Difference between revisions of "Geometry and Destiny"

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.

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 \[a_{\max}=\left(\frac{\Omega_{m0}}{\Omega_{m0}-1}\right)^{1/3}.\]

In this case the Friedman equation can be integrated to yield the following analytical solution \[t=\frac{2}{3H_0\sqrt{\Omega_{m0}-1}} \arcsin\left(\frac{a}{a_{\max}}\right)^{3/2},\] \[t_{collapse} =\frac{2\pi}{3H_0\sqrt{\Omega_{m0}-1}}.\]

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 \[a(t)\approx \exp\left(\sqrt{\Omega_{\Lambda0}}H_0t\right).\]

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.