The antenna strain sensitivity

• The detectors use resonant transducers (and therefore there are two resonance modes coupled to the gravitational field) in order to obtain high coupling and high Q.

  For the discussion on the detector sensitivity and frequency bandwidth it is sufficient to consider the simplest resonant antenna, a cylinder of high Q material, strongly coupled to a non resonant transducer followed by a very low noise electronic amplifier. The equation for the end bar displacement $\xi$ is
  

  $$\ddot{\xi}+2\beta_1\dot{\xi}+\omega_o^2\xi=\frac{f}{m}$$
  
  
  where $f$ is the applied force, $m$ the oscillator reduced mass (for a cylinder $m = \frac{M}{2}$) and $\beta_1=\frac{\omega_o}{2Q}$ is the inverse of the decay time of an oscillation due to a delta excitation.

  We consider here only the noise which can be easily modeled, the sum of two terms: the thermal (Brownian) noise and the electronic noise. The power spectrum due to the thermal noise is
  

  $$S_f=\frac{2\omega_o}{Q}mkT_e$$
  
  
  where $T_e$ is the equivalent temperature which includes the effect of the back-action from the electronic amplifier.

  By referring the noise to the displacement of the bar ends, we obtain the power spectrum of the displacement due to the Brownian noise:
  

  $$S^B_\xi =\frac{S_f}{m^2} \frac{1}{(\omega^2-\omega^2_o)^2 +\frac{\omega^2 \omega^2_o}{Q^2}}$$
  
  
  From this we can calculate the mean square displacement
  
  $$\bar{\xi^2}=\frac{kT_e}{m\omega^2_o}$$
  
  
  that can be also obtained, as well known, from the equipartition of the energy.

  To this noise we must add the wide-band noise due to the electronic amplifier (the contribution to the narrow-band noise due to the amplifier has been already included in $T_e$).

  For sake of simplicity we consider an electromechanical transducer that converts the vibration of the detector in a voltage signal
  

  $$V=\alpha \xi$$
  
  
  with transducer constant $\alpha$ (typically of the order of $10^7$ V/m). Thus the electronic wide-band power spectrum, $S_o$ , is expressed in units of $\frac{V^2}{Hz}$ and the overall noise power spectrum referred to the bar end is given by
  
  $$S^n_\xi =\frac{2kT_e\omega_o}{mQ} \frac{1}{(\omega^2-\omega^2_o)^2 +\frac{\omega^2\omega^2_o}{Q^2}}+\frac{S_o}{\alpha^2}$$
  
  
  We calculate now the signal due to a gravitational wave with amplitude $h$ and optimum polarization impinging perpendicularly to the bar axis. The bar displacement corresponds to the action of a force
  
  $$f=\frac{2}{\pi^2}mL\ddot{h}$$
  
  

• For a GW excitation with power spectrum $S_h(\omega)$, the spectrum of the corresponding bar end displacement is
  
  $$S_\xi =\frac{4L^2\omega^4S_h}{\pi^4} \frac{1}{(\omega^2-\omega^2_o)^2 +\frac{\omega^2\omega^2_o}{Q^2}}$$
  
  

• We can then write the SNR
  
  $$SNR=\frac{S_\xi}{S^n_\xi}=\frac{4L^2\omega^4S_h}{\pi^4\frac{... ...-\frac{\omega^2}{\omega^2_o})^2+ \frac{\omega^2}{\omega^2_o})}$$
  
  
  where the quantity $\Gamma$ is
  
  $$\Gamma=\frac{S_o \beta_1}{\alpha^2\bar{\xi^2}} \sim \frac{T_n}{\beta QT_e}$$
  
  
  $T_n$ is the noise temperature of the electronic amplifier and $\beta$ indicates the fraction of energy which is transferred from the bar to the transducer. It can be readily seen that $\Gamma<<1$.

  The GW spectrum that can be detected with SNR=1 is:
  

  $$S_h(\omega)=\pi^2\frac{kT_e}{MQv^2}\frac{\omega^3_o}{\omega^... ...1-\frac{\omega^2}{\omega^2_o})^2+ \frac{\omega^2}{\omega^2_o})$$
  
  
  where $v$ is the sound velocity in the bar material ($v$=5400 m/s in aluminum). For $\omega=\omega_o$ we obtain the highest sensitivity
  
  $$S_h(\omega_o)=\pi^2\frac{kT_e}{MQv^2}\frac{1}{\omega_0}$$
  
  
  having considered $\Gamma<<1$.

  The $spectral ~amplitude$ -(or strain sensitivity) $1/\sqrt(Hz)$ is the square root of $S_h$:
  

  $$\tilde{h}=\sqrt{S_h}$$
  
  

  We remark that the best spectral sensitivity, obtained at the resonance frequency of the detector, only depends on the temperature T, on the mass M and on the quality factor Q of the detector, provided $T=T_e$. Note that this condition is rather different from that required for optimum pulse sensitivity. The bandwidth of the detector is found by imposing that $S_h(\omega)$ be equal to twice the value $S_h(\omega_o)$. We obtain , in terms of the frequency $f = \frac{\omega}{2\pi}$
  

  $$\triangle f=\frac{f_o}{Q}\frac{1}{\sqrt{\Gamma}}$$
  
  
  The present detector bandwidths are of the order of 1 Hz, but it is expected that the bandwidths will become of the order of 50 Hz, by improving the amplifier noise temperature $T_n$ , the coupling parameter $\beta$ and the quality factor Q.

An example of the Nautilus strain sensitivity is given in this picture