How Loud was the Big Bang?

The other day I was giving a talk about cosmology at Cardiff University’s Open Day for prospective students. I was talking, as I usually do on such occasions, about the cosmic microwave background, what we have learnt from it so far and what we hope to find out from it from future experiments, assuming they’re not all cancelled.

Quite a few members of staff listened to the talk too and, afterwards, some of them expressed surprise at what I’d been saying, so I thought it would be fun to try to explain it on here in case anyone else finds it interesting.

As you probably know the Big Bang theory involves the assumption that the entire Universe – not only the matter and energy but also space-time itself – had its origins in a single event a finite time in the past and it has been expanding ever since. The earliest mathematical models of what we now call the  Big Bang were derived independently by Alexander Friedman and George Lemaître in the 1920s. The term “Big Bang” was later coined by Fred Hoyle as a derogatory description of an idea he couldn’t stomach, but the phrase caught on. Strictly speaking, though, the Big Bang was a misnomer.

Friedman and Lemaître had made mathematical models of universes that obeyed the Cosmological Principle, i.e. in which the matter was distributed in a completely uniform manner throughout space. Sound consists of oscillating fluctuations in the pressure and density of the medium through which it travels. These are longitudinal “acoustic” waves that involve successive compressions and rarefactions of matter, in other words departures from the purely homogeneous state required by the Cosmological Principle. The Friedman-Lemaitre models contained no sound waves so they did not really describe a Big Bang at all, let alone how loud it was.

However, as I have blogged about before, newer versions of the Big Bang theory do contain a mechanism for generating sound waves in the early Universe and, even more importantly, these waves have now been detected and their properties measured.

The above image shows the variations in temperature of the cosmic microwave background as charted by the Wilkinson Microwave Anisotropy Probe about five years ago. The average temperature of the sky is about 2.73 K but there are variations across the sky that have an rms value of about 0.08 milliKelvin. This corresponds to a fractional variation of a few parts in a hundred thousand relative to the mean temperature. It doesn’t sound like much, but this is evidence for the existence of primordial acoustic waves and therefore of a Big Bang with a genuine “Bang” to it.

A full description of what causes these temperature fluctuations would be very complicated but, roughly speaking, the variation in temperature you corresponds directly to variations in density and pressure arising from sound waves.

So how loud was it?

The waves we are dealing with have wavelengths up to about 200,000 light years and the human ear can only actually hear sound waves with wavelengths up to about 17 metres. In any case the Universe was far too hot and dense for there to have been anyone around listening to the cacophony at the time. In some sense, therefore, it wouldn’t have been loud at all because our ears can’t have heard anything.

Setting aside these rather pedantic objections – I’m never one to allow dull realism to get in the way of a good story- we can get a reasonable value for the loudness in terms of the familiar language of decibels. This defines the level of sound (L) logarithmically in terms of the rms pressure level of the sound wave Prms relative to some reference pressure level Pref

L=20 log10[Prms/Pref]

(the 20 appears because of the fact that the energy carried goes as the square of the amplitude of the wave; in terms of energy there would be a factor 10).

There is no absolute scale for loudness because this expression involves the specification of the reference pressure. We have to set this level by analogy with everyday experience. For sound waves in air this is taken to be about 20 microPascals, or about 2×10-10 times the ambient atmospheric air pressure which is about 100,000 Pa.  This reference is chosen because the limit of audibility for most people corresponds to pressure variations of this order and these consequently have L=0 dB. It seems reasonable to set the reference pressure of the early Universe to be about the same fraction of the ambient pressure then, i.e.

Pref~2×10-10 Pamb

The physics of how primordial variations in pressure translate into observed fluctuations in the CMB temperature is quite complicated, and the actual sound of the Big Bang contains a mixture of wavelengths with slightly different amplitudes so it all gets a bit messy if you want to do it exactly, but it’s quite easy to get a rough estimate. We simply take the rms pressure variation to be the same fraction of ambient pressure as the averaged temperature variation are compared to the average CMB temperature,  i.e.

Prms~ a few ×10-5Pamb

If we do this, scaling both pressures in logarithm in the equation in proportion to the ambient pressure, the ambient pressure cancels out in the ratio, which turns out to be a few times 10-5

With our definition of the decibel level we find that waves corresponding to variations of one part in a hundred thousand of the reference level  give roughly L=100dB while part in ten thousand gives about L=120dB. The sound of the Big Bang therefore peaks at levels just a bit less than  120 dB. As you can see in the Figure to the left, this is close to the threshold of pain,  but it’s perhaps not as loud as you might have guessed in response to the initial question. Many rock concerts are actually louder than the Big Bang.

 

A useful yardstick is the amplitude  at which the fluctuations in pressure are comparable to the mean pressure. This would give a factor of about 1010 in the logarithm and is pretty much the limit that sound waves can propagate without distortion. These would have L≈190 dB. It is estimated that the 1883 Krakatoa eruption produced a sound level of about 180 dB at a range of 100 miles. By comparison the Big Bang was little more than a whimper.

PS. If you would like to read more about the actual sound of the Big Bang, have a look at John Cramer’s webpages. You can also download simulations of the actual sound. If you listen to them you will hear that it’s more of  a “Roar” than a “Bang” because the sound waves don’t actually originate at a single well-defined event but are excited incoherently all over the Universe.

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8 Responses to “How Loud was the Big Bang?”

  1. Anton Garrett Says:

    Although Fred Hoyle was no friend of Big Bang cosmology, his coining of the phrase was not derisory but merely an example of his economy of description, as he later made clear; see footnote 1 of the current Wikipedia entry for “Big Bang”.

    John Cramer, like Fred Hoyle, has written science fiction novels and is good at proposing striking scietific ideas; he came up with something called the transactional interpretation of quantum mechanics in the 1970s, in which the wavefunction Psi travels forward in time and the wavefunction Psi* therefore travels backward in time (under time conjugation). They meet and ‘handshake’ whenever an observation is done, which involves \int Psi* (OPERATOR) Psi. I regard this idea as an heroic failure, but it is honest about acausality, which we have no choice but to accept in delayed-choice experiments and in certain Bell-type experiments with appropriate spacetime separation between the observations on the two particles.

    Anton

    • telescoper Says:

      Anton,
      I’m aware that Fred Hoyle later claimed the didn’t mean the term to be pejorative but I’m not sure he wasn’t protesting too much. Professor Roger Tayler, who worked with Fred on a classic paper about Helium production in the Big Bang was in no doubt that Hoyle (who was never the most tactful person) had intended the phrase to be derogatory. We’ll never really know.

  2. Anton Garrett Says:

    Peter,

    This is guesswork as you say, but Hoyle would have seen his description of the theory he was against taken up by its advocates, and the temptation to let people think he meant it derogatorily would consequently have been great. He knew how to make enemies but I do not believe he was a liar, so that provided the report of his explicit denial is accurate then I believe it.

    Hoyle’s autobiography “Home is where the wind blows” was more entertaining reading than some of his novels, principally for the rows he got into…

    Anton

  3. telescoper Says:

    Yes, I read Home is where the wind blows some time ago. In fact, I think I reviewed it for some journal or other. Many of Hoyle’s disputes seem astonishingly petty for someone so obviously intelligent. I met him a few times and I have to say he was as cantankerous as I’d been led to believe.

  4. Mark Whittle also made a fascinating audio representation of the early universe acoustic fluctuations, compressed to 10 seconds, shifted up by 50 octaves and played at constant volume: http://www.astro.virginia.edu/~dmw8f/BBA_web/index_frames.html (click on Introduction & Overview to shortcut to the sound file). It sounds a little different to John Cramer’s earlier rendering for reasons that elude me.

  5. [...] Hur starkt lät Big Bang? L=20 log10[Prms/Pref] .¨- läs mer. [...]

  6. [...] there’s the Big Bang. Well, I don’t need to go on about that because I’ve already posted about it. Suffice to say that the Big Bang wasn’t anywhere near as Big as you’ve been led to [...]

    • John B. Dodson Says:

      While there may have been resonating, or resiliant “sound” microseconds after the “bang,” there could have been absolutely no sound at all the instant of the “bang” because there were no molecules to disturb, nor were there receptors to vibrate. The old question; “if a tree falls in the forest and no one is around to hear it (or electronic receptors, mics, etc.) would it make a sound?” The answer is an unqualified NO! I insist the Big “Bang” was eerily silent at the instant of explosion. Afterward? Who knows?
      John Dodson

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