Cosmology: FRW Metric & the Expanding Universe
Cosmology: FRW Metric & the Expanding Universe
Applying GR to a homogeneous, isotropic universe yields the Friedmann-Robertson-Walker (FRW) metric and the Friedmann equations. These govern cosmic expansion, predicting the Big Bang, distinct matter/radiation domination eras, and the current \(\Lambda\)CDM accelerating expansion driven by dark energy.
FRW Metric & Friedmann Equations
\(ds^2 = -c^2dt^2 + a(t)^2\!\left[\frac{dr^2}{1-kr^2} + r^2 d\Omega^2\right]\), \(k\in\{-1,0,+1\}\). Friedmann equations: \(H^2 = \frac{8\pi G\rho}{3} - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}\); \(\;\frac{\ddot a}{a} = -\frac{4\pi G}{3}\!\left(\rho + \frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}\).
Equation of State & \(\Lambda\)CDM
For \(p = w\rho c^2\): \(\rho\propto a^{-3(1+w)}\). Matter \((w=0)\): \(\rho_m\propto a^{-3}\); radiation \((w=1/3)\): \(\rho_r\propto a^{-4}\); \(\Lambda\) \((w=-1)\): \(\rho_\Lambda=\text{const}\). Current concordance: \(\Omega_m\approx0.31\), \(\Omega_\Lambda\approx0.69\), \(H_0\approx67.4\,\text{km/s/Mpc}\).
Example 1: Hubble Law from FRW
Show that for nearby galaxies, recession velocity \(v = H_0 d\).
Solution: Physical distance \(d = a(t)\chi\) for comoving distance \(\chi\). Then \(\dot d = \dot a\chi = (\dot a/a)\cdot d = Hd\). For small \(z\), \(H\approx H_0\), recovering Hubble's law. This is cosmic expansion, not peculiar velocity.
Example 2: Age of the Universe
Estimate \(t_0\) for flat \(\Lambda\)CDM with \(\Omega_\Lambda=0.7\), \(H_0=70\,\text{km/s/Mpc}\).
Solution: \(t_0 = \frac{1}{H_0}\int_0^1\frac{da}{a\sqrt{\Omega_m a^{-3}+\Omega_\Lambda}}\). Approximating: \(t_0 \approx \frac{2}{3H_0\sqrt{\Omega_\Lambda}}\ln\frac{1+\sqrt{\Omega_\Lambda/\Omega_m}}{1}\approx 13.8\,\text{Gyr}\). Consistent with CMB observations.
Practice
- Derive the Friedmann equation from the \((00)\) EFE with the FRW metric.
- Show that the matter-dominated flat universe scales as \(a(t)\propto t^{2/3}\).
- Compute the comoving particle horizon at matter-radiation equality.
- Explain how \(\Lambda\) with \(w=-1\) drives \(\ddot a > 0\) (accelerated expansion).
Show Answer Key
1. FRW metric: $ds^2 = -c^2dt^2+a^2(t)[\frac{dr^2}{1-kr^2}+r^2d\Omega^2]$. The $(00)$ EFE gives: $(\dot{a}/a)^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}$. This is the Friedmann equation: $H^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2}+\frac{\Lambda c^2}{3}$.
2. Matter-dominated ($\rho \propto a^{-3}$), flat ($k=0$), no $\Lambda$: Friedmann $\to H^2 = (\dot{a}/a)^2 = H_0^2(a_0/a)^3$. Let $a/a_0 = (t/t_0)^\alpha$: $\alpha/t = H_0(t_0/t)^{3\alpha/2}$. Matching: $\alpha = 2/3$, so $a(t) \propto t^{2/3}$. $H = 2/(3t)$, age $t_0 = 2/(3H_0)$.
3. Comoving particle horizon: $d_H(t) = a(t)\int_0^t \frac{c\,dt'}{a(t')}$. At matter-radiation equality ($a_{eq}$): during radiation domination ($a \propto t^{1/2}$), $\int_0^{t_{eq}} dt'/a(t') = 2t_{eq}/a_{eq}$. So $d_H(t_{eq}) = 2ct_{eq}$. Using $t_{eq} \approx 50,000$ yr: $d_H \approx 100,000$ light-years (comoving), corresponding to $\sim 1°$ on the CMB sky today.
4. Friedmann acceleration equation: $\ddot{a}/a = -\frac{4\pi G}{3}(\rho+3P/c^2)+\frac{\Lambda c^2}{3}$. For $\Lambda$-dominated universe ($\rho_{\Lambda}=\Lambda c^2/(8\pi G)$, $P=-\rho_\Lambda c^2$): $\rho+3P/c^2 = \rho_\Lambda - 3\rho_\Lambda = -2\rho_\Lambda < 0$, so $\ddot{a} > 0$. The solution is $a(t) \propto e^{Ht}$ with $H = \sqrt{\Lambda c^2/3}$ (de Sitter expansion).