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EigenState

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Message 6377 - Posted: 27 Jun 2008, 17:14:29 UTC

In “The Cosmic Symphony” by Wanye Hu and Martin White (Scientific American, February 2004, p. 44ff), the authors state:

Furthermore, quantum fluctuations in the inflaton field, magnified by the rapid expansion, provide initial disturbances that are approximately equal on all scales—that is, the disturbances to small regions have the same magnitude as those affecting large regions. These disturbances become fluctuations in the energy density from place to place in the primordial plasma.

Evidence supporting the theory of inflation has now been found in the detailed pattern of sound waves in the CMB. Because inflation produced the density disturbances all at once in essentially the first moment of creation, the phases of all the sound waves were synchronized.


I understand quantum fluctuations and my interpretation of “magnified” is that inflation increased the amplitudes of the initial quantum fluctuations. It is also clear that for the observed harmonics to occur, the initial quantum fluctuations must have been effectively phase coherent.

Several questions remain: Why did the initial quantum fluctuations have to be phase coherent? Why did inflation “magnify” these quantum fluctuations? Why must the initial density disturbances have been approximately equal on all scales?
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Profile Benjamin Wandelt
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Message 6613 - Posted: 18 Jul 2008, 3:06:10 UTC - in response to Message 6377.  


I understand quantum fluctuations and my interpretation of “magnified” is that inflation increased the amplitudes of the initial quantum fluctuations.


Actually, the \"magnified\" refers to stretching the quantum fluctuations in length scale, not increasing it in amplitude.

It is also clear that for the observed harmonics to occur, the initial quantum fluctuations must have been effectively phase coherent.


Good job - this is actually a fairly subtle point...

Several questions remain: Why did the initial quantum fluctuations have to be phase coherent?


I\'ll try to explain this subtle point without going into too much detail:

Actually the initial perturbations do not need to be phase coherent - after a perturbations is stretched beyond the scale where microphysics can affect it, the evolution of this perturbation is very simple and the same for perturbations on all scales larger than a critical scale (the \"horizon\" scale). This \"super-horizon\" evolution phase-focuses the random perturbations. After inflation the horizon grows (because the Universe expands slowly enough, that light and gravitational interactions can outrun the expansion). Because perturbations that are outside the horizon are phase-focused, it means that they all enter into the horizon in a very regular way and begin oscillating under the relativistic photon pressure that affects all sub-horizon evolution.

So phase-focusing on super-horizon scales leads to this very orderly progression of oscillating fluctuations - this prevents the osciallations from being washed out in the power spectrum of the cosmic microwave background fluctuations.

Why did inflation “magnify” these quantum fluctuations?


See above - inflation stretches spaces, so the fluctuations get stretched, too.


Why must the initial density disturbances have been approximately equal on all scales?


This is the prediction of all standard inflation models. It arises physically from the fact that the horizon (the maximum size over which distant events can communicate with one another) remains constant during inflation.

Just like a black hole horizon radiates Hawking radiation (quantum perturbations) as if it had a certain temperature, the horizon during inflation radiates de Sitter radiation also has a certain temperature. The temperature is directly linked to the size of the horizon (again, analogous to black holes where the temperature depends on the mass of the black holes). This link between horizons and quantum perturbations is something very deep about quantum gravity.

So since the horizon stays the same size during inflation (and the perturbations are stretched outside it) the quantum mechanical temperature remains the same and hence the perturbation amplitude remains the same. Hence, scale-invariant perturbations!

Hope this helped...

Ben

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Message 6615 - Posted: 18 Jul 2008, 4:26:17 UTC

Greetings Ben,

Thank you. It certainly has helped but will take much more thought on my part, which is just fine.

One immediate question however. You mentioned...
...This \"super-horizon\" evolution phase-focuses the random perturbations.

The process of phase-focusing sounds very much like constructive interference to me at the moment. Is that at least close to being a reasonable way to think about it?

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EigenState
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Message 6619 - Posted: 18 Jul 2008, 16:21:21 UTC - in response to Message 6615.  


You mentioned...
...This \"super-horizon\" evolution phase-focuses the random perturbations.

The process of phase-focusing sounds very much like constructive interference to me at the moment. Is that at least close to being a reasonable way to think about it?


Ok, this is slightly more detail - I should say that you are asking a fairly advanced topic, so this is for aficionados... :)

Think of the perturbations as a superposition of plane waves (jargon: \"mode\"). The equations describe the evolution of the amplitude and the phase of these plane waves. For those plane waves, where the wavelength is larger than the horizon size (the \"super-horizon modes\") the evolution equations will force the plane wave to have a particular phase. This phase is the same for all plane waves, hence \"phase-focusing.\"

So later on, after inflation, when the horizon size grows beyond the wavelength of a given plane wave (jargon: \"it comes into the horizon\") the phase is the same for all of those waves and the subsequent evolution (compression and rarefaction due to radiation pressure which effects modes inside the horizon) starts at the same angle in the phase diagram for every plane wave. This regular progression of modes coming into the horizon and starting to oscillate gives rise to the peaks and dips in the cmb power spectrum.

If all the modes started off at random phases (which is what happens in models where the perturbations are due to some active sources, like explosions or topological defects like cosmic string), the modes start oscillating with random phases, and they peak and go through zero and random times. This washes out the peak and dips in the cmb power spectrum.

Hence peaks and dips are a sign of perturbations having been on super-horizon scales at some point and hence evidence for inflation (a paper by Neil Turok giving a counterexample notwithstanding - he would be happy to admit that the counterexample is of mathematical interest but would require unknown physics to actually produce).

All the best,

Ben


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Message 6620 - Posted: 18 Jul 2008, 16:58:43 UTC

Greetings Ben,

Thanks again!

Think of the perturbations as a superposition of plane waves (jargon: \"mode\"). The equations describe the evolution of the amplitude and the phase of these plane waves. For those plane waves, where the wavelength is larger than the horizon size (the \"super-horizon modes\") the evolution equations will force the plane wave to have a particular phase. This phase is the same for all plane waves, hence \"phase-focusing.\"

Superpositions are familiar territory--hardly surprising given my screen name. ;-)

I have to struggle more with the concept that the super-horizon modes are forced to share a specific phase. Thus, I am off to the library so to speak. I did manage to find a paper in Phys Rev Letters by Albrecht, et al that might help.

I really appreciate your taking time to reply.

Best regards,
EigenState
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