Cosmology IV « Physics Made Easy

Cosmology IV

Temperature, energy and scale factor

Define H₀=100h kms^-1Mpc^-1

Define energy density for black body radiation $\epsilon_{rad}=\rho_{rad}c^2=\alpha T^4$ where $\alpha=\frac{\pi^2k^4}{15\bar{h}^3c^3}$

Using the observed CMB temperature, $\epsilon_{rad}(t_0)=4.19\times 10^{-14} Jm^{-3}$

Critical density $\rho_c(t_0)=2.78h^{-1}\times 10^{11}M_O (h^{-1} Mpc)^{-3}$

$\Omega_{rad}=\frac{\epsilon_{rad}}{\rho_cc^2}=2.47\times 10^{-5}h^{-2}$

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

We know $\rho_{rad} \propto \frac{1}{a^4}$

$\therefore T \propto \frac{1}{a}$

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 ( $n \propto \frac{1}{a^3}$ . 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 $\bar{E}~3kT~7.05\times 10^{-4} eV$ for T=2.728K

Number density of photons $n_{\gamma}=\frac{\epsilon_rad}{\bar{E}}$

$n_{\gamma}=3.78\times 10^8 m^{-3}$

Baryon number density- from nucleosynthesis, the density parameter for baryons is Ω_B~0.02h^-2

Energy density $\epsilon_B=\rho_Bc^2=\Omega_B\rho_cc^2~3.4\times 10^{-11}Jm^{-3}$

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 ~10⁹ photons for every baryon.

Neutrino energy density is ~0.68x photon energy density, so density parameter for relativistic particles $\Omega_{rel}=4.17\times 10^{-5}h^2$ (scales as $\frac{1}{a^4}$

Non-relativistic Ω scales as $\frac{1}{a^3}$

$\frac{\Omega_{rel}}{\Omega_{non-rel}}=\frac{4.17\times 10^{-5}}{\Omega_0h^2}\frac{1}{a}$

$\Omega_{non-rel}=\Omega_0$

At recombination z~1000

$1+z=\frac{a_0}{a} \Rightarrow a=\frac{1}{1000}$ as a₀=1.

At decoupling $\frac{\Omega_{rel}}{\Omega_{non-rel}}=\frac{0.04}{\Omega_0h^2}$

Unless Ω₀h² 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

$a_{eq}=\frac{1}{24000\Omega_0h^2}$

Thermal history: assume instantaneous transition between radiation domination and matter domination. For matter domination and a flat universe, $a \propto t^{\frac{2}{3}}$ so $T \propto t^{-\frac{2}{3}}$

$\frac{T}{T_0}=(\frac{t_0}{t})^{\frac{2}{3}}$

T₀=2.728K, t₀=3×10¹⁷sec

This holds for $T<T_{eq}=\frac{T_0}{a_eq}=66000\Omega_0h^2 K$

So the time of matter-radiation equality is given by

$t_{eq}=(\frac{T_0}{T})^{\frac{3}{2}}t_0$

$t_{eq} \simeq 8\times 10^{10}\Omega_0^{-\frac{3}{2}}h^{-3} sec$

$t_{eq} \simeq 2500 \Omega_0^{-\frac{3}{2}}h^{-3} yr$

At temeperatures above T_eq, radiation domination takes over, and from the expansion law $a \propto t^{\frac{1}{2}}$ , we get

$\frac{T}{T_{eq}}=(\frac{t_{eq}}{t})^{\frac{1}{2}}$

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

Ignoring dependence on Ω₀ and h, we have

$(\frac{1 sec}{t})=\frac{T}{2\times 10^{10}K}$

When the universe was 1 second old, the temperature would have been about 2×10¹⁰K 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).

$dI_{\lambda}=-\kappa_{\lambda} \rho I_{\lambda}ds$

Where dI_λ is change in intensity at wavelength λ, κ_λ is the absorption coefficient (opacity), ρ is density, I_λ is intensity and s is distance travelled.

$\int_{I_{\lambda}(0)}^ {I_{\lambda}(f)} \frac{dI_{\lambda}} {I_{\lambda}}=-\int_0^s \kappa_{\lambda}\rho ds$

$I_{\lambda}(f)=I_{\lambda}(0)e^{-\int_0^s \kappa_{\lambda}\rho ds}$

or, for a gas of uniform opacity and density

$I_{\lambda}(f)=I_{\lambda}(0)e^{-\kappa_{\lambda}\rho s}$

Characteristic distance $l=\frac{1}{\kappa_{\lambda}\rho}$

For scattered photons, the characteristic distance is the photon mean free path $l=\frac{1}{n\sigma_{\lambda}}$

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

Define optical depth τ_λ as

$d\tau_{\lambda}=\kappa_{\lambda}\rho ds$

$\tau_{\lambda}=\int_0^s \kappa_{\lambda}\rho ds$

$I_{\lambda}=I_{\lambda}(0)e^{-\tau_{\lambda}}$

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_obs=4380Å).

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 (10²² 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 $P \propto N_c^{-0.75}$ , with few systems having N_c=10²⁵ 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~10¹⁶ 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.

$\tau_{SGP}=\int_0^{z_{\rho}}\sigma[v_0(1+z)]n_{H1}(z)\frac{dl}{dz}dz$

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, q₀: 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

$t-t_0=\frac{1}{H_0}\int_0^{z_0}\frac{1}{1+z}\frac{dz}{\sqrt{(1+z)^2(1+\Omega_M)^2-z(2+z)\Omega_{\Lambda}}}$

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 H₀^-1.

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Cosmology IV

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