REVEALING TYPE Ia SUPERNOVA PHYSICS WITH COSMIC RATES AND NUCLEAR GAMMA RAYS

The SN Ia rate at a given time corresponds to the stellar birthrate at earlier epochs, convolved with the DTD, normalized by the efficiency of forming SN Ia progenitor systems. The normalization is

$$\begin{equation} \eta = \int {A}_{\rm Ia}(M) \xi (M)\,dM, \end{equation} \tag{ 1 }$$

where A_Ia(M) is the SN Ia formation efficiency, ξ(M) is the initial mass function (IMF), and the integration is performed over the mass range for SN Ia progenitors. Unless otherwise stated, we present all results for the Salpeter IMF; our results on the SN Ia rate do not strongly depend on this choice because the SN Ia progenitor mass range is sufficiently high. Assuming A_Ia and ξ do not vary with space and time, η is a constant representing the number of SNe Ia produced per M_☉ of stars formed so that

$$\begin{equation} R_{\rm Ia} \left[ z(t_c) \right] = \eta \int _{t_{10}}^{t_c} \Phi (t_c-t^{\prime }_c) \dot{\rho }_\star [z(t^{\prime }_c)] \, dt^{\prime }_c, \end{equation} \tag{ 2 }$$

where Φ(t) is the DTD, its argument is the time difference between progenitor star formation and SN Ia explosion, $\dot{\rho }_{\star }(t_c)$ is the star formation rate, and the integration is performed over cosmic time from t₁₀, the age of the universe when the first stars were being formed, z ≈ 10. Φ(t) is normalized such that its integral over the entire delay time range is unity. We do not consider the effect of metallicity on the rates (e.g., Prantzos & Boissier 2003; Prieto et al. 2008; Boissier & Prantzos 2009; Meng et al. 2009).

In the limit that Φ(t) is large only for small t, we recover a prompt population of SNe, i.e., similar to core-collapse SNe, for which the cosmic rate evolution is the same as for the star formation rate. For SNe Ia, the observation that they occur in both young star-forming galaxies and old elliptical galaxies with little star formation demonstrates the need for a wide range of delay times. Recently, two populations of SNe Ia, a "prompt" population with delay times ∼0.1 Gyr and a "delayed" population with delay times ∼10 Gyr, have been reported (Mannucci et al. 2005; Scannapieco & Bildsten 2005; Mannucci et al. 2006; Sullivan et al. 2006; Maoz et al. 2010a). Whether this represents a truly bimodal distribution, or whether it is the result of two coarsely sampled bins of what is in reality a continuous DTD, remains uncertain.

We adopt a power-law DTD, Φ(t) ∝ t^−α. Power laws have been reported from SN Ia observations: for example, with α = 0.5 ± 0.2 from the SNLS data (Pritchet et al. 2008), with α = 1.08^+0.15_−0.11 from observations of transients in passively evolving galaxies (Totani et al. 2008), with α ∼ 1 from local SNe Ia in the Lick Observatory Supernova Search (LOSS; Maoz et al. 2010a), and most recently α = 1.2 ±  0.3 from cluster SNe Ia (Maoz et al. 2010b). In fact, the number of WDs following an instantaneous star formation burst is also a simple power law in t: for an IMF ξ(M) ∝ M^x (x = −2.35 for Salpeter) and a power-law approximation to the evolutionary timescale of 3–8 M_☉ stars t_age ∝ M^y (y ≈ −2.6), the index is α = (x − y + 1)/y ≈ 0.5. However, this distribution extends only over the lifetimes of progenitors, 30–400 Myr, which is much smaller than the observed range of delay times. The extra time needed for mass transfer before triggering an SN Ia would shift the DTD toward longer delays, but the final DTD is largely dependent on the properties of the binary and its evolution.

We apply our DTD over times 0.1–13 Gyr and adopt α as a parameter. The lower time limit could be as short as the lifetime of an 8 M_☉ star, ∼0.03 Gyr, but recent studies of the spatial distribution of SNe Ia in their host galaxies indicate a lower limit of 0.2 Gyr (Raskin et al. 2009a). We adopt 0.1 Gyr; at the precision of the current SN Ia rate measurements, our results are not very sensitive to small delays. We fix our upper limit to the look-back time to z = 10, which is sufficiently long to accommodate SNe Ia observed in the oldest elliptical galaxies. Allowing a larger value would not change the redshift evolution, and would simply scale the efficiency.

The cosmic star formation rate density is shown in Figure 1. The compilation of Hopkins & Beacom (2006), which includes data from multiple star formation indicators, is shown together with recent data from mid-infrared-emitting galaxies (Rujopakarn et al. 2010), Lyman break galaxies (LBG; Verma et al. 2007; Bouwens et al. 2007, 2008; Reddy & Steidel 2009), and gamma-ray burst (GRB) host galaxies (Yüksel et al. 2008; Kistler et al. 2009; Wanderman & Piran 2010; Butler et al. 2010). Note that Bouwens et al. (2007, 2008) report results integrated down to a chosen luminosity limit; we show instead complete integrations (see Kistler et al. 2009 for details). We also show Hubble Ultra Deep field (UDF) measurements from Yan et al. (2010), similarly integrated to low galaxy luminosities. New UDF results from Bouwens et al. (2010) have just appeared. As these indicate a steeper rise at low galaxy luminosities, the integration for the full star formation rate would need to be carefully regulated, and we do not perform this here. The star formation rate has been checked by independently measured quantities, such as the extragalactic background light (Horiuchi et al. 2009), stellar mass density (Wilkins et al. 2008), and upper limits on the diffuse SN neutrino background (Beacom 2010).

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On the top axis we label cosmic time. It highlights that delay times of Gyr will result in noticeable differences in the slope between the star formation and SN Ia rates. The star formation rate rapidly increases by 1 order of magnitude between redshift 0 and 1, which has been very well measured by a variety of indicators. This serves as an important feature to be compared to the SN Ia rate. At higher redshift, the star formation rate eventually declines, at first slowly after z = 1 and then more rapidly after z ∼ 4. The exact slope above z ∼ 4 remains uncertain, although this does not affect our conclusions, as we discuss in Section 2.3.

Since the first measurement of SNe Ia at cosmological distances (Pain et al. 1996), many surveys have collected homogenous samples of CCD-discovered SNe Ia. These surveys periodically observe the same patch of sky, or the same sample of galaxies, to find transients. Cuts are applied to select the most confidently identified SNe, and corrections for dust and incompleteness are applied. We summarize the present SN Ia rate measurements in Table 3 in the Appendix.

The high-redshift (z ≳ 0.5) rates are the hardest to measure. Dust corrections are non-trivial and uncertain, e.g., the dust corrected data of Dahlen et al. (2004, 2008) are a factor ∼2 higher than their uncorrected data. Even this has been argued to be insufficient and the rates could be treated as lower limits (Greggio et al. 2008). On the other hand, SN Ia surveys are likely contaminated by core-collapse SNe, for which the rate increases from ≈10⁻⁴ yr^-1 Mpc⁻³ at z = 0 to ≈10⁻³ yr^-1 Mpc⁻³ by z = 1, i.e., ∼10 times the SN Ia rate by z = 1. It is therefore important that only the most confidently identified SNe Ia—those that are spectroscopically confirmed—are selected for rate measurements. Indeed, rates derived using photometric classification (Barris & Tonry 2006; Poznanski et al. 2007; Kuznetsova et al. 2008) are generally higher than those based on a higher fraction of spectroscopically confirmed SN Ia (Dahlen et al. 2004, 2008). This is usually reflected in the large systematic errors. Finally, the sample sizes rapidly become small at high redshift, where typically N_Ia= 2–3. These issues make it difficult to use the high-redshift rate measurements (Oda et al. 2008), even though they would be very sensitive to delays due to the small cosmic time difference since the first stars were being formed.

At intermediate redshift (0.1 ≲ z ≲ 0.5), many of the measurements are based solely on spectroscopically confirmed SNe Ia (Hardin et al. 2000; Pain et al. 2002; Tonry et al. 2003; Madgwick et al. 2003; Blanc et al. 2004; Neill et al. 2006; Dilday et al. 2008). Multi-band photometric identification is also now reliable: Dilday et al. (2010) show that in the SDSS-II Supernova Survey only ∼2% of the photometric SNe Ia in z < 0.3 may be misidentified. Furthermore, the effects of dust become less important than at high redshifts (Botticella et al. 2008). However, many surveys target a pre-selected sample of galaxies and, while a large sample is adopted, there would be a bias toward bright galaxies, and SNe that explode in faint galaxies would be missed. Lastly, we note that sample sizes vary largely from survey to survey.

Finally, in the local (D ≲ 100 Mpc) volume, the rate has been calculated from compilations of carefully selected SNe Ia (Cappellaro et al. 1999; Smartt et al. 2009), as well as the local LOSS measurement (Leaman et al. 2010; Li et al. 2010a, 2010b). Note that the rate calculated from Smartt et al. (2009) probably reflects a local enhancement (Section 3.3).

Compared to the star formation rate, the SN Ia rate is further away from a consensus. Although the number of rate measurements has increased, the scatter is large, and at times, measurements in comparable redshifts are in direct disagreement. Bearing in mind the strengths and weaknesses of the data, we group the rate measurements into two categories:

• 1.  
  Filled points: those based on dedicated surveys using only spectroscopically identified SNe Ia. We also include select surveys with less than 100% spectroscopic identification, including the survey by SDSS which includes an order of magnitude more SNe Ia than any other survey (Dilday et al. 2010), and measurements by Dahlen et al. (2004, 2008) which are the most reliable measurements in the high-redshift regime.
• 2.  
  Empty points: those based largely on photometrically identified SNe Ia.

We caution that even within each category, the sample size, observation schedule, limiting magnitude, and other conditions vary, and our description is simply an attempt to appreciate some of the important differences between measurements (Blanc & Greggio 2008).

Now, we can make better sense of the calculated and measured SN Ia rates despite their uncertainties. When we restrict ourselves to the filled points, we find that the SN Ia rate increases by only a factor of ∼2–3 from redshift 0 to 0.8 or so. On the other hand, the cosmic star formation rate increases by a factor of 10 from redshift 0 to 1. This implies that a large fraction of SNe Ia have cosmologically large time delays.

We show in Figures 2 and 3 the calculated and measured SN Ia rates. We fit the calculated cosmic SN Ia rate form to the selection of rate data consisting of filled points and local measurements (excluding that by Smartt et al. 2009). For the power-law DTD we fit for the normalization at redshift zero, R_Ia(0), and the DTD exponent, α. We find best-fit values (R_Ia(0), α) = (0.24, 1.0) with χ² = 5.4 for 17 degrees of freedom. The projections of the Δχ² = 2.30 elliptical contour give R_Ia(0) = 0.24 ± 0.04 and α = 1.0 ±  0.3; these parameters are strongly anti-correlated. The uncertainty is shown as gray shading in the figures. The best-fit efficiency and uncertainty of making SNe Ia is (5 ±  1) ×  10⁻⁴ M⁻¹_☉, or, assuming an SN Ia progenitor mass range of 3–8 M_☉, an SN Ia fraction (which we define as the number of SNe Ia divided by the number of 3–8 M_☉ stars formed; the fraction of stars in SN Ia-producing binaries is twice this) of 2.4% ± 0.5%.

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In addition to the power-law DTD we show the bimodal DTD of Mannucci et al. (2006; dot-dashed), the 3.4 Gyr narrow DTD of Strolger et al. (2004), Dahlen et al. (2008), and Strolger et al. (2010; dot-dot-dashed), and the no-delay case (dotted) for comparison. These DTDs do not fit the data as well, with χ² values of 10, 17, and 21, respectively. All rise too fast compared to the reliable data (filled points). This also means they underpredict the z = 0 rate. The different results in Strolger et al. (2004), Dahlen et al. (2008), and Strolger et al. (2010) are due to the choice of data. The narrow DTD is driven by a declining SN Ia rate at z ∼ 1.4 (see green left-pointing triangles in Figure 3), while our result is driven by the slow rise at low redshift.

At present, the DTD does not clearly identify the SN Ia progenitor scenario, because both the SD and DD scenarios can lead to power-law DTDs with α ≈ 1. In the DD scenario, the delay is approximately dictated by the time taken by a binary to merge by angular momentum loss, which from general relativity scales as the fourth power of the binary separation. For a logarithmically flat distribution of binary separations, one obtains α ∼ 1. Although binary evolution synthesis studies of the SD scenario typically show that the DTD peaks at characteristic scales related to stable mass transfer (e.g., Yungelson & Livio 2000; Belczynski et al. 2005; Meng et al. 2009; Meng & Yang 2010), power-law DTDs have also been predicted: for example, Hachisu et al. (2008) report power-law DTDs with α ≈ 1. Both progenitor scenarios are acceptable fits, but as predictions of SN Ia progenitor scenarios improve and SN Ia rate data accumulate, better testing would become possible. We have assumed for simplicity a power-law DTD; in order to generally test the progenitor scenario, a generic form for the DTD should be statistically tested (e.g., Strolger et al. 2010).

The choice of the IMF does not affect the shape of the SN Ia rate, but changes the number of SN Ia progenitor stars. This introduces a small tens of percent difference in the efficiency. For a modern Baldry–Glazebrook IMF with a low-mass suppression (Baldry & Glazebrook 2003), the SN Ia fraction required is 2.9% ± 0.6%. Since the comparison of cosmic rates is not sensitive to small delays, there could be more short-lived progenitors than indicated by our simple DTD form. Uncertainty in the cosmic star formation rate beyond z ∼ 4 affects the low-redshift SN Ia slope, albeit weakly, through the long tail of the DTD. There is a degeneracy in the high-redshift star formation slope and the value of α, where a higher star formation rate can be compensated by a larger α. However, at the precision of the current SN Ia rate measurements, this uncertainty does not affect our conclusions.

Our SN Ia efficiency is comparable to those from recent studies of volumetric SN Ia rates (Blanc & Greggio 2008), and somewhat lower than those derived by other methods (Greggio 2010; Maoz & Badenes 2010; Maoz et al. 2010a). Note that comparisons must be done with a common assumption for the IMF. As described above, a more modern IMF would increase our efficiency, although a difference remains. The true value of the efficiency remains to be determined; see Maoz (2008) for a detailed discussion and implications of the SN Ia efficiency. For example, our SN Ia fraction contrasts with the ∼0.2% advocated in some DD scenarios, where conservative conditions on the mass ratios are applied (Ruiter et al. 2009). The higher required efficiency that we found suggests the need to relax some assumptions, or to include contributions from SNe Ia of lower mass progenitors. We should note here that our definition for efficiency implicitly assumes binaries that would become an SN Ia within ≈13 Gyr.

It is revealing to discuss the SN Ia rate jointly with the core-collapse SN rate. The ratio of SN Ia to core-collapse SN(Ia/II) at z = 0 is ≈0.2–0.3. Here, we use the z = 0 cosmic SN Ia rate instead of the local 10 Mpc data, as the latter rate is ∼0.1 yr⁻¹; the recent decade, for example, had no SNe Ia (Kistler et al. 2008). Since SNe Ia are delayed, the SN Ia rate should increase less strongly than the star formation rate and the core-collapse rate from z = 0 to z = 1. The Ia/II ratio should therefore decrease with redshift in this range. However, the data do not clearly show such a trend, being consistent with no evolution (Horiuchi et al. 2009; this is true even with updates on the core-collapse rate, e.g., Bazin et al. 2009). A likely possibility is that fainter core-collapse SNe are being missed compared to the brighter SNe Ia (see Horiuchi et al. 2009). The importance of the missing core-collapse SNe spans topics such as the formation of black holes in massive stellar collapse, SN neutrinos, and metal enrichment.

In all SN Ia models, the decay chain ⁵⁶Ni → ⁵⁶Co → ⁵⁶Fe provides the primary source of energy that powers the SN Ia optical display. The ⁵⁶Ni decays by electron capture and the daughter ⁵⁶Co emits gamma rays by the nuclear de-excitation process:

$$\begin{eqnarray} ^{56}{\rm Ni} + e^- &\rightarrow & ^{56}{\rm Co}^* + \nu _e \\ \nonumber ^{56}{\rm Co}^* &\rightarrow & ^{56}{\rm Co} + \gamma \\ \nonumber E_\gamma &=& 158 \, {\rm keV} (99\%), \, 812 \, {\rm keV} (86\%), \end{eqnarray} \tag{ 3 }$$

where percentages express photons per decay and the sum can be larger than 100%. Only the dominant lines are noted here. The daughter ⁵⁶Co decays by electron capture (81%) as well as positron emission (19%), and the daughter ⁵⁶Fe de-excites by emitting gamma rays:

$$\begin{eqnarray} ^{56}{\rm Co} + e^- &\rightarrow & ^{56}{\rm Fe}^* + \nu _e \\ \nonumber ^{56}{\rm Co} &\rightarrow & ^{56}{\rm Fe}^* + e^+ + \nu _e \\ \nonumber ^{56}{\rm Fe}^* &\rightarrow & ^{56}{\rm Co} + \gamma \\ \nonumber E_\gamma &=& 847 \, {\rm keV} (100\%), \, 1238 \, {\rm keV} (67\%), \end{eqnarray} \tag{ 4 }$$

where the percentages include the effects of the 19% branching ratio for positron production by beta decay.

Initially, most of the gamma rays Compton scatter on the SN Ia ejecta and deposit their energy. As the ejecta expands and the matter density decreases, more gamma rays escape. The observed 847 keV line luminosity is

$$\begin{equation} S_\gamma (t) = p_{\rm esc} (t) \left[ \frac{M_{\rm Ni}N_A}{56} \right] \left[ \frac{e^{-t/\tau _{\rm Co}} - e^{-t/\tau _{\rm Ni}} }{\tau _{\rm Co} - \tau _{\rm Ni}} \right] \, {\rm \gamma \, s^{-1} }, \end{equation} \tag{ 5 }$$

where p_esc is the model-dependent probability of escape through the SN Ia ejecta, the quantity in the first square brackets defines the model-dependent nucleosynthesis yield, and the second square brackets reflect the nuclear decay rate. Here, M_Ni is the nickel mass, N_A is the Avogadro number, and τ_Ni = t_1/2/ln(2) where t_1/2 is the half-life: t_1/2 = 6.1 days for ⁵⁶Ni and 77.2 days for ⁵⁶Co. The distribution of M_Ni (over many SNe Ia) and especially the time evolution of p_esc (per SN Ia) can be used to probe SN Ia physics. In deflagration, nuclear burning ignites near the center and burning moves subsonically across the progenitor. In pure detonation, the flame front propagates supersonically as a shock front, but we do not consider this further since the resulting elemental abundances disagree with data (Arnett et al. 1971). In delayed detonation, an initial deflagration becomes a detonation at some critical density that is an unknown parameter. All these models are usually assumed to be initiated from a WD near the Chandrasekhar mass, accreting mass from a non-degenerate star. In He detonation, burning commences in the degenerate helium layer near the surface of a sub-Chandrasekhar mass WD. In DD scenario models, ignition occurs at low densities as mass from the disrupted binary accretes. It is expected that a large material envelope covers the burning sites, so that the escape probability for gamma rays is lower than in SD scenarios, thus probing the progenitors.

Milne et al. (2004) compared the gamma-ray emission from seven transport codes for a selection of SN Ia models. They conclude that differences due to transport codes are 10%–20%, much less than the differences that result from models. The continuum emission is more model dependent than the line emission since it depends on multiple scatterings and the time-integrated continuum differs by up to a factor of ∼2 between codes.

From Milne et al. (2004) we adopt a selection of SN Ia models representative of normal, superluminous, and subluminous SNe Ia, and spanning the deflagration, delayed-detonation, and He-detonation models. The average peak line emission, over a 10⁶ s period, are summarized in Table 1. The light curves for superluminous SNe Ia (solid) and normal SNe Ia (dashed) are shown in Figure 7. We comment on DD SN Ia models in Section 3.6. We adopt two time-integrated gamma-ray spectra: the deflagration model W7 of Nomoto et al. (1984), which yields 0.58 M_☉ of ⁵⁶Ni, and the delayed-detonation model 5p0z22.23 of Höflich et al. (2002), which yields 0.56M_☉ of ⁵⁶Ni. Both models are representative of normal SNe Ia, the most common kind.

The SN Ia contribution to the CGB depends on the cosmic SN Ia rate and the time-integrated gamma-ray number spectrum per SN Ia, f(E), as

$$\begin{equation} E^2 \frac{{d}N}{{d}E} =\frac{c}{4 \pi } \int ^{z_{\rm max}}_0 R_{\rm Ia}(z) \frac{E^{\prime 2} }{(1+z)} f(E^\prime) \left| \frac{{d}t}{{d}z} \right| {d}z, \end{equation} \tag{ 6 }$$

where E is the measured photon energy, and |dz/dt| = H₀(1 + z)[Ω_m(1 + z)³ + Ω_Λ]^1/2. The left-hand side is equivalent to νI_ν, and the redshift factor in Equation (6) comes from the energy scaling.

In Figure 4, we show the resulting SN Ia contribution to the CGB, for the deflagration model W7 (solid) and the delayed-detonation model 5p0z22.23 (dot-dashed). For W7, the shading shows the uncertainty due to the SN Ia rate. We see that the SN Ia contributions are at least a factor of ∼5 smaller than current CGB measurements, and up to ∼20 depending on the SN Ia rate and SN Ia model. Although the spectrum per SN Ia has line features, these are washed out due to redshift. Our results are comparable to those of previous studies. Although early studies showed large contributions from SNe Ia (Clayton & Silk 1969; Clayton & Ward 1975; The et al. 1993; Zdziarski 1996; Watanabe et al. 1999; Ruiz-Lapuente et al. 2001), later studies report contributions to be ∼10 (Strigari et al. 2005) and ≳10 (Ahn et al. 2005) less than the measured intensity.

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If the dominant fraction of the CGB could be subtracted by future detectors, the SN Ia contribution could be detected. The feasibility depends on the true nature of the currently measured CGB. Proposed sources include various populations of AGNs (e.g., Ajello et al. 2009), hot coronae of AGNs (Inoue et al. 2008), and exotic dark matter models (Ahn & Komatsu 2005; Cembranos et al. 2007; Lawson & Zhitnitsky 2008). The measured CGB may also be dominated by detector backgrounds. The true nature is unknown and remains an important quest for future experiments to elucidate. Angular-correlation techniques may help differentiate the various possibilities (Zhang & Beacom 2004).

Knowing the local SN Ia rate is a critical prerequisite for assessing the prospects for detecting gamma rays from individual SN Ia. Many studies have discussed the SN Ia rate, most notably in the 1980s by Gehrels et al. (1987) and in the 1990s by Timmes & Woosley (1997), which provided much needed guidance on gamma-ray detection prospects. Now we have the advantage of systematic SN surveys and greatly improved SN Ia statistics. In addition to SN catalogs, we use our cosmic SN Ia rate to arrive at a consistent picture for the local SN Ia rate.

More than 5000 SNe discovered up to the end of 2009 are listed in the Sternberg Astronomical Institute Supernova Catalog (SAI; Bartunov et al. 2007), the Asiago Supernova Catalog, and the catalog maintained by the Central Bureau for Astronomical Telegrams. In some cases, the catalogs disagree on details, but these discrepancies are increasingly rare for more recent SNe. In the presence of a disagreement, we chose the classification in SAI; only small quantitative, and no qualitative, differences appear if preference is given to the other catalogs. We have used the reported recession velocities in the SAI catalog for their distances with cross checks for the nearest SNe Ia with catalogs of galaxies (Karachentsev et al. 2004).

SNe Ia discovered over the most recent 40 yr (1970–2009) are shown in Figure 5 as a function of distance. We see that while rare, there have nonetheless been a number of SNe Ia within 10 Mpc, at approximately one per decade (unfortunately, none in the recent decade). It is clear that one needs to go only a small factor in distance to observe ≳10 SNe Ia per decade. At larger distance, we see that the commencement of dedicated SN Ia searches in the 1990s dramatically increased the number of SNe Ia discovered. Yet these are still underestimates, due to missing coverage at large distances. Furthermore, the 4π sky is not evenly sampled; the northern hemisphere is more closely observed than the south, resulting in an SN Ia discovery ratio of approximately 1.4:1 (2000–2009, within 100 Mpc).

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The comparison to the cosmic SN Ia rate is also revealing. The red lines show the extrapolated cosmic SN Ia rate, and the incompleteness of the catalog is apparent by ∼30 Mpc. With next-generation surveys such as the Palomar Transient Factory (Law et al. 2009), SN Ia measurements are becoming more complete. Many more SNe Ia are also being discovered pre-maximum, offering more targets for future gamma-ray detectors.

Around 20 Mpc, the catalog shows more SNe Ia than the cosmic extrapolation. The excess is as high as a factor of ∼3 for the 20–22 Mpc bin, although this is not statistically strong. Even considering the uncertainty in the cosmic SN Ia rate this excess persists. There is also some excess from the Virgo galaxy cluster, located at ∼18 Mpc in the northern hemisphere and containing in excess of 1000 galaxies. The SN Ia sample of Smartt et al. (2009), which contains SNe Ia within 28 Mpc, similarly shows a high SN Ia rate.

There are strong upper limits on the gamma-ray emission from individual SN Ia. Among the earliest was SN 1986G, observed by the Solar Maximum Mission (SMM) instrument (Matz & Share 1990), followed by SN 1991T (Lichti et al. 1994; Leising et al. 1995; Morris et al. 1997) and SN 1998bu (Leising et al. 1999), both observed by the Compton Gamma Ray Observatory (CGRO). The derived limits depend on the fact that SN 1986G was subluminous, SN 1991T was superluminous, and SN 1998bu was normal. Furthermore, distance uncertainties weaken the limits. In all cases, limits are compatible with theoretical predictions within uncertainties, though very close (Höflich et al. 1994).

On the top x-axis of Figure 5, the distance scale is converted to a 847 keV number flux, assuming that all SNe Ia yield S_γ = 4.3 × 10⁴⁷ s⁻¹ during the peak 10⁶ s. This is a conservative estimate for a normal SN Ia; subluminous and superluminous SNe Ia would have different axis scalings (Table 1). The in-flight 3σ narrow-line sensitivity of the spectrometer SPI on board the INTEGRAL satellite is 3 × 10⁻⁵ cm⁻² s⁻¹ at ∼1 MeV for a 10⁶ s exposure (Roques et al. 2003). The equivalent for COMPTEL on board the CGRO satellite is 6 × 10⁻⁵ cm⁻² s⁻¹ (Schoenfelder et al. 1993). The Nuclear Compton Telescope (NCT) balloon experiment has a narrow-line sensitivity comparable to INTEGRAL (Bellm et al. 2009), the DUAL gamma-ray mission is 30 times more sensitive (Boggs et al. 2010), and the proposed Advanced Compton Telescope (ACT) satellite is 60 times better (Boggs 2006). These are labeled in Figure 6, where the cumulative number of catalog SNe Ia are shown as functions of the 847 keV flux (bottom, x-axis) and distance (top, x-axis).

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Nearby SNe Ia are labeled in Figure 5, from which we see that those observed by CGRO were just out of range. Note that SN 1991T was a superluminous SN Ia, for which the CGRO horizon is larger than shown in Figure 5, but still insufficient. SN 2003gs was briefly observed by the INTEGRAL satellite, with no line detection (Leising & Diehl 2009).

The INTEGRAL sensitivity horizon to 847 keV gamma rays is ∼11 Mpc, corresponding to 0.16 SN Ia yr⁻¹ (Figure 6). The 812 keV flux is smaller, and is visible out to a maximum of 4–6 Mpc, corresponding to 0.006–0.03 SN Ia yr⁻¹. The 847 keV horizon for superluminous and subluminous SNe Ia are ∼13 Mpc and ∼7 Mpc, respectively. In practice, the energy resolution of SPI is better than the Doppler broadened width of the lines, so that the signal is spread over multiple energy bands. This reduces the effectiveness of detector background rejection with SPI, and the sensitivity horizon decreases by a factor of at most ∼2 (Gómez-Gomar et al. 1998). This issue similarly applies to next-generation detectors which we discuss in the following section. The rates are summarized in Table 2, neglecting the reduced detector background rejection due to line width.

The initial goal of SN Ia gamma-ray studies would be a single long-exposure detection of the 847 keV line from a nearby SN Ia, which, together with the optical data, would provide a robust handle on the amount of ⁵⁶Ni synthesized in the explosion. The prospects for detecting the 847 keV emission from ∼1 SN Ia yr⁻¹ is not too far from current INTEGRAL sensitivity (Table 2).

However, more important in the future is improving beyond a single 847 keV detection and measuring the gamma-ray light curve, as this ultimately holds the power to distinguish between SN Ia models. In particular, the period during which the system transitions from an optically thick to a thin one gives the p_esc(t) in Equation (5) and provides the best opportunity for model discrimination. For example, He detonation produces ⁵⁶Ni nearer to the WD surface compared to other models, resulting in an earlier transition and hence an earlier rise of gamma-ray lines. Deflagration is the other extreme, with ignition occurring in the central regions of the WD, resulting in slower rise of the gamma-ray emission. These points are clear in Figure 7, where the He-detonation models HECD and HED8 are shown together with the delayed-detonation (W7DT and DD202C) and deflagration (W7) models. In addition, detection of the 812 keV line from ⁵⁶Ni is important, given its strong emission during the transition period. In Figure 7, both superluminous SN Ia (solid) and normal SN Ia (dashed) are shown together, but optical observations would distinguish between these two classes, so we focus on model testing within each class.

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First, we show on the right y-axis of Figure 7 the significance that can be achieved by ACT observing an SN Ia at 20 Mpc in 1 day (∼10⁵ s) viewing periods. Since MeV gamma-ray satellites are dominated by detector backgrounds, the significance of the SN Ia signal scales as s = N_sig/N_bkg^1/2 ∝ T^1/2, where N_sig is the signal counts, N_bkg is the background counts, and T is the viewing period. The targeted 3σ sensitivity of ACT is 5 × 10⁻⁷ cm^-2 s⁻¹ (for 10⁶ s), and we scale it to a viewing period of 1 day. We see that ACT will allow detailed reconstruction of the 812 keV and 847 keV light curves with very large significance at this distance.

Second, provided that the SN Ia is close enough for sufficient photon statistics, studying the shape of the light curve does not depend on knowing the exact distance to the SN Ia. The shape is distinctly different between models, as we illustrate more clearly in the bottom panel of Figure 7, where we show the difference counts at ACT, for a viewing period of 1 day, all relative to the W7 model. An uncertain distance would scale the fluxes accordingly, but retain the shape differences between models. For illustration, the square root of the background counts is shown as the horizontal shading. ACT would easily distinguish between normal SN Ia models with high significance at this distance. The difference between superluminous SN Ia is smaller than the square root of the background counts; better testing would be possible with a coarser time binning.

The physics potential can be scaled to a detector with generic properties. The significance shown in Figure 7 (top right y-axis) scales as

$$\begin{eqnarray} s &\approx & 17.6 \, \left(\frac{ \phi _{\rm sig}}{ 1 \times 10^{-5} \, {\rm cm^{-2} \, s^{-1}} } \right) \left(\frac{ T }{ 1 \, {\rm day} } \right)^{1/2}\nonumber \\ && \times \left(\frac{5 \times 10^{-7} \, {\rm cm^{-2} \, s^{-1}} }{\phi _{\rm sens}} \right) \left(\frac{10^6 \, {\rm s} }{ T_0 } \right)^{1/2}, \end{eqnarray} \tag{ 7 }$$

where ϕ_sig is the signal flux, T is the viewing period, and ϕ_sens is the 3σ line sensitivity in a T₀ = 10⁶ s viewing period. Note that significance accumulates with the square root of the time period, i.e., two consecutive 10σ detection implies an overall $10 \sqrt{2} \sigma$ detection. The square root of the background counts (bottom of Figure 7) scales as

$$\begin{eqnarray} \sqrt{N_{\rm bkg}} &\approx & 50 \, \left(\frac{ T }{ 1 \, {\rm day} } \right)^{1/2} \left(\frac{A_{\rm eff}}{10^3 \, {\rm cm^{2}}} \right)\nonumber \\ && \times \left(\frac{ \phi _{\rm sens}}{5 \times 10^{-7} \, {\rm cm^{-2} \, s^{-1}}} \right) \left(\frac{T_0}{10^6 \, {\rm s} } \right)^{1/2}, \end{eqnarray} \tag{ 8 }$$

where A_eff is the effective area of the detector.

The rate of SN Ia within 20 Mpc is ∼1 yr⁻¹ and therefore, ACT will strongly constrain SN Ia models at a rate of at least 1 yr⁻¹. SNe Ia further away also yield information, with statistical significance gained by coarser time binning. At 2–20 times yr⁻¹, ACT would detect the 812 keV emission and build a detailed 847 keV light curve. At 60–80 times yr⁻¹, ACT would detect the 847 keV light curve near peak, and at almost 100 times yr⁻¹, it would detect the 847 keV emission and measure the variation in the ⁵⁶Ni yield of SNe Ia. We summarize these gamma-ray detection prospects in Table 2. The wide-field all-sky nature of ACT could in principle detect SNe Ia independently of optical discoveries, although in practice next-generation optical surveys would bring out the full potential of ACT by providing more targets for joint analysis. DUAL's focusing capabilities would rely on knowledge of the SN Ia location from an optical trigger.

For all the above estimates we have assumed the narrow-line sensitivities of detectors as documented. Since SN Ia gamma-ray lines are expected to be Doppler-broadened by up to 3%, this treatment is optimistic, although it does allow us to compare the maximum performance of detectors. In practice, the broad-line sensitivities are only marginally worse than the narrow-line sensitivities. For example, the 3% broadened line sensitivity of ACT is 1.2 × 10⁻⁶ cm^-2 s⁻¹ (3σ in 10⁶ s; Boggs (2006)), a factor 2 different from the narrow-line sensitivity. The gamma-ray horizon therefore decrease by ${\sim} 1/\sqrt{2}$, and the numbers in the final column of Table 2 decrease to 20–30, 16–21, 1–5, and 0.1–1, respectively. Although smaller, these rates would still allow rapid progress in the understanding of SN Ia. Similarly, the significance and $\sqrt{N_{\rm bkg}}$ for the broad-line case can be calculated to be ≈7.3 and ≈120 in a 1 day bin, respectively. The significance remains high, and $\sqrt{N_{\rm bkg}}$ remains smaller than the model differences, demonstrating the physics potential.

We focused on the 812 keV and 847 keV lines because they are expected to have the highest fluxes. There are other lines for which there are interesting prospects, for example, the 158 keV line from ⁵⁶Ni and the 1238 keV line from ⁵⁶Co. The detectability of these lines is only marginally less than that of the 812 keV and 847 keV lines (Equations (3) and (4)). Additionally, the 511 keV line from the annihilation of positrons produced in ⁵⁶Co decay (Equation (4)) is particularly important for understanding the positron escape fraction, thought to be ∼1%, but quite uncertain (Milne et al. 2001; Lair et al. 2006). It has important implications for the origin of the Galactic 511 keV emission (see, e.g., Beacom & Yüksel 2006 and references therein). It should also be mentioned that other radioactive nuclei such as ⁴⁴Ti and ⁵⁷Ni are expected in SNe Ia. The half-life of ⁴⁴Ti → ⁴⁴Sc is 68 yr, implying historical SNe Ia as prime targets rather than the SNe Ia discussed in this paper. The ⁵⁷Ni → ⁵⁷Co → ⁵⁷Fe decay chain, with half lives of 52 hr and 391 days, would be a more suitable target. However, as the production of ⁵⁷Ni is not as abundant as ⁵⁶Ni in most cases, it would be a useful line only for exceptionally close SNe Ia.

Other model-distinguishing features that we have not mentioned include line shift and line-width evolutions. During the first few weeks, the high optical depth means that much of the gamma-ray emission originates from the approaching ejecta, with photons escaping essentially radially, so that the lines are blueshifted. Together with the line width, these are measures of the expansion velocity, and their time evolution tests SN Ia models. The caveat with the former is that shifts occur mostly during the optically thick regime, so model testing is difficult except for exceptional cases (e.g., the DD scenario we mention next). The line width offers a more promising test, in particular with detectors with spectral resolution of order λ/δλ ∼ 200–300 (see, e.g., Höflich et al. 1998; Milne et al. 2004, for detailed predictions).

The gamma-ray emission for the DD scenario experiences significant suppression due to the envelope resulting from the merging WDs. In the det2env4 model of Höflich et al. (1998), the enveloping matter means that the line shifts remain high for a long period of time, up to ∼100 days, whereas SD scenario models all fall in a few weeks (Höflich et al. 1998). In addition, the envelope strongly suppresses the 812 keV emission from ⁵⁶Ni decay, so that the flux is weaker than even the deflagration model W7. The 847 keV emission also peaks at a later stage. Recently, several authors have explored the collision of two WDs in dense stellar systems as an alternate pathway to SN Ia within the family of DD scenarios (Rosswog et al. 2009; Raskin et al. 2009b). Shock-triggered thermonuclear explosions in such events are predicted to yield ⁵⁶Ni masses that are sufficient to power subluminous SNe Ia. Although detailed gamma-ray light curves have not yet been published, they could be targets for next-generation gamma-ray detectors if their gamma-ray emissions are comparable to those of subluminous SNe Ia.

3.6.1. SN Ia Variants

We first investigate the SN Ia rate, which is a prerequisite for SN Ia gamma-ray detection prospects. It is also interesting since the SN Ia rate with respect to its progenitor formation rate depends on what the SN Ia progenitors are. We jointly analyze the cosmic star formation rate, cosmic SN Ia rate, and the SN Ia rate derived from SN Ia catalogs. We deduce a DTD and SN Ia fraction that fit the data, and discuss the local (<100 Mpc volume) SN Ia rate.

• 1.  
  Delay times: when SN Ia rate measurements are categorized according to sample size and fraction of spectroscopically identified SNe Ia, we find that the more reliable measurements collectively show significantly slower evolution with redshift than the star formation rate. The difference is due to the delay between progenitor formation and SNe Ia: we find that a DTD of the form ∝t^−α with α = 1.0 ± 0.3 provides a good fit. The substantially prompt bimodal DTD of Mannucci et al. (2006) and the narrow Gaussian DTD around 3.4 Gyr of Dahlen et al. (2008) do not fit the global data as well.
• 2.  
  SN Ia efficiency: for our DTD, we find an efficiency of making SN Ia of (5 ± 1) × 10⁻⁴ M⁻¹_☉. Assuming that the SN Ia progenitor mass range is 3–8 M_☉, this equates to an SN Ia fraction of 2.4% ± 0.5% (for the Salpeter IMF; the dependence on this choice is weak).
• 3.  
  Implication for local SN Ia rate: the local SN Ia rate is much higher than previously thought. SN catalog entries between 2000 and 2009 reveal on average 40 SNe Ia per year within 100 Mpc, a significant increase from the ∼2 in the 1980s (Gehrels et al. 1987) and 5.5 in the 1990s (Timmes & Woosley 1997). Even so, discoveries are still severely incomplete outside about 30 Mpc. The expected true rate within 100 Mpc is about ≈100 SN Ia yr⁻¹. The SN Ia rate within 20 Mpc is ∼1 yr⁻¹.

The detection of gamma rays from SNe Ia directly tests the ⁵⁶Ni mass inferred from optical observations, and also provides tomography of the SN Ia interior by measuring the time-dependent escape probability. In the previous section, we quantitatively discussed how SN Ia discoveries are becoming more complete, and highlighted 20 Mpc as the distance for annual SN Ia discovery. Our main results on SN Ia gamma-ray detection prospects and the physics potential of gamma-ray detectors are as follows:

• 1.  
  CGB contribution: the SN Ia contribution to the CGB is at most 10%–20% of the CGB flux published by SMM and COMPTEL, confirming previous results. The origin of the ∼ MeV CGB, and whether the SN Ia contribution can be identified, therefore remain an important task to be clarified by future gamma-ray observations.
• 2.  
  Current local SN Ia prospects: local SN Ia gamma-ray detection prospects are better than thought a decade ago, principally driven by the vastly increased rate of SN Ia discoveries. Current gamma-ray satellites probe SNe Ia in the rare regime: INTEGRAL probes SNe Ia within ∼10 Mpc, occurring at a rate of ≈0.1 SN Ia per year. This is only somewhat larger than previous estimates of ∼0.03 yr⁻¹ (Gehrels et al. 1987; Timmes & Woosley 1997), but we are more confident about the normalization.
• 3.  
  Future local SN Ia prospects: the distance for annual SNe Ia discovery (20 Mpc) is only a factor of 2 further than current horizons (Figures 5 and 6). Future detectors with a line sensitivity of 1 × 10⁻⁵ cm^-2 s⁻¹—only a factor of 3 better than that of INTEGRAL—will cross this threshold. The proposed ACT satellite, with a 60 times better line sensitivity than that of INTEGRAL, will probe SNe Ia out to ∼90 Mpc, translating to a rate of 100 SNe Ia per year. Improved SN surveys are discovering more SNe Ia and at earlier times, making this possible.
• 4.  
  Implication for explosions: nearby SNe Ia will be targets for detecting gamma-ray lines from both ⁵⁶Ni and ⁵⁶Co decays, and for their light-curve reconstruction. The ACT satellite will give a hugely significant >100σ detection of gamma-ray light curves every year from SNe Ia within 20 Mpc. These SNe Ia will allow detailed analysis of SN Ia explosion physics, providing unprecedented understanding of the SN Ia explosion mechanism. A more modest detector with a line sensitivity of 1 × 10⁻⁵ cm^-2 s⁻¹ would measure the 847 keV light curve for tomography once a year, and detect the 812 keV line ∼0.1 yr⁻¹ (Table 2).

Given the importance of SNe Ia from cosmology to nucleosynthesis, the need to understand the mechanisms of SNe Ia will only increase with time. Detecting their gamma ray emission is the only way to probe their inner physics. Judging from the SN Ia rates, even a modest improvement in gamma-ray line sensitivity would reach the distance range with annual detections. Proposed next-generation gamma-ray satellites are in an excellent position to rapidly offer revolutionary results.