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A new analytical solution for the luminosity distance in flat ΛCDM cosmology is derived in terms of elliptical integrals of first kind with real argument. The consequent derivation of the distance modulus allows evaluating the Hubble constant,
H_{0}=69.77±0.33,
Ω_{M}=0.295±0.008, and the cosmological constant,
.

The release of two catalogs for the distance modulus of Supernova (SN) of type Ia, namely, the Union 2.1 compilation, see [

We now focus our attention on the flat Friedmann-Lemaître-Robertson-Walker (flat-FLRW) cosmology. A first fitting formula has been derived by [

This section reviews the adopted statistical framework, the ΛCDM cosmology, and an existing solution for the luminosity distance in flat-FLRW cosmology.

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 , (1)

where N is the number of SNs, ( m − M ) i is the observed distance modulus evaluated at redshift 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 , (2)

where N F = N − k is the number of degrees of freedom, N is the number of SNs, and k is the number of 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 ) , (3)

where GAMMQ is a subroutine for the incomplete gamma function.

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, (4)

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

We follow [

D H ≡ c H 0 . (5)

The first parameter is Ω M

Ω M = 8 π G ρ 0 3 H 0 2 , (6)

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

Ω Λ ≡ Λ c 2 3 H 0 2 , (7)

where Λ is the cosmological constant, see [

Ω M + Ω Λ + Ω K = 1. (8)

The comoving distance, D C , is

D C = D H ∫ 0 z d z ′ E ( z ′ ) (9)

where E ( z ) is the “Hubble function”

E ( z ) = Ω M ( 1 + z ) 3 + Ω K ( 1 + z ) 2 + Ω Λ . (10)

The above integral does not have an analytical solution but a solution in terms of Padé approximant has been found, see [

The first model starts from Equation (2.1) in [

d L ( z ; c , H 0 , Ω M ) = c H 0 ( 1 + z ) ∫ 1 1 + z 1 d a Ω M a + ( 1 − Ω M ) a 4 , (11)

where H 0 is the Hubble constant expressed in km・s^{−1}・Mpc^{−1}, c is the speed of light expressed in km・s^{−1}, z is the redshift and a is the scale-factor. The indefinite integral, Φ ( a ) , is

Φ ( a , Ω M ) = ∫ d a Ω M a + ( 1 − Ω M ) a 4 . (12)

The solution is in terms of F, the Legendre integral or incomplete elliptic integral of the first kind, and is given in [

The luminosity distance is

d L ( z ; c , H 0 , Ω M ) = ℜ ( c H 0 ( 1 + z ) ( Φ ( 1 ) − Φ ( 1 1 + z ) ) ) , (13)

where ℜ means the real part. The distance modulus is

( m − M ) = 25 + 5 log 10 ( d L ( z ; c , H 0 , Ω M ) ) . (14)

The second model for the flat cosmology starts from 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 . (15)

The above formula can be obtained from formula (9) for the comoving distance inserting Ω K = 0 and the variable of integration, t, denotes the redshift.

A first change in the parameter Ω M introduces

s = 1 − Ω M Ω M 3 (16)

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 . (17)

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 . (18)

Up to now we have followed [

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 2 3 + 1 / 4 2 ) − F ( 2 3 4 s ( s + 1 ) s + 1 + s 3 , 1 / 4 2 3 + 1 / 4 2 ) ) , (19)

where s is given by Equation (16) 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 , (20)

see [

( m − M ) = 25 + 5 log 10 ( d L ( z ; c , H 0 , s ) ) , (21)

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 ) , (22)

where

F 1 = F ( 2 s ( s + 1 + z ) 3 4 s 3 + s + z + 1 , 1 / 4 2 3 + 1 / 4 2 ) (23)

and

F 2 = F ( 2 3 4 s ( s + 1 ) s + 1 + s 3 , 1 / 4 2 3 + 1 / 4 2 ) , (24)

with s as defined by Equation (16).

Data AnalysisIn recent years, the extraction of the cosmological parameters from the distance modulus of SNs has become a common practice, see among others [

∂ ( m − M ) ∂ H 0 , which has a simple expression, and the first derivative ∂ ( m − M ) ∂ Ω M ,

which has a complicated expression. A simplification can be introduced by imposing a fiducial value for the Hubble constant, namely H 0 = 70 km ⋅ s − 1 ⋅ Mpc − 1 , see [

Cosmology | SNs | k | parameters | χ 2 | χ r e d 2 | Q |
---|---|---|---|---|---|---|

flat-FLRW | 580 | 2 | H 0 = 69.77 ± 0.33 ; Ω M = 0.295 ± 0.008 | 562.55 | 0.9732 | 0.66 |

flat-FLRW-1 | 580 | 1 | H 0 = 70 ; Ω M = 0.295 ± 0.008 | 563.52 | 0.9732 | 0.669 |

ΛCDM | 580 | 3 | H 0 = 69.81 ; Ω M = 0.239 ; Ω Λ = 0.651 | 562.61 | 0.975 | 0.658 |

JLA compilation is available at the Strasbourg Astronomical Data Centre (CDS), and consists of 740 type I-a SNs for which we have the heliocentric redshift, z, the

apparent magnitude m B ⋆ in the B band, the error in m B ⋆ , σ m B ⋆ , the parameter X1,

the error in X1, σ X 1 , the parameter C, the error in C, σ C , and log 10 ( M s t e l l a r ) . The observed distance modulus is defined by Equation (4) in [

m − M = − C β + X 1 α − M b + m B ⋆ . (25)

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 ⊙ (26)

see line 1 in

σ m − M = α 2 σ X 1 2 + β 2 σ C 2 + σ m B ⋆ 2 . (27)

The parameters as derived from the JLA compilation are presented in

Cosmology | SNs | k | parameters | χ 2 | χ r e d 2 | Q |
---|---|---|---|---|---|---|

flat-FLRW | 740 | 2 | H 0 = 69.65 ± 0.231 ; Ω M = 0.3 ± 0.003 | 627.91 | 0.85 | 0.998 |

As an example the luminosity distance for the Union 2.1 compilation with data as in the first line of

d L ( z ) = 8147.04 ( 1.0 + z ) × ( − 0.637664 F ( 2.63214 3.1188 + 1.33542 z 4.64846 + z ,0.965925 ) + 1.75322 ) Mpc (28)

when 0 < z < 1.5

and the distance modulus is

m − M = 25.0 + 2.17147 ln ( 8147.04 ( 1.0 + z ) × ( − 0.63766 F ( 2.6321 3.1188 + 1.3354 z 4.6484 + z ,0.96592 ) + 1.7532 ) ) . (29)

when 0 < z < 1.5

We now derive some approximate results without Legendre integral for the flat-FLRW case and Union 2.1 compilation with data as in

d L ( z ) = 0.000423646 + 4296.57 z + 3344.13 z 2 − 1186.94 z 3 + 979.403 z 4 − 42078.6 z 5 Mpc (30)

when 0 < z < 0.197 .

The upper limit in redshift, 0.197, is the value for which the percentage error, see Equation (4), is δ = 1.16 % . The asymptotic expansion of the luminosity distance with respect to the variable z to order 5 for the flat-FLRW case and Union 2.1 compilation gives

d L ( z ) ∼ 14283.5 z − 15802 1 z − 1 + 14283.5 − 7901.01 z − 1 + 1975.25 ( z − 1 ) 3 / 2 Mpc (31)

when 1.27 < z < 1.5 .

At the lower limit of z = 1.27 the percentage error is δ = 0.54 % . The two above approximations at low and high redshift have a limited range of existence but does not contain the Legendre integral as solutions (28) and (29) which cover the overall range 0 < z < 1.5 .

A Taylor expansion of order 6 of the distance modulus as given by Equation (22) around z = 0.1 for the flat-FLRW case and Union 2.1 compilation gives

( m − M ) = 36.0051 + 23.1777 z − 109.604 ( z − 0.1 ) 2 + 724.464 ( z − 0.1 ) 3 − 5429.06 ( z − 0.1 ) 4 + 43429.8 ( z − 0.1 ) 5 (32)

when 0.1 < z < 0.197 .

The upper limit in redshift, 0.197, is the value at which the percentage error is δ = 0.14 % .

The asymptotic expansion of the distance modulus with respect to the variable z to order 5 for the flat-FLRW case and Union 2.1 compilation gives

( m − M ) ∼ 45.7741 + 2.17147 ln ( z ) − 2.40231 z − 1 + 0.842625 z − 1 + 0.221081 ( z − 1 ) 3 / 2 − 0.570086 z − 2 − 0.150471 ( z − 1 ) 5 / 2 + 0.357849 z − 3 + 0.491842 ( z − 1 ) 7 / 2 + 0.179989 z − 4 (33)

when 1.27 < z < 1.5 .

The lower limit in redshift, 1.27, is the value at which the percentage error is δ = 0.54 % . The ranges of existence in z for the analytical approximations here derived have the percentage error <2%, see Equation (4).

We now introduce the best minimax rational approximation, see [

m 2,1 ( z ) = a + b z + c z 2 d + e z . (34)

In the case in which the distance modulus is represented by Equation (29) and given the interval [ 0.001,1.5 ] , the coefficients of the best minimax rational approximation are presented in

We have presented an analytical approximation for the luminosity distance in terms of elliptical integrals with real argument. The fit of the distance modulus

Name | value |
---|---|

maximum error | 2.28881836 × 10^{−5} |

a | 0.981622279 |

b | 19.6473351 |

c | 1.08210218 |

d | 0.0309164915 |

e | 0.462291896 |

of SNs of type Ia allows finding the parameters H 0 and Ω M for the two compilations in flat-FLRW cosmology

H 0 = ( 69.77 ± 0.33 ) km ⋅ s − 1 ⋅ Mpc − 1 , Ω M = 0.295 ± 0.008 (35)

flat-FLRW-Union 2.1,

H 0 = ( 69.65 ± 0.23 ) km ⋅ s − 1 ⋅ Mpc − 1 , Ω M = 0.3 ± 0.003 (36)

flat-FLRW-JLA,

A first comparison with [

H 0 = ( 67.27 ± 0.60 ) km ⋅ s − 1 ⋅ Mpc − 1 , Ω M = 0.3166 ± 0.0084 (37)

Planck 2018.

In the case of the Union 2.1 compilation, the percentage error p = 3.71 % for the derivation of H 0 and p = 6.82 % for Ω M . A Taylor expansion at low redshift and an asymptotic expansion are presented both for the luminosity distance and the distance modulus. A simple version of the distance modulus is determined through the best minimax rational approximation. Adopting the cosmological parameters found here, the cosmological constant Λ turns out to be, for the Union 2.1 compilation,

Λ = ( 1.19457 ± 0.017 ) × 10 − 52 1 m 2 (38)

flat-FLRW Union 2.1,

or introducing c = 1 and the Planck time, t p ,

Λ = ( 3.12046 ± 0.0462942 ) × 10 − 122 1 t p 2 (39)

flat-FLRW Union 2.1.

The statistical parameters of the fits are given in

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

Zaninetti, L. (2019) A New Analytical Solution for the Distance Modulus in Flat Cosmology. International Journal of Astronomy and Astrophysics, 9, 51-62. https://doi.org/10.4236/ijaa.2019.91005