Comparing early dark energy and extra radiation solutions to the Hubble tension with BBN

The $\Lambda$CDM cosmological model has been successful in explaining the properties and evolution of our Universe. However, various low-redshift measurements of the Hubble constant $H_{0}$ have reported a significantly larger value than that inferred from the temperature anisotropy of cosmic microwave background (CMB) measured by Planck (2018) $H_{0}=67.36\pm 0.54$ km/s/Mpc Aghanim:2018eyx . SH0ES measures the Hubble constant by using Cepheids and type Ia supernovae as standard candles and has reported $H_{0}=73.45\pm 1.66$ km/s/Mpc in Ref. Riess:2018uxu (R18) and $H_{0}=74.03\pm 1.42$ km/s/Mpc in Ref. Riess:2019cxk (R19). Similarly, the H0LiCOW Collabration obtained the result $H_{0}=73.3\pm 1.7$ km/s/Mpc from gravitational lensing with time delay Wong:2019kwg . Since all local measurements of $H_{0}$ by different methods consistently indicate a larger value of $H_{0}$ than that from Planck, even if there is an unknown systematic error Efstathiou:2013via ; Freedman:2017yms ; Rameez:2019wdt ; Ivanov:2020mfr , this discrepancy cannot be easily solved Bernal:2016gxb .

As this discrepancy seems serious, several ideas for an extension of the $\Lambda$CDM model have been proposed to solve or relax this tension. A modification in the early Universe would be more promising than our during later times, because low-redshift $z$ cosmology is also well constrained by baryon acoustic oscillation (BAO) measurements. One approach to relax the Hubble tension is to introduce a beyond-the-standard-model component Bernal:2016gxb ; Aylor:2018drw . One of the simplest methods is to increase the relativistic degrees of freedom parametrized by $N_{\mathrm{eff}}$ Aghanim:2018eyx . By quoting the value

for (CMB$+$BAO$+$R18) from the Planck Collaboration Aghanim:2018eyx , it has been regarded that $0.2\lesssim\Delta N_{\mathrm{eff}}\lesssim 0.5$ is preferred to relax the $H_{0}$ tension. Several beyond-the-standard-model proposals DEramo:2018vss ; Escudero:2019gzq could accommodate such an extra $N_{\mathrm{eff}}$. Some of them are interesting because they can address not only the Hubble tension, but also other subjects such as the anomalous magnetic moment of the muon Escudero:2019gzq , the origin of neutrino masses Escudero:2019gvw or a sub-GeV weakly interacting massive particle dark matter Okada:2019sbb . Another popular scenario is so-called early dark energy (EDE) models, where a tentative dark energy component somewhat contributes the cosmic expansion around the recombination epoch Poulin:2018zxs ; Poulin:2018dzj ; Poulin:2018cxd ; Agrawal:2019lmo ; Alexander:2019rsc ; Lin:2019qug ; Smith:2019ihp ; Berghaus:2019cls ; Sakstein:2019fmf ; Chudaykin:2020acu ; Braglia:2020bym ; Gonzalez:2020fdy ; Niedermann:2020dwg ; Lin:2020jcb ; Murgia:2020ryi ; Chudaykin:2020igl ; Yin:2020dwl . The main idea of how to relax the Hubble tension by adding an extra energy component is summarized as follows.

Once the cosmic expansion rate is enhanced by introducing a new extra component, the comoving sound horizon for acoustic waves in a baryon-photon fluid at the time of recombination with the redshift $z_{*}$ becomes shorter than that in the standard $\Lambda$CDM model.

The position of the first acoustic peak in the CMB temperature anisotropy power spectrum corresponds to its angular size $\theta_{*}$, which is related to the sound horizon $r_{s*}$ by $\theta_{*}\equiv r_{s*}/D_{M*}$, with the angular diameter distance

For a fixed measured $\theta_{*}$, the reduction of $D_{M*}$ due to the shorter $r_{s*}$ leads to a larger Hubble parameter, because the primary term of the integrand in Eq. (2) at low $z$ does not change much.

The effects of the shorter sound horizon due to new components like those mentioned above can be seen in the power spectrum as a shift of peak positions to higher multipoles $\ell$. Other effects are to let the first and second peaks higher as well as other high $\ell$ peaks lower. The magnitudes of the change of the peak heights and position shifts depend on the specific extra component model. On the other hand, as is well known, generally an increase of the dark matter density shifts the spectra to lower $\ell$ values and reduces the height of the peaks, and an increase of the baryon density extends the height difference between the first and second peaks Hu:2001bc . The scalar spectral index $n_{s}$ controls the spectral tilt of the whole range of the spectrum. In order to compensate the new component effects on the power spectrum, the dark matter density needs to be increased to return the original spectrum that matches with the $\Lambda$CDM, and then the baryon density also needs to be increased to adjust the relative height of the first and second peaks.

This approach is limited due to the resultant modification to the Silk damping scale, i.e., the photon diffusion scale Knox:2019rjx . One way to see the first problem is the fact that, as mentioned above, the phase shift and damping of the amplitude of high-$\ell$ peaks cannot be well recovered by changing only the dark matter and baryon density¹¹1For a scenario free from this diffusion problem, see Refs. Sekiguchi:2020teg ; Sekiguchi:2020igz .. For EDE models, a poor fit to large-scale structure data has also been claimed Hill:2020osr ; Ivanov:2020ril , which can be seen in the value of $\sigma_{8}$ in Ref. Poulin:2018cxd . Despite this limitation, the introduction of extra radiation or EDE is a simple extension of $\Lambda$CDM that addresses the $H_{0}$ discrepancy. The extra radiation energy and EDE contribute differently throughout the whole cosmological history. While EDE significantly contributes to the energy budget only around the epoch of matter-radiation equality to recombination and its energy density decreases quickly, the extra radiation exists throughout the whole history of the Universe. This difference can be seen in its effects on big bang nucleosystheisis (BBN). In fact, a study for $N_{\mathrm{eff}}$ with referring BBN in the context of the Hubble tension was done in Refs. Cuceu:2019for ; Schoneberg:2019wmt . In EDE models, although the negligible energy density of the EDE component at the BBN epoch appears not to change the BBN prediction, the baryon abundance inferred from the CMB would be different from that in the $\Lambda$CDM model to adjust the CMB spectrum and hence the resultant light element abundance could be affected. In this paper, by taking the fit with BBN into account, we evaluate and compare these scenarios of additional relativistic degrees of freedoms and EDE.

This paper is organized as follows. In the next section, we first describe our modeling of the extra radiation and EDE. After we describe the methods and data sets used in our analysis in Sec. III, we show the results and discuss their interpretation in Sec. IV. The last section is devoted to a summary.

The expansion rate of the Universe—the Hubble parameter,—is defined as

where $a(t)$ is the scale factor and a dot denotes a derivative with respect to cosmic time $t$. In the following, we use the scale factor $a$ instead of time $t$ as a “time” variable. Then, we regard the Hubble parameter as a function of $a$, $H(a)$ and normalize the scale factor as $a(t_{0})=a_{0}=1$, with $t_{0}$ being the age of the Universe.

We perform a Markov Chain Monte Carlo (MCMC) analysis on an $N_{\mathrm{eff}}$ model and the EDE model described in the previous section. We use the public MCMC code CosmoMC-planck2018 Lewis:2002ah and implement the above EDE scenarios by modifying its equation file in CAMB. For estimation of light elements, we use PArthENoPE standard Pisanti:2007hk in CosmoMC.

A shorter sound horizon scale at the recombination epoch, arising from introducing extra energy components such as extra radiation or EDE is a simple approach to resolving the so-called Hubble tension. However, then the compatibility with successful BBN would be a concern, because the extra radiation may contribute to the cosmic expansion or the inferred baryon asymmetry would be different from that in the $\Lambda$CDM. We have compared the EDE models, $N_{\mathrm{eff}}$ model, and a coexistence model, paying attention to the fit to BBN. In fact, the EDE models are also subject to the BBN constraints by increasing the order-unity $\chi^{2}$ as in the $N_{\mathrm{eff}}$ model. Our main results are summarized in Fig. 6 and Table 1. By comparing the posteriors based on the CMB$+$BAO$+$Pantheon$+$R19 combined data, both the $N_{\mathrm{eff}}$ and a coexistence models indicate the largest $H_{0}$ between all of the models studied. $H_{0}$ can be as large as $70.5$ km/s/Mpc within $1\sigma$ only for the coexistence model. The goodness of the fits for the models in terms of $\chi^{2}$ are also listed in Table 1. The fitting is good in the order of the $N_{\mathrm{eff}}+$EDE$6$, $N_{\mathrm{eff}}$, and EDE$6$ models and the $\Lambda$CDM. The difference of the best-fit $H_{0}$ values between the $N_{\mathrm{eff}}+$EDE$6$ and $N_{\mathrm{eff}}$ models is tiny, while the $\chi^{2}$ difference is about $2.4$. The extra radiation seems to be more effective at causing a large $H_{0}$ than the EDE model. Thus, the $N_{\mathrm{eff}}$ model is a much simpler and better model than the EDE models.

We also examined the data sets dependence, whether we include BBN or not. The difference on $N_{\mathrm{eff}}$ in the $N_{\mathrm{eff}}$ model is only about $0.06$ in its mean value, however, the including errors indicate

and there is almost $0.1$ difference in the upper. By comparing this with Eq. (1), we can see the impact of the R$19$ data compared to the R$18$ data. For EDE models, if we include the BBN data, a smaller $H_{0}$ and smaller EDE energy density are preferred.

We would like to thank T. Sekiguchi for kind correspondences concerning the use of CosmoMC. This work is supported in part by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grants Nos. 19K03860, 19H05091, and 19K03865 (O.S.).