To construct our mock catalogs, we require a method for generating mock galaxies which are distributed according to some galaxy clustering model, for reasons detailed in Section 3.1. We should therefore consider how best to include such a model in our simulations before constructing them. Ideally, we would incorporate galaxy clustering in our simulations by populating dark matter halos produced in N-body simulations with galaxies in such as way as to reproduce the luminosity and color dependence of clustering (e.g. Skibba & Sheth, 2009). Since we do not have N-body simulations to work with, we take a simpler approach to modeling galaxy clustering. This is accomplished by placing simulated galaxies into “groups” of fixed comoving spherical volume. Each group contains 20 galaxies which are
uniformly distributed within a spherical volume of comoving radius 288kpc/h(i.e.,
the comoving volume of each group is 0.1 (M pc/h)3). Projected onto the sky, the
Figure 3.1 Redshift distribution of generated groups (black histogram) and the expected redshift distribution of groups if they are uniformly distributed in volume (blue dotted curve).
it is 1 arcminute, and at a redshiftz= 0.1 it is 3.4 arcminutes. This model of galaxy clustering is not meant to mimic true galaxy clustering properties; rather, we chose it to allow us to simply illustrate the effects of clustering on our method.
Having chosen to place galaxies into groups of fixed comoving size, we are ready to populate a simulated volume with them. A circular patch of sky 1.56
degrees in radius, spanning the redshift range 0 ≤z ≤ 1, forms the volume which
we populate with mock galaxies. It takes the form of a cone, and is centred on the observer. Groups are deposited at random within this cone and their redshifts are calculated accordingly. The number density of groups is 8.15×10−3 (h/M pc)3.
Before proceeding further, we check that the redshift distribution of our groups matches that of a population of objects which are uniformly distributed in volume. Figure 3.1 shows the redshift distribution, plotted as the black histogram, of our generated groups. If they are indeed distributed uniformly through the survey volume, their redshift distribution should be described by
dN(z) dz = 4π 20 fskyχ 2(z) c H(z) Z ∞ Llim φ(L)dL, (3.1)
where χ(z) is comoving distance, H(z) is the evolution of the Hubble parameter, fsky is the solid angle subtended by the cone,φ(L) is the input luminosity function
(which we define below),Llimis the luminosity of the faintest simulated galaxy (also
defined below), and the factor of 1/20 accounts for the fact that each group contains 20 galaxies. It is clear from figure 3.1 that the redshift distribution of generated galaxies is consistent with equation 3.1.
Now that each group has a sky position and a redshfit, we give its 20
member galaxies a position within a spherical volume which extends 0.1 (M pc/h)3
around the group centre. Galaxy sky positions and redshifts are computed accord- ingly. As the number density of groups is 8.15×10−3
(h/M pc)3, it is easy to see that
the mean density of galaxies is 0.163 (h/M pc)3. Since the density of galaxies within
a group is 200 (h/M pc)3, the groups are 1227 times denser than the background.
To check that our galaxies are uniformly distributed in volume, we compare their redshift distribution to that of a population which is uniformly distributed in volume. The black histogram in Figure 3.2 shows the redshift distribution of our mock galaxies. This should be compared to the redshift distribution of a population of galaxies uniformly distributed in volume, which is plotted as the dotted blue line. Such a distribution is described by
dN(z) dz = 4π fskyχ 2(z) c H(z) Z ∞ Llim φ(L)dL. (3.2)
(All quantities in are as in equation 3.1.) We see from figure 3.2 that the redshift distribution of our galaxies is consistent with the expected one.
Our galaxies have sky positions and redshifts; they now need luminosities. Luminosities for each galaxy are drawn according to a Schechter luminosity function
which has parameters φ∗
= 1.54×10−2
(h/M pc)3, M∗
r =−21.44, andα =−1.04.
Figure 3.2 Redshift distribution of generated galaxies (black histogram) and the expected distribution of galaxies if they are uniformly distributed in volume (blue dashed curve).
(2003b), after having been evolved to the mean redshift (z = 0.594) of our Mg II
absorption sample sample by the prescription of Linet al.(1999). We shall hereafter refer to it as the “input luminosity function.” To ensure that the input luminosity function produces the average density of galaxies quoted above, we set a lower limit for generated luminosities of L/L∗ = 10−4; this corresponds to M
r =−11.4.
Once their luminosities have been assigned, the galaxies’ absolute r−magnitudes
are calculated according to
Mr=M ∗ r −2.5 log10 L L∗ . (3.3)
In the top panel of figure 3.3, we display the absolute magnitude distribution of our
generated galaxies. Note that there is a sharp cut-off in counts at Mr ≈ −11.5;
this is the result of the luminosity cut-off we imposed above. The bottom panel of figure 3.3 compares the luminosity function of the galaxies we generate (shown as the data points) with the input one (shown as the solid black line). We estimate the luminosity function for our generated galaxies from their absolute magnitude distri-
Figure 3.3 Absolute magnitude distribution (top) and luminosity function (bottom) of generated galaxies. In the bottom panel, our input luminosity function is plotted as the solid line.
limit, each generated galaxy could be seen throughout the entire simulated cone; thus each receives the sameVmax weight, which in this case is the cone’s volume.
Once all galaxies have redshifts and absolute magnitudes, their apparent
r−magnitudes can be determined from
mr =Mr+ 5 log10 dL(z) 10 pc , (3.4)
Figure 3.4 Apparent magnitude distribution of generated galaxies.
galaxy generation code. The galaxies’ apparent magnitude distribution is illustrated
in figure 3.4. Galaxies which have mr ≤22.5 have been shaded grey. We have also
included as the smooth curve a plot of the expected apparent magnitude distribution. This prediction is given by (c.f. equation 2.2)
N(m) = 4π fsky Z ∞ 0 c dz H(z)χ 2(z) φ m−5 log10 dL(z) 10 pc , (3.5)
where fsky, χ(z), φ, and dL(z) are as given above. Note that our luminosity cut-
off leads to a drop-off in the distribution for mr > 27.5. The faintest apparent
magnitude we expect ismr = 32; this is the apparent magnitude of a galaxy which
has luminosityL/L∗ = 10−4 and a redshift z= 1. Indeed, we see no galaxy counts
beyond this point.
Each mock galaxy now has a redshift, an angular position on the sky, an
absolute magnitude Mr and an apparent magnitude mr; this is all the data we
need to assemble counterparts to the absorber and reference samples of Chapter 2. The catalog which contains all this information shall be referred to as the “full mock catalog.” We compile a separate catalog which contains information only for
galaxies which have apparent magnitudes less than the SDSS limiting magnitude
of mr = 22.5; this catalog will be referred to as the “apparent magnitude limited
mock catalog.” (The galaxies in the apparent magnitude limited mock catalog have been shaded grey in Figure 3.4.) While the full mock catalog retains all information contained in the apparent magnitude limited mock catalog, we found a separate
catalog of galaxies with mr ≤ 22.5 to computationally inexpensive to create and
keep, and useful when compiling the catalogs to be detailed in the next section. Before moving on to choose either a mock Mg II system or a mock refer- ence QSO and compile catalogs of galaxies projected near them, we check that our apparent magnitude cut has not altered the redshift distribution of galaxies. To do so, we plot in figure 3.5 the redshift distribution (black histogram) of galaxies
having mr ≤ 22.5 and compare it to that expected for a population of uniformly
distributed galaxies (blue dotted curve). This prediction is calculated from
dN(z) dz = 4πfsky c χ2(z) H(z) Z ∞ Llim(z) φ(L)dL, (3.6)
where χ(z), H(z), fsky, and φ(L) are the same as in equation 3.1, and Llim(z)
denotes the luminosity of a galaxy at redshift z which hasmr = 22.5. There is good agreement between the two curves.
We also investigate the impact of an apparent magnitude cut atmr= 22.5
on the absolute magnitude distribution of galaxies, plotted in the top panel of fig- ure 3.6. The cut has dramatically reduced the number of galaxies withMr>−21.0, as would be expected if galaxies with faint apparent magnitudes are removed from
the sample, but leaves the distribution unaltered for Mr ≤ −21.0. We also note
that the peak in absolute magnitude counts occurs at Mr ≈M
∗
r, which one would expect. To check that this absolute magnitude distribution yields the correct lu-
Figure 3.5Redshift distribution of generated galaxies once an apparent magnitude cut at
mr= 22.5 has been applied. The blue dotted curve is the expected redshift distribution of
galaxies if they are uniformly distributed in volume and havemr≤22.5.
assigned to each galaxy will depend on its absolute magnitude; our apparent mag- nitude cut means that faint galaxies can no longer be seen throughout the same
volume as bright ones. For our apparent magnitude limited mock catalog, Vmax(M)
is given by Vmax(M) =fsky 4π 3 χ 3(z max) (3.7)
where χ(z) and fsky are the same as in equation 3.1, and zmax is the smaller of
the maximum redshift to which a galaxy could have been seen or z = 1.0. The
resultant luminosity function is shown in the bottom panel of figure 3.6; the input luminosity function is plotted in this panel as the solid black curve. The good agreement between the estimated and input functions verifies that our apparent magnitude cut has not affected the galaxies’ luminosity function. With our vetted full and apparent magnitude limited mock catalogs in hand, we are ready to chose mock Mg II absorption systems and mock reference QSOs, and find galaxies in the fields surrounding them.
Figure 3.6 Absolute magnitude distribution (top) and luminosity function (bottom) of generated galaxies, once an apparent magnitude cut atmr= 22.5 has been applied. In the