AGN Luminosity and Stellar Age: Two Missing Ingredients for AGN Unification as Seen with iPTF Supernovae

The iPTF is an untargeted wide-field sky survey using the 1.2 m Samuel Oschin telescope (P48) at Palomar Observatory to detect and follow up transient astronomical sources. The scientific scope of the iPTF spans from small solar system objects to extragalactic phenomena. The iPTF project has been running since 2013, and it had a precursor, the Palomar Transient Factory (PTF), which was active during 2009–2012. The technical background of PTF is presented in Law et al. (2009) and the scientific motivation in Rau et al. (2009). The iPTF is described in Kulkarni (2013). For brevity, we will refer to our SN catalog (including SNe found during the PTF period) simply as the iPTF catalog.

In this study, we use the iPTF SN catalog from the time window between 2009 March 2 until 2014 June 17. This catalog contains 2190 extragalactic SNe with known right ascension (α), declination (δ), spectroscopic redshift (z_SN), and spectroscopic classification. This SN catalog contains 1494 thermonuclear SNe (i.e., Type Ia) and 632 c-c SNe. In the c-c category, we include SNe Types Ib, Ic, Ib/c, Ibn, II, IIP, IIL, IIn, and IIb. The remaining 64 SNe are either of unclear spectroscopic type or are superluminous. None of the 11 superluminous SNe with $z\lt 0.2$ in our catalog was found in a galaxy with a spectrum in SDSS DR7. The superluminous SNe can therefore, for our purposes, be included in the remainder category.

The SNe in this sample are located at declinations $-25^\circ \lt \delta \lt 80^\circ$, mostly at galactic latitudes $| b| \gt 20^\circ$. The mean redshift of the sample is $z\approx 0.09$, with 95% of the SNe having $z\lt 0.2$. For the motivation for curtailing the SN catalog at 2014 June 17, see Section 4.3.

An advantage of the iPTF catalog is its cohesive nature. All the SNe have been discovered during an untargeted search with the same telescope, and spectroscopic classifications have been made in a timely fashion. The compatibility with SDSS in sky coverage makes the iPTF SNe sample suitable for our investigation. Our SDSS samples covers $-10^\circ \lt \delta \lt 70^\circ$ and galactic latitudes $| b| \gt 20^\circ$, comparable to the distribution of the SN locations.

Another SN catalog that comes to mind is the SDSS SN catalog (Sako et al. 2014). The 902 confirmed SNe in the SDSS SN survey⁵ represent an SN sample of smaller scope than the iPTF sample. Not all SNe are spectroscopically classified, but the survey is deep and homogenous. Unfortunately, the SNe are all located in the SDSS southern equatorial Stripe 82, which is outside the area of the SDSS DR7 sample from the central region used in this study. This renders the SDSS SNe unsuitable for our purposes.

The Asiago SN catalog (Barbon et al. 1999) is a historically comprehensive compilation of extragalactic SNe. The catalog encompasses 6530 SNe as of 2016 March 27, in both hemispheres. The Asiago catalog is extensive but has uneven quality—some of the SNe in it lack spectroscopic classification, and some lack spectroscopic redshift. The SN catalog compiled by Lennarz et al. (2012) contains data for 5526 extragalactic SNe, whereas the Sternberg SN catalog, presented by Tsvetkov et al. (2004), contains less than 3000 extragalactic SNe. The circumstance that they are compiled from a wide range of sources make them less suitable compared to the cohesive iPTF catalog. The same holds for The Open SN Catalog (Guillochon et al. 2016), with $\approx$37,000 SNe (of which 12% have spectra in the catalog) as of 2016 December, collected from different public sources.

As discussed by Anderson & Soto (2013), earlier SN searches have prioritised SN detection over completeness with respect to SN types or host galaxy types. This bias should affect such compilations as the Asiago catalog, and other catalogs listing SNe found before the start of untargeted SN searches.

The samples of host galaxies were obtained through the SDSS Data Release 7. We selected objects classified as either "quasars" or "galaxies," within redshift $0.03\lt z\lt 0.2$, unless flagged for brightness (flags&0 × 2 = 0), saturation (flags&0 × 40000 = 0), or blending (flags&0 × 8 = 0) (Stoughton et al. 2002, their Table 9).

The emission lines are obtained from the SpecLine table in DR7. We require that the objects have Hα in emission and select Type-1 AGNs, Type-2 AGNs and star-forming galaxies using optical emission line diagnostics. Our Type-1 AGNs are objects with σ(Hα) > 10 Å (or FWHM(Hα) > 1000 km s⁻¹). The Type-2 AGNs have narrow lines σ(Hα) < 10 Å, fulfilling the Kauffmann criterion (Kauffmann et al. 2003):

$$\begin{eqnarray}&&\mathrm{log}([{\rm{O}}\,{\rm{III}}]/{\rm{H}}\beta )\gt 0.61/(\mathrm{log}([{\rm{N}}\,{\rm{II}}]/{\rm{H}}\alpha ))-0.05)+1.3.\end{eqnarray} \tag{ 1 }$$

The star-forming galaxies are defined as all the other narrow-line objects.

In this way, we classify the objects into Type-1s, Type-2s, and star-forming galaxies, referred to as "largest samples," using optical emission line diagnostics. For all objects we search for morphological classifications ("spiral," "elliptical," "uncertain") from the project Galaxy Zoo 1 (Lintott et al. 2008, 2011), emission line measurements, redshifts, and celestial coordinates, leaving "parent samples" of 11,632 Type-1 AGNs (1864 spiral), 77708 Type-2 AGNs (36,720 spiral) and 137,489 star-forming galaxies (49,072 spiral). For the vast majority of these objects we can also find Wide-field Infrared Survey Explorer (WISE) magnitudes.

2.3.1. Refined Samples

The numbers of coherently collected SNe are scarce (2190 in our sample as on 2014 June 17). Thus, we create pairwise matched subsamples of Type-1 AGNs, Type-2 AGNs, and star-forming objects to compare objects as similar to each other as possible in redshift distribution and selected properties e.g., the luminosity L[O iii]5007. We compare (i) Type-1 AGNs to Type-2 AGNs, and (ii) Type-2 AGNs to star-forming galaxies. The aim with the latter test is to probe whether the observed Type-2 AGN properties can be explained by star formation alone.

For two samples of intrinsically similar objects the probability of detecting faint SNe is the same if they have similar redshift distributions. Therefore, for each galaxy in the parent sample, we select a galaxy from the second parent sample having the closest value in redshift and a specific property of interest. After the matching is done, we first discard all matched pairs that differ by more than 20% in the property of interest.

Four types of specific properties and matchings are explored:

• 1.  
  Redshift only. This allows us to remove biases in Galaxy Zoo morphology classifications as well as the Malmquist bias.
• 2.  
  L[O iii]5007 from the narrow-line region (NLR). In the simplest AGN unification, the NLR is believed to be isotropically distributed outside the torus and to be equally strong for AGNs of the same activity level irrespective of the viewing angle. Selecting on L[O iii]5007—meaning one selects all Type-1 and Type-2 AGNs above a selected certain line flux in a sample—should give the same host galaxy properties under the conditions of isotropy (Antonucci 1993). Matching on L[O iii]5007 is similar to selecting on L[O iii]5007 if the two L[O iii]5007 distributions are the same (as predicted by the simplest unification) but can be problematic if the distributions differ at the high-luminosity end. Moreover, we expect the same line width σ[O iii]5007 in matched Type-1 and Type-2 AGN samples.
• 3.  
  WISE M_w4 (22 μm) absolute magnitude. We use this match as our stellar-mass proxy assuming the dust emission traces the stellar mass. The 22 μm magnitude is a good measure of heated-dust emission in the host galaxy, especially in galaxies where the torus contribution to the total 22 μm is negligible in comparison, meaning galaxies having ${m}_{W1}-{m}_{W2}\lt 0.5$ (Wright et al. 2010). A less favorable option is to use dust reddening F(Hα/Hβ), but if dust reddening in the BLR and NLR differs (Gaskell 1984), the matching will be biased. We therefore do not correct L[O iii]5007 for dust reddening. This matching is only done for galaxies in the parent samples that have WISE magnitudes.
• 4.  
  Exponential fit scale radius (r-band). As the star formation history depends on the available gas mass, matching by an apparent measure of the galaxy volume should minimize differences in star formation histories under the assumption of a mass–size relation.

We do a two-sample Kolmogorov–Smirnov test for each matched property to ensure the sample distributions are statistically similar. For the M_w4-matched samples we had to do a finer match by discarding matched pairs differing by more than 5% in M_w4.

The pairwise matched subsamples for examining SN counts are created by matching in three different parameters (redshift z, luminosity L[O iii]5007, and m_W4) as described earlier. The sizes of the pairwise matched subsamples can be found in Table 4 below. An example of the redshift and property distributions in the matched Type-1 and Type-2 samples can be seen in Figures 1 and 2. The similarity of all the matched samples is ensured through the two-sample Kolmogorov–Smirnov test where the null hypothesis (that two matched samples are similar) holds for the nominal value α = 0.05. The procedure is repeated for pairwise matched Type-2 AGNs and star-forming galaxies.

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Finally, the refined samples of face-on hosts with high signal-to-noise are also matched with the same method. In addition, the refined samples are matched by Balmer decrement F(Hα)/F(Hβ).

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A SN is considered to be matched with a galaxy if the following two conditions are both satisfied:

• 1.  
  The projected distance on the plane of the sky between the SN and the galaxy is computed (plane approximation). The search radius d is set by converting the desired physical search radius into a angular radius at a galaxy distance assumed to be ${z}_{\mathrm{AGN}}\cdot c/{h}_{0}$. A Hubble constant ${h}_{0}=72$ km s⁻¹ Mpc⁻¹ is used throughout this work. If the SN is found to lie inside the given search radius, the compliance with a redshift condition is also checked.
• 2.  
  Our redshift condition is $| {z}_{\mathrm{SN}}-{z}_{\mathrm{AGN}}| \lt 0.003$ and accounts for redshifts due to peculiar motions. This is more strict than the redshift matching condition of $| {z}_{\mathrm{SN}}-{z}_{\mathrm{AGN}}| \lt 0.01$ used in the study by Wang et al. (2010). In their study, however, most (97%) of their sample of 620 cross-matches fulfills the $| {z}_{\mathrm{SN}}-{z}_{\mathrm{AGN}}| \lt 0.003$ condition.

If both these conditions are fulfilled, the SN is considered to be associated with the galaxy. The SNe inside a galaxy are collected by the $d\lt 10$ kpc, and those inside a galaxy or in a close companion are found with the $d\lt 100$ kpc criterion. If this is wrong and there is no association between the AGN and the SN, then the distribution of different SNe with different $| {\rm{\Delta }}z|$ = $| {z}_{\mathrm{SN}}-{z}_{\mathrm{AGN}}|$ ought to be a uniform distribution. This can be checked by plotting a histogram with the $| {\rm{\Delta }}z|$ between the AGN and the SNe, see Figure 3. It is clear that the distribution is bottom-heavy and that most of the SNe are associated with the host galaxies.

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It is important to point out that the SN redshift determination method introduces uncertainty regarding the redshift uncertainties. Most of the time the redshift is measured from the host galaxy of the SN and is rather accurate. But sometimes the SN redshift is measured directly from the SN itself and is influenced by Doppler broadening and expansion velocities, being slightly larger (Blondin & Tonry 2007).

The degree of association between SNe and galaxies can also be checked by visually inspecting the SDSS DR7 images of the galaxies, with the SN positions overplotted. We note the subjectivity involved, e.g., in cases where a SN occurs in an interacting pair of galaxies. For the largest sample of Type-2 AGNs (77,708 galaxies), and the 59 SNe matched to them, about 10% of the SNe visually appear to be located in a companion galaxy of a sample galaxy. For the largest sample of star-forming galaxies (137,489), and the 152 SNe matched to them, a comparable fraction (about 9%) appears to be located in a companion galaxy. These results from visual inspection of SN positions complement the conclusions drawn from Figure 3.

We begin our analysis by counting SNe within two different projected distances from the center of the host galaxies: 10 kpc (within the typical radius of a spiral galaxy), and 100 kpc (within the galaxy or a possible close companion). We start with the largest samples; see Table 3.

The first thing to notice is the lack of SNe around Type-1 AGNs at projected separations d < 100 kpc. Only one SN is found. The numbers of SNe around Type-2 AGNs are clearly systematically higher for the largest samples in Table 3 (size of effect ∼5.1σ). We verify that this is no bias due to potential difficulties in detecting SNe in the immediate vicinity of bright or transient AGN by excluding all SNe within 3 arcsec from the host centers, see Section 4.3.2. This still yields one SN near 11,632 Type-1 AGNs (detection fraction ∼8.6 × 10⁻⁵) and 46 SNe near 77,708 Type-2 AGNs (detection fraction ∼5.9 × 10⁻⁴)—a significant difference (size of effect ∼4.1σ, see Section 5.1). Visual inspection shows that the majority of the SNe (>90%) come directly from the AGN hosts, with only a small fraction from the companion galaxies.

For the Type-1 and Type-2 samples matched in redshift or M_w4 the differences are significant. But for the L[O iii]5007 matched samples we only find one SN in each sample, yielding no difference at all. Does this mean the simplest unification is valid and there is no difference in galaxy properties and SN counts in samples selected and matched in L[O iii]5007? If so, also the host morphologies for the matched samples should be the same. But the fraction of Type-1 and Type-2 AGNs in spiral hosts is 20% versus 44%, in disagreement with the simplest unification. This suggests that the lack of significant difference in SN counts stems from poor statistics.

As an alternative test, we select on L[O iii]5007. This should be suitable as in the simplest unification no difference is expected in L[O iii]5007 on the higher-luminosity end (while matching removes potential differences at the high-luminosity end). We do this twice for the largest samples using both F[O iii]$5007\gt \,10$ × 10⁻¹⁷ erg s⁻¹ cm⁻² and F[O iii]5007 $\gt \,30$ × 10⁻¹⁷ erg s⁻¹ cm⁻² as flux limits. The difference in L[O iii]5007-selected Type-1 and Type-2 AGN samples is notable (F[O iii]5007 $\gt \,10$ case: size of effect ∼2.8σ, F[O iii]5007 $\gt \,30$ case: 1.8σ) following the same trend of Table 3. This reflects purely changes in the sample sizes, unless the difference in matched versus selected samples originates in a breakdown of unification at the higher luminosities.

As early-type objects are well-known (Li et al. 2011) for having few SNe setting off, it would be ideal to use only spiral hosts. We do not find a single SN around spiral Type-1 AGNs. But using a statistical hypothesis test (Krishnamoorthy & Thomson 2002) (see Section 5.3) only a borderline-significant difference in SN counts for Type-1 and Type-2 AGNs at $d\lt 100$ kpc (p-value ∼0.06) is found, showing the need for larger SN samples.

For the pairwise matched samples the counts are too small to show any significance individually. The L[O iii]5007 matched samples we commented upon earlier, while for the redshift matched samples a significant difference is found for $d\lt 10$ kpc. The lack of significant difference for $d\lt 100$ kpc therefore stems from the poor statistics. The samples matched in redshift and apparent galaxy size yield small, insignificant differences between Type-1 and Type-2 AGNs. As the size of the galaxy depends on the star formation history, no difference in counts could mean that we have either matched successfully in the star formation history or, more realistically, that the sample sizes are too small. However, the collected results in Table 4 are visually presented in Figure 4 where the left column reinforces our earlier conclusions: Type-2 AGN hosts have higher SN rates than Type-1 AGN hosts, much higher than from the expectations of the simplest unification theory (represented by the gray line). Also the insignificant differences fall into the same area of the plot.

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The larger count of SNe could mean either a difference in stellar age or stellar mass (or both). But samples unmatched in stellar mass lead to differences in clustering on the Mpc scale (Mendez et al. 2016) while we find no differences in SN counts on a large scale, see Sections 4.3.3 and 4.3.4. Moreover, the matching in M_w4—our proxy for stellar mass—still yields significant different SN counts. This suggests that the discrepancy in SN counts is due to differences in stellar age between Type-1 and Type-2 AGNs.

The conclusion is supported by the much higher $w2-w3$ color indices in Type-2s indicating stronger star formation (Coziol et al. 2015) in samples consisting of face-on, spiral-host Type-1 and Type-2 AGNs that are dominated by dust emission from stars with ${m}_{W1}-{m}_{W2}\lt 0.5$ (Wright et al. 2010) and pair-wise matched in L[O iii]5007, see Section 4.1.

Less striking differences are found between Type-2 AGNs and star-forming galaxies. Star-forming spirals show higher c-c SN counts at d < 10 kpc (∼3.3σ) and even in the redshift matched samples a difference is present. The right column in Figure 4 shows a relationship between the two classes, although offset in y-axis following the shape of the gray line.

One may argue that the L[O iii]5007 may be anisotropic or even have different physical origins in Type-1 and Type-2 AGNs. If so, one expects to find differences in the Gaussian line width of the [O iii]5007 emission in Type-1s and Type-2s and/or differences in SN counts.

3.2.1. SN Samples: Anisotropic Sample Selection?

3.2.2. NLR Kinematics: Anisotropic Sample Selection?

If the obscuration is the dominant factor that separates Type-2 from Type-1 AGNs, one may expect that the refined samples of Type-1s and Type-2s matched in the heated-dust emission from their host galaxy are more or less as luminous in L[O iii]5007. For the 137 paired Type-1s and Type-2s in the refined samples matched in redshift and M_w4, we compare the mean L[O iii]5007. The mean log₁₀L[O iii]5007 (erg s⁻¹) is 40.934 ± 0.0455 for Type-1s and 39.9738 ± 0.0889 for Type-2s. This demonstrates that Type-1s are much more luminous (∼9.6σ) than Type-2s in host galaxies with similar dust distributions. The effect is equally convincing if exploring and matching objects with WISE bands dominated by star formation ${m}_{W1}-{m}_{W2}\lt 0.5$.

3.3.1. Refined Samples Dominated by Dust Emission Near Stars

To control several biases at the same time, we use the refined, pairwise matched subsamples. The pairwise matching ensures our samples have the same redshift distribution and also the same distribution in one of the remaining four parameters separately (only z, L[O iii]5007, F(Hα/Hβ), m_w4). The refined sample sizes are shown in Table 2.

One would wish to explore the SN rate in these samples at $d\lt 10$ kpc. However, the currently available SN samples are far too small to allow this kind of investigation. We have to rely on an alternative star formation indicator. We can use WISE $w2-w3$ colors to measure the activity of star formation, assuming that our objects WISE colors are dominated by the same source (either AGNs or the stars): the higher $w2-w3$, the more star formation (Coziol et al. 2015).

For the refined samples, we see a strong difference in the $w2-w3$ color. The $w2-w3$ is significantly larger in Type-2 compared to Type-1 AGNs for all four matchings. An example is the w4-matched Type-1 and Type-2 refined samples: $w2-w3=2.962\pm 0.03$ for Type-1s and $w2-w3\,=3.674\pm 0.029$ for Type-2s. The other matchings give very similar results. This strongly supports the observed larger number of SN detections around Type-2 AGNs.

Also, if we redo the entire pairwise matching and analysis only using objects having the infrared WISE emission dominated by dust emission near stars using either $w1-w2\lt 0.8$ (Assef et al. 2013) and or $w1-w2\lt 0.5$ (Wright et al. 2010), the conclusion stays equally true.

The Galaxy Zoo Data Release 1 morphologies (for those that have, ∼90% of all galaxies) fall into three categories: spiral, elliptical and uncertain. The higher the  redshift, the more difficulties a Galaxy Zoo volunteer has in recognizing a certain morphology. Therefore, it might be difficult for a Galaxy Zoo observer to recognize a spiral at higher z, especially if there is a strong light from the nucleus. Some spirals can fall into the category of "uncertain" due to the strong light in Type-1 AGNs.

However, this bias cannot cause misclassifications in the other direction. While a spiral can be classified as "uncertain" it is very unlikely an "uncertain" galaxy will be classified as a spiral. Therefore, what in Galaxy Zoo is classified as a spiral is very likely to be one, as voted by hundreds of Galaxy Zoo volunteers.

Any redshift-dependent biases such as the morphology classification bias, are removed by the use of redshift matched (all-morphologies) samples. They show significant differences between Type-1 and Type-2 AGNs. We attempted doing the same analysis for redshift-matched Type-1 and Type-2 spiral hosts, but given the extremely small statistics it was not possible. We can only hope that future data releases will permit us to do this final, but very important test. This motivated us to use the alternative star formation indicator as in Section 4.1.

As an additional test we also visually classified the de Vaucoleurs–Buta stage in the de Vaucouleurs Revised Hubble–Sandage Classification System of the Type-1 and Type-2 AGNs in the refined samples to see if any insight could be gained about the relative age of the stellar populations; we did not see any difference.

When matching our AGNs with the iPTF SNe, some biases may be introduced which could affect the results. Two sources of bias are considered:

• 1.  
  On 2014 June 17, automatic filtering was introduced in the iPTF SN scanning software in order to save known AGNs as so called Nuclear objects before a human scanner could begin vetting the candidates. The AGN identification done from 2014 June onwards was based on SDSS DR10 data. In 2015 February, QSOs from SDSS DR 12 were added (Y. Cao 2017, personal communication).
• 2.   
  The match radius used to tie changes in brightness to a certain transient is $1^{\prime\prime}$. This means that if the angular separation between two transient sources exceeds $1^{\prime\prime}$, they are considered to be different sources. This could lead to a potential skewness in SN detections near AGNs, arising from confusing AGNs with SNe if they reside $1^{\prime\prime}$ or less from each other.

We avoid the first bias, throughout this work, by only considering SNe discovered before 2014 June 17. However, we need to perform a test for the second potential bias. The impact of the second bias can be weakened by only considering SNe appearing at angular distances from $\mathrm{AGN}\gt \,3^{\prime\prime}$. This is generously larger than what is called for by the $1^{\prime\prime}$ matching condition.

We will herein present three different tests that probe the two biases. The results support a coherent targeting of SNe around Type-1s and Type-2s. The observed differences in SNe counts presented is thus a physical effect.

4.3.1. Suitability of the iPTF

4.3.2. Test with an $r\gt 3^{\prime\prime}$ Cut

4.3.3. Test Using Foreground and Background Objects, $d\lt 100$ kpc

4.3.4. Test Using Large-scale Environment, $100\lt d\lt 1100$ kpc

Let the random variables ${X}_{1}\sim {Po}({N}_{1}{\lambda }_{1})$ and ${X}_{2}\sim {Po}({N}_{2}{\lambda }_{2})$ denote the number of SNe in galaxy sample no. 1 (Type-1 AGNs) and 2 (Type-2 AGNs), respectively, and let ${N}_{i}\sim {Po}({\mu }_{i})$, be the total number of galaxies observed in sample no. $i,\,i=1,2$. Here, the parameter ${\lambda }_{i}$ denotes the average rate of SNe per galaxy in the Type-i AGNs, while ${\mu }_{i}$ denotes the number count of Type-i AGNs in the observed volume. The random variable

$$\begin{eqnarray}&&Z=\displaystyle \frac{{X}_{1}}{{N}_{1}}-\displaystyle \frac{{X}_{2}}{{N}_{2}},\end{eqnarray} \tag{ 2 }$$

then describes the difference between the rates of SNe in the two samples. In order to put the observed difference in context, we define the ratio

$$\begin{eqnarray}&&\beta =\displaystyle \frac{| {\mathbb{E}}(Z)| }{\sqrt{\mathrm{Var}(Z)}},\end{eqnarray} \tag{ 3 }$$

which is a measure of the expected size of the difference in units of the standard deviation of the distribution of Z. Note that, in general, Z is not normally distributed, whereby a detection at a level of, say, $3\sigma$ cannot directly be compared with a $3\sigma$ detection where the distributions of interest are normal. In our case, however, the difference should be relatively small.

Since the difference in Equation (2) involves ratios of two random variables, the computation of the expectation and variance of Z is not straightforward and we will use Gauss's approximation formulae (first order) to derive an explicit expression of Equation (3). We have that

$$\begin{eqnarray}\mathrm{Var}(Z) & = & \mathrm{Var}\left(\displaystyle \frac{{X}_{1}}{{N}_{1}}-\displaystyle \frac{{X}_{2}}{{N}_{2}}\right)\\ & = & \mathrm{Var}\left(\displaystyle \frac{{X}_{1}}{{N}_{1}}\right)+\mathrm{Var}\left(\displaystyle \frac{{X}_{2}}{{N}_{2}}\right),\end{eqnarray} \tag{ 4 }$$

where it is assumed that ${X}_{1}/{N}_{1}$ and ${X}_{2}/{N}_{2}$ are independent. Now,

$$\begin{eqnarray}\mathrm{Var}\left(\displaystyle \frac{{X}_{i}}{{N}_{i}}\right) & \approx & \displaystyle \frac{1}{{\mu }_{i}^{2}}\cdot \mathrm{Var}({X}_{i})\\ & & +\displaystyle \frac{{\mathbb{E}}{({X}_{i})}^{2}}{{\mu }_{i}^{4}}\cdot \mathrm{Var}({N}_{i})\\ & & -2\cdot \displaystyle \frac{1}{{\mu }_{i}}\displaystyle \frac{{\mathbb{E}}({X}_{i})}{{\mu }_{i}^{2}}\cdot C({X}_{i},{N}_{i}).\end{eqnarray} \tag{ 5 }$$

By the fact that ${X}_{i}| {N}_{i}=n\sim {Bin}(n,{\lambda }_{i})$, the law of total expectation gives that ${\mathbb{E}}({X}_{i})={\mathbb{E}}({\mathbb{E}}({X}_{i}| {N}_{i}))={\mathbb{E}}({N}_{i}{\lambda }_{i})={\mu }_{i}{\lambda }_{i}$. Similarly, for the variance we have that

$$\begin{eqnarray}\mathrm{Var}({X}_{i}) & = & {\mathbb{E}}(\mathrm{Var}({X}_{i}| {N}_{i}))+\mathrm{Var}({\mathbb{E}}({X}_{i}| {N}_{i}))\\ & = & {\mathbb{E}}({N}_{i}{\lambda }_{i}(1-{\lambda }_{i}))+\mathrm{Var}({N}_{i}{\lambda }_{i})\\ & = & {\mu }_{i}{\lambda }_{i}(1-{\lambda }_{i})+{\mu }_{i}{\lambda }_{i}^{2}={\mu }_{i}{\lambda }_{i},\end{eqnarray} \tag{ 6 }$$

while the covariance also amounts to $C({X}_{i},{N}_{i})={\mu }_{i}{\lambda }_{i}$. Hence, we have that

$$\begin{eqnarray}&&\mathrm{Var}\left(\displaystyle \frac{{X}_{i}}{{N}_{i}}\right)\approx \displaystyle \frac{{\lambda }_{i}}{{\mu }_{i}}+\displaystyle \frac{{\lambda }_{i}^{2}}{{\mu }_{i}}-2\cdot \displaystyle \frac{{\lambda }_{i}^{2}}{{\mu }_{i}}=\displaystyle \frac{{\lambda }_{i}(1-{\lambda }_{i})}{{\mu }_{i}}.\end{eqnarray} \tag{ 7 }$$

For the expectation of Z, we have that ${\mathbb{E}}(Z)={\mathbb{E}}({X}_{1}/{N}_{1}-{X}_{2}/{N}_{2})\approx {\lambda }_{1}-{\lambda }_{2}$. It is noted that also for the second-order approximation, ${\mathbb{E}}(Z)\approx {\lambda }_{1}-{\lambda }_{2}$ Thus, we have

$$\begin{eqnarray}\beta & = & \displaystyle \frac{| {\mathbb{E}}(Z)| }{\sqrt{\mathrm{Var}(Z)}}\approx \displaystyle \frac{| {\lambda }_{1}-{\lambda }_{2}| }{\sqrt{\tfrac{{\lambda }_{1}(1-{\lambda }_{1})}{{\mu }_{1}}+\tfrac{{\lambda }_{2}(1-{\lambda }_{2})}{{\mu }_{2}}}}\\ & = & \displaystyle \frac{\sqrt{{\mu }_{1}{\mu }_{2}}| {\lambda }_{1}-{\lambda }_{2}| }{\sqrt{{\lambda }_{1}(1-{\lambda }_{1}){\mu }_{2}+{\lambda }_{2}(1-{\lambda }_{2}){\mu }_{1}}}.\end{eqnarray} \tag{ 8 }$$

Now, a plug-in estimate of the ratio in Equation (8) is given by the expression

$$\begin{eqnarray}&&\widehat{\beta }\approx \displaystyle \frac{\sqrt{{\widehat{\mu }}_{1}{\widehat{\mu }}_{2}}| {\widehat{\lambda }}_{1}-{\widehat{\lambda }}_{2}| }{\sqrt{{\widehat{\lambda }}_{1}(1-{\widehat{\lambda }}_{1}){\widehat{\mu }}_{2}+{\widehat{\lambda }}_{2}(1-{\widehat{\lambda }}_{2}){\widehat{\mu }}_{1}}},\end{eqnarray} \tag{ 9 }$$

where ${\widehat{\lambda }}_{i},{\widehat{\mu }}_{i},\,i=1,2$ are estimated from the observed numbers. For example, for the full samples and $d\lt 100\,\mathrm{kpc}$, we have that ${\widehat{\mu }}_{1}=11632,\,{\widehat{\mu }}_{2}=77708,\,{\widehat{\lambda }}_{1}=1/11632$, and ${\widehat{\lambda }}_{2}=59/77708$ (see Table 3). Thus, we obtain

$$\begin{eqnarray}&&\widehat{\beta }\approx 5.1,\end{eqnarray} \tag{ 10 }$$

i.e., the observed difference is more than $5\sigma$ away from ${\lambda }_{1}-{\lambda }_{2}=0$, as defined by Equation (3).

Table 3.  SN Counts around Different Host Galaxies

Total SN Counts
Max Distance	Type-1 AGN (11632)	f	Type-2 AGN (77708)	f	Star-forming (137489)	f
d < 10 kpc	0	0	39 $(21| 16| 2)$	5 × 10−4	117 $(58| 56| 3)$	$8.5\times {10}^{-4}$
d < 100 kpc	1 $(1| 0| 0)$	$8.6\times {10}^{-5}$	59 $(36| 20| 3)$	7.6 × 10−4	152 $(73| 76| 3)$	$1.1\times {10}^{-3}$
SN Counts Around Spiral Hosts
Max Distance	Type-1 AGN (1864)	f	Type-2 AGN (36720)	f	Star-forming (49072)	f
d < 10 kpc	0	0	25 $(12| 11| 2)$	$6.8\times {10}^{-4}$	59 $(24| 34| 1)$	$1.2\times {10}^{-3}$
d < 100 kpc	0	0	39 $(22| 14| 3)$	$1.1\times {10}^{-3}$	77 $(33| 43| 1)$	$1.6\times {10}^{-3}$

Note. We classify the SNe into three different types: SNe type Ia (marked as "SNIa"), core-collapse ("c-c") and "Unknown," and mark the numbers of SNe of each type in parenthesis in the Form "N ($\mathrm{SNIa}| {\rm{c}}-{\rm{c}}| {\rm{u}}$)" at Each Line. The detection fraction $f={N}_{\mathrm{SN}}/{N}_{\mathrm{gal}}$ indicated for each sample is the ratio between number of observed SNe and the given galaxy sample size. A strong difference in SN counts between Type-1 and Type-2 AGNs at d < 100 kpc is apparent. A two-sample hypothesis test (Krishnamoorthy & Thomson 2002) at significance level $\alpha =0.05$ gives the p-value ∼ $2.1\times {10}^{-5}$. The size of effect is estimated to be ∼5.1σ. The signal of detection ∼2.6σ if using a statistics based on null distribution (Section 5.2). The difference is significant for both maximum projected distances (10 and 100 kpc). In the numbers of SNe around spiral-host Type-2 AGNs and star-forming objects a significant size of effect (∼3.3σ), or a 2.8σ signal of detection, can be found at ${\rm{d}}\lt 10$ kpc in the number of core-collapse SNe. Using a hypothesis test (Krishnamoorthy & Thomson 2002) where the null hypothesis H₀ is that two samples have the same SN counts, We estimate significance levels of statistical differences for the spiral host AGNs. We reject H₀ at the nominal value α = 0.05 and conclude that AGN hosts have fewer SNe than star-forming galaxies at ${\rm{d}}\lt 10$ and 100 kpc. Comparing spiral-host Type-1 and Type-2 AGNs we get p = 0.06, borderline significant but not enough to reject H₀, showing the need for larger samples.

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Since the SN rates generally are very small, i.e., $\lambda \sim {10}^{-4}-{10}^{-3}$, the factor $1-{\lambda }_{i}\simeq 1$ in the expression for Var(Z) and we may take ${N}_{1}={n}_{1}$ and ${N}_{2}={n}_{2}$ as fixed. Then, Z reduces to

$$\begin{eqnarray}&&Z=\displaystyle \frac{{X}_{1}}{{n}_{1}}-\displaystyle \frac{{X}_{2}}{{n}_{2}}\end{eqnarray} \tag{ 11 }$$

and

$$\begin{eqnarray}\widehat{\beta } & = & \displaystyle \frac{| \widehat{{\mathbb{E}}(Z)}| }{\sqrt{\widehat{\mathrm{Var}(Z)}}}=\displaystyle \frac{\sqrt{{n}_{1}{n}_{2}}| {\widehat{\lambda }}_{1}-{\widehat{\lambda }}_{2}| }{\sqrt{{\widehat{\lambda }}_{1}{n}_{2}+{\widehat{\lambda }}_{2}{n}_{1}}}\\ & = & \displaystyle \frac{| {x}_{1}/{n}_{1}-{x}_{2}/{n}_{2}| }{\sqrt{{x}_{1}/{n}_{1}^{2}+{x}_{2}/{n}_{2}^{2}}},\end{eqnarray} \tag{ 12 }$$

where x_i is the total number of SNe observed in the Type-i AGNs. For ${x}_{1}=1,\,{x}_{2}=59,\,{n}_{1}=11632$, and ${n}_{2}=77708$, we have $\widehat{\beta }\approx 5.1$, which is the result of Equation (10). For the hypothesis testing discussed in Section 5.3, we make the assumption that n₁ and n₂ are fixed.

Let us hypothesise that the SN rates of the two AGN samples are equal. At what level will the observed difference then just be due to chance? By taking the null hypothesis to be

$$\begin{eqnarray*}{H}_{0}\,:{\lambda }_{1} & = & {\lambda }_{2}\,(={\lambda }_{0})\,\mathrm{versus}\\ {H}_{1}\,:{\lambda }_{1} & \ne & {\lambda }_{2},\end{eqnarray*}$$

where ${\lambda }_{0}$ is the common SN rate, we have instead that ${X}_{1}\sim {Po}({N}_{1}{\lambda }_{0})$ and ${X}_{2}\sim {Po}({N}_{2}{\lambda }_{0})$. Consequently, the variance of the random variable $Z={X}_{1}/{N}_{1}-{X}_{2}/{N}_{2}$ becomes

$$\begin{eqnarray}&&\mathrm{Var}(Z)\approx {\lambda }_{0}(1-{\lambda }_{0})\left(\displaystyle \frac{1}{{\mu }_{1}}+\displaystyle \frac{1}{{\mu }_{2}}\right).\end{eqnarray} \tag{ 13 }$$

The expression of the ratio β, defined as $\beta =| {\lambda }_{\mathrm{diff}}| /\sqrt{\mathrm{Var}(Z)}$ where ${\lambda }_{\mathrm{diff}}$ denotes the observed difference ${\widehat{\lambda }}_{1}-{\widehat{\lambda }}_{2}$, then equals

$$\begin{eqnarray}&&\beta =\displaystyle \frac{| {\lambda }_{\mathrm{diff}}| }{\sqrt{\mathrm{Var}(Z)}}\approx \displaystyle \frac{| {\lambda }_{\mathrm{diff}}| }{\sqrt{{\lambda }_{0}(1-{\lambda }_{0})\cdot \left(\tfrac{1}{{\mu }_{1}}+\tfrac{1}{{\mu }_{2}}\right)}}.\end{eqnarray} \tag{ 14 }$$

Following the steps is Section 5.1, we have that

$$\begin{eqnarray}&&\widehat{\beta }=\displaystyle \frac{| {\lambda }_{\mathrm{diff}}| }{\sqrt{\widehat{\mathrm{Var}(Z)}}}\approx \displaystyle \frac{| {\widehat{\lambda }}_{1}-{\widehat{\lambda }}_{2}| }{\sqrt{{\widehat{\lambda }}_{0}(1-{\widehat{\lambda }}_{0})\cdot \left(\tfrac{1}{{\widehat{\mu }}_{1}}+\tfrac{1}{{\widehat{\mu }}_{2}}\right)}},\end{eqnarray} \tag{ 15 }$$

where an estimate of ${\lambda }_{0}$ is given by

$$\begin{eqnarray}&&{\widehat{\lambda }}_{0}=\displaystyle \frac{{x}_{1}+{x}_{2}}{{n}_{1}+{n}_{2}}.\end{eqnarray} \tag{ 16 }$$

Hence, for the full, unmatched samples ($d\lt 100$ kpc) we have that ${\widehat{\lambda }}_{0}=(1+59)/(11632+77708)\simeq 6.72\,\times \,{10}^{-4}$ and $\widehat{\beta }\approx 2.6$. This is a more conservative measure of detection, i.e., a difference in the SN rates is detected close to the $2.6\sigma$ level.

Finally, we have that

$$\begin{eqnarray}\widehat{\beta } & = & \displaystyle \frac{| {\widehat{\lambda }}_{1}-{\widehat{\lambda }}_{2}| }{\sqrt{{\widehat{\lambda }}_{0}/{n}_{1}+{\widehat{\lambda }}_{0}/{n}_{2}}}\\ & = & \sqrt{\displaystyle \frac{{n}_{1}{n}_{2}}{{x}_{1}+{x}_{2}}}\cdot | {x}_{1}{/n}_{1}-{x}_{2}{/n}_{2}| ,\end{eqnarray} \tag{ 17 }$$

under the assumption that $Z={X}_{1}/{n}_{1}-{X}_{2}/{n}_{2}$. It is noted that for this approximation, the estimates of β as given by Equations (12) and (17) are equal for ${n}_{1}={n}_{2}$.

We report this estimate of the signal of detection alongside with the size of effect and p-value in the captions to Table 4.

Table 4.  SN Types and Counts

SNe Within 10 kpc
All Hosts	N of Galaxies/Sample	Type-1 AGN*	Type-2 AGN	H0 Rejected ($p\leqslant 0.05$)	N of Galaxies/Sample	Type-2 AGN*	Star-forming	H0 Rejected ($p\leqslant 0.05$)
z	10146	0	4 $(2| 2| 0)$	yes ($p=4.1\times {10}^{-2}$)	⋯	⋯	⋯	⋯
L[O iii]5007	6252	0	0	no (p = 1)	⋯	⋯	⋯	⋯
Mw4	8506	0	6 $(6| 0| 0)$	yes ($p=9.6\times {10}^{-3}$)	⋯	⋯	⋯	⋯
rexp	7218	0	2 $(2| 0| 0)$	no ($p=2.0\times {10}^{-1}$)	⋯	⋯	⋯	⋯
Spiral Hosts	N of Galaxies/Sample	Type-1 AGN*	Type-2 AGN	H0 Rejected ($p\leqslant 0.05$)	N of Galaxies/Sample	Type-2 AGN*	Star-forming	H0 Rejected ($p\leqslant 0.05$)
z	⋯	⋯	⋯	⋯	13067	13 $(6| 7| 0)$	25 $(13| 12| 0)$	no ($p=5.2\times {10}^{-2}$)
L[O iii]5007	⋯	⋯	⋯	⋯	7705	5 $(3| 2| 0)$	9 $(5| 3| 1)$	no ($p=3.0\times {10}^{-1}$)
Mw4	⋯	⋯	⋯	⋯	13108	2 $(2| 0| 0)$	9 $(4| 5| 0)$	yes ($p=4.0\times {10}^{-2}$)
rexp	⋯	⋯	⋯	⋯	10827	6 $(3| 3| 0)$	14 $(5| 9| 0)$	no ($p=7.9\times {10}^{-2}$)
SNe Within 100 kpc
All Hosts	N of Galaxies/Sample	Type-1 AGN*	Type-2 AGN	H0 Rejected ($p\leqslant 0.05$)	N of Galaxies/Sample	Type-2 AGN*	Star-forming	H0 Rejected ($p\leqslant 0.05$)
z	10146	1 $(1| 0| 0)$	5 $(3| 2| 0)$	no ($p=1.3\times {10}^{-1}$)	⋯	⋯	⋯	⋯
L[O iii]5007	6252	1 $(1| 0| 0)$	1 $(1| 0| 0)$	no (p = 1)	⋯	⋯	⋯	⋯
Mw4	8506	1 $(1| 0| 0)$	9 $(8| 1| 0)$	yes ($p=8.9\times {10}^{-3}$)	⋯	⋯	⋯	⋯
rexp	7218	1 $(1| 0| 0)$	3 $(3| 0| 0)$	no ($p=4.4\times {10}^{-1}$)	⋯	⋯	⋯	⋯
Spiral Hosts	N of Galaxies/Sample	Type-1 AGN*	Type-2 AGN	H0 Rejected ($p\leqslant 0.05$)	N of Galaxies/Sample	Type-2 AGN*	Star-forming	H0 Rejected ($p\leqslant 0.05$)
z	⋯	⋯	⋯	⋯	13067	18 $(10| 8| 0)$	28 $(13| 15| 0)$	no ($p=1.4\times {10}^{-1}$)
L[O iii]5007	⋯	⋯	⋯	⋯	7705	9 $(5| 3| 1)$	10 $(5| 4| 1)$	no ($p=8.4\times {10}^{-1}$)
Mw4	⋯	⋯	⋯	⋯	13108	8 $(7| 1| 0)$	12 $(7| 5| 0)$	no ($p=3.9\times {10}^{-1}$)
rexp	⋯	⋯	⋯	⋯	10827	11 $(6| 4| 1)$	15 $(6| 9| 0)$	no ($p=4.4\times {10}^{-1}$)

Note. We compare SNe around host galaxies within two different distances, 10 kpc and 100 kpc. We do this (i) for mixed Hubble types, (ii) for only spiral-hosts, and in four types of matchings. For Type-1 and Type-2 AGNs in spiral hosts the number of objects are too few. We classify the SNe into three different types: SNe Type Ia (marked as "SNIa"), core-collapse ("c-c") and "unknown," and mark the numbers of SNe of each type in parentheses in the form "N ($\mathrm{SNIa}| {\rm{c}}-{\rm{c}}| {\rm{u}}$)" at each line. We assume the null hypothesis H₀, that the total SN rates are the same for each pair of matched samples against the alternative hypothesis ${H}_{1}:{\lambda }_{1}\ne {\lambda }_{2}$, and perform a two-sample test (Krishnamoorthy & Thomson 2002) at the significance level $\alpha =0.05$; see Section 5.3.

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The number of SNe in the host galaxies of AGNs should, within the realm of the unification model, not depend on the type of AGN. We have performed a statistical test of this hypothesis. Recall that the SN rate (read proportion/success probability) λ in a host galaxy population of size n should be binomially distributed. However, since $\lambda \lt 0.003\ll 0.1$ in all observed cases, $\mathrm{Bin}](n,\lambda )\approx {Po}(n\lambda )$ to a very high degree. Therefore, we assume that the number of SNe believed to be associated with the Type-1 and Type-2 AGN samples are, respectively, observations of the random variables ${X}_{1}\sim {Po}({n}_{1}{\lambda }_{1})$ and ${X}_{2}\sim {Po}({n}_{2}{\lambda }_{2})$. Here, ${n}_{i},\,i=1,2$ is the size of the Type-i AGN sample and ${\lambda }_{i}$ is the corresponding rate of SNe in the Type-i AGN population, uniformly corrected for the biases discussed above. Furthermore, it is assumed that X₁ and X₂ are independent. Let the null hypothesis be

$$\begin{eqnarray*}{H}_{0}\,:{\lambda }_{1} & = & {\lambda }_{2},\,\mathrm{against}\,\mathrm{the}\,\mathrm{alternative}\\ {H}_{1}\,:{\lambda }_{1} & \ne & {\lambda }_{2}.\end{eqnarray*}$$

The exact conditional test (C-test) for comparing two Poisson means by Przyborowski & Wilenski (1940) is known to be overly conservative, i.e., the chance of failing to reject a false null hypothesis is higher than the nominal level. We have therefore chosen to perform a test based on estimated p-values instead (Krishnamoorthy & Thomson 2002). The pivot statistics for ${\lambda }_{1}-{\lambda }_{2}=0$ is given by

$$\begin{eqnarray}&&{T}_{{X}_{1},{X}_{2}}=\displaystyle \frac{{X}_{1}/{n}_{1}-{X}_{2}/{n}_{2}}{\sqrt{{\hat{V}}_{{X}_{1},{X}_{2}}}},\end{eqnarray} \tag{ 18 }$$

where

$$\begin{eqnarray}&&{\hat{V}}_{{X}_{1},{X}_{2}}=\displaystyle \frac{{X}_{1}}{{n}_{1}^{2}}+\displaystyle \frac{{X}_{2}}{{n}_{2}^{2}}\end{eqnarray} \tag{ 19 }$$

is the unbiased variance estimator of the standardized difference ${X}_{1}/{n}_{1}-{X}_{2}/{n}_{2}$. The p-value for the two-sided test is then given by

$$\begin{eqnarray}&&p=P(| {T}_{{X}_{1},{X}_{2}}| \geqslant | {T}_{{x}_{1},{x}_{2}}| \,| {H}_{0}),\end{eqnarray} \tag{ 20 }$$

where ${T}_{{x}_{1},{x}_{2}}=({x}_{1}{/n}_{1}-{x}_{2}{/n}_{2})/\sqrt{{\hat{V}}_{{x}_{1},{x}_{2}}}$ is the observed value of ${T}_{{X}_{1},{X}_{2}}$ and x₁ and x₂ are the observed numbers of SNe in the Type-1 and Type-2 sample, respectively. Hence, the p-value is estimated by the expression (see Krishnamoorthy & Thomson 2002)

$$\begin{eqnarray}&&p=\displaystyle \sum _{{k}_{1}=0}^{\infty }\,\displaystyle \sum _{{k}_{2}=0}^{\infty }\displaystyle \frac{{e}^{-{n}_{1}{\hat{\lambda }}_{2}}{({n}_{1}{\hat{\lambda }}_{2})}^{{k}_{1}}}{{k}_{1}!}\displaystyle \frac{{e}^{-{n}_{2}{\hat{\lambda }}_{2}}{({n}_{2}{\hat{\lambda }}_{2})}^{{k}_{2}}}{{k}_{2}!}{I}_{| {T}_{{k}_{1},{k}_{2}}| \geqslant | {T}_{{x}_{1},{x}_{2}}| },\end{eqnarray} \tag{ 21 }$$

where ${I}_{a\geqslant b}$ denotes the indicator function such that

$$\begin{eqnarray}{I}_{a\geqslant b}=\left\{\begin{array}{ll}1, & a\geqslant b,\\ 0, & a\lt b,\end{array}\right.\end{eqnarray} \tag{ 22 }$$

and ${\hat{\lambda }}_{2}=({x}_{1}+{x}_{2})/({n}_{1}+{n}_{2})$ is the estimate of ${\lambda }_{2}$ (see Equation (17)). Note that under the null hypothesis, ${\hat{\lambda }}_{1}={\hat{\lambda }}_{2}$. Also, we have that ${T}_{\mathrm{0,0}}=0$. The null hypothesis is then rejected at the significance level α if $p\leqslant \alpha$. We use the nominal value $\alpha =0.05$.

The results from the hypothesis test are reported in the Table 4.