On the Expansion of Space and How to Measure It

We describe the effect of the expansion of space on the wavelength of the light beam in a Fabry-Pérot interferometer. For an instrument such as the Laser Interferometer Gravitational-Wave Observatory (LIGO), which has high sensitivity and a long period of light storage, the wavelength λ L of laser photons are redshifted due to the expansion of space in each cavity by an amount δ λ given by δ λ / λ L = H 0 τ s ≈ 8 . 8 × 10 - 21 , where H 0 ≈ 2 . 2 × 10 - 18 s - 1 is the Hubble constant and τ s ≈ 4 ms is the light storage time for the cavity. Since τ s is based on the cavity finesse F which depends on the laser beam full width at half maximum (FWHM) δ ω of each cavity, we show that a difference in finesses between the LIGO arm cavities produces a signal h H ( t ) at the anti-symmetric output port given by h H t = 2 a 1 H 0 1 δ ω X t - 1 δ ω Y t , where δ ω X ( t ) and δ ω Y ( t ) are the beam FWHM at time t, respectively, for the X and Y arm cavities and a 1 is a beam proportionality constant to be determined expermentally. Assuming a 1 ≈ 1 , then for cavity beams FWHM of δ ω ( t ) ≈ ( 523 . 2 ± 31 ) rad . s - 1 the output signal has the range ∣ h H ( t ) ∣ ≤ 1 × 10 - 21 , which is detectable by advanced LIGO.