In the following section we consider only the linear in perturbations theory.
§ 1Perturbations on flat background
Problems 7.2.1 – 7.2.4
7.2.1
Problem 7.2.1
problem id: per15
Consider the stability of stationary Universe, filled with matter $(p=0)$ and by substance with $p = w\rho$.
solution
Substituting the expressions $\rho_w = \rho _{w0} + \delta \rho_w;\; a = a_0 + \delta a$ into second Friedman equation (here subscript 0 denotes the unperturbed quantities) and keeping only first order terms, we obtain
$$\delta \ddot a = - \frac{8\pi G}{3} a_0\delta \rho _w.$$
Using
$$
\frac{\delta \rho _w}{\rho _w} = - 3(1 + w)\frac{\delta a}{a},$$
one could obtain
$$
\delta \ddot a - 4\pi G\rho _{w0}(1 + w)(1 + 3w)\delta a = 0.
$$
In the general case the solution of this equation has the form:
$$
\delta a=C_1e^{\sqrt{4\pi G\rho _{w0}(1+w)(1+3w)}\cdot t}+C_2e^{-\sqrt{4\pi G\rho _{w0}(1+w)(1+3w)}\cdot t}.
$$
In the region $-1<w<1/3$, thus, Universe is stable to small perturbations and $\delta \propto \sin(t)$. Outside of this region of $w$, perturbations exponentially increase in time. Such Universe is unstable. The case of Einstein's closed stationary Universe must be considered separately (see Problem 1 in Time-dependent Cosmological Constant in chapter 9).
7.2.2
Problem 7.2.2
problem id: per12
Determine the dependence of density fluctuations on scale factor in the flat Universe when
a) radiation,
b) matter
is dominating.
7.2.3
Problem 7.2.3
problem id: per13
Determine the dependence of density fluctuations on time in the closed Universe with $k=1$.
7.2.4
Problem 7.2.4
problem id: per14
Determine the dependence of density fluctuations on time in the open Universe with $k=-1$.
§ 2Problems 7.2.5 – 7.2.14
7.2.5
Problem 7.2.5
problem id: per18
The inhomogeneities in matter distribution in the Universe generate perturbations of different kinds. In linear approximation these perturbations do not interact with each other and evolve independently. Construct the classifications of perturbations.
solution
$$
\begin{array}{l}
ds^{2} =\left[g_{\mu \nu }^{(0)} +\delta g_{\mu \nu } \right]dx^{\mu } dx^{\nu } \\
{\left|\delta g_{\mu \nu } \right|\ll \left|g_{\mu \nu }^{(0)} \right|} \end{array}
$$
Using the conformal time the unperturbed part of the metrics can be expressed in the following form:
$$
g_{\mu \nu }^{(0)} dx^{\mu } dx^{\nu } =a^{2} (\eta )\left(d\eta ^{2} -\delta _{ij} dx^{i} dx^{j} \right).
$$
Perturbations of the metrics $\delta _{\mu \nu }$ can be scalar, vector and tensor. This classification is based on the invarience of unperturbed metrics under spatial rotations and translations. Scalar component of perturbation is
$$
\delta g_{00} =2a^{2} \varphi ,
$$
where $\varphi (\vec{r},t)$ is a scalar.
Vector component $\delta g_{0i}$ can be expressed as a sum of a gradient of scalar function $B$ and vector $S_{i} $ with zero divergence. This component transforms as
$$
\delta g_{0i} =a^{2} \left(\frac{\partial B}{\partial x^{i} } +S_{i} \right).
$$
Due to the condition $S_{,i}^{i} =0$ this vector has only two independent components.
Component $\delta g_{ij} $ transform as a tensor under three--dimensional rotations and can be expressed as
$$
\delta g_{ij} =a^{2} \left(2\psi \delta _{ij} +2\frac{\partial E}{\partial x^{i} \partial x^{j} } +\frac{\partial F_{i} }{\partial x^{j} } +\frac{\partial F_{j} }{\partial x^{i} } +h_{ij} \right).
$$
Here $\psi $ and $E$ are scalar functions, $F_{i} $ is a vector with zero divergence and $h_{ij} $ is a tensor, which satisfies four conditions:
$$
h_{i}^{i} =0;\; h_{j,i}^{i} =0.
$$
Scalar perturbations describe inhomogeneities in energy (and mass) distribution. Due to Jeans instability these components lead to the formation of structures in the Universe. Vector perturbations are are associated with rotational motions in fluid. Tensot perturbations have no classical analogs and describe the gravitational degrees of freedom and gravitational waves in particular.
7.2.6
Problem 7.2.6
problem id: per19
Determine the number of independent functions required for description of perturbations, considered in the previous problem.
solution
Scalar perturbations describe inhomogeneities in energy distribution and are determined by four functions $\varphi ,\psi ,B,E$. Vector (rotational) perturbations are described by the vectors $S_{i} ,F_{i} $ with corresponding conditions of zero divergence, so that four independent functions describe this class of perturbations. Tensor perturbations are described by two independent functions, since the number of independent components of symmetric order--3 tensor is 6 and four conditions apply:
$$
h_{i}^{i} =0;\quad h_{j,i}^{i} =0.
$$
Thus, ten independent functions is needed in order to describe the whole variety of considered perturbations.
7.2.7
Problem 7.2.7
problem id: per20ntenzor
Assuming, that in the Universe with metrics
\begin{equation}\label{met_ten_per}
ds^2=dt^2-a^2(t)(\delta_{ik}-h_{ik})dx^idx^k\;(i,k=1,2,3),
\end{equation}
the four--velocity has the form
$$u^0=\frac{1}{a}(1+\delta u^0),~u^i=\frac{1}{a} v^i,$$
determine the connection between components of four-velocity and metrics in linear approximation in perturbations. Consider $u^0$ and $v^i$ as first-order terms.
solution
The background and perturbed parts of four--velocity are
\begin{equation}\label{dalta_u}
u^0=\frac{1}{a}(1+\delta u^0),~u^i=\frac{1}{a} v^i.
\end{equation}
Using the general expression
$$g_{\mu\nu}u^\mu u^\nu=1.$$
one obtains
$$(1+h_{00})(1+2\delta u^0)=1$$
and finally
$$\delta u^0=-\frac{1}{2}h_{00}.$$
It's also useful to calculate $u_0$:
$$u_0=a^2(1+h_{00})u^0=a\left( 1+h_{00}\right)$$
Spatial components of $v^i$ remain undetermined. The explanation is that local motions of matter in linear approximation do not influence metrics.
7.2.8
Problem 7.2.8
problem id: per21ntenzor
Demonstrate, that spatial velocity $v^i$ from previous problem is physical.
solution
The four--velocity is expressed as
$$
\delta u^i=\frac{dx^i}{ds}\approx\frac{dx^i}{dt},
$$
which is exact in the first order. Physical velocity is, in turn,
$$
v^i=\frac{\delta l^i}{\delta t}=\frac{a dx^i}{dt},
$$
where $\delta l^i=a dx^i$ is physical length.
Let's note that only in formulation \eqref{dalta_u} $v^i$ is a physical velocity.
7.2.9
Problem 7.2.9
problem id: per22ntenzor
Using the results of two previous problems, calculate the components of energy--momentum tensor perturbation $\delta T^\mu_\nu$, which is induced by the perturbations of density and pressure $$\widetilde{\rho}=\rho+\delta \rho, ~ \widetilde{p}=p+\delta p,$$ where $\rho,p$ are their background values.
solution
By definition
\begin{equation}\label{delta_T}
\delta T^\mu_\nu =\widetilde{T}^\mu_\nu -T^\mu_\nu = T^\mu_\nu \left(\widetilde{\rho},\widetilde{p}\right)-T^\mu_\nu \left(\rho,p\right).
\end{equation}
Substituting the components of four--velocity from previous problem into this expressions and using the expression for unperturbed energy-momentum tensor (see chapter 2), we obtain in linear approximation
\begin{eqnarray}
\delta T^0_0 &=& \delta \rho \\
\delta T^0_i &=& -(\rho+p)v_i \\
\delta T^i_j &=& - \delta^i_j \delta p.
\end{eqnarray}
7.2.10
Problem 7.2.10
problem id: per23ntenzor
Obtain the equations of covariant convervation in linear approximation in components of tensor perturbations $h_{\mu\nu}$. Use the gauge $h_{0i}=0$ for simplicity.
solution
Due to Lorentz covariance of energy--momentum tensor the four non--trivial equations exist: one for zero component of energy--momentum tensor $\nabla_\mu T^\mu_{0}:$
\begin{equation}\label{cov_ten_00}
\delta \rho '+3\mathcal{H}\left( \delta \rho +\delta p\right)+(p+\rho)\left( \partial_i v_i - \frac 12 h'\right)=0
\end{equation}
and three for spatial components $\nabla_\mu T^\mu_{i}:$
\begin{equation}\label{cov_ten_ij}
\partial_i \delta p+ \left( \rho + p\right)\left(4 \mathcal{H} v_i - \frac 12 \partial_i h_{00}\right)+\left[ (\rho + p)v \right ]'=0
\end{equation}
where $h=h_{ii}$, and dash denotes derivative with respect to conformal time.
7.2.11
Problem 7.2.11
problem id: per20
In the first order of $h_{ik}$ calculate the components of energy--momentum tensor for the Universe. filled with ideal fluid with equation of state$p=w\rho$ and metrics \eqref{met_ten_per}
solution
Varying the inverse-metric identity $g^{\mu\rho}g_{\rho\nu}=\delta^{\mu}_{\nu}$ gives the following relation:
$$
h^{\mu \nu } \equiv \delta g^{\mu \nu } = - g^{\lambda \nu }g^{\mu \rho }\delta g_{\rho \lambda},
$$
with components $h^{ij} = - a^{ - 4}h_{ij};\;h^{i0} = a^{ - 2}h_{i0};
\;h^{00} = - h_{00}.$ Energy--momentum tensor for ideal fluid has the form:
$$
T_{\mu \nu } = \left( p + \rho \right)u_\mu u_\nu - pg_{\mu \nu },
$$
where $g^{\mu \nu }u_\mu u_\nu = 1$. Thus,
$$
u_\mu u_\nu \delta g^{\mu \nu } = - 2g^{\mu \nu }u_\mu\delta u_\nu = u^\nu \delta u_\nu.
$$
For ideal fluid filling the Universe ${u_0} = 1,\:{u_i} = 0$, which immediately gives
$$
\delta {u_0} = \delta {u^0} = \frac{1}{2}h_{00},
$$
$$\delta T_{\mu \nu } = u_\mu u_\nu \delta \left( p + \rho \right) + \left( u_\mu \delta u_\nu + u_\nu\delta u_\mu \right)\left( p + \rho\right) - p\delta g_{\mu \nu } - g_{\mu \nu }\delta p.
$$
$$
\begin{gathered}
\delta T_{ij} = - w\left(\rho h_{ij} - a^2\left( t \right)\delta _{ij}\delta \rho \right); \\
\delta T_{i0} = \left( 1 + w \right)\rho \delta u_i - w\rho h_{i0};\; \\
\delta T_{00} = \delta \rho + w\rho h_{00}.\\
\end{gathered}
$$
7.2.12
Problem 7.2.12
problem id: per21
In linear approximation determine the transformation of metrics $g_{\mu\nu}$, which is generated by the space transformation $x^\alpha\rightarrow \tilde{x}^\alpha=x^\alpha+\xi^\alpha$, where $\xi^\alpha$ is infinitesimal scalar function.
solution
The second order tensor is transformed according to
$$
g_{\alpha \beta }\left(\tilde x^\rho \right) = \frac{\partial x^\gamma }
{\partial \tilde x^\alpha }\frac{\partial x^\delta }
{\partial \tilde x^\beta }{g_{\gamma \delta }}\left( {x^\rho } \right) \approx {}^{(0)}g_{\alpha \beta } + \delta g_{\alpha \beta } - {}^{(0)}g_{\alpha \delta }\xi _{,\beta }^\delta - {}^{(0)}g_{\gamma \beta }\xi _{,\alpha }^\gamma ,
$$
where only linear terms in $\delta g_{\alpha \beta }$ è ${\xi ^\rho }$ are holded.
In new coordinates $\tilde x$ the metrics can be expressed as a sum of background and perturbed parts:
$$
\tilde g_{\alpha \beta }\left( \tilde x^\rho \right) = {}^{(0)}g_{\alpha \beta }\left( \tilde x^\rho \right) + \delta g_{\alpha \beta },
$$
where ${}^{(0)}g_{\alpha \beta }$ is an unperturbed metrics, which depends on $\tilde x$.At the same time, it easy to obtainthe following relation:
$$
{}^{(0)}g_{\alpha \beta }\left( x^\rho \right) = {}^{(0)}g_{\alpha \beta }\left( \tilde x^\rho \right) - {}^{(0)}g_{\alpha \beta ,\gamma }{\xi ^\gamma },
$$
Finally, the transformation is
$$
\delta g_{\alpha \beta } \to \delta \tilde g_{\alpha \beta } = \delta g_{\alpha \beta } - {}^{(0)}g_{\alpha \beta ,\gamma }\xi ^\gamma- {}^{(0)}g_{\gamma \beta }\xi _{,\alpha }^\gamma - {}^{(0)}g_{\alpha \delta }\xi _{,\beta }^\delta .
$$
7.2.13
Problem 7.2.13
problem id: per22
Using the results of previous problem determine the transformation of metrics, generated by the transformation $x^\alpha\rightarrow \tilde{x}^\alpha=x^\alpha+\xi^\alpha$. Here the four--vector $\xi^\alpha=(\xi^0,\xi^i)$ satisfies the condition $\xi^i=\xi^i_\bot+\zeta^i$, $\xi^i_\bot$ is a three--vector with zero divergence ($\xi^i_{\bot,i}=0$) and $\zeta^i$ is a scalar function.
solution
In conformal time the metrics of Friedman universe has the form:
$$
g_{00} = a^2\left( \eta \right),\;g_{ij} =-a^2\left( \eta \right)\delta_{ij}.
$$
using the results of previous problem:
$$
\begin{gathered}
\delta \tilde g_{00} = \delta g_{00} - 2a\left( a\xi ^0\right)^\prime,\\
\delta \tilde g_{0i}= \delta g_{0i}+ a^2\left[ {\xi '}_{ \bot i} + \left( \zeta ' - \xi ^0 \right)_{,i} \right],\\
\delta \tilde g_{ij} = \delta g_{ij}+ a^2\left[ 2\frac{a'}
{a}\delta _{ij}\xi ^0 + \zeta _{,ij} + \left( \xi _{ \bot i,j} +\xi _{ \bot j,i} \right)\right], \\
\end{gathered}
$$
where the relation $\xi _{\bot i}\equiv \xi _ \bot ^i$ is used and dash denoted the derivative with the respect to conformal time $\eta$.
7.2.14
Problem 7.2.14
problem id: per22_1
Demonstrate, that non--uniform flat Friedman metrics
$$
ds^2= \left(1-\frac{2}{\sqrt{\lambda}}\dot{f}(t,\vec{r})\right)dt^2-a^2(\delta_{ij}-2\mathcal{B}_{,ij})dx^idx^j,
$$
with arbitrary small function $f(t,\vec{r})$, can be trnsformed to the uniform one.
solution
It is easy to demonstrate by direct substitution, that transformations
$$
t'=t-\lambda^{-1/2}f(t,\vec{x}),~ \vec{y}=\vec{x}-\mathcal{\nabla B},~ \mathcal{B}=\lambda^{-1/2}\int a^{-2}f(t,\vec{x})dt,~a=a_0(t')=a_0\cdot (1-f(t,\vec{x})),
$$
with $f(t,\vec{x}) = k\frac{\vec{x}{\,}^2}{2},$ convert the metrics
$$
ds^2= \left(1-\frac{2}{\sqrt{\lambda}}\dot{f}(t,\vec{r})\right)dt^2-a^2(\delta_{ij}-2\mathcal{B}_{,ij})dx^idx^j,
$$
into the metrics for homogeneous space:
$$
ds'^2=dt'^2-a^2d\vec{y}.
$$