SC4K: M* and UV LFs- Part 3, Weeks 4, 5 and 6

This is the eagerly anticipated third and possibly final instalment of the world renowned SC4K: M* and UV LFs blog.

In the space of time since the last blog post, I have accomplished the goals that I set myself at the end of the last post and more, and this blog post will document those steps.

Firstly, I did in fact managed to edit my code such that, when applying a Schechter fit to a mass function, the upper limits were accounted for by creating ‘invisible’ data points, shown as stars in Figure 1, at negligible phi values (in this case at -4 of whatever the phi value of upper limit), with error bars that extend up to the upper limit; these ‘invisible’ points and errors were then included when fitting, thus allowing us to account for these upper limits without giving them the same weight as actual bins that contain sources.

Figure 1: The SMF for redshift bin z=5.4=+-0.5, now with the upper limits being accounted for when fitting the Schechter function, and with the ‘invisible points depicted as stars.

And then without the ‘invisible’ points, the SMF appears as follows in Figure 2.

Figure 2: The SMF for redshift bin z=5.4=+-0.5, now with the upper limits being accounted for when fitting the schechter function, with the ‘invisible’ points removed.

As stated in the last blog, we desired to create SMFs for each individual redshift slice, an example of this is shown in Figure 3 as the SMF for the filter IA484, or redshift slice z=2.98.

Figure 3: The SMF for redshift bin z=2.98, corresponding to filter IA484, plotted and fitted like the redshift bins shown previously.

It is worth noting that at this point in the internship I was provided with an updated catalogue of mass values for the LAEs, so any changes in the shape of SMFs may be due to that fact. (Credit: Sergio Da Graca Santos)

A problem with fitting to stellar mass functions is that at a point when going to lower and lower masses, a dip is observed; the issue is that this dip is not caused by a scientific or characteristic reduction of the number of LAEs at these masses, but instead by the effect of completeness, i.e. the less massive and fainter LAEs are much less likely to be observed than when compared to their more massive counterparts. Up until now when fitting, we have used the range 9-11 log solar masses, for all redshift bins and slices, but to account for the completeness ‘dip’ in each individual slice/bin, we decided to set a minimum mass, by which each slice/bin shall be fitted to. This minimum mass is depicted in the following figures by a vertical dashed line.

Figure 4: The SMF of redshift slice z=3.15, corresponding to filter IA505, with the minimum mass line now shown by a vertical dashed line, in this case at log(M_star )= 9.4.
Figure 5: The SMF of redshift bin z=2.5=-0.1, with the minimum mass line now shown by a vertical dashed line, in this case at log(M_star)=9.1, the masses have been extended down to 8.5, in order to show the dip.

After this I produced a new stellar mass density vs redshift plot, now including both the new fitted data using the new minimum mass method, and all of the redshift bins and the redshift slices, shown as yellow stars and blue circles respectively in Figure 6.

Figure 6: The initial SMD created, including both redshift bins and redshift slices, showing a strange phenomenon whereby some redshift bin densities, appear to be above the densities of its constituent slices.

As you may have noticed there is an unusual phenomenon for the redshift bins containing more than one redshift slice, as they appear to have a much higher density than the redshift slices they comprise of.

Is this a new breakthrough in how we think of space?

The answer is of course no. In fact it was a simple error in how the calculating of the volume used to create the phi values of the redshift bins that was the problem. For a redshift bin, the volume surveyed for each source is the sum of the surveyed volumes of its constituent redshift slices, but in my divine wisdom I had decided to find the average of the volumes instead. This was a very frustrating error, which puzzled me for hours, as my fellow interns could attest to.

Correcting this mistake I was able to produce a much more realistic SMD, as shown below in Figure 7.

Figure 7: The revised stellar mass density plot for all redshift slices and bins after fixing the issue with the volume use in the phi calculation, thus creating a more realistic plot.

As can be seen, the redshift bins’ density values now for the most part lie more neatly within the centre of the range of its constituting redshift slices’ density values, as was expected. As can be seen the density values for LAEs seemingly stay approximately constant over redshift.

We then decided that it would be useful to also examine how the schechter parameters, M* and phi*, themselves evolved with redshift, and thus we created the following figures.

Figure 8: A plot depicting the evolution with redshift of the Schechter parameter phi*, showing an initial slope from z~2.5 to a plateau at z~4.5.
Figure 9: A plot depicting the evolution of the Schechter parameter M*, showing a seemingly consistent value of ~10.6.

By observing Figures 8 and 9 we observed that M* seems to stay relatively consistent over redshift, and phi* appears to show a negative slope from z~2.5 until z~4.5 whereby it seems to reach a plateau.

Then, it was decided that the best way to display how the SMFs evolve with redshift, was to plot them all together as a grid, similar to those seen in the SC4K paper (see Figure 8). The first attempt at the resulting grid of all of the SMFs for all of the redshift slices and bins is shown in Figure 10.

Figure 10: The first stellar mass function grid plot, showing all 13 redshift slices and the 5 redshift bins.

Figure 10 represents my first attempt at this kind of plot, but after a discussion with my supervisor I realised a few flaws that I could correct in order to make it paper ready. The flaws with this plot include colouring the completeness points when they should really be colourless to show they are much less important, the Schechter fit not extending to the mass minimum line, and the data points to the left of the line being the same colour as those that the Schechter fit accounts for, when fading them could help show that they are not including when fitting.

After addressing a couple of these flaws, the grid produced was as follows in Figure 11, with a more complete edition being shown in Figure 12.

Figure 11: An intermediate SMF grid plot where we also increased the number of bins to better evaluate the source spread, and also the non-completeness points, now have been coloured white, to better illustrate their insignificance, and the Schechter fit extends to the mass minimum line.
Figure 12: The final grid plot including both redshift slices and redshift bins. It was at this point that it was decided that in order to show to data more clearly, the redshift slices and bins should be split.

It was then decided that combining both the redshift slices and the bins was unnecessary in such a large plot, thus I split them, keeping the individual redshift slices displayed as in Figure 12, but for the redshift bins we decided that showing the change in the Schechter fit was the only thing necessary, the resulting plots are shown in Figure 16 and 17.

In the midst of me working on the grid, I also improved the plots of the stellar mass density’s and Schechter parameters’ evolutions with redshift, shown in the following figures. They now clearly show which data points have been found using which fixed values. The shaded region shown in the Schechter parameter plots show the values of these parameters up to redshift, z=5, found by Iary Davidson et al. in 2017 for galaxies in the COSMOS survey (see here for the paper).

Figure 13: The updated log plot of the evolution of the Schechter parameter M* as a function of redshift, showing both which values are fixed in the creation of a data point, and how my data compares to that of Davidson 2017.
Figure 14: Log plot of the evolution of the phi* Schechter parameter over redshift, when compared to literature data obtained from Davidson 2017.

As can be seen quite clearly, my data shows an evolution of the Schechter parameters agrees quite nicely with the results found by Davidson, especially for phi* in Figure 14, as the slope going to z~4.5 is seen in both, and for M* they found a relatively constant M* as well.

The stellar mass density plot stays relatively similar as I have not yet been able to compare it to data found previously, also from the Davidson paper and more, but this is in progress.

Figure 15: The current version of the stellar mass density plot, showing the evolution of the stellar mass density of LAEs over redshift, better illustrating which values are created using fixed M* values.

It was after the completion of these plots that I was provided with the completeness functions for 2 of the 3 filters that I had to exclude previously, NB392, and NB816, this allowed me to create SMFs for them and improve my results. However, while NB816 had a simple completeness function (i.e. completeness against Lyman-alpha flux) NB392 had multiple, depending upon which area of the survey it was observed in, this is due to the fact that observed in the CALYMHA survey (see here), and as such I had to include a clause in the code such that if the source was detected by NB392, the COSMOS ID of the source is used to determine which completeness function must be computed.

Finally, Figures 16 and 17 show the new and improved stellar mass function grid plot, including all the redshift slices possible (including NB392 and NB816), and the SMF for the redshift bins depicting only their respective fits.

In the new SMF grid, the fit line and shaded region are red for plots that use fixed M* values as well as fixed alpha values, and they are blue for fits using only fixed alpha values. 

Figure 16: The current iteration of the SMF grid plot, with the red shaded fits illustrating that these fits were created using a fixed M* value. The additional SMFs for filter bands NB392 (z=2.22) and NB816 (z=5.71) have been included in this version.

You might notice that the difference between the completeness and non-completeness points for the newly added NB392/z=2.22 slice, are at lot less than for the rest of the slices. This issue highlighted a problem in my selection process up until this point. I should have set a completeness limit for sources as a means to eliminate sources that have such a low completeness that they may be skewing the data. After a discussion with my supervisor we decided that a 30% completeness limit would eliminate the most disturbing of sources. As such when observing these last plots, take them with a pinch of salt, as they shall be altered after I apply this limit.

Figure 17 was also created before the addition of the completeness limit, but the general idea of the plot can be gathered, if not the scientific result.

Figure 17: This is the preliminary version of the plot comparing the fits for all the redshift bins, however this plot shall need changing to account for the 30% confidence limit, and as such needs to be taken with a pinch of salt.

The next steps are as follows: Apply the 30% completeness limit to all of the data, include the data for the new redshift slices into the SMFs and thus create Schechter parameter and stellar mass density plot. Also, we decided that an interesting thing to investigate is the SMF of only high luminosity galaxies (i.e L > 1042.8 erg/s) to see whether they behave any differently. I have also recently obtained a dataset of mass values of the SC4K cut except these were determined, not with SED fitting with MAGPHYS, but instead used CIGALE (a python Code Investigating GALaxy Emission), not the best acronym but still very cool, and it will be interesting to see how much their results differ from those I have used.

I am very excited to write my first, first author paper that will be published, and I must say thank you to my supervisor Dr. David Sobral for this opportunity, and also to the Ogden Trust for giving me the means to take it.

Finally, I have to mention the great fellow interns that I have been working alongside during this internship: Emma Dodd, Heather Wade, Harry Baker, Cass Barlow-Hall, and Amaia Imaz Blanco. It has been great to get to know you over the last few weeks!

This may be my last blog or not, it depends on how long it takes me to write my paper alongside my literature review for my Masters.

Thank you very much for reading.

-Josh Butterworth

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