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in which \(\mathcal M\) is a nuisance parameter corresponding to some combination of the absolute magnitude M and \(H_0\). We refer to [78,79,80] for technical details (see also e.g. [81]). Since \(H_0\) is absorbed into \(\mathcal M\) in the analytic marginalization, the Pantheon SNIa sample is free of the Hubble constant \(H_0\).
We further consider the observational data from the baryon acoustic oscillation (BAO). Note that there exist many kinds of BAO data in the literature, such as \(D_V(z)\), \(d_z\equiv r_s(z_d)/D_V(z)\), \(D_A(z)/r_d\), \(D_M(z)/r_d\), \(H(z)\,r_s(z_d)\) and A. However, in the former ones, they will introduce one or more extra model parameters, for instance \(H_0\), and/or \(\Omega _b h^2\). Since the \(f\sigma _8\) data, the SNIa data, the cosmography for \(D_L\), and other data are all free of \(H_0\) and \(\Omega _b h^2\), we choose to avoid introducing extra model parameters here. Thus, in this work, we use the BAO data only in the form of (see e.g. [82, 83])
On the other hand, the free parameter \(\sigma _8,\,0\) cannot be well constrained by using the \(f\sigma _8\) data and the observations of the expansion history. Fortunately, in the literature there are many observational data of the combination \(S_8\equiv \sigma _8,\,0(\Omega _m0/0.3)^0.5\) from the cosmic shear observations [84], which can be used to constrain both the free parameters \(\sigma _8,\,0\) and \(\Omega _m0\). Here, we consider the ten data points given in Table 1. The corresponding \(\chi ^2_S_8= \sum _i \, (S_8,\,obs,\,i-S_8,\,\mathrmmod,\,i)^2 / \sigma _S_8,\,i^2\,\). Note that if the upper and the lower uncertainties of the data are not equal, we choose the bigger one as \(\sigma _S_8,\,i\) conservatively.
In fact, there are other kinds of observational data in the literature. However, we do not use them here, to avoid introducing extra model parameters, as mentioned above. For instance, if we want to use the 51 observation H(z) data compiled in [95] (the largest sample by now to our best knowledge), an extra free parameter \(H_0\) is necessary. So, we give up. On the other hand, since the usual cosmography cannot work well at very high redshift, we also do not consider the observational data from cosmic microwave background (CMB) at redshift \(z\sim 1090\). Otherwise, the cosmographic parameters should be fine-tuned. However, the Padé cosmography works well at very high redshift, and hence we can use the CMB data in this case (see Sect. 4).
Since the constraints become loose if the number of free parameters increases, we only consider the cosmography up to third order. Thus, the dimensionless luminosity distance \(D_L=H_0 d_L/c\) is given by
where the redshift of the recombination \(z_*=1089.92\) from the Planck 2018 data [94], and the angular diameter distance \(d_A\) is related to the luminosity distance \(d_L\) through \(d_A=d_L(1+z)^-2\) (see e.g. the textbooks [28, 29]). Here we adopt the value \(R_obs=1.7502\pm 0.0046\) [108] derived from the Planck 2018 data. Thus, the corresponding \(\chi ^2\) from the latest CMB data is given by \(\chi ^2_R=(R_\mathrmmod-R_obs)^2/\sigma _R^2\). Although the number of data points \(\mathcal N\) and the number of free parameters \(\kappa \) both increase by 1, the degree of freedom \(dof=\mathcalN-\kappa \) is unchanged in this case. It is worth noting that the acoustic scale \(l_A\), and \(\Omega _b h^2\), the scalar spectral index \(n_s\) are commonly used with the shift parameter R in the literature, but they will introduce extra model parameters as mentioned above, and hence we do not use them here.
It is of interest to compare the 9 cases considered here. We adopt several goodness-of-fit criteria used extensively in the literature to this end, such as \(\chi ^2_min/dof\), \(P(\chi ^2>\chi ^2_min)\) (see e.g. [109, 110]), Bayesian Information Criterion (BIC) [111] and Akaike Information Criterion (AIC) [112], where the degree of freedom \(dof=\mathcalN-\kappa \), while \(\mathcal N\) and \(\kappa \) are the number of data points and the number of free model parameters, respectively. The BIC is defined by [111]
It is worth noting that throughout this work, we always consider the growth index \(\gamma \) as a Taylor series with respect to z or y, namely \(\gamma (z)=\gamma _0+\gamma _1\,z+\gamma _2\,z^2+\cdots \), or \(\gamma (y)=\gamma _0+\gamma _1\,y+\gamma _2\,y^2+\cdots \). However, in Sect. 4, we parameterize the dimensionless luminosity distance \(D_L\) by using the Padé approximant, and hence it can still work well at very high redshift \(z\sim 1090\). Obviously, it is better to also parameterize the growth index \(\gamma (z)\) by using the Padé approximant (we thank the referee for pointing out this issue). But the cost is expensive to do this. If we want to catch the arched structure in \(\gamma (z)\), at least a Padé approximant of order \((2,\,2)\) is needed, which has 5 free parameters (n.b. Eq. (16)), and almost double the number of free parameters in a 2nd order Taylor series. So, in the P-2 case, the total number of free model parameters will be ten. It will consume significantly more computation power and time, but the corresponding constraints will be very loose. Therefore, we choose not to do this at a great cost. But one should be aware of the possible artificial results from this choice. For example, \(\gamma (z)=\gamma _0+\gamma _1\,z+\gamma _2\,z^2\) will diverge at \(z\sim 1090\), and hence the values of \(\gamma _1\) and \(\gamma _2\) tend to be zero to fit the high-z CMB data in the P-1, P-2 cases (we thank the referee for pointing out this issue). 2ff7e9595c
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