Big bang nucleosynthesis and early dark energy in light of the EMPRESS $Y_p$ results and the $H_0$ tension

Recent measurements of the primordial ${}^4\mathrm{He}$ abundance $Y_p$ from EMPRESS suggest a cosmological scenario with an effective number of neutrino species that deviates from the standard value and a nonzero lepton asymmetry. We argue that the standard cosmological model would be extended further if the Hubble tension were taken into account, in which the derived baryon density could be somewhat higher than the in the standard $\mathrm{\Lambda }\mathrm{CDM}$ framework. We also discuss the issue by assuming early dark energy whose energy density can have a sizable fraction at the epoch of big bang nucleosynthesis. We show that the existence of early dark energy can reduce some of the tension implied by the EMPRESS $Y_p$ results.

Figure 1

Parameter ranges within $1\sigma$ and $2\sigma$ errors for $Y_p$ [9] (light and dark dotted (blue) regions) and $D_p$ [26] (light and dark magenta regions for $1\sigma$ and $2\sigma$) in the ${\eta }_{10}\text{−}{N}_{\mathrm{eff}}$ plane. Here the $1\sigma$ and $2\sigma$ errors include both observational and theoretical ones. We take the neutrino degeneracy parameter as ${\xi }_e=0$ (left), 0.05 (middle), and 0.08 (right).

Figure 2

$1\sigma$ and $2\sigma$ allowed regions in the $N_{\mathrm{eff}}\text{−}{\xi }_e$ plane from measurements of $Y_p$ and $D_p$ with an ${\eta }_{10}$ prior. For the ${\eta }_{10}$ prior, we take ${\eta }_{10}={\eta }_{10}^{\mathrm{ref},1}\pm {\sigma }_{{\eta }_{10,}1}=6.14\pm 0.038$ (black dotted and solid lines for $1\sigma$ and $2\sigma$ in both panels) obtained from $\mathrm{CMB}+\mathrm{BAO}$ data [2] in the $\mathrm{\Lambda }\mathrm{CDM}$ framework and ${\eta }_{10}={\eta }_{10}^{\mathrm{ref},2}\pm {\sigma }_{{\eta }_{10},2}=6.4\pm 0.060$ (light and dark shaded (magenta) regions for $1\sigma$ and $2\sigma$ in the right panel) and ${\eta }_{10}={\eta }_{10}^{\mathrm{ref},3}\pm {\sigma }_{{\eta }_{10},3}=6.25\pm 0.060$ (light and dark shaded (blue) regions for $1\sigma$ and $2\sigma$ in the left panel), which are suggested by the $H_0$ tension.

Figure 3

Contours of $D_p$ (left), $Y_p$ (middle), and $T_t[\mathrm{MeV}]$ (right) in the $f_{\mathrm{EDE}}\text{−}{\rho }_0$ plane for EDE1 with $n=4$. $1\sigma$ allowed ranges for $Y_p$ and $D_p$, in which both observational and theoretical uncertainties are included, are lightly shaded (magenta). The value of ${\rho }_0$ is shown in units of ${\mathrm{MeV}}^4$. In this figure, we take ${\eta }_{10}=6.14$, $N_{\mathrm{eff}}=2.3$, and ${\xi }_e=0$.

Figure 4

The same as Fig. 3 except that $n=5$ is assumed in this figure.

Figure 5

The same as Fig. 3 except that $n=6$ is assumed in this figure.

Figure 6

The same as Fig. 3 except that ${\eta }_{10}=5.80$ and $n=6$ are assumed in this figure.

Figure 7

$1\sigma$ and $2\sigma$ allowed regions in the ${\eta }_{10}\text{−}{N}_{\mathrm{eff}}$ plane from measurements of $Y_p$ and $D_p$ for EDE1 with $n=4$ (left), 5 (middle), and 6 (right). We take ${\rho }_0={10}^{-6}{\mathrm{MeV}}^4$ and $f_{\mathrm{EDE}}=0.09$ (light and dark dotted (magenta) for $1\sigma$ and $2\sigma$ regions) and 0.50 (dark and light striped (orange) for $1\sigma$ and $2\sigma$ regions) in all panels. For reference, we also show the constraint for the case without EDE with black dotted ($1\sigma$) and solid ($2\sigma$) lines. In this figure, we assume no lepton asymmetry (i.e., ${\xi }_e=0$).

Figure 8

Contours of $D_p$ (left), $Y_p$ (middle), and $T_t[\mathrm{MeV}]$ (right) in the $f_{\mathrm{EDE}}\text{−}{\rho }_0$ plane for EDE2. The $1\sigma$ allowed region for $Y_p$ and $D_p$ is shown with light shade (magenta). The value of ${\rho }_0$ is shown in units of ${\mathrm{MeV}}^4$. In this figure, we take ${\eta }_{10}=6.14$, $N_{\mathrm{eff}}=3.046$, and ${\xi }_e=0$.

Figure 9

$1\sigma$ and $2\sigma$ allowed ranges from observations of $Y_p$ [9] (light and dark dotted (blue)) and $D_p$ [26] (light and dark shaded (magenta)) in the $f_{\mathrm{EDE}}\text{−}{N}_{\mathrm{eff}}$ plane for EDE1 with $n=6$. The priors for ${\eta }_{10}$ assumed in the analysis are ${\eta }_{10}={\eta }_{10}^{\mathrm{ref},1}$ (top), ${\eta }_{10}^{\mathrm{ref},2}$ (middle), and ${\eta }_{10}^{\mathrm{ref},3}$ (bottom), as shown in the figure. The value of ${\rho }_0$ is fixed and is also shown in the figure. In all panels, the electron neutrino degeneracy parameter is fixed as ${\xi }_e=0$.

Figure 10

The same as Fig. 9 except that EDE2 is assumed in this figure.

Figure 11

$1\sigma$ and $2\sigma$ allowed regions (light and dark shade (magenta), respectively) in the ${\rho }_0\text{−}{f}_{\mathrm{EDE}}$ plane from measurements of $Y_p$ and $D_p$ for EDE1 with $n=6$ in the cases of ${\eta }_{10}={\eta }_{10}^{\mathrm{ref},1}$ (left) and ${\eta }_{10}^{\mathrm{ref},2}$ (right). Here we vary $N_{\mathrm{eff}}$ and marginalize it. No lepton asymmetry is assumed (i.e., ${\xi }_e=0$).

Figure 12

The same as Fig. 11 but for the case of EDE2.

Figure 13

$1\sigma$ and $2\sigma$ allowed regions in the ${\eta }_{10}\text{−}{N}_{\mathrm{eff}}$ plane from measurements of $Y_p$ and $D_p$ for the cases of EDE1 (left) and EDE2 (right). In the left panel, we take ${\rho }_0={10}^{-6}{\mathrm{MeV}}^4$ with $f_{\mathrm{EDE}}=0.09$ (light and dark dotted (magenta) for $1\sigma$ and $2\sigma$ regions) and 0.50 (dark and light striped (orange) for $1\sigma$ and $2\sigma$ regions). In the right panel, we take ${\rho }_0={10}^{-2}{\mathrm{MeV}}^4$ with $f_{\mathrm{EDE}}=0.23$ (light and dark dotted (magenta) for $1\sigma$ and $2\sigma$ regions) and 0.44 (dark and light striped (orange) for $1\sigma$ and $2\sigma$ regions). For reference, we also show the constraint for the case without EDE with black dotted ($1\sigma$) and solid ($2\sigma$) lines. In both panels, we assume no lepton asymmetry (i.e., ${\xi }_e=0$).

Figure 14

$1\sigma$ and $2\sigma$ allowed regions in the $N_{\mathrm{eff}}\text{−}{\xi }_e$ plane for the case with EDE1, adopting the priors of ${\eta }_{10}={\eta }_{10}^{\mathrm{ref},1}\pm {\sigma }_{{\eta }_{10},1}$ (left) and ${\eta }_{10}={\eta }_{10}^{\mathrm{ref},2}\pm {\sigma }_{{\eta }_{10},2}$ (right). We take $f_{\mathrm{EDE}}=0.09$ (light and dark dotted (magenta) for $1\sigma$ and $2\sigma$ regions) and 0.44 (dark and light striped (orange) for $1\sigma$ and $2\sigma$ regions) with ${\rho }_0={10}^{-6}{\mathrm{MeV}}^4$. For reference, we also show the constraint for the case without EDE with black dotted ($1\sigma$) and solid ($2\sigma$) lines.

Figure 15

$1\sigma$ and $2\sigma$ allowed regions in the $N_{\mathrm{eff}}\text{−}{\xi }_e$ plane for the case with the EDE2, adopting the priors of ${\eta }_{10}={\eta }_{10}^{\mathrm{ref},1}\pm {\sigma }_{{\eta }_{10},1}$ (left) and ${\eta }_{10}={\eta }_{10}^{\mathrm{ref},2}\pm {\sigma }_{{\eta }_{10},2}$ (right). In the left panel, we take $f_{\mathrm{EDE}}=0.17$ (light and dark dotted (magenta) for $1\sigma$ and $2\sigma$ regions) and 0.44 (dark and light striped (orange) for $1\sigma$ and $2\sigma$ regions). In the right panel, we take $f_{\mathrm{EDE}}=0.41$ (light and dark dotted (magenta) for $1\sigma$ and $2\sigma$ regions) and 0.47 (dark and light striped (orange) for $1\sigma$ and $2\sigma$ regions). In both panels, ${\rho }_0$ is assumed as ${\rho }_0={10}^{-1}{\mathrm{MeV}}^4$. For reference, we also show the constraint for the case without EDE with black dotted ($1\sigma$) and solid ($2\sigma$) lines.