  By post-Newtonian (PN) expanding the well-known, factorized and resummed,
effective-one-body energy flux for circularized binaries we show that: (i)
because of the presence of the resummed tail factor, the 4.5PN-accurate
tails-of-tails-of-tails contribution to the energy flux recently computed by
Marchand et al. [Class. Q. Grav. 33 (2016) 244003] is actually contained in the
resummed expression; this is also the case of the the next-to-leading-order
tail-induced spin-orbit term of Marsat et al. [Class. Q. Grav. 31 (2014)
025023]; (ii) in performing this expansion, we also obtain, for the first time,
the explicit 3.5PN leading-order tail-induced spin-spin flux term; (iii)
pushing the PN expansion of the (nonspinning) EOB flux up to 5.5PN order, we
compute 4PN, 5PN and 5.5PN contributions to the energy flux, though in a form
that explicitly depends on, currently unknown, 4PN and 5PN non-test-mass
corrections to the factorized waveform amplitudes. Within this (parametrized)
4.5PN accuracy, we calculate the TaylorF2 approximant. Focusing for simplicity
on the nonspinning case and using the numerical-relativity calibrated
IMRPhenomD waveform model as benchmark, we demonstrate that it is possible to
reproduce the derivative of the IMRPhenomD phase (say up to the frequency of
the Schwarzschild last-stable-orbit) by flexing only a 4PN "effective" waveform
amplitude parameter. A preliminary analysis also illustrates that similar
results can be obtained for the spin-aligned case provided only the
leading-order spin-orbit and spin-spin terms are kept. Our findings suggest
that this kind of, EOB-derived, (parametrized), higher-order, PN approximants
may serve as promising tools to construct Inspiral-Merger-Ringdown
phenomenological models or even to replace the standardly used 3.5PN-accurate
TaylorF2 approximant in searches of small-mass binaries.
