The Cosmic Microwave Background

Westminster Astronomical Society Library

4) Things were so much simpler back then

It is believed that the first nine inhabitants who had descended from the skies were sexless and sinless and lived on a kind of flavoured earth. Their appetites grew and when they took to eating a sort of huskless rice which cooked itself they became gross and heavy, developed sex and after it crime because they had to work for a living

Kachin Myth

Early universe must have been very simple:

which gives an exact expression for the temp: $$ \color{red}{ T^2 = \frac{1}{t}\sqrt {\frac{{3c^3 }}{{64\pi G\sigma }}} ,t = \frac{\xi }{{T^2 }}} $$

• COsmic Background Explorer
• Wilkinson Microwave Anisotropy Probe

COBE/WMAP results

FIRAS: No distortion from B-B spectrum in any given direction (results have to be divided into pre and post COBE)

Temp. of the microwave sky in a scale in which blue is 0 K and red is 4 K. Note completely uniform on this scale. The actual temperature of the cosmic microwave background is 2.725 Kelvin.

Dipole effect: if we are moving through CMBR we would expect to see it "warmer" in front and "colder" behind. Blue ↔2.721 Kelvin red ↔2.729 Kelvin

so CMBR is blue-shifted in the direction we are going in (note residual effect of galaxy): what do we expect for 600 km/s?

Credit: DMR, COBE, NASA, Four-Year Sky Map

shows we are moving towards Leo at≈ 606 km/s

Except there is a tiny problem: some of the features in the CMBR seem to be aligned with the solar system....

Sunayev-Zeldovich effect:

Temp of B-B increases in direction of galaxy due to Compton scattering from free electrons

Shifts BB spectrum to higher frequencies (greatly exaggerated)

So looks cooler at long wavelengths, hotter at short

Allows one to pick out galaxies clusters that can't be seen otherwise

Recombination

\color{red}{\gamma + e^ - \Rightarrow \gamma + e^ - } dominates. Mean free path

σ_e Thomson X-sect

• Want prob X of H atom being ionized: \color{red}{Xn_H = \left( {1 - X} \right)n_p }
• Assume universe is radiation dominated (can justify this afterwards) Can then combine this with \color{red}{\eta = \frac{{n_p }}{{Xn_\gamma }}} (which doesn't change after decoupling) and the BB equation \color{red}{n_\gamma = .243\left( {\frac{{kT}}{{\hbar c}}} \right)^3 } to give
• \color{red}{n_p = .243X\eta \left( {\frac{{kT}}{{\hbar c}}} \right)^3 } Combining gives
  \color{red}{ \frac{{1 - X}}{{X^2 }} = 3.84\eta \left( {\frac{{kT}}{{m_e c^2 }}} \right)^{3/2} e^{Q/kT} }
  whihc must be solved numerically
• Put X = 1/2, to give
  \color{red}{ kT_{recomb} \approx .32eV \approx \frac{Q}{{42}} \Rightarrow T = 3740K \Rightarrow z \approx 1370 \Rightarrow t_{rec} \approx 240000yr}
• (note importance of 42 yet again!)

• {90\% } at z= 1475
• ⇒ {10\% } at z=1255
• which means that universe went from ionized to bound in ∼70000 years
• Note: not quite the same time as photon freeze-out or decoupling:
• scattering rate was
  \color{red}{ \Gamma \left( z \right) = n_e \left( z \right)\sigma _e c = X\left( z \right)\left( {1 + z} \right)^3 n_{b,0} \sigma _e c = 4.4 \times 10^{ - 21} X\left( z \right)\left( {1 + z} \right)^3 }

• Universe is transparent when it is expanding faster than scattering rate: \color{red}{ \frac{1}{\Gamma } < \frac{1}{H} },
• so put Γ = H ⇒ z_dec ≈1200 and T_dec ≈3000 (need to use non-equilm thermo to do this properly)
• Hence "time of last scattering" t_ls ≈ "decoupling time t_dec" ≈ 350000 yrs is the time from which we see the CMBR.
• Before this, electrons + baryons + photons all in equilm. and form fluid

• Sometimes referred to a primordial soup:

  http://www.lib.uchicago.edu/e/crerar/exhibits/humor5.html

• Note in this model the universe became matter-dominated at 47000 yrs (z ≈3570) so assumpt. that it is rad-dom isn't quite right.

• Well formed galaxies were there at z ≈ 1, t ≈ 3 ×109 years New observations suggest they are there at z ≈ 4.: Lyman forest shows hydrogen clouds present very early

• How did the galaxies form so quickly between t ≈ t_dec (where there are no indications)
  and
  t ≈ 2 × 10⁹ years (where they are well formed and look like today's galaxies)?
• Seeds must have been there ⇒ temp fluctuations.

• Source: Fluctuations in energy density ⇒ changes in grav. potential ⇒ grav red-shift (Sachs-Wolf effect)
  \color{red}{ \varepsilon \left( r \right) = \bar \varepsilon + \delta \varepsilon \left( r \right) \Rightarrow \nabla ^2 \left( {\delta \varphi } \right) = 4\pi G\delta \rho \left( r \right) \Rightarrow \frac{{\delta T}}{T} = \frac{{\delta \varphi }}{{3c^2 }}}
• i.e. measuring temp differences gives us direct measurement of density fluctuations.

• However we don't know r and d_p is not observable! However, for light we have a null geodesic
  \color{red}{ 0 = - c^2 dt^2 + a\left( t \right)^2 dr^2 }
  so
  \color{red}{ r = c\int {\frac{{dt}}{{a\left( t \right)}}} }

• So horizon distance
  \color{red}{ d_{hor} \left( {t_0 } \right) = c\int_0^{t_0 } {\frac{{dt}}{{a\left( t \right)}}} }
• If we take our flat, 1-cpt universe
  \color{red}{ a\left( t \right) = \left( {\frac{t}{{t_0 }}} \right)^{2/\left( {3 + 3w} \right)} }
• so horizon distance ≈ Hubble distance
  \color{red}{ d_{hor} \left( {t_0 } \right) = \frac{c}{{H_0 }}\frac{2}{{1 + 3w}} \approx \frac{c}{{H_0 }}}
  In benchmark model, a bit more complicated:
  \color{red}{ d_{hor} \left( {t_0 } \right) \approx \frac{{3.24c}}{{H_0 }} \approx 14Gpc}

Angular size related to via δ θ \color{red}{ \delta \theta = \frac{l}{{d_A }}}

Hence

Any fluctuations smaller than Hubble distance could be correlated Any larger ones must have a different cause corresponds to angular size now θH ∼ 1° or l ∼ 200 Hence importance of ang. res.

Ryden written Before WMAP

• COBE had angular resolution δθ = 7° WMAP has δθ = 10" COBE and WMAP comparison

• Want to find correlation between temp at different points in the sky \color{red}{ C\left( \theta \right) = \left\langle {\frac{{\delta T_1 }}{T}\frac{{\delta T_2 }}{T}} \right\rangle _{\cos \left( \theta \right)} } Two-point correlation function

• Can make Legendre polyn expansion:
  \color{red}{ C\left( \theta \right) = \frac{1}{{2\pi }}\sum\limits_{}^{} {\left( {2l + 1} \right)} c_l P_l \left( {\cos \left( \theta \right)} \right)}
  (usual expansion in orthog functions, so quite general).
• Actual map has no importance, but info about early universe is in multipole moment c_l: large l corresponds to small ang. separation in sky.

COBE/BOOMERANG

WMAP

• High density ⇔ High δT

• Also, because of dependence of angular size on geometry, the multipole plot gives direct measurement of curvature

• Structure looks smaller, peak shifts to larger l > 180 if universe has κ = -1. Structure looks larger, peak shifts to smaller l < 180 if universe has κ = +1.

Small-scale fluctuations

• Before decoupling, Universe is filled with photon-baryon fluid.

• Normally density fluctuations die away (e.g sound waves) but in massive fluid they get amplified by gravity

• \color{red}{ P = w\varepsilon } with w = 1/3 for photons, w = 0 for matter so 0<w_pb<1/3.
• will give acoustic oscillations : speed of sound
  \color{red}{ v = \sqrt {\frac{{dP}}{{d\rho }}} = \sqrt {\frac{{c^2 dP}}{{d\varepsilon }}} = c\sqrt {w_{pb} } }
• (i.e. in a pure γ or ν gas, v = .58c)

Except that we have a self-gravitating fluid: fluctuations will cause higher density, and will attract fluid: if it gets too dense, pressure will drive it apart.

Different models for gas:

Consistent with κ=0 and Ω0=1 Ωb∼ .04 ⇒ Benchmark Model Will add info from supernova later