as an effective isotropic pressure in the stress-energy tensor
The matter source terms can then be written
with 
.
where R is the scalar Riemann curvature, 
is the interaction
potential, 
is typically assumed to be 
,
and 
is the field-curvature coupling constant
( 
for minimally coupled fields and 
for
conformally coupled fields).
Varying the action yields the Klein-Gordon equation
for the scalar field and
for the stress-energy tensor, where 
.
For a massive, minimally coupled scalar field [41]
and
where
,
and is a general potential which, for example,
can be set to 
in the chaotic inflation model.
The covariant form of the scalar field equation (27
)
can be expanded as in [86] to yield
which, when coupled to (33), determines the
evolution of the scalar field.
where m is the rest mass of the particles, n is the number density
in the comoving frame, and 
is the 4-velocity of each particle.
The matter source terms are
There are two conservation laws: the conservation of particles
,
and the conservation of energy-momentum
,
where 
is the covariant derivative in the full 4-dimensional
spacetime.
Together these conservation laws lead to
,
the geodesic equations of motion for the particles, which can be
written out more explicitly in the
computationally convenient form
where 
is the coordinate position of each particle,
is determined by the normalization 
,
is the Lagrangian derivative, and
is the ``transport'' velocity
of the particles as measured by
observers at rest with respect to the coordinate grid.
where 
is the 4-metric,
is the relativistic enthalpy,
and 
, P, 
and 
are the specific internal
energy (per unit mass), pressure, rest mass density and four-velocity
of the fluid. Defining
as the generalization of the special relativistic boost factor, the matter source terms become
The hydrodynamics equations are derived from the
normalization of the 4-velocity, ,
the conservation of baryon number, 
,
and the conservation of energy-momentum, .
The resulting equations can be written in flux conservative
form as [126
]
where 
, 
, 
,
, 
,
and g is the determinant of the 4-metric satisfying
.
A prescription for specifying an equation of state
(e.g., 
for an ideal gas)
completes the above equations.
When solving
equations (47, 48, 49),
an artificial viscosity method is needed
to handle the formation and propagation of shock
fronts [126, 72, 73]. Although these
methods are simple to implement and are computationally
efficient, they are inaccurate for ultrarelativistic
flows with very high Lorentz factors.
On the other hand, a number of different
formulations [67]
of these equations have been developed to
take advantage of the hyperbolic and
conservative nature of the equations in using high resolution
shock capturing schemes (although strict conservation is only possible
in flat spacetimes - curved spacetimes exhibit source terms due to
geometry).
Such high resolution Godunov techniques [107, 24]
provide more accurate and stable treatments
in the highly relativistic regime.
A particular formulation due to [24] is the following:
where
and
,
,
,
,
,
,
and
.
|
Computational Cosmology: from the Early Universe to the Large Scale Structure Peter Anninos http://www.livingreviews.org/lrr-2001-2 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |
