Diagnosing deceivingly cold dusty galaxies at 3.5<z<6: a substantial population of compact starbursts with high infrared optical depths

Using NOEMA and ALMA 3mm line scans, we measure spectroscopic redshifts of six new dusty galaxies at 3.5<z<4.2 by solidly detecting [CI](1-0) and CO transitions. The sample was selected from the COSMOS and GOODS-North super-deblended catalogs with FIR photometric redshifts z>6, based on template IR spectrum energy distribution (SED) from known submillimeter galaxies at z=4--6. Dust SED analyses explain the photo-z overestimate from seemingly cold dust temperatures (Td) and steep Rayleigh-Jeans (RJ) slopes, providing additional examples of cold dusty galaxies impacted by the Cosmic Microwave Background (CMB). We therefore study the general properties of the enlarged sample of 10 ``cold"dusty galaxies over 3.5<z<6. We conclude that these galaxies are deceivingly cold at the surface but actually warm in their starbursting cores. Several lines of evidence support this scenario: (1) The high infrared surface density and cold Td from optically thin models appear to violate the Stefan-Boltzmann law; (2) the gas masses derived from optically thin dust masses are inconsistent with estimates from dynamics and CI luminosities; (3) the implied high star formation efficiencies would conflict with cold Td; (4) high FIR optical depth is implied even using the lower, optically-thick dust masses. This work confirms the existence of a substantial population of deceivingly cold, compact dusty starburst galaxies at z>~4, together with the severe impact of the CMB on their RJ observables, paving the way for the diagnostics of optically thick dust in the early universe. Conventional gas mass estimates based on RJ dust continuum luminosities implicitly assume an optically thin case, overestimating gas masses by a factor of 2--3 on average in compact dusty star-forming galaxies.