SUBMILLIMETER GALAXY NUMBER COUNTS AND MAGNIFICATION BY GALAXY CLUSTERS

Over the last decade, submillimeter (submm) surveys have yielded significant advances in our understanding of the galaxy population responsible for the high-redshift component of the cosmic infrared background (CIB; Fixsen et al. 1996, 1998; Smail et al. 1997; Hughes et al. 1998; Barger et al. 1998; Dwek et al. 1998; Greve et al. 2004; Pope et al. 2006; Coppin et al. 2006; Devlin et al. 2009). With typical far-infrared (FIR) luminosities >10¹² L_☉, submm galaxies are presumed to be the high-redshift counterparts to (ultra)luminous infrared galaxies ((U)LIRGs). The high luminosity of these galaxies is the result of star formation rates of 100–1000 M_☉ yr⁻¹. Approximately half of these galaxies are located at 1.9 ≲ z ≲ 2.9 (Chapman et al. 2005; Aretxaga et al. 2007), dominating the total star formation rate at this epoch (Pérez-González et al. 2005; Michałowski et al. 2009).

One way to express the results of submm surveys is through number counts of galaxies as a function of flux for each observed wavelength. The shape of these counts has been interpreted as arising from different populations of galaxies whose characteristics evolve over cosmic time (Lagache et al. 2003, 2004; Pearson & Khan 2009; Le Borgne et al. 2009). These empirical models have successfully reproduced the counts. However, they may be masking a simpler explanation for the departure of the counts from a Schechter distribution at the high-flux end: magnification due to high-redshift galaxy clusters and groups (e.g., Blain 1996; Perrotta et al. 2002; Negrello et al. 2007). In fact, the predictions of Perrotta et al. (2002) and Negrello et al. (2007) use the physical models of Granato et al. (2001, 2004), respectively, which explain the presence of a substantial high-redshift population prone to lensing effects.

Millimeter wavelength surveys have also aimed at detecting galaxy clusters via the Sunyaev Zel'dovich (SZ) effect (Hincks et al. 2008; Carlstrom et al. 2009); the first results, including cosmic microwave background power spectra and cluster catalogs, have been released recently (Fowler et al. 2010; Staniszewski et al. 2009; Vanderlinde et al. 2010).

The South Pole Telescope (SPT) has measured number counts of dusty galaxies at wavelengths λ = 1.4 mm and 2.0 mm over an area of 87 deg² (Vieira et al. 2009). The observed numbers at the bright end are higher than expected. The possibility that these galaxies lie at lower redshifts than the bulk of the population of submm galaxies is disfavored by the lack of detected counterparts in other surveys that probed the low-z population (Vieira et al. 2009). Placing these galaxies at high redshifts and assuming they are intrinsically bright would require them to be far more luminous than an underlying Schechter-like luminosity function would permit. Thus, the favored explanation is that they have typical luminosities for high-z galaxies, but have been magnified by foreground galaxies or clusters. In fact, lensing of high-redshift submm galaxies has been observed in a number of systems (Smail et al. 1997, 2002; Wilson et al. 2008; Rex et al. 2009; Gonzalez et al. 2009; Swinbank et al. 2010).

In this Letter, we explore the possibility that the existing observed galaxy number counts over a wide range of wavelengths can be reproduced by a single population of galaxies at high redshift. Foreground galaxy groups and clusters gravitationally lense the background submm population (Lima et al. 2009), leading to significant enhancements to the high-flux end of the galaxy counts. In Section 2, we describe the lensing magnification formalism, which we then apply to a high-z galaxy population and present results in Section 3. We discuss implications for high-z galaxies and cluster searches in Section 4.

Throughout, we use a flat cosmology with parameters based on the Wilkinson Microwave Anisotropy Probe (WMAP) fifth year data results (WMAP5; Komatsu et al. 2009). The cosmological parameters and their values are δ_ζ = 2.41 × 10⁻⁴ at k = 0.02 Mpc⁻¹ (corresponding to σ₈ = 0.8), n = 0.96, Ω_bh² = 0.023, Ω_mh² = 0.13, Ω_DE = 0.74, and w = −1. We also consider changes in σ₈ consistent with the WMAP5 errors of Δσ₈ ≈ 0.03.

In recent papers (Lima et al. 2009; Jain & Lima 2010), we have presented a halo model for calculating the effects of lensing magnification by galaxy groups and clusters. Here, we specialize to the case of steep galaxy counts at high redshifts, where lensing effects are quite dramatic. We assume a Schechter function (Schechter 1976) for the intrinsic number density distribution of a population of galaxies:

$$\begin{eqnarray} &&\frac{dn}{dS} = \frac{n^*}{S^{*}} \left(\frac{S}{S^*}\right)^{\alpha } e^{-S/S^{*}}, \end{eqnarray} \tag{ 1 }$$

where n*, S*, and α are free parameters. Lensing magnification by intervening halos changes the intrinsic dn/dS to its observed counterpart as

$$\begin{eqnarray} \frac{dn_{\rm obs}(S_{\rm obs})}{dS_{\rm obs}} &=& \int d\mu \frac{P(\mu)}{\mu } \frac{dn}{dS}\left(\frac{S_{\rm obs}}{\mu }\right), \end{eqnarray} \tag{ 2 }$$

where μ is the lensing magnification and P(μ) is its probability for a given galaxy population at redshift z_s. Conditional probabilities quantify the effects of different magnification ranges on the observed flux density S_obs. The integrand of Equation (2) defines the probability P(μ|S_obs):

$$\begin{eqnarray} P(\mu |S_{\rm obs}) &=& \left(\frac{dn_{\rm obs}(S_{\rm obs})}{dS_{\rm obs}}\right)^{-1} \frac{P(\mu)}{\mu } \frac{dn}{dS}\left(\frac{S_{\rm obs}}{\mu }\right), \end{eqnarray} \tag{ 3 }$$

which can be interpreted as the relative contribution of a given μ to the total dn_obs/dS_obs at S_obs (Paciga et al. 2009). Similarly, $P(\mu _{\rm min}|S_{\rm obs})= \int _{\mu _{\rm min}}^{\infty } d\mu \ P(\mu |S_{\rm obs})$ measures the integrated contribution from all μ>μ_min. The mean magnification at a given S_obs is defined as

$$\begin{eqnarray} \langle \mu \rangle (S_{\rm obs}) &=& \int _0^{\infty } d\mu \ \mu \ P(\mu |S_{\rm obs})\,. \end{eqnarray} \tag{ 4 }$$

The distribution P(μ) can be estimated either by ray-tracing on N-body simulations (e.g., Hilbert et al. 2007), or by semi-analytical methods (e.g., Perrotta et al. 2002; Lima et al. 2009), integrating halo contributions on the line of sight up to the source redshift:

$$\begin{eqnarray} &&P(>\mu)=\int _{0}^{z_s} dz_l \frac{D_A^2(z_l)}{H(z_l)} \int _{M_{\rm th}}^{\infty } d\ln M \ \frac{dn(z_l,M)}{d\ln M} \ \Delta \Omega _{\mu } \,, \nonumber \\ \end{eqnarray} \tag{ 5 }$$

where D_A is the angular diameter distance, H is the Hubble parameter, dn/dln M is the halo mass-function, and ΔΩ_μ = ΔΩ_μ(z_s, z_l, M) is the cross-section for magnifications larger than μ produced by halos of mass M at redshift z_l on sources at redshift z_s. The integrand,

$$\begin{eqnarray} \frac{d^2P(>\mu)}{d \ln M \ dz_l}&=& \frac{D_A^2(z_l)}{H(z_l)} \ \frac{dn(z_l,M)}{d\ln M} \ \Delta \Omega _{\mu }, \end{eqnarray} \tag{ 6 }$$

gives the range of halo masses and redshifts contributing to the probability of a minimum magnification μ.

In summary, Equations (2) and (5) give the total effect on the counts, Equation (3) indicates which magnifications contribute most to a given S_obs, and Equation (6) tells us which halo masses and redshifts contribute to a given magnification.

We use this halo-model P(μ) and correct it for a number of effects. First, as described in Lima et al. (2009), we match our P(μ) at large magnifications to that of ray-tracing in dark matter simulations (Hilbert et al. 2007) by tuning the ellipticity of our halos. Next, we account for the effect of luminous matter, which can lead to higher densities via gas cooling, using the simulation results of Hilbert et al. (2008). We also correct for the combination of finite source size and multiple image effects. Finally, magnification effects are sensitive to the value of σ₈ since it affects the abundance of cluster halos. In the next section, we discuss how we account for the uncertainties in our model by giving a range for our predictions.

Our analytical calculation of P(μ) has some advantages over the approach of numerical simulations (we can easily study changes in source redshift, σ₈, and the contribution from different halo masses and redshift), but it also has limitations. We only use the one-halo term, which is accurate at the high magnifications relevant for the effects considered here but overestimates the lensing contribution at μ ∼ 1. We do not include a distribution of ellipticities or halo substructure, which can also increase magnification cross-sections. We have instead tuned the average halo ellipticity to match the P(μ) measured in dark matter simulations (see Lima et al. 2009 for a detailed discussion). And whereas we account for the effects of baryons observed in simulations by Hilbert et al. (2008), these authors note that their results still underestimate baryonic effects for halos of smaller masses as they do not predict sufficient numbers of multiple-imaged quasars. Finally, the effect of finite source size is very uncertain given the lack of our knowledge about submm galaxies and the sensitivity to the precise caustic structure of the lenses (e.g., Li et al. (2005). Proper inclusion of all these missing effects would likely increase the magnification probabilities compared with our current model.

In Figure 1, we illustrate the lensing effect on an intrinsic Schechter distribution given by Equation (1) for sources at different redshifts. In all our results, we fix α = −1.0 and n* = 5 × 10³ deg⁻². Changing to α = −1.5 does not have a significant effect; while it matches the faint end behavior of dn/dS for the model of Lagache et al. (2004), we preferred to use α = −1.0 since this better fits the number counts at shorter wavelengths. With these parameter values, our counts are lower than the model of Lagache et al. (2004) at all S, which also ensures that the total flux does not exceed the CIB (Dwek et al. 1998).

Zoom In Zoom Out Reset image size

Figure 2 illustrates our main results: we show predicted number counts that include lensing (gray bands), assuming galaxies at z_s = 3.0, along with measured number counts from submm surveys at different wavelengths. As indicated in the panels, these are BLAST at λ = 500 μm (Devlin et al. 2009), SCUBA at λ = 850 μm (Coppin et al. 2006), AzTEC at 1.1 mm (Austermann et al. 2010), and SPT dusty submm galaxies at λ = 1.4 mm and 2.0 mm (Vieira et al. 2009). In all of our results, the SPT number counts correspond to those of Vieira et al. (2009), after removing both synchrotron emission galaxies as well as low-redshift galaxies that have matches with galaxies in the Infrared Astronomy Satellite (IRAS) survey (Moshir et al. 1992; Fisher et al. 1995; Oliver et al. 1996). Predictions are shown by bands rather than curves to reflect the uncertainties in the model as discussed below. All the data sets can be fit by changing the single parameter S* once lensing magnification is included. This remarkable result implies that, within the measurement and theoretical uncertainties, a single high-z population of galaxies is sufficient to describe all the observations. The high-flux measurements of BLAST and SPT are fit by highly magnified galaxies—if these counts were dominated by a population of a different galaxy type, it would be a coincidence that their relative counts fit the same scaling with wavelength as the fainter (normal) population. Finally, we also show a prediction for MUSTANG at 3.3 mm (Mason et al. 2006), which has begun operating on the Green Bank Telescope.

Zoom In Zoom Out Reset image size

The lower bound for the predictions in Figure 2 uses σ₈ = 0.77, while the upper bound uses σ₈ = 0.83—these reflect the uncertainties in the WMAP5 results. The enhancement due to baryons is factored into these predictions, though it is likely to be an underestimate as discussed in Section 2. The effect of source sizes and multiple imaging is uncertain; we simply assume that, due to the finite source size, the effective magnification is reduced by 50%–25% (for the lower and upper bounds, respectively)

The right panel of Figure 2 shows all points rescaled by plotting

$$\begin{equation} \frac{d\tilde{n}}{d\tilde{S}}=\tilde{S}^\alpha \ e^{-\tilde{S}}, \end{equation} \tag{ 7 }$$

where $\tilde{n}=n/n^*$ and $\tilde{S}=S/S^*$. In Figure 3, we show values of S* used for each wavelength. We compared the frequency scaling of S* with that of a typical spectral energy distribution (SED) of submm galaxies SED(λ) ∝ (λ)B(T, λ), redshifted to z_s = 3 with emissivity (λ) = 1 − exp [(−λ₀/λ)^β] and blackbody spectrum B(T, λ) at temperature T. The values of S* are consistent with β = 1–2 and T = 30–40 K.

Zoom In Zoom Out Reset image size

The fit to the BLAST counts at λ = 500, 350, and 250 μm falls further below the high-flux measurement at shorter wavelengths, suggesting the need for a lower redshift population. Indeed, Eales et al. (2009) have identified the radio and 24 μm counterparts of the bright BLAST sources. Almost all of the bright sources at 250 μm, and one third to one half of the sources in the highest flux bin at 500 μm are identified as being at z < 1. Removing these would lower the corresponding point in Figure 2, in agreement with the model. The remaining sources at 500 μm (and less than a tenth of the sources at 250 μm) are likely the result of lensing. Similar results should be expected with the upcoming release of the large-area Herschel surveys.

Since the lensed distributions at z_s = 3.0 are consistent with SPT data points at both wavelengths, we study the range of magnifications and halo masses that contribute most in this case. As we consider S_obs/S* ≳ 10, the observed sources come from intrinsically low-flux sources which have been significantly magnified. Figure 4 shows P(μ_min|S_obs) and 〈μ〉 as a function of S_obs for SPT (λ = 1.4 mm) and indicates magnifications that contribute the most at each S_obs. For instance, for S_obs = 20–40 mJy, 〈μ〉 ∼ 20–30, and P(μ_min|S_obs)>0.5 for μ_min ∼ 10–20.

Zoom In Zoom Out Reset image size

These results imply that magnifications of 10–30 are necessary to explain the boost in dn/dS at S_obs ∼ 10–40 mJy, if it is due to lensing of an intrinsic Schechter distribution. Note that due to the finite size of submm galaxies, their magnifications must have a cutoff, which has been estimated to be in the range μ ∼ 10–40 (Perrotta et al. 2002) for galaxy lenses, and is probably a factor of 2 or so larger for more massive lenses. Indeed, galaxies have been measured with estimated magnifications of at least ∼45 (Kneib et al. 2004).

In Figure 5, we show d²P/dln Mdz_l as a function of halo mass for different values of μ_min and z_l. This indicates that halo masses above 10¹³h⁻¹ M_☉ contribute significantly to magnifications of 10–30, with most of the contribution coming from ∼10¹⁴ h⁻¹ M_☉. Note that this does not include baryonic effects, which boost the contribution from lower mass (≲10¹³ h⁻¹ M_☉) halos as discussed in Section 2.

Zoom In Zoom Out Reset image size

We have assumed that all source galaxies are at a fixed redshift z_s, whereas in reality they have a redshift distribution which needs to be incorporated in the computation. The curves at different source redshifts shown in Figure 1 provide approximate limits for what we can expect from a redshift distribution of source galaxies. Our results imply that the bulk of the population of submm galaxies is at redshifts z ≳ 2. The alternative explanation for the high-flux measurements is to have a significant fraction of galaxies at very low z, which would easily have been observed in surveys such as IRAS (indeed, the SPT data shown remove the small fraction of such sources). As shown in Figure 1, galaxies at higher redshift but still at z ≲ 1 would not match the measurements as the magnification boost is insufficient (again within the context of an underlying Schechter distribution).

We have considered the possibility that bright millimeter and submm galaxy counts arise largely from a galaxy population at high redshifts that is lensed by intervening galaxy groups and clusters. Our model predictions match the counts from 500 μm to 2 mm from the BLAST, SCUBA, AzTEC, and SPT surveys (see Figure 2). We find that the high-flux SPT number counts can be explained by highly magnified galaxies from this high-z population (once the known, low-z counterparts detected in IRAS are removed). This high-z galaxy population is described by a Schechter luminosity function with L* ∼ 2.5 × 10¹² L_☉ and a source redshift z_s = 3.0. Our model predictions fit the data for 500μm <λ < 2 mm by varying S* with wavelength within the range of typical submm galaxy SEDs (see Figure 3).

Our model has some simplifying assumptions, such as the fixed source redshift z_s = 3, and there are significant measurement and theoretical uncertainties. A complete analysis would require a more detailed treatment of lensing effects and inclusion of the measured galaxy clustering. In addition, it has been established that at the shorter wavelengths probed by BLAST, an increasing fraction of sources lie at low-z—hence, we can expect a smooth variation in the fraction of low-z galaxies with observed wavelength.

Nevertheless, our results in Figure 2 imply that current number counts do not require an additional galaxy population to explain the high-flux measurements. Such a population has been invoked in theoretical models (e.g., Lagache et al. 2003) and suggested as a possible explanation (as well as lensing of the high-z population) for the recent SPT measurements (Vieira et al. 2009). Indeed, adding a significant fraction of a second population could cause the predictions to exceed the measured counts once magnification effects for the high-z population are included. It would also be difficult to explain how the scaling with wavelength of the high-flux number counts is the same as the lower flux counts if they came from different galaxy populations. Since lensing does not depend on frequency, our model naturally follows this common scaling.

Current SZ surveys with sensitivities of (3–7) × 10¹⁴ h⁻¹ M_☉ cannot detect most halos that produce this lensing contribution. Conversely, looking for extremely bright objects in millimeter and submm wavelengths provides a way to find high-z lensing halos associated with galaxy groups and clusters. The number of halos that can be found in this way is only a small fraction of all halos within a given mass range. As discussed above, our model most likely underestimates the contribution from halos with M ≲ 10¹³ M_☉. It is therefore of great interest to investigate the lenses corresponding to the bright sources in current data.

Follow-up observations with optical telescopes should be able to identify lensing group/cluster candidates up to z ≃ 1. These clusters and groups host the brightest cluster galaxies with luminosities L ∼ 10¹¹–10¹² L_☉, based on the low-z results of Johnston et al. (2007). Multi-band optical imaging with limiting magnitude of 23–24 (for the r band) would enable identification of these groups and clusters. Conversely, targeted observations of the high magnification regions of known strong lensing clusters could provide detections of faint submm galaxies (which would lie below the detection threshold without the magnification boost). Similar to the use of clusters as gravitational telescopes in optical imaging, this may also help resolve submm galaxies. Planned observations with AzTEC and the Large Millimeter Telescope have considered such an approach (D. Hughes 2010, private communication).

We thank David Hughes, Roxana Lupu, Joaquin Vieira, Kim Scott, Ian Smail, Eric Switzer, and various members of the ACT collaboration for useful discussions. We benefited from discussions on strong lensing effects with Matthias Bartelmann, Gary Bernstein, Neal Dalal, and Ravi Sheth. We are very grateful to Stefan Hilbert for sharing his simulation results and Eric Switzer for providing the SPT data. This work was supported in part by an NSF-PIRE grant and AST-0607667.