Expressing f_{GW} in Terms of 't' from ODE Solution

Hello everyone,

I’ve been working on an equation describing the spin-up process of a binary system involving Primordial Black Holes (PBHs) surrounded by dark matter. The equation looks like this:
\frac{df}{dt} = C_GW * f^{11/3} + C_DF * f^{3/2}
After solving this Ordinary Differential Equation (ODE), I obtained the following expression:
\frac{{}{2}F{1}(-3/13, 1; 10/13; -C_{r}*f_{0}^{13/6})}{f_{0}^{1/2}} - \frac{{}{2}F{1}(-3/13, 1; 10/13; -C_{r}f_{GW}^{13/6})}{f_{GW}^{1/2}} = \frac{C_DF}{2}(t-t_{0}).

Now, I’m looking to express f_{GW} in terms of ‘t’ from this equation. Any help with this would be greatly appreciated.

Thanks!

Hi Charchit,

I think this question would be more suitable for physics stack exchange or math stack exchange, since it concerns a general ODE solution rather than a GWs specific problem.

Having said that, some quick comments:

• I don’t think there is generic way to invert a hypergeometric function (the $$2F_1$$), so you could just plot it as $f{gw}$ vs $t$ and try a fit.

• Alternatively, you could break your initial ODE to frequency ranges, depending on which term dominates and solve a simpler ODE with an invertable analytic solution. This will depend also on the relative importance of $C_{GW}$ and $C_{DF}$ that you have. For a quick example, assuming $C_{GW} = C_{DF} = 1$ and for a frequency range $f \epsilon (0.01, 100)$ Hz, you get this:

Hope this helps!

Best,
Marios

Hi Marios,

Thanks for your feedback. I used the similar approach to fit a power law because I want to use the Generalized Frequency Hough Transform to analyze the signal and find its parameters. However, I’m wondering if there’s a way to fit a power law with an approximate breaking point for a full fdot.

In my analysis, I tried fitting a combination of two power laws, but I’m realizing it might not be the right approach. This is because I know that the vacuum term follows a power law with a breaking index 11/3, while the dynamical friction term follows a power law with a breaking index 3/2. I want to use the GFHT to account for both of these effects.

Thanks,
Charchit.