Definition of characteristic strain h_c,n and GW strain amplitude h_0,n

[Peggy-Guo]

Hi Tom,

According to your derivation for characteristic strain - if I am understanding it correctly - it comes from [BarackCutler04]

$$h_{c,n}^2 = \frac{1}{(\pi D_L)^2} \left( \frac{2 G}{c^3} \frac{\dot{E_n}}{\dot{f_n}} \right)$$ eq. (1)

But by referring to these paper I found this equation is cited from [FinnThorne00]

where the definitions of h_c,n and h_0,n are (with c=G=1)

$$h_{c, n}^2 \equiv h_{0, n}^2 \left(\frac{2~f_n^2}{\dot{f}_n} \right)$$ eq.(2)

$$h_{0, n} \equiv \sqrt{\langle h_{+,n}^2 + h_{\times,n}^2\rangle} = \frac{2\sqrt{\dot E_n}}{n \omega_{orb} D_L}$$ eq.(3)

Combine eq (2) and eq (3) we can get eq (1).

However, in your code, h_c is calculated via eq.(1), and h_0 calculated with the new definition without the factor of 2:

$$h_{c, n}^2 = \left(h_{0,n}^2 \frac{f_n^2}{\dot{f}_n} \right)$$

So my question is, is the calculation for h_0 in your code missing a factor of $\sqrt{2}$ since eq (1) is actually a definition comes from eq (2)&(3)? The further question is, if the factor of 2 is already included in the sensitivity curve, could you still use eq.(1) to calculate h_c?

I am confused by the factor of $\sqrt{2}$ difference from my own calculation and legwork calculation for a few days, and I think the question I asked above may be the reason for it. But I wanna double check with you in case I miss anything in your derivation that might also cause this problem!

Thanks!

[TomWagg]

Hi Peggy!

Yes so the difference of a factor of 2 in that equation is because of a difference in the definition of the sensitivity curve in Robson+19 vs Finn & Thorne 2000 (check out the footnote to equation 17 in the legwork paper).

You choice on whether to include the factor in the calculation of the strain will depend on how you intend to apply it. If you're working with the Robson+19 sensitivity curve, or curves derived in a similar manner, then you should use the derivation and function in LEGWORK. If there's a different application you have in mind then it may be more relevant to use your alternative equation (but you should be explicit about why you're doing that when you use it to avoid people getting confused 🙂).

[Peggy-Guo]

Hi Tom,

Thanks for your timely response! I noticed that footnote, but I am still confused about your derivation for h_c.
Let me ask my questions in a different way:

1. Is the characteristic strain in legwork the same physical quantity as the characteristic strain in Finn & Thorne 2000? Meaning will they have the same value?
2. And if not - I assume not since the 2 is missing - is it still correct to use equation (18) in the legwork paper to calculate the value for h_c? Since equation (18) is based on $$h_{c, n}^2 = \left(2 * \frac{f_n^2}{\dot{f}_n} \right) h_n^2$$ where the 2 is present.

[katiebreivik]

Hey Peggy -- thanks for your interest in legwork!

To answer your question: if you use the h_c as calculated by legwork, there will be a difference from Finn & Thorne. However, this really only has an impact in the connection between h_c,n and h_n, and not equation 18. By that, I mean that the description of characteristic strain in terms of the energy radiated in GWs [ie. E_dot] as written in equation 18 is 100% correct. The connection between h_c,n and h_n is where the difference comes into play because the characteristic strain is usually considered in the context of building up signal in a specific detector [hence why we apply the factor of 2 as we do since we always use the Robson+19 curve as a default].

All of that is to say: if you have a use case where you would like to plot the characteristic strain alone and cite Finn & Thorne 2000, you should add the factor of 2 appropriately in the conversion. However, if you are plotting the characteristic strain in the context of a LISA curve, you should check to see whether that factor of two is included since the height of the strain above the sensitivity curve means something physically. If you are using Robson+2019 as implemented in legwork, you should use the characteristic strain as implemented in legwork.

[Peggy-Guo]

Hi Katie!

Thanks for your answer! That definitely clear things up a bit.

The reason I asked about equation 18 is that, when I tried to derive the characteristic strain myself, I first derive my h_0,n (defined as < $$h_+^2 + h_x^2$$ >, the rms amplitude average over sky angel and orbital period) with energy flux and got the result of $$h_{0,n}^2 = \frac 1 {(\pi D_L)^2 }\frac{\dot E}{f_n^2}$$. Then I used the relation between h_c,n and h_0,n to calculate h_c,n. And here's where the conflict comes in. It needs the factor of 2 included here to calculate h_c to be consistent with equation 18 - detailed derivation below (with c=G=1):

$$h_{0,n}^2 = \frac 1 {(\pi D_L)^2 }\frac{\dot E}{f_n^2}$$

$$h_{c,n}^2 = 2 \frac{f_n^2}{\dot f_n} h_{0,n}^2$$

Only combine these two equations (with the factor of 2 present) can I get the exact same result as equation 18:

$$h_{c,n}^2 = 2 \frac{f_n^2}{\dot f_n} \frac 1 {(\pi D_L)^2 }\frac{\dot E}{f_n^2} = \frac 1 {(\pi D_L)^2 }\frac{2~\dot E}{\dot f_n}$$

Does this mean that the h_n in legwork calculation is not the same rms amplitude implied in Finn & Thorne? Or do you have other insights in this?

Thanks for your patience in answering this!

[TomWagg]

Hey Peggy, awfully sorry for the slow reply, I've been finished up the PhD and I lost track of this in the maelstrom.

Does this mean that the h_n in legwork calculation is not the same rms amplitude implied in Finn & Thorne? Or do you have other insights in this?

Indeed I believe this indicates we have a factor of 2 difference, which is accounted for in the Robson sensitivity curve but not directly in the strain. As Katie says: if you have a use case where you would like to plot the strain alone and cite Finn & Thorne 2000, you should add the factor of 2 appropriately in the conversion. But any case where you're interacting with the Robson curve too, that factor of 2 is accounted for.