**Temperature, energy and scale factor**

Define *H _{0}*=100

*h*kms

^{-1}Mpc

^{-1}

Define energy density for black body radiation where

Using the observed CMB temperature,

Critical density

CMB radiation is a small but not insignificant fraction of the critical density.

We know

Or to put it simply, the universe cools as it expands. The particle distribution continues to correspond to a thermal distribution, but with a lower temperature.

**Photon to baryon ratio:** if interactions are negligible, particles cannot simply disappear and so particle number densities reduce in inverse proportion to the volume (. This is true for protons and neutrons (baryons) and CMB photons; the ratio of photons to baryons is therefore a constant.

Photon number density- present CMB energy density is at the top of the page.

Mean energy for *T*=2.728K

Number density of photons

Baryon number density- from nucleosynthesis, the density parameter for baryons is *Ω _{B}*~0.02h

^{-2}

Energy density

Rest mass ~ 948 MeV -> number density *n _{B}*=0.22m

^{-3}

Total baryon energy density considerably exceeds that of photons, but when it comes to numbers there are ~10^{9} photons for every baryon.

Neutrino energy density is ~0.68x photon energy density, so density parameter for relativistic particles (scales as

Non-relativistic *Ω* scales as

At recombination z~1000

as *a _{0}*=1.

At decoupling

*Unless Ω _{0}h^{2}* is very much smaller than we currently think it is, the Universe will be matter dominated.

At matter-radiation equality, *Ω _{rel}*=

*Ω*

_{non-rel}**Thermal history:** assume instantaneous transition between radiation domination and matter domination. For matter domination and a flat universe, so

*T _{0}*=2.728K,

*t*=3×10

_{0}^{17 }sec

This holds for

So the time of matter-radiation equality is given by

At temeperatures above *T _{eq}*, radiation domination takes over, and from the expansion law , we get

where the constants of proportionality are fixed at matter-radiation equality.

Ignoring dependence on *Ω _{0}* and

*h*, we have

When the universe was 1 second old, the temperature would have been about 2×10^{10}K and the typical energy would be 2 MeV, almost ready for nucleosynthesis cookery (not as tasty as real cookery- see the earlier graph of *log(T)* vs. *log(t) *in the matter-radiation equality section).

**Absorption and optical depth: **any process which removes photons from a beam will be called absorption here (including scattering).

Where *dI _{λ}* is change in intensity at wavelength

*λ*,

*κ*is the absorption coefficient (opacity),

_{λ}*ρ*is density,

*I*is intensity and

_{λ}*s*is distance travelled.

or, for a gas of uniform opacity and density

Characteristic distance

For scattered photons, the characteristic distance is the photon mean free path

*ρκ _{λ}* or

*nσ*can be thought of as the gas target area encountered by a photon for every unit length it travels.

_{λ}Define optical depth *τ _{λ}* as

If the optical depth of the ray’s starting point is 1, the intensity of the ray will decline by a factor of *e ^{-1}* before escaping.

Optical depth may be thought of as the number of mean free paths from the original position to some other significant point (e.g. surface of star or boundary of gas cloud.

If *τ _{λ}*>>1, material is optically thick.

If *τ _{λ}*<<1, material is optically thin.

**Probing the intergalactic medium** can be achieved with quasar spectra. A quasar at redshift z=2.6 has the Lyman-α line redshifted from the UV into the optical (*λ _{em}*=1216Å,

*λ*=4380Å).

_{obs}**Lyman-α absorption:** intervening clouds of cool natural hydrogen will absorb radiation at the Lyman-α frequency in their rest frame. For example, an intervening cloud at z=2 introduces a broad absorption peak at 3650Å due to resonant Lyman- α scattering.

The column density (10^{22} atoms m^{-2}) of such clouds is comparable to that in a disc of a present-day spiral galaxy, suggesting that these clouds are proto-galaxies. Such clouds are seen up to z~3.

The distribution of column densities, *N _{c}*, seems to follow an approximate power law , with few systems having

*N*=10

_{c}^{25}m

^{-2}.

**Lyman-α forest:** in the spectra discussed above, we also see a prominent forest of narrow absorption lines. These are likely to be cause by Lyman-α scattering in lower surface density clouds along the line of sight. This theory is confirmed by observations of Lyman-β and higher (1s -> 1p) lines in the Lyman series; some heavy elements are also present. These absorption lines are referred to as the Lyman-α forest. The weakest lines here are sensitive to column densities *N _{c}*~10

^{16}m

^{-2}.

**Scheuer-Gunn-Peterson effect:** total optical depth for resonant scattering at the observed frequency is given by the line integral of the cross-section multiplied by the neutral hydrogen proper density.

Up to *z*=5, the intergalactic medium (unlike clouds) appears to contain little neutral hydrogen, only highly ionised gas, but at higher *z* we see a higher density of neutral hydrogen. It appears that the co-moving density of neutral hydrogen is ~100 times higher at *z=*6 than *z=*3 (shows that the intergalactic medium re-ionised between those times).

**Deceleration of the Universe, q_{0}:** the parameter

*q*quantifies the rate at which the expansion of the Universe is being slowed down by the matter within it.

An empty universe would have *q=*0 (no deceleration).

A flat universe would have *q*=0.5.

A cosmological constant *Λ* can produce acceleration in the expansion of the Universe if the cosmic repulsion is strong enough to overcome the self-attractive gravitational forces.

**Redshift and cosmic time**

The age of the Universe decreases with *Ω _{M}*– this comes from the fact that if there is less matter in the Universe, it must have taken longer for the rate of expansion to reach its present value.

Conversely, the age of the Universe increases with *Ω _{Λ}*; this comes from the fact that positive

*Λ*gives more acceleration, i.e. there was a smaller expansion rate at earlier times, hence the Universe needed longer to expand at its current speed.

If a cosmological constant is introduced and the density is low enough, then the age of the universe will be greater than *H _{0}^{-1}*.