_{1}

^{*}

We review the distance modulus in twelve different cosmologies: the ΛCDM model, the wCDM model, the Cardassian model, the flat case, the ?CDM cosmology, the Einstein—De Sitter model, the modified Einstein—De Sitter model, the simple GR model, the flat expanding model, the Milne model, the plasma model and the modified tired light model. The above distance moduli are processed for three different compilations of supernovae and a supernovae + GRBs compilation: Union 2.1, JLA, the Pantheon and Union 2.1 + 59 GRBs. For each of the 48 analysed cases we report the relative cosmological parameters, the chi-square, the reduced chi-square, the AIC and the Q parameter. The angular distance as function of the redshift for five cosmologies is reported in the framework of the minimax approximation.

At the moment of writing, the determination of the Hubble constant is oscillating between a low value as derived by the Planck collaboration [_{0}. The number of supernovae (SNs) of type Ia for which the distance modulus is available has grown with time: 34 SNs in the sample which produced evidence for the accelerating universe [

In the following, we analyze twelve cosmologies. A useful introduction to the distances in cosmology can be found in [

In ΛCDM cosmology the Hubble distance D H is defined as

D H ≡ c H 0 , (1)

where c is the speed of light and H_{0} is the Hubble constant. We then introduce the first parameter Ω M ,

Ω M = 8 π G ρ 0 3 H 0 2 , (2)

where G is the Newtonian gravitational constant and ρ 0 is the mass density at the present time. A second parameter is Ω Λ ,

Ω Λ ≡ Λ c 2 3 H 0 2 , (3)

where Λ is the cosmological constant, see [_{0} are found the numerical value of the cosmological constant is derived, Λ ≈ 1.2 m − 2 .

The two previous parameters are connected with the curvature Ω K by

Ω M + Ω Λ + Ω K = 1. (4)

The comoving distance, D C , is

D C = D H ∫ 0 z d z ′ E ( z ′ ) , (5)

where E ( z ) is the “Hubble function”:

E ( z ) = Ω M ( 1 + z ) 3 + Ω K ( 1 + z ) 2 + Ω Λ . (6)

The above integral cannot be done in analytical terms, except for the case of Ω Λ = 0 , but the Padé approximant, see Appendix 5, allows to derive the approximated indefinite integral, see Equation (10).

The approximate definite integral for (5) is therefore,

D C ,2,2 = D H ( F 2,2 ( z ; a 0 , a 1 , a 2 , b 0 , b 1 , b 2 ) − F 2,2 ( 0 ; a 0 , a 1 , a 2 , b 0 , b 1 , b 2 ) ) , (7)

where F 2,2 is Equation (10). The transverse comoving distance D M is:

D M = { D H 1 Ω K sinh [ Ω K D C / D H ] for Ω K > 0 D C for Ω K = 0 D H 1 | Ω K | sin [ | Ω K | D C / D H ] for Ω K < 0 (8)

and the approximate transverse comoving distance D M ,2,2 computed with the Padé approximant is:

D M ,2,2 = { D H 1 Ω K sinh [ Ω K D C ,2,2 / D H ] for Ω K > 0 D C ,2,2 for Ω K = 0 D H 1 | Ω K | sin [ | Ω K | D C ,2,2 / D H ] for Ω K < 0 (9)

The Padé approximant for the luminosity distance is

D L ,2,2 = ( 1 + z ) D M ,2,2 , (10)

and the Padé approximant for the distance modulus, ( m − M ) 2,2 , is

( m − M ) 2,2 = 25 + 5 log 10 ( D L ,2,2 ) . (11)

As a consequence, M 2,2 , the absolute magnitude of the Padé approximant, is

M 2,2 = m − 25 − 5 log 10 ( D L ,2,2 ) . (12)

The expanded version of the Padé approximant distance modulus is:

( m − M ) 2,2 = 25 + 5 1 ln ( 10 ) ln ( c ( 1 + z ) H 0 Ω K sinh ( 1 / 2 Ω K A b 2 2 4 b 0 b 2 − b 1 2 ) ) , (13)

with

A = ln ( z 2 b 2 + z b 1 + b 0 ) a 1 b 2 4 b 0 b 2 − b 1 2 − ln ( z 2 b 2 + z b 1 + b 0 ) a 2 b 1 4 b 0 b 2 − b 1 2 − ln ( b 0 ) a 1 b 2 4 b 0 b 2 − b 1 2 + ln ( b 0 ) a 2 b 1 4 b 0 b 2 − b 1 2 + 2 a 2 z b 2 4 b 0 b 2 − b 1 2 + 4 arctan ( 2 z b 2 + b 1 4 b 0 b 2 − b 1 2 ) a 0 b 2 2 − 2 arctan ( 2 z b 2 + b 1 4 b 0 b 2 − b 1 2 ) b 1 a 1 b 2 − 4 arctan ( 2 z b 2 + b 1 4 b 0 b 2 − b 1 2 ) a 2 b 0 b 2 + 2 arctan ( 2 z b 2 + b 1 4 b 0 b 2 − b 1 2 ) b 1 2 a 2 − 4 arctan ( b 1 4 b 0 b 2 − b 1 2 ) a 0 b 2 2 + 2 arctan ( b 1 4 b 0 b 2 − b 1 2 ) b 1 a 1 b 2 + 4 arctan ( b 1 4 b 0 b 2 − b 1 2 ) a 2 b 0 b 2 − 2 arctan ( b 1 4 b 0 b 2 − b 1 2 ) b 1 2 a 2

More details can be found in [

In the dynamical dark energy cosmology (wCDM), firstly introduced by [

D H ( z ; Ω M , w , Ω D E ) = 1 ( 1 + z ) 3 Ω M + Ω D E ( 1 + z ) 3 + 3 w , (14)

where w is the equation of state here considered constant, see Equation (3.4) in [

Ω M + Ω D E = 1, (15)

and the Hubble distance becomes

D H ( z ; Ω M , w ) = 1 ( 1 + z ) 3 Ω M + ( 1 − Ω M ) ( 1 + z ) 3 + 3 w . (16)

The indefinite integral in the variable z of the above Hubble distance, I z ≡ D C D H , is

I z ( z ; Ω M , w ) = ∫ D H ( z ; Ω M , w ) d z , (17)

where the new symbol I z underline the mathematical operation of integration. In order to solve for the indefinite integral we perform a change of variable 1 + z = t 1 / 3 .

I z ( t ; Ω M , w ) = 1 3 ∫ 1 − t ( ( − 1 + Ω M ) t w − Ω M ) t 2 / 3 d t . (18)

The indefinite integral is

I z ( t ; Ω M , w ) = − 2 F 2 1 ( 1 2 , − 1 6 w − 1 ; 1 − 1 6 w − 1 ; − t w − ( 1 − Ω M ) Ω M ) Ω M t 6 , (19)

where F 2 1 ( a , b ; c ; z ) is the regularized hypergeometric function, see [

I z ( z ; Ω M , w ) = − 2 F 2 1 ( 1 2 , − 1 6 w − 1 ; 1 − 1 6 w − 1 ; − ( − z 3 + 3 z 2 + 3 z + 1 ) w ( 1 − Ω M ) − Ω M ) Ω M z 3 + 3 z 2 + 3 z + 1 6 . (20)

We denote by F ( z ; Ω M , w ) the definite integral,

F ( z ; Ω M , w ) = I z ( z = z ; Ω M , w ) − I z ( z = 0 ; Ω M , w ) . (21)

The luminosity distance, D L , for wCDM cosmology in the case of the analytical solution is

D L ( z ; c , H 0 , Ω M , w ) = c H 0 ( 1 + z ) F ( z ; Ω M , w ) , (22)

where F ( z ; Ω M , w ) is given by Equation (21) and the distance modulus is

( m − M ) = 25 + 5 log 10 ( D L ( z ; c , H 0 , Ω M , w ) ) . (23)

More details can be found in [

In flat Cardassian cosmology [

D H ( z ; Ω M , w , n ) = 1 ( 1 + z ) 3 Ω M + ( 1 − Ω M ) ( 1 + z ) 3 n , (24)

where n is a variable parameter, and n = 0 means the ΛCDM cosmology, see Equation (17) in [

I z ( z ; Ω M , n ) = ∫ D H ( z ; Ω M , n ) d z . (25)

In order to obtain the indefinite integral we perform a change of variable 1 + z = t 1 / 3 ,

I z ( t ; Ω M , n ) = 1 3 ∫ 1 − t n Ω M + Ω M t + t n t 2 / 3 d t . (26)

The indefinite integral is

I z ( t ; Ω M , n ) = − 2 F 2 1 ( 1 / 2 , − ( 6 n − 6 ) − 1 ; 6 n − 7 6 n − 6 ; t n − 1 ( Ω M − 1 ) Ω M ) Ω M t 6 , (27)

where F 2 1 ( a , b ; c ; z ) is the regularized hypergeometric function. We now return to the original variable z and the indefinite integral is

I z ( z ; Ω M , n ) = − 2 F 2 1 ( 1 / 2 , − ( 6 n − 6 ) − 1 ; 6 n − 7 6 n − 6 ; ( ( 1 + z ) 3 ) n − 1 ( Ω M − 1 ) Ω M ) Ω M ( 1 + z ) 3 6 . (28)

We denote by F c ( z ; Ω M , n ) the definite integral,

F c ( z ; Ω M , n ) = I z ( z = z ; Ω M , n ) − I z ( z = 0 ; Ω M , n ) . (29)

In the case of the Cardassian cosmology, the luminosity distance is

D L ( z ; c , H 0 , Ω M , n ) = c H 0 ( 1 + z ) F c ( z ; Ω M , n ) , (30)

where F c ( z ; Ω M , n ) is given by Equation (29) and the distance modulus is

( m − M ) = 25 + 5 log 10 ( D L ( z ; c , H 0 , Ω M , n ) ) . (31)

In the flat Cardassian cosmology, there are three parameters: H 0 , Ω M and n. More details can be found in [

The starting point is Equation (1) for the luminosity distance in [

D L ( z ; c , H 0 , Ω M ) = c ( 1 + z ) H 0 ∫ 0 z 1 Ω M ( 1 + t ) 3 + 1 − Ω M d t , (32)

where the variable of integration, t, denotes the redshift.

A first change in the parameter Ω M introduces

s = 1 − Ω M Ω M 3 (33)

and the luminosity distance becomes

D L ( z ; c , H 0 , s ) = 1 H 0 c ( 1 + z ) ∫ 0 z 1 ( 1 + t ) 3 s 3 + 1 + 1 − ( s 3 + 1 ) − 1 d t . (34)

The following change of variable, t = s − u u , is performed for the luminosity distance, which becomes

D L ( z ; c , H 0 , s ) = − c H 0 s 2 ( 1 + z ) ( s 3 + 1 ) ∫ s s 1 + z u u 3 + 1 s 3 ( u 3 + 1 ) u 3 ( s 3 + 1 ) d u . (35)

The integral for the luminosity distance is

D L ( z ; c , H 0 , s ) = − 1 / 3 c ( 1 + z ) 3 3 / 4 s 3 + 1 s H 0 × ( F ( 2 s ( s + 1 + z ) 3 4 s 3 + s + z + 1 , 1 / 4 3 + 1 / 4 2 ) − F ( 2 3 4 s ( s + 1 ) s + 1 + s 3 , 1 / 4 2 3 + 1 / 4 2 ) ) , (36)

where s is given by Equation (33) and F ( ϕ , k ) is Legendre’s incomplete elliptic integral of the first kind,

F ( ϕ , k ) = ∫ 0 sin ϕ d t 1 − t 2 1 − k 2 t 2 , (37)

see [

( m − M ) = 25 + 5 log 10 ( D L ( z ; c , H 0 , s ) ) , (38)

and therefore,

( m − M ) = 25 + 5 1 ln ( 10 ) ln ( − 1 3 c ( 1 + z ) 3 3 / 4 ( F 1 − F 2 ) s 3 + 1 s H 0 ) , (39)

where,

F 1 = F ( 2 s ( s + 1 + z ) 3 4 s 3 + s + z + 1 , 1 / 4 2 3 + 1 / 4 2 ) (40)

and

F 2 = F ( 2 3 4 s ( s + 1 ) s + 1 + s 3 , 1 / 4 2 3 + 1 / 4 2 ) , (41)

with s as defined by Equation (33). More details can be found in [

The inflationary universe has been introduced by [

E ( z ; Ω M 0 , Ω f 0 α β ) = ( 1 + z ) 3 Ω M 0 + Ω f 0 ( 1 + z ) α e β z , (42)

where Ω M 0 = ρ m 0 3 H 0 2 is the adimensional present density of matter, Ω f 0 = ρ ϕ 0 3 H 0 2

is the present adimensional density of the scalar field, H 0 is the present value of the Hubble constant, ρ m 0 is the present density of matter, ρ ϕ 0 is the present density of the scalar field, α and β are two parameters which allow to match theory and observations. In absence of curvature we have

Ω M 0 + Ω f 0 = 1, (43)

and therefore,

E ( z ; Ω M 0 , α , β ) = ( 1 + z ) 3 Ω M 0 + ( 1 − Ω M 0 ) ( 1 + z ) α e β z . (44)

The luminosity distance is

D L ( z ; c , H 0 , Ω M 0 , α , β ) = c ( 1 + z ) H 0 ∫ 0 z 1 E ( t ; Ω M 0 , α , β ) d t , (45)

where the variable of integration, t, denotes the redshift. At the moment of writing there is not an analytical solution for the above integral and therefore we implement a numerical solution, D L,num ( z ; c , H 0 , Ω M 0 , α , β ) . The distance modulus is

( m − M ) = 25 + 5 log 10 ( D L,num ( z ; c , H 0 , Ω M 0 , α , β ) ) . (46)

An approximate value of the above integral (45) is obtained with a Taylor expansion of the integrand about z = 1 of order seven denoted by D L,7 ( z ; c , H 0 , Ω M 0 , α , β ) . We report the numerical expression with cosmological parameters as in

D L,7 ( z ) = 4282.7 ( 1 + z ) ( 0.91287 z − 0.16562 z 2 + 0.039001 ( z − 1 ) 3 − 0.003084 ( z − 1 ) 4 − 0.0036858 ( z − 1 ) 5 + 0.0028217 ( z − 1 ) 6 − 0.00115816 ( z − 1 ) 7 + 0.03442 ) . (47)

The approximate distance modulus is

( m − M ) 7 = 25 + 5 log 10 ( D L,7 ( z ; c , H 0 , Ω M 0 , α , β ) ) , (48)

which for the Union 2.1 compilation has the following numerical expression,

( m − M ) 7 = 25 + 5 ln ( 10 ) ( ln ( 4282.7 ( 1 + z ) ( 0.91287 z − 0.16562 z 2 + 0.039001 ( z − 1 ) 3 − 0.0030847 ( z − 1 ) 4 − 0.0036858 ( z − 1 ) 5 + 0.0028217 ( z − 1 ) 6 − 0.0011581 ( z − 1 ) 7 + 0.03442 ) ) ) . (49)

In the Einstein—De Sitter model the luminosity distance, D L , after [

D L = 2 c ( 1 + z − z + 1 ) H 0 , (50)

and the distance modulus for the Einstein—De Sitter model is:

m − M = 25 + 5 1 ln ( 10 ) ln ( 2 c ( 1 + z − z + 1 ) H 0 ) . (51)

There is one free parameter in the Einstein—De Sitter model: H_{0}. The Einstein—De Sitter model has been recently improved by [

m − M = 5 ln ( 5 / 3 R 0 ( 1 + z ) I G ( z ) ) ln ( 10 ) + 25, (52)

where,

R 0 = c H 0 , (53)

and

I G ( z ) = ∫ 0 z 1 1 + 2 3 ( 1 + x ) 3 2 d x . (54)

Evaluating the integral yields:

I G ( z ) = − 3 6 ( arctan ( 2 3 6 2 3 3 − 3 3 ) − arctan ( 2 3 6 2 3 3 ( 1 + z ) 3 2 3 − 3 3 ) ) 2 3 − 12 2 3 12 ( ln ( − 2 3 3 3 + 2 2 3 + 3 2 3 ) − 2 ln ( 2 3 + 3 3 ) + 2 ln ( 2 3 3 2 / 3 ( 1 + z ) 3 / 2 3 + 3 ) − ln ( 2 2 3 3 3 ( ( 1 + z ) 3 2 ) 2 3 − 2 3 3 2 3 ( 1 + z ) 3 2 3 + 3 ) − ln ( 3 ) ) . (55)

The integrand of (54) can be approximated with a Padé approximant with p = 2 , q = 2 ,

I G 22 ( z ) = ∫ 0 z − 3 x 2 + 36 x + 144 67 x 2 + 204 x + 240 d x , (56)

and therefore we have the approximate integral,

I G 22 ( z ) = − 3 z 67 + 1512 ln ( 67 z 2 + 204 z + 240 ) 4489 + 64368 1419 2123297 arctan ( ( 134 z + 204 ) 1419 5676 ) − 1512 ln ( 240 ) 4489 − 64368 1419 arctan ( 17 1419 473 ) 2123297 , (57)

which generates the following approximate distance modulus,

( m − M ) 22 = 5 ln ( 5 / 3 R 0 ( 1 + z ) I G 22 ( z ) ) ln ( 10 ) + 25. (58)

The percent error between the approximate distance modulus as given by Equation (58) and the exact distance modulus as given by Equation (52) is ≈ 0.03 % when z = 4 and H 0 = 69.1 .

In the framework of GR, the received flux, f, is

f = L 4 π D L 2 , (59)

where D L is the luminosity distance, which depends on the cosmological model adopted, see Equation (7.21) in [

The distance modulus in the simple GR cosmology is

m − M = 43.17 − 1 ln ( 10 ) ln ( H 0 70 ) + 5 ln ( z ) ln ( 10 ) + 1.086 ( 1 − q 0 ) z , (60)

see Equation (7.52) in [_{0} and q_{0}.

This model is based on the standard definition of luminosity in the flat expanding universe. The luminosity distance, r ′ L , is

r ′ L = c H 0 z , (61)

and the distance modulus is

m − M = − 5 log 10 + 5 log 10 r ′ L + 2.5 log ( 1 + z ) , (62)

see formulae (13) and (14) in [_{0}.

In the Milne model, which is developed in the framework of SR, the luminosity distance, after [

D L = c ( z + 1 2 z 2 ) H 0 , (63)

and the distance modulus for the Milne model is

m − M = 25 + 5 1 ln ( 10 ) ln ( c ( z + 1 2 z 2 ) H 0 ) . (64)

There is one free parameter in the Milne model: H_{0}.

In a Euclidean static framework from among many possible absorption mechanisms, we have selected a plasma effect which produces the following relation for the distance d,

d = c H 0 ln ( 1 + z ) , (65)

where the distance expressed in lower case underline the difference with the relativistic case, see Equation (50) in [

In the presence of plasma absorption, the observed flux is

f = L ⋅ exp ( − b d − H 0 d − 2 H 0 d ) 4 π d 2 , (66)

where the factor exp ( − b d ) is due to galactic and host galactic extinctions, − H 0 d is the reduction due to the plasma in the IGM and − 2 H 0 d is the reduction due to the Compton scattering, see the formula before Equation (51) in [

m − M = 5 ln ( ln ( z + 1 ) ) ln ( 10 ) + 15 2 ln ( z + 1 ) ln ( 10 ) + 5 1 ln ( 10 ) ln ( c H 0 ) + 25 + 1.086 b , (67)

see Equation (7) in [_{0} when b = 0 . A detailed analysis of this and other physical mechanisms which produce the observed redshift can be found in [

In a Euclidean static universe, the concept of modified tired light (MTL) was introduced in Section 2.2 of [

d = c H 0 ln ( 1 + z ) , (68)

where the distance expressed in lower case underline the difference with the relativistic case. The distance modulus in MTL is

m − M = 5 2 β ln ( z + 1 ) ln ( 10 ) + 5 1 ln ( 10 ) ln ( ln ( z + 1 ) c H 0 ) + 25, (69)

where β is a parameter lying between 1 and 3 which allows matching theory with observations. There are two free parameters in MTL: H_{0} and β .

We first review the statistics involved and then we process the 12 × 4 cosmological cases.

In the case of the distance modulus, the merit function χ 2 is

χ 2 = ∑ i = 1 N [ ( m − M ) i − ( m − M ) ( z i ) t h σ i ] 2 , (70)

where N is the number of SNs, ( m − M ) i is the observed distance modulus evaluated at a redshift of z i , σ i is the error in the observed distance modulus evaluated at z i , and ( m − M ) ( z i ) t h is the theoretical distance modulus evaluated at z i , see formula (15.5.5) in [

χ r e d 2 = χ 2 / N F , (71)

where N F = N − k is the number of degrees of freedom, N is the number of SNs, and k is the number of free parameters. Another useful statistical parameter is the associated Q-value, which has to be understood as the maximum probability of obtaining a better fitting, see formula (15.2.12) in [

Q = 1 − GAMMQ ( N − k 2 , χ 2 2 ) , (72)

where GAMMQ is a subroutine for the incomplete gamma function. The Akaike information criterion (AIC), see [

AIC = 2 k − 2 ln ( L ) , (73)

where L is the likelihood function. We assume a Gaussian distribution for the errors; then the likelihood function can be derived from the χ 2 statistic L ∝ exp ( − χ 2 2 ) where χ 2 has been computed by Equation (70), see [

AIC = 2 k + χ 2 . (74)

The goodness of the approximation in evaluating a physical variable p is evaluated by the percentage error δ ,

δ = | p − p a p p r o x | p × 100, (75)

where p a p p r o x is an approximation of p.

The parameters of the twelve cosmologies here analyzed are found minimizing the χ 2 as given by Equation (70). We now report the adopted numerical techniques:

1) In absence of an analytical solution for the distance modulus we do k (the number of free parameters) nested numerical loops for the evaluation of the χ 2 . The parameters which minimize the χ 2 are selected. This method allows to find, as an example, the parameters of the ΛCDM and ϕCDM cosmologies.

2) In presence of an analytical solution, an approximate Taylor series and a Padé approximant for the distance modulus we derive the parameters through the Levenberg—Marquardt method (subroutine MRQMIN in [

The above techniques allow to derive the cosmological parameters with unprecedented accuracy, as an example, an error of 0.1 km·s^{−1}·Mpc^{−1} can be associated with the Hubble constant. The advantage to have approximate results, i.e. the Padé approximant for the distance modulus ( m − M ) 2,2 as given by Equation (11), is that we can evaluate in an analytical way the first derivative required by the Levenberg-Marquardt method and the numerical integration is not necessary.

In order to avoid the degeneracy in the Hubble constant-absolute magnitude plane we deal only with already calibrated distance modulus. The first astronomical test we perform is on the 580 SNs of the Union 2.1 compilation, see [

The second test we perform is on the joint light-curve analysis (JLA), which contains 740 SNs [

m − M = − C β + X 1 α − M b + m B ⋆ . (76)

cosmology | Equation | k | parameters | χ 2 | χ r e d 2 | Q | AIC |
---|---|---|---|---|---|---|---|

ΛCDM | (11) | 3 | H 0 = 69.56 ± 0.1 ; Ω M = 0.238 ± 0.01 ; Ω Λ = 0.661 ± 0.01 | 562.59 | 0.975 | 0.658 | 569.39 |

wCDM | (23) | 3 | H 0 = 70.02 ± 0.35 ; Ω M = 0.277 ± 0.025 ; w = − 1.003 ± 0.05 | 562.21 | 0.974 | 0.662 | 568.21 |

Cardassian | (31) | 3 | H 0 = 70.15 ± 0.38 ; Ω M = 0.305 ± 0.019 ; n = − 0.081 ± 0.01 | 562.35 | 0.974 | 0.661 | 568.35 |

flat | (39) | 2 | H 0 = 69.77 ± 0.33 ; Ω M = 0.295 ± 0.008 | 562.55 | 0.9732 | 0.66 | 566.55 |

ϕCDM | (46) | 4 | H 0 = 70 ± 0.1 ; Ω M 0 = 0.28 ± 0.02 ; α = − 0.08 ± 0.2 ; β = 0.05 ± 0.02 | 562.23 | 0.976 | 0.65 | 570.23 |

Einstein--De Sitter | (51) | 1 | H 0 = 63.17 ± 0.2 | 1171.39 | 2.02 | 2 × 10^{−42} | 1173.39 |

EdesNa | (52) | 1 | H 0 = 69.04 ± 0.22 | 569.46 | 0.98 | 0.603 | 571.46 |

simple GR | (60) | 2 | H 0 = 73.79 ± 0.024 , q 0 = − 0.1 | 689.34 | 1.194 | 9.5 × 10^{−4} | 693.34 |

flat expanding model | (62) | 1 | H 0 = 66.84 ± 0.22 | 653 | 1.12 | 0.017 | 655 |

Milne | (64) | 1 | H 0 = 67.53 ± 0.22 | 603.37 | 1.04 | 0.23 | 605.37 |

plasma | (67) | 1 | H 0 = 74.2 ± 0.24 | 895.53 | 1.546 | 5.2 × 10^{−16} | 897.5 |

MTL | (69) | 2 | β = 2.37 , H 0 = 69.32 ± 0.34 | 567.96 | 0.982 | 0.609 | 571.9 |

The adopted parameters are α = 0.141 , β = 3.101 and

M b = ( − 19.05 if M s t e l l a r < 10 10 M ⊙ − 19.12 if M s t e l l a r ≥ 10 10 M ⊙ , (77)

where M ⊙ is the mass of the sun, see line 1 in

σ m − M = α 2 σ X 1 2 + β 2 σ C 2 + σ m B ⋆ 2 . (78)

The cosmological parameters with the JLA compilation are reported in see

The third test is performed on the Union 2.1 compilation (580 SNs) + the distance modulus for 59 calibrated high-redshift GRBs, the so called “Hymnium” sample of GRBs, which allows to calibrate the distance modulus in the high redshift up to z ≈ 8 [

cosmology | Equation | k | parameters | χ 2 | χ r e d 2 | Q | AIC |
---|---|---|---|---|---|---|---|

ΛCDM | (11) | 3 | H 0 = 70.71 ± 0.1 ; Ω M = 0.238 ± 0.01 ; Ω Λ = 0.621 ± 0.01 | 626.53 | 0.85 | 0.998 | 632.53 |

wCDM | (23) | 3 | H 0 = 69.38 ± 0.31 ; Ω M = 0.2 ± 0.016 ; w = − 0.8 ± 0.031 | 626.01 | 0.849 | 0.998 | 632.01 |

Cardassian | (31) | 3 | H 0 = 70.03 ± 0.44 ; Ω M = 0.3 ± 0.019 ; n = − 0.055 ± 0.004 | 628.73 | 0.853 | 0.998 | 634.73 |

flat | (39) | 2 | H 0 = 69.65 ± 0.23 ; Ω M = 0.3 ± 0.003 | 627.91 | 0.85 | 0.998 | 631.91 |

ϕCDM | (46) | 4 | H 0 = 69.6 ± 0.1 ; Ω M 0 = 0.24 ± 0.02 ; α = 0.31 ± 0.2 ; β = 0.03 ± 0.02 | 626.52 | 0.851 | 0.998 | 634.52 |

Einstein--De Sitter | (51) | 1 | H 0 = 62.57 ± 0.17 | 1307.75 | 1.76 | 3.27 × 10^{−34} | 1309.75 |

EdesNa | (52) | 1 | H 0 = 68.91 ± 0.19 | 630.46 | 0.853 | 0.998 | 632.46 |

simple GR | (60) | 2 | H 0 = 73.79 ± 0.023 , q 0 = − 0.14 | 749.14 | 1.016 | 0.369 | 755.14 |

flat expanding model | (62) | 1 | H 0 = 66.49 ± 0.18 | 717.3 | 0.97 | 0.709 | 719.3 |

Milne | (64) | 1 | H 0 = 67.19 ± 0.18 | 656.11 | 0.887 | 0.986 | 658.11 |

plasma | (67) | 1 | H 0 = 74.45 ± 0.2 | 1017.79 | 1.377 | 3.59 × 10^{−11} | 1019.79 |

MTL | (69) | 2 | β = 2.36 , H 0 = 69.096 ± 0.32 | 626.27 | 0.848 | 0.998 | 630.27 |

cosmology | Equation | k | parameters | χ 2 | χ r e d 2 | Q | AIC |
---|---|---|---|---|---|---|---|

ΛCDM | (11) | 3 | H 0 = 67.8 ± 0.2 ; Ω M = 0.259 ± 0.02 ; Ω Λ = 0.691 ± 0.02 | 586.04 | 0.921 | 0.922 | 592.04 |

wCDM | (23) | 3 | H 0 = 69.34 ± 0.32 ; Ω M = 0.2 ± 0.016 ; w = − 0.626 ± 0.015 | 592.1 | 0.93 | 0.892 | 598.1 |

Cardassian | (31) | 3 | H 0 = 70.1 ± 0.42 ; Ω M = 0.299 ± 0.019 ; n = − 0.063 ± 0.009 | 585.43 | 0.92 | 0.924 | 591.43 |

flat | (39) | 2 | H 0 = 69.82 ± 0.24 ; Ω M = 0.295 ± 0.003 | 585.74 | 0.919 | 0.927 | 589.74 |

ϕCDM | (46) | 4 | H 0 = 70 ± 0.1 ; Ω M 0 = 0.28 ± 0.02 ; α = − 0.07 ± 0.2 ; β = 0.05 ± 0.02 | 585.41 | 0.922 | 0.92 | 593.41 |

Einstein--De Sitter | (51) | 1 | H 0 = 63.14 ± 0.2 | 1205.2 | 1.88 | 3.58 × 10^{−37} | 1205.21 |

EdesNa | (52) | 1 | H 0 = 69.05 ± 0.22 | 592.79 | 0.929 | 0.899 | 594.79 |

simple GR | (60) | 2 | H 0 = 73.79 ± 0.023 , q 0 = − 0.01 | 809.5 | 1.27 | 3.85 × 10^{−6} | 813.5 |

flat expanding model | (62) | 1 | H 0 = 66.851 ± 0.22 | 676.36 | 1.06 | 0.141 | 678.36 |

Milne | (64) | 1 | H 0 = 67.55 ± 0.22 | 634.27 | 0.994 | 0.534 | 636.27 |

plasma | (67) | 1 | H 0 = 74.25 ± 0.24 | 951.16 | 1.49 | 9.39 × 10^{−14} | 953.16 |

MTL | (69) | 2 | β = 2.35 , H 0 = 69.23 ± 0.34 | 594.69 | 0.933 | 0.883 | 598.69 |

The fourth test is performed on the Pantheon sample of 1048 SN Ia [

In order to see how χ 2 varies around the minimum for the Pantheon sample in the case of the ΛCDM cosmology, _{0} and Ω M are allowed to vary around the numerical values which fix the minimum.

_{0} is fixed and Ω M and w are allowed to vary.

cosmology | Equation | k | parameters | χ 2 | χ r e d 2 | Q | AIC |
---|---|---|---|---|---|---|---|

ΛCDM | (11) | 3 | H 0 = 68.209 ± 0.2 ; Ω M = 0.278 ± 0.02 ; Ω Λ = 0.651 ± 0.02 | 1054.71 | 1.01 | 0.41 | 1060.71 |

wCDM | (23) | 3 | H 0 = 69.8 ± 0.27 ; Ω M = 0.3 ± 0.016 ; w = − 0.989 ± 0.03 | 1053.67 | 1 | 0.419 | 1059.67 |

Cardassian | (31) | 3 | H 0 = 70.01 ± 0.31 ; Ω M = 0.329 ± 0.014 ; n = − 0.091 ± 0.005 | 1054.49 | 1 | 0.412 | 1060.49 |

flat | (39) | 2 | H 0 = 69.94 ± 0.171 ; Ω M = 0.296 ± 0.002 | 1053.53 | 1 | 0.429 | 1057.53 |

ϕCDM | (46) | 4 | H 0 = 69.7 ± 0.1 ; Ω M 0 = 0.28 ± 0.02 ; α = 0.12 ± 0.2 ; β = 0.05 ± 0.02 | 1053.84 | 1 | 0.4 | 1061.84 |

Einstein--De Sitter | (51) | 1 | H 0 = 62.71 ± 0.2 | 2387.62 | 2.28 | 0 | 2389.62 |

EdesNa | (52) | 1 | H 0 = 69.1 ± 0.13 | 1059.84 | 1.01 | 0.384 | 1061.8 |

simple GR | (60) | 2 | H 0 = 73.79 ± 0.015 , q 0 = − 0.063 | 1476.59 | 1.411 | 2.67 × 10^{−17} | 1480.59 |

flat expanding model | (62) | 1 | H 0 = 66.67 ± 0.12 | 1219 | 1.16 | 1.6 × 10^{−4} | 1221 |

Milne | (64) | 1 | H 0 = 67.37 ± 0.12 | 1132.6 | 1.08 | 0.033 | 1134.6 |

plasma | (67) | 1 | H 0 = 74.7 ± 0.14 | 2017.3 | 1.92 | 0 | 2019.3 |

MTL | (69) | 2 | β = 2.31 , H 0 = 68.95 ± 0.222 | 1069.7 | 1.022 | 0.298 | 1073.7 |

In the relativistic models the angular diameter distance, D A [

D A = D L ( 1 + z ) 2 . (79)

We now introduce the minimax approximation. Let f ( x ) be a real function defined in the interval [ a , b ] . The best rational approximation of degree ( k , l ) evaluates the coefficients of the ratio of two polynomials of degree k and l, respectively, which minimizes the maximum difference of:

max | f ( x ) − p 0 + p 1 x + ⋯ + p k x k q 0 + q 1 x + ⋯ + q l x l | , (80)

on the interval [ a , b ] . The quality of the fit is given by the maximum error over the considered range. The coefficients are evaluated through the Remez algorithm, see [

D A ,2,2 = − 0.08126207 + ( 296.9974312 + 2.715947207 z ) z 0.0672056121 + ( 0.0810298760 + 0.02498056665 z ) z Mpc (81)

maximumerror = 0.6911273 Mpc ,

for wCDM cosmology when k = 3 and p = 2 is:

D A ,3,2 = 0.034977336 + ( 287.18685 + ( 1.1871126 + 0.0002567152 z ) z ) z 0.0665238 + ( 0.09134443 + 0.023282807 z ) z Mpc (82)

maximumerror = 0.07 Mpc ,

for Cardassian cosmology when k = 2 and p = 2 is:

D A ,2,2 = − 0.11928613 + ( 273.3160492 + 2.420885784 z ) z 0.0638700712 + ( 0.0750594027 + 0.02611741351 z ) z Mpc (83)

maximumerror = 0.8346776 Mpc ,

for flat cosmology when k = 2 and p = 2 is:

D A ,2,2 = − 0.03653022 + ( 274.6370918 + 2.192330157 z ) z 0.0641307653 + ( 0.0767316787 + 0.02582682170 z ) z Mpc (84)

maximumerror = 0.629004 Mpc ,

and for ϕCDM cosmology when k = 2 and p = 2 is:

D A ,2,2 = − 0.01852238 + ( 278.5646306 + 2.230340777 z ) z 0.0652823706 + ( 0.0768568011 + 0.02575830541 z ) z (85)

maximumerror = 0.6261293 Mpc .

In MTL there is no difference between the distance d, see Equation (68), and the angular distance. We report the numerical value of d in the interval 0 < z < 8 with data as in

d = 4330.383620 ln ( z + 1 ) Mpc . (86)

A promising field of investigation in applied cosmology is the maximum of the angular distance as function of the redshift [

The numerical value of z max is reported in

Another example is given by the ring associated with the galaxy SDP.81, see [

cosmology | z max | radius (kpc) |
---|---|---|

ΛCDM | 1.691 | 13.333 |

wCDM | 1.716 | 11.797 |

Cardassian | 1.607 | 11.938 |

flat | 1.615 | 11.907 |

ϕCDM | 1.632 | 12.05 |

MTL | ∞ | 45.15 |

Compilation | first model | second model | third model | fourth model |
---|---|---|---|---|

Union 2.1 | wCDM Hypergeometric | Cardassian | ϕCDM | flat |

JLA | wCDM Hypergeometric | MTL | ϕCDM | ΛCDM |

Union 2.1 + GRBs | ΛCDM | ϕCDM | Cardassian | flat |

Pantheon | wCDM Hypergeometric | Cardassian | flat | ϕCDM |

Cosmological models: We list according to increasing order of the values of the merit function, χ 2 , the first, second, third, and fourth cosmological models, see

The Einstein—De Sitter, simple GR, and plasma models produce the highest values in the χ 2 and are here considered only for historical reasons.

Physics versus Astronomy: The value of the Newtonian gravitational constant, denoted by G, is derived applying the weighted mean, but the uncertainties were multiplied by a factor of 14, of 11 values available in TableXXIV in [

By analogy, we average the values of H_{0} for the Pantheon sample and we report as error for H_{0} the standard deviation,

H 0 ¯ = ( 69.29 ± 3.18 ) km ⋅ s − 1 ⋅ Mpc − 1 Pantheon sample , (87)

see

The author is grateful to David Jones for information useful for downloading the data of the Pantheon sample.

The author declares no conflicts of interest regarding the publication of this paper.

Zaninetti, L. (2021) Sparse Formulae for the Distance Modulus in Cosmology. Journal of High Energy Physics, Gravitation and Cosmology, 7, 965-992. https://doi.org/10.4236/jhepgc.2021.73057

Given a function f ( z ) , the Padé approximant, after [

f ( z ) = a 0 + a 1 z + ⋯ + a p z p b 0 + b 1 z + ⋯ + b q z q , (1)

where the notation is the same as in [

The coefficients a i and b i are found through Wynn’s cross rule, see [

1 E ( z ) = 1 Ω M ( 1 + z ) 3 + Ω K ( 1 + z ) 2 + Ω Λ , (2)

and the Padé approximant is:

1 E ( z ) = a 0 + a 1 z + a 2 z 2 b 0 + b 1 z + b 2 z 2 , (3)

where,

a 0 = 16 ( 32 Ω K 3 Ω Λ + 16 Ω K 2 Ω Λ 2 + 160 Ω K 2 Ω Λ Ω M + 24 Ω K 2 Ω M 2 + 64 Ω K Ω Λ 2 Ω M + 320 Ω K Ω Λ Ω M 2 + 40 Ω K Ω M 3 + 96 Ω Λ 2 Ω M 2 + 192 Ω Λ Ω M 3 + 15 Ω M 4 ) ( Ω M + Ω K + Ω Λ ) 4 (4)

a 1 = 4 ( 128 Ω K 4 Ω Λ + 32 Ω K 3 Ω Λ 2 + 704 Ω K 3 Ω Λ Ω M − 16 Ω K 2 Ω Λ 2 Ω M + 1456 Ω K 2 Ω Λ Ω M 2 + 32 Ω K 2 Ω M 3 − 64 Ω K Ω Λ 3 Ω M − 384 Ω K Ω Λ 2 Ω M 2 + 1512 Ω K Ω Λ Ω M 3 + 50 Ω K Ω M 4 − 192 Ω Λ 3 Ω M 2 − 288 Ω Λ 2 Ω M 3 + 648 Ω Λ Ω M 4 + 15 Ω M 5 ) ( Ω M + Ω K + Ω Λ ) 3 (5)

a 2 = − ( 256 Ω K 4 Ω Λ Ω M − 64 Ω K 3 Ω Λ 3 + 320 Ω K 3 Ω Λ 2 Ω M + 960 Ω K 3 Ω Λ Ω M 2 − 320 Ω K 2 Ω Λ 3 Ω M + 240 Ω K 2 Ω Λ 2 Ω M 2 + 1440 Ω K 2 Ω Λ Ω M 3 + 16 Ω K 2 Ω M 4 − 1600 Ω K Ω Λ 3 Ω M 2 − 480 Ω K Ω Λ 2 Ω M 3 + 1140 Ω K Ω Λ Ω M 4 + 20 Ω K Ω M 5 − 256 Ω Λ 4 Ω M 2 − 1600 Ω Λ 3 Ω M 3 − 240 Ω Λ 2 Ω M 4 + 380 Ω Λ Ω M 5 + 5 Ω M 6 ) ( Ω M + Ω K + Ω Λ ) 2 (6)

b 0 = 16 ( Ω M + Ω K + Ω Λ ) 9 / 2 ( 32 Ω K 3 Ω Λ + 16 Ω K 2 Ω Λ 2 + 160 Ω K 2 Ω Λ Ω M + 24 Ω K 2 Ω M 2 + 64 Ω K Ω Λ 2 Ω M + 320 Ω K Ω Λ Ω M 2 + 40 Ω K Ω M 3 + 96 Ω Λ 2 Ω M 2 + 192 Ω Λ Ω M 3 + 15 Ω M 4 ) (7)

b 1 = 4 ( Ω M + Ω K + Ω Λ ) 7 / 2 ( 256 Ω K 4 Ω Λ + 96 Ω K 3 Ω Λ 2 + 1536 Ω K 3 Ω Λ Ω M + 96 Ω K 3 Ω M 2 + 336 Ω K 2 Ω Λ 2 Ω M + 3696 Ω K 2 Ω Λ Ω M 2 + 336 Ω K 2 Ω M 3 − 64 Ω K Ω Λ 3 Ω M + 384 Ω K Ω Λ 2 Ω M 2 + 4200 Ω K Ω Λ Ω M 3 + 350 Ω K Ω M 4 − 192 Ω Λ 3 Ω M 2 + 288 Ω Λ 2 Ω M 3 + 1800 Ω Λ Ω M 4 + 105 Ω M 5 ) (8)

b 2 = ( Ω M + Ω K + Ω Λ ) 5 / 2 ( 512 Ω K 5 Ω Λ + 384 Ω K 4 Ω Λ 2 + 3584 Ω K 4 Ω Λ Ω M + 192 Ω K 3 Ω Λ 3 + 1984 Ω K 3 Ω Λ 2 Ω M + 10752 Ω K 3 Ω Λ Ω M 2 + 320 Ω K 3 Ω M 3 + 960 Ω K 2 Ω Λ 3 Ω M + 5136 Ω K 2 Ω Λ 2 Ω M 2 + 17760 Ω K 2 Ω Λ Ω M 3 + 840 Ω K 2 Ω M 4 + 2752 Ω K Ω Λ 3 Ω M 2 + 7392 Ω K Ω Λ 2 Ω M 3 + 15060 Ω K Ω Λ Ω M 4 + 700 Ω K Ω M 5 + 256 Ω Λ 4 Ω M 2 + 2752 Ω Λ 3 Ω M 3 + 3696 Ω Λ 2 Ω M 4 + 5020 Ω Λ Ω M 5 + 175 Ω M 6 ) (9)

The indefinite integral of (3), F 2,2 , is:

F 2,2 ( z ; a 0 , a 1 , a 2 , b 0 , b 1 , b 2 ) = a 2 z b 2 + 1 2 ln ( z 2 b 2 + z b 1 + b 0 ) a 1 b 2 − 1 2 ln ( z 2 b 2 + z b 1 + b 0 ) a 2 b 1 b 2 2 + 2 a 0 4 b 0 b 2 − b 1 2 arctan ( 2 z b 2 + b 1 4 b 0 b 2 − b 1 2 ) − 2 a 2 b 0 b 2 4 b 0 b 2 − b 1 2 arctan ( 2 z b 2 + b 1 4 b 0 b 2 − b 1 2 ) − b 1 a 1 b 2 4 b 0 b 2 − b 1 2 arctan ( 2 z b 2 + b 1 4 b 0 b 2 − b 1 2 ) + b 1 2 a 2 b 2 2 4 b 0 b 2 − b 1 2 arctan ( 2 z b 2 + b 1 4 b 0 b 2 − b 1 2 ) . (10)