The Standard siren tests of viable $f(R)$ cosmologies

Technical Summary: The Standard Siren Tests of Viable f(R) Cosmologies

Problem Statement
Recent cosmological observations, particularly from the Dark Energy Spectroscopic Instrument (DESI), combined with Cosmic Microwave Background (CMB) and Type Ia supernova (SNe Ia) data, suggest potential deviations from a pure cosmological constant, motivating tests of dynamical dark energy. Simultaneously, a significant discrepancy known as the "Hubble tension" persists between local distance ladder measurements and CMB-inferred values of the Hubble constant ($H_0$). While $f(R)$ gravity offers a minimal extension to General Relativity (GR) capable of explaining late-time acceleration without a fundamental cosmological constant, viable $f(R)$ models are notoriously difficult to distinguish from the standard $\Lambda$CDM model when constrained solely by electromagnetic (EM) background expansion data. This paper investigates whether future Standard Siren (SS) observations—gravitational wave (GW) sources providing direct luminosity distance measurements—can break the degeneracy between $f(R)$ gravity and $\Lambda$CDM, and whether they can resolve the Hubble tension.

Methodology
The authors constrain two representative viable $f(R)$ models: the Hu-Sawicki and Starobinsky models. The analysis employs a two-step approach:

1. Electromagnetic Constraints: The models are first constrained using current EM datasets: PantheonPlus (PP) SNe Ia, Cosmic Chronometers (CC), Planck 2018 CMB distance priors, and DESI Data Release 2 (DESI2) Baryon Acoustic Oscillation (BAO) measurements. A Markov Chain Monte Carlo (MCMC) analysis is performed using CosmoMC.
2. Standard Siren Simulations: Based on the best-fit parameters derived from the EM data, the authors generate mock Standard Siren catalogs for the Einstein Telescope (ET). These simulations include 1,000 events (500 binary neutron stars and 500 black hole-neutron star systems) over a 10-year period.
   • Modified GW Propagation: The simulations incorporate the modified GW propagation characteristic of $f(R)$ gravity, where the luminosity distance ($D_L^{SS}$) differs from the EM counterpart ($D_L^{EM}$) due to an effective friction term $\delta(z)$ dependent on the curvature evolution $F_R(z)$.
   • Fiducial Cosmologies: To isolate the impact of SS data, the simulations use the EM best-fit results as fiducial inputs. Notably, for the Hu-Sawicki model, the PP+CC best-fit parameters were found to be unstable; thus, a $\Lambda$CDM-based fiducial was used for the Hu-Sawicki SS simulations to ensure physical consistency.
3. Analysis: The authors combine the EM and simulated SS datasets to derive joint constraints on parameters ($\Omega_{m0}, H_0, b$, etc.). They evaluate model performance using the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) relative to $\Lambda$CDM and quantify the Hubble tension using the $T_1$ statistic against the SH0ES baseline.

Key Contributions and Results

• Discriminating Power: The study demonstrates that simulated SS data provide complementary sensitivity to the modified GW friction term. This enhances the ability to discriminate between $f(R)$ gravity and $\Lambda$CDM compared to EM data alone.
• Hu-Sawicki Model Findings:
  • Several data combinations (particularly PP+CC) favor $F_{RR} < 0$, which implies potential theoretical instabilities in the model.
  • While SS data can mildly reduce the inferred Hubble tension in specific simulation setups (e.g., reducing tension to $\sim 0.01\sigma$ when using PP+CC fiducials), the authors note that this reduction is driven by the adopted fiducial cosmologies rather than an intrinsic resolution of the tension.
  • Information criteria (BIC) suggest positive evidence against $\Lambda$CDM for PP+CC-based simulations, but the authors conclude that the Hu-Sawicki model may be disfavored or ruled out by future high-precision SS data due to the preference for unstable parameter regions ($F_{RR} < 0$).
• Starobinsky Model Findings:
  • The model was split into two branches ($b < 0$ and $b > 0$) to capture bimodal posteriors.
  • EM-only constraints are nearly symmetric between the two branches. However, the inclusion of SS constraints introduces mild asymmetries, revealing the sensitivity of SS observables to the detailed curvature dependence of the theory.
  • For the Starobinsky model, the redshift evolution of the effective equation of state ($w_{eff}$) and the trajectories of dynamical variables ($F_R, F_{RR}, r, m$) in the combined DESI2 dataset clearly distinguish the scenario from $\Lambda$CDM.
  • Unlike the Hu-Sawicki case, the Starobinsky model does not show strong evidence of theoretical instability ($F_{RR} > 0$) in the constraints.
• Hubble Tension: The paper explicitly states that in this analysis, standard sirens do not offer a viable resolution to the Hubble tension. The inferred constraints on $H_0$ are predominantly driven by the fiducial cosmologies adopted in the simulations. Consequently, SS datasets constructed with similar fiducial assumptions yield comparable levels of tension to their EM counterparts.

Significance and Claims
The paper claims that while standard sirens are a powerful tool for distinguishing modified gravity from $\Lambda$CDM through the friction term in GW propagation, their current simulated utility in resolving the Hubble tension is limited by the dependence on assumed fiducial models. The study highlights that future, truly independent standard siren observations are crucial for a definitive assessment of $f(R)$ gravity as an alternative to $\Lambda$CDM. Specifically, the authors suggest that the Hu-Sawicki model faces theoretical challenges that future data may confirm, whereas the Starobinsky model remains a viable candidate that exhibits distinct signatures in SS observables, particularly regarding curvature dependence. The work underscores the necessity of combining EM and SS data to break degeneracies and test the stability of modified gravity theories.