Dynamics of the Universe in terms of redshift and conformal time

Problem 1: the first Friedman equation

Express the first Friedman equation in terms of redshift and analyze the contribution of different terms in different epochs.

solution

Using the relation $a_{0}/a=z+1$ with normalization $a_{0}=1$, we can rewrite the energy conservation law for every non-interacting component as \[\Omega_{i}=\Omega_{i0}a^{-3(1+w_{i})} =\Omega_{i0}(1+z)^{3(1+w_{i})},\] so for the Universe filled with matter, radiation and curvature the first Friedman equation becomes \[H^{2}(z) = H_0^2\left[ \Omega _{r0}(1+z)^4 + \Omega_{m0}(1+z)^3+ \Omega_{k0}(1+z)^2\right].\] At large $z$ (early Universe) the term with the highest power becomes dominating -- that is, the one with radiation. As the term with curvature is very small at present, it was all the more negligible before. On the other hand, it should become dominating in the future ($1+z\to 0$), but in the standard cosmological model, which will be discussed in the corresponding chapter later, the cosmological constant enters the play before that, and the curvature term does not ever have the chance to shine.

Problem 2: $\eta(a)$ in one-component Universe

Find the conformal time as function of the scale factor for a Universe with domination of a) radiation and b) non-relativistic matter.

solution

As \[\eta = \int \frac{dt}{a(t)},\] for the Universe with the radiation component dominating $a(t) \sim t^{1/2}$, so $\eta \sim a$. For the non-relativistic matter $a(t) \sim t^{2/3}$, so $\eta\sim a^{1/2}.$

Problem 3: $t(z)$ for matter domination

Find the relation between time and redshift in the Universe with dominating matter.

solution

Using $H(z)$, we can express the first Friedman equation in the form \[\left(\frac{\dot z}{1 + z} \right)^2 =H_0^2 \sum\limits_i{\Omega_i},\] where $\Omega_i$ are the relative densities of all components filling the Universe (including curvature) with state parameters $w_i$. On separating variables, we arrive to \[t_0(z) = \frac{1}{H_0}\int_0^z\frac{dz'} {(1 + z')\sqrt{\sum \limits_i \Omega _i(1 + z')^{3(1 +w_i)}}}.\] In case $\Omega _{m0} = 1,\;\Omega _{r0} =\Omega _{curv} = 0$ then \[t_0(z) = \frac{2/3}{H_0\sqrt{\Omega _{m0}}} \left(1-\frac{1}{(1 + z)^{3/2}}\right).\]

Problem 4: $a(\eta)$ for radiation domination

Derive $a(\eta)$ for a spatially flat Universe with dominating radiation.

solution

The first Friedman equation takes the form \[\frac{1}{a^4}\left(\frac{da}{d\eta}\right)^2 = H_0^2\frac{\Omega _{r0}}{a^4},\] and its solution is \[a - a_0 = H_0\sqrt{\Omega _{r0}}(\eta-\eta_0 ).\]

Problem 5: $t(\eta)$ for dominating radiation

Express the cosmic time through the conformal time in a Universe with dominating radiation.

solution

Using the result of the previous problem with $a_0 =0$, we have \[t = \int a(\eta )d\eta =\frac{1}{2}H_0\sqrt{\Omega _{r0}}\;\eta^2.\]

Problem 6: $a(\eta)$ for dominating dust

Derive $a(\eta)$ for a spatially flat Universe with dominating matter.

solution

In this case the first Friedman equation is \[\frac{1}{a^4}\left(\frac{da}{d\eta}\right)^2 = \frac{H_0^2\Omega _{m0}}{a^3},\] and we obtain \[a(\eta)-a_{0} = \frac{H_0^2\Omega _{m0}}{4}(\eta-\eta_0)^2.\]

Problem 7: $a(\eta)$ for radiation and dust

Find $a(\eta)$ for a spatially flat Universe filled with a mixture of radiation and matter.

solution

The first Friedman equation is \[\frac{1}{a^4}\left(\frac{da}{d\eta}\right)^2 = H_0^2\left(\frac{\Omega _{m0}}{a^3} + \frac{\Omega _{r0}}{a^4} \right),\]so \[\frac{H_{0}\Omega_{m0}}{2}(\eta-\eta_{0}) =\sqrt{\Omega_{r0}+\Omega_{m0}a} -\sqrt{\Omega_{r0}+\Omega_{m0}a_{0}}.\] Let $\eta_{0}=a_{0}=0$. Then \[a=\frac{H_{0}^{2}\Omega_{m0}}{4}\,\eta^{2} -H_{0}\sqrt{\Omega_{m0}}\;\eta\]

Problem 8: variable EoS parameter

Suppose a component's state parameter $w_i=p_i/\rho_i$ is a function of time. Find its density as function of redshift.

solution

As $w_i =w_{i}(t)$ and $t = t(z)$, we have $w_i = w_{i}(z)$. The conservation equation then is \[\dot\rho_{i}+3\frac{\dot z}{1 + z}[1+w(z)]\rho_{i}= 0.\] Separating the variables, one gets \[\rho_{i}(z) =\rho_{i0}\exp\left\{ -3\int_0^z \big[1 + w_{i}(z')\big] \frac{dz'}{(1 + z')}\right\}.\]

Problem 9: $H(z)$ for dominating dust

Derive the Hubble parameter as a function of redshift in a Universe filled with non-relativistic matter.

solution

From the first Friedman equation \[H^2 = H_0^2\Omega _{m0}(1 + z)^3 \quad\Rightarrow\quad H(z)=H_0\sqrt {\Omega _{m0}} (1 + z)^{3/2}.\]

Problem 10: $\dot z$ for dominating dust

The redshift of any object slowly changes with time due to acceleration (or deceleration) of the Universe's expansion. Find the rate of change of redshift $\dot{z}$ for a Universe with dominating non-relativistic matter.

solution

The equation of null curve (and radial null geodesics) is $ds^{2}=a^{2}(d\eta^{2}-d\chi^{2})=0$, and thus $d\eta=\pm d\chi$. Then the light signal, which comes from a source at comoving distance $\chi$, and which we observe at conformal time $\eta_{0}$, was emitted at conformal time $\eta_{e}=\eta_{0}-\chi$. The observed redshift is thus a function of the time of observation $\eta_0$ and is equal to \[z\left(\eta _0 \right) = \frac{a_0}{a_e} - 1 = \frac{a\left(\eta_0\right)} { a\left(\eta _0 - \chi \right)}-1.\] As $\chi$ does not change (we a looking at the same object), recalling the definition of conformal time $d\eta = dt/a(t)$, we get \begin{align*} \dot z \equiv \frac{dz}{dt_0} &= \frac{1}{a\left( \eta _0\right)} \frac{\partial z}{\partial\eta _0} = \frac{1}{a\left( \eta _0\right)} \frac{\partial }{\partial\eta _0} \left(\frac{a\left(\eta _0 \right)} {a\left( \eta _0- \chi \right)} -1 \right)=\\ &= \frac{1}{a(\eta_0 - \chi)\,a(\eta _0)} \frac{\partial a(\eta _0)}{\partial \eta _0} - \frac{1}{a(\eta _0 - \chi)^2} \frac{\partial a(\eta _0 - \chi)} {\partial(\eta _0- \chi)}=\\ &=\frac{\dot a_0}{a_e} - \frac{\dot a_e}{a_e} = (1+z)\frac{\dot a_0}{a_0} - \frac{\dot a_e}{a_e} = (1 + z)H_0 - H(z) \end{align*} For the Universe with dominating non-relativistic matter then \[\dot z = \left(1 + z\right)H_0 \left(1 - \sqrt{\Omega _{m0}}(1 +z)^{1/2}\right).\]

Problem 11: optical horizon

The Universe is known to have become transparent for electromagnetic waves at $z\approx 1100$ (in the process of formation of neutral hydrogen, recombination), i.e when it was $1100$ times smaller than at present. Thus in practice the possibility of optical observation of the Universe is limited by the so-called optical horizon: the maximal distance that light travels since the moment of recombination. Find the ratio of the optical horizon to the particle one for a Universe dominated by matter.

solution

The comoving distance $\chi$ to the surface we are observing is determined from the condition $ds^2 = dt^2 - a(t)d\chi^2 = 0$. Expressing $\chi$ thought its redshift, we get \[\chi = \int_0^z \frac{dz'}{H(z')}.\] For a model with dominating matter \[\chi = \int_0^z \frac{dz'}{H_0\sqrt{\Omega _{m0} (1 + z')^3}} =\frac{2}{H_0\sqrt{\Omega _{m0}}} \left( 1 - \sqrt {\frac{1}{1 + z}}\right).\] Using $\Omega _{m0} \approx 1$ and the value of redshift at the moment of recombination $z_r \approx 1100$, we arrive to $\chi\approx 7.8$ GPc. Then the particle horizon is \[ L_{p} = \int_0^\infty \frac{dz}{H_0\sqrt {\Omega _{m0}(1 + z)^3}}= \frac{2}{H_0\sqrt{\Omega _{m0}}} \approx 8.02\mbox{ GPc},\] and therefore \[\frac{L_{p}}{\chi} = \Big(1 - \sqrt {\frac{1}{1 + z}}\Big)^{-1} \approx 1+\frac{1}{\sqrt{z}} \approx 1.031.\]

Problem 12: particle horizon

Derive the particle horizon as function of redshift for a Universe filled with matter and radiation with relative densities $\Omega_{m0}$ and $\Omega_{r0}$.

solution

In the same way as for the optical horizon, for the model with domination of radiation and matter we obtain \[L_{p} =\int\limits_{0}^{z} \frac{dz'}{H_0\sqrt{ \Omega _{m0}(1 + z')^3 +\Omega_{r0}(1 + z')^4}} =\frac{2}{\Omega _{m0}H_0} \left(1 - \sqrt{\frac{1+\Omega_{r0}z}{1 + z}}\right).\] We have taken into account here that $\Omega _{m0} + \Omega _{r0} = 1$.

Problem 13: time dilation at cosmological horizon

Show that any signal emitted from the cosmological horizon will arrive to the observer with infinite redshift.