Observational Cosmology and the Standard Model
Connecting the FLRW framework to astronomical observations, from redshift surveys to the cosmic microwave background
The FLRW cosmology becomes a predictive science when connected to astronomical observations. In this section we develop the key observational relations -- redshift, distance measures, and the cosmic microwave background -- and discuss the triumphs and outstanding puzzles of the standard cosmological model.
Cosmological Redshift
As light propagates through the expanding universe, its wavelength is stretched by the same factor as the scale factor. For a photon emitted at time \(t_{\text{emit}}\) and observed at\(t_0\), the cosmological redshift z is defined by:
$$1 + z = \frac{\lambda_{\text{obs}}}{\lambda_{\text{emit}}} = \frac{a(t_0)}{a(t_{\text{emit}})} = \frac{1}{a(t_{\text{emit}})}$$
The redshift directly measures the ratio of the scale factor at emission to today (where \(a_0 = 1\))
This is not a Doppler shift due to the motion of galaxies through space, but rather a stretching of the photon's wavelength due to the expansion of space itself. The distinction is important: nearby galaxies can have additional peculiar velocity Doppler shifts superimposed on the cosmological redshift, but at large distances the cosmological contribution dominates.
The redshift-distance relation for nearby objects (\(z \ll 1\)) reduces to Hubble's law:
$$cz \approx H_0 d \quad \text{(for } z \ll 1\text{)}$$
Hubble's law: the recession velocity is proportional to distance, with proportionality constant \(H_0\)
Cosmological Distance Measures
In an expanding universe, there is no single notion of "distance." Different observational methods probe different distance definitions, all related to the comoving distance \(\chi(z)\):
Luminosity Distance
If we know the intrinsic luminosity L of a source (a "standard candle"), we can define the luminosity distance \(d_L\) through the observed flux F:
$$d_L = \sqrt{\frac{L}{4\pi F}} = (1+z)\,S_k[\chi(z)]$$
The two factors of \((1+z)\) arise from energy redshift and time dilation of the photon arrival rate
For a flat universe (\(k = 0\)), \(S_k(\chi) = \chi\) and the luminosity distance simplifies to\(d_L = (1+z)\chi(z)\). Type Ia supernovae serve as standardizable candles, and their luminosity distance measurements at \(z \sim 0.5-1\) first revealed the accelerating expansion in 1998.
Angular Diameter Distance
If we know the physical size \(\ell\) of an object (a "standard ruler"), we can define the angular diameter distance \(d_A\) through its observed angular size \(\delta\theta\):
$$d_A = \frac{\ell}{\delta\theta} = \frac{S_k[\chi(z)]}{1+z} = \frac{d_L}{(1+z)^2}$$
The Etherington reciprocity relation: \(d_L = (1+z)^2 d_A\) (valid in any metric theory of gravity)
A remarkable feature of the angular diameter distance is that it reaches a maximum and thendecreases at high redshift. Objects at very high z can actually appear largeron the sky than similar objects at moderate z. This occurs because the universe was much smaller when the light was emitted, so objects subtend larger angles.
Expanding the luminosity distance to second order in z gives the relation used in the Hubble diagram:
$$d_L = \frac{c}{H_0}\left[z + \frac{1}{2}(1-q_0)z^2 + \cdots\right]$$
where the deceleration parameter is \(q_0 = -\ddot{a}a/\dot{a}^2|_0 = \frac{1}{2}\Omega_m - \Omega_\Lambda\)
The Cosmic Microwave Background
The cosmic microwave background (CMB) is the thermal radiation left over from the hot, dense early universe. It was emitted at the epoch of recombination when the universe cooled enough for neutral hydrogen to form, at a redshift of \(z_* \approx 1089\), about 380,000 years after the Big Bang.
$$T_{\text{CMB}}(z) = T_0(1+z), \quad T_0 = 2.7255 \pm 0.0006 \text{ K}$$
$$T_* = T_0(1+z_*) \approx 3000 \text{ K}$$
The temperature at recombination was about 3000 K, the temperature at which hydrogen atoms form
The CMB spectrum is an almost perfect blackbody -- the most precise blackbody ever measured in nature. The tiny temperature fluctuations (\(\delta T / T \sim 10^{-5}\)) encode a wealth of cosmological information:
Angular Power Spectrum
The temperature fluctuations are expanded in spherical harmonics:\(\delta T/T = \sum_{\ell m} a_{\ell m} Y_{\ell m}\). The power spectrum\(C_\ell = \langle|a_{\ell m}|^2\rangle\) shows acoustic peaks at specific angular scales, corresponding to sound waves frozen at recombination. The position of the first peak at \(\ell \approx 220\) directly constrains the spatial curvature.
What the Peaks Tell Us
The first peak position gives the total density (curvature). The ratio of odd-to-even peaks determines the baryon density. The overall damping envelope constrains the matter density and Hubble constant. The polarization pattern constrains reionization and potentially primordial gravitational waves.
The Standard Cosmological Parameters (Planck 2018)
The Planck satellite measured the CMB temperature and polarization with unprecedented precision, determining the parameters of the standard ΛCDM model:
Planck 2018 Best-Fit Parameters (ΛCDM)
\(H_0 = 67.36 \pm 0.54\) km/s/Mpc
Present expansion rate
\(\Omega_b h^2 = 0.02237 \pm 0.00015\)
Physical baryon density
\(\Omega_c h^2 = 0.1200 \pm 0.0012\)
Physical cold dark matter density
\(n_s = 0.9649 \pm 0.0042\)
Scalar spectral index (deviation from scale invariance)
\(\Omega_m = 0.3153 \pm 0.0073\)
Total matter density parameter
\(\Omega_\Lambda = 0.6847 \pm 0.0073\)
Dark energy density parameter
\(\tau = 0.0544 \pm 0.0073\)
Optical depth to reionization
\(\sigma_8 = 0.8111 \pm 0.0060\)
Amplitude of matter fluctuations at 8 Mpc/h
Age of the Universe
The age of the universe can be calculated by integrating the Friedmann equation from the Big Bang (\(a = 0\)) to today (\(a = 1\)):
$$t_0 = \int_0^1 \frac{da}{aH(a)} = \frac{1}{H_0}\int_0^\infty \frac{dz}{(1+z)E(z)}$$
where \(E(z) = H(z)/H_0 = \sqrt{\Omega_r(1+z)^4 + \Omega_m(1+z)^3 + \Omega_k(1+z)^2 + \Omega_\Lambda}\)
For the Planck best-fit parameters, this integral gives:
$$t_0 = 13.797 \pm 0.023 \text{ Gyr}$$
Remarkably, the age exceeds the Hubble time \(1/H_0 \approx 14.5\) Gyr because the expansion was decelerating in the past
The Flatness Problem
The observation that \(\Omega_k \approx 0\) today poses a fine-tuning puzzle. The curvature density parameter evolves as:
$$|\Omega_k(t)| = \frac{|k|c^2}{a^2 H^2} \propto \begin{cases} a^2 & \text{(radiation domination)} \\ a & \text{(matter domination)} \end{cases}$$
Curvature grows with time in a decelerating universe -- flatness is an unstable fixed point
Since \(|\Omega_k|\) grows with time during radiation and matter domination, the fact that \(|\Omega_k| < 0.002\) today means it must have been even closer to zero in the early universe. At the Planck time (\(t \sim 10^{-43}\) s), we would need\(|\Omega_k| < 10^{-62}\). Why was the early universe so extraordinarily flat?
The Horizon Problem
The CMB is remarkably uniform -- regions on opposite sides of the sky (separated by 180 degrees) have the same temperature to within \(10^{-5}\). Yet at the time of recombination, the particle horizon subtended only about 2 degrees on the sky:
$$\theta_{\text{horizon}} \approx \frac{d_{\text{particle}}(t_*)}{d_A(z_*)} \approx 2°$$
The last scattering surface consists of roughly 40,000 causally disconnected patches, all at the same temperature
How did these causally disconnected regions "know" to be at the same temperature? In standard Big Bang cosmology, there is no mechanism for them to have exchanged information.
Inflation as a Solution
Cosmic inflation, proposed by Alan Guth in 1981, resolves both the flatness and horizon problems in a single stroke. The idea is that the very early universe underwent a brief period of exponential expansion:
$$a(t) \propto e^{H_{\text{inf}}t}, \quad H_{\text{inf}} \sim 10^{13}\text{--}10^{14} \text{ GeV}$$
During inflation, the scale factor increases by at least a factor of \(e^{60} \sim 10^{26}\)
Inflation solves the problems as follows:
Flatness Problem Resolved
During inflation, \(|\Omega_k| \propto 1/(aH)^2 \propto e^{-2H_{\text{inf}}t}\) decreases exponentially. The comoving Hubble radius shrinks, driving \(\Omega_k \to 0\) regardless of initial conditions. After 60 e-folds, any initial curvature is diluted by a factor of \(e^{120}\).
Horizon Problem Resolved
Before inflation, the entire observable universe was contained within a single causal patch. Inflation stretched this small, causally connected region to enormous size. The uniformity of the CMB is then natural: those regions were in thermal equilibrium before inflation stretched them apart.
Additionally, inflation naturally predicts the nearly scale-invariant spectrum of primordial perturbations observed in the CMB. Quantum fluctuations of the inflaton field are stretched to cosmological scales, providing the seeds for all structure in the universe.
Dark Energy and the Cosmological Constant Problem
The discovery of the accelerating expansion in 1998 (via Type Ia supernovae) revealed that ~69% of the energy density of the universe is in a mysterious component with negative pressure -- dark energy. The simplest candidate is Einstein's cosmological constant \(\Lambda\), corresponding to a vacuum energy density:
$$\rho_\Lambda = \frac{\Lambda c^2}{8\pi G} \approx 5.96 \times 10^{-27} \text{ kg/m}^3 \approx (2.25 \times 10^{-3} \text{ eV})^4/(c^5\hbar^3)$$
A tiny but non-zero energy density of the vacuum
The cosmological constant problem is one of the deepest puzzles in theoretical physics: quantum field theory predicts a vacuum energy density that is at least 60 orders of magnitude larger than the observed value:
$$\rho_{\text{QFT}} \sim \frac{M_{\text{Pl}}^4 c^3}{\hbar^3} \sim 10^{76} \text{ GeV}^4 \quad \gg \quad \rho_\Lambda \sim 10^{-47} \text{ GeV}^4$$
A discrepancy of \(\sim 10^{123}\) -- the largest fine-tuning problem in all of physics
Alternative explanations for dark energy include quintessence (a dynamical scalar field with time-varying equation of state \(w(z)\)), modified gravity theories, and anthropic arguments involving the string theory landscape. Distinguishing between these possibilities requires precise measurements of \(w(z)\) and the growth of cosmic structure.
Key Concepts Summary
FLRW Framework
- FLRW metric: \(ds^2 = -c^2dt^2 + a^2[dr^2/(1-kr^2) + r^2d\Omega^2]\)
- 1st Friedmann: \(H^2 = 8\pi G\rho/3 - kc^2/a^2 + \Lambda c^2/3\)
- 2nd Friedmann: \(\ddot{a}/a = -4\pi G(\rho + 3p/c^2)/3 + \Lambda c^2/3\)
- Continuity: \(\dot{\rho} + 3H(\rho + p/c^2) = 0\)
- EOS scaling: \(\rho \propto a^{-3(1+w)}\)
- Density parameters: \(\Omega_r + \Omega_m + \Omega_\Lambda + \Omega_k = 1\)
Observational Pillars
- Redshift: \(1 + z = 1/a(t_{\text{emit}})\)
- Hubble's law: \(v = H_0 d\) (nearby galaxies)
- CMB: \(T_0 = 2.7255\) K, \(\delta T/T \sim 10^{-5}\)
- Age: \(t_0 = 13.80 \pm 0.02\) Gyr
- Acceleration: \(q_0 < 0\) (discovered 1998)
- Open questions: \(H_0\) tension, dark energy nature, inflation mechanism
The Standard Model of Cosmology
The ΛCDM model -- a spatially flat universe composed of baryons, cold dark matter, and a cosmological constant, with nearly scale-invariant primordial perturbations generated by inflation -- fits an extraordinary range of observations: the CMB power spectrum, baryon acoustic oscillations, Type Ia supernovae, big bang nucleosynthesis abundances, and the growth of large-scale structure. With just six free parameters, it provides a remarkably complete description of the universe from the first fraction of a second to the present day, spanning 14 billion years of cosmic evolution. Yet its two dominant components -- dark matter and dark energy -- remain mysterious, suggesting that our understanding of the cosmos is still profoundly incomplete.