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Comparison between hemisphere comparison method and dipole-fitting method in tracing the anisotropic expansion of the Universe use the Union2 dataset

Type-Ia supernovae (SNe Ia) are often used as the standard candles to probe the anisotropic expansion of the Universe. In this paper, we make a comprehensive comparison between the hemisphere comparison (HC) method and dipole-fitting (DF) method in searching for the cosmological preferred direction using the Union2 dataset, a compilation of 557 well-calibrated SNe Ia. We find that the directions of the faintest SNe Ia derived from these two methods are approximately opposite. Monte Carlo simulations show that the results of the HC method strongly depend on the distribution of the data points in the sky. The coincidence that the HC method and DF method give two completely opposite directions may be due to the extremely nonuniform distribution of the Union2 dataset.


INTRODUCTION
The cosmological principle, which states that the Universe is homogeneous and isotropic on large scale, is one of the foundations of the Modern cosmology. However, the cosmological principle is strongly doubted recently. Observations on the largescale structure of the Universe, such as the large-scale bulk flow (Kashlinsky et al. 2008;Watkins et al. 2009;Lavaux et al. 2010), the alignments of low multipoles in the CMB angular power spectrum (Lineweaver et al. 1996;Tegmark et al. 2003;Bielewicz et al. 2004;Copi et al. 2010;Frommert & Enßlin 2010), the large-scale alignments of the quasar polarization vectors (Hutsemekers et al. 2005(Hutsemekers et al. , 2011, the spatial variation of the fine-structure constant (King et al. 2012;Mariano & Perivolaropoulos 2012), so on, show that the Universe may be anisotropic. Especially, the recent released WMAP data (Bennett et al. 2011(Bennett et al. , 2013 and Planck data (Ade et al. 2013a,b) further arouse great interests in probing the possible anisotropy of the Universe. Bonvin et al. (2006) analyzed a sample of nearby supernovae and find a dipole of luminosity distance consistent with the CMB at a significance of more than 2σ. More interestingly, it is found that the CMB maximum temperature asymmetry axis is aligned with dark energy dipole and bulk flow directions (Antoniou & Perivolaropoulos 2013). All of these hint that there may exist a preferred direction in the Universe.
Type-Ia supernovae (SNe Ia) are the ideal distance indicators to trace the expansion history of the Universe due to their consistent absolute magnitudes. In fact, they have been widely used as the standard candles to search for the preferred direction of the Universe. Generally speaking, there are two different ways to investigate the possible anisotropy from the data. One is directly fitting the data to a specific anisotropic cosmological model (Campanelli et al. 2011;Li et al. 2013;Chang et al. 2013Chang et al. , 2014aSchücker et al. 2014). Many anisotropic cosmological models have be proposed to match the observations. The Bianchi I type cosmological model (Campanelli et al. 2011;Schücker et al. 2014) and the Rinders-Finsler cosmological model (Chang et al. , 2014a are two models which are consistent with the SNe Ia data. A scalar perturbation of the ΛCDM model may also break the spherical symmetry of the Universe such that a preferred axis arises ). Mariano & Perivolaropoulos (2012) proposed the extended topological quintessence model which leads to a spherical inhomogeneous distribution for dark energy density. The large-scale matter density inhomogeneities would naturally introduce accelerating expansion anisotropy of the Universe.
An alternative method is to analyze the SNe Ia data in a model-independent way (Antoniou & Perivolaropoulos 2010;Mariano & Perivolaropoulos 2012;Cai & Tuo 2012;Cai et al. 2013;Kalus et al. 2013;Zhang et al. 2013;Yang et al. 2014). The advantage of this method is that it does not rely on the particular cosmological model. The hemisphere comparison (HC) method and dipole-fitting (DF) method are two most used methods in literatures. The HC method divides the data points into two subsets according to their position in the sky and fit the subsets to an isotropic cosmological model (e.g., ΛCDM model) independently, while the DF method directly fits the data points to a dipole (or dipole plus monopole) model. Antoniou & Perivolaropoulos (2010) found that the SNe Ia are the faintest in the direction pointing to (l, b) = (309 • , 18 • ) in the Union2 dataset using the HC method. The work of Cai & Tuo (2012) further confirms this result. In a later paper, Mariano & Perivolaropoulos (2012) found a dipole direction pointing towards (l, b) = (309.4 • , −15.1 • ) in the same dataset, but using the DF method. At the first glance, these two preferred directions are consistent within uncertainties. However, further investigation shows that the dipole direction corresponds to the brightest SNe Ia, while the faintest SNe Ia are in the opposite direction (see the next section for details). This is to say, the result of the HC method completely conflicts with that of the DF method. This triggers us to further study the discrepancies between these two methods.
This paper aims to addressing the above issues. The rest of the paper is organized as follows: In section 2, we make a comprehensive comparison between the HC method and DF method using the Union2 compilation. It is found that there are obvious discrepancies between these two methods. In section 3, we discuss the reasons that may cause the discrepancies. Finally, section 4 is devoted to a short summary.

COMPARISON BETWEEN THE HC METHOD AND DF METHOD
In this section, we make a comprehensive comparison between the HC method and the DF method using the Union2 dataset. This section is divided into four subsections. In section 2.1, we introduce the Union2 dataset that is used in the analysis and the basic formulae of the ΛCDM model. Section 2.2 shortly reviews the HC method, while section 2.3 describes the DF method. We find that the directions of the faintest SNe Ia obtained using these two methods are approximately opposite. In section 2.4, we apply Monte Carlo simulations to further investigate the discrepancies between these two methods.

The Union2 dataset and ΛCDM model
The Union2 dataset (Amanullah et al. 2010) consists of 557 SNe Ia with well-observed redshift in the range of z ∈ [0.015, 1.4]. The distance moduli and their uncertainties are extracted from the SALT2 light-curve fitter. The directions of SNe Ia are well localized in the sky of the equatorial coordinates system (Blomqvist et al. 2010), which can be easily converted to the galactic coordinates system (Smith 1989). The distribution of the Union2 dataset in the sky of the galactic coordinates system is showed in Fig.1. As can be seen, the SNe Ia are not uniformly distributed. A lot of data points cluster near a line corresponding to the equator of the equatorial coordinates system.
We work in the background of the standard cosmological model, i.e., the ΛCDM model, in which the luminosity distance is written as where c is the speed of light, H0 is the Hubble constant, and ΩM is the normalized matter density. The luminosity distance can be conveniently converted to the dimensionless distance modulus as µ(z) = 5 log 10 DL(z) Mpc + 25.
The best-fit parameters (ΩM , H0) can be obtained using the usual least-χ 2 method. Define χ 2 as where N = 557 is the number of SNe Ia in the Union2 dataset, µ th is theoretical distance modulus calculated from Eqs. (1) and (2), µ obs is the observed distance modulus, and σµ is the uncertainty of µ obs . The sum runs over all the data points. The best-fit values are the ones which can minimize χ 2 , i.e.,  Figure 1. The distribution of the Union2 data points in the galactic coordinates system.

The HC method
To investigate the possible anisotropic expansion of the Universe, we apply the HC method to the Union2 dataset. The main procedures of the HC method are as follows: (i) Generate a random direction (l, b) in the northern hemisphere of the galactic coordinates system, where l ∈ [0 • , 360 • ) and b ∈ [0 • , 90 • ] are random numbers following the uniform distribution.
(ii) The " equator" corresponding to the random direction divides the sky in to two hemispheres, which are called " up" hemisphere and " down" hemisphere, respectively. Correspondingly, the data points are divided into two subsets according to their position in the sky.
(iii) Find the best-fit parameters (ΩM , H0) in each hemisphere. Define the magnitude of anisotropy in that direction as where ΩM,u and Ω M,d are the best-fit parameters in the " up" hemisphere and " down" hemisphere, respectively. (iv) Repeat steps (i) to (iii) n times (for example, n = 400), and find the maximum value of |DΩ| and the corresponding direction (l, b).
We plot the pseudo color map of DΩ(l, b) in Fig.2. When plotting this figure, instead of choosing random axes, as is described in step (i), we divide the sky into 5 • × 5 • grids. The axes are chosen to be towards the center of each grid. This makes the figure to be more smooth, but makes the computation much more time-consuming. The maximum value of |DΩ| is found to be in the direction (Antoniou & Perivolaropoulos 2010) or equivalently, its opposite direction 1 The 1σ errors are derived using the method described in Antoniou & Perivolaropoulos (2010). In this two directions, the magnitude of anisotropy reaches its maximum, i.e., |DΩ|max = 0.43 ± 0.06. To see the possible anisotropy of the Hubble parameter, we apply the anisotropy analysis done for the parameter ΩM to the parameter H0. Define the magnitude of anisotropy of the Hubble parameter as  Following the procedures of the HC method, we find that the maximum value of H0 is in the direction while the minimum value of H0 is in the opposite direction, i.e., Therefore, the maximum anisotropy of the Hubble parameter is |DH |max = 0.03 ± 0.01. This is consistent to the result of Kalus et al. (2013), who obtained ∆H/H < 0.038 at 95% confidence level using four groups of SNe Ia datasets of redshift z < 0.2, although the redshifts of the Union2 dataset extend to z = 1.4. This implies that H0 is not sensitive to the directions. In fact, if we fix H0 to 70.0 km s −1 Mpc −1 (the best-fit value of the ΛCDM model), and implement the HC method to search the anisotropy of ΩM , we find a similar direction of maximum anisotropy, pointing towards (l, b) = (320 . If we assume that the Hubble parameter is isotropic (this is approximately true, as is showed here) and the absolute magnitude of SNe Ia is a constant, smaller ΩM leads to larger luminosity distance so fainter supernova (at a fixed redshift). Therefore, the hemisphere of the faintest SNe Ia is in the direction , while the hemisphere of the brightest SNe Ia is in the opposite direction.

The DF method
To check the reasonableness of the HC method, we apply the DF method to the same dataset. The DF method is a method of directly fitting the data to a dipole (or dipole plus monopole) model. If the Universe is really intrinsically anisotropic and there exists a preferred direction, these two methods should give the similar results. The main steps of the DF method are as follows: (i) Calculate the deviation of the distance modulus of each supernova from its best fit ΛCDM value, i.e., define whereμ is the distance modulus predicted by the ΛCDM model with the best-fit parameters given in Eq.(4), and µ is the observed distance modulus.
(ii) Given any axisn = cos(b) cos(l)î + cos(b) sin(l)ĵ + sin(b)k in the sky, calculate the angle of each supernova with respect to the axis, which is determined by cos θi =n ·p i , wherep i = cos(bi) cos(li)î + cos(bi) sin(li)ĵ + sin(bi)k is the unit vector pointing towards the i-th supernova.
(iii) Rewrite Eq.(11) into the dipole plus monopole form, i.e., where A and B are the magnitudes of the dipole and monopole, respectively. Using the least-χ 2 method, we can find the best-fit parameters (A, B, l, b) which can minimize the χ 2 , where σi = σ The best-fit dipole direction is found to be towards (Mariano & Perivolaropoulos 2012) and the magnitudes of the dipole and monopole are given as We can see that the magnitude of the monopole is one order of magnitude smaller than that of the dipole. On the other hand, the statistical significance of the dipole is about at the 2σ level, while the statistical significance of the monopole is less than 1σ. In fact, if we fit the data with a dipole only, we get a similar direction, pointing towards and the magnitude of the dipole is also about at the 2σ level, i.e., It is interesting that the dipole direction derived from the DF method is close to the direction of the faintest SNe Ia derived from the HC method. The angle between these two directions is about ∆θ ≈ 33 • . Therefore, some authors believe that the result of the DF method is coincident with that of the HC method (Antoniou & Perivolaropoulos 2010;Mariano & Perivolaropoulos 2012). However, an important thing has been ignored. Since the magnitudes of the dipole and monopole are positive, at the dipole direction, cos θ = 0, and the right-hand-side of Eq.(12) is positive. Therefore, the right-hand-side of Eq.(11) is also positive, i.e.,μ > µ. This means that the observed distance modulus (so the luminosity distance) is smaller than that predicted by the ΛCDM model. Thus, in the dipole direction, ΩM is the largest (larger than the best-fit value of the ΛCDM model), since µ decreases as ΩM increases. This further implies that the dipole direction is the direction of the brightest SNe Ia. On the contrary, the faintest SNe Ia are in the opposite direction. This conflicts with the result of the HC method -the direction of the faintest SNe Ia derived from the DF method is almost opposite to that from the HC method! In the following, we will show that the discrepancies between these two methods can be seen more clearly if the data points are uniformly distributed in the sky.

Monte Carlo simulations
This subsection aims to further verifying the discrepancies between the HC method and DF method. To address this issue, we perform Monte Carlo simulations. We generate a dataset which exactly follows the dipole distribution, and then perform the HC method on the dataset to see if the obtained preferred direction is close to the dipole direction or not. The steps of the simulations are given as follows: (ii) The distance modulus of each supernova is replaced by the simulated distance modulus, calculated from µsim(z) = µ(z)[1 − A cos θ], where θ is the angle of the supernova with respect to the dipole direction given by Eq.(16), and A is the magnitude of the dipole given by Eq.(17). We endow each data the same weight, i.e., the uncertainties of all the data points are assumed to be the same. Therefore, the simulated data points always follow the dipole distribution, and performing the DF method on the simulated dataset always leads to the dipole direction (l, b) = (309.0 • , −19.3 • ).
(iii) Find the direction of the faintest SNe Ia (i.e., the direction of minimum ΩM ) for the simulated dataset using the HC method described in section 2.2.
(iv) Repeat steps (i) to (iii) n times (for example, n = 200), we can find n directions of the faintest SNe Ia, as well as n magnitudes of anisotropy. Fig.3 shows the obtained directions of the faintest SNe Ia in 200 simulations. In each simulation, we choose 500 axes randomly in the hemisphere. The blue star is the dipole direction (i.e., the direction of the brightest SNe Ia derived from the DF method) pointing towards (l, b) = (309.0 • , −19.3 • ), and the red star is its opposite direction (i.e., direction of the faintest SNe Ia derived from the DF method). The blue patch is a circular region with " radius" θ 60 • with respective to the brightest SNe Ia direction, while the red patch is a circular region with " radius" θ 40 • with respective to the faintest   SNe Ia direction. The black dots are the faintest SNe Ia directions derived from the HC method. If the HC method and the DF method are consistent, the black dots should cluster near the red star. However, we can see in Fig.3 that the black dots are almost uniformly distributed in the sky, except for the blue region. None of the black dots fall into the blue region. About 13.5% (27 out of 200) black dots fall into the red region. This implies that the HC method and the DF method are inherently different methods and they can lead to very different results in many cases. There are only about 13.5% possibility that they can get similar results (within 40 • uncertainty). It is unlikely that these two methods get two completely opposite directions (within 60 • uncertainty), if the dataset are uniformly distributed in the sky. The coincidence that the faintest SNe Ia directions derived from the HC method and that from the DF method are approximately opposite for the Union2 may be due to the nonuniform distribution of the dataset.
The maximum anisotropy of ΩM , (i.e., |DΩ|max) in the 200 simulations follows the Gauss distribution. The histogram of |DΩ|max is plotted in Fig.4. It can be well fitted by the Gauss function with the best-fit values and their 1σ uncertainties a = 42.8 ± 6.2, b = 0.038 ± 0.002, c = 0.013 ± 0.002.
The average maximum anisotropy of the simulated data, |DΩ|max = 0.038, is about one order of magnitude smaller than that obtained from the Union2 dataset. This shows that both the magnitude of anisotropy and the preferred direction derived from the HC method strongly depend on the distribution of the data points, even though implementing the DF method to the simulated dataset always leads to the same magnitude and direction. The ability of the HC method to detect a signal in the dataset depends sensitively on the distribution of data points in the sky. The distribution-dependence of some methods in searching for the signal of anisotropy has already been noticed by Appleby & Shafieloo (2014).

DISCUSSION
As we have showed, the faintest SNe Ia direction obtained from the HC method strongly depends on the distribution of the data points in the sky. Different distributions will lead to very different directions of the faintest SNe Ia, even if the data    points are exactly follow the dipole distribution. To further show the relation between the faintest SNe Ia direction and the nonuniform distribution of the Union2 dataset, we perform a procedure similar to the HC method, but we take the number difference of SNe Ia in two opposite hemispheres as a diagnostic to quantify the anisotropy level of the distribution. Define where Nu and N d are the numbers of SNe Ia in the " up" and " down" hemispheres, respectively. We divide the sky into 5 • × 5 • grids, and calculate the DN (l, b) value in the center of each grid. The pseudo color map of DN (l, b) is plotted in Fig.5. As is showed, the most sparse direction points towards (l, b) = (312 • , 2 • ), while the most clustered direction is its opposite direction, i.e., (l, b) = (132 • , −2 • ). The numbers of SNe Ia in these two hemispheres are 101 and 456, respectively. In these n n O C C θ θ Figure 6. The celestial sphere.n 1 andn 2 are two directions with separation θ, while C 1 and C 2 are the corresponding equators.
two directions, the maximum number anisotropy is |DN |max = 1.27. This implies that the distribution of the Union2 dataset in extremely nonuniform. Comparing the pseudo color maps of Fig.2 and Fig.5, we can see that they are similar. Especially, it is amazing that the faintest SNe Ia direction derived from the HC method is so close to the most sparse direction (with angle difference ∆θ ≈ 16 • ), while the brightest SNe Ia direction is near the most clustered direction. The nonuniform distribution of the data points may significantly affect the detected signal of anisotropy. In addition, we just use the diagonal errors in the fitting. Considering the non-diagonal components of the covariance matrix may much increase the uncertainties, such that the signal of anisotropy even vanishes (Jiménez et al. 2014).
There is another issue that should be payed specific attention when using the HC method. In order to search for the maximum anisotropy direction of the data points, we can choose an axis randomly and calculate the anisotropy level in this direction. If the axis runs over the sky, the maximum anisotropy direction can be found. Besides the random algorithm, we can divide the whole sky into equal longitude and latitude grids. Alternatively, we can divide the sky into equal area patches using the Healpix method (Gorski et al. 2005). Since the random algorithm is the fast, it is used by most authors. In this paper, we also take the random algorithm to search for the preferred directions. But Fig.2 and Fig.5 are plotted using the equal longitude and latitude grids method, to make them seem more smooth. Some authors argue that it is enough to choose the number of random axes a little larger that the number of SNe Ia in each hemispheres, because change the direction of an axis does not change the corresponding ∆ΩM /ΩM until a data point is crossed by the corresponding equator line (Antoniou & Perivolaropoulos 2010;Cai & Tuo 2012). " Such a crossing is expected to occur when the direction of an axis changes by approximately the mean angular separation between data points" (Antoniou & Perivolaropoulos 2010). In fact, this is a misunderstanding. We will show in the following that much more axes should be chosen in order to ensure that only one data point is crossed when changing an axis to its adjacent one, although increasing the number of axes may not improve the accuracy significantly.
Suppose an axis is changed fromn1 ton2, and the corresponding equator line is changed from C1 to C2 (see Fig.6). The angle betweenn1 andn2 is denoted by θ. The area of spherical surface swept by the equator is given as Assume that there are N data points uniformly distributed in the sphere, then the number of data points swept by the equator is about If we randomly choose n axes in one hemisphere, then the angle between two adjacent axes, ∆θ, can be approximately determined by where we have assumed that ∆θ ≪ 1. For n = 400, as most authors take, we have ∆θ ≈ 4 • . Thus, the number of data points crossed by the equator is about ∆N = N sin ∆θ/2 ≈ 20, where N = 557 for the Union2 dataset. This is to say, if one axis is changed to its adjacent axis, there are about 10 data points go from the " up" hemisphere to the " down" hemisphere, and at the same time, about 10 data points go from the " down" hemisphere to the " up" hemisphere. This is the reason why the obtained ∆ΩM /ΩM values are so sensitive to the directions. If we require that no more than one data point is crossed by the equator, the angle between two adjacent axes should be smaller than ∼ 0.4 • . In other words, the axes in each hemisphere should be as much as n ∼ 4 × 10 4 , two order of magnitudes larger than the number of data points. It should be mentioned that the Hubble parameter derived from the HC method is not exactly isotropic, although the magnitude of anisotropy is very small. It seems that H0 is larger in the hemisphere where ΩM is smaller. Even though a lower ΩM implies fainter SNe Ia in this hemisphere, a higher value of H0 indicates the opposite (i.e., brighter SNe Ia). If the contribution of H0 is more significant than the contribution of ΩM , then this implies that the direction of lower ΩM is also the direction of brighter SNe Ia. But this occurs only if the redshift is small enough. For large redshift, smaller ΩM always leads to fainter SNe Ia.

SUMMARY
In this paper, we compared the HC method and DF method in probing the anisotropic expansion of the Universe using the Union2 dataset. We found that the faintest SNe Ia directions obtained from the HC method and the DF method are approximately opposite. These two directions are either close to the most sparse direction of the data points, or near the most clustered direction. In order to further investigate the discrepancies between these two methods, we performed the Monte Carlo simulations. It was found that the faintest SNe Ia direction obtained from the HC method strongly depends on the distribution of the data points, although using the DF method we get a fixed direction. There is about 13.5% probability that these two methods get the consistent directions, if the data points are uniformly distributed in the sky. It is unlikely that they lead to two completely opposite directions. The coincidence that the faintest SNe Ia directions obtained from these two method are approximately opposite in the Union2 dataset may be because of the extremely nonuniform distribution of the data points. We pointed out that choosing the number of axes a litter larger that the number of data points in each hemisphere is not accurate enough. The maximum accuracy can be achieved only if the number of axes is about two order of magnitudes larger than the number of data points in each hemisphere. However, even if we improve the accuracy, the conflictions between the HC method and DF method still can not be reconciled.