Unveiling early black hole growth with multi-frequency gravitational
  wave observations
  Third Generation ground based Gravitational Wave Interferometers, like the
Einstein Telescope (ET), Cosmic Explorer (CE), and the Laser Interferometer
Space Antenna (LISA) will detectcoalescing binary black holes over a wide mass
spectrum and across all cosmic epochs. We track the cosmological growth of the
earliest light and heavy seeds that swiftly transit into the supermassive
domain using a semi analytical model for the formation of quasars at $z=6.4$, 2
and $0.2$, in which we follow black hole coalescences driven by triple
interactions. We find that light seed binaries of several $10^2$ M$_\odot$ are
accessible to ET with a signal-to-noise ratio ($S/N$) of $10-20$ at $6<z<15$.
They then enter the LISA domain with larger $S/N$ as they grow toa few $10^4$
M$_\odot$. Detecting their gravitational signal would provide first time
evidence that light seeds form, grow and dynamically pair during galaxy
mergers. The electromagnetic emission of accreting black holes of similar mass
and redshift is too faint to be detected even for the deepest future
facilities. ET will be our only chance to discover light seeds forming at
cosmicdawn. At $2<z<8$, we predict a population of "starved binaries",
long-lived marginally-growing light seed pairs, to be loud sources in the ET
bandwidth ($S/N>20$). Mergers involving heavy seeds ($\sim 10^5 M_\odot - 10^6
M_\odot$) would be within reach up to $z=20$ in the LISA frequency domain. The
lower-z model predicts $11.25(18.7)$ ET(LISA) events per year, overall.
