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Gravitational Wave Detection #2: How detectors work Peter Saulson Syracuse University 8 June 2004 Summer School on Gravitational Wave Astronomy 1 Outline 1. How does an interferometer respond to gravitational waves? 2. How does a resonant bar respond to gravitational waves? 3. A puzzle: If light waves are stretched by gravitational waves, how can we use light as a ruler to detect gravitational waves? 8 June 2004 Summer School on Gravitational Wave Astronomy 2 Three test masses 8 June 2004 Summer School on Gravitational Wave Astronomy 3 Solving for variation in light travel time: start with x arm ds 2 c 2 dt 2 1 h11 dx 2 0 First, assume h(t) is constant during light’s travel through ifo. Rearrange, and replace square root with 1st two terms of binomial expansion 1 1 dt c 1 2 h11 dx and integrate from x = 0 to x = L: t h11L / 2c 8 June 2004 Summer School on Gravitational Wave Astronomy 4 Solving for variation in light travel time (II) In doing this calculation, we choose coordinates that are marked by free masses. “Transverse-traceless (TT) gauge” Thus, end mirror is always at x = L. Round trip back to beam-splitter: t h11L / c y-arm (h22 = - h11 = -h): t y hL / c Difference between x and y round-trip times: 2hL / c 8 June 2004 Summer School on Gravitational Wave Astronomy 5 Multipass, phase diff To make the signal larger, we can arrange for N round trips through the arm instead of 1. More on this in a later lecture. 2 NL h h stor c It is useful to express this as a phase difference, dividing time difference by radian period of light in the ifo: h stor 8 June 2004 2c Summer School on Gravitational Wave Astronomy 6 Sensing relative motions of distant free masses Michelson interferometer 8 June 2004 Summer School on Gravitational Wave Astronomy 7 How do we make the travel-time difference visible? In an ifo, we get a change in output power as a function of phase difference. At beamsplitter, light beams from the two arms are superposed. Thus, at the port away from laser Eout E0 cos and at the port through which light enters Erefl E0 sin 8 June 2004 Summer School on Gravitational Wave Astronomy 8 Output power We actually measure the optical power at the output port Pout Pin cos 2 Pin 1 cos 2 2 Note that energy is conserved: Pout Prefl Pin cos 2 sin 2 Pin 8 June 2004 Summer School on Gravitational Wave Astronomy 9 Interferometer output vs. arm length difference 1 0.9 0.8 Output power 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -1.5 8 June 2004 -1 -0.5 0 0.5 1 1.5 Optical Path Difference, modulo n wavelengths (cm) -5 x 10 Summer School on Gravitational Wave Astronomy 10 Ifo response to h(t) Free masses are free to track time-varying h. As long as stor is short compared to time scale of h(t), then output tracks h(t) faithfully. If not, then put time-dependent h into integral of slide 4 before carrying out the integral. Response “rolls off” for fast signals. This is what is meant by interferometers being broadband detectors. But, noise is stronger at some frequencies than others. (More on this later.) This means some frequency bands have good sensitivity, others not. 8 June 2004 Summer School on Gravitational Wave Astronomy 11 Interpretation Recall grav wave’s effect on one-way travel time: hL t 2c Just as if the arm length is changed by a fraction L h L 2 In the TT gauge, we say that the masses didn’t move (they mark coordinates), but that the separation between them changed. Metric of the space between them changed. 8 June 2004 Summer School on Gravitational Wave Astronomy 12 Comparison with rigid ruler, force picture We can also interpret the same physics in a different picture, using different coordinates. Here, define coordinates with rigid rods, not free masses. With respect to a rigid rod, masses do move apart. In this picture, it is as if the grav wave exerts equal and opposite forces on the two masses. 8 June 2004 Summer School on Gravitational Wave Astronomy 13 Two test masses and a spring The effect of a grav wave on this system is as if it applied a force on each mass of 1 Fgw mLh 2 Character of force: Tidal (prop to L,) and gravitational (prop to m) August 10, 1998 SLAC Summer Institute 14 Resonant-mass detector response Impulse excites high-Q resonance Impulse Response (N-kg/m) 7 1 x 10 0.5 0 -0.5 -1 0 0.001 0.002 0.003 200 300 Impulse Response (N-kg/m) 7 1 0.004 0.005 0.006 Time (sec) 0.007 0.008 0.009 0.01 700 800 900 1000 x 10 0.5 0 -0.5 -1 August 10, 1998 0 100 400 500 600 Time (sec) SLAC Summer Institute 15 Strategy behind resonant detectors Ideally, an interferometer responds to an impulse with an impulsive output. Not literally, but good approximation unless signal is very brief. Bar transforms a brief signal into a long-lasting signal. As long as the decay time of the resonant mode. Why? It is a strategy to make the signal stand out against broad-band noise. More on this in a later lecture. 8 June 2004 Summer School on Gravitational Wave Astronomy 16 First generation resonant-mass system PZT Low noise amp Rectifier or lock-in To chart recorder Room temperature, hrms ~ 10-16 August 10, 1998 SLAC Summer Institute 17 Second generation resonant-mass system Tuned resonator Superconducting transducer SQUID amplifier Lock-in amplifier To computer T = 4 K, hrms ~ 10-18 August 10, 1998 SLAC Summer Institute 18 How are tests masses realized? Interferometer: test masses are 10 kg (or so) cylinders of fused silica, suspended from fine wires as pendulums Effectively free for horizontal motions at frequencies above the resonant frequency of the pendulum (~1 Hz) This justifies modeling them as free masses. Bar: “test masses” are effectively the two ends of the bar. Resonant response, due to “spring” made up of the middle of the bar, is key. 8 June 2004 Summer School on Gravitational Wave Astronomy 19 How to find weak signals in strong noise Look for something that looks more like the signal than noise, at suspiciously large level. s1(t) 5 0 -5 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 Time t 60 70 80 90 100 s2(t+tau) 5 0 xcorr(s1,s2) -5 100 0 -100 August 10, 1998 SLAC Summer Institute 20 The “rubber ruler puzzle” If a gravitational wave stretches space, doesn’t it also stretch the light traveling in that space? If so, the “ruler” is being stretched by the same amount as the system being measured. And if so, how can a gravitational wave be observed using light? How can interferometers possibly work? 8 June 2004 Summer School on Gravitational Wave Astronomy 21 Handy coordinate systems In GR, freely-falling masses play a special role, whose motion is especially simple. It makes sense to use freely-falling masses to mark out coordinate systems to describe simple problems in gravitation. In cosmology, we describe the expansion of the Universe using comoving coordinates, marked by freely-falling masses that only participate in the Hubble expansion. For gravitational waves, the transverse-traceless gauge used on previous slide is marked by freely-falling masses as well. 8 June 2004 Summer School on Gravitational Wave Astronomy 22 Cosmological redshift ds2 c2dt 2 R2 t dx 2 0 8 June 2004 1 dt c R t dx 1 Rt1 0 Rt 0 Summer School on Gravitational Wave Astronomy 23 Heuristic interpretation Cosmological redshift: Wavefronts of light are adjacent to different comoving masses. As Universe expands, each wavefront “thinks” it is near a non-moving mass. As separations between masses grow, separations between wavefronts must also grow. Hence, wavelength grows with cosmic scale factor R(t). Especially vivid if we were to imagine the Universe expanded from R(t0) to R(t1) in a single step function. All light suddenly grows in wavelength by factor R(t1)/ R(t0). 8 June 2004 Summer School on Gravitational Wave Astronomy 24 A gravitational wave stretches light Imagine many freely-falling masses along arms of interferometer. Next, imagine that a step function gravitational wave, with polarization h+, encounters interferometer. Along x arm, test masses suddenly farther apart by (1+h). Wavefronts near each test mass stay near the mass. (No preferred frames in GR!) The wavelength of the light in an interferometer is stretched by a gravitational wave. 8 June 2004 Summer School on Gravitational Wave Astronomy 25 Are length changes real? Yes. The Universe does actually expand. Interferometer arms really do change length. We could (in principle) compare arm lengths to rigid rods. It is important not to confuse our coordinate choices with facts about dynamics. “Eppur si muove” 8 June 2004 Summer School on Gravitational Wave Astronomy 26 OK, so how do interferometers actually work? Argument above only proves that there is no instantaneous response to a gravitational wave. Wait. Arm was lengthened by grav wave. Light travels at c. So, light will start to arrive late, as it has to traverse longer distance than it did before the wave arrived. Delay builds up until all light present at wave’s arrival is flushed out, then delay stays constant at h(2NL/c). 8 June 2004 Summer School on Gravitational Wave Astronomy 27 New light isn’t stretched, so it serves as the good ruler In the end, there is no puzzle. The time it takes for light to travel through stretched interferometer arms is still the key physical concept. Interferometers can work. 8 June 2004 Summer School on Gravitational Wave Astronomy 28