Dark energy at early times, the Hubble parameter, and the string axiverse

Precise measurements of the cosmic microwave background (CMB) power spectrum are in excellent agreement with the predictions of the standard $\mathrm{\Lambda }\mathrm{CDM}$ cosmological model. However, there is some tension between the value of the Hubble parameter $H_0$ inferred from the CMB and that inferred from observations of the Universe at lower redshifts, and the unusually small value of the dark-energy density is a puzzling ingredient of the model. In this paper, we explore a scenario with a new exotic energy density that behaves like a cosmological constant at early times and then decays quickly at some critical redshift $z_c$. An exotic energy density like this is motivated by some string-axiverse-inspired scenarios for dark energy. By increasing the expansion rate at early times, the very precisely determined angular scale of the sound horizon at decoupling can be preserved with a larger Hubble constant. We find, however, that the Planck temperature power spectrum tightly constrains the magnitude of the early dark-energy density and thus any shift in the Hubble constant obtained from the CMB. If the reionization optical depth is required to be smaller than the Planck 2016 $2\sigma$ upper bound $\tau \lesssim 0.0774$, then early dark energy allows a Hubble-parameter shift of at most $1.6\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ (at $z_c\simeq 1585$), too small to fully alleviate the Hubble-parameter tension. Only if $\tau$ is increased by more than $5\sigma$ can the CMB Hubble parameter be brought into agreement with that from local measurements. In the process, we derive strong constraints to the contribution of early dark energy at the time of recombination—it can never exceed $\sim 2\%$ of the radiation/matter density for $10\lesssim z_c\lesssim 10^5$.

Figure 1

Shown here are the evolutions of the energy densities of exotic energy (EE; dashed lines) for several critical redshifts $z_c$, matter (solid blue), radiation (solid green), and the cosmological constant (solid red). For each $z_c$ we choose the exotic-energy density ${\mathrm{\Omega }}_{\mathrm{ee}}$ to be the $3\sigma$ upper limit we derive from the Planck temperature power spectrum assuming the reionization optical depth $\tau$ is fixed to the current Planck best-fit value. The energy densities are all shown, relative to the critical density ${\rho }_c$ today, as a function of the scale factor $a$.

Figure 2

The shifts caused in the TT power spectrum due to the addition of EE are shown for various $z_c$. Here, the value of the reionization optical is fixed to the current Planck best-fit value $\tau =0.0596$. The other cosmological parameters are fixed at the values, shown in Table 1, that provide the best fit to the TT power spectrum. Clearly the critical redshift of the EE is important in determining how the EE shifts the TT spectrum. In the upper figure, the value of ${\mathrm{\Omega }}_{\mathrm{ee}}$ chosen for each $z_c$ is the $3\sigma$ upper limit of its best fit. In the lower figure, ${\mathrm{\Omega }}_{\mathrm{ee}}$ is chosen such that it moves ${\theta }_{*}$ by 1%. ${\mathrm{\Omega }}_{\mathrm{ee}}$ is approximately two orders of magnitude greater for the lower plot.

Figure 3

Shown here are the partial derivatives of the TT spectrum with respect to the cosmological parameters, $H_0$ (dark blue), ${\omega }_{\mathrm{b}}$ (green), ${\omega }_{\mathrm{c}}$ (red), $\ln (10^{10}{A}_{\mathrm{s}})$ (light blue) and $n_{\mathrm{s}}$ (pink). These were derived at the best-fit values obtained by setting $\tau ={\tau }_{\mathrm{Pl}}$, shown in Table 1.

Figure 4

The partial derivatives of the TT spectrum with respect to ${\mathrm{\Omega }}_{\mathrm{ee}}$ are shown here for various values of $z_c$.

Figure 5

The best-fit values and errors on ${\mathrm{\Omega }}_{\mathrm{ee}}$ are shown here. The optical depth $\tau$ was fixed at the best-fit Planck-16 value to obtain these constraints.

Figure 6

Shown here are the best-fit values of the Hubble parameter $H_0$ and its $1\sigma$ upper limit obtained by including EEs in the fit to the Planck temperature power spectrum. We also show the central value obtained from local measurements in [3] as well as the values that are $1\sigma$ and $2\sigma$ lower than the best fit.

Figure 7

The best-fit values and errors on ${\mathrm{\Omega }}_{\mathrm{ee}}$ are shown for various critical redshifts of the EE. We fix $\tau$ at ${\tau }_{\mathrm{Pl}}+2{\sigma }_{\tau ,\text{Pl}}$.

Figure 8

The best-fit and best-fit $+1\sigma$ values for $H_0$ (in $\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$) are shown along with its local measurement at various $\sigma$.

Figure 9

The best-fit values and errors on ${\mathrm{\Omega }}_{\mathrm{ee}}$ for various $z_c$ are shown for $\tau$ fixed at ${\tau }_{\mathrm{Pl}}+5{\sigma }_{\tau ,\text{Pl}}$.

Figure 10

The best-fit vales of $H_0$ (in $\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$) are shown with their $1\sigma$ errors. The local measurement is also shown at various $\sigma$.

Figure 11

We plot the $1\sigma$ likelihood contours for the Hubble parameter against ${\mathrm{\Omega }}_{\mathrm{ee}}$ for various critical redshifts, covering the range of critical redshifts that we probe. We fix $\tau ={\tau }_{\mathrm{Pl}}+5{\sigma }_{\tau ,\text{Pl}}$ for these. In each plot, the Planck-16 values for both parameters are marked by the horizontal and vertical dashed black lines. The dashed blue lines mark the value for the best-fit Hubble parameter for just the TT spectrum without EEs. Negative values of ${\mathrm{\Omega }}_{\mathrm{ee}}$ are unphysical but allowed in our analysis. Estimators of ${\mathrm{\Omega }}_{\mathrm{ee}}$ are consistent with zero within $\sim 2\sigma$.

Figure 12

We plot two best fits (blue and black) for the Planck temperature angular power spectrum for $\tau ={\tau }_{\mathrm{Pl}}+2{\sigma }_{\tau ,\text{Pl}}$, and the Planck data (red). In black is the best fit without any EE. In blue we plot the best fit including an EE with $z_c=1259$. This is the EE that increases the best-fit Hubble parameter the most. In the lower panel, we subtract $D_{\ell }^{\mathrm{best}\text{−}\mathrm{fit}}$ from all three spectra and plot the residues. The bottom left and bottom right panels are scaled differently such that the residues may be more easily distinguishable.

Figure 13

The exotic energy density at its critical redshift ${\rho }_{\mathrm{ee}}(z_c)$ is plotted as a fraction of the total energy density ${\rho }_{\mathrm{\Lambda }\mathrm{CMD}}(z_c)$ of a pure $\mathrm{\Lambda }\mathrm{CDM}$ universe for a range of redshifts. This fraction is within an order-unity factor of the greatest contribution of the EE to the energy density of the Universe. This plot was made for $\tau ={\tau }_{\mathrm{Pl}}$ and allowing ${\mathrm{\Omega }}_{\mathrm{ee}}={\sigma }_{{\mathrm{\Omega }}_{\mathrm{ee}}}$.