Hu-Sawicki $f(R)$ Gravity in a Non-Flat Universe: Constraints from DESI-DR2, BBN, and Type Ia Supernovae

Technical Summary: Hu-Sawicki f(R) Gravity in a Non-Flat Universe

Problem Statement
The standard $\Lambda$CDM cosmological model, while successful, faces theoretical challenges such as the cosmological constant problem and the persistent Hubble tension ($H_0$). Alternative gravity theories, specifically $f(R)$ gravity, offer a mechanism to explain late-time cosmic acceleration by modifying the gravitational Lagrangian rather than introducing a dark energy component. However, most analyses of viable $f(R)$ models assume a spatially flat universe ($\Omega_k = 0$). This assumption may be premature given that inflationary models can produce open or closed universes, and recent Planck 2018 data within $\Lambda$CDM have hinted at a preference for a mildly closed universe. Furthermore, strong degeneracies exist between dark energy parameters and spatial curvature. This work addresses the lack of detailed analysis regarding viable $f(R)$ models in a non-flat background, specifically investigating whether allowing spatial curvature to vary freely alters constraints on modified gravity parameters and how these models compare to $\Lambda$CDM using current observational data.

Methodology
The authors analyze the Hu-Sawicki $f(R)$ gravity model, a specific viable extension of General Relativity defined by the function:
$$f(R) = R - c_1 \frac{R_{HS} (R/R_{HS})^p}{c_2 (R/R_{HS})^p + 1}$$
where $p=1$ is fixed, and the model is characterized by a dimensionless modified-gravity parameter $b$ (related to the original parameters $c_1, c_2, R_{HS}$). The limit $b \to 0$ recovers the $\Lambda$CDM model.

The study treats both the modified-gravity parameter $b$ and the spatial curvature density parameter $\Omega_k$ as free parameters. The analysis utilizes the following observational datasets:

1. DESI-DR2: Baryon Acoustic Oscillation (BAO) measurements from the Dark Energy Spectroscopic Instrument (second data release), covering redshifts $0.295 \le z \le 2.330$.
2. BBN: Big Bang Nucleosynthesis constraints based on primordial deuterium and helium abundances.
3. Type Ia Supernovae (SN Ia): Four distinct compilations were used to test robustness:
   • PantheonPlus (PP)
   • PantheonPlus+SH0ES (PPS), calibrated with Cepheid host-galaxy distances.
   • Union3
   • DESY5 (Dark Energy Survey Year 5)

Theoretical predictions were generated using the Boltzmann solver CLASS, interfaced with the Markov Chain Monte Carlo (MCMC) sampler MontePython (modified for $f(R)$ scenarios). Constraints were derived at 68% and 95% confidence levels. Model comparison was performed using the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) to assess whether the improved fit of the $f(R)$ model justifies the additional free parameter $b$.

Key Contributions and Results

• Hubble Constant and Matter Density: Across all four data combinations, the Hubble constant ($H_0$) and matter density ($\Omega_m$) in the Hu-Sawicki model remain close to their $\Lambda$CDM values. A notable shift in $H_0$ occurs only in the PPS combination (calibrated by SH0ES), where $H_0 \approx 71.9$ km/s/Mpc, reflecting the calibration source rather than an intrinsic feature of the $f(R)$ sector.
• Constraints on the Modified-Gravity Parameter ($b$):
  • In three of the four combinations (PP, Union3, DESY5yr), the data prefer $b \neq 0$ at better than $2\sigma$, with DESY5yr showing the most significant departure ($b = 0.501^{+0.11}_{-0.092}$).
  • The PPS combination is an outlier, preferring a negative value ($b = -0.26^{+0.18}_{-0.16}$), which the authors attribute to the specific SH0ES calibration shifting the expansion history requirements.
  • The sign and magnitude of $b$ are not uniquely determined by current data and depend heavily on the SN Ia calibration used.
• Spatial Curvature ($\Omega_k$):
  • When $b$ is left free, $\Omega_k$ is consistent with a flat universe ($\Omega_k = 0$) at the $1\sigma$ level for all four combinations.
  • This contrasts sharply with $\Lambda$CDM fits on the same data, where PPS mildly prefers a closed universe ($\Omega_k = -0.063 \pm 0.020$) and DESY5yr prefers an open one.
  • The authors identify a strong positive correlation among $b$, $H_0$, and $\Omega_k$. This suggests that curvature signatures detected under $\Lambda$CDM can be "reabsorbed" into the modified-gravity sector when the additional freedom of $b$ is allowed.
• Statistical Model Comparison:
  • AIC: Leans toward the Hu-Sawicki model in three of four cases (with DESY5yr showing strong evidence, $\Delta$AIC = 7.78).
  • BIC: Due to its heavier penalty on the extra parameter, BIC favors $\Lambda$CDM in most cases. Only the DESY5yr combination shows a marginal preference for the Hu-Sawicki model under BIC ($\Delta$BIC = 2.26).
  • The PPS combination consistently favors $\Lambda$CDM under both criteria.

Significance and Claims
The paper claims that the Hu-Sawicki $f(R)$ scenario with unconstrained curvature is a statistically feasible alternative to $\Lambda$CDM, though not obviously preferred by current data. The primary significance lies in the finding that curvature constraints are not independent of the underlying gravity model. The apparent preference for non-zero curvature in $\Lambda$CDM fits can be mitigated or eliminated when a modified-gravity parameter is introduced, due to the geometric degeneracy between curvature and the modification of gravity.

The authors conclude that current DESI-DR2, BBN, and SN Ia data do not provide a decisive statistical preference for the Hu-Sawicki model over $\Lambda$CDM across all datasets. They emphasize that the mild preference for $b \neq 0$ found in some combinations may reflect observational limitations or calibration systematics rather than a genuine departure from General Relativity. Future improvements in BAO precision and tighter control over supernova calibration systematics are required to determine if the observed deviations are physical.