User:TallJimbo/WeakLensing/Cosmological

The gravitational lensing by large scale structure also produces an observable pattern of alignments in background galaxies. The thin-lens approximation usually used in cluster and galaxy lensing does not always work in this regime, because structures can be elongated over the line of sight. Instead, the distortion can be derived by assuming that the deflection angle is always small. As in the thin-lens case, the effect can written as a mapping from the unlensed angular position $$\vec{\beta}$$ to the lensed position $$\vec{\theta}$$. The Jacobian of the transform can be written as an integral over the gravitational potential $$\Phi$$ along the line of sight.

$$ \frac{\partial \beta_i}{\partial \theta_j} = A = \delta_{ij} + \int_0^{r_\infty} dr  g(r) \frac{\partial^2  \Phi(\vec{x}(r))}{\partial x^i \partial x^j} $$

where $$r$$ is the comoving distance, $$x^i$$ are the transverse distances, and

$$ g(r) = 2 r \int^{r_\infty}_r \left(1-\frac{r^\prime}{r}\right)W(r^\prime) $$

is the lensing kernel, which defines the efficiency of lensing for a distribution of sources $$W(r)$$.

Just like with the thin-lens approximation, the Jacobian can be decomposed into shear and convergence terms.

Shear Correlation Functions
Because large scale cosmological structures do not have a well-defined location, detecting cosmological gravitational lensing typically involves the computation of shear correlation functions, which measure the mean product of the shear at two points as a function of the distance between those points. Because there are two components to the shear, three different correlation functions can be defined:

$$ \xi_{++}(\Delta\theta) = \langle \gamma_+(\vec{\theta}) \gamma_+(\vec{\theta}+\vec{\Delta\theta}) \rangle $$

$$ \xi_{\times\times}(\Delta\theta) = \langle \gamma_\times(\vec{\theta}) \gamma_\times(\vec{\theta}+\vec{\Delta\theta}) \rangle $$

$$ \xi_{\times +}(\Delta\theta)=\xi_{+ \times}(\Delta\theta) = \langle \gamma_+(\vec{\theta}) \gamma_\times(\vec{\theta}+\vec{\Delta\theta}) \rangle $$

where $$\gamma_+$$ is the component along or perpendicular to $$\vec{\Delta\theta}$$, and $$\gamma_\times$$ is the component at 45°. These correlation functions are typically computed by averaging over many pairs of galaxies. The last correlation function, $$\xi_{\times +}$$, is not affected at all by lensing, so measuring a value for this function that is inconsistent with zero is often interpreted as a sign of systematic error.

The functions $$\xi_{++}$$ and $$\xi_{\times\times}$$ can be related to projections (integrals with certain weight functions) of the dark matter density correlation function, which can be predicted from theory for a cosmological model through it's Fourier transform, the matter power spectrum (see Structure Formation).

Because they both depend on a single scalar density field, $$\xi_{++}$$ and $$\xi_{\times\times}$$ are not independent, and they can be decomposed further into E-mode and B-mode correlation functions. In analogy with electric and magnetic fields, the E-mode is curl-free and the B-mode field is divergence free. Because gravitational lensing can only produce an E-mode field, the decomposition into E and B modes provides yet another test for systematic errors.

Weak Lensing and Cosmology
The ability of weak lensing to constrain the matter power spectrum makes it a potentially powerful probe of cosmological parameters, especially when combined with other observations such as the cosmic microwave background, supernovae, and galaxy surveys. The lensing signal is very weak, however; while the noise due to the intrinsic distribution of galaxy alignments is of order 30%, the ellipticity induced by lensing by large-scale structure is only 0.1%-1%. Because many background galaxies must be used, a weak lensing survey must be both deep and wide, and because these background galaxies are small, the image quality must be very good (the point spread function must be small and well-behaved).

Because of these challenges, while weak lensing of large scale structure was discussed as early as the 1960s (TODO: references), it was not detected until 40 years later, when large CCD cameras enabled surveys of the necessary size and quality. In 2000, four independent groups (TODO: references) published the first detections of cosmic shear, and subsequent observations have started to put constraints on cosmological parameters (particularly the dark matter density $$\Omega_m$$ and power spectrum amplitude $$\sigma_8$$) that are competitive with other cosmological probes.

For current and future surveys, one goal is to use the redshifts of the background galaxies (often approximated using photometric redshifts) to divide the survey into multiple redshift bins. The low-redshift bin will only be lensed by structures very near to us, while the high-redshift bins will be lensed by structures over a wide range of redshift. This technique, dubbed "cosmic tomography" makes it possible to map out the 3D distribution of mass. Because the third dimension involves not only distance but cosmic time, tomographic weak lensing is sensitive not only to the matter power spectrum today, but also to its evolution over the history of the universe, and the expansion history of the universe during that time. This is a much more valuable cosmological probe, and many proposed experiments to measure the properties of Dark Energy and Dark Matter have focused on weak lensing, such as the Dark Energy Survey, Pan-STARRS, and LSST.