A MEASUREMENT OF THE RATE OF TYPE Ia SUPERNOVAE IN GALAXY CLUSTERS FROM THE SDSS-II SUPERNOVA SURVEY

The C4 cluster catalog is based on the main spectroscopic sample of the SDSS and contains clusters in the redshift range 0.03 < z < 0.17. The C4 cluster catalog provides, among other quantities, cluster coordinates, the luminosity of each cluster in the SDSS r band, and the number of galaxies identified as members of each cluster, N_gals. The main spectroscopic sample of galaxies from SDSS is designed to be complete to a limiting magnitude of r ≈ 17.8 (Strauss et al. 2002). For typical cluster galaxy luminosities, this implies that the identification of member galaxies, with L>0.4L_⋆, is complete for clusters at z < 0.11. For clusters above this redshift limit a correction has to be made to the total cluster luminosity. The C4 cluster identification algorithm works by searching for groups of objects that are tightly clustered in a seven-dimensional feature space that includes spatial position, redshift, and observed colors. Note that there is no requirement that the colors for the galaxies be consistent with the colors of early-type galaxies, only that they be consistent with one another. The SDSS main galaxy sample cannot be 100% complete, as the SDSS fiber-spectrograph imposes a minimum angular separation of 55'' (≈100 kpc h⁻¹ at z = 0.15) for objects targeted for the SDSS main spectroscopic galaxy sample (Stoughton et al. 2002). A correction is applied to the C4 cluster luminosities to account for this by including galaxies as members of the cluster when they satisfy the magnitude requirement (r < 17.8) and have similar colors to the spectroscopically determined cluster members. The published C4 catalog is based on the 2nd data release of the SDSS (Abazajian et al. 2004) and contains 748 clusters. The catalog used in this work is an extended version of the C4 catalog that contains 1713 clusters and is based on the 5th data release of the SDSS (Adelman-McCarthy et al. 2007; B. Nichol 2007, private communication). The subset of clusters that we will use in measuring the cluster SN rate are those that lie within stripe 82, and that have −40° < α_J2000 < 50°. There are 71 C4 clusters in this subset.

The maxBCG catalog is based on SDSS photometric measurements, and the cluster identification algorithm relies on the tight relationship between color and redshift for luminous red galaxies, which make up the majority of galaxy cluster composition. The maxBCG algorithm assigns a photometric redshift to each identified cluster that is derived by comparing the cluster member galaxy colors to the expected colors for early-type galaxies, as a function of redshift. Comparison of the photometric redshifts for the maxBCG clusters to the spectroscopically measured redshift of the brightest cluster galaxy (BCG), when available, shows that the residuals for redshift (photometric–spectroscopic) are well described by a Gaussian distribution with a mean of 0 and a standard deviation of σ ≈ 0.015. The observed (g − r) and (r − i) colors for maxBCG member galaxies, as a function of redshift, are shown in Figure 2. Linear functions were fit to the (r − i) and (g − r) colors as a function of redshift, z, and these will be used below for applying k-corrections to the observations. For reference, the functions derived, valid for 0.1 ⩽ z ⩽ 0.3, are,

$$\begin{equation} (r-i) = 0.345 + 0.720 z \end{equation} \tag{ 1 }$$

$$\begin{equation} (g-r) = 0.632 + 3.054 z. \end{equation} \tag{ 2 }$$

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The maxBCG catalog is restricted to the redshift interval 0.1 < z < 0.3, and is thus highly complementary to the C4 catalog. The lower limit for the maxBCG cluster catalog is imposed because the fractional photometric redshift errors for redshifts less than z = 0.1 are significant and have a large systematic effect on derived cluster properties. Furthermore, cluster catalogs based on spectroscopic redshifts for z < 0.1 are already available. The upper limit is imposed because, at z ≈ 0.3, the "4000 Å break" that is responsible for the uniformity in early-type galaxy colors moves into the region between the SDSS observer frame g- and r-band filters. Thus, the accuracy and precision for galaxy photometric redshifts is severely diminished (Koester et al. 2007a). A lower limit for the luminosity of cluster members is imposed so that the definition of the composition of the clusters is consistent across the redshift range. The limit corresponds to an absolute magnitude of ^(0.25)i ≈ −20.25, (the ^(0.25)i notation is explained in Section 3.3) and is such that the catalog is volume limited over the entire redshift range 0.1 < z < 0.3. Measures of the physical extent, r₂₀₀, and of the richness, N₂₀₀ and L₂₀₀, are provided for each cluster. r₂₀₀ is defined as the radius such that the mean density of early-type galaxies contained within is 200 times greater than the mean density of such galaxies (Koester et al. 2007a); N₂₀₀(L₂₀₀) is defined as the number (total luminosity) of early-type galaxies contained within r₂₀₀. The public maxBCG cluster catalog contains 13,823 clusters with N₂₀₀ ⩾ 10. The subset of maxBCG clusters that we will use in measuring the cluster SN rate are those that lie within stripe 82, and that have −50° < α_J2000 < 60°. There are 492 maxBCG clusters in this subset. Additionally, this work makes use of a catalog of the maxBCG member galaxies (B. Koester 2007, private communication).

In addition to the cluster richness and luminosity estimates that are provided with the maxBCG catalog, the maxBCG clusters have been extensively studied by, e.g., Sheldon et al. (2009a, 2009b) and Johnston et al. (2007). Of particular relevance to the study of the SN rate in galaxy clusters are the luminosity functions (LFs) of maxBCG clusters presented by Hansen et al. (2009), and we will make extensive use of these below.

In Section 7, we will present the SN rate in galaxy clusters per unit luminosity. In this section, we present a comprehensive discussion of the luminosity content for the C4 and maxBCG cluster catalogs. In what follows we use the notation ^(z)m to denote the observer frame magnitude, m, for an object that has been k-corrected to a redshift z. All luminosities presented here are for galaxies with L>0.4L_* (L_* is a characteristic luminosity for cluster members; see Equation (3)). Correcting for the faint end of the luminosity distribution is discussed below.

The C4 catalog provides total cluster luminosities in SDSS r band. The total uncorrected r-band luminosity for the galaxies identified as cluster members, for clusters considered in this study, is 2.02 × 10¹³ L_☉ h⁻², where L_☉ is the luminosity of the Sun, and h is the value of the Hubble constant in units of 100 km s⁻¹ Mpc⁻¹. As discussed in Section 3.1, these luminosities have to be corrected to account for incompleteness of the spectroscopic sample due to fiber collisions. The total r-band luminosity after making this correction is also provided by the C4 catalog and has the value 4.08 × 10¹³L_☉ h⁻². As mentioned above, the luminosities for clusters at z>0.11 also have to be corrected to account for the fact that some cluster galaxies with L>0.4L_⋆ will have the observed magnitudes fainter than the completeness limit of the main SDSS galaxy sample (r ≈ 17.8). Of the 71 C4 clusters we are considering, 12 are at z>0.11. Applying a correction to the luminosities of these 12 clusters, assuming the cluster LFs of Hansen et al. (2009), results in a total r-band luminosity for C4 clusters in this study of 4.12 × 10¹³ L_☉ h⁻².

The maxBCG catalog includes the summed luminosities for member galaxies in ^(0.25)i and ^(0.25)r. However, the maxBCG cluster LFs presented by Hansen et al. (2009) represent a more complete study of the luminosity content of maxBCG clusters, and we will use these as the definitive measure of the total maxBCG cluster luminosities. In particular, a background subtraction has been performed that reduces inaccuracies in the cluster luminosities due to interloping foreground and background galaxies that may be counted as maxBCG cluster members. In Hansen et al. (2009), LFs are presented for maxBCG satellites, as a function of the richness measure, N₂₀₀. The LFs account for both red and blue cluster galaxies, but do not include the contribution to the luminosity from the BCG. The LFs are assumed to take the form of a Schechter function,

$$\begin{equation} \phi (L) dL = \phi ^{*} \left(\frac{L}{L_{*}}\right)^{\alpha } e^{-L/L_{*}} \frac{dL}{L_{*}}, \end{equation} \tag{ 3 }$$

where L_* is a characteristic luminosity for cluster members, and ϕ* is a normalization constant with units of inverse volume. The three parameters of the LFs, ϕ*, L_*, α, are each expressed as functions of N₂₀₀, with the general functional form of A × N^β₂₀₀. The values of A (normalization) and β (exponent) for these three LF parameters are given in Table 2. To use the LFs to compute the total luminosity we exploit the identity

$$\begin{equation} \Sigma L = \langle L(N_{200}) \rangle N_{200}, \end{equation} \tag{ 4 }$$

where ΣL denotes the summed luminosity for the cluster, and the average luminosity, 〈L〉, is given by,

$$\begin{equation} \langle L(N_{200}) \rangle = \frac{\int _{0.4 L_{*}}^{\infty } dL L \phi (L)}{\int _{0.4 L_{*}}^{\infty } dL \phi (L)}. \end{equation} \tag{ 5 }$$

Equation (4) gives the total ^(0.25)i luminosity in a cluster, as a function of N₂₀₀. The total luminosity in clusters in the SN survey region is then the sum over N₂₀₀ of 〈L(N₂₀₀)〉N_SN(N₂₀₀), where N_SN(N₂₀₀) is the number of clusters in the survey region for which the number of member galaxies is N₂₀₀. Using this formalism we find that the total luminosity of maxBCG clusters in the survey region is ΣL^0.25_i = 1.096 × 10¹⁴L_☉ h⁻².

3.3.1. Correcting Cluster Luminosities for Faint Galaxies

The luminosities quoted above for the C4 and maxBCG cluster catalogs include only galaxies with L>0.4L_*, which is a conventional way of characterizing cluster luminosities. In measuring the cluster SN rate we do not wish to exclude SNe occurring in faint galaxies, and so it is necessary to estimate the contribution to the total cluster luminosities from galaxies with L < 0.4L_*. The total cluster luminosity can be estimated as L = κL⁺, where L⁺ denotes the luminosity for galaxies with L>0.4L_* and the correction factor, κ, is given by

$$\begin{equation} \kappa = \frac{\int _{0}^{\infty } dL L \phi (L)}{\int _{0.4 L_{*}}^{\infty } dL L \phi (L)}. \end{equation} \tag{ 6 }$$

For a typical maxBCG cluster with N₂₀₀ = 20 the power-law exponent of the LF is α = −0.59 (Table 2). For a cluster with N₂₀₀ = 35 the value for the exponent is α = −0.68. The corresponding correction factors are κ = 1.21 and κ = 1.25. We will assume that the faint end of the LF is a characteristic property of galaxy clusters and that the power-law behavior of the LFs for maxBCG clusters is appropriate for the C4 clusters also.

3.3.2. k-corrections

In order to minimize uncertainties due to k-corrections, the maxBCG magnitudes, and the corresponding luminosities, have been k-corrected to the median redshift of the clusters, z = 0.25. For comparison of the SN rate results based on the maxBCG clusters with previous cluster SN rate measurements, it will be necessary to k-correct the luminosities into more standard filters. In order to determine the appropriate k-corrections we selected a set of galaxies from the SDSS galaxy catalog that satisfies the color versus redshift relations for maxBCG members discussed above. The corresponding set of galaxies contains ≈675, 000 members. We then matched these galaxies with their corresponding records in the photoz database provided by the SDSS catalog archive server (CAS), which provides k-corrections based on the work of Blanton et al. (2003a). We thereby determine an average k-correction as a function of redshift, appropriate to the early-type galaxies that make up the bulk of the maxBCG catalog. The k-corrections so derived, in SDSS r and i bands, are shown in Figure 3. For reference, the functions employed for k-corrections as a function of redshift, z, valid for 0.1 ⩽ z ⩽ 0.3, are

$$\begin{equation} K_{rr}(z) = 9.17 \times 10^{-2} + 1.48 z, \end{equation} \tag{ 7 }$$

$$\begin{equation} K_{ii}(z) = 3.95 \times 10^{-3} + 1.01 z. \end{equation} \tag{ 8 }$$

The functions above were derived by binning the k-corrections in redshift and taking the error on the mean (rms divided by square root of the number of entries) as the uncertainty. With this formalism the error on the fitted parameters is ≲1% and is thus negligible in comparison to the statistical error on our SN Ia rate measurements.

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To transform the maxBCG cluster luminosities from ^(0.25)i to r, we note the following identity:

$$\begin{eqnarray} m_{r} & = & m^{0.25}_{r} - K_{rr}(z=0.25)\\&&\nonumber\\ & = & m^{0.25}_{i} + (r-i)(z=0.25) - K_{rr}(z=0.25). \end{eqnarray} \tag{ 9 }$$

We k-correct the maxBCG cluster luminosities from ^(0.25)i to r because at a redshift z = 0.25 rest-frame r maps closely to observer-frame i, and because the analysis of Hansen et al. (2009) that we take as the definitive measurement of the maxBCG cluster luminosities is performed for ^(0.25)i and not ^(0.25)r. Using the expressions given above for (r − i) and K_rr, evaluated at z = 0.25, we derive the transformation m_r = ^(0.25)m_i + 0.063. For comparison to other cluster SN rate measurements, it is necessary to express the cluster luminosities in units of L^r_☉. The conversion to solar luminosities is given by

$$\begin{equation} \left(\frac{L}{L_{\odot }}\right)_{r} = \left(\frac{L}{L_{\odot }}\right)_{^{0.25}i} \times 10^{-0.4 ((m_{r} - m_{^{0.25}i}) - (M^{r}_{\odot } - M^{^{0.25}i}_{\odot }))}, \end{equation} \tag{ 11 }$$

where M_☉ is the absolute magnitude of the Sun. To compute the absolute magnitude of the Sun in an arbitrary filter, we use a solar spectrum obtained from the CALSPEC²⁸ database, hosted by the Space Telescope Science Institute, and compute synthetic magnitudes using a custom piece of software written for this task. We thereby derive values for the absolute magnitude of the Sun of M^r_☉ = 4.64 (in agreement with the value published by Blanton & Roweis 2007), and $M^{^{0.25}i}_{\odot } = 4.67$. With these values, Equation (11) becomes $(L/L_{\odot })_{r} = 0.918 (L/L_{\odot })_{^{0.25}i}$, giving the r-band luminosity in maxBCG clusters, for galaxies with L>0.4L_*, as ΣL_r = 1.006 × 10¹⁴L_☉.

Any uncertainty on the luminosity of the clusters described above will propagate to an uncertainty on the derived cluster SN Ia rates. Sources of uncertainty on the cluster luminosities might include, e.g., uncertainties in the background subtraction used in defining the maxBCG cluster luminosities, or uncertainty in the correction for fiber collisions in defining the C4 clusters. The cluster catalogs described above do not include uncertainties on the cluster luminosities, and it is beyond the scope of this analysis to determine these uncertainties. However, as an estimate of the order of magnitude of the cluster luminosity uncertainties, we note that Menanteau et al. (2010) give uncertainties on the luminosities of individual clusters, for a cluster catalog that was constructed in a similar way to the maxBCG cluster catalog. These uncertainties are ≈10%, and include both statistical and systematic effects. As discussed above, the SN Ia rate analysis includes 71 and 492 clusters from the C4 and maxBCG catalogs, respectively. If the uncertainties on the individual clusters are ≈10% and are independent, then the uncertainty on the total cluster luminosities will be ≲1%. We conclude that uncertainty on the cluster luminosities is likely to be negligible in comparison to the statistical uncertainty on the cluster SN Ia rate measurement, but we acknowledge that any additional uncertainties on the cluster luminosities, or correlations between the uncertainties on the luminosity of the individual clusters, would impact the precision of our cluster SN Ia rate measurements.

To define the SN sample that is associated with galaxy clusters it is necessary to define selection criteria on both the SN light-curve properties, and the SN spatial correlation with clusters. As was done in our study of the low-redshift SN rate from the first season of the SDSS-II Supernova Survey (Dilday et al. 2008), the SNe in the rate sample are required to meet certain selection criteria on their light-curve fits to the MLCS2k2 light curve model (Jha et al. 2007). For the present analysis the selection criteria are as follows:

• 1.  
  At least 1 observation at <−2 days relative to peak in the supernova rest-frame.
• 2.  
  At least 1 observation at >t_late days relative to peak in the supernova rest-frame.For maxBCG cluster SN candidates t_late = 10 days, and for C4 cluster SN candidates t_late = 5 days. The peak of the SN light curve is defined as the epoch of maximum luminosity in the rest-frame B band.
• 3.  
  Fit probability >10⁻³.The fit probability is computed assuming that the value of the minimum χ² from the MLCS2k2 light curve fit follows a χ² distribution with N_p − 4 degrees of freedom, with N_p the number of photometric observations. All SNe light curves are fit for epoch of maximum light, luminosity parameter (Δ), and extinction parameter (A_V). When the SN redshift is known from spectroscopic measurement, the SN light curve is additionally fit for distance modulus. In the case of photometric SNe, the additional fit parameter is the redshift of the SN.

The requirement of a late-time photometric measurement is used primarily to reduce misidentification of photometric SN Ia candidates. For the low redshifts of the C4 galaxy clusters the majority of SNe have spectroscopic confirmation of type, and so a requirement of t_late>5 days increases the size of the SN sample, without introducing uncertainty from misidentification of photometric SNe. For the maxBCG clusters, the number of photometric cluster SN candidates is significant and a stricter requirement, t_late>10 days must be used. Additionally, the SNe that will be used for determining the SN rate are required to satisfy a set of selection criteria, based on fits of their search photometry to models of Type Ia, Type II, and Type Ib/c light curve models. These additional selection criteria are identical to those discussed in more detail in Dilday et al. (2010). Briefly, the SN candidates must be discovered in at least three epochs of the SN search, must fit the Type Ia model well in relation to the Type II and Type I/bc models, and must not have an overly broad light curve shape, as typified by the peculiar SN Ia 2005gj (Aldering et al. 2006; Prieto et al. 2007). The combination of cuts on light-curve sampling and shape effectively reject active galactic nuclei (AGNs) from the cluster SN Ia sample.

An SN candidate is defined to be associated with a galaxy cluster if it satisfies the following:

• 1.  
  SN is within 1 Mpc h⁻¹ projected distance of the center of a cluster.
• 2.  
  SN redshift is consistent with the cluster redshift.

Projected distance here refers to the orthogonal distance from the SN to the center of the galaxy cluster, assuming the redshift of the cluster. The choice of 1 Mpc h⁻¹ projected distance to define association with a cluster is chosen largely for consistency with previous cluster SN Ia rate measurements. The definition of redshift consistency depends on whether the SN and/or cluster redshifts are determined photometrically or spectroscopically. For the C4 clusters, the cluster redshifts are always precisely determined with a spectroscopic measurement, whereas the maxBCG cluster redshifts are determined photometrically. Consistency between the SN redshift, z_s, and cluster redshift, z_c, is defined in the following way:

• 1.  
  Spectroscopic SN redshift and spectroscopic cluster redshift|z_s − z_c| < 0.015.
• 2.  
  Spectroscopic SN redshift and photometric cluster redshift|z_s − z_c| < 0.025.
• 3.  
  Photometric SN redshift and spectroscopic cluster redshift$| z_{s} - z_{c} | < 2.5 \sqrt{(0.01)^2 + \delta z_{s}^2}$ .
• 4.  
  Photometric SN redshift and photometric cluster redshift$| z_{s} - z_{c} | < 2.5 \sqrt{(0.015)^2 + \delta z_{s}^2}$ ,

where δz_s is the error on the SN photometric redshift. In the case that both the SN and the cluster have spectroscopically determined redshifts, the allowable spread in galaxy redshifts is taken as 0.015. This number was determined from the catalog of maxBCG cluster member galaxies. For each cluster that has at least four member galaxies with spectroscopically determined redshifts, we computed the rms of the difference between the member spectroscopic redshifts and the redshift assigned to the cluster. The mean of the resulting distribution of rms values is ≈0.015. A similar value is obtained if the difference between the cluster redshift and the member galaxy spectroscopic redshifts are instead fit with Gaussian distributions, and the mean of the distribution of rms values for each Gaussian is computed. We note that for the nine C4 cluster SNe that have spectroscopic redshifts for both the SN and the cluster, the deviation in redshift is ≲0.002 and this cut could be made more strict without affecting the SN rate result. In the case that the SN has a spectroscopically measured redshift and the cluster has a photometrically determined redshift (maxBCG clusters only) the tolerance on redshift consistency (0.025) is determined by adding in quadrature the rms spread of maxBCG member galaxy redshifts mentioned above (0.015) and the typical accuracy of the maxBCG cluster photometric redshifts (≈0.015–0.020).

In the case that the SN redshift is determined photometrically, the cuts we apply correspond to a tolerance of 2.5σ_z, where σ_z represents the error on the SN redshift, added in quadrature to a tolerance on the cluster redshift. This is a rather loose tolerance on redshift consistency, and we discuss this issue further in Section 5. There are 9 SNe Ia in C4 clusters and 27 SNe Ia from maxBCG clusters that satisfy the selection criteria, and these are listed in Table 3 (C4 clusters) and Table 4 (maxBCG clusters), respectively. We note that SNe 12979 and 18375 occur in a cluster that is a member of both the C4 and maxBCG catalogs, and thus there are 34 distinct SNe Ia that pass the selection criteria. We note that SNe 16280, 18047, and 18362 are each associated with two distinct clusters, and that SNe 14279 and 16215 are associated with the same cluster. In the cases where a SN is associated with two clusters, we will count it only once in the SN rate calculation, assigning it to the nearest of the two clusters.

The method for determining the SN discovery efficiency is based on the same Monte Carlo (MC) studies discussed in Dilday et al. (2010). However, there is an important modification to be considered for SNe in galaxy clusters. It is a well established result that Type Ia SNe in early-type galaxies are more likely to be intrinsically faint, fast-declining SNe (e.g., Sullivan et al. 2006; Jha et al. 2007; M. Smith et al. 2010, in preparation). Therefore, the assumed distribution for the luminosity parameter of the MLCS2k2 light-curve model, Δ, for the entire SN sample, as employed in Dilday et al. (2010), is not appropriate for considering SNe in galaxy clusters. To determine the SN discovery efficiency for SNe in galaxy clusters, we generated a set of MC SNe with a Δ distribution similar to the observed distribution for cluster SNe. The Δ distribution assumed for cluster SNe Ia is shown in Figure 4. The fiducial Δ distribution has mean 〈Δ〉 = 0.033, with an rms of 0.286, and the cluster SNe Δ distribution has mean 〈Δ〉 = 0.081, with an rms of 0.246. Assuming a larger mean value for Δ does not significantly affect the SN selection efficiency at the low redshifts of the C4 clusters, and has a 2% effect at the redshifts of the maxBCG clusters. The SN selection efficiency at the redshifts of the C4 clusters is approximately constant for each observing season, with the values 0.77, 0.73, and 0.72 for the 2005, 2006, and 2007 observing seasons, respectively. The redshift-dependent efficiencies used for the maxBCG cluster SNe are discussed in Dilday et al. (2010). Briefly, the efficiencies are well described by a function, $\epsilon _{0}/(1+e^{(z - z_0)/s_z})$, with ₀ ≈ 0.7, z₀ ≈ 0.35, and s_z ≈ 0.05. As in Dilday et al. (2008, 2010) the efficiency at low redshift is significantly less than 1 due mainly to the requirements of early and late time observing epochs.

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In Section 4, we have defined a set of criteria on the consistency of the SN and cluster redshifts, which, in the case of photometric redshifts, corresponds roughly to a requirement that the redshifts match to within ≈2.5σ. We chose to apply a loose cut on the photometric redshift consistency because cluster SNe are rare by their nature and we wish to maximize the size of the sample, and because we assert that the requirement of a spatial correspondence with a cluster is already a strong filter on contamination from chance projections. Here we quantify further the expected contamination rate from chance projections, in the presence of finite error on the SN redshift measurement. We note that for the C4 cluster SN sample, all the SNe have spectroscopically measured redshifts (see Table 3), such that the contamination rate due to using SN photometric redshifts is manifestly 0, and we focus exclusively on contamination in the maxBCG SN sample.

To estimate the expected contamination rate we employ the following procedure. For each cluster in the maxBCG catalog we compute the expected contamination,

$$\begin{equation} n^{i}_{\mathrm{contam}} = \int _{0}^{\infty } dz \eta (z) \int _{z_{\mathrm{cl}}-n_{\sigma }\sigma _z}^{z_{\mathrm{cl}}+n_{\sigma }\sigma _z} dz^{\prime } \frac{1}{\sqrt{2\pi }\sigma _z} \exp \left(\frac{-(z-z^{\prime })^2}{2\sigma _z^2}\right), \end{equation} \tag{ 12 }$$

where η(z) is the SN number density, per unit redshift, z_cl is the cluster redshift, σ_z is the assumed error on the SN photometric redshift, and n_σ defines our cut on redshift consistency. The error on the photometric redshift, as a function of redshift, is shown in Dilday et al. (2010) to be σ_z ≈ 0.2 × z^1.5. To compute η(z), we use the power-law rate model from Dilday et al. (2008; r_V = 2.6 × 10⁻⁵(1 + z)^1.5SNe Mpc⁻³h³₇₀), along with the SDSS-II Supernova Survey SN efficiency function. The solid angle used in computing η(z) is the solid angle corresponding to a 1 Mpc h⁻¹ separation from a cluster at redshift z_cl. However, we assume that no SNe will be spatially associated with a cluster and have a redshift within ±0.015 of the cluster unless it is, in fact, associated with the cluster. Therefore, η(z) is assumed to be 0 for −0.015 < z − z_cl < +0.015. The choice of 0.015 as the spread of maxBCG cluster redshifts is discussed in Section 4.1. For each cluster in the maxBCG catalog, we also compute the number of expected cluster SNe Ia, N^T, assuming a cluster SN Ia rate per unit r-band luminosity that is broadly consistent with those given by Sharon et al. (2007a) and Mannucci et al. (2008). The observed number of SNe that would be included in the sample, N^O, is then related to the true number, N^T, by

$$\begin{equation} \frac{N^{O}}{N^{T}} = \mathrm{erf}(n_{\sigma }/\sqrt{(}2)) + \frac{N_{\mathrm{contam}}}{N^{T}}, \end{equation} \tag{ 13 }$$

where erf(·) is the error function, and N_contam is the total expected contamination, Σ_inⁱ_contam. To clarify our approach, we note that the right-hand side of Equation (13) can be computed, with reasonable assumptions about the value of the cluster SN Ia rate. We can then compare this to the observed value of the left-hand side of Equation (13), and thereby infer the contamination rate of the SN Ia sample due to finite redshift uncertainties. For cuts on redshift consistency of n_σ = 1.5 and n_σ = 2.5, N_contam/N^T has the values 0.151(r_cl/SNur)⁻¹ and 0.276(r_cl/SNur)⁻¹, where r_cl is the assumed value of the cluster SN rate, and SNur = (10¹⁰L^r_☉)⁻¹(100 yr)⁻¹.

We first note that using values for n_σ ≈ 1.5–2.5, we consistently arrive at N^T ≈ 5. By way of example, if we choose n_σ = 2.5 and r_cl = 0.6SNur, we then find N^O/N^T = 1.44 (the right-hand side of Equation (13)). There are seven SNe Ia with photometrically determined redshifts that are within ±2.5σ of the cluster redshift (Table 3), and so this implies a value of N^T = N^O/1.44 ≈ 5 (the left-hand side of Equation (13)). Therefore, we apply a bias correction to our maxBCG SN sample by assuming that the number of photometric SNe in the sample is five, rather than the seven listed in Table 4. To estimate the systematic error on the size of the maxBCG SN sample, we vary the assumed power-law SN rate model according to the errors given in Dilday et al. (2008), and vary the assumed cluster rate as r_cl = (0.6 ± 0.2)SNur, which results in an estimated systematic error of ≈(−2 to + 3)% on the maxBCG cluster SN rate measurements, due to employing photometric redshifts for a subset of the cluster SNe.

Here we discuss the effect of incompleteness and contamination in the galaxy cluster catalogs on our cluster SN rate measurements, and on our comparisons of the cluster SN rate to the rate in early-type galaxies in the field. In what follows we assume that the SN rate in both galaxy clusters and the field is proportional to the host luminosity, with proportionality constants α_c and α_f for clusters and the field, respectively. We note that the C4 cluster catalog has ≈10% incompleteness and an ≈5% contamination rate (Miller et al. 2005), and that the maxBCG catalog has ≈10% incompleteness and an ≈10% contamination rate (Koester et al. 2007b).

Cluster incompleteness. With the ansatz that the cluster SN Ia rate is proportional to the stellar luminosity, independent of cluster richness, cluster incompleteness does not affect the derived value of the SN rate. That is, although real clusters may fail to be included in the cluster catalog, the cluster SN sample will be reduced in direct proportion to the luminosity of the missing clusters. Cluster incompleteness, however, will affect the value of the ratio of the rates for cluster versus field early-type galaxies, as SNe occurring in clusters may be erroneously associated with the field SN sample.

In the presence of cluster incompleteness, the inferred ratio of the SN Ia rate in clusters, r_c, to the SN rate in field early-type galaxies, r_f, will be

$$\begin{eqnarray} \frac{r_c}{r_f} & = & \left(\frac{N_c - N_{\delta -}}{L_c - L_{\delta -}}\right) \left(\frac{L_f + L_{\delta -}}{N_f + N_{\delta -}}\right) \end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray} & \approx & \frac{\alpha _{c}}{\alpha _{f}}(1+(1-\alpha _{c}/\alpha _{f})(L_{\delta -}/L_c)(L_c/L_f)), \end{eqnarray} \tag{ 15 }$$

where N(L) represents the true number of SNe (luminosity) in a given subset, and the c, f, and δ− subscripts denote clusters, the field, and the portion of the clusters missed due to incompleteness, respectively. The quantity L_δ−/L_c is the cluster incompleteness. To estimate the quantity L_c/L_f, we compute a cluster luminosity density by dividing the summed r-band luminosity from the maxBCG cluster catalog (k-corrected from ^0.25i as described in Section 3.3), for clusters in the SN survey region, by the corresponding volume. For redshifts z ≳ 0.2 this derived cluster luminosity density has the value ≈0.104 × 10⁸ L_☉ Mpc⁻³ h. In Dilday et al. (2008), we showed that, using the LFs of Blanton et al. (2003b), and an objective classification scheme for early-type galaxies, the ^0.1r-band luminosity density, for early-type galaxies is 0.994 × 10⁸ L_☉ Mpc⁻³ h. Applying a k-correction as described in Section 3.3, this corresponds to a r-band luminosity density, for early-type galaxies, of 1.24 × 10⁸ L_☉ Mpc^-3h. Therefore, we conclude that the ratio of the total r-band light in cluster galaxies versus field early-type galaxies, L_c/L_f, is ≈0.091. The fraction of cluster luminosity contained in early-type galaxies is ≈80% (Hansen et al. 2009), and thus the ratio of the r-band light in cluster early-type galaxies versus field early-type galaxies, (L_c/L_f)_red, is ≈0.073. In turn, this results in an estimated ≈−(1–2)% correction factor, for values of the ratio of the SN rates, α_c/α_f, of ≈2–3.

Cluster contamination. Contamination of the cluster catalogs will affect both the value of the SN rate in clusters, and the ratio of the rate in clusters to the rate in the field.

The inferred cluster SN rate will be

$$\begin{eqnarray} r_c & = & \frac{N_c + N_{\delta +}}{L_c + L_{\delta +}}\\&&\nonumber\\ & \approx & \alpha _{c} (1+(\alpha _{f}/\alpha _{c}-1) L_{\delta +}/L_c), \end{eqnarray} \tag{ 16 }$$

where N and L are defined as in Equation (14), and δ+ denotes field quantities that were erroneously identified with clusters. The quantity L_δ+/L_c represents the cluster contamination rate. The estimated correction factor on the SN Ia cluster rate measurement due to cluster contamination is then ≈+3% for C4 clusters, and ≈+(5–7)% for maxBCG clusters.

The inferred ratio of the SN rate will be

$$\begin{eqnarray} \frac{r_c}{r_f} & = & \left(\frac{N_c + N_{\delta +}}{L_c + L_{\delta +}}\right) \left(\frac{L_f - L_{\delta +}}{N_f - N_{\delta +}}\right)\\&&\nonumber\\ & \approx & \frac{\alpha _{c}}{\alpha _{f}}(1+(\alpha _{f}/\alpha _{c}-1) L_{\delta +}/L_c), \end{eqnarray} \tag{ 18 }$$

with estimated correction factors on the cluster versus field SN rates of ≈+3% for C4 clusters, and ≈+(5–7)% for maxBCG clusters.

To summarize our consideration of necessary corrections to our cluster SN Ia rate measurements, we have considered the effects of the lesser average intrinsic luminosity for SNe Ia in early-type galaxies as compared to a representative galaxy sample, contamination of the cluster SN Ia sample due to the use of SN photometric redshifts, and incompleteness (clusters are not identified by the cluster finding algorithms) and contamination (objects that are not clusters are identified as clusters) of the galaxy cluster catalogs. The lesser average intrinsic luminosity of SNe Ia in early-type galaxies does not affect the selection efficiency for the relatively low-redshift C4 SN sample, and has an ≈2% effect on the maxBCG SN sample. This factor is accounted for through the efficiency function, (z), that is mentioned in Section 7. We estimate that ≈2 of the 7 SNe with photometric redshifts in the maxBCG SN Ia sample are not in fact associated with the cluster, and thus we will apply a correction of f^pz ≈ −2/27 = −7.4% to the maxBCG cluster rate. Cluster contamination results in correction factors of f^c ≈ +3% for the C4 cluster SN Ia rate, and f^c ≈ +(5–7)% for the maxBCG cluster SN Ia rate. Cluster contamination also results in correction factors on the ratio of the SN Ia rate in early-type cluster galaxies to early-type field galaxies of f^c ≈ +3% for the C4 clusters, and f^c ≈ +(5–7)% for the maxBCG clusters. Additionally, cluster incompleteness results in an f^I ≈ −(1–2)% correction factor on the ratio of the SN Ia rate in early-type cluster galaxies to early-type field galaxies for both the C4 and the maxBCG clusters. We note that in Dilday et al. (2010) it is shown that the contamination of the SDSS-II SN Survey photometric SN Ia sample by non-Ia SNe is not more than ≈3%. Photometric SNe Ia make up ≈50% of the maxBCG cluster SN Ia sample, and so the overall contamination of the cluster SN Ia sample is ≲2%. Contamination of the SN Ia sample due to lack of a spectroscopic identification is likely to be correlated with the contamination due to the use of photometric redshifts discussed above, and so the size of the effect is expected to be less than this.

As discussed above, there are nine Type Ia SNe in C4 clusters that satisfy the SN selection criteria. The total r-band luminosity in the SN survey region, after correcting for the faint end of the LF, is 4.99 × 10¹³ L^r_☉ h⁻², and the mean redshift of the C4 clusters is 〈z〉 = 0.0786. The observing time for the SDSS-II Supernova Survey was 89, 90, and 90 days, for the 2005, 2006, and 2007 observing seasons, respectively. The SN selection efficiency is approximately constant over the range of the C4 clusters, with the values 0.77, 0.73, and 0.72 for the 2005, 2006, and 2007 observing seasons, respectively. Using these values, the SN rate in C4 clusters is

$$\begin{equation} r = \frac{N (1+\langle z \rangle)g^{c}}{\Sigma L_{r} \Sigma _i(\epsilon T)} = {0.37}_{-0.12-0.01}^{+0.17+0.01} \mathrm{SNu}r\, h^{2}, \end{equation} \tag{ 20 }$$

where i denotes each observing season, T is the survey observation time, is the SN selection efficiency, N is the number of SNe Ia, ΣL_r is the total cluster luminosity in the r band, and SNur = (10¹⁰ L^r_☉)⁻¹(100 yr)⁻¹. The factor g^c = 1 + f^c = 1.03, where f_c is the correction due to contamination of the cluster catalog, discussed in Section 5. The errors quoted are the 1σ statistical and systematic errors, respectively. We have used the mean survey efficiency, determined from Monte Carlo simulations, as both the positions of the clusters and the Monte Carlo SNe are effectively uniform random samplings of the survey area. Jorgensen (1997; clusters) and Padmanabhan et al. (2004; field) give the average stellar mass to luminosity ratio (r-band) for early-type galaxies as ≈3, which is in good agreement with the M/L conversion employed by Sharon et al. (2007a). Using this assumption, the Type Ia SN rate per unit luminosity quoted above is equal to 0.123^+0.056+0.004_{−0.040−0.003}SNuM h², with SNuM = (10¹⁰ M_☉)⁻¹(100 yr)⁻¹. Using the same average conversion factor from r to B as Sharon et al. (2007a), this corresponds to 0.46^+0.21+0.01_{−0.15−0.01}SNuB h².

As discussed in Section 4, there are 27 SNe in maxBCG clusters from SDSS-II Supernova Survey that satisfy the selection criteria. As the SDSS-II Supernova Survey SN discovery efficiency is not well approximated as constant over the redshift range of the maxBCG catalog, in determining the SN rate in maxBCG clusters we use the more formal definition of the SN rate per unit luminosity, r_L

$$\begin{equation} r_{L} = \frac{N g^{pz} g^{c}}{\widetilde{\epsilon T \Sigma L}} \end{equation} \tag{ 21 }$$

with

$$\begin{equation} \widetilde{\epsilon T \Sigma L} = T_{\oplus } \int _{z_{\mathrm{min}}}^{z_{\mathrm{max}}} dz \frac{\Sigma L(z) \epsilon (z)}{1+z}, \end{equation} \tag{ 22 }$$

where T_⊕ is the earth frame observation time, ΣL(z) is the total cluster luminosity as a function of redshift, and (z) is the SN discovery efficiency. The factors g^pz and g^c represent the corrections due to use of SN photometric redshifts and contamination of the cluster catalog, respectively (Section 5). These factors have the values g^pz = 1 + f^pz = 0.926 and g^c = 1 + f^c = 1.06. For the maxBCG catalog z_min is fixed to 0.1. In Figure 5, we show the value of the maxBCG cluster SN rate as a function of z_max. This figure shows that the derived SN rate is not strongly sensitive to the upper limit on the cluster sample. If the upper limit is chosen as the upper limit of the maxBCG catalog, z = 0.3, then we have N = 27 and $\widetilde{\epsilon T \Sigma L} = 4.85 \times 10^{13}$ yr L^r_☉ h⁻², including the correction for the faint end of the LF. The derived value of the SN rate is thus r_L = 0.55^+0.13+0.02_{−0.11−0.01}SNur h²(=0.68^+0.17+0.02_{−0.14−0.02}SNuB h²=0.18^+0.044+0.006_{−0.036−0.004}SNuM h²). Figure 6 shows the cluster SN rate as a function of the limit on the projected distance from the center of the cluster, in units of Mpc h⁻¹. The fraction of red, early-type galaxies in clusters is larger at small separations from the cluster center, and in this sense the SN rate at smaller separations is a more reliable probe of the component of the SN rate that originates from an old stellar population. Furthermore, the extent of a cluster is not an unambiguously defined quantity, and SNe at smaller separations are more robustly associated with the cluster. Figure 6 shows that the derived SN rate is not strongly dependent on the limiting projected radius that we use to define cluster membership. Figure 7 shows the SN rate as a function of the lower limit on the cluster richness measure, N₂₀₀.

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The SDSS-II Supernova Survey cluster rate results, along with the previous measurements listed in Table 1, in units of SNuB h², are shown in Figure 8. A fit of the data to a linear model of the cluster SN Ia rate as a function of redshift, r = A + Bz, gives best fit values of A = 0.49 ± 0.14SNuB h² and B = 0.81^+0.82_−0.80 (χ²/NDF = 0.88/5). A fit of the data to a constant model of the cluster SN Ia rate as a function of redshift, r = A, gives a best fit value of A = 0.61^+0.95_−0.89SNuB h² (χ²/NDF = 1.9/6).

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In Dilday et al. (2008), the SN rate in low-redshift early-type galaxies was estimated from the first year data of the SDSS-II Supernova Survey as ≈0.17^+0.06_−0.04 SNur h². One SN in the sample described in Dilday et al. (2008) is a cluster SN, and hence the early-type field rate is ≈0.16^+0.06_−0.04 SNur h². Using the same color cut (u − r = 2.4) to differentiate early- and late-type galaxies as Dilday et al. (2008), we find that 6/9 of the C4 cluster SNe reside in early-type galaxies, while 20/27 maxBCG cluster SNe reside in early-type galaxies (18/25 after bias correction to account for SN photometric redshifts). According to Hansen et al. (2009), the fraction of light in early-type galaxies for the maxBCG clusters is ≈80%. Therefore, the cluster SN rate for early-type galaxies is r_L = 0.31^+0.18+0.01_{−0.12−0.01}SNur h² and r_L = 0.49^+0.15+0.02_{−0.11−0.01}SNur h² for C4 and maxBCG clusters, respectively. As discussed in Section 5, the ratio of the cluster early-type to field early-type rates must be multiplied by a factor g^I = 1 − f^I = 0.985 to account for incompleteness of the galaxy cluster catalogs. The C4 and maxBCG cluster SN Ia rates are seen to be larger by factors of 1.94^+1.31+0.043_{−0.91−0.015} and 3.02^+1.31+0.062_{−1.03−0.048} compared to the field SN rate for early-type galaxies, respectively. This is in broad agreement with the enhancement of the Type Ia SN rate in early-type galaxies in galaxy clusters reported by Mannucci et al. (2008).

Of the 32 SNe in our cluster sample (after accounting for duplicates and for contamination due to use of SN photometric redshifts), two are in the BCG of the cluster. These are SNe 12979 and 13905. Another, SN 18890 appears to be in the BCG, although it has a somewhat ambiguous host. All three of these SNe are in C4 clusters, while one is in a maxBCG cluster. The cluster BCGs contain ≈6% of the total r-band cluster luminosity, and so the SN Ia rate, in the cluster BCGs, is r_L = 2.04^+1.99+0.07_{−1.11−0.04}SNur h² for the C4 clusters and r_L = 0.36^+0.84+0.01_{−0.30−0.01}SNur h² for the maxBCG clusters.

For the SNe Ia in our cluster sample, all but two are visually associated with a host galaxy. The two SNe Ia with no host evident in the images are SN 13073 and SN 19001. Additionally, one SN, SN 1782, lies at a distance ≈25 kpc h⁻¹ from its presumed host galaxy, and might reasonably be considered to be hostless. Assuming SN 1782 is in fact a hostless SN, we can constrain the fraction of hostless SNe in galaxy clusters to be ≈9.4%, with 1σ confidence interval (4.3–17.7)%. For comparison, Gal-Yam et al. (2003) reported two hostless SNe from a sample of seven for an estimated fraction of hostless SNe of ≈30%. We note, however, that the classification of these two SNe as hostless by Gal-Yam et al. (2003) relied, in part, on the observation of velocity offsets between the SNe and its potential host of ≈750–2000 km s^-1. In principle a more thorough analysis of SN and galaxy spectra for SNe from the SDSS-II Supernova Survey may reveal additional hostless SNe. The clusters considered by Gal-Yam et al. (2003) were generally more massive than the clusters considered here, and an additional possibility is that the rate of hostless SNe Ia in galaxy clusters is larger for more massive clusters.

Forster & Schawinski (2008) have studied the radial distribution of SNe in early-type galaxies. The sample of SNe includes an unspecified number of SNe from the SDSS-II Supernova Survey. Here we consider the radial distribution for SNe for field and cluster ellipticals.

To construct the sample of early-type galaxies, we employ the following procedure. We first assign a host galaxy to each SNe by locating the nearest host from the SDSS database, in units of the isophotal radius of the host galaxy in the r band. The algorithm is described in detail in Dilday et al. (2008). Early-type galaxies are defined as those host galaxies that satisfy the following criteria:

• 1.  
  u − r>2.4,
• 2.  
  r < 21.5,
• 3.  
  Δr < 0.05,

where u and r are the SDSS model magnitudes and Δr is the error on the r-band magnitude. It is a well established result that the (u − r) color for SDSS galaxies is bimodal (Strateva et al. 2001), with early-type galaxies generally having (u − r)>2.2. As discussed in Dilday et al. (2008), studies of the observed u − r distribution for galaxies from the photometric redshift catalog of Oyaizu et al. (2008) suggest that u − r = 2.4 may provide a more robust separation. The requirement that r < 21.5 is imposed as the separation of stars and galaxies is fairly robust to this limit. The requirement that Δr < 0.05 is imposed to remove outlying, poorly measured galaxies. The distributions of the distance of each SN from its host galaxy, for field and cluster early-type hosts, is shown in Figure 9. For field early-type galaxies, a fit of the data to a Sersic model of the luminosity distribution, $dN/dr = A \rho e^{-\gamma \rho ^{-\lambda }}$, where ρ denotes the distance of the SN in units of the de Vaucouleurs radius of the host galaxy, gives a value for λ of 0.20 ± 0.08, which is consistent with a de Vaucouleurs profile (λ = 0.25). Since the distribution of light in early-type galaxies is known to follow a de Vaucouleurs profile, the result of the fit confirms the results of Forster & Schawinski (2008) that the SN rate in field ellipticals is well represented by a constant rate per unit luminosity. The radial distribution for SNe in cluster early-type galaxies shows an enhancement at small radial separations in comparison to a de Vaucouleurs profile. An enhancement of the SN Ia rate in regions that have undergone recent star formation has been reported by several authors (see, e.g., the review by Mannucci 2009), and our observed enhancement at small radial separations is possible evidence for a component of the cluster SN Ia rate that tracks residual star formation activity in cluster early-type galaxies. Such an enhancement is in qualitative agreement with the larger SN Ia rate in cluster ellipticals, compared with field ellipticals, mentioned above and by Mannucci et al. (2008), as well as with recent independent evidence for some recent star formation activity in early-type galaxies (e.g., Kaviraj et al. 2007). Additionally, Della Valle et al. (2005) and Graham et al. (2010) report an enhancement of the SN Ia rate in radio-loud early-type galaxies, and attribute this to mergers that not only power the radio emission, but also provide a young stellar population that can account for the enhanced SN Ia rate. While we do not consider the radio properties of the early-type galaxies in our sample, it is possible that our results are related to the results of Della Valle et al. (2005) and Graham et al. (2010). While the best fit radial profile for cluster early-type galaxies does not match a de Vaucouleurs profile, a Kolmogorov–Smirnov (KS) test on the radial distributions of SNe Ia in cluster and field ellipticals results in an ≈30% probability that the data are drawn from the same underlying distribution. As discussed in Dilday et al. (2008), all SN selection efficiency calculations are based ultimately on artificial SNe Ia inserted directly into the survey search imaging data. Analysis of these artificial SNe Ia from the three observing seasons of the SDSS-II Supernova Survey does not show evidence for significant loss of efficiency near the cores of galaxies. While the artificial SNe were inserted into random galaxies that on the average may be less luminous in the cores than typical cluster galaxies, any additional inefficiency would only increase the observed enhancement of the SN Ia rate in the cores of cluster early-type galaxies.

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