Towards a Solution to the H₀ Tension: the Price to Pay

Even before the discovery of the accelerated expansion rate of the Universe [2, 3, 4], the quest for determining the rate of expansion of the Universe has occupied a central role in cosmology for decades. However, great technical and observational achievements in recent years have delivered percentage-level precision measurements of the cosmological parameters. The improvement probing the physics of different epochs of the universe has yielded discordance in some measurements. In particular we have an increasing tension among the value of some cosmological parameters as for example the value of H₀.

According to the distance ladder measurements, which uses close by Cepheids to anchor supernovae data and determine their distance [5, 6, 1], the obtained value for the Hubble parameter is up to $3.4\sigma$ higher than the value determined using Cosmic Microwave Backgound (CMB) probes. This distance ladder method to determine H₀ is model-independent, i.e. it does not rely on the underlying cosmological model, and the latest estimated reports a value of H${}_{0}=74.03\pm 1.42$ km s^-1Mpc^-1 [1].

An alternative calibration of the distance ladder uses the tip of the red giant branch (TRGB) method [7]. This method is independent of the Cepheid distance scale and gives a value of H${}_{0}=69.8\pm 1.9$  km s^-1Mpc^-1 [8] which is midway in the range defined by the current Hubble tension. It agrees at the $1.2\sigma$ level with that of the Planck collaboration [9], and at the $1.7\sigma$ level with the SH₀ES (Supernovae H₀ for the Equation of State) measurement of $H_{0}$ based on the Cepheid distance scale [1]. Measurements of lensing time delays [10, 11, 12] between multiple images of background quasars provide a high value for H₀ which is in agreement with the traditional local distance ladder estimation. In [10] they report a value of H${}_{0}=71.9^{+2.4}_{-3.0}$ km s^-1Mpc^-1 , while [11] finds a value H${}_{0}=72.5^{+2.1}_{-2.3}$ km s^-1Mpc^-1 , and the H0LiCOW (H₀ Lenses in CosmoGrail Wellspring) team [12], reports H${}_{0}=73.3^{+1.7}_{-1.8}$ km s^-1Mpc^-1 , by making a joint analysis of six gravitationally lensed quasars with measured time delays. This technique is completely independent of both the supernovae and CMB analyses.

The precise CMB measurements by Planck [P-18] determines a value of the Hubble parameter as H₀ = ($67.27\pm 0.60$)km s^-1Mpc^-1 at 68% confidence level assuming the standard $\Lambda$CDM model, corresponding the particle content of the standard model of particles physics [13] supplemented by cold dark matter and a cosmological constant as dark energy. The value of H₀ from CMB is in conflict with the value of H₀ determined at late cosmological times from local measurements in Riess et al [R-19] [1] given by H₀ = ($74.03\pm 1.42$)km s^-1Mpc^-1 and with an average reported value H₀ =($73.3\pm 0.8$)km s^-1Mpc^-1   [14] from different local measurements projects. A combined analysis of distance measurements for four megamaser-hosting galaxies done by the Megamaser Cosmology Project (MCP) [15] reports a value H${}_{0}=73.9\pm 3.0$ km s^-1Mpc^-1 . A combination of time-delay cosmography and the distance ladder results is in 5.3$\sigma$ tension with Planck CMB determinations of H₀ in flat LCDM.

On the other hand, the latest results from the Planck collaboration report a model dependent $\Lambda$CDM  extrapolated value of H${}_{0}=(67.36\pm 0.54)$ km s^-1Mpc^-1 [9], while the latest results the Atacama Cosmology Telescope [16] CMB probe found a value that agrees with the Planck satellite estimate within $0.3\%$, reporting a value of H${}_{0}=67.6\pm 1.5$ km s^-1Mpc^-1 . In a recent review L.Verde, A. Riess and T.True reported an average value of local measurements of H₀ =$73.03\pm 0.8$ km s^-1Mpc^-1 [14]. Regardless of the exact value of H₀ by local measurements, the significance of the tension between the measurements of early and late time lies in the range 4.0$\sigma$ and 5.7$\sigma$ [14], implying some profound miss-understanding in either the systematic errors of the observational analysis, or the theoretical $\Lambda$CDM  model.

The discrepancy between the early and late-Universe H₀ measurements has gained major attention by the cosmological community and some authors have explored a variety of extensions to the minimal $\Lambda$CDM  model to accommodate the high value of H₀ obtained by local measurements with the precise information encoded in the CMB. Trying to understand the tension between these two values led to a reexamination of possible sources of systematic errors in the observations [17], [18], [19], but it also suggests the need to extend our physical model describing the universe. Any of these theoretical modifications should leave the accurate determination of the angular scale of the acoustic peaks in the CMB power spectrum by Planck unchanged [20].

The H₀   tension has been studied recently in [21, 22, 23, 24, 25] and more recently in [26, 27, 28, 29, 30, 31, 32, 33] where the impact on structure formation has been studied.

The suggestion made by [5] to explore the existence of dark radiation in the early Universe in the range of $\Delta N_{eff}=0.4-1$ to solve this tension was explored in detail by [34] where they explore changing the value of $N_{eff}$ and $c_{s}$. Alternatively, some models explore the possibility of having interactions between the dark sector that can, not only help to solve the cosmic coincidence problem but also solve the H₀ tension [35, 36]. In the logic of exploring alternative models for the dark sector, in [37] investigated the possible scenarios for a phantom crossing dark energy component as another option for solving the Hubble tension. Given the amount of interest invested in this topic, some authors have explored changes to General Relativity in order to accommodate the high value of H₀ with CMB data. For instance, the work by [38] explores a model in which a fifth force between dark matter particles is mediated by a scalar field which plays the role of dark energy. In another work, models which vary the effective gravitational constant and effective number of relativistic degrees of freedom are explored by [39]. In a different approach, [40] explores the possibility of strongly interacting massive neutrinos to alleviate the H₀ tension.

However, perhaps the most widely explored extension to $\Lambda$CDM is known as Early Dark Energy (EDE) [41, 42, 43, 44]. There is no unique or unambiguous definition of EDE. Typically in EDE models there is an early period during which an extra energy component, not contained in $\Lambda$CDM  model, contributes to the expansion rate of the universe H. Even the terminology of "early period" is model and case dependent as it can take place in radiation domination era or at late times as for example at $z\sim 4$. Original Early Dark Energy models where motivated by the evolution of scalar fields (quintessence) to describe the evolution of Dark Energy [45, 46, 41, 42, 47]. This EDE models had in general a none negligible energy density at early times well in radiation domination. The equation of state $w$ of the quintessence scalar field had a period of $w=1/3$ at early times with a later transition to $w\sim 1$, diluting the energy density and becoming subdominant for a long period of time covering most of the matter domination period, to finally reappear dynamically at late time as dark energy and were originally studied in [48, 49] and [50, 51, 52, 53] and [54, 55]. Alternatively, recent EDE models add an extra component to the energy-momentum tensor $\Omega_{\textrm{EDE}}(a)$ at different scales and this EDE dilutes rapidly at scale factor $a_{c}$, which determines the time of the transition, with $\Omega_{\textrm{EDE}}=0$ for $a\gg a_{c}$. This EDE modifies the expansion rate of the universe, the cosmological distances and the density perturbations at different epochs [56, 43] and [57, 21, 58] and have been proposed as deviations from $\Lambda$CDM and possible solutions to H₀ crisis [1].

Furthermore, the increasing statistical tension in the estimated Hubble parameter from early and late times observations [14] has reignited interest in alternative cosmological models, while the surge in clustering data [59] and the percentage precision for cosmic distances [59, 9] allows to search for extensions beyond $\Lambda$CDM by searching for cosmological features in the matter [60, 32, 24, 42, 56, 43, 57] or CMB power spectra, standard distances rulers, or tensions in $\Lambda$CDM model as the recent H₀ crisis [1].

On the other hand a physically motivated Dark Energy model presented in [51, 61, 62, 63] introduces a dark sector, corresponding to a dark gauge group SU(3) similar to the strong QCD interaction in the standard model. The fundamental particles contained in this dark SU(3) are massless and redshift as radiation for $a<a_{c}$ but the underling dynamics of the gauge interaction of this group forms massive bound states once the interaction becomes strong, similar as with protons and neutrons in the strong QCD force, and we refer to this model as "Bound Dark Energy model" (BDE) [51, 61, 62]. The energy of the elementary particles is transferred to the lightest bound state after the phase transition takes place at $a_{c}$ and corresponds to a scalar field $\phi$. Due to the dynamics of $\phi$ the energy density of BDE dilutes at $a_{c}$ and eventually reappears close to present time as Dark Energy [61, 62]. This dilution at $a_{c}$ leaves interesting imprints on the matter power spectrum [63] (for a model independent analysis see for instance [64, 65]). BDE is a particular model of elementary particles physics where an extra gauge group $SU(3)$ is introduced and contains naturally the main characteristics of EDE, namely it accounts for extra relativistic energy density $\rho_{ex}$ at high energies while $\rho_{ex}(a)$ dilutes rapidly for $a>a_{c}$ due to a phase transition of the underlying gauge and forms bounds states [61, 62].

The main goal in this work is study the tension and possible solution in the value of H₀ from low redshift probes with the precise determination of CMB data. This paper is organised as follows. Section 2 we present a brief introduction and the details behind our modifications through a toy model calculations in section 3. The working details and implementation in the Boltzmann code CAMB and COSMOMC are presented in section 4, the results are discussed in section 4 and the analysis are in section 4.1, while we present our conclusions in 5.

The main goal in this work s study the tension and possible solutions in the value of H₀ from low redshift probes and the precise determination of CMB data. We work with two different cosmological models, the first one is simply the standard $\Lambda$CDM  model, corresponding to a content of the standard model of particles physics [13], cold dark matter and a cosmological constant as dark energy, while the second model we denote as $\Lambda$CDM-Nx, corresponding to $\Lambda$CDM  but supplemented with extra relativistic energy density $\rho_{ex}(a)sim1/a^{3}$ present at a scale factor $a\leq a_{c}$, while for $a>a_{c}$ $\rho_{ex}(a)$ dilutes as $\rho_{ex}(a)\sim 1/a^{6}$. This model $\Lambda$CDM-Nx is inspired by BDE [61, 62].

For definiteness in this study we take the recent local measurement $H_{0}=(74.03\pm 1.42)$ km s^-1Mpc^-1   at 68% confidence level from Riess et al [R-19] [1], and the inferred value of H${}_{0}=(67.27\pm 0.60)$ km s^-1Mpc^-1 at one-$\sigma$ level from Planck-2018 [Pl-18] [9] for a $\Lambda$CDM model using (TT,TE,EE+LowE) measurements. We modified CAMB [66, 67] and make a full COSMO-MC (Markov Chains) analysis ²²2http://cosmologist.info/cosmomc [68, 69, 70]. We perform the analysis for $\Lambda$CDM  and $\Lambda$CDM-Nx models. However, besides the one-sigma value $\sigma_{H}=1.42$ from local H₀   measurements [R-19] and we also consider two forecasting one-sigma values $\sigma_{H}$, and for definiteness we choose and we introduce in the analysis a value of $H_{0}=(74.03\pm\sigma_{H}$ km s^-1Mpc^-1   with $\sigma_{H}=1$ and $\sigma_{H}=0.5$ (in units of km s^-1Mpc^-1 ). With these two forecasting values we asses the impact of a more precise local H₀ measurements on the posterior value of H₀   from CMB + local H₀ data. Notice however, that our forecasting one-sigmas $\sigma_{H}=1$ and $\sigma_{H}=0.5$ are similar to the one reported in [14] with H₀ =$(73.03\pm 0.8)$km s^-1Mpc^-1 . The results of our analysis are shown in section 4.1 and we present the conclusions in (5).

The value of H₀ =$74.03\pm 1.42$km s^-1Mpc^-1   measurements determined by A. Riess and his team [R-19] [5, 6, 1] has a discrepancy between 4.0$\sigma$ and 5.8$\sigma$ [14] with the Planck’s CMB (TT,TE,EE+LowE) [9] [P-18] inferred value of H${}_{0}=(67.27\pm 0.60)$ km s^-1Mpc^-1 at one-$\sigma$ level in a $\Lambda$CDM model. The solution to this discrepancy remains an open question in cosmology. Since CMB radiation is generated at an early epoch $a_{\star}=1/1090$, the prediction of the Hubble constant at present time H₀   inferred by Planck data is a consequence of the assumption of the validity of the standard $\Lambda$CDM  model. So either Planck or local H₀   measurements are inaccurate, due to possible systematics, or we need to modify the concordance cosmological $\Lambda$CDM  model. Here we follow this second option and we will attempt to reconcile the value of H₀ of these two observational experiments.

We will work with two different cosmological models. The first one is simply the standard $\Lambda$CDM  model, corresponding to a content of the standard model (sm) particles physics [13] and a cosmological constant as dark energy. We name the second model as $\Lambda$CDM-Nx  and it consists of $\Lambda$CDM  supplemented by an extra relativistic energy density $\rho_{ex}(a)\sim 1/a^{3}$ present at early times for a scale factor $a\leq a_{c}$, where $a_{c}$ denotes the transition scale factor, and we assume that $\rho_{ex}(a)\sim 1/a^{6}$ for $a>a_{c}$, motivated by BDE and EDE models.

We have implemented the cosmological $\Lambda$CDM-Nx   model in CAMB [66, 67] and and we perform a full COSMO-MC (Markov Chains) analysis for several data sets described in section (sec.results) for both $\Lambda$CDM  and $\Lambda$CDM-Nx models and we present the results and conclusions in section (4).

For definiteness we take the (TT,TE,EE+lowE) measurements from Planck-2018 [Pl-18] [9] with H${}_{0}=(67.27\pm 0.60)$ km s^-1Mpc^-1 and the recent local measurement $H_{0}=(74.03\pm 1.42)$ km s^-1Mpc^-1   at 68% confidence level from Riess et al [R-19] [1]. However, besides the one-sigma value $\sigma_{H}=1.42$ (in units of km s^-1Mpc^-1 ) from local measurements [R-19], we also introduce two forecasting one-sigma values and we choose $\sigma_{H}=1$ and $\sigma_{H}=0.5$. With these two forecasting values we want to asses the impact a more precise local H₀ measurements on the posterior value of H₀   combined with the same CMB as before. Notice however, that our forecasting one-sigmas values $\sigma_{H}=1$ and $\sigma_{H}=0.5$ are of the same order as in the average value H₀ =($73.03\pm 0.8$)km s^-1Mpc^-1 reported in [14]. The results of these analysis are shown in section 4.1 and we present the conclusions in (5).

Before discussing the results of the implemented the cosmological models $\Lambda$CDM  and $\Lambda$CDM-Nx models in CAMB and COSMO-MC presented in section 4 we would like to follow up a simple toy model presented in in section 3 illustrating how an extra relativistic energy density $\rho_{ex}(a)$, present only at early times $a<a_{c}$, can account for the same acoustic scale $\theta(a_{\star})$ as measured by Planck [P-18] but with the value of H₀ consistent with Riess [R-19]. We estimate the cosmological constraints analytically in section 3.2 and we study the impact of the extra $\rho_{ex}$ in the growth of the linear matter density and in the matter power spectrum in section 3.3.

We present now a simple toy model illustrating analytically how adding an extra relativistic energy density $\rho_{ex}(a)$, present at early times, can account for having the same acoustic scale $\theta(a_{\star})$ as $\Lambda$CDM model but with a higher value of H₀.

We consider here two models, the first one is simple the standard $\Lambda$CDM model while our second model corresponds to an extension to $\Lambda$CDM, where we add extra relativistic energy density $\rho_{ex}(a)\propto 1/a^{3}$ present only at early times for a scale factor $a$ smaller than $a_{c}$, corresponding to the transition scale factor, and the extra energy density dilutes as $\rho_{ex}\propto a^{-6}$ for $a\gg a_{c}$ and becomes therefore rapidly negligible. We refer to this later model as $\Lambda$CDM-Nx and is motivated by Bound Dark Energy (BDE) model [61, 62] and EDE models [41, 42, 43, 44].

With this rapid dilution we avoid a step function transition at $a_{c}$ in the evolution of $\rho_{ex}(a)$. Clearly $\rho_{ex}$ dilutes faster than radiation for $a>a_{c}$ and its contribution becomes rapidly subdominant. The energy density $\rho_{ex}$ can also be parametrized by the number of extra relativistic degrees of freedom $N_{ex}$, defined in terms of the the neutrino temperature $T_{\nu}$ as $\rho_{ex}=(\pi^{2}/30)N_{ex}T_{\nu}^{4}$. We implement the $\Lambda$CDM-Nx model in the Boltzmann code CAMB [66, 67, 68, 69, 70]) and make a full COSMO-MC (Markov Chains) analysis for $\Lambda$CDM and $\Lambda$CDM-Nx for several data sets and we present the results and conclusions in section (4.1).