**2.1. Cosmological fluid equations**

A fluid is a dense set of particles treated as a continuum. If particle collisions are rapid enough to establish a local thermal equilibrium (e.g., Maxwell-Boltzmann velocity distribution), the fluid is an ideal collisional gas. If collisions do not occur (e.g., a gas of dark matter particles), the gas is called collisionless. (I exclude incompressible fluids, i.e., liquids, from consideration because the gases considered in cosmology are generally very dilute and compressible.) The fluid equations discussed in this lecture apply only for a collisional gas (or a pressureless collisionless gas). They apply, for example, to baryons (hydrogen and helium gas) after recombination, to cold dark matter before trajectories intersect ("cold dust"), and (with relativistic corrections) to the coupled photon-baryon fluid before recombination.

I shall assume a nonrelativistic gas and ignore bulk electric and magnetic forces. These are not difficult to add, but the essential physics of cosmological fluid dynamics does not require them.

The fluid equations consist of mass and momentum conservation laws and
an equation of state. Mass conservation is represented by the
**continuity equation**. In proper coordinates
(** r**,

(2.1) |

We convert to comoving coordinates
=
*dt* /
*a*(*t*),
** x** =

(2.2) |

where ** v** =

(2.3) |

Momentum conservation for an ideal fluid is represented by the
**Euler equation**
(Landau & Lifshitz
1959).
It is most simply obtained
by adding the pressure-gradient force to the equation of motion for a
freely-falling mass element, eq. (1.7). In comoving
coordinates, we find

(2.4) |

The time derivative is taken along the fluid streamline and is known as the convective or Lagrangian time derivative:

(2.5) |

Closing the fluid equations requires an evolution equation for the pressure
or some other thermodynamic variable. Perhaps the most natural is the
entropy. For a collisional gas, thermodynamics implies an **equation
of state** *p* =
*p*(,
*S*) where *S* is the specific entropy. For example,
for an ideal nonrelativistic monatomic gas, for reversible changes we have

(2.6) |

which says that the heat input to a fluid element equals the change in
thermal energy plus the pressure work done by the element, i.e., energy
is conserved. Combining this with the ideal gas law
*p* = *k*_{B}*T* /
*µ* where *µ* is the mean molecular mass and
*k*_{B} is the Boltzmann constant, we obtain

(2.7) |

The equation of state must be supplemented by an evolution equation for the specific entropy. Outside of shock waves, the entropy evolution equation is

(2.8) |

where and
are,
respectively, the proper specific
heating and cooling rates (in erg g^{-1} s^{-1}). They
are determined
by microphysical processes such as radiative emission and absorption,
cosmic ray heating, Compton processes, etc. For the simplest case, adiabatic
evolution, =
= 0. For a
realistic non-ideal gas, it may be
necessary to evolve the radiation field, the ionization fraction, and other
variables specifying the equation of state.

The fluid equations are much harder to solve than Newton's laws for
particles
falling under gravity, for several reasons. First, they are nonlinear
partial differential equations rather than a set of coupled ordinary
differential equations. Second, shock waves (discontinuities in
,
*p*, *S*, and ** v**) prevent intersection of fluid
elements. These
discontinuities must be resolved (on a computational mesh or otherwise)
and followed stably and accurately. Finally, heating and cooling for
realistic gases are complicated and can lead to large temperature or entropy
gradients that are difficult to resolve. An example of the latter is the
sun, whose temperature changes by about 15 million K in a distance that
is minuscule compared with cosmological distance scales.

Computational fluid dynamics is a difficult art but is important for galaxy formation. I shall not summarize the numerical methods here but refer the reader instead to the literature (e.g., Sod 1985, Leveque 1992, Monaghan 1992, Bryan et al. 1994, Kang et al. 1994).

Some of the most important effects of gas pressure can be gleaned from linear perturbation theory, in which we linearize the fluid equations about the uniform solution for an unperturbed Robertson-Walker spacetime. This technique is useful for checking for gravitational and other linear instabilities. Moreover, the linearized fluid equations may provide a reasonable description of large-scale, small-amplitude fluctuations in the (dark+luminous) matter, even if structure is nonlinear on small scales. This is a common assumption in large-scale structure theory. It is supported reasonably well by numerical simulations.

Linearizing the continuity and Euler equations gives

(2.9) |

where an overdot denotes
/
. The pressure gradient
may be obtained from the equation of state *p* =
*p*(,
*S*). For an ideal nonrelativistic monatomic gas,

(2.10) |

Finally, we must linearize the entropy evolution equation. If the time scale
for entropy changes is long compared with the acoustic or gravitational time
scales, eq. (2.8) becomes
*dS* / *d*
0. For the small
peculiar velocities of linear perturbation theory this reduces to
0.

There are five fluid variables
(, *S*,
and three components of ** v**),
hence five linearly independent modes. The general linear perturbation is a
linear combination of these, which we now proceed to examine.