On the Dearth of Ultra-faint Extremely Metal-poor Galaxies

The galaxy LF, Φ(M), is defined as the number of galaxies with absolute magnitude M, per unit volume and unit magnitude. Figure 4(a) shows the LF in the r-band determined by Blanton et al. (2005) from the SDSS-DR2 data. The number of galaxies in a survey fainter than a given limiting magnitude, M_lim, is

$$\begin{eqnarray}&&N(M\gt {M}_{\mathrm{lim}})={\int }_{{M}_{\mathrm{lim}}}^{{M}_{1}}S(M){\rm{\Phi }}(M){dM},\end{eqnarray} \tag{ 6 }$$

with S(M) the selection function that provides the effective volume that is sampled by the survey. Equation (6) assumes S to depend only on the absolute magnitude of the Galaxy, which is reasonable in our case, and simplifies the treatment. This assumption will be relaxed later on. Equation (6) also assumes the existence of a limit for the magnitude of the faintest galaxy, M₁.

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In the case of a volume-limited sample, S is just the volume of the sample, and so independent of the absolute magnitude. In the case of an apparent-magnitude-limited survey, where all galaxies brighter than the apparent magnitude, m_lim, are included, S(M) is the volume where galaxies of magnitude M have apparent magnitude m_lim or brighter, i.e.,

$$\begin{eqnarray}&&S(M)=V(M)=\displaystyle \frac{{d}^{3}}{3}{\rm{\Omega }},\end{eqnarray} \tag{ 7 }$$

with Ω the solid angle covered by the survey and d the maximum distance at which a galaxy of magnitude M can be observed, i.e.,

$$\begin{eqnarray}&&\mathrm{log}(d)=\displaystyle \frac{1}{5}({m}_{\mathrm{lim}}-M)-5,\end{eqnarray} \tag{ 8 }$$

with d in Mpc. The SDSS spectroscopic survey was designed to be apparent-magnitude-limited, with a limit in the r-band given by

$$\begin{eqnarray}&&{m}_{\mathrm{lim}}=17.77.\end{eqnarray} \tag{ 9 }$$

In practice, all surveys have a limit in surface brightness. Blanton et al. (2005) work it out for the SDSS, showing that the completeness of the spectroscopic survey decreases drastically for an average surface brightness, SB_r, fainter than 23 mag arcsec⁻², reaching only 10% completeness at 24 mag arcsec⁻². This bias against low-SB objects is particularly severe for QXMPs. Galaxies fainter than the limit in Equation (2) may, in principle, have any surface brightness. However, faint galaxies tend to have a low SB as well (e.g., Kormendy 1985; Skillman 1999). Blanton et al. (2005) give the following relationship between surface brightness and absolute magnitude in the r-band,

$$\begin{eqnarray}&&{{SB}}_{r}=23.8+0.45({M}_{r}+13.3).\end{eqnarray} \tag{ 10 }$$

This relationship is in close agreement with others found in the literature (e.g., Kormendy 1985; Geller et al. 2012), and appears to be valid down to very low magnitudes (even for M_B > −8; e.g., Karachentsev et al. 2013). Equation (10) implies that galaxies fainter than the QXMP limit (Equation (4)) are fainter than 23.8 mag arcsec⁻², and so potentially subject to severe incompleteness in the SDSS. Completeness is quantified using the completeness function, which gives the fraction of galaxies with a given SB that are detected in the survey, C'(SB). Combining the SDSS completeness function by Blanton et al. (2005, Figure 3) with Equation (10), one finds the completeness function, C(M), in Figure 5 (the symbols joined by a solid line). Including completeness, the selection function turns out to be

$$\begin{eqnarray}&&S(M)=V(M)C(M),\end{eqnarray} \tag{ 11 }$$

which remains a function of the absolute magnitude only. As we show in the Appendix, Equation (11) remains formally valid when the scatter of the relationship between SB and M is taken into account. This scatter is bound to be important in the analysis, since most observed QXMPs are high-SB outliers of the SB_r–M_r relationship (Figure 3). In this case, the completeness, C(M), in Equation (11) must be replaced with an effective completeness,

$$\begin{eqnarray}&&S(M)=V(M){C}_{\mathrm{eff}}(M),\end{eqnarray} \tag{ 12 }$$

where

$$\begin{eqnarray}&&{C}_{\mathrm{eff}}(M)={\int }_{\forall {SB}}P({SB}| M)C^{\prime} ({SB})d{SB}.\end{eqnarray} \tag{ 13 }$$

$P({SB}| M)$ stands for the conditional probability function of having a surface brightness SB when the magnitude of the Galaxy is M. C'(SB) represents the completeness function in terms of SB. $P({SB}| M)$ is also provided by Blanton et al. (2005) (see Appendix A), and the resulting C_eff(M) is represented in Figure 5 by the red solid line. In order to evaluate the integral in Equation (13), we use an erf function fit to the actual discrete completeness measured by Blanton et al. (2005) (i.e., we use the smooth thick solid line in Figure 5, meant to reproduce the symbols). As can be appreciated in Figure 5, the scatter increases the effective completeness of the survey at low SB, and this occurs because many targets happen to have an SB larger than that assigned by Equation (10), and those are the ones that are preferentially selected.

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The integrand of Equation (6) gives the number of galaxies of a given magnitude to be expected in a survey. Figure 4(b) shows this integrand for the SDSS-DR7 spectroscopic survey, which implies employing the m_limit in Equation (9) and Ω = 2.45 srad (Abazajian et al. 2009). We use the LF for extremely low-luminosity galaxies determined by Blanton et al. (2005). The actual LF is shown by the red solid line in Figure 4(a), although we use the double Schechter function fitted to the observations by Blanton et al. (2005) for the calculations, i.e.,

$$\begin{eqnarray}{\rm{\Phi }}(M) & = & (0.4\,\mathrm{ln}10\,{h}^{3}\,{\mathrm{Mpc}}^{-3})\exp [-{10}^{-0.4(M-{M}_{\star })}]\\ & & \times [{\phi }_{\star ,1}\,{10}^{-0.4(M-{M}_{\star })* (1+{\alpha }_{1})}\\ & & +{\phi }_{\star ,2}\,{10}^{-0.4(M-{M}_{\star })* (1+{\alpha }_{2})}]\end{eqnarray} \tag{ 14 }$$

with ϕ_⋆,1 = 0.0134, ϕ_⋆,2 = 0.0086, α₁ = 0.33, α₂ = −1.40, ${M}_{\star }=-19.99-5\,\mathrm{log}h$, and h the Hubble constant normalized to 100 km s⁻¹ Mpc⁻¹. This double Schechter function is shown by the black solid line in Figure 4(a). Thus, our estimates are based on the extrapolation to low luminosity of the observed LF, which yields a function that continues growing toward the region occupied by the QXMPs (to the left of the vertical black line in Figure 4(a)). Assuming the Malmquist bias alone, i.e., using Equation (7) for the selection function, the number of expected QXMPs, N_QXMP, is 149 (see Table 2). This figure results from integrating the solid line in Figure 4(b) up to the QXMP threshold. When the full selection function is considered (Equation (12)), the total number of expected QXMP galaxies turns out to be

$$\begin{eqnarray}&&{N}_{\mathrm{QXMP}}\simeq 42.\end{eqnarray} \tag{ 15 }$$

Table 2.  Number of Galaxies to be Expected in an SDSS-DR7-like Survey

Description	Number or Percentage	Comment
Total numbera, LF with $C\ne 1$	1.1 × 106	dashed line in Figure 4(b)
QXMP, C = 1	149	solid line in Figure 4(b)
QXMP, $C\ne 1$	42	dashed line in Figure 4(b)
	12–73	uncertainties in Section 3.2
% QXMP in the survey	0.004
% QXMP in a volumeb	87
Total numbera, varying fb, $C\ne 1$	1.1 × 106	red dashed line in Figure 6(b)
QXMP, C = 1	20	red solid line in Figure 6(b)
QXMP, $C\ne 1$	6	red dashed line in Figure 6(b)
	3–12	uncertainties like in Section 3.2
% QXMP in the survey	0.0005
% QXMP in a volumeb	45
Total numbera, exponential fb, $C\ne 1$	1.1 × 106	green dashed line in Figure 6(b)
QXMP, C = 1	24	green solid line in Figure 6(b)
QXMP, $C\ne 1$	7	green dashed line in Figure 6(b)
	4–15	uncertainties like in Section 3.2
% QXMP in the survey	0.0006
% QXMP in a volumeb	42
Total numbera, LF for centralsc	0.8 × 106	solid line in Figure 8(b)
QXMP, C = 1	15	solid line in Figure 8(b)
QXMP, $C\ne 1$	4	dashed line in Figure 8(b)
	3–9	uncertainties like in Section 3.2
% QXMP in the survey	0.0005
% QXMP in a volumeb	52

Notes.

^aThe real SDSS-DR7 spectroscopic survey has 0.93 × 10⁶ galaxies (Abazajian et al. 2009). ^bIn an unbiased, purely volume-limited survey. ^cFrom Yang et al. (2009).

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We integrate from M_r = −13.3 to −8. The upper limit is taken from Karachentsev et al. (2013, Figure 10) as the faintest magnitude of the late-type galaxies in the local universe. This upper limit, however, does not affect N_QXMP since the selection function is very small at low luminosities. Even though the number in Equation (15) represents a minuscule fraction of the galaxies to be expected in the full survey (1.1 × 10⁶; Table 2), the actual number exceeds the 9 ± 3 QXMP galaxies that are observed (Section 2). The predicted N_QXMP depends on the completeness function, and it falls off toward low SB, which is uncertain. We work out the error budget in Section 3.2, yielding the number of expected QXMPs to be between 12 and 73, with the high end of range strongly favored. Thus, the discrepancy between observations and predictions remains even when uncertainties are taken into account.

Our detailed description of differences between the number of observed and predicted QXMPs should not override the fact that these two numbers are always very small. QXMPs are, at most, a few tens in a survey such as the SDSS-DR7, which contains almost one million galaxies with spectra. QXMPs outnumber any other type of galaxy, but only a tiny fraction of these are detected: in the prediction described above, 87% of the galaxies are QXMPs, but they constitute only 0.004% of the detected galaxies (Table 2). The actual percentages depend on the specific assumptions (see Table 2), but the vast disproportion between the true number of QXMPs and their paucity in surveys always holds true.

The number of QXMPs depends on several assumptions, as explained in the previous section. Those assumptions are modified here to determine their impact on the estimated number of QXMPs.

• 1.  
  Neglecting the scatter in the SB_r–M_r relationship. The scatter increases the effective completeness quite substantially. If the scatter is neglected, then one is left with the completeness function, C; either the actual completeness determined by Blanton et al. (2005, the symbols in Figure 5) or the erf function fit to this completeness (the smooth solid black line in Figure 5). If the actual completeness is used, then N_QXMP ≃ 12. If the erf fit is used, then N_QXMP ≃ 19. These estimates are closer to (but still larger than) the number of observed QXMPs (Equation (5)). However, they must be regarded as lower limits to the predicted N_QXMP. The scatter in the SB_r–M_r relationship is a key ingredient of the detection process, as proven by the fact that the observed QXMPs are high-SB outliers of the extrapolated SB_r–M_r relation (Figure 5). Consequently, the scatter must be included, and N_QXMP ≥ 12.
• 2.  
  The completeness function. We have used the erf fit to the completeness function by Blanton et al. (2005) to evaluate the effective completeness leading to the limit in Equation (15). If rather than the fit the actual completeness is used (i.e., the symbols in Figure 5), then N_QXMP ≃ 38. The difference between the erf and the completeness function by Blanton et al. (2005) is about ±0.5 mag, in the sense that if the erf fit is shifted by ±0.5 mag then it encompasses all of the points. If the effective completeness is evaluated using the completeness shifted by ±0.5 mag, one obtains the black dashed and dotted lines in Figure 5. They yield an N_QXMP between 36 and 50.
• 3.  
  The absolute magnitude limit. The absolute magnitude limit to be a QXMP, given in Equation (4), depends on a number of factors. The value of this limit is relevant because the majority of the QXMPs are within one magnitude of the cutoff (i.e., the number of QXMPs is dominated by objects with metallicity just below the metallicity cutoff and its corresponding absolute magnitude). If we use exactly one-tenth of the solar abundance by Asplund et al. (2009), then the limit in Equation (1) becomes 7.69, so that the mass–metallicity relationship by Berg et al. (2012) predicts M_B ≥ −12.9, and the limit in the r-band becomes M_r ≥ −13.7. This brighter limit allows for more QXMPs, specifically, N_QXMP ≃ 73. The conversion from Equation (3) to Equation (4) employs both the magnitude transformation by Jester et al. (2005) and a single color $g-r\simeq 0.4$ for all galaxies. If one uses the full range of colors for galaxies in the blue cloud, $0.2\leqslant g-r\leqslant 0.6$ (e.g., Blanton & Moustakas 2009), then M_r goes from −13.0 to −13.5, which increases N_QXMP from 28 to 55. Finally, if the uncertainties in the relation derived by Berg et al. (2012) are propagated into the magnitude cutoff, then M_B ≥ −12.5 ± 0.3, and so M_r ≥ −13.3 ± 0.3, leading to values of N_QXMP between 28 to 64.
• 4.  
  The relationship between absolute magnitude and surface brightness. The relationship between the absolute magnitude and surface brightness in Equation (10) is given by Blanton et al. (2005). In order to test the uncertainty introduced by the use of this relationship, we also employed the law by Geller et al. (2012) for blue objects, ${{SB}}_{r}=29.9+0.46\,{M}_{r}$, and by Kormendy (1985), ${{SB}}_{r}=30.3+0.47\,{M}_{r}$. The latter was digitized from Figure 3 in Kormendy's paper. The magnitudes were then transformed from V and B to r (Jester et al. 2005), and the central surface brightness was converted to the half-light surface brightness assuming an exponential light profile. The use of these two alternative relationships renders an N_QXMP always around 42.
• 5.  
  A combination of the previous assumptions. In the previous items, the ingredients that determine N_QXMP are analyzed independently. We have also checked the combined effect of all of them operating simultaneously. We carried out a Monte Carlo simulation where N_QXMP was estimated, simultaneously varying the completeness function, the magnitude limit, and the mapping between SB_r and M_r. Explicitly, the center of the completeness function and the absolute magnitude limit were randomly changed following Gaussian distributions with standard deviations of 0.5 mag (item #2) and 0.3 mag (item #3), respectively. In addition, the three relations between SB_r and M_r discussed in item #4 were assumed to be equally probable. As a result of 1000 trials, we obtain N_QXMP = 46 ± 20, which is similar to the range of values inferred separately from the individual factors (Table 2).
• 6.  
  The adopted LF. We separate the properties of the LF into two parts: the shape and the normalization. Changes in the shape are analyzed in Section 3.3, and produce significant changes in the number of expected QXMPs. The normalization, however, is fairly well constrained by the total number of galaxies in the SDSS-DR7 spectroscopic sample, which amount to 0.93 × 10⁶ galaxies. By scaling the LF to reproduce the actual number of sources in the DR7 (i.e., to go from 1.1 million to 0.93 million; see Table 2), one finds N_QXMP ≃ 35.

As we have shown above, the number of observed QXMPs (around nine; Equation (5)) is not consistent with the number of QXMPs expected from extrapolating the observed LF to low luminosities (from 12 to 73, with the best value around 42; Table 2). Such an extrapolation of the LF implicitly neglects the quenching of galaxy formation in low-mass halos expected from numerical models of galaxy formation (see Section 1). These predict a rapid fall-off of the baryon fraction, f_b, in low-mass halos due to various physical processes, such as heating of the intergalactic medium by the UV background or stellar feedback (see Section 1). The drop in f_b induces a drop in the gas that fuels star formation, and hence a drop in the number of the low-mass, low-luminosity galaxies affected by the decrease of baryons.

The effect of the baryon fraction on the LF can be modeled by considering that the magnitude of a galaxy is related to the baryon fraction as follows:

$$\begin{eqnarray}&&M-{M}^{\odot }=-2.5\mathrm{log}[{f}_{L}(1-{f}_{g}){f}_{b}\,\mu ],\end{eqnarray} \tag{ 16 }$$

where M^⊙, f_L, and f_g stand for the absolute solar magnitude, the light-to-stellar mass ratio (in solar units), and the gas fraction, respectively. The symbol μ in Equation (16) stands for the total mass, including dark matter, gas, and stars. Equation (16) allows us to express Φ in terms of the LF obtained by assuming the baryon fraction to be constant, Φ₀. In this case, the mapping between M and the magnitude M₀ when f_b is a constant equal to f_b0 turns out to be

$$\begin{eqnarray}&&M-{M}_{0}=-2.5\mathrm{log}[{f}_{b}/{f}_{b0}],\end{eqnarray} \tag{ 17 }$$

so that the LFs for Φ(M) and Φ₀(M₀) are linked according to (e.g., Martin 1971)

$$\begin{eqnarray}&&{\rm{\Phi }}(M)={{\rm{\Phi }}}_{0}({M}_{0})\displaystyle \frac{{{dM}}_{0}}{{dM}}.\end{eqnarray} \tag{ 18 }$$

The ratio between the two LFs at the same magnitude, X(M), quantifies the drop in LF induced by the drop in the baryon fraction, i.e.,

$$\begin{eqnarray}&&X(M)=\displaystyle \frac{{\rm{\Phi }}(M)}{{{\rm{\Phi }}}_{0}(M)}=\displaystyle \frac{{{\rm{\Phi }}}_{0}({M}_{0})}{{{\rm{\Phi }}}_{0}(M)}\,\displaystyle \frac{{{dM}}_{0}}{{dM}}.\end{eqnarray} \tag{ 19 }$$

Neglecting variations of the mass-to-light ratio and the gas fraction with halo mass, then

$$\begin{eqnarray}&&\displaystyle \frac{{{dM}}_{0}}{{dM}}=\displaystyle \frac{{{dM}}_{0}/d\mu }{{dM}/d\mu }=\displaystyle \frac{1}{1+d\mathrm{ln}{f}_{b}/d\mathrm{ln}\mu }.\end{eqnarray} \tag{ 20 }$$

The baryon fraction is usually expressed in terms of the half-fraction mass, so that at mass μ = μ_c, the baryon fraction, f_b, is half the cosmic baryon fraction $\langle {f}_{b}\rangle$. In the parametrization determined by Gnedin (2000), and subsequently adopted by many others,

$$\begin{eqnarray}&&{f}_{b}=\langle {f}_{b}\rangle {[1+{({2}^{a/3}-1)(\mu /{\mu }_{c})}^{-a}]}^{-3/a},\end{eqnarray} \tag{ 21 }$$

with a ≃ 2, as constrained by numerical simulations (Okamoto et al. 2008). Numerical simulations also give μ_c ≃ 9.3 × 10⁹ M_⊙ in the local universe at redshift zero (Okamoto et al. 2008, Figure 3). If the baryon fraction is given by Equation (21), then the transformation between Φ₀ and Φ can be computed analytically, since

$$\begin{eqnarray}&&\displaystyle \frac{d\mathrm{ln}{f}_{b}}{d\mathrm{ln}\mu }=\displaystyle \frac{3({2}^{a/3}-1)}{1+({2}^{a/3}-1){({\mu }_{c}/\mu )}^{a}}{\left(\displaystyle \frac{{\mu }_{c}}{\mu }\right)}^{a}.\end{eqnarray} \tag{ 22 }$$

It is important to realize that, in the limit of very low luminosities, μ ≪ μ_c, so that Equation (22) predicts

$$\begin{eqnarray}&&\displaystyle \frac{d\mathrm{ln}{f}_{b}}{d\mathrm{ln}\mu }\simeq 3.\end{eqnarray} \tag{ 23 }$$

Therefore, there is a drop in the LF associated with the vanishing baryon fraction, but it is not very large,

$$\begin{eqnarray}&&{\rm{\Phi }}(M)/{{\rm{\Phi }}}_{0}({M}_{0})\simeq 0.25.\end{eqnarray} \tag{ 24 }$$

In fact, Φ(M) flattens for very low masses because for a given variation of M, M₀ changes very little, so that Φ₀(M₀) in Equation (24) is approximately constant, and Φ(M) also becomes independent of M. The LFs in Figure 6(a) show this behavior—see the red solid line, which is computed from Equations (18), (17), (20), (21), and (22) with $\langle {f}_{b}\rangle =0.158$ (Planck Collaboration et al. 2016), ${f}_{b0}=\langle {f}_{b}\rangle$, f_L = 1, and f_g = 0.9.

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The damping of the LF caused by the baryon fraction in Equation (21) is never very large. In order to make the drop more pronounced, we also tried a negative exponential parametrization of the baryon fraction,

$$\begin{eqnarray}&&{f}_{b}=\langle {f}_{b}\rangle \,\exp \left(-\displaystyle \frac{{\mu }_{c}\mathrm{ln}2}{\mu }\right),\end{eqnarray} \tag{ 25 }$$

which is hardly distinguishable from Equation (21) in the representation used to compare with numerical simulations (see Figure 7(a)), but which produces a linear drop of the LF (Figure 7(b)), since

$$\begin{eqnarray}&&\displaystyle \frac{d\mathrm{ln}{f}_{b}}{d\mathrm{ln}\mu }={\mu }_{c}\mathrm{ln}2/\mu ,\end{eqnarray} \tag{ 26 }$$

so that at low luminosity, where μ ≪ μ_c,

$$\begin{eqnarray}&&{\rm{\Phi }}(M)/{{\rm{\Phi }}}_{0}({M}_{0})\simeq \mu /({\mu }_{c}\mathrm{ln}2)\longrightarrow 0.\end{eqnarray} \tag{ 27 }$$

The LF resulting from the exponential fall-off of the baryon fraction is shown as a green solid line in Figure 6(a).

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Figure 6(a) is similar to Figure 4(a), except that it reproduces the LFs when including the decrease of baryon fraction toward low-mass halos. The red and the green lines correspond to Equations (21) and (25), respectively, whereas the black solid line is the same as in Figure 4(a), and has been included for reference. Figure 6(b) shows the number of galaxies expected in the SDSS-DR7 spectroscopic survey, considering only the apparent-magnitude threshold (solid lines), and both the apparent-magnitude threshold and the incompleteness (black dashed lines). In the case of the baryon fraction given in Equation (21), and considering the apparent-magnitude limit and incompleteness,

$$\begin{eqnarray}&&{N}_{\mathrm{QXMP}}\simeq 6,\end{eqnarray} \tag{ 28 }$$

which is significantly smaller than the estimate for the LF with the upturn at low luminosity (Equation (15)), and consistent with the observed number of QXMPs (Equation (5)). The agreement with observations is enhanced even further after considering the error budget expounded in the next paragraph. The exponential baryon fraction (Equation (25)) gives N_QXMP ≃ 7, as is reflected in Table 2.

Similar to the estimate in Equation (15), the number in Equation (28) is quite uncertain. We have repeated the exercise in Section 3.2 for the case of LFs with varying baryon fraction. The result is an N_QXMP in the range between 3 and 12 objects. Another source of uncertainty, which we do not treat in Section 3.2 because it does not affect Equation (15), is the mapping between masses and magnitudes. According to Equations (16) and (21), the free parameters of this mapping, f_L, $1-{f}_{g},$ and $\langle {f}_{b}\rangle$, appear in the equations as a single parameter, corresponding to their product. If this product is two times larger or smaller, N_QXMP changes from 5 to 8, respectively. If, on the other hand, the half-baryonic fraction mass is varied from 2 × 10⁹ M_⊙ to 2 × 10¹⁰ M_⊙, then N_QXMP goes from 12 to 4. (The nominal value we use is 9.3 × 10⁹ M_⊙.) These uncertainties in the estimate of N_QXMP are summarized in Table 2.

The number of QXMPs in the SDSS-DR7 is not consistent with the extrapolation to low luminosities of the observed LF. Observations and theory agree much better if a varying baryon fraction is included to flatten the upturn of the observed LF at low luminosities, which implicitly assumes the QXMPs to be central galaxies of their dark matter halos. The baryon fraction of satellite galaxies is determined by interactions with nearby galaxies (tidal stripping and harassment) and with the circumgalactic medium of the central galaxy (ram pressure stripping and starvation; e.g., Combes 2004; Benson 2010). Hence, the baryon fraction depends not only on the halo mass but on many other factors, and expressions for f_b like Equation (21) are no longer valid. Consequently, the consistency of the estimated N_QXMP with observations suggests that the upturn in the observed LF is caused by the presence of satellites. This result is in agreement with the conclusion reached by Lan et al. (2016). They model the LF by Blanton et al. (2005) as the sum of an LF for centrals (i.e., the most massive galaxy in its dark matter halo) plus an LF for satellites (i.e., galaxies sharing a dark matter halo with other more massive galaxies). This decomposition reveals that the LF of field galaxies is dominated by satellite galaxies at M_r > −17, and that only halos more massive than 10¹⁰ M_⊙ contribute to the LF at M_r < −12.

The conjecture that QXMPs are central galaxies rather than satellite galaxies is also qualitatively consistent with the observed N_QXMP. If one uses the empirical LF for central galaxies determined by Yang et al. (2009), and simply extrapolates it to low luminosity, then N_QXMP ≃ 3–9; see Figures 8(a) and (b), and Table 2. This LF for centrals does not show the upturn and stays below the LF by Blanton et al. (2005) (see Figure 8). The lack of an upturn produces a major drop in the number of QXMPs. The LF for centrals also predicts 30% less galaxies than the reference LF by Blanton et al. (2005). This difference is due to low-luminosity (M_r > −17) satellite galaxies, included by Blanton et al. (2005) but separated out by Lan et al. (2016).

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Assuming that our model for the selection function provides a good representation of the SDSS properties, one can investigate which, among the parameters of the survey, are responsible for the limited number of observed QXMPs. In principle, there are three main parameters that control such a number, namely, (1) the completeness, (2) the area coverage of the survey, and (3) the apparent-magnitude limit.

The number of objects of a particular type in the survey linearly scales with the area of the survey. Since SDSS already covers a significant part of the sky (around 20%; Abazajian et al. 2009), this is not a limiting factor in the present case. In other words, no dramatic (tenfold) increase in the number of QXMPs will follow from increasing the area covered by the SDSS-DR7.

Even though completeness is an important factor, it is not the key factor that determines N_QXMP. At the absolute magnitude limit that characterizes the QXMPs, the completeness is around 30% (see Figure 5). Although the completeness can be increased, it cannot exceed the value of one, which in turn implies a moderate increase in N_QXMP. This dependence can be apprised in Figure 9(a), where the expected number of QXMPs is represented for different SB cutoffs of the completeness function. To compute the number, we have used the same completeness of the SDSS-DR7 (Figure 5, the red solid line), shifted in SB_r, with the shift parameterized as one-half of the SB_r completeness. As the SB of the drop increases, N_QXMP also increases. However, it saturates at a value that is only 10 times larger than the value corresponding to the SDSS-DR7 spectroscopic survey (the point of lower SB in Figure 9(a)). The curves in Figures 9(a) and (b) assume the LF shown as a red solid line in Figure 6(a).

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The apparent-magnitude limit turns out to be the critical parameter that limits the number of QXMPs. It determines the volume sampled by the survey, which, in principle, can be increased indefinitely as m_limit increases. The behavior is shown in Figure 9(b). Just to provide an idea of the increase, if one consideres the apparent-magnitude limit of the SDSS-DR7 photometric survey (m_limit = 22.2 in the r-band), then Figure 9(b) predicts the presence of approximately 2800 QXMPs, which should be compared to the prediction of 6 QXMPs in the SDSS-DR7 spectroscopic sample (m_limit = 17.77; see Table 2 and Figure 9(b)).

The formalism developed in Section 3 allows us to estimate the number of expected QXMPs in any magnitude-limited survey, given their magnitude limit, half-completeness SB_r, and coverage area. If the completeness function is assumed to have the functional form of the SDSS completeness, then the half-completeness SB_r fully describes it. We carried out the exercise of estimating N_QXMP for a number of ongoing and forthcoming large area surveys, specifically, for the the Dark Energy Survey (DES; The Dark Energy Survey Collaboration 2005), the Galaxy and Mass Assembly survey (GAMA; Liske et al. 2015), the Kilo-Degree Survey (KIDS; de Jong et al. 2015), and the Large Synoptic Survey Telescope (LSST; Ivezic et al. 2008).The parameters that define the surveys are given in Table 3. LSST, DES, and KIDS do not explicitly give the half-compleness SB. In these cases, we infer it from the depth for point sources as described in Appendix B.

As shown in Table 3, the number of QXMPs expected in the next generation of wide area surveys is very large, reaching up to 10⁷ in the 10 year average LSST. The actual number changes by one order of magnitude, depending on whether the LF increases toward low luminosity (the black solid line in Figure 6(a)) or if it flattens out (the red solid line in Figure 6(a)). Therefore, it should be easy to discriminate both trends using these new surveys. For example, the ongoing GAMA survey predicts 10 or 100 QXMPs, depending on the LF faint end. In agreement with the conclusion in the previous section, the key factor determining N_QXMP is not so much the SB cutoff but the limiting magnitude, m_lim. Note, however, that even if the surveys contain all these new QXMPs, it will be imposible to confirm the XMP nature of many of the faint objects. The spectroscopic follow-up required to determine abundances will be possible only in those QXMPs where star-forming regions are bright enough.