Expansion of cosmological perturbations in helicities
When dealing with perturbations in cosmology one usually doesn't distinguish between original functions and their Fourier transforms. Coordinate and momenetum representations are connected by Fourier transform:
\begin{equation*}
h_{\mu\nu}(\eta,\, \vec{x})= \int d^3ke^{i\vec{k}\vec{x}} h_{\mu\nu}(\vec{k}),
\end{equation*}
which is reduced to the replacement $\partial_i\longleftrightarrow ik_i$ (see problem \ref{per29nnn}), and $\vec{k}$ has a meaning of conformal momentum.
Due to isotropy, metrics is invariant under spatial rotations, while at fixed conformal momentum $\vec{k}$ it is invariant under the rotations around the direction of $\vec{k}$, i.e. posses posses $SO(2)$ symmetry. Arbitrary three--dimensional tensor can be expanded in irreducible representations of $SO(2)$, which have certain helicity (eigenvalues of a rotation operator $L_\alpha = -i\frac{\partial}{d \alpha}$ at angle $\alpha$ ). \\ For example, three--dimensional scalar has zero helicity, since it doesn't transform under rotations around $\vec{k}$. Since for three--dimensional vector $v_i \propto k_i$, it has unit helicity, or more precisely a superposition of helicities +1 and -1. Three--dimensional tensor of the form $h_{ij}\propto v_i v_j$ or $h_{ij}\propto \delta_{ij}$ has the same helicity. Symmetric transverse traceless tensor $h_{ij}^{TT}$ has double helicity. Mathematically this can be expressed as \begin{equation}\label{hijTT} h_{ii}^{TT}=0,~ k_ih_{ij}^{TT}=k_jh_{ij}^{TT}=0. \end{equation} In linear theory, the Einstein's equations and equations of energy--momentum tensor covariance split into independent equations for helical components of cosmological perturbations. Thus, the perturbations are divided into tensor (double helicity), vector (unit helicity) and scalar (zero helicity). Expansion of perturbed metrics in helical components in general case has the form (see, for example \cite{Rubakov2}): \begin{equation}\label{h00_gen} h_{00} = 2\Phi, \end{equation} \begin{equation}\label{h0i_gen} h_{0i} = i k_i Z+ Z_i^T, \end{equation} \begin{equation}\label{hij_gen} h_{ij} = -2\Psi \delta_{ij}-2k_i k_j E+i(k_i W_j^T+k_j W_i^T)+h_{ij}^{TT}, \end{equation} where $\Phi, Z, \Psi, E$ are scalar functions of coordinates,$Z_i^T, W_i^T$ are transverse vectors ($k_i Z_i^T=k_i W_i^T =0$) and $h_{ij}^{TT}$ is a transverse traceless tensor.
In the subsequent problems the all types of perturbations are considered: scalar, vecotr and tensor. Also, we will use the gauge $h_{0i}=0.$
Contents
Scalar perturbations in conformal Newtonian gauge
Problem 1
problem id: per23
Calculate the Christoffel symbols in conformal Newtonian coordinate system with metrics \begin{equation} \label{per_newton_conf} ds^2=a^2(\eta)[(1+2\Phi)d\eta^2-(1-2\Phi)\delta_{ij}dx^idx^j], \end{equation} where $\Phi$ is a scalar function.
Assuming that $\Phi$ and $\Psi$ are small and keeping only linear terms, the Christoffel symbols obtain the form: \begin{equation} \label{Gamma_kris} \begin{gathered} \Gamma _{00}^0 = \mathcal{H}+\Phi ';\;\Gamma _{00}^i = {\Gamma _{00}}_i = {\Phi _{,i}};\\ \Gamma _{ij}^0 = \left[ \mathcal{H}\left( 1 - 4\Phi \right)- \Phi ' \right]\delta _{ij}; \\ \Gamma _{j0}^i = \left( \mathcal{H}- \Phi ' \right){\delta _{ij}};\;\Gamma _{ik}^j = \Phi _{,k}\delta _{ij} - \Phi _{,j}\delta _{ik}- \Phi _{,i}\delta _{jk}, \\ \end{gathered} \end{equation} where we denote $\mathcal{H}=\frac{a'}{a}.$ It is useful to calculate the convolution of the Christoffel symbols to use it in future calculations: \begin{equation} \label{Gamma_conv} \Gamma _{\lambda 0}^\lambda =4\mathcal{H}-\Phi ', \;\;\; \Gamma _{\lambda i}^\lambda=-2\Phi_{,i} \end{equation}
Problem 2
problem id: per24nnn
In conformal Newtonian gauge and perturbed energy--momenetum tensor derive the covariant conservation equations
Covariant conservation equations have the form: \begin{eqnarray} \delta \rho '+3 \mathcal{H}\left( \delta \rho +\delta p\right)+(p+\rho)\left( \Delta v - 3\Phi '\right)&=&0;\\ \delta p+4\mathcal{H} \left( \rho + p\right)v +(p+\rho)\Phi+\left[(\rho + p)v\right ]'=0 \end{eqnarray} and can be obtained from equation in solution of problem by substitution $h_{00}=2\Phi,~h_{ij}=2\Phi\delta_{ij},v_i=\partial _iv$.
Problem 3
problem id: per25nnn
Calculate the components of Ricci tensor in conformal Newtonian metrics \eqref{per_newton_conf}.
The Ricci tensor has the form $$ {\widetilde{R}_{\mu \nu }} = \partial_\lambda \widetilde{\Gamma}_{\mu \nu }^\lambda - \partial _\mu \widetilde{\Gamma} _{\lambda \nu }^\lambda + \widetilde{\Gamma} _{\rho \lambda }^\lambda \widetilde{\Gamma} _{\mu \nu }^\rho - \widetilde{\Gamma} _{\rho \mu }^\lambda \widetilde{\Gamma} _{\nu \lambda }^\rho. $$ Using the expressions for Christoffel symbols \eqref{Gamma_kris} and \eqref{Gamma_conv} from problem \ref{per23}: $$\partial_\lambda \widetilde{\Gamma}_{\mu \nu}^\lambda:~\partial_\lambda \widetilde{\Gamma}_{00}^\lambda=\mathcal{H}{'}+\Phi+\Delta\Phi,~\partial_\lambda \widetilde{\Gamma}_{0i}^\lambda=0,~\partial_\lambda \widetilde{\Gamma}_{ij}^\lambda=\left[\mathcal{H}'(1-4\Phi)-4\mathcal{H}\Phi '-\Phi ''-\Delta\Phi\right]\delta_{ij}-2\partial_i\partial_j; $$ $$ \partial_\mu \widetilde{\Gamma}_{\lambda \nu}^\lambda:~ \partial_0 \widetilde{\Gamma}_{\lambda 0}^\lambda=4\mathcal{H}'-2\Phi '', ~ \partial_0\widetilde{\Gamma}_{\lambda i}^\lambda=-2\partial_i\Phi ',~ \partial_i \widetilde{\Gamma}_{\lambda j}^\lambda=-2\partial_i\partial_j\Phi; $$ $$ \widetilde{\Gamma}_{ \rho\lambda}^\lambda\widetilde{\Gamma}_{ \mu\nu}^\rho:~ \widetilde{\Gamma}_{ \rho\lambda}^\lambda\widetilde{\Gamma}_{ 00}^\rho=2\mathcal{H}\left(2\mathcal{H}+\Phi '\right), ~\widetilde{\Gamma}_{ \rho\lambda}^\lambda\widetilde{\Gamma}_{ 0i}^\rho=2\mathcal{H}\partial_i\Phi,~\widetilde{\Gamma}_{ \rho\lambda}^\lambda\widetilde{\Gamma}_{ ij}^\rho=2\mathcal{H}\left(2\mathcal{H}+3\Phi ' - 8\mathcal{H}\Phi\right)\delta_{ij}; $$ $$ \widetilde{\Gamma}_{ \rho\mu}^\lambda\widetilde{\Gamma}_{\nu\lambda}^\rho:~ \widetilde{\Gamma}_{ \rho 0}^\lambda\widetilde{\Gamma}_{0\lambda}^\rho =4\mathcal{H}\left(\mathcal{H}-\Phi ' \right),~ \widetilde{\Gamma}_{ \rho 0}^\lambda\widetilde{\Gamma}_{i\lambda}^\rho=0,~ \widetilde{\Gamma}_{ \rho i}^\lambda\widetilde{\Gamma}_{j\lambda}^\rho=2\mathcal{H} \left(\mathcal{H}-4\mathcal{H}\Phi ' - 2\Phi '\right)\delta_{ij}; $$ Substituting these relations into the expression for Ricci tensor we finally obtain $$ \widetilde{R}_{ 00}=-3\mathcal{H}' +3\Phi '' +\Delta \Phi +6\Phi '\mathcal{H}; $$ $$ \widetilde{R}_{ i0}=2\partial_i\Phi '+2\mathcal{H}\partial_i\Phi; $$ $$ \widetilde{R}_{ ij}=\delta_{ij}\left(\mathcal{H} '+\Delta \Phi-4 \Phi \mathcal{H} ' - \Phi '' +2\mathcal{H}^2 -8\mathcal{H}^2\Phi-6\mathcal{H}\Phi '\right); $$
Problem 4
problem id: per26nnn
Calculate the components of perturbed Einstein tensor $G_\mu^\nu$ using the components of Ricci tensor.
Using the definition of Einstein tensor $$ G_\mu^\nu =R_\mu^\nu- \frac 12\delta_\mu^\nu R, $$ one can easily obtain \begin{eqnarray} a^2\widetilde{G}_0^0 &=& 3\mathcal{H}^2+ 2\Delta \Phi -6\mathcal{H}\Phi '-6\mathcal{H}^2\Phi; \\ a^2\widetilde{G}_0^i &=& 2\mathcal{H}\partial_i\Phi -2\partial_i\Phi ';\\ a^2\widetilde{G}_i^j&=&\left( 3\mathcal{H}^2+ 2\mathcal{H} '- 2 \Phi '' -6\mathcal{H}\Phi -4\mathcal{H}'\Phi -2\mathcal{H}^2\Phi\right)\delta_{ij}. \end{eqnarray}
Problem 5
problem id: per27nnn
Calculate the components of linearized Einstein tensor using the results of previous problems.
As is well known $$ \delta G_\mu^\nu = \widetilde{G}_\mu^\nu -G_\mu^\nu, $$ and, thus, \begin{eqnarray} \delta G_0^0 &=&\frac{2}{a^2}\left(\Delta \Phi-3\mathcal{H}\Phi- 3\mathcal{H}^2\Phi \right);\\ \delta G_0^i &=& \frac{2}{a^2}\left(\mathcal{H}\partial_i\Phi +\partial_i\Phi '\right);\\ \delta G_0^i &=& \frac{2}{a^2}\left(\Phi ''+ 3\mathcal{H}\Phi '+\left( 2\mathcal{H}'+\mathcal{H}^2\right)\Phi \right)\delta_{ij}. \end{eqnarray}
Problem 6
problem id: per28nnn
Obtain the linearized Friedman equations in conformal Newtonian gauge using the results of previous problems.
In linear approximation the Einstein equations have the form \begin{equation}\label{Ein_lin} \delta G^\mu_\nu = 8\pi G \delta T^\mu_\nu. \end{equation} Using the linearized Einstein tensor $\delta G_\mu^\nu$ from previous problem and linearized energy--momentum tensor $\delta T_\mu^\nu = \widetilde{T}_\mu^\nu -T_\mu^\nu$: \begin{equation} \Delta \Phi - 3\mathcal{H}\Phi ' - 3\mathcal{H}^2\Phi =4\pi G a^2 \sum_i \delta\rho_{i};\label{Ein_1} \end{equation} \begin{equation} \Phi ' + 3\mathcal{H}\Phi = 4\pi G a^2 \sum_i \left(\rho_i - p_i\right) v_i;\label{Ein_2} \end{equation} \begin{equation} \Phi ''+ 3\mathcal{H}\Phi '+ 2\mathcal{H}'+\mathcal{H}^2=4\pi G a^2 \sum_i \delta p_{i}.\label{Ein_3} \end{equation}
Problem 7
problem id: per29nnn
Rewrite the linearized Friedman equations in momentum representation.
It is convenient to search for solutions for perturbations equations in Fourier representation: $$ h_{\mu\nu}(\eta,\, \textbf{x})= \int d^3ke^{i\textbf{kx}} h_{\mu\nu}(\textbf{k}). $$ The similar representation is used for $\delta\rho,\,\, \delta p,\,\, v_i$ and other quantities. Einstein equations for scalar perturbations in momentum representation are obtained by substitution $\partial_i\longleftrightarrow ik_i$: \begin{equation} k^2\Phi +3\frac{a'}{a}\Phi '+3\frac{a'^2}{a^2}\Phi =-4\pi Ga^2\delta\rho \label{kPhi} \end{equation} \begin{equation} \Phi ''+3\frac{a'}{a}\Phi+\left(2\frac{a''}{a}-\frac{a'^2}{a^2}\right) \Phi =4\pi Ga^2\delta p \label{Phi_} \end{equation}
Problem 8
problem id: per30nnn
Using the linearized Friedman equations in momentum representation, obtain the equation, which contains only gravitational potential $\Phi.$
Multiplying the second equation from previous problem by $u_s^2$ and adding it to the first equation from previous problem: $$ \Phi +3\frac{a'}{a}(1+u_s^2)\Phi '+2\left[2\frac{a''}{a}- \frac{a'^2}{a^2}(1-3u^2_s)\right]\Phi + u^2_sk^2\Phi=0. $$ Using the first Friedman equation $\frac{a'^2}{a^4} = \frac{8\pi G}{3}\rho$ and ($ij$)--component of Einstein equation $2\frac{a''}{a^3}-\frac{a'^2}{a^4}=-8\pi G p$, one can rewrite the expression in square brackets through background pressure and energy density: $$ 2\frac{a''}{a}- \frac{a'^2}{a^2}(1-3u^2_s) = -8\pi G a^2(p-u^2_s\rho) $$ For single component Universe, the expansing is governed by this components itself, so that $w=u^2_s$ and $p=u^2_s\rho$. Finally: \begin{equation}\label{Per_Phi} \Phi '' +3\frac{a'}{a}(1+u_s^2)\Phi ' + u^2_sk^2\Phi=0. \end{equation} It is important to note, that relation $w=u^2_s$ holds at all stages of hot Big Bang, except transitional periods. This relation not only violates in transitional periods, but also values in equation become time dependent.
Problem 9
problem id: per31nnn
Determine the evolution of relativistic matter perturbations in single component expansing Universe.
In single component Universe filled with relativistic mater $w=u_s^2=1/3$ and scale factor linearly depends on conformal time $a\propto \eta$. Equation \eqref{Per_Phi} for gravitational potential obtains the form: $$ \Phi'' +\frac{4}{\eta}\Phi' +u_s^2k^2\Phi=0. $$ Multiplying both parts by $\eta^2$ and using the substitution $x=u_sk\eta$ one obtains $$ x^2\Phi'' +4x\Phi' +x^2\Phi=0, $$ where $'$ denotes the derivative with respect to new coordinate $x$. This is the modified Bessel equation. Using the ansatz $\Phi =\Psi x^\alpha$: $$ x^2\Psi'' +2x(\alpha +2)\Psi' +(1+\alpha(\alpha+3))\Psi=0, $$ In case of $\alpha=-\frac{3}{2}$ this equation reduces to ordinary Bessel equation with general solution in from of Bessel functions $J_n(x)$ and $J_{-n}(x)$ with fractional order $n=\frac{3}{2}$. When reverting to initial coordinates: $$\Phi(\eta) = \frac{1}{(u_sk\eta)^{3/2}} \left[C_1J_{3/2}(u_s k\eta)+C_2J_{-3/2}(u_sk\eta)\right]$$ where $C_1$ and $C_2$ are integration constants. Obviously, solution $J_{-3/2}(u_sk\eta)$ is not applicable, so that general solution is $$ \Phi(\eta)=\frac{C_1}{u_s k \eta}J_{3/2}(u_s k\eta) = \frac{1}{(\pi u_s \eta)^2}\left(\frac{1}{\pi u_s \eta}\sin(\pi u_s \eta)-\cos(\pi u_s \eta)\right). $$ After reaching the sound horizon (ie. when $u_sk\eta \ll 1$) this solution describes a wave with decreasing amplitude and well--defined phase $$\Phi(\eta)= -C_1\sqrt{\frac{2}{\pi}} \frac{\cos k\eta}{(k\eta)^2}.$$ When calculating the density perturbations under the horizon, first term in left--hand side of \eqref{kPhi} is dominating, so that \begin{equation}\label{delta_rho_rad} \delta\rho_r = -\frac{1}{4\pi}\frac{k^2}{a^2}\Phi(\eta) \end{equation} At stage of relativistic matter domination the Friedman equation has the form $$H^2=\frac{1}{a^2\eta^2},$$ and, thus \begin{equation}\label{delta_rho_rad} \delta_r =\frac{\delta\rho_r}{\rho_r}= \frac{6C_1}{u_s^2} \end{equation} It is evident, that relativistic matter density perturbations under the horizon experience acoustic oscillations with non--increasing (but also non--decreasing) amplitude at the stage of matter domination. Because of the expansion, phenomena like Jeans instability do not appear in relativistic matter dominated Universe.
Problem 10
problem id: per32nnn
Derive the continuity equation and Euler equation for single component relativistic medium.
For dingle component medium $v^2_s=w=1/3$ and using the equations from problem \ref{cov_newt}, one can obtain \begin{eqnarray} \delta ' -\frac{4}{3}k^2v &=&4 \Phi '\nonumber\\ v ' + \frac 14 \delta &=&- \Phi \nonumber \end{eqnarray}
Problem 11
problem id: per34nnn
Calculate the fluctuations (perturbations) of density and velocity of non--relativistic matter in the Universe dominated by non--relativistic matter.
Since the pressure of non--relativistic matter is negligible, the speed of sound can be set to zero. All modes, thus, are beyond the sound horizon. Using the solution of problem \ref{F=const}: $$ \Phi(\eta)=\Phi=const $$ Using the equation \eqref{kPhi} and taking into account that scale facto at the non--relativistic matter domination stage is $a(\eta)=const\cdot\eta^2$ (see problem \ref{dyn63}), one obtains the density perturbation: $$ \delta\rho=-\frac 1{4\pi Ga^2}\left(k^2+\frac {12}{\eta^2} \right)\Phi $$ Since $k^2\ll\frac{a'}{a}$beyond the horizon, the second term in brackets dominates, so that $$ \delta\rho=-\frac {3\Phi}{\pi Ga^3}\propto a^{-3}. $$ Energy density is also proportional to $a^{-3}$ and, thus, relative perturbations are constant beyond the horizon. using the first Friedman equation and $H=2/(a\eta)$, one obtains $$ \delta=\delta_{(i)}=-2\Phi,\,\,\,k\eta\ll 1 $$ The first term in $\left(k^2+\frac {12}{\eta^2} \right)$ is dominating below the horizon and contrast auickly increases in time: $\delta\propto a(\eta)$. This is the main mechanism of perturbations growth, which plays a part of Jeans instability in expanding Universe. From the equation for $\delta\rho$ and Friedman equations one can conclude, that at dust stage below the horizon perturbations obtain the form: $$ \delta = -\frac 23\frac{q^2(\eta)}{H^2(\eta)}\Phi= \frac 13\frac{q^2(\eta)}{H^2(\eta)}\delta_{(i)}, \,\,\, k\eta\gg 1. $$ Using equation \eqref{Ein_2} and assuming $\Phi(\eta)=const$, one can derive the fluctuation of velocity: $$ kv=-\frac\Phi 3 k\eta. $$ These fluctuations are small for modes below the horizon, since $kv\ll\Phi_{(i)}$. As the density perturbations $\delta\propto a(\eta)$, velocity perturbations grow at dust stage, but slower: $kv\propto\sqrt{a}$.
Problem 12
problem id: per35nnn
Determine the evolution of non--relativistic matter density perturbations in hypothetical expanding Universe, dominated by negative spatial curvature.
Problem 13
problem id: per24
Consider the Universe filled with ideal fluid with the equation of state $p=w\rho$ with metrics from previous (?) problem. Derive the equations, corresponding to condition $T^\beta_{\alpha,\beta}=0$.
Zero components of four--velocity in the first order approximation in $\Phi$ have the form: $$ u_0 = a\left( 1 + \Phi \right);\,u^0 = \frac{1}{a}\left( 1 - \Phi\right). $$ The divergence of the second--order tensor is $$ T_{\mu ;\nu }^\nu = T_{\mu ,\nu }^\nu - T_\alpha ^\nu \Gamma _{\mu \nu }^\alpha + T_\mu ^\alpha \Gamma _{\alpha \nu }^\nu . $$ Spatial components are ($\mu = i = 1,2,3$): $$ T_{i;\nu }^\nu = \frac{\partial T_i^\nu} {\partial {x^\nu }} - \left( T_0^0\Gamma _{i0}^0 + - p\delta _k^j\Gamma _{ij}^k \right) + \left( T_i^0\Gamma _{0\nu }^\nu - p\delta _i^j\Gamma _{jk}^s\delta _s^k \right) = 0, $$ and also the following relation holds due to homogeneity of space: $$\frac{\partial T_i^\nu} {\partial {x^\nu }} = \frac{\partial T_i^0} {\partial {x^0}} + \frac{\partial T_i^j} {\partial {x^j}} = 0. $$ Component $\mu = 0$ reads: $$ T_{0;\nu }^\nu = \frac{\partial T_0^\nu } {\partial {x^\nu }} - T_\alpha ^\nu \Gamma _{0\nu }^\alpha + T_0^\alpha \Gamma _{\alpha \nu }^\nu = \dot \rho + \left( T_0^0\Gamma _{00}^0 - p\delta _i^j\Gamma _{0j}^i \right) + \left( T_0^0\Gamma _{00}^0 + \rho \delta _i^j\Gamma _{0j}^i\right), $$ $$ T_{0;\nu }^\nu = \rho ' + p\left( \mathcal{H} + \Psi '\right)\delta _j^i\delta _i^j + \rho \left( \mathcal{H}+ \Psi ' \right)\delta _i^k\delta _k^i = 0, $$ $$ \rho ' + 3\left( \mathcal{H} + \Psi '\right)\left( p + \rho \right) = 0. $$
Problem 14
problem id: per27
Using the metrics \eqref{per_newton_conf}, derive the equation of motion of scalar field $\varphi$ from variational princile.
Using the same arguments as in problem \ref{inf5} in Chapter 8 and denoting $\mathcal{H} = \frac{a'}{a} = \frac{1}{a}\frac{da}{d\eta}$ , one obtains: $$ \varphi '' + 3\mathcal{H}\varphi ' + V_{,\varphi }a^2= 0. $$ Also, the equation of motion for scalar field in FRW metrics and conformal time has the form: $$ \varphi '' + 2\mathcal{H}\varphi ' + V_{,\varphi }a^2= 0. $$
Problem 15
problem id: per30
Derive the equation for fluctuations of scalar field $\delta \phi(t)$ using the metrics \eqref{per_newton_conf} and assuming linear approximation in scalar potential $\Phi$.
\begin{equation} \ddot{\delta \phi} + 3\frac{\dot{a}}{a}\dot{\delta \phi} + \frac{k^2 \delta \phi}{a^2} + 2\Phi V'(\phi) -4\dot{\Phi}\dot{\phi} + V''(\phi)\delta \phi = 0 \label{eqn::pertkg} \end{equation}
Problem 16
problem id: Tmunu_inf
Using the conformal Newtonian gauge, determine the components of energy--momentum tensor of scalar field, which is represented as a sum of unifom background field $\varphi(t)$ and perturbation part $\psi(\vec{x},t)$.
Problem 17
problem id: v_inf
Considering the scalar field from previous problem, determine which quantity plays a part of velocit of medium.
Problem 18
problem id: per31
Using the conformal Newtonian gauge determine the scalar of three--dimensional curvature in co--moving reference frame.
The components of perturbed Ricci tensor were obtained in problem #per23. Its spatial component is $$ \widetilde{R}_{ ij}=\delta_{ij}\left(\mathcal{H} '+\Delta \Phi-4 \Phi \mathcal{H} ' - \Phi '' +2\mathcal{H}^2 -8\mathcal{H}^2\Phi-6\mathcal{H}\Phi '\right). $$ Unperturbed components, at the same time, have the form $ R_{ ij}=\delta_{ij}\left(\mathcal{H}' +2\mathcal{H}^2 \right). $ Linearized part of Ricci tensor is $$ \delta R_{ ij}=\delta_{ij}\left(\Delta \Phi- 4 \Phi \mathcal{H} ' - \Phi '' -8\mathcal{H}^2\Phi-6\mathcal{H}\Phi '\right). $$ Using the definition of scalar Ricci curvature, one can express its spatial part as $$\delta R^{(3)}= g^{ij} \delta R_{ ij}. $$ Using the linearized Einstein equations from problem #per28nnn: \begin{equation}\label{R3phi} R^{(3)}= -\frac{4}{a^2}\Delta\widetilde{\Phi}, \end{equation} $$\widetilde{\Phi}=-\Phi+\frac{\delta \rho_{tot}}{3(\rho+p)_{tot}}.$$ it is also convenient to introduce the following quantity \begin{equation}\label{calR_def} \mathcal{R}=-\Phi+\frac{a'}{a}v_{tot}, \end{equation} where \begin{equation}\label{v_tot_def} v_{tot} =\frac{\sum_i(\rho_i+p_i)v_i} {\sum_i(\rho_i+p_i)} \equiv \frac{\left[(\rho+p)v\right]_{tot}}{\left[\rho+p\right]_{tot}} \end{equation} The physical meaning of $\mathcal{R}$ is the following: it is proportional to the curvature of spatial hypersurfaces in co--moving reference frame, i.e. hypersurfaces $v_{tot} =0:$ \begin{equation}\label{R3phi} R^{(3)}= -\frac{4}{a^2}\Delta\mathcal{R}. \end{equation} This is the general property of scalar perturbations.
Evolution of vector and tensor perturbations
Problem 19
problem id: per20h
Calculate the components of the Christoffel symbols in a coordinate system for which the metric is given in the form \begin{equation} \label{per_gen_conf} ds^2=a^2(\eta)[(\eta_{\mu\nu}+h_{\mu\nu}(\vec{x}))dx^\mu dx^\nu], \end{equation} where $\eta_{\mu\nu}$ is a standard Minkowski metrics and $h_{\mu\nu}(\vec(x))$ -- function of the coordinates describing the perturbations over a flat background.
Christoffel symbols for this metrics have the form \begin{equation}\label{Cristoff_gen_per} \begin{split} \Gamma^{0}_{00}&=\frac{a'}{a}+\frac{1}{2} h'_{00},~~ \Gamma^{0}_{0i}= \Gamma^{i}_{00}=\frac{1}{2}\partial_i h_{00};\\ \Gamma^{i}_{0j}&=\frac{a'}{a}\delta_{ij}-\frac{1}{2} h'_{ij} \\ \Gamma^{0}_{00}&=\frac{a'}{a}(1-h_{00})\delta_{ij}-\frac{a'}{a}h_{ij}-\frac{1}{2} h'_{ij} \\ \Gamma^{i}_{jk}&=- \frac{1}{2}\left(\partial_j h_{ik}+\partial_k h_{ij}-\partial_i h_{jk}\right). \end{split} \end{equation} As was said above, we consider the linear theory and, therefore, keep only linear terms in $ h_ {ij} $.
Problem 20
problem id: per20R
Obtain the components of the Ricci tensor for perturbed metrics of the form \eqref{per_gen_conf}. Calculate also the Ricci scalar curvature.
Substituting the Christoffel symbols \eqref {Cristoff_gen_per} obtained in the previous problem into the expression for the Ricci tensor $$ {R_{\mu \nu }} = \partial _\lambda \Gamma _{\mu \nu }^\lambda - \partial _\mu \Gamma _{\lambda \nu }^\lambda + \Gamma _{\rho \lambda }^\lambda \Gamma _{\mu \nu }^\rho - \Gamma _{\rho \mu }^\lambda \Gamma _{\nu \lambda }^\rho. $$ In linear order in $h_{\mu \nu }$ we obtain \begin{equation}\label{Cristoff_gen_per} \begin{split} R^0_0&=\frac{1}{2}\left(h''+\Delta h_{00}\right),\\ R^0_i&=\frac{1}{2}\left(\partial_i h'+\partial_j h'_{ij}\right),\\ R^i_j&=\frac{1}{2}\left(\partial_i \partial_k h_{jk}+\partial_j \partial_k h_{ik} +h''_{ij}- \Delta h_{ij} +\partial_i \partial_j h_{00}-\partial_i \partial_j h \right),\\ \end{split} \end{equation} where the notation $h=h_{ii}$ is used. Using the obtained components, one can obtain an expression for the scalar Ricci curvature: \begin{equation}\label{ricci_scal_per} R=h''_{ij}+ \Delta h_{00}+ \partial_i \partial_j h_{ij}-\Delta h. \end{equation}
Problem 21
problem id: per20G
Calculate the components of linearized Einstein tensor in metrics \eqref{per_gen_conf}.
\begin{equation}\label{Ein_gen_per1} a^2\delta G^0_0=-3h_{00}\left(\frac{a'}{a}\right)^2-\frac{1}{2}\partial_i \partial_j h_{ij}+\frac{1}{2}\Delta h +\frac{a'}{a}h', \end{equation} \begin{equation}\label{Ein_gen_per2} a^2\delta G^0_i=\frac{1}{2}\partial_i h'-\frac{1}{2} \partial_j h'_{ij}+\frac{a'}{a}\partial_i h_{00}, \end{equation} \begin{equation}\label{Ein_gen_per_fin} \begin{split} a^2\delta G^i_j=&\frac{1}{2}\partial_i \partial_k h_{jk}+\frac{1}{2}\partial_j \partial_k h_{ik}+\frac{1}{2}h''_{ij}-\frac{1}{2}\Delta h_{ij}+\frac{1}{2}\partial_i \partial_j h_{00}-\frac{1}{2}\partial_i \partial_j h+\frac{a'}{a}h'_{ij}-\\ -&\delta^i_j\left[\frac{1}{2} h''+\frac{1}{2}\Delta h_{00}+\frac{1}{2}\partial_l \partial_k h_{lk}-\frac{1}{2}\Delta h+2\frac{a''}{a}h_{00}-\left(\frac{a'}{a}\right)^2h_{00}+\frac{a'}{a}(h_{00}+h') \right]. \end{split} \end{equation}
Problem 22
problem id: gr_wave_eq
Construct the equation describing the evolution of tensor perturbations in the expanding Universe.
Substituting the expansion of the tensor perturbation \eqref{hij_gen} into the expression, defining the spatial part of the Einstein tensor (it is the only non-trivial under such expansion) from problem #per20G and keeping only the terms linear in $ h_ {ij} ^ {TT} $ (since they are responsible for the tensor perturbations), one obtain \begin{equation}\label{GijTT} 2 a^2\delta G^i_j = \widetilde{h}''_{ij} + 2\frac{a'}{a} \widetilde{h}'_{ij}- \Delta \widetilde{h}_{ij}=0, \end{equation} Since all terms in this equation have double helicity (Freud is crying?:))), we use the simplified notation: $h_{ij}^{TT} = \widetilde{h}_{ij}$. As it was shown in problem per22ntenzor, energy-momentum tensor has components with a double helicity, hence the equation for gravitational waves is \begin{equation}\label{widetildehij} \widetilde{h}''_{ij} + 2\frac{a'}{a} \widetilde{h}'_{ij}- \Delta \widetilde{h}_{ij}=0, \end{equation} or in the momentum representation: \begin{equation}\label{widetildehij} \widetilde{h}''_{ij} + 2\frac{a'}{a} \widetilde{h}'_{ij}+k^2\widetilde{h}_{ij}=0. \end{equation}
Problem 23
problem id: gr_wave_eq_action
Construct the expression for the action of gravitational waves in the expanding Universe.
In problem \ref{equ25} in Chapter 2, the expression for the variation of action of gravitational field was constructed in the case of arbitrary metrics \eqref{delta_S_m_g }: $$ \delta S =- \frac{1}{16\pi G}\int \left( R_{\mu \nu } - \frac{1}{2}Rg_{\mu \nu } - 8\pi GT_{\mu \nu } \right) \delta g^{\mu \nu }\sqrt { - g} d\Omega. $$ Using the general expression for the perturbed metrics $g^{\mu \nu } = a^{-2}(\eta^{\mu \nu }-h^{\mu \nu })$, one obtains the expressions for variation: \begin{equation}\label{delta_S_m_g} \delta S = \frac{1}{16\pi G}\int \left(G_{\mu \nu } - 8\pi GT_{\mu \nu }\right) \delta h^{\mu \nu }a^2 dx^4. \end{equation} Using the perturbation theory, we make the following replacement: $$ G_{\mu \nu } \to G_{\mu \nu }+ \delta G_{\mu \nu }, T_{\mu \nu } \to T_{\mu \nu }+ \delta T_{\mu \nu },$$ òîãäà and obtain \begin{equation}\label{delta_S_m_g1} \delta S =\frac{1}{16\pi G}\int \left[ (G_{\mu \nu } - 8\pi GT_{\mu \nu }) + (\delta G_{\mu \nu } - 8\pi G \delta T_{\mu \nu }) \right] \delta h^{\mu \nu }a^2 dx^4. \end{equation} The expression in the first parenthesis is unperturbed and satisfies undisturbed Einstein equations and, therefore, identically equals zero. The expression in the second brackets can be rewritten as: $$ \delta G_{\mu \nu } - 8\pi G \delta T_{\mu \nu }=a^2\eta_{\nu\alpha} (\delta G_\mu^\alpha - 8\pi G \delta T_\mu^\alpha)=a^2\eta_{\nu\alpha}\delta G_\mu^\alpha, $$ where we have taken into account that in the case of the Universe filled with perfect fluid perturbed energy-momentum tensor $ T_\mu ^ \nu$ has no components with a double helicity. Thus we obtain the following expression for the tensor perturbations: \begin{equation}\label{delta_S_m_g2} \delta S =-\frac{1}{16\pi G}\int \delta G_i^j \delta \widetilde{h}_{ij}a^4 dx^4. \end{equation} Since $\delta G_i^j$ is linear in $\widetilde{h}_{ij}$, the corresponding term $\propto \widetilde{h}_{ij}\delta \widetilde{h}_{ij}$ should be obtained by variation of unknown action, which is quadratic in $\widetilde{h}_{ij}$. It follows that the unknown action of gravitational perturbations is half of the right--hand side of \eqref{delta_S_m_g2} with $\delta \widetilde{h}_{ij}$ replaced by $\widetilde{h}_{ij}$: \begin{equation}\label{S_GravW0} S =-\frac{1}{32\pi G}\int \delta G_i^j \widetilde{h}_{ij}a^4 dx^4. \end{equation} Using the results of the previous problem we obtain \begin{equation}\label{S_GravW} \delta S =-\frac{1}{64\pi G}\int dx^4 \widetilde{h}_{ij}a^2 \left(\widetilde{h}''_{ij} + 2\frac{a'}{a} \widetilde{h}'_{ij}- \Delta \widetilde{h}_{ij}\right), \end{equation} or in momentum representation: \begin{equation}\label{S_GravW_k} \delta S =-\frac{1}{64\pi G}\int dx^4 \widetilde{h}_{ij}a^2 \left(\widetilde{h}''_{ij} + 2\frac{a'}{a} \widetilde{h}'_{ij}+k^2\widetilde{h}_{ij}\right). \end{equation} It is easy to see that the equation for the action of gravitational perturbations has exactly the same form as the equation of motion and action for a free scalar field. \begin{equation}\label{series_spirality} \widetilde{h}_{ij} = \sum_S e_{ij}^{(T)} h^{(T)}, \end{equation} where $e_{ij}^{(S)}$ is a unit antisymmetric traceless tensor with double helicity and subscript $S$ takes the values, which correspond to positive and negative helicity $S=+,-.$. In this expansion the action \eqref{S_GravW} becomes \begin{equation}\label{S_GravW1} \delta S =- \sum_T \frac{1}{64\pi G}\int dx^4 h^{(T)} a^2 \left( \partial_\eta^2 h^{(T)} + 2\frac{a'}{a} \partial_\eta h^{(T)}- \Delta h^{(T)}\right). \end{equation} This action describes the perturbation of two polarizations, independently evolving in the expanding Universe. The corresponding equation takes the form \begin{equation}\label{eq_g_wave_s} \partial_\eta^2 h^{(T)} + 2\frac{a'}{a} \partial_\eta h^{(T)}- \Delta h^{(T)}=0. \end{equation}
Problem 24
problem id: gr_wave_eq_Mimk
Construct the equation for gravitational perturbations in the Minkowski background metrics.
The transition to the Minkowski background space is performed by replacement $ a (\eta) \to 0. $ In such transition $ \eta$, $ x $ have a meaning of ordinary time and space coordinates. Hence, equation \eqref{eq_g_wave_s} transforms into \begin{equation}\label{eq_g_wave_s1} \Box^{4} h^{(T)} =0. \end{equation} where $\Box^{4}$ -- four-dimensional d'Alembert operator in Minkowski space: $\Box^{4} = \eta^{\mu\nu}\partial_{\mu}\partial_{\nu}$. This equation describes propagation of gravitational waves with the speed of light and two possible polarizations.
Problem 25
problem id:
Find the solution for the equation describing the evolution of tensor modes beyond the horizon.
The modes beyond the horizon satisfy the condition $$k\ll \frac{a'}{a},$$ hence the equation describing their evolution becomes \begin{equation} h''+2\frac{a'}{a}h' =0. \end{equation} One of the solutions to this equation is constant in time, \begin{equation} h =h_{(i)}=const, \end{equation} and is called constant mode; subscript $(i)$ denoted the initial value of amplitude. Second solution is a decreasing mode: \begin{equation} h =const\int\frac{d\eta}{a^2(\eta)}. \end{equation} This mode descreases with increasing scale factor: $h\propto a^{-1}$ at the relativistic matter domination stage and $h\propto a^{-3/2}$ at the stage of non--relativistic matter domination. If the decreasing mode is not too large at early stages of the hot Big Bang era, it quickly becomes negligible. Conversely, if at the time of wave's entry under the horizon decreasing mode is not negligible, it was large in the early stages, that is, the Universe was once a highly inhomogeneous and anisotropic.
Problem 26
problem id:
Determine the amplitude of the tensor perturbations in the Universe filled with relativistic matter. Consider the case $\eta \to 0$.
At the stage of relativistic matter domination the following relation holds: $\frac{a'}{a}=\eta^{-1}$. Hence, the equation \eqref{eq_g_wave_s} transforms into Bessel equation: $$ h''+\frac{2}{\eta}h'+k^2h =0.$$ This equation is a modified Bessel equation and its solution is written as a superposition of the Bessel function of the first $ J_{\nu} (k \eta) $ and second $Y_{\nu}(k\eta)$ kinds: \begin{equation}\label{h_k_grav_rd} h(\eta)= (k\eta)^{-1/2}\left[C_1J_{-1/2}(k\eta)+C_2Y_{-1/2}(k\eta)\right], \end{equation} where $C_1$ and $C_2$ are integration constants. In the limit $\eta \to 0$: $$J_{-1/2}(k\eta)= \sqrt{\frac{2}{\pi k\eta }}\cos k\eta,~~Y_{-1/2}(k\eta)= \sqrt{\frac{2}{\pi k\eta }}\sin k\eta.$$ Discarding the the infinite mode under $\eta \to 0$, one obtains $$h(\eta)= C \frac{\sin k\eta}{k\eta}.$$
Problem 27
problem id:
Determine the amplitude of the tensor perturbations in the Universe filled with relativistic matter. Consider the case $\eta \to 0, ~\eta \to \infty.$
The modes, which enter below the horizon at the stage of non--relativistic matter domination $a(\eta)\propto \eta^2$ the equation \eqref{eq_g_wave_s} takes the form $$ h''+\frac{4}{\eta}h'+k^2h =0.$$ This equation also reduces to the Bessel equation whose solution is of the form \begin{equation}\label{h_k_grav_md} h(\eta)= (k\eta)^{-3/2}\left[C_1J_{-3/2}(k\eta)+C_2Y_{-3/2}(k\eta)\right], \end{equation} where $C_1$ and $C_2$ are integration constants. Asymptotics under $h(\eta \to 0) \to h_{(i)}=const$ has the form $$h(\eta)= h_{(i)} (k\eta)^{-3/2}Y_{-3/2}(k\eta) = \sqrt{\frac{2}{\pi}}\frac{h_{(i)}}{(k\eta)^{2}}\left(\frac{\sin k\eta}{k\eta} - \cos k\eta\right)$$ At $k\eta \ll1$ solution behaves as $$h(\eta)= -h_{(i)}\sqrt{\frac{2}{\pi}} \frac{\cos k\eta}{(k\eta)^2}.$$
