A comparison of the $$R_{\mathrm{h}}=ct$$ Rh=ct and $$\varLambda $$ Λ CDM cosmologies based on the observed halo mass function

The growth of structure may be traced via the redshift-dependent halo mass function. This quantity probes the re-ionization history and quasar abundance in the Universe, constituting an important probe of the cosmological predictions. Halos are not directly observable, however, so their mass and evolution must be inferred indirectly. The most common approach is to presume a relationship with galaxies and halos. Studies based on the assumption of a constant halo to stellar mass ratio $$M_h/M_*$$ (extrapolated from $$z \lesssim 4$$ ) reveal significant tension with $$\varLambda $$ CDM – a failure known as “The Impossibly Early Galaxy Problem”. But whether this ratio evolves or remains constant through redshift $$4 \lesssim z \lesssim 10$$ is still being debated. To eliminate the tension with $$\varLambda $$ CDM, it would have to change by about 0.8 dex over this range, an issue that may be settled by upcoming observations with the James Webb Space Telescope. In this paper, we explore the possibility that this major inconsistency may instead be an indication that the cosmological model is not completely correct. We study this problem in the context of another Friedmann–Lemaitre–Robertson–Walker (FLRW) model known as the $$R_{\mathrm{h}}=ct$$ universe, and use our previous measurement of $$\sigma _8$$ from the cosmological growth rate, together with new solutions to the Einstein–Boltzmann equations, to interpret these recent halo measurements. We demonstrate that the predicted mass and redshift dependence of the halo distribution in $$R_{\mathrm{h}}=ct$$ is consistent with the data, even assuming a constant $$M_h/M_*$$ throughout the observed redshift range ( $$4\lesssim z\lesssim 10$$ ), contrasting sharply with the tension in $$\varLambda $$ CDM. We conclude that – if $$M_h/M_*$$ turns out to be constant – the massive galaxies and their halos must have formed earlier than is possible in $$\varLambda $$ CDM.