Rock 'n' Roll Solutions to the Hubble Tension

TL;DR

The paper tackles the Hubble tension by proposing a localized energy injection near recombination from a scalar field with a monomial potential $V \propto \phi^{2n}$, analyzing both rolling and oscillatory solutions via the Emden-Fowler mapping and emphasizing exact treatment of background and perturbations over coarse-grained fluid approximations. By fitting to Planck, BAO, SH0ES, and Pantheon data, it finds that a rolling solution with $n=2$ provides the best overall improvement, delivering a larger $H_0$ while maintaining consistency with CMB constraints, though it predicts a higher $\sigma_8$ and some tension with late-time measurements. The study compares these scalar-field models to the $N_{\rm eff}$ extension, showing similar CMB fits but different implications for dark matter, BAO, and $S_8$, and uses AIC to indicate a modest preference for the $n=2$ case. It also demonstrates that coarse-grained approaches can mischaracterize the dynamics near the energy-injection peak, underlining the need for exact evolution in robust cosmological analyses. Future data on the late-time amplitude of matter fluctuations and reionization history could help distinguish these scenarios from competing resolutions of the Hubble tension.

Abstract

Local measurements of the Hubble parameter are increasingly in tension with the value inferred from a $Λ$CDM fit to the cosmic microwave background (CMB) data. In this paper, we construct scenarios in which evolving scalar fields significantly ease this tension by adding energy to the Universe around recombination in a narrow redshift window. We identify solutions with scalar field potential $V \propto φ^{2n}$ that have simple asymptotic behavior, both oscillatory (rocking) and rolling. These solutions consistently describe both the field evolution and its fluctuations without approximation. Our findings differ qualitatively from some of the existing literature, which rely upon a coarse-grained fluid description. Combining CMB data with low-redshift measurements, the best fit model has $n=2$ with a significantly higher value of the Hubble constant as compared to a $Λ$CDM fit to the same data. Future measurements of the late-time amplitude of matter fluctuations and of the reionization history could help distinguish these models from competing solutions.

Figures (11)

• Figure 1: Top panel: Evolution of the energy density of the scalar field $\phi$ relative to total energy density of the universe in a matter-dominated toy cosmology ($w_b=0$). Here, $\rho_{\rm tot}$ includes the contribution from the scalar field, and $a_c$ denotes the scalar factor at which the field starts rolling. We show the evolution for different choices of $n$ in the scalar potential $V\propto \phi^{2n}$, as well as an example of exponential potential, $V\propto e^{-\lambda\phi}$. We choose values of $n$ to show an example each of an oscillating ($n=2$) and a rolling solution ($n=5$). The asymptotics of the numerical solutions is shown to match well with the analytical results derived in the text, neglecting the back-reaction. Note that the presence of slow oscillations around the rolling solution for $n=5$ is due to the transition from $w_{\phi} = -1$ to the asymptotic solution; the stability of the rolling solution guarantees that these oscillations are quickly damped. We note that the width of the energy injection grows larger as $n$ is increased. Bottom panel: Equation of state of the scalar field as a function of the scale factor for the same potentials as in the left panel. For $n=2$, the cycle-average of $w_\phi$ (dark thick blue line) rapidly approaches $w_{\rm osc}=1/3$, while for $n=5$, it asymptotes to the rolling solution with $w_{\phi}=1/4$ instead of the cycle-averaged approximation ($w_{\rm osc}=2/3$). For the exponential potential, the scalar field approaches the well-known tracking behavior with $w_\phi=w_b$.
• Figure 2: Marginalized posterior distributions for $V\propto\phi^{2n}$ models for three different values of $n$. Results are shown here for the data combination "Planck + BAO + SH0ES + Pantheon".
• Figure 3: Left panel: Energy injection profile for the best-fit models for each value of $n$ as a function of redshift. Results are shown here for the data combination "Planck + BAO + SH0ES + Pantheon". For reference, we also show the amount of energy injected as compared to standard $\Lambda$CDM for the best fit $N_{\rm eff}$ model using the same data combination (corresponding to $\Delta N_{\rm eff}=0.27$). To guide the eye, we have indicated by vertical dashed lines the matter-radiation equality and baryon drag epochs in the standard $\Lambda$CDM model. Right panel: The scalar field equation of state as a function of redshift for each value of $n$.
• Figure 4: High-$\ell$ CMB residuals between the "Planck + BAO + SH0ES + Pantheon" best fit models for $\Lambda$CDM, $N_{\rm eff}$, and $\phi^{2n}$, and a best-fit $\Lambda$CDM reference model obtained using the data combination "Planck + BAO + Pantheon". The left panel shows the temperature residuals, while the right panel shows the E-mode polarization residuals.
• Figure 5: Marginalized posterior in the $f_\phi$-$n$ plane (left panel), and the $H_0$-$n$ plane (right panel). The gray band shows the SH0ES measurement Riess:2018byc. Results are shown here for the data combination "Planck + BAO + SH0ES + Pantheon." Clearly, larger values of the Hubble constant require lower values of $n$.