We discussed in the previous section that the effect of gravitational waves can be measured by tracking the time of flight of electromagnetic signals between free-falling observers. One technical challenge in designing such a light time gravitational wave detector is that the time of flight fluctuations must be measured with extreme precision, which requires very stable clocks at both ends of the detector. A way around this is to use only a single timing reference, typically in the form of a laser source of coherent light, which is split into two seperate, orthogonal paths, reflected at mirrors acting as test-masses and then recombined at the point of origin. Such a differential detector is called a Michelson interferometer, and is depicted in fig.2.3. Since the light output from the laser comes from the same origin and travels almost equal paths, any fluctuations in the lasers frequency are heavily suppressed at the output port of this configuration. This allows us to measure relative distance fluctuations between the two arms with the required precision to detect GWs. For example, the advanced Laser Interferometer Gravitational-Wave Observatory (aLIGO) [3] can resolve relative changes in its 4 km arms to better than a factor 10−23, which corresponds to an absolute distance change of less than 4×10−20m. For comparison, this is roughly a hundred-thousand times smaller than the
Thecharge radiusis a measure of the size of an atomic nucleus, and is determined by scattering electrons around it.
charge radius of a proton [60].
4 And/or be seperated by extremely large distances. In addition, any matter between the ob-servers can affect the light propagation time as well, potentially further limiting the achievable performance [73].
2.3 direct gravitational wave detection 19
2.3.1 Principle of a Michelson interferometer
We formulate here the basic operating principle of a Michelson interferometer, as depicted in fig.2.3. A single laser beam is split at a central beam splitter.
The two resulting beams are send along two seperate arms, reflected off of mirrors at the end and finally recombined at the same beam splitter. Since the two arms differ in length, the two recombined beams have a relative phase shift and interfere. The resulting interferometric signal can be read out using a photo diode.
Formally, the electrical field of the laser before the beam is split can be written as
E(t) =2Acos(Φ(t)) =2Acos(ωt+φ(t)), (2.46) where we assume the amplitudeAto be constant whileφ(t)accounts for any intrinsic imperfections in the lasers phase. We assume the two arms to be of unequal length, such that the light travel time for one round-trip is given as τ+δτN for the north arm andτ+δτE for the east arm. Therefore, the two arms differ in round trip time by∆τ=δτE−δτN.
The electrical field after recombining the two beams after each round trip We ignore here any other phase shifts of then given as a superposition of the two recombined beams,
EBS(t) = A(cos(Φ(t−τN)) +cos(Φ(t−τE))). (2.47) The photodiode detects a signal proportional to the time averaged squared magnitude of the electrical field:
P=h|EBS|2i
= A2(1+cos(Φ(t−τN)−Φ(t−τE)))).
= A2(1+cos(ω∆τ+φ(t−τN)−φ(t−τE))).
(2.48)
Ground-based Michelson interferometers such as aLIGO are usually designed
such that the nominal ∆τ Interferometers with a
nominally constant readout signal are calledhomodyne in-terferometers.
is constant. Note that the cosine in eq. (2.48) can take values between 1 and -1, such that the overall detected power at the photodiode can take any value between 2A2 and 0. By intentionally moving one of the mirrors, it is possible to adjust ∆τin such a way that the interferometer is tuned to a nominal operating point. In the case of aLIGO, this operating point is close to the dark fringe, at which no light arrives at the photodiode. This ultimately allows a readout of the phase fluctuations [44],
ω∆τ+φ(t−τN)−φ(t−τE). (2.49)
Assuming that the arm length mismatch ’Small’ means here that
fmax1/∆τ, with fmaxas the highest signal frequency we
20 gravitational wave astronomy
Figure 2.3:A simple Michelson interfer-ometer. A single laser beam is split up at a beam splitter, send to two mirrors, reflected, and recombined at the same beam splitter.
Any pathlength fluc-tuations in either of the two arms creates a phase difference be-tween the two beams, which creates the in-terferometric signal.
τ+δτE
τ+δτN
for the terms related to the inherent laser instability. Here, we usedν(t) =
2π1 φ˙(t)for the lasers frequency fluctuations. These directly couple into the phase readout of a Michelson interferometer. Since they enter scaled by the armlength mismatch, a Michelson interferometer with perfectly equal arms is completely insensitive to laser frequency noise.
On the other hand, the term ω∆τ means that Michelson interferometers are very sensitive to differential changes in the arm length, which makes them ideal gravitational wave detectors. As discussed in section2.2.6, a plus polarized monochromatic GW propagating in the x3 direction periodically changes relative distances in thex1 andx2directions, with opposing signs. If our Michelson interferometer is oriented in such a way that the north arm is aligned withx1direction and the east arm is aligned with thex2direction the effect of the gravitational wave directly translates into a differential armlength change.
Keeping the same orientation, a cross-polarized wave, on the other hand, doesn’t create any differential armlength change, such that the interferometer is insensitive to it. It is therefore possible to determine the polarization of a passing GW by utilizing multiple detectors with different orientations.
Multiple detectors working in tandem have a number of additional benefits.
For one, they allow discrimination of instrumental glitches by rejecting signals which are only visible in one detector at a time. In addition, since GWs propagate at the speed of light, multiple detectors can triangulate the direction the wave is coming from, since the same signal will be seen in different detectors with a small time delay.
2.3.2 First detections
On the14th of September2015, aLIGO made the first direct detection of a gravitational wave signal [5]. They observed the merging of two black holes