Probing dynamical embeddings in a five-dimensional spacetime in light of DESI BAO

Technical Summary: Probing Dynamical Embeddings in a Five-Dimensional Spacetime in Light of DESI BAO

Problem Statement
The standard $\Lambda$CDM cosmological model faces increasing pressure from observational tensions, most notably the discrepancy in the inferred value of the Hubble constant ($H_0$) between early- and late-universe measurements, and the tension regarding the amplitude of matter fluctuations ($S_8$). While various modifications to General Relativity (GR) have been proposed, many suffer from theoretical degeneracies or require the introduction of additional fields. Furthermore, traditional braneworld scenarios (e.g., Randall-Sundrum) often rely on specific junction conditions (like the Israel condition) that may not be fundamental or universally applicable to higher-dimensional embeddings. This paper investigates "Nash Gravity," a model based on Nash's embedding theorem, as a geometric alternative to $\Lambda$CDM that addresses these tensions without introducing new scalar fields or ghost instabilities.

Methodology
The authors formulate a five-dimensional (5D) bulk spacetime ($V_5$) containing an embedded four-dimensional (4D) geometry ($V_4$). Unlike conventional braneworld models where matter is confined and gravity propagates in the bulk via specific junction conditions, this framework utilizes Nash's embedding theorem to treat the extrinsic curvature as a dynamical variable.

• Theoretical Framework: The model introduces orthogonal perturbations to the background metric via variations in the extrinsic curvature ($k_{\mu\nu}$). The background dynamics are governed by the Einstein equations in the bulk, projected onto the 4D brane. The authors derive a modified Friedmann equation where the extrinsic curvature contributes an effective energy density term ($\Omega_{ext}$) dependent on a bending function $b(t) \propto a(t)^{\alpha_0}$.
• Perturbation Theory: The study derives the full set of linear perturbation equations in the conformal Newtonian gauge. Crucially, the extrinsic curvature perturbations are shown to be geometrically induced, avoiding the ghost instabilities common in Dvali-Gabadadze-Porrati (DGP) models. The perturbations lead to a scale-independent effective gravitational constant, $G_{eff}(a) = G / (1 - \beta_0 a^{2\alpha_0})$, where $\beta_0$ is a dimensionless coupling parameter representing residual extrinsic effects.
• Observational Analysis: The model is tested against a comprehensive set of cosmological datasets using the CLASS Boltzmann solver and MontePython MCMC sampler. The datasets include:
  • Cosmic Microwave Background (CMB) temperature and polarization data from Planck 2018.
  • Baryon Acoustic Oscillation (BAO) measurements from the Dark Energy Spectroscopic Instrument (DESI) Data Release 2 (DR2).
  • Type Ia Supernovae (SN Ia) compilations: PantheonPlus (PP), PantheonPlus&SH0ES (PPS), Union 3.0, and DESY5.
• Statistical Comparison: The authors perform joint analyses to constrain parameters ($\omega_b, \omega_c, A_s, n_s, \tau_{reio}, H_0, w_0, \beta_0$) and compare the model's performance against standard $\Lambda$CDM using the Akaike Information Criterion (AIC) and $\Delta \chi^2$.

Key Contributions

1. Geometric Perturbations: The paper reformulates 5D embedding dynamics directly in terms of the extrinsic curvature scalar, providing a transparent link between geometry and cosmology without auxiliary fields.
2. Ghost-Free Dynamics: The authors demonstrate that the model avoids ghost instabilities because the extrinsic curvature enters quadratically and satisfies Gauss-Codazzi constraints, unlike DGP-like models where scalar modes associated with brane bending can become ghosts.
3. Modified Growth of Structure: The derivation of a scale-independent effective gravitational constant ($G_{eff}$) that depends on the scale factor and the coupling $\beta_0$, offering a mechanism to suppress the growth of structure naturally.
4. DESI-DR2 Constraints: This work presents one of the first comprehensive constraints on Nash Gravity using the latest DESI DR2 BAO data combined with CMB and SN Ia datasets.

Results

• Hubble Constant ($H_0$): The joint analysis of CMB and DESI-DR2 yields $H_0 = 69.32 \pm 0.72$ km/s/Mpc. This is slightly higher than the standard $\Lambda$CDM prediction ($68.17 \pm 0.28$ km/s/Mpc), offering a modest alleviation of the $H_0$ tension. The shift is driven by a degeneracy between $H_0$ and the equation of state parameter $w_0$.
• Structure Growth ($S_8$): The model predicts a suppressed growth of structure, with $S_8 \approx 0.76$ across various joint analyses (e.g., $0.759 \pm 0.029$ for CMB+DESI+DESY5). This is lower than typical $\Lambda$CDM predictions and aligns with the "low $S_8$" trend observed in some large-scale structure measurements, potentially alleviating the $S_8$ tension.
• Model Fit: The parameter $\beta_0$ is constrained to be positive ($0.22^{+0.24}_{-0.20}$ in CMB+DESI-DR2), indicating a departure from pure GR.
• Statistical Preference:
  • In the CMB+DESI-DR2+DESY5 joint analysis, Nash Gravity shows a statistically significant preference over $\Lambda$CDM, with $\Delta \chi^2_{min} = -6.20$ and $\Delta AIC = -4.20$ (where $\Delta AIC < -2$ indicates positive evidence and $<-5$ strong evidence).
  • However, the authors note that other dynamical dark energy models (e.g., CPL) achieve even better fits ($\Delta \chi^2 \approx -20$) in similar analyses.
  • When accounting for the penalty of extra parameters via AIC, the extended model is not universally favored over $\Lambda$CDM with strong significance across all dataset combinations, though it remains a viable alternative.

Significance and Claims
The paper claims that Nash Gravity offers a physically motivated, observationally consistent extension of standard cosmology. Its primary significance lies in its ability to:

1. Address Tensions: Naturally produce a higher $H_0$ and a lower $S_8$ through geometric modifications rather than ad-hoc dark energy components.
2. Theoretical Consistency: Provide a framework where metric perturbations arise directly from geometry, avoiding the need for new fields and the ghost instabilities plaguing other extra-dimensional models.
3. Viability: Demonstrate that the model fits current data (including the latest DESI BAO) as well as or better than $\Lambda$CDM in specific joint analyses, particularly when including DESY5 supernova data.

The authors conclude that while Nash Gravity does not decisively rule out $\Lambda$CDM, it provides a compelling alternative that warrants further investigation, particularly regarding scale-dependent effects on the effective gravitational coupling and future large-scale structure surveys.