Testing cosmic anisotropy with Pantheon sample and quasars at high redshifts
AAstronomy & Astrophysics manuscript no. proof c (cid:13)
ESO 2020August 31, 2020
Testing cosmic anisotropy with Pantheon sample and quasars athigh redshifts
J. P. Hu , Y. Y. Wang , and F. Y. Wang , School of Astronomy and Space Science, Nanjing University, Nanjing 210093, Chinae-mail: [email protected] Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210093, ChinaReceived date; accepted date
ABSTRACT
In this paper, we investigate the cosmic anisotropy from the SN-Q sample, consisting of the Pantheon sample and quasars, by em-ploying the hemisphere comparison (HC) method and the dipole fitting (DF) method. Compared to the Pantheon sample, the newsample has a larger redshift range, a more homogeneous distribution, and a larger sample size. For the HC method, we find that themaximum anisotropy level is AL max = . ± .
026 in the direction ( l , b ) = (316 . ◦ + . − . , 4 . ◦ + . − . ). The magnitude of anisotropyis A = ( − + . − . ) × − and the corresponding preferred direction points toward ( l , b ) = (29 . ◦ + . − . , 71 . ◦ + . − . ) for the quasarsample from the DF method. The combined SN and quasar sample is consistent with the isotropy hypothesis. The distribution of thedataset might impact the preferred direction from the dipole results. The result is weakly dependent on the redshift from the redshifttomography analysis. There is no evidence of cosmic anisotropy in the SN-Q sample. Though some results obtained from the quasarsample are not consistent with the standard cosmological model, we still do not find any distinct evidence of cosmic anisotropy in theSN-Q sample. Key words. supernovae: general – quasar – large-scale structure of Universe
1. Introduction
The Lambda cold dark matter model ( Λ CDM) is consistentwith most astronomical observations (Reid, et al. 2010; Trujillo-Gomez, et al. 2011; Bennett, et al. 2013; Hinshaw, et al. 2013;Planck Collaboration, et al. 2014b). The basis of the Λ CDMmodel assumes that the universe is homogeneous and isotropicon a large scale (Weinberg 1972, 2008). This hypothsis is knownas the cosmological principle, and Clarkson & Maartens (2010)discuss the requirements in order to probe it. But several analy-ses of observations indicate that the universe may be anisotropic,for instance, these include quasar polarization vectors (Hut-semékers, et al. 2005), the fine-structure constant (Webb, et al.2011; King, et al. 2012), the direct measure of the Hubble pa-rameter (Bonvin, Durrer & Kunz 2006), the anisotropic darkenergy (Koivisto & Mota 2008), the cosmic microwave back-ground (CMB) (Tegmark, de Oliveira-Costa & Hamilton 2003;Bielewicz, Górski & Banday 2004; Eriksen, et al. 2004; Kim& Naselsky 2010b; Gruppuso, et al. 2011; Zhao 2014; Copi,et al. 2015; Akrami, et al. 2019), the large dipole of radiosource counts (Singal 2011; Gibelyou & Huterer 2012; Rubart& Schwarz 2013; Tiwari & Nusser 2016; Colin, et al. 2017;Bengaly, Maartens & Santos 2018; Singal 2019), the quasar vec-tor polarization aligment (Pelgrims & Hutsemékers 2016; Tiwari& Jain 2019), and the galaxy number counts in optical and IRwavelengths (Alonso, et al. 2015; Javanmardi & Kroupa 2017;Bengaly, et al. 2018). These works hint that the universe mayhave a preferred expanding direction (Perivolaropoulos 2014).In recent years, type Ia supernovae (SNe Ia) (Amanullah, etal. 2010; Suzuki, et al. 2012; Betoule, et al. 2014; Scolnic, et al.2018) have been widely employed to test cosmic isotropy. Anto-niou & Perivolaropoulos (2010) searched for the preferred direc- tion of anisotropy for the Union2 sample by adopting the hemi-sphere comparison (HC) method (Schwarz & Weinhorst 2007).They found a maximum accelerating expansion rate, which cor-responds to a preferred direction of anisotropy. After that, Mar-iano and Perivolaropoulos (Mariano & Perivolaropoulos 2012)found a possible preferred anisotropic direction at the 2 σ levelusing the Union2 sample, but by employing the dipole fitting(DF) method. Since then, these two methods have been widelyused to explore the cosmic anisotropy (Cai & Tuo 2012; Cai, etal. 2013; Zhao, Wu & Zhang 2013; Li et al. 2013; Heneka, Marra& Amendola 2014; Bengaly, Bernui & Alcaniz 2015; Andrade,et al. 2018; Sun & Wang 2019) by investigating observationaldata of, for instance, the Union2.1 sample (Yang, Wang & Chu2014; Javanmardi, et al. 2015; Lin, Li & Chang 2016), the JointLight-Curve Analysis (JLA) sample (Lin et al. 2016; Chang, etal. 2018; Wang & Wang 2018), the Pantheon sample (Sun &Wang 2018), gamma-ray bursts (GRBs; Wang & Wang 2014),galaxies (Zhou, Zhao & Chang 2017), as well as gravitationalwave and fast radio bursts (Qiang, Deng & Wei 2019; Cai, et al.2019). Using HC and DF methods, Zhao, Zhou & Chang (2019)studied the cosmic anisotropy via the Pantheon sample. Theyfound that the SDSS sample plays a decisive role in the Pan-theon sample. It may imply that the inhomogeneous distributionhas a significant e ff ect on the cosmic anisotropy (Chang, et al.2018). This opinion was also presented by Sun & Wang (2019).Their conclusions show that the e ff ect of redshift on the result isweak and there is a negligible anisotropy when making a redshifttomography. Deng & Wei (2018) tested the cosmic anisotropywith the Pantheon sample, but by using the following three meth-ods: the HC method, the DF method, and Healpix (Górski, et http: // healpix.sourceforge.net Article number, page 1 of 16 a r X i v : . [ a s t r o - ph . C O ] A ug & A proofs: manuscript no. proof al. 2005). They also performed a cross check. There are twopreferred directions from the HC method. In adopting the DFmethod and Healpix, they found no noticeable anisotropy. Theyalso compared the HC method with the DF method by using theJLA sample (Deng & Wei 2018) and found that the results ofthese two methods have not always been approximately coinci-dent with each other. In order to better test the cosmic isotropy,the best way would be to add new samples with a relatively ho-mogeneous distribution.Quasars can be regarded as quasi-standard candles (Lusso& Risaliti 2017; Salvestrini, et al. 2019) by using the nonlin-ear relation (Avni & Tananbaum 1986; Lusso & Risaliti 2016)between the UV and X-ray monochromatic luminosities. Thus,the nonlinear relation can be used for cosmological purposes(Khodyachikh 1989; Qin, et al. 1997; Risaliti & Lusso 2015;Bisogni, Risaliti & Lusso 2017; Melia 2019; Khadka & Ratra2020; Velten & Gomes 2020; Wei & Melia 2020; Lusso 2020).The quasar sample that consists of 1598 sources was built byRisaliti & Lusso (2019). They employed this sample to verifycosmological constraints. (Lusso, et al. 2019) and found that adeviation from the standard cosmological model emerges, witha statistical significance of 4 σ . Whether this deviation will ap-pear in cosmic isotropy is unclear. In this paper, we test the cos-mological anisotropy from the SN-Q sample, which consists ofthe Pantheon sample and the quasar sample, by adopting the HCand DF methods. A fiducial value of H =
70 km s − Mpc − isadopted in this paper. The structure of the paper is as follows. InSection 2, we present the quasar sample and discuss the advan-tages of the SN-Q sample over the single Pantheon sample. Thebest-fitting values of the quasar and SN-Q samples are obtainedby using the Markov chain Monte Carlo (MCMC) method. InSection 3, the HC and DF methods are given. By using thesetwo methods, we investigate the cosmic anisotropy by employ-ing the SN-Q sample. Finally, a brief summary is presented inthe last section.
2. The observational data and MCMC fitting
In this paper, we adopt the MCMC method provided by emcee to explore the whole parameter space and investigate the cos-mic anisotropy by using the Pantheon and quasar samples. ThePantheon sample, which was compiled by Scolnic, et al. (2018),contains 1048 SNe Ia covering the redshift range of 0.01 < z < < z < < z < https: // emcee.readthedocs.io / en / stable / ple distributes within z = .
3. The distributions and correspond-ing densities of these three samples in the galactic coordinatesare illustrated in Figure 2.The cosmic anisotropy obtained from the SNe sample mightrely on the inhomogeneous distribution of SNe Ia. The distribu-tions of SNe Ia from the Pantheon sample are shown in panel(a) of Figure 2. It is obvious that the belt part (the SDSS sam-ple) plays a major role in the full Pantheon sample. To makeit easier to comprehend this focus, we plotted the density dis-tribution of the Pantheon sample, which is shown in panel (b)of Figure 2. Zhao, Zhou & Chang (2019) analyzed the e ff ect ofthe inhomogeneous distribution of the Pantheon sample on thecosmic anisotropy and found that the SDSS sample plays themost important role in the Pantheon sample. From panels (c)and (d) of Figure 2, we find that the distribution of the quasarsample is more homogeneous and its maximum density valueis smaller than that of the Pantheon sample. Compared with thePantheon sample, adding the quasar sample reduces the uneven-ness of the overall sample and weakens the decisive role of theSDSS subsample as shown in panels (e) and (f) of Figure 2. Bycombining Figures 1 and 2, we find that the SN-Q sample is bet-ter than the Pantheon sample in terms of quantity, redshift range,and the uniform of distribution. However, the unevenness maynot have been eliminated. It is obvious that the data number ofthe North Galactic Hemisphere is larger than the South Galac-tic Hemisphere in the quasar sample. Through statistics, the datanumber of the South and North Galactic Hemispheres in the Pan-theon sample, quasar sample, and SN-Q sample are (644, 404),(438, 983), and (1082, 1387), respectively. The same case alsoappears in the SN-Q sample. In order to neutralize the inhomoge-neous Pantheon sample, it might be necessary to obtain a largerand more homogeneous sample. For the standard cosmological model, the theoretical distancemodulus can be written as µ th = d L Mpc + , (1)where d L is the luminosity distance. In the flat Λ CDM model, d L can be calculated from d L = c (1 + z ) H (cid:90) z dz (cid:48) (cid:112) Ω m (1 + z (cid:48) ) + (1 − Ω m ) , (2)where c is the speed of light, H is the Hubble constant, and Ω m is the matter density. For SNe Ia, the best fitting value of Ω m isachieved by minimizing the value of χ χ S N = (cid:88) i = ( µ obs ( z i ) − µ th ( Ω m , z i )) σ , (3)where σ i ( z i ) is the observational uncertainty of the distance mod-ulus.For quasars, the relation between UV and X-ray luminositiescan be parameterized as (Avni & Tananbaum 1986)log ( L X ) = γ log ( L UV ) + β, (4)where L X is the rest-frame monochromatic luminosity at 2 keVand L UV is the luminosity at 2,500 Å. We note that γ and β aretwo free parameters. Considering L = π d L F , equation (4) canbe written aslog (4 π d L F X ) = γ log (4 π d L F UV ) + β. (5) Article number, page 2 of 16. P. Hu et al.: Testing cosmic anisotropy with Pantheon sample and quasars at high redshifts
From equation (5), we derive the theoretical X-ray flux (Khadka& Ratra 2020), φ ([ F UV ] i , d L [ z i ]) = log ( F X ) = γ (log F UV ) + ( γ −
1) log 4 π + γ −
1) log d L + β. (6)For the quasar sample, the form of χ Q related to the X-ray flux F X of the quasar is given as χ Q = (cid:88) i = ( (log ( F X ) i − φ ([ F UV ] i , d L [ z i ])) s i + ln(2 π s i )) , (7)where the variance s i consists of the global intrinsic δ and themeasurement error σ i in ( F X ) i , that is, s i ≡ δ + σ i . The func-tion φ corresponds to the theoretical X-ray flux (equation (4)).Compared to δ and σ i , the error of ( F UV ) i is negligible.In substituting equation (6) for equation (7) and minimiz-ing the value of χ Q , we obtain the best-fit parameters: Ω m = + . − . , δ = + . − . , γ = + . − . , and β = + . − . . Itis important to note that Ω m is in 4 σ tension with the Λ CDMmodel (Risaliti & Lusso 2019). By combining equations (1), (3),(6), and (7), the χ statistic for the SN-Q sample is χ Total = χ S N + χ Q = (cid:88) i = ( µ obs ( z i ) − µ th ( Ω m , z i )) σ + (cid:88) i = ( (log ( F X ) i − φ ([ F UV ] i , d L [ z i ])) s i + ln(2 π s i )) . (8)By using equation (8), the best-fit parameters are Ω m = + . − . , δ = + . − . , γ = + . − . , and β = + . − . from the SN-Q sample, which is shown in Figure 3. The result isconsistent with the Λ CDM model.From the best-fit parameters for the quasar and SN-Q sam-ples, we find that when only considering the quasar sample, thematter density Ω m = + . − . . This is a significant departurefrom the Ω m given by the standard cosmological model. But Ω m = + . − . from the SN-Q sample is consistent with the stan-dard cosmological model. This may imply that the quasars couldnot be used for the cosmological probe independently at present(Velten & Gomes 2020). It can be combined with others probes,for instance, SNe Ia, CMB, GRBs (Wang, Dai & Liang 2015),and fast radio bursts (Yu & Wang 2017).
3. Testing the cosmic anisotropy with the HC andDF methods
The HC method was first proposed by Schwarz & Weinhorst(2007); it is widely used to investigate the cosmic anisotropy.For example, the anisotropy of cosmic expansion, the tempera-ture anisotropy of the CMB (Bennett, et al. 2013; Eriksen, et al.2004; Planck Collaboration, et al. 2014a; Akrami, et al. 2014;Hansen, Banday & Górski 2004; Quartin & Notari 2015), andthe acceleration scale of modified Newtonian dynamics (Zhou,Zhao & Chang 2017; Chang, et al. 2018; Chang & Zhou 2019).Firstly, we briefly introduce this approach. Its goal is to identify the direction, which corresponds to the axis of maximal asym-metry from the dataset, by comparing the accelerating expan-sion rate. In the spatially flat Λ CDM model, it is convenient toemploy Ω m to replace the accelerating expansion rate consider-ing the relationship between the deceleration parameter q and Ω m . The most important step is to produce random directionsˆ V ( l , b ), which are used to split the dataset into two parts (de-fined as "up" and “down"), where l ∈ (0 ◦ , 360 ◦ ) and b ∈ ( − ◦ ,90 ◦ ) are the longitude and latitude in the galactic coordinate sys-tem, respectively. According to “up" and “down" subdatasets,the corresponding best-fit values of Ω m , u and Ω m , d are obtainedadopting the MCMC method. The nuisance parameters ( δ , γ and β ) are marginalized along with Ω m in the MCMC process. Theanisotropy level (AL) made up of Ω m , u and Ω m , d can be used todescribe the accelerating expansion rate.In this section, we adopt the HC method and use the SN-Qsample to study the cosmic anisotropy. The AL is defined as AL = (cid:52) Ω m ¯ Ω m = × Ω m , u − Ω m , d Ω m , u + Ω m , d , (9)where Ω m , u and Ω m , d are the best-fit Ω m of the “up" subset and“down" subset, respectively. These two subsets are distinguishedfrom the full SN-Q sample by a random direction ˆ V ( l , b ). The 1 σ uncertainty σ AL is σ AL = (cid:113) σ Ω max m , u + σ Ω max m , d Ω max m , u + Ω max m , d . (10)During the calculation, we repeated 3000 random directionsˆ V ( l , b ) for the HC method and the results are Ω max m , u = Ω max m , d = σ Ω max m , d = σ Ω max m , d = σ uncertainty σ AL = . σ uncertainty is AL max = . ± . , (11)and the corresponding direction is( l , b ) = (316 . ◦ + . − . , . ◦ + . − . ) . (12)The distribution of AL( l , b ) is shown in Figure 4. This preferreddirection is inconsistent with the previous results with the Pan-theon sample from the HC method. For instance, by employingthe HC method, Deng & Wei (2018) obtained two preferred di-rections (138.8 ◦ , − ◦ ) and (102.36 ◦ , −
28 .58 ◦ ) from the Pan-theon sample. In using the same sample, Sun & Wang (2018) andZhao, Zhou & Chang (2019) achieved the HC preferred direc-tion ( l , b ) = (37 ◦ , 33 ◦ ) and ( l , b ) = (123.05 ◦ , 4.78 ◦ ), respectively.These four directions are plotted in Figure 5. From the result ofthe HC method, it shows that the quasar sample has an obviousimpact on the HC results.We also assessed statistical significance of the results foundby means of this test with simulated datasets. The simulated SN-Q datasets have the same direction in the sky as real data, buta di ff erent distance modulus (SNe Ia) and X-ray flux (quasar).Then the corresponding distance modulus and X-ray flux wereconstructed by the Gaussian function with the mean determinedby equation (8), where Ω m = δ = γ = β = σ error. There-fore, we constructed 200 simulated datasets and obtain the corre-sponding AL max by employing the HC method. The distributionof AL max in these datasets was fitted by a Gaussian function with Article number, page 3 of 16 & A proofs: manuscript no. proof the mean value 0.104 and the standard deviation 0.032, as shownin Figure 6. From Figure 6, we can find that the statistical sig-nificance of the maximum anisotropy level of AL max , which isabout 1.23 σ and 14 percent of AL max obtained from the simu-lated datasets, is greater than that of the real data. Therefore, thevalue of AL max is not large enough. In addition, it is necessaryto examine whether the maximum AL from the SN-Q sample isconsistent with statistical isotropy. In order to determine this, weevenly redistributed the original data-sets across the sky and re-peated it 200 times. The result of the simulated isotropic datasetsis shown in Figure 7. The mean value is 0.099 and the standarddeviation is 0.024. We note that 7 percent of AL max obtainedfrom the simulated isotropic datasets is larger than that of thereal data. The statistical significance of AL max is about 1.65 σ .The results of the random and isotropic datasets are smaller than2 σ , which still supports the absence of significant anisotropy.In the end of this subsection, we compare the HC preferreddirection of the SN-Q sample with those derived in various ob-servational datasets. In Figure 5, we plotted the preferred direc-tions ( l , b ) found from various observational datasets by the HCmethod in the galactic coordinate system. From Figure 5, wefind that the HC preferred direction in this paper is the devia-tion from that of the Pantheon sample and the SPARC galaxiessample, but it is generally consistent with those in the Union2sample (Antoniou & Perivolaropoulos 2010; Cai & Tuo 2012;Chang & Lin 2015), the Union2.1 sample (Sun & Wang 2019;Lin, Li & Chang 2016), the Constitution sample (Sun & Wang2019; Kalus, et al. 2013), and the JLA sample (Deng & Wei2018). Overall, the distribution of the HC preferred direction ob-tained from a di ff erent sample is di ff use. As discussed by Sun& Wang (2019) and Zhao, Zhou & Chang (2019), the cosmicanisotropy found in the supernova sample significantly relies onthe inhomogeneous distribution of SNe Ia in the sky. The pre-ferred directions in various observational datasets are also shownin Table 1. From Table 1, we find that the HC preferred direc-tion in this paper is also in agreement with the results of theCMB dipole (Lineweaver, et al. 1996), velocity flows (Watkins,Feldman & Hudson 2009; Feldman, Watkins & Hudson 2010),quasar alignment (Hutsemékers, et al. 2005, 2011; Hutsemék-ers & Lamy 2001), the CMB quadrupole (Bielewicz, Górski &Banday 2004; Frommert & Enßlin 2010), the CMB octopole(Bielewicz, Górski & Banday 2004), (cid:52) α / α (Webb, et al. 2011;King, et al. 2012), and infrared galaxies (Yoon, et al. 2014; Ben-galy, et al. 2017). The DF method is also used to test the cosmic anisotropy. Con-sidering the dipole magnitude A and monopole term B , the theo-retical distance modulus should be rewritten as˜ µ th = µ th × (1 + A ( ˆ n · ˆ p ) + B ) , (13)where ˆ n and ˆ p correspond to the dipole direction and the unitvector pointing to the position of the SN Ia or quasar, respec-tively. In the galactic coordinate, ˆ n is written asˆ n = cos ( b ) cos ( l )ˆ i + cos ( b ) sin ( l ) ˆ (j) + sin ( b ) ˆ k . (14)For any i th ( l i , b i ) data points, ˆ p is given byˆ p i = cos ( b i ) cos ( l i )ˆ i + cos ( b i ) sin ( l i ) ˆ (j) + sin ( b i ) ˆ k . (15)We were able to obtain the best-fit dipole direction ( l , b ) bysubstituting equation (13) for equation (3) and minimizing χ . Equation (13) can be directly used for the SNe Ia sample. Forthe quasars, according to equation (1), the theoretical distancemodulus can also be written as˜ µ th = d L Mpc + . (16)In substituting equation 16 for equation 13, we obtain˜log d L Mpc = (log d L Mpc + × (1 + A ( ˆ n · ˆ p ) + B ) − . (17)Thus the theoretical X-ray flux with dipole and monopole cor-rections is given by˜ φ ([ F UV ] i , d L [ z i ]) = γ (log F UV ) + ( γ −
1) log 4 π + β + γ − d L Mpc + × (1 + A ( ˆ n · ˆ p ) + B ) − . (18)Adding equations (13) and (18) to (8) and minimizing the valueof χ Total , the marginalized posterior distribution for the SN-Qsample is shown in Figure 8. In total, there are eight parametersin the DF method, four of which are used to describe the univer-sal anisotropy, that is, l , b , A , and B . The results show that thedipole direction ( l , b ) is (327 . ◦ + . − . , 48 . ◦ + . − . ), the dipolemagnitude A = ( − + . − . ) × − and the monopole term B = ( − + . − . ) × − . The dipole direction is generally consistentwith the preferred direction (316 . ◦ + . − . , . ◦ + . − . ) of theHC method. Both the dipole magnitude A and monopole term B are approximately equal to zero, indicating that there is nosignificant anisotropy in the SN-Q sample. Next, we carry outa redshift tomography analysis to discuss the e ff ect of redshift.The results of the redshift tomography analysis were calculated,as shown in Table 2. Based on analyses of Figure 1, the numberof low redshift sources is relatively large, so the redshift inter-vals are not uniform in the redshift tomography analysis. Fromthe results of redshift tomography, it can be found that the dipoledirections are distributed in a relatively small range. The maxi-mum of monopole term A and the dipole magnitude B are nearzero. There is no significant change in the dipole direction andanisotropic level with di ff erent redshift ranges. We note that | A | and | B | of the first bin are larger than that of others bins. This in-dicates that the anisotropy level of the low redshift range mightbe relatively higher.At the end of this section, we make a comparison betweenthe dipole directions of the SN-Q sample with that derived fromother samples and compare the results between the HC methodand the DF method for the same sample. At first, we marked thedipole directions obtained from di ff erent samples in the galacticcoordinate system, as shown in Figure 5. The dipole directionsobtained from di ff erent samples are mostly located in a relativelysmall part of the South Galactic Hemisphere. The dipole direc-tion in this paper is inconsistent with them. The longitude l isclose, but the deviation of latitude b is large. It might be causedby the inhomogeneous distribution of SNe Ia, as discussed inSection 2.1. The data number of the SN-Q sample in the NorthGalactic Hemisphere is larger than that in the South GalacticHemisphere. But the dipole direction is in agreement with theresults derived for the CMB dipole, velocity flows, quasar align-ment, the CMB quadrupole, the CMB octopole, and (cid:52) α / α . Forthe same sample, we find that the results of these two methodsare not always consistent with each other. For instance, thesetwo preferred directions obtained by the HC and DF methods inthe SPARC galaxies sample (Zhou, Zhao & Chang 2017; Chang, Article number, page 4 of 16. P. Hu et al.: Testing cosmic anisotropy with Pantheon sample and quasars at high redshifts et al. 2018), the Union2 sample (Antoniou & Perivolaropoulos2010; Mariano & Perivolaropoulos 2012; Cai & Tuo 2012; Cai,et al. 2013; Chang & Lin 2015), and the JLA sample (Bengaly,Bernui & Alcaniz 2015; Sun & Wang 2019; Lin et al. 2016;Deng & Wei 2018), are consistent. However, the HC and DF pre-ferred directions obtained from the Constitution sample (Sun &Wang 2019; Kalus, et al. 2013) are inconsistent. The same case isalso exhibited in the Pantheon sample (Sun & Wang 2018; Zhao,Zhou & Chang 2019; Deng & Wei 2018). This may be due to thesensitivity of the two methods (Chang & Lin 2015).In Table 1, we summarize the preferred directions ( l , b ) foundin di ff erent cosmological models using di ff erent methods and ob-servational datasets. The parts of the DF method and HC methodhave been discussed at the beginning of this section. In addi-tion, we also summarize the research results obtained by variousmethods for di ff erent models ( Λ CDM, w CDM, and CPL). Look-ing through the results from the same sample, we find that theresults are almost independent of these three models.
4. Summary
The cosmic principle assumes that the universe is homoge-neous and isotropic on cosmic scales. The research on cosmicanisotropy from SNe Ia also shows that there is no obviousanisotropy. In this work, we test the cosmic anisotropy with anew sample, which consists of SNe Ia and the quasars, by us-ing the HC and DF methods. Nevertheless, the results show thatthere is no obvious anisotropy.Firstly, we briefly investigate the Pantheon sample andquasar sample. By assessing the results, we find that addingthe quasar sample reduces the unevenness of the overall sam-ple. Compared with the Pantheon sample, the new sample (SN-Q sample) has a larger size, wider range of redshift, and a moreuniform distribution. But the distribution of the SN-Q sampleis still inhomogeneous in the North Galactic Hemisphere andSouth Galactic Hemisphere. We also briefly discuss the e ff ect ofinhomogeneous distribution on the dipole preferred in the SN-Qsample. By adopting the MCMC method, we obtain the best fit-ting values of Ω m , δ , γ , β , and H and find that 4 σ tensions existbetween the best-fit value of Ω m from the quasar sample and thatof the Λ CDM model. Then, by adding the Pantheon sample intothe quasar sample, the 4 σ tension disappears.For the HC method, the preferred direction that corre-sponds to the maximum accelerating expansion is 316 . ◦ + . − . ,4 . ◦ + . − . . The anisotropy level AL max is equal to 0.142 ± σ . In ad-dition, we also examine whether the AL max from the SN-Q sam-ple is consistent with statistical isotropy by employing the sim-ulated isotropic datasets. The statistical significance is about1.65 σ . The results show that it is hardly reproduced by simulateddatasets or isotropy simulated datastes. For the DF method, wefind that the preferred direction in the SN-Q sample points to-ward (327 . ◦ + . − . , 48 . ◦ + . − . ) with an anisotropy level of A = ( − + . − . ) × − , which is marginally consistent with theresult of the HC method. There is a considerable deviation forthe latitude direction l from those obtained from other SNe Iasamples, which may be caused by including the quasar sample.The preferred directions obtained by using the HC method andthe DF method from the SN-Q sample are both in agreementwith the results of the CMB dipole, velocity flows, quasar align-ment, the CMB quadrupole, the CMB octopole, and (cid:52) α / α . Theresults of the redshift tomographic analysis show that the dipole direction is weakly dependent on redshift. Comparing this withprevious studies of the Pantheon sample, the preferred directionsin the SN-Q sample have an obvious divergence. There does notexist an obvious anisotropy from the SN-Q sample.Although the SN-Q sample is better than the Pantheon sam-ple in some aspects, such as the quantity, redshift range, and uni-form of distribution, there are still some shortcomings. For ex-ample, the distribution of the SN-Q sample is not uniform in thewhole sky. Next-generation X-ray surveys, such as eROSITA,will provide us with larger and more precise luminosity dis-tance determinations of quasars, so that we should be able toreduce the uncertainties obtained in our analysis. By employinga simulated e-ROSITA quasar sample, predictions for e-ROSITAfrom a quasar Hubble diagram have been made by Lusso (2020).Colin, et al. (2019) point out that the cosmic acceleration is dueto a non-negligible dipole anisotropy by analyzing the JLA sam-ple. We think that the same research can be performed by a com-bined sample of SNe Ia and quasars, considering the quasar sam-ple enables us to talk about this study in a higher redshift rangeand to test whether the same result will still be obtained. Thiswill be pursued in future work. Acknowledgements.
We thank the anonymous referee for valuable comments.This work is supported by the National Natural Science Foundation of China(grant U1831207).
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Article number, page 6 of 16. P. Hu et al.: Testing cosmic anisotropy with Pantheon sample and quasars at high redshifts
Fig. 1.
Redshift cumulative distributions. Red, blue, and black lines correspond to the Pantheon, quasar, and SN-Q samples, respectively. The SNeIa account for 42% of the SN-Q sample and the quasars for 58%. Article number, page 7 of 16 & A proofs: manuscript no. proof(a) Pantheon sample (b) Pantheon sample(c) The quasar sample (d) The quasar sample(e) SN-Q sample (f) SN-Q sample
Fig. 2.
Distributions and density contours in the galactic coordinate system. Panels (a), (c), and (e) are the coordinate distributions of the Pantheon,quasar, and SN-Q samples in the galactic coordinate system, respectively. The corresponding density contours are described in panels (b), (d), and(f).Article number, page 8 of 16. P. Hu et al.: Testing cosmic anisotropy with Pantheon sample and quasars at high redshifts . . . . . . . . . . . . m . . . . . . . . . . . . . . . . Fig. 3.
Confidence contours (1 σ, σ , and 3 σ ) and the marginalized likelihood distributions for the space of the parameters ( Ω m , δ , γ , β ) from theSN-Q sample in the spatially flat Λ CDM model. Article number, page 9 of 16 & A proofs: manuscript no. proof
Fig. 4.
Distribution of the AL in the galactic coordinate system. The triangle marks the direction of the largest of the AL in the sky.Article number, page 10 of 16. P. Hu et al.: Testing cosmic anisotropy with Pantheon sample and quasars at high redshifts Fig. 5.
Distribution of the preferred directions ( l , b ) in the various observational datasets. The blue color and red color correspond to the HCmethod and the DF method, respectively. Article number, page 11 of 16 & A proofs: manuscript no. proof max P r o p o r t i o n Real datasimulation
Fig. 6.
Distribution of AL max in 200 simulated datasets. The black curve is the best fitting result to the Gaussian function. The solid black andvertical dashed lines are commensurate with the mean and the standard deviation, respectively. The vertical red line shows the maximum ALderived from the actual SN-Q sample. max P r o p o r t i o n Real dataisotropy
Fig. 7.
Distribution of AL max in 200 simulated isotropic datasets. The black curve is the best fitting result to the Gaussian function. The solid blackand vertical dashed lines are commensurate with the mean and the standard deviation, respectively. The vertical red line shows the maximum ALderived from the actual SN-Q sample.Article number, page 12 of 16. P. Hu et al.: Testing cosmic anisotropy with Pantheon sample and quasars at high redshifts . . . . . . . . . . . . . . l b . . . . A .
24 0 .
27 0 .
30 0 . m . . . . . B .
22 0 .
23 0 .
24 0 .
25 0 .
60 0 .
62 0 .
64 0 .
66 0 .
68 6 . . . . . l
80 40 0 40 80 b . . . . A . . . . . B Fig. 8.
Confidence contours (1 σ, σ, and 3 σ ) and marginalized likelihood distributions for the parameters space ( Ω m , δ , γ , β , l , b , A , B ) to theSN-Q sample in the dipole-modulated Λ CDM model. Article number, page 13 of 16 & A proofs: manuscript no. proof
Table 1.
Preferred directions ( l , b ) found in di ff erent cosmological models using di ff erent methods and observational datasets. Cosmological Obs. Model Method l ( ◦ ) b ( ◦ ) Ref.Union2 Λ CDM HC 309 ◦ + − ◦ + − Antoniou &Perivolaropoulos (2010) Λ CDM - 309 ◦ ◦ Colin, et al. (2011) Λ CDM HC 314 ◦ + − ◦ + − Cai & Tuo (2012) Λ CDM DF 309.4 ◦ ± ◦ ± Λ CDM HC 334 ◦ + − ◦ + − Chang & Lin (2015) Λ CDM DF 309 ◦ ± ◦ ± Λ CDM AM 126 ◦ + − ◦ + − Cai, et al. (2013)67GRB Λ CDM AM 336 ◦ + − -5 ◦ + − Cai, et al. (2013) w CDM AM 340 ◦ + − -4 ◦ + − Cai, et al. (2013)CPL AM 339 ◦ + − -6 ◦ + − Cai, et al. (2013)Union2 + Λ CDM AM 129 ◦ + − ◦ + − (Cai, et al. 2013) w CDM AM 129 ◦ + − ◦ + − Cai, et al. (2013)CPL AM 131 ◦ + − ◦ + − Cai, et al. (2013)Union2 + SN FACTORY Λ CDM DF (z,0.015-0.035) 298 ◦ ±
25 15 ◦ ±
20 Feindt, et al. (2013) Λ CDM DF (z,0.035-0.045) 302 ◦ ±
48 -12 ◦ ±
26 Feindt, et al. (2013) Λ CDM DF (z,0.045-0.060) 359 ◦ ±
32 14 ◦ ±
27 Feindt, et al. (2013) Λ CDM DF (z,0.060-0.100) 285 ◦ ±
234 -23 ◦ ±
112 Feindt, et al. (2013)Union2.1 Λ CDM HC – – Yang, Wang & Chu(2014) Λ CDM DF 307.1 ◦ ± ◦ ± Λ CDM HC 241.9 ◦ -19.5 ◦ (Lin, Li & Chang 2016) Λ CDM DF 310.6 ◦ ± ◦ ± Λ CDM Hubble map 326.25 ◦ ◦ Bengaly, Bernui & Al-caniz (2015) Λ CDM q map 354.38 ◦ ◦ Bengaly, Bernui & Al-caniz (2015) Λ CDM HC 352 ◦ -9 ◦ Sun & Wang (2019) Λ CDM DF 309.3 ◦ + . − . -8.9 ◦ + . − . Sun & Wang (2019)Union2.1 + Λ CDM DF 309.2 ◦ ± ◦ ± + VLT Λ CDM DF 320.5 ◦ ± ◦ ± Λ CDM HC -35 ◦ -19 ◦ Kalus, et al. (2013) Λ CDM HC 141 ◦ -11 ◦ Sun & Wang (2019) Λ CDM DF 67.0 ◦ + . − . -0.6 ◦ + . − . Sun & Wang (2019)JLA Λ CDM Hubble map 58.00 ◦ -60.43 ◦ Bengaly, Bernui & Al-caniz (2015) Λ CDM q map 225.00 ◦ ◦ Bengaly, Bernui & Al-caniz (2015) Λ CDM HC(max) 23.49 ◦ ◦ Deng & Wei (2018) Λ CDM HC(submax) 299.47 ◦ ◦ Deng & Wei (2018) Λ CDM DF 185 ◦ − ◦ + . − . Deng & Wei (2018) Λ CDM DF 316 ◦ + − -5 ◦ + − Lin et al. (2016) w CDM DF 320 ◦ + − -4 ◦ + − Lin et al. (2016)
Article number, page 14 of 16. P. Hu et al.: Testing cosmic anisotropy with Pantheon sample and quasars at high redshifts
CPL DF 318 ◦ + − -8 ◦ + − Lin et al. (2016) Λ CDM DF 94.4 ◦ -51.7 ◦ Sun & Wang (2019)Pantheon Λ CDM HC 37 ◦ ±
40 33 ◦ ±
16 Sun & Wang (2018) Λ CDM DF 329 ◦ + − ◦ + − Sun & Wang (2018) Λ CDM HC(max) 138.08 ◦ + . − . -6.8 ◦ + . − . Deng & Wei (2018) Λ CDM HC(submax) 102.36 ◦ + . − . -28.58 ◦ + . − . Deng & Wei (2018) Λ CDM DF - - Deng & Wei (2018) Λ CDM HC 123.05 ◦ + . − . ◦ + . − . Zhao, Zhou & Chang(2019) Λ CDM DF 306.00 ◦ + . − . -34.20 ◦ + . − . Zhao, Zhou & Chang(2019) Λ CDM HC 286.93 ◦ ± ◦ ◦ ± ◦ Kazantzidis &Perivolaropoulos (2020) Λ CDM DF 210.25 ◦ ± ◦ ◦ ± ◦ Kazantzidis &Perivolaropoulos (2020) Λ CDM DF 306.00 ◦ + . − . -23.41 ◦ + . − . Chang & Zhou (2019) w CDM DF 298.81 ◦ + . − . -19.80 ◦ + . − . Chang & Zhou (2019)CPL DF 313.20 ◦ + . − . -27.00 ◦ + . − . Chang & Zhou (2019)Finslerian DF 298.80 ◦ + . − . -23.41 ◦ + . − . Chang & Zhou (2019)CMB Dipole - - 263.99 ◦ ± ◦ ± ◦ ◦ Watkins, Feldman& Hudson (2009),Watkins, Feldman &Hudson (2009), Feld-man, Watkins & Hudson(2010)Quasar Alignment - - 267 ◦ ◦ Hutsemékers & Lamy(2001), Hutsemékers, etal. (2005), Hutsemékers,et al. (2011)CMB Octopole - - 308 ◦ ◦ Bielewicz, Górski &Banday (2004)CMB Quadrupole - - 240 ◦ ◦ Bielewicz, Górski &Banday (2004), From-mert & Enßlin (2010) (cid:52) α/α - - 330 ◦ -13 ◦ Webb, et al. (2011) ,King, et al. (2012)SPARC Galaxies - HC(max) 175.5 ◦ -6.5 ◦ Zhou, Zhao & Chang(2017)- HC(submax) 114.5 ◦ ◦ Zhou, Zhao & Chang(2017)- DF 171 ◦ -15 ◦ Chang, et al. (2018)Galaxy Cluster Λ CDM - 303 ◦ -27 ◦ Migkas, et al. (2020)Infrared galaxies - - 310 ◦ -15 ◦ Yoon, et al. (2014)- HC 323 ◦ -5 ◦ Bengaly, et al. (2017)Pantheon + Λ CDM HC 316.08 ◦ + . − . ◦ + . − . this paper Λ CDM DF 327.55 ◦ ± ◦ ± Article number, page 15 of 16 & A proofs: manuscript no. proof T a b l e . R e d s h i f tt o m og r a phy r e s u lt s u s i ngd i po l e fi tti ng m e t hod f o r t h e S N - Q s a m p l e . N S N + Q u a s a r Ω m δ γ β l ( ◦ ) b ( ◦ ) AB z < . + . + . − . . + . − . . + . − . . + . − . . ◦ + . − . . ◦ + . − . - . + . − . e - - . + . − . e - z < . + . + . − . . + . − . . + . − . . + . − . . ◦ + . − . . ◦ + . − . - . + . − . e - - . + . − . e - z < . + . + . − . . + . − . . + . − . . + . − . . ◦ + . − . . ◦ + . − . - . + . − . e - - . + . − . e - z < . + . + . − . . + . − . . + . − . . + . − . . ◦ + . − . . ◦ + . − . - . + . − . e - - . + . − . e - z < . + . + . − . . + . − . . + . − . . + . − . . ◦ + . − . . ◦ + . − . - . + . − . e - - . + . − . e - z < . + . + . − . . + . − . . + . − . . + . − . . ◦ + . − . . ◦ + . − . - . + . − . e - - . + . − . e - z < . + . + . − . . + . − . . + . − . . + . − . . ◦ + . − . . ◦ + . − . - . + . − . e - - . + . − . e - z < . + . + . − . . + . − . . + . − . . + . − . . ◦ + . − . . ◦ + . − . - . + . − . e - - . + . − . e -5