If we want to use gravitational waves (GW) to determine the Hubble constant, we need to find the source in the electromagnetic spectrum (EMS). However, we need to be lucky to ‘see’ it simultaneously in EMS and GW. This is a problem, but do BBH not ‘emit’ more than one GW? E.g. every time they spiral? Because than we have time to find it in the EMS, no? What makes it so difficult? Thanks in advance!

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    $\begingroup$ Electromagnetic Spectrum $\endgroup$ – PrincepsMaximus Jun 27 '20 at 11:27

The duration of a gravitational wave detection is not particularly important in detecting electromagnetic counterparts, although the fact that they are not recurrent or repeating sources is.

Binary systems continually emit gravitational waves, up until the time that they merge, predominantly at twice the orbital frequency. At the same time, the power emitted in gravitational waves, which is what drives the system towards merger, also increases dramatically with the orbital frequency.

This means that as a binary system spirals inwards towards merger, the frequency of the signal rises and the power of the signal rises - known as a "chirp". This is a one-way process; once the binary has finished merging, the gravitational wave emission essentially stops.

Gravitational wave detectors are able to detect merging binaries once their frequency enters the sensitive range of the instrument (roughly 20 Hz to 2 kHz) and the gravitational wave is "loud" enough to be detected. The rate of development of the "chirp" increases with increasing mass. A massive black hole binary will scan through frequencies from 20 Hz to perhaps 200 Hz (when it merges) in less than a second. A lower mass neutron star binary might be detected from 20 Hz to 1 kHz over tens of seconds.

Detecting the electromagnetic counterpart does not have to be simultaneous. Whilst some EM signatures are likely to be prompt (e.g. Gamma ray bursts) on timescales of seconds, the development of a kilonova from merging neutron stars, takes hours or even days (Smartt et al. 2017). Recent work on possible EM counterparts to merging black hole binaries, embedded in accretion disks, even suggests there may be a delay of tens of days before seeing any EM counterpart (Graham et al. 2020).

The key is not so much the time over which the gravitational wave source is detected, as being able to determine its direction and distance well and hence narrow down the field of view (and volume of space) to be searched by EM telescopes. To do this effectively requires the signal to be detected by multiple instruments (e.g. the two LIGO detectors and VIRGO). It is true though that if a gravitational wave source were recurrent, it could be better located in the sky.

EDIT: To address mmeent's interesting comment. The duration of the GW signal does become a factor if it enables the source to be located more precisely. This will happen if the orientation of the interferometer changes with respect to the source position during the observation. For the current ground-based interferometers, this means the rotation of the Earth changes the detector orientation in space, so means the GW duration would need to be an hour or longer.

Assuming a circular orbit, the duration of a merger event, starting from a binary with period $T_0$, with a total mass $M$ and a reduced mass of $\mu$ is given by $$ \tau = \left(\frac{5c^5}{256(4\pi)^{4/3}G^{5/3}}\right) M^{-2/3} (T_0^{8/3}-T_{\rm min}^{8/3})\mu^{-1},$$ where $T_{\rm min}$ is the shortest orbital period before merger. The way to increase $\tau$ is to have small masses, long orbital periods and a very unequal mass ratio.

On the other hand, in order to be detectable, the GW frequency (twice the orbital frequency) needs to be $20<f<2000$ Hz, which puts an upper limit of $T_0=0.1$ s and a lower limit of $T_{\rm min}=10^{-3}$ s (or the period at merger, whichever is longer). Assuming $T_0 =0.1$ s, $\mu=M/4$ (equal mass components), and $\tau >3600$ s, we can rearrange the equation above to get $M<0.43M_{\odot}$, which is too small to be merging neutron stars. In order to get a larger $M$ we could change the mass ratio. For example if $M=1.5 M{\odot}$ then a mass ratio of $\sim 30$ would be required. (i.e. a neutron star of mass $\sim 1.45M_{\odot}$ and a companion of mass $\sim 0.05M_{\odot}$. (More typical merging neutron stars could not be observable in the required frequency window for more than an hour).

Leaving aside the question of what the lower mass companion could be, then if the merging binary object is to provides an EM counterpart that can be used to constrain the Hubble constant, it needs to be close enough to be detected at $f=20$ Hz. The strain of the binary at the Earth (for an optimal face-on orientation) is approximately $$ h \simeq \left(\frac{4(4\pi)^{1/3} G^{5/3}}{c^4}\right) \mu M^{2/3} T^{-2/3} r^{-1} ,$$ where $r$ is the distance to the source.

In order to be detectable, the "characteristic strain" (which takes account of accumulating a signal over many orbital cycles) $h_c \sim \sqrt{2 \tau f}h$ must be greater than about $10^{-22}$ for detection by LIGO. Setting $\mu \sim M/30$, $M=1.5M_{\odot}$, $f=20$ Hz, $T=0.1$ s, $\tau=3600$ s and $h_c \sim 10^{-22}$, then to be detectable $r<17$ Mpc. This is too close to be used as a reliable probe of the Hubble constant, since the recession velocity of any host galaxy would be comparable to typical magnitudes of peculiar velocity with respect to the Hubble flow.

(NB: There is ample room for numerical errors in the calculation above, so feel free to check it!)

  • $\begingroup$ Could you share a link of the 'recent work on possible EM counterparts)? $\endgroup$ – PrincepsMaximus Jun 27 '20 at 17:47
  • $\begingroup$ @PrincepsMaximus arxiv.org/abs/2006.14122 $\endgroup$ – ProfRob Jun 27 '20 at 20:19
  • $\begingroup$ It should be noted that typically, it is easier to localize a GW source if the signal is longer. In particular, if the signal is long enough that the detector sensitivity patterns change over the course of the detection. (i.e. hours for ground based detectors, months for detectors in solar orbit). $\endgroup$ – mmeent Jun 29 '20 at 6:46
  • $\begingroup$ @RobJeffries, you state "This is too close to be used as a reliable probe of the Hubble constant". Is it possible to use this as a valid method in de future, with for example LISA and/or improvements on the current detectors. When we can detect more distant GW's, it's a very reliable method to detect H0, or am I wrong? $\endgroup$ – PrincepsMaximus Jun 29 '20 at 13:34
  • $\begingroup$ @PrincepsMaximus The edit simply addresses the point mmeent made. For current ground based detectors there is little prospect of detecting realistic inspiralling binary systems over >1hour duration, and even if you allow weird mass ratios they would be too close to be useful. A much more sensitive (order of magnitude) detector would help with the latter but not the former. $\endgroup$ – ProfRob Jun 29 '20 at 14:00

We can currently only detect gravitational radiation when it is extremely intense: in the last fraction of a second. For example the first gravitational wave detection lasted less 0.15 seconds. The black holes are releasing gravitational radiation with every orbit, but that radiation is too weak for us to detect. It takes a colossal amount of energy being released for it to be detected by current technology. The inspiral of GW150914 released 3 solar masses of energy, nearly all in the final 0.15 seconds before merger. Even so this only distorted space by less than 1/10000 of the diameter of a proton. It is this that makes it so difficult.


Just a supplement to @JamesK's excellent answer. The image below (from Caltech/MIT by way of New Sciencist) gravitational waveform shows what was detected for one collision. On the left (at the start) the blackholes orbit one another about every 0.03 seconds, but the waveform is too faint to detect. At about 0.3 seconds on the Time axis the waves start being detectable and the increase in strength and decrease in duration as the black holes spiral closer over the next 0.12s. The merge happens at about 0.42 and then there is a short, rapidly fading patten called "ringdown" as the black hole settles to its final form. So yes, there are multiple waves (about 8 detectable ones in this example) but they all arrive at almost the same time.


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