Perturbation Representation

A general metric perturbation can be broken up into the normal modes of scalar (m=0), vector ($m=\pm 1$) and tensor ($m=\pm 2)$ types,

$$\begin{array} {rcl}h_{00} &=& - \sum_m 2 A^{(m)} Q^{(m)} \,,... ...)} Q^{(m)} \gamma_{ij}+2 H_T^{(m)} Q_{ij}^{(m)} \,. \end{array}$$

Note that scalar quantities cannot be formed from vector and tensor modes so that A^(m)=0 and H_L^(m)=0 for $m\ne 0$; likewise vector quantities cannot be formed from tensor modes so that B^(m)=0 for |m| = 2.

There remains gauge freedom associated with the coordinate choice for the metric perturbations (see Appendix A2). It is typically employed to eliminate two out of four of these quantities for scalar perturbations and one of the two for vector perturbations. The metric is thus specified by four quantities. Two popular choices are the synchronous gauge, where

$$\begin{array} {rcl}H_L^{(0)} = h_L, &\qquad& H_T^{(0)} = h_T , \nonumber\\ H_T^{(1)} = h_V, &\qquad& H_T^{(2)} = H,\end{array}$$

and the generalized (or conformal) Newtonian gauge, where

$$\begin{array} {rcl}A^{(0)} = \Psi\,, &\qquad& B^{(1)}=V \,, ... ...ber \\ H_L^{(0)} = \Phi\,, &\qquad& H_T^{(2)}=H \, .\end{array}$$

Here and below, when only the $m\ge 0$ expressions are displayed, the m < 0 expressions should be taken to be identical unless otherwise specified.

The stress energy tensor can likewise be broken up into scalar, vector, and tensor contributions. Furthermore one can separate fluid (f) contributions and seed (s) contributions. The latter is distinguished by the fact that the net effect can be viewed as a perturbation to the background. Specifically $T_{\mu\nu} = \bar T_{\mu\nu} + \delta T_{\mu\nu}$ where $\bar T^0_{\hphantom{0}0} = -\rho_f$,$\bar T^0_{\hphantom{0}i} = \bar T_0^{\hphantom{i}i} =0$and $\bar T^i_{\hphantom{i}j} = p_f \delta^i_{\hphantom{i}j}$is given by the fluid alone. The fluctuations can be decomposed into the normal modes of §IIA as  

$$\begin{array} {lcl} \delta T^0_{\hphantom{0}0} &=& - \sum_m... ...om{i}j} \vphantom{\displaystyle{\dot a \over a}}\, .\end{array}$$ (11)

Since $\delta^{(m)}_f=\delta p^{(m)}_f= 0$ for $m\ne 0$, we hereafter drop the superscript from these quantities.

A minimally coupled scalar field $\varphi$ with Lagrangian

$${\cal L} = -{1 \over 2} \sqrt{-g} \left[ g^{\mu \nu} \partial_\mu \varphi \partial_\nu \varphi + 2V(\varphi) \right]$$ (12)

can be treated in the same way with the associations  

$$\rho_\phi = p_\phi + 2{\cal V}= {1 \over 2} a^{-2} \dot \phi^2 + {\cal V}\,,$$ (13)

for the background density and pressure. The fluctuations $\varphi = \phi+ \delta\phi$are related to the fluid quantities as [17] 

$$\begin{array} {rcl}\delta \rho_\phi =\delta p_\phi + 2{\cal ... ...hi \nonumber\,,\\ p_\phi \pi_\phi^{(0)} & = & 0\,.\end{array}$$

The evolution of the matter and metric perturbations follows from the Einstein equations $G_{\mu\nu} = 8\pi G T_{\mu\nu}$ and encorporates the continuity and Euler equations through the implied energy-momentum conservation $T^{\mu\nu}{}_{;\nu} = 0$.We give these relations explicitly for the scalar, vector and tensor perturbations in both Newtonian and synchronous gauge in Appendix A (see also [7]).

These equations hold equally well for relativistic matter such as the CMB photons and the neutrinos. However in that case they do not represent a closed system of equations (the equation of motion of the anisotropic stress perturbations $\pi_f^{(m)}$ is unspecified) and do not account for the higher moments of the distribution or for momentum exchange between different particle species. To include these effects, we require the Boltzmann equation which describes the evolution of the full distribution function under collisional processes.