In ΛCDM cosmology, the Hubble distance $D_{\text{H}}$ is defined by

$D_{\text{H}}\equiv \frac{c}{H_0},$(1)

where $c$ is the speed of light and $H_0$ is the Hubble constant. We then introduce a parameter ${\Omega }_{\text{M}}$

${\Omega }_{\text{M}}=\frac{8\pi G{\rho }_0}{3H_0^2},$(2)

where $G$ is the Newtonian gravitational constant and ${\rho }_0$ is the mass density at the present time. Another is ${\Omega }_{\Lambda }$

${\Omega }_{\Lambda }\equiv \frac{\text{Λ}{c}^2}{3H_0^2},$(3)

where Λ is the cosmological constant, see [9] . Once ${\Omega }_{\Lambda }$ and $H_0$ are set, the numerical value of the cosmological constant is derived, $\text{Λ}\approx 1.2\frac{1}{{\text{m}}^2}$.

The two previous parameters are connected with the curvature ${\Omega }_K$ by

${\Omega }_{\text{M}}+{\Omega }_{\Lambda }+{\Omega }_K=1.$(4)

The comoving distance, $D_{\text{C}}$, is

$D_{\text{C}}=c{\displaystyle {\int }_0^z\frac{\text{d}{z}^{\prime }}{E\left(z^{\prime }\right)}},$(5)

where $c$ is the speed of light and $E\left(z\right)$ is the “Hubble function”

$E\left(z\right)=\sqrt{{\Omega }_{\text{M}}{\left(1+z\right)}^3+{\Omega }_K{\left(1+z\right)}^2+{\Omega }_{\text{Λ}}}.$(6)

The above integral cannot be done in analytical terms, except for the case of ${\Omega }_{\Lambda }=0$, but the Padé approximant, see Appendix A in [10] , allows us to derive an approximation for the indefinite integral.

The approximate definite integral for (5) is therefore

$D_{\text{C},2,2}=D_{\text{H}}\left(F_{2,2}\left(z;a_0,a_1,a_2,b_0,b_1,b_2\right)-F_{2,2}\left(0;a_0,a_1,a_2,b_0,b_1,b_2\right)\right),$(7)

where $F_{2,2}$ is from equation (A10) in [10] . The transverse comoving distance $D_{\text{M}}$ is

$D_{\text{M}}=\{\begin{array}{ll}{D}_{\text{H}}\frac{1}{\sqrt{{\Omega }_K}}\sinh \left[\sqrt{{\Omega }_K}{D}_{\text{C}}/D_{\text{H}}\right]\hfill & \text{for}{\Omega }_K>0\hfill \\ D_{\text{C}}\hfill & \text{for}{\Omega }_K=0\hfill \\ D_{\text{H}}\frac{1}{\sqrt{\left|{\Omega }_K\right|}}\text{sin}\left[\sqrt{\left|{\Omega }_K\right|}{D}_{\text{C}}/D_{\text{H}}\right]\hfill & \text{for}{\Omega }_K<0\hfill \end{array}$(8)

The approximate transverse comoving distance, $D_{\text{M},2,2}$, computed with the Padé approximant, is

$D_{\text{M},2,2}=\{\begin{array}{ll}{D}_{\text{H}}\frac{1}{\sqrt{{\Omega }_K}}\text{sinh}\left[\sqrt{{\Omega }_K}{D}_{\text{C},2,2}/D_{\text{H}}\right]\hfill & \text{for}{\Omega }_K>0\hfill \\ D_{\text{C},2,2}\hfill & \text{for}{\Omega }_K=0\hfill \\ D_{\text{H}}\frac{1}{\sqrt{\left|{\Omega }_K\right|}}\text{sin}\left[\sqrt{\left|{\Omega }_K\right|}{D}_{\text{C},2,2}/D_{\text{H}}\right]\hfill & \text{for}{\Omega }_K<0\hfill \end{array}$(9)

The Padé approximant for the luminosity distance is

$D_{\text{L},2,2}=\left(1+z\right)D_{\text{M},2,2},$(10)

and the Padé approximant for the distance modulus, ${\left(m-M\right)}_{2,2}$, is

${\left(m-M\right)}_{2,2}=25+5{\log }_{10}\left(D_{\text{L},2,2}\right).$(11)

In the dynamical dark energy cosmology (wCDM), first introduced by [11] , the Hubble distance is

$D_H\left(z;{\Omega }_{\text{M}},w,{\Omega }_{DE}\right)=\frac{1}{\sqrt{{\left(1+z\right)}^3{\Omega }_{\text{M}}+{\Omega }_{DE}{\left(1+z\right)}^{3+3w}}},$(12)

where $w$ is the equation of state, here considered constant, see equation (3.4) in [12] or equation (18) in [13] for the luminosity distance. Here we assumed $w$ to be constant but also the case of $w$ as a function of $z$ can be considered, see equation (19) in [13] . In the above cosmology, the cosmological constant is absent. In flat cosmology,

${\Omega }_{\text{M}}+{\Omega }_{DE}=1,$(13)

and the Hubble distance becomes

$D_H\left(z;{\Omega }_{\text{M}},w\right)=\frac{1}{\sqrt{{\left(1+z\right)}^3{\Omega }_{\text{M}}+\left(1-{\Omega }_{\text{M}}\right){\left(1+z\right)}^{3+3w}}}.$(14)

The indefinite integral in the variable $z$ of the above Hubble distance, $Iz\equiv \frac{D_C}{D_H}$, is

$Iz\left(z;{\Omega }_{\text{M}},w\right)={\displaystyle \int D_H\left(z;{\Omega }_{\text{M}},w\right)\text{d}z},$(15)

where the new symbol $Iz$ underlines the mathematical operation of integration. In order to obtain the indefinite integral, we perform a change of variable $1+z=t^{1/3}$

$Iz\left(t;{\Omega }_{\text{M}},w\right)=\frac{1}{3}{\displaystyle \int \frac{1}{\sqrt{-t\left(\left(-1+{\Omega }_{\text{M}}\right)t^w-{\Omega }_{\text{M}}\right)}{t}^{2/3}}\text{d}t}.$(16)

The indefinite integral is

$Iz\left(t;{\Omega }_{\text{M}},w\right)=\frac{-2{}_{2}F{}_1\left(\frac{1}{2},-\frac{1}{6}{w}^{-1};1-\frac{1}{6}{w}^{-1};-\frac{t^w-\left(1-{\Omega }_{\text{M}}\right)}{{\Omega }_{\text{M}}}\right)}{\sqrt{{\Omega }_{\text{M}}}\sqrt[6]{t}},$(17)

where ${}_{2}F{}_1\left(a,b;c;z\right)$ is the regularized hypergeometric function, see [14] - [18] . We now return to the variable $z$, the redshift and then the indefinite integral becomes

$Iz\left(z;{\Omega }_{\text{M}},w\right)=\frac{-2{}_{2}F{}_1\left(\frac{1}{2},-\frac{1}{6}{w}^{-1};1-\frac{1}{6}{w}^{-1};-\frac{{\left(-z^3+3z^2+3z+1\right)}^w\left(1-{\Omega }_{\text{M}}\right)}{-{\Omega }_{\text{M}}}\right)}{\sqrt{{\Omega }_{\text{M}}}\sqrt[6]{z^3+3z^2+3z+1}}.$(18)

We denote by $F\left(z;{\Omega }_{\text{M}},w\right)$ the definite integral

$F\left(z;{\Omega }_{\text{M}},w\right)=Iz\left(z=z;{\Omega }_{\text{M}},w\right)-Iz\left(z=0;{\Omega }_{\text{M}},w\right).$(19)

The luminosity distance, $D_{\text{L}}$, for wCDM cosmology in the case of the analytical solution is

$D_{\text{L}}\left(z;c,H_0,{\Omega }_{\text{M}},w\right)=\frac{c}{H_0}\left(1+z\right)F\left(z;{\Omega }_{\text{M}},w\right),$(20)

where $F\left(z;{\Omega }_{\text{M}},w\right)$ is given by Equation (19) and the distance modulus is

$\left(m-M\right)=25+5{\log }_{10}\left(D_{\text{L}}\left(z;c,H_0,{\Omega }_{\text{M}},w\right)\right).$(21)

More details can be found in [19] .

We are interested in the astrophysical applications of the $f\left(Q,L_m\right)$ gravity model as presented in [8] . The basic astrophysical parameter is $\omega$, which represents the equation of state (EoS) of dark matter. The comoving distance, $D_{\text{C}}$, is

$D_{\text{C}}=c{\displaystyle {\int }_0^z\frac{\text{d}{z}^{\prime }}{E\left(z^{\prime }\right)}},$(22)

where $E\left(z\right)$ is the ‘Hubble function’ and the luminosity distance is

$D_{\text{L}}=\left(1+z\right)D_{\text{C}}.$(23)

The distance modulus $\mu \left(z\right)$ is

$\mu \left(z\right)=\left(m-M\right)=25+5{\log }_{10}\left(D_{\text{L}}\left(z;c,H_0,w\right)\right),$(24)

where $m$ and $M$ are the apparent and absolute magnitude. In the $f\left(Q,L_m\right)$ gravity model three models are presented.

The definite integral, which represents the “Hubble function”, can be evaluated in the following different ways:

1) analytical integration;

2) numerical integration;

3) approximate integration using a series for the integrand;

4) approximate integration using a Padé approximant $R_{k,l}\left(x\right)=p\left(x\right)/q\left(x\right)$ for the integrand where $p\left(x\right)$ is a polynomial of degree $k$ and $q\left(x\right)$ is a polynomial of degree $l$.

The first model is a linear dependence for the EoS

$\omega =z{\omega }_1+{\omega }_0,$(25)

which corresponds to

$H\left(z\right)=H_0{\left(1+z\right)}^{\frac{3{\omega }_0}{2}+\frac{3}{2}-\frac{3{\omega }_1}{2}}{\text{e}}^{\frac{3z{\omega }_1}{2}}.$(26)

In this case, the integral (22) can be evaluated and is

$D_{\text{C}}=\frac{N}{A},$(27)

where

$\begin{array}{c}N=-(\text{Γ}\left(-\frac{3{\omega }_0}{2}+\frac{3{\omega }_1}{2}+\frac{3}{2},\frac{3{\omega }_1\left(1+z\right)}{2}\right)\\ -\text{Γ}\left(-\frac{3{\omega }_0}{2}+\frac{3{\omega }_1}{2}+\frac{3}{2},\frac{3{\omega }_1}{2}\right)){\text{e}}^{\frac{3{\omega }_1}{2}}{\omega }_1^{\frac{1}{2}+\frac{3{\omega }_0}{2}-\frac{3{\omega }_1}{2}}\sqrt{1+z}{3}^{\frac{1}{2}+\frac{3{\omega }_0}{2}-\frac{3{\omega }_1}{2}}{2}^{-\frac{3{\omega }_0}{2}+\frac{3{\omega }_1}{2}+\frac{3}{2}}\\ +6{\left(1+z\right)}^{-\frac{3{\omega }_0}{2}+\frac{3{\omega }_1}{2}}{\text{e}}^{-\frac{3{\omega }_{1}z}{2}}\left(\left(2+z\right){\omega }_1-{\omega }_0+\frac{1}{3}\right)+6\sqrt{1+z}\left({\omega }_0-2{\omega }_1-\frac{1}{3}\right),\end{array}$(28)

and

$A=\sqrt{1+z}\left(1+3{\omega }_0-3{\omega }_1\right)\left(-1+3{\omega }_0-3{\omega }_1\right)H_0,$(29)

where $\Gamma \left(a,z\right)$ is the incomplete Gamma function defined by

$\Gamma \left(a,z\right)={\displaystyle {\int }_z^{\infty }{t}^{a-1}{\text{e}}^{-t}\text{d}t},$(30)

see [18] . The luminosity distance is given by Equation (23) and its derivative with respect to the redshift is

$\begin{array}{c}\frac{\partial D_{\text{L}}}{\partial z}=\frac{1}{\sqrt{1+z}\left(1+3{\omega }_0-3{\omega }_1\right)\left(-1+3{\omega }_0-3{\omega }_1\right)H_0}\\ \times 2c({\omega }_1^{\frac{1}{2}+\frac{3{\omega }_0}{2}-\frac{3{\omega }_1}{2}}\sqrt{1+z}{\text{e}}^{\frac{3{\omega }_1}{2}}{2}^{\frac{1}{2}-\frac{3{\omega }_0}{2}+\frac{3{\omega }_1}{2}}{3}^{\frac{1}{2}+\frac{3{\omega }_0}{2}-\frac{3{\omega }_1}{2}}\\ \times \left(\text{Γ}\left(-\frac{3{\omega }_0}{2}+\frac{3{\omega }_1}{2}+\frac{3}{2},\frac{3{\omega }_1}{2}\right)-\text{Γ}\left(-\frac{3{\omega }_0}{2}+\frac{3{\omega }_1}{2}+\frac{3}{2},\frac{3{\omega }_1\left(1+z\right)}{2}\right)\right)\\ +3{\left(1+z\right)}^{-\frac{3{\omega }_0}{2}+\frac{3{\omega }_1}{2}}{\text{e}}^{-\frac{3{\omega }_{1}z}{2}}\left(\frac{3{\omega }_1^2}{2}+\left(z-3{\omega }_0+2\right){\omega }_1+\frac{3{\left(-\frac{1}{3}+{\omega }_0\right)}^2}{2}\right)\\ \stackrel{}{}+\left(3{\omega }_0-6{\omega }_1-1\right)\sqrt{1+z}).\end{array}$(31)

A more compact numerical relationship is derived with the best minimax rational approximation of degree (2, 2) for the luminosity distance, $D_{L22}$, on the interval in redshift [0, 2.4]

$D_{L22}=\frac{-0.138341+\left(3104.76+2838.17z\right)z}{0.745533+\left(0.203583+0.00470683z\right)z}.$(32)

The above relationship can be inverted in order to have the redshift, $z_{22}$, as a function of the luminosity distance

$\begin{array}{c}{z}_{22}=\frac{1}{4.70682\times {10}^9{D}_{L22}-2.83817\times {10}^{15}}\times 10(1.01791\times {10}^{10}{D}_{L22}-1.55238\times {10}^{14}\\ +\sqrt{6.85244\times {10}^{19}{D}_{L22}^2+1.79991\times {10}^{25}{D}_{L22}+2.41027\times {10}^{28}}).\end{array}$(33)

The derivative of the approximate luminosity distance is

$\frac{\partial D_{\text{L}22}}{\partial z}\frac{1.91019\times {10}^{8}z+2.54212\times {10}^7{z}^2+1.04482\times {10}^8}{{\left(z+39.2133\right)}^2{\left(z+4.03927\right)}^2}.$(34)

Figure 1 shows the difference between the analytical and the numerical relationship for the luminosity distance as a function of the redshift.

The distance modulus $\mu \left(z\right)$ has the following analytical expression

$\mu \left(z\right)=\left(m-M\right)=25+\frac{5\ln \left(\frac{\left(1+z\right)cN}{A}\right)}{\ln \left(10\right)}.$(35)

According to [8] , the EoS for the second model is

$\omega ={\omega }_0+\frac{{\omega }_{1}z\left(1+z\right)}{z^2+1},$(36)

which corresponds to

$H\left(z\right)={\left(z^2+1\right)}^{\frac{3{\omega }_1}{4}}{H}_0{\left(1+z\right)}^{\frac{3{\omega }_0}{2}+\frac{3}{2}}.$(37)

In this case, the integral (22) can be evaluated but has a complicated expression and therefore we shift to the numerical analysis.

According to [8] , the EoS for the third model is

$\omega ={\omega }_0+\frac{{\omega }_1{z}^2}{z^2+1},$(38)

which corresponds to

$H\left(z\right)=H_0{\left(1+z\right)}^{\frac{3{\omega }_0}{2}+\frac{3{\omega }_1}{4}+\frac{3}{2}}{\left(z^2+1\right)}^{\frac{3{\omega }_1}{8}}{\text{e}}^{-\frac{3{\omega }_1\arctan \left(z\right)}{4}}.$(39)

In this case the integral (22) cannot be evaluated and therefore we shift to the numerical analysis.

The fourth model (new) for the EoS is

$\omega =z^2{\omega }_1+{\omega }_0,$(40)

which corresponds to

$H\left(z\right)={\left(1+z\right)}^{\frac{3{\omega }_0}{2}+\frac{3}{2}+\frac{3{\omega }_1}{2}}{H}_0{\text{e}}^{\frac{3{\omega }_{1}z\left(z-2\right)}{4}}.$(41)

At the moment of writing, the integral (22) cannot be evaluated. We now explore two approximations. The first approximation of $1/H\left(z\right)$ is given by the following series

$\frac{1}{H\left(z\right)}=\frac{{\left(1+z\right)}^{-\frac{3{\omega }_0}{2}-\frac{3{\omega }_1}{2}-\frac{3}{2}}\left({\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\frac{{\left(\frac{3}{4}\right)}^k{\left(-{\omega }_1\left(z-2\right)z\right)}^k}{k!}}\right)}{H_0}.$(42)

An approximate expression of order 4 for the distance modulus can be obtained by inserting the data of Table 2 .

$\begin{array}{c}\mu \left(z\right)=\left(m-M\right)\\ =\frac{1}{\ln \left(2\right)+\ln \left(5\right)}(-25\ln \left(2\right)+25\ln \left(5\right)+5\ln (-495.452((49446.5-6.95944z^5\\ +173.172z^3-0.048045z^8+0.49566z^7-2.6263z^4-0.70507z^6\\ -3972.99z+247.988z^2){\left(1+z\right)}^{0.406328}-49446.5\sqrt{1+z})\sqrt{1+z})).\end{array}$(43)

The percent error of the above distance modulus in the interval $0.01<z<2.4$ is shown in Figure 2 .

The second approximation of $1/H\left(z\right)$ is given by the Padé approximant evaluated around $z=1$

$\frac{1}{H\left(z=1\right)}=H_{2,1}.$(44)

A Padé approximation of the distance modulus is obtained by inserting the data of Table 2 :

$\begin{array}{l}\mu {\left(z\right)}_{2,1}={\left(m-M\right)}_{2,1}\\ =25+\frac{5\ln \left(1300.5\left(1+z\right)\left(1.9229\ln \left(7.2813+6.262z\right)-3.8177+0.69578\left(-0.41053z+2.1251\right)z\right)\right)}{\ln \left(10\right)}.\end{array}$ (45)

The maximum percent error of the Padé distance modulus in $0.01<z<2.4$ is ≈0.12. In order to invert the above relationship, we first apply the minimax algorithm, which transforms the above relationship to

$\mu {\left(z\right)}_{2,1}={\left(m-M\right)}_{2,1}=\frac{1.89433+\left(33.4185+1.37875z\right)z}{0.058972+0.780935z},$(46)

from which the redshift as a function of the modulus is derived

$z=0.28320\mu -12.119+1.4505\times {10}^{-10}\sqrt{3.8116\times {10}^{18}{\mu }^2-3.2418\times {10}^{20}\mu +6.9146\times {10}^{21}}.$ (47)

The above inverse relationship is displayed in Figure 3 .

The fifth model (new) for the EoS is

$\omega ={\omega }_0+z^3{\omega }_1,$(48)

which corresponds to

$H\left(z\right)=H\left(z\right)={\left(1+z\right)}^{\frac{3{\omega }_0}{2}+\frac{3}{2}-\frac{3{\omega }_1}{2}}{H}_0{\text{e}}^{\frac{{\omega }_{1}z\left(2z^2-3z+6\right)}{4}}.$(49)

The basic integral can be approximated by adopting a series expansion around $z=1$ up to order 4, and therefore, the luminosity distance is

$\begin{array}{c}{D}_L\left(z\right)=c\left(1+z\right)\times {\displaystyle {\int }_0^z\frac{\text{d}{z}^{\prime }}{E\left(z^{\prime }\right)}}\\ =-\frac{1}{512}{L}_1{z}^5+\frac{1}{512}{L}_2{z}^4-\frac{1}{256}{L}_3{z}^3-\frac{1}{256}{L}_4{z}^2+\frac{1}{128}{L}_{5}z\end{array}$(50)

where

$L_1=\frac{c\left(9{\omega }_0^3+27{\omega }_0^2{\omega }_1+27{\omega }_0{\omega }_1^2+9{\omega }_1^3+45{\omega }_0^2-18{\omega }_0{\omega }_1-63{\omega }_1^2+71{\omega }_0+11{\omega }_1+35\right)}{{\text{e}}^{\frac{3\left({\omega }_0+1-{\omega }_1\right)\ln \left(2\right)}{2}}{H}_0{\text{e}}^{\frac{5{\omega }_1}{4}}},$(51a)

$L_2=\frac{c\left(27{\omega }_0^3+81{\omega }_0^2{\omega }_1+81{\omega }_0{\omega }_1^2+27{\omega }_1^3+183{\omega }_0^2+42{\omega }_0{\omega }_1-141{\omega }_1^2+341{\omega }_0-31{\omega }_1+185\right)}{{\text{e}}^{\frac{3\left({\omega }_0+1-{\omega }_1\right)\ln \left(2\right)}{2}}{H}_0{\text{e}}^{\frac{5{\omega }_1}{4}}},$(51b)

$L_3=\frac{c\left(9{\omega }_0^3+27{\omega }_0^2{\omega }_1+27{\omega }_0{\omega }_1^2+9{\omega }_1^3+93{\omega }_0^2+78{\omega }_0{\omega }_1-15{\omega }_1^2+295{\omega }_0+43{\omega }_1+211\right)}{{\text{e}}^{\frac{3\left({\omega }_0+1-{\omega }_1\right)\ln \left(2\right)}{2}}{H}_0{\text{e}}^{\frac{5{\omega }_1}{4}}},$(51c)

$L_4=\frac{c\left(9{\omega }_0^3+27{\omega }_0^2{\omega }_1+27{\omega }_0{\omega }_1^2+9{\omega }_1^3+45{\omega }_0^2-18{\omega }_0{\omega }_1-63{\omega }_1^2-25{\omega }_0-85{\omega }_1-317\right)}{{\text{e}}^{\frac{3\left({\omega }_0+1-{\omega }_1\right)\ln \left(2\right)}{2}}{H}_0{\text{e}}^{\frac{5{\omega }_1}{4}}},$(51d)

$L_5=\frac{c\left(9{\omega }_0^3+27{\omega }_0^2{\omega }_1+27{\omega }_0{\omega }_1^2+9{\omega }_1^3+81{\omega }_0^2+54{\omega }_0{\omega }_1-27{\omega }_1^2+263{\omega }_0+59{\omega }_1+319\right)}{{\text{e}}^{\frac{3\left({\omega }_0+1-{\omega }_1\right)\ln \left(2\right)}{2}}{H}_0{\text{e}}^{\frac{5{\omega }_1}{4}}}.$(51e)

An approximate distance modulus is now easily derived using Equation (24) and the parameters of Table 2

$\begin{array}{l}\mu \left(z\right)=\left(m-M\right)\\ =25+\frac{5\ln \left(-\frac{24.6562\left(1+z\right)\left(-212.831z+56.2073z^2-13.1375z^3+2.95741z^4\right)}{{\text{e}}^{0.412976\ln \left(2\right)}}\right)}{\ln \left(10\right)}.\end{array}$ (52)

The percent error of the above distance modulus in the interval $0.01<z<2.4$ is given in Figure 4 .

The sixth model (new) for the EoS is

$\omega ={\omega }_0+\frac{{\omega }_1{z}^3}{z^3+1},$(53)

which corresponds to

$H\left(z\right)=H_0{\left(z^2-z+1\right)}^{\frac{{\omega }_1}{4}}{\left(1+z\right)}^{\frac{3{\omega }_0}{2}+{\omega }_1+\frac{3}{2}}{\text{e}}^{-\frac{{\omega }_1\left(6\sqrt{3}\left(1+z\right)\arctan \left(\frac{\left(2z-1\right)\sqrt{3}}{3}\right)+\pi \left(1+z\right)\sqrt{3}+18z\right)}{36+36z}}.$(54)

Also here the basic integral is approximated by a series expansion around $z=1$ up to order 3. Therefore the luminosity distance is

$D_L\left(z\right)=25+\frac{5\ln \left(\frac{1}{128}{L}_1{z}^4-\frac{1}{64}{L}_2{z}^3+\frac{1}{16}{L}_3{z}^2+\frac{1}{128}{L}_{4}z\right)}{\ln \left(10\right)},$(55)

where

$L_1=\frac{c\left(12{\omega }_0^2+12{\omega }_0{\omega }_1+3{\omega }_1^2+32{\omega }_0+4{\omega }_1+20\right)}{H_0{\text{e}}^{\frac{\left(3+2{\omega }_1+3{\omega }_0\right)\ln \left(2\right)}{2}}{\text{e}}^{-\frac{\sqrt{3}\left(2\pi +3\sqrt{3}\right){\omega }_1}{36}}},$(56a)

$L_2=\frac{c\left(12{\omega }_0^2+12{\omega }_0{\omega }_1+3{\omega }_1^2+56{\omega }_0+16{\omega }_1+44\right)}{H_0{\text{e}}^{\frac{\left(3+2{\omega }_1+3{\omega }_0\right)\ln \left(2\right)}{2}}{\text{e}}^{-\frac{\sqrt{3}\left(2\pi +3\sqrt{3}\right){\omega }_1}{36}}}$(56b)

$L_3=\frac{c\left(6{\omega }_0+3{\omega }_1+22\right)}{H_0{\text{e}}^{\frac{\left(3+2{\omega }_1+3{\omega }_0\right)\ln \left(2\right)}{2}}{\text{e}}^{-\frac{\sqrt{3}\left(2\pi +3\sqrt{3}\right){\omega }_1}{36}}},$(56c)

$L_4=\frac{c\left(36{\omega }_0^2+36{\omega }_0{\omega }_1+9{\omega }_1^2+192{\omega }_0+60{\omega }_1+284\right)}{H_0{\text{e}}^{\frac{\left(3+2{\omega }_1+3{\omega }_0\right)\ln \left(2\right)}{2}}{\text{e}}^{-\frac{\sqrt{3}\left(2\pi +3\sqrt{3}\right){\omega }_1}{36}}}.$(56d)

The distance modulus is now easily derived using Equation (24)

$\begin{array}{l}\mu \left(z\right)=\left(m-M\right)\\ =25+\frac{5\ln \left(-\frac{1}{4096}{L}_1{z}^5+\frac{1}{4096}{L}_2{z}^4-\frac{1}{2048}{L}_3{z}^3-\frac{1}{2048}{L}_4{z}^2+\frac{1}{1024}{L}_{5}z\right)}{\text{ln}\left(10\right)}.\end{array}$(57)

A numerical expression for the distance modulus is derived using the parameters of Table 2

$\mu \left(z\right)=25+\frac{5\ln \left(\frac{32.5588\left(1+z\right)\left(6.6977z^3-50.198z^2+208.302z\right)}{{\text{e}}^{1.0041\ln \left(2\right)}{\text{e}}^{-0.00703688\sqrt{3}\left(2\pi +3\sqrt{3}\right)}}\right)}{\ln \left(10\right)}.$(58)

The percent error of the above distance modulus in the interval $0.01<z<2.4$ is given in Figure 5 .

In the case of the distance modulus, the merit function ${\chi }^2$ is

${\chi }^2=\underset{i=1}{\overset{N}{{\displaystyle \sum }}}{\left[\frac{{\left(m-M\right)}_i-\left(m-M\right){\left(z_i\right)}_{th}}{{\sigma }_i}\right]}^2,$(59)

where $N$ is the number of SNs, ${\left(m-M\right)}_i$ is the observed distance modulus evaluated at a redshift of $z_i$, ${\sigma }_i$ is the error in the observed distance modulus evaluated at $z_i$, and $\left(m-M\right){\left(z_i\right)}_{th}$ is the theoretical distance modulus evaluated at $z_i$, see formula (15.5.5) in [20] . The reduced merit function ${\chi }_{red}^2$ is

${\chi }_{red}^2={\chi }^2/NF,$(60)

where $NF=N-k$ 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 [20] :

$Q=1-\text{GAMMQ}\left(\frac{N-k}{2},\frac{{\chi }^2}{2}\right),$(61)

where GAMMQ is a subroutine for the incomplete gamma function. The Akaike information criterion (AIC), see [21] , is defined by

$\text{AIC}=2k-2\ln \left(L\right),$(62)

where $L$ is the likelihood function. We assume a Gaussian distribution for the errors; then the likelihood function can be derived from the ${\chi }^2$ statistic $L\propto \text{exp}\left(-\frac{{\chi }^2}{2}\right)$ where ${\chi }^2$ has been computed by Equation chi-square), see [22] [23] . Now the AIC becomes

$\text{AIC}=2k+{\chi }^2.$(63)

The goodness of the approximation in evaluating a physical variable $p$ is evaluated by the percentage error $\delta$

$\delta =\frac{\left|p-p_{approx}\right|}{p}\times 100,$(64)

where $p_{approx}$ is an approximation of $p$.

The cosmological parameters of the two cosmologies of reference are given in Table 1 ; the six models of the $f\left(Q,L_m\right)$ gravity model are given in Table 2 .

Figure 6 shows the best fit in the ΛCDM cosmology for the Pantheon+ compilation.

The different behaviors of some of the cosmologies here analysed is evident when high values of the redshift are considered, for example the interval [2, 2.4], see Figure 7 .

We processed the SDSS Photometric Catalogue DR 12, see [24] , which contains 10450256 galaxies (elliptical + spiral) with redshifts. In the following, we will use the generic term “galaxy” without distinguishing between the two types, elliptical and spiral.

We used the Schechter function, see [25] , as a luminosity function (LF) for galaxies

$\Phi \left(L\right)\text{d}L=\left(\frac{{\Phi }^{*}}{L^{*}}\right){\left(\frac{L}{L^{*}}\right)}^{\alpha }\exp \left(-\frac{L}{L^{*}}\right)\text{d}L,$(65)

here $\alpha$ sets the slope for low values of luminosity, $L$, $L^{\text{*}}$ is the characteristic luminosity and ${\Phi }^{*}$ is the normalisation. The equivalent distribution in absolute magnitude is

$\Phi \left(M\right)\text{d}M=0.921{\Phi }^{*}{10}^{0.4\left(\alpha +1\right)\left(M^{*}-M\right)}\exp \left(-{10}^{0.4\left(M^{*}-M\right)}\right)\text{d}M,$(66)

where $M^{\text{*}}$ is the characteristic magnitude as derived from the data. The scaling with $h$ is $M^{*}-5{\log }_{10}h$ and ${\Phi }^{*}{h}^3\left[{\text{Mpc}}^{-3}\right]$.

The flux density or irradiance, $f$, is

$f=\frac{L}{4\pi r^2},$(67)

where $r$ is the luminosity distance. The flux density in SI units is expressed in Watt/m² and in astrophysical units in $L_{\odot }/{\text{Mpc}}^2$. The luminosity distance is

$r=D_{L22},$(68)

see Equation (32) and the relationship between $\text{d}r$ and $\text{d}z$ is

$\frac{\text{d}r}{\text{d}z}=\frac{\partial D_{\text{L}22}}{\partial z},$(69)

see Equation (34).

The joint distribution in z and f for the number of galaxies, see formula (5.133) in [9] or formula (1.104) in [26] or formula (1.117) in [27] , is

$\frac{\text{d}N}{\text{d}\Omega \text{d}z\text{d}f}=\frac{1}{4\pi }{\displaystyle {\int }_0^{\infty }4\pi r^2\text{d}r}\Phi \left(L;L^{*},\sigma \right)\delta \left(z-\left(z_{22}\right)\right)\delta \left(f-\frac{L}{4\pi r^2}\right),$(70)

where $\delta$ is the Dirac delta function, $\Phi \left(L;L^{*},\sigma \right)$ has been defined in Equation (65) and $z_{22}$ has been defined in Equation (33). An explicit version is

$\frac{\text{d}N}{\text{d}\Omega \text{d}z\text{d}f}=\frac{NN}{DD},$(71)

where

$\begin{array}{c}NN=567218{\left(-0.138330+3104.76z+2838.17z^2\right)}^4\\ \times {\left(\frac{4\pi f{\left(-0.138330+\left(3104.76+2838.17z\right)z\right)}^2}{{\left(0.745533+\left(0.203583+0.0047068z\right)z\right)}^2{L}^{*}}\right)}^{\alpha }\\ \times {\text{e}}^{-\frac{4\pi f{\left(-0.138330+\left(3104.76+2838.17z\right)z\right)}^2}{{\left(0.74553+\left(0.203583+0.00470684z\right)z\right)}^2{L}^{*}}}\times {\Phi }^{*}\left(4231.9z+563.19z^2+2314.73\right),\end{array}$(72)

and

$DD={\left(0.745533+0.203583z+0.00470684z^2\right)}^4{L}^{*}{\left(z+39.2133\right)}^2{\left(z+4.03927\right)}^2.$(73)

The total number of galaxies comprised between a minimum value of flux, $f_{\min }$, and a maximum value of flux, $f_{\max }$, can be computed through the integral

$\frac{\text{d}N}{\text{d}\Omega \text{d}z}={\displaystyle {\int }_{f_{\min }}^{f_{\max }}\frac{NN}{DD}\text{d}f}.$(74)

This integral has a complicated analytical solution in terms of the Whittaker function $M_{\kappa ,\mu }\left(z\right)$, see [18] . Figure 8 presents all of the galaxies of SDSS DR12 and also the theoretical curve.