The Extragalactic Radio Background

I have recently published an article with colleagues from Manchester University on an updated estimate of the Extragalactic Radio Background (EGRB) or Cosmic Radio Background. The article can be found here: https://arxiv.org/abs/2004.13596. What follows is a brief summary of what the EGRB is, how we have estimated it and the updates we made to a previous estimate by Protheroe and Biermann (1996) which can be found at https://arxiv.org/abs/astro-ph/9605119. Further details on our method and the mathematics can be found in the paper.

The EGRB is the total contribution of radio emission from Star Forming Galaxies (SFG) and Radio Galaxies to the radio sky averaged into units of per steradian. It is difficult to measure the EGRB across a wide range of radio frequencies, $10^3 - 10^{11}$Hz, because the ionosphere prevents measurements at low frequency and emission from our own Galaxy acts as a foreground. In order to take measurements at high frequency the Galactic foreground has to be modelled and removed which is a non-trivial task.

Estimates of the EGRB at discrete high frequencies can be made by integrating source counts. Source counts are the total number of sources with a given flux density in the universe at a given frequency. Hence, the integral underneath source counts is the total intensity from the population of $\textit{observed}$ sources at a given frequency and this can easily be given per steradian. I have emphasised the word $\textit{observed}$ in the previous sentence to highlight that estimating the EGRB using this method is limited by the extent of radio surveys which are often incomplete with limits on the minimum flux density, sky coverage or source angular size. You can read more about source counts through the references in both papers linked at the top of this article and in 'Galaxy Formation' by Malcolm Longair.

The importance of accurately estimating the EGRB is apparently obvious from our inability to access it via measurement. It is also important for studies of the early universe as it acts as a foreground to measurements of for example the Global 21-cm Signal ( see S. Fulanetto et al. 2006, https://arxiv.org/abs/astro-ph/0608032). It also causes the attenuation of Ultra High Energy Photons (UHEP) with the attenuation length determined by the intensity of the background (see papers linked at the beginning of the article and references therein). The Global 21-cm Signal can help us understand the conditions of the early universe and the period in which the first stars and galaxies began to form and UHEPs if detected will help us understand the origins of Ultra High Energy Cosmic Rays.

In order to begin to estimate the EGRB we need to estimate the spectra, across the band of interest, of typical galaxies in both dominant populations, SFGs and RGs. The radio spectra of SFGs is dominated by synchrotron and free-free processes where as the spectra of RGs is dominated by synchrotron emission and absorption alone. A detailed discussion of the mathematics used to determine the relative contributions of each radiative process in each source can be found in our paper and in 'Radio Astrophysics' by A. G. Pacholczyk. 

It is not enough simply to know the mathematics used to produce the typical spectra for the populations. We also need estimates of typical values for variables like the magnetic field, electron density and spectral index that inform the spectra. An accurate model of the typical geometry of galaxies in each population is also needed.

For our typical SFG spectrum we use the spiral galaxy M51 and it's associated parameters taking this to be representative of the population. Where Protheroe and Biermann took informed estimates of SFG properties we use direct observations of the spiral galaxies magnetic field, spectral index and other parameters to produce our typical spectrum. We also use a radially dependent model of M51 where Protheroe and Biermann took their galaxy to be uniform in structure. The radial dependence of our model and of the parameters we use allow us to account for, with our geometry, the bulge and spiral arms commonly found in SFGs. Protheroe and Biermann's model of the geometry was, on the other hand, a flat cylindrical model with a fixed radius. The resultant spectrum we have calculated for M51 is shown below with an image of the galaxy itself.

Also shown in the plotted spectra are observational measurements of the radio luminosity of M51 with which our theoretical model agrees well. The large errors at low frequency are as a result of large errors in the thermal electron density we used for M51 and the strong dependence of free-free absorption, which is dominant at these frequencies, on this variable. Again a more detailed exploration of this can be found in our paper.

For our typical RG spectrum we took a different approach. Rather than basing our estimate on radial measurements of one galaxy we took all of the galaxies in the 3CRR radio galaxy survey and averaged their spectra into a typical spectrum representative of the population. Since each galaxy in the survey has it's own spectral index measurement this allows us to account for variation of the spectral index across the population. As with SFGs Protheroe and Biermann took informed estimates of RG properties to produce their spectrum. They also approximated the geometry of a typical RG to a sphere of a given area. In contrast we used a double cone structure representative of an FRII type RG which are thought to be the most common and brightest sub-population of RGs. The geometry we used and our typical RG spectrum are shown below. The typical RG spectrum is shown as a black line amongst our calculated spectra for each of the 170+ 3CRR galaxies. The errors on our typical RG spectrum are given simply by the process of averaging.

With our typical spectra both calculated we can go on to calculat the EGRB. In doing this we are imagining that the spectrum of all SFGs and all RGs follow the shape of our typical spectra. We use luminosity functions adapted for luminosity evolution to expand our models to high redshift and sum up the contributions from each population to the EGRB. We have updated the luminosity functions used in the calculation to be more representative of the currently accepted theories of cosmology and adapted the luminosity evolution appropriately.

The luminosity function tells us the number of sources of a given luminosity at a given redshift per unit co-moving volume per unit luminosity for a discrete frequency. We therefore multiply this by a value of luminosity at the relevant discrete frequency and a normalised spectrum with it's shape determined by our typical spectrum redshifted to $z$ for each population. We then integrate this product over luminosity at the discrete frequency of the luminosity function. This is followed by translating this back to redshift zero, multiplying by the differential co-moving volume, translating the resultant luminosity to an intensity and integrating over redshift. The redshift limit is determined by considerations of the Epoch of Re-ionization and is set at 10. The final step is to divide this integral by 4$\pi$ to get the intensity per steradian on the sky. If we do this for each population, RG and SFG, then the sum of these components will give us the total background and this is shown below.

The above graph shows the background as calculated in our work as a black line with the one sigma error as a grey shaded region. We also show the contribution as we calculated it from RGs and SFGs as blue and red lines with associated one sigma errors. The results presented in Protheroe and Biremann's work are also plotted as dashed and dotted lines. The various symbols shown are estimates of the EGRB at high frequencies from integration of independent models to source counts data. The details of this are discussed in the paper but we find much better agreement with the source counts than can be found with Protheroe and Biermann's models.

In the paper we discuss in some detail the use of this spectrum in estimating the attenuation length of UHEPs and we go into detail on some of the approximations we have made. It can clearly be seen that our improved models for the typical spectra of RGs and SFGs as well as our efforts to improve of the cosmology used have led to improvements in the estimate of the EGRB.

Thanks for reading!