Power and hybrid expansion
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Let us consider a general class of power-law cosmology described by the scale factor \[a(t)=a_0\left(\frac t{t_0}\right)^\alpha\] where $\alpha$ is a dimensionless positive parameter.
Problem 6
problem id: 150_dp5
Obtain deceleration parameter in the power-law cosmology.
\[q\equiv=-\frac{\ddot a}{aH^2}=\frac1\alpha-1.\] The positivity of $\alpha$ leads to $q>-1$.
Problem 7
problem id: 150_dp6
Express the scale factor $a(t)$ and Hubble parameter $H(z)$ in terms of the deceleration parameter.
\[a(t)=a_0\left(\frac t{t_0}^{\frac1{1+q}}\right),\] \[H(z)=H_0(1+z)^{1+q}.\]
Problem 8
problem id: 150_dp7
Obtain the comoving distance $r(z)$ in the power-law cosmology.
\[r(z)=\frac1{\sqrt{|\Omega_c|}}F\left(|\Omega_c|\int\limits_0^zdz'\frac{H_0}{H(z')}\right),\quad \Omega_c\equiv-\frac k{a_0^2H_0^2}.\] \[F(x)\equiv\left\{\begin{array}{ll} \sinh(x), & k=-1;\\ x, & k=0;\\ \sin(x), & k=+1. \end{array}\right.\] Using result of the previous problem, one finally obtains \[r(z)=\frac1{\sqrt{|\Omega_c|}} F\left(|\Omega_c|q^{-1}\left[1-(1+z)^{-q}\right]\right).\]
Problem 9
problem id: 150_dp8
Find the comoving distance in the Milne model.
The Milne model (empty Universe with $k=-1$) can be viewed as a power-law cosmology with $\Omega_c=1$, $\alpha=1$ or $q=0$. For this model therefore \[r(z)=\sinh\left[\ln(1=z)\right].\]
Problem 10
problem id: 150_dp9
In the power-law cosmology, find time dependence of the CMB temperature in terms of the deceleration parameter.
\[\frac{T_0}T=\frac a{a_0}=\left(\frac t{t_0}\right)^\alpha =\left(\frac t{t_0}\right)^{\frac1{1+q}},\] \[T(t)=T_0\left(\frac t{t_0}\right)^{-\alpha} =T_0\left(\frac t{t_0}\right)^{-\frac1{1+q}}.\]
Let us consider now a simple generalization of power-law cosmology, called the hybrid expansion law [O. Akarsu et. al. Cosmology with hybrid expansion law: scalar field reconstruction of cosmic history and observational constraints, arXiv:gr-qc/1307.4911] \[a(t)=a_0\left(\frac t{t_0}\right)^\alpha e^{\beta\left(\frac t{t_0}-1\right)}\] where $\alpha$ and $\beta$ are non-negative constants.
Problem 11
problem id: 150_dp10
Find Hubble parameter, deceleration parameter and jerk parameter for the hybrid expansion law.
\begin{align} \nonumber H & = \frac\alpha t + \frac\beta{t_0},\\ \nonumber q & = \frac{\alpha t_0^2}{(\beta t+\alpha t_0)^2} -1,\\ \nonumber j & = \frac{\alpha t_0^2}{(\beta t+\alpha t_0)^2}(2t_0-3\beta t-3\alpha t_0)+1. \end{align}
Problem 12
problem id: 150_dp11
Find asymptotes of the scale factor and of the cosmographic parameters for the hybrid expansion law at $t\to0$ and $t\to\infty$.
For $t\to0$ the cosmological parameters approximate to the following: \[a\to a_0\left(\frac t{t_0}\right)^\alpha,\quad H\to\frac\alpha t,\quad q\to-1+\frac1\alpha,\quad j\to1-\frac3\alpha+\frac2{\alpha^2}.\] In the limit $t\to\infty$ we have \[a\to a_0 e^{\beta\left(\frac t{t_0}-1\right)},\quad H\to\frac\beta{t_0},\quad q\to-1,\quad j\to1.\]
Problem 13
problem id: 150_dp12
Find the moment of time $t_{tr}$ when the transition from deceleration to acceleration takes place.
\[q=0\to\frac{t_{tr}}{t_0}=\frac{\sqrt\alpha-\alpha}{\beta}.\]
Problem 14
problem id: 150_dp13
Find the range of the parameter $\alpha$ variation for the hybrid expansion law.
Use results of the previous problem. From the condition $t_{tr}>0$ it follows that \[\sqrt\alpha-\alpha>0\Rightarrow0<\alpha<1.\]
Problem 15
problem id: 150_dp14
In 1983, Berman [M. Berman, A special law of variation for Hubble's parameter. Nuovo Cimento B 74, 182 (1983)] proposed a special law of variation of Hubble parameter in FLRW space-time, which yields a constant value of DP, \[H=Da^{-n}.\] Find time dependence of the scale factor providing the constant deceleration parameter.
Integration of the relation \(H=Da^{-n}\) leads to \[a(t)=(nDt+c_1)^{\frac1n}\] where $c_1$ is a constant of integration.
Problem 16
problem id: 150_dp15
For Berman's law of variation for Hubble's parameter \(H=Da^{-n}\) find the deceleration parameter and analyze what values of the parameter $n$ correspond to accelerated expansion, and which --- to decelerated one.
The value of deceleration parameter in this case is \[q=n-1,\] which is a constant. As usual, sign of $q$ determines the expansion regime. A positive sign of $q$, i.e., $n>1$ corresponds to the standard decelerating model whereas the negative sign of $q$, i.e. $0<n<1$, corresponds to accelerating model. The expansion of the Universe at a constant rate corresponds to $n=1$, that is, $q=0$.
Linearly varying deceleration parameter
Inspired by O. Akarsu et al. Probing kinematics and fate of the Universe with linearlytime-varying deceleration parameter, arXiv:gr-qc/1305.5190 A general approach is to expand the deceleration parameter in Taylor series is \[q(x)=q_0+q_1\left(1-\frac x{x_0}\right)+q_2\left(1-\frac x{x_0}\right)^2+\ldots\] where $x$ is a some cosmological parameter as cosmic scale factor $a$, cosmic redshift $z$, cosmic time $t$ etc. As the first step one can take the following linear approximation \[q(x)=q_0+q_1\left(1-\frac x{x_0}\right).\]
Problem 17
problem id: 150_dp16
Consider the linearly varying deceleration parameter in terms of cosmic redshift $z$. Analyze advantages and defeats of such interpretation.
In this case \[\frac x{x_0}=\frac{a_0}a=1+z\] and \[q=q_0+q_1z.\] We note that the deceleration parameter grows monotonically with no limits as we go back to earlier times of the Universe. On the other hand, the model is well behaved in the future such that the Universe either expands forever (provided that $q_0\ge q_1-1$ or ends with a Big Rip in finite future (provided that $q_0<q_1-1$). However, the predicted future of the Universe using such parametrization cannot be considered so reliable since the observational constraints obtained using it with high redshift data are not reliable.
Problem 18
problem id: 150_dp17
Unlimited growth of the deceleration parameter for large $z$ in the parametrization, used in the previous problem, forces us to consider the linearly varying deceleration parameter in terms of scale factor. Make transition to such parametrization.
Substituting \[\frac x{x_0}=\frac{a_0}a\] we obtain \[q=q_0+q_1\left(1-\frac a{a_0}\right).\] Such parametrization can be written in terms of redshift as follows: \[q=q_0+q_1\frac z{1+z}.\]
Problem 19
problem id: 150_dp19-problem-1
Treating the Universe as a dynamical system it is useful to consider the parametrization of the deceleration parameter directly in terms of cosmic time $t$. Make transition to such parametrization.
Such parametrization can be obtained by substitution \[\frac x{x_0}=\frac t{t_0},\quad q=q_0+q_1\left(1-\frac t{t_0}\right).\] In contrast to the two previous problems, this model cannot be used for observational analysis directly. One should first solve it for the scale factor explicitly, and then obtain the time red-shift relation. This procedure can be facilitated if one first solves the equation with separating variables \[\frac{dH}{dt}=-H^2(1+q).\] For the chosen parametrization \[H(t)=2\frac{t_0}t\left[2(1+q_0+q_1)t_0-q_1t\right]^{-1}.\] Using the obtained expression for $H(t)$ to integrate the equation \[\frac{dz}{dt}=-(1+z)H(t),\] we obtain \[q=q_0+q_1\left(1-\frac{2(1+q_0+q_1)}{q_1+(2+2q_0+q_1)(1+z)^{1+q_0+q_1}}\right).\]
Problem 20
problem id: 150_dp19-problem-2
Consider a single-component flat Universe with a fluid described by an EoS parameter expressed as a first order Taylor expansion in cosmic time: \[w=w_0+w_1(1-t),\] where $w_0$ and $w_1$ are real constants and $t$ is the normalized time. Find the corresponding parametrization for the deceleration parameter.
\[q=\frac\Omega2+\frac32\sum\limits_iw_i\Omega_i=\frac12+\frac32\left[w_0+w_1(1-t)\right].\] Such an approach enables us to use the cosmographic (model-free) constraints to test the EoS for different components.
Problem 21
problem id: 150_dp20
In a Universe filled with a fluid characterized by the EoS parameter $w=w_0+w_1(1-t/t_0)$, find time dependence of the scale factor. [S. Kumar, Probing the matter and dark energy sources in a viable Big Rip model of the Universe, arXiv:1404.1910]
Using \[3H^2=\rho,\quad (8\pi G=1),\] \[2\dot H+3H^2=-p,\] find the following solution for the scale factor: \[a(t)=a_0\left[\frac{w_1t}{2(1+w_0+w_1)t_0-w_1t}\right]^{\frac2{3(1+w_0+w_1)}},\] where $a_1$ is a constant of integration, the first constant of integration being assumed zero.
Problem 22
problem id: 150_dp21
In a Universe filled with a fluid characterized by the EoS parameter $w=w_0+w_1(1-t/t_0)$, find the Hubble parameter, deceleration parameter and the jerk parameter.
Using expression for the scale factor obtained in the previous problem, one finds \begin{align} \nonumber H(t) & =\frac{4t_0}{3t[2(1+w_0+w_1)t_0-w_1t]},\\ \nonumber q(t) & =\frac12 +\frac32\left[w_0+w_1\left(1-\frac t{t_0}\right)\right],\\ \nonumber j(t) & =\frac{27w_1^2t^2}{8t_0^2} -\frac{9w_1Wt}{4t_0} +\frac{W(1+W)}2,\quad W\equiv1=3w_0+3w_1. \end{align}
Problem 23
problem id: 150_dp22
In a Universe filled with a fluid characterized by the EoS parameter $w=w_0+w_1(1-t/t_0)$, find the pressure and energy density.
\begin{align} \nonumber \rho(t) & =\frac13H^2= \frac{16t_0^2}{3t^2[2(1+w_0+w_1)t_0-w_1t]^2},\\ \nonumber \rho(t) & =w(t)\rho(t)= \frac{16t_0[-w_1t+(w_0+w_1)t_0]}{3t^2[2(1+w_0+w_1)t_0-w_1t]^2}. \end{align}
Problem 24
problem id: 150_dp23
Show that the Universe filled with a fluid characterized by the EoS parameter $w=w_0+w_1(1-t/t_0)$ achieves de Sitter phase ($q=-1$) at the end of its half life.
As we have seen in the previous problems, that the parameters $a,H,p,\rho$ diverge to infinity at two distinct epochs $t=0$ and $t=2(1+w_0+w_1)t_0/w_1$. On the other hand, the parameters $w,q$ and $j$ are well behaved from former to the later epoch. It implies that in this model, the Universe begins with Big Bang at $t=0$ and ends in Big Rip at $t_{BR}=2(1+w_0+w_1)t_0/w_1$. The de Sitter time $t_{dS}$, which is the solution of $q(t=t_{dS})=-1$, is $t_{dS}=(1+w_0+w_1)t_0/w_1$. Thus, the Universe achieves de Sitter phase at the end of its half life.