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## Main Question or Discussion Point

I'd like to numerically calculate the power spectra of the scalar perturbation at the Hubble crossing in

$$P_S = \Bigg( \frac{H_*}{\dot\phi_*} \Bigg)^2 \delta\phi_*^2 = \Bigg( \frac{H_*}{\dot\phi_*} \Bigg)^2 \Bigg(\frac{\sqrt{3(1+Q)} H_*T_*}{2\pi^2}\Bigg)$$

where ##H## is the Hubble parameter, ##\phi## is the inflaton field, ##T## is the temperature, and ##Q = \frac{\Gamma}{3H}## is the ratio of the effectiveness of the dissipation ##\Gamma##. The "##_*##" denotes the quantities are evaluated at the horizon crossing.

How do I solve for the quantities AT the horizon crossing? Can I plot out an evolution of some quantity and identify that at some point on the plot, that is the horizon crossing? Or how should I proceed in solving this? Does anyone know of any resources/ material that I can look into to be able to know how to do this?

*warm inflation*, my problem is that I don't know how to do it. As I know, the Hubble crossing happens at the onset of*warm inflation*where the different modes become larger than the Hubble length. Now suppose I have solved the dynamical equations of*warm inflation*with respect to time. So given the scalar power spectra at the Hubble crossing,$$P_S = \Bigg( \frac{H_*}{\dot\phi_*} \Bigg)^2 \delta\phi_*^2 = \Bigg( \frac{H_*}{\dot\phi_*} \Bigg)^2 \Bigg(\frac{\sqrt{3(1+Q)} H_*T_*}{2\pi^2}\Bigg)$$

where ##H## is the Hubble parameter, ##\phi## is the inflaton field, ##T## is the temperature, and ##Q = \frac{\Gamma}{3H}## is the ratio of the effectiveness of the dissipation ##\Gamma##. The "##_*##" denotes the quantities are evaluated at the horizon crossing.

How do I solve for the quantities AT the horizon crossing? Can I plot out an evolution of some quantity and identify that at some point on the plot, that is the horizon crossing? Or how should I proceed in solving this? Does anyone know of any resources/ material that I can look into to be able to know how to do this?