# Supernova Rates and Luminosity Functions from ASAS-SN I:
2014–2017 Type Ia SNe and Their Subtypes

D. D. Desai1, C. S. Kochanek2,3, B. J. Shappee1, T. Jayasinghe4,†, K. Z.
Stanek2,3, T. W.-S. Holoien5, T. A. Thompson2,6,3, C. Ashall7, J. F.
Beacom6,2,3, A. Do1, Subo Dong8, J. L. Prieto9,10
1 Institute for Astronomy, University of Hawai‘i at Mānoa, 2680 Woodlawn
Drive, Honolulu, HI 96822, USA
2 Department of Astronomy, The Ohio State University, 140 West 18th Avenue,
Columbus, OH 43210, USA
3 Center for Cosmology and Astro-Particle Physics, The Ohio State University,
191 West Woodruff Avenue, Columbus, OH 43210, USA
4Department of Astronomy, University of California, Berkeley, CA 94720
5 Carnegie Observatories, 813 Santa Barbara Street, Pasadena, CA 91101, USA
6 Department of Physics, The Ohio State University, 191 West Woodruff Avenue,
Columbus, OH 43210, USA
7 Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA
8 Kavli Institute for Astronomy and Astrophysics, Peking University, Yi He
Yuan Road 5, Hai Dian District, Beijing 100871, China
9 Núcleo de Astronomía de la Facultad de Ingeniería y Ciencias, Universidad
Diego Portales, Av. Ejército 441, Santiago, Chile
10 Millennium Institute of Astrophysics, Santiago, Chile
† NASA Hubble Fellow E-mail<EMAIL_ADDRESS>0000-0002-2164-859X
0000-0003-4631-1149 0000-0002-6244-477X 0000-0001-9206-3460
0000-0003-2377-9574 0000-0002-5221-7557 0000-0002-0005-2631
0000-0003-3429-7845 0000-0002-1027-0990

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present the volumetric rates and luminosity functions (LFs) of Type Ia
supernovae (SNe Ia) from the $V$-band All-Sky Automated Survey for Supernovae
(ASAS-SN) catalogues spanning discovery dates from UTC 2014-01-26 to UTC
2017-12-29. Our standard sample consists of 404 SNe Ia with
$m_{\mathrm{\textit{V},peak}}<17\,\mathrm{mag}$ and Galactic latitude
$|b|>15\degr$. Our results are both statistically more precise and
systematically more robust than previous studies due to the large sample size
and high spectroscopic completeness. We make completeness corrections based on
both the apparent and absolute magnitudes by simulating the detection of SNe
Ia in ASAS-SN light curves. We find a total volumetric rate for all subtypes
of $R_{\mathrm{tot}}=2.28^{+0.20}_{-0.20}\times
10^{4}\,\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,h^{3}_{70}$ for
$M_{\mathrm{\textit{V},peak}}<-16.5\,\mathrm{mag}$
($R_{\mathrm{tot}}=1.91^{+0.12}_{-0.12}\times
10^{4}\,\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,h^{3}_{70}$ for
$M_{\mathrm{\textit{V},peak}}<-17.5\,\mathrm{mag}$) at the median redshift of
our sample, $z_{\mathrm{med}}=0.024$. This is in agreement ($1\sigma$) with
the local volumetric rates found by previous studies. We also compile
luminosity functions (LFs) for the entire sample as well as for subtypes of
SNe Ia for the first time. The major subtypes with more than one SN include
Ia-91bg, Ia-91T, Ia-CSM, and Ia-03fg with total rates of
$R_{\mathrm{Ia-91bg}}=1.4^{+0.5}_{-0.5}\times
10^{3}\,\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,h^{3}_{70}$,
$R_{\mathrm{Ia-91T}}=8.5^{+1.6}_{-1.7}\times
10^{2}\,\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,h^{3}_{70}$, $R_{\mathrm{Ia-
CSM}}=10^{+7}_{-7}\,\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,h^{3}_{70}$, and
$R_{\mathrm{Ia-03fg}}=30^{+20}_{-20}\,\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,h^{3}_{70}$,
respectively. We estimate a mean host extinction of $E(V-r)\approx
0.2\,\mathrm{mag}$ based on the shift between our $V$ band and the Zwicky
Transient Facility $r$-band LFs.

###### keywords:

methods: data analysis – surveys – supernovae: general

††pubyear: 2023††pagerange: Supernova Rates and Luminosity Functions from
ASAS-SN I:  2014–2017 Type Ia SNe and Their Subtypes–A

##  1 Introduction

Type Ia supernovae (SNe Ia) are thermonuclear explosions of carbon/oxygen
(C/O) white dwarfs (WDs; e.g., Hoyle & Fowler, 1960). SNe Ia have been widely
used as standardizable candles (e.g., Pskovskii, 1977; Phillips, 1993) to
measure relative cosmological distances (e.g., Riess et al., 1998; Perlmutter
et al., 1999). Although the progenitor systems of SNe Ia are uncertain, all
likely scenarios include the interaction of a WD with another star (for
reviews see Maoz et al., 2014; Ruiter, 2020). The two basic models are the
single-degenerate (SD) scenario, where the WD accretes from a non-degenerate
companion star (e.g., Whelan & Iben, 1973; Nomoto, 1982), and the double-
degenerate (DD) scenario, where both objects are WDs (e.g., Tutukov &
Yungelson, 1979; Iben & Tutukov, 1984; Webbink, 1984; Shen et al., 2018). Most
observational studies disfavor the SD scenario for the majority of SNe (e.g.,
Nugent et al., 2011; Chomiuk et al., 2012; Shappee et al., 2013, 2018; Tucker
et al., 2022b). However, the uncertain nature of the SNe Ia progenitor systems
and explosion mechanisms remains a substantial problem for understanding
potential systematic errors in using SNe Ia for cosmology (Betoule et al.,
2014).

It is also becoming apparent that SNe Ia are not a uniform class of objects;
rather, there is a growing number of subtypes. These include the overluminous
91T-like SNe Ia (e.g., Filippenko et al., 1992b; Phillips et al., 1992); the
subluminous 91bg-like SNe Ia (e.g., Filippenko et al., 1992a; Leibundgut et
al., 1993; Turatto et al., 1996); 03fg-like SNe Ia, which are brighter in the
near-infrared and have long rise times (e.g., Howell et al., 2006; Hicken et
al., 2007; Hsiao et al., 2020; Ashall et al., 2021; Jiang et al., 2021; Lu et
al., 2021); 02cx-like SNe Ia, which show light curves that are both broad and
faint (e.g., Li et al., 2003; Foley et al., 2013); the subluminous 02es-like
SNe Ia, with the broad, slowly declining light curves seen in overluminous SNe
Ia, yet lacking a prominent secondary maximum in the $i$ band as seen in
subluminous SNe Ia (e.g., Foley et al., 2010; Ganeshalingam et al., 2012).
There are still additional transitional subtypes (for a review of subtypes,
see Taubenberger, 2017).

Different progenitor scenarios operate over a wide range of timescales,
ranging from $\sim$100 Myr to the Hubble time. The rate as a function of the
time span $\tau$ between a burst of star formation and the resulting SNe Ia is
known as the delay-time distribution (DTD). Measurements of the DTD of SNe Ia
(Maoz & Mannucci, 2012) can be used to constrain progenitor models and the
convolution of the DTD with the cosmic star formation history (SFH) gives the
redshift evolution of SN Ia rate. Most DD models predict that the DTD is a
$\sim\tau^{-1}$ power-law (e.g. Ruiter et al., 2009; Horiuchi & Beacom, 2010;
Mennekens et al., 2010) for longer delay times and observations of SN Ia rates
are broadly consistent with such a model (e.g. Scannapieco & Bildsten, 2005;
Maoz et al., 2012; Graur & Maoz, 2013; Graur et al., 2014). However, the SD
scenario predicts a broad range of DTDs which fail to account for long delay
times (e.g., Graur et al., 2014).

The shorter delay times are attributed to 91T-like SNe Ia, which are
predominantly found in late-type, star-forming galaxies (Howell, 2001; Li et
al., 2011b) and are therefore likely associated with young stellar
populations. The longer delay times are associated with 91bg-like SNe Ia,
which are mostly found in massive early-type galaxies with star-formation
rates below $\sim 10^{-9}\,\mathrm{M_{\odot}\,yr^{-1}}$ (Howell, 2001; Neill
et al., 2009; González-Gaitán et al., 2011). Similar to their 91bg-like
cousins, 02es-like events have a tendency to preferentially, but not
exclusively, explode in massive, early-type host galaxies (White et al.,
2015). To better understand the diversity in the subtypes of SNe Ia and their
contribution to the DTD, we must understand the rates of the SNe Ia subtypes.

Large surveys have made it possible to obtain SN rate measurements and
luminosity functions (LFs) to constrain the DTD and probe SNe Ia physics.
Surveys such as the Lick Observatory Supernova Search (LOSS; Li et al., 2000),
the Palomar Transient Factory (PTF; Rau et al., 2009; Law et al., 2009), the
Sloan Digital Sky Survey (SDSS)-II Supernova Survey (Frieman et al., 2008),
the Panoramic Survey Telescope & Rapid Response System (Pan-STARRS; Flewelling
et al., 2020; Chambers et al., 2016), the All-Sky Automated Survey for
Supernovae (ASAS-SN; Shappee et al., 2014; Kochanek et al., 2017), the
Asteroid Terrestrial-impact Last Alert System (ATLAS; Tonry et al., 2018;
Smith et al., 2020), and the Zwicky Transient Facility (ZTF; Masci et al.,
2019; Bellm et al., 2019) have discovered thousands of SNe. Hundreds of these
SNe are spectroscopically classified (e.g., Smartt et al., 2015; Tucker et
al., 2022a), especially the more local ones.

From these and other samples, volumetric rates of normal Type Ia SNe have been
measured over a wide range of redshifts. Local ($z<0.1$) rates were measured
by Cappellaro et al. (1999) using 70 SNe Ia from heterogeneous sources, by Li
et al. (2011b) using 274 SNe Ia from LOSS, by Frohmaier et al. (2019) using 90
SNe Ia from PTF, by Perley et al. (2020) using 875 SNe from the ZTF Bright
Transient Survey (BTS) sample, and by Sharon & Kushnir (2022) using a lower
redshift volume-limited subset of 298 SNe from the ZTF sample. The PTF and the
ZTF BTS samples have the highest spectroscopic completeness ($93\%$) among the
previous studies. They all agree at the $\sim 1\sigma$ level. At higher
redshifts, Dilday et al. (2010) measured SN Ia rates within $z<0.3$ using 270
spectroscopically classified SNe Ia from SDSS-II SN survey and Perrett et al.
(2012) measured the rates up to $z\sim 1.1$ using 286 spectroscopically
classified SNe Ia from the SuperNova Legacy Survey (SNLS). Other studies
include the Institute for Astronomy Deep Survey in the redshift range
$0.1<z<1.05$ (Rodney & Tonry, 2010) and the Subaru Deep Field out to $z\sim 2$
(Graur et al., 2011). These observed SN rates have been used to compare
different DTD models and show that a $\sim\tau^{-1}$ DTD successfully
describes the observed rates (e.g., Horiuchi & Beacom, 2010; Maoz et al.,
2012; Graur et al., 2014).

The creation of ASAS-SN was largely motivated by the incompleteness of the
local census of supernovae. Prior to that, discoveries were dominated by
amateurs, favored large galaxies, and showed year-to-year fluctuations that
were too large to be random. In Holoien et al. (2017a, b, c, 2019), ASAS-SN
cataloged the supernovae found in its first five years (2013–2017). This
sample includes 704 Type Ia SNe with 97% spectroscopically classified. The
limiting magnitude of ASAS-SN in the $V$ band ($\sim$17 mag) made
spectroscopic follow-up possible without the use of large telescopes, leading
to high spectroscopic completeness. In addition to confirming the bias of the
amateurs towards luminous hosts, Holoien et al. (2017a, b, c, 2019) found that
both amateur and other professional surveys were strongly biased against
finding SNe close to the centers of galaxies. The median radial offset of the
2013-2017 ASAS-SN sample was $2.4\,\mathrm{kpc}$ compared to
$5.7\,\mathrm{kpc}$ and $4.5\,\mathrm{kpc}$ for the amateur and other
professional surveys, respectively.

The first statistical SNe Ia study with ASAS-SN (Brown et al., 2019) extended
the finding by Li et al. (2011a) that the specific SNe Ia rate increases for
lower mass galaxies from $\sim 3$ decades in mass to $\sim 6$
($6.3\leq\log(M_{*}/M_{\odot})\leq 12.3$). Brown et al. (2019) found that the
rate per unit stellar mass $M_{*}$ scales roughly as $M_{*}^{-1/2}$ for the
Bell et al. (2003) stellar mass function ($M_{*}^{-1/3}$ for the Baldry et al.
2012 stellar mass function, see Gandhi et al. 2022). For the Li et al. (2011a)
mass range, this could be explained by lower mass galaxies having younger
stellar populations (Kistler et al., 2013; Graur & Maoz, 2013), but this
solution does not work for even lower masses. Using Feedback In Realistic
Environments (FIRE-2) cosmological zoom-in simulations, Gandhi et al. (2022)
found that including an SN Ia rate that increases with decreasing metallicity
($Z^{-0.5}$ to $Z^{-1}$) significantly improves agreement with observations.
Johnson et al. (2022) then used simple numerical calculations using mean star
formation histories from the UniverseMachine (Behroozi et al., 2019), a
$\tau^{-1}$ SN Ia DTD (e.g., Maoz & Mannucci, 2012), and the mass-metallicity
relation for galaxies (e.g., Tremonti et al., 2004; Andrews & Martini, 2013;
Zahid et al., 2011, 2014) and found that a $\sim Z^{-0.5}$ scaling is
required. Specifically, Johnson et al. (2022) found that the combination of
younger ages and lower metallicities for lower-mass galaxies can explain the
scaling over the full mass range of Brown et al. (2019). Both Gandhi et al.
(2022) and Johnson et al. (2022) proposed that a likely explanation for the SN
rate-metallicity trend is the rapid rise in the binary fraction towards lower
metallicity (Badenes et al., 2018; Moe et al., 2019; Wyse et al., 2020).

In this study, we use Type Ia SNe from the ASAS-SN catalogues to measure the
local volumetric rate and LF of Type Ia SNe, including, for the first time,
LFs for several major spectroscopic subtypes. In Section 2, we describe the
supernova sample and refit the data to update the peak apparent magnitudes in
ASAS-SN. In Section 3, we outline our approach to making completeness
corrections using simulations and the method for calculating the rates. In
Section 4, we present our local volumetric rates and LFs for Type Ia SNe and
several spectroscopic subtypes and compare them to earlier results. We also
provide the LFs corrected assuming various global values of the host-galaxy
extinction. Finally, in Section 5, we summarize the results and discuss future
projects. Throughout this paper we adopt a flat Lambda cold dark matter
cosmology with a Hubble constant
$H_{0}=70\,\mathrm{km\,s}^{-1}\,\mathrm{Mpc}^{-1}$ and a matter density
$\Omega_{m,0}=0.3$.

##  2 The Supernova Sample

Figure 1: ASAS-SN $V$-band light curves (blue points) of four SNe
(ASASSN-15hx, ASASSN-15lp, SN 2017fax, and ASASSN-16jw) spanning a range in
peak magnitudes along with the best fits using the Nugent et al. (2002)
$V$-band SN Ia templates (purple line). The phase is given in the observed
frame.

We use the Type Ia supernova sample of 704 SNe Ia from the $V$-band ASAS-SN
Bright Supernova Catalogues (Holoien et al., 2017a, b, c, 2019) spanning
discovery dates from UTC 2014-01-26 to UTC 2017-12-29. We perform our analysis
on the 578 SNe Ia discovered or recovered in ASAS-SN. SNe discovered in the
$g$ band but never recovered in the $V$ band are excluded from the analysis.

The ASAS-SN catalogues defined the peak magnitude of a supernova by taking the
brighter value between the brightest point in the light curve and the peak of
a parabolic fit to magnitudes. However, we found that this method
systematically biases the peak magnitudes to be too bright since random
fluctuations often make the brightest point brighter than the peak of a fit.
This is primarily a problem for the faintest SNe in the catalogue and,
unfortunately, most SNe are faint, with the median peak magnitude being 16.4
mag.

To obtain more accurate values for the peak magnitudes, we refit all ASAS-SN
light curves in flux instead of magnitude using the $V$-band SN Ia templates
from Nugent et al. (2002). Using the templates, varying the stretch in time,
time of peak, and peak magnitude, and using the fluxes instead of magnitudes
leads to more robust fits for fainter SNe. The median peak magnitude of our
new fits is $0.3\,\mathrm{mag}$ fainter than for the original approach.
Unfortunately, the stretch is generally poorly constrained from the ASAS-SN
light curves alone, as the fainter SNe are too noisy. In Table 1, we report
the updated peak magnitudes for all SNe Ia with ASAS-SN $V$-band light curves.
We experimented with the Spectral Adaptive Light Curve Template (SALT2; Guy et
al., 2007) templates and found consistent estimates of the peak magnitudes.

Table 1: The $V$-band Sample of Type Ia SNe Discovered or Recovered by ASAS-
SN.

SN Name | IAU Name | RA | Dec. | Redshift | Type |  $m_{\mathrm{\textit{V},peak}}$ a |  $M_{\mathrm{\textit{V},peak}}$ b |  $s$ c | $t_{\mathrm{peak}}$ | Host Name | Disc./Rec. d
---|---|---|---|---|---|---|---|---|---|---|---
|  | [J2000] | [J2000] |  |  | [mag] | [mag] |  | [JD] |  |
ASASSN-14ad | — | 12:40:11.10 |  $+$18:03:32.8 | $0.0264$ | Ia | $16.15$ | $-18.65$ | $1.04$ | $2456696.9$ | KUG 1237+183 | D
ASASSN-14ar | — | 09:09:41.68 |  $+$37:36:07.6 | $0.0230$ | Ia-91bg | $16.41$ | $-18.65$ | $0.96$ | $2456769.4$ | IC 527 | D
ASASSN-14as | — | 12:57:34.11 |  $+$35:31:35.8 | $0.0374$ | Ia | $17.13$ | $-18.99$ | $0.80$ | $2456775.0$ | MGC +06-29-001 | D
ASASSN-14ax | — | 17:10:00.68 |  $+$27:06:20.1 | $0.0330$ | Ia | $16.99$ | $-18.94$ | $1.07$ | $2456792.4$ | SDSS J171000.69+270619.5 | D
ASASSN-14ba | — | 10:21:31.72 |  $+$08:24:18.6 | $0.0327$ | Ia-91T | $16.77$ | $-19.08$ | $0.81$ | $2456798.1$ | SDSS J102131.91+082419.8 | D
ASASSN-14bb | — | 12:14:11.35 |  $+$38:39:40.8 | $0.0230$ | Ia | $16.14$ | $-18.91$ | $0.98$ | $2456801.0$ | 2MASX J12141125+3839400 | D
ASASSN-14ay | — | 15:57:02.99 |  $+$37:24:56.4 | $0.0309$ | Ia | $16.52$ | $-19.20$ | $0.87$ | $2456796.4$ | 2MASX J15570268+3725001 | D
ASASSN-14bd | — | 12:52:44.85 |  $+$26:28:13.1 | $0.0214$ | Ia-91bg | $16.93$ | $-17.94$ | $0.84$ | $2456802.7$ | IC 831 | D
iPTF14bdn | — | 13:30:44.88 |  $+$32:45:42.4 | $0.0156$ | Ia-91T | $14.80$ | $-19.38$ | $1.10$ | $2456824.8$ | UGC 08503 | R
ASASSN-14bt | — | 10:19:19.74 |  $+$58:30:19.7 | $0.0289$ | Ia | $16.48$ | $-19.06$ | $0.91$ | $2456814.9$ | UGC 5566 | D
ASASSN-14cb | — | 13:08:14.45 |  $+$62:02:02.5 | $0.0336$ | Ia | $16.98$ | $-18.90$ | $0.80$ | $2456823.2$ | 2MASXi J1308145+620200 | D
ASASSN-14co | — | 15:57:29.75 |  $+$01:06:34.0 | $0.0333$ | Ia | $16.95$ | $-19.13$ | $1.10$ | $2456819.7$ | CGCG 023-005 | D
LSQ14cnm | — | 16:05:24.50 |  $+$01:12:58.7 | $0.0326$ | Ia | $16.89$ | $-19.22$ | $1.10$ | $2456829.7$ | 2MASX J16052452+0113000 | R
ASASSN-14cu | — | 12:47:02.61 |  $-$24:14:41.7 | $0.0248$ | Ia | $17.36$ | $-18.07$ | $0.80$ | $2456831.6$ | 2MASX J12470274-2414435 | D
2014bv | 2014bv | 12:24:30.98 |  $+$75:32:08.6 | $0.0056$ | Ia | $14.07$ | $-17.94$ | $0.81$ | $2456840.9$ | NGC 4386 | R
ASASSN-14db | — | 22:02:01.83 |  $-$70:02:27.9 | $0.0375$ | Ia | $17.02$ | $-19.15$ | $1.02$ | $2456827.7$ | ESO 075-G049 | D
ASASSN-14dc | — | 02:18:38.06 |  $+$33:36:58.4 | $0.0440$ | Ia-CSM | $15.93$ | $-20.70$ | $1.10$ | $2456839.0$ | 2MASX J02183825+3336556 | D
ASASSN-14dz | — | 15:05:54.52 |  $+$12:44:43.3 | $0.0222$ | Ia | $16.03$ | $-18.99$ | $0.99$ | $2456853.8$ | Mrk 842 | D
2014by | 2014by | 14:27:49.00 |  $+$11:33:40.0 | $0.0248$ | Ia | $16.35$ | $-18.90$ | $0.82$ | $2456859.9$ | UGC 09267 | R

Note: This table includes 578 Type Ia SNe discovered or recovered in ASAS-SN
from the catalogues (Holoien et al., 2017a, b, c, 2019) with updated peak
apparent magnitudes $m_{\mathrm{\textit{V},peak}}$ and peak absolute
magnitudes $M_{\mathrm{\textit{V},peak}}$. This table is available in its
entirety in a machine-readable form in the online journal. A portion is shown
here for guidance regarding its form and content.
a Peak apparent magnitudes reported here are the values after refitting the
ASAS-SN light curves using the $V$-band SN Ia templates from Nugent et al.
(2002).
b Peak absolute magnitudes are computed using $m_{\mathrm{\textit{V},peak}}$
and Eq. 1.
c Model stretch parameter related to $\Delta m_{15}(B)$ by Eq.3. $s=1.0$
corresponds to $\Delta m_{15}(B)=1.05$.
d Indicates whether the SN was discovered by ASAS-SN (D) or independently
recovered in ASAS-SN data (R).

Figure 1 shows a few examples of the light-curve fits. Figure 1a and Figure 1d
demonstrate the ability to fit one of the brightest (ASASSN-15hx) and one of
the faintest (ASASSN-16jw) SNe in the sample. ASASSN-15lp, which only has a
declining light curve (Figure 1b), would have had obvious issues for the peak
magnitude based on either the brightest point or a parabolic fit near the
peak. However, the templates encapsulate the shape of the decline and thus are
able to predict the peak magnitude reasonably well. In the case of SN 2017fax
(Figure 1c), with only two points near peak, a parabolic fit is poorly
constrained, whereas the templates cover a larger time range leading to a
better overall fit and peak magnitude.

We compute absolute magnitudes for all SNe using

$M_{V}=m_{V}-\mu(z)-A_{\mathrm{\textit{V},MW}}-K(z)$ (1)

where $m_{V}$ is the $V$-band apparent magnitude, $\mu(z)$ is the distance
modulus as a function of redshift obtained using the Python package
astropy.cosmology (Astropy Collaboration et al., 2013, 2018),
$A_{\mathrm{\textit{V},MW}}$ is the Milky Way extinction assuming a
$7000\,\mathrm{K}$ source and $R_{V}=3.1$ from Table 6 of Schlafly &
Finkbeiner (2011), and $K(z)$ is the $K$-correction (Hogg et al., 2002) as a
function of redshift computed with SNooPy (Burns et al., 2011), which uses the
SN Ia templates from Hsiao et al. (2007). We include no correction for the
host-galaxy extinction here but explore its effects in Section 4.3.

Figure 2: Peak apparent magnitude and redshift distribution of the 578 SNe Ia
discovered or recovered by ASAS-SN. The filled circles and filled histograms
show our standard sample of 404 SNe Ia. The open points are the SNe excluded
due to the cuts on the peak absolute magnitude $M_{\mathrm{\textit{V},peak}}$
(open squares), Galactic latitude $b$ (open triangles), peak apparent
magnitude $m_{\mathrm{\textit{V},peak}}$ and redshift $z$ (open circles). The
gray dashed lines mark our choice of the limiting peak apparent magnitude at
$m_{\mathrm{\textit{V},peak}}=17\,\mathrm{mag}$ and minimum redshift at
$z_{\mathrm{min}}=0.005$. The cosmic rate (solid black line) shows that the
sample is volume limited up to a redshift of $z\sim 0.02$.

We explore the statistics of the 578 SNe Ia discovered or recovered by ASAS-SN
in the $V$ band (Holoien et al., 2017a, b, c, 2019) whose distribution in
$m_{\mathrm{\textit{V},peak}}$ and redshift is shown in Figure 2. The black
line in the top histogram of Figure 2 displays the cosmic rate assuming a
uniform distribution of SNe in comoving volume. It shows that the sub-sample
with $M_{\mathrm{\textit{V},peak}}\lesssim 17.5\,\mathrm{mag}$ (which
constitutes the majority of the sample) is volume-limited up to a redshift of
$z\sim 0.02$ and magnitude-limited beyond that. This sample does not include
the 13 SNe discovered in 2013 due to the high incompleteness of the survey and
systematic errors during the early operations of ASAS-SN. We restrict our
standard analysis to SNe more luminous than
$M_{\mathrm{\textit{V},peak}}<-16.5\,\mathrm{mag}$, which reduces the sample
to 574 SNe. For our standard analysis we use a limiting Galactic latitude at
$|b|>15\degr$, which leaves 541 SNe, and a limiting peak apparent magnitude at
$m_{\mathrm{\textit{V},peak}}<17\,\mathrm{mag}$, where our completeness is
$\sim 50\%$ (see Section 3). These limits exclude the two SNe Iax from our
standard sample; SN 2015H because we found a peak magnitude
$>17\,\mathrm{mag}$ and SN 2017gbb because it was not recovered by ASAS-SN.
Finally, we restrict the redshift range to be from $z_{\mathrm{min}}=0.005$,
to eliminate systems where peculiar velocities can significantly affect
distance estimates, to $z_{\mathrm{max}}=0.08$, which includes all systems.
This leaves us with 404 SNe in our standard sample, where the reduced number
is almost entirely due to the magnitude limit. We explore the consequences of
varying these limits in Section 4.1.

We update the subtype classification of three SNe in our standard sample to be
consistent with the classification scheme of Taubenberger (2017). ASASSN-15us
was classified as Ia-06bt by Holoien et al. (2017b), which falls under the
broader class of Ia-02es. ASASSN-15hy and ASASSN-16ex were classified as
Ia-07if and Ia-09dc, respectively, both of which fall under the broader class
of Ia-03fg (Ashall et al., 2021).

##  3 Rate Computations

Magnitude-limited surveys like ASAS-SN must correct the observed sample for
the probability of detecting the SNe. Incompleteness is driven by the survey
cadence, seasonal gaps, survey magnitude limit, and observing conditions. As
in previous studies, we use simulations to estimate the completeness
corrections as a function of peak apparent and absolute magnitudes.

Based on all ASAS-SN observations of supernovae (discovered, recovered, and
missed), we found that we could describe well the probability of a detection
($p$) in any given epoch as a simple function of the signal-to-noise ratio
($SNR$) of the observation

$p=\left\\{\begin{array}[]{ll}0,&\text{for}\,\,SNR\leq 5\\\
0.65\left(1-\frac{12-SNR}{7}\right),&\text{for}\,\,5\leq SNR\leq 12\\\
0.65,&\text{for}\,\,SNR\geq 12.\\\ \end{array}\right.$ (2)

There was no apparent dependence on other variables. The saturation at
$p=0.65$ is due to a broad range of systematic problems associated with how
the overall processing system tries to minimize the number of false positives.
However, this is a probability per observation, and with a $V$-band cadence of
3–4 days, the probability of actually missing a bright SN is very low. It is
likely that Eq. 2 is underestimating $p$ for bright SNe, but this also has no
important consequences since bright SNe will have many detection trials ($n$)
and the overall probability of detection $1-(1-p)^{n}$ converges rapidly to
unity. For example, if $n=5$, the probability of detection is $99.5\%$ for
$p=0.65$ versus $100.0\%$ for $p=0.95$.

Using Eq. 2, we perform injection recovery simulations on the ASAS-SN light
curves. We obtain the ASAS-SN $V$-band light curves for random positions
uniformly distributed on the sky with a mean density of four points per square
degree. These light curves include the magnitudes, fluxes, and their
uncertainties for each epoch and consequently include the survey cadence and
seasonal gaps. Next, we inject simulated Type Ia SNe onto the light curves
using the $V$-band templates from Nugent et al. (2002) and ask which ones
would be detected. We use the same magnitude and Galactic latitude limits as
for the observed SN sample in Section 2. For bright SNe in bright galaxies,
the presence of the galaxy will affect the $SNR$, but the $SNR$ is so high
that neglecting the host does not matter for the detection probability. For
faint SNe, the noise is dominated by the sky, so the presence of the host flux
has little effect on the detection probability. Any systematic errors created
by host galaxies will only become important once the statistical errors are
smaller.

The time of peak $t_{0}$ for each simulated SN is drawn randomly from a
uniform distribution over the time span covering the discovery dates of all
SNe with a padding of 15 days on both ends. The redshift $z$ is drawn randomly
assuming our standard cosmology and a constant comoving density with a maximum
redshift of $z_{\mathrm{lim}}$ where the SN would have the limiting magnitude
of $m_{\mathrm{\textit{V},peak}}=17.0\,\mathrm{mag}$ given no extinction. The
trial is kept if a uniform deviate is $<1/(1+z)$ to account for the time
dilation of the rates. The templates are stretched in time by the stretch
parameter $s$ which is related to $\Delta m_{15}(B)$, the light curve decline
rate parameter (Phillips, 1993), by

$s=-0.397\Delta m_{15}(B)^{3}+1.767\Delta m_{15}(B)^{2}-3.034\Delta
m_{15}(B)+2.698.$ (3)

This relation is obtained by fitting a third-order polynomial to the stretch
factor as a function of $\Delta m_{15}(B)$ using the $B$-band templates from
Nugent et al. (2002), where $s=1.0$ corresponds to $\Delta m_{15}(B)=1.05$.
The $\Delta m_{15}(B)$ for each simulated SN is given by the $\Delta
m_{15}(B)$ – $M_{\mathrm{\textit{V},peak}}$ relation of

$\Delta
m_{15}(B)=1.1+0.290\log\left[1+\frac{M_{\mathrm{\textit{V},peak}}+19.389-5\log
h_{70}}{0.096}\right]$ (4)

from Garnavich et al. (2004), where
$H_{0}=70\,\mathrm{km\,s^{-1}\,Mpc^{-1}}\,h_{70}$. Note that $\Delta
m_{15}(B)$ is only an intermediate parameter to go from
$M_{\mathrm{\textit{V},peak}}$ to the stretch $s$ that is applied to the
$V$-band templates used to fit the light curves. The templates are further
stretched in time by a factor of $(1+z)$ to account for the cosmological time
dilation.

To account for the measurement uncertainty for the simulated SN light curve,
we approximate the noise by adding a Gaussian deviate of dispersion

$\sigma_{F}=\sqrt{\frac{F_{0}F}{g}},$ (5)

where $F$ is the flux for a given simulated point, $g$ is the gain for the
camera used for observation at that epoch, and $F_{0}=Z_{0}10^{-m_{0}/2.5}$,
where $Z_{0}$ and $m_{0}$ are the flux and magnitude zero-points,
respectively.

Figure 3: Differential distribution of peak apparent magnitudes
$m_{\mathrm{\textit{V},peak}}$ for our observed SNe Ia sample (solid blue
line) compared with the prediction from our detection model (dashed purple
line). The model is normalized to have the same total number of SNe with
$m_{\mathrm{\textit{V},peak}}<17.0\,\mathrm{mag}$. The shaded region shows the
$1\sigma$ Poisson uncertainty given the expected number of SNe.

We carry out 100 random trials for each of the $N_{\mathrm{LC}}=164{,}191$
random light curves, for a total of $M=100\,N_{\mathrm{LC}}$ trial SNe. This
is done for each peak luminosity over the range $-21\,\mathrm{mag}\leq
M_{\mathrm{\textit{V},peak}}\leq-16.5\,\mathrm{mag}$ in intervals of
$0.5\,\mathrm{mag}$. For each of these simulated SN, we use the $K$-correction
and distance modulus appropriate to the random redshift, add the Galactic
extinction correction associated with the light curve coordinates, and then
add the flux of the SN to the random ASAS-SN light curve. The trial is logged
as a detection if at least one epoch of the simulated SN satisfies the $SNR$
criterion in Eq. 2.

The result from applying the detection model of Eq. 2 to our simulated sample
is shown as the dashed purple line in Figure 3 and it predicts the observed
magnitude distribution of the sample very well despite having made no use of
this information. The model shown here is normalized to the total number of
SNe in the observed sample. The simulated sample of SNe not only reproduces
the bright end of the observed distribution, but also well models the turnover
at the faint end ($m_{\mathrm{\textit{V},peak}}\gtrsim 16.5\,\mathrm{mag}$).

Next, we compute the completeness as a function of peak absolute and apparent
magnitudes. If a total of $N$ out of $M$ trials are detections, then the
completeness is $F_{1}=N/M$. Figure 4 shows the completeness as a function of
apparent magnitude for the different absolute magnitudes after binning the
trials by their apparent magnitude. The completeness flattens out at the
bright end at $\sim 80$ – $85\%$, limited by the seasonal gaps. The
completeness slowly declines but then begins to drop rapidly from $\sim 60\%$
at $m_{\mathrm{\textit{V},peak}}=16\,\mathrm{mag}$ to $\sim 0\%$ at
$m_{\mathrm{\textit{V},peak}}=18\,\mathrm{mag}$. The completeness is also
lower for less luminous SNe because they spend less time near their peak
luminosity. We choose a limit of
$m_{\mathrm{\textit{V},lim}}=17\,\mathrm{mag}$ for our standard analysis since
that is where the completeness is $\sim 50\%$.

Figure 4: Completeness as a function of peak apparent magnitude
$m_{\mathrm{\textit{V},peak}}$ with the peak absolute magnitudes
$M_{\mathrm{\textit{V},peak}}$ shown in different colours. The vertical dashed
red line marks our standard choice for the limiting magnitude,
$m_{\mathrm{\textit{V},lim}}=17\,\mathrm{mag}$, where the completeness is
$\sim 50\%$. More luminous SNe are luminous for a longer time, leading to
higher completeness.

Since we adjust $z_{\mathrm{lim}}$ with the peak absolute magnitude to avoid
wasting trials, we need to correct the completeness to a common volume for all
SNe. Our choice of $z_{\mathrm{max}}=0.08$ defines the maximum redshift. Thus
given the comoving volume $V(z)$, there is a second completeness factor of
$F_{2}(M_{\mathrm{\textit{V},peak}})=V(z_{\mathrm{lim}}(M_{\mathrm{\textit{V},peak}}))/V(z_{\mathrm{max}})$
to correct for the differences in volume between $z_{\mathrm{lim}}$ and
$z_{\mathrm{max}}$. The final statistical weight for the $i^{\mathrm{th}}$
observed SN is $w_{i}=\left(F_{1,i}\,F_{2,i}\right)^{-1}$.

Given the statistical weights, the volumetric rate $R$ for SNe Ia is
calculated by summing $N$ SNe within a time span $\Delta t$ and a fixed
comoving volume $V$. Each SN is weighted by the factor $w_{i}$ that accounts
for the incompleteness given its peak apparent and absolute magnitudes. The
volumetric SN rate is then

$R=\frac{\sum_{i=1}^{N}w_{i}}{V\,\Delta
t\,\left(1-\sin{b_{\mathrm{lim}}}\right)},$ (6)

where $\Delta t=4.0\,\mathrm{yr}$ is the time span between UTC 2014-01-01 and
UTC 2017-12-31,
$V=\frac{4}{3}\pi\left(d_{\mathrm{max}}^{3}-d_{\mathrm{min}}^{3}\right)$ is
the total comoving volume corresponding to the maximum and minimum redshift
limits, and $(1-\sin{b_{\mathrm{lim}}})$ corrects for our Galactic latitude
limit. Including the lower redshift limit of $z_{\mathrm{min}}=0.005$ changes
the volume by only $\sim 0.03\%$. The effects of time dilation are already
included in the computation of the weights. Therefore, the rate $R$, as given
by Eq. 6, provides an estimate for the number of SNe that occur per unit
comoving volume and time.

We estimate the total volumetric rate using Eq. 6. We also compute an LF by
splitting the sample into absolute magnitude bins of width 0.5 mag and use Eq.
6 to calculate the rate in each bin divided by the bin size to obtain the rate
per magnitude. Because of the sample size and high spectroscopic completeness,
we are also able to compute LFs for some of the major subtypes of SNe Ia. The
subtypes that have more than one object include the overluminous Ia-91T
($N=30$), the subluminous Ia-91bg ($N=9$), the extremely bright Ia-CSM ($N=3$)
and Ia-03fg ($N=2$) classes.

The statistical errors on all rates are estimated using bootstrapping. We
first randomly draw the sample size as a Poisson deviate with an expected
number equal to the observed sample ($N=404$ for our standard sample). Then we
randomly draw that number of SNe from the observed sample with replacement.
This also approximates the error in the completeness corrections coming from
the SN weights through the random selection of SNe. The standard error on the
rate is then given by the $16^{\mathrm{th}}$ and the $84^{\mathrm{th}}$
percentiles of the bootstrapped rate distribution. Errors for the bins in the
LFs with more than one object are obtained in the same manner as the total
sample. Errors for the bins with only one object are estimated using only the
Poisson uncertainties.

We did not explicitly correct for the five SNe in the $V$-band catalogues
without spectroscopic classification that could have been selected in our
final sample. Given the relative numbers of SNe Ia and core-collapse SNe in
Holoien et al. (2019), we would expect $\sim 3$ of these to be SNe Ia.
Crudely, this implies an underestimate of the rates by $\sim 3/404$, or less
than $1\%$, which is much smaller than the statistical or other systematic
uncertainties.

##  4 Results & Discussion

###  4.1 Volumetric SN Ia Rates

Figure 5: Left: Rate as a function of the Galactic latitude cut. Middle: Rate
as a function of the limiting apparent magnitude. Right: Rate as a function of
maximum redshift cut. In all three panels, the numbers in red are the number
of SNe in the samples. The red arrows point to the values used for our
standard sample.

Figure 5 shows how our total rate estimates depend on the choice of the
minimum Galactic latitude, limiting apparent magnitude, and maximum redshift.
A fainter limiting magnitude or working closer to the Galactic plane includes
more SNe in the sample, thereby reducing the statistical errors, but requires
larger completeness corrections, which increases the systematic uncertainties.
The total volumetric rate is roughly constant for a latitude cut of
$|b_{\mathrm{lim}}|>15\degr$ and for a limiting magnitude of
$m_{\mathrm{\textit{V},lim}}<17\,\mathrm{mag}$, indicating that the
completeness corrections are performing as expected given the uncertainties.
As a compromise between the number of SNe and systematic uncertainty, we
choose our standard values of $|b_{\mathrm{lim}}|=15\degr$ and
$m_{\mathrm{\textit{V},lim}}=17\,\mathrm{mag}$. The total rate approaches a
constant value with larger volume as $z_{\mathrm{max}}$ increases. There are
no more SNe in our sample beyond $z=0.08$ so the rate is constant for higher
redshifts. We choose $z_{\mathrm{max}}=0.08$ as our standard limit.

Our SNe Ia sample has a median redshift of $z_{\mathrm{med}}=0.024$ and the
total rate calculated using the method described in Section 3 is

$R_{\mathrm{tot}}=2.28^{+0.20}_{-0.20}\times
10^{4}\,\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,h^{3}_{70}.$ (7)

The fractional uncertainty ($\sigma_{R}/R$) of 9% is significantly more than
the Poisson uncertainties from having 404 sources (5%) in part because the
weight factors $w_{i}$ are not uniform. For example, if a small fraction of
the sources have very high weights, the statistical uncertainties are
ultimately controlled by the Poisson variations in the numbers of these highly
weighted sources rather than the fluctuations in the larger number of overall
sources. In our case, this is driven by the minimum luminosity
$M_{\mathrm{\textit{V},peak}}$ used to define the sample. While the weight
factor depends weakly on absolute magnitude (Figure 4), the volume correction
basically scales as $\log w_{i}\propto-0.6\,M_{\mathrm{\textit{V},peak}}$,
which varies by a factor of $500$ between
$M_{\mathrm{\textit{V},peak}}=-16.5\,\mathrm{mag}$ and $-21\,\mathrm{mag}$. As
shown in Table 2, increasing the minimum luminosity limit by a magnitude
excludes few SNe from the sample. The total rate decreases as a result, since
a smaller luminosity range is considered. But, excluding the lower luminosity
SNe leads to uncertainties that approach the Poisson statistics limit.

Table 2: Volumetric SNe Ia Rates for Varying Minimum Luminosities. $M_{\mathrm{\textit{V},peak}}$ | $N_{\mathrm{Ia}}$ | Rate | $\sigma_{R}/R$
---|---|---|---
[mag] |  | $[10^{4}\,\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,h^{3}_{70}]$ |
$[-16.5,-21.5]$ | 404 | $2.28^{+0.20}_{-0.20}$ | $9\%$
$[-17.0,-21.5]$ | 402 | $2.09^{+0.15}_{-0.15}$ | $7\%$
$[-17.5,-21.5]$ | 397 | $1.91^{+0.12}_{-0.12}$ | $6\%$

After converting the rates from various studies to a consistent value of
$H_{0}=70\,\mathrm{km\,s}^{-1}\,\mathrm{Mpc}^{-1}$, we first compare our total
volumetric rate to other studies at low redshift ($z<0.1$). The rates at low
redshift are summarized in Table 3 and are shown in Figure 6. Rates from both
Cappellaro et al. (1999) and Li et al. (2011b) use a galaxy-targeted sample,
which introduces large systematics in the observed sample. In particular,
there was a bias towards more luminous galaxies, thus biasing the SN sample
due to the correlation of SNe Ia and host properties (e.g., Sullivan et al.,
2010). However, due to their large uncertainties, these rate measurements
agree with our total rate within $1\sigma$.

Untargeted surveys provide a means of reducing the galaxy bias, although there
is still the question of the radial SN distribution. Holoien et al. (2017a, b,
c, 2019) showed that between 2014 and 2017, amateurs and surveys other than
ASAS-SN were less effective in discovering SNe close to the centers of their
hosts. Nonetheless, untargeted surveys have fewer systematics and show an
improvement over targeted samples.

Using the 270 spectroscopically confirmed SNe Ia from the SDSS-II Supernova
Survey (Frieman et al., 2008), Dilday et al. (2010) computed the rates out to
a redshift of $z\sim 0.3$. Their lowest redshift bin ($0.025\leq z\leq 0.050$)
contains only four SNe, while their sample with $z<0.12$ contains 37 SNe. Due
to a small sample size, their statistical uncertainties dominate, making their
rates consistent with our total rate. Frohmaier et al. (2019) used a larger
sample of 90 spectroscopically confirmed SNe Ia in the redshift range $z<0.09$
from the untargeted PTF survey. The increase in the low-redshift sample size
over the SDSS sample significantly reduces their statistical uncertainties.
The PTF rate is consistent with the SDSS rate, and agrees with our total rate
within $1\sigma$.

Figure 6: Volumetric SN Ia rate as a function of redshift $z$. Our rate is the
filled red square at the median redshift of our sample, with other results as
listed in the figure are shown as the open symbols. The dashed and dotted-
dashed lines show a power-law fits of the form $(1+z)^{\alpha}$ from Dilday et
al. ($\alpha=2.04^{+0.90}_{-0.89}$; 2010) and Perrett et al. ($\alpha=2.11\pm
0.28$; 2012), respectively. Dilday et al. (2010) includes all subtypes of SNe
Ia, whereas Perrett et al. (2012) only includes ‘normal’ SNe Ia. All displayed
uncertainties include statistical and systematic.

The largest untargeted sample is the ZTF BTS SNe Ia sample, going out to a
redshift of $z\sim 0.1$. Perley et al. (2020) were able to benefit from the
faint magnitude limit ($m\approx 18.5\,\mathrm{mag}$) of ZTF BTS to build a
larger sample of 875 SNe. However, there are likely larger systematics since
they used an average completeness correction rather than weighting individual
SNe. Nonetheless, the total ZTF BTS rate is consistent with our total rate.

Srivastav et al. (2022) computed the volumetric rate of SNe Ia within a
redshift of $z<0.024$ using the ATLAS local volume survey and their estimate
is consistent with ours. With 269 classified SNe Ia in their sample, they
perform an assessment of recovery of their simulated light curves given the
history of ATLAS observations and quality metrics. They also compute a
slightly higher rate of $(2.83\pm 0.29)\times
10^{4}\,\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,h^{3}_{70}$ including the
spectroscopically unclassified SNe Ia by assuming they have the same relative
fractions of SN subtypes as the classified sample.

Using a subset of 298 SNe Ia within a redshift range $0.01\leq z\leq 0.04$
from the ZTF BTS sample, Sharon & Kushnir (2022) computed the volumetric rate
after correcting for host-galaxy extinction. They obtained the intrinsic
luminosities, corrected for host-galaxy extinction, based on the colour
stretch $s_{gr}$ calibrated from the Carnegie Supernova Project SNe Ia sample
(Contreras et al., 2010; Stritzinger et al., 2011; Krisciunas et al., 2017;
Burns et al., 2018; Ashall et al., 2020). Since they corrected for the host-
galaxy extinction, the rate from Sharon & Kushnir (2022) is higher than our
rate at a similar median redshift. We discuss host extinction in Section 4.3.

Table 3: Local Volumetric SNe Ia Rates.

$z$ | $N_{\mathrm{Ia}}$ | Rate | Reference
---|---|---|---
|  | $[10^{4}\,\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,h^{3}_{70}]$ |
$\sim$0 | 70 | $2.8\pm 0.9$ | Cappellaro et al. (1999)
0.025 – 0.050 | 4 | $2.78^{\,+1.13}_{\,-0.83}$ | Dilday et al. (2010)
< 0.12 | 37 | $2.35^{\,+0.47}_{\,-0.39}$ | Dilday et al. (2010)
$\sim$0 | 274 | $3.01\pm 0.62$ | Li et al. (2011b)
0.015 – 0.090 | 90 | $2.43^{\,+0.44}_{\,-0.35}$ | Frohmaier et al. (2019)
< 0.1 | 875 | $2.35\pm 0.24$ | Perley et al. (2020)
0.01 – 0.04 | 298 | $2.91^{\,+0.58}_{\,-0.45}$ | Sharon & Kushnir (2022)
< 0.024 | 269 | $2.41\pm 0.25$ | Srivastav et al. (2022)
0.005 – 0.08 | 404 | $2.28^{\,+0.20}_{\,-0.20}$ | This Work

Figure 7: Luminosity functions of Type Ia SNe. Each magnitude bin has a width
of $0.5\,\mathrm{mag}$ for the ASAS-SN data. $R$\- and $r$-bands of Perley et
al. (2020), Sharon & Kushnir (2022), and Li et al. (2011a) are corrected to
the $V$-band using Eq. 8. The LOSS LF is scaled to match the total ZTF BTS
rate and shifted so their maxima coincide. All others are observed LFs, but
the LF from Sharon & Kushnir (2022) is an intrinsic LF, corrected for host-
galaxy extinction. The two most luminous bins only contain SNe Ia-CSM.

Figure 6 also shows the rate as a function of redshift. Using SDSS data,
Dilday et al. (2010) computed the rates in a broad range of redshifts,
$0.025\leq z\leq 0.325$, and fitted the redshift evolution with a power law of
the form $(1+z)^{\alpha}$, with best-fit $\alpha=2.0^{+0.9}_{-0.9}$ (dashed
line in Figure 6) where the normalization is set by the rate at $z=0$. Perrett
et al. (2012) measured the rates within a redshift range $0.1\leq z\leq 1.1$
using SNe Ia from the SNLS. Their evolution of the rate with redshift is
similarly fitted with a $(1+z)^{\alpha}$ power law with $\alpha=2.1\pm 0.3$
(dotted dashed line in Figure 6). However, their extrapolated rate at $z=0$ is
lower than that of Dilday et al. (2010) because Perrett et al. (2012) did not
include SNe that are subluminous Ia-91bg, super-Chandrasekhar, or extremely
rare events. They limited their sample to ‘normal’ SNe Ia for better
modelling. The result is a reduction in the overall normalization of the
redshift evolution by $\sim 15$ – $20\%$. Our rate measurement, which includes
all subtypes, is consistent with the fit from Dilday et al. (2010) and not
with that from Perrett et al. (2012). For both Dilday et al. (2010) and
Perrett et al. (2012), the DTD fit is consistent with $\propto\tau^{-1}$. The
redshift evolution of rate from Rodney & Tonry (2010) and Graur et al. (2011)
agrees with that of Perrett et al. (2012) at higher redshifts but both have
significantly larger uncertainties. Another approach, as shown in Horiuchi &
Beacom (2010), is to use a star formation rate (SFR) to constrain the exponent
of the DTD. However, this method requires an assumption for the SFR.

###  4.2 Luminosity Functions

In Figure 7, we compare our LF to the $r$\- or $R$-band LFs from Li et al.
(2011a), Perley et al. (2020), and Sharon & Kushnir (2022). We convert these
LFs to the $V$ band using

$M_{V}=M_{0,V}+\frac{b_{V}}{b_{r}}\left(M_{r}-M_{0,r}\right)$ (8)

where the values of $M_{0,V}=-19.12\pm 0.01\,\mathrm{mag}$, $M_{0,r}=-19.03\pm
0.01\,\mathrm{mag}$, $b_{V}=0.95\pm 0.11$, and $b_{r}=1.02\pm 0.11$ are from
Table 9 of Folatelli et al. (2010). Sharon & Kushnir (2022) compute the
observed LF for the LOSS survey shown in Figure 7 by scaling the rates to
match the total BTS rate at the peak of the LF. The two most luminous bins in
our LF only include SNe Ia-CSM, which are not present in the LF from Perley et
al. (2020), causing the steep decline in rate at the luminous end. The low-
luminosity end of our LF is consistent with the LF from Perley et al. (2020),
however, we see a significant difference for the higher luminosities. In their
rate calculations, Perley et al. (2020) used an average completeness
correction factor of $f_{\mathrm{rec}}=0.6$ for the recovery efficiency of
transients. As seen in Figure 4, the completeness is a function of absolute
magnitude because more luminous SNe are brighter for longer. The use of a
single correction factor would result in an underestimate of the completeness
for high luminosity SNe leading to an overestimate of the associated rates.
The rates from Perley et al. (2020) also do not seem to account for the time
dilation of the rates, but this effect would be less important since the
redshifts are modest. The magnitude shifts predicted by Eq. 8 are uncertain by
roughly $0.1\,\mathrm{mag}$, which is not large enough to explain the factor
of $\sim 5$ difference in the LFs near $M_{V}=-19.5\,\mathrm{mag}$.

A larger effect that can explain the difference is host-galaxy extinction. We
can make the luminous end of the LFs agree if we have an average extra
contribution to the right side of Eq. 8 of
$\left(R_{V}-\frac{b_{V}}{b_{r}}R_{r}\right)E(V-r)$. If we use a typical
Galactic extinction law with $R_{V}=3.1$ and $R_{r}=2.3$ (Cardelli et al.,
1989), then a mean extinction of $E(V-r)\approx 0.21\,\mathrm{mag}$ provides
the necessary shift. If we use an empirical extinction law derived from
observations of SNe Ia but of uncertain interpretation, such as $R_{V}=1.74$
and $R_{r}=0.89$ from Folatelli et al. (2010), then a mean extinction of
$E(V-r)\approx 0.22\,\mathrm{mag}$ again provides the necessary shift. These
are roughly consistent with the estimates of $E(B-V)\approx 0.1$ and
$E(B-V)\approx 0.17$ found by Sharon & Kushnir (2022) and Burns et al. (2014),
respectively.

Sharon & Kushnir (2022) attempted to correct for the host-galaxy extinction
and obtain an intrinsic LF by using the colour stretch $s_{gr}$ from the $g$\-
and $r$-band light curves with the luminosities calibrated using the CSP SNe
Ia sample (Contreras et al., 2010; Stritzinger et al., 2011; Krisciunas et
al., 2017; Burns et al., 2018). The resulting LF is more sharply peaked
probably due to ignoring the scatter in the peak magnitude – $s_{gr}$
relation, which leads to a narrower distribution in magnitudes. The Perley et
al. (2020) and Sharon & Kushnir (2022) rates are based on the same ZTF BTS
sample but Sharon & Kushnir (2022) find higher rates. An additional
contribution to the rate difference may be that Sharon & Kushnir (2022) use a
shorter survey duration compared to the BTS analysis. Using the longer
duration of Perley et al. (2020) lowers the Sharon & Kushnir (2022) rate to
$2.56^{+0.58}_{-0.46}\,\times
10^{4}\,\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,h^{3}_{70}$, which is within
$10\%$ of the rate from Perley et al. (2020).

Figure 8: Luminosity functions for various subtypes of SNe Ia. The total LF
for all subtypes is shown in black. Ia-norm+other includes normal SNe Ia as
well as all SNe not classified into a specific subtype, only having the
classification of ‘Ia.’ For clarity, LFs for Ia-91T and Ia-CSM are shifted to
the left by $0.05\,\mathrm{mag}$, Ia-91bg by $0.07\,\mathrm{mag}$, and Ia-
norm+other to the right by $0.05\,\mathrm{mag}$. The sample size for each
subtype is given in parentheses. Table 4: Luminosity Functions and Total Rates
for Each Major subtype of SNe Ia.

$M_{\mathrm{\textit{V},peak}}$ | $R_{\mathrm{all}}$ | $R_{\mathrm{Ia-norm}}$ | $R_{\mathrm{Ia-91T}}$ | $R_{\mathrm{Ia-91bg}}$ | $R_{\mathrm{Ia-CSM}}$ | $R_{\mathrm{Ia-03fg}}$ | $R_{\mathrm{Ia-02es}}$ | $R_{\mathrm{Ia-00cx}}$
---|---|---|---|---|---|---|---|---
[mag] | $[\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,\mathrm{mag}^{-1}\,h^{3}_{70}]$
$[-16.5,-17.0]$ | $4^{+3}_{-3}\times 10^{3}$ | $4^{+3}_{-3}\times 10^{3}$ | — | — | — | — | — | —
$[-17.0,-17.5]$ | $3.6^{+1.8}_{-1.4}\times 10^{3}$ | $3.6^{+1.8}_{-1.4}\times 10^{3}$ | — | — | — | — | — | —
$[-17.5,-18.0]$ | $4.4^{+1.3}_{-1.3}\times 10^{3}$ | $2.3^{+1.0}_{-1.0}\times 10^{3}$ | — | $2.1^{+1.0}_{-0.9}\times 10^{3}$ | — | — | — | —
$[-18.0,-18.5]$ | $8.0^{+1.4}_{-1.3}\times 10^{3}$ | $7.2^{+1.3}_{-1.4}\times 10^{3}$ | — | $5.0^{+0.3}_{-0.3}\times 10^{2}$ | — | — | $2.0^{+2.0}_{-1.3}\times 10^{2}$ | —
$[-18.5,-19.0]$ | $1.59^{+0.14}_{-0.13}\times 10^{4}$ | $1.52^{+0.14}_{-0.13}\times 10^{4}$ | $5^{+2}_{-2}\times 10^{2}$ | $2.5^{+1.5}_{-1.3}\times 10^{2}$ | — | — | — | —
$[-19.0,-19.5]$ | $9.0^{+0.8}_{-0.8}\times 10^{3}$ | $7.8^{+0.7}_{-0.7}\times 10^{3}$ | $1.1^{+0.3}_{-0.2}\times 10^{3}$ | — | — | $40^{+50}_{-30}$ | — | —
$[-19.5,-20.0]$ | $9.7^{+1.7}_{-1.6}\times 10^{2}$ | $8.3^{+1.7}_{-1.5}\times 10^{2}$ | $1.1^{+0.6}_{-0.5}\times 10^{2}$ | — | — | — | — | $29^{+30}_{-19}$
$[-20.0,-20.5]$ | $60^{+30}_{-30}$ | $28^{+15}_{-15}$ | $20^{+20}_{-13}$ | — | — | $11^{+12}_{-7}$ | — | —
$[-20.5,-21.0]$ | $14^{+8}_{-7}$ | — | — | — | $14^{+8}_{-7}$ | — | — | —
$[-21.0,-21.5]$ | $6^{+7}_{-4}$ | — | — | — | $6^{+7}_{-4}$ | — | — | —
| Total Rates $[\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,h^{3}_{70}]$
$[-16.5,-21.5]$ | $2.28^{+0.20}_{-0.20}\times 10^{4}$ | $2.04^{+0.20}_{-0.19}\times 10^{4}$ | $8.5^{+1.6}_{-1.7}\times 10^{2}$ | $1.4^{+0.5}_{-0.5}\times 10^{3}$ | $10^{+7}_{-7}$ | $30^{+20}_{-20}$ | $1.0^{+1.0}_{-0.8}\times 10^{2}$ | $14^{+16}_{-9}$
$[-17.0,-21.5]$ | $2.09^{+0.15}_{-0.15}\times 10^{4}$ | $1.85^{+0.14}_{-0.14}\times 10^{4}$ | $8.5^{+1.6}_{-1.7}\times 10^{2}$ | $1.4^{+0.5}_{-0.5}\times 10^{3}$ | $10^{+7}_{-7}$ | $30^{+20}_{-20}$ | $1.0^{+1.0}_{-0.8}\times 10^{2}$ | $14^{+16}_{-9}$
$[-17.5,-21.5]$ | $1.91^{+0.12}_{-0.12}\times 10^{4}$ | $1.67^{+0.11}_{-0.11}\times 10^{4}$ | $8.5^{+1.6}_{-1.7}\times 10^{2}$ | $1.4^{+0.5}_{-0.5}\times 10^{3}$ | $10^{+7}_{-7}$ | $30^{+20}_{-20}$ | $1.0^{+1.0}_{-0.8}\times 10^{2}$ | $14^{+16}_{-9}$

Note: Results shown in this table are for a host-galaxy extinction of
$0\,\mathrm{mag}$. The column of $R_{\mathrm{Ia-norm}}$ includes normal SNe Ia
as well as SNe not classified into a specific subtype.

Because of our high spectroscopic completeness, we are also able to compute
the LF of SNe Ia for different subtypes for the first time. Specifically, the
subtypes include Ia-norm (& other), Ia-91bg, Ia-91T, Ia-CSM, Ia-03fg, Ia-02es,
and Ia-00cx. Ia-norm (& other) includes the normal SNe Ia as well as the SNe
Ia that were not classified into a subtype. In total there are 358 Ia-norm (&
other), 9 Ia-91bg, 30 Ia-91T, 3 Ia-CSM, 2 Ia-03fg, 1 Ia-02es, and 1 Ia-00cx.
Figure 8 shows the LF for each subtype of SNe Ia and Table 4 gives the rates.
The LF of SNe Ia-91bg peaks at a fainter magnitude
($M_{\mathrm{\textit{V},peak}}\sim-17.75\,\mathrm{mag}$) than that of SNe
Ia-91T ($M_{\mathrm{\textit{V},peak}}\sim-19.25\,\mathrm{mag}$), illustrating
that SNe Ia-91bg are intrinsically fainter than SNe Ia-91T. From the total
rates $R_{\mathrm{Ia-91bg}}$ and $R_{\mathrm{Ia-91T}}$, we also note that SNe
Ia-91bg are intrinsically more common than SNe Ia-91T, but since SNe Ia-91bg
are faint and more difficult to classify, we do not observe as many of them.
Finally, as expected, we see that only SNe Ia-CSM contribute to the two most
luminous bins and are the only SNe Ia more luminous than
$M_{\mathrm{\textit{V},peak}}=-21\,\mathrm{mag}$ due to their CSM interaction.

The rates and LFs for each subtype shown here serve as a lower limit, since it
is likely that some SNe Ia classified without a subtype fall in this category.
Li et al. (2011a) found the observed fractions of subtypes of SNe Ia in a
volume-limited sample; SNe Ia-norm are $\sim 70\%$ of the total, SNe Ia-91bg
are $\sim 15\%$, and SNe Ia-91T are $\sim 9\%$. In comparison, we find a
larger fraction of 89% for SNe Ia-norm (& other), and lower fractions of 6%
for SNe Ia-91bg, and 4% for SNe Ia-91T. The fractions for the rarer subtypes
are $0.4\%$ for SNe Ia-02es, $0.1\%$ for SNe Ia-03fg, $0.06\%$ for SNe
Ia-00cx, and $0.04\%$ for SNe Ia-CSM. We note that our SNe Ia-CSM rate is
consistent with the recent ZTF-BTS rate determination of $\sim 0.02\%-0.2\%$
(Sharma et al., 2023). For our sample, the classification of SNe Ia-norm (&
other) includes SNe Ia that are not classified into any specific subtype in
addition to the ‘normal’ SNe Ia, thus artificially increasing the fraction of
‘normal’ SNe Ia and reducing the fractions of subtypes. A more careful,
uniform spectroscopic classification is needed in the future to classify SNe
Ia into appropriate subtypes.

###  4.3 Correcting for Host-Galaxy Extinction

Correcting for the host-galaxy extinction in each SN requires multi-band light
curves, which we do not have for this SN sample. Sharon & Kushnir (2022) used
the colour stretch to correct for host-galaxy extinction and found a mean
selective extinction of $E(B-V)\approx 0.1$. From their reanalysis of the LOSS
data, they found a mean extinction of $A_{V}\sim 0.5\,\mathrm{mag}$ with a
tail out to $A_{V}\sim 2-3\,\mathrm{mag}$. Similarly, the ZTF-BTS sample of
host galaxies shows a mean extinction of $A_{r}\sim 0.25\,\mathrm{mag}$ and
$A_{g}\sim 0.25\,\mathrm{mag}$, both tailing out to values of $\sim
2\,\mathrm{mag}$.

Here, we provide LFs as a function of host-galaxy extinction values which can
be weighted to provide any ‘desired’ correction. We show the LFs for different
values of host-galaxy extinction $A_{V}=$ ($0.0,\,0.25,\,0.5,\,1.0$, and
$1.5\,\mathrm{mag}$) in Figure 9. The overall effect is nearly identical to
simply adding a constant extinction to all SNe leading to a shift in the rates
of $10^{0.6A_{V}}$ modulo small second-order effects. As outlined in Section
4.2, the LFs in two different filters can also be used to estimate the host-
extinction. Assuming that our LF and the LF from Perley et al. (2020) are both
correct, they must agree with each other after accounting for the host-galaxy
extinction term when converting from one filter to another. Using this method,
we estimate a mean host extinction of $E(V-r)\approx 0.2\,\mathrm{mag}$.

Figure 9: Luminosity functions for various assumed values of average host-
galaxy extinction $A_{V}$.

##  5 Summary

We use a nearly spectroscopically complete catalogue of SNe Ia from ASAS-SN
(Holoien et al., 2017a, b, c, 2019) to estimate the volumetric SN Ia rate in
the local universe ($z<0.08$). We start by refitting all ASAS-SN light curves
using the $V$-band SN Ia templates from Nugent et al. (2002) to improve the
estimates of the peak magnitudes (see Section 2). We use randomly drawn light
curves and injected SNe to estimate completeness corrections as a function of
the peak apparent and absolute magnitudes of SNe. After weighing each observed
SN according to its peak apparent and absolute magnitudes, we compute the
volumetric rate (see Eq. 6).

After considering the effect of various limiting cuts (see Section 4.1), we
use a sample of 404 SNe Ia at a median redshift of $z_{\mathrm{med}}=0.024$
for our standard results. The choice of $|b_{\mathrm{lim}}|=15\degr$ and
$m_{\mathrm{\textit{V},lim}}=17\,\mathrm{mag}$ provides a balance between
statistical and systematic uncertainties for our sample. The standard sample
yields a total volumetric rate of $R_{\mathrm{tot}}=2.28^{+0.20}_{-0.20}\times
10^{4}\,\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,h^{3}_{70}$. This rate is in
agreement with rates from other studies at low redshifts (Cappellaro et al.,
1999; Dilday et al., 2010; Li et al., 2011b; Frohmaier et al., 2019; Perley et
al., 2020; Srivastav et al., 2022), but has smaller uncertainties.

We construct the observed LF and compare it with LFs from Li et al. (2011a),
Perley et al. (2020), and Sharon & Kushnir (2022). We use a completeness
correction as a function of both apparent and absolute magnitudes, rather than
an average completeness. Assuming that our LF and that from (Perley et al.,
2020) are both correct, we estimate a mean host extinction of $E(V-r)\approx
0.2\,\mathrm{mag}$ based on the magnitude shift between the LFs in the
different filters ($V$ and $r$). We also compute, for the first time, LFs for
the major subtypes of SNe Ia, finding that the less luminous SNe Ia-91bg are
more numerous than the more luminous SNe Ia-91T (see Section 4.2). Finally, we
provide the LFs corrected for several values of average host-galaxy extinction
(see Section 4.3).

In the upcoming papers, we plan on updating these SNe Ia rates using the newer
$g$-band ASAS-SN data from Neumann et al. (2023) spanning SNe discovered from
2018 to 2020. The ASAS-SN $g$-band observations are sensitive to objects
$\sim$1 mag fainter than the $V$-band data and will roughly double the sample
size. This will allow improving the estimates for the rates and LFs. Such a
large sample will also allow us to obtain the rates as a function of other
parameters such as galaxy type, local SN environments, or separation of an SN
from its host. We will employ a more careful uniform spectroscopic
classification to classify SNe Ia into appropriate subtypes and improve the
subtype rates. We will also be able to calculate rates for other populations,
in particular, the core-collapse (CC) SNe and the tidal disruption events
(TDEs). This paper is the first in a series that will measure rates and LFs
for SNe Ia, CC SNe, TDEs, and other transients from ASAS-SN.

## Acknowledgements

We thank Federica Chiti, Jason Hinkle, Willem Hoogendam, and Michael Tucker
for helpful discussions.

We thank Las Cumbres Observatory and its staff for their continued support of
ASAS-SN. ASAS-SN is funded in part by the Gordon and Betty Moore Foundation
through grants GBMF5490 and GBMF10501 to the Ohio State University, and also
funded in part by the Alfred P. Sloan Foundation grant G-2021-14192.
Development of ASAS-SN has been supported by NSF grant AST-0908816, the Mt.
Cuba Astronomical Foundation, the Center for Cosmology and AstroParticle
Physics at the Ohio State University, the Chinese Academy of Sciences South
America Center for Astronomy (CAS-SACA), and the Villum Foundation.

We wish to extend our special thanks to those of Hawaiian ancestry on whose
sacred mountains of Maunakea and Haleakalā, we are privileged to be guests.
Without their generous hospitality, the observations presented herein would
not have been possible.

Support for TJ was provided by NASA through the NASA Hubble Fellowship grant
HF2-51509 awarded by the Space Telescope Science Institute, which is operated
by the Association of Universities for Research in Astronomy, Inc., for NASA,
under contract NAS5-26555. KZS is supported by the 2022 Guggenheim Fellowship,
and his stay at the UCSB KITP was supported in part by the National Science
Foundation under NSF Grant No. PHY-1748958. JFB was supported by NSF Grant No.
PHY-2012955. SD acknowledges support by the National Natural Science
Foundation of China (Grant No. 12133005).

## Data Availability

The data used in this paper are available in a machine-readable format from
the online journal as supplementary material. A portion is shown in Table 1
for guidance regarding its form and content.

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##  Appendix A Rates and Luminosity Functions for Non-Zero Host-Galaxy
Extinction

Table 5: Luminosity Functions and Total Rates for a Mean Host-Galaxy
Extinction of $A_{V}=0.25\,\mathrm{mag}$.

$M_{\mathrm{\textit{V},peak}}$ | $R_{\mathrm{all}}$ | $R_{\mathrm{Ia-norm}}$ | $R_{\mathrm{Ia-91T}}$ | $R_{\mathrm{Ia-91bg}}$ | $R_{\mathrm{Ia-CSM}}$ | $R_{\mathrm{Ia-03fg}}$ | $R_{\mathrm{Ia-02es}}$ | $R_{\mathrm{Ia-00cx}}$
---|---|---|---|---|---|---|---|---
[mag] | $[\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,\mathrm{mag}^{-1}\,h^{3}_{70}]$
$[-16.5,-17.0]$ | $1.2^{+0.7}_{-0.8}\times 10^{4}$ | $1.2^{+0.7}_{-0.8}\times 10^{4}$ | — | — | — | — | — | —
$[-17.0,-17.5]$ | $4^{+3}_{-2}\times 10^{3}$ | $4^{+3}_{-2}\times 10^{3}$ | — | — | — | — | — | —
$[-17.5,-18.0]$ | $7^{+2}_{-2}\times 10^{3}$ | $5.1^{+1.9}_{-1.9}\times 10^{3}$ | — | $1.5^{+1.2}_{-0.8}\times 10^{3}$ | — | — | — | —
$[-18.0,-18.5]$ | $8.8^{+2.1}_{-1.8}\times 10^{3}$ | $6.6^{+1.6}_{-1.5}\times 10^{3}$ | — | $1.9^{+1.0}_{-0.9}\times 10^{3}$ | — | — | $2.8^{+3.2}_{-1.9}\times 10^{2}$ | —
$[-18.5,-19.0]$ | $1.28^{+0.16}_{-0.16}\times 10^{4}$ | $1.22^{+0.16}_{-0.16}\times 10^{4}$ | — | $6^{+3}_{-3}\times 10^{2}$ | — | — | — | —
$[-19.0,-19.5]$ | $2.12^{+0.15}_{-0.15}\times 10^{4}$ | $1.95^{+0.15}_{-0.14}\times 10^{4}$ | $1.7^{+0.4}_{-0.4}\times 10^{3}$ | — | — | — | — | —
$[-19.5,-20.0]$ | $4.8^{+0.5}_{-0.5}\times 10^{3}$ | $4.2^{+0.5}_{-0.5}\times 10^{3}$ | $5.2^{+1.7}_{-1.5}\times 10^{2}$ | — | — | $60^{+70}_{-40}$ | — | $40^{+40}_{-30}$
$[-20.0,-20.5]$ | $2.4^{+0.8}_{-0.8}\times 10^{2}$ | $1.8^{+0.7}_{-0.7}\times 10^{3}$ | $60^{+30}_{-30}$ | — | — | — | — | —
$[-20.5,-21.0]$ | $25^{+19}_{-15}$ | — | — | — | $10^{+11}_{-6}$ | $15^{+17}_{-10}$ | — | —
$[-21.0,-21.5]$ | $16^{+10}_{-10}$ | — | — | — | $16^{+10}_{-10}$ | — | — | —
| Total Rates $[\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,h^{3}_{70}]$
$[-16.5,-21.5]$ | $3.5^{+0.4}_{-0.4}\times 10^{4}$ | $3.0^{+0.4}_{-0.4}\times 10^{4}$ | $1.1^{+0.2}_{-0.2}\times 10^{3}$ | $2.0^{+0.7}_{-0.7}\times 10^{3}$ | $13^{+8}_{-8}$ | $40^{+30}_{-30}$ | $1.4^{+1.6}_{-1.0}\times 10^{2}$ | $20^{+20}_{-15}$
$[-17.0,-21.5]$ | $2.9^{+0.2}_{-0.2}\times 10^{4}$ | $2.6^{+0.2}_{-0.2}\times 10^{4}$ | $1.1^{+0.2}_{-0.2}\times 10^{3}$ | $2.0^{+0.7}_{-0.7}\times 10^{3}$ | $13^{+8}_{-8}$ | $40^{+30}_{-30}$ | $1.4^{+1.6}_{-1.0}\times 10^{2}$ | $20^{+20}_{-15}$
$[-17.5,-21.5]$ | $2.72^{+0.19}_{-0.18}\times 10^{4}$ | $2.39^{+0.17}_{-0.17}\times 10^{4}$ | $1.1^{+0.2}_{-0.2}\times 10^{3}$ | $2.0^{+0.7}_{-0.7}\times 10^{3}$ | $13^{+8}_{-8}$ | $40^{+30}_{-30}$ | $1.4^{+1.6}_{-1.0}\times 10^{2}$ | $20^{+20}_{-15}$

Note: The column of $R_{\mathrm{Ia-norm}}$ includes normal SNe Ia as well as
SNe not classified into a specific subtype.

Table 6: Luminosity Functions and Total Rates for a Mean Host-Galaxy
Extinction of $A_{V}=0.5\,\mathrm{mag}$.

$M_{\mathrm{\textit{V},peak}}$ | $R_{\mathrm{all}}$ | $R_{\mathrm{Ia-norm}}$ | $R_{\mathrm{Ia-91T}}$ | $R_{\mathrm{Ia-91bg}}$ | $R_{\mathrm{Ia-CSM}}$ | $R_{\mathrm{Ia-03fg}}$ | $R_{\mathrm{Ia-02es}}$ | $R_{\mathrm{Ia-00cx}}$
---|---|---|---|---|---|---|---|---
[mag] | $[\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,\mathrm{mag}^{-1}\,h^{3}_{70}]$
$[-16.5,-17.0]$ | $3.7^{+0.3}_{-0.3}\times 10^{4}$ | $3.7^{+0.3}_{-0.3}\times 10^{4}$ | — | — | — | — | — | —
$[-17.0,-17.5]$ | $9^{+6}_{-5}\times 10^{3}$ | $9^{+6}_{-5}\times 10^{3}$ | — | — | — | — | — | —
$[-17.5,-18.0]$ | $7^{+3}_{-3}\times 10^{3}$ | $7^{+3}_{-3}\times 10^{3}$ | — | — | — | — | — | —
$[-18.0,-18.5]$ | $9^{+3}_{-3}\times 10^{3}$ | $4.5^{+2.1}_{-1.7}\times 10^{3}$ | — | $4.0^{+2.0}_{-1.9}\times 10^{3}$ | — | — | — | —
$[-18.5,-19.0]$ | $1.5^{+0.3}_{-0.2}\times 10^{4}$ | $1.4^{+0.3}_{-0.3}\times 10^{4}$ | — | $9^{+6}_{-6}\times 10^{2}$ | — | — | $4^{+4}_{-3}\times 10^{2}$ | —
$[-19.0,-19.5]$ | $2.9^{+0.3}_{-0.2}\times 10^{4}$ | $2.7^{+0.2}_{-0.2}\times 10^{4}$ | $8.2^{+0.4}_{-0.3}\times 10^{2}$ | $5^{+2}_{-2}\times 10^{2}$ | — | — | — | —
$[-19.5,-20.0]$ | $1.62^{+0.13}_{-0.13}\times 10^{4}$ | $1.41^{+0.12}_{-0.11}\times 10^{4}$ | $2.0^{+0.5}_{-0.4}\times 10^{3}$ | — | — | $80^{+90}_{-60}$ | — | —
$[-20.0,-20.5]$ | $1.9^{+0.4}_{-0.3}\times 10^{3}$ | $1.6^{+0.3}_{-0.3}\times 10^{3}$ | $2.2^{+1.1}_{-1.0}\times 10^{2}$ | — | — | — | — | $60^{+60}_{-40}$
$[-20.5,-21.0]$ | $1.1^{+0.6}_{-0.5}\times 10^{2}$ | $50^{+30}_{-30}$ | $40^{+40}_{-30}$ | — | — | $21^{+24}_{-14}$ | — | —
$[-21.0,-21.5]$ | $27^{+14}_{-13}$ | — | — | — | $27^{+14}_{-13}$ | — | — | —
$[-21.5,-22.0]$ | $8^{+9}_{-5}$ | — | — | — | $8^{+9}_{-5}$ | — | — | —
| Total Rates $[\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,h^{3}_{70}]$
$[-16.5,-22.0]$ | $6.2^{+1.6}_{-1.6}\times 10^{4}$ | $5.3^{+1.5}_{-1.5}\times 10^{4}$ | $1.5^{+0.3}_{-0.3}\times 10^{3}$ | $2.7^{+1.0}_{-1.0}\times 10^{3}$ | $18^{+11}_{-11}$ | $50^{+40}_{-40}$ | $2.0^{+2.0}_{-1.5}\times 10^{2}$ | $30^{+30}_{-20}$
$[-17.0,-22.0]$ | $4.3^{+0.4}_{-0.4}\times 10^{4}$ | $3.8^{+0.4}_{-0.4}\times 10^{4}$ | $1.5^{+0.3}_{-0.3}\times 10^{3}$ | $2.7^{+1.0}_{-1.0}\times 10^{3}$ | $18^{+11}_{-11}$ | $50^{+40}_{-40}$ | $2.0^{+2.0}_{-1.5}\times 10^{2}$ | $30^{+30}_{-20}$
$[-17.5,-22.0]$ | $3.9^{+0.3}_{-0.3}\times 10^{4}$ | $3.4^{+0.3}_{-0.3}\times 10^{4}$ | $1.5^{+0.3}_{-0.3}\times 10^{3}$ | $2.7^{+1.0}_{-1.0}\times 10^{3}$ | $18^{+11}_{-11}$ | $50^{+40}_{-40}$ | $2.0^{+2.0}_{-1.5}\times 10^{2}$ | $30^{+30}_{-20}$

Note: The column of $R_{\mathrm{Ia-norm}}$ includes normal SNe Ia as well as
SNe not classified into a specific subtype.

Table 7: Luminosity Functions and Total Rates for a Mean Host-Galaxy
Extinction of $A_{V}=1.0\,\mathrm{mag}$.

$M_{\mathrm{\textit{V},peak}}$ | $R_{\mathrm{all}}$ | $R_{\mathrm{Ia-norm}}$ | $R_{\mathrm{Ia-91T}}$ | $R_{\mathrm{Ia-91bg}}$ | $R_{\mathrm{Ia-CSM}}$ | $R_{\mathrm{Ia-03fg}}$ | $R_{\mathrm{Ia-02es}}$ | $R_{\mathrm{Ia-00cx}}$
---|---|---|---|---|---|---|---|---
[mag] | $[\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,\mathrm{mag}^{-1}\,h^{3}_{70}]$
$[-17.5,-18.0]$ | $3^{+5}_{-3}\times 10^{4}$ | $3^{+5}_{-3}\times 10^{4}$ | — | — | — | — | — | —
$[-18.0,-18.5]$ | $1.7^{+0.9}_{-0.7}\times 10^{4}$ | $1.7^{+0.9}_{-0.7}\times 10^{4}$ | — | — | — | — | — | —
$[-18.5,-19.0]$ | $1.6^{+0.5}_{-0.5}\times 10^{4}$ | $8.7^{+0.4}_{-0.4}\times 10^{3}$ | — | $8^{+4}_{-3}\times 10^{3}$ | — | — | — | —
$[-19.0,-19.5]$ | $2.7^{+0.5}_{-0.5}\times 10^{4}$ | $2.5^{+0.5}_{-0.4}\times 10^{4}$ | — | $1.7^{+1.2}_{-1.1}\times 10^{3}$ | — | — | $7^{+8}_{-5}\times 10^{2}$ | —
$[-19.5,-20.0]$ | $5.3^{+0.5}_{-0.4}\times 10^{4}$ | $5.1^{+0.4}_{-0.4}\times 10^{4}$ | $1.6^{+0.7}_{-0.6}\times 10^{3}$ | $8^{+4}_{-4}\times 10^{2}$ | — | — | — | —
$[-20.0,-20.5]$ | $3.2^{+0.3}_{-0.3}\times 10^{4}$ | $2.8^{+0.3}_{-0.2}\times 10^{4}$ | $4.0^{+0.9}_{-0.9}\times 10^{3}$ | — | — | $1.6^{+1.8}_{-1.1}\times 10^{2}$ | — | —
$[-20.5,-21.0]$ | $3.7^{+0.6}_{-0.6}\times 10^{3}$ | $3.1^{+0.6}_{-0.6}\times 10^{3}$ | $4.2^{+0.2}_{-0.2}\times 10^{2}$ | — | — | — | — | $1.1^{+1.2}_{-0.7}\times 10^{2}$
$[-21.0,-21.5]$ | $2.2^{+1.2}_{-1.0}\times 10^{2}$ | $1.0^{+0.6}_{-0.5}\times 10^{2}$ | $70^{+80}_{-50}$ | — | — | $40^{+50}_{-30}$ | — | —
$[-21.5,-22.0]$ | $50^{+30}_{-30}$ | — | — | — | $50^{+30}_{-30}$ | — | — | —
| Total Rates $[\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,h^{3}_{70}]$
$[-17.5,-22.0]$ | $8.6^{+2.8}_{-2.7}\times 10^{4}$ | $7.9^{+2.8}_{-2.6}\times 10^{4}$ | $3.0^{+0.6}_{-0.6}\times 10^{3}$ | $5.1^{+1.9}_{-1.9}\times 10^{3}$ | $26^{+13}_{-13}$ | $1.0^{+0.8}_{-0.8}\times 10^{2}$ | $4^{+4}_{-3}\times 10^{2}$ | $50^{+60}_{-30}$

Note: The column of $R_{\mathrm{Ia-norm}}$ includes normal SNe Ia as well as
SNe not classified into a specific subtype.

Table 8: Luminosity Functions and Total Rates for a Mean Host-Galaxy
Extinction of $A_{V}=1.5\,\mathrm{mag}$.

$M_{\mathrm{\textit{V},peak}}$ | $R_{\mathrm{all}}$ | $R_{\mathrm{Ia-norm}}$ | $R_{\mathrm{Ia-91T}}$ | $R_{\mathrm{Ia-91bg}}$ | $R_{\mathrm{Ia-CSM}}$ | $R_{\mathrm{Ia-03fg}}$ | $R_{\mathrm{Ia-02es}}$ | $R_{\mathrm{Ia-00cx}}$
---|---|---|---|---|---|---|---|---
[mag] | $[\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,\mathrm{mag}^{-1}\,h^{3}_{70}]$
$[-18.5,-19.0]$ | $1.5^{+2.4}_{-1.0}\times 10^{4}$ | $1.5^{+2.4}_{-1.0}\times 10^{4}$ | — | — | — | — | — | —
$[-19.0,-19.5]$ | $3.6^{+1.2}_{-1.1}\times 10^{4}$ | $1.9^{+0.9}_{-0.7}\times 10^{4}$ | — | $1.7^{+0.8}_{-0.8}\times 10^{4}$ | — | — | — | —
$[-19.5,-20.0]$ | $5.1^{+0.9}_{-0.9}\times 10^{4}$ | $4.6^{+0.9}_{-0.7}\times 10^{4}$ | — | $3^{+2}_{-2}\times 10^{3}$ | — | — | $1.3^{+1.5}_{-0.9}\times 10^{3}$ | —
$[-20.0,-20.5]$ | $1.06^{+0.09}_{-0.08}\times 10^{5}$ | $1.01^{+0.09}_{-0.08}\times 10^{5}$ | $3.1^{+1.4}_{-1.2}\times 10^{3}$ | $1.7^{+0.9}_{-0.8}\times 10^{3}$ | — | — | — | —
$[-20.5,-21.0]$ | $6.3^{+0.5}_{-0.5}\times 10^{4}$ | $5.5^{+0.5}_{-0.4}\times 10^{4}$ | $7.7^{+1.7}_{-1.7}\times 10^{3}$ | — | — | $3^{+4}_{-2}\times 10^{2}$ | — | —
$[-21.0,-21.5]$ | $7.2^{+1.3}_{-1.3}\times 10^{3}$ | $6.2^{+1.3}_{-1.1}\times 10^{3}$ | $8^{+4}_{-4}\times 10^{2}$ | — | — | — | — | $2.1^{+2.4}_{-1.4}\times 10^{2}$
$[-21.5,-22.0]$ | $4.3^{+2.1}_{-1.9}\times 10^{2}$ | $2.0^{+1.0}_{-1.0}\times 10^{2}$ | $1.5^{+1.7}_{-1.0}\times 10^{2}$ | — | — | $80^{+90}_{-50}$ | — | —
| Total Rates $[\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,h^{3}_{70}]$
$[-17.5,-22.0]$ | $1.33^{+0.15}_{-0.15}\times 10^{5}$ | $1.16^{+0.15}_{-0.14}\times 10^{5}$ | $5.9^{+1.1}_{-1.2}\times 10^{3}$ | $1.1^{+0.4}_{-0.4}\times 10^{4}$ | — | $2.0^{+1.6}_{-1.6}\times 10^{2}$ | $7^{+7}_{-5}\times 10^{2}$ | $1.1^{+1.1}_{-0.8}\times 10^{2}$

Note: The column of $R_{\mathrm{Ia-norm}}$ includes normal SNe Ia as well as
SNe not classified into a specific subtype.