Constraining deviations from CDM in the Hubble expansion rate

The Λ CDM cosmological 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 Λ CDM framework. We explore a phenomenological model that quantifies the deviation of the Hubble expansion rate from the standard scenario, which is expressed as H²(z) = H²_ CDM( ₘ, z)[1+δ (z)] H 2 ( z ) = H Λ CDM 2 ( Ω m , z ) [ 1 + δ ( z ) ] . We consider three distinct forms for the deviation parameter δ (z) δ ( z ) : in model I, δ (z)=δ c δ ( z ) = δ c ; in model II, δ (z)=δ cz/(1+z) δ ( z ) = δ c z / ( 1 + z ) , and in model III, δ (z)=δ cln(1+z) δ ( z ) = δ c ln ( 1 + z ) . Here, δ c δ 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, δ c δ 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 Λ </jats:inline-formul