Pad\'e Approximant and Minimax Rational Approximation in Standard Cosmology

The luminosity distance in the standard cosmology as given by $\Lambda$CDM and consequently the distance modulus for supernovae can be defined by the Pad\'e approximant. A comparison with a known analytical solution shows that the Pad\'e approximant for the luminosity distance has an error of $4\%$ at redshift $= 10$. A similar procedure for the Taylor expansion of the luminosity distance gives an error of $4\%$ at redshift $=0.7 $; this means that for the luminosity distance, the Pad\'e approximation is superior to the Taylor series. The availability of an analytical expression for the distance modulus allows applying the Levenberg--Marquardt method to derive the fundamental parameters from the available compilations for supernovae. A new luminosity function for galaxies derived from the truncated gamma probability density function models the observed luminosity function for galaxies when the observed range in absolute magnitude is modeled by the Pad\'e approximant. A comparison of $\Lambda$CDM with other cosmologies is done adopting a statistical point of view.


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In order to obtain astronomical observables such as the distance modulus and the absolute magnitude 18 for supernovae (SN) of type Ia in the standard cosmological approach, as given by the ΛCDM model, for the observed distance modulus for SNs of type Ia. We use the same symbols as in [9], where the Hubble distance D H is defined as We then introduce a first parameter Ω M where G is the Newtonian gravitational constant and ρ 0 is the mass density at the present time. A second parameter is Ω Λ where Λ is the cosmological constant, see [10]. The two previous parameters are connected with the curvature Ω K by Ω M + Ω Λ + Ω K = 1 .
The comoving distance, D C , is where E(z) is the 'Hubble function' The above integral does not have an analytical formula, except for the case of Ω Λ = 0, but the Padé approximant, see Appendix B, give an approximate evaluation and the indefinite integral is (B.3) where the coefficients a j and b j can be found in Appendix A. The approximate definite integral for (5) is therefore D C,2,2 = 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 ) .
The transverse comoving distance D M is for Ω Λ = 0.
This expression is useful for calibrating the numerical codes which evaluate D M when Ω Λ = 0.

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The luminosity distance is which in the case of Ω Λ = 0 becomes and the distance modulus when Ω Λ = 0 is (13)   where D L (z) is the exact luminosity distance when Ω Λ = 0, see Eqn. (11) and D L,app (z) is the Taylor or 50 Padé approximate luminosity distance, see also formula (2.12) in [1].  The integrand of (5) contains poles or singularities for a given set of parameters, see Figure 3. 56 The equation which models the poles is The exact solution of the above equation z(Ω Λ ; Ω K = 0.11) is shown in Figure 4 together with the 57 Padé approximated solution z 2,2 (Ω Λ ; Ω K = 0.11). Is therefore possible to conclude that the Padé   Astronomical Data Center (CDS) and consists of SNe (type I-a) for which we have a heliocentric redshift, z, apparent magnitude m B in the B band, error in m B , σ m B , parameter X1, error in X1, σ X1 , parameter C, error in the parameter C, σ C and log 10 (M stellar ). The observed distance modulus is defined by Eq. (4) The adopted parameters are α = 0.141, β = 3.101 and where M is the mass of the sun, see line 1 in The three astronomical parameters in question, H 0 , Ω M and Ω Λ , can be derived trough the Levenberg-Marquardt method (subroutine MRQMIN in [12]) once an analytical expression for the derivatives of the distance modulus with respect to the unknown parameters is provided. As a practical example, the derivative of the distance modulus, (m − M ) 2,2 , with respect to H 0 is This numerical procedure minimizes the merit function χ 2 evaluated as where N = 480, (m−M ) i is the observed distance modulus evaluated at z i , σ i is the error in the observed distance modulus evaluated at z i , and (m − M )(z i ) th is the theoretical distance modulus evaluated at z i , see formula (15.5.5) in [12]. A reduced merit function χ 2 red is evaluated by where N F = n − k is the number of degrees of freedom, n is the number of SNe, 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 [12]: where GAMMQ is a subroutine for the incomplete gamma function. The Akaike information criterion (AIC), see [13], is defined by where L is the likelihood function. We assume a Gaussian distribution for the errors and the likelihood function can be derived from the χ 2 statistic L ∝ exp(− χ 2 2 ) where χ 2 has been computed by Eq. (26), see [14], [15]. Now the AIC becomes  Table 1. Numerical values of χ 2 , χ 2 red , Q, and the AIC of the Hubble diagram for two compilations, k stands for the number of parameters.  The Padé approximant distance modulus has a simple expression when the minimax rational approximation is used, as an example p = 3, q = 2, see Appendix C for the meaning of p and q. In the case of the Union 2.1 compilation, the approximation of formula (17) with the parameters of Table 1 over the range in z ∈ [0, 4] gives the following minimax equation   Table  1.   p and q is shown in Table 2.

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In the case of the JLA compilation, the minimax equation is the maximum error being 0.003.

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The maximum difference between the two minimax formulas which approximate the distance modulus, Eqs. (31) and (32), is at z = 4, and is 0.0584 mag. In the case of the luminosity distance as given by the Padé approximation, see Eq. (14), the minimax approximation gives The Schechter LF, after [16], is the standard LF for galaxies: Here, α sets the shape, L * is the characteristic luminosity, and Φ * is the normalization. The distribution in absolute magnitude is where M * is the characteristic magnitude. The gamma LF is where Ψ * is the total number of galaxies per unit Mpc 3 , is the gamma function, L * > 0 is the scale and c > 0 is the shape, see formula (17.23) in [17]. Its expected value is The change of parameter (c − 1) = α allows obtaining the same scaling as for the Schechter LF (34). 3.3. The truncated gamma luminosity function 78 We assume that the luminosity L takes values in the interval [L l , L u ] where the indices l and u mean lower and upper; the truncated gamma LF is where Ψ * is the total number of galaxies per unit Mpc 3 , and the constant k is where is the upper incomplete gamma function, see [18,19]. Its expected value is   Table 3 and the upper magnitude-z relationship is given in Table 4.
m, the limiting magnitude of the considered catalog. We now outline how to build an observed LF for a 87 galaxy in a consistent way; the selected catalog is zCOSMOS, which is made up of 9697 galaxies up to

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Here we analyse the distance modulus for SNe in other cosmologies in the framework general 103 relativity (GR), expanding flat universe, special relativity (SR) and Euclidean static universe.   In the framework of GR the received flux, f, is where d L is the luminosity distance which depends from the cosmological model adopted, see Eq. (7.21) 106 in [24] or Eq. (5.235) in [25].

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The distance modulus in the simple GR cosmology is

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This model is based on the standard definition of luminosity in the flat expanding universe. The luminosity distance, r L , is and the distance modulus is m − M == −5 log 10 +5 log 10 r L + 2.5 log(1 + z) , see formulae (13) and (14) and the distance modulus for the Einstein-De Sitter model is The number of free parameters in the Einstein-De Sitter model is one: H 0 .

Milne universe in SR
In the Milne model, which is developed in the framework of SR, the luminosity distance, after [29-31], is and the distance modulus for the Milne model is The number of free parameters in the Milne model is one: H 0 . In an Euclidean static framework among many possible absorption mechanisms we selected a photo-absorption process between the photon and the electron in the IGM. This relativistic process produces a nonlinear dependence between redshift and distance where < n e > is expressed in cgs units. A second mechanism is a plasma effect which produces the following relationship see Eq. (50) in [33]. Also this second mechanism produces the same nonlinear d-z dependence as our Eq. (59). In presence of plasma absorption the observed flux is where the factor exp (−bd) is due to Galactic and host galactic extinctions, −H 0 d is reduction to the plasma in the IGM and −2H 0 d is the reduction due to Compton scattering, see formula before Eq. (51) in [33]. The resulting distance modulus in the plasma mechanism is m − M = 5 ln (ln (z + 1)) ln (10) + 15 2 ln (z + 1) ln (10) + 5 1 ln (10) ln see Eq. (7) in [34]. The number of free parameters in the plasma cosmology is one: H 0 when b = 0.

Modified tired light
In an Euclidean static framework the modified tired light (MTL) has been introduced in Section 2.2 in [35]. The distance in MTL is The distance modulus in the modified tired light (MTL) is β ln (z + 1) ln (10) + 5 1 ln (10) ln ln (z + 1) c H 0 + 25 .
Here β is a parameter comprised between 1 and 3 which allows to match theory with observations. The  The statistical parameters for the different cosmologies here analysed can be found in Table 5 in the 122 case of the Union 2.1 compilation and in Table 6 for the JLA compilation.

Conclusions
It is generally thought that in the case of the luminosity distance the Padé approximant is more 126 accurate than the Taylor expansion. As an example, at z = 1.5, which is the maximum value of the 127 redshift here considered, the percentage error of the luminosity distance is δ = 0.036% in the case of the 128 Padé approximation. In the case of of the Taylor expansion, δ = 0.036% for the luminosity distance is 129 reached z = 0.322 which means a more limited range of convergence than for the Padé approximation. 130 Once a precise approximation for the luminosity distance was obtained, see Eq. (11) means a percentage error δ = 5.9%, for the JLA compilation, see Table 1.  The simple model (GR), the Einstein-De Sitter model (SR), the Milne model (SR) and the plasma 156 model (Euclidean) are rejected because the reduced merit function χ 2 red is smaller than one, see Table 5.
rejected because the reduced merit function χ 2 red is smaller than one, see  Y/N indicates if the item is treated or not and the columns identifies the paper in question, LF means 170 luminosity function for galaxies.

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A. The Padé approximant Given a function f (z), the Padé approximant, after [37], is f (z) = a 0 + a 1 z + · · · + a p z p b 0 + b 1 z + · · · + b q z q , (A.1) where the notation is the same as in [19]. We now present the indefinite integral of (5) for different values of p and q.