> For the complete documentation index, see [llms.txt](https://slowdiveptg.gitbook.io/notes/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://slowdiveptg.gitbook.io/notes/basics-of-cosmological-physics/chapter-10.md).

# 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
    * $&#x6D;*{\ce H}\approx m*{\ce{p}}$
    * $Q=\left(&#x6D;*{\mathrm{p}}+m*{\mathrm{e}}-m\_{\mathrm{H}}\right) c^{2}$
  * 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 $\Omeg&#x61;*{\mathrm{k}, 0}=0, \Omega*{\mathrm{m}, 0}+\Omega\_{\Lambda, 0}=1$ 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&#x20;
    * 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