5.2 Temperature anisotropies and their significance

Cosmic Microwave Background (CMB) Temperature Anisotropies

The cosmic microwave background (CMB) carries a snapshot of the universe roughly 380,000 years after the Big Bang. Its tiny temperature variations, just 1 part in 100,000 relative to the mean temperature of 2.725 K, trace quantum fluctuations from the inflationary period that eventually grew into all the large-scale structure we see today.

The CMB power spectrum organizes these fluctuations by angular scale. Its peaks and troughs encode the universe's composition, geometry, and evolution, making it one of the most powerful tools for measuring cosmological parameters and testing models of the early universe.

Temperature anisotropies in CMB radiation

Temperature anisotropies are the minuscule variations in CMB temperature observed across the sky. Their magnitude is roughly $\frac{\Delta T}{T} \sim 10^{-5}$, or about 1 part in 100,000 compared to the mean temperature of 2.725 K.

These anisotropies span a wide range of angular scales:

• Large angular scales represent fluctuations stretched across vast distances in the primordial universe. The most prominent large-scale feature is the dipole anisotropy (180-degree angular scale), caused by Earth's motion relative to the CMB rest frame. This dipole is subtracted before analyzing the cosmologically interesting fluctuations.
• Small angular scales correspond to more compact regions in the early universe. The peaks in the CMB power spectrum at angular scales of around 1 degree and smaller arise from acoustic oscillations in the photon-baryon plasma before recombination.

Evidence for quantum fluctuations

The prevailing theory traces CMB temperature anisotropies back to quantum fluctuations during the inflationary period. Inflation was a brief phase of exponential expansion that stretched microscopic quantum fluctuations to macroscopic, cosmological scales.

Specifically, quantum fluctuations in the inflaton field (the scalar field driving inflation) produced slight inhomogeneities in the density and temperature of the primordial plasma. Overdense regions had stronger gravitational pull, attracting more matter over time. This gravitational collapse ultimately formed the galaxies, galaxy clusters, and cosmic web we observe today.

The statistical properties of the CMB anisotropies match inflationary predictions in several ways:

• The fluctuations are nearly Gaussian, as inflation predicts
• The power spectrum is nearly scale-invariant, with a slight tilt (spectral index $n_s \approx 0.965$), consistent with slow-roll inflation
• The amplitude and shape of the angular power spectrum provide strong evidence that cosmic structure originated from quantum processes

Significance of the CMB power spectrum

The CMB power spectrum plots the variance (power) of temperature fluctuations as a function of angular scale, expressed through the multipole moment $\ell$. Low $\ell$ values correspond to large angular scales; high $\ell$ values correspond to small angular scales.

The position, amplitude, and width of the peaks are sensitive to key cosmological parameters:

• $\Omega_b$ (baryon density): affects the relative heights of odd versus even peaks
• $\Omega_m$ (total matter density): influences the overall peak structure and the matter-radiation equality epoch
• $\Omega_\Lambda$ (dark energy density): affects the geometry of the universe and thus the angular scale of the peaks
• $H_0$ (Hubble constant): shifts peak positions through its effect on the sound horizon

By comparing the measured power spectrum against theoretical predictions from different cosmological models, researchers can constrain these parameter values and evaluate competing theories of the early universe.

Two results from the power spectrum stand out:

• The first peak at $\ell \approx 200$ (about 1 degree on the sky) indicates a spatially flat universe. In a positively curved universe, this peak would shift to lower $\ell$ (larger angular scales); in a negatively curved universe, to higher $\ell$.
• The relative amplitudes of odd and even peaks reveal the baryon-to-dark-matter ratio. Odd peaks are enhanced by baryons (which compress with gravity), while even peaks correspond to rarefaction phases. The pattern requires a substantial dark matter component that provides gravitational potential wells without participating in the pressure oscillations.

Effects shaping CMB anisotropies

Three main physical effects shape the anisotropy pattern at different angular scales:

• Sachs-Wolfe effect (large scales, $\ell \lesssim 100$): Photons climbing out of gravitational potential wells at the last scattering surface lose energy, producing a gravitational redshift. Photons from deeper wells appear cooler. On these large scales, the fluctuations directly reflect the primordial density perturbations because there wasn't enough time for acoustic oscillations to develop at such large wavelengths.
• Acoustic oscillations / Doppler effect (intermediate scales, $100 \lesssim \ell \lesssim 1000$): Before recombination, the photon-baryon plasma underwent sound waves driven by the competition between gravitational compression and radiation pressure. The Doppler shifts from the plasma's bulk motion, combined with the intrinsic temperature variations from compression and rarefaction, produce the characteristic series of acoustic peaks in the power spectrum.
• Silk damping (small scales, $\ell \gtrsim 1000$): On small scales, photons diffuse out of overdense regions before recombination, smoothing out temperature fluctuations. This photon diffusion erases anisotropies on scales smaller than the photon mean free path, creating a damping tail in the power spectrum. The shape of this tail provides information about the thickness of the last scattering surface and the primordial helium abundance (since helium affects the number of free electrons and thus the photon mean free path).

Implications of CMB observations

The observed CMB anisotropies carry deep implications for our understanding of the universe's geometry and large-scale structure.

The remarkable uniformity of the CMB temperature, with fluctuations at only 1 part in 100,000, provides strong evidence for the homogeneity and isotropy of the universe on large scales. This supports the cosmological principle: the universe looks statistically the same from every location and in every direction on sufficiently large scales.

The first acoustic peak at $\ell \approx 200$ is consistent with a spatially flat universe. Quantitatively, the total density parameter $\Omega_\text{total} = \Omega_m + \Omega_\Lambda + \Omega_r \approx 1$, where $\Omega_r$ is the radiation density. A value of exactly 1 corresponds to flat spatial geometry. This flatness is itself a prediction of inflation: the exponential expansion during inflation drove the universe's curvature extremely close to zero, regardless of its initial value.

Together, CMB observations have established the $\Lambda$CDM model as the standard cosmological model, pinning down parameters like the age of the universe, its matter-energy composition, and the geometry of space with percent-level precision.