Part IV: CMB Physics | Chapter 1

Cosmic Microwave Background (CMB) Physics

The relic radiation from the epoch of recombination: blackbody spectrum, temperature anisotropies, acoustic oscillations, polarization, and precision constraints on cosmological parameters

1. Introduction and Historical Context

Discovery of the CMB

In 1965, Arno Penzias and Robert Wilson at Bell Laboratories detected an excess antenna temperature of approximately 3.5 K at a wavelength of 7.35 cm using a horn reflector antenna originally built for satellite communication. This isotropic, unpolarized signal could not be attributed to any terrestrial or astronomical source. Simultaneously, Robert Dicke, Jim Peebles, Peter Roll, and David Wilkinson at Princeton had independently predicted that the hot Big Bang should have left behind a thermal radiation bath — the cosmic microwave background (CMB).

The theoretical prediction traces back further: George Gamow, Ralph Alpher, and Robert Herman in the late 1940s had estimated a present-day temperature of approximately 5 K from Big Bang nucleosynthesis calculations. Penzias and Wilson received the 1978 Nobel Prize in Physics for their discovery.

The CMB is the oldest electromagnetic signal in the universe. It was emitted at the epoch of recombination, when the universe was approximately 380,000 years old (redshift $z \approx 1100$), and the temperature had dropped to roughly 3000 K, allowing electrons and protons to combine into neutral hydrogen. Before recombination, Thomson scattering kept photons tightly coupled to the baryon-electron plasma. After recombination, photons streamed freely — the universe became transparent. The CMB we observe today is a snapshot of the last scattering surface.

Key CMB Missions

COBE (1989–1993)

The FIRAS instrument measured the CMB spectrum to extraordinary precision, confirming a perfect blackbody at $T_0 = 2.725 \pm 0.002$ K. The DMR instrument detected temperature anisotropies at the $\Delta T/T \sim 10^{-5}$ level. Mather and Smoot shared the 2006 Nobel Prize.

WMAP (2001–2010)

The Wilkinson Microwave Anisotropy Probe mapped the full sky with angular resolution of ~0.2° across five frequency bands (23–94 GHz), precisely measuring the first three acoustic peaks and constraining cosmological parameters to percent-level accuracy.

Planck (2009–2013)

ESA's Planck satellite achieved 5-arcminute resolution across nine frequency bands (30–857 GHz), measuring the angular power spectrum to $\ell \sim 2500$ and constraining the six-parameter $\Lambda$CDM model to sub-percent precision.

2. Blackbody Spectrum of the CMB

The CMB has the most perfect blackbody spectrum ever observed in nature. The spectral radiance (energy per unit area, per unit time, per unit frequency, per unit solid angle) is given by the Planck distribution:

Planck Blackbody Spectrum

$$B_\nu(\nu, T) = \frac{2h\nu^3}{c^2}\,\frac{1}{e^{h\nu/(k_B T)} - 1}$$

where $h$ is Planck's constant, $k_B$ is Boltzmann's constant, $\nu$ is frequency, and $T$ is temperature.

The COBE FIRAS instrument measured this spectrum over the range 60–600 GHz with deviations from a perfect blackbody smaller than 50 parts per million. The current best-fit temperature is:

$$T_0 = 2.7255 \pm 0.0006 \;\text{K}$$

(Fixsen 2009, from combined COBE FIRAS data)

From this temperature we can derive fundamental properties of the CMB photon field. The photon number density is obtained by integrating the Bose-Einstein distribution:

$$n_\gamma = \frac{2\zeta(3)}{\pi^2}\left(\frac{k_B T}{\ \hbar c}\right)^3 \approx 410.7 \;\text{cm}^{-3}$$

$$\rho_\gamma = \frac{\pi^2}{15}\frac{(k_B T)^4}{(\hbar c)^3 c^2} \approx 4.64 \times 10^{-34}\;\text{g/cm}^3$$

where $\zeta(3) \approx 1.202$ is the Riemann zeta function. The energy density parameter is $\Omega_\gamma h^2 \approx 2.47 \times 10^{-5}$.

The CMB temperature scales with redshift as $T(z) = T_0(1+z)$, a direct consequence of the adiabatic expansion of the universe. At recombination ($z \approx 1100$), the temperature was $T_{\rm rec} \approx 3000$ K.

3. Temperature Anisotropies

While the CMB is extraordinarily uniform, tiny temperature fluctuations of order$\Delta T / T \sim 10^{-5}$ encode a wealth of information about the early universe. These anisotropies are decomposed into a hierarchy of multipole moments.

3.1 Monopole, Dipole, and Higher Multipoles

Monopole ($\ell = 0$)

The sky-averaged temperature $\bar{T} = 2.7255$ K. This is the mean blackbody temperature, set by the photon-to-baryon ratio at the epoch of recombination.

Dipole ($\ell = 1$)

The largest anisotropy ($\Delta T \approx 3.36$ mK) is a dipole pattern caused by our peculiar motion relative to the CMB rest frame. The Solar System moves at$v \approx 370$ km/s toward the constellation Leo, producing a Doppler shift $\Delta T / T = v/c \approx 1.2 \times 10^{-3}$.

Higher Multipoles ($\ell \geq 2$)

After subtracting the monopole and dipole, the remaining fluctuations at the level of$\Delta T / T \sim 10^{-5}$ ($\sim 30\;\mu\text{K}$) arise from primordial density perturbations amplified by acoustic oscillations in the photon-baryon plasma. These are the cosmologically interesting anisotropies.

3.2 Spherical Harmonic Decomposition

The CMB temperature field on the sky is a function of angular position$(\theta, \varphi)$ and is expanded in spherical harmonics:

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

The coefficients $a_{\ell m}$ encode the amplitude and phase of each mode. For Gaussian random fields (predicted by inflation), the $a_{\ell m}$ are independent Gaussian random variables.

The multipole moment $\ell$ corresponds roughly to an angular scale$\theta \sim 180°/\ell$. The coefficients are extracted by:

$$a_{\ell m} = \int \frac{\Delta T}{T}(\theta,\varphi)\,Y_{\ell m}^*(\theta,\varphi)\,d\Omega$$

3.3 Angular Power Spectrum

Statistical isotropy of the universe implies that the two-point correlation of the$a_{\ell m}$ coefficients depends only on $\ell$, not on$m$. We define the angular power spectrum:

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

The second equality follows from statistical isotropy and is used as an estimator from the data. The variance of this estimator is the cosmic variance:$\text{Var}(C_\ell) = 2C_\ell^2/(2\ell+1)$.

The conventional plotting quantity is the dimensionless combination:

$$\mathcal{D}_\ell = \frac{\ell(\ell+1)}{2\pi}\,C_\ell$$

This quantity is approximately constant on large scales (Sachs-Wolfe plateau) and displays the characteristic acoustic peak structure for $\ell \gtrsim 100$.

4. Physics of CMB Anisotropies

The observed temperature anisotropy at a given direction $\hat{n}$ receives contributions from several physical effects evaluated at the last scattering surface (LSS) and along the photon path. The complete expression is:

Full Temperature Anisotropy

$$\frac{\Delta T}{T}(\hat{n}) = \left[\frac{1}{3}\Phi + \frac{1}{4}\delta_\gamma + \hat{n}\cdot\mathbf{v}_b\right]_{\text{LSS}} + \int_{\eta_{\rm rec}}^{\eta_0}\left(\dot{\Phi} + \dot{\Psi}\right)d\eta$$

where $\Phi$ and $\Psi$ are the Bardeen gravitational potentials,$\delta_\gamma$ is the photon density contrast, and$\mathbf{v}_b$ is the baryon velocity. Dots denote conformal time derivatives.

Let us examine each contribution in detail.

4.1 The Sachs-Wolfe Effect

On large angular scales ($\ell \lesssim 30$), the dominant contribution comes from the gravitational redshift experienced by photons climbing out of potential wells on the last scattering surface. Consider a photon emitted from an overdense region with gravitational potential $\Phi < 0$ (a potential well). The photon loses energy climbing out, producing a redshift. However, the photon also starts in a denser, hotter region. The net effect, for adiabatic perturbations in a matter-dominated universe, is:

$$\left(\frac{\Delta T}{T}\right)_{\text{SW}} = \frac{\Phi}{3}$$

The intrinsic temperature perturbation is $\delta_\gamma/4 = -2\Phi/3$(adiabatic condition), while the gravitational redshift contributes $+\Phi$. The net effect: $-2\Phi/3 + \Phi = +\Phi/3$. Overdense regions (deeper potential wells, $\Phi < 0$) appear colder.

The Sachs-Wolfe effect gives a nearly scale-invariant spectrum on large scales. For a Harrison-Zeldovich spectrum ($n_s = 1$), the primordial power spectrum$P_\Phi(k) \propto k^{n_s - 4}$ produces a flat plateau in$\mathcal{D}_\ell$:

$$\frac{\ell(\ell+1)}{2\pi}C_\ell^{\text{SW}} \approx \text{const} \approx \frac{1}{9}\frac{H_0^4}{(2\pi)^2}\frac{\Delta_{\mathcal{R}}^2}{k_0^{n_s-1}}\cdot \ell^{n_s - 1}$$

4.2 The Integrated Sachs-Wolfe (ISW) Effect

When gravitational potentials evolve in time, photons traversing them experience a net energy change. This occurs in two regimes:

Early ISW

Just after recombination, the universe transitions from radiation domination to matter domination. During this transition, potentials decay (they are constant in pure matter domination but decay in radiation domination). This enhances the first acoustic peak at $\ell \sim 200$.

Late ISW

At late times ($z \lesssim 1$), dark energy dominance causes potentials to decay. This contributes extra power at low $\ell$ and is a direct signature of dark energy. The effect is:

$$\left(\frac{\Delta T}{T}\right)_{\text{ISW}} = \int_{\eta_{\rm rec}}^{\eta_0}(\dot{\Phi} + \dot{\Psi})\,d\eta$$

4.3 Doppler Effect

The bulk velocity of the baryon-photon fluid at last scattering produces a Doppler shift. If the baryons at a given point on the LSS have velocity $\mathbf{v}_b$, the line-of-sight component contributes:

$$\left(\frac{\Delta T}{T}\right)_{\text{Doppler}} = \hat{n}\cdot\mathbf{v}_b$$

The Doppler contribution is 90° out of phase with the density perturbation in Fourier space. It fills in the troughs between acoustic peaks, producing the characteristic shape of the power spectrum.

4.4 Silk Damping (Photon Diffusion Damping)

On small angular scales ($\ell \gtrsim 1000$), anisotropies are exponentially suppressed by photon diffusion. Before recombination, photons undergo a random walk due to Thomson scattering. In a finite time, they diffuse over a characteristic comoving length$\lambda_D$, washing out perturbations on scales smaller than this damping scale:

$$\lambda_D^2 \approx \int_0^{t_{\rm rec}} \frac{c^2}{n_e \sigma_T a^2}\cdot\frac{1}{6}\left(\frac{R^2 + \frac{16}{15}(1+R)}{(1+R)^2}\right)dt$$

$$C_\ell \propto C_\ell^{\text{undamped}} \cdot e^{-(\ell/\ell_D)^2}$$

where $R = 3\rho_b/(4\rho_\gamma)$, $\sigma_T$ is the Thomson cross-section, and $\ell_D \approx 1400$ for the standard cosmological model. Joseph Silk first calculated this damping in 1968.

5. Acoustic Oscillations in the Photon-Baryon Fluid

The most striking feature of the CMB power spectrum is the series of acoustic peaks arising from sound waves in the photon-baryon plasma before recombination. These oscillations are the cosmological analogue of standing waves in a resonant cavity.

5.1 The Photon-Baryon Fluid

Before recombination, Thomson scattering tightly couples photons and baryons into a single fluid. In the tight-coupling approximation, the photon-baryon fluid has a well-defined sound speed:

Sound Speed of the Photon-Baryon Fluid

$$c_s = \frac{c}{\sqrt{3(1+R)}}, \qquad R \equiv \frac{3\rho_b}{4\rho_\gamma} = \frac{3\Omega_b}{4\Omega_\gamma}\,a$$

In the early universe ($R \ll 1$), $c_s \to c/\sqrt{3}$. As the baryon density increases, the sound speed decreases because the baryons add inertia to the fluid without contributing radiation pressure.

5.2 Wave Equation for Perturbations

In Fourier space, the photon temperature perturbation $\Theta_0 = \delta_\gamma/4$(the monopole of the photon distribution) satisfies, in the tight-coupling limit:

$$\ddot{\Theta}_0 + \frac{\dot{R}}{1+R}\dot{\Theta}_0 + k^2 c_s^2\,\Theta_0 = -\frac{k^2}{3}\Phi - \frac{\dot{R}}{1+R}\dot{\Phi} - \ddot{\Phi}$$

This is a forced, damped harmonic oscillator. The $\dot{R}/(1+R)$ term acts as a friction (the baryon loading increases with time). The right-hand side represents the gravitational driving force from the potential $\Phi$.

For constant $R$ and constant $\Phi$ (valid on sub-horizon scales in the matter-dominated era), the solution is:

$$\Theta_0(\eta) + \Phi = \left[\Theta_0(0) + (1+R)\Phi\right]\cos(k r_s) + \frac{1}{kc_s}\dot{\Theta}_0(0)\sin(k r_s) - R\Phi$$

where $r_s(\eta)$ is the sound horizon defined below. For adiabatic initial conditions from inflation, the cosine mode dominates.

5.3 The Sound Horizon

The comoving distance a sound wave can travel from the Big Bang to recombination defines the sound horizon:

$$r_s = \int_0^{t_{\rm rec}} \frac{c_s}{a}\,dt = \int_0^{a_{\rm rec}} \frac{c_s}{a^2 H(a)}\,da$$

For standard cosmological parameters, $r_s \approx 147$ Mpc (comoving). This is the fundamental ruler of CMB physics, also responsible for the baryon acoustic oscillation (BAO) feature in the galaxy power spectrum.

5.4 Acoustic Peak Positions

Standing waves reach maximum compression (or rarefaction) when$k r_s = n\pi$ for integer $n$. The multipole moment corresponding to wavenumber $k$ is $\ell \approx k\,d_A$, where$d_A$ is the comoving angular diameter distance to the last scattering surface. Therefore, the $n$-th acoustic peak is located at:

$$\ell_n \approx n\pi\,\frac{d_A}{r_s}$$

The first peak at $\ell_1 \approx 220$ corresponds to$\theta_s = r_s/d_A \approx 0.°82 \approx 1°$. This angular scale is a direct measure of the geometry of the universe: it confirms spatial flatness ($\Omega_k \approx 0$).

5.5 Odd vs. Even Peaks: Baryon Loading

The baryon loading parameter $R$ introduces an important asymmetry between odd and even peaks:

Odd peaks (n = 1, 3, 5, ...): These correspond to maximum compression of the photon-baryon fluid inside potential wells. The baryons add extra gravitational infall, enhancing the compression. Odd peaks are therefore boosted.

Even peaks (n = 2, 4, 6, ...): These correspond to maximum rarefaction (the fluid has bounced back out of the potential well). The baryon inertia opposes the rarefaction. Even peaks are therefore suppressedrelative to odd peaks.

The ratio of odd to even peak heights is directly sensitive to the baryon density: a higher $\Omega_b h^2$ increases the odd/even asymmetry. Planck measures$\Omega_b h^2 = 0.02237 \pm 0.00015$.

6. The Angular Power Spectrum: What Each Feature Tells Us

The CMB angular power spectrum is our most powerful probe of cosmological parameters. Each feature in the spectrum encodes specific physical information:

Low-$\ell$ Plateau ($\ell \lesssim 30$)

The Sachs-Wolfe plateau. Its amplitude measures the primordial power spectrum amplitude$A_s$. Deviations from flatness at $\ell \lesssim 10$ probe the late ISW effect, sensitive to $\Omega_\Lambda$. Cosmic variance fundamentally limits measurements here: we have only $2\ell+1$ modes per$\ell$.

Peak Positions

The ratio $d_A/r_s$ determines $\ell_1$. For a flat universe,$\ell_1 \approx 220$. A closed universe ($\Omega_k < 0$) shifts peaks to higher $\ell$ (smaller angles); an open universe shifts them to lower$\ell$. Current constraint: $\Omega_k = 0.001 \pm 0.002$.

Peak Heights

The relative heights of the first, second, and third peaks constrain the baryon density$\Omega_b h^2$ (odd/even ratio) and the total matter density$\Omega_m h^2$ (overall peak height suppression via the early ISW effect and the matter-radiation transition epoch).

Damping Tail ($\ell \gtrsim 1000$)

The exponential Silk damping at high $\ell$ constrains the damping scale$\ell_D$, which depends on the baryon density and the expansion rate at recombination. The slope of the damping tail also constrains the scalar spectral index$n_s$, with $n_s < 1$ indicating a slight red tilt as predicted by inflation.

Overall Amplitude and Optical Depth

Reionization at $z \sim 7\text{-}8$ partially rescatters CMB photons, suppressing power at $\ell \gtrsim 10$ by a factor$e^{-2\tau}$, where $\tau$ is the optical depth to reionization. Planck measures $\tau = 0.054 \pm 0.007$. The CMB constrains the combination$A_s e^{-2\tau}$ from the peak amplitudes, and $\tau$independently from large-scale polarization.

7. CMB Polarization

The CMB is linearly polarized at the ~5% level (relative to the temperature anisotropies) due to Thomson scattering of anisotropic radiation at the last scattering surface. Polarization measurements provide independent cosmological constraints and, crucially, offer a unique window into primordial gravitational waves from inflation.

7.1 Origin of CMB Polarization

Thomson scattering polarizes radiation when the incoming radiation has a quadrupolar anisotropy. Consider an electron at the last scattering surface: if the radiation field incident on it is hotter along one axis than the perpendicular axis (a quadrupole), the scattered radiation acquires a net linear polarization. Only the quadrupole moment ($\ell = 2$) of the local radiation field produces polarization — this is why the polarization signal is much smaller than the temperature anisotropy.

7.2 Stokes Parameters

The polarization state of electromagnetic radiation is described by the Stokes parameters$(I, Q, U, V)$. For the CMB (which is linearly polarized, so$V = 0$):

$I$: Total intensity (the temperature map).

$Q$: Difference in intensity between two orthogonal linear polarization directions (e.g., vertical minus horizontal).

$U$: Same as $Q$but rotated by 45°.

$Q$ and $U$ are spin-2 quantities: under a rotation of the coordinate system by angle $\psi$, they transform as$(Q \pm iU) \to e^{\mp 2i\psi}(Q \pm iU)$.

7.3 E-mode and B-mode Decomposition

The polarization field on the sky can be uniquely decomposed into two scalar fields with distinct parity properties:

E-modes (gradient pattern)

Even-parity polarization patterns that are aligned radially or tangentially around hot/cold spots. E-modes arise from scalar (density) perturbations and are correlated with temperature anisotropies. They were first detected by DASI in 2002.

The TE cross-correlation spectrum shows characteristic features that constrain the optical depth to reionization $\tau$ and the matter density.

B-modes (curl pattern)

Odd-parity polarization patterns with a handedness (they cannot be produced by scalar perturbations in linear theory). B-modes have two sources:

Primordial gravitational waves (tensor perturbations from inflation) — peak at $\ell \sim 80$
Gravitational lensing of E-modes by large-scale structure — peak at $\ell \sim 1000$

7.4 The Tensor-to-Scalar Ratio

The amplitude of primordial B-modes is parameterized by the tensor-to-scalar ratio:

$$r = \frac{\Delta_t^2}{\Delta_s^2} = \frac{P_t(k_0)}{P_s(k_0)}$$

where $\Delta_t^2$ and $\Delta_s^2$ are the tensor and scalar primordial power spectrum amplitudes at the pivot scale $k_0$.

Detection of primordial B-modes would be a “smoking gun” signature of cosmic inflation, directly probing the energy scale of inflation:

$$V_{\rm inf}^{1/4} \approx 1.06 \times 10^{16}\;\text{GeV}\left(\frac{r}{0.01}\right)^{1/4}$$

The current best upper limit from BICEP/Keck (2021) is $r < 0.036$ at 95% confidence. This already rules out several inflation models including$V(\phi) \propto \phi^2$.

8. Planck Mission Results: Precision Cosmology

The Planck satellite (2018 data release) provided the definitive measurement of the CMB temperature and polarization power spectra. The six-parameter $\Lambda$CDM model fits the data with remarkable precision over the range $2 \leq \ell \leq 2500$.

Planck 2018 Best-Fit $\Lambda$CDM Parameters

Parameter	Symbol	Best-Fit Value	Physical Meaning
Baryon density	$\Omega_b h^2$	$0.02237 \pm 0.00015$	Odd/even peak ratio
CDM density	$\Omega_c h^2$	$0.1200 \pm 0.0012$	Peak heights, matter-radiation equality
Hubble constant	$H_0$	$67.36 \pm 0.54\;\text{km/s/Mpc}$	Derived from peak positions and heights
Optical depth	$\tau$	$0.054 \pm 0.007$	Large-scale polarization (EE spectrum)
Scalar spectral index	$n_s$	$0.9649 \pm 0.0042$	Tilt of primordial spectrum (damping tail slope)
Amplitude	$\ln(10^{10}A_s)$	$3.044 \pm 0.014$	Overall normalization of power spectrum

From these six primary parameters, many derived quantities follow:

$\Omega_m$

0.315 &pm; 0.007

$\Omega_\Lambda$

0.685 &pm; 0.007

$\sigma_8$

0.811 &pm; 0.006

Age

13.797 &pm; 0.023 Gyr

$z_{\rm rec}$

1089.92 &pm; 0.25

$r_s$

144.43 &pm; 0.26 Mpc

The Hubble Tension

The Planck CMB value of $H_0 = 67.4 \pm 0.5$ km/s/Mpc is in significant tension (4–6$\sigma$) with local distance-ladder measurements, notably the SH0ES collaboration's value of $H_0 = 73.0 \pm 1.0$ km/s/Mpc based on Cepheid-calibrated Type Ia supernovae. This “Hubble tension” may point to new physics beyond $\Lambda$CDM, such as early dark energy, additional relativistic species, or modified recombination physics. It remains one of the most important open questions in cosmology.

9. Numerical Exploration: Simplified CMB Power Spectrum

The following Python code computes a simplified model of the CMB angular power spectrum using the photon-baryon fluid equations. It demonstrates the acoustic oscillation pattern including the effects of baryon loading, Silk damping, and the Sachs-Wolfe contribution.

Python

cmb_power_spectrum.py138 lines

#!/usr/bin/env python3
"""
cmb_power_spectrum.py
Compute a simplified CMB angular power spectrum showing the acoustic peak structure.
Uses the photon-baryon fluid equations in the tight-coupling approximation.
"""
import numpy as np
import matplotlib.pyplot as plt

# ============================================
# Cosmological parameters (Planck 2018 LCDM)
# ============================================
H0       = 67.36          # km/s/Mpc
h        = H0 / 100.0
Omega_b  = 0.02237 / h**2 # Baryon density parameter
Omega_c  = 0.1200 / h**2  # CDM density parameter
Omega_m  = Omega_b + Omega_c
Omega_r  = 9.15e-5        # Radiation (photons + neutrinos)
Omega_g  = 2.47e-5 / h**2 # Photon density only
T_cmb    = 2.7255         # CMB temperature [K]
n_s      = 0.9649         # Scalar spectral index
z_rec    = 1089.92        # Redshift of recombination
a_rec    = 1.0 / (1.0 + z_rec)

# Sound horizon and angular diameter distance (precomputed, Mpc)
r_s      = 144.43         # Sound horizon at recombination
d_A      = 12.8e3         # Comoving angular diameter distance to LSS

# ============================================
# Compute simplified C_ell
# ============================================
ell = np.arange(2, 2501)

# Baryon-to-photon ratio at recombination
R_rec = 3.0 * Omega_b / (4.0 * Omega_g) * a_rec  # ~ 0.6

# Sound speed
c_s = 1.0 / np.sqrt(3.0 * (1.0 + R_rec))

# Acoustic scale: k_A = pi / r_s
# ell_A = pi * d_A / r_s
ell_A = np.pi * d_A / r_s  # ~ 279

# Map ell to wavenumber k (Mpc^-1)
k = ell / d_A

# ============================================
# Photon-baryon oscillation solution
# (Adiabatic initial conditions, tight-coupling)
# Theta_0 + Phi = (1+3R)Phi/3 * cos(k*r_s) - R*Phi
# For scale-invariant Phi(k) ~ k^(ns-4)/2
# ============================================

# Primordial power spectrum (dimensionless)
k_pivot = 0.05  # Mpc^-1
Delta_R2 = 2.1e-9 * (k / k_pivot)**(n_s - 1)

# Effective gravitational potential on LSS (transfer function approximation)
# Simple approximation: Phi ~ const on large scales, suppressed inside horizon
k_eq = 0.01  # Matter-radiation equality scale ~ 0.01 Mpc^-1
T_k = 1.0 / (1.0 + (k / k_eq)**2)**0.5  # Simplified transfer function

# Sachs-Wolfe + acoustic oscillation solution
# SW: DeltaT/T = Phi/3 (large scales)
# Acoustic: DeltaT/T ~ (1+3R)*Phi/3 * cos(k*r_s) - R*Phi + Doppler
# Doppler ~ v_b . n_hat ~ Phi/(k*c_s) * sin(k*r_s)

Phi_k = np.sqrt(Delta_R2) * T_k

# Temperature transfer function (monopole + SW + Doppler)
cos_term = (1.0 + 3.0 * R_rec) / 3.0 * np.cos(k * r_s)
sin_term = np.sqrt(1.0 + R_rec) / np.sqrt(3.0) * np.sin(k * r_s)

# Effective temperature perturbation squared
Theta_sq = Phi_k**2 * (cos_term**2 + sin_term**2)

# Silk damping: exponential suppression
ell_D = 1380.0  # Silk damping multipole
damping = np.exp(-(ell / ell_D)**2)

# C_ell (proportional to)
C_ell = Theta_sq * damping

# D_ell = ell*(ell+1)*C_ell/(2*pi) — the conventional plotting quantity
D_ell = ell * (ell + 1) * C_ell / (2.0 * np.pi)

# Normalize to approximately match Planck (peak ~ 5800 muK^2)
D_ell_norm = D_ell / np.max(D_ell) * 5800.0

# ============================================
# Plot
# ============================================
fig, axes = plt.subplots(2, 1, figsize=(10, 10))

# Top: D_ell vs ell
ax1 = axes[0]
ax1.plot(ell, D_ell_norm, 'b-', lw=1.2, label='Simplified model')
ax1.set_xlabel(r'Multipole moment $\ell$', fontsize=13)
ax1.set_ylabel(r'$\mathcal{D}_\ell = \ell(\ell+1)C_\ell/2\pi\;[\mu K^2]$',
               fontsize=13)
ax1.set_title('CMB Temperature Angular Power Spectrum (Simplified Model)',
              fontsize=14)
ax1.set_xlim(2, 2500)
ax1.set_ylim(0, 7000)
ax1.axvline(x=220, color='r', ls='--', alpha=0.5, label=r'$\ell_1 \approx 220$')
ax1.axvline(x=540, color='orange', ls='--', alpha=0.5, label=r'$\ell_2 \approx 540$')
ax1.axvline(x=810, color='green', ls='--', alpha=0.5, label=r'$\ell_3 \approx 810$')
ax1.legend(fontsize=11)
ax1.grid(True, alpha=0.3)

# Bottom: Log scale
ax2 = axes[1]
ax2.semilogy(ell, D_ell_norm, 'b-', lw=1.2)
ax2.set_xlabel(r'Multipole moment $\ell$', fontsize=13)
ax2.set_ylabel(r'$\mathcal{D}_\ell\;[\mu K^2]$', fontsize=13)
ax2.set_title('CMB Power Spectrum (Log Scale) — Showing Damping Tail',
              fontsize=14)
ax2.set_xlim(2, 2500)
ax2.grid(True, alpha=0.3)

# Annotate regions
ax2.axvspan(2, 30, alpha=0.15, color='purple', label='Sachs-Wolfe plateau')
ax2.axvspan(100, 1000, alpha=0.1, color='cyan', label='Acoustic peaks')
ax2.axvspan(1200, 2500, alpha=0.1, color='red', label='Silk damping tail')
ax2.legend(fontsize=11, loc='lower left')

plt.tight_layout()
plt.savefig('cmb_power_spectrum.png', dpi=150, bbox_inches='tight')
plt.show()

print("=== Acoustic Peak Locations (approximate) ===")
print(f"  ell_A = pi * d_A / r_s = {ell_A:.0f}")
print(f"  Sound speed at recombination: c_s = {c_s:.4f} c")
print(f"  Baryon loading R_rec = {R_rec:.3f}")
print(f"  Silk damping scale ell_D ~ {ell_D:.0f}")
print(f"\nExpected peak positions (n * ell_A):")
for n in range(1, 8):
    print(f"  Peak {n}: ell ~ {n * ell_A:.0f}")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Code Output

The script generates a plot showing the characteristic acoustic peak structure:

The Sachs-Wolfe plateau at low $\ell$ from large-scale gravitational redshifts
The series of acoustic peaks with the first peak at $\ell \approx 220$
The odd/even peak asymmetry from baryon loading ($R \approx 0.6$ at recombination)
The exponential Silk damping suppressing power at $\ell \gtrsim 1400$

Note: This simplified model captures the qualitative features but differs from the full Boltzmann-solver output (e.g., CAMB or CLASS) which includes photon polarization, neutrino free-streaming, reionization, and the full radiation transfer hierarchy.

10. Summary of Key Results

Planck Blackbody Spectrum

$$B_\nu = \frac{2h\nu^3}{c^2}\,\frac{1}{e^{h\nu/k_B T} - 1}, \qquad T_0 = 2.7255\;\text{K}, \qquad n_\gamma = 410.7\;\text{cm}^{-3}$$

Spherical Harmonic Decomposition

$$\frac{\Delta T}{T}(\theta,\varphi) = \sum_{\ell m} a_{\ell m}\,Y_{\ell m}(\theta,\varphi), \qquad C_\ell = \langle|a_{\ell m}|^2\rangle$$

Sachs-Wolfe Effect

$$\left(\frac{\Delta T}{T}\right)_{\text{SW}} = \frac{\Phi}{3}$$

Photon-Baryon Sound Speed

$$c_s = \frac{c}{\sqrt{3(1+R)}}, \qquad R = \frac{3\rho_b}{4\rho_\gamma}$$

Sound Horizon at Recombination

$$r_s = \int_0^{t_{\rm rec}}\frac{c_s}{a}\,dt \approx 147\;\text{Mpc}$$

Acoustic Peak Location

$$\ell_n \approx n\pi\,\frac{d_A}{r_s}, \qquad \ell_1 \approx 220 \;\Rightarrow\; \theta_s \approx 1° \;\Rightarrow\; \text{flat universe}$$

Silk Damping

$$C_\ell \propto e^{-(\ell/\ell_D)^2}, \qquad \ell_D \approx 1400$$

Tensor-to-Scalar Ratio (Inflation Probe)

$$r = \frac{P_t}{P_s}, \qquad V_{\rm inf}^{1/4} \approx 1.06 \times 10^{16}\;\text{GeV}\left(\frac{r}{0.01}\right)^{1/4}, \qquad r < 0.036\;\text{(95\% CL)}$$

The CMB as a Cosmological Probe: Key Connections

1. Blackbody spectrum → confirms thermal equilibrium in the early universe

2. Temperature anisotropies $\sim 10^{-5}$ → seeds of all cosmic structure

3. First peak at $\ell \approx 220$ → spatial flatness ($\Omega_k \approx 0$)

4. Odd/even peak ratio → baryon density $\Omega_b h^2$

5. Peak heights → matter density $\Omega_m h^2$

6. Damping tail slope → scalar spectral index $n_s$ (inflation constraint)

7. Large-scale polarization → optical depth $\tau$ (reionization epoch)

8. B-mode polarization → gravitational waves from inflation (energy scale of inflation)

9. Low-$\ell$ ISW → dark energy ($\Omega_\Lambda$)

10. Six parameters determine everything → extraordinary predictive power of $\Lambda$CDM

Bibliography

Textbooks & Monographs

Dodelson, S. & Schmidt, F. (2020). Modern Cosmology, 2nd ed. Academic Press. — Excellent derivation of the CMB power spectrum, Boltzmann hierarchy, and parameter estimation.
Weinberg, S. (2008). Cosmology. Oxford University Press. — Rigorous treatment of CMB anisotropies, polarization, and acoustic oscillations.
Durrer, R. (2020). The Cosmic Microwave Background, 2nd ed. Cambridge University Press. — Dedicated comprehensive monograph on all aspects of CMB physics.
Mukhanov, V. (2005). Physical Foundations of Cosmology. Cambridge University Press. — Detailed treatment of perturbation theory and CMB anisotropy generation.
Ryden, B. (2017). Introduction to Cosmology, 2nd ed. Cambridge University Press. — Accessible introduction to the CMB and its role in precision cosmology.

Key Papers

Penzias, A.A. & Wilson, R.W. (1965). “A Measurement of Excess Antenna Temperature at 4080 Mc/s,” Astrophysical Journal 142, 419–421. — Discovery of the CMB radiation (Nobel Prize 1978).
Mather, J.C. et al. (1994). “Measurement of the cosmic microwave background spectrum by the COBE FIRAS instrument,” Astrophysical Journal 420, 439–444. — Definitive measurement of the CMB blackbody spectrum (Nobel Prize 2006).
Smoot, G.F. et al. (1992). “Structure in the COBE differential microwave radiometer first-year maps,” Astrophysical Journal 396, L1–L5. — First detection of CMB temperature anisotropies by COBE/DMR (Nobel Prize 2006).
Sachs, R.K. & Wolfe, A.M. (1967). “Perturbations of a Cosmological Model and Angular Variations of the Microwave Background,” Astrophysical Journal 147, 73–90. — The Sachs-Wolfe effect relating density perturbations to CMB anisotropies.
Silk, J. (1968). “Cosmic Black-Body Radiation and Galaxy Formation,” Astrophysical Journal 151, 459–471. — Prediction of photon diffusion damping (Silk damping) of small-scale perturbations.
Sunyaev, R.A. & Zel’dovich, Ya.B. (1970). “Small-Scale Fluctuations of Relic Radiation,” Astrophysics and Space Science 7, 3–19. — Theory of acoustic oscillations and their imprint on the CMB.
Hu, W. & Dodelson, S. (2002). “Cosmic Microwave Background Anisotropies,” Annual Review of Astronomy and Astrophysics 40, 171–216. arXiv:astro-ph/0110414. — Authoritative review of CMB physics and parameter extraction.
Planck Collaboration (2020). “Planck 2018 results. I. Overview,” Astronomy & Astrophysics 641, A1. arXiv:1807.06205; “VI. Cosmological parameters,” A6. arXiv:1807.06209. — Final Planck mission results establishing six-parameter concordance cosmology.
Zaldarriaga, M. & Seljak, U. (1997). “All-sky analysis of polarization in the microwave background,” Physical Review D 55, 1830–1840. — Formalism for E-mode and B-mode CMB polarization.

← Cosmic Inflation Structure Formation →

Parameter	Symbol	Best-Fit Value	Physical Meaning
Baryon density	\(\Omega_b h^2\)	\(0.02237 \pm 0.00015\)	Odd/even peak ratio
CDM density	\(\Omega_c h^2\)	\(0.1200 \pm 0.0012\)	Peak heights, matter-radiation equality
Hubble constant	\(H_0\)	\(67.36 \pm 0.54\;\text{km/s/Mpc}\)	Derived from peak positions and heights
Optical depth	\(\tau\)	\(0.054 \pm 0.007\)	Large-scale polarization (EE spectrum)
Scalar spectral index	\(n_s\)	\(0.9649 \pm 0.0042\)	Tilt of primordial spectrum (damping tail slope)
Amplitude	\(\ln(10^{10}A_s)\)	\(3.044 \pm 0.014\)	Overall normalization of power spectrum

Cosmology

Cosmic Microwave Background (CMB) Physics

1. Introduction and Historical Context

Discovery of the CMB

Key CMB Missions

COBE (1989–1993)

WMAP (2001–2010)

Planck (2009–2013)

2. Blackbody Spectrum of the CMB

Planck Blackbody Spectrum

3. Temperature Anisotropies

3.1 Monopole, Dipole, and Higher Multipoles

Monopole (\(\ell = 0\))

Dipole (\(\ell = 1\))

Higher Multipoles (\(\ell \geq 2\))

3.2 Spherical Harmonic Decomposition

3.3 Angular Power Spectrum

4. Physics of CMB Anisotropies

Full Temperature Anisotropy

4.1 The Sachs-Wolfe Effect

4.2 The Integrated Sachs-Wolfe (ISW) Effect

Early ISW

Late ISW

4.3 Doppler Effect

4.4 Silk Damping (Photon Diffusion Damping)

5. Acoustic Oscillations in the Photon-Baryon Fluid

5.1 The Photon-Baryon Fluid

Sound Speed of the Photon-Baryon Fluid

5.2 Wave Equation for Perturbations

5.3 The Sound Horizon

5.4 Acoustic Peak Positions

5.5 Odd vs. Even Peaks: Baryon Loading

6. The Angular Power Spectrum: What Each Feature Tells Us

Low-\(\ell\) Plateau (\(\ell \lesssim 30\))

Peak Positions

Peak Heights

Damping Tail (\(\ell \gtrsim 1000\))

Overall Amplitude and Optical Depth

7. CMB Polarization

7.1 Origin of CMB Polarization

7.2 Stokes Parameters

7.3 E-mode and B-mode Decomposition

E-modes (gradient pattern)

B-modes (curl pattern)

7.4 The Tensor-to-Scalar Ratio

8. Planck Mission Results: Precision Cosmology

Planck 2018 Best-Fit \(\Lambda\)CDM Parameters

The Hubble Tension

9. Numerical Exploration: Simplified CMB Power Spectrum

Code Output

10. Summary of Key Results

Planck Blackbody Spectrum

Spherical Harmonic Decomposition

Sachs-Wolfe Effect

Photon-Baryon Sound Speed

Sound Horizon at Recombination

Acoustic Peak Location

Silk Damping

Tensor-to-Scalar Ratio (Inflation Probe)

The CMB as a Cosmological Probe: Key Connections

Bibliography