Constraints on the metallicity of supernovae progenitors forming throughout the cosmic history

The observed stellar mass function of star forming galaxies gives the number density of objects of different masses.

The mass − metallicity relation allows us to assign metallicities to galaxies of different stellar masses.

The star formation - mass relation allows us to specify the contribution of galaxies of different masses (metallicities) to the total star formation rate density at a certain redshift.

We vary the assumptions about those relations (low mass end slope of the GSMF, normalization and shape of the MZR, high mass end of the SFMR) to cover the range of possibilities present in the literature.

We account for the intrinsic scatter  present in the relations (σ ₀ , σ _{S FR} ) and the observed anti-correlation between the SFR and metallicity. On top of that we introduce scatter in metallicity to account for the internal distribution of metallicities in the star forming gas within galaxies (σ _{∇Z O/H} ).

All relations evolve with redshift.
Combining all relations we obtain the distribution of the cosmic star formation rate density over metallicities and redshifts SFRD(Z,z).

See Chruslinska & Nelemans (2019) for more details.

The figure shown above summarizes our main result.

Upper panels: distribution of the star formation rate density (SFRD) at different metallicities and redshift (z). Left – the model variation that leads to the highest fraction of mass formed at high metallicities, right – the model variation that leads to the highest fraction of mass formed at low metallicities. Difference between the two extremes depicts the uncertainty of the SFRD(Z,z) based on the observational properties of star forming galaxies. Bottom panels: the ratio of the total SFRD(z) calculated in the two variations of our model and as inferred from observations.

Figure from Chruslinska & Nelemans (2019)

The results of our calculations are publicly available and can be used in studies focusing on the properties of populations of systems composed of stars and their remnants, stellar-evolution related transients and their likely environments. In particular, they can be applied to calculate the rates of those transients and to evaluate their uncertainty due to the assumed distribution of the cosmic star formation rate density across metallicities and redshifts. It can also serve for comparison for the cosmological simulations.

The contribution of low mass galaxies (~<10⁸ M_o) to the total star formation rate density is particularly uncertain. Those galaxies are relatively faint and difficult to observe, especially at higher redshifts.

The figure above shows a variation of our model calculated with the low mass end slope of the galaxy stellar mass function that evolves with redshift in such a way that the number density of low mass galaxies increases with redshift, as suggested by some authors (see Sec. 2.1 in Chruslinska & Nelemans 2019 and references therein).

In this model variation low mass galaxies start to completely dominate the picture at high redshifts and the SFR shifts to very low metallicities. Note that at high redshifts z>4 (shaded area) the cosmic star formation rate density calculated in this variation of our model is significantly higher than estimated from observations. At lower redshifts the effect of this assumption on the final distribution is minor.

Our results can be applied to look at the distribution of various transients of stellar origin at different metallicities. The most straightforward example application is to look at the local core-collapse supernovae (CCSN). The figure above shows specific CCSN rate as a function of metallicity for different model variations (star formation-mass relation - SFMR - with no flattening – thick solid line, moderate flattening – dashed line, sharp flattening – dotted line) and observations (Graur et al.  2017). Metallicity scale assumes the Tremonti et al. (2004) calibration (but the result of the comparison is not affected by this choice) . The thin solid line shows the case with no flattening and without taking into account the metallicity distribution within galaxies. The blue shading shows the additional uncertainty (−0.24 dex) due to the fact that the oxygen abundance in the observed sample was measured in the central region of the SN host galaxy and not at the SN location. Our model shows satisfying agreement with the data, although the variations with sharp flattening in SFMR may underestimate the local high metallicity star formation. Our results can be used to investigate the CCSN rate as a function of metallicity also beyond the local Universe.

A valuable consistency check is to compare the observational estimates of the CCSN rate density at different redshifts with the CCSN rate density resulting from our calculations for different variations of our observation-based model.

The figure above shows the core-collapse supernovae rate density calculated using our method (short dashed red line − high metallicity extreme; solid green line − moderate variation; thick blue long dashed line − low metallicity extreme; thin gray dashed line − low metallicity extreme with the low mass end of the GSMF evolving with redshift) compared with the observational estimates (see legend). In case of the model results we assume that CCSN progenitors come from the initial mass range 8-25 M_o. For the reference, we also plot the SFRD(z) from Madau & Fragos (2017) (MF17, thick gray solid line) and the high redshift SFRD estimate by Bouwens et al. (2015) (B+15, magenta line; converted to our default Kroupa (2001) IMF) multiplied by the efficiency of formation of CCSN progenitors (k _CCSN ∼ 0.009 M_o⁻¹  ; assuming CCSN mass range 8−25 M_o   and Kroupa 2001 IMF).

Within the observational uncertainties, all the model variations provide satisfying match to the data (but note that this comparison does not discriminate between the assumptions about metallicity). We note that there is a number of factors that make this comparison not straightforward, as discussed in Sec. 5 of Chruslinska & Nelemans (2019).

Recent observational and theoretical studies indicate that the stellar initial mass function (IMF) varies systematically with the environment (SFR, metallicity).  Although the exact dependence of the IMF on those properties is likely to change with the improving observational constraints , the reported trend in the shape of the IMF appears robust: in high SFR/low metallicity environments the IMF is more top heavy (i.e. implies more massive stars) and in low SFR environments the IMF appears top light (i.e. implies less massive stars ) than the IMF estimated in the Milky Way.

Such systematic variations may have non trivial effect on the inferred distribution of the cosmic SFRD over metallicity and redshift. This effect will also be different for stars of different masses. We discuss that in Chruslinska et al. (2020) 

We apply the empirically driven model of IMF variations described by the integrated galactic IMF (IGIMF) theory as presented in Jerabkova et al. (2018) and revise the cosmic SFRD distribution over metallicity and redshift that was obtained under the assumption of a universal IMF by Chruslinska & Nelemans (2019).

Observationally inferred SFR depends on the assumed IMF. The IMF is typically assumed to be universal (i.e. independent of the conditions within the observed galaxy or its redshift) and to resemble the one inferred for the Milky Way. When we take into account the IMF dependence on the environment, we need to determine the error associated with the observation-based galaxy SFR when assuming a universal IMF and then correct for it.

Different assumptions about the IMF also affect the estimated composition of the young stellar population, i.e. the relative number of low mass and massive stars that form within the galaxy.

Upper panels: Distribution of the cosmic SFRD at different metallicities and redshifts (z). Left – Background colour and brown contours show the high metallicity extreme assuming the non-universal IMF, and white contours show the high metallicity extreme assuming a universal IMF. Right – Same as in the left panel, but for the low metallicity extreme. To plot the white contours,  SFRD in each Z-z bin for the universal IMF cases has been re-scaled in such a way that the total SFRD at each redshift matches that of the corresponding variation with the non-universal IMF (see the bottom panel). This reveals the presence of the more extended low-metallicity tail present in the non-universal IMF variations.
Bottom panel: Ratio of the total SFRD at each redshift as calculated under the assumption of the non-universal to universal IMF for the two SFRD (Z,z) variations from Chruslinska et al. (2020) (dashed line – low-Z extreme, solid line – high-Z extreme). The assumption of the universal IMF leads to a factor of ~2 higher SFRD at high redshifts than estimated for the non-universal IMF case.

Figure from Chruslinska et al. (2020)

In the local Universe, our calculation applying the IGIMF theory to describe the IMF variations suggests more white dwarf and neutron star progenitors in comparison with the universal IMF scenario, while the number of black hole progenitors remains unaffected. The fraction of metal-poor stars being formed is higher than obtained under the assumption of the universal IMF. The effect (both in terms of the number and metallicity) is the strongest for white dwarf progenitors forming in the mass range relevant for the progenitors of type Ia supernovae.

Left: Ratio of the number of stars forming in different mass ranges using the environment-dependent IMF to that using the universal IMF, as a function of redshift.

Right: ratio of the fraction of stars forming at low metallicity (<0.1 solar) in different mass ranges using the environment-dependent IMF to that using the universal IMF, as a function of redshift. At redshift z=1, about 2.5 times more stars with M∗<1Mo  form at low metallicity in the considered non-universal IMF scenario than in the universal IMF case.

Figures from Chruslinska et al. (2020)

The knowledge of the star formation rate density distribution over metallicity and cosmic time/redshift (and its uncertainty!) is particularly important if one wants to estimate the properties of the populations of transients that form with long delays with respect to the star formation episode in which the progenitor stars/systems were born.

Examples of such transients are type Ia supernovae and mergers of double compact objects that can be observed with ground-based gravitational wave detectors.

The population of such transients that we observe involves a mixture of binaries that formed at different times and with different metallicities. Chruslinska et al. (2019) show that different assumptions about the birth metallicities of stars formed across the cosmic history can significantly affect the theoretical estimates of e.g. BHBH merger rate (see the summary in the enclosed pdf ->).

In case of SN Ia, little is known about the potential metallicity dependence of their formation efficiency and the nature of the progenitor is still debated. Different SN Ia progenitor models invoke white dwarfs originating from stars with masses between ~3-8 M_o.

As discussed in Chruslinska et al. (2020), the biggest impact of the transition from the universal to the environment dependent IMF is on the formation of stars in the WD progenitor mass range. Chruslinska et al. (2020) find  2-4 times more WD forming at low metallicity in the mass range relevant for the SN Ia progenitors if the IMF is assumed to depend on SFR and metallicity instead of being universal. This increase may be particularly important if the efficiency of formation of type Ia SN depends on metallicity.

Therefore, it is important to consider the IMF non-universality when estimating the properties of the populations of the progenitors of SN Ia.

We show the result of a illustrative calculation of SN Ia rate performed under different IMF assumptions.  The relative rate of type Ia SN to CCSN, as well as type Ia SN rate as shown in the above figure is lowered in the non-universal IMF scenario by a factor of a few, but we stress that the exact magnitude of this effect depends e.g. on the assumed delay time distribution of type Ia SN.  We note that the formation efficiency for type Ia SN is not known theoretically but inferred from the observations. Therefore, in order to explain the observed type Ia SN rate with a given progenitor model, this efficiency would simply need to be higher in the non-universal IMF than in the universal IMF scenario. 

The SN Ia rate shown above assumes no metallicity dependence of the formation efficency of SN Ia, t^-1 delay time distribution with t_min=40 Myr and binary progenitor composed of stars with initial masses between 3-8 Mo. See Chruslinska et al. (2020) for more details.

• we obtain the distribution of the cosmic star formation rate density over metallicity and redshift SFRD(Z,z) based on observations and discuss its  uncertainty (we consider: metallicity calibrations, GSMF at low M -- esp. important at z>4, SFMR flattening, IMF)
• it is particularly important to know SFRD(Z,z) and its uncertainty when estimating the properties of transients with long delay times with respect to star formation (e.g. GW sources, SN Ia) and/or strong dependence on metallicity (e.g. BBH mergers, long GRB)
• our result is less extreme in terms of the amount of low/high metallicity star formation than some SFRD(Z,z) used in the literature to estimate the properties of GW sources

• the results of our calculations show good agreement with the local specific CCSN rate as a function of metallicity from Graur et al. (2017) and observational estimates of the CCSN rate density at different redshifts.
• our results are available and can be used to evaluate the uncertainty due to assumed SFRD(Z,z) e.g. of the BHBH merger rates; serve as a reference for cosmological simulations; estimate the properties of populations of various transients

The assumption of the universal IMF may lead to:

• underestimate of the fraction of NS and WD progenitors forming at low metallicity, especially at low redshifts.
• overestimate in the number of NS and WD progenitors forming at z>1 and underestimate of the number of those stars at lower z.
• overestimate of the cosmic SFRD at high z

-> it has the strongest effect on the number and metallicity of the WD progenitors with the masses >1M_o, relevant for SN Ia progenitors.

Questions? Comments? Ideas? Contact me! m.chruslinska [at] astro.ru.nl