3.2.1
Calculation of Number Counts
When calculating the galaxy number counts, it is standard practice to plot galaxy num- ber counts relative to the predictions for a spatially flat, non-evolving Universe; also known as Euclidean normalised number counts. Number counts are also known as source counts. A Euclidean space is the three dimensional coordinate system used to describe the Universe at non-relativistic speeds.
Galaxy number counts are counting the number of galaxies in a given survey with fluxes between S and S + dS, where dS is some small increment of S. The flux of one star/galaxy is given by Equation 3.1, where S is the flux of the source, L is the luminosity of the source and r is the distance from the source to the observer.
S= L
4πr2 (3.1)
Taking the derivative with respect to r gives:
dS dr = − L 2πr3 (3.2) dS dr ∝ L r3 (3.3)
Equation 3.3 can be re-written as:
dr dS ∝
r3
L (3.4)
Also from Equation 3.1
3.2. Number Counts Theory 71
r ∝ S−1/2 (3.6)
To find the number of galaxies (with identical flux) within a given radial shell from the observer, we need the volume of the shell. The area of the sphere of the shell is given by 4πr2, where r is the radius of the sphere. The thickness of the shell is given by dr. The volume can be approximated by the area multiplied by the thickness of the shell. Thus the volume of the shell is 4πr2dr. Hence the number of galaxies in the shell is given by Equation 3.7. where ρ is the density of the galaxies.
dN= ρ×4πr2dr (3.7)
dN ∝ ρr2dr (3.8)
We need to know the number of galaxies between S and S + dS, hence we need the equation in terms of dNdS. Multiplying both sides of Equation 3.8 by dSdr gives:
dNdr dS = ρr
2drdr
dS (3.9)
From Equation 3.4 we know dSdr ∝ r
3 L dNdr dS = ρr 2drr3 L (3.10)
Dividing both sides by dr
dN dr dr dS ∝ ρ r5 L (3.11)
The two dr terms on the LHS panel cancel. Using Equation 3.6 where r ∝ S−1/2gives:
dN dS ∝ S
−5/2
The above equation gives the number of galaxies in the flux range S to S + dS, for a flat Euclidean Universe; and is known as Euclidean differential source counts (Serjeant, 2010).
For galaxy number counts the flux S is plotted against dNdSS2.5 per flux per unit area. If we live in a flat Euclidean Universe with non-evolving galaxies, then the differential galaxy number counts, as described above, when plotted on a log-log plot, should all lie along a horizontal line.
As the Universe is expanding, one would expect a small deviation from a horizontal line. What observational cosmologists are testing for is a much larger deviation from a horizontal line, which is caused by galaxies evolving. Galaxy evolution models are tested against the observed galaxy number counts. Galaxy evolution models are made up of percentages of different types of galaxies. By comparing observation with model, the evolution of the different types of galaxies can be probed (Patanchon et al., 2009).
3.2.2
Reliability
When performing source extraction on a mosaicked image, some non-galaxy sources will be mis-identified as real sources. Fine tuning the parameters in the source ex- traction will minimise fake source detection, but some are likely to remain. These mis-identified galaxies are normally due to noise in the image. Galaxy number counts need to be corrected for these false positive sources, which have been mis-identified as galaxies. These false positive sources will increase the number counts in the flux bins with the fake sources. There are several ways in which reliability can be tested. The most common way is to assume that there are as many negative fake sources as there are positive fake sources. By multiplying the mosaicked image by −1, the negative sources become positive and vice-versa. This negative image is then run through the source extraction software with the same settings as for the positive image. Though simple to perform, this method does rely on the assumption that there are as many negative fake sources as positive fake sources. If there are some particular artefacts
3.2. Number Counts Theory 73
in the image, this assumption may not be true (Clements et al. (2011), Pearson et al. (2010), Pearson et al. (2014) and Murata et al. (2014a)). A second method is to use Monte Carlo simulations by creating a simulated noise realisation of the mosaicked image. Each simulated noise realisation is run through the source extraction software, again with the same settings. This is performed multiple times. This method assumes that all fake sources are caused by noise but will not find those fake sources caused by instrumental effects (Hatsukade et al., 2011). A third method is to visually inspect the mosaicked image at the positions of the point sources, which have a point source probability below a certain level. This method would be quite time consuming for large survey areas, and/or multi-wavelength filter sets for galaxy catalogues, as well as including a human error into the point source detection (Scott et al., 2010a). A fourth method is to create a difference image from two different observations of the same area. Then this difference image is run through the source extraction software. Fake sources should be detected as sources in the difference image (Oliver et al., 2010). This requires an absence of astrometry shifts between the observations. For the reliability correction of the IRAC Dark Field, ELAIS North 1 and AKARI Deep Field South number counts for the work in this thesis, the first method has been used. The second method was not used, due to the fact that it assumes that all fake sources are caused by noise, this is known not to be the case for AKARI e.g. AKARI is known to have Earth- shine light residual. The third method was rejected due to it being time consuming and because it could include a human error. The fourth method was not employed because it relies on the fact that there is no astrometry error between observations. It is known that AKARI data have astrometry error. Section 3.7.3 gives the reliability correction for both extragalactic fields.
3.2.3
Completeness
A further correction which is required to be performed on the number counts is to cor- rect for the galaxies that have not been detected during the source extraction. This is
called a completeness correction. This correction normally raises the faint end number count slope, because more faint galaxies will be un-detected in the source extraction, than brighter galaxies. It will particularly increase the numbers of galaxies close to the detection limit of the survey. By far the most common way to correct for completeness is to inject fake sources of a known flux and location into the mosaicked image and then run the image through the source extraction software, with the same settings as those for the original source extraction. By running Monte Carlo simulations of this i.e. re- peating this multiple time for a specific flux range, the percentages of injected sources detected can be calculated. The simplest flux range is to use the range of each flux bin used to create the number counts. One problem with this method is the detected source may not be the injected source, but a true source, which was already present in the image. There are several slightly different variations of this method. The first method is to inject just one artificial fake source into the mosaicked image and run this through the source extraction. This is repeated many times for each flux range (Hatsukade et al., 2011). The second method is to inject many fake sources simultaneously into the mosaicked image and then run this image through the source extraction. Again, this is repeated for each flux range. To inject many fake sources simultaneously, one requires a large survey area (Clements et al. (2011), Pearson et al. (2010), Pearson et al. (2014) and Murata et al. (2014a)). A third method is similar to the second method, but injects the fake sources at a distance greater than a specific distance (normally some multiple of the FWHM, of the instrument) away from any real point sources. This is done to avoid creating a fake source/true source blend, but will not correct for a two-source blend, as the previous methods do (Scott et al. (2010a) and Hopwood et al. (2010)). A fourth method does not use a simulated fake source; instead, a real source is created empirically from the mosaicked image, and the flux is scaled to be that of the flux range. This method is useful when the Point Spread Function (PSF) of the detector is highly irregular (Fazio et al., 2004). After completeness correction has been per- formed, i.e. the number counts have been scaled accordingly, it must be decided at what completeness percentage the number counts will be cut.