Measuring distances in an expanding universe (or, more generally, in GR) is fraught with ambiguities, so we have to be careful to define exactly what we mean. Some possibilities can be dreamt up immediately:
comoving distance ${l}_{0}$ -- the metric distance to an object today. When it actually emitted its light towards us, this object would have been closer (due to the expansion of the universe^{17}^{17} 17 This assumes that we and the objects that we observe are moving with the overall Hubble flow. That will be an assumption we hold throughout the first part of the course, and in any case is a very good approximation to the real universe on large scales.).
physical distance $l(t)$ at time $t$. At time $t$, by definition, it’s equal to $l(t)=a(t){l}_{0}$.
angular diameter distance – the distance ${d}_{A}$ according to which the angular separation on the sky of two distant objects scales $\propto {d}_{A}^{-1}$
luminosity distance – the distance ${d}_{L}$ according to which the received light intensity of a source scales $\propto {d}_{L}^{-2}$
All of these definitions give precisely the same answer in every-day situations (i.e. Minkowski space), of course! It’s the expansion of the universe that makes them differ. The first two are relatively easy to calculate, but the latter three are more closely connected to observations. We need a toolkit to evaluate them efficiently.
The fundamental measure from which all others may be calculated is the distance on the comoving grid. If the universe is flat, as we will assume throughout most of these lectures, computing distances on the comoving grid is easy. One very important comoving distance is the distance travelled by light since $t=0$ (in the absence of interactions). Recalling that we are working in units with $c=1$, in time $dt$, light travels a distance $dx=\frac{dt}{a}$; so, the total comoving distance light travels is:
$$\eta (t)\equiv {\int}_{0}^{t}\frac{d{t}^{\prime}}{a({t}^{\prime})}.$$ | (176) |
No information could have propagated faster than $\eta $ on the comoving grid since the beginning of time; thus $\eta $ is called the causal horizon or comoving horizon. A related concept is the particle horizon ${d}_{H}$, the proper radius travelled by light since $t=0$:
$${d}_{H}\equiv a(t){\int}_{0}^{t}\frac{d{t}^{\prime}}{a({t}^{\prime})}=a(\eta )\eta .$$ | (177) |
Regions separated by comoving distances $>\eta $ (or, equivalently, physical distances $>{d}_{H}$) cannot have sent each other any information; we say that “they are causally disconnected”, meaning there is no physical way for one object to effect the other. Conversely, we say that regions separated by comoving distances $$ are causally connected, or “in causal contact”, meaning that in principle they could have affected one another.
We can also think of $\eta $ (which increases monotonically) as a time variable and call it the conformal time. In terms of $\eta $, the FRW metric becomes
$$d{s}^{2}={a}^{2}(\eta )\left[d{\eta}^{2}-\frac{d{r}^{2}}{1-\kappa {r}^{2}}-{r}^{2}d{\mathrm{\Omega}}^{2}\right].$$ | (178) |
Just like $t$, $a$ and $z$, $\eta $ can be used to discuss the evolution of the universe. Despite its slightly strange definition, $\eta $ is actually the most useful time variable for many purposes. In the analysis of the evolution of perturbations, we will use it instead of $t$. Note that we have to be careful when working with a metric of the form (178) that we update normalizations appropriately, as in the following example:
Quite often, $\eta $ can be expressed analytically in terms of $a$. In particular, during radiation domination (RD), matter domination (MD) and for matter with equation-of-state $w$,
$\mathrm{RD}:$ | $\rho \propto {a}^{-4},\eta \propto a$ | (179) | |||
$\mathrm{MD}:$ | $\rho \propto {a}^{-3},\eta \propto \sqrt{a}$ | (180) | |||
$\mathrm{Generally}:$ | $\rho \propto {a}^{-3(1+w)},\eta \propto {a}^{(3w+1)/2}\mathit{\hspace{1em}}(w\ne -1/3).$ | (181) |
In the later part of the courses we will use conformal time extensively, so an update to Table 1 to include this new information is given in Table 2. Helpfully, we can also get an exact solution for $\eta $ in a mixed fluid case:
☞ Exercise 6B
Show that the conformal time as a function of scale-factor in a flat universe containing only matter and radiation is
$$\frac{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\eta $}}{{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\eta $}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$0$}}}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$=$}\sqrt{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$a$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$+$}{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$a$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\mathrm{EQ}$}}}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$-$}\sqrt{{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$a$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\mathrm{EQ}$}}}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$.$}$$ | (182) |
where ${\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$a$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\mathrm{EQ}$}}$ denotes the epoch of matter-radiation equality and ${\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\eta $}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$0$}}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$=$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$2$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$$}{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$($}{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$H$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$0$}}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$$}\sqrt{{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\mathrm{\Omega}$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$m$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$,$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$0$}}}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$)$}}^{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$-$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$1$}}$. Show that, for $\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$a$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\gg $}{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$a$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\mathrm{EQ}$}}$ and $\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$a$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\ll $}{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$a$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\mathrm{EQ}$}}$, you can recover the appropriate limits as given by equations (179) and (180).
Show that when $\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$a$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$=$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$1$}$ (i.e. at the present day), $\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\eta $}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\approx $}{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\eta $}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$0$}}$. Under what cirumstances would the equality be exact?
${w}_{i}$ | $\rho (a)$ | $a(t)$ | $a(\eta )$ | $a(\eta )H(\eta )$ | |
Matter | $0$ | $\propto {a}^{-3}$ | $\propto {t}^{2/3}$ | $\propto {\eta}^{2}$ | $2/\eta $ |
Radiation | $\frac{1}{3}$ | $\propto {a}^{-4}$ | $\propto {t}^{1/2}$ | $\propto \eta $ | $1/\eta $ |
$\mathrm{\Lambda}$ | $-1$ | constant | $\propto \mathrm{exp}({H}_{0}t)$ | $\propto -{\eta}^{-1}$ | $-1/\eta $ |
Curvature | $-\frac{1}{3}$ | $\propto {a}^{-2}$ | $\propto t$ | $\propto \mathrm{exp}{H}_{0}\eta $ | ${H}_{0}$ |
Consider the distance between us and a distant emitter. The comoving distance to an object at scale factor $a$ (or redshift $z=\frac{1}{a}-1$) is:
$$d(a)={\int}_{t(a)}^{{t}_{0}}\frac{d{t}^{\prime}}{a({t}^{\prime})}=\eta ({t}_{0})-\eta ({t}_{a})\text{.}$$ | (183) |
We can see objects out to $z\lesssim 10$. Throughout these times, radiation can be ignored. Let’s temporarily also ignore dark energy; then we have
$$d(a)=\frac{2}{{H}_{0}}\left[1-\sqrt{a}\right]=\frac{2}{{H}_{0}}\left[1-\frac{1}{\sqrt{1+z}}\right]\mathit{\hspace{1em}}\text{(flat, matter dominated).}$$ | (184) |
where I’ve used equation (180).
☞ Exercise 6C
Show that equation (184) follows from equation (180). Show that, when $\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$z$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\ll $}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$1$}$, $\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$d$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\approx $}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$z$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$/$}{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$H$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$0$}}$, which is the Hubble law.
Similarly, one can define the lookback time, elapsed between now and when light from redshift $z$ was emitted:
$${t}_{\mathrm{lookback}}(a)={\int}_{t(a)}^{{t}_{0}}\mathit{d}{t}^{\prime}={\int}_{a}^{1}\frac{d{a}^{\prime}}{{a}^{\prime}H({a}^{\prime})}.$$ | (185) |
For a flat, matter-dominated universe, the lookback time to redshift $z$ is:
$${t}_{\mathrm{lookback}}(z)=\frac{2}{3{H}_{0}}\left[1-{(1+z)}^{-3/2}\right]\mathit{\hspace{1em}}\text{(flat, matter dominated).}$$ | (186) |
The total age of a matter-dominated universe is obtained by letting $z\to \mathrm{\infty}$:
$${t}_{0}=\frac{2}{3{H}_{0}}\mathit{\hspace{1em}}\text{(flat, matter dominated).}$$ | (187) |
For universes that are not totally matter-dominated, the factor of $\frac{2}{3}$ will not be quite right, but for reasonable values of the cosmological parameters, we usually get ${t}_{0}\approx {H}_{0}^{-1}$.
The above measures are helpful theoretical constructions but can’t be measured directly. A classic way of actually measuring distances in astronomy is to measure the flux from an object of known luminosity, a standard candle. Let us neglect expansion for a moment, and consider the observed flux $F$ at a distance ${d}_{L}$ from a source of known luminosity $L$:
$$F=\frac{L}{4\pi {d}_{L}^{2}}.$$ | (188) |
This definition comes from the fact that in flat non-expanding space, for a source at distance $d$, the flux over the luminosity is just the inverse of the area of a sphere centred around the source, $1/A(d)=1/4\pi {d}^{2}$. In an FRW universe, however, the flux will be diluted. Conservation of photons tells us that all of the photons emitted by the source will eventually pass through a sphere at a comoving distance $\chi $ from the emitter. But the flux is diluted by two additional effects: the individual photons redshift by a factor $(1+z)$, and the photons hit the sphere less frequently, since two photons emitted a time $\delta t$ apart will be measured at a time $(1+z)\delta t$ apart.
Overall we will have
$$\frac{F}{L}=\frac{1}{{(1+z)}^{2}A}.$$ | (189) |
The area $A$ of a sphere centred at a comoving distance $\chi $ can be derived from the coefficient of $d{\mathrm{\Omega}}^{2}$ in (127), yielding
$$A=4\pi {S}_{\kappa}^{2}(\chi ),$$ | (190) |
where we have set $a(t)=1$ because the photons are being observed today. Comparing with (188), we obtain the luminosity distance:
$${d}_{L}=(1+z){S}_{\kappa}(\chi ).$$ | (191) |
Here, we must point out a caveat: the observed luminosity is related to emitted luminosity at a different wavelength. Here, we have assumed that the detector measures the incoming energy from all photons.
The luminosity distance ${d}_{L}$ is something we might hope to measure, since there are some astrophysical sources which are standard candles. But $\chi $ is not directly measurable, so we should substitute for $\chi $ in terms of $z$, which is something we can measure independently of ${d}_{L}$. On a radial null geodesic, we have
$$0=\mathrm{d}{s}^{2}=\mathrm{d}{t}^{2}-{a}^{2}\mathrm{d}{\chi}^{2}={a}^{2}\left(\mathrm{d}{\eta}^{2}-\mathrm{d}{\chi}^{2}\right),$$ | (192) |
which implies that
$$\chi =\mathrm{\Delta}\eta ={\eta}_{0}-{\eta}_{e}={\int}_{{a}_{e}}^{1}\frac{\mathrm{d}a}{{a}^{2}H(a)}.$$ | (193) |
where ${\eta}_{0}$ and ${\eta}_{e}$ are the conformal time at which the photon is received and emitted, respectively. The integral expression in terms of scalefactor follows directly from the integral for conformal time, (176). It’s conventional to convert the scale factor at the time of emission to redshift of the source using ${a}_{e}=1/(1+z)$, so we have
$$\chi (z)={\int}_{0}^{z}\frac{\mathrm{d}{z}^{\prime}}{H({z}^{\prime})},$$ | (194) |
leading to the luminosity distance,
$${d}_{L}=(1+z){S}_{\kappa}\left[{\int}_{0}^{z}\frac{\mathrm{d}{z}^{\prime}}{H({z}^{\prime})}\right].$$ | (195) |
Recall from equation (126) that ${S}_{\kappa}(\chi )=\chi $ for $k=0$. When $k\ne 0$, we more normally think in terms of the density parameter ${\mathrm{\Omega}}_{\kappa ,0}=-k/{H}_{0}^{2}$, which can either be determined though measurements of the spatial curvature, or by measuring the matter density and using ${\mathrm{\Omega}}_{\kappa ,0}=1-{\mathrm{\Omega}}_{\mathrm{m},0}$. Thus we can write the luminosity distance in terms of measurable cosmological parameters as
$${d}_{L}=(1+z)\frac{1}{{H}_{0}\sqrt{|{\mathrm{\Omega}}_{\kappa ,0}|}}{\mathcal{S}}_{k}\left[{H}_{0}\sqrt{|{\mathrm{\Omega}}_{\kappa ,0}|}{\int}_{0}^{z}\frac{\mathrm{d}{z}^{\prime}}{H({z}^{\prime})}\right],$$ | (196) |
where now
$$ | (197) |
and the integral can be evaluated by making use of the Friedmann equation. Though it appears unwieldy, equation (196) is of fundamental importance in cosmology. Given the observables ${H}_{0}$ and ${\mathrm{\Omega}}_{i,0}$, we can calculate ${d}_{L}$ to an object any redshift $z$; conversely, we can measure ${d}_{L}(z)$ for objects at a range of redshifts, and from that extract ${H}_{0}$ or the ${\mathrm{\Omega}}_{i,0}$.
Another classic distance measurement in astronomy is to measure the angle $\delta \theta $ subtended by an object of known physical size $\mathrm{\ell}$, known as a standard ruler. The angular diameter distance is then defined as,
$${d}_{A}=\frac{\mathrm{\ell}}{\delta \theta},$$ | (198) |
where $\delta \theta $ is small. At the time when the light was emitted, when the universe had scale factor $a$, the object was at redshift $z$ at comoving coordinate $\chi $ (assuming again that we are at $\chi =0$). Hence, from the angular part of the metric, $\mathrm{\ell}=a{S}_{\kappa}(\chi )\delta \theta $, and comparing with (198) we have the angular diameter distance
$${d}_{A}=\frac{{S}_{\kappa}(\chi )}{1+z}.$$ | (199) |
Fortunately, the unwieldy dependence on cosmological parameters is common to all distance measures, and we are left with a simple dependence on redshift:
$${d}_{L}={(1+z)}^{2}{d}_{A}.$$ | (200) |
The overall pattern of behaviour that you uncover in the exercise above is not significantly changed when $\mathrm{\Lambda}$ is included. However, both distances ${d}_{A},{d}_{L}$ are larger in a universe with a cosmological constant than in one without. (To be precise, here we are talking about observing an object of a known redshift $z$, fixing ${H}_{0}$ to its observed value today, and taking ${\mathrm{\Omega}}_{\kappa ,0}=0$ so that ${\mathrm{\Omega}}_{m,0}=1$.) The increased distances follow since the energy density extrapolated backwards in time, and hence the expansion rate at earlier times, is smaller in a universe with $\mathrm{\Lambda}$ than one with only matter. The universe was therefore expanding more slowly early on, and light had more time to travel from objects at any given redshift. Distant objects therefore appear fainter in a $\mathrm{\Lambda}$-dominated universe than if the universe were MD today. This observation (using Type Ia supernovae as standardizable candles) is exactly what lead to the confirmation of dark energy in the 1990s.
In an expanding Universe, there are multiple definitions of ‘distance’ between two widely separated objects;
The clearest of these theoretically speaking are the comoving distance and physical distance, which are both measured at a single moment in time (which in turn corresponds to measuring over a spatial slice with constant density);
The comoving horizon $\eta $ – i.e. the comoving distance travelled by a beam of light since the big bang – is an exceptionally useful quantity in theoretical derivations;
Objects separated by a comoving distance greater than $\eta $ are said to be “causally disconnected”; conversely objects separated by a comoving distance less than $\eta $ are “in causal contact” since they can in principle have physically affected each other;
Because $\eta $ increases monotonically with time, it can actually be used as a time variable and in this context is known as conformal time, with the metric taking the form (178); in many cases this actually simplifies derivations, so we will use it a lot;
Observationally speaking we often work with either the angular diameter distance ${d}_{A}$ or luminosity distance ${d}_{L}$ which are defined with reference to the angular extent of an object in the sky, or incoming flux from an object.