Effect of Cosmic Mean Metallicity on the Supernovae Cosmology

Research on cosmological evolution requires a reliable standard to measure galaxy distances. The normal Ia Supernovae (SNe Ia) could serve as a cosmological standard candle by using the empirical standardization procedures based on the light-curve shape and color (Phillips 1993; Riess et al. 1996; Perlmutter et al. 1997; Guy et al. 2007; Jha et al. 2007). It further leads to the discovery of the accelerating expansion of the universe and provides an evidence of the existence of dark energy (Riess et al. 1998; Schmidt et al. 1998; Perlmutter et al. 1999). Nevertheless, the light curves of SNe Ia still have nonnegligible dispersion at the peak brightness after the standardization, which may have an significant influence on our understanding of the cosmological evolution.

In the last few decades, a growing number of studies showed that the standardized brightness of SNe Ia may evolve with look-back time and is likely correlated with host morphology (Riess et al. 1998; Schmidt et al. 1998; Perlmutter et al. 1999; Hicken et al. 2009; Suzuki et al. 2012), host mass (Kelly et al. 2010; Sullivan et al. 2010; Childress et al. 2013; Johansson et al. 2013), local star formation rate (Rigault et al. 2015, 2020; Kim et al. 2018; Roman et al. 2018; Barkhudaryan et al. 2019; Rose et al. 2019), host ages (Jones et al. 2018; Kang et al. 2020; Lee et al. 2020; Rose et al. 2020; Murakami et al. 2021), and metallicity (Timmes et al. 2003; Bravo et al. 2010). These studies have revealed that SNe Ia is fainter in the late-type, less massive, higher star formation rate, younger and higher metallicity host galaxies. However, the host morphology, host mass, and local star formation rate cannot directly effect SNe Ia luminosity. Kang et al. (2020) shed new light on the origin of the previously reported correlations of SNe Ia luminosity with host morphology, host mass, and local star formation rate and argued that they are most likely originated from either population age or metallicity.

The initial metallicity of progenitor, Z, as an important factor of the SNe Ia luminosity evolution has been studied extensively. The light of SNe Ia explosion originates from the decay of synthesized radioisotopes and the maximum luminosity is dominated by the amount of ⁵⁶Ni formed during the nucleosynthesis (Arnett 1982). The ⁵⁶Ni yield is influenced by Z since the larger the Z the more neutron-rich is the environment to form more stable nuclides. Besides, the ⁵⁶Ni yield is also related to the progenitor system and the explosion mechanism of SNe Ia (Kobayashi et al. 2020; Palla 2021). Now the widely accepted scenario is that SNe Ia is produced by a thermonuclear explosion of a carbon-oxygen (CO) white dwarf (WD) in a binary system, while the progenitor system, accretion rate and explosion mechanism are still unclear (Hillebrandt & Niemeyer 2000; Hillebrandt et al. 2013; Maoz et al. 2014; Ruiter 2019). The near-Chandrasekhar-mass (near-M_ch) model is one of the most promising progenitor scenarios of SNe Ia. By assuming that most of the ⁵⁶Ni is synthesized in material that burns fully to nuclear statistical equilibrium (NSE) and the neutron excess is not modified during the explosion, Timmes et al. (2003) obtained a linear relationship between the mass of ⁵⁶Ni synthesized, M(⁵⁶Ni), and Bravo et al. (2010) argued that the case where a sizeable amount of ⁵⁶Ni is synthesized during incomplete Si burning would provide a stronger linear dependence between M(⁵⁶Ni). Leung & Nomoto (2018) and Seitenzahl et al. (2013) presented 1D, 2D, and 3D hydrodynamics simulations of near-M_ch CO WD for SNe Ia using the turbulent deflagration model with deflagration-detonation transition (DDT) and get the ⁵⁶Ni yield at different metallicity, respectively. In addition, the sub-Chandrasekhar-mass (sub-M_ch) model is another promising progenitor scenario of SNe Ia. The C detonation is triggered by a surface detonation of He in most of sub-M_ch models. Shen et al. (2018) and Leung & Nomoto (2020) presented 1D and 2D hydrodynamics simulations of sub-M_ch CO WD for SNe Ia with the double-detonation (DD) model and also get the ⁵⁶Ni yield at a different metallicity. Based on these studies, the relationship between the SNe Ia luminosity and the initial metallicity of progenitor can be established.

Further estimation of the luminosity correction requires reliable measurements of the initial metallicity of progenitors. However, the measurements are difficult and the results are uncertain whether Z are estimated from the supernova remnants or their environments (Badenes et al. 2008, 2009a, 2009b; Bravo & Badenes 2011; Foley & Kirshner 2013). Since the initial metallicity of a main-sequence star is inherited from its ambient interstellar medium, the studies of the cosmic mean metallicity (CMM) provide a possibility to search for the evolution of Z. Prochaska et al. (2003) analyzed the metallicity measurements for 125 damped Lyα (DLA) systems at 0.5 < z < 5 and obtained a relationship between the CMM and redshift. Rafelski et al. (2012) got a more accurate relationship by using the expanded 242 DLA systems at 0.09 < z < 5.06, which confirmed the relationship obtained by Prochaska et al. (2003). Based on the chemical evolution models of galaxies, Vincoletto et al. (2012) and Gioannini et al. (2017) also got the CMM evolution by investigating the evolution of the cosmic star formation rate (CSFR). By considering the delay time distribution (DTD) of SNe Ia, the redshift at the birth of progenitor could be deduced from that at the explosion of SNe Ia (Greggio 2005; Totani et al. 2008; Vanbeveren et al. 2010; Maoz & Mannucci 2012; Tsujimoto 2013; Maoz et al. 2014; Heringer et al. 2017; Soker 2019). These studies will be used to determined the initial metallicity of progenitors with the cosmological evolution.

In the present work, the corrections of SNe Ia luminosity with redshift could be quantitatively estimated in large statistics. Then several important cosmological parameters such as the current proportion of dark energy and the age of the universe will be updated using the numerical method. In addition, we will examine the influences of the different progenitor channels and the CMM evolution rates on the corrections.

2.1. The Correction of Distance Modulus by Luminosity Evolution

For a source of absolute luminosity L, the observed luminosity is ${L}_{\mathrm{obs}}=L/{\left(1+z\right)}^{2}$ since photons lose energy and arrive less frequently with the expansion of the universe. According to the Gaussian flux relationship, the observed flux ${ \mathcal F }$ of a source is

$$\begin{eqnarray}&&{ \mathcal F }=\displaystyle \frac{L}{4\pi {r}_{h}^{2}{\left(1+z\right)}^{2}},\end{eqnarray} \tag{ 1 }$$

where r_h is the comoving distance between two points

$$\begin{eqnarray}&&{r}_{h}={\int }_{0}^{z}\displaystyle \frac{1}{H(z^{\prime} )}{dz}^{\prime} .\end{eqnarray} \tag{ 2 }$$

H(z) is the Hubble parameter at redshift z. The luminosity distance is defined as

$$\begin{eqnarray}&&{d}_{L}\equiv \sqrt{\displaystyle \frac{L}{4\pi { \mathcal F }}}.\end{eqnarray} \tag{ 3 }$$

Therefore, the relationship of the luminosity distance and the comoving distance is d_L = (1 + z)r_h . The luminosity distance can be further expressed as a function of redshift z and the current proportion of dark energy Ω_Λ based on Λ cold dark matter (ΛCDM) model,

$$\begin{eqnarray}&&{H}_{0}{d}_{L}=(1+z){\int }_{0}^{z}{\left[(1-{{\rm{\Omega }}}_{{\rm{\Lambda }}}){\left(1+z^{\prime} \right)}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}\right]}^{-\tfrac{1}{2}}{dz}^{\prime} ,\end{eqnarray} \tag{ 4 }$$

where H₀ is the Hubble constant.

The apparent magnitude is usually used instead of the observed flux in astronomical observations. It is defined as

$$\begin{eqnarray}&&{m}_{B}=5\mathrm{log}{d}_{L}-2.5\mathrm{log}L+C,\end{eqnarray} \tag{ 5 }$$

where C is a constant. The absolute magnitude is the magnitude at the 10 pc distance

$$\begin{eqnarray}&&{M}_{B}=5-2.5\mathrm{log}L+C.\end{eqnarray} \tag{ 6 }$$

Then, the distance modulus is given as follows

$$\begin{eqnarray}&&\mu ={m}_{B}-{M}_{B}=5\mathrm{log}{d}_{L}-5.\end{eqnarray} \tag{ 7 }$$

The term of luminosity is canceled and the distance modulus is only a function of the luminosity distance.

If the maximum luminosity of SNe Ia is considered as a fixed value across all the redshifts, the absolute magnitude of SNe Ia would be a constant. By taking the maximum luminosity of local SNe Ia L₀ as a benchmark, the absolute magnitude is

$$\begin{eqnarray}&&{M}_{B0}=5-2.5\mathrm{log}{L}_{0}+C.\end{eqnarray} \tag{ 8 }$$

Then, the experimental distance modulus can be obtained by using Equation (5) and (8)

$$\begin{eqnarray}&&{\mu }_{B}^{\exp }={m}_{B}-{M}_{B0}=5\mathrm{log}d{{\prime} }_{L}-5,\end{eqnarray} \tag{ 9 }$$

where

$$\begin{eqnarray}&&d{{\prime} }_{L}={d}_{L}{\left(\displaystyle \frac{L}{{L}_{0}}\right)}^{-\tfrac{1}{2}}\end{eqnarray} \tag{ 10 }$$

is the distance measured with the fixed absolute magnitude. It is related to the luminosity evolution of SNe Ia. The deviation of the actual distance modulus and the experimental distance modulus is

$$\begin{eqnarray}&&{\rm{\Delta }}\mu ={\mu }_{B}-{\mu }_{B}^{\exp }=2.5\mathrm{log}\displaystyle \frac{L}{{L}_{0}}.\end{eqnarray} \tag{ 11 }$$

It varies with the luminosity evolution of SNe Ia.

2.2. Relationships between (M⁵⁶Ni) and Z

The maximum luminosity of SNe Ia is determined directly by the mass of ⁵⁶Ni formed during the nucleosynthesis, L ∝ M(⁵⁶Ni) (Arnett 1982). The ⁵⁶Ni yield during the explosion is closely related to the initial metallicity of SNe Ia progenitor, which has been investigated in Timmes et al. (2003), Bravo et al. (2010), Seitenzahl et al. (2013), Leung & Nomoto (2018), Shen et al. (2018), Leung & Nomoto (2020), and Palla (2021).

For the near-M_ch scenario, Timmes et al. (2003) found that M(⁵⁶Ni) depends linearly on Z based on the basic nuclear physics,

$$\begin{eqnarray}&&M{(}^{56}\mathrm{Ni})\propto f(Z)=1-b\displaystyle \frac{Z}{{Z}_{\odot }},\end{eqnarray} \tag{ 12 }$$

where Z_⊙ denotes the solar metallicity and the slope b denotes the strength of the dependence. Then, the relationship between the relative maximum luminosity of SNe Ia and the metallicity is L/L₀ = f(Z)/f(Z₀). Z₀ denotes the initial metallicity of progenitor for the SNe Ia that explodes at present. If most of the ⁵⁶Ni is synthesized in material that burns fully to NSE and the neutron excess is not modified during the explosion, the slope is b = 0.057 (Timmes et al. 2003). The detailed postprocessing of W7-like models (Nomoto et al. 1984) has confirmed this linear dependence.

Bravo et al. (2010) researched the relationship in 1D DDT model. They considered that a sizeable fraction of ⁵⁶Ni is synthesized during incomplete Si burning and got a steeper slope b = 0.075 with a central density of WD ρ_c = 3 × 10⁹ g cm⁻³. The stronger dependence is because the free neutrons provided by the quickly photodissociation of the neutron-rich isotopes within the Si-group are efficiently captured by nuclei in the Fe-peak group, which favors their neutron-rich isotopes (Bravo et al. 2010).

Leung & Nomoto (2018) simulated the nucleosynthesis yields at different metallicity in 2D DDT model. And the adopted 2D DDT model is selected as their benchmark model for the near-M_ch scenario in their study. Figure 1 shows that M(⁵⁶Ni) (units of solar mass) varies with Z (units of solar metallicity) with a central density ρ_c = 3 × 10⁹ g cm⁻³. The blue points are the ⁵⁶Ni yield at metallicity Z/Z_⊙ = 0, 0.1, 0.5, 1, 2, 5 which are taken from Table 15 in Leung & Nomoto (2018). It can be seen that M(⁵⁶Ni) and Z/Z_⊙ are approximately linear. The read line is the best fitting, which gives the slope b = 0.072 ± 0.003. The relationship obtained from 1D DDT model is confirmed by the result of 2D DDT model.

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Seitenzahl et al. (2013) simulated the nucleosynthesis yields at different metallicity in 3D DDT model. Figure 2 shows that M(⁵⁶Ni) varies with Z with a central density ρ_c = 2.9 × 10⁹ g cm⁻³. The blue points are the ⁵⁶Ni yield at metallicity Z/Z_⊙ = 0.01, 0.1, 0.5, 1, which are taken from Table 3 in Seitenzahl et al. (2013). M(⁵⁶Ni) and Z/Z_⊙ are still approximately linear. The best fitting gives the slope b = 0.078 ± 0.002, which is also consistent with the results of 1D and 2D DDT models.

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For the sub-M_ch scenario, Shen et al. (2018) simulated the ⁵⁶Ni yield at different metallicity in 1D DD model. They found a ∼ 10% decrease in M(⁵⁶Ni) for a 1.0 M_⊙ WD detonation when the initial metallicity is changed from 0 to 2 Z_⊙. It is equivalent to the slope b = 0.05. This relationship is close to that obtained by Timmes et al. (2003).

As a continuation of their previous work on 2D DDT model, Leung & Nomoto (2020) further simulated the ⁵⁶Ni yield at different metallicity in 2D DD model. The red, green, and blue points in Figure 3 are the ⁵⁶Ni yield at metallicity Z/Z_⊙ = 0, 0.1, 0.5, 1, 2, 5, which are taken from Tables 7, 9, and 11 in Leung & Nomoto (2020) based on the models 110-100-2-50 (X), 110-050-2-B50 (Y), and 100-050-2-S50 (S), respectively. The three models are selected as the benchmark models for the sub-M_ch scenario in their study since all of them have a ⁵⁶Ni production at solar metallicity in line with standard SNe Ia, i.e., ∼0.6M_⊙. The names of the models imply the selected parameters. For example, model 110-100-2-50 (X) stands for a WD with total mass M = 1.10M_⊙, the He envelope mass M_He = 0.10M_⊙, the initial metallicity Z = 0.02, and the initial He detonation triggered at 50 km above the core-envelope interface. The endings "-B50" and "-S50" stand for different initial He detonations: a belt (ring) detonation around the "equator" of the WD and a spherical detonation triggered at 50 km above the He/CO interface, respectively. The "X" and "Y" in the brackets stand for the detonation that is first started along the rotation axis and symmetry axis, while the "S" in the brackets stand for the detonation seeds that have spherical symmetry. The best fittings give the slopes b = 0.050 ± 0.008, 0.051 ± 0.007, 0.056 ± 0.001 for the models 110-100-2-50 (X), 110-050-2-B50 (Y), and 100-050-2-S50 (S), respectively. The relationships based on the three models indicate that the different conditions have marginal effect on their M(⁵⁶Ni) − Z relationships, which are consistent within the error range of the fittings. In addition, the relationships obtained from 1D and 2D DD models are also consistent.

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The f(Z) varying with progenitor metallicity in different models are summarized in Figure 4. For the near-M_ch scenario, 1D, 2D, and 3D DDT models give similar relationships between M(⁵⁶Ni) and Z, b = 0.075, 0.072, 0.078, respectively. The dependences obtained from the DDT models are stronger than that obtained from the NSE model. It means that a sizeable fraction of ⁵⁶Ni is likely synthesized during incomplete Si burning in this scenario. The relationship obtained from 3D DDT model (b = 0.078) is chosen as the description of the general relationship. For the sub-M_ch scenario, all the relationships are close to that obtained by Timmes et al. (2003). It means that almost all the ⁵⁶Ni burns fully to NSE in this scenario. We choose the slope b = 0.050 as a general description. The dependence of M(⁵⁶Ni) on metallicity in near-M_ch model is stronger than that in sub-M_ch model, which means that the sub-M_ch SNe Ia is less affected by the initial metallicity of progenitors.

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2.3. The CMM Evolution with Redshift

The relationships between M(⁵⁶Ni) and Z in different progenitor channels for SNe Ia have been studied in the last subsection. The next step is to estimate the initial metallicity of progenitors. Since the initial metallicity of a main-sequence star is inherited from its ambient interstellar medium, it is reasonable to consider that the initial metallicity of progenitors roughly follow the CMM evolution.

Many studies of the CMM evolution have shown or implied that $\mathrm{log}{(Z/{Z}_{\odot })}_{\mathrm{CMM}}$ is approximately linear with redshift at least in the range of z < 5 (Prochaska et al. 2003; Rafelski et al. 2012; Vincoletto et al. 2012; Gioannini et al. 2017),

$$\begin{eqnarray}&&\mathrm{log}{(Z/{Z}_{\odot })}_{\mathrm{CMM}}={az}+{a}_{0}.\end{eqnarray} \tag{ 13 }$$

The absolute value of the parameter a indicates the evolution rate of the CMM and the parameter a₀ indicates the CMM at present.

Prochaska et al. (2003) got the evolution rate of the CMM a = −0.26 ± 0.07 by analyzing 125 DLA systems at 0.5 < z < 5. Rafelski et al. (2012) expanded the DLA systems to 242 samples at 0.09 < z < 5.06 and got a more accurate evolution rate a = −0.22 ± 0.03, which confirmed the relationship obtained by Prochaska et al. (2003). The two relationships are shown as the black dashed line and solid line in Figure 5.

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Vincoletto et al. (2012) and Gioannini et al. (2017) also got the CMM evolution by investigating the CSFR evolution. The blue and orange points in Figure 5 are taken from Figure 10 in Vincoletto et al. (2012) and the bottom panel of Figure 7 in Gioannini et al. (2017), respectively. Since the two results both show an approximately linear relationship between $\mathrm{log}{(Z/{Z}_{\odot })}_{\mathrm{CMM}}$ and z at z < 7, we use linear functions to fit the data points. The best fittings of the two sets of data points (green line and red line) give very consistent slopes, which are both about a = −0.15. It should be noted that the differences in the metallicity between galaxies of different morphologies at the same redshift are naturally expected, but we only focus on the SNe Ia luminosity evolution with redshift in this paper. So the adopted CMM is the average interstellar metallicity in galaxies of different morphological types including the spirals, ellipticals, and irregulars.

The evolution rate of the CMM given by the DLA systems (a = −0.22) is about 1.5 times of that obtained from the CSFR evolution (a = −0.15). We will use the two evolution rates of the CMM to further estimate the initial metallicity of progenitors and examine their influence on the luminosity evolution of SNe Ia. In addition, the CMM is different at the same redshift in different methods. The studies of the DLA systems and the CSFR evolution imply a distinctly different CMM at present. So the effect of the current CMM with a sub- or supersolar metallicity will also be discussed later.

2.4. The Effect of the SNe Ia Delay Time

In the last subsection, Equation (13) gives the CMM at the redshift of SNe Ia explosion instead of the initial metallicity of progenitors. Actually, the delay times of SNe Ia, the time elapsed between the birth of a SNe Ia progenitor and its explosion, have a wide range from the minimum value of 0.04 Gyr, which is the nuclear lifetime of the most massive stars that produce a WD ( ∼8M_⊙ star) to the maximum value on the order of a Hubble time. So the effect of the SNe Ia delay time should be considered in the estimation of the initial metallicity of progenitors.

The age of the universe at redshift z can be derived from the Friedmann equation based on ΛCDM model,

$$\begin{eqnarray}&&t=\displaystyle \frac{2}{3{H}_{0}\sqrt{{{\rm{\Omega }}}_{{\rm{\Lambda }}}}}\mathrm{ln}\left(\sqrt{\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{\Lambda }}}}{{{\rm{\Omega }}}_{M}{\left(1+z\right)}^{3}}+1}+\sqrt{\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{\Lambda }}}}{{{\rm{\Omega }}}_{M}{\left(1+z\right)}^{3}}}\right),\end{eqnarray} \tag{ 14 }$$

where Ω_M = 1 − Ω_Λ is the current proportion of matter in the universe, assuming that the time of the birth of a SNe Ia progenitor and its explosion are $\widetilde{t}$ and t, and the corresponding redshifts are $\widetilde{z}$ and z, respectively. Then the delay time of SNe Ia can be expressed as

$$\begin{eqnarray}&&\tau =t-\widetilde{t}=\displaystyle \frac{2}{3{H}_{0}\sqrt{{{\rm{\Omega }}}_{{\rm{\Lambda }}}}}\mathrm{ln}\left(\displaystyle \frac{\sqrt{\tfrac{{{\rm{\Omega }}}_{{\rm{\Lambda }}}}{{{\rm{\Omega }}}_{M}{\left(1+z\right)}^{3}}+1}+\sqrt{\tfrac{{{\rm{\Omega }}}_{{\rm{\Lambda }}}}{{{\rm{\Omega }}}_{M}{\left(1+z\right)}^{3}}}}{\sqrt{\tfrac{{{\rm{\Omega }}}_{{\rm{\Lambda }}}}{{{\rm{\Omega }}}_{M}{\left(1+\widetilde{z}\right)}^{3}}+1}+\sqrt{\tfrac{{{\rm{\Omega }}}_{{\rm{\Lambda }}}}{{{\rm{\Omega }}}_{M}{(1+\widetilde{z})}^{3}}}}\right).\end{eqnarray} \tag{ 15 }$$

The redshift of the birth of SNe Ia progenitor $\widetilde{z}$ can be solved from Equation (15) if the delay time is known.

The delay time distribution as a fundamental function for the purpose of the progenitor question has been widely investigated (Greggio 2005; Totani et al. 2008; Vanbeveren et al. 2010; Maoz & Mannucci 2012; Tsujimoto 2013; Maoz et al. 2014; Heringer et al. 2017; Soker 2019). Greggio (2005) studied the DTD function of SNe Ia and obtained the timescale over which 50% of the total SNe Ia explosions from an instantaneous burst of star formation have occurred. The typical timescales are about 0.5 Gyr and 1 Gyr for sub-M_ch and near-M_ch models, respectively. For simplicity, we adopt the two typical timescales as the average delay times for sub-M_ch and near-M_ch models to obtain the $\widetilde{z}$.

The expression of $\widetilde{z}$ solved from Equation (15) can be expanded near the H₀ τ = 0 due to H₀ τ ≪ 1, about 0.035 for sub-M_ch model and 0.069 for near-M_ch model with H₀ = 67.4 km s⁻¹ Mpc⁻¹ (Aghanim et al. 2020). By keeping the first order of H₀ τ, $\widetilde{z}$ has a simple approximation

$$\begin{eqnarray}&&\widetilde{z}\approx z+(1+z)H\tau ,\end{eqnarray} \tag{ 16 }$$

where $H={H}_{0}\sqrt{{{\rm{\Omega }}}_{{\rm{\Lambda }}}+{{\rm{\Omega }}}_{M}{\left(1+z\right)}^{3}}$ is the Hubble parameter. So the initial metallicity of progenitors for the SNe Ia which explodes at redshift z and at present are

$$\begin{eqnarray}&&Z/{Z}_{\odot }={10}^{a\widetilde{z}+{a}_{0}}={10}^{a[z+(1+z)H\tau ]+{a}_{0}},\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{Z}_{0}/{Z}_{\odot }=Z/{Z}_{\odot }{| }_{z=0}={10}^{{{aH}}_{0}\tau +{a}_{0}}.\end{eqnarray} \tag{ 18 }$$

2.5. The Correction of Distance Modulus

Putting all the relationships together, we obtain the following expression for the correction of distance modulus Δμ as a function of redshift z,

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}\mu & = & 2.5\mathrm{log}\displaystyle \frac{L}{{L}_{0}}=2.5\mathrm{log}\displaystyle \frac{f(Z)}{f({Z}_{0})}=2.5\mathrm{log}\displaystyle \frac{1-b\tfrac{Z}{{Z}_{\odot }}}{1-b\tfrac{{Z}_{0}}{{Z}_{\odot }}}\\ & = & 2.5\mathrm{log}\displaystyle \frac{1-b\times {10}^{a[z+(1+z)H\tau ]+{a}_{0}}}{1-b\times {10}^{{{aH}}_{0}\tau +{a}_{0}}},\end{array}\end{eqnarray} \tag{ 19 }$$

where the parameter a ∈ {−0.22, −0.15} represent the faster and slower CMM evolutions, respectively. The parameter b ∈ {0.050, 0.078} represent the strength of the dependences between M(⁵⁶Ni) and Z obtained from the sub-M_ch and near-M_ch models, respectively. The parameter τ ∈ {0.5, 1} Gyr represent the typical delay times for the sub-M_ch and near-M_ch models, respectively.

We can roughly analyze the effects of the different progenitor channels and CMM evolutions in this subsection. Equation (19) can be expanded near the b = 0 due to b ≪ 1. By only taking the first order, Equation (19) has a simple approximation

$$\begin{eqnarray}&&{\rm{\Delta }}\mu \propto {10}^{{a}_{0}}\left({10}^{{{aH}}_{0}\tau }-{10}^{a[z+(1+z)H\tau ]}\right)b.\end{eqnarray} \tag{ 20 }$$

Since the Δμ in Equation (20) is proportional to b, it is easy to estimate that the correction of distance modulus in the near-M_ch model is about 1.6 times as much as that in the sub-M_ch model for the same CMM evolution rate. While the estimation of the effect of different CMM evolution rates is a little more complicated due to the dependence on the SNe Ia delay time. Approximately, the Δμ with the faster CMM evolution is about 1.3 ∼ 1.4 times as much as that with the slower CMM evolution at z < 2 for the same progenitor channel. In addition, the current CMM with a sub- or supersolar metallicity will reduce or enlarge the Δμ by the same ratio ${10}^{{a}_{0}}$ at any redshift. In Figure 5, the studies of the DLA systems and the CSFR evolution imply that the current CMM may be a subsolar metallicity ∼10^−0.6 or a supersolar metallicity ∼10^0.3. The corresponding Δμ will be one fourth or two times as much as that obtained by setting the current CMM to be the solar metallicity. But it does not change the relative corrections of distance modulus with the different progenitor channels or CMM evolution rates. For this reason, in the subsequent analysis we will consider solar metallicity as the current CMM.

If the solar metallicity is taken as the current CMM, a₀ = 0, then Equation (19) is simplified as

$$\begin{eqnarray}&&{\rm{\Delta }}\mu =2.5\mathrm{log}\displaystyle \frac{1-b\times {10}^{a[z+(1+z)H\tau ]}}{1-b\times {10}^{{{aH}}_{0}\tau }}.\end{eqnarray} \tag{ 21 }$$

Since the correction from the SNe Ia delay time is related to the Ω_Λ, it is necessary to research the influence of the SNe Ia delay time with the change of Ω_Λ.

Figure 6 shows the Δμ varying with redshift z and Ω_Λ by taking the parameters a = −0.22, b = 0.078, and τ = 1 Gyr in Equation (21). It can be seen that the Δμ increases with redshift z for the same Ω_Λ. The dispersions of Δμ are 1.8 × 10⁻³ and 3.5 × 10⁻³ in the range Ω_Λ ∈ [0.6, 0.8] at the redshift z = 1 and z = 2, respectively. They are an order of magnitude less than the Δμ typical variations with redshift. So the influence of the SNe Ia delay time with the change of Ω_Λ is negligible. We can safely fix the current proportion of dark energy with the Planck 2018 result Ω_Λ = 0.6889 (Aghanim et al. 2020) in Equation (21). The more detailed analysis of the SNe Ia luminosity evolution with redshift will be shown in the next section.

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The relationships presented in the previous section show that the CMM evolution leads to an overall variation on the mass of ⁵⁶Ni synthesized and the maximum luminosity of SNe Ia with the cosmological evolution. In this section, we will quantitatively analyze the corrections of the distance modulus Δμ varying with redshift z in different models. Some important cosmological parameters will be updated by numerical calculation with the corrections.

Panel (a) in Figure 7 shows that the Δμ varies with redshift z in different models described by Equation (21). The green and red lines are the results in the sub-M_ch (b = 0.050) and near-M_ch (b = 0.078) models, respectively. The solid and dashed lines are the results with the faster (a = −0.22) and slower (a = −0.15) CMM evolutions, respectively. By taking the maximum luminosity of local SNe Ia L₀ as a reference, the Δμ increases with z and the values at z = 2 are about 1.7 times as much as that at z = 1 in different models. The blue solid line and dashed line in panel (b) represent the ratios of the Δμ in the near-M_ch to that in the sub-M_ch models with the faster and slower CMM evolutions, respectively. As per the predictions in Equation (20), the corrections in the near-M_ch model are about 1.7 times as much as that in the sub-M_ch model at the same redshift for the same CMM evolution rate. So the distance modulus in sub-M_ch SNe Ia is less affected by the initial metallicity of progenitors. The ratio R_b with the slower CMM evolution is slightly bigger than that with the faster CMM evolution, which means that the corrections of the distance modulus with the slower CMM evolution is more sensitive to the progenitor models. The green and red dotted lines in panel (c) represent the ratios of the Δμ with the faster CMM evolution to that with the slower CMM evolution in the sub-M_ch and near-M_ch models, respectively. As the previous discussion, the faster CMM evolution gives larger revisions of distance modulus which are about 1.2 ∼ 1.5 times of that with the slower CMM evolution at z < 2 for the same progenitor channel. The ratio R_a in the sub-M_ch model is slightly bigger than that in the near-M_ch model, which means that the corrections of the distance modulus in the sub-M_ch model are more sensitive to the CMM evolution rates.

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Then, we intend to further estimate the corrections of the current proportion of cosmological components. In order to extract the metallicity effect purely, we assume that it is the only factor that affects the distance measurements. Assuming that the distance with and without the metallicity correction are d_L and $d{{\prime} }_{L}$, respectively. If they both follow the equation Equation (4), the corresponding parameters are Ω_Λ and ${\rm{\Omega }}{{\prime} }_{{\rm{\Lambda }}}$. Since the integral in Equation (4) cannot be integrated into an analytical expression, we will use numerical method to calculate the correction. The specific steps are as follows:

• (i)  
  We calculate ${H}_{0}d{{\prime} }_{L}$ by using Equation (4) with the interval of redshift 0.1 in the redshift z < 2 and take ${\rm{\Omega }}{{\prime} }_{{\rm{\Lambda }}}=0.6889$ published by the Planck 2018 (Aghanim et al. 2020). The 21 points are used as the artificial data based on the assumption of a fixed absolute magnitude of SNe Ia.
• (ii)  
  According to Equations (10), (12), (17), and (18), the relationship between d_L and $d{{\prime} }_{L}$ is

  $$\begin{eqnarray}&&{H}_{0}{d}_{L}={H}_{0}d{{\rm{{\prime} }}}_{L}{\left(\displaystyle \frac{L}{{L}_{0}}\right)}^{{\textstyle \tfrac{1}{2}}}={H}_{0}d{{\rm{{\prime} }}}_{L}{\left(\displaystyle \frac{1-b\times {10}^{a[z+(1+z)H\tau ]}}{1-b\times {10}^{{aH}_{0}\tau }}\right)}^{{\textstyle \tfrac{1}{2}}}.\end{eqnarray} \tag{ 22 }$$

  Then, ${H}_{0}d{{\prime} }_{L}$ is corrected to H₀ d_L by considering the luminosity evolution with Equation (22) at each point.
• (iii)  
  We tune the Ω_Λ to fit the corrected points with Equation (4) and the best fitting of the Ω_Λ is determined by minimizing the standard deviation

  $$\begin{eqnarray}&&\sigma =\sqrt{\displaystyle \frac{1}{N}\sum _{i}{\left(y{{\prime} }_{i}-{y}_{i}\right)}^{2}},\end{eqnarray} \tag{ 23 }$$

  where N is the number of points. y_i and $y{{\prime} }_{i}$ are H₀ d_L and the corresponding fitting values at each point, respectively. The best-fit value of the Ω_Λ is the corrected current proportion of dark energy.

As an example, Figure 8 shows that the corrected values H₀ d_L (red points) varies with redshift by taking the parameters a = −0.22 and b = 0.078. The red curve is the best fitting by tuning the Ω_Λ in Equation (4) and the minimal σ is 1.4 × 10⁻³. All the other fittings for the different models are also pretty well.

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Figure 9 shows the corrected Ω_Λ obtained from the different modification models. The black columns in the histogram represent the Ω_Λ without the metallicity corrections, which are 0.6889. For the faster CMM evolution, the corrected Ω_Λ's are 0.7086 and 0.7210 in the sub-M_ch and near-M_ch models, respectively. The corrected Ω_Λ's are bigger than the original value, which means that the acceleration of cosmological expansion is greater than previously estimated. The increases of Ω_Λ in the near-M_ch model are about 1.7 times as much as that in the sub-M_ch model for the same CMM evolution rate, and they with the faster CMM evolution are about 1.3 times as much as that with the slower CMM evolution for the same progenitor channel. The influences of the different progenitor channels and CMM evolution rates on Ω_Λ are similar to that on Δμ.

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Having obtained Ω_Λ, we can derive other related quantities. The fractional density of dark energy is

$$\begin{eqnarray}&&\displaystyle \frac{{\rho }_{{\rm{\Lambda }}}}{{\rho }_{\mathrm{tot}}}=\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{\Lambda }}}}{{{\rm{\Omega }}}_{M}{\left(1+z\right)}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}}=\displaystyle \frac{1}{({{\rm{\Omega }}}_{{\rm{\Lambda }}}^{-1}-1){\left(1+z\right)}^{3}+1}.\end{eqnarray} \tag{ 24 }$$

The evolutions of the dark energy proportion with redshift in different models are shown in Figure 10. It can be seen that the ρ_Λ/ρ_tot corrected by the near-M_ch model is growing faster than that corrected by the sub-M_ch model with either the faster or slower CMM evolutions. The corrected ρ_Λ/ρ_tot both are growing faster than that without correction. By comparing the above and below panels, a faster CMM evolution rate will promote the growth of the ρ_Λ/ρ_tot.

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The current age of the universe is corresponding to the redshift z = 0 in Equation (14) with the updated Ω_Λ (Li & Li 2021),

$$\begin{eqnarray}&&{t}_{0}=\displaystyle \frac{2}{3{H}_{0}\sqrt{{{\rm{\Omega }}}_{{\rm{\Lambda }}}}}\mathrm{ln}\left(\displaystyle \frac{1+\sqrt{{{\rm{\Omega }}}_{{\rm{\Lambda }}}}}{\sqrt{1-{{\rm{\Omega }}}_{{\rm{\Lambda }}}}}\right).\end{eqnarray} \tag{ 25 }$$

This formula states that t₀ increases with Ω_Λ.

The t₀ obtained from the different modification models is shown in Figure 11. The t₀ without the metallicity corrections is about 13.82 Gyr with the Ω_Λ = 0.6889 and H₀ = 67.4 km s⁻¹ Mpc⁻¹ (Aghanim et al. 2020). For the faster CMM evolutions, the metallicity corrections causes the t₀ to increase by 0.26 and 0.43 Gyr in the sub-M_ch and near-M_ch models, respectively. The corrections with the faster CMM evolution are about 1.3 times as much as that with the slower CMM evolution for the same progenitor channel. In addition, the corrections in the near-M_ch model are about 1.7 times as much as that in the sub-M_ch model with either the faster or slower CMM evolutions.

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With Equation (14) the cosmological age at which the matter density is equal to the dark energy density, ${{\rm{\Omega }}}_{M}{\left(1+z\right)}^{3}={{\rm{\Omega }}}_{{\rm{\Lambda }}}$, is

$$\begin{eqnarray}&&{t}_{e}=\displaystyle \frac{2\mathrm{ln}(\sqrt{2}+1)}{3{H}_{0}\sqrt{{{\rm{\Omega }}}_{{\rm{\Lambda }}}}}.\end{eqnarray} \tag{ 26 }$$

This time is early with the increasing of Ω_Λ.

The t_e obtained from the different modification models is shown in Figure 12. The point is at about 10.25 Gyr without the metallicity corrections. For the faster CMM evolutions, the metallicity corrections bring the t_e forward by 0.14 and 0.23 Gyr in the sub-M_ch and near-M_ch models, respectively. The influences of the different progenitor channels and CMM evolution rates on t_e are same to that on t₀.

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The CMM effect on the luminosity evolution of SNe Ia is investigated in this paper. By comparing the ⁵⁶Ni yield simulated in different models, we derived the relationships between M(⁵⁶Ni) and Z in different progenitor channels and found that the strength of the dependence of M(⁵⁶Ni) on Z in the near-M_ch model is about 1.6 times of that in the sub-M_ch model. In addition, we derived a faster and a slower CMM evolution based on the studies of the DLA systems and the CSFR evolution. With the consideration of the SNe Ia delay times, the initial metallicity of progenitors is determined by using the CMM evolution. By combining these analysis, the metallicity corrections of SNe Ia luminosity increase with redshift z and the Δμ at z = 2 are about 1.7 times as much as that at z = 1 in different models. The adoptions of different progenitor channels and CMM evolution rates give similar results on the revisions of μ, Ω_Λ, t₀ and t_e . The revisions based on the near-M_ch model are about 1.7 times as much as that based on the sub-M_ch model for the same CMM evolution rate, which means that the sub-M_ch SNe Ia is less affected by the initial metallicity of progenitors. The revisions with the faster CMM evolution are about 1.3 times as much as that with the slower CMM evolution for the same progenitor channel. According to the metallicity corrections of SNe Ia luminosity, the updated cosmological parameters implied that the acceleration of cosmological expansion may be greater than previously estimated and our universe is older than previously thought.

This work was supported by the National Natural Science Foundation of China under grant No. 11490563, the Continuous Basic Scientific Research Project under grant No. WDJC-2019-13, the National Key Research and Development Project under grant No.2016YFA0400502, and the China Postdoctoral Science Foundation under grant No. 12BSH2001.