In the previous section, we demonstrated that the full population of galax- ies in our mock Mg II and mock reference neighbour catalogs have the luminosity function originally assigned to them. Since the galaxies which remain in the sample
after background subtraction should be a subset of these galaxies, we expect them to have a luminosity function consistent with the input one as well. We now apply the technique described in Chapter 2 to the two catalog pairs compiled in Section 3.3 to test whether or not this is in fact the case.
We first use our background subtraction procedure to estimate the ab- solute magnitude distribution of mock Mg II neighbours. Recall from Chapter 2 that to implement this method, we must calculate absolute magnitudes and pro- jected comoving distances for our mock galaxies based on the redshift of the mock absorber (or ghost mock absorber, in the case of the mock reference neighbour cata- log) in whose field they were found. Also recall that, due to the broad redshift range
spanned by absorbers (∆z = 0.45), our 3 arcminute search radius corresponds to
different projected comoving distances; therefore, we only keep galaxies which lie within the circle fully sampled by all absorbers. This circle has a radius of 878kpc/h. (Because we do not need to worry about seeing or blending effects in our mock cat- alogs, there is no inner annulus within which galaxies cannot be detected. This was not the case in Chapter 2 for our SDSS data.) By subtracting the resulting absolute magnitude distributions of the mock Mg II neighbour and mock reference neighbour catalogs, our method should isolate the true neighbours of mock absorbers.
The results of applying our technique to the two mock Mg II neighbour— mock reference neighbour catalog pairs compiled in Section 3.3 are displayed in Figure 3.9. The estimated absolute magnitude distribution of neighbours surround- ing absorbers chosen using the nearest neighbour and random selection methods are plotted in the top and bottom panels, respectively. The solid line in both panels plots our analytical prediction from equation 2.10, which we repeat here for clarity.
Nξ(M)≈Vξφ(M)
Z zmax(M) zmin(M)
Figure 3.9Background subtracted absolute magnitude distributions for each of the catalog pairs described in Section 3.3. The measured distribution for the nearest neighbour selection method catalog pairs is plotted in the top panel; in the bottom panel, it is plotted for the random selection method catalog pairs. In each panel, the solid line is the expected distribution and is calculated as described in the text.
Since we place galaxies into fixed spherical volumes of fixed comoving radius,Vξ for our mock catalogs is given by
Vξ = ∆tophat
Z 2R
0
drprpf(rp) (3.9)
where ∆tophat = 200/0.163, R = 288h−1 kpc (the radius of the spherical volume of
a group), and f(rp) = 3R 64 " p2(p2−16) ln 2 + p 4−p2 p ! + p4−p2(16 + 2p2) # . (3.10)
In this equation, p = rp/2R and rp denotes projected comoving distance. Equa-
tion 3.10, in turn, is found by evaluating
f(rp) = Z πmax 0 1− 3r 4R + 1 16 r R 3 dπ; (3.11)
the term in brackets is the correlation function of a spherical top-hat distribution (which has a value of 0 for r > 2R), and π = qr2−r2
p. Both of the measure-
ments plotted in Figure 3.9 are noisy, which is not very surprising given that the two catalog pairs contain 1/10 the number of simulated systems as exist in the full data catalog. We can clearly see, however, that the absolute magnitude dis- tribution of neighbours of randomly chosen absorbers is in much better agreement with our predictions than is the distribution of neighbours of absorbers chosen using the nearest neighbour selection method. Though the shape of the latter distribu- tion is smoother, its amplitude is much larger than expected. This implies that we find more galaxies surrounding absorbers chosen by the nearest neighbour selection method than we expect. In contrast, though neighbours of randomly chosen ab- sorbers have an absolute distribution whose shape is noisier, its amplitude matches the expected one.
We now apply ourVmax(M) weighting scheme of Chapter 2 to the absolute
magnitude distributions shown in Figure 3.9. Recall from Section 2.3.1 that the
Vmax(M) weight appropriate for our mock catalog pairs is given by equation 2.17,
which we repeat here for clarity:
Vmax(M) =π(r2max−r2min)
Z χmax χmin dχ Z zmax(M) zmin(M) dzabs dN dzabs . (3.12)
We saw in Section 2.3.1 that, when weighting by equation 2.17, the resulting distri- bution is proportional to the luminosity function. The constant of proportionality is ratio of the effective and full survey volumesVξ/Vsurvey. Our estimated luminos- ity functions are plotted in Figure 3.10. As in Figure 3.9, the top panel plots the luminosity function of neighbours of absorbers chosen using the nearest neighbour selection method; in the bottom panel, the luminosity function of neighbours of absorbers chosen using the random selection method is plotted. In both panels, the input luminosity function is included as the solid line. Aside from the noisiness of the estimates shown in Figure 3.10, we see that the luminosity function of galaxies surrounding absorbers chosen using the nearest neighbour selection method has a higher than expected amplitude. It is, however, the less noisy of the two measure- ments. While the estimated luminosity function of neighbours of randomly selected absorbers is noisier, its amplitude is in good agreement with the input one.
In Chapter 2, we applied our background subtraction method to data in a range of annuli which were smaller than our fully sampled one; this was done to probe how the signal-to-noise of the measurement depends on scale. Thus to fully test the reliability of our method, we should perform our analysis on our mock catalog pairs for a range of circles which fall within the fully sampled one. (We do not need to worry about seeing or blending effects in our mock catalogs, so there is no inner annulus boundary.) If our method is robust, we will recover the correct
Figure 3.10Estimated luminosity functions for each of the catalog pairs described in Sec- tion 3.3. The estimated function for the nearest neighbour selection method catalog pairs is plotted in the top panel; in the bottom panel, it is plotted for the random selection method catalog pairs. In each panel, the input luminosity function is plotted as the solid line.
absolute magnitude distribution and luminosity function for all scales considered. When considering these smaller scales, however, there is a subtlety which we must account for. Recall that, whether chosen via the nearest neighbour selection method or the random selection method, our mock absorbers were chosen to lie at the center of the field. We therefore know at all times the location of one of the galaxies in the absorber’s group: the absorber itself, which lies atrp = 0. This means that, as the size of our circle shrinks to zero, we find some non-zero number of galaxies, though none would be expected. Our analytical calculations must account for this if they are to provide accurate predictions.
If we shrink the size of our circle to zero, we will detect only the mock ab- sorbers themselves, since they were chosen to lie at the center of the field. They will have an absolute magnitude distribution which is described by (c.f. equation 2.10)
Nmock absorbers(M) =φ(M)Veff
Z zmax(M) zmin(M) dzabs dNabs dzabs (3.13)
wherezmin(M) andzmax(M) are the minimum and maximum redshifts, respectively,
to which a galaxy with absolute magnitudeM could be detected,dNabs/dzabs is the
redshift distribution of the mock absorbers, andVeff = 6.135 (M pc/h)3. Veff, in turn,
is the effective volume associated with a single galaxy in the simulation; it can be
derived from 1/n, where n is the average density of galaxies in our simulations.
Recall from Section 3.2 thatn= 0.163 (h/M pc)3.
As we increase the size of the circle, we include more galaxies which lie in the same group as the mock absorber. These neighbouring galaxies are correlated with the mock absorbers; this correlation is described by the correlation function
ξ(r). In the case of our mock catalogs, ξ(r) has the form of a spherical tophat
these neighbouring galaxies is (c.f. equation 2.10) Nneighbours(M) =φ(M)Vnbr Z zmax(M) zmin(M) dzabs dNabs dzabs . (3.14)
Herezmin(M),zmax(M), and dNabs/dzabs are as in equation 3.13; Vnbr is given by
Vξ= ∆tophat
Z 2R
0
drprpf(rp), (3.15)
where ∆tophat = 190/0.163, R = 288h−1 kpc, and f(rp) is given by equation 3.10. Essentially, this expression is identical to equation 3.8, but with ∆tophat = 190/0.163
instead of 200/0.163. ∆tophat = 190/0.163 here because we have considered the ab-
sorbers themselves separately, leaving 19 galaxies to account for with equation 3.14.
Obviously, as we increase the radius of our search circle, we detect more neighbours of mock absorption systems. However, the contribution from the ab- sorbers themselves has not gone away. Thus the total absolute magnitude distribu- tion we expect is given by
Nneighbours(M) = [Vnbr+Veff]φ(M) Z zmax(M) zmin(M) dzabs dNabs dzabs . (3.16)
If we had neglected to consider the mock absorbers separately, the absolute magni- tude distribution we predict would be given by
Nξ(M) = ∆tophat Z 2R 0 drprpf(rp)φ(M) Z zmax(M) zmin(M) dzabs(dN/dzabs) (3.17)
with ∆tophat = 200/0.163. (This is the expression that was plotted in Figure 3.9 and described at the beginning of this section.) Note that the two expressions 3.17 and 3.16 yield identical expressions forrp ≥576kpc/h, but very different expressions on scales whererp is closer to zero.
Figure 3.11 Background subtracted absolute magnitude distributions for each of the cata- log pairs described in Section 3.3, for multiple scales. From top to bottom, results for scales of 72, 144, 288 and 576kpc/hare shown. Panels on the left hand side plot the distributions from the catalog pairs compiled for absorbers chosen using the nearest neighbour selection method; on the right hand side, the distributions from the catalog pairs compiled for ran- domly chosen absorbers are plotted. In each panel, the solid line is the expected distribution and is calculated as described in the text.
In Figure 3.11 we show the results of applying our background subtraction
technique to our mock catalog pairs on scales of 72 (top), 144, 288 and 576 kpc/h
(bottom). In each panel, we plot our theory prediction (equation 3.16) as the solid curve. As one might expect based on our earlier findings, the estimated absolute magnitude distributions of neighbours of randomly selected absorbers best match our predictions. Note that the distribution at a scale of 576kpc/his less noisy than the distribution at a scale of 880kpc/hwas; this is because the quantityC (defined in Section 2.3.3) is higher and therefore the signal-to-noise is higher. We also note from Figure 3.11 that the estimated absolute magnitude distributions of neighbours of absorbers chosen using the nearest neighbour selection method have amplitudes much higher than predicted. Thus we find that, all scales considered, there are more galaxies surrounding absorbers chosen using the nearest neighbour selection method than we expect.
Figure 3.12 plots the estimated luminosity function of galaxies in our catalog pairs for the same scales as above. These estimates are derived by ap-
plying our Vmax(M) weighting scheme to the distributions in Figure 3.11; note
that here, though, the constant of proportionality for the weighting scheme is (Vnbr +Veff)/Vsurvey rather than Vξ/Vsurvey. We see from Figure 3.12 that our method recovers the input luminosity function when it is applied to the catalog pairs compiled for absorbers which were randomly chosen. The estimated luminos- ity function of galaxies surrounding absorbers chosen using the nearest neighbour selection method yields the correct shape—at least for the faint end—but not the correct amplitude. This is in agreement with our previous findings.