Chapter 10 Recombination and CMB

• Early universe - high electron density - Thomson scattering

  $$\sigma_{\mathrm{T}}=\frac{1}{6 \pi \epsilon_{0}^{2}}\left(\frac{\mathrm{e}^{2}}{m_{\mathrm{e}} c^{2}}\right)^{2}=6.6 \times 10^{-25}\ \mathrm{cm}^{2}$$

  Mean free path

  $$\lambda=\frac{1}{n_{\mathrm{e}} \sigma_{\mathrm{T}}}$$

  The rate at which a photon undergoes scattering

  $$\Gamma_{\mathrm{T}, \mathrm{e}}=\frac{c}{\lambda}=n_{\mathrm{e}} \sigma_{\mathrm{T}} c$$

  Optical depth

  $$\tau=\int \Gamma_{\mathrm{T}, \mathrm{e}}(t) d t$$
• Assuming the universe is totally ionized today that $n{\mathrm{e}} \simeq n{\mathrm{b}}=n_{\mathrm{b}, 0}(1+z)^{3}$

  $$\Gamma_{\mathrm{T}, \mathrm{e}} \simeq5 \times 10^{-21}(1+z)^{3} \mathrm{s}^{-1}$$

Matter Domination

• Occurred at $z{\text{eq}}=3380$, $T{\mathrm{eq}}=2.7255 \times 3381=9215 \mathrm{K}$

• The expansion is then be driven by pressure less matter until either the curvature or the dark energy donimates

Recombination

• Cooling universe - ions (protons and $\ce{He^2+}$) and electrons combine and form neutral atoms

• $n\rm{e}$ decreases rapidly - $\Gamma{\mathrm{T}, \mathrm{e}}$ drops below the expansion rate $H$ - the photons decouple from the electrons and can stream freely

• The temperature at which recombination takes place

  • Baryon-to-photon ratio $\eta$

  • Ionization potential of the species involved ($\ce{H}$ : $Q=13.6\text{ eV}$, neglect $\ce{He+}$ and $\ce{He^2+}$)

    \ce{H + \gamma <-> p + e-}
  • The distribution of particles with different masses (MB distribution)

    $$n_{\mathrm{x}}=g_{\mathrm{x}}\left(\frac{m_{\mathrm{x}} k T}{2 \pi \hbar^{2}}\right)^{3 / 2} \exp \left[-\frac{m_{\mathrm{x}} c^{2}}{k T}\right]$$

    Then for $\ce{H}$ atoms, protons and free electrons

    $$\frac{n_{\mathrm{H}}}{n_{\mathrm{p}} n_{\mathrm{e}}}=\frac{g_{\mathrm{H}}}{g_{\mathrm{p}} g_{\mathrm{e}}}\left(\frac{m_{\mathrm{H}}}{m_{\mathrm{p}} m_{\mathrm{e}}}\right)^{3 / 2}\left(\frac{k T}{2 \pi \hbar^{2}}\right)^{-3 / 2} \exp \left[\frac{\left(m_{\mathrm{p}}+m_{\mathrm{e}}-m_{\mathrm{H}}\right) c^{2}}{k T}\right]$$

    • The ratio of statistical weighs is 1

    • $m{\ce H}\approx m{\ce{p}}$

    • $Q=\left(m{\mathrm{p}}+m{\mathrm{e}}-m_{\mathrm{H}}\right) c^{2}$

  • Saha equation

    $$\frac{n_{\mathrm{H}}}{n_{\mathrm{p}} n_{\mathrm{e}}}=\left(\frac{m_{\mathrm{e}} k T}{2 \pi \hbar^{2}}\right)^{-3 / 2} \exp \left[\frac{Q}{k T}\right]$$

    Ionization fraction

    $$X \equiv \frac{n_{\mathrm{p}}}{n_{\mathrm{p}}+n_{\mathrm{H}}}=\frac{n_{\mathrm{p}}}{n_{\mathrm{b}}}=\frac{n_{\mathrm{e}}}{n_{\mathrm{b}}}\Rightarrow \frac{1-X}{X}=n_{\mathrm{p}}\left(\frac{m_{\mathrm{e}} k T}{2 \pi \hbar^{2}}\right)^{-3 / 2} \exp \left[\frac{Q}{k T}\right]$$

    \eta=\frac{n_\rm{b}}{n_\gamma}=\frac{n_\rm{p}}{Xn_\gamma}

    For a blackbody spectrum

    $$n_{\gamma}=\frac{2.404}{\pi^{2}}\left(\frac{k T}{\hbar c}\right)^{3}=0.244\left(\frac{k T}{\hbar c}\right)^{3}\Rightarrow n_{\mathrm{p}}=0.244 X \eta\left(\frac{k T}{\hbar c}\right)^{3}$$

    Solving the equation of $X$

    $$X=\frac{-1+\sqrt{1+4 S}}{2 S}$$

    where

    $$S(T, \eta)=3.84 \eta\left(\frac{k T}{m_{\mathrm{e}} c^{2}}\right)^{3 / 2} \exp \left[\frac{Q}{k T}\right]$$
  • When $kT\gg Q$, $X\sim1$

  • Once $kT<Q$, $X\to0$ - difficult to achieve for $\eta$ and $\left(k T / m_{\mathrm{e}} c^{2}\right)^{3 / 2}$ are very small

  • When $X=0.5$ and for $\eta=6.1\times10^{-10}$

    $$k T_{\mathrm{rec}}=0.323 \mathrm{eV}=\frac{Q}{42}$$

    $$T_{\mathrm{rec}}=0.323 \mathrm{eV} \equiv 3750 \mathrm{K},\ \left(1+z_{\mathrm{rec}}\right)=1375,\ t_{\mathrm{rec}}=251000\ \mathrm{yr}$$

Photon Decoupling

• Recombination was not an instantaneous process - $X=0.9\to X=0.1$ takes $\sim 70000\text{ yr}$

• The time when photons and baryons decoupled follows soon

• Overionization ($X$ is larger than what the Saha equation predicts) - the photons emitted in recombination are so easily absorbed by other $\ce{H}$ atoms

• Two-photon emission

  • Highly forbidden

  • Energy of photons too low to excite an atom from the ground state

  • $z_\text{dec}=1090$

• CMB

  • Last scattering layer

  • Strong evidence for the Big Bang

CMB

• $\langle h \nu\rangle= 6.3 \times 10^{-4} \mathrm{eV}$, $\sim$ vibrational and rotational levels of molecules

• Solution to Olbers' paradox - the sky at night is bright everywhere, but at the milimeter wavelength

Isotropy

• Closest approximation to an ideal blackbody

  $$\langle T\rangle=\frac{1}{4 \pi} \int T(\theta, \phi) \sin \theta d \theta d \phi=2.7255 \pm 0.0006 \mathrm{K}$$
• Highly isotropic

  $$\frac{\delta T}{T}(\theta, \phi)=\frac{T(\theta, \phi)-\langle T\rangle}{\langle T\rangle}\Rightarrow \left\langle\left(\frac{\delta T}{T}\right)^{2}\right\rangle^{1 / 2}=1.1 \times 10^{-5}$$

Horizon Problem

• Comoving horizon

  $$s_{\text {hor}, \operatorname{com}(t)}=\int_{0}^{t} \frac{c \mathrm{d} t}{a(t)}=\int_{0}^{a} \frac{c \mathrm{d} a}{a^{2} H(a)}$$
• Early in the matter-dominated era

  $$H(a) \simeq H_{0} \sqrt{\Omega_{\mathrm{m}, 0}} a^{-3 / 2}\Rightarrow S_{\text {hor,}\text {com}}(a) \simeq \frac{c}{H_{0}} \Omega_{\mathrm{m}, 0}^{-1 / 2} \int_{0}^{a} \frac{1}{a^{1 / 2}} \mathrm{d} a$$

  $$s_{\mathrm{hor}, \operatorname{com}}(z) \simeq 2 \frac{c}{H_{0}} \Omega_{\mathrm{m}, 0}^{-1 / 2}(1+z)^{-1 / 2}$$

  Then the proper horizon distance at decoupling

  $$S_{\text {hor,prop}}\left(z_{\text {dec}}\right)= aS_{\text {hor,com}}\left(z_{\text {dec}}\right)\simeq 2 \frac{c}{H_{0}} \Omega_{\mathrm{m}, 0}^{-1 / 2}\left(1+z_{\mathrm{dec}}\right)^{-3 / 2}$$
• The angle on the sky subtended by the proper horizon

  $$\theta_{\text {hor,dec}}=\frac{s_{\text {hor}, \text {prop}}\left(z_{\text {dec}}\right)}{d_{\mathrm{A}}\left(z_{\text {dec}}\right)}$$

  where $d_\rm{A}$ is the angular diameter distance

  $$d_{\mathrm{A}}(z)=\frac{c}{H_{0}} \frac{1}{(1+z)} \int_{0}^{z} \frac{d z}{\left[\Omega_{\mathrm{m}, 0}(1+z)^{3}+\Omega_{\Lambda, 0}\right]^{1 / 2}}$$

  In an open universe with no dark energy, the so-called Mattig relation applies

  $$d_{\mathrm{A}}(z)=2 \frac{c}{H_{0}} \frac{1}{\Omega_{\mathrm{m}, 0}^{2}(1+z)^{2}} \left[\Omega_{\mathrm{m}, 0} z+\left(\Omega_{\mathrm{m}, 0}-2\right)\left(\sqrt{1+\Omega_{\mathrm{m}, 0} z}-1\right)\right]\approx2 \frac{c}{H_{0}} \frac{1}{\Omega_{\mathrm{m}, 0} z}$$

  for $z\gg1$, then

  $$\theta_{\mathrm{hor}, \mathrm{dec}} \approx\left(\frac{\Omega_{\mathrm{m}, 0}}{z_{\mathrm{dec}}}\right)^{1 / 2}=\left(\frac{0.312}{1090}\right)^{1 / 2}=0.017\ \mathrm{radians} \sim 1^{\circ}$$
• Under the cosmic model in consensus, $\theta_{\mathrm{hoor}, \mathrm{dec}} \approx 1.8^{\circ}$

• CMB photons coming to us from two directions separated by more than $\sim 2^\circ$ originated from regions which were not in causal contact at $z_\rm{dec}$

• Inflation

• After decoupling, the photo-baryon fluid cecame a pair of gases

  • Baryons - free gravitational collapse

  • Gravity turned the tiny temperature fluctuations into large scale structure

  • The anisotropies in the temperature of the CMB radiation encode a host of cosmological parameters

Statistical Description of the Fluctuations

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

• The correlation function

  $$C(\theta)=\left\langle\frac{\delta T}{T}(\mathbf{r}) \frac{\delta T}{T}\left(\mathbf{r}^{\prime}\right)\right\rangle_{\mathbf{r} \cdot \mathbf{r}^{\prime}=\cos \theta}=\frac{1}{4 \pi} \sum_{\ell=0}^{\infty}(2 \ell+1) C_{\ell} P_{\ell}(\cos \theta)$$

  • $C_\ell$ is a measure of $\delta T/T$ on the angular scale $\theta\sim180^\circ/\ell$

  • $\ell=0$ (monopole) - vanish

  • $\ell=1$ (dipole) - the motion of the Earth through space

  • $\ell\ge2$ - the fluctuations present at the time of last scattering

• Power spectrum

  $$\Delta_{\mathrm{T}}^{2} \equiv \frac{\ell(\ell+1)}{2 \pi} C_{\ell}\langle T\rangle^{2}$$

Dipole

• Earth's motion relative to the local comoving frame of reference

  $$T(\theta) \approx\langle T\rangle\left(1+\frac{v}{c} \cos \theta\right)$$

Higher Multipoles

photon-baryon fluid acoustic oscillations

• $\theta\sim1^\circ$ - Sound horizon

• Baryon-photon fluid - relativistic

  • As gravity tries to compress the fluid, radiation pressure resists

    • Sound is a travelling change of pressure

    • Stops oscillating at decoupling - the pattern of maxima and minima in the density is frozen - temperature fluctuation

    • Maximum compression - highest density - hottest

  • Modes caught at extrema of their oscillations - the peaks in the CMB power spectrum - Acoustic peaks or Doppler peaks

    • First peak - compressed once inside potential wells before recombination

    • Second peak - compressed and then rarefied

    • Third peak - compressed then rarefied then compressed

    • ...

• The angular scales and amplitudes of the acoustic peaks are the main route to determining the cosmological parameters encoded in the temperature anisotropy of the CMB

The First Doppler Peak: a Measure of the Curvature of the Universe

• EoS

  $$p=w \rho c^{2}$$

  Sound speed

  $$c_{\mathrm{s}}^{2}=\frac{d p}{d \rho}=\omega c^2$$

  For radiation, $\omega=1/3\Rightarrow c_\mathrm{s}=c/\sqrt{3}$

• Sound horizon

  $$s_{\mathrm{hor}, \mathrm{s}} \simeq \frac{2}{\sqrt{3}} \frac{c}{H_{0}} \Omega_{\mathrm{m}, 0}^{-1 / 2}\left(1+z_{\mathrm{dec}}\right)^{-3 / 2}$$

  For $z\gg1$ and $\Omega_\Lambda=0$

  $$\theta_{\mathrm{hor}, \mathrm{s}} \simeq \frac{1}{\sqrt{3}}\left[\frac{\left(1-\Omega_{\mathrm{k}, 0}\right)}{z_{\mathrm{dec}}}\right]^{1 / 2}$$
• Larger $\Omega{\text{k},0}$ gives smaller $\theta{\mathrm{hor}, \mathrm{s}}$, thus the first acoustic peak moves to larger $\ell$ values

• In a $\Omega{\mathrm{k}, 0}=0, \Omega{\mathrm{m}, 0}+\Omega_{\Lambda, 0}=1$ cosmology, we expect the first peak is located at

  $$\theta_{\mathrm{hor}, s} \approx \frac{1.8^{\circ}}{\sqrt{3}} \simeq 1^{\circ}$$

Higher Doppler Peaks

• Inside the sound horizon at decoupling - subject to physical effects acting on baryon-photon fluid

Baryon Loading: a Measure of $\Omega_{\mathrm{b}, 0}$

• Baryon loading - take the gravitational and inertial mass of the baryons into account

  • Higher amplitude of compression peaks (odd peaks) than rarefaction peaks (even peaks)

    • The fluid compress further inside before rarefactions

  • Decrease the frequency of oscillations

    • Slowed down by baryons - the wavelength decays as the velocity stays a constant

    • All the peaks get slightly higher $\ell$ (smaller $\theta$)

The Damping Tail

• As we approach the epoch of decoupling, the coupling between baryons and photons is not perfect

  • Anisotropies removed by diffusion of photons (with higher mean free path)

  • The acoustic oscillations are exponentially damped on scales smaller than the distance photons random walk

• The shape of the damping tail

  • Increasing the baryon density

    • The photon-baryon fluid more tightly coupled at recombination

    • The mean free path of the photons is shorter

    • The damping tail shifts to smaller angular scales

  • Total matter density

    • The age of the Universe at $z_\text{rec}$

    • Angular diameter distance $d_\rm{A}$

    • Both are smaller for larger matter density - more damping at a fixed multiple moment

Super-horizon Scales

• The principal source of temperature fluctuations are the intrinsic inhomogeneities in the distribution of matter

Sachs-Wolfe Effect

• Variations in the gravitational potential - temperature fluctuation

  $$\left(\frac{\delta T}{T}\right)_{\mathrm{S}-\mathrm{W}}=\frac{1}{3} \frac{\delta \Phi}{c^{2}}$$
• Two competing effects

  • Photons climbing out of potential well experience a gravitational redshift, and lose energy in the process

    • The potential wells appear slightly colder than the mean in the CMB map

  • Photons scattered from regions of higher density than average and received today were scattered at slightly earlier times, when the CMB temperature was slightly higher - what we see is earlier CMB

    • The potential wells appear slightly hotter than the mean in the CMB map

• No scale dependence

  • A constant $\Delta_{\mathrm{T}}^{2}$ in power spectrum

• Cosmic variance

  • Only $2\ell+1$ independent sampling can be made of our CMB sky

  • Limit of precision

    $$\left(\frac{\Delta C_{\ell}}{C_{\ell}}\right)^{2}=\frac{2}{2 \ell+1}$$

Peculiar Velocities

• Density fluctuations are always related to peculiar velocities of matter

• Photons last scattered by gas receding from us with a speed slightly larger than the average Hubble expansion will experience an additional redshift which reduces the temperature measured in that direction

  $$\left(\frac{\delta T}{T}\right)_{\mathrm{v}, \mathrm{pec}}=\frac{1}{3} \frac{\delta \Phi}{c^{2}} \frac{\theta_{\mathrm{hor}, \mathrm{rec}}}{\theta}\Rightarrow \Delta_{\mathrm{T}}^{2} \propto \theta^{-1}$$

Secondary Fluctuations

Thomson Scattering

• Free electrons in IGM, following the so-called epoch of reionization

• The fluctuation amplitude decreases by a factor of $e^{-\tau}$ - help deduce $z_\text{reion}\approx8.5\pm1.3$

Gravitational Lensing

• The gravitational field of the cosmic density fluctuations leads to changes in the photon direction

• The correlation function of the temperature fluctuations is slightly smeared out on small angular scales

Integrated Sachs-Wolfe effect

• The gravitational potential of the large-scale structure changes over timescales comparable to the travel time of CMB photons through the structures

  • Important over the largest angular scales and when dark energy dominates the expansion

• The blueshift that a CMB photon undergoes as is travels down a potential well is more significant than the corresponding redshift as it climbs out

The Sunyaev-Zel’dovich (S-Z) Effect

• Inverse Compton scattering of CMB photons by electrons in the intracluster gas (ICM) of massive galaxy clusters

• The isotropy of the cosmic background ensures that, on average, the total number of CMB photons reaching us is unchanged

• Frequency distribution

  • Raleigh-Jeans part ($\lambda\ge1\text{ mm}$) - removed

  • Wien part - boosted

• Intensity - related to the physical properties of the cluster

  $$\frac{\Delta I_{\nu}^{\mathrm{RJ}}}{I_{\nu}^{\mathrm{RJ}}}=-2 y$$

  where

  $$y=\int \frac{k T}{m_{\mathrm{e}} c^{2}} \sigma_{\mathrm{T}} n_{\mathrm{e}} \mathrm{d} l$$

  • Independent of redshift and of the details of the gas distribution within the cluster - identification of clusters at high redshifts