Is the $H_0$ tension suggesting a 4th neutrino's generation?

TL;DR

This study investigates whether the tension in the Hubble constant measurements suggests the existence of a fourth neutrino, using cosmological data and models, and finds evidence favoring an extra relativistic species.

Contribution

The paper performs a joint analysis of multiple cosmological distance measurements to test the existence of a fourth neutrino species, finding support for $N_{eff} eq 3$.

Findings

Best-fit $N_{eff} \\approx 4$ in both models.

Standard $N_{eff} \\approx 3$ is ruled out at about 3 sigma.

Chaplygin gas parameter slightly negative, $\\alpha \\approx -0.04$.

Abstract

Flavour oscillations experiments are suggesting the existence of a sterile, $4$th neutrino's generation with a mass of an eV order. This would mean an additional relativistic degree of freedom in the cosmic inventory, in contradiction with recent results from the Planck satellite, that have confirmed the standard value $N_{eff} \approx 3$ for the effective number of relativistic species. On the other hand, the Planck best-fit for the Hubble-Lema\^itre parameter is in tension with the local value determined with the Hubble Space Telescope, and adjusting $N_{eff}$ is a possible way to overcome such a tension. In this paper we perform a joint analysis of three complementary cosmological distance rulers, namely the CMB acoustic scale measured by Planck, the BAO scale model-independently determined by Verde {\it et al.}, and luminosity distances measured with JLA and Pantheon SNe Ia surveys. Two Gaussian priors were imposed to the analysis, the local expansion rate measured by Riess {\it et al.}, and the baryon density parameter fixed from primordial nucleosynthesis by Cooke {\it et al.}. For the sake of generality, two different models are used in the tests, the standard $\Lambda$CDM model and a generalised Chaplygin gas. The best-fit gives $N_{eff} \approx 4$ in both models, with a Chaplygin gas parameter slightly negative, $\alpha \approx -0.04$. The standard value $N_{eff} \approx 3$ is ruled out with $\approx 3\sigma$.

Figures

Tables

Table 1. Table 1: Lower and higher limits of the flat priors used in the analysis. The last four rows are related to supernovae nuisance parameters.

Parameter	Uniform Prior
$N_{\mathrm{eff}}$	$𝒰$[0.05,10.00]
${\Omega }_{dm0}{h}^2$	$𝒰[0.001,1.000]$
$\alpha$	$𝒰[-0.99,2.00]$
${\alpha }_s$	$𝒰[0,1]$
${\beta }_s$	$𝒰[0,4]$
${ℳ}_B$	$𝒰[-22,-16]$
${\Delta }_M$	$𝒰[-1,1]$

Table 2. Table 2: Best-fit values and 2 σ 𝜎 \sigma regions of some cosmological parameters for generalised Chaplygin gas and Λ Λ \Lambda CDM models using both JLA and Pantheon SNe compilation sets.

Model & Data	${\chi }_{\mathrm{bf}}^2$	${\chi }_{\nu }^2$	$H_0^{\mathrm{bf}}$		$N_{\mathrm{eff}}^{\mathrm{bf}}$		${\Omega }_{m0}^{\mathrm{bf}}$		${\alpha }^{\mathrm{bf}}$
Chaplygin
${\theta }_{\ast }$+$r_d^h$+JLA	683.04	0.927	73.37	${73.59}_{-2.83}^{+2.81}$	4.02	${4.17}_{-0.80}^{+0.85}$	0.30	$0.30\pm 0.10$	-0.04	${0.02}_{-0.31}^{+0.41}$
	683.22	0.926	68.89	${70.22}_{-1.37}^{+1.63}$	$3.046$ (fixed)		0.30	$0.26\pm 0.08$	-0.01	${0.09}_{-0.28}^{+0.36}$
${\theta }_{\ast }$+$r_d^h$+Pantheon	1026.89	0.983	72.63	${73.62}_{-2.55}^{+2.51}$	3.91	${4.17}_{-0.72}^{+0.74}$	0.30	$0.29\pm 0.07$	0.00	${0.06}_{-0.26}^{+0.33}$
	1026.96	0.982	68.74	${70.1}_{-1.09}^{+1.38}$	$3.046$ (fixed)		0.29	$0.27\pm 0.06$	0.02	${0.04}_{-0.22}^{+0.27}$
$\Lambda$CDM
${\theta }_{\ast }$+$r_d^h$+JLA	682.93	0.925	74.74	${73.59}_{-2.62}^{+2.57}$	4.29	${4.14}_{-0.74}^{+0.78}$	0.30	$0.30\pm 0.03$	0 (fixed)
	683.19	0.924	69.13	${70.33}_{-1.17}^{+1.53}$	$3.046$ (fixed)		0.29	$0.28\pm 0.02$
${\theta }_{\ast }$+$r_d^h$+Pantheon	1026.88	0.982	74.11	${73.64}_{-2.68}^{+2.61}$	4.27	${4.16}_{-0.72}^{+0.75}$	0.30	${0.30}_{-0.02}^{+0.03}$	0 (fixed)
	1026.89	0.981	68.97	${70.07}_{-1.12}^{+1.46}$	$3.046$ (fixed)		0.30	$0.28\pm 0.02$

Table 3. Table 3: Mean values and credible intervals of the free cosmological parameters and some derived ones for the Λ Λ \Lambda CDM model, using the full CMB likelihood with the JLA SNe compilation set and r d h superscript subscript 𝑟 𝑑 ℎ r_{d}^{h} , Ω b ​ h 2 subscript Ω 𝑏 superscript ℎ 2 \Omega_{b}h^{2} and h ℎ h priors.

parameter	mean	$1\sigma$	$2\sigma$	$3\sigma$
${\Omega }_b{h}^2$	$0.0228$	$\pm 0.0002$	$\pm 0.0005$	${}_{-0.0007}{}^{}{}_{}{}^{+0.0008}$
${\Omega }_c{h}^2$	$0.124$	${}_{-0.003}{}^{}{}_{}{}^{+0.004}$	$\pm 0.007$	${}_{-0.010}{}^{}{}_{}{}^{+0.011}$
$100{\theta }_{MC}$	$1.0406$	$\pm 0.0005$	$\pm 0.0010$	${}_{-0.0016}{}^{}{}_{}{}^{+0.0017}$
$\tau$	$0.10$	$\pm 0.02$	$\pm 0.04$	$\pm 0.06$
$N_{\mathrm{eff}}$	$3.556$	${}_{-0.192}{}^{}{}_{}{}^{+0.189}$	${}_{-0.372}{}^{}{}_{}{}^{+0.379}$	${}_{-0.551}{}^{}{}_{}{}^{+0.610}$
$\ln \left({10}^{10}{A}_s\right)$	$3.15$	$\pm 0.04$	$\pm 0.08$	${}_{-0.12}{}^{}{}_{}{}^{+0.11}$
$n_s$	$0.991$	$\pm 0.008$	$\pm 0.016$	$\pm 0.025$
$H_0$	$72.05$	${}_{-1.39}{}^{}{}_{}{}^{+1.45}$	${}_{-2.78}{}^{}{}_{}{}^{+2.77}$	${}_{-4.20}{}^{}{}_{}{}^{+4.40}$
${\Omega }_m$	$0.284$	${}_{-0.010}{}^{}{}_{}{}^{+0.011}$	${}_{-0.019}{}^{}{}_{}{}^{+0.021}$	${}_{-0.028}{}^{}{}_{}{}^{+0.032}$
${\sigma }_8$	$0.858$	$\pm 0.019$	${}_{-0.036}{}^{}{}_{}{}^{+0.038}$	${}_{-0.058}{}^{}{}_{}{}^{+0.056}$

Equations

$$r_s(z)={\int }_z^{\infty }\frac{c_s}{H(z^{\prime })}d{z}^{\prime },$$

$$\frac{c_s}{c}={\left[3+\frac{9{\Omega }_{b0}}{4{\Omega }_{\gamma 0}}(1+z{)}^{-1}\right]}^{-1/2},$$

$$H(z)=H_0\sqrt{(1-{\Omega }_{m0})+{\Omega }_{m0}(1+z{)}^3+{\Omega }_{R0}(1+z{)}^4}.$$

$${\Omega }_{R0}={\Omega }_{\gamma 0}\left[1+0.68(N/3)\right],$$

$$r_d^h\propto \frac{h}{\sqrt{2.47\left[1+0.68(N/3)\right]}},$$

$$\frac{N}{3}=2.47\left(\frac{h^2}{{\tilde{h}}^2}\right)-\mathrm{1.47.}$$

$${\theta }_{\ast }=\frac{r_{\ast }}{D_A(z_{\ast })},$$

$$D_A(z_{\ast })={\int }_0^{z_{\ast }}\frac{c}{H(z)}dz.$$

$${\left[\frac{H(z)}{H_0}\right]}^2$$

$$\log {\mathcal{L}}_{\mathrm{CMB}}(\mathbf{p})=-0.5{\left(\frac{100{\theta }_{\ast }(\mathbf{p})-1.04109}{0.00030}\right)}^2,$$

$$\log {\mathcal{L}}_{\mathrm{BAO}}(\mathbf{p})=-0.5{\left(\frac{r_d^h(\mathbf{p})-101.2}{2.3}\right)}^2,$$

$$\log {\mathcal{L}}_{\mathrm{SNe}}(\mathbf{p})=$$

$$m_B^{\mathrm{mod}}=5{\log }_{10}{d}_L(z_{\mathrm{CMB}},z_{\mathrm{hel}})-{\alpha }_s{X}_1+{\beta }_s\mathcal{C}+{\mathcal{M}}_B,$$

Full text

I Introduction

The panorama on neutrino flavour oscillation experiments is very robust. Data from different experimental setups converge into a concise explanation, in which neutrino flavour oscillations are driven by two large and one small mixing angles and two hierarchical mass differences Esteban:2016qun . Such framework provides a precise prediction on flavour transitions of atmospheric, solar, reactor and accelerator neutrinos, in an energy range that varies from sub-MeV to several GeV, and distances that vary from few meters to astrophysical distances. These predictions have been corroborated by different experimental results on the last decades.

However, experiments that find neutrino flavour conversion signals that are not easily accommodated in the 3-neutrino mixing framework are piling up. More than 15 years ago the LSND experiment Aguilar:2001ty observed an appearance of electron anti-neutrinos in a muon anti-neutrinos flux, which if explained through mass-driven flavour oscillations would suggest a mass scale incompatible with others oscillation experiments results. Recently MiniBoone Aguilar-Arevalo:2018gpe confirmed the main features of LSND results, both in neutrino and anti-neutrino channels, strengthening the hypothesis that there is a fourth neutrino family, which does not couple with weak gauge bosons (hence, sterile neutrinos), but participates in flavour neutrino oscillations with a mass scale of order $\sim 1$ eV.

Although the above mentioned results can be well explained by a fourth neutrino family, it seems that they are incompatible with disappearance experiments, such as Minos/Minos+ Adamson:2017uda , NEOS Ko:2016owz and Daya Bay An:2016luf (see for instance Dentler:2018sju for a comprehensive comparison between experiments). Therefore, assuming that these experimental results should be explained by new physics on neutrino sector, it seems that such new physics would have to go beyond the simple addition of an extra neutrino family. As stated in Liao:2018mbg , the neutrino sector seems quite baroque.

Nevertheless, most of the solutions proposed to accommodate all oscillation neutrino experiments results would add an extra degree of freedom in the relativistic species that would be produced in the early universe. It is then worthwhile to revisit the cosmological results on this subject Hamann ; Giusarma ; bernal ; suecos . In the present contribution we analyse two distance rulers that are sensitive to the number of relativistic species, namely the CMB and BAO acoustic scales, complemented by SNe Ia luminosity distances observations and by the current priors on the local expansion rate and baryonic density.

The paper is organised as follows. In the next section we discuss why the tension between Planck and HST measurements of $H_{0}$ can be alleviated with a higher $N_{\mathrm{eff}}$ value. In section 3 we describe the tests to be performed, and in sections 4 and 5 we show the results of our joint analysis. In section 6 some conclusions are outlined.

II The acoustic horizon

The acoustic horizon, given by

[TABLE]

has two important values in the context of cosmological data. When we are dealing with the CMB acoustic scale $\theta_{*}$, the acoustic horizon is evaluated at the redshift of last scattering ($z_{*}\approx 1090$), so that $r_{*}\equiv r_{s}(z_{*})$. In the case of BAO, the acoustic horizon is evaluated at the drag epoch ($z_{d}\approx 1060$), which we will refer to as $r_{d}\equiv r_{s}(z_{d})$. The sound speed is given by

[TABLE]

and the Hubble-Lemaître function of the spatially-flat standard model by

[TABLE]

In the above expressions, $\Omega_{m0}=\Omega_{dm0}+\Omega_{b0}$ and $\Omega_{R0}=\Omega_{\nu 0}+\Omega_{\gamma 0}$ are, respectively, the density parameters of total matter (dark matter + baryons) and radiation (neutrinos + photons), and $H_{0}=100\,h$ km/s Mpc*-1* is the Hubble-Lemaître parameter. The radiation density parameter can be expressed as

[TABLE]

where $N$ is the number of neutrinos species. In a rough estimation, neglecting the contribution of the baryonic and dark sectors for $z\gg 1$, and taking the observed $\Omega_{\gamma 0}h^{2}=2.47\times 10^{-5}$ Planck , we have

[TABLE]

where $r_{d}^{h}\equiv r_{d}h$. Let us consider a hypothetical observational probe of the acoustic scale, and let $\tilde{h}$ be the value obtained when the number of species is fixed in $\tilde{N}=3$. For an arbitrary $N$, the same probe will give a Hubble-Lemaître parameter $h$ such that

[TABLE]

Using for $\tilde{h}$ the Planck value $\tilde{h}=0.68$ Planck , and for $h$ the local value $h=0.73$ Riess , it follows that $N\approx 4.1$.

III Standard rulers

We will consider two standard rulers in our analysis. The first is given by the position of the first peak in the CMB spectrum of anisotropies, more precisely the angular scale

[TABLE]

where $D_{A}$ is the comoving angular diameter distance to the last scattering surface,

[TABLE]

Its observed value is $100\,\theta_{*}=1.04109\pm{0.00030}$ Collaboration:2018va . The second ruler comes from BAO observations, and can be encompassed, in an approximately model-independent way, in the acoustic horizon derived by Verde et al., $r_{d}^{h}=101.2\pm 2.3$ Verde . We will complement the analysis by fitting the luminosity distances to supernovae Ia of the JLA compilation sne:betoule14 . Compared to other surveys, it has the advantage of allowing the light-curve recalibration with the model under test. Although it was also used to derive the Verde et al. acoustic horizon at the drag epoch Verde , this fitting is insensitive to $N_{\mathrm{eff}}$, and will be used for better constraining the matter density. Anyway, in order to control the effect of a double counting, we will also use the Pantheon SNe Ia compilation Scolnic:2017caz in the analysis, which contains supernovae not used in the Verde et al. fitting. As Gaussian priors of our analysis, we will take the Riess et al. local value of the Hubble-Lemaître parameter Riess , $h=0.7348\pm 0.0166$, and the Cooke et al. value for the baryonic density parameter, $\Omega_{b0}h^{2}=0.02226\pm 0.00023$, which comes from nucleosynthesis constraints Cooke .

For the sake of generality, our tests will be performed with two different models. The first is the standard model, for which the indication of a 4th neutrino generation will already be manifest. The robustness of this possibility will be verified by testing an extension of the standard model given by the generalised Chaplygin gas gCg1 ; gCg2 ; gCg3 ; gCg4 ; gCg5 ; gCg6 ; gCg7 ; gCg8 , with a Hubble function given, with the addition of radiation, by

[TABLE]

In the binomial expansion of the brackets, we have a leading term $\Omega_{m0}(1+z)^{3}$, which shows that, for the present purpose of background tests, the baryonic content can be absorbed in the above defined gas. For $\alpha=0$ we recover the standard $\Lambda$CDM model. Perturbative tests are outside the scope of this paper, but let us comment that, although the adiabatic generalised Chaplygin gas is ruled out by the observed matter power spectrum waga , non-adiabatic versions with negative $\alpha$ present a good concordance when tested against background and LSS observations wands21 ; wands22 ; wands23 ; non-adiabatic1 ; non-adiabatic2 ; wang ; cassio .

IV Joint analysis and results

On the basis of Bayesian Statistics, we defined the joint log-likelihood as a function of the parameter array p, adding to the CMB log-likelihood,

[TABLE]

a log-likelihood for $r_{d}^{h}$,

[TABLE]

and the log-likelihood of supernovae,

[TABLE]

.

For the Chaplygin gas the set of free cosmological parameters were ${\bf p_{c}}=\{H_{0},\Omega_{b0}h^{2},\Omega_{dm0}h^{2},\alpha,N_{\rm{eff}}\}$, with free nuisance parameters due to corrections on SNe light-curves, ${\bf p_{s}}=\{\alpha_{s},\beta_{s},\mathcal{M}_{B},\Delta_{M}\}$ for JLA SNe likelihood sne:betoule14 ; guy10 or ${\bf p_{s}}=\{\mathcal{M}_{B}\}$ for Pantheon SNe likelihood Scolnic:2017caz , so that ${\bf p}=\{{\bf p_{c}},{\bf p_{s}}\}$.

The supernovae theoretical apparent magnitude $m_{B}^{\mathrm{mod}}$ is written as

[TABLE]

and so JLA supernovae light-curves are standardised along with cosmological parameters of the tested model, finding the best stretch ($\alpha_{s}$) and color ($\beta_{s}$) corrections, and also fitting the absolute magnitude $\mathcal{M}_{B}$ with a step $\Delta_{M}$ for more massive host galaxies. Even the covariance matrix $\mathbf{C_{SN}}$ is a function of $\alpha_{s}$ and $\beta_{s}$. For the Pantheon sample, $X_{1}$ and $\mathcal{C}$ values are not available, and the apparent magnitude and covariance matrix already calibrated for the standard $\Lambda$CDM model is given, so that it is allowed to adjust only the absolute magnitude $\mathcal{M}_{B}$.

We explored the parameter space via the PyMultiNest multinest01 ; multinest02 ; multinest03 ; pymultinest module for Python, setting 1500 live points and ‘parameter’ sampling efficiency. Besides the Gaussian priors previously mentioned, all other parameters had uniform priors presented on Table 1.

The results of our joint analysis are summarised in Fig. 1. Previous results cassio with $N_{\mathrm{eff}}=3.046$, and JLA dataset only, favoured negative values of $\alpha$, which is not obtained in the present scenario. Also, even in the standard model case, a 4th neutrino generation is suggested by the data. The $2\sigma$ confidence intervals for some parameters are presented on Table 2. The standard value $N_{\mathrm{eff}}=3$ is marginally ruled out with $99\%$ of confidence in all considered scenarios.

V Joint analysis with full CMB

In spite of the above results, which show a better agreement between the Planck and local values of the Hubble-Lemaître parameter when an extra relativistic degree of freedom is added to the cosmic inventory, the performed tests do not involve the full CMB spectrum of anisotropies, but only a joint analysis of distance ladders as the CMB acoustic scale (that is, the position of the first peak in the anisotropy spectrum), the characteristic scale distances to the BAO peaks, and the luminosity distances to type Ia supernovae. The fit of the full CMB data with the $\Lambda$CDM model leads, in fact, to lower values of $N_{\text{eff}}$ and $H_{0}$ chineses ; heavens . In addition, higher values of $N_{\text{eff}}$ usually require higher values for the scalar spectral index $n_{s}$, which challenges the standard inflationary models heavens2 ; micol ; micol2 . In this section we present a joint analysis of the full CMB data, obtained with the CosmoMC engine cosmomc ; cosmomc2 ; camb and the Planck 2015 likelihood Planck , including the probes used above, namely the JLA compilation of SNe Ia sne:betoule14 and the Gaussian priors given by Verde et al. to the BAO scale $r_{d}$ Verde , by Cooke et al. to the physical baryon density Cooke , and by Riess et al. to the local value of $H_{0}$ Riess . Our analysis was performed only for the $\Lambda$CDM model, while the generalised Chaplygin gas case will be presented in a forthcoming publication. The resulting probability density functions for some free and derived parameters are shown in Fig. 2, together with the $1\sigma$ and $2\sigma$ 2D confidence regions. Dashed lines refer to public available 2015 Planck chains (plikHM_TT_lowTEB and plikHM_TT_lowTEB_post_JLA), and solid lines to the joint analysis, plikHM_TT_lowTEB_JLA including the aforementioned priors. The corresponding confidence intervals are shown in Table III. As expected, there is a positive correlation between $N_{\text{eff}}$ and $H_{0}$, as well as between $N_{\text{eff}}$ and $n_{s}$. The joint analysis marginally rules out the standard scenario of $3$ relativistic species, with $99\%$ of confidence. In contrast, an additional species is marginally within the $2\sigma$ confidence interval. The scalar spectral index results to be $n_{s}\approx 0.99$, close to a Harrison-Zeldovich spectrum but still in the region $n_{s}<1$ allowed by the inflation paradigm.

VI Concluding Remarks

The above results show that overcoming the $H_{0}$ tension between the CMB and HST observations may require a number of relativistic species that corroborates current experimental results in the neutrinos section of the standard model of particle physics. Indeed, the obtained best-fit $N_{\mathrm{eff}}\approx 4$ might be a clear signature of an additional, sterile, neutrino’s family. We should stress, however, that the analysis we have performed includes only background tests, involving measurements of angular diameter and luminosity distances. The number of relativistic species also affects observations in the perturbative sector of cosmology, because the ratio between the matter and radiation densities defines, for example, the turnover of the matter power spectrum through the horizon scale value at the time of matter-radiation equality. Although the data do not determine this turnover precisely enough, a joint analysis of background and LSS observations would be complementary to the present results. Furthermore, a sterile neutrino with $1$eV mass would contribute with $\approx 8\%$ of a warm component in the present dark matter Dodelson . On the other hand, despite the possibility presented here of conciliating the CMB acoustic scale with local $H_{0}$ measurements by adding a relativistic degree of freedom, the best-fit of the full CMB spectrum with the $\Lambda$CDM model in fact leads to a lower value of $N_{\mathrm{eff}}$ chineses ; heavens . It is also worth of note that a higher $N_{\mathrm{eff}}$ may be correlated to a higher spectral index of primordial fluctuations heavens2 ; micol ; micol2 . The tests performed here, using distance rulers that are approximately model-independent, are complementary to other constraints, but the definite value of $N_{\mathrm{eff}}$ remains a subject for further investigation.

Acknowledgements

We are thankful to J.S. Alcaniz and G. Gambini for helpful suggestions. Work partially supported by CNPq (SC, grant no. 307467/2017-1 and PCH, grant no. 310952/2018-2) and FAPESP (grant no. 2014/19164-6).