Expansion of cosmological perturbations in helicities

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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 $h_{0i}=0.$

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.

Problem 2

problem id: per24nnn

In conformal Newtonian gauge and perturbed energy--momenetum tensor derive the covariant conservation equations

Problem 3

problem id: per25nnn

Calculate the components of Ricci tensor in conformal Newtonian metrics \eqref{per_newton_conf}.

Problem 4

problem id: per26nnn

Calculate the components of perturbed Einstein tensor $G_\mu^\nu$ using the components of Ricci tensor.

Problem 5

problem id: per27nnn

Calculate the components of linearized Einstein tensor using the results of previous problems.

Problem 6

problem id: per28nnn

Obtain the linearized Friedman equations in conformal Newtonian gauge using the results of previous problems.

Problem 7

problem id: per29nnn

Rewrite the linearized Friedman equations in momentum representation.

Problem 8

problem id: per30nnn

Using the linearized Friedman equations in momentum representation, obtain the equation, which contains only gravitational potential $\Phi.$

Problem 9

problem id: per31nnn

Determine the evolution of relativistic matter perturbations in single component expansing Universe.

Problem 10

problem id: per32nnn

Derive the continuity equation and Euler equation for single component relativistic medium.

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.

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 $T^\beta_{\alpha,\beta}=0$.

Problem 14

problem id: per27

Using the metrics \eqref{per_newton_conf}, derive the equation of motion of scalar field $\varphi$ from variational princile.

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 $\Phi$.

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.

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.

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.

Problem 21

problem id: per20G

Calculate the components of linearized Einstein tensor in metrics \eqref{per_gen_conf}.

Problem 22

problem id: gr_wave_eq

Construct the equation describing the evolution of tensor perturbations in the expanding Universe.

Problem 23

problem id: gr_wave_eq_action

Construct the expression for the action of gravitational waves in the expanding Universe.

Problem 24

problem id: gr_wave_eq_Mimk

Construct the equation for gravitational perturbations in the Minkowski background metrics.

Problem 25

problem id:

Find the solution for the equation describing the evolution of tensor modes beyond the horizon.

Problem 26

problem id:

Determine the amplitude of the tensor perturbations in the Universe filled with relativistic matter. Consider the case $\eta \to 0$.

Problem 27

problem id:

Determine the amplitude of the tensor perturbations in the Universe filled with relativistic matter. Consider the case $\eta \to 0, ~\eta \to \infty.$