \ \ \ \ \ g^{rr} = - \frac{1-k r^2}{{a}^2(t)} \ \ \ \ \ \ \ \ \ \ \ \ g^{\theta\theta} = - \frac{1}{{a}^2(t) r^2} \ \ \ \ \ \ \ \ \ \ \ \ g^{\phi\phi} = - \frac{1}{{a}^2(t) r^2 \sin^2 \left( \theta \right)} \ \ \ \ \ \ \ \ \ \ \ \ \mu \neq \nu \implies g^{\mu\nu} = 0 $$

For The Temporal Tensor


\frac{-G_{tt}}{g_{tt}} = $$

g^{rr}g^{\theta\theta}\left(\frac{1}{2}g_{rr,\theta\theta} + \frac{1}{2}g_{\theta\theta,rr} - \frac{1}{4} g^{rr} ( g_{rr,\theta} )^2 - \frac{1}{4} g^{\theta\theta} ( g_{\theta\theta,r} )^2 - \frac{1}{4} g^{rr}g_{rr,r}g_{\theta\theta,r} - \frac{1}{4} g^{\theta\theta}g_{rr,\theta}g_{\theta\theta,\theta} + \frac{1}{4} g^{tt}g_{rr,t}g_{\theta\theta,t} + \frac{1}{4} g^{\phi\phi}g_{rr,\phi}g_{\theta\theta,\phi} \right) $$



g^{rr}g^{\phi\phi}\left( \frac{1}{2}g_{rr,\phi\phi} + \frac{1}{2}g_{\phi\phi,rr} - \frac{1}{4} g^{rr} ( g_{rr,\phi} )^2 - \frac{1}{4} g^{\phi\phi} ( g_{\phi\phi,r} )^2 - \frac{1}{4} g^{rr}g_{rr,r}g_{\phi\phi,r} - \frac{1}{4} g^{\phi\phi}g_{rr,\phi}g_{\phi\phi,\phi} + \frac{1}{4} g^{tt}g_{rr,t}g_{\phi\phi,t} + \frac{1}{4} g^{\theta\theta}g_{rr,\theta}g_{\phi\phi,\theta} \right) $$



g^{\theta\theta}g^{\phi\phi}\left( \frac{1}{2}g_{\theta\theta,\phi\phi} + \frac{1}{2}g_{\phi\phi,\theta\theta} - \frac{1}{4} g^{\theta\theta} ( g_{\theta\theta,\phi} )^2 - \frac{1}{4} g^{\phi\phi} ( g_{\phi\phi,\theta} )^2 - \frac{1}{4} g^{\theta\theta}g_{\theta\theta,\theta}g_{\phi\phi,\theta} - \frac{1}{4} g^{\phi\phi}g_{\theta\theta,\phi}g_{\phi\phi,\phi} + \frac{1}{4} g^{tt}g_{\theta\theta,t}g_{\phi\phi,t} + \frac{1}{4} g^{rr}g_{\theta\theta,r}g_{\phi\phi,r} \right) $$

After eliminating the the terms that are null in a Friedmann Lemaitre Robertson Walker metric when expressed in polar co-ordinates.


\frac{-G_{tt}}{g_{tt}} = $$

g^{rr}g^{\theta\theta}\left( \frac{1}{2}g_{\theta\theta,rr} - \frac{1}{4} g^{\theta\theta} ( g_{\theta\theta,r} )^2 - \frac{1}{4} g^{rr}g_{rr,r}g_{\theta\theta,r} + \frac{1}{4} g^{tt}g_{rr,t}g_{\theta\theta,t} \right) $$



g^{rr}g^{\phi\phi}\left( \frac{1}{2}g_{\phi\phi,rr} - \frac{1}{4} g^{\phi\phi} ( g_{\phi\phi,r} )^2 - \frac{1}{4} g^{rr}g_{rr,r}g_{\phi\phi,r} + \frac{1}{4} g^{tt}g_{rr,t}g_{\phi\phi,t} \right) $$



g^{\theta\theta}g^{\phi\phi}\left( \frac{1}{2}g_{\phi\phi,\theta\theta} - \frac{1}{4} g^{\phi\phi} ( g_{\phi\phi,\theta} )^2 + \frac{1}{4} g^{tt}g_{\theta\theta,t}g_{\phi\phi,t} + \frac{1}{4} g^{rr}g_{\theta\theta,r}g_{\phi\phi,r} \right) $$

Then substituting


\frac{-G_{tt}}{g_{tt}} = $$

g^{rr}g^{\theta\theta}\left( \frac{1}{2}g_{\theta\theta} \frac{2}{r^2} - \frac{1}{4} g^{\theta\theta} \left( g_{\theta\theta} \frac{2r}{r^2} \right)^2 - \frac{1}{4} g^{rr}g_{rr} \frac{-(1-kr^2)^{-2}( -2kr )}{(1-kr^2)^{-1}} g_{\theta\theta} \frac{2r}{r^2} + \frac{1}{4} g^{tt}g_{rr} \frac{2a\left( \mathrm{d}{a}/\mathrm{d}{t} \right)}{a^2} g_{\theta\theta} \frac{2a\left( \mathrm{d}{a}/\mathrm{d}{t} \right)}{a^2} \right) $$



g^{rr}g^{\phi\phi}\left( \frac{1}{2}g_{\phi\phi} \frac{2}{r^2} - \frac{1}{4} g^{\phi\phi} \left( g_{\phi\phi} \frac{2r}{r^2} \right)^2 - \frac{1}{4} g^{rr}g_{rr} \frac{-(1-kr^2)^{-2}( -2kr )}{(1-kr^2)^{-1}} g_{\phi\phi} \frac{2r}{r^2} + \frac{1}{4} g^{tt}g_{rr} \frac{2a\left( \mathrm{d}{a}/\mathrm{d}{t} \right)}{a^2} g_{\phi\phi} \frac{2a\left( \mathrm{d}{a}/\mathrm{d}{t} \right)}{a^2} \right) $$



g^{\theta\theta}g^{\phi\phi}\left( \frac{1}{2}g_{\phi\phi} \frac{2\cos(\theta)\cos(\theta) - 2\sin(\theta)\sin(\theta)}{\sin^2(\theta)} - \frac{1}{4} g^{\phi\phi} \left( g_{\phi\phi} \frac{2\sin(\theta)\cos(\theta)}{\sin^2(\theta)} \right)^2 + \frac{1}{4} g^{tt}g_{\theta\theta} \frac{2a\left( \mathrm{d}{a}/\mathrm{d}{t} \right)}{a^2} g_{\phi\phi} \frac{2a\left( \mathrm{d}{a}/\mathrm{d}{t} \right)}{a^2} + \frac{1}{4} g^{rr}g_{\theta\theta} \frac{2r}{r^2} g_{\phi\phi} \frac{2r}{r^2} \right) $$

and simplifying.


\frac{-G_{tt}}{g_{tt}} = $$

g^{rr}g^{\theta\theta}\left( g_{\theta\theta} \frac{1}{r^2} - g_{\theta\theta} \frac{1}{r^2} - \frac{ k }{1-kr^2} g_{\theta\theta} + g^{tt}g_{rr} g_{\theta\theta} \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2} \right) $$



g^{rr}g^{\phi\phi}\left( g_{\phi\phi} \frac{1}{r^2} - g_{\phi\phi} \frac{1}{r^2} - \frac{ k }{1-kr^2} g_{\phi\phi} + g^{tt}g_{rr} g_{\phi\phi} \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2} \right) $$



g^{\theta\theta}g^{\phi\phi}\left( - g_{\phi\phi} \frac{\sin^2(\theta) - \cos^2(\theta)}{\sin^2(\theta)} - g_{\phi\phi} \frac{\cos^2(\theta)}{\sin^2(\theta)} + g^{tt}g_{\theta\theta} g_{\phi\phi} \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2} + g^{rr}g_{\theta\theta} g_{\phi\phi} \frac{1}{r^2} \right) $$



\frac{-G_{tt}}{g_{tt}} = - \frac{ k }{1-kr^2} g^{rr} + g^{tt} \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2} - \frac{ k }{1-kr^2} g^{rr} + g^{tt} \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2} - g^{\theta\theta} + g^{tt} \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2} + g^{rr} \frac{1}{r^2} $$



\frac{-G_{tt}}{g_{tt}} = - \frac{ 2k }{1-kr^2} g^{rr} + 3g^{tt} \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2} - g^{\theta\theta} + g^{rr} \frac{1}{r^2} $$



\frac{-G_{tt}}{g_{tt}} = \frac{ 2k }{1-kr^2} \frac{1-kr^2}{a^2} + 3g^{tt} \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2} + \frac{1}{a^2 r^2} - \frac{1-kr^2}{a^2} \frac{1}{r^2} $$



\frac{-G_{tt}}{g_{tt}} = \frac{ 2k }{a^2} + 3g^{tt} \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2} + \frac{1}{a^2 r^2} - \frac{1-kr^2}{a^2 r^2} $$



\frac{-G_{tt}}{g_{tt}} = \frac{ 2k }{a^2} + 3g^{tt} \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2} + \frac{k}{a^2} $$



\frac{-G_{tt}}{g_{tt}} = 3g^{tt} \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2} + 3\frac{k}{a^2} $$



-G_{tt} = 3 \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2} + 3\frac{k}{a^2}g_{tt} $$



-G_{tt} = 3 \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2} + 3\frac{kc_{*}^2}{a^2} $$



-G_{tt} = 3 \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2 + kc_{*}^2}{a^2} $$

For The Spatial Tensors


\frac{-G_{\theta\theta}}{g_{\theta\theta}} = \frac{-G_{\phi\phi}}{g_{\phi\phi}} = \frac{-G_{rr}}{g_{rr}} $$



\frac{-G_{rr}}{g_{rr}} = $$

g^{\theta\theta}g^{\phi\phi}\left( \frac{1}{2}g_{\theta\theta,\phi\phi} + \frac{1}{2}g_{\phi\phi,\theta\theta} - \frac{1}{4} g^{\theta\theta} ( g_{\theta\theta,\phi} )^2 - \frac{1}{4} g^{\phi\phi} ( g_{\phi\phi,\theta} )^2 - \frac{1}{4} g^{\theta\theta}g_{\theta\theta,\theta}g_{\phi\phi,\theta} - \frac{1}{4} g^{\phi\phi}g_{\theta\theta,\phi}g_{\phi\phi,\phi} + \frac{1}{4} g^{tt}g_{\theta\theta,t}g_{\phi\phi,t} + \frac{1}{4} g^{rr}g_{\theta\theta,r}g_{\phi\phi,r} \right) $$

+ g^{tt}g^{\theta\theta}\left( \frac{1}{2}g_{tt,\theta\theta} + \frac{1}{2}g_{\theta\theta,tt} - \frac{1}{4} g^{tt} ( g_{tt,\theta} )^2 - \frac{1}{4} g^{\theta\theta} ( g_{\theta\theta,t} )^2 - \frac{1}{4} g^{tt}g_{tt,t}g_{\theta\theta,t} - \frac{1}{4} g^{\theta\theta}g_{tt,\theta}g_{\theta\theta,\theta} + \frac{1}{4} g^{rr}g_{tt,r}g_{\theta\theta,r} + \frac{1}{4} g^{\phi\phi}g_{tt,\phi}g_{\theta\theta,\phi} \right) $$

+ g^{tt}g^{\phi\phi}\left( \frac{1}{2}g_{tt,\phi\phi} + \frac{1}{2}g_{\phi\phi,tt} - \frac{1}{4} g^{tt} ( g_{tt,\phi} )^2 - \frac{1}{4} g^{\phi\phi} ( g_{\phi\phi,t} )^2 - \frac{1}{4} g^{tt}g_{tt,t}g_{\phi\phi,t} - \frac{1}{4} g^{\phi\phi}g_{tt,\phi}g_{\phi\phi,\phi} + \frac{1}{4} g^{rr}g_{tt,r}g_{\phi\phi,r} + \frac{1}{4} g^{\theta\theta}g_{tt,\theta}g_{\phi\phi,\theta} \right) $$

After eliminating the null terms


\frac{-G_{rr}}{g_{rr}} = $$

g^{\theta\theta}g^{\phi\phi}\left(  \frac{1}{2}g_{\phi\phi,\theta\theta} - \frac{1}{4} g^{\phi\phi} ( g_{\phi\phi,\theta} )^2 + \frac{1}{4} g^{tt}g_{\theta\theta,t}g_{\phi\phi,t} + \frac{1}{4} g^{rr}g_{\theta\theta,r}g_{\phi\phi,r} \right) $$

+ g^{tt}g^{\theta\theta}\left(  \frac{1}{2}g_{\theta\theta,tt} - \frac{1}{4} g^{\theta\theta} ( g_{\theta\theta,t} )^2 - \frac{1}{4} g^{tt}g_{tt,t}g_{\theta\theta,t} \right) $$

+ g^{tt}g^{\phi\phi}\left(  \frac{1}{2}g_{\phi\phi,tt} - \frac{1}{4} g^{\phi\phi} ( g_{\phi\phi,t} )^2 - \frac{1}{4} g^{tt}g_{tt,t}g_{\phi\phi,t} \right) $$

Then substituting


\frac{-G_{rr}}{g_{rr}} = $$

g^{\theta\theta}g^{\phi\phi}\left(  \frac{1}{2}g_{\phi\phi} \frac{2\cos(\theta)\cos(\theta) - 2\sin(\theta)\sin(\theta)}{\sin^2(\theta)} - \frac{1}{4} g^{\phi\phi} \left( g_{\phi\phi} \frac{2\sin(\theta) \cos(\theta)}{\sin^2(\theta)} \right)^2 + \frac{1}{4} g^{tt}g_{\theta\theta} \frac{2a\left( \mathrm{d}{a}/\mathrm{d}{t} \right)}{a^2} g_{\phi\phi} \frac{2a\left( \mathrm{d}{a}/\mathrm{d}{t} \right)}{a^2} + \frac{1}{4} g^{rr}g_{\theta\theta} \frac{2r}{r^2} g_{\phi\phi} \frac{2r}{r^2} \right) $$

+ g^{tt}g^{\theta\theta}\left(  \frac{1}{2}g_{\theta\theta} \frac{2\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2 + 2a\left( \mathrm{d^2}{a}/\mathrm{d}{t}^2 \right)}{a^2} - \frac{1}{4} g^{\theta\theta} \left( g_{\theta\theta} \frac{2a\left( \mathrm{d}{a}/\mathrm{d}{t} \right)}{a^2} \right)^2 - \frac{1}{4} g^{tt}g_{tt} \frac{2c_{*}\left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}^2} g_{\theta\theta} \frac{2a\left( \mathrm{d}{a}/\mathrm{d}{t} \right)}{a^2} \right) $$

+ g^{tt}g^{\phi\phi}\left(  \frac{1}{2}g_{\phi\phi} \frac{2\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2 + 2a\left( \mathrm{d^2}{a}/\mathrm{d}{t}^2 \right)}{a^2} - \frac{1}{4} g^{\phi\phi} \left( g_{\phi\phi} \frac{2a\left( \mathrm{d}{a}/\mathrm{d}{t} \right)}{a^2} \right)^2 - \frac{1}{4} g^{tt}g_{tt} \frac{2c_{*}\left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}^2} g_{\phi\phi} \frac{2a\left( \mathrm{d}{a}/\mathrm{d}{t} \right)}{a^2} \right) $$

and simplifying.


\frac{-G_{rr}}{g_{rr}} = $$

g^{\theta\theta}g^{\phi\phi}\left(  g_{\phi\phi} \frac{ \cos^2(\theta) - \sin^2(\theta) }{\sin^2(\theta)} - g_{\phi\phi} \frac{\cos^2(\theta)}{\sin^2(\theta)} + g^{tt}g_{\theta\theta} g_{\phi\phi} \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2} + g^{rr}g_{\theta\theta} g_{\phi\phi} \frac{1}{r^2} \right) $$

+ g^{tt}g^{\theta\theta}\left(  g_{\theta\theta} \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2 + a\left( \mathrm{d^2}{a}/\mathrm{d}{t}^2 \right)}{a^2} - g_{\theta\theta} \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2}  - \frac{\mathrm{d}{c_{*}}/\mathrm{d}{t}}{c_{*}} g_{\theta\theta} \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) $$

+ g^{tt}g^{\phi\phi}\left(  g_{\phi\phi} \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2 + a\left( \mathrm{d^2}{a}/\mathrm{d}{t}^2 \right)}{a^2} - g_{\phi\phi} \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2}  - \frac{\mathrm{d}{c_{*}}/\mathrm{d}{t}}{c_{*}} g_{\phi\phi} \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) $$



\frac{-G_{rr}}{g_{rr}} = $$

g^{\theta\theta}g^{\phi\phi}\left(  g_{\phi\phi} \frac{ \cos^2(\theta) - \sin^2(\theta) }{\sin^2(\theta)} - g_{\phi\phi} \frac{\cos^2(\theta)}{\sin^2(\theta)} + g^{tt}g_{\theta\theta} g_{\phi\phi} \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2} + g^{rr}g_{\theta\theta} g_{\phi\phi} \frac{1}{r^2} \right) $$

+ g^{tt}g^{\theta\theta}\left(  g_{\theta\theta} \frac{ a\left( \mathrm{d^2}{a}/\mathrm{d}{t}^2 \right)}{a^2} - \frac{\mathrm{d}{c_{*}}/\mathrm{d}{t}}{c_{*}} g_{\theta\theta} \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) $$

+ g^{tt}g^{\phi\phi}\left(  g_{\phi\phi} \frac{ a\left( \mathrm{d^2}{a}/\mathrm{d}{t}^2 \right)}{a^2} - \frac{\mathrm{d}{c_{*}}/\mathrm{d}{t}}{c_{*}} g_{\phi\phi} \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) $$



\frac{-G_{rr}}{g_{rr}} = g^{\theta\theta}\left(  \frac{ \cos^2(\theta) - \sin^2(\theta) }{\sin^2(\theta)} - \frac{\cos^2(\theta)}{\sin^2(\theta)} + g^{tt}g_{\theta\theta} \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2} + g^{rr}g_{\theta\theta} \frac{1}{r^2} \right) + 2g^{tt}\left(  \frac{ \mathrm{d^2}{a}/\mathrm{d}{t}^2 }{a} - \frac{\mathrm{d}{c_{*}}/\mathrm{d}{t}}{c_{*}} \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) $$



\frac{-G_{rr}}{g_{rr}} = - g^{\theta\theta} + g^{rr} \frac{1}{r^2} + g^{tt} \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2} + 2g^{tt} \frac{\left( \mathrm{d^2}{a}/\mathrm{d}{t}^2 \right)}{a} - 2g^{tt} \frac{\mathrm{d}{c_{*}}/\mathrm{d}{t}}{c_{*}} \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} $$



\frac{-G_{rr}}{g_{rr}} = \frac{1}{a^2r^2} - \frac{1-kr^2}{a^2} \frac{1}{r^2} + \frac{1}{c_{*}^2} \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2} + \frac{2}{c_{*}^2} \frac{\left( \mathrm{d^2}{a}/\mathrm{d}{t}^2 \right)}{a} - \frac{2}{c_{*}^2} \frac{\mathrm{d}{c_{*}}/\mathrm{d}{t}}{c_{*}} \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} $$



\frac{-G_{rr}}{g_{rr}} = \frac{k}{a^2} + \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2c_{*}^2} + 2\frac{ \mathrm{d^2}{a}/\mathrm{d}{t}^2 }{ac_{*}^2} - 2\frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)\left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{ac_{*}^3} $$



\frac{-G_{rr}}{g_{rr}} = \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2 + kc_{*}^2}{a^2c_{*}^2} + 2\frac{ \mathrm{d^2}{a}/\mathrm{d}{t}^2 }{ac_{*}^2} - 2\frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)\left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{ac_{*}^3} $$

Apendix B: Re-calculating The Friedmann Equations
$$\text{For the Friedmann Lemaitre Robertson Walker Metric the Stress Energy Tensor is given by}$$

$$ T_{tt} = g_{tt}\rho \ \ \ \ \ \ \ \ \ \ \ \ T_{rr} = -g_{rr} p \ \ \ \ \ \ \ \ \ \ \ \ T_{\theta\theta} = -g_{\theta\theta} p \ \ \ \ \ \ \ \ \ \ \ \ T_{\phi\phi} = -g_{\phi\phi} p \ \ \ \ \ \ \ \ \ \ \ \ \mu \neq \nu \implies T_{\mu\nu} = 0 $$

$$\text{Where } \rho \text{ is energy density and } p \text{ is pressure.}$$

$$-G_{\mu \nu} - \Lambda(c_{*})g_{\mu \nu} = \kappa(c_{*})T_{\mu \nu}$$

$$-G_{tt} - \Lambda(c_{*})g_{tt} = \kappa(c_{*})T_{tt}$$

$$-G_{tt} = \Lambda(c_{*})g_{tt} + \kappa(c_{*})g_{tt}\rho$$

$$-G_{tt} = c_{*}^2 \Lambda(c_{*}) + c_{*}^2\kappa(c_{*})\rho$$

$$3\frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2+kc_{*}^2}{a^2} = c_{*}^2 \Lambda(c_{*}) + c_{*}^2\kappa(c_{*})\rho$$

$$\frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2+kc_{*}^2}{a^2} = \frac13 c_{*}^2 \Lambda(c_{*}) + \frac13 c_{*}^2\kappa(c_{*})\rho$$

$$\frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2} + \frac{kc_{*}^2}{a^2} = \frac13 c_{*}^2 \Lambda(c_{*}) + \frac13 c_{*}^2\kappa(c_{*})\rho$$

$$ \left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = -\frac{kc_{*}^2}{a^2} + \frac13 c_{*}^2 \Lambda(c_{*}) + \frac13 c_{*}^2\kappa(c_{*})\rho$$

$$-G_{\mu \nu} - \Lambda(c_{*})g_{\mu \nu} = \kappa(c_{*})T_{\mu \nu}$$

$$-G_{rr} - \Lambda(c_{*})g_{rr} = \kappa(c_{*})T_{rr} = - \kappa(c_{*})g_{rr}p$$

$$-G_{rr} = \Lambda(c_{*})g_{rr} - \kappa(c_{*})g_{rr}p$$

$$-\frac{G_{rr}}{g_{rr}} = \Lambda(c_{*}) - \kappa(c_{*})p$$

$$-\frac{G_{rr}}{g_{rr}} = \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2 + kc_{*}^2}{a^2c_{*}^2} + 2\frac{\left( \mathrm{d^2}{a}/\mathrm{d}{t}^2 \right)}{ac_{*}^2} - 2\frac{ \left( \mathrm{d}{a}/\mathrm{d}{t} \right) \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{ac_{*}^3} = \Lambda(c_{*}) - \kappa(c_{*})p$$

$$ \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2 + kc_{*}^2}{a^2} + 2\frac{\left( \mathrm{d^2}{a}/\mathrm{d}{t}^2 \right)}{a} -2\frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a}\frac{ \mathrm{d}{c_{*}}/\mathrm{d}{t} }{c_{*}} = c_{*}^2 \Lambda(c_{*}) - c_{*}^2\kappa(c_{*})p $$

$$3\frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2+kc_{*}^2}{a^2} = c_{*}^2 \Lambda(c_{*}) + c_{*}^2\kappa(c_{*})\rho$$

$$ \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2 + kc_{*}^2}{a^2} + 2\frac{\left( \mathrm{d^2}{a}/\mathrm{d}{t}^2 \right)}{a} -2\frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a}\frac{ \mathrm{d}{c_{*}}/\mathrm{d}{t} }{c_{*}} = 3\frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2+kc_{*}^2}{a^2} - c_{*}^2\kappa(c_{*})\rho - c_{*}^2\kappa(c_{*})p $$

$$2\frac{\left( \mathrm{d^2}{a}/\mathrm{d}{t}^2 \right)}{a} -2 \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2 + kc_{*}^2}{a^2} -2\frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a}\frac{ \mathrm{d}{c_{*}}/\mathrm{d}{t} }{c_{*}} =  - c_{*}^2\kappa(c_{*})p - c_{*}^2\kappa(c_{*})\rho   $$

$$\frac{\left( \mathrm{d^2}{a}/\mathrm{d}{t}^2 \right)}{a} - \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2 + kc_{*}^2}{a^2} -\frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a}\frac{ \mathrm{d}{c_{*}}/\mathrm{d}{t} }{c_{*}} =  - \frac12 c_{*}^2\kappa(c_{*})p - \frac12 c_{*}^2\kappa(c_{*})\rho   $$

$$\frac{\left( \mathrm{d^2}{a}/\mathrm{d}{t}^2 \right)}{a} - \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2} - \frac{kc_{*}^2}{a^2} -\frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a}\frac{ \mathrm{d}{c_{*}}/\mathrm{d}{t} }{c_{*}} =  - \frac12 c_{*}^2\kappa(c_{*}) \left( p + \rho \right) $$

$$\frac{\left( \mathrm{d^2}{a}/\mathrm{d}{t}^2 \right)}{a} - \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2} = \frac{kc_{*}^2}{a^2} + \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a}\frac{ \mathrm{d}{c_{*}}/\mathrm{d}{t} }{c_{*}} - \frac12 c_{*}^2\kappa(c_{*}) \left( p + \rho \right)   $$

$$ \frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) =  \frac{\left( \mathrm{d^2}{a}/\mathrm{d}{t}^2 \right)}{a} - \frac{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)^2}{a^2}$$

$$ \frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = \frac{kc_{*}^2}{a^2} + \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \frac{ \mathrm{d}{c_{*}}/\mathrm{d}{t} }{c_{*}} - \frac12 c_{*}^2\kappa(c_{*}) \left( p + \rho \right) $$

first friedmann equation
$$ \left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = -\frac{kc_{*}^2}{a^2} + \frac13 c_{*}^2 \Lambda(c_{*})   + \frac13 c_{*}^2\kappa(c_{*})\rho $$

$$ \left( \frac{ \mathrm{d}{a}/\mathrm{d}{\tau} }{a} \right)^2 \left( \frac{ \mathrm{d}{\tau}} {\mathrm{d}{t} } \right)^2 = -\frac{kc_{*}^2}{a^2} + \frac13 c_{*}^2 \Lambda(c_{*})   + \frac13 c_{*}^2\kappa(c_{*})\rho $$

$$ \left( \frac{ \mathrm{d}{a}/\mathrm{d}{\tau} }{a} \right)^2 \left( \frac{ c_{*} } { c_{} } \right)^2 = -\frac{kc_{*}^2}{a^2} + \frac13 c_{*}^2 \Lambda(c_{*})   + \frac13 c_{*}^2\kappa(c_{*})\rho $$

$$ \left( \frac{ \mathrm{d}{a}/\mathrm{d}{\tau} }{a} \right)^2 = -\frac{kc_{}^2}{a^2} + \frac13 c^2 \Lambda(c_{*})   + \frac13 c_{}^2\kappa(c_{*})\rho $$

$$ H^2 = -\frac{kc^2}{a^2} + \frac13 c_{}^2 \Lambda(c_*) + \frac13 c_{}^2\kappa(c_*)\rho$$

second friedmann equation
$$ \frac{ \mathrm{d} } {\mathrm{d}{t} } \left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = \frac{kc_{*}^2}{a^2} + \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \ \frac{ \mathrm{d}{c_{*}}/\mathrm{d}{t} }{c_{*}} - \frac12 c_{*}^2\kappa(c_{*}) \left( p + \rho \right) $$

$$ \frac{ \mathrm{d} } {\mathrm{d}{\tau} } \left( \frac{ \mathrm{d}{a}/\mathrm{d}{\tau} }{a} \ \frac{ \mathrm{d}{\tau}} {\mathrm{d}{t} } \right) \frac{ \mathrm{d}{\tau}} {\mathrm{d}{t} } = \frac{kc_{*}^2}{a^2} + \frac{ \mathrm{d}{a}/\mathrm{d}{\tau} }{a} \ \frac{ \mathrm{d}{\tau}} {\mathrm{d}{t} } \

\frac{ \mathrm{d}{c_{*}}/\mathrm{d}{\tau} }{c_{*}} \ \frac{ \mathrm{d}{\tau}} {\mathrm{d}{t} }

- \frac12 c_{*}^2\kappa(c_{*}) \left( p + \rho \right) $$

$$ \frac{ \mathrm{d} } {\mathrm{d}{\tau} } \left( \frac{ \mathrm{d}{a}/\mathrm{d}{\tau} }{a} \ \frac{ c_{*} } { c_{} } \right) \frac{ c_{*} } { c_{} } = \frac{kc_{*}^2}{a^2} + \frac{ \mathrm{d}{a}/\mathrm{d}{\tau} }{a} \ \frac{ c_{*} } { c_{} } \ \frac{ \mathrm{d}{c_{*}}/\mathrm{d}{\tau} }{c_{*}} \ \frac{ c_{*}} { c_{} }

- \frac12 c_{*}^2\kappa(c_{*}) \left( p + \rho \right) $$

$$ \frac{ \mathrm{d} } {\mathrm{d}{\tau} } \left( \frac{ \mathrm{d}{a}/\mathrm{d}{\tau} }{a}  \right) \frac{ c_{*}^2} { c_{}^2 } + \frac{ \mathrm{d}{a}/\mathrm{d}{\tau} }{a} \ \frac{  \mathrm{d} } {\mathrm{d}{\tau} } \left( \frac{ c_{*} } { c_{} } \right) \frac{ c_{*} } { c_{} } = \frac{kc_{*}^2}{a^2} + \frac{ \mathrm{d}{a}/\mathrm{d}{\tau} }{a} \ \frac{ c_{*} } { c_{} } \ \frac{ \mathrm{d}{c_{*}}/\mathrm{d}{\tau} }{c_{}}

- \frac12 c_{*}^2\kappa(c_{*}) \left( p + \rho \right) $$

$$ \frac{ \mathrm{d} } {\mathrm{d}{\tau} } \left( \frac{ \mathrm{d}{a}/\mathrm{d}{\tau} }{a}  \right) \frac{ c_{*}^2} { c_{}^2 } + \frac{ \mathrm{d}{a}/\mathrm{d}{\tau} }{a} \ \frac{ \mathrm{d}{c_{*}}/\mathrm{d}{\tau} } { c_{} } \ \frac{ c_{*} } { c_{} } = \frac{kc_{*}^2}{a^2} + \frac{ \mathrm{d}{a}/\mathrm{d}{\tau} }{a} \ \frac{ c_{*} } { c_{} } \ \frac{ \mathrm{d}{c_{*}}/\mathrm{d}{\tau} }{c_{}}

- \frac12 c_{*}^2\kappa(c_{*}) \left( p + \rho \right) $$

$$ \frac{ \mathrm{d} } {\mathrm{d}{\tau} } \left( \frac{ \mathrm{d}{a}/\mathrm{d}{\tau} }{a}  \right) \frac{ c_{*}^2} { c_{}^2 } = \frac{kc_{*}^2}{a^2} - \frac12 c_{*}^2\kappa(c_{*}) \left( p + \rho \right) $$

$$ \frac{ \mathrm{d} } {\mathrm{d}{\tau} } \left( \frac{ \mathrm{d}{a}/\mathrm{d}{\tau} }{a}  \right) = \frac{kc_{}^2}{a^2} - \frac12 c_{}^2\kappa(c_{*}) \left( p + \rho \right) $$

$$ \dot{H} = \frac{kc^2}{a^2} - \frac12 c_{}^2\kappa(c_*) \left( p + \rho \right) $$

Conservation Of Energy
What does "conservation of energy" mean in an expanding universe?

The expanding universe seems to have its own arcane view of what "conservation of energy" actually means. That the energy density of matter should fall as the third power of the spatial scale factor seems to be fairly straight forward. That the energy density of radiation should fall as the fourth power of the spatial scale factor is harder to deal with, and what about vacuum energy?

To understand these matters properly we need to turn to the Continuity Equation. The Continuity Equation in the absence of heat transfer will surfice for our purposes, since under the assumptions of the Friedmann Lemaitre Robertson Walker model there is no heat transfer.

Energy Density And Pressure As Functions Of The Two Scale Factors
$$ \text{ } $$

$$ \text{The two scale factors are functions of co-ordinate time. It will be useful to view the other functions of co-ordinate time as functions of the two scale factors. } $$

$$ \frac{d}{d{t}} = \frac{d{a}}{d{t}}\frac{\partial}{\partial{a}} + \frac{d{c_{*}}}{d{t}}\frac{\partial}{\partial{c_{*}}} = \left( \mathrm{d}{a}/\mathrm{d}{t} \right)\frac{\partial}{\partial{a}} + \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)\frac{\partial}{\partial{c_{*}}} $$

$$ \text{Energy is conserved for changes in the spatial scale factor but not for changes in the temporal scale factor.} $$

$$ \text{The continuity equation in the absence of heat transfer, where }p \text{ is pressure, }V \text{ is volume, and }U \text{ is the energy content of that volume is:} $$

$$ p d{V} + d{U} = 0 \ \ \ \ $$

$$ \text{Using partial derivatives in the spatial scale factor, where energy is conserved. } $$

$$ p\frac{\partial{V}}{\partial{a}} + \frac{\partial{U}}{\partial{a}} = 0 $$

$$ \text{If }a^3 \text{ is used for the volume then the energy is }\rho a^3 \text{ giving } $$

$$ p\frac{\partial}{\partial{a}} \left( a^3 \right) + \frac{\partial}{\partial{a}} \left( \rho a^3 \right) = 0 $$

$$ 3 a^2p + 3a^2\rho  + \frac{\partial{\rho}}{\partial{a}} a^3 = 0 $$

$$ 3 \left( p + \rho \right) + \frac{\partial{\rho}}{\partial{a}} a = 0 $$

$$ 3H \left( p + \rho \right) + \frac{\partial{\rho}}{\partial{a}} \left( \mathrm{d}{a}/\mathrm{d}{t} \right) = 0 $$

Applying this to the Friedmann Equations
$$\frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = \frac{kc_{*}^2}{a^2} + \left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \frac12 c_{*}^2\kappa(c_{*}) \left( p + \rho \right)   $$

$$2H\frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = 2H\frac{kc_{*}^2}{a^2} + 2H^2\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) c_{*}^2\kappa(c_{*})\left( p + \rho \right)   $$

$$ \left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = -\frac{kc_{*}^2}{a^2} + \frac13 \left( c_{*}^2 \kappa(c_{*}) - c_{*}^2\kappa(c_{})  \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 c_{*}^2\kappa(c_{*})\rho$$

$$ 2H\frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = \frac{d}{d{t}} \left( -\frac{kc_{*}^2}{a^2} + \frac13 \left( c_{*}^2 \kappa(c_{*}) - c_{*}^2\kappa(c_{}) \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 c_{*}^2\kappa(c_{*})\rho \right) $$

$$ 2H\frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = \left( \left( \mathrm{d}{a}/\mathrm{d}{t} \right)\frac{\partial}{\partial{a}} + \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)\frac{\partial}{\partial{c_{*}}} \right) \left( -\frac{kc_{*}^2}{a^2} + \frac13 \left( c_{*}^2 \kappa(c_{*}) - c_{*}^2\kappa(c_{}) \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 c_{*}^2\kappa(c_{*})\rho \right) $$

$$ 2H\frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) =

2\left( \mathrm{d}{a}/\mathrm{d}{t} \right)\frac{kc_{*}^2}{a^3} + \frac13 c_{*}^2\kappa(c_{*}) \frac{\partial{\rho}}{\partial{a}} \left( \mathrm{d}{a}/\mathrm{d}{t} \right) + \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)\frac{\partial}{\partial{c_{*}}} \left( -\frac{kc_{*}^2}{a^2} + \frac13 \left( c_{*}^2 \kappa(c_{*}) - c_{*}^2\kappa(c_{}) \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 c_{*}^2\kappa(c_{*})\rho \right) $$

$$ 2H\frac{kc_{*}^2}{a^2} + 2H^2\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} - \left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) c_{*}^2\kappa(c_{*})\left( p + \rho \right) = 2\left( \mathrm{d}{a}/\mathrm{d}{t} \right)\frac{kc_{*}^2}{a^3} + \frac13 c_{*}^2\kappa(c_{*}) \frac{\partial{\rho}}{\partial{a}} \left( \mathrm{d}{a}/\mathrm{d}{t} \right) + \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)\frac{\partial}{\partial{c_{*}}} \left( -\frac{kc_{*}^2}{a^2} + \frac13 \left( c_{*}^2 \kappa(c_{*}) - c_{*}^2\kappa(c_{}) \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 c_{*}^2\kappa(c_{*})\rho \right) $$

$$ 2H^2\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} - \left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) c_{*}^2\kappa(c_{*})\left( p + \rho \right) = \frac13 c_{*}^2\kappa(c_{*}) \frac{\partial{\rho}}{\partial{a}} \left( \mathrm{d}{a}/\mathrm{d}{t} \right) + \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)\frac{\partial}{\partial{c_{*}}} \left( -\frac{kc_{*}^2}{a^2} + \frac13 \left( c_{*}^2 \kappa(c_{*}) - c_{*}^2\kappa(c_{}) \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 c_{*}^2\kappa(c_{*})\rho \right) $$

$$ 2H^2\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} = \left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) c_{*}^2\kappa(c_{*})\left( p + \rho \right) + \frac13 c_{*}^2\kappa(c_{*}) \frac{\partial{\rho}}{\partial{a}} \left( \mathrm{d}{a}/\mathrm{d}{t} \right) + \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)\frac{\partial}{\partial{c_{*}}} \left( -\frac{kc_{*}^2}{a^2} + \frac13 \left( c_{*}^2 \kappa(c_{*}) - c_{*}^2\kappa(c_{}) \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 c_{*}^2\kappa(c_{*})\rho \right) $$

$$ 2H^2\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} = \frac13 c_{*}^2\kappa(c_{*}) \left( 3H \left( p + \rho \right) + \frac{\partial{\rho}}{\partial{a}} \left( \mathrm{d}{a}/\mathrm{d}{t} \right) \right) + \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)\frac{\partial}{\partial{c_{*}}} \left( -\frac{kc_{*}^2}{a^2} + \frac13 \left( c_{*}^2 \kappa(c_{*}) - c_{*}^2\kappa(c_{}) \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 c_{*}^2\kappa(c_{*})\rho \right) $$

$$ 3H \left( p + \rho \right) + \frac{\partial{\rho}}{\partial{a}} \left( \mathrm{d}{a}/\mathrm{d}{t} \right) = 0 \ \ \ \ \text{Because of conservation of energy. Therefore} $$

$$ 2H^2\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} = \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)\frac{\partial}{\partial{c_{*}}} \left( -\frac{kc_{*}^2}{a^2} + \frac13 \left( c_{*}^2 \kappa(c_{*}) - c_{*}^2\kappa(c_{}) \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 c_{*}^2\kappa(c_{*})\rho \right) $$

$$ \frac{2}{c_{*}} \left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = \frac{\partial}{\partial{c_{*}}} \left( -\frac{kc_{*}^2}{a^2} + \frac13 \left( c_{*}^2 \kappa(c_{*}) - c_{*}^2\kappa(c_{}) \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 c_{*}^2\kappa(c_{*})\rho \right) $$

$$ \frac{2}{c_{*}} \left( -\frac{kc_{*}^2}{a^2} + \frac13 \left( c_{*}^2 \kappa(c_{*}) - c_{*}^2\kappa(c_{}) \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 c_{*}^2\kappa(c_{*})\rho \right) = \frac{\partial}{\partial{c_{*}}} \left( -\frac{kc_{*}^2}{a^2} + \frac13 \left( c_{*}^2 \kappa(c_{*}) - c_{*}^2\kappa(c_{}) \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 c_{*}^2\kappa(c_{*})\rho \right) $$

$$ \frac{2}{c_{*}} \left( \frac13 c_{*}^2 \kappa(c_{*}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \frac13 c_{*}^2\kappa(c_{*})\rho \right) = \frac{\partial}{\partial{c_{*}}} \left( \frac13 c_{*}^2 \kappa(c_{*}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \frac13 c_{*}^2\kappa(c_{*})\rho \right) $$

$$ \frac{2}{c_{*}} \left( \frac13 c_{*}^2 \kappa(c_{*}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \frac13 c_{*}^2\kappa(c_{*})\rho \right) = \frac{2}{c_{*}} \left( \frac13 c_{*}^2 \kappa(c_{*}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \frac13 c_{*}^2\kappa(c_{*})\rho \right) +

\frac13 c_{*}^2 \frac{\partial}{\partial{c_{*}}} \kappa(c_{*}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \frac13 c_{*}^2 \frac{\partial}{\partial{c_{*}}} \left( \kappa(c_{*})\rho \right) $$

$$ 0 = \frac13 c_{*}^2 \frac{\partial}{\partial{c_{*}}} \left( \kappa(c_{*}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho}  + \kappa(c_{*})\rho \right) $$

$$ \frac{\partial}{\partial{c_{*}}} \left( \kappa(c_{*}) \overset{{\ }_{\boldsymbol{\circ}}}{\rho}  + \kappa(c_{*})\rho \right) = 0 $$

$$ \frac{\partial{\left(\kappa(c_{*})\right)}}{\partial{c_{*}}} \left(  \overset{{\ }_{\boldsymbol{\circ}}}{\rho}  + \rho \right) + \kappa(c_{*}) \frac{\partial{\rho}}{\partial{c_{*}}} = 0 $$

$$ \frac{\partial {\kappa(c_{*})} / \partial{c_{*}}}{\kappa(c_{*})} \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right) \left(  \overset{{\ }_{\boldsymbol{\circ}}}{\rho}  + \rho \right) +  \frac{\partial{\rho}}{\partial{c_{*}}} \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right) = 0 $$

$$ \frac{\frac{\mathrm{d} {\kappa(c_{*})} / \mathrm{d}{t}}{\kappa(c_{*})} }{\kappa(c_{*})} \left(  \overset{{\ }_{\boldsymbol{\circ}}}{\rho}  + \rho \right) +  \frac{\partial{\rho}}{\partial{c_{*}}} \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right) = 0 $$

$$ \frac{\frac{\mathrm{d} {\kappa(c_{*})} / \mathrm{d}{t}}{\kappa(c_{*})} }{\kappa(c_{*})} \left(  \overset{{\ }_{\boldsymbol{\circ}}}{\rho}  + \rho_{max} \right) +  \frac{\partial{\rho_{max}}}{\partial{c_{*}}} \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right) = 0 $$

$$ \frac{d{\rho_{max}}}{d{t}} = \frac{\partial{\rho_{max}}}{\partial{a}} \left( \mathrm{d}{a}/\mathrm{d}{t} \right) + \frac{\partial{\rho_{max}}}{\partial{c_{*}}} \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right) = 0 $$

$$ \frac{\partial{\rho_{max}}}{\partial{c_{*}}} \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right) = - \frac{\partial{\rho_{max}}}{\partial{a}} \left( \mathrm{d}{a}/\mathrm{d}{t} \right) $$

$$ \text{From the continuity equation we know that} \ \ \ \ 3H \left( p_{max} + \rho_{max} \right) + \frac{\partial{\rho_{max}}}{\partial{a}} \left( \mathrm{d}{a}/\mathrm{d}{t} \right) = 0 \ \ \ \ \text{therefore} $$

$$ \frac{\partial{\rho_{max}}}{\partial{c_{*}}} \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right) = 3H \left( p_{max} + \rho_{max} \right) $$

$$ \frac{\frac{\mathrm{d} {\kappa(c_{*})} / \mathrm{d}{t}}{\kappa(c_{*})} }{\kappa(c_{*})} \left(  \overset{{\ }_{\boldsymbol{\circ}}}{\rho}  + \rho_{max} \right) +  3H \left( p_{max} + \rho_{max} \right) = 0 $$

$$ \left(  \overset{{\ }_{\boldsymbol{\circ}}}{\rho}  + \rho_{max} \right) \frac{\frac{\mathrm{d} {\kappa(c_{*})} / \mathrm{d}{t}}{\kappa(c_{*})} }{\kappa(c_{*})} = -3 \left( p_{max} + \rho_{max} \right)\frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} $$

$$ \left(  \overset{{\ }_{\boldsymbol{\circ}}}{\rho}  + \rho_{max} \right) \int \frac{\frac{\mathrm{d} {\kappa(c_{*})} / \mathrm{d}{t}}{\kappa(c_{*})} }{\kappa(c_{*})} d{t} = -3 \left( p_{max} + \rho_{max} \right) \int \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} d{t} $$

$$ \left(  \overset{{\ }_{\boldsymbol{\circ}}}{\rho}  + \rho_{max} \right) \ln{\left( \kappa(c_{*}) \right)} = -3 \left( p_{max} + \rho_{max} \right) \ln{\left( a \right)} + C $$

$$ \kappa(c_{*}) ^ \left(  \overset{{\ }_{\boldsymbol{\circ}}}{\rho}  + \rho_{max} \right) \propto a ^ {-3 \left( p_{max} + \rho_{max} \right)} $$ $$

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JUNK


mbox{ 1 \over 2 }



JUNK2


\frac{ 1 } {\sqrt{ c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} $$

$$ \text{is a constant with the dimensions of time and a value close to one fundamental unit.} $$

$$\int \csc_{*}^n{ax} \, dx = -\frac{\csc_{*}^{n-2}{ax} \cot{ax}}{a(n-1)} \,+\, \frac{n-2}{n-1}\int \csc_{*}^{n-2}{ax} \, dx \qquad \mbox{ (for }n \ne 1\mbox{)}$$

$$\int \csc_{*}^{-n}{ax} \, dx = -\frac{\csc_{*}^{-n-2}{ax} \cot{ax}}{a(-n-1)} \,+\, \frac{-n-2}{-n-1}\int \csc_{*}^{-n-2}{ax} \, dx \qquad \mbox{ (for }-n \ne 1\mbox{)}$$


 * $$\int \sin^{n}{ax} \, dx = -\frac{\sin^{n+2}{ax} \cot{ax}}{a(-n-1)} \,+\, \frac{-n-2}{-n-1}\int \sin^{n+2}{ax} \, dx \qquad \mbox{ (for }-n \ne 1\mbox{)}$$


 * $$\int \sin^{n}{ax} \, dx = \frac{\sin^{n+2}{ax} \cot{ax}}{a(n+1)} \,+\, \frac{n+2}{n+1}\int \sin^{n+2}{ax} \, dx \qquad \mbox{ (for }n \ne -1\mbox{)}$$


 * $$\int \sin^{n}{ax} \, dx = \frac{\sin^{n+1}{ax} \cos{ax}}{a(n+1)} \,+\, \frac{n+2}{n+1}\int \sin^{n+2}{ax} \, dx \qquad \mbox{ (for }n \ne -1\mbox{)}$$



{}_1^2\!\Omega_3^4  \ \ \ \ \tau \tau \ \!\tau \ \ \tau\ \ {}_1^2\Omega_3^4   \ \}\{    \$\# $$

JUNK2


\frac{ 1 } {\sqrt{ c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} $$

$$ \text{is a constant with the dimensions of time and a value close to one fundamental unit.} $$

$$\int \csc_{*}^n{ax} \, dx = -\frac{\csc_{*}^{n-2}{ax} \cot{ax}}{a(n-1)} \,+\, \frac{n-2}{n-1}\int \csc_{*}^{n-2}{ax} \, dx \qquad \mbox{ (for }n \ne 1\mbox{)}$$

$$\int \csc_{*}^{-n}{ax} \, dx = -\frac{\csc_{*}^{-n-2}{ax} \cot{ax}}{a(-n-1)} \,+\, \frac{-n-2}{-n-1}\int \csc_{*}^{-n-2}{ax} \, dx \qquad \mbox{ (for }-n \ne 1\mbox{)}$$


 * $$\int \sin^{n}{ax} \, dx = -\frac{\sin^{n+2}{ax} \cot{ax}}{a(-n-1)} \,+\, \frac{-n-2}{-n-1}\int \sin^{n+2}{ax} \, dx \qquad \mbox{ (for }-n \ne 1\mbox{)}$$


 * $$\int \sin^{n}{ax} \, dx = \frac{\sin^{n+2}{ax} \cot{ax}}{a(n+1)} \,+\, \frac{n+2}{n+1}\int \sin^{n+2}{ax} \, dx \qquad \mbox{ (for }n \ne -1\mbox{)}$$


 * $$\int \sin^{n}{ax} \, dx = \frac{\sin^{n+1}{ax} \cos{ax}}{a(n+1)} \,+\, \frac{n+2}{n+1}\int \sin^{n+2}{ax} \, dx \qquad \mbox{ (for }n \ne -1\mbox{)}$$



{}_1^2\!\Omega_3^4  \ \ \ \ \tau \tau \ \!\tau \ \ \tau\ \ {}_1^2\Omega_3^4   \ \}\{    \$\# $$

Co-ordinate Curvature
$$ \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} = \left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = \sqrt{\frac13 c_{}^2 \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\cos{(\chi)}$$

$$ \frac{d{\chi}}{dt} = \frac{\frac12 c_{}^2\kappa(c_{})\left( p_{max} + \rho_{max} \right)}{\sqrt{\frac13 c_{}^2 \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }}\sin{(\chi)}  $$

$$ \frac{\left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} = \frac{ \cos(\chi)}{ \sin(\chi)} \frac{d{\chi}}{dt} $$

$$ \frac{\left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} = \frac{ \cos(\chi)}{ \sin(\chi)} \frac{\frac12 c_{}^2\kappa(c_{})\left( p_{max} + \rho_{max} \right)}{\sqrt{\frac13 c_{}^2 \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }}\sin{(\chi)} $$

$$ \frac{\left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} =  \frac{\frac12 c_{}^2\kappa(c_{})\left( p_{max} + \rho_{max} \right)}{\sqrt{\frac13 c_{}^2 \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }}\cos(\chi) $$

$$ \frac{\left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)/c_{*}}{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)/a} =  \frac{\frac{\frac12 c_{}^2\kappa(c_{})\left( p_{max} + \rho_{max} \right)}{\sqrt{\frac13 c_{}^2 \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }}\cos(\chi)} {\sqrt{\frac13 c_{}^2 \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\cos{(\chi)}}$$

$$ \frac{\left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)/c_{*}}{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)/a}  =  \frac{\frac12 c_{}^2\kappa(c_{})\left( p_{max} + \rho_{max} \right)} {\frac13 c_{}^2 \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) } $$

$$ \frac{\left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)/c_{*}}{\left( \mathrm{d}{a}/\mathrm{d}{t} \right)/a}  =  \frac32 \frac{ p_{max} + \rho_{max}} { \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max}   } $$

$$ \frac{\frac{d}{dt}\ln{c_{*}}}{\frac{d}{dt}\ln{a}}  =  \frac32 \frac{ p_{max} + \rho_{max}} { \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max}   } $$

The Vacuum Catastrophe -- Full Blast
the superluminal expansion will effectively wash out any curvature term in the metric

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The Radiation Dominated Era
The universe now expands subluminally thus consuming excess scale factor.

as the expansion continues radiation loses its dominance and matter takes over

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The Matter Dominated Era
The universe expands more quickly but still subluminally and continues to use up excess scale factor

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The Present Era
Dark Energy.

Re-write for Unruh Energy.

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Fundamental Units Revisited
Mks units are formed from metres, Kilograms, and seconds. Planck Units are formed from Maxwell's Constant, Planck's Constant, and Newton's gravitational Constant. Compton Schwarzschild Units are formed from Maxwell's Constant, Planck's Constant, and Schwarzschild's gravitational Constant. But is there a better way. Are there three quantities that are causes and all the rest are effects. It would seem that the most likely candidates for causes would be the zero point values. It is hard to see them being effects. So let us take the Zero Point Action Flow Density Of The Vacuum as one of the causes and the Zero Point Curvature Of Space as another. (Space rather than space-time because we expect the fourth dimension to be non-uniform.)

What then is to be the third leg of our stool. The ratio of the Zero Point Curvature Of Space and the Zero Point Action Flow Density Of The Vacuum forms some sort of gravitational constant but this cannot be the third leg of our stool because it is not independent. What would seem to fit the bill is something that differs from this ratio by a factor of a power of c_{*}

The plan of action then is to use these three values and both Friedmann equations to determine the evolutions of the scale factor and the time factor, the aspect ratio of space and the fourth dimension.

$$NUM\kappa.c_{*}^n = \frac{\overset{{}_{\boldsymbol{\circ}}}{\Lambda}}{ \overset{{\ }_{\boldsymbol{\circ}}}{\rho}} $$

c_{*} is the value of the Time Factor( $$ \tilde{c_{*}} $$ ) where the two zero point values cancel each other out.

\kappa is the third leg of our stool, c_{*} is Maxwell's Constant, NUM is a simple numerical value 8pi 4pi or some other, and n is the power of c_{*} by which the gravitational constant differs from the ratio of The ratio of the Zero Point Curvature Of Space to the Zero Point Action Flow Density Of The Vacuum

It is written as n in the hope that it is a whole number.

Fundamental units
$$\text{What are fundamental units and what is the numerical value of } \kappa $$

I. R. Kenyon, in his book, "General Relativity" (Chapter 12, The path to quantum \kappaity), attempts a derivation of fundamental units by finding a mass which has its Compton Wavelength equal to its schwarzschild Radius, and thereby arrives at Planck Units. However, he only does this by omitting a factor of two. Something done without explanation. The only justification I can find for this action is in order to derive the aforementioned Planck Units. The basic Planck Units are found by taking Maxwell's constant(c_{*}), Planck's reduced constant($$\hbar$$), and Newton's gravitational Constant(G), and then using them in combination to get quantaties of the right dimensions. Now I have nothing against Newton. Smashing bloke, don't know where we'd be without him, but I can't help noticing that his theory of \kappaition has been superceded, and I therefore ask if it is appropriate to use his gavitational constant to derive fundamental units.

$$$$

Compton Schwartchild Units.
So, following Dr. Kenyon's programme, but this time keeping the factor of two.

The Compton Wavelength is given by $$\frac{\hbar}{Mc}$$

and the schwarzschild Radius by $$\frac{2GM}{c_{*}^2}$$

Where the Compton Wavelength meets the schwarzschild Radius.

$$\frac{2GM_{cs}}{c_{*}^2}=\frac{\hbar}{M_{cs}c_{*}}$$

$$M_{cs}^2=\frac{\hbar c_{*}}{2G}$$

$$M_{cs}=\sqrt{\frac{\hbar c_{*}}{2G}}$$

$$l_{cs}=\sqrt{\frac{2G\hbar}{c_{*}^3}}$$

$$t_{cs}=\sqrt{\frac{2G\hbar}{c_{*}^5}}$$

$$E_{cs}=\sqrt{\frac{\hbar c_{*}^5}{2G}}$$

Compton Schwarzschild Units can be constructed from: Maxwell's Constant $$c_{*}$$, Planck's Reduced Constant $$\hbar$$, and what I shall call Schwarzschild's gravitational Constant. Schwarzschild's gravitational Constant is twice the size of Newton's gravitational Constant and to avoid confusion I will write it as "$$2G$$".

Compton Schwarzschild units give $$\kappa$$(kappa) a numerical value of $$4 \pi$$. If we consider the electrostatic force then the permittivity of free space$$ (\epsilon_0)$$ is defined in terms of the forces between parallel plates and when it comes to the forces between point charges a factor of $$4 \pi$$ is involved. If we think of \kappaity as being analogous to point charges then acceleration is analogous to parallel plates.

The Gaussian curvature of the surface of a sphere of radius $$r$$ is $$\frac{1}{r^2}$$. Perhaps a more natural measure of curvature would be $$\frac{1}{4\pi r^2}$$ i.e. the reciprocal of the area. This would seem to be particularly apt for the curvature of space time because the radius of curvature of the cuvature of space time is area radius.

A Possible Beginning
$$\text{For no or negligible curvature} $$

If we make the assumption that the Left Over Vacuum Energy starts out large enough to max out the vacuum then we may make the simplifying assumption that in the earliest epoch \rho and p were constant.

$$ \left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = \frac13 \left( c_{*}^2. \kappa(c_{*}) - c_{*}^2\kappa(c_{})  \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 c_{*}^2\kappa(c_{*})\rho_{max}$$

$$ \frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = \frac{ \left( \mathrm{d}{a}/\mathrm{d}{t} \right)}{a}\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \frac12 c_{*}^2\kappa(c_{*}).p_{max} - \frac12 c_{*}^2\kappa(c_{*})\rho_{max}   $$

note the Free Lunch term in the alternative form of the second Friedmann equation.

$$\text{In the simplest case of dependence } c_{*}^2\kappa(c_{*}) \text{ would be constant and would therefore be equal to } c_{}^2\kappa(c_{})$$

$$ \frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = \left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \frac12 c_{}^2\kappa(c_{}).p_{max} - \frac12 c_{}^2\kappa(c_{})\rho_{max}   $$

$$ \left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = \frac13 \left( c_{}^2. \kappa(c_{}) - c_{*}^2\kappa(c_{})  \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 c_{}^2\kappa(c_{})\rho_{max}$$

$$ \left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = \frac13 \kappa(c_{})\left( c_{}^2\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + c_{}^2\rho_{max} - c_{*}^2\overset{{\ }_{\boldsymbol{\circ}}}{\rho}  \right) $$

$$ \left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = \frac13 c_{}^2 \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) \left( 1 -  \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max}}\frac{c_{*}^2}{c_{}^2} \right)$$

$$ \text{If we define} \ \ \chi \ \text{such that} \ \ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max}}\frac{c_{*}^2}{c_{}^2} = \sin^2{(\chi)} \ \ \text{ then }$$

$$ \left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = \frac13 c_{}^2 \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) \cos^2{(\chi)}$$

$$ \left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = \sqrt{\frac13 c_{}^2 \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\cos{(\chi)}$$

$$ \frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = -\sqrt{\frac13 c_{}^2 \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt}$$

$$ \frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = \left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \frac12 c_{}^2\kappa(c_{})\left( p_{max} + \rho_{max} \right)   $$

$$ -\sqrt{\frac13 c_{}^2 \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} = \sqrt{\frac13 c_{}^2 \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\cos{(\chi)}\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} - \frac12 c_{}^2\kappa(c_{})\left( p_{max} + \rho_{max} \right)   $$

$$ -\sqrt{\frac13 c_{}^2 \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} = \sqrt{\frac13 c_{}^2 \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\cos{(\chi)}\frac{ \cos(\chi)}{ \sin(\chi)} \frac{d{\chi}}{dt} - \frac12 c_{}^2\kappa(c_{})\left( p_{max} + \rho_{max} \right)   $$

$$ \frac12 c_{}^2\kappa(c_{})\left( p_{max} + \rho_{max} \right) = \sqrt{\frac13 c_{}^2 \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\cos{(\chi)}\frac{ \cos(\chi)}{ \sin(\chi)} \frac{d{\chi}}{dt} + \sqrt{\frac13 c_{}^2 \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt}  $$

$$ \frac12 c_{}^2\kappa(c_{})\left( p_{max} + \rho_{max} \right) = \sqrt{\frac13 c_{}^2 \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\frac{1}{\sin{(\chi)}} \frac{d{\chi}}{dt}  $$

$$ \frac{dt}{d{\chi}}  = \frac{\sqrt{\frac13 c_{}^2 \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }}{\frac12 c_{}^2\kappa(c_{})\left( p_{max} + \rho_{max} \right)}\frac{1}{\sin{(\chi)}}  $$


 * $$\int \frac{1}{\sin{(\chi)}} \, d\chi= -\ln{\left| \frac{1+\cos{(\chi)}}{\sin{(\chi)}}\right|}+C$$


 * $$\int \frac{1}{\sin{(\chi)}} \, d\chi= \ln{\left| \tan{(\chi/2)}\right|}+C$$


 * $$\int \frac{1}{\sin{(\chi)}} \, d\chi= \ln{\left| \tan{\left(\frac{\chi}{2}\right)}\right|}+C$$

$$ c_{}d\tau = cd\mathbf{t} $$

$$ \frac{c_{*}}{c_{}} = \sqrt{\frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max}}{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \sin{(\chi)} $$

$$\frac{d\tau}{d{\chi}} = \frac{d\tau}{d\mathbf{t}}\frac{d\mathbf{t}}{d{\chi}} = \frac{c_{*}}{c_{}}\frac{d\mathbf{t}}{d{\chi}} = \sqrt{\frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max}}{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \sin{(\chi)} \frac{\sqrt{\frac13 c_{}^2 \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }}{\frac12 c_{}^2\kappa(c_{})\left( p_{max} + \rho_{max} \right)} \frac{1}{\sin{(\chi)}} =  \frac{1 } {\sqrt{\frac34 c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \frac{\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }{\left( p_{max} + \rho_{max} \right)}$$


 * $$\int_{0}^{\tau} d\tau = \int_{0}^{\frac{\pi}{2} }

\frac{1 } {\sqrt{\frac34 c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \frac{\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }{\left( p_{max} + \rho_{max} \right)} d\chi $$


 * $$\tau =

\frac{\frac{\pi}{2} } {\sqrt{\frac34 c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \frac{\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }{\left( p_{max} + \rho_{max} \right)} $$



\frac{ 1 } {\sqrt{ c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} $$

A Possible Beginning2
$$\text{For no or negligible curvature} $$

If we make the assumption that the Left Over Vacuum Energy starts out large enough to max out the vacuum then we may make the simplifying assumption that in the earliest epoch \rho and p were constant.

$$ \left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = \frac13 \left( c_{*}^2. \kappa(c_{*}) - c_{*}^2\kappa(c_{})  \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 c_{*}^2\kappa(c_{*})\rho_{max}$$

$$ \frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = \frac{ \left( \mathrm{d}{a}/\mathrm{d}{t} \right)}{a}\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \frac12 c_{*}^2\kappa(c_{*}).p_{max} - \frac12 c_{*}^2\kappa(c_{*})\rho_{max}   $$

$$\text{In this case of dependence } c_{*}^{2n}\kappa(c_{*}) \text{ would be constant and would therefore be equal to } c_{}^{2n}\kappa(c_{})$$

$$ c_{*}^{2n-2}\frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = c_{*}^{2n-2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \frac12 c_{}^{2n}\kappa(c_{}).p_{max} - \frac12 c_{}^{2n}\kappa(c_{})\rho_{max}   $$

$$ c_{*}^{2n-2}\left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = \frac13 \left( c_{}^{2n}. \kappa(c_{}) - c_{*}^{2n}\kappa(c_{})  \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 c_{}^{2n}\kappa(c_{})\rho_{max}$$

$$ c_{*}^{2n-2}\left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = \frac13 \kappa(c_{})\left( c_{}^{2n}\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + c_{}^{2n}\rho_{max} - c_{*}^{2n}\overset{{\ }_{\boldsymbol{\circ}}}{\rho}  \right) $$

$$ c_{*}^{2n-2}\left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = \frac13 c_{}^{2n} \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) \left( 1 -  \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max}}\frac{c_{*}^{2n}}{c_{}^{2n}} \right)$$

$$ \text{If we define} \ \ \chi \ \text{such that} \ \ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max}}\frac{c_{*}^{2n}}{c_{}^{2n}} = \sin^2{(\chi)} \ \ \text{ then }$$

$$ c_{*}^{2n-2}\left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = \frac13 c_{}^{2n} \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) \cos^2{(\chi)}$$

$$ c_{*}^{n-1}\left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = \sqrt{\frac13 c_{}^{2n} \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\cos{(\chi)}$$

$$ ({n-1})c_{*}^{n-1}\left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} + c_{*}^{n-1}\frac{  \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = -\sqrt{\frac13 c_{}^{2n} \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt}$$

$$ c_{*}^{n-1}\frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = -\sqrt{\frac13 c_{}^{2n} \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} - ({n-1})c_{*}^{n-1}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} $$

$$ c_{*}^{2n-2}\frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = c_{*}^{n-1}\left(-\sqrt{\frac13 c_{}^{2n} \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} - ({n-1})c_{*}^{n-1}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} \right)$$

$$ c_{*}^{2n-2}\frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = c_{*}^{2n-2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \frac12 c_{}^{2n}\kappa(c_{})\left( p_{max} + \rho_{max} \right)   $$

$$ c_{*}^{n-1}\left(-\sqrt{\frac13 c_{}^{2n} \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} - ({n-1})c_{*}^{n-1}\left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} \right) = c_{*}^{2n-2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \frac12 c_{}^{2n}\kappa(c_{})\left( p_{max} + \rho_{max} \right)   $$

$$ -c_{*}^{n-1}\sqrt{\frac13 c_{}^{2n} \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} - ({n-1})c_{*}^{2n-2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  = c_{*}^{2n-2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \frac12 c_{}^{2n}\kappa(c_{})\left( p_{max} + \rho_{max} \right)   $$

$$ -c_{*}^{n-1}\sqrt{\frac13 c_{}^{2n} \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} = ({n-1})c_{*}^{2n-2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} + c_{*}^{2n-2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \frac12 c_{}^{2n}\kappa(c_{})\left( p_{max} + \rho_{max} \right)   $$

$$ -c_{*}^{n-1}\sqrt{\frac13 c_{}^{2n} \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} =  n c_{*}^{2n-2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \frac12 c_{}^{2n}\kappa(c_{})\left( p_{max} + \rho_{max} \right)   $$

$$ -c_{*}^{n-1}\sqrt{\frac13 c_{}^{2n} \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt}  = n c_{*}^{n-1}\sqrt{\frac13 c_{}^{2n} \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\cos{(\chi)}\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \frac12 c_{}^{2n}\kappa(c_{})\left( p_{max} + \rho_{max} \right)   $$

$$ -c_{*}^{n-1}\sqrt{\frac13 c_{}^{2n} \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} = n c_{*}^{n-1}\sqrt{\frac13 c_{}^{2n} \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\cos{(\chi)} \frac1n\frac{ \cos(\chi)}{ \sin(\chi)} \frac{d{\chi}}{dt} - \frac12 c_{}^{2n}\kappa(c_{})\left( p_{max} + \rho_{max} \right)  $$

$$ -c_{*}^{n-1}\sqrt{\frac13 c_{}^{2n} \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} = c_{*}^{n-1}\sqrt{\frac13 c_{}^{2n} \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\cos{(\chi)} \frac{ \cos(\chi)}{ \sin(\chi)} \frac{d{\chi}}{dt} - \frac12 c_{}^{2n}\kappa(c_{})\left( p_{max} + \rho_{max} \right)  $$

$$ \frac12 c_{}^{2n}\kappa(c_{})\left( p_{max} + \rho_{max} \right) = c_{*}^{n-1}\sqrt{\frac13 c_{}^{2n} \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\cos{(\chi)} \frac{ \cos(\chi)}{ \sin(\chi)} \frac{d{\chi}}{dt} + c_{*}^{n-1}\sqrt{\frac13 c_{}^{2n} \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} $$

$$ \frac12 c_{}^{2n}\kappa(c_{})\left( p_{max} + \rho_{max} \right) = c_{*}^{n-1}\sqrt{\frac13 c_{}^{2n} \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) } \left( \cos{(\chi)} \frac{ \cos(\chi)}{ \sin(\chi)} \frac{d{\chi}}{dt} + \sin{(\chi)}\frac{d{\chi}}{dt} \right) $$

$$ \frac12 c_{}^{2n}\kappa(c_{})\left( p_{max} + \rho_{max} \right) = c_{*}^{n-1}\sqrt{\frac13 c_{}^{2n} \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) } \frac{1}{\sin{(\chi)}} \frac{d{\chi}}{dt} $$

$$ \frac{dt}{d{\chi}}  = \frac{\sqrt{\frac13 c_{}^{2n} \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }}{\frac12 c_{}^{2n}\kappa(c_{})\left( p_{max} + \rho_{max} \right)}\frac{c_{*}^{n-1}}{\sin{(\chi)}}  $$

$$ \frac{dt}{d{\chi}}  = \frac{\sqrt{\frac13 c_{}^2 \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }}{\frac12 c_{}^2\kappa(c_{})\left( p_{max} + \rho_{max} \right)}\frac{1}{\sin{(\chi)}}  $$


 * $$\int \frac{1}{\sin{(\chi)}} \, d\chi= -\ln{\left| \frac{1+\cos{(\chi)}}{\sin{(\chi)}}\right|}+C$$

$$ c_{}d\tau = cd\mathbf{t} $$

$$ \frac{c_{*}}{c_{}} = \sqrt{\frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max}}{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \sin{(\chi)} $$

$$\frac{d\tau}{d{\chi}} = \frac{d\tau}{d\mathbf{t}}\frac{d\mathbf{t}}{d{\chi}} = \frac{c_{*}}{c_{}}\frac{d\mathbf{t}}{d{\chi}} = \sqrt{\frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max}}{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \sin{(\chi)} \frac{\sqrt{\frac13 c_{}^2 \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }}{\frac12 c_{}^2\kappa(c_{})\left( p_{max} + \rho_{max} \right)} \frac{1}{\sin{(\chi)}} =  \frac{1 } {\sqrt{\frac34 c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \frac{\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }{\left( p_{max} + \rho_{max} \right)}$$


 * $$\int_{0}^{\tau} d\tau = \int_{0}^{\frac{\pi}{2} }

\frac{1 } {\sqrt{\frac34 c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \frac{\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }{\left( p_{max} + \rho_{max} \right)} d\chi $$


 * $$\tau =

\frac{\frac{\pi}{2} } {\sqrt{\frac34 c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \frac{\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }{\left( p_{max} + \rho_{max} \right)} $$



\frac{ 1 } {\sqrt{ c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} $$

A Possible Beginning3
$$\text{For no or negligible curvature} $$

If we make the assumption that the Left Over Vacuum Energy starts out large enough to max out the vacuum then we may make the simplifying assumption that in the earliest epoch \rho and p were constant.

$$ \left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = \frac13 \left( c_{*}^2. \kappa(c_{*}) - c_{*}^2\kappa(c_{})  \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 c_{*}^2\kappa(c_{*})\rho_{max}$$

$$ \frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = \frac{ \left( \mathrm{d}{a}/\mathrm{d}{t} \right)}{a}\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \frac12 c_{*}^2\kappa(c_{*}).p_{max} - \frac12 c_{*}^2\kappa(c_{*})\rho_{max}   $$

$$\text{In we define } f \text{ such that }f^{2}(c_{*})\kappa(c_{*}) \text{ is constant and therefore equal to } f^{2}(c_{})\kappa(c_{}) \text{ then }$$

$$ \frac{f^2(c_{*})}{c_{*}^2}\frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \frac12 f^{2}(c_{})\kappa(c_{}).p_{max} - \frac12 f^{2}(c_{})\kappa(c_{})\rho_{max}   $$

$$ \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = \frac13   \left(   f^{2}(c_{}). \kappa(c_{})    - f^{2}  (c_{*})  \kappa(c_{})    \right)  \overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 f^{2}(c_{})\kappa(c_{})\rho_{max}$$

$$ \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = \frac13 \kappa(c_{})\left( f^{2}(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + f^{2}(c_{})\rho_{max} - f^{2}(c_{*})\overset{{\ }_{\boldsymbol{\circ}}}{\rho}  \right) $$

$$ \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = \frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) \left( 1 -  \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max}}\frac{f^{2}(c_{*})} {f^{2}(c_{}) }\right)$$

$$ \text{If we define} \ \ \chi \ \text{such that} \ \ \sqrt{ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max}} } \frac{f(c_{*})}{f(c_{})} = \sin{(\chi)} \ \ \text{ then }$$

$$ \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = \frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) \cos^2{(\chi)}$$

$$ \frac{f(c_{*})}{c_{*}}\left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = \sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\cos{(\chi)}$$

$$ \frac{f(c_{*})}{c_{*}}\left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) \left( \frac{\dot{f}(c_{*})}{f(c_{*})} - \frac{\left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} \right) +\frac{f(c_{*})}{c_{*}}\frac{  \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = -\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt}  $$

$$ \frac{f(c_{*})}{c_{*}}\frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = -\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} - \frac{f(c_{*})}{c_{*}}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\left( \frac{\dot{f}(c_{*})}{f(c_{*})} - \frac{\left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} \right) $$

$$ \frac{f^2(c_{*})}{c_{*}^2}\frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = \frac{f(c_{*})}{c_{*}}\left(-\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} \right) - \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\left( \frac{\dot{f}(c_{*})}{f(c_{*})} - \frac{\left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} \right) $$

$$ \frac{f^2(c_{*})}{c_{*}^2}\frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)   $$

$$ \frac{f(c_{*})}{c_{*}}\left(-\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} \right) - \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\left( \frac{\dot{f}(c_{*})}{f(c_{*})} - \frac{\left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} \right) = \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)   $$

$$ -\frac{f(c_{*})}{c_{*}}\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} - \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\left( \frac{\dot{f}(c_{*})}{f(c_{*})} - \frac{\left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} \right) = \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)   $$

$$ -\frac{f(c_{*})}{c_{*}}\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} = \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\left( \frac{\dot{f}(c_{*})}{f(c_{*})} - \frac{\left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} \right) + \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)   $$

$$ -\frac{f(c_{*})}{c_{*}}\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} = \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) \frac{\dot{f}(c_{*})}{f(c_{*})}  - \frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)   $$

$$ -\frac{f(c_{*})}{c_{*}}\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt}  = \frac{f(c_{*})}{c_{*}} \frac{\dot{f}(c_{*})}{f(c_{*})} \frac{f(c_{*})}{c_{*}}\left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)   - \frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)   $$

$$ -\frac{f(c_{*})}{c_{*}}\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt}  = \frac{f(c_{*})}{c_{*}} \frac{\dot{f}(c_{*})}{f(c_{*})} \sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\cos{(\chi)} - \frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)   $$

$$ -\frac{f(c_{*})}{c_{*}}\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt}  = \frac{f(c_{*})}{c_{*}} \sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\cos{(\chi)} \frac{\dot{f}(c_{*})}{f(c_{*})} - \frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)  $$

$$ -\frac{f(c_{*})}{c_{*}}\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} = \frac{f(c_{*})}{c_{*}}\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\cos{(\chi)} \frac{ \cos(\chi)}{ \sin(\chi)} \frac{d{\chi}}{dt} - \frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)  $$

$$ \frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right) = \frac{f(c_{*})}{c_{*}}\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\cos{(\chi)} \frac{ \cos(\chi)}{ \sin(\chi)} \frac{d{\chi}}{dt} + \frac{f(c_{*})}{c_{*}}\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} $$

$$ \frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right) = \frac{f(c_{*})}{c_{*}}\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) } \left( \cos{(\chi)} \frac{ \cos(\chi)}{ \sin(\chi)} \frac{d{\chi}}{dt} + \sin{(\chi)}\frac{d{\chi}}{dt} \right) $$

$$ \frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right) = \frac{f(c_{*})}{c_{*}}\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) } \frac{1}{\sin{(\chi)}} \frac{d{\chi}}{dt} $$

$$ \frac{dt}{d{\chi}}  = \frac{\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }}{\frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)}\frac{{f(c_{*})}/{c_{*}}}{\sin{(\chi)}}  $$

$$ c_{}d\tau = cd\mathbf{t} $$

$$\frac{d\tau}{d{\chi}} = \frac{d\tau}{d\mathbf{t}}\frac{d\mathbf{t}}{d{\chi}} = \frac{c_{*}}{c_{}}\frac{d\mathbf{t}}{d{\chi}} =  \frac{\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }}{\frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)}\frac{{f(c_{*})}/{c_{*}}}{\sin{(\chi)}} \frac{c_{*}}{c_{}}$$

$$\frac{d\tau}{d{\chi}} = \frac{d\tau}{d\mathbf{t}}\frac{d\mathbf{t}}{d{\chi}} = \frac{c_{*}}{c_{}}\frac{d\mathbf{t}}{d{\chi}} =  \frac{\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }}{\frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)}\frac{{f(c_{*})}/{c_{}}}{\sin{(\chi)}} $$

$$ \frac{f(c_{*})}{f(c_{})} = \sqrt{\frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max}}{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \sin{(\chi)} $$

$$ \frac{f(c_{*})/c_{}}{\sin{(\chi)}} = \sqrt{\frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max}}{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \frac{f(c_{})}{c_{}} $$

$$\frac{d\tau}{d{\chi}} =  \frac{\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }}{\frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)}\sqrt{\frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max}}{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \frac{f(c_{})}{c_{}} $$

$$\frac{d\tau}{d{\chi}} =   \frac{1 } {\sqrt{\frac34 c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \frac{\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }{\left( p_{max} + \rho_{max} \right)}$$


 * $$\int_{0}^{\tau} d\tau = \int_{0}^{\frac{\pi}{2} }

\frac{1 } {\sqrt{\frac34 c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \frac{\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }{\left( p_{max} + \rho_{max} \right)} d\chi $$


 * $$\tau =

\frac{\frac{\pi}{2} } {\sqrt{\frac34 c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \frac{\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }{\left( p_{max} + \rho_{max} \right)} $$

x



\frac{ 1 } {\sqrt{ c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} $$

$$ \frac{dt}{d{\chi}}  = \frac{\sqrt{\frac13 c_{}^2 \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }}{\frac12 c_{}^2\kappa(c_{})\left( p_{max} + \rho_{max} \right)}\frac{1}{\sin{(\chi)}}  $$


 * $$\int \frac{1}{\sin{(\chi)}} \, d\chi= -\ln{\left| \frac{1+\cos{(\chi)}}{\sin{(\chi)}}\right|}+C$$


 * $$\int \csc_{*}^n{ax} \, dx = -\frac{\csc_{*}^{n-2}{ax} \cot{ax}}{a(n-1)} \,+\, \frac{n-2}{n-1}\int \csc_{*}^{n-2}{ax} \, dx \qquad \mbox{ (for }n \ne 1\mbox{)}$$


 * $$\int \csc_{*}^{-n}{ax} \, dx = -\frac{\csc_{*}^{-n-2}{ax} \cot{ax}}{a(-n-1)} \,+\, \frac{-n-2}{-n-1}\int \csc_{*}^{-n-2}{ax} \, dx \qquad \mbox{ (for }-n \ne 1\mbox{)}$$


 * $$\int \sin^{n}{ax} \, dx = -\frac{\sin^{n+2}{ax} \cot{ax}}{a(-n-1)} \,+\, \frac{-n-2}{-n-1}\int \sin^{n+2}{ax} \, dx \qquad \mbox{ (for }-n \ne 1\mbox{)}$$


 * $$\int \sin^{n}{ax} \, dx = \frac{\sin^{n+2}{ax} \cot{ax}}{a(n+1)} \,+\, \frac{n+2}{n+1}\int \sin^{n+2}{ax} \, dx \qquad \mbox{ (for }n \ne -1\mbox{)}$$


 * $$\int \sin^{n}{ax} \, dx = \frac{\sin^{n+1}{ax} \cos{ax}}{a(n+1)} \,+\, \frac{n+2}{n+1}\int \sin^{n+2}{ax} \, dx \qquad \mbox{ (for }n \ne -1\mbox{)}$$

A Theory Of Inflation
$$ \text{From appendix B The two Friedmann equations are} $$

$$ \left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = -\frac{kc_{*}^2}{a^2} + \frac13 c_{*}^2 \Lambda(c_{*}) + \frac13 c_{*}^2\kappa(c_{*})\rho$$

$$ \frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = \frac{kc_{*}^2}{a^2} + \left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) \frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} - \frac12 c_{*}^2\kappa(c_{*}) \left( p + \rho \right) $$

$$ \text{The first Friedmann equation is standard. The second Friedmann equation has an extra term due to the temporal scale factor.} $$

$$ \text{The following assumptions are now made.} $$

$$ \text{Firstly, that the co-ordinate curvature is zero or has a negligible effect and can therefore be ignored.} $$

$$ \text{Secondly, that the zero point energy density of the vacuum is much larger than the carrying capacity of the vacuum } $$ $$ \text{i.e. the energy density and pressure of the maximum amount of matter and radiation that the vacuum can hold. } $$ $$ \text{This seems to be a quite reasonable assumption. The mass density at the centre of a neutron star is approximately }10^{18} Kg/m^3. $$ $$ \text{If all the quarks were replaced with X and Y bosons then the mass density might be increased by 15 orders of magnitude. } $$ $$ \text{This would still be 63 orders of magnitude short of the Planck density. } $$

$$ \text{Thirdly, that throughout the period of inflation, energy density and pressure are held constant at their maximum values.} $$

$$ \text{Adjusting the equations accordingly the results are.} $$

$$ \left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = \frac13 c_{*}^2 \Lambda(c_{*}) + \frac13 c_{*}^2\kappa(c_{*})\rho_{max}$$

$$ \frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = \left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) \frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} - \frac12 c_{*}^2\kappa(c_{*}) \left( p_{max} + \rho_{max} \right) $$

$$ \text{Please note } p_{max} \text{ and } \rho_{max} \text{ both exclude vacuum energy. }$$

$$ \text{In the first equation a separate term deals with vacuum energy. } $$

$$ \text{In the second equation vacuum energy can be ignored because } p_{vac} \text{ and } \rho_{vac} \text{ add up to zero. }$$

$$\text{If we replace } \Lambda(c_{*}) \text{ in the first equation.}$$

$$ \left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = \frac13 \left( c_{*}^2 \kappa(c_{*}) - c_{*}^2\kappa(c_{})  \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 c_{*}^2\kappa(c_{*})\rho_{max}$$

$$\text{If } f \text{ is defined such that }f^{2}(c_{*})\kappa(c_{*}) \text{ is constant and therefore equal to } f^{2}(c_{})\kappa(c_{}) \text{ then }$$

$$ \frac{f^2(c_{*})}{c_{*}^2}\frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \frac12 f^{2}(c_{*})\kappa(c_{*}) \left( p_{max} + \rho_{max} \right)  $$

$$ \frac{f^2(c_{*})}{c_{*}^2}\frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \frac12 f^{2}(c_{})\kappa(c_{}) \left( p_{max} + \rho_{max} \right)  $$

$$ \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = \frac13   \left(   f^{2}(c_{*}) \kappa(c_{*})    - f^{2}  (c_{*})  \kappa(c_{})    \right)  \overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 f^{2}(c_{*})\kappa(c_{*})\rho_{max}$$

$$ \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = \frac13   \left(   f^{2}(c_{}) \kappa(c_{})    - f^{2}  (c_{*})  \kappa(c_{})    \right)  \overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 f^{2}(c_{})\kappa(c_{})\rho_{max}$$

$$ \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = \frac13 \kappa(c_{})\left( f^{2}(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + f^{2}(c_{})\rho_{max} - f^{2}(c_{*})\overset{{\ }_{\boldsymbol{\circ}}}{\rho}  \right) $$

$$ \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = \frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) \left( 1 -  \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max}}\ \frac{f^{2}(c_{*})} {f^{2}(c_{}) }\right)$$

$$ \text{If } \ \ \chi \ \text{ is defined such that} \ \ \sqrt{ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max}} } \ \frac{f(c_{*})}{f(c_{})} = \sin{(\chi)} \ \ \text{ then }$$

$$ \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)^2 = \frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) \cos^2{(\chi)}$$

$$ \frac{f(c_{*})}{c_{*}}\left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = \sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\cos{(\chi)}$$

$$ \frac{f(c_{*})}{c_{*}}\left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) \left( \frac{\dot{f}(c_{*})}{f(c_{*})} - \frac{\left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} \right) +\frac{f(c_{*})}{c_{*}}\frac{  \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = -\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt}  $$

$$ \frac{f(c_{*})}{c_{*}}\frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = -\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} - \frac{f(c_{*})}{c_{*}}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\left( \frac{\dot{f}(c_{*})}{f(c_{*})} - \frac{\left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} \right) $$

$$ \frac{f^2(c_{*})}{c_{*}^2}\frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = \frac{f(c_{*})}{c_{*}}\left(-\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} \right) - \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\left( \frac{\dot{f}(c_{*})}{f(c_{*})} - \frac{\left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} \right) $$

$$ \frac{f^2(c_{*})}{c_{*}^2}\frac{ \mathrm{d} } {\mathrm{d}{t} }\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) = \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)   $$

$$ \frac{f(c_{*})}{c_{*}}\left(-\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} \right) - \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\left( \frac{\dot{f}(c_{*})}{f(c_{*})} - \frac{\left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} \right) = \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)   $$

$$ -\frac{f(c_{*})}{c_{*}}\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} - \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\left( \frac{\dot{f}(c_{*})}{f(c_{*})} - \frac{\left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} \right) = \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)   $$

$$ -\frac{f(c_{*})}{c_{*}}\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} = \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\left( \frac{\dot{f}(c_{*})}{f(c_{*})} - \frac{\left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}} \right) + \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)\frac{ \left( \mathrm{d}{c_{*}}/\mathrm{d}{t} \right)}{c_{*}}  - \frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)   $$

$$ -\frac{f(c_{*})}{c_{*}}\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} = \frac{f^2(c_{*})}{c_{*}^2}\left( \frac{  \mathrm{d}{a}/\mathrm{d}{t} }{a} \right) \frac{\dot{f}(c_{*})}{f(c_{*})}  - \frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)   $$

$$ -\frac{f(c_{*})}{c_{*}}\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt}  = \frac{f(c_{*})}{c_{*}} \frac{\dot{f}(c_{*})}{f(c_{*})} \frac{f(c_{*})}{c_{*}}\left( \frac{ \mathrm{d}{a}/\mathrm{d}{t} }{a} \right)   - \frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)   $$

$$ -\frac{f(c_{*})}{c_{*}}\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt}  = \frac{f(c_{*})}{c_{*}} \frac{\dot{f}(c_{*})}{f(c_{*})} \sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\cos{(\chi)} - \frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)   $$

$$ -\frac{f(c_{*})}{c_{*}}\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt}  = \frac{f(c_{*})}{c_{*}} \sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\cos{(\chi)} \frac{\dot{f}(c_{*})}{f(c_{*})} - \frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)  $$

$$ -\frac{f(c_{*})}{c_{*}}\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} = \frac{f(c_{*})}{c_{*}}\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\cos{(\chi)} \frac{ \cos(\chi)}{ \sin(\chi)} \frac{d{\chi}}{dt} - \frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)  $$

$$ \frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right) = \frac{f(c_{*})}{c_{*}}\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\cos{(\chi)} \frac{ \cos(\chi)}{ \sin(\chi)} \frac{d{\chi}}{dt} + \frac{f(c_{*})}{c_{*}}\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }\sin{(\chi)}\frac{d{\chi}}{dt} $$

$$ \frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right) = \frac{f(c_{*})}{c_{*}}\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) } \left( \cos{(\chi)} \frac{ \cos(\chi)}{ \sin(\chi)} \frac{d{\chi}}{dt} + \sin{(\chi)}\frac{d{\chi}}{dt} \right) $$

$$ \frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right) = \frac{f(c_{*})}{c_{*}}\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) } \frac{1}{\sin{(\chi)}} \frac{d{\chi}}{dt} $$

$$ \frac{dt}{d{\chi}}  = \frac{\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }}{\frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)}\frac{{f(c_{*})}/{c_{*}}}{\sin{(\chi)}}  $$

$$ \text{If } \tau \text{ is defined to be cosmological time, i.e. the proper time for any geodesic with constant spatial co-ordinates, then } $$

$$ c_{}d \tau = c_{*}dt $$

$$\frac{d\tau}{d{\chi}} = \frac{d\tau}{dt}\frac{dt}{d{\chi}} = \frac{c_{*}}{c_{}}\frac{dt}{d{\chi}} =  \frac{\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }}{\frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)}\frac{{f(c_{*})}/{c_{*}}}{\sin{(\chi)}} \frac{c_{*}}{c_{}}

=  \frac{\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }}{\frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)}\frac{{f(c_{*})}/{c_{}}}{\sin{(\chi)}} $$

$$ \frac{f(c_{*})}{f(c_{})} = \sqrt{\frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max}}{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \sin{(\chi)} $$

$$ \frac{f(c_{*})/c_{}}{\sin{(\chi)}} = \sqrt{\frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max}}{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \frac{f(c_{})}{c_{}} $$

$$\frac{d\tau}{d{\chi}} =  \frac{\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }}{\frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)}\sqrt{\frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max}}{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \frac{f(c_{})}{c_{}} $$

$$\frac{d\tau}{d{\chi}} =   \frac{1 } {\sqrt{\frac34 c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \frac{\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }{\left( p_{max} + \rho_{max} \right)}$$

$$ \text{This value is constant. Cosmological time is proportional to the parameter } \chi \text{, which has a finite range. } $$

$$ \text{The maximum range of } \chi \text{ is zero to just under } \pi/2 $$

$$ \text{In terms of proper time the period of inflation is finite, and has a maximum value short of:} $$


 * $$\int_{0}^{\tau} d\tau = \int_{0}^{\frac{\pi}{2} }

\frac{1 } {\sqrt{\frac34 c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \frac{\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }{\left( p_{max} + \rho_{max} \right)} d\chi $$


 * $$\tau =

\frac{\frac{\pi}{2} } {\sqrt{\frac34 c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \frac{\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }{\left( p_{max} + \rho_{max} \right)} $$


 * $$\tau =

\frac{\frac{\pi}{\sqrt{3}} } {\sqrt{c_{}^2\kappa(c_{})\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }} \frac{\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }{\left( p_{max} + \rho_{max} \right)} $$

To Illustrate With A Possible Example.
$$ $$

$$ \text{If it is assumed that } { f(c_{*}) \propto c_{*} } \text{ then } f(c_{*})/c_{*} \text{ would be constant and since } $$

$$ \frac{dt}{d{\chi}}  = \frac{\sqrt{\frac13 f^{2}(c_{}) \kappa(c_{})\left( \overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho_{max} \right) }}{\frac12 f^{2}(c_{})\kappa(c_{})\left( p_{max} + \rho_{max} \right)}\frac{{f(c_{*})}/{c_{*}}}{\sin{(\chi)}}  $$

$$ \text{ this would mean } $$

$$ \frac{dt}{d{\chi}}  \propto \frac{1}{\sin{(\chi)}}  $$

$$\text{therefore}$$

$$\int dt \propto \int \frac{d\chi}{\sin{(\chi)}} \, = \ln{\left( \tan{\left(\chi/2 \right)} \right)}+C$$

$$\text{co-ordinate time could go back infinitely far.}$$

$$\text{A universe beginning infinitely far back in co-ordinate time but finitely far back in proper time would be a White Hole solution.}$$

XXXXXtemp A Theory Of Inflation
$$ \text{From appendix B The two Friedmann equations are} $$

$$ H^2 = -\frac{kc^2}{a^2} + \frac13 c_{}^2 \Lambda(c_*) + \frac13 c_{}^2\kappa(c_*)\rho$$

$$ \dot{H} = \frac{kc^2}{a^2} - \frac12 c_{}^2\kappa(c_*) \left( p + \rho \right) $$

$$\text{If } \Lambda(c_*) \text{ is replaced by zero point values.}$$

$$ H^2 = -\frac{kc^2}{a^2} + \frac13 \left( c_{}^2 \kappa(c_*) - c_{}^2\kappa(c) \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 c_{}^2\kappa(c_*)\rho $$

$$ \dot{H} = \frac{kc^2}{a^2} - \frac12 c_{}^2\kappa(c_*) \left( p + \rho \right) $$

$$ \text{It the assumtion is made that the co-ordinate curvature is zero, or has a negligible effect, then the equations can be adjusted accordingly} $$

$$ H^2 = \frac13 \left( c_{}^2 \kappa(c_*) - c_{}^2\kappa(c) \right)\overset{{\ }_{\boldsymbol{\circ}}}{\rho}    + \frac13 c_{}^2\kappa(c_*)\rho $$

$$ \dot{H} = - \frac12 c_{}^2\kappa(c_*) \left( p + \rho \right) $$

full space
$$ \overset{{}_{\boldsymbol{\circ}}}{\Lambda} \ \ \overset{{\ }_{\boldsymbol{\circ}}}{\rho} \ \ \bold{f} \ \ \mbox{abcdefg} \ \ abcdefg \ \ \bold{ \overset{{}_{\boldsymbol{\circ}}}{\Lambda} } $$

$$ \overset{{}_{\boldsymbol{\circ}}}{\Lambda} \ \ \overset{{\ }_{\boldsymbol{\circ}}}{\rho} \ \ \boldsymbol{\circ} \bold{\boldsymbol{\circ}} \ \ \ \bold{\overset{{}_}{\Lambda}}\ \ \bold{\overset{{}_{\boldsymbol{\circ}}}{\Lambda}} \ \ \bold{\overset{{\ }_}{\rho}}\ \ \bold{\overset{{\ }_{\boldsymbol{\circ}}}{\rho}} $$

$$ \overset{{}_{\boldsymbol{\circ}}}{\Lambda} \ \ \overset{{}_{\boldsymbol{\circ}}}{\rho} \ \ \boldsymbol{\circ} \bold{\boldsymbol{\circ}} \ \ \ \bold{\overset{{}_{\boldsymbol{\circ}}}{\Lambda}}\ \ \bold{\overset{{\ }_{\boldsymbol{\circ}}}{\rho}} $$

$$ \rho\ \ \bold{\rho}\ \ $$

$$ \text{Secondly, that the zero point energy density of the vacuum is much larger than the carrying capacity of the vacuum } $$ $$ \text{i.e. the energy density and pressure of the maximum amount of matter and radiation that the vacuum can hold. } $$ $$ \text{This seems to be a quite reasonable assumption. The mass density at the centre of a neutron star is approximately }10^{18} Kg/m^3. $$ $$ \text{If all the quarks were replaced with X and Y bosons then the mass density might be increased by 15 orders of magnitude. } $$ $$ \text{This would still be 63 orders of magnitude short of the Planck density. } $$

$$ \text{Thirdly, that throughout the period of inflation, energy density and pressure are held constant at their maximum values.} $$

$$ \mbox{Secondly, that the zero point energy density of the vacuum is much larger than the carrying capacity of the vacuum } $$ $$ \mbox{i.e. the energy density and pressure of the maximum amount of matter and radiation that the vacuum can hold. } $$ $$ \mbox{This seems to be a quite reasonable assumption. The mass density at the centre of a neutron star is approximately }10^{18} Kg/m^3. $$ $$ \mbox{If all the quarks were replaced with X and Y bosons then the mass density might be increased by 15 orders of magnitude. } $$ $$ \mbox{This would still be 63 orders of magnitude short of the Planck density. } $$

$$ \mbox{Thirdly, that throughout the period of inflation, energy density and pressure are held constant at their maximum values.} $$

$$ \text{Please note } p_{max} \text{ and } \rho_{max} \text{ both exclude vacuum energy. }$$

$$ \text{In the first equation a separate term deals with vacuum energy. } $$

$$ \text{In the second equation vacuum energy can be ignored because } p_{vac} \text{ and } \rho_{vac} \text{ add up to zero. }$$

test space
$$ \overset{{}_{\boldsymbol{\circ}}}{\Lambda} \ \ \overset{{\ }_{\boldsymbol{\circ}}}{\rho} \ \ \bold{f} \ \ \mbox{abcdefg} \ \ abcdefg \ \ \bold{ \overset{{}_{\boldsymbol{\circ}}}{\Lambda} } $$

$$ \overset{{}_{\boldsymbol{\circ}}}{\Lambda} \ \ \overset{{\ }_{\boldsymbol{\circ}}}{\rho} \ \ \boldsymbol{\circ} \bold{\boldsymbol{\circ}} \ \ \ \bold{\overset{{}_}{\Lambda}}\ \ \bold{\overset{{}_{\boldsymbol{\circ}}}{\Lambda}} \ \ \bold{\overset{{\ }_}{\rho}}\ \ \bold{\overset{{\ }_{\boldsymbol{\circ}}}{\rho}} $$

$$ \overset{{}_{\boldsymbol{\circ}}}{\Lambda} \ \ \overset{{}_{\boldsymbol{\circ}}}{\rho} \ \ \boldsymbol{\circ} \bold{\boldsymbol{\circ}} \ \ \ \bold{\overset{{}_{\boldsymbol{\circ}}}{\Lambda}}\ \ \bold{\overset{{\ }_{\boldsymbol{\circ}}}{\rho}} $$

Draft
We seem to be in the paradoxical situation that non-uniform time is an inherent property of the manifold and uniform time is no more than an imposed perspective. Further, that this non-uniformaty is largely indeterminate, but never-the-less necessary. On the positive side this avoids the problem of a singularity in the manifold and despite uniform time only being an imposed perspective it is within this perspective that energy is conserved.

$$ \text{The following example shows how it is possible to have a begining to time without having a singularity in the manifold.}$$

$$ \text{The foundations of this theory need three fundamental values that are dimensionally independent and do not change under any circumstances. }$$

$$ \text{ Two obvious candidates are the zero point values. }$$

$$ \text{Because Einstein's field equations can be derived from the principle of least action it seems necessary to have as the third value a fundamental unit of action. }$$

$$ \text{The three values are therefore: the zero point energy density of the vacuum, the zero point curvature of space, and Planck's reduced constant. }$$

$$ \ \ \text{With the three values represented respectively by: }$$ $$ \text{rho zero point, Lambda zero point, and aitch bar. }$$

$$ \ \ \overset{{\ }_{\boldsymbol{\circ}}}{\rho} \overset{{}_{\boldsymbol{\circ}}}{\Lambda} \hbar $$

2 $$ \text{These values together with the Temporal Scale Factor can be aranged to produce a dimensionless quantity }$$ $$ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho}} {\left( \overset{{}_{\boldsymbol{\circ}}}{\Lambda} \right)^2 \hbar c_*} $$

$$ \text{ }$$

4 $$ \text{The arbitary function kappa can be re-writen in terms of an arbitary dimensionless funtion }$$ $$ f: x \in \mathbb{R} \mapsto f(x) \in \mathbb{R}   $$

$$ \kappa(c_{*}) \equiv f\left( \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho}} {\left( \overset{{}_{\boldsymbol{\circ}}}{\Lambda} \right)^2 \hbar c_*}  \right) \frac{\overset{{}_{\boldsymbol{\circ}}}{\Lambda}} {\overset{{\ }_{\boldsymbol{\circ}}}{\rho}} $$

5  $$ \text{Please note that the above implies } f\left( \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho}} {\left( \overset{{}_{\boldsymbol{\circ}}}{\Lambda} \right)^2 \hbar c}  \right) \equiv 1 $$ $$ \text{ It does not imply that }$$ $$ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho}} {\left( \overset{{}_{\boldsymbol{\circ}}}{\Lambda} \right)^2 \hbar c}  \equiv 1 $$

$$ \ \text{Because }\ \   f: x  \mapsto constant \ \  \text{  is ruled out, }$$ $$ \text{the simplest available option for } f \text{ would be }\ \   f: x  \mapsto x \ \ \ \text{i.e.}\ \ \ f(x)\equiv x  $$

9 $$ \text{Taking this as our example. }$$

$$ \kappa(c_{*}) \equiv \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho}} {\left( \overset{{}_{\boldsymbol{\circ}}}{\Lambda} \right)^2 \hbar c_*} \frac{\overset{{}_{\boldsymbol{\circ}}}{\Lambda}} {\overset{{\ }_{\boldsymbol{\circ}}}{\rho}} \equiv \frac{1} {\overset{{}_{\boldsymbol{\circ}}}{\Lambda} \hbar c_*} $$

11 $$ \text{From the definition of } \ \ \chi \ \text{ (chi) } \ \ $$

$$\kappa(c_{*}) = \kappa(c_{}) \ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}  \  \sin^{-2}{(\chi)} \ \ $$

12 $$ \text{Substituting. }$$

$$ \frac{1} {\overset{{}_{\boldsymbol{\circ}}}{\Lambda} \hbar c_*} = \kappa(c_{}) \ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}  \  \sin^{-2}{(\chi)} \ \ $$

13 $$ \text{Integrating with respect to cosmological time. }$$

$$ \int \frac{1} {\overset{{}_{\boldsymbol{\circ}}}{\Lambda} \hbar c_*} \  \mathrm{d}{\tau} = \int \kappa(c_{}) \ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}  \  \sin^{-2}{(\chi)} \ \mathrm{d}{\tau} \ $$

$$ \text{ }$$ $$ \int \frac{1} {\overset{{}_{\boldsymbol{\circ}}}{\Lambda} \hbar c_*} \ \frac{\mathrm{d}{\tau} } { \mathrm{d}t } \ \mathrm{d}t = \int \kappa(c_{}) \ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}  \  \sin^{-2}{(\chi)} \ \frac{\mathrm{d}{\tau} } { \mathrm{d}\chi } \ \mathrm{d}\chi $$

15 $$ \text{from the definition of cosmological time } \ \ c\mathrm{d}\tau = c_*\mathrm{d}t \ \ \text{ we find the substitution }$$

$$ \int \frac{1} {\overset{{}_{\boldsymbol{\circ}}}{\Lambda} \hbar c_*} \ \frac{ c_* } { c } \ \mathrm{d}t = \int \kappa(c_{}) \ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}  \  \sin^{-2}{(\chi)} \ \frac{\mathrm{d}{\tau} } { \mathrm{d}\chi } \ \mathrm{d}\chi $$

$$ \text{ }$$ $$ \int \frac{1} {\overset{{}_{\boldsymbol{\circ}}}{\Lambda} \hbar c} \ \ \mathrm{d}t = \int \kappa(c_{}) \ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}  \  \sin^{-2}{(\chi)} \ \frac{\mathrm{d}{\tau} } { \mathrm{d}\chi } \ \mathrm{d}\chi $$

17 $$ \text{Rearranging to take all of the constants outside the integrals. }$$

$$ \frac{1} {\overset{{}_{\boldsymbol{\circ}}}{\Lambda} \hbar c} \ \ \int \mathrm{d}t = \kappa(c_{}) \ \frac{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} }{\overset{{\ }_{\boldsymbol{\circ}}}{\rho} + \rho^{full}}  \ \frac{\mathrm{d}{\tau} } { \mathrm{d}\chi } \int \sin^{-2}{(\chi)} \ \ \mathrm{d}\chi $$

18 $$ \text{Then just using proportionality  }$$

$$ \int \mathrm{d}t \propto\int \sin^{-2}{(\chi)} \ \ \mathrm{d}\chi $$

$$ \text{ }$$

$$ \int \mathrm{d}t \propto -\cot{\left( \chi \right)} +C $$

20 $$ \text{We find the result that although cosmological time is finite into the past, co-ordinate time could go back infinitely far.}$$

$$\text{A universe beginning infinitely far back in co-ordinate time but finitely far back in proper time would be a White Hole solution.}$$

$$\kappa(c_{*}) = \frac{8\pi G}{c_*^{4}}$$

$$\kappa(c_{*}) = \frac{8\pi G}{{c_*}^{4}}$$

In order to avoid a fine-tuning problem it seems necessary that Maxwell's Constant be given no other signicance than it be the limiting value of the Temporal Scale Factor.

the gravitational parameter is inversely proportional to the temporal scale factor or in other words the the inverse gravitational parameter is directly proportional to the temporal scale factor.

Rather perversely proper time is an emergent property and co-ordinate time is a an inherent property of the manifold.

The fourth dimension is never a dimension of space. It is time like at all points except the begining of time where it is NULL. It starts off Null and after that is time-like.

Draft2
Here we see the Friedmann equations in their more cannonical form. The overdots may be taken to be derivatives with respect to uniform time.


 * $$\left(\frac{\dot a}{a}\right)^{2} + \frac{kc^{2}}{a^2} - \frac13{\Lambda c^{2}} = \frac13\frac{8\pi G}{c^{2}}\rho$$
 * $$2\frac{\ddot a}{a} + \left(\frac{\dot a}{a}\right)^{2} + \frac{kc^{2}}{a^2} - \Lambda c^{2} = -\frac{8\pi G}{c^{2}} p$$

If we take the first Frienmann Equation, multiply all the terms by a^{3}, and differentiate

then multiply all the terms in the second Freidmann equation by dot{a} a^2, the Left Hand Sides of each will match exactly.



\left(\frac{\dot a}{a}\right)^{2} a^3 + \frac{kc^{2}}{a^2} a^3 - \frac13{\Lambda c^{2}} a^3 = \frac13\frac{8\pi G}{c^{2}}\rho a^3 $$



{\dot a}^{2} a + kc^{2} a - \frac13{\Lambda c^{2}} a^3 = \frac13\frac{8\pi G}{c^{2}}\rho a^3 $$



2\ddot{a}{\dot a} a + {\dot a}^{3} + kc^{2} {\dot a} - {\Lambda c^{2}} {\dot a}a^2 = \frac{8\pi G}{c^{2}} \frac13 \frac{ d\left( {\rho a^3} \right) }{dt} $$


 * $$2\frac{\ddot a}{a} + \left(\frac{\dot a}{a}\right)^{2} + \frac{kc^{2}}{a^2} - \Lambda c^{2} = -\frac{8\pi G}{c^{2}} p.$$



2\frac{\ddot a}{a} \dot{a} a^2 + \left(\frac{\dot a}{a}\right)^{2} \dot{a} a^2 + \frac{kc^{2}}{a^2} \dot{a} a^2 - \Lambda c^{2} \dot{a} a^2 = -\frac{8\pi G}{c^{2}} p \dot{a} a^2 $$



2\ddot a} \dot{a} a + {\dot a}^{3} + {kc^{2} \dot{a} - \Lambda c^{2} \dot{a} a^2 = -\frac{8\pi G}{c^{2}} p \dot{a} a^2 $$



\frac{8\pi G}{c^{2}} \frac13 \frac{ d\left( {\rho a^3} \right) }{dt} = -\frac{8\pi G}{c^{2}} p \dot{a} a^2 $$



\frac13 \frac{ d\left( {\rho a^3} \right) }{dt} = - p \dot{a} a^2 $$



\frac{ d\left( {\rho a^3} \right) }{dt} = - 3p \dot{a} a^2 $$



\frac{ d\left( {\rho a^3} \right) }{dt} = - p \left( 3 \dot{a} a^2 \right) $$



\frac{ d\left( {\rho V} \right) }{dt} = - p \frac{dV}{dt} $$



\frac{ dU}{dt} = - p \frac{dV}{dt} $$