Observational Cosmology

Introduction

Observational cosmology provides the empirical foundation for our understanding of the universe. Through careful measurements of cosmic phenomena, we determine the fundamental parameters of cosmology and test theoretical predictions.

Modern cosmology is remarkably well-described by the ΛCDM model with just six parameters, determined to unprecedented precision by multiple independent observations.

1. Cosmic Microwave Background (CMB)

Discovery and Temperature

The CMB is relic radiation from the Big Bang, discovered by Penzias and Wilson in 1965. It has a nearly perfect blackbody spectrum at temperature:

$$T_{\text{CMB}} = 2.7255 \pm 0.0006 \text{ K}$$

Temperature Anisotropies

The CMB temperature varies across the sky at the level of $\Delta T/T \sim 10^{-5}$. These anisotropies are expanded in spherical harmonics:

$$\frac{\Delta T}{T}(\theta, \phi) = \sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell} a_{\ell m} Y_{\ell m}(\theta, \phi)$$

Angular Power Spectrum

The angular power spectrum characterizes the temperature fluctuations:

$$C_\ell = \frac{1}{2\ell+1}\sum_{m=-\ell}^{\ell}|a_{\ell m}|^2$$

Key features of the power spectrum:

  • • Sachs-Wolfe plateau ($\ell < 100$): Gravitational potential fluctuations
  • • Acoustic peaks ($\ell \sim 200-1000$): Sound waves in photon-baryon fluid
  • • Damping tail ($\ell > 1000$): Silk damping from diffusion

Physical Interpretation of Peaks

The first acoustic peak location determines the spatial curvature:

$$\ell_1 \approx \frac{\pi}{\theta_*} = \frac{\pi D_A(z_*)}{{r_s(z_*)}}$$

where $D_A$ is the angular diameter distance to recombination and $r_s$ is the sound horizon:

$$r_s(z) = \int_z^{\infty} \frac{c_s(z')}{H(z')} dz'$$

The sound speed in the photon-baryon plasma:

$$c_s = \frac{c}{\sqrt{3(1 + R)}}$$

where $R = 3\rho_b/4\rho_\gamma$ is the baryon-to-photon density ratio.

Polarization

Thomson scattering produces linear polarization, decomposed into E-modes and B-modes:

  • E-modes: Gradient-type patterns from density perturbations
  • B-modes: Curl-type patterns from gravitational waves (primordial) or lensing (secondary)

2. Cosmic Distance Ladder

Standard Candles

Objects with known intrinsic luminosity $L$ allow distance determination via:

$$d_L = \sqrt{\frac{L}{4\pi F}}$$

where $F$ is the observed flux and $d_L$ is the luminosity distance.

Type Ia Supernovae

SNe Ia result from white dwarf explosions near the Chandrasekhar limit, providing standardizable candles. After empirical corrections (stretch and color):

$$m_B = M_B + 5\log_{10}(d_L/10\text{ pc}) + \alpha X - \beta C$$

where $X$ is the light curve stretch factor and $C$ is the color excess.

Key discovery (1998): Distant SNe Ia are dimmer than expected in a decelerating universe, revealing cosmic acceleration!

Cepheid Variables

Period-luminosity relation:

$$M_V = -2.43(\log_{10} P - 1) - 4.05$$

where $P$ is the pulsation period in days.

3. Baryon Acoustic Oscillations (BAO)

Physical Origin

Sound waves in the pre-recombination plasma create a characteristic scale in the matter distribution. The sound horizon at drag epoch provides a "standard ruler":

$$r_d = \int_0^{z_d} \frac{c_s(z)}{H(z)} dz \approx 147 \text{ Mpc}$$

Two-Point Correlation Function

The galaxy correlation function $\xi(r)$ shows a bump at $r \approx r_d$:

$$\xi(r) = \frac{1}{2\pi^2}\int_0^{\infty} P(k) \frac{\sin(kr)}{kr} k^2 dk$$

where $P(k)$ is the matter power spectrum.

Cosmological Constraints

BAO measurements constrain the combination:

$$D_V(z) = \left[(1+z)^2 D_A^2(z) \frac{cz}{H(z)}\right]^{1/3}$$

This provides geometric constraints complementary to SNe Ia and CMB.

4. Hubble Constant and Tensions

Local Measurements

SH0ES collaboration (Cepheids + SNe Ia):

$$H_0 = 73.04 \pm 1.04 \text{ km s}^{-1}\text{ Mpc}^{-1}$$

CMB-based Inference

Planck 2018 (assuming ΛCDM):

$$H_0 = 67.4 \pm 0.5 \text{ km s}^{-1}\text{ Mpc}^{-1}$$

Hubble Tension

The $\sim 5\sigma$ discrepancy between early-universe and late-universe measurements presents a major challenge. Possible explanations:

  • • Systematic errors in distance ladder or CMB analysis
  • • New physics: early dark energy, modified gravity, extra relativistic species
  • • Late-time modifications: evolving dark energy $w(z)$

5. Gravitational Lensing

Weak Lensing

Cosmic shear from large-scale structure distorts background galaxy images. The convergence and shear are:

$$\kappa(\theta) = \int_0^{\chi_H} d\chi \, W(\chi) \delta(\chi\theta, \chi)$$

where the lensing weight function is:

$$W(\chi) = \frac{3H_0^2\Omega_m}{2c^2}\frac{\chi}{a(\chi)}\int_{\chi}^{\chi_H}d\chi' \, n(\chi')\frac{\chi' - \chi}{\chi'}$$

Strong Lensing

The lens equation relates source and image positions:

$$\vec{\beta} = \vec{\theta} - \vec{\alpha}(\vec{\theta})$$

where $\vec{\alpha}$ is the deflection angle. For a point mass:

$$\alpha = \frac{4GM}{c^2 b}$$

where $b$ is the impact parameter.

Time Delay Cosmography

Multiple images of lensed quasars arrive at different times. The time delay constrains $H_0$:

$$\Delta t = \frac{D_{\Delta t}}{c}[(1+z_L)\Delta\phi - \frac{1}{2}\Delta|\vec{\theta}|^2]$$

where $D_{\Delta t} = \frac{D_L D_S}{D_{LS}}$ is the time-delay distance.

6. Large-Scale Structure

Matter Power Spectrum

The power spectrum quantifies clustering as a function of scale:

$$P(k) = \langle|\delta_k|^2\rangle = A k^{n_s} T^2(k)$$

where $n_s \approx 0.96$ is the spectral index and $T(k)$ is the transfer function.

Growth of Structure

Linear growth is characterized by the growth function $D(a)$:

$$\delta(k, a) = D(a) \delta(k, a_i)$$

The growth rate parameter:

$$f(a) = \frac{d\ln D}{d\ln a} \approx \Omega_m(a)^{0.55}$$

Redshift-Space Distortions

Peculiar velocities distort clustering measurements in redshift space:

$$P^s(k, \mu) = \left(1 + \beta\mu^2\right)^2 P^r(k)$$

where $\beta = f/b$, with $b$ the galaxy bias.

7. Modern Galaxy Surveys

Major Surveys

  • SDSS (Sloan Digital Sky Survey): Mapped millions of galaxies, foundational for BAO
  • DES (Dark Energy Survey): Weak lensing, galaxy clusters, SNe Ia
  • DESI (Dark Energy Spectroscopic Instrument): 40 million galaxy redshifts
  • Euclid: Space-based weak lensing and galaxy clustering
  • LSST/Vera Rubin: 20 billion galaxies, time-domain astronomy

21cm Cosmology

Neutral hydrogen 21cm line traces matter at high redshift:

$$T_b(z) \approx 27 x_{HI}(1+\delta)\left(\frac{H(z)}{dv_{\parallel}/dr_{\parallel}}\right)\left(\frac{1+z}{10}\right)^{1/2} \text{ mK}$$

This probes the Dark Ages and Epoch of Reionization.

8. ΛCDM Best-Fit Parameters

Planck 2018 + BAO + Pantheon SNe (TT,TE,EE+lowE+lensing):

$\Omega_b h^2 = 0.02237 \pm 0.00015$
$\Omega_c h^2 = 0.1200 \pm 0.0012$
$H_0 = 67.4 \pm 0.5$ km/s/Mpc
$\tau = 0.054 \pm 0.007$
$n_s = 0.9649 \pm 0.0042$
$\ln(10^{10}A_s) = 3.044 \pm 0.014$

Derived Parameters

$\Omega_m = 0.315 \pm 0.007$
$\Omega_\Lambda = 0.685 \pm 0.007$
$\sigma_8 = 0.811 \pm 0.006$
$t_0 = 13.80 \pm 0.02$ Gyr

These parameters describe a spatially flat universe dominated by dark energy (68.5%), with dark matter (26.8%) and ordinary matter (4.9%). The universe is 13.8 billion years old.