Horizons, Redshift, and Distance Measures
Conformal time, causal structure, cosmological redshift, and the zoo of distance measures used in observational cosmology
Overview
In an expanding universe, the notion of “distance” is not unique. Light signals propagate on null geodesics of the FLRW metric, and the finite age of the universe creates causal horizons. We introduce conformal time, derive the particle and event horizons, and define the key distance measures that connect theory to observation.
1. Conformal Time
It is convenient to define conformal time \(\eta\) by factoring out the scale factor from the time coordinate:
Conformal Time
$$\eta = \int_0^t \frac{dt'}{a(t')}$$
In conformal time the FLRW metric becomes conformally flat:
$$ds^2 = a^2(\eta)\left[-d\eta^2 + \frac{dr^2}{1-kr^2} + r^2\,d\Omega^2\right]$$
Light rays travel at 45-degree angles in \((\eta, r)\) coordinates, just as in Minkowski spacetime. This makes conformal time ideal for drawing causal (Penrose) diagrams.
2. The Particle Horizon
The particle horizon is the maximum comoving distance from which light could have reached an observer since \(t = 0\):
Particle Horizon
$$d_H(t) = a(t) \int_0^t \frac{dt'}{a(t')} = a(t)\,\eta(t)$$
This is the proper distance to the edge of the observable universe at time \(t\).
For a matter-dominated universe (\(a \propto t^{2/3}\)), we find \(d_H = 3t = 2/H\). For radiation domination (\(a \propto t^{1/2}\)), \(d_H = 2t = 1/H\). The Hubble radius \(1/H\) is always of order the particle horizon but they are not identical.
3. Cosmological Redshift
A photon emitted at scale factor \(a_e\) and received at \(a_0\)is stretched by the expansion. From the null geodesic condition \(ds^2 = 0\):
Cosmological Redshift
$$\boxed{1 + z = \frac{a_0}{a_e} = \frac{\lambda_{\text{obs}}}{\lambda_{\text{em}}}}$$
This is purely a stretching of wavelength by expansion — it is not a Doppler shift, though at low \(z\) the two are indistinguishable: \(z \approx v/c\)for \(z \ll 1\).
4. Comoving Distance
The comoving distance to an object at redshift \(z\) is:
$$r(z) = \int_0^z \frac{dz'}{H(z')}$$
where \(H(z)\) is given by the Friedmann equation expressed in terms of density parameters:
$$H^2(z) = H_0^2\left[\Omega_r(1+z)^4 + \Omega_m(1+z)^3 + \Omega_k(1+z)^2 + \Omega_\Lambda\right]$$
5. Luminosity Distance
The luminosity distance is defined so that the inverse-square law holds: an object of absolute luminosity \(L\) observed with flux \(F\) is at luminosity distance \(d_L = \sqrt{L/(4\pi F)}\). In FLRW:
Luminosity Distance
$$\boxed{d_L = (1+z)\,r(z)}$$
The two factors of \((1+z)\) arise from the redshifting of photon energy and the time-dilation of the arrival rate.
6. Angular Diameter Distance
The angular diameter distance is defined so that an object of proper size \(\ell\)subtending angle \(\delta\theta\) satisfies \(\ell = d_A\,\delta\theta\):
Angular Diameter Distance
$$\boxed{d_A = \frac{r(z)}{1+z} = \frac{d_L}{(1+z)^2}}$$
Remarkably, \(d_A\) is not monotonic: it reaches a maximum near \(z \sim 1.6\)in our universe, then decreases. Objects at very high redshift appear larger than objects at intermediate redshift.
7. Critical Density and Density Parameters
The critical density is the total energy density required for a spatially flat universe (\(k = 0\)):
$$\rho_c = \frac{3H^2}{8\pi G} \approx 1.88 \times 10^{-29}\,h^2\;\text{g cm}^{-3}$$
The density parameter for each component is \(\Omega_i = \rho_i / \rho_c\). Current observations give:
Concordance Cosmology Parameters
$$\Omega_m \approx 0.31, \qquad \Omega_\Lambda \approx 0.69, \qquad \Omega_r \approx 9 \times 10^{-5}$$
The curvature parameter \(\Omega_k = 1 - \Omega_{\text{tot}}\) is consistent with zero to within \(\pm 0.002\), confirming spatial flatness.
With these parameters the first Friedmann equation becomes a sum rule:
$$\Omega_m + \Omega_r + \Omega_\Lambda + \Omega_k = 1$$
8. The Etherington Reciprocity Relation
The luminosity and angular diameter distances are not independent. In any metric theory of gravity with photon number conservation, they satisfy the exact relation:
Distance Duality
$$\boxed{d_L = (1+z)^2\,d_A}$$
Any violation of this relation would signal either photon non-conservation (e.g., mixing with axion-like particles in magnetic fields) or a breakdown of metric theories of gravity. Current observations confirm distance duality to within \(\sim 5\%\).
This relation is fundamental to cosmological tests: Type Ia supernovae measure \(d_L(z)\), while baryon acoustic oscillations and the CMB measure \(d_A(z)\). Their consistency provides a powerful cross-check of the cosmological model.