Probing the isotropy of cosmic acceleration using different supernova samples

Recent studies indicated that an anisotropic cosmic expansion may exist. In this paper, we use three data sets of type Ia supernovae (SNe Ia) to probe the isotropy of cosmic acceleration. For the Union2.1 data set, the direction and magnitude of the dipole are (l=309.3∘-15.7∘+15.5∘,b=-8.9∘-9.8∘+11.2∘)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(l=309.3^{\circ } {}^{+ 15.5^{\circ }}_{-15.7^{\circ }} ,\ b = -8.9^{\circ } {}^{ + 11.2^{\circ }}_{-9.8^{\circ }} )$$\end{document}, and A=(1.46±0.56)×10-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ A=(1.46 \pm 0.56) \times 10^{-3}$$\end{document} from dipole fitting method. The hemisphere comparison results are δ=0.20,l=352∘,b=-9∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta =0.20,l=352^{\circ },b=-9^{\circ }$$\end{document}. For the Constitution data set, the results are (l=67.0∘-66.2∘+66.5∘,b=-0.6∘-26.3∘+25.2∘)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(l=67.0^{\circ }{}^{+ 66.5^{\circ }}_{-66.2^{\circ }},\ b=-0.6^{\circ }{}^{+ 25.2^{\circ }}_{-26.3^{\circ }})$$\end{document}, and A=(4.4±5.0)×10-4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ A=(4.4 \pm 5.0) \times 10^{-4}$$\end{document} for dipole fitting and δ=0.56,l=141∘,b=-11∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta = 0.56,l=141^{\circ },b=-11^{\circ }$$\end{document} for hemisphere comparison. For the JLA data set, no significant dipolar or quadrupolar deviation is found. We find previous works using (l, b, A) directly as fitting parameters may get improper results. We also explore the effects of anisotropic distributions of coordinates and redshifts on the results using Monte-Carlo simulations. We find that the anisotropic distribution of coordinates can cause dipole directions and make dipole magnitude larger. Anisotropic distribution of redshifts is found to have no significant effect on dipole fitting results.


Introduction
Type Ia supernovae (SNe Ia) are ideal standard candles (Phillips 1993).In 1998, the accelerating expansion of the Universe was discovered using the luminosity-redshift relation of SNe Ia (Riess et al. 1998;Perlmutter et al. 1999).The cosmological principle assumes that the Universe is homogeneous and isotropic at large scales.Based on the cosmological principle and numerous observational facts, the standard ΛCDM model has been established.It can be used to explain various observations.However, it is worthy to examine the validity of the standard ΛCDM model (Kroupa et al. 2012;Kroupa 2012;Perivolaropoulos 2014;Koyama 2016) and its assumptions, namely the cosmological principle.Deviation from cosmic isotropy with high statistical confidence level would lead to a major paradigm shift.At present, the standard cosmology confronts some challenges.Observations on the large-scale structure of the Universe, such as "great cold spot" on cosmic microwave background (CMB) sky map (Vielva et al. 2004), alignment of lower multipoles in CMB power spectrum (de Oliveira-Costa et al. 2004;Tegmark et al. 2003), alignment of polarization directions of quasars in large scale (Hutsemékers et al. 2005), handedness of spiral galaxies (Longo 2009), and spatial variation of the fine structure constant (King et al. 2012;Mariano & Perivolaropoulos 2012), show that the Universe may be anisotropic.
The isotropy of the cosmic acceleration has been widely tested using SNe Ia.Generally, there are two different ways to study the possible anisotropy from SNe Ia.The first one is directly fitting the data to a specific anisotropic model (Campanelli, L et al. 2011;Li et al. 2013;Wang & Wang 2018).Many anisotropic cosmological models have been proposed to match the observations, including the Bianchi I type cosmological model (Campanelli, L et al. 2011;Aluri et al. 2013) and the Rinders-Finsler cosmological model (Chang et al. 2014).
The extended topological quintessence model with a spherical inhomogeneous distribution for dark energy density is also proposed (Mariano & Perivolaropoulos 2012).
An alternative method is directly analysing the SNe Ia data in a model-independent way (Antoniou & Perivolaropoulos 2010;Cai & Tuo 2011;Cai et al. 2013;Mariano & Perivolaropoulos 2012;Zhao et al. 2013;Yang et al. 2013;Wang & Wang 2014;Javanmardi et al. 2015;Jimenez et al. 2015;Lin et al. 2016), which does not depend on the specific cosmological model.The hemisphere comparison (HC) method and dipole fitting (DF) method are usually used in literature.The hemisphere comparison method divides samples into two hemispheres perpendicular to a polar axis, then fits cosmological parameters using samples in each hemisphere independently and compares their differences.The dipole fitting (DF) method assumes a dipolar deviation on redshift-distance modulus relation, then derives the dipole's direction and magnitude using statistic approaches.Meanwhile, low-redshift SNe Ia are used to estimate the direction and amplitude of the local bulk flow (Bonvin et al. 2006;Schwarz & Weinhorst 2007;Gordon et al. 2008;Colin et al. 2011;Turnbull et al. 2012;Kalus et al. 2013;Appleby & Shafieloo 2014;Huterer et al. 2015).So far, no study has been able to rule out the isotropy at more than 3σ.The gravitational wave as standard siren has also been proposed to probe cosmic anisotropy (Cai et al. 2017).
The directions and magnitudes of anisotropy from previous works are shown in table 1.It's obvious that different results are derived from different authors.In this paper, we compare the DF fitting results of different SNe Ia samples and try to find the reason for the differences.This paper is organized as follows.In section 2, SNe Ia datasets and DF method are introduced.The fitting results are shown in section 3. We discuss the possible reasons for the differences in section 4. Finally, we summarize in section 5.

Dipole fitting method
Firstly, we briefly introduce the dipole fitting method.For Union2.1 and Constitution datasets, the luminosity distance could be expanded by Hubble series parameters: Hubble parameter H, deceleration parameter q, jerk parameter j and snap parameter s.These parameters can be expressed as functions of the scale factor a and its derivatives, (1) Taylor expansion of luminosity distance could be made in terms of redshift and Hubble series parameters (Visser 2004).However, this expansion diverges at z > 1.Thus, another parameter y = z/(1 + z) is introduced to overcome this problem.The luminosity distance can be expanded as a function of y (Cattoën & Visser 2007;Wang, F Y et al. 2009) (2) The distance modulus is defined as Then the χ 2 can be calculated as where µ obs and σ i are observational values of distance moduli and their errors, respectively.The best-fitting values of parameters could be obtained by minimizing χ 2 (H 0 , q 0 , j 0 , s 0 ).
For the JLA sample, the observational values of distance moduli are not directly given.Therefore, we use the values obtained in the ΛCDM model to avoid fitting too many free parameters simultaneously.The theoretical luminosity distance in ΛCDM model can be expressed as Union2.1 and Constitution datasets already give µ obs as a part of the released data.For JLA dataset, µ obs can be derived from light curve parameters of SN Ia from (Betoule et al. 2014) where m * B is the observed peak magnitude in the rest-frame of the B band, X 1 describes the time stretching of light-curve, C describes the supernova color at maximum brightness and M B is the absolute B-band magnitude, which depends on the host galaxy properties.α, β are nuisance parameters.M B can be fitted by a simple step function related with M stellar (Johansson et al. 2013), where M 1 B and ∆ M are nuisance parameters.C(α, β) is the total covariance matrix, which can be obtained with JLA data.χ 2 is defined as By minimizing χ 2 JLA , all free parameters mentioned above can be fitted.Our best-fitting results are consistent with those of Betoule et al. (2014).
In order to quantify the anisotropic deviations on luminosity distance, we define the distance moduli with dipole A and monopole B as where μth is the theoretical value of distance modulus with dipolar direction dependence, and n is the unit vector pointing at the corresponding SN Ia.A The best-fitting dipole and monopole parameters can be derived with the following steps: 1. Substitute µ th with μth in the expression of χ 2 as is shown in equation ( 4) and equation (8).
2. Fit the dipole (A x , A y , A z ) and the monopole component B by minimizing χ 2 .
Finally, we analyze the likelihood of the fitted parameters and the significance of dipole magnitude utilizing Markov Chain Monte Carlo (MCMC) sampling.In order to obtain the significance of dipole anisotropy precisely, we use the Monte Carlo simulation method (Mariano & Perivolaropoulos 2012;Yang et al. 2013;Wang & Wang 2014).To be specific, we construct a type of synthetic samples based on original datasets by assuming that theoretical values of distance moduli are "real" values.We refer to these synthetic samples as "isotropic" samples.Applying MCMC sampling on these samples, probability distributions of the fitted parameters can be obtained.They are also used to probe the effect of anisotropic factors, which we will discuss in section 4.

Results
Fitting results are shown in Table 2.The confidence level is defined as the probability P(|A iso | < |A fit |), where |A iso | is the dipole magnitude of an arbitrary dataset in "isotropic" samples, and |A fit | is the best-fitting dipole magnitude.Best-fitting dipole directions and 1σ errors for Union2.1,Constitutio, and JLA datasets, along with dipole fitting results of samples are plotted in Figures 1, 2 and 3, respectively.
We generate 2 × 10 6 effective samples for each dataset for MCMC sampling.Probability distributions of dipole and monopole parameters for Union2.1,Constitution, and JLA datasets are shown in Figures 4, 5 and 6, respectively.Note that the best-fitting parameters do not coincide with most probable values for some parameters.This is caused by transformation from rectangular coordinates (A x , A y , A z ) used in MCMC sampling to polar coordinates (l, b) and dipole magnitude A.
For Union2.1 dataset, the direction and magnitude of the dipole are (l = 309.3 , and A = (1.46 ± 0.56) × 10 −3 .The confidence level of dipolar anisotropy is 98.3%.For Constitution dataset, these parameters are (l = 67.0 It is worth mentioning that, JLA dataset gives null results in dipole fitting.The 1σ error range of dipole direction covers the whole celestial sphere.The confidence level of dipole magnitude is merely 0.23%.Furthermore, there is no significant difference between the likelihood of simulation results in 1σ error range and full results.The same is true for the likelihood of parameters of "isotropic" samples and original samples.In addition, significant deviations exist in best-fitting values and most probable values of fitted parameters.Thus, no significant dipolar anisotropy of redshift-distance modulus relation is found in the JLA dataset. For Union2.1 dataset, we get similar results as previous works (Antoniou & Perivolaropoulos 2010;Cai & Tuo 2011;Cai et al. 2013;Mariano & Perivolaropoulos 2012;Yang et al. 2013;Wang & Wang 2014;Lin et al. 2016).For Constitution datasets, our results are different from those of Kalus et al. (2013).Considering different methods, and the weak signal of the dipole in this dataset, the difference is reasonable.
For JLA dataset, we get different likelihood distributions as Lin et al. (2015), which can be attributed to different fitting parameters used in the MCMC estimation.In this paper, we fit the dipole by fitting its rectangular components (A x , A y , A z ), then convert the fitting results to spherical coordinate.As shown in Figure 7, non-zero likelihood at b = ±90 • forms an unreasonable 'spike' at poles, which depends on the choice of coordinate system.By comparison, posteriors used in this paper are in accordance with best-fitting values, and joint likelihood contours are smooth oval shape, as shown in Figure 8.

Effects of anisotropy in data distribution
As shown in Figures 4, 5 and 6, even if no redshift-distance anisotropy in the input data, fitting results are still distributed an-isotropically (green dotted lines).This indicates other reasons, such as anisotropic coordinate or redshift distribution would affect the results.
In order to determine whether the coordinate distribution of samples would affect the results, we introduce two types of synthetic datasets.
Type A datasets substituting the coordinates in the original dataset with random coordinates uniformly distributed in the whole sky.µ obs are replaced with synthetic data.
Type B datasets substituting thr coordinates in the original dataset with random coordinates, but only uniformly distributed in the eastern hemisphere of the celestial sphere.Distance moduli are substituted the same manner as type A datasets.
Using MCMC sampling method introduced in section 2, we find the dipoles in type A datasets are uniformly distributed in the whole sky.However, the dipoles in Type B datasets tend to concentrate in (l, b) = (90 • , 0 • ), (l, b) = (270 • , 0 • ), as shown in Figure 9.Meanwhile, dipole magnitudes are generally larger than it is in Type A datasets, as shown in Figure 10.This indicates that anisotropy of coordinates of samples does affect the results of dipole fitting.For a better understanding of this case, we examine the anisotropy of datasets with the following method: 1. Define the "sample count" of a given coordinate as the number of samples in 90 • range on the celestial sphere.
2. Calculate the "mean absolute deviation" (MAD) of sample counts.It is defined as where N sc is the sample count defined before, N sc is the mean value of sample counts.
3. Draw contours of MAD of sample counts for three datasets, respectively.Then compare them with dipole coordinates in "isotropic" samples.
We find that dipole directions tend to concentrate in places where sample counts deviate from the average most, as shown in Figures 11, 12 and 13.It is also in accordance with the results of previous simulations.
The spatial distribution of redshifts can be anisotropic, i.e., the redshift of samples in one patch of the sky may be generally smaller than another patch of sky.This may also cause influence on fitting results.To extract the effects of anisotropy in redshift distribution from other factors, we introduce another kind of synthetic dataset.
Type C datasets shuffle the coordinates in the original dataset, making distance-related data of every sample correspond with a coordinate of another random sample in the dataset.Keep the 'shuffled coordinates' order unchanged, then substitute distance moduli as type A datasets.
Using the generating method described above, we can alternate the spatial distribution of redshifts without changing the distribution of coordinates.We find that the fitting results of type C datasets are fairly consistent with "isotropic" samples, as shown in Figure 14.Therefore, the anisotropy of redshift distribution does not cause a significant influence on fitting results.

Conclusions
In this paper, we study three different datasets of SNe Ia, namely Union2.1,Constitution, and JLA, to find possible dipolar anisotropy in redshift-distance relation.We fit the dipole and monopole parameters by minimizing χ 2 , then run MCMC sampling to determine the error range and confidence level of the fitting parameters.
We also study the effects of anisotropy of coordinate and redshift distribution in dipole fitting method, and find that anisotropy distribution of coordinates can cause fitted dipole direction to concentrate in places where the mean absolute deviation is larger, cause dipole magnitude to become larger.However, anisotropy distribution of redshifts does not have a significant influence on fitting results.
In future, the next generation cosmological surveys, such as LSST (Ivezic et al. 2008), Euclid (Amendola et al. 2016), and WFIRST (Hounsell et al. 2017) will observe much larger SNe Ia datasets with enhanced light-curve calibration, which may shed light on the anisotropy in redshift-distance relation.
98.3% 19.7% 0.23% • n represents the projected dipole magnitude in the direction of the given SNe Ia sample.n can be represented in galactic coordinate as n = cos(b) cos(l) î + cos(b) sin(l) ĵ + sin(b) k. (10) Then the projection is A • n = cos(b) cos(l)A x + cos(b) sin(l)A y + sin(b)A z .

Fig. 3 .
Fig. 3.-Best-fitting dipole direction (star) of JLA dataset.Scatter points represents dipole directions and magnitudes generated by MCMC sampling of original samples.

Fig. 4 .
Fig. 4.-Blue lines show marginalized likelihoods of dipole A, monopole B and (l, b) for Union2.1 dataset, and black vertical lines represent best-fitting values.Green dotted lines represent results of "isotropic" samples.

Fig. 7 .
Fig. 7.-Probability distribution functions of rectangular components of dipole A and monopole B for JLA dataset directly using (l, b, A) as fitted parameters.

Fig. 8 .
Fig. 8.-Probability distribution functions of rectangular components of dipole A and monopole B for JLA dataset directly using (A x , A y , A z ) as fitted parameters.

Fig. 9 .
Fig. 9.-Distribution of dipole directions of type B samples.Dipole directions concentrate near crossed positions.

Fig. 10 .
Fig. 10.-Likelihood of dipole magnitude A of different samples.Blue solid line indicates result of synthetic samples with isotropically distributed coordinates, which tend to reduce the dipole magnitude.Green dotted line indicates result of synthetic samples with extremely an-isotropically distributed coordinates, which increase the dipole magnitude.

Fig. 11 .
Fig. 11.-Sample count and dipole distribution of Union2.1 dataset.The contour shows mean absolute deviations, which represent how far the sample density near a specific point differs from the average density.

Table 2 :
Best-fitting results of dipole and monopole for three datasets