Riemann surface
matlab

Consider a universe in which $\Omega_M=0.3$, but which is otherwise empty.
a) In units of the Hubble time ( $\left.1 / H_0\right)$, how old is the universe now, $t_0$ ? You will need to do a numerical integration, so please include your code.

a) The Friedmann equation for a universe that contains matter, but is otherwise empty is given by equation 5.85 from the textbook.
$$\begin{gathered} \frac{1}{H_0^2}\left(\frac{\dot{a}}{a}\right)^2=\frac{\Omega_M}{a^3}+\frac{1-\Omega_M}{a^2} \ \frac{\dot{a}}{H_0}=\sqrt{\frac{\Omega_M}{a}+1-\Omega_M} \end{gathered}$$
To calcualate the age of this universe, $t_0$, we integrate by separation of variables.
$$H_0 t_0=\int_0^1\left(\frac{\Omega_M}{a}+1-\Omega_M\right)^{-1 / 2} \mathrm{~d} a$$
This must be computed numerically. For example, this can be done with a scheme as simple as Riemann integration, which gives
$$H_0 t_0 \approx 0.8$$

b) Now adjust your model so that it contains a matter component $\Omega_M=0.3$, and a “cosmological constant” component, $\Omega_{\Lambda}=0.7$ (with $w<-1$ ). How old is this universe?

b) If we instead assume that the universe is flat, and that there is then the Friedmann equation becomes:
$$\begin{gathered} \frac{\dot{a}^2}{H_0^2}=\frac{\Omega_M}{a}+\left(1-\Omega_M\right) a^2 \ \frac{\dot{a}}{H_0}=\sqrt{\frac{\Omega_M}{a}+\left(1-\Omega_M\right) a^2} \ H_0 t_0=\int_0^1\left(\frac{\Omega_M}{a}+\left(1-\Omega_M\right) a^2\right)^{-1 / 2} \mathrm{~d} a \end{gathered}$$
Again applying Riemann integration, we find that
$$H_0 t_0 \approx 0.96$$
In this case, the presence of a cosmological constant hastens the expansion, cuasing the universe to appear older than a case with only matter.

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