Padé Approximant and Minimax Rational Approximation in Standard Cosmology

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1 Article Padé Approximant and Minimax Rational Approximation in Standard Cosmology Lorenzo Zaninetti Dipartimento di Fisica, Università degli Studi di Torino, via P.Giuria, I-05 Turin, Italy; Academic Editor: Emilio Elizalde Received: Novemer 05/ Accepted: January 06/ Pulished: 8 Feruary 06 Astract: The luminosity distance in the standard cosmology as given y ΛCDM and, consequently, the distance modulus for supernovae can e defined y the Padé approximant. A comparison with a known analytical solution shows that the Padé approximant for the luminosity distance has an error of 4% at redshift 0. 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 Padé approximation is superior to the Taylor series. The availaility of an analytical expression for the distance modulus allows applying the Levenerg Marquardt method to derive the fundamental parameters from the availale compilations for supernovae. A new luminosity function for galaxies derived from the truncated gamma proaility density function models the oserved luminosity function for galaxies when the oserved range in asolute magnitude is modeled y the Padé approximant. A comparison of ΛCDM with other cosmologies is done adopting a statistical point of view. Keywords: cosmology; oservational cosmology; distances; redshifts; radial velocities; spatial distriution of galaxies; magnitudes and colors; luminosities PACS: k; Es; 98.6.Py; 98.6.Qz. Introduction In order to otain astronomical oservales, such as the distance modulus and the asolute magnitude for supernovae (SN) of Type Ia in the standard cosmological approach, as given y the ΛCDM model, we need the evaluation of the luminosity distance, which is derived from the comoving distance. At the moment of writing, there is no analytical expression for the integral of the comoving distance in ΛCDM, and a numerical integration should e implemented. An analytical expression for the integral of the comoving distance in ΛCDM can e otained y adopting the technique of the Padé approximant; see [ 3]. Once an approximate solution is otained for the luminosity distance, we can evaluate the distance modulus and the asolute magnitude for SNs. Furthermore, the minimax rational approximation can provide a compact formula for the two aove astronomical oservales as functions of the redshift. From an oservational point of view, the progressive increase in the numer of supernova (SN) of Type Ia for which the distance modulus is availale, 34 SNein the sample, which produced evidence for the accelerating universe (see [4]), 580 SNe in the Union. compilation (see [5]) and 740 SNe in the joint light-curve analysis (JLA) (see [6]), allows analyzing oth the ΛCDM and other cosmologies from a statistical point of view. The statistical approach to cosmology is not new and has een recently adopted y [7,8]. In order to cover the previous arguments, Section introduces the Padé approximant and determines the asic integral of the ΛCDM, which allows deriving the approximate luminosity distance. The approximate magnitude here derived is applied to parametrize a new luminosity function for galaxies at high redshift; see Section 3. The distance modulus in different Galaxies 06, 4, 4; doi:0.3390/galaxies

2 Galaxies 06, 4, 4 of 0 cosmologies is reviewed, and the main statistical parameters connected with the distance modulus are derived; see Section 4.. The Standard Cosmology This section introduces the Hule distance, the dark energy density, the curvature, the matter density and the comoving distance (which is presented as the integral of the inverse of the Hule function). In the asence of a general analytical formula for the comoving distance, we introduce the Padé approximation. As a consequence, we deduce an approximate solution for the transverse comoving distance, the luminosity distance and the distance modulus. The shift that the Padé approximation introduces in the relationship for the poles is discussed. The caliration of the Padé approximation for the distance modulus on two astronomical catalogs allows deducing the minimax polynomial approximation for the oserved distance modulus for SNs of Type Ia... The Padé Approximant We use the same symols as in [9], where the Hule distance D H is defined as: D H c H 0 () We then introduce the first parameter: Ω M Ω M 8π G ρ 0 3 H 0 () 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 3 H 0 where Λ is the cosmological constant; see [0]. The two previous parameters are connected with the curvature Ω K y: Ω M ` Ω Λ ` Ω K (4) The comoving distance, D C, is: where Epzq is the Hule function : Epzq D C D H ż z 0 dz Epz q Ω M p ` zq 3 ` Ω K p ` zq ` Ω Λ (6) The aove integral does not have an analytical formula, except for the case of Ω Λ 0, ut the Padé approximant (see Appendix B) gives an approximate evaluation, and the indefinite integral is (B3), where the coefficients a j and j can e found in Appendix A. The approximate definite integral for (5) is therefore: D C,, F, pz; a 0, a, a, 0,, q F, p0; a 0, a, a, 0,, q (7) (3) (5) The transverse comoving distance D M is: $ D H? sinh? Ω K D C {D H for Ω K ą 0 & ΩK D M D C for Ω K 0 a ΩK ı % D H a ΩK sin D C {D H for Ω K ă 0 (8)

3 Galaxies 06, 4, 4 3 of 0 and the approximate transverse comoving distance D M,, computed with the Padé approximant is: $ D H? sinh? Ω K D C,, {D H for Ω K ą 0 & ΩK D M,, D C,, for Ω K 0 a ΩK ı % D H a ΩK sin D C,, {D H for Ω K ă 0 (9) An analytic expression for D M can e otained when: Ω Λ 0: D M D H r Ω M p zq p Ω M q? ` Ω M zs Ω M p ` zq for Ω Λ 0 (0) This expression is useful for calirating the numerical codes, which evaluate D M when Ω Λ 0. The luminosity distance is: D L p ` zq D M () which in the case of Ω Λ 0 ecomes: and the distance modulus when Ω Λ 0 is: D L c ` Ω M p zq p Ω M q? zω M ` H 0 Ω M m M 5 ` 5 ln`0 ln` c ` ΩM ` z ` ΩM? zωm ` H 0 Ω M () (3) The Padé approximant luminosity distance when Ω Λ 0 is: D L,, p ` zq D M,, (4) and the Padé approximant distance modulus, pm Mq,, in its compact version, is: pm Mq, 5 ` 5 log 0 pd L,, q (5) and, as a consequence, the Padé approximant asolute magnitude, M,, is: with: M, m 5 5 log 0 pd L,, q (6) The expanded version of the Padé approximant distance modulus is: pm Mq, 5 ` 5 ln p0q ln c p ` zq? sinh { H 0 ΩK? ΩK A (7) 4 0 A ln z ` z ` 0 a 4 0 ln z ` z ` 0 a 4 0 ln p 0 q a 4 0 ` ln p 0 q a 4 0 ` a z 4 0 `4 arctan z ` 4 0 a 0 arctan z ` a 4 0

4 Galaxies 06, 4, 4 4 of 0 4 arctan z ` a 0 ` arctan z ` a arctan a 0 ` arctan a `4 arctan a 0 arctan 4 0 a 4 0 The aove procedure can also e applied when the argument of the integral (5) is expanded aout z = 0 in a Taylor series of order six. The resulting luminosity distance, D L,6, is: where: ˆ? c p ` zq ΩK zc D L,6? sinh T ΩK H C T 35 Ω M 5 z 5 ` 350 Ω M 4 z 5 40 Ω M 4 z 4 ` 400 Ω M 3 z Ω M 3 z 4 ` 480 Ω M z 5 `600 Ω M 3 z Ω M z 4 ` 640 z 5 Ω M ` 70 Ω M z z 4 Ω M ` 80 z Ω M z `960 z 3 Ω M 536 z 4 80 z Ω M ` 90 z 3 ` 90 zω M 560 z ` 3840 z 7680 (9) The goodness of the approximation is evaluated through the percentage error, δ, which is: δ ˇ ˇDL pzq D L,app pzqˇˇ D L pzq (8) ˆ 00 (0) where D L pzq is the exact luminosity distance when Ω Λ 0 (see Equation ()) and D L,app pzq is the Taylor or Padé approximate luminosity distance; see also Formula (.) in []. Figure. Percentage error, δ, relative to the Taylor approximated luminosity distance (see Equation (8)) when H km s Mpc and Ω M 0.9.

5 Galaxies 06, 4, 4 5 of 0 Figure. Percentage error, δ, relative to the Padè approximated luminosity distance (see Equation (4)) when H km s Mpc and Ω M 0.9. Figures and report the percentage error as a function of the redshift z for the Taylor and Padé approximations, respectively. The Padé approximation is superior to the truncated Taylor expansion, ecause δ «4 is reached at z 0 for the Padé approximant and at z 0.7 for the Taylor expansion... The Presence of Poles The integrand of (5) contains poles or singularities for a given set of parameters; see Figure 3. Figure 3. Behavior of Epzq as a function of z and Ω Λ in the neighorhoods of the poles when Ω K 0..

6 Galaxies 06, 4, 4 6 of 0 The equation which models the poles is: Epzq 0 () The exact solution of the aove equation zpω Λ ; Ω K 0.q is shown in Figure 4, together with the Padé approximated solution z, pω Λ ; Ω K 0.q. Is therefore possile to conclude that the Padé approximation shifts the locations of the poles y z; this shift expressed as a percentage error is δ «7% in the considered interval Ω Λ r.5,.85s. Figure 4. The exact solution for the zero in E(z), full red line, and Padé approximated solution, dashed lue line, when Ω K An Astrophysical Application We now have a Padé approximant expression for the distance modulus as a function of of H 0, Ω M and Ω Λ. We now perform an astronomical test on the 580 SNe in the Union. compilation (see [5]) and on the 740 SNe in the joint light-curve analysis (JLA). The JLA compilation is availale at the Strasourg 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 and, error in m B, σ m, parameter X, error in B X, σ X, parameter C, error in the parameter C, σ C and log 0 pm stellar q. The oserved distance modulus is defined y Equation (4) in [6]: The adopted parameters are α 0.4, β 3.0 and: m M Cβ ` X α M ` m B () M # 9.05 if Mstellar ă 0 0 M 9. if M stellar ě 0 0 M (3) see Line in Tale 0 of [6]. The uncertainty in the oserved distance modulus, σ m M, is found y implementing the error propagation equation (often called the law of errors of Gauss) when the covariant terms are neglected; see Equation (3.4) in [], σ m M α σ X ` β σ C ` σ m B (4)

7 Galaxies 06, 4, 4 7 of 0 The three astronomical parameters in question, H 0, Ω M and Ω Λ, can e derived trough the Levenerg Marquardt method (suroutine MRQMINin []) 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, pm Mq,, with respect to H 0 is: dpm Mq, 5 dh 0 H 0 ln p0q (5) This numerical procedure minimizes the merit function χ evaluated as: χ Nÿ j pm Mqi pm Mqpz i q th (6) i where N 480, pm Mq i is the oserved distance modulus evaluated at z i, σ i is the error in the oserved distance modulus evaluated at z i and pm Mqpz i q th is the theoretical distance modulus evaluated at z i ; see Formula (5.5.5) in []. A reduced merit function χ red is evaluated y: σ i χ red χ {NF (7) where NF n k is the numer of degrees of freedom, n is the numer of SNe and k is the numer of parameters. Another useful statistical parameter is the associated Q-value, which has to e understood as the maximum proaility of otaining a etter fitting; see Formula (5..) in []: Q GAMMQp N k, χ q (8) where GAMMQ is a suroutine for the incomplete gamma function. The Akaike information criterion (AIC) (see [3]) is defined y: AIC k lnplq (9) where L is the likelihood function. We assume a Gaussian distriution for the errors, and the likelihood function can e derived from the χ statistic L9 expp χ q, where χ has een computed y Equation (6); see [4,5]. Now, the AIC ecomes: AIC k ` χ (30) Tale reports the three astronomical parameters for the two catalogs of SNs, and Figures 5 and 6 display the est fits. Tale. Numerical values of χ, χ red, Q, and the AIC of the Hule diagram for two compilations; k stands for the numer of parameters. SN, supernoval; JLA, joint light-curve analysis. Compilation SNs k Parameters χ χ red Q AIC Union H 0 = 69.8; Ω M 0.39; Ω Λ JLA H 0 = ; Ω M 0.8; Ω Λ In order to see how χ varies around the minimum found y the Levenerg Marquardt method, Figure 7 presents a D color map for the values of χ when H 0 and Ω M are allowed to vary around the numerical values, which fix the minimum.

8 Galaxies 06, 4, 4 8 of 0 Figure 5. Hule diagram for the Union. compilation. The solid line represents the est fit for the approximate distance modulus as represented y Equation (7); parameters as in Tale. Figure 6. Hule diagram for the JLA compilation. The solid line represents the est fit for the approximate distance modulus as given y Equation (7); parameters as in Tale. Figure 7. Color contour plot for χ of the Hule diagram for the Union. compilation when H 0 and Ω M are variales and Ω Λ 0.65.

9 Galaxies 06, 4, 4 9 of 0 The Padé approximant distance modulus has a simple expression when the minimax rational approximation is used, as an example p 3, q ; see Appendix C for the meaning of p and q. In the case of the Union. compilation, the approximation of Formula (7) with the parameters of Tale over the range in z P r0, 4s gives the following minimax equation: pm Mq 3, ` z ` z ` z ` z ` z Union. compilation (3) the maximum error eing The maximum error of the polynomial approximation as a function of p and q is shown in Tale. Tale. The maximum error in the minimax rational approximation for the distance modulus in the case of the Union. compilation. p q Maximum Error In the case of the JLA compilation, the minimax equation is: pm Mq 3, ` z ` z ` z ` z ` z JLA compilation (3) the maximum error eing The maximum difference etween the two minimax formulas, which approximate the distance modulus, Equations (3) and (3), is at z 4 and is mag. In the case of the luminosity distance as given y the Padé approximation (see Equation (4)), the minimax approximation gives: D L,3, z z z z ` z Mpc Union. (33a) D L,3, z z z z ` z Mpc JLA (33) 3. Application at High Redshift This section introduces a new luminosity function (LF) for galaxies, which has a lower and an upper ound. The presence of a lower ound for the luminosity of galaxies allows one to model the evolution of the LF as a function of the redshift. 3.. The Schechter Luminosity Function The Schechter LF, after [6], is the standard LF for galaxies: Φp L Φ qdl p qp L qα exp` L dl (34) Here, α sets the shape, is the characteristic luminosity and Φ is the normalization. The distriution in asolute magnitude is: ΦpMqdM 0.9Φ 0 0.4pα`qpM Mq exp` 0 0.4pM Mq dm (35) where M is the characteristic magnitude.

10 Galaxies 06, 4, 4 0 of The Gamma Luminosity Function The gamma LF is: f pl; Ψ,, cq Ψ where Ψ is the total numer of galaxies per unit Mpc 3, ˆ L c L e Γ pcq (36) ż 8 Γpzq e t t z dt (37) 0 is the gamma function, ą 0 is the scale and c ą 0 is the shape; see Formula (7.3) in [7]. Its expected value is: EpΨ,, cq Ψ c (38) The change of parameter pc q α allows otaining the same scaling as for the Schechter LF (34) The Truncated Gamma Luminosity Function We assume that the luminosity L takes values in the interval rl l, L u s where the indices l and u mean lower and upper; the truncated gamma LF is: f pl; Ψ,, c, L l, L u q Ψ k p L L qc e (39) where Ψ is the total numer of galaxies per unit Mpc 3, and the constant k is: k c ˆ Lu e Lu Γ ˆ ` c, L u c ` Γ ˆ ` c, L ˆ c l Ll e Ll (40) where: ż 8 Γpa, zq t a e t dt (4) is the upper incomplete gamma function; see [8,9]. Its expected value is: EpΨ,, c, L l, L u q Ψ z ˆ ˆ c Γ ` c, L ˆ u Γ ` c, L l ˆ c Lu e Lu ˆ Γ ` c, L u ˆ ` Γ ` c, L l ˆ c Ll e Ll More details on the truncated gamma PDF can e found in [0,]. The four luminosities L, L l, and L u are connected to the asolute magnitude M, M l, M u and M through the following relationship: (4) L 0 0.4pMd Mq L, l 0 0.4pM d M u q, 0 0.4pM d M q, L u 0 0.4pM d M l q L d L d L d L d (43) where the indices u and l are inverted in the transformation from luminosity to asolute magnitude, and M d is the asolute magnitude of the Sun in the considered and. The gamma truncated LF in magnitude is:

11 Galaxies 06, 4, 4 of 0 where ΨpMqdM 0.4 c M 0.4 M c e M 0.4 M Ψ pln pq ` ln p5qq D e M l `0.4 M M l`0.4 M c e M 0.4 Mu Γ D M 0.4 M u c ` c, M l`0.4 M ` Γ ` c, M 0.4 M u (44) (45) The first test on the reliaility of the truncated gamma LF was performed on the data of the Sloan Digital Sky Survey (SDSS) (see []) in the and z. The numer of variales can e reduced to two once M u and M l are identified with the maximum and minimum asolute magnitude of the considered sample. The LFs considered here are displayed in Figure 8. A second test is represented y the ehavior of the LF at high z. We expect a progressive decrease of the low luminosity galaxies (high magnitude) when z is increasing. A formula that models the previous statement can e otained y Equation (6), which models the asolute magnitude, M, as a function of the redshift, inserting as the apparent magnitude, m, the limiting magnitude of the considered catalog. We now outline how to uild an oserved LF for a galaxy in a consistent way; the selected catalog is zcosmos, which is made up of 9697 galaxies up to z 4; see [3]. The oserved LF for zcosmos can e uilt y employing the following algorithm. () The minimax approximation for the luminosity distance in the case of the JLA compilation parameters (see Equation (33)) allows fixing the distance, in the following r, once z is given. () A value for the redshift is fixed, z, as well as the thickness of the layer, z. (3) All of the galaxies comprised etween z and z are selected. (4) The asolute magnitude is computed from Equation (6). (5) The distriution in magnitude is organized in frequencies versus asolute magnitude. (6) The frequencies are divided y the volume, which is V Ωπr r, where r is the considered radius, r is the thickness of the radius and Ω is the solid angle of ZCOSMOS. (7) The error in the oserved LF is otained as the square root of the frequencies divided y the volume. Figure 8. The luminosity function data of the Sloan Digital Sky Survey (SDSS) (z ) are represented with error ars. The continuous line fit represents our truncated gamma luminosity function (LF) (44) with parameters M l = 3.73, M u = 7.48, M =., Ψ 0.04 Mpc 3 and c 0.0. The dotted line represents the Schechter LF with parameters Φ 0.03 Mpc 3 and α.07.

12 Galaxies 06, 4, 4 of 0 Figures 9 present the LF of zcoosmos, as well as the fit with the truncated eta LF at z 0., z 0.4 and z 0.6, respectively. Figure 9. The luminosity function data of zcosmos are represented with error ars. The continuous line fit represents our gamma truncated LF (44); the chosen redshift is z 0. and z The parameters independent of the redshift are given in Tale 3, and the upper magnitude-zrelationship is given in Tale 4. Figure 0. The luminosity function data of zcosmosare represented with error ars. The continuous line fit represents our gamma truncated LF (44); the chosen redshift is z 0.4 and z = Parameters in Tales 3 and 4. Tale 3. Parameters of the gamma truncated LF independent of z when c 0.0. M l M c Tale 4. Upper magnitude, M u (mag), and normalization, Ψ Mpc 3, dependence on z when c 0.0. z Ψ M u

13 Galaxies 06, 4, 4 3 of 0 Figure. The luminosity function data of zcosmos are represented with error ars. The continuous line fit represents our gamma truncated LF (44); the chosen redshift is z 0.6 and z Parameters in Tales 3 and Different Cosmologies Here, we analyze the distance modulus for SNe in other cosmologies in the framework of general relativity (GR), an expanding flat universe, special relativity (SR) and a Euclidean static universe. 4.. Simple GR Cosmology In the framework of GR, the received flux, f, is: f L 4 πd L (46) where d L is the luminosity distance, which depends on the cosmological model adopted; see Equation (7.) in [4] or Equation (5.35) in [5]. The distance modulus in the simple GR cosmology is: m M 43.7 ln p0q ln ˆH0 ` 5 70 ln pzq ln p0q `.086 p q 0q z (47) see Equation (7.5) in [4]. The numer of free parameters in the simple GR cosmology is two: H 0 and q Flat Expanding Universe This model is ased on the standard definition of luminosity in the flat expanding universe. The luminosity distance, r L, is: r L c H 0 z (48) and the distance modulus is: m M 5 log 0 `5 log 0 r L `.5 logp ` zq (49) see Formulae (3) and (4) in [6]. The numer of free parameters in the flat expanding model. is one: H 0.

14 Galaxies 06, 4, 4 4 of Einstein De Sitter Universe in SR In the Einstein De Sitter model, which is developed in SR, the luminosity distance, after [7,8], is: d L c ` ` z?z ` (50) H 0 and the distance modulus for the Einstein De Sitter model is: m M 5 ` 5 ln p0q ln c ` ` z?z ` H 0 (5) The numer of free parameters in the Einstein De Sitter model is one: H Milne Universe in SR In the Milne model, which is developed in the framework of SR, the luminosity distance, after [9 3], is: ˆ c z ` z d L (5) H 0 and the distance modulus for the Milne model is: ˆ m M 5 ` 5 ln p0q ln c z ` z H 0 (53) The numer of free parameters in the Milne model is one: H Plasma Cosmology In a Euclidean static framework, among many possile asorption mechanisms, we selected a photo-asorption process etween the photon and the electron in the IGM. This relativistic process produces a nonlinear dependence etween redshift and distance: z pexpph 0 dq q (54) see Equation (4) in [3]. The previous equation is identical to our Equation (59). The Hule constant in this first plasma model is: H ă n e ą km s Mpc (55) where <n e > is expressed in cgsunits. The second mechanism is a plasma effect, which produces the following relationship: d c H 0 lnp ` zq (56) see Equation (50) in [33]. Furthermore, this second mechanism produces the same nonlinear d-z dependence as our Equation (59). In the presence of plasma asorption, the oserved flux is: f L exp p d H 0d H 0 dq 4πd (57) where the factor exp p dq is due to Galactic and host galactic extinctions, H 0 d is reduction to the plasma in the IGM and H 0 d is the reduction due to Compton scattering; see the formula efore Equation (5) in [33]. The resulting distance modulus in the plasma mechanism is:

15 Galaxies 06, 4, 4 5 of 0 m M 5 ln pln pz ` qq ln p0q ` 5 ln pz ` q ln p0q ` 5 ˆ c ln p0q ln ` 5 `.086 (58) H 0 see Equation (7) in [34]. The numer of free parameters in the plasma cosmology is one: H 0 when Modified Tired Light In a Euclidean static framework, the modified tired light (MTL) has een introduced in Section. in [35]. The distance in MTL is: d c H 0 lnp ` zq (59) The distance modulus in the modified tired light (MTL) is: m M 5 β ln pz ` q ln p0q ` 5 ln p0q ln ˆln pz ` q c H 0 ` 5 (60) Here, β is a parameter comprised etween one and three, which allows one to match theory with oservations. The numer of free parameters in MTL is two: H 0 and β Results for Different Cosmologies The statistical parameters for the different cosmologies here analyzed can e found in Tale 5 in the case of the Union. compilation and in Tale 6 for the JLA compilation. Tale 5. Numerical values of χ, χ red, Q and the AIC of the Hule diagram for the Union. compilation; k stands for the numer of parameters; H 0 is expressed in km s Mpc. Cosmology Equation k Parameters χ χ red Q AIC simple (GR) (47) H , q flat expanding model (49) H Einstein De Sitter (SR) (5) H Milne (SR) (53) H plasma (Euclidean) (58) H MTL (Euclidean) (60) β.37, H Tale 6. Numerical values of χ, χ red, Q and the AIC of the Hule diagram for the JLA compilation; k stands for the numer of parameters; H 0 is expressed in km s Mpc. Cosmology Equation k Parameters χ χ red Q AIC simple (GR) (47) H , q flat expanding model (49) H Einstein De Sitter (SR) (5) H Milne (SR) (53) H plasma (Euclidean) (58) H MTL (Euclidean) (60) β.36, H Conclusions 5.. Padé approximant It is generally thought that in the case of the luminosity distance, the Padé approximant is more accurate than the Taylor expansion. As an example, at z.5, which is the maximum value of the redshift here considered, the percentage error of the luminosity distance is δ 0.036% in the case of the Padé approximation. In the case of of the Taylor expansion, δ 0.036% for the luminosity distance is

16 Galaxies 06, 4, 4 6 of 0 reached z 0.3, which means a more limited range of convergence than for the Padé approximation. Once a precise approximation for the luminosity distance was otained (see Equation ()), we derived an approximate expression for the distance modulus (see Equation (7)) and the asolute magnitude (see Equation (6)). 5.. Astrophysical Applications The availaility of the oserved distance modulus for a great numer of SNs of Type Ia allows deducing H 0, Ω M and Ω Λ for two catalogs; see Tale. In order to derive the aove parameters, the Levenerg Marquardt method was implemented, and therefore, the first derivative of the distance modulus (see Equation (7)) with respect to three parameters is provided. The value of H 0 is a matter of research rather than a well-defined constant. As an example, a recent evaluation with a sample of Cepheids gives H km s Mpc ; see [36]. Once the aove value is considered the true value, we have found, adopting the Padé approximant, H km s Mpc, which means a percentage error δ 5.4%, for the Union. compilation, and H km s Mpc, which means a percentage error δ 5.9%, for the JLA compilation; see Tale Evolutionary Effects The evolution of the LF for galaxies as a function of the redshift is here modeled y an upper and lower truncated gamma PDF. This choice allows modeling the lower ound in luminosity (the higher ound in asolute magnitude) according to the evolution of the asolute magnitude; see Equation (6). According to the LF here considered (see Equation (44)), the evolution with z of the LF is simply connected to the evolution of the higher ound in asolute magnitude; see Figures 9. Is not necessary to modify the shape parameters of the LF, which are c and M, ut only to calculate the normalization Ψ at different values of the redshift Statistical Tests for Union. In the case of the Union. compilation, the est results for χ red are otained y the ΛCDM cosmology (GR), χ red 0.975, against χ red 0.98 of the MTL cosmology (Euclidean), ut the situation is inverted when the AIC is considered: the AIC is 57.9 for the MTL cosmology and for the ΛCDM cosmology (GR); see Tales and 5. The simple model (GR), the Einstein De Sitter model (SR), the Milne model (SR) and the plasma model (Euclidean) are rejected ecause the reduced merit function χ red is smaller than one; see Tale 5. The est performing one-parameter model is that of Milne, χ red.04, followed y the flat expanding model, χ red.; see Tale Statistical Tests for JLA In the case of the JLA compilation, the est results for χ red are otained y the MTL cosmology (Euclidean), χ red 0.848, against χ red for the ΛCDM cosmology (GR); see Tales and 6. The simple model (GR), the Einstein De Sitter model (SR) and the plasma model (Euclidean) are rejected ecause the reduced merit function χ red is smaller than one; see Tale 6. In the case of the JLA, the test on the Milne model is positive ecause χ red is smaller than one. The est performing one-parameter model is that of Milne, χ red 0.887, followed y the flat expanding model, χ red 0.97; see Tale Different Approaches Tale 7 reports six items connected to the use of the Padé approximant in Cosmology: the letters Y/N indicate if the item is treated or not, and the columns identifies the paper in question; LF means the luminosity function for galaxies.

17 Galaxies 06, 4, 4 7 of 0 Tale 7. Arguments treated in papers on Padé approximants and here. Prolem Aviles 04 Wei 04 Adachi 0 Here luminosity distance Y Y Y Y distance modulus Y Y Y Y empty eam N N Y N distance modulus minimax N N N Y poles N N N Y LF=f(z) N N N Y Conflicts of Interest: The authors declare no conflict of interest. Appendix A. The Padé Approximant Given a function f pzq, the Padé approximant, after [37], is: f pzq a 0 ` a z ` ` a p z p 0 ` z ` ` q z q (A) where the notation is the same as in [9]. The coefficients a i and i are found through Wynn s cross rule (see [38,39]), and our choice is p and q. The choice of p and q is a compromise etween precision, high values for p and q and the simplicity of the expressions to manage low values for p and q; Appendix B gives three different approximations for the indefinite integral for three different cominations in p and q. In the case in which 0 0, we can divide oth the numerator and denominator y 0, reducing y one the numer of parameters; see as an example [40]. The integrand of Equation (5) is: Epzq a ΩM p ` zq 3 ` Ω K p ` zq ` Ω Λ (A) and the Padé approximant gives: where: Epzq a 0 ` a z ` a z 0 ` z ` z (A3) a 0 a 6 `3 Ω K 3 Ω Λ ` 6 Ω K Ω Λ ` 60 Ω K Ω Λ Ω M ` 4 Ω K Ω M ` 64 Ω K Ω Λ Ω M `30 Ω K Ω Λ Ω M ` 40 Ω K Ω M3 ` 96 Ω Λ Ω M `9 Ω Λ Ω M3 ` 5 Ω M 4 `Ω M ` Ω K ` Ω Λ 4 4 `8 Ω K 4 Ω Λ ` 3 Ω K 3 Ω Λ ` 704 Ω K 3 Ω Λ Ω M 6 Ω K Ω Λ Ω M `456 Ω K Ω Λ Ω M ` 3 Ω K Ω M3 64 Ω K Ω Λ 3 Ω M 384 Ω K Ω Λ Ω M `5 Ω K Ω Λ Ω M3 ` 50 Ω K Ω M4 9 Ω Λ 3 Ω M 88 Ω Λ Ω M3 ` 648 Ω Λ Ω M 4 `5 Ω M 5 `Ω M ` Ω K ` Ω Λ 3 (A4) (A5)

18 Galaxies 06, 4, 4 8 of 0 a `56 Ω 4 K Ω Λ Ω M 64 Ω 3 K Ω Λ3 ` 30 Ω 3 K Ω Λ Ω M ` 960 Ω 3 K Ω Λ Ω M 30 Ω K Ω 3 Λ Ω M ` 40 Ω K Ω Λ Ω M ` 440 Ω K Ω Λ Ω M3 ` 6 Ω 4 K Ω M 600 Ω K Ω 3 Λ Ω M 480 Ω K Ω 5 Λ Ω M3 ` 40 Ω K Ω Λ Ω M4 ` 0 Ω K Ω M 56 Ω 4 Λ Ω M 600 Ω 3 Λ Ω M3 40 Ω 5 Λ Ω M4 ` 380 Ω Λ Ω M `5 Ω M 6 `Ω M ` Ω K ` Ω Λ (A6) 0 6 `Ω M ` Ω K ` Ω Λ 9{`3 ΩK 3 Ω Λ ` 6 Ω K Ω Λ ` 60 Ω K Ω Λ Ω M `4 Ω K Ω M ` 64 Ω K Ω Λ Ω M ` 30 Ω K Ω Λ Ω M ` 40 Ω K Ω M 3 `96 Ω Λ Ω M ` 9 Ω Λ Ω M3 ` 5 Ω M 4 4 `Ω M ` Ω K ` Ω Λ 7{`56 ΩK 4 Ω Λ ` 96 Ω K 3 Ω Λ ` 536 Ω K 3 Ω Λ Ω M `96 Ω K 3 Ω M ` 336 Ω K Ω Λ Ω M ` 3696 Ω K Ω Λ Ω M `336 Ω K Ω M3 64 Ω K Ω Λ 3 Ω M ` 384 Ω K Ω Λ Ω M ` 400 Ω K Ω Λ Ω M 3 `350 Ω K Ω M4 9 Ω Λ 3 Ω M ` 88 Ω Λ Ω M3 ` 800 Ω Λ Ω M4 ` 05 Ω M 5 `ΩM ` Ω K ` Ω Λ 5{`5 ΩK 5 Ω Λ ` 384 Ω K 4 Ω Λ ` 3584 Ω K 4 Ω Λ Ω M `9 Ω K 3 Ω Λ3 ` 984 Ω K 3 Ω Λ Ω M ` 075 Ω K 3 Ω Λ Ω M ` 30 Ω K 3 Ω M 3 `960 Ω K Ω Λ 3 Ω M ` 536 Ω K Ω Λ Ω M ` 7760 Ω K Ω Λ Ω M3 ` 840 Ω K Ω M 4 `75 Ω K Ω Λ 3 Ω M ` 739 Ω K Ω Λ Ω M3 ` 5060 Ω K Ω Λ Ω M4 ` 700 Ω K Ω Λ 5 `56 Ω Λ 4 Ω M ` 75 Ω Λ 3 Ω M3 ` 3696 Ω Λ Ω M4 ` 500 Ω Λ Ω M5 ` 75 Ω M 6 (A7) (A8) (A9) B. The Integrals as Functions of p and q We now present the indefinite integral of (5) for different values of p and q. In the case p, q, F, pz; a 0, a, 0, q a z ` ln pz ` 0 q a 0 ln pz ` 0 q 0 a (B) In the case p, q, F, pz; a 0, a, a, 0, q { a z ` az z 0a ` ln pz ` 0 q a 0 ln pz ` 0 q 0 a ` ln pz ` 0 q a 0 3 (B) In the case p, q, F, pz; a 0, a, a, 0,, q a z ln ` `z ` z ` 0 a ln `z ` z ` 0 a a 0 ` arctan z ` a 0 arctan z ` a arctan z ` ` a arctan z ` (B3)

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