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1.2. Dynamics

Everything discussed so far has been "geometrical", relying only on the form of the Robertson-Walker metric. To make further progress in understanding the evolution of the universe, it is necessary to determine the time dependence of the scale factor a(t). Although the scale factor is not an observable, the expansion rate, the Hubble parameter, H = H(t), is.

Equation 8 (8)

The present value of the Hubble parameter, often referred to as the Hubble "constant", is H0 ident H(t0) ident 100 h km s-1Mpc-1 (throughout, unless explicitly stated otherwise, the subscript "0" indicates the present time). The inverse of the Hubble parameter provides an expansion timescale, H0-1 = 9.78 h-1 Gyr. For the HST Key Project (Freedman et al. 2001) value of H0 = 72 km s-1 Mpc-1 (h = 0.72), H0-1 = 13.6 Gyr.

The time-evolution of H describes the evolution of the universe. Employing the Robertson-Walker metric in the Einstein equations of General Relativity (relating matter/energy content to geometry) leads to the Friedmann equation

Equation 9 (9)

It is convenient to introduce a dimensionless density parameter, Omega, defined by

Equation 10 (10)

We may rearrange eq. 9 to highlight the relation between matter content and geometry

Equation 11 (11)

Although, in general, a, H, and Omega are all time-dependent, eq. 11 reveals that if ever Omega < 1, then it will always be < 1 and in this case the universe is open (kappa < 0). Similarly, if ever Omega > 1, then it will always be > 1 and in this case the universe is closed (kappa > 0). For the special case of Omega = 1, where the density is equal to the "critical density" rhocrit ident 3H2 / 8pi G, Omega is always unity and the universe is flat (Euclidean 3-space sections; kappa = 0).

The Friedmann equation (eq. 9) relates the time-dependence of the scale factor to that of the density. The Einstein equations yield a second relation among these which may be thought of as the surrogate for energy conservation in an expanding universe.

Equation 12 (12)

For "matter" (non-relativistic matter; often called "dust"), p << rho, so that rho / rho0 = (a0 / a)3. In contrast, for "radiation" (relativistic particles) p = rho / 3, so that rho / rho0 = (a0 / a)4. Another interesting case is that of the energy density and pressure associated with the vacuum (the quantum mechanical vacuum is not empty!). In this case p = -rho, so that rho = rho0. This provides a term in the Friedmann equation entirely equivalent to Einstein's "cosmological constant" Lambda. More generally, for p = w rho, rho / rho0 = (a0 / a)3(1+w).

Allowing for these three contributions to the total energy density, eq. 9 may be rewritten in a convenient dimensionless form

Equation 13 (13)

where Omega ident OmegaM + OmegaR + OmegaLambda.

Since our universe is expanding, for the early universe (t << t0) a << a0, so that it is the "radiation" term in eq. 13 which dominates; the early universe is radiation-dominated (RD). In this case a propto t1/2 and rho propto t-2, so that the age of the universe or, equivalently, its expansion rate is fixed by the radiation density. For thermal radiation, the energy density is only a function of the temperature (rhoR propto T4).

1.2.1. Counting Relativistic Degrees of Freedom

It is convenient to write the total (radiation) energy density in terms of that in the CMB photons

Equation 14 (14)

where geff counts the "effective" relativistic degrees of freedom. Once geff is known or specified, the time - temperature relation is determined. If the temperature is measured in energy units (kT), then

Equation 15 (15)

If more relativistic particles are present, geff increases and the universe would expand faster so that, at fixed T, the universe would be younger. Since the synthesis of the elements in the expanding universe involves a competition between reaction rates and the universal expansion rate, geff will play a key role in determining the BBN-predicted primordial abundances.

Accounting for all of the particles present at a given epoch in the early (RD) evolution of the universe,

Equation 18 (18)

For example, for the standard model particles at temperatures Tgamma approx few MeV there are photons, electron-positron pairs, and three "flavors" of lefthanded neutrinos (along with their righthanded antiparticles). At this stage all these particles are in equilibrium so that Tgamma = Te = Tnu where nu ident nue, nuµ, nutau. As a result

Equation 19 (19)

leading to a time - temperature relation: t = 0.74 T-2Mev sec.

As the universe expands and cools below the electron rest mass energy, the e± pairs annihilate, heating the CMB photons, but not the neutrinos which have already decoupled. The decoupled neutrinos continue to cool by the expansion of the universe (Tnu propto a-1), as do the photons which now have a higher temperature Tgamma = (11/4)1/3 Tnu (ngamma / nnu = 11/3). During these epochs

Equation 20 (20)

leading to a modified time - temperature relation: t = 1.3 T-2Mev sec.

1.2.2. "Extra" Relativistic Energy

Suppose there is some new physics beyond the standard model of particle physics which leads to "extra" relativistic energy so that rhoR -> rho'R ident rhoR + rhoX; hereafter, for convenience of notation, the subscript R will be dropped. It is useful, and conventional, to account for this extra energy in terms of the equivalent number of extra neutrinos: Delta Nnu ident rhoX / rhonu (Steigman, Schramm, & Gunn 1977 (SSG); see also Hoyle & Tayler 1964, Peebles 1966, Shvartsman 1969). In the presence of this extra energy, prior to e± annihilation

Equation 21 (21)

In this case the early universe would expand faster than in the standard model. The pre-e± annihilation speedup in the expansion rate is

Equation 22 (22)

After e± annihilation there are similar, but quantitatively different changes

Equation 23 (23)

Armed with an understanding of the evolution of the early universe and its particle content, we may now proceed to the main subject of these lectures, primordial nucleosynthesis.

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