Geometry and Destiny
See L.Krauss, M. Turner, arXiv:9904020; B.Ryden, Introduction to cosmology, Addison Wesley; J.E.Felten,R.Isaacman, Rev.Mod.Phys. 58,689 , 1986
Presence of the cosmological constant reevaluates standard notions about the connection between geometry and the fate of the Universe. The traditional philosophy of General Relativity is that "Geometry is Destiny", with "geometry" in this context implying openness or closure of the three-space of constant cosmological time. If energy content is provided by "ordinary" matter (nonrelativistic matter or radiation) this slogan transforms into "Density is Destiny". If the density of matter is less or equal than the critical value (and the Universe is open), then the destiny of Universe is eternal expanding; if the density is greater, and the Universe is closed, then the destiny is recollapse. If the Universe contains cosmological constant (or any energy component with $w<-1/3$) the situation changes radically: an open Universe can recollapse, while a closed Universe can expand forever. Geometry no longer determines the fate of the Universe.
Problem 1
As $\Omega_{tot}-1=k/(H^2a^2)$ the sign of $k$ is determined by whether $\Omega_{tot}$ is greater or less than $1$. A measurement of $\Omega_{tot}$ at any epoch---including the present---determines the geometry of the Universe. However, as opposed to situation with only non-relativistic matter, we can no longer claim that the magnitude of $\Omega_{tot}$ uniquely determines the fate of the Universe. Explain decoupling between $\Omega_{tot}$ and destiny using deceleration parameter.
We know that (see problems of chapter 2) \[q=\frac{\Omega_{tot}}{2}+\frac32\sum\limits_i w_i\Omega_i.\] The sign of deceleration parameter at any given epoch depends upon the equation of state and not on alone. While in the presence of a cosmological constant ($w=-1$), $\Omega_{tot}$ no longer determines the ultimate fate of the Universe.
Problem 2
Show that decoupling between $\Omega_{tot}$ and destiny of the Universe is due to violation of strong energy condition.
Problem 3
Find the maximum value of scale factor for a hypothetical flat Universe with $\Omega_{\Lambda0}<0$.
The Friedman equation for the considered Universe reads \[\frac{H^2}{H_0^2}=\Omega_{m0}a^{-3}+\Omega_{\Lambda0}.\] Condition $H=0$ transforms into \[a_{\max}=\left(\frac{\Omega_{m0}}{\Omega_{m0}-1}\right)^{1/3}.\]
Problem 4
For the Universe considered in the previous problem, find the analytical solution $t(a)$ and time of the collapse from $a=a_{\max}$ back down to $a=0$.
In this case the Friedman equation can be integrated to yield the following analytical solution \[t=\frac{2}{3H_0\sqrt{\Omega_{m0}-1}} \arcsin\left(\frac{a}{a_{\max}}\right)^{3/2},\] \[t_{collapse} =\frac{2\pi}{3H_0\sqrt{\Omega_{m0}-1}}.\]
Problem 5
In the flat Universe with $\Omega_{m0}<1$ and positive cosmological constant find the late time asymptotic for the scale factor.
In this case the Friedman equation can be integrated to yield the following analytical solution \[t=\frac{2}{3H_0\sqrt{\Omega_{m0}-1}} \ln\left[\left(\frac{a}{a_{eq}}\right)^{3/2} +\sqrt{1+\left(\frac{a}{a_{eq}}\right)^3}\right].\] Here \[a_{eq}=\left(\frac{\Omega_{m0}}{\Omega_{\Lambda0}}\right)^{1/3}\] is the scale factor corresponding to equality of densities of matter and cosmological constant. For $a\gg a_{eq}$ this expression is reduced to \[a(t)\approx \exp\left(\sqrt{\Omega_{\Lambda0}}H_0t\right).\]
Problem 6
Show that eternal expansion is inevitable if and only if \[\Omega_\Lambda>4\Omega_m\left\{\cos\left[\frac13\arccos(\Omega_m^{-1}-1)+\frac{4\pi}{3}\right]\right\}.\]
Problem 7
Show that if $\Omega_{m0}+\Omega_{\Lambda0}>1$ (positive spatial curvature) and $\Lambda>0$, then it is possible to have a Universe that expands at late times, but without the initial Big Bang ($a=0,\ t=0$).
Friedman equation for a curved Universe with both matter and cosmological constant is \[\frac{H^2}{H^2_0}=\frac{\Omega_{m0}}{a^3}+\frac{1-\Omega_{m0}-\Omega_{\Lambda0}}{a^2}+\Omega_{\Lambda0}.\] If $\Omega_{m0}+\Omega_{\Lambda0}>1$, the value of $H^2$ is positive for small and large values of scale factor $a$. But for intermediate values of $a$ we have $H^2<0$, thus this range is forbidden. Let us start out with $a\gg1$, $H<0$ (contracting phase of low density, $\Lambda$-dominated state). As the Universe contracts, negative curvature term becomes dominant, causing the contraction to stop at a minimum scale factor $a_{\min}$, equal to the real root of the equation \[\frac{\Omega_{m0}}{a^3}+\frac{1-\Omega_{m0}-\Omega_{\Lambda0}}{a^2}+\Omega_{\Lambda0}=0.\] Thus it is possible to have a Universe that expands at late time, but without a Big Bang.
Problem 8
Show that collapse of the Universe is possible if the following conditions are satisfied:
1. Closed ($k>0$) Universe: $\rho_m>2\rho_\Lambda$ when $H=0$.
2. Open, flat ($k\le0$) or closed ($k>0$) Universe: $\rho_\Lambda\le0$.
Problem 9
Show that eternal expansion of the Universe is possible if the following conditions are satisfied
1. Closed ($k>0$) Universe: $\rho_m<2\rho_\Lambda$ before $H=0$.
2. Open or flat ($k\le0$) Universe: $\rho_\Lambda\ge0$.
