The Hubble constant tension with next-generation galaxy surveys

TL;DR

Next-generation galaxy surveys like Euclid and SKA can measure the Hubble parameter across a wide redshift range with high precision, potentially resolving the current tension in Hubble constant measurements by providing independent constraints.

Contribution

This paper demonstrates how upcoming spectroscopic galaxy surveys can precisely measure H(z) and extrapolate to H0, offering a new method to address the Hubble tension.

Findings

Euclid-like surveys can achieve ~3% precision on H0

SKA-like surveys can reach ~2% precision on H0

Combining multiple surveys can improve precision to ~1% and resolve the tension

Abstract

The rate at which the universe is expanding today is a fundamental parameter in cosmology which governs our understanding of structure formation and dark energy. However, current measurements of the Hubble constant, $H_0$, show a significant tension ($\sim 4-6\sigma$) between early- and late-Universe observations. There are ongoing efforts to check the diverse observational results and also to investigate possible theoretical ways to resolve the tension~-- which could point to radical extensions of the standard model. Here we demonstrate the potential of next-generation spectroscopic galaxy surveys to shed light on the Hubble constant tension. Surveys such as those with Euclid and the Square Kilometre Array (SKA) are expected to reach sub-percent precision on Baryon Acoustic Oscillation (BAO) measurements of the Hubble parameter, with a combined redshift coverage of $0.1<z<3$. This wide redshift range, together with the high precision and low level of systematics in BAO measurements, mean that these surveys will provide independent and tight constraints on $H(z)$. These $H(z)$ measurements can be extrapolated to $z = 0$ to provide constraints on $H_0$ using a non-parametric regression. To this end we deploy Gaussian processes and we find that Euclid-like surveys can reach $\sim$3\% precision on $H_0$, with SKA-like intensity mapping surveys reaching $\sim$2\%. When we combine the low-redshift SKA-like Band 2 survey with either its high-redshift Band 1 counterpart, or with the non-overlapping Euclid-like survey, the precision is predicted to be close to 1\% with 40 $H(z)$ data points. This would be sufficient to rule out the current early- or late-Universe measurements at a $\sim$5$\sigma$ level.

Tables

Table 1. Table 1: Uncertainty of reconstructed H 0 subscript 𝐻 0 H_{0} and resulting H 0 subscript 𝐻 0 H_{0} tension ( 6 ), for simulated data from surveys ( 7 ) and ( 8 ).

Survey	$N$	${\sigma }_{H_0}/H_0(\%)$	$T_{\mathrm{rec}}$
	10	$3.993$	$1.669$
	15	$3.705$	$1.798$
Euclid-like	20	$3.482$	$1.914$
	25	$3.303$	$2.017$
	30	$3.155$	$2.112$
	10	$2.482$	$2.684$
	15	$2.321$	$2.871$
SKA-like B1	20	$2.178$	$3.059$
	25	$2.057$	$3.239$
	30	$1.954$	$3.410$
	5	$2.498$	$2.664$
SKA-like B2	8	$2.175$	$3.063$
	10	$2.038$	$3.270$
	20	$1.362$	$4.895$
	25	$1.302$	$5.118$
SKA-like B1+B2	30	$1.260$	$5.288$
(combined)	35	$1.227$	$5.432$
	40	$1.198$	$5.561$
P18	-	$0.802$	$4.390$
R19	-	$1.981$	$4.390$

Equations

$$H_0^{P18}=67.36\pm 0.54{\mathrm{km}\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1},$$

$$H_0^{R19}=74.03\pm 1.42{\mathrm{km}\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}.$$

$$H^{\rm fid}(z)=H^{\rm fid}_{0}\big{[}\Omega_{\rm m}(1+z)^{3}+(1-\Omega_{\rm m})\big{]}^{1/2},$$

$${\Omega }_{\mathrm{m}}=0.3166\pm 0.0084.$$

$$T={\Delta h\over\sigma}=\frac{\big{|}h_{1}-h_{2}\big{|}}{\left[\sigma(h_{1})^{2}+\sigma(h_{2})^{2}\right]^{1/2}}\,,$$

$$T_{\rm rec}={\Delta h_{\rm rec}\over\sigma_{\rm rec}}=\frac{\big{|}h^{\rm R19}_{\rm rec}-h^{\rm P18}_{\rm rec}\big{|}}{\sigma_{\rm rec}}\,,$$

$$0.6<z<2.0,N=10,15,20,25,30.$$

$$0.35<z<3.06,N=10,15,20,25,30,$$

$$0.1<z<0.5,N=5,8,10,$$

$$N_1=10,15,20,25,30\text{mbox}and{N}_2=10,$$

$$\frac{{\sigma }_{H_0}}{H_0}(\text{mbox}SKAB2+Euclid)=\{\begin{array}{ll}1.37\%& N_{\mathrm{E}}=10,\\ 1.32\%& N_{\mathrm{E}}=20,\\ 1.29\%& N_{\mathrm{E}}=30.\end{array}$$

$$\frac{{\sigma }_{H_0}}{H_0}\simeq \{\begin{array}{ll}10\%& \text{mbox}DESI,0.6<z<2,N=30,\\ 5\%& \text{mbox}MeerKAT,0.1<z<0.58,N=10,\\ 5\%& \text{mbox}SKAGS,0.1<z<0.5,N=10.\end{array}$$

Full text

I Introduction

The Hubble constant $H_{0}=100h\,\mathrm{km\,s}^{-1}\,\mathrm{Mpc}^{-1}$ is a fundamental cosmological parameter requiring precise measurement. However, there is a significant tension between the Planck measurement from cosmic microwave background (CMB) anisotropies, assuming a concordance model planck18 (see also Ade:2015xua ),

[TABLE]

and measurements using type Ia supernovae (SNIa) calibrated with Cepheid distances riess19 (see also riess16 ; riess18a ),

[TABLE]

Recent measurements using time delays from lensed quasars wong19 obtained $H_{0}=73.3^{+1.7}_{-1.8}\,\mathrm{km\,s}^{-1}\,\mathrm{Mpc}^{-1}$, while Freedman:2019jwv found $H_{0}=69.8\pm 1.9\,\mathrm{km\,s}^{-1}\,\mathrm{Mpc}^{-1}$ using the tip of the red giant branch applied to SNIa, which is independent of the Cepheid distance scale - in contrast with the $H_{0}=72.4\pm 1.9\,\mathrm{km\,s}^{-1}\,\mathrm{Mpc}^{-1}$ measurement obtained by yuan19 . Analysis of a compilation of these and other recent high- and low-redshift measurements shows Verde:2019ivm that the discrepancy between P18 and any three independent late-Universe measurements is between 4 and $6\sigma$.

Here our focus is not on ways to explain the tension via possible observational systematics or theoretical modifications to the cosmological model, but on the potential of next-generation spectroscopic surveys to provide an independent way of ruling out the P18 or R19 measurement. We take R19 as the representative late-Universe measurement, but the method and results apply to other such recent measurements or combinations of them that are in tension with P18 at a level $\gtrsim 4\sigma$.

Next-generation spectroscopic surveys will measure the redshift and angular extents of the baryon acoustic oscillation (BAO) feature, $\Delta z$ and $\Delta\theta$. The BAO radial and transverse physical scales are $\Delta R_{\|}(z)={\Delta z(z)/(1+z)H_{\|}(z)}$ and $\Delta R_{\perp}(z)=d_{A}(z)\Delta\theta(z)$, where $z$ is the observed redshift, $H_{\|}$ is the radial rate of expansion of matter Jimenez:2019cll and $d_{A}$ is the angular diameter distance. These expressions apply in a general cosmology.

In a perturbed Friedmann model, $H_{\|}=H$, so that $\Delta R_{\|}(z)$ directly determines $H(z)$, while $(1+z)d_{A}$ is an integral of $H^{-1}$, so that $\Delta R_{\perp}(z)$ also contains information about $H(z)$. If we have independent determinations of the radial and transverse scales, we can find $H(z)$ from BAO measurements. The radial and transverse BAO scales should be equal, after accounting for projection and Alcock-Paczynski effects Lepori:2016rzi : $\Delta R_{\|}=\Delta R_{\perp}\equiv R$. The physical BAO scale at decoupling $R_{\rm d}=R(z_{\rm d})$ is the sound horizon, which is estimated with high precision by Planck. This estimate is extremely insensitive to physics at low redshifts, since $R_{\rm d}$ is determined by the physical matter densities $\Omega_{\rm c}h^{2},\Omega_{\rm b}h^{2}$, which are fixed mainly by the relative heights of the CMB acoustic peaks. The estimate of $R_{\rm d}$ assumes the $\Lambda$CDM cosmology at high redshifts Lemos:2018smw .

Spectroscopic surveys with Euclid euclid18 and SKA1 (using 21cm intensity mapping) ska18 are forecast to deliver errors on $H(z)$ that are sub-percent for $0.5\lesssim z\lesssim 2$ and $O(1)\%$ for lower and higher $z$ once systematics like foreground cleaning, in the case of intensity mapping, are properly taken into account bull16 ; villaescusa-navarro16 . We take the forecast errors on $H(z)$, over the redshift ranges of Euclid-like and SKA-like surveys, from ska18 (see the left panel of their Figure 10). Then we use a non-parametric Gaussian process to estimate $H_{0}$ from a regression analysis on the mock $H(z)$ data points, assuming the standard flat $\Lambda$CDM model (see also busti14 ; wang17b ; gomezvalent18 ). The regression produces errors on $H_{0}$, which we compare to the errors from P18 and R19 in (1) and (2).

In summary, we aim to answer the following questions:

How precise are the $H_{0}$ estimates that Euclid- and SKA-like surveys can obtain? Can these surveys rule out P18 or R19?

II Data Analysis

A Gaussian process is a distribution over functions, rather than over variables as in the case of a Gaussian distribution. This allows us to reconstruct a function from data points without assuming a parametrization. We use the GaPP (Gaussian Processes in Python) code seikel12 (see also shafieloo12 ) in order to reconstruct $H(z)$ from data. (For other applications of GaPP in cosmology, see e.g. yahya13 ; gonzalez16 ; pinho18 ; vonmarttens19 .)

We simulate $H(z)$ data assuming the fiducial model,

[TABLE]

where $H^{\rm fid}_{0}$ is chosen as either the P18 or R19 best-fit in (1) or (2). We fix the matter density to the P18 best-fit (TT, TE, EE+lowE+lensing):

[TABLE]

The tension between two $H_{0}$ measurements is defined following camarena18 ; bengaly19a as

[TABLE]

which is valid for an one-dimension Gaussian distribution. The current tension between the measured values in (1) and (2) is $T_{h}=4.4$, corresponding to $\Delta h=4.4\sigma$.

When we apply (5) to the reconstructed $h(z)$ from mock data, with fiducials given by $h^{\rm P18}$ and $h^{\rm R19}$, the uncertainties do not depend on the fiducial, and so they are the same. Then (5) becomes

[TABLE]

where $h^{\rm P18}_{\rm rec}$ and $h^{\rm R19}_{\rm rec}$ are the $h$ measurements reconstructed from the mock data, each with the same uncertainty $\sigma_{\rm rec}$.

For the surveys, we use the redshift ranges given in euclid18 ; ska18 , and we assume a range of values for $N$, the number of $H(z)$ data points, as follows.

Euclid-like galaxy survey:

[TABLE]

SKA-like intensity mapping survey:

[TABLE]

where Band 1+2 delivers the combined constraining power of Band 2 with 10 data points and Band 1 with $N_{1}$ points.

The $H(z)$ measurement uncertainties are taken from the interpolated curves in Figure 10 (left) of ska18 .

III Results

III.1 Measurements of the Hubble Constant

The Gaussian-process reconstructed $H(z)$ for Euclid-like and SKA-like B1+2 surveys is shown in Figure 1 by the 2 and $3\sigma$ regions of uncertainty on the reconstruction. The data points and their forecast $1\sigma$ error bars are also shown – where the errors are increased by 10 (Euclid-like) and 6 (SKA-like) to enhance visibility. We show $N=30$ for Euclid-like, and $N=30$ for SKA-like B1, with a fixed $N=10$ for SKA-like B2. In these Figures, $H_{0}^{\rm P18}$ is the fiducial; using $H_{0}^{\rm R19}$ only shifts the reconstructed $H_{0}$ upward, but has no effect on its uncertainty.

The reconstructed $H_{0}$ and its uncertainty follow from the intersection of the reconstruction region with $z=0$ in Figure 1, in the case of $H_{0}^{\rm P18}$ as fiducial (and similarly for $H_{0}^{\rm R19}$ as fiducial).

The results for the uncertainty and the tension are shown in Table 1 and illustrated in Figure 2. The individual SKA-like B1 and B2 surveys perform better than the Euclid-like survey, given that the former includes lower and higher redshifts than the latter. With 10 low-$z$ (B2) data points and 30 high-$z$ (B1) points, SKA-like surveys in B1 and B2 separately can provide $H_{0}$ measurements as precise as R19 ($\sim 2.0\%$ precision). Nonetheless, they are only able to discriminate between R19 and P18 at $\sim$3$\sigma$.

On the other hand, the combination of B1 + B2 can push $\sigma_{H_{0}}/H_{0}$ down to $1.2\%$, which is close to the P18 precision. This means that SKA-like B1+2 combined is predicted to be able to discriminate between P18 and R19 with $\sim$5$\sigma$ precision.

The results for SKA- and Euclid-like surveys are competitive with future standard siren measurements from gravitational wave events, whose forecasts predict a $H_{0}$ measurement with few percent precision chen18 ; nair18 ; vitale18 ; mortlock18 ; shafieloo18 ; zhang19 . It is estimated that 50 binary neutron star standard sirens could resolve the P18–R19 tension feeney19 , comparable to the 10 + 30 $H(z)$ measurements needed by SKA-like B1+2 combined surveys.

Other model-independent methods are: $\gamma$-ray attenuation data dominguez19 , giving a measurement with $\sigma_{H_{0}}/H_{0}=6.2\%$; Gaussian process regression on galaxy age determination of $H(z)$ together with SNIa data, giving gomezvalent18 $\sigma_{H_{0}}/H_{0}=7.1\%$; HII galaxy data fernandezarenas18 , giving $\sigma_{H_{0}}/H_{0}=4.9\%$. Methods that depend on assuming a cosmological model can deliver greater precision, but at the cost of losing model-independence. These include: using SDSS and eBOSS (quasars) BAO data, giving a direct H0 measurement (marginalising over $\Omega_{m}$) wang17a with $\sigma_{H_{0}}/H_{0}=3.4\%$; using DES clustering combined with weak lensing provides $\sigma_{H_{0}}/H_{0}=1.8\%$. Figure 3 displays these and other measurements, including our forecasts for SKA-like B1+B2 combined surveys with P18 and R19 fiducials.

III.2 Robustness of results

The non-parametric Gaussian process regression that we use is not significantly sensitive to the cosmology assumed to perform the $H_{0}$ estimate. We verified this for dynamical dark energy extensions of the standard model, using $w$CDM and $(w_{0},w_{a})$CDM models. For example, the SKA-like B1+2 combined survey with 40 data points gives $\sigma_{H_{0}}/H_{0}=1.29\%$ with $(w_{0},w_{a})=(-1.1,-0.20)$ and $1.26\%$ with $(w_{0},w_{a})=(-0.9,+0.2)$. This is compatible with $\sigma_{H_{0}}/H_{0}=1.20\%$ obtained from the fiducial $\Lambda$CDM model (see Table 1). These results are consistent with the findings of keeley19 , whose reconstructed cosmological parameters from GP are found to be unbiased with respect to the cosmological model assumed – unlike parameter inference using methods like Monte Carlo Markov Chain.

We varied the cosmological parameters $\bm{p}=(\Omega_{\rm m},H_{0})$ according to a Gaussian distribution $\mathcal{N}(\bm{p},\sigma_{\bm{p}})$, where the parameters and their uncertainties are given by (1), (2) and (4). This test checks whether variations around the fiducial model within the limits imposed by state-of-the-art observations can bias our results. We find a negligible effect on the reconstructed $H_{0}$ error, with an extra variation of only $\Delta(\sigma_{H_{0}}/H_{0})\lesssim 0.1\%$ for SKA-like B1, which produces a change of only $\Delta T_{\rm rec}\sim 0.03\sigma_{\rm rec}$. The results are qualitatively similar for the other surveys.

We verified the robustness of our results with respect to changes of the GP covariance function. By changing the squared exponential kernel that we used to the Matérn(5/2), (7/2) and (9/2) kernels seikel12 ; shafieloo12 , we obtained $\sigma_{H_{0}}/H_{0}=1.62\%,1.37\%,1.29\%$, for SKA-like B1+B2 combined with 25 data points. This is comparable to $1.30\%$ obtained in Table 1.

As a final illustration of the effectiveness of our method, we show in Fig. 4 the recovered $H_{0}$ using standard parametric methods, assuming a $\Lambda$CDM and the CPL dark energy parameterization [$w(z)=w_{0}+w_{a}(1-a)$], for simulations of a SKA-like IM B1+B2 surveys with 40 data points total. Assuming 3 different fiducial models ($\Lambda$CDM as the baseline model, while CPL1 has $w_{0}=-0.7,w_{a}=+0.4$ and CPL2 $w_{0}=-1.0,w_{a}=+0.4$), we see quite different results. For a $\Lambda$CDM parameterization, the errors are small ($\sigma H_{0}/H_{0}=0.56\%$), but can be significantly biased. Fitting a CPL model does recover the fiducial $H_{0}$ but at the expense of much larger errors ($\sigma_{H_{0}}/H_{0}=2.67\%$ for a $\Lambda$CDM simulation and $\sigma_{H_{0}}/H_{0}=5.95\%$ for CPL1, for instance), and fitting a “wrong” model (like fitting a $\Lambda$CDM model on a CPL simulation) yields biased results. The GP method recovers the correct value to within 1$\sigma$ irrespective of model, with the errors quoted above.

IV Conclusions

Next-generation spectroscopic surveys are expected to provide high precision BAO measurements, delivering Hubble rate $H(z)$ data over a wide range of $z$ with a low level of systematics. We estimated the potential precision from such measurements with Euclid-like and SKA-like surveys in estimating $H_{0}$ from $H(z)$, using non-parametric reconstruction and regression. We simulated $H(z)$ data sets following the expected specifications for both surveys, and carried out a Gaussian-process reconstruction of $H(z)$ from these data, allowing for regression down to $z=0$. We checked the robustness of our results with respect to changes in the cosmological model and in the GP covariance functions.

We found that SKA intensity mapping in Bands 1 and 2 separately can measure $H_{0}$ with $2.0-2.5\%$ precision, better than Euclid-like surveys with $3.2-4.0\%$ precision. Although these measurements are not able distinguish between the $H_{0}$ values from CMB and standard candles at higher than $3.5\sigma$, we found that the combination of SKA-like Band 1+2 can reach a precision of $1.2\%(1.4\%)$ with 20 (40) total data points. This leads to a $4.9(5.6)\sigma$ precision in distinguishing between the current P18 and R19 measurements.

The successful combination of low- and high-redshift data in SKA-like surveys suggests an alternative: a combination of SKA-like Band 2 with Euclid-like surveys, which have no overlap between them. With $N_{2}=10$ Band 2 data points, and $N_{\rm E}$ Euclid data points, we find that the combined constraining power is:

[TABLE]

This corresponds to a $4.9(5.2)\sigma$ precision in distinguishing between the current P18 and R19 measurements with 20 (40) total data points. In other words, the combination of low-$z$ SKA- and high-$z$ Euclid-like surveys delivers precision that is almost as high as SKA-like B1+2.

For comparison, we also computed the precision predicted for other spectroscopic surveys: the DESI galaxy survey desi16 , a MeerKAT L-band intensity mapping survey Santos:2017qgq ; Santos2019 , and an SKA1 HI galaxy survey ska18 . We find that:

[TABLE]

We conclude that Euclid-like galaxy and SKA-like intensity mapping surveys are forecast to provide the best $H_{0}$ estimates from GP regression of $H(z)$ measurements, allowing in the best cases (SKA-like B1+2, Euclid-like + SKA-like B2) for resolution of the tension between the $H_{0}$ measured from early- and late-Universe probes.

Acknowledgments – The authors acknowledge the anonymous referee for constructive criticism. CB and RM were supported by by the South African Radio Astronomy Observatory (SARAO) and the National Research Foundation (Grant No. 75415). CB also acknowledges support from the Swiss National Science Foundation at the late stage of this work. CC and RM were supported by the UK Science & Technology Facilities Council Consolidated Grants ST/P000592/1 (CC) and ST/N000668/1 (RM).