Inflationary perturbations redux [Archive]

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d70yxj

Oct11-04, 06:41 PM

Dear Forum, I posted this in sci.phys.research but with no replies. If anyone here can help, much appreciated....

During inflation, as a wavenumber k crosses the horizon, the quantum field
mode associated with momentum k is supposed to "freeze in" as a classical
perturbation, right?

So, it seems like that mode of the field operator collapses into one
particular eigenstate. And, when the mode re-crosses the horizon later on,
it doesn't return to behaving as a quantum fluctuation - so the freezing
seems to be irreversible.

Does this mean that the freezing-in is like a measurement (in the sense of
quantum mechanics), or is there another way to understand it? If it is a
measurement, what does the measuring and when does it happen? If it isn't, then what am I getting wrong, above?

Lubos Motl

Oct12-04, 03:11 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Mon, 11 Oct 2004, d70yxj wrote:\n\n> So, it seems like that mode of the field operator collapses into one\n> particular eigenstate. And, when the mode re-crosses the horizon later on,\n> it doesn\'t return to behaving as a quantum fluctuation - so the freezing\n> seems to be irreversible.\n\nI would like to hear a consistent and comprehensive description of this\nsituation myself. In some sense, it seems likely to me that Redbull\'s\ndescription will be confirmed.\n\nWhen the mode crosses the horizon for the first time, it amounts to some\nsort of measurement that "freezes" the operator and converts it into a\nclassical value. Once it\'s classical, it behaves more or less\ndeterministically and it can\'t "unfreeze", so it\'s not shocking that this\nprocess is irreversible.\n\nI also agree with Redbull that the key process is the "freezing" at the\nbeginning, which is a sort of measurement. Is it legitimate to describe it\nin terms of decoherence - much like other quantum quantities that are\nbecoming classical, and quantum probabilities (that interfere) becoming\nclassical probabilities (that don\'t)?\n\nThe quantum field is in a superposition before its wavelength expands, but\nsome chosen classical values of the field are picked once the wavelength\nreaches the critical point. This seems to mean that the density matrix\ndescribing these modes should rapidly become pseudo-diagonal (in the basis\nof the "classical states") near the critical moment, right?\n\nOtherwise we would face the usual Schrodinger cat questions - why is not\nthe field projected into a superposition of classically distinct states,\nwould not we?\n\nNow, such rapid diagonalization of the density matrix is usually justified\nby intense enough interactions with the environment - the environment\nperforms a long series of "measurements", and only when we trace over the\ndegrees of freedom of the environment - that are uninteresting for us - we\nget a more or less diagonal density matrix (it is diagonal in the basis of\nstates that can "self-reproduce" their information by evolution).\n\nIt seems that in order to justify the measurement, we need to find some\ndegrees of freedom of the "environment" - they are unlikely to be\ngeometrically outside the system because we study a mode of the quantum\nfield that fills the whole Universe. What are they? Well, they should\nprobably be some modes inside the Universe, and moreover, the interaction\nbetween the "freezing mode" and the "environmental degrees of freedom"\nmust be fast enough to make the measurement.\n\nThe other possible scheme to explain the situation would be based on some\nkind of complementarity - the quantum field behind the horizon is not\nquite independent from the quantum field inside the horizon volume, which\nwould modify something and allow the field to decohere, or something like\nthat.\n\nThanks for your comments in advance,\nLubos, http://motls.blogspot.com/\n________________________________________________ ______________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Mon, 11 Oct 2004, d70yxj wrote:

> So, it seems like that mode of the field operator collapses into one
> particular eigenstate. And, when the mode re-crosses the horizon later on,
> it doesn't return to behaving as a quantum fluctuation - so the freezing
> seems to be irreversible.

I would like to hear a consistent and comprehensive description of this
situation myself. In some sense, it seems likely to me that Redbull's
description will be confirmed.

When the mode crosses the horizon for the first time, it amounts to some
sort of measurement that "freezes" the operator and converts it into a
classical value. Once it's classical, it behaves more or less
deterministically and it can't "unfreeze", so it's not shocking that this
process is irreversible.

I also agree with Redbull that the key process is the "freezing" at the
beginning, which is a sort of measurement. Is it legitimate to describe it
in terms of decoherence - much like other quantum quantities that are
becoming classical, and quantum probabilities (that interfere) becoming
classical probabilities (that don't)?

The quantum field is in a superposition before its wavelength expands, but
some chosen classical values of the field are picked once the wavelength
reaches the critical point. This seems to mean that the density matrix
describing these modes should rapidly become pseudo-diagonal (in the basis
of the "classical states") near the critical moment, right?

Otherwise we would face the usual Schrodinger cat questions - why is not
the field projected into a superposition of classically distinct states,
would not we?

Now, such rapid diagonalization of the density matrix is usually justified
by intense enough interactions with the environment - the environment
performs a long series of "measurements", and only when we trace over the
degrees of freedom of the environment - that are uninteresting for us - we
get a more or less diagonal density matrix (it is diagonal in the basis of
states that can "self-reproduce" their information by evolution).

It seems that in order to justify the measurement, we need to find some
degrees of freedom of the "environment" - they are unlikely to be
geometrically outside the system because we study a mode of the quantum
field that fills the whole Universe. What are they? Well, they should
probably be some modes inside the Universe, and moreover, the interaction
between the "freezing mode" and the "environmental degrees of freedom"
must be fast enough to make the measurement.

The other possible scheme to explain the situation would be based on some
kind of complementarity - the quantum field behind the horizon is not
quite independent from the quantum field inside the horizon volume, which
would modify something and allow the field to decohere, or something like
that.

Thanks for your comments in advance,
Lubos, http://motls.blogspot.com/
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Robert C. Helling

Oct13-04, 07:03 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Tue, 12 Oct 2004 16:11:58 -0400, Lubos Motl <motl@feynman.harvard.edu> wrote:\n\n> On Mon, 11 Oct 2004, d70yxj wrote:\n>\n>> So, it seems like that mode of the field operator collapses into one\n>> particular eigenstate. And, when the mode re-crosses the horizon later on,\n>> it doesn\'t return to behaving as a quantum fluctuation - so the freezing\n>> seems to be irreversible.\n>\n> I would like to hear a consistent and comprehensive description of this\n> situation myself.\n\nMe too. I just spent two years in Cambridge with lots of cosmologists\nbut still have no good understanding of this process.\n\n> When the mode crosses the horizon for the first time, it amounts to some\n> sort of measurement that "freezes" the operator and converts it into a\n> classical value. Once it\'s classical, it behaves more or less\n> deterministically and it can\'t "unfreeze", so it\'s not shocking that this\n> process is irreversible.\n>\n> I also agree with Redbull that the key process is the "freezing" at the\n> beginning, which is a sort of measurement. Is it legitimate to describe it\n> in terms of decoherence - much like other quantum quantities that are\n> becoming classical, and quantum probabilities (that interfere) becoming\n> classical probabilities (that don\'t)?\n\nThat I would like to see in more details. Especially since the\ncosmological models are usually very crude and do not contain any\nother degrees of freedom that the fluctuations could dissipate to.\n\nMy impression is rather that this whole "freezing" business is just a\nmetaphor and that there is nothing fancy between quantum and classical\nphysics that can be understood in a deeper way. Maybe it all boils\ndown to the following simple observation (which in a sense is purely\nclassical field theory, or even just mechanics):\n\nYou start with the Lagrangian of a massless scalar\n\nL = phi box phi\n\nand evaluate this in a FRW spacetime\n\nds^2 = -dt + a(t)^2 (dx^2+dy^2+dz^2).\n\nFurthermore you do Fourier transformation in the spacial\ndirections. Then the field equation for the mode of phi with momentum\nk becomes something like\n\nphi_k\'\' + 2 H phi_k\' + k^2 phi_k = 0\n\nwhere the prime is the time derivative and H = a\'/a is the Hubble\nparameter. If H is constant this is a damped harmonic oscillator with\nk playing the role of mass and H being related to the damping.\n\nThen you know from physics 101 that there are two regimes: If the\ndamping is small (H<<k) phi oscillates whereas for large damping\n(H>>k) the motion is over-damped and phi_k slowly creeps back to 0. It\nmight be that cosmologists call the first regime the quantum and the\nover-damped one the classical, frozen regime. Note that 1/H is the size\nof the horizon (or related to it) and 1/k is the wavelength. This\nwould justify the figure of speech that the mode freezes once it\ncrosses the horizon.\n\nRobert\n\n--\n..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO o.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO\nRobert C. Helling School of Science and Engineering\nInternational University Bremen\nprint "Just another Phone: +49 421-200 3574\nstupid .sig\\n"; http://www.aei-potsdam.mpg.de/~helling\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Tue, 12 Oct 2004 16:11:58 -0400, Lubos Motl <motl@feynman.harvard.edu> wrote:

> On Mon, 11 Oct 2004, d70yxj wrote:
>
>> So, it seems like that mode of the field operator collapses into one
>> particular eigenstate. And, when the mode re-crosses the horizon later on,
>> it doesn't return to behaving as a quantum fluctuation - so the freezing
>> seems to be irreversible.
>
> I would like to hear a consistent and comprehensive description of this
> situation myself.

Me too. I just spent two years in Cambridge with lots of cosmologists
but still have no good understanding of this process.

> When the mode crosses the horizon for the first time, it amounts to some
> sort of measurement that "freezes" the operator and converts it into a
> classical value. Once it's classical, it behaves more or less
> deterministically and it can't "unfreeze", so it's not shocking that this
> process is irreversible.
>
> I also agree with Redbull that the key process is the "freezing" at the
> beginning, which is a sort of measurement. Is it legitimate to describe it
> in terms of decoherence - much like other quantum quantities that are
> becoming classical, and quantum probabilities (that interfere) becoming
> classical probabilities (that don't)?

That I would like to see in more details. Especially since the
cosmological models are usually very crude and do not contain any
other degrees of freedom that the fluctuations could dissipate to.

My impression is rather that this whole "freezing" business is just a
metaphor and that there is nothing fancy between quantum and classical
physics that can be understood in a deeper way. Maybe it all boils
down to the following simple observation (which in a sense is purely
classical field theory, or even just mechanics):

You start with the Lagrangian of a massless scalar

L = \phi[/itex] box \phi

and evaluate this in a FRW spacetime

ds^2 = -dt + a(t)^2 (dx^2+dy^2+dz^2).

Furthermore you do Fourier transformation in the spacial
directions. Then the field equation for the mode of \phi with momentum
k becomes something like

[itex]\phi_k'' + 2 H \phi_k' + k^2 \phi_k =

where the prime is the time derivative and H = a'/a is the Hubble
parameter. If H is constant this is a damped harmonic oscillator with
k playing the role of mass and H being related to the damping.

Then you know from physics 101 that there are two regimes: If the
damping is small (H<<k) \phi oscillates whereas for large damping
(H>>k) the motion is over-damped and \phi_k slowly creeps back to . It
might be that cosmologists call the first regime the quantum and the
over-damped one the classical, frozen regime. Note that 1/H is the size
of the horizon (or related to it) and 1/k is the wavelength. This
would justify the figure of speech that the mode freezes once it
crosses the horizon.

Robert

--
..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo. oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO
Robert C. Helling School of Science and Engineering
International University Bremen
print "Just another Phone: +49 421-200 3574
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling

John Gonsowski

Oct13-04, 07:04 AM

Urs Schreiber

Oct14-04, 03:23 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Robert C. Helling" <helling@ariel.physik.hu-berlin.de> schrieb im\nNewsbeitrag news:slrncmq137.dli.helling-100000@localhost.localdomain...\n> On Tue, 12 Oct 2004 16:11:58 -0400, Lubos Motl <motl@feynman.harvard.edu>\n> wrote:\n>\n>> On Mon, 11 Oct 2004, d70yxj wrote:\n>>\n>>> So, it seems like that mode of the field operator collapses into one\n>>> particular eigenstate. And, when the mode re-crosses the horizon later\n>>> on,\n>>> it doesn\'t return to behaving as a quantum fluctuation - so the freezing\n>>> seems to be irreversible.\n>>\n>> I would like to hear a consistent and comprehensive description of this\n>> situation myself.\n>\n> Me too. I just spent two years in Cambridge with lots of cosmologists\n> but still have no good understanding of this process.\n\n\n>> I also agree with Redbull that the key process is the "freezing" at the\n>> beginning, which is a sort of measurement. Is it legitimate to describe\n>> it\n>> in terms of decoherence - much like other quantum quantities that are\n>> becoming classical, and quantum probabilities (that interfere) becoming\n>> classical probabilities (that don\'t)?\n>\n> That I would like to see in more details. Especially since the\n> cosmological models are usually very crude and do not contain any\n> other degrees of freedom that the fluctuations could dissipate to.\n\nHi Robert -\n\nI like your explanation concerning the damped harmonic oscillator dynamics\nof the classical modes and it looks as if this really is what most\ncosmologists have in mind when talking about freezing.\n\nBut in the golden age of quantum cosmology about 10 years ago there also\nhave been lots of studies of decoherence effects in cosmology using\n"midi-superspace" instead of mini-superspace, which means (as you will know\nbut let me say it for completeness) that not just a finite number of degrees\nof freedom of GR are retained, but an infinite number of modes of their\npertrubations.\n\nI don\'t have a good list of literature at hand right now, but chasing\ncitations of Claus Kiefer\'s work should turn up everything of relevance. For\ninstance there is\n\nC. Kiefer:\nQuantum cosmology and the emergence of a classical world\ngr-qc/9308025\n\nwith the abstract:\n\n"This paper gives an elementary introduction to some of the conceptual\nproblems of quantum cosmology. Contents: 1. Why quantum cosmology? 2. Time\nin quantum gravity 3.Decoherence and the recovery of the Schrodinger\nequation 4. The direction of time "\n\nthen\n\nC. Kiefer, D. Polarski, A.A. Starobinsky:\nQuantum-to-classical transition for fluctuations in the early Universe\ngr-qc/9802003\n\nwith the abstract\n\n"According to the inflationary scenario for the very early Universe, all\ninhomogeneities in the Universe are of genuine quantum origin. On the other\nhand, looking at these inhomogeneities and measuring them, clearly no\nspecific quantum mechanical properties are observed. We show how the\ntransition from their inherent quantum gravitational nature to classical\nbehaviour comes about -- a transition whereby none of the successful\nquantitative predictions of the inflationary scenario for the present-day\nuniverse is changed. This is made possible by two properties. First, the\nquantum state for the spacetime metric perturbations produced by quantum\ngravitational effects in the early Universe becomes very special (highly\nsqueezed) as a result of the expansion of the Universe (as long as the\nwavelength of the perturbations exceeds the Hubble radius). Second,\ndecoherence through the environment distinguishes the field amplitude basis\nas being the pointer basis. This renders the perturbations presently\nindistinguishable from stochastic classical inhomogeneities. "\n\nSimilarly:\n\nC. Kiefer, D. Polarski:\nEmergence of classicality for primordial fluctuations: Concepts and\nanalogies\ngr-qc/9805014\n\nand\n\nClaus Kiefer, Julien Lesgourgues, David Polarski, Alexei A. Starobinsky:\nThe Coherence of Primordial Fluctuations Produced During Inflation\ngr-qc/9806066 .\n\nThere is probably much more literature than that.\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Robert C. Helling" <helling@ariel.physik.hu-berlin.de> schrieb im
Newsbeitrag news:slrncmq137.dli.helling-100000@localhost.localdomain...
> On Tue, 12 Oct 2004 16:11:58 -0400, Lubos Motl <motl@feynman.harvard.edu>
> wrote:
>
>> On Mon, 11 Oct 2004, d70yxj wrote:
>>
>>> So, it seems like that mode of the field operator collapses into one
>>> particular eigenstate. And, when the mode re-crosses the horizon later
>>> on,
>>> it doesn't return to behaving as a quantum fluctuation - so the freezing
>>> seems to be irreversible.
>>
>> I would like to hear a consistent and comprehensive description of this
>> situation myself.
>
> Me too. I just spent two years in Cambridge with lots of cosmologists
> but still have no good understanding of this process.

>> I also agree with Redbull that the key process is the "freezing" at the
>> beginning, which is a sort of measurement. Is it legitimate to describe
>> it
>> in terms of decoherence - much like other quantum quantities that are
>> becoming classical, and quantum probabilities (that interfere) becoming
>> classical probabilities (that don't)?
>
> That I would like to see in more details. Especially since the
> cosmological models are usually very crude and do not contain any
> other degrees of freedom that the fluctuations could dissipate to.

Hi Robert -

I like your explanation concerning the damped harmonic oscillator dynamics
of the classical modes and it looks as if this really is what most
cosmologists have in mind when talking about freezing.

But in the golden age of quantum cosmology about 10 years ago there also
have been lots of studies of decoherence effects in cosmology using
"midi-superspace" instead of mini-superspace, which means (as you will know
but let me say it for completeness) that not just a finite number of degrees
of freedom of GR are retained, but an infinite number of modes of their
pertrubations.

I don't have a good list of literature at hand right now, but chasing
citations of Claus Kiefer's work should turn up everything of relevance. For
instance there is

C. Kiefer:
Quantum cosmology and the emergence of a classical world
http://www.arxiv.org/abs/gr-qc/9308025

with the abstract:

"This paper gives an elementary introduction to some of the conceptual
problems of quantum cosmology. Contents: 1. Why quantum cosmology? 2. Time
in quantum gravity 3.Decoherence and the recovery of the Schrodinger
equation 4. The direction of time "

then

C. Kiefer, D. Polarski, A.A. Starobinsky:
Quantum-to-classical transition for fluctuations in the early Universe
http://www.arxiv.org/abs/gr-qc/9802003

with the abstract

"According to the inflationary scenario for the very early Universe, all
inhomogeneities in the Universe are of genuine quantum origin. On the other
hand, looking at these inhomogeneities and measuring them, clearly no
specific quantum mechanical properties are observed. We show how the
transition from their inherent quantum gravitational nature to classical
behaviour comes about -- a transition whereby none of the successful
quantitative predictions of the inflationary scenario for the present-day
universe is changed. This is made possible by two properties. First, the
quantum state for the spacetime metric perturbations produced by quantum
gravitational effects in the early Universe becomes very special (highly
squeezed) as a result of the expansion of the Universe (as long as the
wavelength of the perturbations exceeds the Hubble radius). Second,
decoherence through the environment distinguishes the field amplitude basis
as being the pointer basis. This renders the perturbations presently
indistinguishable from stochastic classical inhomogeneities. "

Similarly:

C. Kiefer, D. Polarski:
Emergence of classicality for primordial fluctuations: Concepts and
analogies
http://www.arxiv.org/abs/gr-qc/9805014

and

Claus Kiefer, Julien Lesgourgues, David Polarski, Alexei A. Starobinsky:
The Coherence of Primordial Fluctuations Produced During Inflation
http://www.arxiv.org/abs/gr-qc/9806066 .

There is probably much more literature than that.

d70yxj

Oct17-04, 02:31 PM

Robert Helling wrote:
-------------------------------------------------------------------------
My impression is rather that this whole "freezing" business is just a
metaphor and that there is nothing fancy between quantum and classical
physics that can be understood in a deeper way. Maybe it all boils
down to the following simple observation (which in a sense is purely
classical field theory, or even just mechanics):

\phi_k'' + 2 H \phi_k' + k^2 \phi_k =[/tex]

Then you know from physics 101 that there are two regimes: If the
damping is small (H<<k) [itex]\phi oscillates whereas for large damping
(H>>k) the motion is over-damped and \phi_k slowly creeps back to . It
might be that cosmologists call the first regime the quantum and the
over-damped one the classical, frozen regime. Note that 1/H is the size
of the horizon (or related to it) and 1/k is the wavelength. This
would justify the figure of speech that the mode freezes once it
crosses the horizon.
--------------------------------------------------------------------------

Well, I agree with the analysis above as far as classical field theory goes, and these are indeed the equations on sees when inflation is introduced in a cosmology course. However, I think that the randomness of the freezing process is important to explain large-scale structure today, right? In a classical field theory any perturbations would have to be primordial, i.e. there from the big bang encoded in some complicated initial conditions. I get the impression from cosmologists that a quantum field `freezing in' (whatever this really means) is desirable precisely because it introduces this randomness `for free', even if the initial conditions are in some sense homogeneous.

Does anyone agree/disagree?