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 1 Fabry-Pérot interferometer. For an instrument such as the Laser Interferometer Gravitational-Wave 2 Observatory (LIGO), which has high sensitivity and a long period of light storage, the wavelength λL 3 of laser photons are redshifted due to the expansion of space in each cavity by an amount δλ given 4 by δλ/λL = H0τs ≈ 8.8× 10−21, where H0 ≈ 2.2× 10−18 s−1 is the Hubble constant and τs ≈ 4 ms 5 is the light storage time for the cavity. Since τs is based on the cavity finesse F which depends on 6 the laser beam full width at half maximum (FWHM) δω of each cavity, we show that a difference in 7 finesses between the LIGO arm cavities produces a signal hH(t) at the anti-symmetric output port 8 given by hH (t) = 2a1H0 ( 1 δωX(t) − 1 δωY(t) ) , where δωX(t) and δωY(t) are the beam FWHM at time t, 9 respectively, for the X and Y arm cavities and a1 is a beam proportionality constant to be determined 10 expermentally. Assuming a1 ≈ 1, then for cavity beams FWHM of δω(t) ≈ (523.2± 31) rad. s−1 the 11 output signal has the range | hH(t) | ≤ 1× 10−21, which is detectable by advanced LIGO. 12


Introduction
The Hubble expansion of the universe [1] should be observable in a local frame via the expansion of space.Although gravity in the Galaxy is strong enough that it keeps all massive objects bounded, and material objects are held together by gravitational and electromagnetic forces, photons have a vacuum speed c ≈ 3 × 10 10 cm s −1 which is greater than the escape velocity for the Galaxy (v e = √ 2GM/R ≈ 2G × 1.5 × 10 12 M / 8 kpc ≈ 1.3 × 10 8 cm s −1 ).Therefore it is reasonable to inquire whether the wavelength of photons moving freely in a vacuum anywhere in the Galaxy will be redshifted by the expansion of space.Because General Relativity (GR) is based on fields, all phenomena are determined by local effects alone.Thus the expansion of the universe, a global effect, is described by the local expansion of space at each point in the universe.In the Friedman-Lemaître-Robertson-Walker (FLRW) [2][3][4][5] cosmological model, the isotropic expansion of space is measured by the scale factor R which is a solution to the Friedman equation where Ṙ = dR/dt is the time derivative of R, G is Newton's gravitation constant, ρ is the local matter density, k = {−1, 0, 1} is the curvature constant and Λ is the cosmological constant.In an interferometer cavity, such as in the Laser Interferometer Gravitational-Wave Observatory (LIGO) [6], there is a high vacuum, so the matter density composed of molecules and photons is assumed negligible, implying ρ ≈ 0, and the curvature is assumed flat with k = 0. Then only the Λ term remains in (1) and its solution takes the form where t e ≤ t, t e is the time of emission of the light and t is the time of observation of the light.It can be shown [3] that the cosmological redshift z of the wavelength of light is given by ( 2), 1 + z = R(t)/R(t e ).
For t − t e = ∆t a small time interval, we drop all second order and higher terms from (2), and the expression for the cosmological redshift takes the form It can be shown [7,8] that the Hubble constant H 0 is related to Λ by From ( 3) and ( 4), we can expect a change in light wavelength δλ given by where λ L is the initial wavelength (laser nominal wavelength).From cosmological observations [9] we use a value of H 0 ≈ 67.8 km s −1 Mpc −1 .

How the expansion of space affects the light in an interferometer
The two perpendicular arms L x and L y of a LIGO facility [6] are each of length L = 4 km.At the ends of each arm are suspended test mirrors which are highly reflective, allowing photons from the input laser to be stored coherently in the cavity between the mirrors.The system has a finesse [10] of F ≈ 450, implying an amplification of A ≈ 300, equivalent to increasing the effective length of each arm to The effective travel time τ s of a photon traveling at vacuum speed c within an arm cavity, undergoing multiple reflections, ignoring latencies at the reflection surface of each test mirror, is given by If the cavity field is not replenished, τ s is the time for the field intensity to be attenuated by 1/e.Substituting ∆t = τ s from (7) into (5) we obtain the maximum wavelength change δλ max given by When a gravitational wave interacts with the LIGO instrument the strain h of the wave is related to the change in LIGO arm lengths L x and L y given by[6] From the measured strain of the GW150914 event, (9) implies that However, from (8), in any 4 ms time interval the cavity photons will have a range of drift given by 0 ≤ δλ/λ L ≤ 8.8 × 10 −21 with an average drift of ≈ 4.4 × 10 −21 .The average drift in phase of cavity photons is more than 4 times larger than the reported strain associated with the GW150914 event.
Just prior to the gravitational wave interaction with the LIGO instrument, the photons stored in the arm cavities will be out of phase with the detector reference frequency ω L = 2πc/λ L , where λ L is the nominal laser wavelength.From (5) we get, for 0 ≤ t ≤ τ s , which is a linear sweep in the cavity frequency each time period τ s .When a gravitational wave modulates the instrument, the measurement of the disturbance at the output detector will have both the cavity and wave signals intermixed.

Condition for resonance in a Fabry-Pérot cavity
From (Ref.[11], Eq. 9 and Eq.10), the equation for the dynamics in a Fabry-Pérot cavity is given by where t is the time, t a is the transmissivity of the input test mirror a, r a is the reflectivity of input test mirror a, r b is the reflectivity of end test mirror b, E in (t) is the input field amplitude, E(t) is the cavity field amplitude, i = √ −1, k = ω/c is the wave number where ω is the nominal laser frequency, δL(t) is the variation in the cavity length and is the time for light to travel once along the nominal distance L between the cavity test mirrors a and b.
Assuming δL(t) = 0 and Laplace transforming (12) yields the basic cavity response function where over-bars denote Laplace transforms.If we consider the effect of the universe expansion upon the frequency of the laser light in the cavity, for one round trip time 2T, the shift is toward lower frequency given by Including this shift effect into the cavity frequency control system, which corrects the laser frequency to compensate for changes in the cavity such as movements of the mirrors , from (Ref.[11], Eq. 20), for one round trip of the light in the cavity, where v(t) = d δL(t)/dt is the relative velocity of the mirrors.Use (13) to substitute for c = L/T in (16) and transform it to the Laplace domain, yielding where is the normalized frequency-to-length transfer function.The cavity free spectral range determines the spacing between zeros of C(s).A control system maintains resonance in the cavity by adjusting the input frequency ω by δω(t).If these frequency changes are much less than the cavity free spectral range then C(s) ≈ 1 and (17) reduces to

A toy model for an output signal from LIGO due to the expansion of space
We introduce a toy model of the laser light in the LIGO cavities which can produce a signal at the anti-symmetric port due to phase differences between the LIGO arms caused by the control system as it compensates for changes in the cavity laser frequency due to mirror movements.The control system maintains cavity resonance by shifting the cavity frequency by δω adj (t) according to (20), neglecting the expansion term, at regular time intervals based on the cavity round trip time 2T = 2L/c and the storage time τ s .By (11), the maximum frequency shift during the storage time due to the universe expansion is where where the finesse is given by [12] F = δω fsr δω Here the cavity free spectral range is given by ( 19) and δω is the laser beam full width at half maximum (FWHM).Assume that the frequency variation δω Hadj in a cavity given by ( 21) creates a coherent sub-beam which is imbedded in the main beam, having a subbeam central frequency and a FWHM beamwidth of where a 1 is a beam proportionality constant to be determined experimentally.Since the cavity laser power is constantly being maintained for resonance by adjustments δω adj (t) to the cavity input beam, the subbeam centered on ω H is also indirectly being maintained by effective adjustments δω Hadj (t) as older photons exiting the mirrors are replenished by newer photons from the main beam.
The phase change associated with the cavity subbeam beamwidth δω H , from (21), ( 22) and ( 25), for a period of one round trip time 2T in the cavity is expressed by Although both LIGO arms are locked simultaneously [13], it is reasonable to assume that each cavity is tweaked with a different frequency change δω(t) to stabilize the cavity resonance.This changes the cavity beamwidth FWHM, which affects the cavity finesse (23), resulting in frequency shifts in the subbeam central frequency ω H and the subbeam beamwidth δω H due to the expansion of space, of δω HX for arm X and δω HY for arm Y.For arm X, by (26), the phase change for the subbeam beamwidth δω HX (t) per round trip cycle time 2T expressed as a function of time t is given by where F X (t) is the finesse of the arm X cavity at time t.Similarly, for arm Y, the phase change for the subbeam beamwidth δω HY (t) per round trip cycle is given by where F Y (t) is the finesse of the arm Y cavity at time t.Taking the differential of the phase differences between arms X and Y by subtracting (28) from ( 27) we obtain From ( 19) and ( 23) we get the difference between the cavity finesses Substituting ( 30) into (29) we obtain for the subbeam phase differential between the cavities Dividing both sides of (31) by (2Lω L /c) yields the output signal due to the expansion of space Since the LIGO anti-symmetric port is zero when the interferometer is in its resonant "locked" state, if the control system is not compensating for the frequency sweep due to the expansion of space then the subbeam induced signal (32) would be detectable at the anti-symmetric port.

Discussion
Since the control system maintains the cavity resonance by correcting the input laser frequency, the signal output given by (32) would be proportional to the control system servoing of the cavity frequency in each LIGO arm.This suggests that by servoing the laser beam frequency for each cavity in a known manner by δω Xadj (t) for the X cavity and δω Yadj (t) for the Y cavity, generating laser beamwidths δω X (t) and δω Y (t), respectively, a predictable signal h H (t) due to the expansion of space would be generated at the output port.In Figure 1 is shown the predicted output h H (t) when the beamwidths in the X and Y cavities are modulated at a rate of f = 1/2τ s ≈ 130.8 Hz by the sin(x) function where ∆ω = 523.2rad.s −1 , φ 0 = 0 for the X arm beam and φ 0 = π for the Y arm beam.
where φ 0 = 0 for the X arm subbeam and φ 0 = π for the Y arm subbeam.
Whether or not the local expansion of space is real is uncertain.But an experimental facility such as LIGO can provide answer.