Equations

This section derives and explains the equations solved in LATTICEEASY. Although occasional reference will be made to program files or variables, the focus here is on the equations themselves; their implementation in the program is discussed in the other sections of the documentation.

All of the equations are presented in two forms. First they are derived in their usual physical forms. The metric, units, and other conventions used are explained in section 2. In the program itself, however, the field and spacetime variables are rescaled in ways that make the equations simpler to solve. The form of these rescalings is discussed in section 6.1, and from there on all equations are given both in terms of physical and program variables.

Section 6.1 gives the field equations and uses them to motivate the variable rescalings used in the program. Section 6.2 derives the evolution equation for the scale factor, which the program solves for self-consistently using the Einstein equations. There is also an option in the program to use a fixed power-law expansion and the relevant equations are derived and discussed in section 6.2 as well. Section 6.3 discusses the setting of initial conditions, including initial values for the fields, field derivatives, and Hubble constant. Finally, section 6.4 discusses the ``staggered leapfrog'' method used by the program for solving differential equations.

- Field Equations and Coordinate Rescalings

- Scale Factor Evolution

- Initial Conditions on the Lattice
- Homogeneous Field and Derivative Values
- Initial Conditions for Field Fluctuations
- Initial Conditions for Field Derivative Fluctuations
- Standing Waves: Preserving Isotropy
- Initial Conditions in One and Two Dimensions
- The Initial value of the Hubble Constant
- The Adiabatic Approximation

- The Staggered Leapfrog Method

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