User:Carolingfield/Cold Big Bang Cosmology

Cold Big Bang Cosmology is a designation used in Cosmology to specify a cosmological model characterized by an absolute zero temperature at the beginning of the Universe. See also: Big Bang and Cold Big Bang.

In an attempt to understand the origin of atoms, Georges Lemaître proposed (by 1927) that before the expansion of the universe started all the matter in the universe formed a gigantic ball of nuclear liquid at very low temperature. This low temperature was required to provide a sufficient cohesion within the Lemaître's primaveral atom. In 1966, David Layzer proposed a variant on Lemaître's cosmology in which the initial state of the universe was near absolute zero. Layzer argued that, rather than in an initial high entropy state, the primordial universe was in a very low entropy state near absolute zero.

A Model: Hypotheses and Solution
In, beginning with the Cosmological Principle and the Weyl's postulate, the author revisits the the basic arguments used in (GR) cosmology. Invoking a theorem from the differential geometry, namely the Schur's theorem, the author assumes the Riemann tensor of a constant curvature 3-space, leading to the spacetime geometry, for a comoving frame observer, as given by the Robertson-Walker metric.

From the Einstein's (GR) field equations in natural units, the author reachs the Friedmann equations in the following form


 * $$\frac{\dot{R}^{2}+kc^{2}}{R^{2}}=\frac{8\pi G}{3c^{2}}\left(\rho+\tilde{\rho}\right)$$


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

The author refers to $$R$$ as being the magnification lenght scale of the cosmological dynamics, depending on the cosmological time. In author's words: this measures how an unitary length of the pervading cosmological substratum becomes stretched as the universe goes through a time evolution between two cosmological instants. The author includes the cosmological constant $$\Lambda$$ within the vacuum density $$\tilde{\rho}$$ and vacuum Pressure $$\tilde{p}$$.

The author seems to refer to the following local energy conservation criteria, since, in author's words: the question of energy conservation in cosmology is weakened, supported by the known lack of scope of the Noether's theorem in cosmology


 * $$\nabla_{\mu}T^{\mu}_{\,\,\, t}=\partial_{\mu}T^{\mu}_{\,\,\, t}+\Gamma^{\mu}_{\,\,\mu\nu}T^{\nu}_{\,\,\, t}-\Gamma^{\nu}_{\,\,\mu t}T^{\mu}_{\,\,\,\nu}=0$$

where $$T_{\mu\nu}$$ is the stress-energy tensor. This local covariant implementation leads to the first law of thermodynamics


 * $$\frac{\partial}{\partial t}\left(\rho+\tilde{\rho}\right)+3\frac{\dot{R}}{R}\left(\rho+\tilde{\rho}+p+\tilde{p}\right)=0$$

The author writes down the equation of state $$\rho-3p=0$$ for the Weyl's fluid within the ultrarelativistic limit, firstly considering an universe dominated by radiation.

Integrating the Friedmann equations within this scenario, the author takes into consideration the absolute value, leading to the following result for the pressure


 * $$|p|=\frac{e^{C'}}{R^{4}}$$

where $$C^{'}$$ is an arbitrary constant of integration.

The author neglects the vacuum terms, neglecting the vacuum density $$\tilde{\rho}$$ and the vacuum pressure, in relation to the radiation; applies the initial conditions at the beginning of the universe:


 * $$R=R\left(0\right)$$


 * $$\dot{R}=0$$

at $$t=0$$, obtaining for the pressure


 * $$p(R)=k\frac{c^{4}R_{0}^{2}}{8\pi GR^{4}}$$

where $$k$$ is the normalized curvature.

Invoking internal consistency, robustness, the author obtains $$k=-1$$, implying in an open universe.

Within the appendix, the author raises an argument from Quantum mechanics, using the Heisenberg uncertainty principle, and postulates : "The actual energy content of the universe is a consequence of the increasing indeterminacy of the primordial era. Any origin of a co-moving reference frame within the cosmological substratum has an inherent indeterminacy. Hence, the indeterminacy of the energy content of the universe may create the impression that the universe has not enough energy, raising illusions as dark energy and dark matter speculations. In other words, since the original source of energy emerges as an indeterminacy, we postulate this indeterminacy continues being the energy content of the observational universe: $$\delta E(t)=E^{+}(t)=E^{+}_{0}/\sqrt{1-\dot{R}^{2}/c^{2}}$$

From the Bose-Einstein statistics, the author reachs the following expression for the Cosmic background radiation (MBR) temperature as a function of R


 * $$T^{4}=\frac{15c^{7}h^{3}}{16\pi^{6}Gk_{B}^{4}}\frac{1}{R^{2}}\sqrt{1-\frac{2Gh}{c^{3}R^{2}}}$$

from which the author obtains an absolute zero temperature at the Big Bang, 2.7 K for the actual cosmic background radiation temperature. Also, the value $$10^{51}$$$m/s^{2}$ for the Big Bang ignition.

In a recently published derivation, in which this cosmologically persistent Heisenberg indeterminacy is derived from a quantization criteria, a discreteness criteria :


 * $$N_{t}(\delta E_{\rho})=\dfrac{E^{+}_{0}}{\sqrt{1-\dot{R}^{2}/c^{2}}}$$,

the same previously postulated amount of energy indeterminacy is obtained, where $$N_{t}$$, an increasingly function on the cosmological time $$t$$, is the number of elementary discrete fluctuations of energy $$\delta E_{\rho}$$ within a spherical shell full of cosmological fluid at the cosmological instant $$t$$ of the cosmological time. The points within this shell have got an inherent position indeterminacy, also quantized, such that these points pertain to the cosmological fluid in its simultaneity hypersurface at the instant $$t$$, i.e., the points inside this shell are cosmologically simultaneous.