Observational Constraints on Emergent Fractional Fractal Cosmology

We constrain the Emergent Fractional Fractal (EFF) cosmological model through a joint likelihood analysis of recent cosmological observations at the background and perturbation levels. In this framework, an effective fractal dimension $d$ is introduced to parameterize possible fractional deviations from the standard cosmological model. We consider three combinations of datasets: (i) late-time (LT) observations including PantheonPlus Type Ia supernovae, $H(z)$ measurements, and growth-rate measurements $f\sigma_8$; (ii) LT combined with DESI DR2 BAO and Big Bang nucleosynthesis (BBN); and (iii) LT combined with DESI DR2 BAO and CMB distance priors. With the inclusion of CMB distance priors, the fractal dimension is constrained to $d = 2.0004^{+0.0006}_{-0.0003}$ at the $1\sigma$ confidence level. Model comparison using the Akaike Information Criterion (AIC) shows that the EFF and $\Lambda$CDM models fit the observational data equally well, while the Bayesian Information Criterion (BIC) favors the simpler $\Lambda$CDM model because of its smaller parameter space. These results show that current cosmological observations place strong constraints on fractal extensions of the standard cosmological framework and possible deviations from the $\Lambda$CDM model.