The power spectrum of thermal noise in the case of Finite test masses differ slightly from that of the Infinite Test Mass. The expression for the PSD in the infinite test mass case is not a closed form solution but an infinite sum. In this post, a comparison has been made to check how fast does the sum (and as a result, the PSD) converge. The numerical values used for the calculation have been taken from the paper by Levin and that by Liu and Thorne.

>The linearized PSD plots are created for the case of thermal noise in finite test mass (Sec. V) of Liu and Thorne.

S = 8*kb*T*phi*(U0 + delta_U)/omega

>The maximum energy due to stress is considered by an infinite sum here. A comparison has been made regarding the convergence of the sum.

>The two plots correspond to the cases of considering 10 and 100 terms in the sum respectively.

>The plot shows that the difference is not much and hence convergence is fast.

>The relative difference is plotted w.r.t  S_100, the PSD considering 100 terms in the sum.

 >The relative difference goes like abs(S_100 - S_10)/S_100, where 10 and 100 represent the number of terms considered in the sum.

>The algorithm used to evaluate the sum involving Bessel functions was the one by GWINC. (http://nodus.ligo.caltech.edu:8080/COMSOL/10).