Resumming Post-Minkowskian and Post-Newtonian gravitational waveform expansions

referees 1 & 2

1) We displayed in tables 2,3, pages 18,19, the contributions of higher (ell, m) modes and observe that, for large ℓ

our results converge quickly than those obtained by MST methods and based on a PN expansion. We thank the referee for drawing our attention on this limit.

2) We stressed along the paper that our techniques apply to the probe limit of the binary system. See abstract, 2 paragraph of page 2, first paragraph of Sections 4 and 5.

3) As suggested by the referee we added a "convention" section after the plan that summarises our conventions. The introduction has been revisited to avoid repetitions.

4) We stressed along the paper that x->0 corresponds to the soft limit, see for example the paragraph after 2.8.

5) The coordinates introduced in (4.10) are defined in the text before 4.10 and after.

6) The relation between t and z is given in (2.6)

7) The content of section 5 has been moved to an appendix.

8) We added in the introduction some references on the initial works on PM tree level scattering waveform.

9) Aligment in 4.13 has been fixed.

10) Grammar typos at the beginning of section 2.1 are corrected.

referee 3

we added 2.9 to clarify what the meaning of a as eigenvalue of the monodromy matrix, see also 2.13 for the relation between a and u

. Solutions are eigenvectors of the monodromy action, so any two solutions with the same eigenvalue should be proportional to each other since the eigenvector space is one dimensional.

we write 2.20 in terms of F_inst to make its meaning more straightforward

we changed the sentence after 3.7 to clarify the meaning of R_alpha . We denote by Rα

the two solutions of the Teukolsky equation 3.1, related to G_alpha via 3.8

typo has been corrected

major points

We added a sentence after (2.13) to clarify the relation between a and u. We can think of u(a) as an expansion in x with coefficients u_i(a) depending of a, or viceversa, inverting this relation as a(u)=\sqrt{u}+sum_i a_i(u) x^i. As explained after (2.13), the coefficients u_i(a) are determined imposing the differential equation order by order in x.

We have reworded the sentence before (2.20)

(3.12) is given in the ancillary file PNexpansion.nb, in the section R_+. We added also 3.13 at the end of this ancillary file.

We have checked it explicitly. We do not have a general proof. we have just checked it up to the orders we have reached.

the position of the singularities in the non-confluent case is arbitrary so there is nothing special in the new singularity (in the confluent case two singularities merge). The connection formulas in the non-confluent case take a similar form than the one in the confluent case, after replacing B_{alpha alphap } by its non confluenti analog, and F_{rm inst} by the pre potential of a gauge theory with Nf=4 flavours, that it is a bit more complicated than that of Nf=3.

We cannot say anything about the k->infty limit. The convergence properties of PN expansions are not known. This is an interesting question, but it goes beyond the scope of this paper.

-The connection formulas are summarised in formula (2.38), that follows immediately from the hypergeometric relations (2.22) and (2.31). We added some explanations around 2.38.