Is the Expansion of the Universe Accelerating? All Signs Still Point to Yes: A Local Dipole Anisotropy Cannot Explain Dark Energy

The discovery of the accelerated expansion of the universe (Riess et al. 1998; Perlmutter et al. 1999) was originally made through the use of Type Ia supernovae (SNe Ia). With measurements of light-curve shape and color, the luminosity can be standardized so that the apparent magnitude can give the distance. Today, multiple cosmological probes combine together to provide a largely consistent picture of the universe on large scales (Planck Collaboration et al. 2018).

A recent study (Colin et al. 2019, hereafter C19) claims that this picture is wrong. They investigate the Joint Light-curve Analysis (JLA) SN Ia compilation (Betoule et al. 2014) and find evidence of a dipole anisotropy in the local deceleration parameter on ∼100 Mpc scales. C19 claims that this dipole is larger and more statistically significant than the monopole acceleration, for which they find only weak evidence. C19 alleges that this dipole gives rise to falsely strong evidence of the accelerated expansion.

We reevaluate C19's analysis and find four significant problems, each affecting their results or interpretation.

• 1.  
  C19's primary analysis makes the plainly incorrect assumption that SN Ia light-curve shape and color distributions are constant as a function of redshift (after selection cuts). That incorrect assumption comes from Nielsen et al. (2016) (hereafter N16), which C19 builds on. However, Rubin & Hayden (2016) (hereafter RH16) have already demonstrated the inaccuracy of this assumption, discussing the need for redshift-dependent observed populations. We summarize that discussion, C19's new (since N16) arguments in favor of this incorrect assumption, and counter C19's objections using statistical significance and physically motivated reasoning. Incorrectly assuming redshift-independent observed distributions has little effect on inferring the dipole parameters, but causes large biases in the monopole cosmological parameters. The net result of these biases weakens a portion of the standardization of distant SNe, moving these SNe brighter and thus biasing the inferred cosmology toward less acceleration.
• 2.  
  Shockingly, C19 use heliocentric redshifts to compute their comoving model distances, leaving the well-established motion of the solar system with respect to the cosmic microwave background (CMB) to imprint on the SN redshifts. They further use SNe as close as redshift 0.01, where the decision to use heliocentric or CMB redshifts affects the distance modulus at up to (5/log(10))(370 km s⁻¹)/(0.01c) ≈ ± 0.27 mag (see, e.g., Davis et al. 2011). This distance modulus difference is dominant over per-SN distance uncertainty and is correlated across the sky. In addition to using heliocentric redshift, C19 removed the peculiar-velocity covariance terms from the JLA distance-modulus-uncertainty covariance matrix. This is in keeping with the spirit of using heliocentric redshifts, as C19 did not apply a peculiar-velocity model. However, the removal of these covariance terms has a significant impact. When the full JLA peculiar-velocity covariance matrix is included in the analysis, we find no statistically significant anisotropy at 2σ, even when using C19's preferred heliocentric redshift. We show that these two related decisions are the primary driver of C19's claimed dipole anisotropy. C19's justification of the heliocentric choice is weak, citing evidence that bulk flows may exist on larger than expected scales. However, the choice to work in heliocentric redshifts is arbitrary; C19 shows no evidence that the heliocentric frame is closer to the average reference frame of nearby SN hosts than the CMB frame, let alone that it is better than employing the CMB frame plus corrections for known peculiar velocities (as the JLA analysis did). If the authors of C19 are concerned about the impact of peculiar velocities, then it makes very little sense to remove the peculiar-velocity-uncertainty covariances from the analysis.
• 3.  
  C19 calls out the hemispheric imbalance of surveys included in the JLA compilation; most SNe included are from Northern Hemisphere telescopes. However, C19 completely ignores consistent cosmological results from SNe published more recently than JLA that have been obtained with Southern Hemisphere telescopes (Carnegie Supernova Project; Krisciunas et al. 2017 and the Dark Energy Survey; Abbott et al. 2019). All of these data are public. C19 also ignores their own analysis's failure to find a strong correlation between the dipole term and the monopole term (their Figure 3), which suggests that choice of hemisphere does not affect the JLA analysis.
• 4.  
  The C19 preferred model (Equation (8), which they use for all their quoted results) includes a dipole anisotropy suppressed with an exponential falloff in redshift, and is thus pathological when the scale of the exponential is poorly measured. As the SN measurements are at finite redshift, extrapolating back to the redshift 0 dipole yields a strong degeneracy between the scale of the exponential (S) and the dipole value. We find that we must restrict the range of S around the best-fit range found by C19 to achieve reasonable frequentist coverage (e.g., 68% of the time, the true answer is in the 68% credible interval).

Our response is structured as follows. Section 2 discusses our version of the C19 analysis, based heavily on the work of RH16. Section 3 shows our cosmological constraints. We run analyses with heliocentric redshifts, CMB-centric redshifts, and CMB-centric redshifts with the peculiar-velocity model JLA used (Hudson et al. 2004; Conley et al. 2011) removed. We further investigate the impact of the peculiar-velocity covariances that C19 removed. Section 4 summarizes and concludes.

This section begins by reviewing kinematic parameters and outlining the C19 cosmological model (Section 2.1). Section 2.2 describes our implementation of the analysis. Finally, Section 2.3 discusses the importance of allowing the observed SN population distributions (after selection) to change with redshift, and dismantles C19's claim that constant population distributions are preferred.

2.1. Kinematic Parameters

The origin of the kinematic parameters is a series expansion of the scale factor a(t) around the present time t = t₀:

$$\begin{eqnarray}\begin{array}{rcl}a(t) & = & {a}_{0}\left[1+{H}_{0}(t-{t}_{0})-\displaystyle \frac{1}{2!}{q}_{0}{H}_{0}^{2}{\left(t-{t}_{0}\right)}^{2}\right.\\ & & +\left.\displaystyle \frac{1}{3!}{j}_{0}{H}_{0}^{3}{\left(t-{t}_{0}\right)}^{3}+{ \mathcal O }({\left[t-{t}_{0}\right]}^{4})\right].\end{array}\end{eqnarray} \tag{ 1 }$$

a₀ is frequently defined to be 1. H₀ is the Hubble constant (or the Hubble parameter evaluated at t = t₀), given by

$$\begin{eqnarray}&&{H}_{0}\equiv \displaystyle \frac{\dot{a}(t)}{a(t)}{| }_{t={t}_{0}}.\end{eqnarray} \tag{ 2 }$$

q₀ is the deceleration parameter and is given by

$$\begin{eqnarray}&&{q}_{0}\equiv -\displaystyle \frac{\ddot{a}(t)a(t)}{{\dot{a}}^{2}(t)}{| }_{t={t}_{0}};\end{eqnarray} \tag{ 3 }$$

the negative sign is a historical convention from when the expansion of the universe was assumed to be decelerating, making $\ddot{a}(t)$ negative. j₀ is the jerk parameter and is given by

$$\begin{eqnarray}&&{j}_{0}\equiv \displaystyle \frac{\dddot{a}(t){a}^{2}(t)}{{\dot{a}}^{3}(t)}{| }_{t={t}_{0}}.\end{eqnarray} \tag{ 4 }$$

C19 follows Visser (2004) in using the following series expansion for luminosity distance

$$\begin{eqnarray}\begin{array}{rcl}{d}_{L}(z) & = & \displaystyle \frac{{cz}}{{H}_{0}}\left[1+\displaystyle \frac{1}{2}(1-{q}_{0})z\right.\\ & & -\left.\displaystyle \frac{1}{6}(1-{q}_{0}-3{q}_{0}^{2}+{j}_{0}-{{\rm{\Omega }}}_{k}){z}^{2}\right].\end{array}\end{eqnarray} \tag{ 5 }$$

As d_L = (1 + z_heliocentric) (comoving distance), we slightly modify Equation (5) so that it scales as (1 + z_heliocentric):

$$\begin{eqnarray}&&\begin{array}{l}{d}_{L}(z,{z}_{\mathrm{heliocentric}})=\displaystyle \frac{1+{z}_{\mathrm{heliocentric}}}{1+z}\displaystyle \frac{{cz}}{{H}_{0}}\\ \,\times \,\left[1+\displaystyle \frac{1}{2}(1-{q}_{0})z-\displaystyle \frac{1}{6}(1-{q}_{0}-3{q}_{0}^{2}+{j}_{0}-{{\rm{\Omega }}}_{k}){z}^{2}\right],\end{array}\end{eqnarray} \tag{ 6 }$$

where z nominally is CMB-centric with peculiar-velocity corrections, but we also show comparisons using heliocentric and CMB-centric redshifts. We form the distance modulus as

$$\begin{eqnarray}&&\mu (z,{z}_{\mathrm{heliocentric}})=5{\mathrm{log}}_{10}\left[\displaystyle \frac{{d}_{L}(z,{z}_{\mathrm{heliocentric}})}{10\ \mathrm{pc}}\right].\end{eqnarray} \tag{ 7 }$$

The C19 model separates q₀ into a monopole q_0m and a dipole q_0d which C19 points in the direction of the CMB dipole (${{\boldsymbol{n}}}_{\mathrm{CMB}}$) in the their baseline analysis.

$$\begin{eqnarray}&&{q}_{0}({z}_{\mathrm{SN}},{{\boldsymbol{n}}}_{\mathrm{SN}})={q}_{0m}+{q}_{0d}({{\boldsymbol{n}}}_{\mathrm{SN}}\cdot {{\boldsymbol{n}}}_{\mathrm{CMB}})\ {e}^{-{z}_{\mathrm{SN}}/S},\end{eqnarray} \tag{ 8 }$$

where S is the scale (in redshift) over which the dipole term falls off.

2.2. Parameter Inference

2.3. Population Distributions

To illustrate the necessity of modeling the changes in the observed light-curve-parameter distributions with redshift, we simulate a very simple survey with just color and magnitude. The intrinsic color population is assumed to be Gaussian with width 0.1 mag and mean 0 (constant in redshift). We generate redshifts with $P(z)\propto {z}^{2}$ (following a constant volumetric rate), then construct magnitudes using 5log₁₀(distance) + 3.1 color+arbitrary constant, assuming distance ∝ redshift. For illustrative purposes, we apply a hard magnitude limit, in contrast to actual selection probabilities, which are smooth with magnitude (e.g., Perrett et al. 2010). Figure 1 shows the results of this simulation and qualitative agreement with the actual color trend with redshift in the SuperNova Legacy Survey (SNLS) in the JLA analysis. This tendency to select more luminous SNe out to larger redshifts is frequently referred to as a Malmquist bias (Malmquist 1922). If one incorrectly assumes a redshift-independent population distribution (when the populations after selection do depend on redshift), this biases the latent values of light-curve shape and color for each SN toward the population mean (Wood-Vasey et al. 2007; Kessler et al. 2009; Karpenka 2015; Rubin et al. 2015; Rubin & Hayden 2016). For color, this means closer SNe are biased bluer, and more distant SNe are biased redder. After applying the color standardization, closer SNe are biased fainter and more distant SNe are biased brighter. Such a bias thus undoes some of the evidence for the accelerated expansion.

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RH16 introduced 12 additional parameters: for the nearby SNe, the Sloan Digital Sky Survey SNe, and the SNLS SNe, mean color and light-curve shape (after selection) are modeled as linear functions in redshift. Of course, the averages are not quite linear with redshift, but this is close enough over most of the redshift range of most of the SNe in each survey that more redshift-flexible models do not significantly change the cosmological inference (as noted in RH16).⁴ C19 criticize the inclusion of these 12 additional parameters on three grounds, all of which are demonstrably false.

C19 claim that the additional fit parameters are a posteriori, in other words that these parameters were deliberately added by RH16 to the N16 analysis to recover the concordance cosmology. However, the importance of redshift-dependent color population distributions was well established much earlier than the N16 paper, e.g., in Wood-Vasey et al. (2007), Kessler et al. (2009), Karpenka (2015), and Rubin et al. (2015). The intrinsic light-curve shape population distribution is expected to be redshift dependent as well, as there is a correlation between light-curve shape and host-galaxy properties (Hamuy et al. 1995).

C19 further claims that the additional parameters are not justified by the Bayesian information criterion (Schwarz 1978), a frequently used method that penalizes −2 log(likelihood) with $k\mathrm{log}(n)$ for k parameters and n observations. As discussed above, we have a strong expectation that the observed light-curve population distributions should be redshift dependent, negating this objection to the 12 additional parameters of RH16 even if these parameters were not favored by the Bayesian information criterion. The C19 implementation of the RH16 model gives a −2 log(likelihood) that ranges from −298 to −332, depending on other fit parameters. The C19 constant-populations-in-redshift model gives a −2 log(likelihood) that ranges from −190 to −217.⁵ The Bayesian information criterion relative penalty for 740 measurements (as there are 740 SNe in the JLA compilation, so 740 light-curve-shape measurements for the six additional light-curve-shape parameters and 740 color measurements for the six additional color parameters) and 12 additional parameters is $12\mathrm{log}(740)=79$, much less than the −2 log(likelihood) difference. Even using $12\mathrm{log}(740\times 3)=92$ (×3 for light-curve shape, color, and magnitude), it is still much less than the observed difference. We further note that a χ² improvement of 115 for 12 degrees of freedom is a ∼9σ improvement, again showing strong evidence for the RH16 model. Thus C19's own values show that the RH16 model is actually preferred using the information criterion that C19 claims to be using.

Finally, C19 criticizes RH16 for focusing on the light-curve shape and color parameters to the exclusion of magnitude. This claim is also false, as described in RH16. As part of the JLA analysis, Betoule et al. (2014) included simulations of the impact of selection effects on each SN sample, removed its impact on magnitude, and propagated the uncertainties into the covariance matrix (much of this work was built on Conley et al. 2011). This bias is different from the deficient standardization that one gets from ignoring the redshift dependence of the population distributions, which is due to biasing the inferred latent values of light-curve shape and color. Intrinsic magnitude differences with redshift are controlled for by matching SNe in bins of host-galaxy stellar mass (as an easier-to-measure proxy for SN progenitor age or metallicity, e.g., Sullivan et al. 2010). In any case, the systematic uncertainties due to such effects are a small fraction of the signature of acceleration in the SN Ia data, and a fraction of the bias caused by ignoring the redshift dependence of the population distributions.

Figures 2 and 3 show the 68.3% credible regions and intervals for the cosmological parameters. We compute the credible regions by interpolating the posterior samples using kernel density estimation (KDE) and setting the contour level to enclose 68.3% of the interpolated posterior.⁶ Above the credible regions, we show the one-dimensional probability density functions (1D PDFs). Here, we perform the analogous procedure in 1D: use a KDE (now 1D instead of 2D), and compute and fill the minimum distance that encloses 68.3% of the PDF. Above the KDE results, we compute and show more conventional 1D constraints for each parameter that use the median, the 15.9th percentile, and the 84.1th. The KDE-based credible intervals will agree well with the percentile-based intervals for nearly Gaussian distributions. In contrast, these intervals can be seen to diverge for S, which shows a decreasing distribution.

Figure 2 shows constraints for three cases: heliocentric redshifts, CMB-centric redshifts, and CMB-centric redshifts with peculiar-velocity corrections. To match C19, we remove the peculiar-velocity covariances from the JLA covariance matrix. We see that the choice of redshift frame mostly affects q_0d. Figure 3 shows the same cosmological constraints with the JLA peculiar-velocity covariance matrix included. In contrast to the Figure 2 results (without this part of the covariance matrix), q_0d is consistent with zero at 2σ for all three types of redshifts. The N16/C19 assumption of redshift-independent observed light-curve population distributions underestimates cosmological uncertainties as well as shifting central values. This is primarily due to the increase in average uncertainty on SN color with redshift, from ∼0.02 mag at low redshift to ∼0.06 mag at higher redshift. Both our analysis and C19 find that the observed color population distribution has an intrinsic width of 0.07 mag, so when the mean of the color population is allowed to be redshift dependent, this mean has significant uncertainty. When the color population distribution is incorrectly modeled as redshift independent, the uncertainty on the mean is small (as the distribution is trained on many well-measured low-redshift SNe), resulting in smaller uncertainties on the high-redshift distance moduli. The impact is most evident in j₀, which is effectively inferred over a longer redshift baseline than q_0m. Figures 2 and 3 show ∼50–60% larger uncertainties on j₀ when correctly using redshift-dependent population distributions.

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It is worth briefly discussing the interpretation of series expansions (kinematic expansions) of the expansion history (Equation (1)) to understand the strong correlation between q₀ and j₀ − Ω_k. Frequently, we speak imprecisely about measuring H₀, q₀, or j₀ from SNe Ia (possibly combined with other probes). However, SNe measure distances at finite redshift and thus effectively measure these quantities at nonzero (but low) redshift. For example, the (statistically disfavored) not-currently-accelerating (q₀ forced to 0) models of C19 (in Table A.1) have j₀ − Ω_k ∼ −1.35, and so would have experienced several Gyr of acceleration in the recent past (j₀ < 0), even though the acceleration goes to zero today. See RH16 for a discussion of q₀ and j₀ − Ω_k JLA constraints using other (physical) models.

This work reimplements the C19 cosmological analysis to investigate their claims of a dipole term in the deceleration parameter (q_0d) and a statistically weak monopole (q_0m).

We show that the weak monopole finding is identical to the finding from the related N16 analysis, which was rebutted by RH16 for their incorrect use of constant-in-redshift SN populations (after selection). We find the same criticism still applies, and counter the C19 arguments against RH16, including finding an apparent miscalculation in the C19 Bayesian information criterion.

We find that the C19 result of a significant (3.9σ) q_0d depends strongly on both failing to remove our motion from the redshifts of SNe (that is, working in the heliocentric frame) and failing to include the JLA peculiar-velocity covariances. Changing either of these brings the evidence for a dipole below 2σ. Despite the inclusion of the dipole term, we see virtually the same constraints on q_0m as RH16 saw on q₀ (this work: ${q}_{0m}=-{0.425}_{-0.122}^{+0.116}$, RH16: ${q}_{0}=-{0.425}_{-0.117}^{+0.115}$) when using the same type of redshift model and using the same covariance matrix. We thus conclude that concerns over the value of q_0d have little effect on the strength of the evidence for acceleration.

The technical support and advanced computing resources from the University of Hawaii Information Technology Services–Cyberinfrastructure are gratefully acknowledged. We thank Greg Aldering, Benjamin Rose, Brian Hayden, and the anonymous referee for careful feedback.

Software: Matplotlib (Hunter 2007), Numpy (van der Walt et al. 2011), pystan (Riddell et al. 2018), Python, SciPy (Virtanen et al. 2020), Stan (Carpenter et al. 2017).