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LIGO-T070043-00-Z The definition of H(f) through Eq.(2) follows the normalization of Allen and Romano [2]. Maggiore [3] u
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LASER INTERFEROMETER GRAVITATIONAL WAVE OBSERVATORY - LIGO CALIFORNIA INSTITUTE OF TECHNOLOGY MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Technical Note
LIGO-T070043-00-Z
2007-02-05
Geometric Acceptance of the LIGO Interferometers at the FSR Frequencies T. Fricke, S. Giampanis and A. C. Melissinos University of Rochester Stochastic Analysis Group
Distribution of this document: Stochastic Analysis Group
This is an internal working note of the LIGO project
California Institute of Technology LIGO Project, MS 18-34 Pasadena, CA 91125 Phone (626) 395-2129 Fax (626) 304-9834 E-mail: [email protected]
Massachusetts Institute of Technology LIGO Project, Room NW17-161 Cambridge, MA 02139 Phone (617) 253-4824 Fax (617) 253-7014 E-mail: [email protected]
WWW: http://www.ligo.caltech.edu/
LIGO-T070043-00-Z
1
Definitions
We define the measured cross-correlation (power) spectrum in the frequency domain by
Sm (f ) = hh∗1 (f )h2 (f )i
(1)
The h1,2 (f ) are the amplitude spectral densities of the interferometer output. They are ˜ ), of the detected signal obtained by suitable normalization of the Fourier transform, h(f time series, h(t),
h(f ) =
p
˜ ) 2/T h(f
The average in Eq.(1) is over repeated segments of data, each of duration T [1]. The interpretation of Sm (f ) follows from the defining properties of a stochastic background (assumed isotropic and unpolarized). Namely
hhαβ (t)i = 0 Z
αβ
hhαβ (t)h (t)i = 32π
∞
H(f )df
(2)
0
where H(f ) is a real, non-negative function describing the spectrum of the stochastic background
H(f ) = |h+ (f )|2 + |h× (f )|2
(3)
For interferometers with orthogonal arms
Sm (f ) =
8π γ(f )D(f )H(f ) 5
(4)
γ(f ) is the overlap reduction function. For co-located and co-aligned interferometers γ(f ) = 1. D(f ) is the correction to the acceptance (pattern function) of the interferometer as a function of frequency. It is derived in section 3, but at frequencies f < ∼ 1 kHz we can set D(f ) = 1. page 1 of 6
LIGO-T070043-00-Z The definition of H(f ) through Eq.(2) follows the normalization of Allen and Romano [2]. Maggiore [3] uses instead Sh (f ) = 8πH(f ). As shown in the following section the local energy density in the gravitational wave is 4π 2 c2 ρG = G
∞
Z
f 2 H(f )df
(5)
0
Using Eq.(4) and Eq.(5) we can express the energy density per unit frequency interval in terms of the measured cross-correlation spectrum dρG 20πc2 2 = f Sm (f )/[γ(f )D(f )] df 8G
(6)
Note that dρG /df is not a derivative but the integrand in Eq.(5), [4]. Finally we define
Ω(f ) ≡
1 dρG f ρc df
(7)
The convenient notation
hrms (f ) =
p Sm (f )
(8)
is often encountered.
2
Calculation of the local energy density
The local energy density is given by Eq.(35.23) of Misner, Thorne and Wheeler [5] c2 ˙ ρG = hhαβ (t, ~x), h˙ αβ (t, ~x)i 32πG
(9)
The amplitudes hαβ (t, ~x) can be uniquely expressed by a plane wave expansion [2]
hαβ (t, ~x) =
XZ A
∞
−∞
Z df s2
ˆ x/c) A ˆ ˆ A (f, Ω)e ˆ 2πif (t−Ω·~ dΩh αβ (Ω)
page 2 of 6
(10)
LIGO-T070043-00-Z For a stochastic signal (as defined in the previous section), the following holds in the frequency domain [2] ˆ A0 (f 0 , Ω ˆ 0 )i = δ(f − f 0 )δ 2 (Ω, ˆ Ω ˆ 0 )δAA0 H(f ) hh∗A (f, Ω)h
(11)
Using the expansion of Eq.(10) in Eq.(9) and expressing the ensemble average by Eq.(11) ˆ 0 , df 0 and the summation over A0 , to we can immediately perform the integrations over dΩ obtain 4π 2 c2 2 X ρG = f 32πG A
Z
∞
Z df s2
−∞
Aαβ ˆ A dΩ H(f ) αβ
(12)
ˆ equals 4π. The summation over the polarization tensors equals 4 and the integral over dΩ We also limit the integration of df to the range 0 to ∞ (because H(−f ) = H(f )) yielding a further factor of 2, so that 4π 2 c2 ρG = G
Z
∞
f 2 H(f )df
(13)
0
This proves Eq.(5) and the similar steps lead to Eq.(2). Of course both equations are evaluated at the same point in space, ~x. Some subtle issues related to infinitely long time series are discussed in [6]. Note that Weinberg [7] gives the local energy density in the gravitational wave [his Eq.(10.37)] as
htµν i =
kµ kν (|h+ |2 + |h× |2 ) 16πG
(14)
with c = 1, k = ω = 2πf . If we write
ρG =
XZ
t00 dΩdf
(15)
A
the summation over polarizations and the integration over solid angle introduce a factor of 16π in agreement with Eq.(13). Finally for convenience in evaluating Ω(f ) in Eq.(7) we give
page 3 of 6
LIGO-T070043-00-Z
3c2 H02 8πG
ρc =
3
(16)
The interferometer acceptance as a function of frequency
The gravitational wave signal appears at the antisymmetric (dark) port of the interferometer. It is proportional to the difference in the phase of the carrier light exiting from the two arms. We follow Sigg [8] in calculating the phase shift induced by a gravitational wave of arbitrary incidence and polarization. The interferometer arms are taken along the x and y axes, and the gravitational wave vector is ~k, Ω = c|~k|. In spherical coordinates kˆx = sin θ cos φ
kˆy = sin θ sin φ
kˆz = cos θ
(17)
and k = |~k|. We work in the T T gauge and designate the metric perturbation amplitudes by h+ and h× for the two polarizations of the wave. After rotating the gravitational wave tensor into the interferometer coordinate system we find
hxx = − cos θ sin 2φh× + cos2 θ cos2 φ − sin2 φ h+ hyy = cos θ sin 2φh× + cos2 θ sin2 φ − cos2 φ h+
(18)
The proper time for a light signal is zero
dτ 2 = dxµ gµν dxν = 0
with
gµν = ηµν + hµν
The change in the phase of the carrier is obtained by integrating the proper time dτ over a round trip along the arm Z ∆Φ =
2L
Z ωdτ =
0
0
2L
1 ω [1 + hxx (t)] 2 dx c
or
page 4 of 6
(19)
LIGO-T070043-00-Z
ω ∆Φx (t0 ) = c
Z
L
n h io 12 ˆ 1 + hxx cos Ωt0 + k(1 − kx )x dx +
0
ω + c
Z
L
n h io 21 1 + hxx cos Ωt0 + k 2L − (1 + kˆx )x dx
(20)
0
Since hxx , hyy 1 we expand the square roots, convert to exponential notation, discard the time independent term and shift to the time of arrival to find
∆Φx =
hxx Lω −iΦΩ sin ΦΩ + ikˆx cos ΦΩ − ikˆx eikx ΦΩ e c ΦΩ 1 − kˆ2
(21)
x
and a corresponding expression for ∆Φy . Here
ΦΩ = LΩ/c with L the length of the arm and Ω = 2πfG . The magnitude of Eq. (21) gives the amplitude of the change in carrier phase and the modulus gives the phase shift with respect to the gravitational wave. For normal incidence kx = 0 and we recover the usual expression for the carrier phase. We must form the phase difference between the two arms for each of the two polarizations separately, and we combine the results in quadrature
H = ∆Φx − ∆Φy H+ = H (h× = 0, h+ = 1) H× = H (h× = 1, h+ = 0) p H = |H+ |2 + |H× |2
(22)
|H+ |, |H× | and H are shown as a function of the angles of incidence θ, φ, of the gravitational wave in Fig. 1 for fG = 0 and in Fig. 2 for ΦΩ = π (this corresponds to fG = 37.52 kHz for L = 4 km). We refer to H (θ, φ, fG ) as the antenna pattern at frequency fG . To evaluate the acceptance of the interferometer, i.e., the function D(f ) introduced in Eq.(4) we must average over θ, and φ. We do this by integrating
Z V =
2π
Z dφ
0
π
Z sin θdθ
0
page 5 of 6
0
|H(θ,φ)|
r2 dr
(23)
LIGO-T070043-00-Z and setting 1
D(f ) = [V (f )/V (f = 0)] 3
(24)
Our calculation so far involved a single traversal in the arm. Since the carrier undergoes multiple reflections in the Fabry-Perot cavity we must add the contribution of these repeated traversals. It was shown by Schilling [9] that the result for a single traversal, which we designate H1 , is multiplied by the usual transfer function for a resonant optical cavity. Namely
HF P =
H1 |1 − r1 r2 ei2ΦΩ |
(25)
where r1 , r2 are the reflectivities of the input and end mirrors. Thus the result of Eq.(25) is not modified by the multiple traversals in the arms.
References 1. W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes in C++ ,(Cambridge 2002). 2. B. Allen and J. Romano, Phys. Rev. D59, 102001 (1999). 3. M. Maggiore, Physics Reports 331, 283 (2000). 4. L. Grishchuk and the LSC Stochastic Group, LIGO internal report T060270-00-Z (2006). 5. C. Misner, K. Thorne and J. Wheeler, Gravitation, (Freeman 1973). 6. W. Butler and A. Melissinos, Measurement of a Stochastic Gravitational Background with a Single Laser Interferometer, arXiv:gr-qc/0501089 v1.(2005). 7. S. Weinberg, Gravitation and Cosmology,(Wiley 1972). 8. D. Sigg, Strain Calibration in LIGO, T970101-B-D (2003). 9. R. Schilling, Class. Quantum Gravity 14, 1513 (1997).
page 6 of 6
LIGO-T070043-00-Z
h+ Polarization
h× Polarization
RMS Average Polarization
Figure 1: The acceptance pattern at f < 1 kHz for the plus and cross polarizations and their rms average
h Polarization +
h× Polarization
RMS Average Polarization
Figure 2: The acceptance pattern at f = 37.52 kHz for the plus and cross polarizations and their rms average page 7 of 6