How does an incomplete sky coverage affect the Hubble Constant variance?
Abstract
We address the \(\simeq 4.4\sigma \) tension between local and the CMB measurements of the Hubble Constant using simulated Type Ia Supernova (SN) datasets. We probe its directional dependence by means of a hemispherical comparison through the entire celestial sphere as an estimator of the \(H_0\) cosmic variance. We perform Monte Carlo simulations assuming isotropic and nonuniform distributions of data points, the latter coinciding with the real data. This allows us to incorporate observational features, such as the sample incompleteness, in our estimation. We obtain that this tension can be alleviated to \(3.4\sigma \) for isotropic realizations, and \(2.7\sigma \) for nonuniform ones. We also find that the \(H_0\) variance is largely reduced if the datasets are augmented to 4 and 10 times the current size. Future surveys will be able to tell whether the Hubble Constant tension happens due to unaccounted cosmic variance, or whether it is an actual indication of physics beyond the standard cosmological model.
1 Introduction
Precise measurements of the Hubble Constant (\(H_0 = 100h_0\) \(\mathrm {{km \, s^{1} \, Mpc^{1}}}\)) are of great interest not only to improve our understanding of the current cosmic evolution but also to provide new insights into some fundamental questions of the physics of the early Universe (see e.g., [1] for a broad discussion). Currently, there is a \(\sim 4.4\sigma \) tension between measurements of \(h_0\) performed in the local universe using type Ia supernovae (SN) observations calibrated with Cepheid distances to SN host galaxies, \(H_0 = 74.03 \pm 1.42 \, \mathrm {{km \, s^{1} \, Mpc^{1}}}\) (hereafter R19 [2]; see also [3, 4]), and the estimates of \(H_0\) obtained from the Cosmic Microwave Background (CMB) data in the context of the standard \(\Lambda \)CDM model, \(H_0 = 67.36 \pm 0.54 \, \mathrm {{km \, s^{1} \, Mpc^{1}}}\) (hereafter P18 [5]; see also [6]). Many previous works attempted to address this problem using different approaches, e.g., in light of the cosmic variance due to nearby inhomogeneities [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], recalibrating the distance ladder [20, 21, 22, 23, 24, 25, 26, 27, 28, 29], or looking at extensions to the standard model of Cosmology [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43]. Metastudies of the \(H_0\) measurements and tension also confirmed a \(H_0\) tension above \(3\sigma \) [44]. New Hubble Constant measurements from strongly lensed quasars time delays [45], and using the tip of the red giant branch [46, 47], could not solve this issue [48].
Our goal in this paper is to quantify how the incompleteness of current SN data can affect the \(H_0\) variance, i.e., a shot noise variance estimator. Rather than reassessing these measurements, we focus on quantifying how does the tension between the \(H_0\) measurements change due to it. To do this, we map the anisotropy of the Hubble Constant across the sky using lowz SN (\(0.023< z < 0.150\)) from simulations based on the latest compilation available, namely, the Pantheon data [49]. We produce sets of Monte Carlo (MC) realizations assuming different sky coverage configurations which coincide with isotropic and the real data (nonuniform) sky distributions. This allows to assess how the SN sampling due to the uneven redshift distribution, in addition to their distance uncertainties and incomplete celestial distribution of data points, impacts the \(H_0\) measurements. We also discuss how much does the \(H_0\) cosmic variance decrease when larger, more homogeneous SN data sets are considered. If the cosmic variance becomes much lower than the current values, we can rule it out as a possible explanation for this result, thereby strengthening the hypothesis of physics beyond the concordance model.
2 Data analysis

MCiso1: The SN original positions in the sky are changed according to an isotropic distribution, but with the galactic plane (\(\mathrm {b}<10^{\circ }\)), and the highest declinations (\(\mathrm {DEC}>30^{\circ }\)) excised;

MCiso2: The SN original positions in the sky are changed according to an isotropic distribution, but with the galactic plane (\(\mathrm {b}<10^{\circ }\)) excised only;

MCpantheon: The SN original positions are maintained as the original Pantheon sample.
Case  \(\Delta h_\mathrm{med}\)  \(\Delta h\) (99.7% CL)  \(T_{h_0}\)  \(T^{}_{h_0}\)  \(T^{+}_{h_0}\) 

MCiso1  0.013  [0.008, 0.022]  3.336  3.885  2.495 
MCiso2  0.012  [0.007, 0.021]  3.445  3.988  2.573 
MCpantheon  0.020  [0.012, 0.031]  2.656  3.445  1.932 
Respectively: Same as Table 1, but for 200 MCs assuming 4 and 10 times more data points than the current sample instead
Case (\(\simeq \) 950 SN)  \(\Delta h^\mathrm{med}\)  \(\Delta h\) (99.7% CL)  \(T_{h_0}\)  \(T^{}_{h_0}\)  \(T^{+}_{h_0}\) 

MCiso1  0.008  [0.005, 0.014]  3.885  4.170  3.229 
MCiso2  0.007  [0.005, 0.011]  3.988  4.170  3.556 
Case (\(\simeq \) 2400 SN)  \(\Delta h^\mathrm{med}\)  \(\Delta h\) (99.7% CL)  \(T_{h_0}\)  \(T^{}_{h_0}\)  \(T^{+}_{h_0}\) 

MCiso1  0.005  [0.004, 0.009]  4.170  4.246  3.777 
MCiso2  0.005  [0.002, 0.007]  4.170  4.352  3.988 
3 Results
In Fig. 2, we show the results obtained from 1000 MCs with 237 data points for each sky coverage configuration. The blue histogram gives the \(\Delta h\) distribution for the MCiso1 case, whereas the red and black ones provide, respectively, the MCiso2 and MCpantheon results. The light blue and gray shades correspond to the \(\Delta h^\mathrm{tension}\) quantity for 2 and \(3\sigma \) CL, and the pink vertical line is the \(\Delta h\) obtained from the actual Pantheon data as reported in [51]. We note that the \(\Delta h\) obtained from all the MCs overlap with this \(3\sigma \) shade, but the MCpantheon gives the largest values among all of them, illustrating that the nonuniform sky coverage increases the \(h_0\) cosmic variance
In Table 1, we provide the median, upper and lower values (at 99.7% CL) for \(\Delta h\), in addition to \(T_{h_0}\) and their respective upper and lower limits. We readily note that the MCpantheon realizations perform the best, as \(T_{h_0}\) decreases from \(\simeq 4.4\sigma \) to \(2.7\sigma \) for the median \(\Delta h\), and \(1.9\sigma \, (3.4\sigma )\) for \(T^{+}_{h_0}\) (\(T^{}_{h_0}\)). Both isotropic sets of MCs can alleviate the tension to \(3.4\sigma \) for the median case, and \(2.5\sigma \) for the most optimistic (\(T^{+}_{h_0}\)) one. We also note that the median \(\Delta h\) from both isotropic MCs are consistent with the estimates from [7, 36], i.e., \(\Delta h \simeq 0.009\), but in our case we can naturally incorporate other observational features of incomplete celestial coverage and SN distance measurement uncertainties, which contributes to reduce the tension even further. We also tested whether the individual uncertainties of the bestfitted \(h_0\) values affect our analyses. We found that their average \(1\sigma \) uncertainties for isotropic MCs lead to an additional (average) spread of 0.004 in \(\Delta h\) for the isotropic MCs, and 0.005 for the MCpantheon case. Since these values are smaller than the \(\Delta h\) values presented in Table 1, we stress that they do not impact our analysis.
Moreover, we estimate the \(h_0\) variance of future SN compilations. We do this by producing 200 MCiso1 and MCiso2 realizations according to the same redshift distribution of SN as the Pantheon sample for two cases: one assuming datasets 4 times larger than the Pantheon sample (nearly 950 objects), and another assuming 10 times its size (nearly 2400 objects). If \(\Delta h\) reduces in these cases, this would imply that cosmic variance cannot alleviate the \(\Delta h^\mathrm{tension}\)  and hence increase the evidence for physics beyond \(\Lambda \)CDM as its explanation.
We present the results of these analyses in Fig. 3. The \(\Delta h\) values and \(T_{h_0}\) are provided in Table 2. We can clearly note that these \(\Delta h\) are much smaller compared to the previous sets of MCs, so that \(T_{h_0} > 3.2\sigma \) for the realizations with 950 data points, and \(T_{h_0} > 3.8\sigma \) for those with 2400 objects. Given the advent of forthcoming, nearly allsky distance datasets as expected from LSST [58] and WALLABY [59], along with future \(H_0\) measurements from standard sirens [60, 61], and new assessments of the cosmic distance ladder [48], we should be able to determine whether the \(H_0\) tension arises due to its cosmic variance from uneven sky coverage and limited observational data, or whether it actually indicates physics beyond the concordance model.
4 Conclusions
The tension between the R19 and P18 estimates of \(H_0\) are one of the greatest puzzles of the concordance model of Cosmology at present. Because the Hubble constant is fundamental to our understanding of the cosmic evolution, it is necessary to seek explanations for this tension. The main ones correspond to an additional error budget to the \(H_0\) measurement due to local inhomogeneities and sample incompleteness, and to extensions of \(\Lambda \)CDM model such as timeevolving dark energy, interacting dark energy or new contributions to the effective number of relativistic species \(N_\mathrm{eff}\), among others. None of them could fully account for the \(H_0\) tension, but only alleviate it to \(\simeq 3\sigma \).
We focused on the former as a possible explanation for such a tension. This time around, we estimate the \(H_0\) cosmic variance from a hemispherical comparison estimator, so we can quantify how does the incomplete data sampling affect the \(H_0\) tension. We produced realizations of the Pantheon SN in the interval \(0.023< z < 0.150\) uniformly redistributed across the sky (MCiso1 and MCiso2), besides those assuming the actual distribution of objects (MCpantheon). We also replaced the actual SN distance moduli to values drawn from a normal distribution assuming the \(\Lambda \)CDM model, but the original distance errors. Hence, we quantified the \(H_0\) variance due to the sample incompleteness within the concordance model framework.
We found that the \(H_0\) tension could be alleviated from nearly \(4.4\sigma \) to \(2.5\sigma \), at best, for the isotropic simulations, but the MCs assuming the actual SN distribution gave a median value of \(\simeq 2.7\sigma \) for a lower (upper) bound of \(1.9\sigma \, (3.4\sigma )\). These results significantly improves previous analyses that pointed out \(T_{h_0} \simeq 3\sigma \) due to the \(h_0\) cosmic variance, as we could incorporate further SN sample limitations, such as uneven sky coverage and distance measurement uncertainties. Furthermore, we verified that the \(\Delta h\) will be significantly smaller when larger samples becomes available, and so we can rule out cosmic variance as an explanation for this tension.
Our conclusion is that the cosmic variance can alleviate the \(H_0\) tension  specially if the nonuniform celestial coverage is taken into account. It is worth mentioning that there might also be unidentified systematics in the observations that lead to such tension, as discussed in [57]. As mentioned earlier, the forthcoming, nearly allsky distance data make the prospects very good for solving this issue in the near future.
Footnotes
 1.
We note that changing the value of the deceleration parameter within a reasonable interval (that includes the value \(q_0 = 0.574\) adopted in the analysis) does not appreciably change our results.
Notes
Acknowledgements
The authors thank Valerio Marra for useful discussions. UA acknowledges financial support from CAPES. CAPB acknowledges financial support from the South African SKA Project. JSA acknowledges support from CNPq (Grants no. 310790/20140 and 400471/20140) and FAPERJ (Grant no. 204282). Some of our analyses used the HEALPix software package.
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