The name dark energy started to be used by  in order to explain both the expansion and both the acceleration of the universe. In a few years the dark energy was widely used as a cosmological model to be tested. Many review papers have been written; we select among others a general review by  and a theoretical review by  . The term wCDM has been introduced to classify the case of constant equation of state and we will use in the following wzCDM to classify the variable equation of state. The Cardassian cosmology started with  and was introduced in order to model both the expansion and the acceleration of the universe, the name from a humanoid race in Star Trek. As an example  derived the cosmological parameters for the original Cardassian expansion and the modified polytropic Cardassian expansion. The cosmological theories can be tested on the samples of Supernova (SN) of type Ia. The first sample to be used to derive the cosmological parameters contained 7 SNs, see  , the second one contained 34 SNs, see  and the third one contained 42 SNs, see  . The above historical samples allowed to derive the cosmological parameters for the expanding and accelerating universe. At the moment of writing the astronomical research is focused on value of the distance modulus versus the redshift: the Union 2.1 compilation contains 580 SNs, see  , and the joint light-curve analysis (JLA) contains 740 SNs, see  . The above observations can be done up to a limited value in redshift , we, therefore, speak of evaluation of the distance modulus at low redshift. This limited range can be extended up , the high redshift region, analyzing the Gamma-Ray Burst (GRB) and, as an example,  has derived the distance modulus for 59 calibrated high-redshift GRBs, the so-called “Hymnium” GRBs sample. This paper reviews in Section 2.1. The ΛCDM cosmology evaluates the basic integral of wCDM cosmology in Section 3, introduces a Taylor expansion for the basic integral of wzCDM cosmology in Section 4 and analyzes the Cardassian model in Section 5. The parameters which characterize the three cosmologies are derived via the Levenberg-Marquardt method in Section 6.
This section reviews the ΛCDM cosmology and the adopted statistics.
2.1. The Standard Cosmology
In ΛCDM cosmology the Hubble distance is defined as
The first parameter is
where G is the Newtonian gravitational constant, is the Hubble constant and is the mass density at the present time. The second parameter is
where is the cosmological constant, see  . These two parameters are connected with the curvature by
The comoving distance, , is
where is the “Hubble function”
In the case of , we have the flat case.
2.2. The Statistics
The adopted statistical parameters are the percent error, , between theoretical value and approximated value, the merit function evaluated as
where and represent the observed value and its error at position i and the theoretical value at position i, the reduced merit function , the Akaike information criterion (AIC), the number of degrees of freedom where n is the number of bins and k is the number of parameters and the goodness of the fit as expressed by the probability Q.
3. Constant Equation of State
In dark matter cosmology, wCDM, the Hubble radius is
where w parametrizes the dark energy and is constant, see Equation (3.4) in  or Equation (18) in  for the luminosity distance.
In flat cosmology
and the Hubble radius becomes
The indefinite integral in the variable z of the above Hubble radius, , is
3.1. The Analytical Solution
In order to solve the indefinite integral we perform a change of variable
The indefinite integral is
where is the regularized hypergeometric function, see Appendix B. This dependence of the above integral upon the hypergeometric function has been recognized but not developed by  .
We now return to the variable z, the redshift, and the indefinite integral becomes
We denote by the definite integral
3.2. The Taylor Expansion
We evaluate the integrand of the integral (11) with a first series expansion, about , denoted by I and a second series expansion, , about , denoted by . The order of expansion for the two series is 7. The integration of in z is denoted by and gives
and the coefficients, , are reported in Appendix A. The integral, of the second Taylor expansion about , is complicated and we limit ourselves to order 2, , see Appendix A. The two definite integrals, and are
The percent error, , between the analytical integral F and the two approximations, and is evaluated as
The above value in z will, therefore, be the boundary between region I and region II for the Taylor approximation of the definite integral
Table 1. Numerical values from the Union 2.1 compilation of , and Q, where k stands for the number of parameters.
Figure 1. Numerical values of (full red line) and (dashed blue line) as function of the redshift, parameters as in Table 1.
4. Variable Equation of State
The dark energy as function of the redshift is assumed to be
where and are two parameters to be fixed by the fit. The Hubble radius in wzCDM cosmology is
which is the same as Equation (20) in  . The above integral does not yet have an analytical expression and we evaluate the integrand with a first series expansion about and a second series expansion about . Also here the order of the two series expansion is 7. The integration in z is denoted by and gives
and the first five coefficients, , are reported in Appendix C. The integral, of the second Taylor expansion about is complicated and we limit ourselves to order 2, , see Appendix C. The two definite integrals, and are
Finally the definite integral, , is
The above definite integral can also be evaluated in a numerical way,
5. Cardassian Cosmology
In flat Cardassian cosmology the Hubble radius is
where n is a variable parameter, n = 0 means ΛCDM cosmology, see Equation (17) in  . The indefinite integral in the variable z of the above Hubble radius, , is
Also here in order to solve the indefinite integral we perform a change of variable
The indefinite integral is
where is the regularized hypergeometric function. We now return to the original variable z as function of z which is
We denote by the definite integral
6. The Distance Modulus
The luminosity distance, , for wCDM cosmology in the case of the analytical solution is
where is given by Equation (15) and in the case of the Taylor approximation is
where is given by Equation (21). The distance modulus in the case of the analytical solution for wCDM is
and in the case of the Taylor approximation
In the case of variable equation of state, wzCDM, the numerical luminosity distance is
where is the definite numerical integral and the Taylor approximation for the luminosity distance is
where is given by Equation (27). In wzCDM, the numerical distance modulus is
and the Taylor approximated distance modulus is
In the case of Cardassian cosmology the luminosity distance is
where is given by Equation (33) and the distance modulus is
The cosmological parameters unknown are three, and w, in the case of wCDM and four, and , in the case of wzCDM. In flat Cardassian cosmology the number of parameters is three, and n. In the presence of a given sample for the distance modulus, we can map the chi-square as given by Formula (7), see Figure 2 in the case of wCDM with hypergeometric solution. The above cosmological parameters are obtained by a fit of the astronomical data for the distance modulus of SNs via the Levenberg-Marquardt method (subroutine MRQMIN in  ) which minimizes the chi-square as given by Formula (7). Table 1 presents the above cosmological parameters for the Union 2.1 compilation of SNs and Figure 3 reports the best fit. As a practical example of the utility of the cosmological parameters determination, we report the distance modulus in an explicit form for the Union 2.1 compilation in wCDM.
Figure 2. Map of the in wCDM cosmology when .
Figure 3. Hubble diagram for the Union 2.1 compilation. The solid line represents the best fit for the exact distance modulus in wCDM cosmology as represented by Equation (36). Parameters as in third line of Table 1; Union 2.1 compilation.
And in flat Cardassian cosmology
The extension of the Hubble diagram to the GRBs, as an example, has been implemented in      .
Figure 4. Hubble diagram for the JLA compilation. The solid line represents the best fit for the exact distance modulus in wCDM cosmology as represented by Equation (36). Parameters as in the third line of Table 2; JLA compilation.
Figure 5. Hubble diagram for the Union 2.1 compilation + the “Hymnium” GRBs sample. The solid line represents the best fit for the exact distance modulus in wCDM cosmology as represented by Equation (36). Parameters as in second line of Table 3.
Table 2. Numerical values for the JLA compilation of , and Q, where k stands for the number of parameters.
Table 3. Numerical values from the Union 2.1 compilation + the “Hymnium” GRBs sample of , and Q, where k stands for the number of parameters.
Constant equation of state
In the case of wCDM cosmology, we found a new analytical expression for the Hubble distance in terms of the hypergeometric function, see Equation (13). As a consequence an analytical expression for the luminosity distance and the distance modulus is derived. Two approximate Taylor expansions for the Hubble distance about and of order 7 are also derived. The derivation of the value of w, and , here considered as a parameter to be found, is given for the Union 2.1 compilation, the JLA compilation and the Union 2.1 compilation plus the “Hymnium” GRBs sample, see Tables 1-3. As an example, in the case of the Union 2.1 compilation, we have derived , and .
Variable equation of state
In the case of wzCDM cosmology the Hubble distance, Equation (23) is evaluated numerically and with a Taylor expansion of order 7, see Equation (24). The four parameters , , and are reported in Tables 1-3. As an example, in the case of the Union 2.1 compilation, we have found , , , and .
The inclusion of the “Hymnium” GRBs sample allows to extend the calibration of the distance modulus up to (see Table 3). As an example, the Union 2.1 compilation + the “Hymnium” GRBs sample gives , , , and .
A new solution for the Hubble radius for Cardassian cosmology is presented in terms of the hypergeometric function, see Equation (reficardz). As an example, in the case of the Union 2.1 compilation, we have derived , and .
A. Taylor Expansion When W Is Constant
The coefficients of the Taylor expansion of about
The integral of the Taylor expansion of order 2 about is
B. The Hypergeometric Function
The regularized hypergeometric function, , as defined by the Gauss series, is
where , is the Pochhammer symbol
is the Gamma function defined as
z is a complex variable defined on the disk that should not be confused with the redshift, see      . The following relationship
connect the the hypergeometric function with x in (−1, 1) to one with x in , see more details in  .
C. Taylor Expansion When W Is Variable
The coefficients of the Taylor expansion of about
The integral of the Taylor expansion of order 2 about in the case wzLCDM cosmology