Download Localized wave packet around R 0

Do we really understand the quantum
origin of the seeds cosmic structure?
OQSCM, Imperial College, March 29 2007
Daniel Sudarsky
ICN- UNAM , Mexico
Work in collaboration with:
A. Perez (Marseille) y H.Sahlmann ( Penn State-Utrecht)
gr-qc/0508100, C QG 23, 2317 (2006).
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The last decade is considered as a big success,
for inflationary cosmology.
•
The Universe seems to be spatially flat (i.e. to have a total
density equal, or close, to the critical density)
• Theoretical predictions of the spectrum of primordial in homogeneities resulting from quantum fluctuations of the
inflaton in agreement with such predictions. AMAZING!
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However
• There is something very odd in our understanding of the problem:
The Universe ``starts” as a homogeneous and isotropic space-time
(H&I), and there is a scalar field (the inflaton) which is in a vacuum
state, which is of course also H&I. How is it that we end up with a
situation that is not H & I, given that the dynamics preserves these
symmetries?
• Is this just Quantum Mechanics? Not exactly! The Universe is the
only real example of a QM closed system. Orthodox interpretation
of QM requires an external, classical, measuring apparatus ( This is
why J. Hartle argues for a Generalized Quantum theory applicable
to cosmology).
• Nevertheless, most people in the Field: “There is no problem”.
However you will receive different precise answers from different
cosmologists, which indicates some discomfort with the views of
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the others.
Most popular solution: Decoherence
1) Divide DOF : system + environment.
( identify inaccessible or irrelevant DOF !!!)
2) Compute Reduced Density matrix.
(trace over environment DOF)
3) Perform suitable time average so that the off-diagonal
matrix elements vanish.
4) Regard the diagonal density matrix as describing a
statistical ensemble.
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Decoherence
A B C …..
D E F …..
G H I …..
……………
Interpretation:
i) choice (our wish) vs.
ii) coexistence ( QM).
A 0 0 ….
0 E 0…..
0 0 I ….
………
?
0 0 0…
0 E 0…
0 0 0…
……….
Diagonalization disappears upon change of basis!
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Example
State symmetric under rotations by  around Z
Z
Localized wave packet
around R0
Now consider taking the trace over the spin DOF.
The resulting density matrix is diagonal.
Is the situation classical? Is the state still invariant?
Can a mathematical manipulation with NO physical 6
process counterpart ever change the state of the system?
In the Cosmological Setting
• We seek a historical ( time development)
description of comic evolution that follows the
laws of physics ( Perhaps it is too much to ask? ).
• Such description should explain how did WE arise
( primordial density fluctuations, galaxies, stars,
planets, living organisms, etc).
• Should NOT rely on the measurement (in) abilities
of the late evolved creatures to explain the
emergence of conditions that make them possible
( one can not justify identifying some DOF as
irrelevant environment).
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You might instead be asked to accept:
• Q.M. does not describe Our Universe, as it was never H&I (the
ensemble was) ( Only the superposition of many U is represented by
the H & I Quantum state . Beware:This is not QM!).
• Our Universe is Still H&I.
• That “it does not matter when the Universe stopped being H&I”,
without being able to even address the issues ( when , why, due to
what..?).
• In particular: You can not take the initial state and evolve it using the
Schroedinger equation to conclude that our U is H& I, but you do not
know why not!
• This is not ``JUST PHILOSOPHY”, rather the early Universe offers
the “Lab” where some of the issues can be ( at least in principle)
studied. ( In fact Inflation itself is often attacked as having
motivations that are “too philosophical”).
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R. Penrose has for a long time
argued that QM is incomplete
• If QM is a description of reality (not just of
our information about it).
• The U (unitary evolution) and R ( reduction
or collapse) time evolution processes in
QM must be real.
• QM should not rely on an “ external
classical world”, and should serve to
describe closed systems.
• The R process might have QG origins!
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Desiderata: A scheme that
• Permits, in principle, the assignment of a Quantum
State to the system, at ``every time”.
• Views QM as a theory about the description of the
system, and not just, of our knowledge about it.
• Allows consideration of issues such as “When did
the lack of H&I at such and such scale originate?”
• Take the view that the marriage of GR and QM
might involve changes in both! ( Penrose).
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The standard scheme augmented with the ``self induced
collapse” hypothesis.
• Collapse: Is the only known mechanism, capable of taking a symmetric
state into an asymmetric state while the dynamics preserves the symmetry.
The NEW PHYSICS lies in the fact that in this case we can not rely on an
external agent to induce the collapse.
• SEMICLASSICAL GRAVITY (corrected) ( Note that Gravity is treated
very differently than the matter, is just “effective theory”) & coupled to the
inflaton according to:
– Quantum State subject to :
• Motivation: “The QG DOF are not excited”, except at the jumps.
• Q reflects the jumps in the geometry that must accompany the collapse in
the state. It is assumed to vanish before and after the collapse.
• Goal: To extract characteristics of the NEW PHYSICS from the
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observations.
Pre and Post Collapse Cosmology
• Metric
• Scalar Field :
• Quantize
• Einstein’s Eqs.:
Introduce
&
and
where
and
which is 0 for ``slow roll”.
• Semi-classical version:
• The Fourier Decomposition :
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The state before & after the Collapse
Before the Collapse Y=0:
Assume that at time h the mode k collapses according to (Scheme 1) :
Where the x’s are random ( around 0 and with spread 1).
(Scheme 2: as above but with <y> =0
)
Then, after the collapse
we have:
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•
Study of the Observational
quantities
Metric perturbation :”the Newtonian potential ”(observed in
)
– Einstein’s Semi-Class Eqs.
– U(k) late time physics , F(k) depends on details of the post collapse state .
– The Quantity of interest is
• Is then the result of ``a random walk”.
• Its ``most likely” magnitude:
• In the continuum it becomes :
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The Structure of C(k)
• C(k) contains information about the collapse (through F)
– In scheme 1 we find :
– With
•
and
In general, we will have :
• Agreement with observations requires, =0 or h(k)k
independent of k, (or very small corrections).
• (Ignoring U(k) ), the result is independent of RD ( scale
invariant) iff C(k) is independent of k , and = 0 (“slow
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roll”).
• If C=1,
Thus the relevant amplitude
• Where the “slow roll” parameter
– Could eliminate the need of ``fine tuning” (in Scheme of
collapse #2: <>
– As the quantity :
=0 ):
 = ( a˙/a ) <> / ˙ + Y
(Mukhanov) is constant in the regime of interest.
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•
•
•
•
A Penrose Inspired Model : Collapse occurs when the energy of
gravitational interaction among alternatives reaches M PLANCK.
For individual modes EI(k,h) = (hG /9HI2 )(a/k)(V’)2
– Then
=Z/k con Z = ( /9 MP3HI3 )(hV’)2 independent
of k IT WORKS !
– The collapse for the physically relevant modes occurs about
tens of e-folds before the end of inflation.
In fact the relation between the scale factor at the time of Horizon
exit, of a mode to its size at the time of collapse is a constant
akc /akH = (16/) (63)(1/2)(V h3/MP4)(-1/2)
In general a large, number so in this scheme collapse occurs long
after horizon exit of the mode.
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Conclusions
• Something (related to QM) is missing in our
understanding of the origin of cosmic structure.
• Something like a self induced collapse is required to
take us from a H&I state to another one lacking such
symmetry.
• The present analysis seems offer a path to alleviate the
need of fine tuning .
• In this scheme there would not be Tensor Modes (GW)
contributions to the CMB anisotropies ( except
secondary ones)! The Ideas are testable.
• NEW PHYSICS might be there, just waiting to be
explored
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