Constraining deviations from $\Lambda$CDM in the Hubble expansion rate

The $\Lambda$CDM cosmolgical model has long been regarded as highly successful in accurately describing a wide range of astronomical observations. However, numerous observational findings have also provided hints of discrepancies from the predictions of the $\Lambda$CDM framework. We explore a phenomenological model that quantifies the deviation of the Hubble expansion rate from the standard scenario, which is expressed as $H^{2}(z) = H^{2}_{\rm \Lambda CDM}(\Omega_m, z)[1+\delta(z)]$. We consider three distinct forms for the deviation parameter $\delta(z)$: in model I, $\delta(z)=\delta_c$; in model II, $\delta(z)=\delta_{c}z/(1+z)$, and in model III, $\delta(z)=\delta_{c}{\rm ln}(1+z)$. Here, $\delta_c$ represents a constant value. We utilize a comprehensive set of observational data to constrain the models. Our results show that for most combined datasets, $\delta_c$ tends to take on negative values for models I and II, while consistently taking positive values in model III. Furthermore, we find that both models I and II remain consistent with the standard $\Lambda$CDM model across all datasets examined. In contrast, model III exhibits a significant deviation from the $\Lambda$CDM model, exceeding $2\sigma$ for the full combined datastes. The AIC indicates that models I and II are consistent with the $\Lambda$CDM model, whereas model III is preferred over the standard $\Lambda$CDM model, with the $\Lambda$CDM model being disfavored for the combined datasets DESI BAO + CMB + CC + DESY5. These results suggest that the Hubble expansion rate likely deviates from the standard $\Lambda$CDM prediction.