Cosmic Expansion, Redshifts & Relativistic Cosmology

In the 1910s and 1920s, Vesto Slipher obtained optical spectra of spiral nebulae, discovering that nearly all were systematically shifted toward longer wavelengths (receding from Earth at hundreds of kilometers per second). In 1927, Georges Lemaître theoretically predicted expanding universe solutions from Einstein's general relativity. In 1929, Edwin Hubble combined Slipher's radial velocities with Cepheid variable distance estimates, publishing the empirical linear velocity-distance relation known today as the Hubble-Lemaître Law:

where $v$ is the recession velocity in $\text{km s}^{-1}$, $d$ is physical distance in megaparsecs ($\text{Mpc}$), and $H_0$ is the Hubble constant (the current expansion rate of the universe).

Modern Values of $H_0$ & The Hubble Tension

Measuring $H_0$ is one of the most critical endeavors in modern astrophysics:

• Late-Universe Distance Ladder (SH0ES, Riess et al.): Cepheid-calibrated Type Ia supernovae yield $H_0 = 73.04 \pm 1.04\text{ km s}^{-1}\text{Mpc}^{-1}$.
• Early-Universe CMB Angular Spectrum (Planck 2018): Extrapolating the $\Lambda\text{CDM}$ model from the surface of last scattering ($z \approx 1100$) yields $H_0 = 67.36 \pm 0.54\text{ km s}^{-1}\text{Mpc}^{-1}$.

This persistent $> 5\sigma$ discrepancy is known as the Hubble Tension, possibly hinting at new physics beyond the standard model (e.g., early dark energy or sterile neutrinos).

Hubble Time & Hubble Distance

Dimensional analysis of $H_0$ ($[\text{velocity}/\text{distance}] = \text{time}^{-1}$) yields fundamental cosmic scales:

• The Hubble Time ($t_H$): The characteristic age of the universe assuming constant expansion speed:
  $$t_H \equiv \frac{1}{H_0} = \frac{1}{70\text{ km s}^{-1}\text{Mpc}^{-1}} = \frac{3.0857 \times 10^{19}\text{ km}}{70\text{ km/s}} \approx 4.408 \times 10^{17}\text{ s} \approx 13.97\text{ Gyr}$$
• The Hubble Distance ($D_H$): The distance at which recession velocity equals the speed of light ($v = c$):
  $$D_H \equiv \frac{c}{H_0} = \frac{299{,}792\text{ km/s}}{70\text{ km s}^{-1}\text{Mpc}^{-1}} \approx 4{,}283\text{ Mpc} \approx 4.28\text{ Gpc} \approx 13.97\text{ billion light-years}$$

Objects beyond $D_H$ recede from us faster than light! This does not violate Special Relativity, because galaxies are stationary relative to their local comoving space; it is the metric fabric of space itself that is expanding between them.

In observational cosmology, spectral shifts are described by the dimensionless redshift parameter $z$:

Fundamental Distinction from Doppler Motion

In Special Relativity, a source moving through static space at velocity $v = \beta c$ produces a relativistic Doppler shift:

In cosmology, however, galaxies are not moving through static space; space itself is expanding. Cosmological redshift is a metric phenomenon.

Derivation of Cosmological Redshift from the Scale Factor

Let $a(t)$ denote the dimensionless cosmic scale factor, describing the relative spatial separation between comoving coordinate points as a function of cosmic time $t$, normalized such that today $a(t_0) = 1$.

Consider a light wave emitted at time $t_{\text{emit}}$ with wavelength $\lambda_{\text{emit}}$ and observed today at $t_0$ with wavelength $\lambda_{\text{obs}}$. Light travels along null geodesics ($ds^2 = 0$). For radial propagation in a flat universe: $c\,dt = a(t) dr$.

A first wave crest is emitted at $t_{\text{emit}}$ and arrives at $t_0$:

The subsequent crest is emitted at $t_{\text{emit}} + \Delta t_{\text{emit}}$ and arrives at $t_0 + \Delta t_{\text{obs}}$:

Equating the two integrals and subtracting the overlapping interval $\int_{t_{\text{emit}}+\Delta t_{\text{emit}}}^{t_0} \frac{c\,dt}{a(t)}$:

Since $\Delta t$ is the period of a light wave ($10^{-15}\text{ s}$), $a(t)$ is essentially constant over the interval:

Because $\lambda = c \Delta t$, this yields the foundational relation of relativistic cosmology:

When we observe a galaxy at $z = 1$, the light was emitted when the universe was exactly half its current linear size ($a = 0.5$). For the CMB at $z \approx 1100$, the universe was $1/1101$ of its current size.

Modern cosmology is founded upon the Cosmological Principle, which asserts that on sufficiently large scales ($\gtrsim 100\text{ Mpc}$), the Universe is:

1. Homogeneous: Spatially invariant under translations—no preferred locations exist; every region looks statistically identical to every other region.
2. Isotropic: Invariant under spatial rotations—no preferred directions exist; the sky appears identical in every direction.

The Maximally Symmetric Robertson-Walker Metric

Howard Robertson (1935) and Arthur Walker (1936) proved mathematically that the only four-dimensional spacetime metric compatible with spatial homogeneity and isotropy is the Robertson-Walker metric:

where:

• $t$ is cosmic time, measured by clocks comoving with the cosmic fluid.
• $r, \theta, \phi$ are dimensionless comoving coordinates, fixed to galaxies that move solely with the expansion.
• $a(t)$ is the time-dependent cosmic scale factor (dimensions of length).
• $k$ is the spatial curvature index:
  • $k = 0$: Flat Euclidean Space ($E^3$). Spatial slices are infinite flat 3D planes ($d\sigma^2 = dr^2 + r^2 d\Omega^2$).
  • $k = +1$: Closed Spherical Space ($S^3$). A finite, unbounded 3D hypersphere of radius $a(t)$ embedded in 4D space with positive constant Gaussian curvature. Sum of triangle angles $> 180^\circ$.
  • $k = -1$: Open Hyperbolic Space ($H^3$). An infinite 3D saddle space with negative constant curvature. Sum of triangle angles $< 180^\circ$.

Using the radial coordinate substitution $r = S_k(\chi)$ where $S_k(\chi) = \sin\chi$ ($k=+1$), $\chi$ ($k=0$), and $\sinh\chi$ ($k=-1$), the metric takes the elegant form:

While the Robertson-Walker metric specifies the geometry of spacetime, the dynamic time evolution of the scale factor $a(t)$ is dictated by Einstein's Field Equations of General Relativity:

where $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu}$ is the Einstein tensor and $\Lambda$ is the cosmological constant.

1. Stress-Energy Tensor of the Cosmic Fluid

By the cosmological principle, the matter-energy content of the universe is described by a homogeneous, isotropic perfect fluid:

where $\rho(t)$ is total mass-energy density and $P(t)$ is isotropic pressure.

2. Computing the Christoffel Symbols and Ricci Tensor

For the FLRW metric $g_{00} = -c^2$, $g_{rr} = \frac{a^2}{1-kr^2}$, $g_{\theta\theta} = a^2 r^2$, $g_{\phi\phi} = a^2 r^2 \sin^2\theta$, the non-zero Christoffel symbols $\Gamma^\lambda_{\mu\nu} = \frac{1}{2}g^{\lambda\sigma}(\partial_\mu g_{\nu\sigma} + \partial_\nu g_{\mu\sigma} - \partial_\sigma g_{\mu\nu})$ yield the non-vanishing Ricci tensor components:

The Ricci scalar is $R = g^{\mu\nu} R_{\mu\nu} = \frac{6}{c^2}\left(\frac{\ddot{a}}{a} + \frac{\dot{a}^2}{a^2} + \frac{k c^2}{a^2}\right)$.

3. The First Friedmann Equation

Substituting into the time-time ($00$) component of Einstein's equations $G^0_0 + \Lambda = \frac{8\pi G}{c^4} T^0_0$:

Rearranging and defining the Hubble parameter $H(t) \equiv \frac{\dot{a}}{a}$ gives the First Friedmann Equation:

4. The Second Friedmann (Acceleration) Equation

Combining the spatial ($ii$) component $G^i_i + \Lambda = \frac{8\pi G}{c^4} T^i_i = \frac{8\pi G}{c^4} P$ with the first equation yields the Acceleration Equation:

Notice that ordinary matter and radiation ($\rho > 0, P \ge 0$) exert a negative acceleration, slowing down the expansion ($\ddot{a} < 0$). To produce an accelerating expansion ($\ddot{a} > 0$), we require an energy component with large negative pressure: $P < -\frac{1}{3}\rho c^2$, or a positive cosmological constant $\Lambda > 0$.

5. The Fluid Continuity Equation

Conservation of energy-momentum $\nabla_\mu T^{\mu\nu} = 0$ yields the continuity equation:

The equation of state relating pressure to density is parametrized by the dimensionless parameter $w$:

Substituting into the continuity equation $\frac{\dot{\rho}}{\rho} = -3(1+w)\frac{\dot{a}}{a}$ yields the scaling of density with scale factor:

Cosmic Components & Scaling Behaviors

1. Non-Relativistic Matter ($w = 0$): Dust, cold dark matter, stars, and baryonic gas have negligible thermal velocities ($k T \ll m c^2$), so $P \ll \rho c^2$. $$\rho_m(a) = \rho_{m,0} a^{-3} = \rho_{m,0} (1+z)^3$$ Density drops inversely with volume ($V \propto a^3$).
2. Relativistic Radiation ($w = 1/3$): Photons (CMB) and relativistic neutrinos have $P = \frac{1}{3}u = \frac{1}{3}\rho c^2$. $$\rho_r(a) = \rho_{r,0} a^{-4} = \rho_{r,0} (1+z)^4$$ In addition to volume dilution ($a^{-3}$), each photon's energy is redshifted ($E = h\nu \propto a^{-1}$), yielding an extra factor of $1/a$.
3. Cosmological Constant / Vacuum Energy ($w = -1$): $$P_\Lambda = -\rho_\Lambda c^2 \implies \rho_\Lambda = \frac{\Lambda c^2}{8\pi G} = \text{constant!}$$ Vacuum energy density does not dilute as space expands.

Critical Density & Dimensionless Density Parameters ($\Omega$)

For a spatially flat universe ($k = 0$) without cosmological constant, the First Friedmann equation gives $H^2 = \frac{8\pi G}{3}\rho_c$. The critical density is:

Dividing the First Friedmann Equation by $H_0^2$ defines the dimensionless density parameters today:

The First Friedmann Equation simplifies to the master cosmic balance relation:

If $\Omega_{\text{tot}} = 1$, space is exactly flat ($k=0, \Omega_k = 0$). If $\Omega_{\text{tot}} > 1$, space is spherical ($k=+1, \Omega_k < 0$). If $\Omega_{\text{tot}} < 1$, space is hyperbolic ($k=-1, \Omega_k > 0$).

The Hubble parameter evolves dynamically with redshift as:

Solving the Friedmann equation for single- and multi-component universes reveals how the geometry and cosmic inventory dictate the expansion history.

Analytical Single-Component Universes

• Radiation-Dominated Universe ($\Omega_r = 1$): $$\dot{a} \propto a^{-1} \implies a(t) = \left(\frac{t}{t_0}\right)^{1/2}, \quad t_0 = \frac{1}{2 H_0}$$
• Einstein-de Sitter Flat Matter Universe ($\Omega_m = 1$): $$\dot{a} \propto a^{-1/2} \implies a(t) = \left(\frac{t}{t_0}\right)^{2/3}, \quad t_0 = \frac{2}{3 H_0} \approx 9.3\text{ Gyr}$$
• de Sitter Vacuum Universe ($\Omega_\Lambda = 1$): $$\frac{\dot{a}}{a} = H_0 = \text{const} \implies a(t) = e^{H_0 (t - t_0)}, \quad t_0 = \infty$$
• Milne Empty Universe ($\Omega = 0, k = -1$): $$\dot{a} = c \implies a(t) = \frac{t}{t_0}, \quad t_0 = \frac{1}{H_0} \approx 14\text{ Gyr}$$

The Deceleration Parameter ($q_0$)

The deceleration parameter quantifies the second derivative of the scale factor:

From the acceleration equation $\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}(\rho + 3P/c^2) + \frac{\Lambda c^2}{3}$:

In our modern universe where radiation is negligible ($\Omega_r \approx 10^{-4}$):

Because $q_0 < 0$, the expansion of our Universe is accelerating!

Exact Age of the $\Lambda\text{CDM}$ Universe

For a flat universe with matter and dark energy ($\Omega_m + \Omega_\Lambda = 1, \Omega_k = 0$):

Evaluating this integral analytically via substitution $u = a^{3/2} \sqrt{\Omega_\Lambda/\Omega_m}$:

For Planck parameters ($H_0 = 67.4\text{ km s}^{-1}\text{Mpc}^{-1}, \Omega_m = 0.315, \Omega_\Lambda = 0.685$):

In expanding curved spacetime, the concept of 'distance' bifurcates into distinct operational definitions.

1. Comoving Distance ($\chi$)

The comoving distance between an observer at $z=0$ and a source at redshift $z$ is:

2. Luminosity Distance ($d_L$)

The luminosity distance is defined such that the inverse-square law holds: $F = \frac{L}{4\pi d_L^2}$. In an FLRW universe, two factors dilute the observed photon flux:

• Each photon's energy is redshifted by $(1+z)^{-1}$.
• The arrival rate of photons is time-dilated by $(1+z)^{-1}$.

Consequently, the received flux is $F = \frac{L}{4\pi [a_0 S_k(\chi)]^2 (1+z)^2}$. For a flat universe ($S_k(\chi) = \chi$):

3. Angular Diameter Distance ($d_A$)

The angular diameter distance relates physical transverse source diameter $D$ to observed angular diameter $\theta$: $\theta = D / d_A$. Since the physical diameter was $D = a(t) r \theta = \frac{r \theta}{1+z}$:

Connecting the two is Etherington's Reciprocity Theorem:

Notice that as $z \to \infty$, $d_A(z)$ reaches a maximum near $z \sim 1.6$ and then decreases! Highly distant galaxies appear angularly larger on the sky because their light was emitted when they were physically closer to us!

The Type Ia Supernova Discovery of Cosmic Acceleration

Type Ia supernovae result from the thermonuclear detonation of carbon-oxygen white dwarfs near the Chandrasekhar limit. Because their progenitor masses are identical, they act as magnificent standard candles with peak absolute magnitude $M_B \approx -19.3$.

In 1998, two competing teams—the High-Z Supernova Search Team (Brian Schmidt and Adam Riess) and the Supernova Cosmology Project (Saul Perlmutter)—measured the distance moduli $\mu(z) = m_B - M_B = 5\log_{10} d_L(z) - 5$ of high-redshift supernovae ($z \sim 0.3 - 1.0$).

They discovered that distant supernovae were systematically $\approx 0.25\text{ magnitudes}$ fainter than predicted by a decelerating matter-dominated universe ($\Omega_m = 1$). To produce faint flux at a given redshift requires a larger luminosity distance $d_L(z)$, which requires an accelerating scale factor driven by negative-pressure Dark Energy ($\Omega_\Lambda \approx 0.7$, Nobel Prize in Physics 2011).