Comments on the Observation of the Tides in the fsr channelof the LIGO Gravitational Wave Interferometers

A. C. MelissinosSeptember 4, 2012

1 Introduction

It is experimentally observed that the spectrum of the integrated Power Spectral Density (PSD)of the fsr channel (the fsr frequency is νfsr = 37.520 kHz) contains the tidal frequencies to withinthe resolution of the measurement [1, 2]. The normal channels, ASQ, DARMCTRL, and the Tidalcorrection channel, contain frequencies near the tidal values but they are not exact. The presenceof a signal at the fsr frequency can be attributed to the macroscopic asymmetry in the length ofthe two arms, which is of order ∼ 2 cm, but this does not explain the observed modulation of theintegrated power, for which we have no satisfactory answer. At times we have considered the effectof the horizontal tidal gravity gradient [2, 3], but this leads to inconsistencies.

We begin by discussing the data acquisition and analysis at the fsr frequency and compare thedata to a simulation of the interferometer. In the following section we present the results of thespectral analysis, and discuss the effect of the Earth tides on the operation of the interferometer.In section 4 we address the possible cause of the signal, by considering macroscopic length changes,and the effect of the tidal gradient. We also show that there is a strong correlation between thesignals seen at H1 and L1. In the last section we ask whether the LIGO interferometers can beused to improve the upper limit on the anisotropy of the propagation of light, as in the Michelson-Morley experiment. We also show the spectrum of the dominant twice yearly modulation of thedata. Three Appendices follow: In App.1 it is shown that when the carrier is locked, the phase shiftat the fsr remains fixed, irrespective of the presence of a time-dependant gravity gradient alongthe arms. App.2 gives the parameters used for the Matlab code that simulates the interferometer.Finally App.3 gives a general discussion of the response of the LIGO interferometers to very lowfrequency gravitational waves.

2 Spectrum at the fsr frequency

The fsr (free spectral range) frequency of the interferometer arms is given by νfsr = c/2L = 37.520kHz, and is not directly accessible to the main data acquisition system which samples at 16.384kHz, thus restricting the spectral content to ν < 8.192 kHz. Therefore, an extra signal from thephotodiodes at the ASQ (dark port), followed by a high-pass filter with a knee at ∼ 100 Hz, wassent to the demodulation boards. After mixing with the rf modulation frequency, νrf = 24.468750MHz, and appropriate filtering, the demodulated signal was digitized at 262.144 kHz, shifted to a

1

central frequency of 37.504 kHz, decimated to a sampling rate of 2,048 Hz, and filtered to coverthe frequency range νfsr = 37, 504 ± 1, 024 Hz. A second channel covering 0 < ν < 2, 048 Hz wasalso recorded for calibration purposes.

The data that we discuss were taken with the interferometer in lock, and were written to 64 slong frames, as a time series at the sampling frequency, νs = 2, 048 Hz. We Fourier analyzed 8 slong segments, and averaged the results of the 8 segments within the frame, to obtain amplitudedensity spectra with resolution BW = 0.125 Hz. The spectrum averaged over one month of datataking (∼ 40,000 entries) is shown in Fig.1. This spectrum has several interesting features: (1) thefree spectral range channel is clearly present as an enhancement above noise at the fsr frequencyand has the expected width of the “cavity pole”, FWHM ∼ 250 Hz, (2) the sharp lines correspondto the excitation of mechanical resonances in the test masses, which in turn generate sidebandson the carrier, and (3) a destructive interference feature appears at the exact fsr frequency with awidth typical of the “double cavity pole”, FWHM ∼ 2 Hz.

The Fabry-Perot cavities in the arms of the interferometer have a spectrum of discrete excita-tions spaced by the fsr frequency. Given the length of the arms, ∼ 4 km, and the carrier wavelengthλ0 = 1.064 µm, the occupied level is n0 ∼ 4× 109; if the adjacent levels, n0 ± 1, are partially pop-ulated, they will give rise to sidebands, at ν± = ν0 ± νfsr. The sidebands have a non-zero phaseshift at the AS port when the interferometer is locked, that is even though the carrier phase shiftis zero.

∆ϕ±2π

= 2(Lx − Ly)ν±c

= 2(Lx − Ly)ν0c

± 2(Lx − Ly)νfsrc

=∆ϕ0

2π± ∆L

L, (1)

where ∆ϕ0/2π is an integer. In contrast to the carrier, the sideband fields are broadly distributedaround their central value ν± with a width typical of the arm cavity resonance, ∆ν± ∼ 250 Hz.From the data shown in Fig.1 we can infer the amplitude density of the sideband fields to be (attheir peak) E±(ν) ∼ 10−7E0/

√Hz, with E0 the amplitude of the carrier field. The sidebands fields

originate from carrier phase noise and are sustained because they resonate in the arms; they arealso partially driven by parametric amplification from a test mass vibration mode at a frequencyvery close to νfsr as seen in Fig.1. Typically ∆L was of order 2 cm, so that for a single traversal

∆ϕsingle± /2π = ∆L/L = 5× 10−6.

To include the effect of multiple traversals, ∆ϕsingle± /2π must be multiplied by the effective number

of round trips in the arms, which we approximate as B = 100, yielding ∆ϕmultiple± /2π = 5× 10−4.

This phase shift is ∼ 107 times larger than a typical phase shift for a gravitational wave of ampli-tude h = 10−22. The DAQ channels are not saturated because the amplitude of the fields E± ismuch smaller than the carrier field E0.

The upper and lower sideband have opposite phase shifts, as can be seen from Eq.1. Howeversince they correspond to “conjugate” frequencies (with respect to the carrier), after demodulationthey both contribute additive signals at the fsr frequency. This was also confirmed by the simulationof the interferometer which is shown in Fig.2. Here we introduced sidebands at the X-arm end-mirror, at a frequency covering the fsr region, and we plot the demodulated quadrature signal.

2

3 The 16 month record

As discussed above, from the data we could form the PSD (averaged over 8 measurements) for every64 s of data taking. We then integrate the power in the band of ±200 Hz around νfsr and we applya filter (proportional to the arm transfer function) to emphasize the central region of the spectrum.As a result we obtain a time series of the power at the fsr, sampled at 64 s intervals. The data forthe H1 interferometer are shown in Fig.3 for the period from April 2006 to July 2007, and wereselected to be of “good quality”, and of course only with the interferometer in lock. The dashedlines represent monthly intervals [3]. The inset in the plot shows the variation of the time series ina 2-day period and indicates that the signal follows the daily tides. The mean value of the signalover the 16 month period is stable and there is clear large-scale modulation, which is found to be atexactly twice the rotation frequency of the earth around the sun. When the interferometer looseslock the signal yield at the AS port increases exponentially, but these data have been removed bythe selection cuts. In spite of the large gaps in the time series, the data can be spectrally analyzedby using the Lomb-Scargle algorithm1[4]. Before we present the resulting spectra we discuss theEarth tides and how they are compensated for in the LIGO interferometers.

The Earth is subject to tidal forces from both the moon and the sun [5]. There include a diurnal(once a day) and a semi-diurnal (twice a day) component due to the Earth’s rotation on its axis,as well as considerable fine structure and some long term components. The tidal forces can becalculated as a function of time exactly, using the ephemeris for the sun and the moon [6] and thelocation of the observer. However it is not possible to accurately predict the resulting deformationof the Earth’s surface, because the elastic properties are not well known and are location dependent.The tidal deformation induces both a differential mode change in the arm lengths (similar to theeffect of a gravitational wave), and a common mode change, in which case both arms are length-ened or shortened simultaneously. As an order of magnitude estimate, for the LIGO interferometersthe change in arm length can be as large as 400 µm. This is well beyond the dynamic range ofthe control servo and therefore must be corrected independently by a “feed-forward” mechanismacting on PZT transducers. At Hanford, the common mode change was partially compensated bychanging the carrier frequency. This system worked well and allows the interferometer to stay inlock for periods in excess of 24 hours.

The spectra of the integrated power at the fsr are shown in Fig.4 for the diurnal frequenciesand in Fig.5 for the semi-diurnal frequencies. The results are listed in Table 1, where they arealso compared with the predicted values [5]. The agreement is excellent within the resolution ofthe measurement which is ∆νres = 1/4Ttotal = 6 × 10−9, with Ttotal = 4.2 × 107 seconds, and thefactor of 4 is included because in the spectral analysis the data was oversampled by a factor of four.Comparing the observed frequencies to the predicted ones, and using ∆νres as the measurementerror, yields χ2/DF = 1.86. The table also lists the observed long-term, twice yearly, component.

The comparison of the amplitude of the tidal components is uncertain because, as discussed inthe following section, we do not understand the origin of the modulation.

1The procedure is to fit a set of sinusoids to the data.

3

Table I. Observed and known frequencies of the tidal components (Hz)

Symbol Measured Predicted Origin, L=lunar; S=solar

Long period

Ssa 6.536× 10−8 6.338× 10−8 S declinational

DirnalO1 1.07601× 10−5 1.07585× 10−5 L principal lunar waveP1 1.15384× 10−5 1.15424× 10−5 S solar principal waveS1 1.15741× 10−5 1.15741× 10−5 S elliptic wave of sK1

mK1,sK1 1.16216× 10−5 1.16058× 10−5 L,S declinational waves

Twice-daily

N2 2.19240× 10−5 2.19442× 10−5 L major elliptic wave of M2

M2 2.23639× 10−5 2.23643× 10−5 L principal waveS2 2.31482× 10−5 2.31481× 10−5 S principal wave

mK2,sK2 2.31957× 10−5 2.32115× 10−5 L,S declinational waves

4 Cause for the modulation of the fsr integrated power

The observed modulation can be due to a variation in either the phase or the amplitude of the opticalfield at the fsr frequency. Suppose that a time-dependent phase shift δϕt, is present in addition thethe biasing phase shift ∆ϕbias = 2π(∆L/L); then the integrated power will be modulated as

P =

∫(PSD)dν =

[(∆ϕbias)

2 + 2∆ϕbiasδϕt + (δϕt)2]|C ′|2

∫|E±(ν)|2dν, (2)

Since δϕt is less than a few percent of ∆ϕbias, we can neglect the last term. In that case thetime-dependent relative modulation of the power is given by

∆P

P= 2

|δϕt||∆ϕbias|

(3)

The ratio (∆P/P )for the principal tidal components can be extracted from the spectral analysis, asgiven in the following Table II, but is subject to significant uncertainty. In particular the amplitudeof the principal daily component S1, is probably corrupted by human activity.

Table II. Amplitude modulation of the power of the principal tidal components in %

Symbol ∆P/P

DiurnalO1 0.70P1 0.57

Twice-daily

M2 1.03S2 0.70

4

Using an entry from Table II we can estimate δϕt when ∆P/P = 0.4%, and remembering that

∆ϕmultiplebias /2π = 5× 10−4,

δϕmultiplet

2π=

1

2

∆ϕmultiplebias

2π× ∆P

P= 5× 10−4 × (4× 10−3) = 10−6. (4)

The “time-dependant” phase shift of Eq.4 can be due either to (a) a macroscopic length changeδ∆L in the arm length difference, which then through Eq.1 generates δϕt, or (b) through a micro-

scopic length or frequency change. In the macroscopic case we find δ∆L = Lδϕsinglet ≈ 4× 10−5 m,

or 40 µm; this is of the order of a fraction of the actual shift of the arm lengths due to the tides andwould seem probable except that the data were taken with the interferometer locked, and there-fore while the data were taken, the arm length difference was maintained to an accuracy of order10−19 m. One would have to argue that the tidal feed-forward was over- or under-compensating,leading to a differential arm length change of up to 40 µm, while the control system kept the micro-scopic lengths on resonance and thus the interferometer remained in lock at the carrier frequency.For this to happen the control system would have to “jump” over one or more wavelengths

In case (b) one notes that the tidal acceleration has a horizontal component along the arms,typically ghor ≈ 10−7g ≈ 10−6 m s−2, and this component is time-dependant at the tide frequencies.The horizontal gravity gradient leads to a frequency shift of the light propagating along the arms,and thus to a cumulative phase shift for every traversal. In the weak field approximation, thepresence of a gravitational potential Φ modifies the g00 metric coefficient to

g00 = −(1 + 2Φ/c2) (5)

The departure of g00 from its flat space value gives rise to time dilation, or equivalently to a shiftin the frequency of light propagating through that gravitational field [7, 8].

νA − νB = −ΦA − ΦB

c2ν or

δν

ν=

δn

n=

δΦ

c2, (6)

where we also introduced the refractive index of the light n = c′/c, which is often used in theliterature.

A constant gradient ghor along the x-direction can be described by a potential, Φ = ghorx. Thuslight executing a single round trip in an arm of length L acquires a phase shift (as compared tolight traveling in a field-free region) equal to

δϕsinglet = 2

∫δωdt = 4πν0

∫ L

0

δν

ν

dx

c=

4π

λ0

∫ L

0

Φ

c2dx =

2π

λ0

ghorL2

c2. (7)

Numerically, and after accounting for the multiple traversals, we find

δϕmultiplet

2π∼ 2× 10−8.

This estimate of δϕmultiplet (while still a very large phase shift) is two orders of magnitude too small

to account for the observed phase shift given by Eq.4. There is also the problem that the gravita-tional frequency shift that was invoked to reach the result of Eq.7, affects not only the sideband

5

frequencies, but also the carrier; therefore it would be compensated by the control system. It isshown in Appendix 1, that when the interferometer is locked, the phase shift at the sideband fre-quency, remains at the static value ∆ϕbias/2π = ∆L/L even in the presence of a gravity gradient,and would not lead to a time-dependent signal. A similar argument was advanced in [9].

We conclude this section by including several figures. Figs.6,7 show the fsr data in the timedomain for the periods December 2-7, 2006 and March 2-7, 2007, with the predicted tidal signal inblue. There is qualitative, but not quantitative agreement. The periods when the interferometerwas out of lock can be clearly seen. The dashed black curve is the differential tidal correction signal,and the times at which it was reset are evident. Fig.8 is the result of the spectral analysis of the fsrchannel for the L1 interferometer in the twice daily frequency region, showing that the dominanttidal components are present. Finally Fig.9, is the spectral analysis of the DARM-Control signal forone month of data. It shows that differential arm changes at the tidal frequencies did occur duringdata taking and were corrected by the control system. In contrast, the DARM-Error signal showsno evidence for tidal frequencies indicating a highly accurate correction. Note that the frequenciesin the DARM-Control spectrum deviate slightly from those in Table I.

5 Isotropy of Space

The configuration of the LIGO interferometers begs the question whether they can be used toimprove the upper limit on any anisotropic propagation of light through space, as in the Michelson-Morley experiment [10]. The difference of course is that the arms in the LIGO interferometers arenot fixed; the mirrors are suspended and free to move along the arms under the control system,but also their supporting structure is being continuously shifted due to the Earth tides. When theinterferometer is in lock, the length difference between the arms is kept fixed by the control systemto high accuracy, using the information from the carrier frequency. One could think that a differentvelocity of light in a particular direction of space would manifest itself as a daily modulation of thefsr signal. The problem, of course, is that the arm length difference is maintained by observing thephase shift of the carrier, which would be equally affected by the anisotropic propagation of thelight, and would lead to an arm length difference so as to compensate for the effect.

The dominant modulation of the fsr signal over the long term is evident from Fig.3 and spectralanalysis yields the result shown in Fig.10. The frequency ν = (6.5 ± 0.6) × 10−8 Hz, agrees withthe twice yearly frequency of the Earth’s rotation around the sun, 2νsolar = 6.34 × 10−8 Hz. Thetidal component at this frequency is referred to as the ”zonal, solar declinational” wave, with apredicted amplitude about 1/10 of that of the diurnal component. Instead the observed amplitudeis ∼ 20 times larger than the diurnal component. Furthermore a component of similar origin, dueto the moon, and of twice the solar amplitude, should be present at 2νmoon = 8.47× 10−7 Hz, butit is obviously absent. The origin of this dominant modulation is not understood. One possibleexplanation is that, given the very long period, the control system does not compensate for it, sothat it induces a large macroscopic change in arm length difference. Conversely one can considerthat the tidal correction feed-forward generates a large twice-yearly correction signal which changesthe 2 cm arm length difference by almost 1 mm in three months.

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6 Conclusions

The integrated power at the fsr frequency follows the tides faithfully as can be seen in Figs 6,7 andis also evident from the exact frequencies observed in the spectrum of the 16-month long record.The most probable mechanism, is a change in macroscopic arm-length difference which occurs whilethe interferometer remains in lock. In principle this is an inconsistent statement, since with theinterferometer in lock, the differential distance should remain fixed. However, part of the differen-tial control signal is sent to the piezoelectric actuators that compensate the tidal motion, and havesufficient range to cause the observed effect, if the servo under- or over-compensates the tidal Earthmotion. That such a signal is present at the tidal frequencies can be seen in Fig.9 which is thespectrum of DARM-Control for the period April 2006 to July 2007. The low frequency modulationseen in the fsr signal is much cleaner than the modulation seen in the ASQ-channel. This may berelated to the presence of the high pass filter in the fsr channel. The possibility of a microscopiclength change can not be excluded, but it is less probable.

An interesting conclusion from this analysis is that the horizontal gravity gradient generatedby the tides is equivalent to a strain h = 5 × 10−20 which is well within the range of sensitivityof the interferometers. The problem in detecting the gradient is that the gradient affects boththe fsr and the carrier frequency, and therefore does not lead to an observable modulation of thefsr signal; unless the very low frequencies are not corrected at the carrier frequency. Even then,the observed modulation is ∼ 100 times stronger than expected from the tidal horizontal gravitygradient. This same argument shows that it is not possible to test the isotropic propagation oflight (Michelson-Morley experiment) by using the fsr channel, when the carrier is used to keep theinterferometer on lock.

Finally we remark on the fact that low, and very-low frequency signals can be extracted from theinterferometer data in spite of the large vibrational noise below ∼ 40 Hz. This is possible given longintegration time, as in the present analysis. However low frequency gravitational waves can not bedetected because the tidal force is too weak to displace the test masses (mirrors) from their verticalposition. The direct coupling of the gw to the light in the interferometer is of course present, but itis many orders of magnitude weaker than the indirect effect that involves the motion of the mirrors.

The appearance of the very strong twice annual modulation, (at the exact frequency) raisesthe question whether it has a real physical origin. This was already discussed in Section 5, andthe preferred explanation is that it is due to a (very large) macroscopic variation in arm-lengthdifference occurring at that frequency. This macroscopic motion could be generated, possibly, bythe tidal feed-forward system rather than by tidal Earth motion.

7 Acknowledgements

The hardware for the fsr channel was designed and implemented by Dr. Daniel Sigg. The dataused in this analysis could not have been acquired without the dedicated effort of the staff and op-erators of LHO, and of many LSC colleagues. I am particularly indebted to Chad Forrest, StefanosGiampanis, Tobin Fricke and Bill Butler for their many contributions to this work.

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Appendix 1. FSR phase shift when the carrier is locked

Let δϕ0x and δϕ0y be the phase shifts on the carrier due to the gravitational gradient, along theX-, Y- arms, as given by Eq.7; we assume that δϕ0x = δϕ0y = δϕ0/2. The total phase shift of thecarrier, including the effect of the arm length difference will be

∆ϕ0

2π=

(2

cν0Lx +

δϕ0x

4π

)−

(2

cν0Ly −

δϕ0y

4π

)=

2

cν0∆L+

δϕ0

2π= n−m. (8)

When the IFO is locked this (normalized, i.e. divided by 2π) phase shift is always the differenceof two integers because the light must resonate in the arms. Thus ∆ϕ0/2π = 0, modulo 2π. Thedark fringe condition is then imposed by adjusting the “Michelson” lengths.

Let us now examine the upper sideband frequency ν1 = ν0 + νfsr. The phase shift at the exitof the arms will be

∆ϕ1

2π=

(2

cν1Lx +

δϕ1x

4π

)−

(2

cν1Ly −

δϕ1y

4π

)=

2

cν1∆L+

δϕ1

2π(9)

From Eq.7 it follows that δϕ1/δϕ0 = ν1/ν0, and therefore

∆ϕ1

2π=

[2

c(ν1 − ν0)∆L+

(δϕ1 − δϕ0)

2π

]+

[2

cν0∆L+

δϕ0

2π

]=

[(ν1 − ν0)

ν0

2ν0c

∆L+(ν1 − ν0)

ν0

δϕ0

2π

]+ [n−m] . (10)

Setting ν1 − ν0 = νfsr = c/2L and making use of Eq.8 we find

∆ϕ1

2π=

[1 +

νfsrν0

][n−m]. (11)

We see that when the carrier is locked, the fsr phase shift remains constant, and can not be thecause of the experimentally observed modulation.

8

Appendix 2. Details of running the simulation

The program that simulates the interferometer in Matlab is the function “tw2” and was writtenby Bill Butler. It is similar to “Twiddle” that is used by the LIGO group. It can give any of the23 fields in the interferometer, incoming or outgoing from the mirrors. A typical call is

rf = 24.468750e6;f = rf + [-500:1:500];yifo = [1;zeros(22,1)];Efi = [1:1:23];Fields(Efi,:)=tw2(f,yifo,0,0,0,0,0,0,Efi);

The fields can be displayed by calling

Fields(Efi,:)=tw2(f,yifo,0,0,0,0,0,0,Efi);bodeplot((f+rf),Fields(23,:))angle(Fields(23,501))*180/pi

The code has been tuned to keep the AS port dark. It uses arm lengths of 4,000 m, Schnuppasymmetry = 2 × 0.161 m and lin = 3.0 m. The reflectivities of the optics are those effective forinitial LIGO. The r.f. sidebands reach the AS port un-attenuated. The amplitude gain for thecarrier in the recycling cavity is 8.65 and the gain in the arms it is also 8.5. To generate a signalwe “shake” one of the end mirrors and call the demodulation function as follows

shakxtest = [zeros(13,1);-1;zeros(9,1)];f=[36500:2:38500];[A,G] = ag2(f,rf,shakxtest,0,0,0,0,0,0)bodeplot(f,G)bodeplot(f,A)

Arm length asymmetry ∆L and other parameters can be introduced through the variable argu-ments when calling the function. A plot of the demodulated signal when shaking wa shown in Fig.2.

9

Appendix 3. Tutorial on Interferometer response to VLF Gravitational Waves

The observable is the phase shift at the AS port. This arises because the presence of a gwmodifies the transit time in the IFO arm. The phase shift is most directly calculated in the TTgauge. For a gw of angular frequency Ω, incident normally on the IFO plane and polarized alongone of the arms, the round trip time (in a single arm) is

∆trt(t) = (L/c)h(t)sinΩT

ΩTeiΩT ,

Where T = L/c is the one way travel time in the absence of the gw. Thus the phase shift imposedon a carrier of angular frequency ω0 and wavenumber k0 = ω/c is

∆ϕrt(t) = k0Lh(t− T )sinΩT

ΩT. (12)

For a properly polarized gw the change in transit time is opposite in the two arms, so that thedetected phase shift is twice that of Eq. (12). Use of the TT gauge, however, implies that the testmasses (the mirrors) are free. This assumption depends on the mirror suspension and the frequencyof the gw.

To clarify this point it is necessary to work in the laboratory frame of reference, and calculatethe phase shift in the LL gauge. We then find two contributions to the phase shift:

(1) The change in travel time due to change in the position of the mirrors in response tothe tidal forces exerted by the gw.

(2) The change in travel time due to the modification of the “coordinate” velocity of thelight induced by the gravity gradient in the gw.

We refer to (1) as the “indirect” coupling of the gw to the em field in the IFO, and in generalit is the major contribution to the observable phase shift. The contribution (2) is referred to asthe “direct” coupling and becomes relevant at high frequencies. The individual contributions of(1) and (2) when evaluated in the LL gauge are not translation invariant. However, as discussed insection 4, their sum, including the change in the clock rate at the beam splitter [11], result in thetranslation invariant Eq. (12).

Mirror motion

The tidal acceleration (force/unit mass) is given by [12]

x =1

2h(t)x , (13)

where x is the position of the test mass and h(t) = h0e−iΩ(t+z/c) is the metric perturbation induced

by the gw, along the x axis, and we can set z = 0. Eq. (13) is peculiar in that the accelerationappears to depend on the position x of the mirror in the plane of the gw. This occurs becausethe acceleration of Eq. (13) is tidal, namely it implies a relative acceleration between two pointslocated at xa and xb = xa + L. Thus

∆x = xa − xb =1

2h(t)(xa − xb) =

1

2h(t)L . (14)

10

The solution to this equation is

∆x(t) =1

2Lh(t− T ) or, ∆ϕrt = k0Lh(t− T ) (15)

provided ∆x(t) = LΩh0 ≪ c , which is easily satisfied. Since the round trip time ∆t = 2∆x/cfoundin Eq. (15) agrees with Eq. (12) apart from the form factor sin(ΩT )/(ΩT ). The form factor → 1when ΩT ≪ 1; therefore Eq. (15) is valid when the gw frequency multiplied by the travel time inthe arms is much less than one.

In writing Eq. (13) we imply that no other forces act on the mirror. This is not true becausethe suspension exerts a restoring acceleration

xr ≈ −θg = −(g/l)δx (16)

where δx is the linear displacement of the mirror and l the length of the suspension. From Eq. (14)the differential tidal acceleration is ∆x = (1/2)Ω2Lh, and this must be larger than xr in order todisplace the mirror by δx = (1/2)Lh0,

1

2Ω2Lh0 >

1

2ω2sLh0

or Ω2 > ω2s , (17)

where ωs is the natural frequency of the suspension ∼ 1 Hz. When the condition of Eq. (17) isfulfilled and assuming no damping, the mirror displacement is

∆x =∆x

|Ω2 − ω2s |

→ ∆x

Ω2=

1

2Lh0 . (18)

which is the result we had previously found in Eq. (15). Thus when the gw frequency exceeds thenatural frequency of the suspension the mirrors can be considered to be free. Conversely, for gwfrequencies below the suspension’s natural frequency the mirrors act as if they are fixed.

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References

[1] A. Melissinos (for the LSC), “The effect of the Tides on the LIGO Interferometers”, TwelfthMarcel Grossman Meeting on General Relativity, World Scientific, p.1718 (2012). See alsoarXiv:1001.0558.

[2] C. Forrest, T. Fricke, S. Giampanis and A. Melissinos “Determination of the Frequency ofthe Earth Tides Using a LIGO Interferometer”, LIGO Document T080002-00-Z (2008); C.Forrest and A. Melissinos “The Effect of the Earth Tides on the LIGO Interferometers”,LIGO Document T080215-00-Z (2008).

[3] Chad J. Forrest “Tidal effects on laser gravitational wave detectors”, Thesis, University ofRochester, and LIGO Document P0900003-v1 (2009).

[4] J. D. Scargle, ApJ 263,et. W. Press, W. Vetterling, S. Teukolsky and B. Flannery, “Numeri-calendfigure Recipes in C++”, Cambridge University Press, 1988

[5] P. Melchior “The Tides of the Planet Earth” Pergamon Press, 1978.

[6] JPL “Caltech horizon ephemeris program” (2008).

[7] J. B. Hartle “Gravity: an introduction to Einstein’s general relativity” Addison Wesley, SanFrancisco, 2003.

[8] S. Weinberg “ Gravitation and Cosmology” John Wiley and Sons, NY, 1972.

[9] A. V. Gusev and V. N. Rudenko, “ Gravitational Modulation of the Optical Response of LongBaseline Laser Interferometers”, English translation, JETP Letters, 91, 495 (2010).

[10] A. A. Michelson and E. W. Morley, Am. J. Sci. 34, 333 (1887). G. Joos, Ann. Phys. 7, 385(1930).

[11] M. Rakhmanov, Class. Quantum Grav. 26, 155010 (2009).

[12] C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation W.H.Freeman, NY (1973).

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3.66 3.68 3.7 3.72 3.74 3.76 3.78 3.8 3.82 3.84

x 104

10−1

100

101

102

Frequency (Hz)

Figure 1: Uncalibrated amplitude spectral density for the H1 interferometer in the region of itsfree spectral range.

13

36.5 37 37.5 38 38.50

2

4

6

Frequency (kHz)

Line

ar M

agni

tude

36.5 37 37.5 38 38.5−200

−150

−100

−50

0

Frequency (kHz)

Unw

rapp

ed P

hase

Figure 2: Result of the simulation for the demodulated signal when the X-arm end mirror is shakenat the indicated frequencies

Figure 3: Integrated power in the free spectral range (fsr) region as a function of time, April 2006to July 2007. The data are for the H1 interferometer and are sampled every 64 s. Note the dailyand twice-daily modulation that can be seen in the inset.

14

1.05 1.1 1.15 1.2 1.25 1.3

x 10−5

0

500

1000

1500

2000

2500

3000

3500

Frequency (Hz)

X: 1.157e−005Y: 3268

X: 1.076e−005Y: 1067

Figure 4: Frequency spectrum of the integrated fsr power in the diurnal region. Note the finestructure.

2.1 2.15 2.2 2.25 2.3 2.35 2.4

x 10−5

0

500

1000

1500

2000

2500

X: 2.315e−005Y: 990.2

Frequency (Hz)

X: 2.236e−005Y: 2322

X: 2.192e−005Y: 263.4

Figure 5: Frequency spectrum of the integrated fsr power in the the twice-daily region. Note thefine structure.

15

8.491 8.4915 8.492 8.4925 8.493 8.4935 8.494 8.4945

x 108

−0.1

−0.05

0

0.05

0.1

GPS Time

Figure 6: Modulation of the integrated fsr power for five days in December 2006. The blue curveis calculated from the frequency shift imposed on the light circulating in the arm cavities, by thehorizontal component of the local acceleration of gravity. No amplitude or phase normalization hasbeen introduced.

16

8.5685 8.569 8.5695 8.57 8.5705 8.571 8.5715 8.572 8.5725

x 108

−0.1

−0.05

0

0.05

0.1

GPS Time

Figure 7: Modulation of the integrated fsr power for five days in March 2007. The blue curve iscalculated from the frequency shift imposed on the light circulating in the arm cavities, by thehorizontal component of the local acceleration of gravity. No amplitude or phase normalization hasbeen introduced.

17

2.05 2.1 2.15 2.2 2.25 2.3 2.35 2.4 2.45 2.5

x 10−5

0

200

400

600

800

1000

1200

1400

1600

Frequency (Hz)

Pow

er

X: 2.237e−005Y: 1336

X: 2.315e−005Y: 832.3

X: 2.09e−005Y: 270.1

Figure 8: Frequency spectrum of the integrated fsr power for the L1 interferometer for the periodFebruary to August 2007.

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

x 10−5

0

500

1000

1500

2000

2500

3000

X: 2.328e−05Y: 509.7

DARM−CTRL (L−S redone 3/30/2009)

X: 2.249e−05Y: 2536

X: 1.168e−05Y: 1126

X: 1.082e−05Y: 1059

Figure 9: Frequency spectrum of the DARM-control signal for the period April 06 to July 07.

18

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10−6

0

1

2

3

4

5

6

7x 10

4

Frequency (Hz)

Po

we

r

X: 6.536e−008Y: 6.577e+004

Figure 10: Frequency spectrum of the integrated fsr power in the the twice yearly region. Thepredicted tidal component at this frequency is ∼ 0.003 times weaker than the observed peak.

19