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Gravity in the Era of Equality: Towards solutions to the Hubble problem without fine-tuned initial conditions

Miguel Zumalacarregui

TL;DR

This work investigates early modified gravity within viable Horndeski theories as a route to resolve the $H_0$ tension without fine-tuned initial conditions. It focuses on three mechanisms—Imperfect Dark Energy at Equality ($\text{IDEE}$), Enhanced Early Gravity ($\text{EEG}$), and Late-Universe Phantom Expansion ($\text{LUPE}$)—implemented in a coupled cubic Galileon model with exponential coupling to curvature. Planck+BAO data, with and without a SH0ES prior, show that IDEE alone cannot relieve the tension due to strong CMB constraints, while EEG can raise the inferred $H_0$ to about $68.7\pm1.5$ km/s/Mpc (reducing the Planck+BAO–SH0ES tension to roughly $2.6\sigma$) and LUPE can further contribute when combined with $\Lambda$. The results demonstrate that early- and late-time modifications of gravity offer a viable, testable set of possibilities beyond General Relativity, though challenges from BBN, local gravity tests, and gravitational-wave phenomenology require further model-building and data-driven constraints. Overall, EEG emerges as the most promising pathway among the explored scenarios for mitigating cosmological tensions while remaining compatible with current observations.

Abstract

Discrepant measurements of the Universe's expansion rate ($H_0$) may signal physics beyond the standard cosmological model. Here I describe two early modified gravity mechanisms that reconcile the value of $H_0$ by increasing the expansion rate in the era of matter-radiation equality. These mechanisms, based on viable Horndeski theories, require significantly less fine-tuned initial conditions than early dark energy with oscillating scalar fields. In Imperfect Dark Energy at Equality (IDEE), the initial energy density dilutes slower than radiation but faster than matter, naturally peaking around the era of equality. The minimal IDEE model, a cubic Galileon, is too constrained by the cosmic microwave background (Planck) and baryon acoustic oscillations (BAO) to relieve the $H_0$ tension. In Enhanced Early Gravity (EEG), the scalar field value modulates the cosmological strength of gravity. The minimal EEG model, an exponentially coupled cubic Galileon, gives a Planck+BAO value $H_0=68.7 \pm 1.5$ (68\% c.l.), reducing the tension with SH0ES from $4.4σ$ to $2.6σ$. Additionally, Galileon contributions to cosmic acceleration may reconcile $H_0$ via Late-Universe Phantom Expansion (LUPE). Combining LUPE, EEG and $Λ$ reduces the tension between Planck, BAO and SH0ES to $2.5σ$. I will also describe additional tests of coupled Galileons based on local gravity tests, primordial element abundances and gravitational waves. While further model building is required to fully resolve the $H_0$ problem and satisfy all available observations, these examples show the wealth of possibilities to solve cosmological tensions beyond Einstein's General Relativity.

Gravity in the Era of Equality: Towards solutions to the Hubble problem without fine-tuned initial conditions

TL;DR

This work investigates early modified gravity within viable Horndeski theories as a route to resolve the tension without fine-tuned initial conditions. It focuses on three mechanisms—Imperfect Dark Energy at Equality (), Enhanced Early Gravity (), and Late-Universe Phantom Expansion ()—implemented in a coupled cubic Galileon model with exponential coupling to curvature. Planck+BAO data, with and without a SH0ES prior, show that IDEE alone cannot relieve the tension due to strong CMB constraints, while EEG can raise the inferred to about km/s/Mpc (reducing the Planck+BAO–SH0ES tension to roughly ) and LUPE can further contribute when combined with . The results demonstrate that early- and late-time modifications of gravity offer a viable, testable set of possibilities beyond General Relativity, though challenges from BBN, local gravity tests, and gravitational-wave phenomenology require further model-building and data-driven constraints. Overall, EEG emerges as the most promising pathway among the explored scenarios for mitigating cosmological tensions while remaining compatible with current observations.

Abstract

Discrepant measurements of the Universe's expansion rate () may signal physics beyond the standard cosmological model. Here I describe two early modified gravity mechanisms that reconcile the value of by increasing the expansion rate in the era of matter-radiation equality. These mechanisms, based on viable Horndeski theories, require significantly less fine-tuned initial conditions than early dark energy with oscillating scalar fields. In Imperfect Dark Energy at Equality (IDEE), the initial energy density dilutes slower than radiation but faster than matter, naturally peaking around the era of equality. The minimal IDEE model, a cubic Galileon, is too constrained by the cosmic microwave background (Planck) and baryon acoustic oscillations (BAO) to relieve the tension. In Enhanced Early Gravity (EEG), the scalar field value modulates the cosmological strength of gravity. The minimal EEG model, an exponentially coupled cubic Galileon, gives a Planck+BAO value (68\% c.l.), reducing the tension with SH0ES from to . Additionally, Galileon contributions to cosmic acceleration may reconcile via Late-Universe Phantom Expansion (LUPE). Combining LUPE, EEG and reduces the tension between Planck, BAO and SH0ES to . I will also describe additional tests of coupled Galileons based on local gravity tests, primordial element abundances and gravitational waves. While further model building is required to fully resolve the problem and satisfy all available observations, these examples show the wealth of possibilities to solve cosmological tensions beyond Einstein's General Relativity.

Paper Structure

This paper contains 29 sections, 69 equations, 17 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Summary and guide for the busy reader
  3. Galileon Gravity after GW170817
  4. Coupled Cubic Galileon
  5. Cosmological Dynamics
  6. Cosmological Dynamics
  7. General Considerations
  8. Equations & Definitions
  9. Canonical vs Accelerating Galileon
  10. Coupling & Vainshtein Mechanism
  11. Imperfect Dark Energy at Equality
  12. Enhanced Early Gravity
  13. Late-Universe Dynamics
  14. Cosmological Constraints
  15. Overview of Models, Datasets and Analysis
  16. ...and 14 more sections

Figures (17)

  • Figure 1: Galileons, early modified gravity and the Hubble problem. Model-independent constraints on $H_0$ (dotted bands) prefer a lower acoustic scale $r_s$ than $\Lambda$CDM. Filled contours show the model-dependent Planck+BAO constraints for Galileon models implementing IDEE, EEG and LUPE. In IDEE-only models (dashed) the stringent constraints limit the impact on $r_s$. Coupled EEG models (solid) relax the bounds considerably, extending the degeneracy across the BAO+SNe direction. Uncoupled/coupled LUPE models with $\Lambda=0$ (red/orange) predict a high central value of $H_0$ compared to the canonical $\Lambda\neq 0$ cases (purple, dark green), but have a worse fit and' are ruled out by other observations. LUPE models with $\Lambda\neq0$ (magenta) provide an intermediate case. (Figure adapted from Knox:2019rjx).
  • Figure 2: Scalar field energy density in scenarios reconciling early and late-universe values of $H_0$. LUPE acts at low $z$, while IDEE and EEG reduce $r_s$. An early quintessence model is shown for comparison (Agrawal et al., Ref. Agrawal:2019lmo).
  • Figure 3: Kinetic structure of cubic Galileons. The relation between the shift-charge density (\ref{['eq:shift_current_gal']}) and the field derivative Eq. (\ref{['eq:xi_def']}) is shown for canonical ($c_2>0$, thick) and accelerating ($c_2<0$, thin) models. Absence of ghosts requires a positive slope for the curve (\ref{['eq:kinetic_term']}), with the minimum of $\mathcal{J}$ corresponding to the transition to instability. Stable accelerating/canonical models tend to $\xi\neq0$, $\xi=0$ respectively (\ref{['eq:gal_solutions']}) A positive coupling strength $\beta>0$ sources ${\cal J}$, delaying the approach to the asymptotic solution. Negative coupling strength $\beta<0$ drive the field towards the ghost region.
  • Figure 4: Imperfect Dark Energy at Equality (IDEE) in canonical uncoupled models. The initial energy density of the scalar field dilutes faster than radiation but more slowly than matter (left panel). By virtue of this scaling, the relative scalar field abundance peaks around the era of matter-radiation equality (middle panel), lowering $r_s$ and increasing $H_0$ for fixed $\theta_\star$. Energy contributions of additional relativistic particles and an early quintessence model Agrawal:2019lmo are shown for comparison. The equation of state of the scalar remains in the range $w_\phi\in(0,1/3)$ until the kination phase at low $z$ (right panel).
  • Figure 5: Enhanced Early Gravity (EEG) in canonical coupled models. The effective contribution to the expansion history Eq. (\ref{['eq:M2_hubble_de']}), including effect of $M_*^2$ on cosmic expansion, follows the dominant component at early times (left panel). If $M_{*,i}^2<1$ the strengthening of gravity $\Delta\rho/\rho$ increases the expansion rate before recombination, lowering the acoustic scale and increasing $H_0$ for fixed $\theta_\star$. Energy contributions of additional relativistic particles and the quintessence early dark energy model Agrawal:2019lmo are shown for comparison. Right panel: effective Planck mass evolution (top), the coupling $\beta$ is chosen to fix the effective Planck mass today $M_{*,0}^2 = 1$ from the initial value $M_{*,i}^2$. Reduced scalar field density (bottom), $\hat{\Omega}_\phi$ excludes the contributions of $\Lambda$ and the effect of $M_*^2$ on the expansion. $H_0$ values are for fixed $\theta_\star$.
  • ...and 12 more figures