Conflation between \chi and \chieff

[ilyamandel]

The intro currently says: "There is essentially no posterior support for $\chi \gtrsim 0.5$ as inferred for the majority of the electromagnetically-detected stellar-mass black hole population (see Section \ref{sec:discussion})" and similarly in the caption of Figure 1. In fact, Figure 1 shows an absence of support for $\chieff \gtrsim 0.5$. It does not directly say anything about $\chi$. In fact, Figure 3 makes it quite clear that $\chi$ can have support beyond 0.5 as long as spins are isotropic. Meanwhile, the EM observations describe $\chi$, not $\chieff$, since only one component is a BH in those X-ray binaries. In a way, this whole paper is about the difference between $\chi$ and $\chieff$ (depending on the mis-alignment angles), so conflating them here and in the caption of Figure 1 is quite confusing.

[SimonStevenson]

I have commented some on this in issue #7 . I think some confusion has arisen here. We were using $\chi$ for the projection of the BH spin on the orbital angular momentum, and $a$ for the BH spin magnitude. Now we seem to use $\chi$ for both. We should use $\chi$ as the spin magnitude, or the projection, but not both.

[farr]

Yes, I agree with Simon's comment. We will be using $\chi$ for the projection of the (dimensionless) spin of one black hole onto the $\hat{z}$ axis, and $a$ for the dimensionless spin magnitude. I am going through the paper and making sure this is consistent now.

With this understanding, and noting that all mergers are between comparable-mass objects, I think the statement in the introduction stands: electromagnetic observations generally find spin-orbit alignment, so $a \simeq \chi$, and both quantities are $\gtrsim 0.5$, while this is not really possible with our constraints on $\chieff$.

[ilyamandel]

Use of \chi (rather than, say, \chi^z) for the z component of \chi is sufficiently not obvious (got me confused, for one :) ), that you probably want to explicitly comment on the difference between \chi and a in your notation.

Even if that convention is used, I am still not entirely happy with the statement "There is essentially no posterior support for $\chi \gtrsim 0.5$ as inferred for the majority of the electromagnetically-detected stellar-mass black hole population (see Section \ref{sec:discussion})." I know what you are trying to say, but I am not convinced this is the right place and way to say it, before we discuss anything about expectations for the secondary to also be aligned, etc. At the very least, if you can just conclude this visually from Figure 1, what's the whole paper about? ;-) I'll see if I can make a better suggestion.

[farr]

Well, the remainder of the paper is about what you can infer if you don't assume that all black holes have high spin, but rather allow for the possibility of smaller spins.... I still think it is worth pointing out the inconsistency with the naive interpretation of the EM observations here, but I'm happy for this to be accomplished with different wording---so go ahead and suggest something.

[ilyamandel]

I propose the following text (checked into git): None of the $\chieff$ posteriors are consistent with two black holes with large aligned spins, $\chi_{1,2} \gtrsim 0.5$; this contrasts with the large spins inferred for the majority of black holes in X-ray binaries (see Section \ref{sec:discussion}).

[farr]

I'm very happy with that text---thanks, Ilya.