Correlation Matrix Simulation Tools

This repository contains two complementary tools for producing simulated correlation matrices of the CMB temperature anisotropy for a set of candidate oblique torus topologies:

1. An eigenmode generator for producing a csv file of the k wave vector values of the planar wave eigenmodes for a given topology, set of compactification lengths, and set of oblique angles.

2. A mode-mode correlation matrix function which returns a 837 x 837 square matrix as a Numpy array from a given csv file of eigenmodes.

Together, these tools make a convenient workflow for computing reproducible mode-mode correlation matrices for cosmic topology searches using the correlation matrix of the Planck CMB.

Usage

The eigenmode production tool, eigenmodes.py, and the correlation matrix computation tool, matrix.py, are ran on the command line separately.

First, run eigenmodes.py with the number of eigenmodes N you would like produced, the topology under consideration, three distance values (Lx, Ly, and Lz) for the size of the fundamental domain (In units of Hubble length (4,228 Gpc).) and three angle values for the slants of the oblique fundamental domain (In units of radians). The candidate topologies are called by an index of numbers

• 1 : Untwisted 3-torus
• 2 : Half-turn twisted 3-torus
• 3 : Quarter-turn twisted 3-torus
• 4 : Third-turn twisted hexagonal prism
• 5 : Sixth-turn twisted hexagonal prism

For example, to produce the first 200 eigenmodes of a quarter-turn space universe with sides of 1 Hubble Length and 3 slanted angles of 0.5 radians (28.6 degrees), we would call:

python eigenmodes.py 200 3 1 1 1 0.5 0.5 0.5 .

This will output a space separated value file named

200_Top3_1.0_1.0_1.0_0.5_0.5_0.5.csv ,

which is named by "N_Top(I)_Lx_Ly_Lz_A1_A2_A3" where I is the index of the topology.
Note that the factor of 2x(pi) is removed from the k terms in this file and is added back in the correlation matrix.

Each line of the file has the same format:

n j xi_j kx ky kz .

So, each line of the file represents a planewave term of an eigenmode, indexed by 'j'.
The matrix.py file reads each line of this to compute a mode-mode correlation matrix.

Dependencies

The only dependencies not in the Python standard library are the Numpy and Scipy libraries.
Instructions for installing the entire 'Scipy stack' are through the link above.

Notes and Issues

The inverse oblique transformation matrix is singular for unoblique rectangular prism fundamental domains. So, the code catches this case and simply multiplies the wave vector by the identity matrix.

Some simple functions for writing and reading complex numbers to file are defined in complexIO.py for writing complex coefficients of eigenmode terms. Not all of them are used in eigenmodes.py as of early June 2015.

Acknowledgements

This work is part of John Dulin's senior thesis project for the Case Western Reserve University Physics Department. It was done under the tutelage of Prof. Glenn Starkman of the CWRU Physics Department and Institute for the Science of Origins, and with the assistance of Joshua Osborne to expand correlation matrix searches for cosmic topology to the cases of oblique torus topologies.