This blog post comes off the back of a relatively productive week where I made steady progress towards the goals expressed at the end of the last blog post (see here, if you missed it).
Firstly, as I stated last week, I decided to re-create my stellar mass functions neglecting mass values < 9 due to their large completeness, an example of which is shown below in Figure 1.
I also applied markers to display upper limits on mass bins which contain no sources, to better illustrate the distribution of LAE masses, an example given in Figure 2.
From this point on I converted from using the filters the sources were discovered by, to using the redshift bins described in Sobral et al 2017 (z=2.5, 3.1, 3.9, 4.7, 5.4), as my method of separating my sources into separate plots. I also decided to stop the plot at mass values equal to 11, as there were only a few sources detected above this value that survived the purge of the SED examination, and those that remain have a relatively high chance of being ‘lower redshift’ interlopers that appear to be more ‘massive’ than they actually are. An example of my new SMF is Figure 3, for z~2.5.
The next step of my project was to fit Schechter functions to my data (the logarithmic version used is shown below), and thus obtain the values of alpha (the faint end slope parameter) and the Schechter parameters, characteristic mass and the normalisation point that best fits my SMFs at each redshift. The Schechter parameters were obtained, and the fit completed, using the curve_fit function of the python package SciPy. Since my plots are logarithmic in nature, the parameter values obtained were the logarithmic equivalents, i.e log(phi_0) for phi_0, however the desired schechter values were obtained via simple manipulation.
As an example, for redshift bin z=2.5+-0.1 schechter fit is as shown in Figure 4, it is worth noting that the slight change in shape of the data points from Figures 3 and 4, is due to the change in the number of bins from 10 to 15. The parameter alpha was chosen to be -1.4 for this plot, as previous fit attempts of higher values were less accurate. The schechter parameters values obtained from this particular fit were log(phi_0)~-4.26, and log(M_star)~10.1.
It is worth noting a flaw in my plotting technique at this point in time, as currently my schechter fit neglects the upper limits points of bins which contain no sources. This will hopefully be rectified in the next week, by setting a ‘invisible data point’ at a negligible mass function value e.g -9, with error bars that extend up to to the upper limit, this should allow the schechter fit to account for these upper limits without their existence having a drastic effect. An example of this flaw can be seen in the following figure of the SMF of redshift bin, z=5.4+-0.5.
The next property of the LAEs I wished to investigate was their stellar mass density (SMD), and how this evolves with redshift. Stellar mass density can be obtained by integrating the schechter fit of an SMF, the schechter fit can be integrated analytically using the following equation, which uses the incomplete gamma function, and the parameters obtained from the schechter fit.
Using this integration technique on the respective alpha, and schechter parameters of each redshift bin, I was able to produce the following stellar mass density against redshift plot.
The next step after obtaining this stellar mass density plot, is to get errors for each point. After a failed error propagation attempt, I decided to use the Monte Carlo method to obtain errors. When I obtained the schechter parameter values from the Curve_fit function, I was also provided with the 1 sigma standard deviation values in the each of the parameters. Using these standard deviations, and the actual values of each parameters (the parameter values used as the mean), I was able to model both parameters, or the logs of the parameters in this case, as gaussians (see examples below) and by taking the 16th, 84th, and 50th percentiles respectively I was able to obtain estimates of the best value for each parameter as well as a 1 sigma error, using the statistical 68 rule.
I also applied an error to my alpha parameter value due to its relatively arbitrary status of my just picking its value by how I deemed the fit to look. The error was obtained by modelling the alpha value as a uniform distribution between -1.3 and -1.5, as to keep -1.4 as the mean but also to include -1.3, which was my initial alpha value.
Using all of the errors my final stellar mass density plot appeared as follows:
As a preliminary examination of this plot we can observe that the density of LAEs remains approximately constant across all redshift, the actual behaviour shall become clearer once I split the redshift bins back into the individual redshifts of the filter bands used to detect each source, which is something I plan to do. Also, the catalogue which I had used initially has been recently updated, and so I shall be running these new mass values through my code, however it is unlikely that this will have much of an effect.
The aims for the following week, will be to edit my schechter fits such that they account for the upper limit points in my SMFs. Also, I plan to split up the redshift bins into their respective filter band redshift values, as this will provide us with more data points on our SMD plots, and thus will provide a more detailed outlook on how Lyman-alpha emitter stellar mass density evolves with redshift. Following on from that will be comparing these stellar mass densities at each redshift to the stellar mass density of all galaxies at these redshifts, and thus will allow me to explore how the proportion of LAEs to all galaxies has evolved in this time period.
Thank you for reading this blog post, this internship is a great opportunity for me, and it gives me great pleasure to be able to share my progress.