The ALMA REBELS survey: obscured star formation in massive Lyman-break galaxies at z = 4-8 revealed by the IRX-$β$ and $M_{\star}$ relations

We investigate the degree of dust obscured star formation in 49 massive (${\rm log}_{10}(M_{\star}/{\rm M}_{\odot})>9$) Lyman-break galaxies (LBGs) at $z = 6.5$-$8$ observed as part of the ALMA Reionization Era Bright Emission Line Survey (REBELS) large program. By creating deep stacks of the photometric data and the REBELS ALMA measurements we determine the average rest-frame UV, optical and far-infrared (FIR) properties which reveal a significant fraction ($f_{\rm obs} = 0.4$-$0.7$) of obscured star formation, consistent with previous studies. From measurements of the rest-frame UV slope, we find that the brightest LBGs at these redshifts show bluer ($β\simeq -2.2$) colours than expected from an extrapolation of the colour-magnitude relation found at fainter magnitudes. Assuming a modified blackbody spectral-energy distribution (SED) in the FIR (with dust temperature of $T_{\rm d} = 46\,{\rm K}$ and $β_{\rm d} = 2.0$), we find that the REBELS sources are in agreement with the local ''Calzetti-like'' starburst Infrared-excess (IRX)-$β$ relation. By reanalysing the data available for 108 galaxies at $z \simeq 4$-$6$ from the ALPINE ALMA large program using a consistent methodology and assumed FIR SED, we show that from $z \simeq 4$-$8$, massive galaxies selected in the rest-frame UV have no appreciable evolution in their derived IRX-$β$ relation. When comparing the IRX-$M_{\star}$ relation derived from the combined ALPINE and REBELS sample to relations established at $z < 4$, we find a deficit in the IRX, indicating that at $z > 4$ the proportion of obscured star formation is lower by a factor of $\gtrsim 3$ at a given a $M_{\star}$. Our IRX-$β$ results are in good agreement with the high-redshift predictions of simulations and semi-analytic models for $z \simeq 7$ galaxies with similar stellar masses and SFRs.