3 Distances in Cosmology

Measuring the expansion history of the Universe involves measuring distances. In this section, we consider the various distance measures in cosmology. Our starting point is the FLRW metric, here written as \[ ds^{2}=-dt^{2}+a(t)^{2}\left[d\chi^{2} + S_{k}(\chi)^{2} d\Omega^{2}\right] \]

3.1 Comoving Radial Distance

Our starting point is the comoving radial distance $\chi$ between points connected by a light ray emitted at $t_{1}$ and measured at $t_{0}$. Reading from the metric, we have \[ d\chi = c \frac{dt}{a(t)} \] or \[ \chi = c \int_{t_{1}}^{t_{0}} \frac{dt}{a(t)} \] Note that given $a(t)$, we can easily calculate $\chi$. It is useful to change variables \[ \begin{aligned} a = \frac{1}{1+z} \\ da = -\frac{1}{(1+z)^{2}} dz \\ \implies -\frac{da}{a^{2}} = dz \end{aligned} \] Changing variables in our definition of $\chi$, we have \[ \chi = c \int \frac{da}{\dot{a}} \frac{1}{a} = c \int_{a}^{1} \frac{da}{a^2 H(a)} = c \int_{0}^{z} \frac{dz}{H(z)} \] where we now just need knowledge of $H(a)$ to compute these distances.

3.2 Proper Distance

The proper distance (calculated at the same time) is \[ r_{p} = a \chi \] which is however not directly measurable. When considering angles, it is also convenient to consider \[ d_{M} = S_{k}(\chi) \] which is the factor that shows up in the angular part of the metric. Recall that, for $k=0$, $S_{k}(\chi) = \chi$.

3.3 Luminosity Distance

Recall that in Euclidean space, the flux $F$ is related to the luminosity $L$ by \[ F = \frac{L}{4\pi r^{2}} \] By analogy, in cosmology, we define \[ F = \frac{L}{4 \pi d_{L}^{2}} \] where $d_{L}$ is the luminosity distance. The formula for the flux is modified from the simple Euclidean case by three modifications:

$r \to d_{M}$
Arrival rate is reduced by a factor of $a(t)$
Photon energy is redshifted by a factor of $a(t)$

Putting this together, we get \[ F = \frac{L}{4 \pi d_{M}^{2} (1+z)^{2}} \] which gives us \[ d_{L} = (1+z) d_{M} \]

If one has a set of astronomical objects with a known luminosity (standard candles) at different redshifts, then by measuring the flux to these objects, one can measure the luminosity distance as a function of redshift, which in turn can be used to determine the Hubble parameter (and as we will see later), dark energy. This technique is what allowed Type 1a SNe to be used to measure dark energy.

3.4 Angular Diameter Distance

Similarly, an object of a physical size $D$ subtends an angle $\delta \theta$ given by \[ \delta \theta = \frac{D}{d_{A}} \] where $d_{A}$ is the angular diameter distance. Reading off the metric, we have \[ \delta \theta = \frac{D}{a(t) d_{M}} \] or \[ d_{A} = \frac{d_{M}}{(1+z)} \] This also implies \[ d_{L} = d_{A} (1+z)^{2} \]

Objects of a fixed physical size are standard rulers - measuring their angular extent allows one to measure the angular diameter distance. This forms the basis of the source of the dominant information in the CMB and the BAO technique in dark energy.

3.5 References

An excellent pedagogical reference is

Hogg, D. W. 1999, , astro-ph/9905116. doi:10.48550/arXiv.astro-ph/9905116

# Distances in Cosmology Measuring the expansion history of the Universe involves measuring distances. In this section, we consider the various distance measures in cosmology. Our starting point is the FLRW metric, here written as $$ ds^{2}=-dt^{2}+a(t)^{2}\left[d\chi^{2} + S_{k}(\chi)^{2} d\Omega^{2}\right] $$ ## Comoving Radial Distance Our starting point is the comoving radial distance $\chi$ between points connected by a light ray emitted at $t_{1}$ and measured at $t_{0}$. Reading from the metric, we have $$ d\chi = c \frac{dt}{a(t)} $$ or $$ \chi = c \int_{t_{1}}^{t_{0}} \frac{dt}{a(t)} $$ Note that given $a(t)$, we can easily calculate $\chi$. It is useful to change variables $$ \begin{aligned} a = \frac{1}{1+z} \\ da = -\frac{1}{(1+z)^{2}} dz \\ \implies -\frac{da}{a^{2}} = dz \end{aligned} $$ Changing variables in our definition of $\chi$, we have $$ \chi = c \int \frac{da}{\dot{a}} \frac{1}{a} = c \int_{a}^{1} \frac{da}{a^2 H(a)} = c \int_{0}^{z} \frac{dz}{H(z)} $$ where we now just need knowledge of $H(a)$ to compute these distances. ## Proper Distance The proper distance (calculated at the same time) is $$ r_{p} = a \chi $$ which is however not directly measurable. When considering angles, it is also convenient to consider $$ d_{M} = S_{k}(\chi) $$ which is the factor that shows up in the angular part of the metric. Recall that, for $k=0$, $S_{k}(\chi) = \chi$. ## Luminosity Distance Recall that in Euclidean space, the flux $F$ is related to the luminosity $L$ by $$ F = \frac{L}{4\pi r^{2}} $$ By analogy, in cosmology, we define $$ F = \frac{L}{4 \pi d_{L}^{2}} $$ where $d_{L}$ is the **luminosity distance**. The formula for the flux is modified from the simple Euclidean case by three modifications: - $r \to d_{M}$ - Arrival rate is reduced by a factor of $a(t)$ - Photon energy is redshifted by a factor of $a(t)$ Putting this together, we get $$ F = \frac{L}{4 \pi d_{M}^{2} (1+z)^{2}} $$ which gives us $$ d_{L} = (1+z) d_{M} $$ If one has a set of astronomical objects with a known luminosity (standard candles) at different redshifts, then by measuring the flux to these objects, one can measure the luminosity distance as a function of redshift, which in turn can be used to determine the Hubble parameter (and as we will see later), dark energy. This technique is what allowed Type 1a SNe to be used to measure dark energy. ## Angular Diameter Distance Similarly, an object of a physical size $D$ subtends an angle $\delta \theta$ given by $$ \delta \theta = \frac{D}{d_{A}} $$ where $d_{A}$ is the angular diameter distance. Reading off the metric, we have $$ \delta \theta = \frac{D}{a(t) d_{M}} $$ or $$ d_{A} = \frac{d_{M}}{(1+z)} $$ This also implies $$ d_{L} = d_{A} (1+z)^{2} $$ Objects of a fixed physical size are standard rulers - measuring their angular extent allows one to measure the angular diameter distance. This forms the basis of the source of the dominant information in the CMB and the BAO technique in dark energy. ## References An excellent pedagogical reference is - Hogg, D. W. 1999, , astro-ph/9905116. doi:10.48550/arXiv.astro-ph/9905116