** These are the instructions for recreating plots of the Density Density correlation function 
found in the paper "Correlation patterns from massive phonons in 1 + 1 dimensional acoustic black holes:  
A toy model", Physical Review D 98, 124011 (2018), and in Richard Dudley's Dissertation**


The region denoted 41 ( Which indicates 1 point in the massive exterior region and 1 point
in the massive interior region) is the only region plotted for these results.

1.) Run X1.nb and X4.nb to generate the input position and time coordinates for the 4,1 region. 

2.) Run 41Simplifications.nb to generate input simplifications for the terms in the integral for 
the density-density correlation function in the 4,1 region.

3.) Run DDToyHLower.m and DDToyUpper.m, which compute the lower and upper portions of the integral 
 of the density-density correlation function. 

The output data from DDToyHLower.m and DDToyUpper.m is used to generate the plots.