T_{to}^2 (G_{ta} \! + \! G_{tb}) \frac{ (1 \! + \! \beta)^{2}}{\alpha^{2} \frac{I_{to}^{2}R_{to}}{(G_{ta}+G_{tb})T_{to}}}\!
               {\left( 1 \! + \! \frac{j \omega}{\frac{\omega_\mathit{eff}}{{\mathcal{L}_{2D}}+1}}\right)}^{2}
                                             \end{split}
\label{eq:Sp_Rt}$$ Again, we find that as long as $\alpha$ is large and heating of the TES via external heating ($Q$) or a bath temperature ($T_{b}$) near $T_{c}$ does not suppress $I_{to}^{2} R_{to}$ too significantly, then the thermal fluctuations across $G_{tb}$ dominate the noise at low frequency. However, this is clearly not true at high frequencies due to the zero at $\frac{\omega_\mathit{eff}}{{\mathcal{L}_{2D}}+1}$. In fact, it is this TES Johnson noise term that sets the bandwidth for optimal energy estimators of $\delta P_{t}$, which we can estimate by finding the frequency beyond which $S_{P_{t}\,R_{t}} > S_{P_{t}\,G_{tb}}$. This frequency, which we will designate as the signal-to-noise bandwidth, is given by: $$\omega_{S/N \, \delta \! P_{t}} \sim \frac{G_{ta}+G_{tb}}{C_{t}} \frac{\alpha}{1+\beta} \sqrt{{\mathcal{F}}\frac{G_{tb}}{G_{tb}+G_{ta}}  \frac{I_{to}^{2}R_{to}}{(G_{ta}+G_{tb})T_{to}} }
\label{eq:w_OF_Pt}$$

![[]{data-label="fig:Spt-Sit"}](figs/CRESST_PDet/Spt_v_CRESSTPo.pdf "fig:"){width="3.3in"} ![[]{data-label="fig:Spt-Sit"}](figs/CRESST_PDet/Si_v_CRESSTPo.pdf "fig:"){width="3.3in"}

For electronics designs with large electro-thermal feedback, $\omega_{S/N \, \delta P_{t}} \sim \omega_\mathit{eff}$. However, for CRESST where ${\mathcal{L}_{2D}}$ has purposely been set to near 0 by choosing $R_{l} \sim R_{to}$, $\omega_{S/N \, \delta \! P_{t}} \gg \omega_\mathit{eff}$. To reiterate, when referencing both the signal and noise to current (Fig. \[fig:Spt-Sit\] bottom), both the current signal and 