Galactic and cosmic Type Ia supernova (SNIa) rates: is it possible to impose constraints on SNIa progenitors?

Abstract

We compute the Type Ia supernova (SNIa) rates in typical elliptical galaxies by varying the progenitor models for SNeIa. To do that a formalism which takes into account the delay distribution function of the explosion times and a given star formation history is adopted. Then the chemical evolution for ellipticals with baryonic initial masses 10¹⁰, 10¹¹ and 10¹² M_⊙ is computed, and the mass of Fe produced by each galaxy is precisely estimated. We also compute the expected Fe mass ejected by ellipticals in typical galaxy clusters (e.g. Coma and Virgo), under different assumptions about SNIa progenitors. As a last step, we compute the cosmic SNIa rate in a unitary volume of the Universe by adopting several cosmic star formation rates and compare it with the available and recent observational data. Unfortunately, no firm conclusions can be derived only from the cosmic SNIa rate, neither on SNIa progenitors nor on the cosmic star formation rate. Finally, by analysing all our results together, and by taking into account previous chemical evolution results, we try to constrain the best Type Ia progenitor model. We conclude that the best progenitor models for SNeIa are still the single degenerate model, the double degenerate wide model and the empirical bimodal model. All these models require the existence of prompt SNeIa, exploding in the first 100 Myr since the beginning of star formation, although their fraction should not exceed 15–20 per cent in order to fit chemical abundances in galaxies.

supernovae: general, galaxies: evolution, cosmology

1 INTRODUCTION

The study of the mechanisms of supernova (SN) explosions and the analysis of their nucleosynthesis products are two key ingredients for understanding the chemical evolution of galaxies. Type Ia supernovae (SNeIa) are thought to be the main contributors to the chemical enrichment in iron in the Universe and they have also a significant influence on the early and late evolution of galaxies. In elliptical galaxies, in fact, after the occurrence of the galactic wind and the end of the star formation, they are the only SNe occurring. They contribute to eject continuously energy and iron which eventually will reach the intracluster medium (ICM). SNeIa are also used to track the Hubble law and therefore they are fundamentally important in cosmology.

Therefore, the description of the evolution of the Type Ia supernova (SNIa) rate, in galaxies and in a unitary volume of the Universe (cosmic rate), is a crucial information for galaxy evolution and cosmology. The computation of the SNIa rate is related to the nature of the progenitor systems, which unfortunately is still poorly known. The observed features of SNeIa suggest that these objects may originate from the thermonuclear explosion of CO white dwarfs (WD) of Chandrasekhar mass and the two theoretical main scenarios which have been proposed so are (a) the single degenerate (SD) scenario and (b) the double degenerate (DD) scenario. Recently, it has been suggested also a double detonation in sub-Chandrasekhar masses as a possible mechanism for explaining some of the SNeIa. In fact, in the last years it has become more and more evident the existence of a variety of SNeIa. Here we will analyse only the classical DD and SD scenarios (see Hillebrandt et al. 2013 for a recent review on all possible scenarios).

In the original SD scenario there is a binary system in which the primary component has mass in the range (2–8) M_⊙ while the secondary component is a non-degenerate companion, a red giant or a main-sequence star (e.g. Whelan & Iben 1973), that has mass in the range (0.8–8) M_⊙. The lower limit, in the range of masses, is due to the fact that the only systems of interest are those capable of generating a SNIa in a Hubble time to explain the existence of SNeIa in ellipticals. The upper limit instead is imposed by the fact that single stars with masses M > 8 M_⊙ ignite carbon in a non-degenerate core and do not end their lives as CO WDs. When the secondary star evolves and fills its Roche lobe, the WD accretes material. Thanks to the accretion of matter, via mass transfer from the non-degenerate companion, the primary star reaches the Chandrasekhar mass and explodes. For many years the only suggested explosion mechanism was C-deflagration, but recently it has been suggested that in SD scenario SNe explode via a detonation, after deflagration has been initiated. On the other hand, for the DD scenario the explosion occurs via a prompt detonation or a double detonation (e.g. Pakmor et al. 2013). One of the limitations of the SD scenario is that the accretion rate should be defined in a quite narrow range of values. To avoid this problem, Kobayashi et al. (1998) had proposed a similar scenario, based on the model of Hachisu, Kato & Nomoto (1999), where the companion can be either a red giant or a main-sequence star, but including a metallicity effect which suggests that no Type Ia systems can form for [Fe/H] < −1.0 dex. This is due to the development of a strong radiative wind from the C–O WD which stabilizes the accretion from the companion, and allowing the WD to reach the Chandrasekhar limit for a wide binary parameter space than the previous scenario. This scenario will not be considered here since models of galactic chemical evolution (e.g. Matteucci & Recchi 2001) have demonstrated that the SNIa rate in the Galaxy reaches a maximum when [Fe/H] = −1.0 dex, thus making the long delay due to the metallicity effects unrealistic.

In the original DD scenario the binary system is composed of two CO WDs that, because of the emission of gravitational wave radiation, lose angular momentum and merge achieving the Chandrasekhar mass (e.g. Iben & Tutukov 1984) and explode as mentioned above. The progenitor masses are defined in the range (5–8) M_⊙ to ensure two WDs of 0.7 M_⊙ and then reach the Chandrasekhar mass. The time of the explosion is the lifetime of the secondary star plus the time necessary to merge. The validity of this scenario requires that the two CO WDs have an initial separation less than 3 R_⊙, condition that can be reached by means of two different precursor systems: a close binary and a wide binary system. The two scenarios differ for their efficiency of the common envelope phase during the first mass transfer, and therefore for the separation attained at the end of the first common envelope phase. The analysis of the SNIa rates in the past years has been the subject of several works, as the pioneering works of Greggio & Renzini (1983), Iben & Tutukov (1984), Tornambé & Matteucci (1986) and Matteucci & Greggio (1986), and of the most recent works like those of Matteucci & Recchi (2001), Strolger et al. (2004), Greggio (2005), Mannucci, Della Valle & Panagia (2006), Pritchet, Howell & Sullivan (2008), Totani et al. (2008) and Valiante et al. (2009). An extensive review on the subject can be found in Maoz & Mannucci (2012). We aim at testing various distributions of explosion times and also different star formation (SF) histories for elliptical galaxies. In fact, once established the nature of the SNIa progenitors, the SNIa rate is the convolution of the distribution of the explosion times, usually called the delay time distribution (DTD) function, with the star formation rate (SFR) of the studied galaxy. The adopted DTDs, which describe the rate at which SNeIa explode as a function of time in a simple stellar population (a starburst), refer to the SD model (white dwarf plus red giant companion) and to the DD model (two white dwarfs) as well as to empirical DTDs, as suggested by various authors. A large convergence is found for an empirical DTD proportional to t⁻¹ which provides a behaviour very similar to that predicted by the DD scenario (see later). The purpose of this work is also the computation of the cosmic Type Ia supernova rates, that is the rate as a function of redshift of SNIa explosions in a unitary volume of the Universe. The cosmic Type Ia supernova rate is computed using the same DTDs and different cosmic SF histories. In this case we need to use a cosmic star formation rate (CSFR) that is usually expressed in M_⊙ yr⁻¹ Mpc⁻³. Such histories are partly derived from a fit of observational data and partly from theoretical models making different assumptions about the number density evolution of galaxies. In fact, in the pure luminosity evolution scenario, the number density of galaxies is considered constant as a function of redshift. In the hierarchical galaxy formation scenario, instead, the number density of galaxies is assumed to change with the cosmic time. Therefore, one can in principle, put constraints on the mechanisms of galaxy formation by comparing theoretical results with data of CSFR. In this paper we will compare our predicted cosmic SNIa rates with the latest compilation of data for the SNeIa. Previous works (Maoz & Gal-Yam 2004; Förster et al. 2006; Valiante et al. 2009; Maoz, Sharon & Gal-Yam 2010) have already tried to infer constraints on SNIa progenitors from the cosmic SNIa rate. In particular, Maoz et al. (2010) studied SNIa rates in clusters of galaxies and concluded that a DTD ∝ t^−1/2 can best reproduce the data. However, no firm conclusions were suggested. One difference with the previous works is that here we test the various DTDs also in the chemical enrichment of the ICM. The paper is organized as follows. In Section 2 we describe how to compute the SNIa rate in galaxies, as well as the different DTD functions for the explosion times. In Sections 3 and 4 we present the results for the SNIa rates in a typical elliptical under different assumptions about the SNIa progenitor models. In Section 5 we show the effects of different DTDs in galaxy models on the predicted Fe and gas mass produced by ellipticals in galaxy clusters. In Section 6 we describe how to compute the cosmic SNIa rate by means of different CSFRs and we compare our model results to data. In Section 7 some conclusions are drawn.

2 TYPE IA SN RATE: THE FORMALISM

The SNIa rate, from a theoretical point of view, is difficult to derive because of the uncertainty in the nature of progenitors. Assuming that the SD scenario is valid, the SNIa rate can be written as (Greggio & Renzini 1983)

\begin{equation} R_{{\rm SNIa}}={A_{{\rm Ia}}}\int _{M_{{\rm Bm}}}^{M_{{\rm BM}}}{\varphi (m)\bigg [\int _{\mu _{\rm Bmin}}^{0.5} {f(\mu _{\rm B})\psi (t-\tau _{m2})\mathrm{d}\mu _{\rm B} \bigg ]}{\rm d}m} \end{equation}

(1)

where

• ψ(t − τ_m2) is the SFR at the time of the birth of the binary system;

• A_Ia is the fraction of binary systems which can give rise to SNeIa only relative to the mass range (3–16) M_⊙;

• M_Bm and M_BM are the maximum and minimum total mass of the binary systems able to reproduce a SNIa explosion. The value of the upper limit M_BM = 16 M_⊙ is due to the assumption that the more massive system should be made of two stars of 8 M_⊙ each. Instead the minimum total mass of the binary system is assumed to be M_Bm = 3 M_⊙ to ensure that the companion of the WD is large enough to allow the WD with the minimum possible mass (∼0.5 M_⊙) to reach the Chandrasekhar mass limit (∼1.44 M_⊙) after accretion;

• the function f(μ_B) is the distribution of the mass fraction of the secondary stars in binary systems, namely, |${\mu _{\rm B}}=\frac{M2}{(M1+M2)}$|⁠, with M1 and M2 being the primary and secondary masses of the system, respectively. This function is derived observationally and in the literature it has often been written (see Greggio & Renzini 1983; Matteucci & Greggio 1986) using the following expression:

  \begin{equation} f(\mu _{\rm B})=2^{1+\gamma }(1+\gamma ){\mu _{\rm B}}^\gamma \end{equation}

  (2)

  for |$0>\mu _{\rm B}\ge \frac{1}{2}$|⁠;

• τ_M2 is the lifetime of the secondary star and represents the time elapsed between the formation of the binary system and its explosion.

If we analyse the DD model, the supernova rate can be computed in the following way (Tornambé & Matteucci 1986):

\begin{eqnarray} R_{{\rm SNIa}} = Cq\!\!\int _{M_{{\rm min}}}^{M_{{\rm max}}}\varphi (M)\bigg [\int _{S_{{\rm B}_{{\rm min}}}}^{S_{{\rm B}_{{\rm max}}}}\Psi (t-\tau _{M2} -\tau _{{\rm grav}})\, {\rm d}\log S_{\rm B}\bigg ]{\rm d}M, \nonumber\\ \end{eqnarray}

(3)

where

• C is a normalization constant;

• |$q=\frac{M_2}{M_1}=1$| is the ratio between the secondary and the primary mass which, in this scenario, is assumed, for simplicity, to be equal to unity;

• S_B is the initial separation of the binary system at the beginning of the gravitational wave emission;

• τ_grav is the gravitational time-delay, that for systems which can give rise to SNe of Type Ia, varies from 10⁶ to 10¹⁰ yr and more (Landau & Lifshitz 1962):

\begin{equation} \tau _{{\rm grav}}=1.48\times 10^8{\frac{ \left({S_{\rm B} \over {\rm r}_{{\odot }}}\right)^4}{ \left({M_1 \over \mathrm{M}_{{\odot }}}\right ) \left({M_2 \over \mathrm{M}_{{\odot }}} \right) \left({M_1 \over \mathrm{M}_{{\odot }}}+ {M_2 \over \mathrm{M}_{{\odot }}} \right)}}\,{\rm yr}. \end{equation}

(4)

 It is now possible to use a new more general formulation for the SNIa rate, as proposed by Greggio (2005, hereafter G05), in which the SNIa rate is defined like the convolution between

• the distribution of the explosion times of SNIa progenitors, that is the DTD function and characterizes the progenitor model;

• the SFR, namely the amount of gas turning into stars per unit time, that usually is expressed in |$\frac{\mathrm{M}_{\odot }}{\rm yr}$|⁠.

The peculiarity of this formulation is that it allows us to include any DTD that can represent a different scenario with respect to the SD and DD:

\begin{equation} R_{{\rm SNIa}}(t)={k_\alpha }\int _{\tau _i}^{{\rm min}(t,\tau _x)}{A_{{\rm Ia}}}(t-\tau )\psi (t-\tau )\, {\rm DTD}(\tau )\,{\rm d}\tau , \end{equation}

(5)

where

• A_Ia(t − τ) is the fraction of binary systems which can give rise to SNeIa and in principle it can vary in time. In this new formulation A_Ia is relative to the whole range of star masses (0.1–100) M_⊙, not only relative to the mass range (3–16) M_⊙, as it is in the old formulation. This is unfortunately a free parameter which is fixed by reproducing the present time SNIa rate;

• The DTD is defined in the range (τ_i, τ_x) and normalized as

  \begin{equation} \int _{\tau _i}^{\tau _x}{\rm DTD}(\tau )\,{\rm d}\tau =1; \end{equation}

  (6)

• τ_i is the time at which the first SNeIa start exploding, namely is the minimum delay time for the occurrence of SNeIa, while τ_x is the maximum delay time;

• k_α is the number of stars per unit mass in a stellar generation and contains the initial mass function (IMF); in particular,

  \begin{equation} k_\alpha =\int _{M_L}^{M_U}\varphi (m)\, \mathrm{d}m, \end{equation}

  (7)

  with M_L = 0.1 M_⊙ and M_U = 100 M_⊙.

The IMF is the mass distribution of stars at birth and its most common parametrization is that of Salpeter (1955) which is a simple one-slope power law generally defined in the mass range (0.1–100) M_⊙:

\begin{equation} \varphi (m)=am^{-(1+x)}, \end{equation}

(8)

where

• a is the normalization constant derived by imposing that

  \begin{equation} \int _{0.1}^{100}m\varphi (m)\,{\rm d}m=1. \end{equation}

  (9)

In this paper we have analysed different DTDs that are (i) the DTD of the SD scenario as proposed by Matteucci & Recchi (2001); (ii) the DTD of the wide DD scenario as proposed by G05, and four empirical DTDs that are (i) a bimodal DTD proposed by Mannucci et al. (2006); (ii) a Gaussian DTD proposed by Strolger et al. (2004); (iii) two power-law DTDs proposed by Pritchet et al. (2008) and Totani et al. (2008). It should be noted that in the literature there are many DTDs calculated by means of detailed binary evolution calculations (e.g. Yungelson & Livio 2000; Han & Podsiadlowski 2004; Belczynski, Bulik & Ruiter 2005; Ruiter, Belczynski & Fryer 2009; Mennekens et al. 2010; Wang, Li & Han 2010). However, these detailed calculations are not always easy to use for the kind of semi-analytical calculations performed in this paper.

2.1 The DTD of the SD scenario

The SD model (Whelan & Iben 1973), as described by Matteucci & Recchi (2001, hereafter MR01), is computed adopting the following formalism:

\begin{equation} {\rm DTD}(\tau )\propto \widetilde{\phi }(M_2)\frac{\mathrm{d}M_2}{\mathrm{d}t}, \end{equation}

(10)

which corresponds to the SNIa rate for an instantaneous starburst. The function |$\widetilde{\phi }(M_2)$| is the mass function of the secondary component that, for this DTD, is equal to

\begin{equation} \widetilde{\phi }(M_2)=2^{(1+\gamma )}(1+\gamma ){M_2^\gamma }\frac{\left({M_{\rm b}^{(-s-\gamma )}}-{M_{\rm B}^{(-s-\gamma )}}\right)}{(-s-\gamma )}, \end{equation}

(11)

where s = 1 + x with x being the Salpeter index. The derivative |$\frac{{\rm d}M_2}{{\rm d}t}$| was obtained adopting the inverse of the formula (Padovani & Matteucci 1993)

\begin{equation} \tau (M)=10^{\frac{1.338-\sqrt{1.79-0.2232(7.764-\log (M))}}{0.1116}}, \end{equation}

(12)

which defines the relation between the stellar mass M expressed in M_⊙ and the main-sequence lifetime τ expressed in year. The values of the parameters we will adopt in this study are γ = 0.5, s = 2.35, M_b = max (2M₂, M_min), M_B = M₂ + 0.5(M_max), M_min = 3 M_⊙, M_max = 16 M_⊙. The fraction of prompt SNe in this scenario is ∼10 per cent. It is worth noting that this MR01 DTD behaves like t^−1.6 at late times, in agreement with the empirical DTD found for galaxy clusters derived by Sand et al. (2012).

2.2 The DTD of the DD scenario

The analytic formulation to describe the DTD in the DD scenario was proposed by G05. In that paper there is the analytic formulation of the DTD for two different schemes. (i) Close DD scheme, in which the close binary evolution leads to a narrow distribution of the separations, so that the initially closest binaries merge in a short time, and the initially widest binaries tend to populate the long τ_grav tail of the distribution. In addition, the most massive binaries tend to end up with the smallest final separation, hence merge more quickly. (ii) Wide DD scheme, in which the close binary evolution produces a wide distribution of separations and total binary masses, and these two variables are virtually independent. We considered only the wide DD scheme because it is more in agreement with the observations suggesting the existence of SNeIa exploding at the present in elliptical galaxies, which stopped to form stars several Gyr ago:

\begin{eqnarray} {\rm DTD}(\tau )= \int _{\tau _{n,{\rm inf}}}^{{\rm min}({\tau _{n,x}},\tau )}n(\tau _n)S^{\rm C}(\tau , \tau _n)\, {\rm d}\tau _n \end{eqnarray}

(13)

\begin{eqnarray} \hphantom{{\rm DTD}(\tau )= \int _{\tau _{n,{\rm inf}}}^{{\rm min}({\tau _{n,x}},\tau )}}{\rm for}\ {\rm close\ DD,} \end{eqnarray}

(14)

\begin{eqnarray} \hphantom{{\rm DTD}(\tau )= }\int _{\tau _{n,i}}^{{\rm min}({\tau _{n,x}},\tau )}n(\tau _n)S^{\rm W}(\tau , \tau _n)\,{\rm d}\tau _n \end{eqnarray}

(15)

\begin{eqnarray} \hphantom{{\rm DTD}(\tau )= \int _{\tau _{n,{\rm inf}}}^{{\rm min}({\tau _{n,x}},\tau )}}{\rm for}\ {\rm wide\ DD}, \end{eqnarray}

(16)

where

\begin{eqnarray} S^{\rm C}(\tau , \tau _n)= \frac{(\tau -\tau _n)^\beta _g}{{\tau _{gw,x}^{(1+\beta _g)}}-{\tau _{gw,i}^{(1+\beta _g)}}} \end{eqnarray}

(17)

\begin{eqnarray} {\rm for} \,\, \tau _n \le \tau -\tau _{gw,i} \end{eqnarray}

(18)

\begin{eqnarray} =0 \,\, {\rm for} \,\, \tau _n \ge \tau -\tau _{gw,i}, \end{eqnarray}

(19)

\begin{eqnarray} S^{\rm W}(\tau , \tau _n)= f_{1,2}^{\rm W}(\tau , \tau _n)^{(-0.75+0.25\beta _a)} \end{eqnarray}

(20)

\begin{eqnarray} {\rm for} \,\, \tau _n \le \tau -\tau _{gw,i} \end{eqnarray}

(21)

\begin{eqnarray} =0 \,\,{\rm for} \,\, \tau _n \ge \tau -\tau _{gw,i}. \end{eqnarray}

(22)

The functions S^C(τ, τ_n) and S^W(τ, τ_n) are |$\propto \frac{\mathrm{\partial} g(\tau , \tau _n)}{\mathrm{\partial} \tau }$|⁠, where g(τ, τ_n) is the fraction of systems which having a nuclear delay equal to τ_n have also a total delay shorter than τ:

\begin{eqnarray} \tau _{n,{\rm inf}}= \tau _{n,i} & \quad {\rm for} & \tau <\tau _{n,i}-{\tau _{gw,x}}(\tau _{n,i}), \end{eqnarray}

(23)

\begin{eqnarray} \hphantom{\tau _{n,{\rm inf}}= {i}}\tau _n^\star & \quad {\rm for} & \tau \ge \tau _{n,i}-{\tau _{gw,x}}(\tau _{n,i}) \end{eqnarray}

(24)

and

1. |$\tau _n^\star$| is the solution of the equation τ = τ_n + τ_{gw, x}(τ_n);

2. τ_n is the nuclear lifetime of the secondary;

3. τ_i is the absolute minimum delay of SNIa progenitors;

4. |$\tau _{gw}=\frac{0.15 A^4}{({m_{1R}}+{m_{2R}}){m_{1R}}{m_{2R}}}\,\mathrm{Gyr}$| with A, m_1R, m_2R, respectively, the separation and component masses of DD system, in solar units.

2.3 Empirical DTDs

2.3.1 Bimodal DTD

The bimodal DTD proposed by Mannucci et al. (2006, hereafter MVP06) is derived empirically thanks to the observations of supernovae in radio galaxies. The bimodal trend is given by the sum of two distinct functions:

• a prompt Gaussian centred at 5 × 10⁷ yr;

• a much slower function, either another Gaussian or an exponentially declining function.

This DTD describes a situation where a percentage form 35 to 50 per cent of all SNeIa explode during the first 100 Myr since the beginning of SF (prompt SNeIa), while the rest explodes with larger delays as long as the Hubble time and more (tardy SNeIa). This DTD can be analytically approximated by the following expressions (Matteucci et al. 2006):

\begin{eqnarray} \log {\rm DTD}(\tau ) = 1.4-50(\log \tau -7.7)^2 & \quad {\rm for} & \quad \tau <85\,\mathrm{Myr}, \nonumber\\ \end{eqnarray}

(25)

\begin{eqnarray} -0.8-0.9(\log \tau -8.7)^2 & \quad {\rm for} & \quad \tau >85 \,\mathrm{Myr.} \nonumber\\ \end{eqnarray}

(26)

This bimodal DTD can be associated with the SD scenario.

2.3.2 Gaussian DTD

The Gaussian DTD proposed by Strolger et al. (2004, hereafter S04) is derived thanks to the analysis of the data obtained with Hubble Space Telescope. These observations have allowed to discover 42 SNe in the redshift range 0.2 < z < 1.6. As these data span a large range in redshift, they are ideal for testing the validity of Type Ia supernova progenitor models with the distribution of expected delay times. The result suggests that the models that requires a large fraction of prompt SNeIa poorly reproduces the observed redshift distribution and are rejected at greater than 95 per cent confidence. The conclusion is that Gaussian models best fit the observed data for mean delay times in the range of (2–4) Gyr. The formula of this DTD is

\begin{equation} {\rm DTD}(\tau ,t_{\rm d})=\frac{1}{\sqrt{2\pi \sigma _{\tau }^2}}{\rm e}^{\frac{-(\tau -t_{\rm d})^2}{2\sigma _{t_{\rm d}}^2}}, \end{equation}

(27)

where |$\sigma _{t_{\rm d}}=0.5 t_{\rm d}$| and t_d is the characteristic delay time, here assumed to be 4 Gyr.

2.3.3 Power-law DTDs

The two power-law DTDs are proposed by Totani et al. (2008) and Pritchet et al. (2008). The relation relative to these functions is

\begin{equation} {\rm DTD}(\tau )\propto {\tau ^{\alpha }}, \end{equation}

(28)

with α = −1 in the Totani et al. case, and α = −0.5 in the Pritchet et al. case.

The DTD of Totani et al. (2008) was suggested on the basis of faint variable supernovae detected in the Subaru/XMM–Newton Deep Survey (SXDS). The sample used for the definition of this DTD was composed of 65 SNe showing significant spatial offset from the nuclei of the host galaxies having an old stellar population at z ∼ 0.4–1.2 out of more than 1000 SXDS variable objects. This DTD supports the idea of the DD model (Maoz & Mannucci 2012). The DTD proposed by Pritchet et al. (2008) instead was obtained by the data derived from the Supernova Legacy Survey (SNLS) (Sullivan et al. 2006).

2.4 Properties of the different DTD functions

All the trends of the six different DTDs analysed here are reported in Fig. 1.

Figure 1.

Illustration of the various DTD functions normalized to their own maximum value. The solid red line is the DTD of MVP06; the dashed blue line is the DTD of MR01; the short dashed–dotted magenta line is the DTD of G05; the dotted green line is the DTD of S04; the black short dashed–dotted black line is the DTD of Pritchet et al. (2008); the long dashed yellow line is the DTD of Totani et al. (2008).

The SD and DD formulations follow from general considerations on the evolutionary behaviour of stars in binary systems. Some parameters play a key role in shaping the SD and DD DTDs, most notably:

• the mass range of the secondaries in systems which provide SNeIa events;

• the minimum mass of the primary which yields a massive enough CO WD to ensure the explosion;

• the efficiency of the common envelope process;

• the efficiency of accretion for the SD model;

• the distribution of the separation for the DD system at their birth.

The empirical DTDs instead are derived directly from observations.

3 THE PREDICTED TYPE IA RATES IN A TYPICAL ELLIPTICAL

We have predicted the SNIa rate for an elliptical galaxy having a baryonic mass of 10¹¹ M_⊙. The assumed cosmology, through all the paper, is the Λ cold dark matter (ΛCDM) cosmology with Ω_M = 0.3, Ω_Λ = 0.7 and H₀ ∼ 67 km s^{− 1} Mpc^{− 3}. The SN rate can be computed like the convolution of a given SFR, that it defined as the amount of gas turning into stars per unit time, and a particular DTD, that is the distribution of the explosion times, as shown in equation (5).

Initially, with the purpose of verifying mostly the dependence of the SN rates on the DTD, we have computed the SNIa rate using only a given SFR and the six different distribution functions studied previously. The trend of this SFR is shown in Fig. 2. This SFR is obtained using the Pipino & Matteucci (2004) model applied to an elliptical galaxy of 10¹¹ M_⊙ of initial luminous mass, when the SD model for SNIa progenitors is assumed. In Pipino & Matteucci (2004) model the SFR stops when a galactic wind, triggered mainly by SNe II and Ib,c and partly by SNeIa, occurs. Other mechanisms exist to devoid galaxies from gas such as ram pressure stripping, tidal stirring and strangulation, but we do not consider them here. In any case, there should be a mechanism that quenches SF in ellipticals and galactic winds seem the most reasonable one. Moreover, as shown by Pipino & Matteucci (2004), galactic winds occurring first in massive than in small ellipticals can very well explain the increase of the [α/Fe] ratio with galactic mass in ellipticals. Whether such a wind continues for the whole galaxy lifetime is not clear and probably the other mechanisms will be at action as well. However, what matters here is that the gas is soon or later lost into the ICM.

Figure 2.

The SFR for an elliptical galaxy of 10¹¹ M_⊙. The occurrence of a galactic wind at t_GW = 0.67 Gyr stops the SF.

As it can be seen from the Fig. 2 the SFR has a simple form given by the Schmidt–Kennicut law, |$\psi (t)=\nu \sigma _{\rm gas}^k$|⁠, with an efficiency of SF ν = 10 Gyr⁻¹ and k = 1. The SFR is halted by the occurrence of a galactic wind at t_GW = 0.67 Gyr (see also Valiante et al. 2009).

An important parameter, introduced in the calculation of the rate, is the constant A_Ia that represents the fraction of systems which are able to originate a SNeIa explosion. The value of this constant is calculated a posteriori and it is chosen so as to ensure that the predicted present day SNIa rate is reproduced. The assumed SNIa rates is given by Cappellaro, Evans & Turatto (1999), that is 0.18 ± 0.06SNu, where 1SNu = 1SN/10¹⁰ L_⊙B century^{− 1}.

It is necessary to consider that an elliptical galaxy of initial luminous mass of 10¹¹ M_⊙ at the present time has a lower luminous mass, because of the presence of the galactic winds. Then the value of the constant A_Ia, that we have computed, is relative to a galaxy with a stellar mass of ∼3.5 × 10¹⁰ M_⊙, that is the final mass of the galaxy. As demonstrated by Valiante et al. (2009), to obtain the present time SNIa rate, in units of SNe century⁻¹, for an elliptical with a stellar mass of 3.5 · 10¹⁰ M_⊙, one should multiply the Cappellaro et al. rate by the blue luminosity predicted for a such galaxy, obtaining a SNIa rate of 0.072 SNe century⁻¹ (see Valiante et al. 2009). So we have considered this value for the SNIa rate as reference.

3.1 Properties of different Type Ia SN rates

Fig. 3 represents the predicted SNIa rates as functions of redshift for the different DTDs. From this figure it is possible check that all the curves have a similar trend, except for the rate of S04. This difference in the behaviour of the rate of S04 is due to the fact that this DTD does not contain any prompt SNeIa and then the peak of this curve is shifted at longer times (several Gyr). Also the curve for the rate of Pritchet has a particular trend; in fact, it is possible to verify that the peak of this is much lower compared to all the other curves. The trend of the rate of Totani et al. (2008) is very similar to the rate of G05 for wide binaries, in agreement with the conclusions of Totani et al. (2008).

Figure 3.

The SNIa rate, as a function of time, for an elliptical galaxy of 10¹¹ M_⊙. Symbols are same as in Fig. 1.

Figure 4.

Illustration of the SFR, as a function of time, for an elliptical galaxy of 10¹¹ M_⊙. As one can see, the shape of the SFR is the same for all the cases but the SFR is truncated at different times in the different cases. Symbols are same as in Fig. 1.

In Table 1 we report the values derived from equation (5) of the various values of the parameter A_Ia, that we considered constant in time, used in the different DTD models in order to obtain a present time SNIa rate in agreement with observations for a typical elliptical. However, these values for A_Ia are only indicative since they change if the history of SF changes.

Table 1.

Values of the normalization constant cost_dtd and A_Ia used in the different models. The models refer to the same SF history, the one of Fig. 2, but to different DTDs.

Model	AIa
Mannucci et al. (2006)	0.0053
Matteucci & Recchi (2001)	0.015
Pritchet et al. (2008)	0.00025
Totani et al. (2008)	0.0013
Strolger et al. (2004)	0.028
Greggio (2005)	0.0002

4 A MORE SELF-CONSISTENT COMPUTATION OF THE TYPE Ia SNe RATE IN A TYPICAL ELLIPTICAL

The above analysis has thus allowed us to emphasize how the DTD function may affect the calculation of the SNIa rate. With the purpose of having a more precise estimate of the SNIa rate, we have recalculated the various SNIa rates by adopting the SFR predicted by the chemical evolution model for the various DTDs. In fact, according to the various DTDs, the galactic wind occurs at different times, thus changing the history of SF by truncating it at different times. We recall that a galactic wind occurs when the energy injected into the interstellar medium (ISM) by SNe equates the gas binding energy. This approach is clearly more self-consistent relative to previous one and it allows us to compute in detail the Fe production in ellipticals, as we will see in Section 5. This fact was not considered in Valiante et al. (2009).

Because of the high similarity of Totani et al. (2008) DTD and that of G05 for wide binaries, we will present since now on only the model with the Totani et al. (2008) DTD (∝t⁻¹).

4.1 Star formation rates

The histories of SF dictated by the various DTDs are shown in Fig. 4: one can immediately see how the only SFR that predicts the occurrence of the galactic wind before ∼0.67 Gyr, that is the value relative to the previous SFR relative to the DTD of MR01 (SD scenario), is the SFR relative to the bimodal DTD of MVP06. In this model, in fact, the birth of a galactic wind occurs at ∼0.46 Gyr, due to the larger number of prompt SNeIa in this DTD. All the other SFRs are lasting for a longer time. On the other hand, the system in which the galactic wind occurs latest is the one related to Pritchet et al. (2008) DTD; in this system, in fact, the SFR vanishes at ∼2.64 Gyr, because of the absence of prompt SNeIa. In this case, a large fraction of the Fe produced by SNeIa remains trapped into stars. In the SFR with Totani et al. (2008) DTD the galactic wind occurs at ∼1.6 Gyr, while the galactic wind time obtained with S04 is ∼1.75 Gyr.

The values predicted by the different models for the occurrence of the galactic wind, the total number of SNeIa exploded during the Hubble time and the total mass of Fe they produced are shown in Table 2. The total mass of Fe produced by SNeIa has been computed by assuming that each SNIa produces 0.6 M_⊙ of Fe (Iwamoto et al. 1999).

Table 2.

Values of the t_GW, number of SNeIa and total mass of Fe produced by SNeIa [M_Fe (M_⊙)] in the different models for a typical elliptical, as described in the text.

Model	tGW	SNIa	MFe
	(Gyr)		(M⊙)
Mannucci et al. (2006)	0.46	2.12 × 108	1.27 × 108
Matteucci & Recchi (2001)	0.67	1.47 × 108	8.80 × 107
Pritchet et al. (2008)	2.64	1.5 × 107	9.00 × 106
Totani et al. (2008)	1.60	4.74 × 107	2.84 × 107
Strolger et al. (2004)	1.75	1.68 × 109	1.00 × 109

4.2 Type Ia SN rates

The results of the SNIa rates that we have obtained by exploiting the SFRs determined by the different DTDs are reported in Fig. 5. Clearly the SNIa rates so derived present differences relative to those of Fig. 3 and they are due to the different duration of the SF in the various cases. Different DTDs influence differently the onset of the galactic wind, as we have discussed before, and therefore the behaviour of the SNIa rate. Moreover, the value of the constant A_Ia undergoes some changes. Varying the SFR, in fact, it is again necessary to ensure that the rate calculated is able to reproduce the current value by introducing the appropriate value of this constant. For the computation of the amount of Fe ejected into the ICM, that we will discuss in the next section, is the time of the occurrence of the galactic wind which influences the amount of Fe remaining trapped in stars versus the amount of Fe which can be ejected into the ICM (namely, all the Fe produced after SF has stopped).

Figure 5.

Illustration of the different SNIa rates, as a function of time, for an elliptical galaxy of 10¹¹ M_⊙. Symbols are same as in Fig. 1.

5 CHEMICAL ENRICHMENT OF THE ICM WITH DIFFERENT DTDs

As a test for the different DTDs, we computed the expected Fe enrichment in galaxy clusters. In particular, we considered only two clusters: Virgo and Coma, taken as prototypes of a poor and a rich cluster, respectively. The method we adopted is similar to that of Matteucci & Vettolani (1988), Matteucci & Gibson (1995) and Pipino et al. (2002), where the masses produced in the form of Fe and total gas by ellipticals of different masses were integrated over the mass function of the clusters. We remind that the main results of these previous papers were that the Fe mass in clusters is easily reproduced whereas the total gas mass ejected by galaxies is far lower than observed. The logic conclusion from that was that most of the gas in clusters is primordial. We find the same result here. However, here we are interested in comparing the total Fe masses produced by ellipticals evolving with different DTDs. We have considered only four DTDs: (i) the MR01 DTD; (ii) the Mannucci et al. DTD; (iii) the Totani et al. DTD which is similar to the DTD relative to the DD wide scenario and (iv) the S04 DTD. We did not include the Pritchet DTD since it is clear from Table 2 that it predicts a very low number of SNeIa and therefore of Fe to be ejected into the ICM. In Table 3 we show the masses of Fe and gas, ejected into the ICM by ellipticals of baryonic masses 10¹⁰, 10¹¹ and 10¹² M_⊙. The chemical evolution model adopted is the one described in Section 2: the assumed efficiencies of SF are 3, 10 and 20 Gyr⁻¹ for 10¹⁰, 10¹¹ and 10¹² M_⊙, respectively (see Pipino & Matteucci 2004).

Table 3.

Masses ejected into the ICM by ellipticals in clusters.

DTD	MG	Mgas (M⊙)	MFe (M⊙)
Mannucci	1010	2.26 × 109	8.24 × 106
	1011	1.08 × 1010	3.69 × 107
	1012	8.32 × 1010	3.30 × 108
Totani	1010	9.66 × 108	1.94 × 106
	1011	7.13 × 109	1.39 × 107
	1012	6.00 × 1010	1.22 × 108
Matteucci & Recchi	1010	1.12 × 109	2.40 × 107
	1011	1.04 × 1010	3.25 × 108
	1012	8.62 × 1010	3.05 × 109
Strolger et al.	1010	1.37 × 109	4.31 × 107
	1011	8.44 × 109	4.61 × 108
	1012	7.54 × 1010	4.79 × 109

In the first column of Table 3 is indicated the DTD adopted in the chemical evolution model, in column 2 the initial baryonic mass of the galaxy, in column 3 the ejected total gas (H, He plus heavier elements) and in column 4 the total ejected Fe mass.

Clearly, the Fe masses ejected into ICM in the different cases are quite different, because the time at which the galactic wind occurs and SF stops is different in different galaxies. In Tables 4 and 5 we show the total mass of Fe and gas ejected into the ICM after integrating the single galactic contributions over the cluster mass function for Virgo and Coma, respectively. In the same tables are shown the observed values for M_gas and M_Fe. For Virgo we assumed the following values of the parameters necessary for the integration (see Matteucci & Vettolani 1988 for details): M/L = 10 (typical mass to light ratio of ellipticals), α = −1.25 (slope of the Schechter 1976 function), f = 0.43 (fraction of ellipticals in the cluster), n* = 20 (cluster richness) and M_* = −22 (absolute magnitude of the galaxy at the break of the luminosity function). For Coma we have adopted the following: M/L = 10, α = −1.25, f = 0.82, n* = 115 and M_* = −22. The assumed Hubble constant is 67 km s^{− 1} Mpc^{− 1}.

Table 4.

Virgo: integrated gas and Fe masses.

DTD	Mgas (M⊙)	MFe (M⊙)
Mannucci	6.16 × 1011	2.33 × 109
Totani	3.42 × 1011	6.90 × 108
Matteucci & Recchi	4.45 × 1011	1.29 × 1010
Strolger et al.	4.55 × 1011	2.12 × 1010
Observed values	2 × 1013	1.6 × 1010

Table 5.

Coma: integrated gas and Fe masses.

DTD	Mgas (M⊙)	MFe (M⊙)
Mannucci	6.75 × 1012	2.56 × 1010
Totani	3.74 × 1012	7.56 × 109
Matteucci & Recchi	4.88 × 1012	1.41 × 1011
Strolger et al.	4.99 × 1012	2.33 × 1011
Observed values	(4.4 ± 1.2) × 1014	3.1 × 1011

From Tables 4 and 5 we can see that in the Mannucci et al. DTD case, the Fe ejected into the ICM is less than in the SD case (MR01 DTD): this is because the number of prompt SNeIa, in this DTD, is quite large and 50 per cent of all SNeIa explode inside 100 Myr, before the galactic wind. As a consequence, most of the Fe produced by these SNe will be incorporated into stars at variance with what happens for the DTDs of MR01 and Totani, where the fraction of prompt SNe is much lower. However, in the Totani DTD(∝t⁻¹), although the number of prompt SNe is lower, the total number of SNeIa is also lower than in the other cases. So, the smallest amount of Fe ejected into the ICM is the one from Totani's DTD and is not enough to explain the Fe observed in both clusters. On the other hand, the integrated values for Fe in the MR01 DTD and in the S04 DTD produce Fe masses in agreement with observations. It is worth noting that the S04 DTD predicts the smallest masses of total gas ejected into the ICM; this is due to the fact that the galactic winds in ellipticals occur latest with this DTD. Therefore, the amount of residual gas at the time of the wind is lower than in the other cases. If we compute the Fe abundance in the ICM by dividing the total Fe mass predicted for each cluster by the observed mass of gas we obtain, for the MR01 DTD, values in very good agreement with observations ( ∼ 0.3 Fe_⊙; Renzini 2004): in particular, for Virgo we obtain Fe_Virgo ∼ 0.4–0.5 Fe_⊙ and for Coma Fe_Coma ∼ 0.2–0.3 Fe_⊙, having assumed the solar Fe abundance by Asplund et al. (2009) (Fe_⊙ ∼ 1.34 × 10^{− 3} by mass).

6 THE COSMIC TYPE Ia SN RATE

Another goal that we have set is to test the six different DTDs, previously analysed, in the computation of the cosmic SNIa rate. The determination of the cosmic SNIa rate, as well as the calculation of the SNIa rate, is obtained as the convolution between the assumed CSFR and the DTD function. The CSFR is the SFR in a unitary comoving volume of the Universe and it is expressed in units of M_⊙ yr⁻¹ Mpc⁻³. For each of the DTD, that we will use, we will adopt five different CSFRs, that is (i) Cole et al. (2001) modified (see later) CSFR; (ii) Madau, Della Valle & Panagia (1998) MDP01 and MDP02 CSFRs; (iii) S04 CSFR and (iv) Grieco et al. (2012) CSFR. The assumed cosmology, as stated before, is the ΛCDM cosmology.

6.1 The cosmic star formation rate

The physical meaning of the CSFR is of cumulative SFR owing to galaxies of different morphological types present in a unitary comoving volume of the Universe. The CSFR is not a directly measurable quantity and it can be computed only from the measurement of the luminosity density in different wavebands, which are then transformed into SFR by a suitable calibration. At high redshift, in fact, it is difficult to distinguish galaxy morphology but it is only possible to trace the luminosity density of galaxies.

6.1.1 Cole et al. (2001) CSFR

The CSFR density proposed by Cole et al. (2001) has the form

\begin{equation} \dot{\rho }_{{\rm Cole}}=\frac{a+bz}{1+(\frac{z}{c})^d}, \end{equation}

(29)

where (a, b, c, d) = (0.0, 0.0798, 1.658, 3.105) and after the correction for the dust extinction are (a, b, c, d) = (0.0166, 0.1848, 1.9474, 2.6316). This parametric fit of the CSFR was obtained thanks to the analysis of the data of Two Micron All Sky Survey (2MASS) Extended Source Catalog and the 2dF Galaxy Redshift Survey. Cole et al. (2001) used these data to estimate the galaxy luminosity function and to infer the total mass fraction in stars. In the computation of this CSFR, Cole et al. (2001) have assumed that no mass goes into forming brown dwarfs and they have multiplied the SFR by (1 − R), where R is the recycled fraction, as defined in the simple model of chemical evolution (Tinsley 1980), they have obtained an estimate of the mass locked up in stars. Here we have done the best fit of all the data on CSFR previously used by Cole et al. (2001) plus some new ones (see Vincoletto et al. 2012 for the data and references). The obtained best fit is very similar to that of Cole et al. (2001) and it is shown in Fig. 6 together with the other CSFRs adopted in this paper. We will refer to it as the modified Cole CSFR. For the modified Cole CSFR we have adopted the following parameters: a = 0.00904, b = 0.1122, c = 3.325 and d = 4.143.

Figure 6.

Illustration of the various cosmic SF history. The dotted blue line is the CSFR of Cole et al. (2001) whereas the black dotted line is the modified Cole CSFR (see text); the solid yellow line is the CSFR of MDP01; the long dashed magenta line is the CSFR of MDP02; the dashed–dotted green line is the CSFR of S04 (2004); the short dashed red line is the CSFR of Grieco et al. (2012).

6.1.2 MDP (1998) CSFR

The two cosmic SFR models MDP01 and MDP02 computed in Madau et al. (1998) are

\begin{eqnarray} \dot{\rho }_{{\rm MDP01}}(t)=a_1\left[t_9^{a_2}{\rm e}^{-{\frac{t_9}{a_3}}}+a_4\left(1-{\rm e}^{-{\frac{t_9}{a_3}}}\right)\right], \end{eqnarray}

(30)

\begin{eqnarray} \dot{\rho }_{{\rm MDP02}}(t)=a_1{\rm e}^{-{\frac{t_9}{a_6}}}+a_4\left(1-{\rm e}^{-{\frac{t_9}{a_3}}}\right)+a_5t_9^{a_2}{\rm e}^-{\frac{t_9}{a_3}}, \end{eqnarray}

(31)

where

1. |$t_9= 13(1+z)^{-(\frac{3}{2})}$| is the Hubble time at redshift z;

2. a₁ = 0.049 in MDP01, a₁ = 0.336 in MDP02;

3. a₂ = 5 in both cases;

4. a₃ = 0.64 in both cases;

5. a₄ = 0.2 in MDP01, a₄ = 0.0074 in MDP02;

6. a₅ = 0.00197;

7. a₆ = 1.6.

6.1.3 Strolger et al. (2004) CSFR

This CSFR is given by

\begin{equation} \dot{\rho }_{{\rm S04}}(t)=a_1\left(t^{a_2}{\rm e}^{-{\frac{t}{a_3}}}+{a_4}{\rm e}^{\frac{d(t-t_0)}{a_3}}\right), \end{equation}

(32)

where

1. t₀ = 13.47 Gyr is the age of the Universe;

2. a₁ = 0.182;

3. a₂ = 1.260;

4. a₃ = 1.865;

5. a₄ = 0.071.

These parameters are determined by fitting the collection of measurements of the CSFR of Giavalisco et al. (2004). This CSFR is a model based on a modified version of the parametric form suggested by Madau et al. (1998), taking into account dust extinction.

6.1.4 Grieco et al. (2012) CSFR

To compute the CSFR Grieco et al. (2012) have adopted the following relation:

\begin{equation} \dot{\rho }_{{\rm Grieco}}=\sum _{k}\psi _k(t) n_k^{\star } (\mathrm{M}_{{\odot }}\, {\rm yr}^{-1} \, {\rm Mpc}^{-3}), \end{equation}

(33)

where

1. k identifies a particular galaxy type, that is elliptical, spiral or irregular;

2. ψ_k(t) represents the history of SF in each galaxy, tested by model of galactic chemical evolution;

3. |$n_k^\star$| is the galaxy number density, expressed in units of Mpc⁻³ for each morphological galaxy type, and it is been assumed to be constant and equal to the present time one; models assuming number density evolution can be found in Vincoletto et al. (2012).

For the computation of this CSFR Grieco et al. (2012) have assumed that all galaxies started forming stars at the same time, an oversimplified hypothesis but useful to understand the behaviour of the cosmic rates in extreme conditions.

6.1.5 Trend of different CSFRs

To make a comparison between all the CSFR that we analysed we list all the trends in Fig. 6.

Among the five different curves only two have a different behaviour, that of Grieco et al. (2012) and that of MDP02. The Grieco et al. (2012) CSFR, in fact, shows a quick rise at very high redshift followed by a sharp decline and a following smooth decline until the present time. The rising CSFR for z > 5 depends on the fact that in this model all the ellipticals are assumed to be present since the beginning (no number density evolution). On the other hand, the MDP02 CSFR has only one continuous phase of slow growth and it mimics a monolithic collapse scenario. In both CSFRs a Salpeter (1955) IMF is assumed. These two curves therefore do not have the usual trend that is common to the other three curves, that is an initial phase of growth, the reach of a peak and finally a descending phase. All the other models assume a hierarchical galaxy formation where the most massive objects form later.

6.2 Observational data

The benefit of these new calculations is that, in this case, we can compare the results of our models with a recent compilation of observational data as function of redshift. The data that we have used are reported in Appendix A (Tables A1 and A2), where we indicate the references, the redshift and the values of the cosmic SNIa rate, expressed in 10⁻⁴ Mpc⁻³ yr⁻¹. The most recent ones are from Graur et al. (2011) and come from the Subaru Deep Field search. The data with their error bars are also shown in Fig. 7.

Figure 7.

Illustration of the observational data: Barris & Tonry (2006) (black filled hexagons); Dahlen et al. (2004) (magenta filled hexagons); Kuznetsova et al. (2008) (magenta open stars); Poznanski et al. (2007) (green filled hexagons); Cappellaro et al. (1999) (blue open stars); Hardin et al. (2000) (yellow open triangles); Blanc et al. (2004) (green open triangles); Mannucci et al. (2005) (green open hexagons); Madgwick et al. (2003) (magenta filled triangles); Strolger et al. (2003) (red filled hexagons); Neill et al. (2007) (yellow open hexagons); Horesh et al. (2008) (green open stars); Botticella et al. (2008) (black open stars); Tonry et al. (2003) (red filled squares); Neill et al. (2006) (blue filled pentagons); Pain et al. (2002) (yellow filled hexagons); Graur et al. (2011) (red open hexagons); Rodney & Tonry (2010) (blue open hexagons); Li et al. (2011) (red open pentagons); Dilday et al. (2010) (yellow asterisks); Dahlen et al. (2008) (black open hexagons).

6.3 Computation of the cosmic Type Ia rate

The computation of the cosmic Type Ia rate is, as already said, similar to the computation of the SNIa rate; so, also in this case, it is necessary to introduce the constant A_Ia to reproduce the actual value of the cosmic SNIa rate. The actual value (z = 0) of the cosmic Type Ia rate is 0.301(10⁻⁴ Mpc⁻³ yr⁻¹), as proposed by Li et al. (2011).

6.3.1 Properties of the different cosmic Type Ia SN rate

The model results compared to the data are shown in Figs 8–10. In Figs 8 and 9, each model was obtained by the convolution between a given DTD and five different CSFRs, shown in Fig. 6. These models are compared to the observed cosmic SNIa rate. From this figure we can see that the models that generally agree with most of the data are unable to reproduce the data of Cappellaro et al. (1999) and Blanc et al. (2004). On the other hand, most of the models reproduce the data of Graur et al. (2011), Li et al. (2011) and Rodney & Tonry (2010). Moreover, it is worth noticing that the adopted data suggest a minimum of the cosmic SNIa rate at z ∼ 0.2–0.3 (see Fig. 7). This is confirmed by a more quantitative analysis. The average SNIa rate for the redshift bin 0 < z ≤ 0.1 is 0.273, i.e. it is lower than the present cosmic rate (0.301) determined by Li et al. (2011). The average SNIa rate further decreases in the redshift bins 0.1 < z ≤ 0.2 (0.244) and 0.2 < z ≤ 0.3 (0.195). Then it starts growing for z > 0.3 (it is 0.448 in the redshift bin 0.3 < z ≤ 0.4). None of the adopted combinations of CSFRs and DTDs is able to reproduce this minimum. In order to reproduce it, we should either assume a peculiar CSFR with a minimum at high z (but available data do not corroborate it) or a DTD with a minimum at large delay times (but from a theoretical point of view this is not easy to justify).

Looking at the Figs 8 and 9 it is possible to see how the models computed using the Pritchet et al. (2008) DTD are all too low compared to the data. The same effect seems to be present for the S04 DTD. However, it is very difficult to select a particular DTD and a particular CSFR as the best fit to the data. First of all because of the large error bars present in the data and the different way in which the errors have been computed in different papers. Because of this, any conclusion concerning DTD and CSFR should be taken with care. Concerning the CSFR, we can see from Figs 8 and 9 that in all models, even in those which are not able to fit very well the data, the curve relative to the modified Cole et al. (2001) CSFR predicts the best trend. With the aim of studying this CSFR more in detail, we made a complementary analysis in which we used only the CSFR of Cole et al. (2001) and the different DTDs. The result obtained is shown in Fig. 10. The same considerations made for Figs 8 and 9 hold for this figure. Although, the modified Cole CSFR is probably the best CSFR since it represents the fit to the observed CSFR, it is still not possible to extract the best DTD from this diagram. In fact, just looking by eye one can conclude that the DTDs of Pritchet and S04 are too low relative to the average of the data, whereas the DTDs of Mannucci, MR01 and Totani are probably too high. For choosing the best DTD, it is better to rely to chemical evolution results which adopt elemental abundances measured with higher precision. We adopted a fitness test, as described in Calura et al. (2010) (see their equation 12), to select the best combination DTD/CSFR and the results are shown in Fig. 11. From this figure it appears that the modified Cole and the Grieco et al. (2012) CSFR are the best together with S04, MR01 and Totani DTDs. One can see that in the Fig. 11, where the size of the circle is inversely proportional to the fitness function, as defined in Calura et al. (2010). The fact that MR01 and Totani 's DTDs are the best is not surprising since they correspond to the two most popular and tested SNIa scenarios, the SD and the DD ones. However, it is surprising that also the DTD of S04 is good since it is very different from the other two and it does not reproduce the chemical evolution of the solar vicinity (see Matteucci et al. 2009). The abundance patterns in the solar neighbourhood, in fact, demand the presence of some prompt SNeIa and a majority of tardy SNeIa. Therefore, no firm conclusions can be derived from the cosmic SNIa rate, since the uncertainties in the measured rates prevent a reliable analysis relative to the DTDs.

Figure 8.

Comparison between model and data. The two top panels are related to the MVP06 DTD, one on the left, and the other to the MR01 DTD plied to the five different CSFRs of Fig. 6 (the symbols are the same as in Fig. 7). In the last two panels there are the Pritchet et al. DTD, on the left, and on the right the Totani et al. DTD, also applied to the five CSFRs of Fig. 6.

Figure 9.

Comparison between model and data. The two panels are related to the S04 DTD and to G05 DTD.

Figure 10.

Comparison between the model and the data relative to the cosmic SNIa rate computed using the modified Cole & al (2001) CSFR and all the DTD. The solid red line is the cosmic SNIa rate of MVP06; the short dashed blue line is the cosmic SNIa rate of MR01; the dotted green line is the cosmic SNIa rate of S04; the short dashed–dotted black line is the cosmic SNIa rate of Pritchet et al. (2008); the long dashed yellow line is the cosmic SNIa rate of Totani et al. (2008). The cosmic SNIa rate of G05 is not reported because is identical to the one of Totani et al. (2008).

Figure 11.

Results of the fitness test for finding the best combination of CSFR and DTD. In the Y-axis we report the various DTDs while in the X-axis the various CSFRs. The size of the circle is inversely proportional to the calculated fitness function, as defined in Calura et al. (2010), therefore better CSFR/DTD combinations appear as the largest circles.

7 CONCLUSIONS

We have computed the SNIa rates in elliptical galaxies of various baryonic masses, under different assumptions about SNIa progenitors. To take into account the different SNIa progenitors, we adopted different DTD functions describing the distribution of the explosion times as a function of time. We considered both theoretical and empirical DTD functions. The different DTDs have all being normalized to reproduce the present day SNIa rate in typical ellipticals but they contains a different total number of SNeIa and a different proportion of prompt (exploding in the first 100 Myr since the beginning of star formation) and tardy SNeIa. The bimodal DTD contains ∼50 per cent of prompt SNe, the SD DTD (MR01) contains ∼13–15 per cent, the DD DTD and the DTD ∝ t⁻¹ (Totani et al. 2008) contain ∼10 per cent, the DTD ∝ t^−0.5 (Pritchet et al. 2008) contains ∼4 per cent and the DTD of S04 contains zero. Then, we have calculated the integrated Fe and gas mass in two galaxy clusters (Coma and Virgo) by means of the models for ellipticals including different DTDs. Finally, we have studied the cosmic SNIa rate by adopting the same DTDs as for the elliptical galaxies and by varying the assumed CSFR. We considered CSFRs obtained either as best fits of data or theoretically, and containing different assumptions about galaxy formation mechanisms. We compared the predicted cosmic SNIa rates with the most recent compilation of data relative to the cosmic Type Ia rate observed up to redshift z = 1.75. We have then compared our model results with observations. It is worth noting that the results of this investigation should integrate those obtained by Matteucci et al. (2009). In that paper we tested the effects of different DTDs on the chemical evolution of the Milky Way and we obtained clear suggestions; in particular, we found that only the SD and DD DTDs together with the bimodal DTD, but with less than 50 per cent prompt SNeIa, can fit the abundance patterns observed in the stars of the Galaxy. Abundance measurements are very accurate these days and certainly more accurate than cosmic SN rate ones. In Matteucci et al. (2009) we also concluded that prompt SNeIa are necessary to reproduce the abundance data.

Our main conclusions can be summarized as follows.

• We have found that a different number of prompt SNeIa affects substantially the time for the occurrence of a galactic wind, which then quenches SF and gives rise to the passive evolutionary phase for such galaxies. Clearly, the time for a galactic wind influences the amount of Fe locked up in stars as well as that ejected into the ICM.

• The best DTD in order to obtain the right amount of Fe in the ICM is the one relative to the SD scenario in the formulation of Greggio & Renzini (1983) and MR01. The DTD obtained by Totani et al. (2008), which is similar to the DTD of G05 DD wide scenario, does not produce enough Fe to be ejected into the ICM. This DTD, in fact, contains less SNeIa than the MR01 SD scenario and therefore the galactic wind occurs later. As a consequence of this, the Fe ejected into the ICM is less than in the MR01 SD scenario. The S04 DTD instead produces masses of Fe compatible with observations, but the amount of the total gas ejected is the lowest. This is due to the late galactic winds occurring in galaxies if this DTD is assumed.

• The cosmic SNIa rate is not a good tracer of the DTD nor of the CSFR. In spite of the fact that we adopted the largest data set, the observed SNIa rates are still quite uncertain and limited to a redshift z ≤ 1.75, and the error bars are quite large, especially at high z. We have performed a statistical analysis and found that the best DTDs seems to be those relative to the SD (MR01) and DD (G05) scenarios plus the DTD of S04. This last DTD was in fact deduced from a fit to the observed cosmic SNIa rates. However, given the lack of prompt Type Ia SNe, this DTD does not reproduce correctly the chemical evolution of galaxies, as shown by Matteucci et al. (2006, 2009). In particular, the lack of prompt SNeIa produces a long plateau in the [α/Fe] ratios in the Galaxy, at variance with observations. The opposite effect is obtained when the bimodal DTD is assumed with 50 per cent of prompt SNe. In this case [α/Fe] ratios decrease too steeply with [Fe/H], at variance with observations. The CSFRs adopted here are derived from both hierarchical and monolithic models of galaxy formation; in particular, different assumptions about the galaxy number density evolution underline these cases. In hierarchical models the galaxy number density varies with the cosmic time whereas in monolithic models the galaxy number density is assumed constant (pure luminosity evolution). Moreover, we took into consideration the fit of Cole et al. (2001) of observational data and we also added other points and derived a revised best fit. From our analysis we suggest that the best ones to reproduce the cosmic SNIa rate are that derived as a best fit of observational data (Cole et al. 2001 and this paper) and a theoretical CSFR derived from chemical evolution models (e.g. Grieco et al. 2012) in the framework of pure luminosity evolution.

• In summary, taking all of our results together we suggest that the best DTDs are those relative to the SD, DD and bimodal scenarios. All these DTDs contain prompt SNeIa. We also suggest that the ideal number of prompt SNeIa should not exceed 15–20 per cent of the total SNeIa. However, these results are mainly supported by chemical evolution considerations rather than by the galactic and cosmic SNIa rates. Therefore, the answer to the title of the paper is still rather negative. More precise data on SNIa rates in galaxies and as a function of redshift will help in the future to find a more precise answer.

We thank E. Cappellaro for providing the data on the SN cosmic rates. FM acknowledges financial support from PRIN MIUR 2010-2011, project ‘The Chemical and Dynamical Evolution of the Milky Way and Local Group Galaxies’, prot. 2010LY5N2T. Finally, we thank an anonymous referee for careful reading of the manuscript and important suggestions.

APPENDIX A

Table A1.

A compilation of the observational cosmic Type Ia rates at different redshifts up to z ∼ 1.75.

Reference	z	SN rate (10−4 Mpc−3 yr−1)
Cappellaro et al. (1999)	0.01	0.20 ± 0.059
Hardin et al. (2000)	0.14	|$0.22^{+0.17}_{-0.22}$|
Pain et al. (2002)	0.55	|$1.53^{+0.28}_{-0.25}$|
Magdwick et al. (2003)	0.10	0.32 ± 0.15
Strolger et al. (2003)	0.11	0.37 ± 0.10
Tonry et al. (2003)	0.46	1.4 ± 0.5
Blanc et al. (2004)	0.13	|$0.14^{+0.05}_{-0.035}$|
Dahlen et al. (2004)	0.4	|$0.69^{+0.34}_{-0.27}$|
Dahlen et al. (2004)	0.8	|$1.57^{+0.44}_{-0.25}$|
Dahlen et al. (2004)	1.2	|$1.15^{+0.47}_{-0.26}$|
Dahlen et al. (2004)	1.6	|$0.44^{+0.32}_{-0.25}$|
Mannucci et al. (2005)	0.03	0.28 ± 0.11
Barris & Tonry (2006)	0.25	|$0.17^{+0.17}_{-0.16}$|
Barris & Tonry (2006)	0.35	0.53 ± 0.24
Barris & Tonry (2006)	0.45	0.73 ± 0.24
Barris & Tonry (2006)	0.55	2.04 ± 0.38
Barris & Tonry (2006)	0.65	1.49 ± 0.31
Barris & Tonry (2006)	0.75	1.78 ± 0.34
Neill et al. (2006)	0.47	|$0.42^{+0.09}_{-0.13}$|
Neill et al. (2007)	0.32	0.23 ± 0.06
Neill et al. (2007)	0.50	0.48 ± 0.15
Neill et al. (2007)	0.7	0.60 ± 0.20
Poznanski et al. (2007)	0.75	|$0.43^{+0.36}_{-0.32}$|
Poznanski et al. (2007)	1.25	|$1.05^{+0.45}_{-0.56}$|
Poznanski et al. (2007)	1.75	|$0.81^{+0.79}_{-0.60}$|

Table A2.

A compilation of the observational cosmic Type Ia rates at different redshifts up to z ∼ 1.75.

Reference	z	SN rate (10−4 Mpc−3 yr−1)
Botticella et al. (2008)	0.30	|$0.22^{+0.10}_{-0.08}$|
Dahlen et al. (2008)	0.47	|$0.80^{+1.66}_{-0.27}$|
Dahlen et al. (2008)	0.83	|$1.30^{+0.73}_{-0.51}$|
Dahlen et al. (2008)	1.21	|$1.32^{+0.32}_{-0.38}$|
Dahlen et al. (2008)	1.61	|$0.42^{+0.39}_{-0.23}$|
Dilday et al. (2008)	0.09	|$0.29^{+0.09}_{-0.07}$|
Horesh et al. (2008)	0.2	0.189 ± 0.042
Kuznetsova et al. (2008)	0.4	|$0.53^{+0.39}_{-0.17}$|
Kuznetsova et al. (2008)	0.8	0.93 ± 0.25
Kuznetsova et al. (2008)	1.2	|$0.75^{+0.35}_{-0.30}$|
Kuznetsova et al. (2008)	1.55	|$0.12^{+0.58}_{-0.119}$|
Dilday et al. (2010)	0.12	|$0.269^{+0.034}_{-0.030}$|
Li et al. (2011)	0	|$0.301^{+0.062}_{-0.061}$|
Rodney & Tonry (2010)	0.15	0.32 ± 0.32
Rodney & Tonry (2010)	0.35	0.34 ± 0.19
Rodney & Tonry (2010)	0.45	0.31 ± 0.15
Rodney & Tonry (2010)	0.55	0.32 ± 0.14
Rodney & Tonry (2010)	0.65	0.49 ± 0.17
Rodney & Tonry (2010)	0.75	0.68 ± 0.21
Rodney & Tonry (2010)	0.85	0.78 ± 0.22
Rodney & Tonry (2010)	0.95	0.76 ± 0.25
Rodney & Tonry (2010)	1.05	0.79 ± 0.28
Graur et al. (2011)	0.74	|$0.79^{+0.33}_{-0.41}$|
Graur et al. (2011)	1.23	|$0.84^{+0.25}_{-0.28}$|
Graur et al. (2011)	1.69	|$1.02^{+0.54}_{-0.36}$|