**2.2. Dynamics: The Friedmann Equations**

As mentioned, the RW metric is a purely kinematic consequence of
requiring homogeneity and isotropy of our spatial sections. We next
turn to dynamics, in the form of differential equations governing the
evolution of the scale factor *a*(*t*). These will come from
applying Einstein's equation,

(13) |

to the RW metric.

Before diving right in, it is useful to consider the types of
energy-momentum tensors
*T*_{µ} we
will typically encounter in
cosmology. For simplicity, and because it is consistent with much we
have observed about the universe, it is often useful to adopt the
perfect fluid form for the energy-momentum tensor of cosmological
matter. This form is

(14) |

where *U*^{µ} is the fluid four-velocity,
is the energy
density in the rest frame of the fluid and *p* is the pressure in that
same frame. The pressure is necessarily isotropic, for consistency
with the RW metric. Similarly, fluid elements will be comoving in the
cosmological rest frame, so that the normalized four-velocity in the
coordinates of (7) will be

(15) |

The energy-momentum tensor thus takes the form

(16) |

where *g*_{ij} represents the spatial metric (including the
factor of *a*^{2}).

Armed with this simplified description for matter, we are now ready to apply Einstein's equation (13) to cosmology. Using (7) and (14), one obtains two equations. The first is known as the Friedmann equation,

(17) |

where an overdot denotes a derivative with respect to cosmic time *t*
and *i* indexes all different possible types of energy in the
universe. This equation is a constraint equation, in the sense that we
are not allowed to freely specify the time derivative
; it is
determined in terms of the energy density and curvature. The second
equation, which is an evolution equation, is

(18) |

It is often useful to combine (17) and (18) to obtain the
*acceleration equation*

(19) |

In fact, if we know the magnitudes and evolutions of the different
energy density components
_{i},
the Friedmann equation (17) is sufficient to solve for the evolution
uniquely. The acceleration equation is conceptually useful, but rarely
invoked in calculations.

The Friedmann equation relates the rate of increase of the scale factor, as encoded by the Hubble parameter, to the total energy density of all matter in the universe. We may use the Friedmann equation to define, at any given time, a critical energy density,

(20) |

for which the spatial sections must be precisely flat (*k* = 0).
We then define the density parameter

(21) |

which allows us to relate the total energy density in the universe to its local geometry via

(22) |

It is often convenient to define the fractions of the critical energy density in each different component by

(23) |

Energy conservation is expressed in GR by the vanishing of the covariant divergence of the energy-momentum tensor,

(24) |

Applying this to our assumptions - the RW metric (7) and perfect-fluid energy-momentum tensor (14) - yields a single energy-conservation equation,

(25) |

This equation is actually not independent of the Friedmann and
acceleration equations, but is required for consistency. It implies
that the expansion of the universe (as specified by *H*) can lead to
local changes in the energy density. Note that there is no notion of
conservation of "total energy," as energy can be interchanged
between matter and the spacetime geometry.

One final piece of information is required before we can think about
solving our cosmological equations: how the pressure and energy
density are related to each other. Within the fluid approximation
used here, we may assume that the pressure is a single-valued function
of the energy density *p* =
*p*(). It
is often convenient to define an equation of state parameter, *w*, by

(26) |

This should be thought of as the instantaneous definition of the
parameter *w*; it need represent the full equation of state, which
would be required to calculate the behavior of fluctuations.
Nevertheless, many useful cosmological matter sources do
obey this relation with a constant value of *w*. For example,
*w* = 0 corresponds to pressureless matter, or dust - any collection of
massive non-relativistic particles would qualify. Similarly,
*w* = 1/3 corresponds to a gas of radiation, whether it be actual
photons or other highly relativistic species.

A constant *w* leads to a great simplification in solving our
equations. In particular, using (25), we see
that the energy density evolves with the scale factor according to

(27) |

Note that the behaviors of dust (*w* = 0) and radiation (*w* =
1/3) are consistent with what we would have obtained by more heuristic
reasoning. Consider a fixed *comoving* volume of the universe -
i.e. a volume specified by fixed values of the coordinates, from which
one may obtain the physical volume at a given time *t* by multiplying
by *a*(*t*)^{3}. Given a fixed number of dust
particles (of mass *m*)
within this comoving volume, the energy density will then scale just
as the physical volume, i.e. as *a*(*t*)^{-3}, in
agreement with (27), with *w* = 0.

To make a similar argument for radiation, first note that the
expansion of the universe (the increase of *a*(*t*) with time)
results in a shift to longer wavelength
, or a *redshift*, of
photons propagating in this background. A photon emitted with
wavelength _{e}
at a time *t*_{e}, at which the scale factor is
*a*_{e}
*a*(*t*_{e}) is observed today (*t* =
*t*_{0}, with scale factor
*a*_{0}
*a*(*t*_{0})) at wavelength
_{o}, obeying

(28) |

The redshift *z* is often used in place of the scale factor. Because
of the redshift, the energy density in a fixed number of photons in a
fixed comoving volume drops with the physical volume (as for dust) and
by an extra factor of the scale factor as the expansion of the
universe stretches the wavelengths of light. Thus, the energy density
of radiation will scale as *a*(*t*)^{-4}, once again
in agreement with (27), with *w* = 1/3.

Thus far, we have not included a cosmological constant in the gravitational equations. This is because it is equivalent to treat any cosmological constant as a component of the energy density in the universe. In fact, adding a cosmological constant to Einstein's equation is equivalent to including an energy-momentum tensor of the form

(29) |

This is simply a perfect fluid with energy-momentum tensor (14) with

(30) |

so that the equation-of-state parameter is

(31) |

This implies that the energy density is constant,

(32) |

Thus, this energy is constant throughout spacetime; we say that
the cosmological constant is equivalent to *vacuum energy*.

Similarly, it is sometimes useful to think of any nonzero spatial curvature as yet another component of the cosmological energy budget, obeying

(33) |

so that

(34) |

It is not an energy density, of course;
_{curv} is
simply a convenient way to keep track of how much energy density
is lacking, in comparison to a flat universe.