FLRW: A numerical exploration of Friedmann-Lemaitre-Robertson-Walker Metrics

Author: Jonah Miller (jonah.maxwell.miller@gmail.com) Time-stamp: <2013-12-20 15:53:59 (jonah)>

INTRODUCTION

This is my program to numerically solve for homogeneous, isotropic solutions to Einstein's equations. To this end, I assume the Friedmann-Lemaitre-Robertson-Walker metric and numerically solve the Friedmann eqautons (see, e.g., Wald, or just look at Wikipedia: https://en.wikipedia.org/wiki/Friedmann_equations).

For a brief derivation of the equations I solve, see the included pdf file. I use my implementation for the 4(5) Runge-Kutta-Fehlberg method with adaptive step sizes. The required libraries should be included in the github repository. However, if you want to play with my RKF integrator all by itself, you can find it here: https://github.com/Yurlungur/runge_kutta

INSTALLATION and USE

To download, just clone the repo:

git clone https://github.com/Yurlungur/FLRW.git
cd FLRW

This program uses the linux make utility for installation and configuration. It is not configured for Windows, allthough it could probably be easily modified ot run in that environment.

There are a number of small, pre-configured programs that use the main libraries that you can build to run a simulation. Here are the commands:

• make test This makes a unit test driver for the libraries in question.
• make radiation_dominated_universe --- Simulate a universe filled mostly with photons. This matches the state of the universe right after reheating (or the big bang).
• make matter_dominated_universe --- Simulate a universe filled with non-interacting dust, much like the stars and galaxies that fill the universe today.
• make dark_energy_dominated_universe --- Simulate the universe as it would be if only dark energy filled the universe. It is likely the universe will look like this in the far future.
• make multi-regime --- Simulate the universe as it transitions from radiation-dominated to matter-dominated to dark energy-dominated. We examine four possible ways the universe could transition by selecting an equation of state relating density and pressure of matter in the universe.
• make our_future_universe --- Recent experiments show that the universe may have an equation of state variable omega(rho) equal to -1.2. The type of matter required to create this is strange indeed. It doesn't dilute. Rather energy density increases as the scale factor increases. This simulation explores that possibility.
• make all --- Generates all of the above simulations
• make clean --- Cleans the directory, removing all binary files.

Once you've run make, you should have a *.bin file in your directiory. execute it with

chmod a+x *.bin
./<file name>.bin

The program will then run a number of simulations and output a number of data files ending in *.dat. Each file has three columns, each line is one time step. They show the following quantities: time a(t) rho(t) p(t)

You can plot these quantities with four convenient python programs included, which require python 2.7, numpy, scipy, and matplotlib. Call them with

python2 program.py filename1.dat filename2.dat

they can take an arbitrary number of files.

The python programs:

• plot_scale_factor.py --- plots the scale factor as a function of time for each included *.dat file
• plot_all_variables.py --- plots a(t),rho(t), and p(t) on the same plot.
• plot_all_variables_big_a.py --- The same as plot_all_variables.py but a(t) has been rescaled for easier viewing.
• MultiRegime.py --- Plots omega(rho) for every equation of state used in the multi-regime simulation. Takes no input data.

PHYSICS DETAILS

We use geometrized units, which means that c=G=hbar=1. The scale factor is not normalized. It has no absolute meaning.

Although the cosmological constant LAMBDA can be set to a non-zero value, this is not recomended. The preferred way to include a cosmological constant is manipulate the equation of state function omega(rho). A constant value of -1.0 corresponds to a positive cosmological constant.

A nonzero curvature can be included by setting K=-1 or K=+1. This sets the normalized curvature to a negative or positive value respectively. However, this feature is untested. Treat it with caution.