# ReHILAE: Is Re-ionisation of Hydrogen due to Lyman-alpha emitters (LAEs)?

### Week 2 Research Update:

Our first task today was setting up a GitHub repository (here) to host our project. This will give us much more flexibility when we want to add functionality to our code, as well as allowing us to work on multiple aspects of the code simultaneously.

After that, we started implementing a new equation relating cosmic time and redshift. This equation is derived in Carmeli, M., Harnett, J. and Oliveira, F. (2005) and shown below:

$t \approx \frac{28}{1+(1+z)^2}Gyr$

This can be simply rearranged to make redshift the subject. The plot below shows the plots for the original cosmic time – redshift equation and the approximation above.

It can be seen that both of these equations are very similar, however the approximated equation does decay quicker. The ratio of the approximated redshift to the cosmological redshift has been plotted below:

These plots show how the redshifts given by the two equations differ. From the first graph it can be seen at large values of cosmic time (redshifts close to zero) the equations begin to differ greatly. In the second plot, the large cosmic time values have been omitted. This allows us to much more accurately see how the redshift equations differ during the epoch of re-ionisation; the ratio varies between 0.8 – 1.0 .

After we had the new redshift equation implemented we moved onto finishing our process of creating a file called MPhys_model.py . This file contains all the previous equations as their own functions. Having these as separate, callable functions will make adding additional functionality much easier in the coming weeks. Today we added the differential equation as a function in our MPhys_model.py and plotted it against redshift.

This is very similar to previous plot in Week 3’s update, the variation is due to different starting parameters.

From Sobral, D. and Matthee, J. (2019) we can use the following equation to calculate the escape fraction.

$f_{esc,Ly\alpha} = 0.0048^{+0.0007}_{-0.0007} EW_0 \pm 0.05$

Next week will then look at converting this escape fraction of Lyman alpha photons to the escape fraction for Lyman continuum photons, which we can then sub into our $Q_{ion,Ly\alpha}$ equation. Using data provided in a fits file we were able to produce a plot of equivalent width against redshift.

The plot above shows the equivalent widths for several different red shifts. We then used a Pylab linear fit to generate a function of equivalent width against redshift, shown by the red line.