Did the LIGO gravitational waves originate from primordial black holes?

Discussion in 'Astronomy, Exobiology, & Cosmology' started by paddoboy, Oct 27, 2016.

  1. danshawen Valued Senior Member

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    If the principle of equivalence assumption of GR holds (and it probably does), all black hole mergers should fall at the same rate, but the frequency and length of the chirp may be unrelated or related in a more complex manner. I'm sure someone will come up with a way to test this. A sample size more than just two would help.
     
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  3. danshawen Valued Senior Member

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    nanoHertz gravity waves might also be related to events much closer and much weaker. With the right processing, we should be able to observe a GW signal from the orbit of the moon around the Earth, or Jupiter around the Sun. Only tens of Watts, but it's much closer than primordial black hole mergers and the inverse square law evidently does apply.

    These would closely resemble the inspiral GWs they seem to be looking for.
     
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  5. Q-reeus Banned Valued Senior Member

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    History repeating itself. We discussed this way back in at least one earlier thread. If inverse square law is involved you are not talking GW's but merely quasi-static gravitational fields. And actually 'detection' would be via 1/r^3 tidal fields. You live near a beach? Then check out the surf - there's your local 'detector' at work. Detecting tidal gravity owing to moon and sun. (Well ok there is also wind to factor in, but why spoil things with complications?)
    If you say so Dan.

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    Last edited: Oct 31, 2016
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  7. danshawen Valued Senior Member

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    We did discuss this. So, no gravity waves at all are associated with orbiting bodies in our own vicinity? Don't forget, these detectors could even pick up spurious signals from local logging operations before their sensitivity upgrades.

    Why would this instrument not be sensitive enough to detect all of the tides from all of the planets or moons in our own solar system if it can detect a merger of a 50 solar mass black hole a billion light years away, or for that matter, a tectonic shift deep inside of Saturn? No way to distinguish one of these from noise I guess.

    If a gravity wave is produced and no one detects it, does it sill make GW noise? I didn't previously understand that they were combing the nanoHertz frequency range for signals.
     
    Last edited: Oct 31, 2016
  8. Q-reeus Banned Valued Senior Member

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    What we also discussed or rather what I brought up back then was the matter of 'zones'. Just like for an EM Hertzian dipole oscillator: https://en.wikipedia.org/wiki/Dipole_antenna#Hertzian_dipole
    there will be 3 zones to consider - near field, induction zone, and far-field i.e. radiation zone. As a quadrupole radiator, binaries will radiate at double the orbital frequency, but that is small beer difference. Do the sums and convince yourself that for any sun-planet or planet-planet or moon-planet binary system in solar system, it's ALWAYS the case one is in the very near-field zone. How would you ever hope to separate the far tinier radiation field component? Even if some super technology was sensitive enough to pick up the fantastically weak fields?
     
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  9. danshawen Valued Senior Member

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    3,951
    EM is of course very different from gravitational radiation and also in frequency range from mechanical / acoustical vibration.

    But your answer (and also Einstein's) is deeper here I think than you intended. The coupling to the detector is indeed very much like EM, and I was not giving this consideration nearly enough weight, as you already had. Lots of great physics is here to ponder now that we know for certain that we aren't just chasing a flock of wild geese.
     
  10. rpenner Fully Wired Valued Senior Member

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    Assume \(f(t) = a \left( b (t_0 - t) \right) ^{- \frac{3}{8}} \) with a, b having units of \(\textrm{s}^{-1}\),
    The time to go from \(f_0\) to \(f_1 > f_0\) is \(t(f_1) - t(f_0) = a^{ \frac{8}{3}} b^{-1} \left( f_0^{- \frac{8}{3} } - f_1^{- \frac{8}{3} } \right)\). Call this the chirp time.

    Then \(f'(t) = \frac{3 a b}{8} \left( b (t_0 -t) \right) ^{- \frac{11}{8}} \)
    Then \(f(t)^{-\frac{11}{3}} f'(t) = \frac{3}{8} a^{-\frac{8}{3}} b \) is a constant.

    Then \(\frac{c^3}{G} \left( \frac{5}{96} \pi^{- \frac{8}{3}} f(t)^{-\frac{11}{3}} f'(t) \right)^{\frac{3}{5}} = \frac{c^3}{G} \left( \frac{5}{256} \pi^{- \frac{8}{3}} \right)^{\frac{3}{5}} \sqrt[5]{ \frac{b^3}{a^8} }\) has units of mass. Call this the chirp mass, \(M_{\textrm{chirp}}\). At first order this quantity governs GW frequency for inspiral where \(M_{\textrm{chirp}} = \frac{m_1^{\frac{3}{5}}m_2^{\frac{3}{5}} }{ \left( m_1 + m_2 \right)^{\frac{1}{5}}}\).

    We can rewrite our original equation as
    \(f(t) = \frac{1}{8 \pi} \left( \frac{125 c^{15}}{ G^5 M_{\textrm{chirp}}^5} \right)^{\frac{1}{8}} \left(t_0 - t \right)^{- \frac{3}{8}} \).

    So the time to go from \(f_0\) to \(f_1\) is proportional to \(M_{\textrm{chirp}}^{-\frac{5}{3}} \), specifically \( \frac{5 c^5}{256} \sqrt[3]{ \frac{1}{ \pi^8 G^5 M_{\textrm{chirp}}^5 }} \left( f_0^{- \frac{8}{3} } - f_1^{- \frac{8}{3} } \right)\).

    The bigger the mass, the shorter the chirp time, according to https://en.wikipedia.org/wiki/Chirp_mass

    The same result is gotten by taking equation 6 and the leading term from equation 8 for orbital angular frequency from https://arxiv.org/abs/gr-qc/9602024 and correcting by \(\frac{2}{2 \pi}\) to convert to Hertz for GW.
     
    Last edited: Nov 21, 2016
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  11. Q-reeus Banned Valued Senior Member

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    Standing by everything I brought up this thread re GW chirp frequency/period, beginning with #8, then #11 (that was supposed to be a common-sense reality check), #13, #14, then a simple but physically justified analysis in #16, completely in accord with e.g. Cole Miller's general considerations as per article linked to in #11.

    When starting from assumptions that finishes up with a ludicrous result - predicting that an 'inspiralling' pair of SMBH's, each of perhaps light-days in size, will chirp with a frequency perhaps in the microwave range or even higher, it really, really behooves to start again. I put 'inspiralling' in quotes to emphasize the absurdity - a prediction of chirp frequency rising with scaled system mass inevitably has some 'crisis mass size', beyond which the notion of 'chirp frequency' becomes physically meaningless. To go over again what in #11 was obviously being hinted at. Do it right, and the dilemma vanishes.

    A good thing that even the wildest claims about still hot-topic 'GW astronomy' have nil impact on everyday lives and everyday technology.
     
    Last edited: Nov 21, 2016
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  12. danshawen Valued Senior Member

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    A nice synopsis of formulae used by the GW analysis industry is here:

    http://www.physics.usu.edu/Wheeler/GenRel2013/Notes/GravitationalWaves.pdf

    And an excerpt for circular orbits:

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    Which in the current astrophysical lack of direct corroborative observation, is also used to determine GW luminosity, as well as luminosity distances.

    And don't forget that we will eventually have astrophysical GW corroboration when the Hulse-Taylor binary completely collapses. Thanks to this work, whenever that happens, we will now be ready for it.
     
  13. Q-reeus Banned Valued Senior Member

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    Dan, you might notice the annoying feature in that HUGE FONT formula you have so kindly reproduced, is that frequency and mass are mixed up in the same expression. What do you imagine will happen when one reduces it down to a simple relation between f (and/or f') on one side, and mass on the other (as per my #16, radius is taken care of by mass for final inspiral scenario)? Particularly if one scales by imposing a fixed ratio of M1 to M2.
     
    Last edited: Nov 21, 2016
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  14. danshawen Valued Senior Member

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    3,951

    I'm accustomed to seeing frequency related to mass through Planck's constant and energy relationships. Nothing untoward there. It is of course waiting for a single physically observable corroboration of a GW detection to bolster our confidence in taking General Relativity's predictions of quadrupole gravitational wave emanations to the limit of our instrument's capability. Lots of confounding factors are sure to crop up until or unless that happens.

    I'm willing to suspend any doubts for now and predict that if this is real, that it will evidence in abundance sooner than anyone thinks. Someone needs to digitally sample the belch of a pair of binary BH's and compose a sequel to Tomita the planets or something. This drama needs to be set to the music of the information EH.
     
    Last edited: Nov 21, 2016
  15. Q-reeus Banned Valued Senior Member

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    Err...ok, I guess sort of. Somehow the point I was trying to make has got lost there, but not to worry.
     
  16. rpenner Fully Wired Valued Senior Member

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    Since the relation is \( M_{\textrm{chirp}} \propto f^{-11/5} f'^{3/5} \) or \(f' \propto M_{\textrm{chirp}}^{5/3} f^{11/3} \) or \(f \propto M_{\textrm{chirp}}^{-5/11} f'^{3/11} \), I'm not sure what you want here. Radius isn't an observable, is it?

    Dan's source agrees with my own, as the first order relation between frequency and time before coalescence is a power law.
    Since \( t_0 - t = (5c^5/256) (\pi^8 G^5 M_{\textrm{chirp}}^5)^{-1/3} f^{-8/3} \) and \( f' = (3/8) 1/(8 \pi) (125 c^{15} / ( G^5 M_{\textrm{chirp}}^5 ) )^{1/8} (t_0 - t)^{-11/8} \) it follows that \(f' = (96/5) \pi^{8/3} c^{-5} G^{5/3} M_{\textrm{chirp}}^{5/3} f^{11/3} = (96/5) c^3 f / ( G M_{\textrm{chirp}} ) (\pi G M_{\textrm{chirp}} f / c^3 )^{8/3} \)

    But as I calculated in #27, the time it takes for the frequency to sweep through the bandwidth of a particular detector, where \(f_0\) and \(f_1\) are fixed, varies in proportion to \(M_{\textrm{chirp}}^{-5/3}\) which should settle one argument. That's the technical definition of “The bigger the mass, the shorter the chirp time” because we are speaking of the observable chirp time given a certain detector.
     
    Last edited: Nov 21, 2016
  17. quantum_wave Contemplating the "as yet" unknown Valued Senior Member

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    In regard to the first two posts of the thread, the following comes to mind. At the beginning of a cosmological model, observations and the premises of the model go hand in hand. Once a model gets accepted, new observations can either be seen to support the model, or detract from it. In practice, both cases may come about.

    Some will be quick to say, look at the new observation; it falsifies the model. Others will add new physics to expand the model to fit the new observations. In the minds of the observers, either the old model survives in its revised form, or a new model emerges with new premises. You still have to same universe, and our modeling of it will continue to unfold as science advances.

    In the case of gravitational waves; they are found, they are real. Every model of cosmology must explain them, and there will be changes to the models as the most reasonable and responsible explanations evolve. The laymen among us are in a perpetual Big Wait mode while the scientific professionals sort it out for us

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    .
     
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  18. Q-reeus Banned Valued Senior Member

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    That link in #11 to article by Cole Miller again: https://www.astro.umd.edu/~miller/teaching/astr498/lecture25.pdf
    The 2nd and 3rd paragraphs on p1 cover the essentials. There can be no questioning the fundamental result fmax ~ 1/M for merging BH binaries, with M1/M2 fixed, and M = M1 + M2 (or any such formula e.g. chirp mass) the sole parameter varied.
    I easily found in #16 that, at a given inspiral relative separation r/r_s, fractional mass loss to GW's per orbital cycle is independent of M. Hence scale invariant. Hence the number of inspiral cycles between any two relative GW amplitude markers is scale invariant. The chirp frequency pattern is scale invariant wrt M.
    The only modulation is thus absolute frequency at any point in the chirp profile, which modulation is f ~ 1/M. Thus f'/f ~ 1/M. Conversely, period T ~ M. Thus T'/T ~ M.

    Therefore, the inspiral chirp elapsed time between any two frequencies f0 and f1 (for a given fractional difference δf/f), or alternately any two GW relative amplitudes, will be directly proportional to M. Recalling we are scaling wrt a system where all other parameters such as binary mass ratios, orbital eccentricity, relative spins, are fixed, and only net system mass M varies. Just how a bizarre inversion of such basic results can apply to aLIGO detection window is quite beyond me.
     
    Last edited: Nov 21, 2016
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  19. Q-reeus Banned Valued Senior Member

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    In #35 where it reads:
    "Thus f'/f ~ 1/M. Conversely, period T ~ M. Thus T'/T ~ M."
    , that should have been
    "Thus f' = [f(t+dt)-f(t)]/dt ~ 1/M. Conversely, period T ~ M. Thus T' = [T(t+dt)-T(t)]/dt ~ 1/M."

    I will remind that apart from the baffling chirp time issue, which is supposed to be simply THE interval between two chirp frequencies that are fractionally separated, there is the claim here:
    http://www2.physics.umd.edu/~pshawhan/gw/Shawhan_UMD_Feb2016.pdf p24
    (and similar contradictory statements in http://www.sr.bham.ac.uk/gwastro/what-are-we-looking-for/2 as per #13, #14)
    that chirp frequency increases with system mass M. Physically, both frequency f and it's time rate of change f' for BH binary inspiral must be inversely proportional to M at the same stage of chirp profile, all else being equal.
     
  20. quantum_wave Contemplating the "as yet" unknown Valued Senior Member

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    Chirp analysis could sure be interesting. Should we be able to determine the angle of the plane of the inswirling event relative to the observation point? If the event was observed perpendicular to the plane of the swirling, would there be any chirping?
     
  21. rpenner Fully Wired Valued Senior Member

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    Our sources agree. From your sources for \(f\) and \(R'\) we can compute \(f'\) and how it relates to chirp mass.

    \(f \propto M^{1/2} R^{-3/2}\) p1, right before eqn. 1
    \(R' \propto \mu M^2 R^{-3} \) p2, eqn. 4
    \(f' \propto M^{1/2} R^{-5/2} R' = \mu M^{5/2} R^{-11/2} \) chain rule
    \(f^{-11/5} f'^{3/5} \propto \mu^{3/5} M^{2/5} = \frac{m_1 ^{3/5} m_2 ^{3/5}}{\left( m_1 + m_2 \right)^{3/5}} \left( m_1 + m_2 \right)^{2/5} = \frac{m_1 ^{3/5} m_2 ^{3/5}}{\left( m_1 + m_2 \right)^{1/5}} = M_{\textrm{chirp}}\)


    If \(M_{\textrm{chirp}} = \mu^{3/5} M^{2/5}\) is known to us, what remains of the information that can be squeezed from \(f\) and \(f'\) ? Given just f and f' the only physical quantity one can compute from those observables is the chirp mass. R, M and µ remain unknown to us. But you can extract this quantity: \( f^{-5} f' \propto \mu R^2 \).

    R is not known (at this level of modeling), but since R decays with time, we can compute that time dependence, because \(R' R^3\) is a constant, so the problem of knowing what is R gets replaced by the more tractable problem of figuring out how long before coalescence.

    Starting from \(R' \propto \mu M^2 R^{-3} \) we have a differential equation with solution \(R(t) \propto \mu^{1/4} M^{1/2} \left( t_0 - t \right)^{1/4}\), \(R'(t) \propto \mu^{1/4} M^{1/2} \left( t_0 - t \right)^{-3/4} \)
    with \(f(t) \propto \mu^{-3/8} M^{-2/8} \left( t_0 - t \right)^{-3/8} = M_{\textrm{chirp}}^{-5/8} \left( t_0 - t \right)^{-3/8}\) and \(f'(t) \propto \mu^{-3/8} M^{-2/8} \left( t_0 - t \right)^{-11/8} = M_{\textrm{chirp}}^{-5/8} \left( t_0 - t \right)^{-11/8}\) following by substitution.

    But is that fair? We never get to choose (or know a priori) M, µ or R, but we can know f and f' when f is in the range of our detectors, say between \(f_0\) and \(f_1\).
     
    Last edited: Nov 22, 2016
  22. Q-reeus Banned Valued Senior Member

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    How to sort this out? To reinforce my point that merger waveforms & inspiral trajectories are scale invariant wrt M, one need only examine the plots in the various figures shown here:
    https://arxiv.org/abs/gr-qc/0602026v2
    Notice the merger waveforms all have timescales shown as t/M. Similarly, the inspiral plots in fig.4 have separations as x/M, y/M. All completely consistent with f, f' at any given 'universal' inspiral stage R/R_s being inversely proportional to M. Directly implying |t(f1)-t(f0)| ~ M, when spanning a frequency band at any given inspiral stage.

    What can then be drastically skewing that fundamental merger physics relation re detection? Obviously aLIGO and similar are limited by a GW amplitude detection threshold. Which selects for the higher frequencies at the final chirp, merger, and ringdown stages. But that will be true regardless of the binary system mass M, owing to the universality of the merger waveform. A glance at the log-log plot of GW strain sensitivity vs frequency, fig. 1A, p25 here: https://arxiv.org/pdf/1408.0740
    shows that all such detectors have an overall similar and rather complex profiles. Which for any given detector will yield markedly different amplitude-frequency response depending just where on it's frequency curve band a detection is made.
    Do any of the mixed parameter expressions such as that given in #29 incorporate such detector frequency-amplitude sensitivities? I don't think so.
     
  23. rpenner Fully Wired Valued Senior Member

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    From \(f \propto M^{1/2} R^{-3/2}\) if we make (by choice of units) \(R \propto M\) then we get \(f \propto M^{-1}\) which is to say there is an inherent scale invariance in Newtonian Gravity and low order General Relativity. This simulation adopts that, but then tweaks it a bit in analysis of the ringdown as ~ 3.5% of the mass is radiating away from the system, so the last orbit fits better if \(M_{\textrm{final}} \times \omega_{\textrm{GW}}\) is constant. So by measuring time in units of \(G M / c^3\) and space in units of \(G M / c^2\), they agree with the analysis. Figure 11 shows a good correspondence between analytic GR and simulation for \(-400 M_f < t < -50 M_f\), with the analytic solutions as good or superior at early times (\(t < -100 M_f \) for 1.5PN and \(t < -50 M_f\) for 2PN).

    Since for these simulations \(\mu = \frac{1}{4} M\), when we look at my expression in light of our space units we have \(f' \propto \mu M^{5/2} R^{-11/2} \propto M^{-2}\) which is exactly the the same scale independence for our choice of time units.

    And when we look at my first-order analytic solution \(f \propto M_{\textrm{chirp}}^{-5/8} ( t_0 - t)^{-3/8} \propto M^{-5/8} M^{-3/8} \propto M^{-1}\).

    So if \(R \propto M\) then \(f \propto M^{-1}\) but \(f' \propto M^{-2}\).


    That does not follow. In a dynamical system, \(R \propto M\) is distinct from saying \(R = M\). So if you introduce the constant of proportionality: \(R = r M\) and fix \(\mu = M/4\) then \(f \propto M^{1/2} R^{-3/2}\) becomes \(f \propto M^{-1} r^{-3/2}\), \(R' \propto \mu M^2 R^{-3}\) becomes \(r' M \propto r^{-3}\). Now if \(r' \propto M^{-1} r^{-3}\) that has solution \(r(t) \propto M^{-1/4} (t_0 - t)^{1/4}\) so \(f(t) \propto M^{-5/8} (t_0 - t)^{-3/8} \) and if we replace \(t = s M\) we get \(f(s) \propto M^{-1} (s_0 - s)^{-3/8} \) which implies \( t(M f_1) - t(M f_0) \propto M\).
     

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