Solutions of Friedman equations in the Big Bang model

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One component dominated Universe

Problem 1: matter or radiation in a flat Universe

Derive $\rho(a)$, $\rho(t)$ and $ a(t)$ for a spatially flat$^*$ ($k=0$) Universe that consists of only

a) radiation,

b) non-relativistic matter$^{**}$.

$^*$ The "spatial" part is often omitted when there cannot be any confusion (and even when there can be), generally and hereatfer.

$^{**}$ In this context and below also quite often called just "matter" or dust.

Problem 2: RD and MD regimes

Consider two spatially flat Universes. One is filled with radiation (thus radiation dominated, or RD), the other with dust (thus matter dominated, MD). The current energy density is the same. Compare energy densities when both of them are twice as old.

Problem 3: RD$\to$MD transition

Find the scale factor and density of each component as functions of time in a flat Universe which consists of dust and radiation, for the case one of the components is dominating. Present the results graphically.

Problem 4: Hubble parameter

Derive the time dependence of the Hubble parameter for a flat Universe in which either matter or radiation is dominating.

Matter and radiation

Problem 5: exact solutions for matter and radiation

Derive the exact solutions of the Friedman equations for the Universe filled with matter and radiation. Plot the graphs of scale factor and both densities as functions of time.

Problem 6: equipartition time

At what moment after the Big Bang did matter's density exceed that of radiation for the first time?

Problem 7: age of the Universe

Determine the age of the Universe in which either matter or radiation has always been dominating.

Non-interacting components, effective potential

Problem 8: constant EoS parameter

Derive the dependence $a(t)$ for a spatially flat Universe that consists of matter with equation of state $p=w\rho$, assuming that the parameter $w$ does not change throughout the evolution.

Problem 9: $H(t)$

Find the Hubble parameter as function of time for the previous problem.

Problem 10: effective potential

Using the first Friedman equation, construct the effective potential $V(a)$, which governs the one-dimensional motion of a fictitious particle with coordinate $a(t)$, for the Universe filled with several non-interacting components.

Problem 11: effective potential for radiation and dust

Construct the effective one-dimensional potential $V(a)$ (using the notation of the previous problem) for the Universe consisting of non-relativistic matter and radiation. Show that motion with $\dot{a}>0$ in such a potential can only be decelerating.

Problem 12: exact solutions with dust, radiation and curvature

Derive the exact solutions of the Friedman equations for the Universe with arbitrary curvature, filled with radiation and matter.

Deceleration parameter

Problem 13: one component, flat case

Show that for a spatially flat Universe consisting of one component with equation of state $p = w\rho$ the deceleration parameter $q\equiv -\ddot{a}/(aH^2)$ is equal to $\frac{1}{2}(1 + 3w)$.

Problem 14: relation between the total pressure and the deceleration parameter

Find the relation between the total pressure and the deceleration parameter for the flat Universe.

Problem 15: $\frac{d^k p}{dt^k }$

The expansion of pressure by the cosmic time is given by \[p(t) = \sum\limits_{k = 0}^\infty {\left. {\frac{1}{k!}\frac{d^k p}{dt^k }} \right|} _{t = t_0 } (t - t_0 )^k. \] Using cosmography parameters (see Chapter 2), evaluate the derivatives $\frac{d^k p}{dt^k }$ up to the fourth order

Problem 16: $w(t)$

Сonsider the universe filled with an ideal fluid with the time-dependent equation of state parameter $w(t)$, and find $\rho (t),H(t)$.

Problem 17: $\rho = f(a)$

Find the parameter $w$ in the equation of state if the dependence $\rho = f(a)$ is known. Check the obtained formula for the known special cases.

Problem 18: age of the Universe

Express the age of the spatially flat Universe filled with a single component with equation of state $p=w\rho$ through the deceleration parameter $q=-\ddot{a}/(aH^2)$.

Problem 19: non-flat generalization

Find the generalization of relation $q = \frac{1}{2}(1 + 3w)$ for the non-flat case.

Problem 20: $q[\rho(a)]$

Show that for a one-component Universe filled with ideal fluid of density $\rho$ \[q=-1-\frac{1}{2}\,\frac{d\ln\rho}{d\ln a}.\]

Problem 21: several components

Show that for a Universe consisting of several components with equations of state $p_{i} = w_{i} \rho_{i}$ the deceleration parameter is \[q = \frac{\Omega }{2} + \frac{3}{2}\sum\limits_i {w_i \Omega_i },\] where $\Omega$ is the total relative density.

Problem 22: acceleration or deceleration

For which values of the state parameter $w$ the rate of expansion of a one-component flat Universe increases with time?

Closed Universe

Problem 23: closed dusty Universe, exact solution

Show that for a spatially closed ($k=1$) Universe that contains only non-relativistic matter the solution of the Friedman equations can be given in the form \[a(\eta)=a_{\star}(1-\cos\eta); \qquad t(\eta)=a_{\star}(\eta -\sin\eta); \qquad a_{\star}=\frac{4\pi G\rho_0}{3}; \quad 0<\eta<2\pi.\]

Problem 24: size and lifetime

Find the relation between the maximum size and the total lifetime of a closed Universe filled with dust.

Problem 25: future through observables

Suppose we know the current values of the Hubble constant $H_0$ and the deceleration parameter $q_0$ for a closed Universe filled with dust only. How many times larger will it ever become?

Problem 26: age of the Universe

In a closed Universe filled with non-relativistic matter the current values of the Hubble constant is $H_0$, the deceleration parameter is $q_0$. Find the current age of this Universe.

Problem 27: around the Universe

Suppose in the same Universe radiation is dominating during a negligibly small fraction of total time of evolution. How many times will a photon travel around the Universe during the time from its "birth" to its "death"?

Problem 28: different volumes in dynamical Universe

In a closed Universe filled with dust the current value of the Hubble constant is $H_0$ and of the deceleration parameter $q_0$.

a) What is the total proper volume of the Universe at present time?

b) What is the total current proper volume of space occupied by matter which we are presently observing?

c) What is the total proper volume of space which we are directly observing?

Different problems

Problem 29: open dusty Universe

Find the solution of Friedman equations for spatially open ($k=-1$) Universe filled with dust in the parametric form $a(\eta)$, $t(\eta)$.

Problem 30: density evolution

Suppose the density of some component in a spatially flat Universe depends on scale factor as $\rho(t) \sim a^{-n}(t)$. How much time is needed for the density of this component to change from $\rho_1$ to $\rho_2$?

Problem 31: $q[H(t)]$

Using the expression for $H(t)$, calculate the deceleration parameter for the cases of domination of

a) radiation,

b) matter.

Problem 32: effective EoS for multi-component Universe

Consider a Universe consisting of $n$ components, with equations of state $p_{i}=w_{i}\rho_{i}$, and find $w_{tot}$, the parameter of the equation of state $p_{tot}=w_{tot}\rho_{tot}$.

Problem 33: relative densities

Derive the equations of motion for relative densities $\Omega_{i}=\rho_{i}/\rho_{cr}$ of the two components comprising a spatially flat two-component Universe, if their equations of state are $p=w_i\rho$, $i=1,2$.

Problem 34: EoS for non-relativistic gas

Suppose a Universe is initially filled with a gas of non-relativistic particles of mass density $\rho_{0}$, pressure $p_{0}$, and $c_p/c_v=\gamma$. Construct the equation of state for such a system.

Problem 35: Newtonian derivation of critical density

Derive the expression for the critical density $\rho_{cr}$ from the condition that Hubble's expansion velocity equals the second cosmic velocity (escape velocity) $v=\sqrt{2gR}$.


Problem 36: dust plus something else

Suppose the Universe is filled with non-relativistic matter and some substance with equation of state $p_X=w\rho_X$. Find the evolution equation for the quantity $r \equiv \frac{\rho_m}{\rho _X}$.

Problem 37: deceleration parameter

Express the deceleration parameter through the ratio $r$ for the conditions of the previous problem.

Problem 38: $\rho_{X}\propto H^2$

Let a spatially flat Universe be filled with non-relativistic dust and a substance with equation of state $p_{X}= w\rho_{X}$. Show that in case $\rho_{X}\propto H^2$, the ratio $r =\rho_{m}/\rho_{X}$ does not depend on time.

Problem 39: $r(q)$

Show that for the model of the Universe described in the previous problem the parameter $r$ is related with the deceleration parameter as \[\dot{r}=-2H\frac{\Omega_{curv}}{\Omega_X}q.\]

Problem 40: $r$ in open and closed cases

Show that in the model of the Universe of problem, in case $k=+1$ and $q>0$ (decelerated expansion) $r$ increases with time, in case $k=+1$ and $q<0$ (accelerated expansion) $r$ decreases with time, and for $k=-1$ vice-versa.

Power-law cosmology

The following problems on power-law cosmology are inspired by Kumar [1].

Problem 41: power-law cosmologies

Let us consider a general class of power-law cosmologies described by the scale factor \[a(t) =a_{0}\Big(\frac{t}{t_0}\Big)^\alpha,\] where $t_0$ is the present age of theUniverse and $\alpha$ is a dimensionless positive parameter. Show that:

1) the scale factor in terms of the deceleration parameter may be written as \[a(t) =a_{0}\Big(\frac{t}{t_0}\Big)^{1/1 + q}, \quad\text{i.e.}\quad \alpha=\frac{1}{1+q}.\]

2) the expansion of the Universe is described by Hubble parameter \[H=\frac{1}{(1+q)\;t}\] or in terms of redshift \[H(z)=H_{0}(1+z)^{1+q}.\]

Problem 42: age of the Universe

In the power-law cosmology find the age of the Universe at redshift $z$.

Problem 43: luminosity distance

For the power-law cosmology find the luminosity distance between the observer and the object with redshift $z$.

Problem 44: a "softer" version of the cosmological evolution

The power law cosmological evolution $a(t)\propto t^{\beta }$ leads to the Hubble parameter $H(t)\propto 1/t$. Consider a "softer" version of the cosmological evolution given by the law \[H(t)=\frac{S}{t^{\alpha } } \] where $S$ is a positive constant and $0<\alpha <1$. Analyze the dynamics of such model at $t\to 0$.
(F.Cannata, A.Kamenshchik, D.Regoli, arXiv:0801.2348).