\frac{\partial P_{t}}{\partial E_{\gamma}} + \frac{\partial I_{t}}{\partial P_{a}} \frac{\partial P_{a}}{\partial E_{\gamma}}\\
                                &= \frac{\partial I_{t}}{\partial P_{t}} \left( \frac{\partial P_{t}}{\partial E_{\gamma}} + \frac{G_{ta,a}}{G_{ta,a} \!+ \!G_{ab}}
                                    \frac{1}{1+j\omega/\omega_{ta}} \frac{\partial P_{a}}{\partial E_{\gamma}}\right)
\end{split}
\label{eq:dItdE}$$ where the benefit of being able to write $\frac{\partial I}{\partial P_{a}}$ in terms of $\frac{\partial I}{\partial P_{t}}$ plus additional factors is readily apparent.

Noise Estimation
----------------

With the dynamical response of the detector now modeled, we can estimate the magnitude of noise from thermal power fluctuations across $G_{tb}$, $G_{ab}$, and $G_{ta}$, the Johnson noise across $R_{t}$ and $R_{l}$ and finally the first stage squid noise and compare their relative sizes by referencing them to a thermal power signal flowing directly into the TES ($\delta P_{t}$). This reference point was chosen purposely so that intuition from simpler 1 DOF thermal systems could be used and to suppress differences due to CRESST’s use of a non-standard electronics readout scheme with a large $R_{l}$ to suppress electro-thermal feedback.

With this choice of reference, thermal fluctuation noise (TFN) across $G_{tb}$ is flat and can be estimated as $$S_{P_{t} \, G_{tb}}(\omega) = 4k_{b}T_{to}^{2}G_{tb} {\mathcal{F}}(T_{to},T_{b},\mbox{diffusive}) 
\label{eq:Spt-Gtb}$$ where ${\mathcal{F}}$ is a noise suppression term with a value between and 1 to account for the fact that the power noise 