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

Abstract The $$\varLambda $$ 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 $$\varLambda $$ 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 $$H^{2}(z) = H^{2}_\mathrm{\varLambda CDM}(\varOmega _m, z)[1+\delta (z)]$$ H 2 ( z ) = H Λ CDM 2 ( Ω m , z ) [ 1 + δ ( z ) ] . We consider three distinct forms for the deviation parameter $$\delta (z)$$ δ ( z ) : in model I, $$\delta (z)=\delta _c$$ δ ( z ) = δ c ; in model II, $$\delta (z)=\delta _{c}z/(1+z)$$ δ ( z ) = δ c z / ( 1 + z ) , and in model III, $$\delta (z)=\delta _{c}\textrm{ln}(1+z)$$ δ ( z ) = δ c ln ( 1 + z ) . Here, $$\delta _c$$ 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, $$\delta _c$$ 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 $$\varLambda $$ Λ </jats:inline-formul