[Ushiba, Komori]

Abstract:
We conducted further investigations into suspension thermal noises, including horizontal, vertical, pitch, and roll degrees of freedom.
The sum of these noises aligns closely with the current DARM sensitivity.
To refine these estimates, we need to measure the Q-values of pitch, vertical, and roll modes of the test masses, as well as the vertical-horizontal and roll-horizontal coupling factors.

Detail:
The suspension system exhibits six degrees of freedom, which can be classified into two categories based on the fiber loss mechanisms: pendulum-like and bounce-like modes.

Among the three pendulum-like modes, the translational suspension thermal noise is much smaller than the length mode, and the yaw noise is negligible unless the beam spot is near the edge of the test mass.
Hence, we primarily focus on the length thermal noise.

The measured average violin Q-value is 1.7e4, corresponding to a loss angle of 5.9e-5.
In contrast, the thermo-elastic loss angle alone is estimated to be 4.0e-5.
This implies that the diluted Brownian loss angle is 5.9e-5 - 4.0e-5 = 1.9e-5.
With a dilution factor of 2.2 for the first violin mode, the intrinsic loss angle is estimated to be 1.9e-5 * 2.2 = 4.2e-5.
Substituting this loss angle into the sensitivity calculation, the horizontal thermal noise is plotted as the blue line in the attached figure.

The three bounce-like degrees of freedom—vertical, pitch, and roll—must also be considered.
The pitch-horizontal coupling corresponds to the beam spot mis-centering, measured to be 9.75 mm in klog:31876.

The roll-horizontal coupling mechanism is more complex but evident in the DARM spectrum at the resonant frequencies indicated by roll resonances listed in Ushiba-san's document.
One possible explanation is the birefringence of the test mass.
In the plot (cyan dotted line), a roll-horizontal coupling of 3e-5 m is assumed to match the thermal noise with the roll resonance in the DARM spectrum.
To improve accuracy, we need to measure this coupling and the Q-value of the roll resonance.

The vertical thermal noise (green dotted line) was estimated with a vertical-horizontal coupling factor of 0.02.
As with the roll mode, both the coupling factor and the Q-values of the vertical mode need to be measured.

The horizontal and pitch thermal noise estimations are robust, so the thick black line in the plot provides a reliable representation of the total thermal noise.
The roll and vertical noise estimations, represented by the dotted black line, are less precise but still within reasonable bounds.
In conclusion, the overall suspension thermal noise is very close to the observed DARM sensitivity.