A tomographic test of cosmic anisotropy with the recently-released quasar sample

We test the cosmic anisotropy in the dipole-modulated Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document}CDM model and Finslerian cosmological model with the recently-released quasar sample. Based on the redshift tomographic method, the quasar sample is divided into two subsets z≤zcut\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z \le z_{cut}$$\end{document} and z>zcut\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z > z_{cut}$$\end{document} by different cutting redshifts. The dipole amplitudes of the two cosmological models from the subsets z≤zcut\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z \le z_{cut}$$\end{document} are very weak. We find that quasars at a higher redshift range may provide more detailed information about the dipole amplitude AD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_D$$\end{document}. The dipole directions of each cosmological model from the subsets z≤1.1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z \le 1.1$$\end{document} and z>1.1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z > 1.1$$\end{document} are deviated by 1σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\sigma $$\end{document} level. The Pantheon sample is combined with the two subsets. The dipole amplitude from the two combined datasets is also very weak. In the dipole-modulated Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document}CDM model, the dipole direction from the combined dataset quasar at z≤1.1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z \le 1.1$$\end{document} and Pantheon sample is far away from the one given by the Pantheon sample. In the Finslerian cosmological model, the dipole directions from the two combined datasets are close to the one in the Pantheon sample.

As the most luminous and persistent energy source, quasars have a lot of potential in exploring our Universe. Risaliti and Lusso (et al.) have been attempting to test cosmological models with the non-linear relation between the UV and X-ray luminosity of quasars since 2015 [77][78][79]. More recently, Lusso and Risaliti (et al.) released a new catalogue of optically selected quasars [79]. The 2020 compilation of quasars contains 2421 quasars that over the redshift range 0.009≤ z ≤7.5413. Due to the host-galaxy contamination, the 2500 Å UV fluxes of quasars at the redshift range z < 0.7 are less reliable. So the 2023 quasars at the redshift range z > 0.7 and 15 local quasars with higher quality at the redshift range z < 0.7 are used in the test of cosmic anisotropy.
In this work, we will analyze the quasar sample by using a redshift tomographic method. The quasar sample is divided into two subsets by different cutting redshifts. We will use the subsets to constrain the dipole-modulated CDM model and Finslerian cosmological model. The Pantheon sample is combined with the subsets in our analysis. The rest of this paper is organized as follows. In Sect. 2, we briefly introduce the quasar sample and the two cosmological models. We show our results in Sect. 3. Finally, discussions and conclusions are given in Sect. 4.

The quasar sample
The recently-released quasar compilation used in our work was presented by Lusso et al. [79]. The 2020 compilation consists of 2421 quasars over the redshift range 0.009≤ z ≤7.5413. Because of the host-galaxy contamination at low redshift, the 2500 Å UV fluxes of quasars at the redshift range z < 0.7 are less reliable. We choose the quasars at the redshift range z > 0.7 and 15 local quasars with higher quality at the

The methodology
The non-linear relation between UV and X-ray luminosity of quasars can be parameterized as [111] log where log denotes log 10 . β and γ are two free parameters determined by the quasar sample. L X is the monochromatic luminosity at 2 keV and L UV is the luminosity at 2500 Å. The luminosities and fluxes of quasars are related by the luminosity distance, so we rewrite the Eq. (1) as where F X and F U V are the X-ray and UV fluxes of quasars, respectively. The luminosity distance d L has the form where z is the redshift, c is the speed of light, and H 0 is the Hubble constant. The expression of E(z) varies with cosmological models.
In the Finslerian cosmological model, the scale factor a takes the form [112] where A D is the dipole amplitude. θ is the angle between the preferred direction in the Finslerian cosmological model and the positions of quasars. Correspondingly, E(z) in the Finslerian cosmological model takes the form The dipole fitting (DF) method proposed by Mariano and Perivolaropoulos [10] is widely used to test the cosmic anisotropy. In this paper, we consider a dipole modulation to the theoretical distance modulus in the CDM model, namelỹ where A D is the dipole amplitude.n andp are the dipole direction and the unit vector pointing to a quasar, respectively. In the galactic coordinate system, the dipole directionn takes the form whereî,ĵ , andk are the unit vectors along the axis in the Cartesian coordinates system. l is the galactic longitude and b is the galactic latitude. The unit vectorp i pointing to the position of ith quasar can be written aŝ Considering the expression of distance modulus in the CDM model, We can write the dipole-modulated distance modulus as Plugging Eq. (6) into Eq. (10), we obtaiñ We combine Eqs. (11) and (2) and obtain the theoretical Xray flux in the dipole-modulated CDM model, To explore the whole cosmological parameter space, we employ the likelihood function where s 2 i = σ 2 i + δ 2 . σ i is the error of the observed flux F obs X,i and δ is the global intrinsic dispersion. F th X,i represents the theoretical flux at the redshift z i in the dipole-modulated CDM model or Finslerian cosmological model.

Results
In this paper, we use the Markov chain Monte Carlo (MCMC) method implemented in emcee 1 [113] to explore the whole parameters space. Emcee is an Affine Invariant Markov chain Monte Carlo Ensemble sampler, which is widely used to constrain parameters in astrophysics and cosmology. In the dipole-modulated CDM model and Finslerian cosmological model, the free parameters are the matter density m , the dipole amplitude A D , the dipole direction (l, b), and the three parameters β, γ , and δ related to the quasar sample. The flat priors of the free parameters are m ∼ [0, 1], β ∼ [5,16] We use a tomographic method to study the evolution of dipole parameters with redshift. The cutting redshifts are We divide the quasar sample into two subsets, i.e., z ≤ z cut and z > z cut by different cutting redshifts z cut . The results of the dipole-modulated CDM model and Finslerian cosmological model are summarized in Tables 1 and 2.
From Tables 1 and 2, we can see that the results of the two cosmological models are similar except for the dipole amplitude A D . The results of parameters β, γ , and δ from most subsets z ≤ z cut are consistent within statistical uncertainties, respectively. The constraints of matter density m are very weak from all the subsets. In the Finslerian cosmological model, the results of matter density m from the subsets z ≤ 1.0, 1.1 are consistent with ones from the Pantheon sample and CMB, but with large uncertainties.
The dipole amplitudes A D of the two cosmological models from the subsets z ≤ z cut are very weak. The upper limit of dipole amplitude A D in the Finslerian cosmological model is about a hundred times larger than that in the CDM model. Though the constraints of matter density m are very weak from the subsets z > z cut , the results of the dipole amplitude A D show that quasars at a higher redshift range may provide more detailed information about the dipole amplitude A D .
Because of the large uncertainties, the results of dipole parameter l from all the subsets z ≤ z cut are consistent. But the central values of parameter l from some subsets have a sudden change. For the dipole-modulated CDM are more than 1σ level. The central value of parameter l has a sudden change in some subsets may be due to the big deviations of parameters β, γ , and δ.
The dipole directions derived from two subsets divided by different cutting redshifts are consistent within statistical uncertainties except for the subsets z ≤ 1.1 and z > 1.1. For brevity, the subset z ≤ 1.1 is shortened to Q1 and the subset z > 1.1 is shortened to Q2. In Fig. 2  . In each cosmological model, the difference between the two directions from subsets Q1 and Q2 is about 1σ level. We summarize the results from the subsets Q1 and Q2 in Table 3 for clarity. We show the marginalized posterior distribution for each cosmological model in Figs. 3 and 4.
From Table 3, we can see that the results of the two cosmological models are similar except for the parameter A D . The constraints of matter density m in the two cosmological models are very weak. For the parameters β and γ in each cosmological model, the results from subsets Q1 and Q2 are consistent within 1σ uncertainties, respectively. For the parameter δ in each cosmological model, the differences between the results from subset Q1 and Q2 are about 2σ level.
In the dipole-modulated CDM model, the dipole amplitude A D is constrained as A D < 1.632 × 10 −3 at 95% CL with subset Q1, and the dipole amplitude A D is 0.435 +0.264 −0.222 × 10 −3 with subset Q2. In the Finslerian cosmological model, the dipole amplitude A D is constrained as A D < 0.273 at 95% CL with subsets Q1, and the dipole amplitude A D is 0.040 +0.024 −0.020 with subsets Q2. The dipole amplitude A D in the Finslerian cosmological model is about a hundred times larger than that in the CDM model.
The Pantheon sample is combined with the two subsets Q1 and Q2 to test the cosmological models. In Fig. 2, We show the distributions of Pantheon sample in the galactic coordinate system. For brevity, the dataset Q1 + Pantheon is shortened to Q1-P and the dataset Q2 + Pantheon is short-  Table 3. In Figs. 3 and 4, we show the marginalized posterior distribution for each cosmological model. In Table 3, we show the 68% CL constraints on the parameters in each cosmological model. If the upper or lower limit of a parameter exists, we show the 95% CL limit of the parameter.
In the dipole-modulated CDM model, the dataset Q1-P gives a dipole direction points to It is necessary to check whether the inhomogeneous distributions of quasars and SNe Ia have some influence on the Table 3 The 68% confidence level constraints on the parameters in the dipole-modulated CDM model and Finslerian cosmological model. The datasets are Q1, Q2, Q1-P, and Q2-P. If the upper or lower limit of a parameter exists, we show the 95% confidence level of the parameter   In the end, we make some comparisons between the dipole directions shown in Table 3 with preferred directions derived from other samples. The preferred directions derived from different samples are summarized in Table 4 and shown in Fig. 6. Most of the preferred directions derived from different samples are located in a relatively small part of the south galactic hemisphere and they are consistent within sta-tistical uncertainties. The dipole direction derived from the dataset Q2-P in the dipole-modulated CDM model and the dipole directions derived from the datasets Q1-P and Q2-P in the Finslerian cosmological model are consistent with most preferred directions within statistical uncertainties. The dipole directions derived from the dataset Q2 in the dipole-modulated CDM model and Finslerian cosmological model are close to each other. The two directions are also close to the preferred directions derived from CMB octopole, quasar alignment, and the combination of Pantheon sample and 1421 quasars. The dipole directions derived from the dataset Q1 in the dipole-modulated CDM model and Finslerian cosmological model are also close to each other. There are only two preferred directions around them. One is the preferred direction derived from the Pantheon sample with the HC method and the other is the preferred direction derived from the combination of Union2 and 67 GRBs with the DF method. The dipole direction derived from the dataset Q1-P in the dipole-modulated CDM model is far away from the one derived from the dataset Q1-P in the Finslerian cosmological model. This may be due to the difference between the two cosmological models.

Discussions and conclusions
In this paper, we tested the cosmic anisotropy in the dipole-modulated CDM model and Finslerian cosmological model with the recently-released quasar sample by using a redshift tomographic method. The quasar sample is divided into two subsets by different cutting redshifts. The results of the two cosmological models are similar except for the dipole amplitude A D . The upper limit of dipole amplitude in the Finslerian cosmological model is about a hundred times larger than that in the CDM model. The constraints of matter density m are very weak from all the subsets.
In the Finslerian cosmological model, the results of matter density m from the subsets z ≤ 1.0, 1.1 are consistent with ones from the Pantheon sample and CMB, but with large uncertainties. The results of parameters β, γ , and δ from most subsets z ≤ z cut are consistent within statistical uncertainties, respectively. Though the constraints of matter density m are very weak from the subsets z > z cut , the results of the dipole amplitude A D show that quasars ) in the Pantheon sample [106]. In the Finslerian cosmological model, the dipole directions from the two combined datasets are close to each other. The two directions are also close to the one given by the Pantheon sample in the Finslerian cosmological model.
Our results also show that the inhomogeneous distributions of quasar and Pantheon samples in the sky have significant effects on dipole directions. We need more homogeneous distribution of datasets to search for a convincing cosmic anisotropy. In the future, the surveys such as J-PAS, Euclid, SKA, LSST, and eROSITA, will provide us with larger and more precise observational data of SNe Ia and quasars. Therefore the test of cosmological principle will be more rigorous by employing more precise datasets that have homogeneous distribution in the sky.