Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# On the Expansion of Space and How to Measure It

Version 1 : Received: 25 February 2018 / Approved: 27 February 2018 / Online: 27 February 2018 (03:41:54 CET)

How to cite: Oliveira, F.J. On the Expansion of Space and How to Measure It. Preprints 2018, 2018020170 (doi: 10.20944/preprints201802.0170.v1). Oliveira, F.J. On the Expansion of Space and How to Measure It. Preprints 2018, 2018020170 (doi: 10.20944/preprints201802.0170.v1).

## Abstract

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 ${\lambda }_{L}$ of laser photons are redshifted due to the expansion of space in each cavity by an amount $\delta \lambda$ given by $\delta \lambda /{\lambda }_{L}={H}_{0}{\tau }_{s}\approx 8.8×{10}^{-21}$ , where ${H}_{0}\approx 2.2×{10}^{-18}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ is the Hubble constant and ${\tau }_{s}\approx 4\phantom{\rule{0.166667em}{0ex}}\mathrm{ms}$ is the light storage time for the cavity. Since ${\tau }_{s}$ is based on the cavity finesse $\mathcal{F}$ which depends on the laser beam full width at half maximum (FWHM) $\delta \omega$ of each cavity, we show that a difference in finesses between the LIGO arm cavities produces a signal ${h}_{H}\left(t\right)$ at the anti-symmetric output port given by ${h}_{H}\left(t\right)=2{a}_{1}{H}_{0}\left(\frac{1}{\delta {\omega }_{X}\left(t\right)}-\frac{1}{\delta {\omega }_{Y}\left(t\right)}\right),$ where $\delta {\omega }_{X}\left(t\right)$ and $\delta {\omega }_{Y}\left(t\right)$ 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 ${a}_{1}\approx 1$ , then for cavity beams FWHM of $\delta \omega \left(t\right)\approx \left(523.2±31\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{rad}.\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ the output signal has the range $\mid {h}_{H}\left(t\right)\mid \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\le 1×{10}^{-21}$ , which is detectable by advanced LIGO.

## Subject Areas

universe expansion; Hubble constant; cavity finesse; cosmological redshift; strain

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