The luminosity distance to an astronomical light source is given by $d_{L}=\chi / a(t)$, where $a(t)$ is the expansion scale factor and $\chi$ is the comoving distance in the universe defined by $d t=a(t) d \chi$. A zero-curvature Friedmann universe containing pressure-free matter and a cosmological constant with density parameters $\Omega_{m}$ and $\Omega_{\Lambda} \equiv 1-\Omega_{m}$, respectively, obeys the Friedmann equation

$$H^{2}=H_{0}^{2}\left(\frac{\Omega_{m 0}}{a^{3}}+\Omega_{\Lambda 0}\right)$$

where $H=(d a / d t) / a$ is the Hubble expansion rate of the universe and the subscript 0 denotes present-day values, with $a_{0} \equiv 1$.

If $z$ is the redshift, show that

$$d_{L}(z)=\frac{1+z}{H_{0}} \int_{0}^{z} \frac{d z^{\prime}}{\left[\left(1-\Omega_{\Lambda 0}\right)\left(1+z^{\prime}\right)^{3}+\Omega_{\Lambda 0}\right]^{1 / 2}}$$

Find $d_{L}(z)$ when $\Omega_{\Lambda 0}=0$ and when $\Omega_{m 0}=0$. Roughly sketch the form of $d_{L}(z)$ for these two cases. What is the effect of a cosmological constant $\Lambda$ on the luminosity distance at a fixed value of $z$ ? Briefly describe how the relation between luminosity distance and redshift has been used to establish the acceleration of the expansion of the universe.