**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.

(8) |

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

(9) |

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

(10) |

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

(11) |

Although, in general, *a*, *H*, and
are all
time-dependent, eq. 11 reveals that if ever
< 1, then it
will always be < 1 **and** in this case the universe is open
( < 0).
Similarly, if ever
> 1, then it
will always be > 1 **and** in this case the universe is closed
( > 0). For the special
case of = 1, where
the density is equal to the "critical density"
_{crit}
3*H*^{2} / 8
*G*, is always
unity and the universe is flat (Euclidean 3-space sections;
= 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.

(12) |

For "matter" (non-relativistic matter; often called "dust"),
*p* <<
, so that
/
_{0}
= (*a*_{0} / *a*)^{3}. In contrast,
for "radiation" (relativistic particles)
*p* = /
3, so that /
_{0} =
(*a*_{0} / *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* =
-, so that
=
_{0}.
This provides a term in the Friedmann equation entirely equivalent
to Einstein's "cosmological constant"
. More generally,
for *p* = *w*
,
/
_{0} =
(*a*_{0} / *a*)^{3(1+w)}.

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

(13) |

where
_{M} +
_{R} +
_{}.

Since our universe is expanding, for the early universe (*t*
<< *t*_{0})
*a* << *a*_{0}, so that it is the "radiation"
term in eq. 13 which dominates; the early universe is
radiation-dominated (RD). In this case
*a*
*t*^{1/2} and
*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
(_{R}
*T*^{4}).

**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

(14) |

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

(15) |

If more relativistic particles are present, *g*_{eff}
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, *g*_{eff} will play
a key role in determining the BBN-predicted primordial abundances.

*Photons*Photons are vector bosons. Since they are massless, they have only two degress of freedom:

*g*_{eff}= 2. At temperature*T*their number density is*n*_{}= 411(*T*/ 2.726*K*)^{3}cm^{-3}= 10^{31.5}*T*^{3}_{MeV}cm^{-3}, while their contribution to the total radiation energy density is_{}= 0.261(*T*/ 2.726*K*)^{4}eV cm^{-3}. Taking the ratio of the energy density to the number density leads to the average energy per photon <*E*_{}> =_{}/*n*_{}= 2.70*kT*. All other relativistic**bosons**may be simply related to photons by(16) The

*g*_{B}are the boson degrees of freedom (1 for a scalar, 2 for a vector, etc.). In general, some bosons may have decoupled from the radiation background and, therefore, they will not necessarily have the same temperature as do the photons (*T*_{B}*T*_{}).- Relativistic Fermions
Accounting for the difference between the Fermi-Dirac and Bose-Einstein distributions, relativistic fermions may also be related to photons

(17) *g*_{F}counts the fermion degrees of freedom. For example, for electrons (spin up, spin down, electron, positron)*g*_{F}= 4, while for neutrinos (lefthanded neutrino, righthanded antineutrino)*g*_{F}= 2.

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

(18) |

For example, for the standard model particles at temperatures
*T*_{}
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
*T*_{} = *T*_{e} =
*T*_{} where
_{e},
_{µ},
_{}. As a result

(19) |

leading to a time - temperature relation:
*t* = 0.74 *T*^{-2}_{Mev} 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
(*T*_{}
*a*^{-1}), as do the photons which now have a higher
temperature
*T*_{} = (11/4)^{1/3}
*T*_{}
(*n*_{} /
*n*_{} = 11/3).
During these epochs

(20) |

leading to a modified time - temperature relation:
*t* = 1.3 *T*^{-2}_{Mev} 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
_{R}
*'*_{R}
_{R} +
_{X};
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:
*N*_{}
_{X} /
_{}
(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

(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

(22) |

After *e*^{±} annihilation there are similar, but
quantitatively different changes

(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.