Cosmological Distance Measures: Luminosity & Angular Diameter

1. The Challenge of Measuring Cosmic Distances

In a static, flat space, the distance to an object is unambiguous. You could measure it with a ruler (proper distance), infer it from its apparent angular size, or deduce it from its brightness, and you would get the same answer. In the universe described by the FRW metric, this is no longer true for two main reasons:

1. Expansion: The space between us and a distant galaxy has stretched while the light was in transit. The distance to the galaxy "when the light was emitted" is different from its distance "now".
2. Curvature: The spatial geometry itself might not be Euclidean. A positive curvature (like a sphere) or negative curvature (like a saddle) distorts the relationship between an object's distance and its apparent size.

To handle this, cosmologists use several distance definitions, each tied to a specific measurement method. Before we define them, we must first establish the most fundamental distance: the comoving distance.

2. Comoving Distance: The Unchanging Grid

Recall that the FRW metric describes a set of comoving coordinates that are carried along with the cosmic expansion. The comoving distance, often denoted by $\chi$, is the distance between two objects (say, us at the origin and a galaxy at $\chi$) measured on this fixed grid. It does not change with time.

For a photon travelling along a null radial geodesic ($d{s}^2=0$, $d{\Omega }^2=0$), we have:

The total comoving distance to a galaxy whose light we observe today ($t_o$) that was emitted at time $t_e$ is:

Using the relations $1+z=1/a(t)$ (with $a_0=1$) and $H=\dot{a}/a$, we can change the integration variable to redshift $z$:

So, the comoving distance to an object at redshift $z$ is:

This integral is the backbone of all cosmological distance measures. It tells us how far light has travelled on the comoving grid, and it depends on the entire expansion history of the universe, encapsulated in $H(z^{\prime })$.

3. Angular Diameter Distance ($d_A$)

The angular diameter distance is defined operationally: if you know an object has a proper physical size $\delta s$ (a "standard ruler") and you measure its angular size on the sky to be $\delta \theta$, then:

Let's derive this from the FRW metric. In the form that makes curvature most explicit, the metric is:

Here, $\chi$ is the comoving radial distance, and $S_k(\chi )$ is a function that depends on the spatial curvature $k$:

where $R$ is the radius of curvature. In our notation, we often set $R=1$ and absorb it into the definition of $k$, so $S_k(\chi )$ becomes $\sin (\chi )$, $\chi$, or $\sinh (\chi )$.

Consider an object of physical size $\delta s$ at comoving distance $\chi$, oriented perpendicular to our line of sight. The measurement is made at the time the light was emitted, $t_e$. At that time, its proper size is given by the metric with $dt=0,d\chi =0,d\varphi =0$:

Rearranging this to match our definition of $d_A$:

Using $a(t_e)=a_0/(1+z)=1/(1+z)$ (since $a_0\equiv 1$), we get the final expression for angular diameter distance as a function of redshift:

The following resource provides a formal derivation that aligns with our discussion.

This distance has a fascinating and counter-intuitive property. At low redshift, $d_A\approx \chi \approx cz/H_0$, so things look smaller as they get farther away. However, due to the $(1+z)$ factor in the denominator, at high redshift $d_A(z)$ reaches a maximum and then begins to decrease. This means an object of a fixed size will appear to get smaller as you move it away from you, but then beyond a certain redshift, it will start to look larger again!

The most famous "standard ruler" in cosmology is the characteristic size of the hot and cold spots in the Cosmic Microwave Background (CMB). By measuring their angular size ($\delta \theta \approx 1^{\circ }$), we can calculate $d_A$ to the last scattering surface ($z\approx 1100$) and powerfully constrain the universe's geometry.

4. Luminosity Distance ($d_L$)

The luminosity distance is defined based on the apparent brightness of "standard candles"—objects of known intrinsic luminosity $L$. If we measure a flux $F$ from such an object, its luminosity distance is defined by the inverse-square law:

To derive the cosmological formula for $d_L$, we must account for all the effects that diminish the flux in an expanding universe.

1. Geometric Spreading: At the time of observation ($t_0$), the photons emitted from the source are spread over a sphere of proper radius $S_k(\chi )$ and hence proper area $4\pi [a(t_0)S_k(\chi ){]}^2=4\pi S_k(\chi {)}^2$ since $a_0=1$.
2. Redshift of Energy: Each photon's energy is reduced by the redshift. A photon emitted with energy $E_e$ is observed with energy $E_o=E_e/(1+z)$.
3. Time Dilation: The rate of photon arrival is also slowed. If photons are emitted over a time interval $\delta t_e$, they are observed over an interval $\delta t_o=(1+z)\delta t_e$. This reduces the number of photons arriving per second.

Combining these effects, the observed flux $F$ is related to the intrinsic luminosity $L$ by:

Comparing this with the definition $F=L/(4\pi d_L^2)$, we find:

The Distance Duality Relation

By comparing the expressions for $d_A$ and $d_L$, we find a simple and profound relationship between them:

This is known as Etherington's distance duality relation (or the reciprocity theorem). It is a remarkably general result, relying only on the conservation of photon number and the fact that light travels on null geodesics in a metric theory of gravity. A test confirming this relation is a fundamental check on our understanding of spacetime.

5. Distances in Practice

The distance formulas depend on the integral of $1/H(z)$, which itself depends on the cosmological parameters through the Friedmann equation:

By measuring the luminosity distance to a set of "standard candles" like Type Ia supernovae over a range of redshifts, astronomers can fit the observed $d_L(z)$ data to the theoretical curve and solve for the values of ${\Omega }_m,{\Omega }_{\Lambda }$, and $H_0$. This is precisely how the accelerated expansion of the universe was discovered in 1998.

The following reading shows the full equations and plots of these distances for different cosmological models.

The different distance measures are summarized beautifully in the following plot, which shows how they diverge as redshift increases.

Conclusion

We have now demystified the concept of "distance" in cosmology and equipped ourselves with the practical tools used to measure our universe.

Key Takeaways:

• In an expanding, curved universe, there is no single definition of distance. Different measurement methods yield different results.
• The comoving distance ($\chi$) is the fundamental coordinate distance on the expanding grid, obtained by integrating $c/H(z)$ over redshift.
• The angular diameter distance ($d_A=S_k(\chi )/(1+z)$) is used with "standard rulers" and is sensitive to spatial curvature. It counter-intuitively decreases at high redshift.
• The luminosity distance ($d_L=(1+z)S_k(\chi )$) is used with "standard candles" and was key to discovering the universe's accelerated expansion.
• These distances are connected by the fundamental distance duality relation: $d_L=(1+z{)}^2{d}_A$.
• Measuring these distances as a function of redshift allows us to determine the cosmological parameters (${\Omega }_m,{\Omega }_{\Lambda },{\Omega }_k,H_0$).

Preview of Next Lesson:

The very success of these measurements has revealed deep puzzles about the initial state of our universe. Why is the spatial curvature so close to zero (${\Omega }_k\approx 0$)? Why are regions of the CMB that were causally disconnected in the early universe at almost the exact same temperature? In our next lesson, we will explore these horizon and flatness problems, which challenge the standard Big Bang model and motivate the need for a new ingredient: cosmic inflation.