The correlation of rate of Type Ia Supernovae with the parent galaxy properties: lights and shadows

The correlation of rate of Type Ia Supernovae with the parent galaxy properties: lights and shadows

L. Greggio INAF - Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, Padova, 35122 Italy
email:laura.greggio@inaf.it
   E. Cappellaro INAF - Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, Padova, 35122 Italy
Key Words.:
SNIa rate – SNIa DTD– galaxy evolution
Abstract

Context:The identification of the progenitors of Type Ia Supernovae (SNIa) is extremely important in several astrophysical contexts, ranging from stellar evolution in close binary systems, to evaluating cosmological parameters. Determining the distribution of the delay times (DTD) of SNIa progenitors can shed light on their nature. The DTD can be constrained by analyzing the correlation between the SNIa rate and those properties of the parent galaxy which trace the average age of their stellar populations.

Aims:We investigate on the diagnostic capabilities of this correlation by examining its systematics with the various parameters at play: simple stellar population models, the adopted description for the star formation history in galaxies, and the way in which the masses of the galaxies are evaluated.

Methods:We compute models for the diagnostic correlations for a variety of input ingredients, and for a few astrophysically motivated DTD laws, appropriate for a wide range of possibilities for the SNIa progenitors. The models are compared to the results of three independent observational surveys.

Results:The scaling of the SNIa rate with the properties of the parent galaxy is sensitive to all input ingredients mentioned above. This is a severe limitation on the possibility to discriminate alternative DTDs. In addition, current surveys show some discrepancies for the reddest and bluest galaxies, likely due to limited statistics and inhomogeneity of the observations. For galaxies with intermediate colors the rates are in agreement, leading to a robust determination of the productivity of SNIa from stellar populations of 0.8 events per 1000 .

Conclusions:Large stastistics of SNIa events along with accurate measurements of the SFH in the galaxies are required to derive firm constraints on the DTD. LSST will achieve these results by providing the homogeneous, unbiased and vast database on both SNIa and galaxies.

1 Introduction

The identification of the progenitors of Supernovae of Type Ia (SNIa) is of great importance in several astrophysical contexts, such as constraining the evolutionary paths of close binary systems (e.g. Claeys et al., 2014), measuring the cosmological parameters (Riess et al., 1998; Perlmutter et al., 1999), studying the chemical evolution of galaxies (Matteucci & Greggio, 1986; Thomas et al., 1999; Kobayashi et al., 2015) and of the intergalactic medium (Maoz & Gal-Yam, 2004; Blanc & Greggio, 2008), the evolution of galactic winds (Ciotti et al., 1991), as well as modelling the gravitational waves emission from binary stars (Farmer & Phinney, 2003). Yet, in spite of the great efforts of the last decades on both theoretical and observational sides, the question is still far from settled (Maoz & Mannucci, 2012; Maeda & Terada, 2016). While there is general consensus that the events originate from the thermonuclear explosion of a White Dwarf (WD) member of a close binary system, there are many possible evolutionary paths leading to a successful explosion, each involving different ranges of initial masses and separations, and different delay times, that is the time between the formation of the primordial binary and the SNIa explosion. In general, the evolutionary paths can be accommodated in two classes, depending on the nature of the companion of the exploding WD, which can be either an evolving star (Single Degenerate, SD, Whelan & Iben, 1973) or another WD (Double Degenerate, DD, Webbink, 1984; Iben & Tutukov, 1984). In both scenarios the exploding star is a Carbon-Oxygen (CO) WD which ignites nuclear fuel under degenerate conditions, following accretion from a close companion. Explosion may occur either because the CO WD manages to reach the Chandrasekhar limit, or because a sufficiently massive Helium layer has accumulated on top of the WD, and detonates. The diversity of SNIa light curves indicates that both kinds of explosions occur in nature see (see Hillebrandt et al., 2013, for a comprehensive review). In addition, the correlation of the light curve characteristics with the host galaxy properties suggest that different kinds of events may pertain to different ages and/or metallicities of the parent stellar population (e.g. Childress et al., 2013).

Independent of the explosion mechanism, the delay time essentially reflects the evolutionary channel: while for SD progenitors accretion starts when the secondary component of the binary evolves off the Main Sequence, in the DD scenario at this point of the evolution a common envelope (CE) forms around the two stars, and is eventually expelled from the system, hindering the mass growth of the WD (see Ruiter et al. (2013) for a more detailed view). During the CE phase, the system looses orbital energy and shrinks, so that the final configuration consists of two WDs with a separation smaller than that at the beginning of the CE phase. Hereafter, the DD system looses angular momentum because of gravitational waves radiation, and eventually the two WDs merge. Traditionally these two channels are considered in mutual competition, but there are evidences which support that both paths are at work (Patat et al., 2007; Greggio et al., 2008; Li et al., 2011a; González Hernández et al., 2012; Cao et al., 2015; Hosseinzadeh et al., 2017; Dimitriadis et al., 2019, and references therein). Information on the relative contribution of the various channels requires collecting big samples of data to be able to relate the properties of individual explosions to the specific evolutionary path, as well as to build up a significant statistics. A less ambitious goal consists in determining the distribution of the delay times of SNIa explosions (DTD), which is proportional to the time evolution of the SNIa rate following an instantaneous burst of Star Formation (SF). Indeed, the DTD provides clues on the evolutionary channels of SNIa progenitors, insofar they correspond to different proportions of events at different delay times. In addition, the DTD is interesting in itself because it regulates the energetic and chemical input in an evolving stellar system, and is thus a fundamental ingredient to model the evolution of a variety of astrophysical objects, e.g. star clusters, galaxies, clusters of galaxies, interstellar and intergalactic medium.

Population synthesis codes provide theoretical renditions of the DTD which include the variety of different channels following from the predictions of stellar evolution in close binary systems (e.g. Ruiter et al., 2009; Mennekens et al., 2010; Wang & Han, 2012; Claeys et al., 2014). These models offer a comprehensive scenario of the evolutionary outcome, but their results are sensitive to many assumptions, some of which regarding poorly constrained astrophysical processes. Particularly critical is the description of the CE phase, and especially its efficiency in shrinking the system. In this respect we notice that in order to produce merging within a Hubble time, the separation of the DD system should be reduced down to a few solar radii, or few percent of its value at the first Roche Lobe Overflow (cf. Greggio, 2010).

Alternatively, the DTD can be derived empirically from the analysis of the relation of the SNIa rate with the properties of the parent stellar population (e.g. Mannucci et al., 2006; Scannapieco & Bildsten, 2005). This can be understood considering the following equation which links the supernova rate at a time , , to the DTD, :

(1)

in which is the delay time, is the star formation rate (SFR), and is the number of SNIa per unit mass of one stellar generation, usually assumed constant in time (cf. Greggio, 2005, 2010). Following Eq. (1), the SNIa rate in a galaxy results from the convolution of the DTD and the past star formation history (SFH) over the whole galaxy lifetime, since the delay times of SNIa progenitors span a wide range, possibly up to the Hubble time and beyond. Provided that the DTD is not flat, Eq. (1) implies a correlation of the SNIa rate with the shape of the SFH, modulated by the DTD. It is then possible to derive information on the DTD by analyzing the relation between the SNIa rate and those properties of the parent galaxy which trace the SFH. Since the occurrence of a supernova is a rare event, the correlations are usually constructed by considering large samples of galaxies and averaging the rates and galaxy properties in wide bins. To do this one needs to scale the SNIa rate according to a parameter which describes the galaxy size, e.g. the total luminosity, or the total mass. Depending on the chosen parameter the slope of the correlation changes, so that, for example, the trend of the SNIa rate with the color of the parent galaxy is flatter when normalizing the rate to the total B-band luminosity, rather than to the K-band luminosity (Greggio & Cappellaro, 2009), or to the galaxy mass (Mannucci et al., 2005). From Eq. (1) it is clear that the most direct normalization is obtained scaling the SNIa rate to the integral of the SFH over the galaxy lifetime.

Many different observed correlations have been considered in the literature to derive information on the DTD: (i) the rate per unit mass (and/or luminosity) trend with the color (and/or morphological type) of the parent galaxy (Greggio, 2005; Mannucci et al., 2006); (ii) the rate per unit mass dependence on the specific SFR (sSFR) (Pritchet et al., 2008); (iii) the rate per unit mass versus the mass of the parent galaxy (Graur & Maoz, 2013). Correlations (i) and (ii) have been widely interpreted as showing that the rate depends on the age distribution of the stars in the parent galaxy. Correlation (iii) may indirectly reflect the same property, through the observed galaxy downsizing phenomenon (e.g. Gallazzi et al., 2005): the SNIa rate is expected higher in the less massive galaxies because of the younger average age of their inhabiting stellar populations, compared to the stars in the more massive galaxies (Graur & Maoz, 2013). All observed correlations are compatible with the notion that the rate per unit mass is higher in younger systems, while the rendition of this can be described through different tracers of the average stellar age. This supports the notion of a DTD which is more populated at the short delay times (Greggio, 2005).

In order to analyze the physics behind the above empirical correlations we need to specify the galaxy SFH. So far, most works in this field have adopted some analytic description of the SFH, e.g., an exponentially declining, a delayed exponential, or a power law, all including a free parameter encoded. By varying this parameter and the age of the model, both the predicted galaxy properties (mass, colors, current star formation rate, etc.) and SNIa rate change, with their correlation being modulated by the DTD (e.g. Greggio, 2005; Mannucci et al., 2006; Pritchet et al., 2008). By comparing the predicted correlation to the observations, the shape of the DTD remains constrained.

Maoz et al. (2011) adopt a different approach, in which individual galaxies are examined with the VESPA code to derive their stellar age distribution in a few wide bins by fitting their spectral energy distribution (SED), while the DTD is described as a generic power law. Depending on an assumed value of the power law exponent, a probability of occurrence of a SNIa event in each galaxy is computed, and the analysis of the statistical properties of the sample leads to the determination of a best fit DTD slope. This method presents the advantage of avoiding averaging the galaxy properties in wide bins, and traces the capability of each galaxy to produce SNIa events. On the other hand, it requires an accurate knowledge of the SFH in each galaxy, which is hard to achieve. Also, there it is no guarantee that the global SFH which results from the galaxy mixture reproduces the observed cosmic SFH. In contrast, the general correlations between observed quantities reflect global properties of the stellar populations, averaging out the individual evolutionary paths followed by each galaxy.

In all these cases the SFH is constrained by the galaxy SED, in one way or another. More recently, in the context of galaxy evolution studies a novel methodology has been developed, in which the SFH law is constrained by the observed global properties of large samples of galaxies, e.g. the correlation between the SFR and the galaxy mass, and its evolution with redshift; the existence of a dichotomy in the color of galaxies, which populate the blue cloud and the red sequence; the evolution of the cosmic star formation rate density (Renzini, 2009; Peng et al., 2010; Oemler et al., 2013; Renzini, 2016; Chiosi et al., 2017; Eales et al., 2018). These properties can be arranged to construct a consistent picture of the general features of the evolution of the SFR in galaxies over cosmic times. We consider two formulations of the SFH obtained under this view (Peng et al., 2010; Gladders et al., 2013), both of which describe well the evolution with redshift of the galaxy luminosity functions (Peng et al., 2010; Abramson et al., 2016). Different from the SED fitting of individual galaxies, this approach aims at accounting for the general behaviour of the star formation history of galaxies in the evolving universe, and are well suited to model the trend of the SNIa rate with galaxy properties, which is necessarily measured in big galaxy samples. In the following we address this kind of description of the SFH as cosmological, to distinguish it from the other, more standard one, which considers each galaxy independently. In this paper we will show that the choice of the SFH description has an impact on the predicted correlations and hence it adds a significant contribution to the systematic uncertainties that has been neglected or underestimated so far.

In Botticella et al. (2017) we discussed the constraints on the DTD which could be obtained from the correlations of the SNIa rate with the parent galaxy properties found in the SUDARE survey, adopting a standard description of the SFH. In this paper we expand on those results, including the cosmological SFH laws, as well as a comparison with more observational data, i.e. the rates measured in local galaxies from the LOSS search (Li et al., 2011b; Graur et al., 2017) and from the Cappellaro et al. (1999) search (hereafter CET99). For the three surveys we use the original SN discovery list 111We notice that for the LOSS survey the detailed SN sub-type classification is available which however we do not consider. For our analysis we separate the SNe in two main groups: type Ia and core collapse. and control times to compute supernova rates for different galaxy parameters and binning with respect to the published values. This allows us to optimize the description of the trend of the rate with the properties of the parent galaxy, as well as the comparison of the results from the three independent data sets. For the LOSS survey we use the control times published by Graur et al. (2017), while the SUDARE and CET99 control times were already available to us. We checked that we recover the results published in Li et al. (2011b) and Mannucci et al. (2005) when using the same parameters and binning. Using the original data for the surveys we were also able to test different approaches for the estimate of galaxy masses (cf. Sect. 5). We remark that the SUDARE survey targets galaxies in the Chandra Deep Field South (CDFS) and the Cosmic Evolution Survey (COSMOS) fields, limited to objects with redshift . The vast majority of galaxies in the SUDARE sample is at intermediate redshift (), and their redshift distribution does not show any prominent peak. We focus on the correlations of the SNIa rate with the parent galaxy colors, and along with different descriptions of the SFH, we test different sets of simple stellar population models to compute galaxy colors, which impact on this kind of analysis.

The paper is organized as follows. In Section 2 we describe the ingredients of the computations of the theoretical correlations between the SNIa rate and the parent galaxy colors (DTD and SFH). The model colors are compared to the data of the SUDARE galaxy sample in Section 3 to test the adequacy of our approach. In Section 4 we present the expected correlation of the SNIa rate with the color to illustrate the interplay between the DTD and the SFH. In Section 5 we present the observational scaling of the SNIa rate with the colors of the parent galaxy and discuss the effect of the assumptions on the galaxy mass-to-light ratio. In Section 6 we illustrate the diagnostic capabilities of these correlations by comparing a set of models to literature data; in Section 7 we summarize the results and draw some conclusions concerning future surveys.

2 Model Ingredients

2.1 The DTD

For the distribution of the delay times we adopt a selection of models from Greggio (2005). These analytic formulations, described below, are based on general arguments which take into account the clock of the explosions and the range of initial masses of the stars in systems which can provide successful explosions. For comparison we also consider an empirical DTD akin to the results in Totani et al. (2008) and Maoz et al. (2012), which we describe as a power law with an index of from a minimum delay time of 40 Myr. We notice that by construction this DTD is not motivated by a specific astrophysical scenario and the physical interpretation of the results is left to subsequent analysis.

For the analytic models we consider three options: the Single Degenerate (SD) model, and two versions of the Double Degenerate progenitor model, labelled DD Wide (DDW) and DD Close (DDC) (cf. Fig. 1). As mentioned above, it is likely that SNIa’s arise from both single and double degenerate progenitors; the DTD of mixed models is expected to be intermediate between that of single models depending on the fractional contribution of the two components. Since we want to explore the possibility of discriminating specific channels, in this paper we do not consider mixed models.

In the SD model the time delay is virtually equal to the evolutionary lifetime of the secondary in the core Hydrogen burning phase (), so that early explosions are provided by systems with more massive secondaries, while late epoch events occur in systems with low mass secondaries. There are two discontinuities in the DTD of the SD model (cf. Fig. 1) due to requirements on the mass of the progenitor: the first, at = 1 Gyr, comes from the lower limit of 2   for the mass of the primary, since stars with lower mass in a close binary are more likely to produce a He, rather than a CO, WD. The second discontinuity, at 8 Gyr, comes from the requirement that the WD mass and the envelope mass of the evolving secondary sum up to the Chandrasekhar limit. At this age the mass of the secondary is and the envelope mass is . In order to reach the Chandrasekhar limit the accreting WD must be more massive than 0.6 , the descendant of a primary. At later delay times, the lower mass secondary has a smaller envelope, which demands a primary more massive than 2  to meet the Chandrasekhar limit. The limitation on the primary masses in the progenitor systems, whose range progressively shrinks as the secondary mass decreases, causes the rapid drop of the DTD at late delay times which characterizes the SD model. Notice that the delay time at which this limitation sets in depends on the fraction of envelope mass of the secondary which is actually accreted and burned on top of the CO-WD. The SD model DTD in Fig. 1 assumes that all of the envelope of the secondary is accreted and burned on top of the companion; if only a fraction of the envelope of the secondary is used to grow the WD, the minimum primary mass in successful systems becomes larger, and the drop sets it at earlier delay times (see Greggio, 2010).

In the DD model the delay time is the sum of the evolutionary lifetime of the secondary () plus the time it takes to the gravitational wave radiation to shrink the system up to merging of the DD components (). In Greggio (2005) only the double CO WD channel is considered to lead to a successful SNIa explosion, which implies an upper limit of Gyr to . The value of  is very sensitive to the initial separation of the double degenerate system (), which results after one or more common envelope phases. We point out that, in order to merge within a Hubble time, the separation of the DD system at birth must be smaller than a few solar radii, while the separation of the primordial binary should be at least of a few tens of solar radii, in order to avoid premature merging and allow the formation of a CO WD from the primary. Therefore, a high degree of shrinkage is necessary to construct a successful SNIa progenitor in the close binary evolution. If the process is such that the more massive the binary the more it shrinks, there is little room for massive systems to explode on long delay times: both their delays  and  are short. In this case (dubbed DD Close, DDC) at long delay times the DTD is populated by the less massive systems which manage to keep a large enough separation when they emerge from the CE phase. Conversely, if there is no dependence of the shrinkage from the binary mass, massive systems can also emerge from the CE with large values of , and the DTD in this case (DD Wide, DDW) is flatter compared to the DDC option. The DTDs for the DD models are characterized by a plateau at short delay times, followed by a (close to a) power law decline. In these analytic formulations, two parameters control the DTD of the DD models: the lower limit to the secondary mass in SNIa progenitors, which corresponds to an upper limit to the lifetime of the secondary (), and the distribution of the separations of the DD systems at birth, arbitrarily described as a power law. At a delay time equal to  the DTD presents a cusp due to the setting in of a sharp upper limit to the evolutionary lifetime of the secondary, so that at delay times there is a lower limit to which increases with  increasing. Thus, the minimum mass of the secondary in SNIa progenitors controls the width of the DTD plateau. The slope of the decline at delay times longer than  is instead controlled by the distribution of the separations: the steeper this distribution, the higher the fraction of systems with short , and therefore the steeper the decline of the DTD.

In spite of the approximations introduced to derive these analytic DTDs, they compare well with the results of population synthesis codes, when taking into account the appropriate mass ranges and kind of progenitors (Greggio, 2005, 2010). By construction, these analytic formulations provide a means to explore how the predicted rates and correlations change when varying the DTD under astrophysically motivated arguments concerning the masses, the separations, and their distributions, of the progenitor systems.

Fig. 1 shows the four DTD distributions at the basis of the computations presented in this paper. Besides the empirical law and the SD model discussed above, we consider a DD Close model in which the minimum mass of the secondary in the progenitor systems is of 2.5 , implying = 0.6 Gyr, and a distribution of the separations of the DD systems , i.e. close to that of unevolved binary systems (Kouwenhoven et al., 2007) . The selected DD Wide model, instead, adopts a minimum mass of the secondary of 2 , and a flat distribution of the separations . These values encompass a wide range of possibilities for the progenitor systems.

Figure 1: DTDs for Single Degenerate (SD, solid line), and Double Degnerate models (DD Close, long dashed, DD Wide, short dashed) used in this paper. Because of the choice of parameters (see text) the DD Close DTD accommodates a large fraction of prompt events, while the DD Wide DTD is relatively flat. The dot-dashed line is a a power law . All DTDs considered here are equal to zero at delay times shorter than 40 Myr (the evolutionary lifetime of a 8  star), and are normalized to 1 in the range 40 Myr 13 Gyr.
Figure 2: Selection of functions used to compute the models under the cosmological SFH proposed in Gladders et al. (2013). The color encodes the values of the parameters and as labelled. Solid and dotted lines of the same color share the value of , but differ in , which is varied between a minimum and a maximum value in steps of 0.05 Gyr. The four curves plotted as dashed lines describe galaxies which fall outside the most populated region in Fig. 9 of Gladders et al. All time variables are in Gyr. For each function, the maxima occur when the age of the Universe is .
Figure 3: Age distribution for the inflating SFH model at 2 cosmic epochs: 7 Gyr (magenta) and 13.5 Gyr (black). The three line types refer to the three values for the parameter , as labelled.
Figure 4: Two color diagram for galaxy models, based on the BC03, solar metallicity SSPs, under different prescriptions for their SFH (colored symbols), superposed to the distribution of restframe colors of galaxies of the SUDARE (COSMOS + CDFS) sample (grey scale). The four panels show the models computed with exponentially decreasing SFH (top left), delayed exponential SFH (top right), log-normal SFH (bottom left), and inflating + quenched models (bottom right). The color and symbols encoding is labelled in each panel. For the log-normal SFH, models with the same but different value of the parameter are plotted with the same symbol (shape and color). In all panels, filled symbols show models with sSFR /yr, eligible to be classified as star forming objects, while empty symbols show models with sSFR /yr, which would be classified as passive galaxies. The arrow in the bottom right panel shows the direction of the reddening vector.

2.2 The SFH

In the standard approach a galaxy is viewed as a collection of stellar populations of different ages with the age distribution reflecting the star formation history described by a parametric analytic relation, e.g. exponentially decreasing functions with different e-folding times. The SFH in each galaxy is characterized by two parameters: the age of its oldest stars and the parameter of the adopted analytic relation. The packages currently used to derive age and age distributions in galaxies from their SEDs usually implement this kind of description.

Both cosmological SFHs considered here were constructed to reproduce the observed relation between the SFR and stellar galaxy mass (galaxy Main Sequence), though under different conceptions. In the Peng et al. (2010) model all galaxies evolve along this galaxy Main Sequence locus while they are star forming, until quenching sets in and they rapidly become red, passive objects. Conversely, in the Gladders et al. (2013) model galaxies spend most of their lifetime on the Main Sequence, but do not necessarily evolve along it. In addition, no abrupt quenching occurs; rather, at some point in the evolution, the SFR undergoes a gradual downturn, and the galaxies turn red, while their SFR extinguishes.

In order to test the sensitivity from the SFH of the constraints on the DTD, as derived from the correlation of the SNIa rate with the parent galaxy properties, we consider the following alternative SFH descriptions.

Standard:

  • exponentially decreasing SFH:

    with ranging from 0.1 to 8 Gyr;

  • delayed exponential (Gavazzi et al., 2002):

    with ranging from 0.1 to 20 Gyr.

Cosmological:

  • log-normal SFH :

    regulated by the two parameters and , respectively controlling the width of the distribution and the cosmic epoch at which the SFR peaks (Gladders et al., 2013);

  • inflating+quenched SFH, following Renzini (2016) with two regimes:

    during the active phase, which lasts up to an abrupt quenching occurring at some cosmic epoch , and is thereafter followed by pure passive evolution. In this equation is an adjustable parameter of the order of unity, and is the galaxy stellar mass at cosmic time .

In all these relations time variables are expressed in Gyr. We point out that, while in the standard descriptions the independent variable () is the time since the beginning of star formation in a galaxy, which can occur at any epoch up to the current age of the Universe, in the cosmological formulations the independent variable () is the age of the Universe. In the following we detail the implementation in our modelling of the cosmological SFH models.

2.2.1 The log-normal SFH model

In this alternative, the galaxy SFR grows to a maximum, and then decreases, following a log-normal law. The cosmic epoch at which the maximum occurs, and the width of the log-normal relations are peculiar to individual galaxies. Gladders et al. (2013) use the observed distributions of the specific star formation rates of local galaxies, together with the cosmic star formation rate density in the redshift range , to infer the distribution of the two parameters. We have considered the results presented in Fig. 9 of Gladders et al., constrained by objects up to redshift 1. In this figure most galaxies occupy a triangular region in the ( , ) space, which we sample with a number of discrete values. The variety of SFHs considered here for this model is shown in Fig. 2. As increases the star formation is more and more delayed, and at fixed several possible values of the parameter modulate the distribution of the stellar age. At each cosmic epoch, galaxies with a variety of SFHs coexist, implying a wide distribution of galaxy colors.

2.2.2 The inflating+quenched SFH model

The formulation for this model is taken from Renzini (2016), who states its applicability for Gyr. Presumably, galaxies form stars at earlier epochs, but at about 3.5 Gyr they are found on the galaxy Main Sequence, and from that time they evolve according to the given law, until quenching occurs. For galaxies forming stars with this SFH, at any cosmic time , the fraction of mass in stars with is given by

(2)

which, at fixed , only depends on the cosmic time . In other words, in this model, all star-forming galaxies at fixed redshift have the same age distribution, and their colors span a narrow range, reflecting only a spread in metallicity. Additional color variance at fixed redshift could be attributed to very different contributions from the stellar populations formed at Gyr ; however, this is not consistent with the fact that in the inflating model, most of the star formation in star forming galaxies occurs at epochs later than that, when galaxies are seen on the Main Sequence. Rather, we explore the effect of assuming different values of , under the hypothesis that all galaxies do follow the Main Sequence up to quenching, but with different stamina. Figure 3 shows the age distributions of the inflating SFH models at different cosmic epochs: we can see that when increases, at given cosmic time the galaxy hosts younger stellar populations. However, at late cosmic epochs even the most active star forming galaxies have a significant fraction of relatively old stars, since they have been sitting on the Main Sequence for a long time.

Figure 5: Integrated (top) and (bottom) color as function of the sSFR of model stellar populations with different SFHs: log-normal (blue circles), exponentially declining (red stars), delayed exponential (black crosses), and inflating SFH (cyan asterisks). There is a close correspondence between the integrated color and the sSFR, which is largely independent of the SFH law. Solar metallicity BC03 SSP models have been used to compute the colors.
Figure 6: Models for the scaling of the SNIa rate per unit mass of formed stars with the color of the parent galaxy, for different assumptions on the SFH and the DTD. Vertical panels share the SFH law as labelled on top, and horizontal panels share the adopted DTD, which are labelled in the left panels together with adopted values for  in . The symbols and color encoding is the same as in Fig. 4, with star forming and passive models shown as filled and empty symbols respectively. In the right panel relative to the DD Close DTD we show the evolutionary line of a model which undergoes quenching at , when the universe was 7 Gyr old. In the same panel the vertical dashed line connects models with the same color, but different SNIa rates (see text).

3 Integrated colors

In order to analyze the correlation between the SNIa rate and the colors of the parent galaxies we need to use models which well represent the spectrophotometric properties of the observed sample. The SFH description gives the age distribution for a given galaxy but to obtain its colors we need to combine this information with the predictions for evolutionary models of simple stellar populations (SSP ). SSP models present their own systematic differences due to different input stellar tracks, and bolometric corrections - temperature transformation. A discussion of the related uncertainties is beyond the aims of this paper. However, we performed a basic check of the results obtained with three frequently used SSP models sets, namely: (i) the Bruzual & Charlot (2003) library (hereafter BC03), specifically the models based on the Padova 1994 tracks coupled with the STELIB spectra; (ii) the Maraston et al. (2010) SSP models; (iii) the Padova SSP models based on the Marigo et al. (2008) set of isochrones (hereafter MG08), as obtained with the CMD web tool222stev.oapd.inaf.it/cgi-bin/cmd. A brief report of this analysis is included in Appendix A and the results will be summarized in Sect. 7. All these sets model the properties of stellar populations of single stars.

We found that the BC03 set with solar metallicity well represents the distribution of the galaxies in the versus rest frame color (cf. Fig. 4) for all the different SFH models. Therefore, in the following we adopt BC03 as the reference SSP model set to calculate the galaxy integrated color.

Inspection of Fig. 4 shows that these models nicely reproduce the general features of the observed galaxy colors, (i) by showing a correlation between the two colors which well matches the main axis of the observed distributions, and (ii) by reproducing the location on this plane of the star forming galaxies when the model sSFR is high, and that of passive galaxies when it is low. We point out that the galaxy colors in Fig. 4 have not been corrected for reddening. The direction of the reddening vector, derived from the Cardelli et al. (1989) extinction curve, is shown as a blue arrow in the bottom right panel, and a similar direction applies for the Calzetti (2001) extinction law. On this plane, the effect of both the Milky Way and of the internal absorption is that of shifting the galaxies to the red along the main axis of their distribution. As is well known, reddening is likely responsible for the red extension of the blue galaxy sequence, but it hardly produces a scatter around the main locus. On the other hand such scatter can be ascribed to some differences in the metallicity, given the high sensitivity of these colors to Z (cf. Fig. 18).

Although the main features of the models distribution on this plane are quite similar in the four panels, some differences can be noticed. Some models computed with the standard SFH laws (upper panels) fall in a region of red for where observed galaxies are scarce (e.g. the blue triangles). These models have short timescales , so that their colors evolve fast, taking only 3 Gyrs to reach the region of passive galaxies. Therefore, if galaxies with short star formation timescales formed at high redshift, by z they will already have reached the passive region of the two color diagram. In other words, it is quite possible to reconcile the color distribution of the galaxy population with the standard SFH laws adopting an adequate distribution of the (AGE,) parameters. Similarly, the quenched models in the bottom right panel of Fig. 4 at fall in an underpopulated region, but it only takes 1 Gyr to all quenched models to reach the region of passive galaxies, so that the transition region between the blue and red galaxies indeed is expected to be underpopulated. The log-normal SFH models (bottom left) reproduce very well the high density regions of the observational plane, with a good match of both star forming and passive galaxies. For this class of models the distribution of stellar ages in individual galaxies is wide, so that on this plane they behave like the standard models with long timescales.

In summary, all the SFH laws considered here appear consistent with the distribution of galaxies on the two color plane, and in order to obtain firm constraints on the SFH in galaxies a more extended and punctual analysis of the galaxies distribution is required. This is beyond our scope; we rather focus on the ability of our models to reproduce the general features of the colors of the sample galaxies.

For each of the SFH formulations considered in this paper, Fig. 5 shows that there is a nice correlation between integrated colors and the sSFR. Furthermore, the different SFH laws remarkably describe the same correlation between sSFR and color, except for the exponentially declining and the delayed exponential SFH with short timescales. For these latter models, the decline of the sSFR as time increases is fast, compared to the growth rate of the color. On the other hand, Fig. 4 shows that the exponential SFH models with short timescale fall in an underpopulated region of the two color diagram, unless their . In other words, this SFH description could be adequate for passive galaxies, with very red colors and very low sSFR. We conclude that using the age tracer is equivalent to using , and largely equivalent to using the sSFR parameter. The predicted correlation will be used in Sect. 7 to infer sSFR from color of local galaxies.

4 The SNIa rate versus galaxy colors

In this section we illustrate the model predictions for the correlation between the SNIa rate and the colors of the parent stellar population, used as a diagnostic tool for the DTD. Fig. 6 shows the model correlations of the SNIa rate with the color of the parent galaxy for various formulations of the SFH and the four DTD options displayed in Fig. 1. Delayed exponential SFHs, not shown here, provide trends very similar to the exponentially declining models. Clearly both the DTD and the SFH bear upon the resulting trend, since both the rate and the color depend on the age distribution of the stellar population, but with different sensitivities. Thus, galaxies with different SFHs may have the same color, but different SNIa rate. In other words, the SNIa rate and the color trace the average age of the parent stellar population, but they do so in different ways.

Irrespective of the SFH, some features of this correlation are peculiar to the DTD: the single degenerate channel exhibits a fast drop at red colors, for old passive galaxies, where only events with long delay time take place. We recall here that in the SD model, such events are produced in systems with a high mass primary, because of the need to reach the Chandrasekhar limit by accreting the envelope of the low mass secondary, which fills the Roche Lobe at late delays. The P.L. and the DD Close DTDs produce a relatively narrow correlation, steeper for the DD Close model with respect to the P.L. model. The DD Wide DTD produces a quite flat correlation, with a drop of only a factor of 10 between the bluest and the reddest galaxies.

Some features of the correlation between the SNIa rate and the color of the parent galaxy are instead peculiar to the SFH: exponentially decreasing laws with different e-folding times produce a relatively large variance of the rate at fixed color, especially at , which is typical of Spiral galaxies. Compared to the standard SFH models, the log-normal laws result in a tighter correlation, with models with very different width and peak time describing almost the same scaling of the SNIa rate with the color of the parent galaxy.

The inflating+quenched SFH models predict a quite distinctive behaviour, with two separate sequences, one for star forming galaxies (at bluer colors) and one for passive galaxies. In this class of models, galaxies evolve along the star forming sequence until quenching occurs, at which point they rapidly become red. In Fig. 6 the models adopting the inflating+quenched SFH are plotted for Gyr. In a galaxy sample which includes objects up to high redshift there should appear galaxies with the same color, but lower SNIa rates if star forming, larger SNIa rates if passive, irrespective of the DTD. For illustration, the panel relative to the DD Close model shows an evolutionary line for galaxies which underwent quenching when the universe was 7 Gyr old. The gap between the two sequences in the right panels of Fig. 6 corresponds to a 0.1 Gyr time elapsed from quenching, a process assumed instantaneous. A less abrupt quenching would produce a more gentle evolution from the blue to the red sequence. However, it is expected that, due to the large contribution of events at delay times of a few hundreds Myr, the drop of the SNIa rate lags behind the color evolution, and there should be some relatively red galaxies with a high SNIa rate. In a galaxy sample which includes high statistics for objects up to redshift 1 (when the universe was 6 Gyr old) we may detect these two sequences, with galaxies with the same color showing a higher SNIa rate if quenched, compared to galaxies which are still star forming. The size of the effect is illustrated in the right panel of Fig. 6 of the DD Close delay time distribution, where a vertical line connects two model galaxies with the same color, but different SFH: one star forming in the local universe, and one quenched at , in the very early post quenched life. The two galaxies happen to have the same color ( = 1.6), while the SNIa rate in the quenched galaxy is higher because its average age ( Gyr) is younger than that of the local still star forming galaxy ( 3.9 Gyr).

Figure 7: Model predictions for the ratio between the mass of formed stars and the band luminosity (both in solar units) as a function of the color (in VEGAMAG) of the parent stellar population for different SFH laws, as labelled on top. An absolute magnitude of has been used to compute the mass-to-light ratio of the stellar populations. Filled red circles highlight models with Gyr, in panels (a) and (b), and models with Gyr, in panels (c) and (d). The lines show various options to derive galaxy stellar masses from the color and -band luminosity: the dotted line in panel (a) shows the BJ01 relation for a Salpeter IMF; in all panels the dot-dashed line shows the BJ01-a regression, while the solid and the dashed line are combined in the piece-wise relation. See text for more details.
Figure 8: Observed correlations between the SNIa rate and the parent galaxy (left) and (right) color in the SUDARE , CET99 and LOSS surveys. For the local surveys the galaxies’ stellar mass has been evaluated with the piece-wise relation discussed in the text. For the LOSS survey we also show as filled symbols the rate normalized to the mass derived from the BJ01-a relation. Colors are in the AB magnitude system. The horizontal error bars show the 1 width of the galaxy distribution within each color bin; the vertical error bars show the uncertainty on the rate from the statistics of the events.

5 Observed rates and Mass-to-Light ratios.

The observational counterpart of the correlations plotted on Fig. 6 are the rates per unit mass in galaxies of different color. To derive constraints on the DTD one needs to collect a big sample of galaxies to provide a significant number of SNIa events within each color bin. For each galaxy, the rest-frame, dereddened color has to be determined, as well as the stellar mass.

The mass of a stellar population evolves with time, due to the progress of star formation, and to the mass return from stellar winds and supernovae. Rather than the current stellar mass, it is more appropriate to consider the total mass of formed stars (), that is the integral of the SFR over the total galaxy lifetime, writing Eq. (1) as:

(3)

When normalizing the rate to the current mass of the stellar population, the mass reduction should be taken into account, a factor which is a function of time, of the SFH, of the IMF, and of the initial-final mass relation of single stars (see Greggio & Renzini, 2011). Thus the scaling between the DTD and the observed rate would be less straightforward, and more prone to the choice of the input ingredients of the modelling. Some modelling is required also to apply Eq. (3), since the integral of the SFR is not directly measured, and rather derived from the galaxy luminosity. This is a critical issue as we show in the following.

In this paper we consider SNIa rates measured on different searches and galaxy samples. Galaxies in the SUDARE sample were analyzed using the EAZY code to determine the redshift, and the FAST code (Kriek et al., 2009) to determine the dereddened colors and the mass of formed stars by fitting their SED from the UV to the IR. In general, the best fit SFH, and then its integral (), depend on the specific SFH description adopted by FAST to fit the SED. In practice, we verified that in the SUDARE sample the observational correlation remains the same when using the exponentially decreasing or the delayed SFH law.

The photometric data available for the local samples (LOSS and CET99) are too sparse to perform the SED fitting and derive the SFH of all galaxies. Therefore, to estimate the stellar mass of each galaxy we adopt a relation between the ratio and the color of the parent stellar population, as often done in the literature (e.g. Mannucci et al., 2005; Li et al., 2011b). However, the calibration of this relation requires some discussion.

Fig. 7 shows the ratios of the mass of formed stars to the band luminosity of model galaxies computed with a Salpeter IMF (from 0.1 to 100 ), as function of their color. The four panels refer to the different kinds of SFH considered here. Since we use these models to derive the stellar mass of galaxies in the local universe, we highlight with red filled circles those with old ages, i.e. models with Gyr for the standard SFHs (panels (a) and (b)), and models with Gyr for the cosmological SFH laws (panels (c) and (d)). We recall that in the standard models the parameter is the time elapsed since the beginning of star formation, and even in the local universe this could be short for galaxies of the latest types. Nevertheless, in panels (a) and (b) we highlight the models with old age for a direct comparison to the Bell and De Jong (2001, hereafter BJ01) relation which has been often adopted in the literature to derive the stellar mass from the galaxies luminosity.

The dotted line in panel (a) of Fig. 7 shows the BJ01 regression for exponentially decreasing SFH models with different e-folding time, all at an age of 12 Gyr, and assuming the same IMF and set of SSP models as in our computations (coefficients are taken from Table 4 in BJ01). Its slope well matches our 12 Gyr old models in panel (a), but the Bell and De Jong masses are systematically lower then ours by a factor of (see dashed line in Fig. 7). This shift is largely due to a different definition of the stellar mass: while we consider , BJ01 refers to the current mass of the stellar population, i.e. taking into account the mass recycling during the evolution of the galaxy. The mass reduction factor of an SSP with a Salpeter IMF at an age of 12 Gyr is (Greggio & Renzini, 2011); the small residual discrepancy likely results from other different choices in the ingredients of the computations. Nevertheless, we stress that the slope of the relation between the mass to light ratio and the integrated color of the stellar population is the same in ours and in the BJ01 models, when referring to the same SFH and age. The dot-dashed lines in Fig. 7 show the regression from BJ01 used in Mannucci et al. (2005) to convert band luminosity into stellar mass (herefter BJ01-a). This is the preferred regression in BJ01 for 12 Gyr old galaxies; it is based on a somewhat different assumption on the SFH of the model galaxies, but, most importantly, it assumes a modified Salpeter IMF, flattened at the low mass end. The slope of this latter regression is virtually the same as for the exponentially declining models, but at given luminosity it implies a stellar mass smaller by a factor 0.7 (0.15 dex), due to the different IMF.

When analyzing the correlation between the SNIa rate and the color of the parent galaxy to constrain the DTD,a zero point shift of the mass to light ratio results in a different value for , but has no effect on the shape of the DTD. On the other hand, changing the slope of the regression may have important consequences. Actually, in the local universe there are galaxies whose blue color demand young ages, represented as grey circles in the four panels of Fig. 7. In fact in the RC3 catalogue almost half of the galaxies are bluer that = 3, with a fair number of objects as blue as . For these blue galaxies, the log-normal SFH models cluster along a very steep relation, which is also consistent with the inflating SFH models, during the active phase. The quenched models follow a parallel relation, shifted to redder colors; they describe a very different trend compared to the old and red models resulting from the other SFH laws.

In general, Fig. 7 shows that the color may not be a good tracer of the mass to light ratio in local galaxies as instead usually assumed: this applies to standard SFH laws when allowing for an age spread, as well as to the models constructed with the cosmological SFH. Nonetheless, with the aim to highlight the impact of the adopted mass-luminosity relation, we construct a piece-wise relation tailored to the log-normal SFH models with Gyr (red dots in panel (c) of Fig. 7): we use the combination of the solid (for blue galaxies) and dashed (for the red galaxies) lines shown in all panels. Although this representation is not spotless for the red galaxies, it improves the estimate of the mass-to-light ratio for the blue galaxies with respect to the BJ01 relation. We checked that for the SUDARE galaxies we find an overall consistency between the mass of formed stars obtained from the SED fitting and from the use of this piece-wise relation.

Fig. 8 shows the measured rates versus the galaxy colors for the three independent surveys mentioned above. For the LOSS survey we show the effect of adopting different relations to evaluate the mass-to-light ratio at given color: filled symbols assume the BJ01-a relation while empty symbols are derived with the piece-wise relation. It turns out that the slope of the observed correlation is very sensitive to the adopted scaling of the mass to light ratio with the galaxy color. For the blue galaxies, the mass-to-light ratios from the BJ01-a and from the piece-wise regressions are similar (see Fig. 7), and so are the derived rates per unit mass in the bluest bin. As the color becomes redder, the masses derived at fixed () from the BJ01-a relation are systematically smaller than those estimated with the piece-wise relation, and the rates per unit mass are evaluated higher, an effect which becomes more prominent as the color gets redder. This systematic trend is found also for the correlation of the rate with the color because red galaxies in are also red in .

For the two local surveys, the rates per unit mass as a function of are consistent within the statistical errors when the same mass-to-light ratio regression is used; the same is true for the rates as a function of but for galaxies in the bluest and the reddest bins, with the CET99 values pointing at a much steeper correlation with respect to LOSS. This difference could be related to the poor statistics of the CET99 sample, which counts only 10 and 6 events in the bluest and in the reddest bins, to be compared to the 14 and 73 events in the LOSS survey in the same color bins. Good statistics of events is clearly the first requirement to derive useful information from these correlations.

As mentioned above, the galaxy masses in the SUDARE survey are derived from the multicolor SED fitting, and are not prone to the ambiguity affecting the mass to light ratio as traced by the color. Actually the SED fitting is designed to determine the age distribution, yielding a star formation history for each galaxy. On the other hand, other systematic effects may introduce an uncertainty in the derived masses, e.g. errors in the estimate of the redshift and of extinction correction, but also some degeneracy in disentangling the contribution of the different stellar population components. In addition, for the SUDARE galaxy SED fitting we use the FAST program with a standard SFH description, and we can not check whether using a, e. g., log-normal description of the SFH would lead to a different estimate of the galaxy mass. Yet, as we can see in Fig. 8 the rates from the SUDARE survey appear largely consistent to those of the local surveys. The agreement is very good when binning the galaxies in color, while when binning in the SUDARE rates tend to be systematically higher than those measured in the local surveys. Given statistical and systematic uncertainties, we conclude that overall there is no tension among the results from the three surveys.

6 Which diagnostic from the observed correlation?

As can be appreciated from Fig. 6, the unambiguous interpretation of the correlation requires knowledge of the SFH. The data in hand do not allow us to discriminate among the various SFH models for the galaxies in our data sample, but in this section we proceed comparing models to observations in order to illustrate the diagnostic capabilities of this correlation, once the SFH variable is fixed. To do this we consider the predictions of the log-normal laws in combination with the BC03, solar metallicity SSP set, which appear to well represent the SUDARE galaxy sample (see Fig. 4). We remark that we do not mean to favour any specific SFH description or SSP set of models. To assess the scenario which best describes the photometric properties of galaxies one needs to perform a much more thorough analysis than done here.

As a consistency check, Fig. 9 shows the comparison between models constructed with the log-normal SFH and the rate of core collapse (CC) events measured in the three galaxy samples. The theoretical rates have been computed multiplying the specific SFR of our selection of log-normal SFH models by the number of CC progenitors per unit mass of the parent stellar population (). For a Salpeter IMF and progenitors with mass between 10 and 40 , ; with this value there is a very good agreement between the models and the data. Assuming a smaller value of the minimum mass of the CC supernovae progenitor (see e.g. Smartt, 2009)  would be higher and the observed rates would be lower than the expectations. This may suggest that a fraction of the events goes undetected, or equivalently, that the detection efficiency of the search is overestimated. Notice that the trend of the model rates with the color of the parent galaxy fits the observations very well, so this effect should be approximately the same in blue and in red galaxies. A relatively low value of  was found in paper II for the SUDARE sample, and is now found also for the local surveys. On the other hand, the volumetric rates presented in paper I from the SUDARE sample support a mass range between 8 and 40  for the CC supernovae progenitors (=0.007) in combination with the cosmic SFR of Madau & Dickinson (2014). We do not have a ready explanation for this discrepancy, which could be related to the intrinsic uncertainty in the measurement of the galaxy masses, on the one hand, and of the cosmic SFR on the other.

Fig. 9 shows that the log-normal SFH models adequately describe the scaling of the specific SFR with the color of the parent galaxy. This applies also to the other formulations of SFH considered here, which is not surprising given the tight relation between the specific SFR and the color of the parent galaxy shown in Fig. 5, followed by all the SFH laws considered here.

Figure 9: The rate of CC supernovae as a function of the color of the parent stellar population for the log-normal SFH laws (filled dots), compared to the rates measured in the three independents surveys SUDARE (empty red circles), LOSS (green squares) and CET99 (blue pentagons). Error bars have the same meaning as in Fig. 8. The theoretical rates are obtained as the plain product of the specific SFR and .
Figure 10: Models computed with the log-normal SFH and the four options for the DTD considered here (filled symbols) compared to data (empty symbols). Predictions obtained with different DTDs are plotted with different colors as labelled. For each DTD we adopt the best fitting value for , namely =(0.74,0.77,0.89 and 0.69) 1/ respectively for the SD, P.L., DD CLOSE and DD WIDE DTDs (see Appendix B). The symbols encoding of the observed rates is the same as in Fig. 8.
Figure 11: Same as Fig. 10 but for the galaxies binned according to their color.
Figure 12: Correlation between the rate of SNIa and the specific SFR of the parent galaxy. Filled circles show our models computed with the log-normal SFH; the color encoding is the same as in Fig. 10. Open symbols show our measurements on the SUDARE, CET99 and LOSS data. For the local surveys we adopt the piece-wise relation between the M/L ratio and the color. The filled triangle, pentagon and diamond show literature data (see legend in the upper left corner). The grey stripe shows the result of the simulations in Graur et al. (2015).

Figures 10 and 11 show the comparison between the models and the measurements of the SNIa rate versus the and versus the color for the three independent surveys. The observed rates for the local samples are normalized to the stellar mass derived using the piece-wise relation. The model rates are obtained using Eq. (3) and adopting a value of  which fits the observed rates in galaxies with intermediate color. Indeed, in galaxies with a flat age distribution ( = const.) the SNIa rate at late epochs is equal to the ratio between  and the age of the galaxy, since the DTD is normalized to unity (see Greggio, 2005). The colors of such stellar populations are and . In Appendix B we describe in detail the procedure adopted to derive ; here we only remark that the productivity determined in this way turns out of , with small variations depending on the DTD model. In an SSP of 100 , there are 3 stars with mass between 2.5 and 8 , for a Salpeter IMF; this implies that the probability that one such star ends up as a SNIa should be of 3 % to account for the observed level of the SNIa rate in galaxies.

This estimate is in excellent agreement with the result in Paper I from the analysis of the cosmic SNIa rate, while it is % lower than what derived in Paper II, where we found . We ascribe this discrepancy to the different procedure employed: in paper II we only considered the correlation in the SUDARE data with the color, while here we average over the three surveys, and further average the results from the correlation with both and .

Once the productivity is scaled to galaxies with intermediate colors, the bluest and the reddest bins constrain the shape of the DTD. Inspection of Figures 10 and 11 shows that the available data do not allow us to draw a reliable conclusion on the DTD. The LOSS survey tends to favour a flat shape, while the CET99 and the SUDARE surveys support steeper DTDs. The origin of this discrepancy is unclear: for the red galaxies, the higher rate measured in LOSS compared to the other surveys could be ascribed to the superior statistics, but the discrepancy for the blue galaxies is puzzling. We need to understand the origin of this discrepancy in order to investigate on the shape of the DTD, e.g. with a more precise characterization of the properties of the reddest and bluest galaxies, concerning their intrinsic colors and SNIa rates.

To get further insight on this question we consider the correlation between the SNIa rate and the specific SFR of the parent galaxy. This allows us to include data from other independent surveys, as well as to minimize the systematics introduced by the SSP modelling. In Fig. 12 we show several determinations of such correlation, made on the three surveys SUDARE, CET99 and LOSS, plus measurements in the literature. For the galaxies in the SUDARE survey we use the sSFR determination output of the FAST code. For the galaxies in the LOSS and CET99 samples estimates of the SFR are not readily available, so that the corresponding points in Fig. 12 have been placed exploiting the relation between color and sSFR shown in Fig. 5.

For the literature values we have to use measurements as they have been published on the original papers as far as the choice of galaxy parameter and binning are concerned. Sullivan et al. (2006) obtained a measurement of the rate per unit mass as a function of the galaxy sSFR using data of the Supernova Legacy Survey. Similar measurements were later provided using data from the Sloan Digital Sky Survey (SDSS) by Smith et al. (2012) (imaging search) and by Graur & Maoz (2013) (spectroscopic search). The literature rates were obtained under the assumption of an IMF with a flat slope in the low mass range; we rescale them to our adopted Salpeter IMF by multiplyig the quoted values by a factor of 0.76.

The observations of the SNIa rate versus the specific SFR are compared in Fig. 12 to our models for the different DTDs. There is a reasonable agreement among the various data sets though for the star burst galaxies, the rate of the LOSS survey is relatively low whereas for the passive galaxies the CET99 rate appears particularly low. Actually, the difference between the LOSS and the CET99 rate on this plot mirrors what found from the correlation of the rate with the parent galaxy colors. The discrepancy between the literature values (filled symbols) and our determination for the passive galaxies in the LOSS survey may instead arise from the different evaluation of the galaxy masses used to normalize the rate. Indeed, as shown in Fig. 8, the way in which the galaxy masses are determined induces a systematic effect in the rate estimates, which appears to affect also the relation with the specific SFR. In Fig. 12 we also show the results of simulation by Graur et al. (2015) based on a power law DTD. The model was tailored to the galaxies in the Graur et al. (2015) dataset, and indeed they fit the filled diamonds in Fig.  12. However, also for this model the difference between the various observational measurements weakens the conclusion.

To summarize, once allowed for statistical and systematic uncertainties, the various measurements appear to be in broad agreement, but at present no robust conclusion on the DTD can be drawn from the comparison between the models and the data.

Figure 13: Theoretical correlation between the rate of SNIa and the color of the parent galaxy for the three sets of SSP models as labelled, adopting the log-normal SFHs plotted in Fig. 2 and the SD DTD.
Figure 14: Theoretical correlation between the rate of SNIa and the color of the parent galaxy for the four SFH laws and the SD DTD. Colors are based on adopting the BC03 solar metallicity SSP.

7 Summary and conclusions

In this paper we presented a detailed investigation of the systematic effects which hamper the derivation of the delay times distribution of SNIa progenitors from the analysis of their observed rate correlations with the properties of the parent galaxies. Specifically, we considered the effect of different sets of SSP models to compute the galaxy colors, that of different SFH laws in the galaxy population, and of different ways to evaluate the galaxy masses when computing the rate per unit mass in observational samples. We summarize our results as follows.

  • Different SSP sets provide different galaxy predicted colors for a given SFH. As a consequence, the correlation between the SNIa rate and the color of the parent galaxy depends on the set of SSP models adopted, as illustrated in Fig. 13 for one particular DTD model. The effect also depends on the considered color. In general, for a given observed correlation, both the derived SNIa productivity and slope of the DTD vary with the adopted set of SSP models. It is then very important to ensure that the chosen SSP set consistently describes the properties of the galaxy sample.

  • For a given DTD and SSP set, the models provide different renditions of the correlation between the SNIa rate and the parent galaxy color depending on the adopted SFH , e.g. the log-normal SFHs populate a narrower strip on this plane compared to the other options considered here (cf. Fig. 14).

  • Further uncertainty comes from the normalization of the observed rate to the galaxy mass. When estimating the galaxy mass, systematic differences arise from (i) different choices for the IMF, (ii) whether one considers the total mass of formed stars or the current stellar mass, (iii) the way in which one accounts for the dependence of the M/L ratio on the age distribution in each galaxy. We emphasize that to the end of deriving a robust estimate of  and of the slope of the DTD, the total mass of formed stars is the best tracer of the galaxy size, since the reduction factor due to the mass recycling in the evolution of the galaxy depends on details of the modelling (see Eq. (3)). In addition, the more accurate the assessment of the age distribution in the galaxy, the more reliable the determination of the mass-to-light ratio. Therefore, a procedure based on SED fitting on a wide color baseline is to be preferred to a plain relation between integrated color and the mass-to-light ratio.

  • The above systematic uncertainties have to be compared with the size of the effect we want to measure. As shown in Figures 10 and 11, at fixed scenario for the SFH and SSP model set, the dependence of the correlation on the DTD is not so dramatic to produce radically different trends. This follows from the similarity of the considered DTDs (see Fig. 1), combined with the relatively wide age distribution of stars in galaxies. However, steeper DTD’s do predict steeper correlations, so that accurate rates measured in the bluest and in the reddest galaxies allow us to to discriminate the different models.

The computations presented here are based on DTDs characteristic of either SD or DD progenitors. However, as mentioned in the Introduction, it is more likely that both evolutionary channels are at work in nature. In Greggio (2010) two extreme possibilities for mixed models are considered: the Solomonic mixture, in which both SD and DD channels contribute 50 % of the explosions at any delay time; and the segregated mixture, in which SD (DD) explosions contribute all events with delay times shorter (longer) than 0.15 Gyr, and both channels provide half of the total events from one stellar generation over a Hubble time. The theoretical correlations with the colors of the parent galaxies obtained with these two mixtures are very similar to those shown here, but for the segregated mixture in very blue galaxies. Infact, this mixture provides a very high fraction of prompt explosions, which reflects into a very large rate in stellar populations with very blue colors (). At present there are no data for such blue galaxies to constrain this possibility for the DTD. In this respect, we point out that, rather than validating a specific scenario for the DTD, our aim is to illustrate the limits of the discriminating power of the correlations at present. To this end we consider the representative DTD shapes derived in Greggio (2005), and neglect other propositions in the literature, e.g. the realizations of Binary Populations Synthesis models. Even in this simplified framework the constraining power of the correlations is subject to systematic effects which have not been considered so far.

We remark that our study is limited to considering how the correlation between the SNIa rate and the parent galaxy properties vary for different SFH laws, while neglecting the impact of the distribution of the parameters characterizing the sample galaxies within each SFH scenario (e.g. AGE and for the standard laws). Our analysis rests on the implicit assumption that the galaxy population is characterized by a flat distribution of these parameters. As it can be appreciated from Fig.,e.g., 14, the average rate in color (or equivalently sSFR) bins will be sensitive to the distribution of the SFH parameters within the galaxy population under consideration. Thus, a refined analysis requires constraining these parameters by, e.g., fitting the distribution of the sample galaxies on the two-color diagram. Alternatively, fitting the SFH to each galaxy to determine the expected SNIa rate would tailor the predictions to the specific galaxy sample. Although we did not perform such fit, our analysis shows that the actual distribution of the SFH parameters introduces one further systematic in the derivation of the DTD.

Figure 15: Theoretical and observational relative uncertainties on the value of the SNIa rate per unit mass as a function of the specific SFR. Open squares and dots show the statistical uncertainties for the LOSS survey and expected for the WFD and DDF LSST surveys, as labelled. The blue and grey stripes show the variance of the theoretical rates associated with different DTD models and different SFHs, respectively.

The discussion above leads to a key question: given the many statistical and systematic uncertainties, what are the prospects for the use of the existing rate measurements and of those expected from the next wide field surveys, to discriminate among alternative DTD scenarios ?

To evaluate the constraining power of current data set, we have used the information from three SN surveys (SUDARE, LOSS and CET99). As reference model, we choose the log-normal SFH description, the BC03 SSP models at solar metallicity and a single slope Salpeter IMF from 0.1 to 100 . With these parameters we find that adopting an average value broadly accounts for the observed correlations. This value is in excellent agreement with what found in Paper I from the analysis of the cosmic SNIa rate. Yet, the current data do not allow us to discriminate among the different DTD because of the discrepancies among the three data set. The LOSS data indicate a flat DTD, compatible with the DD Wide model; the CET99 and the SUDARE data indicate steeper DTDs. We conclude that at present all the DTDs considered here, which span a great variety of models for the SNIa progenitors, are consistent with the observations, once allowing for both the statistical and the systematic errors.

In an attempt to quantify the effect we want to measure and compare it with the different contributions to the uncertainties, in Fig. 15 we plot the variance (standard deviation) of the rates predicted by the different DTD models (SD,DDC,DDW,P.L.) in bins of specific SFR (blue strip). This is intended to represent the average difference between DTD models. Clearly, if we want to be able to discriminate between the various DTDs, the observational errors need to be smaller that this level. As we mentioned above, at present this is not the case. Even for the LOSS survey, which is the one with largest number of events, the statistical errors alone have the same magnitude than the predicted model differences (see empty squares in Fig. 15).

However, there are a few ongoing or planned searches that have the potential to strongly increase the number of detected events and hence lower the statistical error. Among them, the upcoming LSST survey will provide the largest database for the SNIa ever, possibly up to a million events in the whole 10 years of operation. Yet, not all the transients will have sufficient data to be properly classified. We use the conservative estimates derived in Astier et al. (2014) for the number of SNIa discovered and reliably classified in the LSST Wide Field Survey (WFS) and in the Deep Drilling Fields (DDF) which counts about events each. Fig. 15 shows the statistical uncertainties in the different sSFR bins associated to the Astier et al. (2014) estimates (red and blue dots). It can be seen that the predicted number of events will be so high in all sSFR bins to bring the statistical uncertainty much below the requirements to discriminate different DTDs. It is worth noting that the recent simulations of the LSST DESC collaboration carried out with the baseline survey strategy show that the survey will provide a superb statistics, with 100000 SNIa with well sampled light curves and spectroscopic redshift (The LSST Dark Energy Science Collaboration et al., 2018). At this point we need to turn our attention to the systematic errors.

According to Li et al. (2011b) the systematic error affecting their observed rates is of a similar magnitude as the statistical error, but it is mostly related to the choice of the global parameters of the computation (cf. Sect. 3.4). Therefore, while this error component is relevant for the determination of , it should not impact on the trend of the rate as function of the galaxy parameters, for which only the statistical errors should be considered.

On the other hand, the grey strip in Fig. 15, shows the variance of the rates for the four different SFH prescriptions explored here. It appears that differencies in the SFH and in the DTD induce a comparable variance of the rate, so that ambiguities on the SFH can potentially blur the chance to identify the dominant SNIa model.

Therefore, in spite of the excellent statistics which will be available with LSST data, the conclusions on the DTD will be affected by systematic uncertainties related to the SFH in the galaxies, and robust results will be achievable only with well constrained age distributions in the sample objects. The LSST dataset will also contain homogeneous multicolor data for the survey galaxies which will allow a fair characterization of the stellar populations, providing an ideal dataset for this kind of investigation.

To conclude, in order to reach firm constraints on the slope of the DTD from the correlation of the SNIa rate with the properties of the parent galaxy we need:

  • large galaxy samples, especially at the bluest and reddest colors; this will decrease the statistical uncertainty at the ends of the correlation, improving the leverage over the slope of the DTD;

  • accurate measurement of redshift and extinction to be able to analyze the correlations in the rest frame dereddened colors;

  • well constrained age distributions and masses of the sample galaxies, from multiband photometry ranging from the UV to the IR, i. e. sensitive to the whole range of stellar ages.

The LSST survey appears very promising in this respect, providing an unbiased, homogeneous and vast database. In parallel, we need to develop theoretical and observational studies on the star formation history in galaxies which account for the various properties of big samples, both in the local and in the distant universe. This will enable us to describe at best the distribution of stellar ages of the sample objects, a necessary ingredient to the end of constraining the DTD from the analysis of the correlation of the SNIa rate with the parent galaxy properties.

Acknowledgements.
We warmly thank Maria Teresa Botticella for fruitful discussions on the scientific content of the paper, and for a careful reading of the manuscript.

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Appendix A Dependence of the model galaxy colors on the SSP set

Figures 16 and 17 show, in the same fashion as Fig. 4 in the main text, the models computed with the various SFH prescriptions, respectively adopting the Maraston and the MG08 sets of SSP models. It can be noticed that the shape of the correlation between the and colors of galaxies with different age distributions depends on the adopted set of SSPs, and indeed it reflects the behaviour of pure SSP models on the two color diagram. For example, in , old solar metallicity SSPs in Maraston are bluer than in BC03, which are bluer than MG08 models, a trend reproduced by the colors of passive galaxies in Figs. 16, 4 and 17. The main axis of the distribution of star forming galaxies is not well matched by Maraston models, which show a too rapid redward evolution of the color at the red end of the distribution. A similar steep trend is also exhibited by the solar metallicity models based on the MG08 SSPs.

However, the colors of model galaxies are sensitive to the metallicity, so that the distribution of the data on this diagram could be met with a suitable distribution of chemical composition among the galaxies. Fig. 18 shows the colors of log-normal SFH models based on the Maraston and on the MG08 sets for subsolar and supersolar metallicities. Using Maraston models, the star forming galaxies could be matched with subsolar metallicity, while the passive galaxies would be better represented with solar abundances. When using MG08 models, the colors of passive galaxies are better matched by the subsolar metallicity set, while the star forming galaxies require a metallicity lower than .

Still, all in all, the BC03 solar metallicity models appear to capture in the best way the general features of the galaxies distribution on this plane, especially the slope of the main axis of the distribution.

Figure 16: The same as in Fig. 4 but adopting Maraston SSP models with solar metallicity.
Figure 17: The same as in Fig. 4 but adopting MG08 SSP models with solar metallicity.
Figure 18: Two color diagram for log-normal SFHs using the Maraston (left panel) and the MG08 (right panel) SSP models with subsolar (upper loci) and supersolar (lower loci) metallicities. The models are superimposed on the distribution of the rest frame color of the SUDARE galaxy sample. The symbol and color encoding is the same as in the bottom left panel of Fig. 4.

Appendix B Determining the SNIa productivity

By construction, the SNIa rate at late epochs in a system with a flat age distribution is equal to the SNIa productivity () divided by the age of the system (see Eq. (3)), independent of the DTD. Stellar populations constructed with a constant SFR for 13 Gyr have and , for BC03 solar metallicity SSP models. Thus the most robust calibration of  is obtained from the SNIa rate in systems with intermediate colors. In practice, however, galaxies with the same intermediate color, may have quite different SNIa rates (see Fig. 10) due to different individual SFHs. Therefore, the rate measured in, e.g., the color bin centered at will, to some extent, be sensitive to the specific distribution of the SFHs of the galaxies populating the bin. This effect can be taken into account only by fitting the age distribution in the individual galaxies, which we deem as a too refined procedure given the heterogeneous datasets in hand. We adopt instead a strategy which aims at averaging over a wide parameter space, by considering the model predictions in the two central color bins for the LOSS and CET99 samples, and in the two reddest color bins for the SUDARE sample. Operationally we proceed as follows: we compute all the models shown in Fig. 2 with a fixed time step of 0.1 Gyr, mimicking a galaxy population evenly distributed among the parameters values in Fig. 2, and compute the average of the theoretical rate (for =1) of all models which fall in a specific color bin. The ratio between the rate measured in the color bin and the computed average model rate yields a value for . We repeat the procedure in the adjacent color bin to construct an average value for  which best represents the measured rate in the galaxies with intermediate color. We perform this computation for each DTD and each observational sample, and show the results in Fig. 19. The figure shows that the various determinations are in broad agreement. Compared to LOSS, the SUDARE and CET99 datasets yield systematically higher values of the productivity in both colors, but the discrepancy is lower when calibrating on the rate vs correlation. For each DTD, the average values of  derived from the two colors are in excellent agreement, and the dependence of the final average values for the different DTDs is negligible. These features support the conclusion that the derived value of the productivity is quite robust.

Figure 19: SNIa productivity resulting from the calibration of the models on the rates measured on galaxies with intermediate color. The point type encodes the observational survey as in the legend. Each of the four groups refers to a different DTD labelled on the bottom axis. Within each group, points to the left (in blue) result from the calibration on galaxies binned in (as in Fig. 10); points to the right (in red) from the calibration on galaxies binend in (as in Fig. 11). The error bars reflect the statistical uncertainties of the measurements of the rates. For each DTD, the  values determined from the three surveys are combined to provide a weighted average  shown as a blue (left) and red (right) star. Finally the asterisked circle (black) shows the value of  obtained combining the determinations from the two colors in a weighted mean. The error bars on the average values show the relative 1 dispersions.
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