### Waiting for the most wanted wave in the physics world ... and then

**A quantitative forecast for the advanced LIGO detector**

Tonight at the hour I types these lines the wind blows strongly behind the window and I know there is a storm warning on the Atlantic French coast. I wish all the best to mariners and people by the sea. The wave I have in mind in this post is quite different; nevertheless it has already started to make some fuss but I want to keep my heads on straight looking for sound estimations when one could watch it for real. Here is the most specific piece of information I could find:

Recent advances in gravitational-wave astronomy make the direct detection of gravitational waves from the merger of two stellar-mass compact objects a realistic prospect. Evolutionary scenarios towards mergers of double compact objects generally invoke common envelope evolution which is poorly understood, leading to large uncertainties in merger rates. We explore the alternative scenario of massive overcontact binary (MOB) evolution, which involves two very massive stars in a very tight binary which remain fully mixed due to their tidally induced high spin. We use the public stellar-evolution code MESA to systematically study this channel by means of detailed simulations.We find that, at low metallicity, MOBs produce double-black-hole (BH+BH) systems that will merge within a Hubble time with mass ratios close to one, in two mass ranges, ∼ 25 . . . 60 M and≥130 M , with pair instability supernovae (PISNe) being produced in-between. Our models are also able to reproduce counterparts of various stages in the MOB scenario in the local Universe, providing direct support for it. We map the initial parameter space that produces BH+BH mergers, determine the expected chirp mass distribution, merger times, Kerr parameters and predict event rates.We typically find that for [metallicity] Z<Z_{☉}/10, there is one BH+BH merger for ∼1,000 core-collapse supernovae. The advanced LIGO (aLIGO) detection rate is more uncertain and depends on the metallicity evolution. Deriving upper and lower limits from a local and a global approximation for the metallicity distribution of massive stars, we estimate aLIGO detection rates (at design limit) of ∼ 19 − 550 yr^{-1}for BH+BH mergers below the PISN gap and of ∼ 2.1 − 370 yr^{-1}above the PISN gap[the first quoted number of the ranges corresponds to a local approximation, the second one to a global approximation to the metallicity distribution].Even with conservative assumptions, we find that aLIGO should soon detect BH+BH mergers from the MOB scenario and that these could be the dominant source for aLIGO detections.

The standard formation scenario of BH+BH binaries involves a number of highly uncertain aspects of binary interactions (Figure B.1). The main uncertainties include, in particular, the treatment of common-envelope (CE) evolution (Ivanova et al. 2013) and the efficiency of accretion and spin-up from mass transfer.These lead to uncertainties in the expected merger rates of several orders of magnitude (Abadie et al. 2010). In contrast, the MOB scenario presented here relies mostly on reasonably well understood physics of the evolution of massive stars [Figure 2], although there are still significant uncertainties in the treatment of, e.g., stellar winds (Langer 2012), rotational mixing (Maeder & Meynet 2012) and the BH formation itself (Heger et al. 2003; Ugliano et al. 2012; Pejcha & Thompson 2015).

...

Even for the ongoing first science run (O1) of aLIGO the prospects for detection should be promising. Given that the sensitivity of aLIGO is currently about 1/3 of the design sensitivity, the expected detection rate is ∼4 % of our calculated values.

...

In fact, the most massive mergers, which probe a large fraction of the whole Universe, could well be the dominant source for aLIGO detections (Abadie et al. 2010). Another factor that helps the detection of BH+BH mergers from low-metallicity populations, that is not taken into account in the above estimates, is that, because of the possibly long merger delay times (see Figure 4), even systems which were formed at rather high redshift may merge at low redshift(cf., Mandel & de Mink 2016).

A remaining caveat of concern for our estimated detection rates is related to the relatively low gravitational-wave frequencies of the more massive BH+BH binaries. The emitted frequencies during in-spiral are expected to peak approximately at the innermost stable circular orbit (ISCO) before the plunge-in phase and the actual merging. However, even determining the ISCO for a merging binary system it is non-trivial and depends, for example, on the spins of the BHs (Balmelli & Damour 2015), which requires numerical or sophisticated (semi-)analytical calculations within general relativity and cannot simply be estimated using a test particle in a Kerr field. For the BH+BH mergers above the PISN gap, the emitted frequencies are most likely ≤ 100 Hz, and with redshift corrections the frequencies to be detected are easily smaller by a factor of two or more. Such a low frequency is close to the (seismic noise) edge of the detection window of aLIGO. However, the waveform amplitudes of the more massive BH+BH binaries are larger (for a given distance) and are also enlarged further by a factor of (1+z). Finally, it may be possible that higher frequency signals from the ringdown of the single, rapidly spinning BH produced could be detectable, despite their expected smaller wave amplitudes. An important question to address is whether the first generation of LIGO should have detected such massive BH+BH merger events.Given that the sensitivity of the first generation of LIGO was about 10 times lower, the number of detections should have been 1000 times lower. Therefore even for our upper limits, it is not surprising that there have been no detections during the previous science runs of the first generation LIGO detectors (Abadie et al. 2012).

(Submitted on 14 Jan 2016)

**A Gargantua's mouthful of quanta**

Time will tell if the former estimations were correct... Now assuming that the direct detection of gravitational waves is achieved sooner or later then comes the next natural question that fits better the editorial line of this blog : is a quantum of gravitational wave detectable?

In the LIGO experiment, if it is successful, we shall detect a classical gravitational wave, not an individual quantum of gravity. A classical wave may be considered to be a coherent superposition of a large number of gravitons. LIGO is supposed to detect a wave with a strain amplitude f of the order of10^{-21}. According to [Landau and Lifshitz, 1975], page 370, the energy density of this wave is

E = (c^{2}/32πG)ω^{2}f^{2},(2)

where G is Newton’s constant of gravitation and ω is the angular frequency. For a wave with angular frequency 1 Kilohertz and amplitude 10^{-21}, Eq. (2) gives an energy density of roughly 10^{-10}ergs per cubic centimeter. A single graviton of a given angular frequency ω cannot be confined within a region with linear dimension smaller than the reduced wavelength (c/ω). Therefore the energy density of a single graviton of this frequency is at most equal to the energy of the graviton divided by the cube of its reduced wave-length, namely

Es = (ℏω^{4}/c^{3}). (3)

For an angular frequency of 1 Kilohertz, the single graviton energy density is at most 3.10...^{-47}ergs per cubic centimeter. So any gravitational wave detectable by LIGO must contain at least 3.10^{37}gravitons

Poincare Prize Lecture International Congress of Mathematical Physics Aalborg, Denmark, August 6, 2012Freeman Dyson, Institute for Advanced Study, Princeton, New Jersey

**Undetected gravitons have you a mass?**

NonCommutative Geometry (NCG) is a natural extension to our familiar notions of Riemannian geometry, that has the additional benefit of producing the action of all the Standard Model fields in addition to gravity terms, purely through geometrical considerations. Thus NCG treats both gravity and matter on an equal footing and provides us with concrete relationships between matter and gravitational couplings.The gravitational sector of (the asymptotic expansion of) this theory produces modifications to General Relativity and in this paper we explore the ramifications of these modifications on the formation and evolution of gravitational waves.

We have shown that the theory contains both massive and massless gravitons and that the requirement that the mass of these gravitons be positive fixes the sign of one of the couplings in the theory(for a given choice of sign conventions). We also show that both these modes are sourced by the quadrupole moment of a system (just as in standard GR) and that the retarded Green’s function is not restricted to the past light cone of the observer (unlike GR), as one would expect for a system with massive modes. We have explicitly calculated the energy loss for a circular binary system and compared the results to those of standard GR.We have demonstrated that the amplitude of these NCG modifications is suppressed by the distance between the observer and the source of the gravitational waves and hence will typically be small.

Despite the extremely small amplitude of deviations from standard results, we have shown that NCG produces several distinctive features. Firstly, the amplitude of the energy lost by a binary pair can be higher or lower than the expected value, depending on the orbital period of the pair and the distance to the observer. This opens up the possibility that the observed energy loss from such a pair would be seen to oscillate as the binary moves with respect to the Earth. Whilst such effects are likely to be beyond current observational resolution, they allow for an unexpected beat phenomenon, which would be a concrete signature of NCG.

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One can immediately use the results of this paper to examine circular binary systems, in order to constrain the value of β[10]. Similarly, one can include eccentricity which may result in more restrictive constraints on the theory.An alternative avenue would be to use the gravitational wave-forms given here to deduce the consequences for direct gravity wave searches (LIGO, VIRGO, LISA, etc). In particular, to extend these result to the large field regime and look for modifications to the chirp that develops at the end of in-spiral events.

(Submitted on 24 May 2010)

One can find in the literature several binary pulsars for which the rate of change of the orbital frequency is well-known and the predictions of GR agree with the data to a high accuracy. Since the magnitude of the [NCG] deviations from GR must be less than the allowed uncertainty in the data,we are able to constrain [8] β, namely β>7.5×10.^{-13}m^{-1}. (24) This constraint is not too strong but since it is obtained from systems with high orbital frequencies, future observations of rapidly orbiting binaries, relatively close to the Earth, could improve it by many orders of magnitude...It is particularly encouraging that by studying graviational waves propagation we were able to constrain one of the free parameters of the theory, namely the one related to the coupling constants at unification.

(Submitted on 20 Jan 2013 (v1), last revised 22 Jan 2013 (this version, v2))

**Update (on the upper limit of graviton mass :-)**

Warning: This is not a blind-injection

... assuming a modified dispersion relation for gravitational waves [97],our observations constrain the Compton wavelength of the graviton to be λ_{g}>10^{13}km, which could be interpreted as a bound on the graviton mass m_{g}<1.2×10^{-22}eV/c^{2}. This improves on Solar System and binary pulsar bounds [98, 99] by factors of a few and a thousand, respectively, but does not improve on the model dependent bounds derived from the dynamics of Galaxy clusters [100] and weak lensing observations [101].

B. P. Abbott et al.*(LIGO Scientific Collaboration and Virgo Collaboration)(Received 21 January 2016; published 11 February 2016)

The observation of GW150914 has given us the opportunity to perform quantitative tests of the genuinely strong-field dynamics of GR. We investigated the nature of GW150914 by performing a series of tests devised to detect inconsistencies in the predictions of GR.With the exception of the graviton Compton wavelength and the test for the presence of a non-GR polarization, we did not perform any study aimed at constraining parameters that might arise from specific alternative theories to GR [13, 14, 85], such as Einstein- æther theory [97] and dynamical Chern–Simons [98], or from compact-object binaries composed of exotic objects such as boson stars [99] or gravastars [100]. Studies of this kind are not possible yet, since we lack predictions for what the inspiral-merger-ringdown GW signal should look like in those cases. We hope that the observation of GW150914 will boost the development of such models in the near future.

We will attempt to measure more than one damped sinusoid from the data after GW150914’s peak, thus extracting the quasi-normal-modes and inferring the final black-hole’s mass and spin. We will, thus, be able to test the no-hair theorem [65, 66] and the second law of black-hole dynamics [70, 71]. However, signals louder than GW150914 might be needed to achieve these goals. GR predicts the existence of only two transverse polarizations for GWs. In the future, we plan to investigate whether an extended detector network will allow the measurement of non-transverse components [13] in further GW signals.

...The detection of GW150914 ushers in a new era in the field of experimental tests of GR: within the limits set by our sensitivity, all the tests we have performed provided no evidence against the predictions of GR.

(LIGO Scientific Collaboration and Virgo Collaboration)

//last edit 13 Feb

The hypertext link to the blind injection has been changed to a more relevant one.

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