A measured hum, a proposed ancestry
The nanohertz gravitational-wave background is the measured part of this story: pulsar timing arrays see it, and the leading reading assigns it to binaries of supermassive black holes above a billion solar masses. Ghodla and Ilie ask what could have seeded those black holes early enough. Under the WIMP dark-matter picture they follow two candidate channels: dark stars, primordial objects held up by dark-matter heating rather than fusion, which can pass a million solar masses before collapsing, and direct-collapse black holes formed from pristine gas. At a comoving density of about one seed per thousand cubic megaparsecs, the dark-star descendants dominate the signal; direct-collapse seeds, expected near one per million cubic megaparsecs, stay sub-dominant.[1]
Between that seed and this hum sits a chain of models, and each link is a choice. Primordial density fluctuations become halos through the extended Press-Schechter mass function; a halo hosts a seed with a probability set by a threshold mass; the seed grows by accretion at an efficiency of 0.1, tuned so the calculation reproduces the black hole to halo mass relation seen locally; halo mergers become black hole mergers after a delay; the binaries radiate through phenomenological inspiral waveforms. What comes out at the end is an amplitude, and that amplitude scales as the square of the seed density and as roughly the 2.8 power of the assumed host halo mass. Two unknowns, one number.[1]
The number that survives the trade-off
That trade-off is why the firmest output here is an upper bound rather than an abundance. Because the amplitude grows as the square of the seed density, a population much denser than the data allow would overproduce the hum. For a mean threshold halo mass of five hundred million solar masses the authors put the ceiling near one seed per twenty cubic megaparsecs, and it falls as that halo mass rises: at a billion solar masses it is closer to one per hundred cubic megaparsecs. Both expected densities sit far below the ceiling, so neither channel is ruled out.[1]
The bound is only as good as the assumption that these binaries reach the pulsar timing band at all. The calculation lets them inspiral under gravitational radiation alone and adopts a mean merger delay of one billion years; the authors say delays of order ten billion years would largely keep supermassive black holes out of the band, and that environmental interactions, left out here, can change the spectrum. An alternative reading fits the same data: the hum may be dominated by black holes assembled late through ordinary galaxy mergers, with the early-seed contribution a minor addition, in which case the ceiling constrains a population that was never the main source.[1]
What would move the ceiling?
One measurable quantity separates the ceiling from an abundance: the typical mass of the halos that host these seeds. If a coming pulsar timing array release holds the background amplitude within its present range while high-redshift observations place that typical halo near a billion solar masses, the limit on seed density tightens by about a factor of five, to roughly one seed per hundred cubic megaparsecs, close enough to the number density of bright galaxies that the two can be compared directly. If instead the amplitude moves, or the inferred slope departs from the two-thirds power an inspiral-only population predicts, the environmental physics this calculation omits is the first place to look.[1]
A limit is a modest thing to be left with after a signal this hard to hear. It is also the kind of result that survives: it does not require dark stars to have existed, and it will still stand when the next data release either narrows it or breaks it. The universe assembled its first supermassive black holes somewhere in its first few hundred million years, and the honest sentence about that epoch, for now, is an upper limit on how many seeds it can have made.[1]