Initial Conditions for Field Fluctuations

Although the field equations are solved in configuration space with
each lattice point representing a position in space, the initial
conditions are set in momentum space and then Fourier transformed to
give the initial values of the fields and their derivatives at each
grid point. As mentioned above, all of the expressions in this and the following two sections will be derived for a three dimensional lattice. Section 6.3.5 will explain how the results are altered in other dimensions. The Fourier transform in three dimensions is defined by

It is assumed that no significant particle production has occurred
before the beginning of the program, so quantum vacuum fluctuations
are used for setting the initial values of the modes. The
probability distribution for the ground state of a real scalar field in a FRW
universe is given by [1,2]

(6.52) |

(6.53) |

Note that this distribution gives the mean-squared value

To derive the expressions used for setting field values on the lattice we must modify equation (6.54) to account for a finite, discrete space, then account for the rescalings of field and spacetime variables, and finally discuss how to implement the Rayleigh distribution. In the rest of this section we do each of these in turn.

There are two steps involved in normalizing these modes on a finite,
discrete lattice. First this definition has to be adjusted to account
for the finite size of the box. This is necessary in order to keep the
field values in position space independent of the box size. To see
this consider the spatial average .

(6.56) |

(6.57) |

Accounting for the discretization of the lattice is even easier.
From the definition of a discrete Fourier transform (denoted here
as ) in three dimensions

(6.58) |

Putting these effects together gives us the following expression for
the rms magnitudes, which we denote by .

(6.59) |

At a point on the Fourier transformed lattice the
value of is given by

(6.60) |

Next we rescale to program variables. The , , and
rescalings are determined by the rescaling of in equation
(6.2), i.e.

(6.61) |

Meanwhile the rescaled mass is given by

(6.63) |

Finally it remains to implement the Rayleigh distribution

(6.64) |

To generate this distribution from a uniform deviate (i.e. a random number generated with uniform probability between and ) first integrate it and then take the inverse (see [3]), which gives

where is a uniform deviate.

There are two more points to note in setting the initial conditions
for the fluctuations. The first is simply that the scale factor is set
to at the beginning of the calculations and may thus be dropped
from the equations. The second is that the phases of all modes are
random and uncorrelated, so they are each set randomly. The
expression for the field modes is thus

(6.67) |

and is set randomly between and . The frequency for a given point on the momentum space lattice is given by

(6.69) |

Go to The LATTICEEASY Home Page

Go to Gary Felder's Home Page

Send email to Gary Felder at gfelder@email.smith.edu

Send email to Igor Tkachev at Igor.Tkachev@cern.ch

This documentation was generated on 2008-01-21