## 2 TES Physics and Models
### TES Noise
It can be demonstrated that when passing through the power-to-current transfer function of the electro-thermal system (see for example [52]), the ITFN contribution to the current noise has a shape similar to the Nyquist-Johnson noise described here below, and it is the highest in the kHz range. The third intrinsic noise contribution is the Nyquist-Johnson noise of the TES itself biased in the transition and it is generalized in the form of a voltage noise \(e_{int}=\sqrt{4k_{B}TR_{C}^{\prime}(I)}\). The response of the TES current to \(e_{int}\) is suppressed at low frequency by the electro-thermal feedback [19] and becomes significant in the detector electrical band at kHz. The function \(\zeta(I)\), introduced in Equation (2), accounts for the non-linear and non-equilibrium nature of the TES resistance, which is strongly dependent on the TES current. For linear resistors at equilibrium \(\zeta(I)=1\). [PERSON] [18] assumed the TES to be a simple Markovian system with no hidden variables such as internal temperature gradients and fluctuating current paths. Applying the [PERSON]'s nonequilibrium Markovian fluctuation-dissipation relations, he calculated the first order, near-equilibrium, non-linear correction term to the noise to be \(\zeta_{ne}(I)=1+2\beta\). A solution for the higher order terms cannot be found from known TES properties since they contain dissipationally indeterminable parameters [18]. In this form, the noise model is known in the literature as non-equilibrium Johnson noise (neJN) and is extensively used to model the, \(\beta\) dependent, Nyquist-Johnson voltage noise observed in the TES. The broadband unexplained or excess noise, typically observed at frequencies larger than the thermal bandwidth of the TES [74], could only partially be explained after the introduction of the correction term \(\zeta_{ne}(I)\) and only at relatively low \(\beta\) values [34, 50, 75]. As shown in [52, 54, 76], the characterization of the noise in the electrical bandwidth is complicated by the fact that both the non-equilibrium Johnson noise and the internal thermal fluctuation noise give a similar contribution in the measured TES current noise, after passing through the system transadmittance. [PERSON] et al. [34], however, excluded the presence of a significant contribution of the ITFN in the excess noise observed with their low resistance, large \(\beta\), high thermal conductance devices. Noise mitigation normal metal bars and stripes have been proposed and successfully, widely introduced in the TES design of microcalorimeters to suppress the excess noise [74, 75]. [PERSON] et al. [77] argued that the observed excess Johnson noise in TES-based detectors could have a natural explanation within the RSJ theory. Following the work of [PERSON] and [PERSON] [78], they calculated the power spectral density of the voltage fluctuations across the TES (considered as a resistively shunted junction) \(S_{V}(\omega)\), averaged over the period of the Josephson oscillations. They obtained  
\[S_{V}(\omega)=\frac{4k_{B}T}{R_{N}}\sum_{m}|Z_{0m}|^{2}, \tag{20}\]  
where \(Z_{mm^{\prime}}(\omega)\) are the components of the impedance matrix of the biased TES with the index \(m\) standing for the \(m_{th}\) harmonic of the Josephson oscillation at \(\omega_{I}\). This approach was used in [79] to develop the quantum-noise theory of a resistively shunted Josephson junction and it was proved that, to calculate the total low frequency voltage fluctuations, one needs to take into account the _mixing down_ of high harmonics of the Josephson frequency. In [77], it was shown that the spectral density of voltage noise in the RSJ model has a unique analytical structure that cannot be reduced to the expression for the neJN from [18] and that the only source of noise is the equilibrium Johnson normal current noise [78]. Moreover, the RSJ model predicts a significant excess noise, with respect to the neJN, for the lower part of the resistive transition, as generally observed in many experiments.