We assume that the frequency $\nu$ of a photon which travels through intergalactic space decreases according to the following ordinary differential equation (ODE), called the Beer-Lambert law, after [3] [8] [9] ,

$\frac{\text{d}}{\text{d}x}\nu \left(x\right)=-k{n}_e\nu \left(x\right),$(6)

where $n_e$ is the number density of matter in 1/m³ and k an attenuation coefficient in m². This ODE is solved while assuming the initial condition $\nu \left(0\right)={\nu }_0$

$\nu \left(x\right)={\nu }_0{\text{e}}^{-k{n}_{e}x}.$(7)

We now define the redshift as a function of the wavelength $\lambda$ and the initial wavelength ${\lambda }_0$ by

$z=\frac{\lambda -{\lambda }_0}{{\lambda }_0}.$(8)

Making use of

$\nu \lambda =c,$(9)

where c is the speed of light, we obtain

$z={\text{e}}^{k{n}_{e}x}-1,$(10)

or

$\text{ln}\left(z+1\right)=k{n}_{e}x.$(11)

A Taylor expansion around $z=0$ gives

$z=k{n}_{e}x,$(12)

and the expansion velocity, v, in the radial Doppler framework can be obtained by multiplying both sides by c,

$v=k{n}_{e}xc.$(13)

The above equation is expressed in terms of Hubble’s Law [1]

$v=H_{0}x\frac{\text{km}}{\text{Mpc}},$(14)

which gives

$H_0=k{n}_{e}c.$(15)

The distance modulus for tired light is

$\left(m-M\right)=25+\frac{5\ln \left(\frac{\ln \left(z+1\right)c}{H_0}\right)}{\ln \left(10\right)}.$(16)

We assume that the frequency $\nu$ of a photon decreases according to the following non-linear law

$\frac{\text{d}}{\text{d}x}\nu \left(x\right)=-a{n}_e\nu {\left(x\right)}^{\alpha },$(17)

where $n_e$ is the number density of matter in 1/m³ and a the attenuation coefficient in ${\text{Hz}}^{1-\alpha }\cdot {\text{m}}^2$. We call the above ODE the generalized tired light (GTL) model. Imposing the initial condition $\nu \left(0\right)={\nu }_0$, the solution is

$\nu \text{}\left(x\right)=\frac{1}{{\left(a{n}_{e}x\alpha -a{n}_{e}x+{\text{e}}^{-\alpha \ln \left({\nu }_0\right)}{\nu }_0\right)}^{\frac{1}{-1+\alpha }}}.$(18)

We now continue inserting

$a=\frac{{\nu }_0^{1-\alpha }{H}_0}{c{n}_e},$(19)

and as a consequence the redshift is

$z=c^{-\frac{1}{-1+\alpha }}{\left(c+\left(-1+\alpha \right)x{H}_0\right)}^{\frac{1}{-1+\alpha }}-1.$(20)

The inversion of the above formula gives

$x=-\frac{\left(-{\left(z+1\right)}^{\alpha }+z+1\right)c}{H_0\left(-1+\alpha \right)\left(z+1\right)},$(21)

and the distance modulus in GTL is

$\left(m-M\right)=25+\frac{5\ln \left(-\frac{\left(-{\left(z+1\right)}^{\alpha }+z+1\right)c}{H_0\left(-1+\alpha \right)\left(z+1\right)}\right)}{\ln \left(10\right)}.$(22)

We now introduce a modification to the Bouguer-Lambert-Beer as suggested in Equation (1) of [10]

$A=\text{log}\left(\frac{I_{in}}{I_{out}}\right)=ϵC^{\alpha }{x}^{\beta },$(23)

where A is the apparent absorbance, $I_{in}$ the input light intensity, $I_{out}$ the output light intensity transmitted through the sample, $ϵ$ the effective specific absorbance, C the sample concentration, $\alpha$ a positive correction coefficient of the concentration, x the path length and $\beta$ a positive correction coefficient of the path length. The frequency decreases in accordance with the following law

$\nu ={\nu }_0{\text{e}}^{-ϵC^{\alpha }{x}^{\beta }}.$(24)

We now continue posing

$ϵ=\frac{H_0}{C^{\alpha }c},$(25)

and the equation which defines the redshift is

$\text{ln}\left(z+1\right)=\frac{H_0{x}^{\beta }}{c}.$(26)

The above equation can be inverted in order to give the distance

$x=\text{ln}{\left(z+1\right)}^{\frac{1}{\beta }}{c}^{\frac{1}{\beta }}{H}_0^{-\frac{1}{\beta }},$(27)

and therefore the distance modulus for the spectroscopic model is

$\left(m-M\right)=\text{ln}{\left(z+1\right)}^{\frac{1}{\beta }}{c}^{\frac{1}{\beta }}{H}_0^{-\frac{1}{\beta }}.$(28)

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

$D_{\text{H}}\equiv \frac{c}{H_0},$(29)

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

${\Omega }_{\text{M}}=\frac{8\pi G{\rho }_0}{3H_0^2},$(30)

where G is the Newtonian gravitational constant and ${\rho }_0$ is the mass density at the present time. We now introduce another parameter, ${\Omega }_{\Lambda }$:

${\Omega }_{\text{Λ}}\equiv \frac{\text{Λ}{c}^2}{3H_0^2},$(31)

where Λ is the cosmological constant, see [11] . Once ${\Omega }_{\Lambda }$ and $H_0$ are found the numerical value of the cosmological constant is derived, $\Lambda \approx 1.2\frac{1}{{\text{m}}^2}$.

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

${\Omega }_{\text{M}}+{\Omega }_{\Lambda }+{\Omega }_K=1.$(32)

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

$D_{\text{C}}=D_{\text{H}}{\displaystyle {\int }_0^z\frac{\text{d}{z}^{\prime }}{E\left(z^{\prime }\right)}},$(33)

where $E\left(z\right)$ is the “Hubble function”

$E\left(z\right)=\sqrt{{\Omega }_{\text{M}}{\left(1+z\right)}^3+{\Omega }_K{\left(1+z\right)}^2+{\Omega }_{\Lambda }}.$(34)

Details on how to derive the distance modulus in ΛCDM cosmology through the Padé approximant can be found in formula (11) in [12] .

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

${\chi }^2=\underset{i=1}{\overset{N}{{\displaystyle \sum }}}{\left[\frac{{\left(m-M\right)}_i-\left(m-M\right){(z_i)}_{th}}{{\sigma }_i}\right]}^2,$(35)

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 [13] . The reduced merit function ${\chi }_{red}^2$ is

${\chi }_{red}^2={\chi }^2/NF,$(36)

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 [13] :

$Q=1-\text{GAMMQ}\left(\frac{N-k}{2},\frac{{\chi }^2}{2}\right),$(37)

where GAMMQ is a subroutine for the incomplete gamma function. The Akaike information criterion (AIC), see [14] , is defined by

$\text{AIC}=2k-2\ln \left(L\right),$(38)

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 (35), see [15] [16] . Now the AIC becomes

$\text{AIC}=2k+{\chi }^2.$(39)

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

$\delta =\frac{\left|p-p_{approx}\right|}{p}\times 100,$(40)

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

The first astronomical test we perform is on the 580 SNs of the Union 2.1 compilation, see [17] , which is available at http://supernova.lbl.gov/Union/figures/SCPUnion2.1_mu_vs_z.txt : in this compilation a calibrated distance versus redshift is provided. The cosmological parameters are given in Table 1 .

Figure 1 gives the best fit in the ΛCDM cosmology for the Union 2.1 compilation.

Figure 2 compares the ΛCDM cosmology, GTL, and the spectroscopic model for the Union 2.1 compilation. The goodness of the distance modulus for the generalized tired light versus ΛCDM cosmology as function of the redshift can be evaluated with formula (40), see Figure 3 .

The second test we perform is on the the joint light-curve analysis (JLA), which contains 740 SNs [18] with data available on CDS at http://cdsweb.u-strasbg.fr/ . The cosmological parameters with the JLA compilation are given in Table 2 .

Figure 4 gives the best fit in the ΛCDM cosmology for the JLA compilation.

Figure 5 compares the ΛCDM cosmology, GTL, and the spectroscopic model for the JLA compilation. The goodness of the distance modulus for GTL versus the ΛCDM cosmology as a function of the redshift is presented in Figure 6 .

The third test is performed on the Pantheon sample of 1048 SN Ia [19] [20] with calibrated data available at https://archive.stsci.edu/prepds/ps1cosmo/jones_datatable.html , see Table 3 .

Figure 7 reports the best fit in the ΛCDM cosmology for the Pantheon compilation.

Figure 8 compares the ΛCDM cosmology, GTL, and the spectroscopic model for the Pantheon compilation. The goodness of the distance modulus for GTL versus the ΛCDM cosmology as a function of the redshift is given in Figure 9 .