PROBING THE ISOTROPY OF COSMIC ACCELERATION TRACED BY TYPE Ia SUPERNOVAE

SNe Ia are not perfect standard candles; their peak brightnesses correlate with their colors, light-curve widths, and the masses of their host galaxies. In the Union2.1 sample, individual light curves are analyzed with the SALT2 fitter which provides estimates for three parameters: the peak magnitude, m_obs, in the rest-frame B band, the deviation, x₁, from the average light-curve shape, and the deviation, c, from the mean $B$–$V$ color. The color and light-curve shape-corrected distance modulus is then written in terms of four unknown parameters ($\alpha ,\beta ,\delta$ and M_B) so that

$$\begin{eqnarray}&&{\mu }_{B}={m}_{\mathrm{obs}}+\alpha \cdot {x}_{1}-\beta \cdot c+\delta \cdot {P}_{\mathrm{host}}-{M}_{B},\end{eqnarray} \tag{ 1 }$$

where M_B is the absolute B-band magnitude at the maximum of an SN Ia and P_host denotes the probability that the SN Ia is hosted by a galaxy with a stellar mass ${M}_{*}\lt {10}^{10}{M}_{\odot }$. This probability is estimated differently for untargeted and targeted surveys.

In the context of the theory of general relativity, homogeneous and isotropic universes are described by Friedmann–Lemaître–Robertson–Walker models. For simplicity we only consider flat models in which the density parameters for the matter and the cosmological constant satisfy the relation ${{\rm{\Omega }}}_{m}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}=1$. Following standard practice, we write the magnitude–redshift relation of SNe Ia in terms of the distance modulus

$$\begin{eqnarray}&&\mu (z)=5\;{\mathrm{log}}_{10}{d}_{{\rm{L}}}(z,{{\rm{\Omega }}}_{{\rm{\Lambda }}})+5\;{\mathrm{log}}_{10}(\displaystyle \frac{{D}_{{\rm{H}}}}{\mathrm{Mpc}})+25,\end{eqnarray} \tag{ 2 }$$

where

$$\begin{eqnarray}&&{d}_{{\rm{L}}}={\int }_{0}^{z}\displaystyle \frac{(1+z){dq}}{{[{(1+q)}^{2}(1+{{\rm{\Omega }}}_{m}q)-q(2+q){{\rm{\Omega }}}_{{\rm{\Lambda }}}]}^{1/2}}\end{eqnarray} \tag{ 3 }$$

is the dimensionless "Hubble-constant-free" luminosity distance and ${D}_{{\rm{H}}}=c/{H}_{0}$ is the Hubble radius defined in terms of the speed of light and the present-day value of the Hubble constant.

Classically, the SN data are fitted with a cosmological model assuming Gaussian errors and following a maximum likelihood approach (e.g., Astier et al. 2006). For N SNe Ia, this corresponds to minimizing the target function

$$\begin{eqnarray}&&{\chi }^{2}={{\boldsymbol{V}}}^{T}\;{{\boldsymbol{C}}}^{-1}\;{\boldsymbol{V}}\end{eqnarray} \tag{ 4 }$$

where ${\boldsymbol{V}}$ is an N-dimensional vector with elements ${V}_{i}={\mu }_{B,i}(\alpha ,\beta ,\delta ,{M}_{B})-\mu ({z}_{i};{H}_{0},{{\rm{\Omega }}}_{{\rm{\Lambda }}})$ and ${\boldsymbol{C}}$ is the covariance matrix of the errors in the observed distance moduli. For the Union compilations, this matrix is publicly available. Its off-diagonal elements include several contributions due to the light-curve fits, galactic extinction, gravitational lensing, peculiar velocities, and sample-dependent systematics. The nuisance parameters $\alpha ,\beta ,\delta$ and M_B are fitted simultaneously with the cosmological parameters. Actually, the best-fit values for M_B and H₀ are completely degenerate as only the combination ${\mathcal{M}}={M}_{B}+5{\mathrm{log}}_{10}({D}_{{\rm{H}}}/\mathrm{Mpc})$ appears in Equation (4). Using the whole data set gives the following best-fit values (Suzuki et al. 2012) α = 0.121, β = 2.47, δ = −0.032, and ${M}_{B}=-19.321$ (for ${H}_{0}=70$ km s⁻¹ Mpc⁻¹).

The parameters α, β, and δ describe correlations between different SN observables and might vary for the different surveys of a compilation. Karpenka et al. (2015) found inconsistencies between the values of these correction parameters in the Union2 catalog. In order to account for possible direction-dependent systematics, when we consider localized sub-samples of the Union2.1 data, we should in principle allow them to vary freely. However, this would require knowledge of the covariance between m_obs, x₁, and c for individual SNe. Regrettably this information is not provided in the Union2.1 catalog. We therefore adopt a simplified approach by assuming a constant correction, μ_cor, for the distance modulus of all the SNe lying within a cone. In other words, we keep the quantities α, β, δ, and ${\mathcal{M}}$ fixed at their global best-fit value (hereafter denoted with a hat) but we write

$$\begin{eqnarray}&&{V}_{i}={\mu }_{B,i}(\hat{\alpha },\hat{\beta },\hat{\delta },{\hat{M}}_{B})-5{\mathrm{log}}_{10}{d}_{{\rm{L}}}({z}_{i};{{\rm{\Omega }}}_{{\rm{\Lambda }}})-{{\rm{\Delta }}}_{0}\end{eqnarray} \tag{ 5 }$$

with ${{\rm{\Delta }}}_{0}=5{\mathrm{log}}_{10}({{cH}}_{0}^{-1})+25-{\mu }_{\mathrm{cor}}$. Note that Δ₀ accounts for both an "anisotropic Hubble constant" and for the mean effect of variations in $\alpha ,\beta ,\delta$ and M_B due to systematic errors. We are left with a two-dimensional problem. For each pixel on the sky, we then determine the best-fitting values of the cosmological parameter ${{\rm{\Omega }}}_{{\rm{\Lambda }}}$ and of the correction parameter Δ₀ by minimizing the χ² target function (covariances are extracted from ${\boldsymbol{C}}$ after identifying the SNe in the cone). However, the model parameters anticorrelate: directions associated with large values of ${{\rm{\Omega }}}_{{\rm{\Lambda }}}$ provide low values of Δ₀ (and vice versa). In order to minimize this effect, we use the luminosity distance, d_L, evaluated at the mean redshift of the sample ($\bar{z}=0.36$) as a pivot point and define

$$\begin{eqnarray}&&{V}_{i}={\mu }_{B,i}(\hat{\alpha },\hat{\beta },\hat{\delta },{\hat{M}}_{B})-5{\mathrm{log}}_{10}(\displaystyle \frac{{d}_{{\rm{L}}}({z}_{i};{{\rm{\Omega }}}_{{\rm{\Lambda }}})}{{d}_{{\rm{L}}}(\bar{z};{{\rm{\Omega }}}_{{\rm{\Lambda }}})})-{\rm{\Delta }}\end{eqnarray} \tag{ 6 }$$

where ${\rm{\Delta }}={{\rm{\Delta }}}_{0}+5{\mathrm{log}}_{10}{d}_{{\rm{L}}}(\bar{z};{{\rm{\Omega }}}_{{\rm{\Lambda }}})$ and ${{\rm{\Omega }}}_{{\rm{\Lambda }}}$ are our free parameters.

Sky maps of the best-fit values for ${{\rm{\Omega }}}_{{\rm{\Lambda }}}$ (left) and Δ (right) are shown in Figure 3 for three different cone opening angles (from top to bottom: $\theta =\frac{\pi }{2}$, $\frac{\pi }{3}$, and $\frac{\pi }{6}$ radians). These have been obtained using all the Union2.1 SNe with redshift $z\geqslant 0.2$. White pixels indicate the directions (mostly located around the Galactic equator) in which the corresponding cone contains less than 25 SNe Ia. These directions are excluded from all statistical analyses because they are associated with extremely large errors in the fitted parameters. Of course their number increases with decreasing the opening angle of the sampling cone. Similarly, the size of fluctuations in the best-fit values for ${{\rm{\Omega }}}_{{\rm{\Lambda }}}$ and Δ increases with reducing θ.

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Although Figure 3 gives a first visual impression of the local best-fit parameters, it does not take into account the non-uniform sky coverage of the Union2.1 data set. For a given opening angle, different directions on the celestial sphere are generally associated with very different numbers of SNe Ia. This strongly influences the uncertainty of the best-fit values.

In order to single out the most discrepant directions in a statistically meaningful way, we assume the null hypothesis that the universe follows the cosmological principle and there are no angle-dependent systematic effects plaguing the Union2.1 sample. For each pixel we then evaluate the χ² target function fixing the free parameters at the values ${\hat{{\rm{\Omega }}}}_{{\rm{\Lambda }}}=0.70$ and $\hat{{\rm{\Delta }}}=41.428$ which provide the best-fit solution for the complete Union2.1 sample. However, only the SNe within the sampling cone are used to calculate the χ² value that we denote by ${\hat{\chi }}^{2}$. Finally, we estimate the probability P that random noise could generate a χ² value exceeding ${\hat{\chi }}^{2}$. Assuming Gaussian errors, this probability coincides with the cumulative chi-square distribution function evaluated at ${\hat{\chi }}^{2}$:

$$\begin{eqnarray}&&P=\displaystyle \frac{1}{{2}^{\nu /2}{\rm{\Gamma }}(\nu /2)}{\int }_{{\hat{\chi }}^{2}}^{\infty }{t}^{\nu /2-1}{e}^{-t/2}{dt},\end{eqnarray} \tag{ 7 }$$

where ν is the number of degrees of freedom—i.e., the number of SNe Ia used in the fitting procedure minus two (the number of free parameters). We adopt the P value as a measure of how well the all-SNe best-fit parameters also describe the SN data in a specific direction on the sky. Consequently we identify the most discrepant direction (i.e., the direction showing the most statistically significant deviation from isotropy) with the pixel showing the smallest P value. It is worth stressing that this is not necessarily the direction in which the universe (or the SN data) might present the strongest intrinsic anisotropy but only the direction in which, given the current data, we can measure the most meaningful deviation in terms of the signal-to-noise ratio.

Maps of the P value are plotted in Figure 4 for the three different cone opening angles. The most (second-most) discrepant directions are highlighted with a star (circle) in each panel. Further information is provided in Table 2 which gives the P value, the coordinates, and the number of SNe Ia associated with the most and the second-most discrepant directions together with the local best-fit parameters. The motivation for showing two directions per map is as follows: (i) the difference between their P values is small (see Table 2), (ii) the covariance matrix provided by the SCP is likely to be a noisy estimate, and (iii) neglecting off-diagonal covariances switches the order between these directions for $\theta =\frac{\pi }{3}$.

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Table 2.  Most Discrepant Directions

θ	(l,b)	${{\rm{\Omega }}}_{{\rm{\Lambda }}}$	Δ	${\hat{\chi }}^{2}/\nu$	P	N
⋆ $\pi /2$	(33.7, −19.5)	${0.58}_{-0.13}^{+0.11}$	${41.424}_{-0.072}^{+0.077}$	1.09	0.192	128
• $\pi /2$	(0.0, −30.0)	${0.61}_{-0.10}^{+0.09}$	${41.441}_{-0.067}^{+0.066}$	1.08	0.197	161
⋆ $\pi /3$	(112.5, −9.6)	${0.60}_{-0.31}^{+0.22}$	${41.439}_{-0.110}^{+0.130}$	1.20	0.086	74
• $\pi /3$	(56.2, −41.8)	${0.59}_{-0.15}^{+0.11}$	${41.424}_{-0.072}^{+0.081}$	1.15	0.101	118
⋆ $\pi /6$	(67.5, −66.4)	${0.58}_{-0.15}^{+0.12}$	${41.419}_{-0.074}^{+0.084}$	1.18	0.081	100
• $\pi /6$	(101.2, −41.8)	${0.55}_{-0.29}^{+0.21}$	${41.408}_{-0.106}^{+0.124}$	1.20	0.085	73

Notes. Galactic coordinates (l, b) and P values characterizing the most (stars) and the second-most (circles) discrepant directions for different cone opening angles, θ. Also reported are the number of SNe Ia in the cones, N, the best-fit values for ${{\rm{\Omega }}}_{{\rm{\Lambda }}}$, and Δ (in mag) and the ratio ${\hat{\chi }}^{2}/\nu$.

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Intriguingly, the most discrepant directions obtained with the three cone opening angles lie close to each other. Also the best-fit parameters are quite similar (let us not forget, however, that the maps with different θ are not independent as they use the same SNe and that there is significant overlap between the most discrepant cones). In Figure 5 we compare the formal⁷ 1σ confidence regions (${\rm{\Delta }}{\chi }^{2}\leqslant 2.30$) obtained from the all-SNe fit against those derived from the local fits along the most discrepant directions. In all cases, the tension between the local and the global fits is marginal and the formal 1σ regions always overlap.

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Visual inspection of Figure 4 shows a striking contrast between the P values measured in the northern and the southern Galactic hemispheres (hereafter NGH and SGH, respectively), although there is no tension between the luminosity–distance relation in the two hemispheres (see Table 3). The discrepancy in the P values is mainly due to the fact that the Union2.1 uncertainties in the distance moduli are on average 30% larger in the NGH. Consequently, the reduced chi-square ${\hat{\chi }}^{2}/\nu$ tends to be smaller in the NGH although there are many more SNe in the SGH (227 versus 123) to drive the fit results for SNe with $z\geqslant 0.2$ closer to the SGH results.

Table 3.  Results of NGH and SGH Fitting

	${{\rm{\Omega }}}_{{\rm{\Lambda }}}$	Δ	${\chi }^{2}/\nu$	N	${\bar{\sigma }}_{\mu }$
NGH	${0.70}_{-0.09}^{+0.07}$	${41.443}_{-0.084}^{+0.088}$	0.82	123	0.30
SGH	${0.68}_{-0.08}^{+0.06}$	${41.441}_{-0.056}^{+0.060}$	1.01	227	0.23

Notes. Best-fit values obtained using SNe in the NGH and SGH, separately. Also reported are the corresponding reduced chi-square, ${\chi }^{2}/\nu$, the number of SNe Ia with $z\geqslant 0.2$ considered for the fit, N, and their average distance-modulus uncertainty, ${\bar{\sigma }}_{\mu }$.

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Taken at face value, the probabilities P associated with most discrepant directions (see Table 2) are moderately significant. However, assuming Gaussian errors is a strong limiting factor. Also, the size of the error bars in the distance modulus (and the off-diagonal covariances) provided in the Union2.1 catalog might be inaccurate and, as a consequence, inferences based on the χ² statistic might be biased. For these reasons, we re-evaluate the statistical significance of the anisotropies using a more robust Monte Carlo method.

In order to assess the impact of random errors and account for the non-uniform angular distribution of the Union2.1 sample, we build 1000 mock catalogs by randomly shuffling the distance moduli of the Union2.1 SNe. In practice, we assign the distance modulus, its uncertainty, and the redshift of an SN Ia to the angular position of another (random) SN Ia. Each mock catalog thus contains exactly the same number of SNe as the original Union2.1 sample and has exactly the same SN sky distribution. Moreover, all possible anisotropies should be erased by the shuffling procedure while the statistical properties of the distance moduli and their uncertainties are unchanged with respect to the observational data. Therefore, our mock catalogs form an ensemble of realizations mimicking an isotropic universe but having the same statistical properties as the actual Union2.1 data.

We treat the mock catalogs as the real data and identify the two most discrepant directions in each of them using the P-value-based method for the three different cone opening angles. We then compute the fraction, f₀, of the realizations in which the most (or the second-most) discrepant direction is associated with a P value which is smaller than the observed ones reported in Table 2. For the most (second-most) discrepant directions we find that ${f}_{0}=0.569,0.623$ and 0.674 (0.473, 0.550, 0.490) for $\theta =\frac{\pi }{2}$, $\frac{\pi }{3}$, and $\frac{\pi }{6}$, respectively. Purely based on this signal-to-noise criterion, we conclude that no statistically significant anisotropy can be detected in the Union2.1 sample.

Although the Monte Carlo test shows that random chance in an isotropic universe can easily produce the most discrepant directions with lower P values than we found analyzing the actual data, the observed anisotropies present a characteristic feature that is worth being discussed.

The temperature distribution in the CMB presents a strong dipole anisotropy which is usually interpreted as due to our motion with respect to the CMB rest frame toward the direction with Galactic coordinates $(263\buildrel{\circ}\over{.} 99\pm 0.14,48\buildrel{\circ}\over{.} 26\pm 0.03)$ (Planck Collaboration et al. 2014b but see Wiltshire et al. 2013 for a different interpretation). Figure 6 shows that the most discrepant directions we obtained from the Union2.1 sample closely align with the axis of the CDP in the SGH opposite to our motion with respect to the CMB rest frame (hereafter CDP-South). Assuming that the redshifts of the SNe Ia in the Union2.1 compilation have been correctly transformed to the CMB rest frame, there is no obvious reason for explaining the origin of this alignment. As already mentioned in the Introduction, the CMB quadrupole (CQP) and octopole (COP) are also closely aligned with the CDP (Schwarz et al. 2004; Copi et al. 2010b, 2015; Planck Collaboration et al. 2014a, see Figure 6). It is yet unclear whether these alignments are a statistical fluke or a signature of new physics. Anyway, our study shows that the magnitude–redshift relation of SNe Ia with $z\geqslant 0.2$ tends to be different in the same direction (albeit the difference is detected with low signal-to-noise ratio). Other authors have reported similar results using the Union compilations (Antoniou & Perivolaropoulos 2010; Cooke & Lynden-Bell 2010; Li et al. 2013).

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The debate on the physical relevance of the CMB anomalies opened up a discussion in the literature about the legitimacy and validity of "a posteriori" analyses in which tailored statistical tests are designed and hand-picked after noticing peculiarities in the data. A widespread point of view states that in a large data set it is always possible to isolate some "strange" features (e.g., Bennett et al. 2011). To minimize the pitfalls of a posteriori reasoning, we focus on the well established CDP and do not consider the CQP and COP any further.

We thus proceed to quantify the probability that the most discrepant directions (defined in terms of the P value as above) form a given angle with the CDP under the null hypothesis of an isotropic magnitude–redshift relation. In order to account for the non-uniform sky distribution of the Union2.1 sample (especially for the SDSS-II stripe which is close to the CDP-South) we use the Monte Carlo realizations introduced in Section 4.2.3. Figure 7 shows the resulting probability distribution for the cosine of the angle between the most discrepant direction and the axis of the CDP-South. Our measurements from the Union2.1 data are indicated by vertical dashed lines. The second column in Table 4 reports the fraction of Monte Carlo realizations, f₁, showing a better alignment than our measurement. Our results suggest that it is rather unlikely to get an alignment as strong as the observed one under the null hypothesis of an isotropic magnitude–redshift relation. In fact, considering the most discrepant direction for $\theta =\frac{\pi }{2}$, only 8.5% of the Monte Carlo realizations show a smaller separation angle than observed and this reduces to 4.5% for $\theta =\frac{\pi }{3}$. Note that for $\theta =\frac{\pi }{3}$ and $\frac{\pi }{6}$, the second-most discrepant directions are even better aligned with CDP-South. In these cases ${f}_{1}\simeq 0.01$.

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Table 4.  Alignment with the CMB Dipole

θ	α	f1	f2
⋆ $\pi /2$	$49\buildrel{\circ}\over{.} 4$	0.085	0.027
• $\pi /2$	$64\buildrel{\circ}\over{.} 3$	0.152	0.052
⋆ $\pi /3$	$45\buildrel{\circ}\over{.} 5$	0.045	0.021
• $\pi /3$	$20\buildrel{\circ}\over{.} 5$	0.008	0.002
⋆ $\pi /6$	$20\buildrel{\circ}\over{.} 1$	0.064	0.045
• $\pi /6$	$13\buildrel{\circ}\over{.} 7$	0.010	0.006

Notes. Angular separation, α, between the direction of the CMB dipole in the SGH (CDP-South) and the most (stars) and the second-most (circles) discrepant directions for the maps based on the Union2.1 data with different cone opening angles, θ. The probability of measuring a value smaller than α in random realizations of an isotropic magnitude–redshift relation is indicated with f₁ while f₂ also accounts for the condition where the most (second-most) discrepant direction is associated with a smaller P value than for the Union2.1 measurement. Both probabilities have been estimated with a Monte Carlo method (see the main text for details).

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The test above is blind to the statistical significance of the most discrepant directions. In order to account for this, we compute the fraction of Monte Carlo realizations, f₂, for which the most discrepant directions are at least as significant as the measured ones (in terms of the P value) and are also better aligned with the CDP-South. The third column in Table 4 shows that for the most discrepant directions this probability is smaller than 4.5% for all the opening angles which means the null hypothesis of an isotropic magnitude–redshift relation should be rejected at the 95% confidence level. The value of f₂ reduces to a fraction of a percent when considering the second-most discrepant direction for $\theta =\frac{\pi }{3}$ and $\frac{\pi }{6}$.

The measured anisotropy could be due to a statistical fluke, to systematics in the SNe data (or error bars), to the presence of localized large-scale structures, or even a sign of the failure of the cosmological principle. To further investigate its properties, we repeat the analysis along the most discrepant directions after slicing the SNe data in five redshift bins ($0.2\leqslant z\lt 0.3$, $0.3\leqslant z\lt 0.4$, $0.4\leqslant z\lt 0.6$, $0.6\leqslant z\lt 0.9$, and $z\geqslant 0.9$). Regrettably, due to the low number of SNe in each bin, the formal $1\sigma$ errors span most, if not all, the parameter space $0\leqslant {{\rm{\Omega }}}_{{\rm{\Lambda }}}\leqslant 1$. Therefore no meaningful statements can be made regarding the variations of the best-fit cosmological parameters along the most discrepant directions. In terms of signal-to-noise ratio, however, the redshift range $0.4\leqslant z\lt 0.6$ clearly emerges as the most discrepant one for all the cone opening angles ($P=0.033,0.021$ and 0.016 for $\theta =\pi /2,\pi /3$ and $\pi /6$, respectively).

Given the current sparsity of the data, no firm conclusion can be drawn except from the fact that there seems to be a moderately statistically significant (2–3σ) anisotropy in the magnitude–redshift relation of SNe Ia close to the direction opposite to our motion with respect to the CMB rest frame. It is worth remembering that, in the Union2.1 compilation, most of the SNe Ia surrounding the CDP-South come from the SDSS-II stripe. Further investigations are thus needed to clarify the relation between the CDP axis, our motion, and the way SNe data around this direction are treated.