Constraining temperature distribution inside LIGO test masses from frequencies of their vibrational modes

Thermal distortion of test masses, as well as thermal drift of their vibrational mode frequencies, present a major challenge for operation of the Advanced LIGO and Advanced VIRGO interferometers, reducing optical efficiency, which limits sensitivity and potentially causing instabilities which reduce duty-cycle. In this paper, we demonstrate that test-mass vibrational mode frequency data can be used to overcome some of these difficulties. First, we derive a general expression for the change in a mode frequency as a function of temperature distribution inside the test mass. Then we show how the mode frequency dependence on temperature distribution can be used to identify the wave function of observed vibrational modes. We then show how monitoring the frequencies of multiple vibrational modes allows the temperature distribution inside the test mass to be strongly constrained. Finally, we demonstrate using simulations, the potential to improve the thermal model of the test mass, providing independent and improved estimates of important parameters such as the coating absorption coefficient and the location of point absorbers.