Probing the anisotropic expansion from supernovae and GRBs in a model-independent way

# Probing the anisotropic expansion from supernovae and GRBs in a model-independent way

J. S. Wang and F. Y. Wang
School of Astronomy and Space Science, Nanjing University, Nanjing 210093, China
Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210093, China
jiesh.wang@gmail.com(JSW)fayinwang@nju.edu.cn(FYW)
###### Abstract

In this paper, we study the anisotropic expansion of the universe using type Ia supernovae Union 2.1 sample and 116 long gamma-ray bursts. The luminosity distance is expanded with model-independent cosmographic parameters as a function of directly. Thus the results are independent of cosmology model. We find a dipolar anisotropy in the direction (, ) in galactic coordinates with a significant evidence (more than ). The magnitude is for the dipole, and for the monopole, respectively. This dipolar anisotropy is more significant at low redshift from the redshift tomography analysis. We also test whether this preferred direction is caused by bulk flow motion or dark energy dipolar scalar perturbation. We find that the direction and the amplitude of the bulk flow in our results are approximately consistent with the bulk flow surveys. Therefore, bulk flow motion may be the main reason for the anisotropic expansion at low redshift, but the effect of dipolar distribution dark energy can not be excluded, especially at high redshift.

###### keywords:
cosmology: theory, dark energy, Type Ia supernovae

## 1 Introduction

The Universe is homogeneous and isotropic on cosmic scales on the basis of the cosmological principle. It is the foundation in modern cosmology. This principle is well confirmed by the precise measurements of cosmic microwave background (CMB) from Wilkinson Microwave Anisotropy Probe (WMAP) (Hinshaw et al., 2013) and (Planck Collaboration et al., 2013). However, in the processing of CMB data, the motion of our Local Group of galaxies should be deducted. Kogut et al. (1993) obtained that the peculiar velocity is km s towards () using the COBE Differential Microwave Radiometers first year data. Bulk flow velocity on the scales around Mpc is found to be km s towards (), roughly close to CMB dipole (Watkins, Feldman & Hudson, 2009). But, it’s much larger than the expected bulk flow velocity on the same scale, which is approximately 110 km s in the standard CDM normalized with WMAP5 (). This hints that the universe may have a preferred expanding direction.

Additional evidences for such dipolar anisotropy have been obtained by low multipoles alignment in CMB angular power spectrum (Lineweaver et al., 1996; Tegmark et al., 2003; Bielewicz, Górski & Banday, 2004; Frommert & Enßlin, 2010), large scale alignments of quasar polarization vectors (Hutsemékers et al., 2005, 2011), dark energy dipole in type Ia supernovae (SNe Ia) (Antoniou & Perivolaropoulos, 2010; Mariano & Perivolaropoulos, 2012; Yang, Wang & Chu, 2014), and the spatial variation in fine-structure constant (Webb et al., 2011; King et al., 2012). The significances of these dipoles anisotropy are around . Indeed, many studies using SNe Ia data to test if the universe accelerates isotropically have been done (Kolatt & Lahav, 2001; Bonvin, Durrer & Kunz, 2006; Gordon, Land & Slosar, 2007; Schwarz & Weinhorst, 2007; Gupta, Saini & Laskar, 2008; Koivisto & Mota, 2008a, b; Blomqvist, Mörtsell & Nobili, 2008; Cooray, Holz & Caldwell, 2010; Gupta & Saini, 2010; Cooke & Lynden-Bell, 2010; Antoniou & Perivolaropoulos, 2010; Campanelli et al., 2011; Koivisto et al., 2011; Colin et al., 2011; Mariano & Perivolaropoulos, 2012; Turnbull et al., 2012; Cai & Tuo, 2012; Li et al., 2013; Yang, Wang & Chu, 2014). These studies of the anisotropic effects are mainly considered to be caused by bulk flow motion or dark energy dipolar distribution on the basis of CDM.

Cai et al. (2013) examined the dark energy anisotropy deviations using the SNe Ia of Union 2 sample and 67 gamma-ray bursts (GRBs) from Liang et al. (2008) and Wei (2010). However, their results show that the anisotropic evidence in CDM doesn’t improve much compared to the results from SNe Ia data alone obtained by Mariano & Perivolaropoulos (2012). Thereby, the significance of anisotropy needs to be studied again with the joint of more high-redshift GRBs. On the other hand, dipolar anisotropy can be caused by many mechanisms, for instance, the cosmic bulk flow motion (Colin et al., 2011; Turnbull et al., 2012; Feindt et al., 2013; Li et al., 2013; Rathaus, Kovetz & Itzhaki, 2013) and dark energy anisotropy (Koivisto & Mota, 2008b; Antoniou & Perivolaropoulos, 2010; Perivolaropoulos, 2014). Therefore, it’s also important to distinguish which mechanism is dominant in the deviation of isotropy.

Cosmological models are assumed in the previous studies, thus, their results of anisotropic expansion are model-dependent. In this paper, we use a model-independent method to study the anisotropic expansion from standard candles, i.e., expanding the luminosity distance using fourth order Hubble series parameters as a function of directly (Cattoen & Visser, 2007; Wang & Dai, 2011). This expansion is only dependent on the cosmological principle and the Friedmann-Robertson-Walker (FRW) metric. The Union 2.1 SNe Ia sample (Suzuki et al., 2012) and 116 GRBs (Wang, Qi & Dai, 2011) are used in our study.

The structure of this paper is organized as follows: in the next section, we give brief introductions of observational data. We then introduce the method for quantifying the anisotropic expansion effects on luminosity distances and give the significance through Monte Carlo simulation. In section 3, we divide the data set into several portions with two approaches: redshift bins and variable redshift limits, then we analyze the anisotropic expansion in different redshift ranges. In section 4, we test the bulk flow dipole and simplified dark energy dipolar perturbation model as possible mechanisms for anisotropy. Conclusions and discussions are given in section 5.

## 2 Dipolar anisotropic expansion with cosmography parameters

### 2.1 Observational data

In analysis, we use the latest Union 2.1 sample (Suzuki et al., 2012) to constrain the dipolar anisotropy, which contains 580 SNe Ia and covers the redshift range . To avoid the lack of high redshift data, we also combine the 116 GRB samples, which are compiled and calibrated by Wang, Qi & Dai (2011) and Wang & Dai (2011) (see detailed information including equatorial coordinates in Table 5). The redshift of GRBs reaches up to . The equatorial coordinates of these GRBs are taken from NASA/IPAC Extragalactic Database .

We expand the luminosity distance in terms of Hubble series parameters: Hubble parameter (), deceleration (), jerk () and snap () parameter. These four parameters are the first, second, third and fourth derivatives of the scale factor in the Taylor expansion, respectively. They are model-independent and obtained only from the FRW metric. The definitions of the cosmography parameters can be expressed as follows,

 H=˙aa,       q=−1H2¨aa, j=1H3˙¨aa,            s=1H4¨¨aa. (1)

Visser (2004) expands the luminosity distance as a function of with the cosmography parameters, which have been studied using observational data (Wang, Dai & Qi, 2009a, b). However, it diverges at high redshift, and the GRB data reaches up to a high redshift . To avoid this problem, Cattoen & Visser (2007) recast the with improved parameter . Therefore, the redshift range can be mapped into . The luminosity distance can be expanded as a function of as following on the assumption of flat Universe (Cattoen & Visser, 2007),

 dL(y)=cH0{y−12(q0−3)y2+16[11−5q0−j0]y3 +124[50−7j0−26q0+10q0j0+21q20−15q30+s0]y4+O(y5)}, (2)

where , , , are the current values. Then the distance modulus can be derived,

 μth=5logdLMpc+25. (3)

The best-fit cosmography parameters can be obtained by minimizing the , which is constructed as follow,

 χ2(H0,q0,j0,s0)=580∑i=1[μSNe(zi)−μth(zi)]2σ2μ,i+116∑i=1[μGRB(zi)−μth(zi)]2σ2μ,i, (4)

where and are the observed distance modulus and error bars, and are taken from Wang & Dai (2011).

### 2.2 Anisotropic deviation effects on luminosity distance

We convert the equatorial coordinates of each SNe Ia and GRB sample to galactic coordinates (see in Figure 1), then we find their unit vectors in Cartesian coordinates

 ^ni=cos(bi)cos(li)^i+cos(bi)sin(li)^j+sin(bi)^k. (5)

In order to quantify the anisotropic deviations on luminosity distance, we define the deviations of distance modulus from the best fit isotropic configuration as follows,

 Δμ(z)¯μ(z)≡¯μ(z)−μobs(z)¯μ(z), (6)

where are the distance modulus in the context of best-fit cosmography parameters, which are calculated in section 2.1, that is .

We use a dipole model in the direction, and a monopole ,

 (Δμ(z)¯μ(z))i=^ni⋅→D−B=Acosθ−B, (7)

where and are the magnitudes of the dipole and monopole, respectively. To fit the models with the SNe Ia and GRB data, we construct the ,

 χ2(→D,B)=696∑i=1[(Δμ(z)¯μ(z))i−Acosθi+B]2σ2i, (8)

where are the errors in data sets.

We find the dipole points to the direction (, ), which is shown in Figure 1. The black star is the dipolar expansion direction, and the dark blue blob is the error region. The magnitudes of the dipole and monopole are and , respectively. It’s approximately consistent with the results from Mariano & Perivolaropoulos (2012), Cai et al. (2013), and Yang, Wang & Chu (2014), which are based on CDM model.

### 2.3 Significance of dipolar anisotropy

Our results show that the monopole is not significant, while implies the dipolar anisotropy, around in the relative errors. To obtain the confidence level of dipole anisotropy precisely, we use the Monte Carlo (MC) simulations.

We define new distance modulus () through a Gaussian random selection function, i.e. new distance modulus () will be obtained by the normal distribution with mean values and standard deviations from the observed data. We then take place of the observed distance modulus with the newly constructed , while use the same observed redshift, standard deviations and coordinates in the observed data.

The analysis method is similar to the method in section 2.2. Then, we obtain a new magnitude of the dipolar anisotropy in each simulation. We do MC simulations in total, and divide them into 47 bins. Figure 2 illustrates the probability of each bin value . The x-axis is the simulated dipole magnitude in units of , and the y-axis is the count of each bin. The arrow points to the dipole magnitude obtained with observed data. The results show that the probability that we can observe the magnitude at is , i.e. the confidence level of the dipolar anisotropy is , larger than . It’s more significant than the results from SNe Ia Union 2 data (Mariano & Perivolaropoulos, 2012) and Union 2.1 data (Yang, Wang & Chu, 2014) alone, which give the probability and , respectively. Therefore, our result shows the significance of dipolar expansion amplitude grows larger with the combination of GRB sample. We also show the evolution of the confidence level with the increasing MC simulations in Figure 3. It illustrates that MC simulations are enough to converge.

## 3 Redshift Tomography

In this section, we focus on the anisotropic effects in different redshift ranges. We use two approaches to study these effects and compare the results with respect to error bar sizes, which relate to the confidence level. The first approach is changing the redshift upper or lower limits, and the second one is dividing the data into 6 redshift bins. The same analysis procedure presented in section 2 are used in each redshift range . The number of data points are approximately equal in each redshift bin, and we define an average redshift of each bin. The variable upper limits method starts from the upper limit , approximately Mpc. Then we increase the upper limit within six steps. The variable lower limit method starts from , then we increase it in three steps.

Our results in different redshift ranges are shown in Table 1. The results show that the Union 2.1 data constraints are more stringent than GRB data. This is obvious because of the smaller error bars of SNe Ia luminosity distances comparing with GRBs. The results from variable redshift upper limits method show that the monopole, dipole magnitudes, and the direction converge with the increasing data points. Most of the results are consistent with the full data, except the lowest redshift range. For the variable lower limits way, the magnitudes of dipole is and in redshift ranges and , respectively. Therefore, these results don’t show significant anisotropy at these high redshift ranges, because of their large relative errors.

The redshift bins methods show the anisotropy direction changes randomly with redshift. The magnitudes of monopole and dipole don’t show significant evolution with the redshift (see Figures 4 and 5), except for the redshift range , but this bin only contains 67 sample, while covers a large redshift range. We find that the lowest redshift bin () show the most significant evidence for dipolar anisotropy with the smallest relative errors, namely, . While other bins show weaker evidences for anisotropy. Because of their random directions, the total effects are even much weaker, which can be obtained from their magnitude: in the range of and in . Thus, the significant dipolar anisotropy of the full data is mainly caused by the low redshift sample.

## 4 Possible mechanism for dipolar anisotropy

We have studied the anisotropic expansion with SNe Ia and GRB luminosity distances. We find that the probability of such a dipolar anisotropy is more than , and it mainly origins from the low redshift data. While the monopole is not significant. Thus, in this section, we try to study two possible mechanisms for dipolar anisotropy. We use bulk flow motion model and simplified scalar perturbation metric model caused by dark energy dipolar distributions to fit the same data. However, our methods to quantify the magnitudes of bulk flow motion and dark energy perturbation are simplified. For the careful study of these effects, we need to use the velocity field (Koivisto & Mota, 2008a; Li et al., 2013) and anisotropy dark energy model (Koivisto & Mota, 2006, 2008b; Mariano & Perivolaropoulos, 2012). Since the magnitudes of anisotropy are very small, our results are still reliable.

### 4.1 Bulk flow motion

Bulk flow motion can affect the Hubble Parameter directly, where is the comoving distance. Many methods have been taken to analyze this effect on SNe Ia data (Bonvin, Durrer & Kunz, 2006; Colin et al., 2011; Feindt et al., 2013; Rathaus, Kovetz & Itzhaki, 2013). We choose one method of them to reconstruct the luminosity distance (Bonvin, Durrer & Kunz, 2006) as follows

 d′L(z)=dL(z)+vBF⋅^ni(1+z)2H(z), (9)

where is the velocity of the bulk flow, is the luminosity distance defined in Eq.(2), and is defined in Eq.(5). The is

 χ2(→vBF)=696∑i=1[μ(d′L)−μi]2σ2i. (10)

The results are shown in Table 2. For bulk flow motion, the effects at low redshift ranges are much more attractive. The velocity and direction are km s and () for the full data. On the scale of Mpc, i.e. , the velocity is km s, and the direction points to (). They are approximately consistent with other peculiar velocity surveys shown in Table 3 (Hudson et al., 2004; Sarkar, Feldman & Watkins, 2007; Watkins, Feldman & Hudson, 2009; Feldman, Watkins & Hudson, 2010; Ma, Gordon & Feldman, 2011). Their average velocity and direction are 344.5 km s and on the scale 82.5 Mpc. But all the velocities are larger than the expected velocity in CDM.

The direction and velocity of redshift range are consistent with the results from . Thus, for this low redshift range, the anisotropy is mainly caused by the bulk flow velocity. The origin for this motion is thought to be the attraction of Shapley Super Cluster (Colin et al., 2011). But this effect could be much weaker at high redshift, because of the larger Hubble flow. Therefore, we can not excluded the dipolar dark energy effects, especially at high redshift.

### 4.2 Simplified dark energy dipolar scalar perturbation

Another possible anisotropic mechanism is the dark energy dipolar distribution, resulting in dipolar scalar perturbation. For simplification, we use an affected metric imitating the Schwarzschild metric instead of FRW metric,

 ds2=(1−2ϕ(→x))dt2−a2(t)(1+2ϕ(→x))δijdxidxj. (11)

We assume the scalar perturbation field , where is the angular between the dipole direction and the observed sample and is the magnitude of the perturbation.

To determine the perturbed energy-momentum tensor, we base on CDM. Then the luminosity distance can be obtained by solve the Einstein equation (Li et al., 2013),

 d′L(z)=c(1+z)H0∫z0(1−dcosθ)dx√Ωm0(1+x)3+1−Ωm0−4dcosθ(1+x)53H20d2L(z) , (12)

where is defined in Eq.(2) and is the current matter density. We use the Eq.(10), and analyze the data in the same way. The results show that the anisotropy amplitude is very small (shown in Table 4). The magnitude of scalar perturbation is , the direction is (). The dipolar evidence in the redshift range is insignificant with large error bar size . But we cannot draw an exact conclusion that the dark energy distributes isotropically or not, because the high redshift sample is sparse.

## 5 conclusions and discussions

In this paper, we study the anisotropic cosmic expansion in a model-independent way. The data we use are the combination of SNe Ia Union 2.1 and 116 GRB samples. The luminosity distance is expanded with model-independent cosmography parameters: Hubble (), deceleration (), jerk () and snap () parameters. These cosmographic parameters obtained from the FRW metric are only based on the cosmological principle.

The magnitudes of dipole and monopole are and . Our results show that the dipolar anisotropy is significant. The confidence level is , more than , by doing MC simulations. It’s more significant than the results from SNe Ia Union 2 (Mariano & Perivolaropoulos, 2012) and Union 2.1 data (Yang, Wang & Chu, 2014) alone, which give out the probability and , respectively. Our results are also much more significant than the results from Cai et al. (2013), who used a combination of SNe Ia Union 2 and 67 GRBs from Liang et al. (2008) and Wei (2010). The dipolar direction in our study points to (, ) in galactic coordinates for the full data. This direction is consistent with the results from Mariano & Perivolaropoulos (2012), Cai et al. (2013) and Yang, Wang & Chu (2014).

To study the anisotropy in different redshift ranges, we used two approaches: changing the redshift ranges upper or lower limits and dividing the full data into six bins. The results are show in Table 1, and these imply that the anisotropy is more significant at low redshift ranges. The magnitude is in the redshift range , while in the bin of , the magnitude becomes to . The relative error of the latter is very large. Thus, the significant dipolar anisotropy of the full data is mainly caused by the low redshift sample. We also find that the magnitudes of anisotropy do not evolve with redshift, while the directions change randomly with redshift.

Since the monopole is not conspicuous, we focus on the dipolar anisotropy, and try to study its possible mechanisms. We consider two possible mechanisms: bulk flow motion model and simplified scalar perturbation metric model caused by dark energy distributions. We show their results in Table 2 and 4. Since both models can help to explain the dipolar effect, we compare our results to bulk flow surveys to break the degeneracy. We find the directions of the dipole from the bulk flow surveys are very close to our results, the average velocity and direction of the bulk flow surveys are 344.5 km s and around the scale 82.5 Mpc (Hudson et al., 2004; Sarkar, Feldman & Watkins, 2007; Watkins, Feldman & Hudson, 2009; Feldman, Watkins & Hudson, 2010; Ma, Gordon & Feldman, 2011). Our results from SNe Ia and GRB data are 271 km s and () on the scale of Mpc. Therefore, the anisotropic expansion at low redshift should be mainly caused by bulk flow motion. But the velocity km s is too small comparing with the Hubble flow at high redshifts. Thus, bulk flow motion can be ignored at high redshift. Therefore, we can not excluded the dipolar dark energy effects, especially at high redshift.

The dark energy dipolar scalar perturbation can affect the SNe and GRB luminosity distance on larger scales. But the redshift tomography results show the significance of anisotropy is insignificant at high redshift. The magnitude of dipole is in redshift ranges . Because the high-redshift sample is sparse, we cannot draw an exact conclusion that the dark energy distributes isotropically or not. Further study will need more high-redshift GRBs, since the SNe Ia cannot reach to higher than 2.0, GRBs are good probes to study cosmology at high redshift (Basilakos & Perivolaropoulos, 2008; Wang & Dai, 2011; Wang, Qi & Dai, 2011).

## Acknowledgements

We thank the referee for detailed and very constructive suggestions that have allowed us to improve our manuscript. We have benefited from reading the publicly available codes of Mariano & Perivolaropoulos (2012). This work is supported by the National Basic Research Program of China (973 Program, grant 2014CB845800) and the National Natural Science Foundation of China (grants 11373022, 11103007, and 11033002). This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.

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