Observational Constraints on Emergent Fractional Fractal Cosmology

Technical Summary: Observational Constraints on Emergent Fractional Fractal Cosmology

Problem Statement
The paper addresses the need to test extensions of the standard $\Lambda$CDM cosmological model that arise from quantum gravity approaches, specifically those utilizing fractional calculus and fractal spacetime geometries. While theories such as asymptotic safety, loop quantum gravity, and string theory aim to unify gravity and quantum mechanics, alternative frameworks propose that spacetime may exhibit non-smooth, fractal properties at the Planck scale. The Emergent Fractional Fractal (EFF) cosmological model, formulated by introducing fractional derivatives into the Wheeler–DeWitt equation, posits that the Universe's expansion is driven by an effective fractal dimension $d$. In this framework, the standard $\Lambda$CDM model is recovered only in the limit where $d=2$. For $d \neq 2$, the continuity equation is modified, altering the scaling of matter and radiation densities and consequently affecting the expansion history and the growth of cosmic structures. The central problem is to determine whether current high-precision cosmological observations can detect deviations from the standard model or if they strictly constrain the fractal dimension to the standard limit.

Methodology
The authors perform a joint likelihood analysis to constrain the parameters of the flat EFF model using a combination of background (geometric) and perturbation (growth) level observations. The analysis employs the Markov Chain Monte Carlo (MCMC) technique using the Metropolis–Hastings algorithm (via the emcee ensemble sampler) to explore the parameter space defined by the baryon density ($\Omega_{b0}$), cold dark matter density ($\Omega_{dm0}$), Hubble constant ($H_0$), effective fractal dimension ($d$), matter fluctuation amplitude ($\sigma_8$), and supernova absolute magnitude ($M_B$).

Three distinct dataset combinations are analyzed to assess the impact of different observational probes:

1. Late-Time (LT): Includes PantheonPlus Type Ia supernovae, direct $H(z)$ measurements from cosmic chronometers and radial BAO, and growth-rate measurements ($f\sigma_8$).
2. LT + DESI DR2 BAO + BBN: Adds Baryon Acoustic Oscillation (BAO) data from the Dark Energy Spectroscopic Instrument (DESI) Data Release 2 and Big Bang Nucleosynthesis (BBN) constraints on the primordial helium abundance.
3. LT + DESI DR2 BAO + CMB: Replaces BBN constraints with Planck 2018 CMB distance priors (shift parameter $R$, acoustic scale $\ell_A$, and baryon density $\omega_b$).

Theoretical predictions for the Hubble parameter $H(z)$, distance modulus $\mu(z)$, deceleration parameter $q(z)$, and the growth rate $f\sigma_8(z)$ are computed by solving the modified Friedmann and Raychaudhuri equations derived from the EFF framework. The modified continuity equation, $\dot{\rho}_i = -\frac{3d}{2}(\rho_i + p_i)H$, dictates the evolution of energy densities, while the growth of structure is governed by a modified differential equation for matter density perturbations. Model performance is evaluated using the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC).

Key Results

• Constraints on Fractal Dimension: The inclusion of CMB distance priors yields the tightest constraint on the fractal dimension: $d = 2.0004^{+0.0006}_{-0.0003}$ at the $1\sigma$ confidence level. This result indicates that any deviation from the standard integer dimension ($d=2$) is extremely small, effectively limiting the model to the $\Lambda$CDM limit.
• Parameter Degeneracies: In the absence of early-Universe data (LT only), the parameter $d$ exhibits mild degeneracies with $H_0$ and $\sigma_8$. Larger values of $d$ allow for higher values of $H_0$. However, the addition of BAO and CMB data breaks these degeneracies, pulling the posterior distributions toward standard $\Lambda$CDM values (e.g., $H_0 \approx 67.3$ km s$^{-1}$ Mpc$^{-1}$ with CMB inclusion).
• Dark Matter: Despite the modified dynamics, the analysis confirms that the dark matter density parameter $\Omega_{dm0}$ remains non-zero and consistent with standard values (e.g., $0.2722 \pm 0.0036$ with CMB), suggesting that the fractal modification does not eliminate the need for non-baryonic dark matter.
• Model Comparison:
  • AIC: The differences in AIC ($\Delta$AIC) between the EFF and $\Lambda$CDM models are small ($\approx 1.35 - 2.00$), indicating that both models fit the observational data equally well.
  • BIC: The differences in BIC ($\Delta$BIC) range from $\approx 6.8$ to $7.5$. According to the Jeffreys' scale, this provides strong evidence favoring the simpler $\Lambda$CDM model due to the penalty for the additional free parameter in the EFF model.
• Evolutionary Behavior: The EFF model tracks the $\Lambda$CDM predictions closely across the redshift range $0 \le z \le 2.5$. Deviations in the Hubble parameter, deceleration parameter, and distance modulus remain below the 1% level. The growth rate $f\sigma_8$ shows slightly lower values at low redshifts for the EFF model compared to $\Lambda$CDM when CMB data is excluded, but these differences vanish when CMB priors are included.

Significance and Claims
The paper concludes that current cosmological observations place stringent constraints on fractal extensions of the standard cosmological framework. While the EFF model is theoretically motivated by fractional quantum gravity and the concept of emergent cosmic space, the data does not support significant deviations from the standard $\Lambda$CDM paradigm. The authors assert that the model is observationally viable only when the fractal dimension is constrained to be extremely close to 2. Consequently, the EFF framework does not currently offer a statistically preferred alternative to $\Lambda$CDM, as the Bayesian Information Criterion favors the standard model's parsimony. The study demonstrates that precision measurements of the expansion history and structure growth are powerful tools for testing non-local and memory-dependent quantum effects in cosmology, effectively ruling out large-scale fractal deviations in the current epoch. The authors note that while $d$ was treated as a constant, future work could investigate redshift-dependent fractal dimensions.