It is believed that a type 1a supernova is formed when a white dwarf captures matter from its neighbor in a binary system, as a result of which its mass increases to a possible limit—the Chandrasekhar limit, when already degraded electrons cannot resist gravitational pressure and the star passes into an unstable stage. An increase in the temperature and density of the star makes it possible for carbon and oxygen to be converted into ⁵⁶Ni, which is accompanied by a thermonuclear explosion [1] . The brightness of the star increases so much that sometimes it exceeds the brightness of the host galaxy, and it can be seen for several thousand megaparsecs. The mass of the exploded star is always near the Chandrasekhar limit, so in the case of such explosions the absolute magnitude can vary within small limits. This allows these stars to be used as distance indicators [2] .

This feature of type 1a supernovae makes it possible to study the behavior of the Universe at considerable distances and evaluate the validity of one or another cosmological model.

At the beginning of the 20th century, Hubble obtained a very interesting result, which led to the conclusion that galaxies are moving away from us, and the speed of receding is directly proportional to the distance from us. Hubble’s work is based on the fact discovered by Slipher, that the spectral lines in the spectra of galaxies are shifted towards the long wavelength [3] . Hubble found that this shift increases with increasing distances to galaxies [4] . Another method for determining distance is based on the modulus of distance.

M = m − 5 lg D L − 25 , (1)

where m is the apparent magnitude, M is the absolute magnitude, D_L is the Luminosity distance. When calculating the distance using this method, it is necessary to accurately estimate the value of the apparent magnitude of the object (take into account the galactic extinction, K-correction, spectral region, etc.). The absolute magnitude should be known either from theoretical approaches (for example, for type 1a supernova stars) or from empirical relationships (for example, in the case of Cepheids).

[5] and [6] , in order to study the properties of the universe, made two assumptions:

1) Assume that type 1a supernovae are indicators of distances, that is, their absolute magnitudes can be considered constant.

2) That the Friedman-Robertson-Walker (FRW) cosmological model, for the case of a flat universe, accurately describes the Universe.

Taking into account quite accurately the phenomena that can influence the result, they calculated the apparent stellar magnitudes, and compared them with the values obtained from the cosmological model. It turned out that the apparent brightnesses were weaker than those obtained from the theory, that is, these objects are further away than they would be, based on Hubble’s law. This led to the idea that the universe is expanding at an accelerating rate. In this regard, the idea of “dark energy” was introduced.

To avoid the idea of dark energy, various attempts have been made to explain the discrepancy between the theoretical and observed supernova luminosities by other phenomena. Let’s list some of them.

1) The weakening of the apparent magnitude of a star occurs due to the absorption or scattering of light by matter in the path of light.

2) There is an evolution in the luminosity of a white dwarf, depending on the chemical composition of the host galaxy over time.

3) Gravity lenses.

4) The reason is the uneven distribution of matter in the Universe.

5) It is assumed that there are two types of supernovae 1a in nature. The second type is not numerous and is formed from the merger of two white dwarfs. As a result of the merger, the mass of the exploding star is no longer fixed.

6) Observational errors may also increase due to the fact that the brightness curves of various supernovae are recorded under different conditions (on Earth and in space).

The degree of influence of these phenomena has been discussed in various studies, showing that many of these inaccuracies cannot be considered satisfactory for refuting the results obtained by [5] and [6] . They can be found in [7] .

However, in our opinion, there are two observational facts that cannot be ignored.

First, this is a rather large width of the distribution of the absolute magnitudes of type 1a supernovae (Figure 1). This issue was studied in the article by [8] . The average absolute magnitude of 115 studied stars was obtained 〈 M B 〉 = − 19.04 ± 0.07 , standard deviation σ M B = 0.70 . 89 of them have late host galaxies (Sa-Irr or star-forming galaxies, S-F), for which 〈 M B 〉 = − 19.20 ± 0.05 , σ M B = 0.49 , and 26 have early host galaxies (E-SO or passive galaxies), respectively 〈 M B 〉 = − 18.48 ± 0.19 , σ M B = 0.98 .

Such large standard deviations in the absolute magnitude distributions of type 1a supernovae allow us to conclude that when estimating the values of cosmological parameters, it is wrong to take as a basis the absolute magnitude determined by several stars. In [9] (Paper I) showed that in this case the obtained cosmological parameters lead to a violation of the initial assumption that the absolute magnitudes of type 1a supernovae do not change with distance. This violation disappears when the absolute magnitude of supernovae is estimated in the course of estimating the cosmological parameters. Thus, when estimating cosmological parameters, the absolute magnitude of supernovae should also be an estimated parameter. The absence of such an approach can be considered a shortcoming in the works of other authors related to this topic. Note that this approach also improves the fit between the observational data and the theory. Assuming that the absolute magnitude of supernovae is constant with distance, we get that the share of dark energy in a flat universe does not exceed 50%. In [9]

also obtained another important result that the cosmological model with zero cosmological parameter describes the universe no worse than the Friedmann-Robertson Walker model.

Second, the correlation between the absolute magnitude of supernovae and the age of the stellar population of host galaxies indicates that there is an evolution in the absolute magnitude of supernovae [10] . It is known that the absolute magnitude of type 1a supernovae correlates with the characteristics of the host galaxy. For example, in [11] found a systematic difference in the absolute magnitude of supernovae of ~0.14 magnitude between very early and very late galaxies. [12] and [13] found that SNe1a in less massive galaxies (by a factor of 10) by ~0.08 magnitudes are weaker than in more massive galaxies. [14] showed that SNe1a in environments with local star formation (higher local SFR) is about 0.16 magnitude weaker than in locally passive environments (lower local SFR). [10] noted features of the host galaxies (morphology, mass and local SFR) were converted to age differences with methods known in the literature. Table 1 is taken from [10] . The table shows the correlation of the absolute magnitude of supernovae 1a with the properties of the parent galaxies. The last column of Table 1 shows the estimated absolute magnitude evolution over 5.3 Gyr, which corresponds to the difference in age at z = 0 and z = 1 (see [10] , for each of the

four different studies. The average of these values is ~0.25 mag/5.3Gyr. In this range of redshifts, the observed decrease in supernova brightness in the Hubble diagram is approximately comparable to this value (see, for example, [5] . And so, this effect may be associated with the evolution of the luminosity of supernovae and has nothing to do with the accelerated receding of distant supernovae.

Thus, when estimating the cosmological parameters, it is also important to estimate the possible evolution of the absolute magnitudes of supernovae. In this article, we study models of the universe under the assumption of the existence of an evolution of the luminosities of type 1a supernovae and try to find those values of the cosmological parameters for which there will be the best fit between theory and observation.

In this paper, we will discuss two models: the ΛCDM model used by [5] and [6] for the case of a flat universe, and the model with a zero cosmological constant, which was widely used before these works (until 1999). The first model assumes the existence of dark energy; in the second model, such a hypothesis is not necessary.

In the case of the ΛCDM model, the dependence of the luminosity distance on redshift is given by the following formula:

D L ( z , Ω M , Ω Λ , Ω K ) = C H 0 − 1 ( 1 + z ) | Ω K | − 1 2 s i n n { | Ω K | 1 2 ∫ 0 z     d z [ ( 1 + z ) 2 ( 1 + Ω M z ) − z ( 2 + z ) Ω Λ ] − 1 2 } (2)

where z is the redshift of the object. H 0 is the Hubble constant. Ω K is related to the curvature of space and in the case of flat universe it is 0 [15] : Ω K = 1 − Ω M − Ω Λ , s i n n = s i n h , when Ω K ≥ 0 and s i n n = s i n , when Ω K ≤ 0 . In the case of Ω K = 0 , we will have:

D L ( z , Ω M , Ω Λ ) = C ( 1 + z ) H 0 ∫ 0 z     d z [ ( 1 + z ) 2 ( 1 + Ω M z ) − z ( 2 + z ) Ω Λ ] − 1 2 (3)

or

D L = C ( 1 + z ) H 0 ∫ 0 z     d z [ ( 1 + z ) 3 Ω M + Ω Λ ] − 1 2

If we assume that Ω Λ = 1 , and Ω M = 0 , we will have [7]

D L ( z ) = C H 0 ( z + z 2 ) (4)

If Ω Λ = 0 , and Ω M = 1 , we have

D L ( z ) = 2 C H 0 [ ( 1 + z ) − 1 + z ] (5)

It should be noted that in 1998, prior to the work of [5] and [6] commonly used the equations of general relativity (GR) with zero cosmological constant ( Λ = 0 ). Using this model, [16] integrated these equations exactly and obtained the luminosity distance as a function of redshift.

D L ( z , q 0 ) = C H 0 q 0 2 [ q 0 z + ( q 0 − 1 ) ( 1 + 2 q 0 z − 1 ) ] (6)

where q 0 is the deceleration parameter, in this case:

q 0 = Ω M 2

(6) with q 0 = 0.5 coinciding with (5).

For the luminosity distance in the case of a flat universe we will use Formula (3), for the luminosity distance in the model with zero cosmological constant ( Λ = 0 ) we will use Formula (6).

We will also discuss the general case (2) with nonzero space curvature. We also assume that the dependence of the absolute magnitude of the supernova on z is linear. Then we can assume M z = M 0 + ε c z and Formula (1) can be written as follows:

M 0 + ε c z = m − 5 log D L − 25 (7)

where M 0 is the absolute magnitude of the supernova at z = 0 ( M 0 = M ( z = 0 ) ), ε c is the evolution coefficient of the absolute magnitude.

Our approach was as follows: to develop a computer model where one can load the observed data of supernovae 1a and easily obtain cosmological parameters by achieving the best fit of observations with theory. As the search variables of the computer model, both the cosmological parameters of the Friedmann-Robertson-Walker model Ω Λ , Ω M , Ω K and the parameters M 0 and ε c were used. The value of the Hubble constant H 0 is assumed to be 72.305. Solutions were sought in the ranges of variables:

0 ≤ Ω M ≤ 1,

0 ≤ Ω Λ ≤ 1,

0 ≤ Ω K ≤ 1,

− 19.5 ≤ M 0 ≤ − 18,

− 1 ≤ ε c ≤ 1,

under the condition of a flat universe ( Ω M + Ω Λ = 1 ) and without it ( Ω M + Ω Λ + Ω K = 1 ). A model with a zero cosmological constant is also investigated, in which the search parameters are q 0 , M 0 and ε c . The solution was searched in the ranges:

0 ≤ q 0 ≤ 0.5,

− 19.5 ≤ M 0 ≤ − 18,

− 1 ≤ ε c ≤ 1.

Solver Excell, Macroses and SciDAVIs were used as optimization decision tools.

For the study, we use a subsample from SNe1a “Union2” [17] . The sample consists of 719 supernovae identified in 17 papers [5] [6] [17] - [31] . Following several principles, the authors [17] cleared the sample and retained 557 supernovae for further study. We will also use the observational material of these 557 stars without making any changes.

We analyze the Hubble diagram and find those values of the parameters present in the discussed model of the universe, which provide the best fit between the model and observation. On the Hubble diagram, the theoretical curve can be represented by the following relationship:

B m a g t h ( z , Ω M , Ω Λ , Ω K ) = M 0 + ε c z + 5 log D L ( z , Ω M , Ω Λ , Ω K ) + 25 (8)

for the Friedmann-Robertson-Walker model, or

B m a g t h ( z , q 0 ) = M 0 + ε c z + 5 log D L ( z , q 0 ) + 25 (9)

for the model with zero cosmological constant.

We need to find those values of the parameters Ω M , Ω Λ , Ω K , M 0 , ε c in the first case and q 0 , M 0 , ε c in the second case, so that the sum of squares ( B o b s − B m a g t h ( z ) ) would be minimal:

C h i 2 = ∑ ( B o b s − B m a g t h ( z ) ) 2 = m i n .

For the luminosity distance D L we use Formulas (2) and (3) for the Friedmann-Robertson-Walker model and Formula (6) for the model with zero cosmological constant. We will investigate the following cases:

1) The flat universe of Friedmann-Robertson-Walker ( Ω Λ + Ω M = 1 ). We will look for Ω Λ , Ω M and M 0 , ε c for the best fit between theory and observation. We will discuss two cases:

a) There is no evolution of the absolute magnitudes of supernovae ( ε c = 0 ) and

b) There is an evolution of the absolute magnitudes of supernovae ( ε c ≠ 0 ).

2) Friedmann-Robertson-Walker universe without restriction on space curvature ( Ω Λ + Ω M + Ω K = 1 ). We will look for Ω Λ , Ω M , Ω K , and M 0 for the best fit between theory and observation. We will also discuss two cases:

a) ε c = 0 and

b) ε c ≠ 0 .

3) Universe with zero cosmological constant. We will look for q 0 and M 0

For the best fit between theory and observation. We will also discuss two cases:

a) ε c = 0 and

b) ε c ≠ 0 .

All cases are tested by the “absolute magnitude test” we proposed in Paper I.

In this section, using the created computer model, we also study the influence of the absolute magnitude of supernovae 1a on the obtained cosmological parameters. It turned out that the cosmological parameters are very sensitive to even a small change in this value.

This issue is very important because, when estimating cosmological parameters, researchers use the absolute magnitude of supernovae obtained by only a few stars. As we noted above, the distribution of the absolute magnitude of type 1a supernovae is very wide, and at first glance it can be seen (you can compare Figure 2 and Figure 3) that the value of the cosmological parameters strongly depends on the absolute magnitude of supernovae. Let’s try to study this issue in more detail.

Figure 8 shows a graph of dependence Ω Λ , Ω M on M. This graph is constructed for the ΛCDM model for a flat universe ( Ω Λ + Ω M = 1 ) and no evolution ( ε c = 0 ). The graph shows the values of M corresponding to three combinations of cosmological parameters:

1) Ω Λ = 0.7 , Ω M = 0.3 obtained at M = − 19.11 ;

2) Ω Λ = 0 , Ω M = 1 obtained at M = − 18.71 ;

3) Ω Λ = 0.397 , Ω M = 0.603 obtained at M = − 18.90 ;

As shown above, the best solution for a flat universe, without taking into account evolution, was obtained in the latter case (see Table 2).

The difference in the absolute magnitude of supernovae 1a with combinations of 1) and 2) is: 19.11 − 18.71 = 0.4 magnitudes, while the standard deviation of the distribution of absolute magnitudes of SNe1a is 0.7 magnitudes. At the same time, the difference in magnitude between the combinations, 1) and 3) is only 0.2.

Table 12 shows an excerpt from the table from the simulation. The first line shows the case of the best fit between theory and observation. The rest of the lines show the results in the vicinity of the point Ω Λ = 0.7 and Ω M = 0.3 . It can be seen that when M changes by only 0.1 in one direction or another, we obtain the value of Ω Λ from 0.6 to 0.8. Such a difference in the values of Ω Λ significantly changes the idea of cosmology. The change in M by 0.1 is well below the range of this value used in various papers.

Thus, the dependence of the values of the parameters Ω Λ and Ω M on the accepted absolute magnitude M SNe1a is very strong, and therefore we must be extremely careful when determining M. According to the authors of this article, the determination of the absolute magnitude of supernovae should be the subject of a simulation using the entire sample of type 1a supernovae. As we saw above, this approach does not violate the original assumption that the absolute magnitudes of supernovae depend on the redshift and, therefore, gives the correct value of the parameters Ω Λ and Ω M .

In the previous article, we studied the value of cosmological parameters in the ΛCDM and CDM models with the assumption that the absolute magnitude of type 1a supernovae is independent of distance. The values of the cosmological parameters were estimated on the basis of the Hubble diagram. The values of these parameters were determined for which the best fit between the theoretical curve of the Hubble diagram and observational data was obtained. Pearson’s goodness-of-fit test or Chi² (Chi-square) test was used.

In this article, we study the case with the assumption of the evolution of the absolute magnitude. We accept that the dependence of the absolute magnitude on the redshift is linear.

It turns out that when the evolution of the absolute magnitudes of supernovae is taken into account, a better fit between theory and observation is obtained.

The main difference between the approaches in our work and the works of other authors is that we estimate the average absolute magnitude of supernovae in the course of simulation, while the authors of other works take into account the average absolute magnitude of these stars, previously obtained by several well-studied stars. We pay attention to the fact that the distribution of the absolute magnitude of supernovae 1a is very wide [8] . This makes it incorrect to use the latter approach. In addition, as the simulation shows, the result strongly depends on the assumed absolute magnitude of the stars.

In the case of a flat universe ( Ω Λ + Ω M = 1 ), the best fit between theory and observation is given by the value of the evolution coefficient ε c = 0.304 . In this case, for the cosmological parameters we obtain Ω Λ = 0.000 , Ω M = 1.000 . And for the absolute magnitude of supernovae 1a, −18.875 was obtained. Naturally, this result exactly matches the simulation result for the model with zero cosmological constant ( ε c = 0.304 , q 0 = 0.500 , M 0 = − 18.875 ).

The Friedmann-Robertson-Walker model is also studied, without restriction on space curvature ( Ω Λ + Ω M + Ω K = 1 ). Within the framework of this model, we obtain the following values ε c = 0.304 , Ω Λ = 0.000 , Ω M = 1.000 , Ω K = 0.000 , M 0 = − 18.875 . Those, the case also leads to a flat Universe model ( Ω K = 0.000 ).

In the framework of this work, the degree of influence of the absolute magnitude M of supernovae of type 1a on the cosmological parameters is also investigated. In particular, it was found that a change in this value by only 0.4 m (from −19.11 to −18.71) leads to a change in the parameters from Ω Λ = 0.7 and Ω M = 0.3 to Ω Λ = 0 and Ω M = 1 . As we noted above, the distribution of the absolute magnitudes of supernovae 1a has a rather large width, which leads us to think that we must be very careful when accepting the absolute magnitude of supernovae. We have come to the conclusion that the absolute magnitude of supernovae must also be subject to simulation. This is the main reason that leads to a discrepancy between our results and the results obtained by other authors, who took as a basis the value of the absolute magnitude of supernovae calculated from a small number of stars.

The validity of our results is substantiated by the absolute magnitude test we proposed in the previous article (Paper I). In essence, this is a test proving that the simulation does not violate the initially accepted dependence of the absolute magnitudes of supernovae on the redshift. This test can also be called a model adequacy test.

The main results of this work are the following:

1) Under the assumption of the evolution of supernovae SNe1a, the ΛCDM model describes the observational data better than under the assumption that the absolute magnitudes of SNe1a are independent of redshift. In this case, a small evolution is obtained ( Δ M = 0.304 during the time of the corresponding z = 1). Young supernovae are dimmer. Evolution is observed for both nearby and distant stars.

2) The universe turns out to be flat, even if this constraint is not initially introduced.

3) There is only gravitational matter in the universe.

4) Objects in the Universe move away from each other with deceleration.

The main difference between our and other authors’ approaches is that we pay attention to two facts in the nature of SNe1a. The first is that there is a critical dependence of the values of cosmological parameters obtained during the simulation on the absolute magnitude of SNe1a. The second is that the distribution of absolute magnitudes of supernovae is very wide.

These facts lead us to the conclusion that the found cosmological parameters based on the absolute magnitude of supernovae 1a predetermined from several stars are not accurate and that it is necessary to find the cosmological parameters and absolute magnitude of supernovae simultaneously in the simulation process using the full sample of supernovae 1a.

The observational data underlying this article are taken from [17] (link below).

The authors declare no conflicts of interest regarding the publication of this paper.

Mahtessian, A.P., Karapetian, G.S., Hovhannisyan, M.A. and Mahtessian, L.A. (2023) The Evolving Absolute Magnitude of Type 1a Supernovae and Its Critical Impact on the Cosmological Parameters. International Journal of Astronomy and Astrophysics, 13, 39-60. https://doi.org/10.4236/ijaa.2023.132003