Use the relation \[\rho_{curv}=-\frac3{8\pi G}\frac{k}{R^2},\] where $R$ is the (comoving) radius of curvature of the open or closed Universe, to find \[\Omega_{k0}=-\frac{k}{RH_0}.\] It is easy to see that the expression \[d_L=\frac{1+z}{H_0\sqrt{\Omega_{k0}}}\sinh\left({H_0\sqrt{\Omega_{k0}}}\int\limits_0^z\frac{dz}{H(z)}\right)\] reproduces the commonly known expression for the luminosity distance \[ d_L(z)=(1+z)\left\{ \begin{array}{lr} R\sinh\left[\frac1{H_0R}\int\limits_0^z\frac{dz'}{E(z')}\right], & open\\ H_0^{-1}\int\limits_0^z\frac{dz'}{E(z')}, & flat\\ R\sin\left[\frac1{H_0R}\int\limits_0^z\frac{dz'}{E(z')}\right], & closed \end{array} \right.\]

The energy density has time dependence determined by the conservation equation \[\dot{\rho}=-3H\rho-3Hp.\] The two terms determine behavior of the homogeneous fluid which contains the energy in a dynamic Universe. The $H$ term provides the "friction", while the density term tracks the reduction in density due to the volume increase during expansion and the pressure term tracks the reduction in pressure energy during expansion.

From the second Friedmann equation \[\frac{\ddot a}{a}=-\frac{4\pi G}{3}(\rho+3p)\] clearly shows that both for matter and for radiation domination, the acceleration is negative; an expanding Universe dominated by the energy density of matter or radiation will decelerate. The acceleration in a matter dominated phase goes as $a^{-2}$ while in a radiation dominated phase it goes as $a^{-3}$. In either case the deceleration is large when $a(t)$ is small and then slows as the scale factor grows.

\begin{tabular}{|l|l|l|l|l|} \hline & UNIQ-MathJax8-QINU & UNIQ-MathJax9-QINU & UNIQ-MathJax10-QINU & UNIQ-MathJax11-QINU\\ \hline matter & UNIQ-MathJax12-QINU & UNIQ-MathJax13-QINU & UNIQ-MathJax14-QINU & UNIQ-MathJax15-QINU\\ \hline radiation & UNIQ-MathJax16-QINU & UNIQ-MathJax17-QINU & UNIQ-MathJax18-QINU & UNIQ-MathJax19-QINU\\ \hline UNIQ-MathJax20-QINU & constant & UNIQ-MathJax21-QINU & UNIQ-MathJax22-QINU & constant\\ \hline \end{tabular}

Indeed, in the case of the power-paw expansion with $\alpha<1$ (decelerated expansion) the Hubble radius grows faster than the Universe expands: $R_H=H^{-1}\propto t$ while $a(t)\propto t^\alpha$. In power law situations the Hubble radius has expansion velocity greater than the light speed \[\frac{d}{dt}\left(\frac1H\right)=\frac1\alpha.\] This behavior is true only for a decelerating Universe composed of matter and radiation. This is not a physical velocity, violating special relativity, but the velocity of expansion of the metric itself.

The cosmological constant do provide the accelerated expansion of Universe needed to realize the inflation: $a(t)\propto e^{Ht}$, $q=-1$. However, in approximation of the cosmological constant, $H$ is constant for all time. Therefore a dynamical mechanism for the limited time of inflation is needed. The physical mechanism for the existence of an approximately constant value of $H$ which lasts for a limited time is given by a scalar field. For a large initial potential energy of scalar field the state equation parameter $w\approx-1$ and the scalar field in process of the "slow roll" imitates the cosmological constant for sufficiently long period of time to solve the flatness and causal problems. Then due to shape of the potential the scalar field exits from the slow-roll regime, oscillates about its' potential minimum decaying into less massive particles insuring that inflation time is finite.

Using definition of the parameter $\varepsilon$ one finds \[\varepsilon=-\frac{\dot H}{H^2}.\] In the slow-roll approximation \[H^2=\frac{8\pi}{3M_P^2V},\] \[3H\dot\varphi=-\frac{dV}{d\varphi}.\] Therefore \[\frac{\ddot a}a=H^2(1-\varepsilon).\] The condition $\varepsilon=1$ is equivalent to $\ddot a=0$. When the value $\varepsilon=1$ is reached due to variation of the potential shape the Universe exits the regime of the accelerated expansion (inflation). Around tend, the inflaton field(s) typically begin oscillating around the minimum of the potential. (see Inflation and the Higgs Scalar, DANIEL GREEN (1412.2107).)

\[N\propto\int\limits_{\varphi_{end}}^{\varphi_{initial}}d\varphi\frac{V}{dV/d\varphi} \propto\int\limits_{\varphi_{end}}^{\varphi_{initial}}d\varphi/\sqrt\varepsilon.\] The number of $e$-folds depends on how fast the field is $f$-decreasing. The number of e-folds, $N$ , is inversely proportional to the square root of the slow roll parameter $\varepsilon$ or proportional to the inverse fractional change of the potential with the field, $V/(dV/d\varphi)$.

\[1+q=\frac{d}{dt}\left(\frac1H\right),\quad \frac{dq}{dt}=f(q)\to dt=\frac{dq}{f(q)},\] \[\frac1{H(q)}=\int\frac{1+q}{f(q)}dq.\] (see S. Carloni et al., A new approach to reconstruction methods in f(R) gravity, arXiv:1005.1840)

The exact velocity-proper distance relation (Hubble law) reads \[V_{exp}=HR.\] For the observer on the Earth living at the time $t=t_0$ the observed redshift serves as a measure of distance at this epoch. For photons moving along the comoving coordinate $r$, $cdt=\pm adr$. Consequently \[a_0dr=\frac{cdz}{H(z)}.\] Integration over the redshift gives the proper distance-redshift relation \[R(t_0,z)\equiv R(z)=\int\limits_0^z\frac{cdz'}{H(z')}.\] Using the $R(z)$ relation one gets the exact velocity-redshift relation \[V_{exp}(z)=H_0 R(z)=H_0\int\limits_0^z\frac{cdz'}{H(z')}.\] where the expansion velocity is taken for an object with the redshift $z$ observed at the time $t=t_0$.

Consider a nearby galaxy and the change of the scale factor during the small time interval $dt$ when the light travels from this galaxy to the observer: \[a(t_{obs})=a(t_{em})+\left.\frac{da}{dt}\right|_{t=t_{em}}dt.\] When the distance is short and the expansion speed much less than the speed of light, then the distance $R$ measured by the astronomer using any of the available ways is practically equal to the metric distance $a(t)r$ and the time interval can be approximated as $dt\approx a(t)r/c=R/c$. From this it follows that \[a(t_{obs})/a(t_{em})=1+z=1+\frac{\dot a}a \frac R c.\] and finally \[cz=HR.\]

\[(1+z)_{Dop}=\sqrt{\frac{c+V}{c-V}}.\] The exact relativistic Doppler velocity-redshift relation is \[V_{Dop}(z)=c\frac{2z+z^2}{2+2z+z^2}.\] For $z\to\infty$, the velocity $V_{Dop}(z)\to c$ corresponding to the limit $V_{Dop}le c$. \item \label{2501_04o} {\bf Show, that $V_{Dop}$ and $V_{exp}$ give the same results only in the first order of $V/c$.} \item \label{2501_05o} {\bf Find $V_{exp}(z)$ for the three cosmological models: Einstein-de Sitter, Milne and de Sitter.} For $\Omega_m=1$, $\Omega_\Lambda=0$: \[V_{exp}(z)=H_0 R(z)=H_0\int\limits_0^z\frac{cdz'}{H(z')}=\frac{2(\sqrt{1+z}-1)}{\sqrt{1+z}}c.\] For $\Omega_m=0$, $\Omega_\Lambda=0$: \[V_{exp}(z)=\frac{z(1+z/2)}{(1+z)}c.\] For $\Omega_m=0$, $\Omega_\Lambda=1$: \[V_{exp}(z)=zc.\]

From the definitions of redshift $1+z=1/a$ we have \[\frac{dz}{dt}=-\frac{dot a}{a^2}=-H(z)(1+z)\] or \[dt=-\frac{dt}{H(z)(1+z)}.\] The lookback time is defined as \[t_0-t=\int\limits_t^{t_0}dt=\int\limits_0^z\frac{dz'}{H(z')(1+z')}=\frac1{H_0}\int\limits_0^z\frac{dz'}{E(z')(1+z')}\] where \[E(z)=\sqrt{\Omega_r(1+z)^4+\Omega_m(1+z)^3+\Omega_k(1+z)^2+\Omega_w\exp\left( 3\int\limits_0^zdz'\frac{1+w(z')}{1+z'}\right)}.\] From the definition of lookback time it is clear that the cosmological time or the time back to the Big Bang, is given by \[t(z)=\int\limits_z^\infty\frac{dz'}{H(z')(1+z')}.\]

Solving the corrected first Friedmann equation and the conservation equation for a matter dominated Universe \[H^2=\frac\rho3\left(1-\frac\rho{\rho_c}\right),\quad \dot\rho+3H\rho=0;\] one obtains the following quantities \[a(t)=\left(\frac34\rho_ct^2+1\right)^{1/3},\quad \rho(t)=\frac{\rho_c}{\frac34\rho_ct^2+1},\quad H(t)=\frac{\frac12\rho_ct}{\frac34\rho_ct^2+1}.\] For small values of the energy density ($\rho\ll\rho_c$) the solution of standard Friedmann equations is recovered.

\[\frac{d^2a^3}{dt^2}=3\left(2H^2+\frac{\ddot a}a\right)a^3=3\left[\frac23\rho-\frac16(\rho+3p)\right]a^3=\frac32(\rho-p)\varphi.\]

\[f'(R)R_{\mu\nu}-\frac12f(R)g_{\mu\nu}-\left(\nabla_\mu\nabla_\nu-g_{\mu\nu}\square\right)f'(R)=\kappa T_{\mu\nu}.\] Here \[T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta S_m}{\delta g^{\mu\nu}},\] $\nabla_\mu$ is covariant derivative associated with the Levi-Civita connection of the metric, and $\square=\nabla_\mu\nabla^\mu$.

\begin{equation}\label{f_r_2_e_1} f'(R)R-2f(R)+3\square f'(R)=\kappa T, \end{equation} where $T=g^{\mu\nu}T_{\mu\nu}$. $R$ and $T$ are related in differential rather in algebraic way like $R=-kT$ in general relativity. Note that equality $T=0$ does not imply that $R$ equals to zero or a constant.

For $R=const$ and $T_{\mu\nu}=0$ the equation \ref{f_r_2_e_1} obtained in the previous problem takes on the form \[f'(R)R-2f(R)=0.\] For given function $f(R)$ this equation represents an algebraic equation for $R$. If $R=0$ is a root of this equation then it immediately follows from the field equations that $R_{\mu\nu}=0$ and the maximally symmetric solution is the Minkowski space-time. If the equation has a root $R=const=C$ then \[R_{\mu\nu}=\frac C4 g_{\mu\nu}\] and the maximally symmetric solution corresponds to the de- Sitter or anti-de Sitter (or to the cosmological constant in usual General Relativity) depending on sign of $C$.

\[H^2=\frac\kappa{3f'}\left[\rho+\frac12(Rf'-f)-3H\dot Rf''\right];\] \[2\dot H+H^2=-\frac\kappa{f'}\left[p+\dot R^2f'''+2H\dot Rf''+\ddot Rf''+\frac12(f-Rf')\right].\]

To reproduce (imitate) the de Sitter equation of state (i.e. the cosmological constant) $w_{eff}=-1$ the following condition is required \[\frac{f'''}{f''}=\frac{\dot RH-\ddot R}{\dot R^2}.\]