The very first galaxies that started the cosmic dawn likely resided in so-called minihaloes, with masses of $sim10^5$-$10^8mathrm{M}_odot$, accreting their gas from the intergalactic medium through H$_2$ cooling. Such molecularly cooled galaxies (MCGs) mostly formed in pristine environments, hosted massive, metal-free stars, and were eventually sterilized by the build-up of a disassociating (Lyman-Werner; LW) background. Therefore, their properties might be very different from the galaxies we see in the later Universe. Although MCGs are probably too faint to be observed directly, we could nevertheless infer their properties from the imprint they leave in the cosmic 21-cm signal. Here we quantify this imprint by extending the public simulation code 21cmFAST to allow for a distinct population of MCGs. We allow MCGs to have different properties from other galaxies, including unique scaling relations for their stellar-to-halo mass ratios, ionizing escape fractions, and spectral energy distributions. We track inhomogeneous recombinations, disassociative LW feedback, and photoheating from reionization. After demonstrating how MCGs can shape the 21-cm signal, we explore to what extent current observations can already place constraints on their properties. The cosmic microwave background optical depth from Planck sets an upper limit on the product of the ionizing escape fraction and the stellar mass in MCGs. When including also the timing of the putative EDGES absorption signal, we find an additional strong degeneracy between the stellar mass and the X-ray luminosity of MCGs. If proven to be of cosmic origin, the timing of the EDGES signal would have been set by MCGs.
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