THE GRACEFUL EXIT FROM INFLATION AND DARK ENERGY

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THE GRACEFUL EXIT FROM INFLATION AND DARK ENERGY
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By Tomislav Prokopec
Publications Tomas Janssen and T. Prokopec,
arXiv0707.3919 Tomas Janssen, Shun-Pei Miao
T. Prokopec, in preparation.
Nikhef, Amsterdam, 18 Dec 2007
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The cosmological constant problem
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Vacuum fluctuates and thereby contributes to the
stress-energy tensor of the vacuum (Casimir
1948)
? A finite volume V L³ in momentum space
constitutes reciprocal lattice each point of the
lattice is a harmonic oscillator with the ground
state energy E/2, where E²(cp)²(mc²)².
Through Einsteins equation this vacuum energy
curves space-time such that it induces an
accelerated expansion
COSMOLOGICAL CONSTANT PROBLEM The expected
energy density of the vacuum
is about 122 orders of magnitude larger than the
observed value
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Cosmic inflation
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?a period of accelerated expansion of the
primordial Universe
EVIDENCE for inflation
?a nearly scale invariant spectrum of
cosmological perturbations
?a near spatial flatness of the Universe
?gaussianity of CMBR fluctuations
Temperature fluctuations of CMBR
CMBR power spectrum (WMAP 3year, 2006)
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Scalar inflationary models
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Guth 1981, Starobinsky 1980
?EOM for a classical scalar field ?(t) in an
expanding Universe
H expansion rate, Vscalar potential
? in the slow roll paradigm d²?/dt²can be
neglected. Take Vm²?, then the FRIEDMANN
EQUATION
SOLUTION
V(?)
SCALAR FIELD TRAJECTORY
?
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Graceful exit problem
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Guth 1981, Linde 1982
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?Inflation realised in de Sitter space with
cosmological term ?1, which after tunnelling
reduces to ?0 ? 0.
tunneling
?0
THE GRACEFUL EXIT PROBLEM
Upon tunnelling, bubbles form and grow, but
INFLATION does not complete the growth of the
false vacuum ?1 wins over that of the true vacuum
?0 ? 0.
The graceful exit problem would be solved if ?
would be (in part) compensated by quantum
effects resulting in a decreasing effective
?eff?(t).
I SHALL ARGUE The one loop scalar field
fluctuations do precisely that!
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Scalar field one loop effective action
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ONE LOOP (MASSLESS) SCALAR FIELD EFFECTIVE ACTION
DIAGRAMMATICALLY 1 LOOP (vacuum bubble)
When the determinant is evaluated in a FLRW
space, it leads to a backreaction that
compensates ?.
NB Can be calculated from knowing the relevant
propagator.
NB Propagators are not known for general spaces
now known for FLRW spaces.
Janssen Prokopec 2007
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Scalar backreaction in FLRW spaces
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The quantum Friedmann equations from 1 loop
scalar field fluctuations
Janssen Prokopec 2007
? When solved for the expansion rate H (with
?0), one gets
? At late times t?? (today), H drops as
Classical (de Sitter) attractor
Quantum behaviour
NB1 ?eff (probably) does not drop fast enough to
explain dark energy
NB2 Minkowski space is the late time attractor
(NOT the classical H²?/3)
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Validity of the backreaction calculation
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Our approximation is valid when -d?/dtltlt?H
?(dH/dt)/H²
Quantum (Minkowski space) attractor
w?/p0
Classical (de Sitter space) attractor
NB The condition -d?/dt ltlt ?H is met (uniformly)
when wlt-1/3
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Gaviton backreaction in FLRW spaces
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The quantum Friedmann equations from 1 loop
graviton fluctuations
Janssen, Miao Prokopec 2007
? When solved for the expansion rate H (with
?0), one gets

• at early times t?0 (Big Bang), H is limited by
  approximately Planck mass
• (probably a perturbation theory artefact).

• at late times t?? (e.g. today), H gets slightly
  reduced. H²??/3 is still late time attractor,
  albeit slightly increased.

quantum

• The scale factor a approaches the de Sitter
  exponential expansion, albeit it gets slightly
  reduced (there is a small delay time).

classical
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Dark energy and acceleration
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Perlmutter Riess 1998
? causes a (tiny) repulsive force which
increases with distance must be measured at
cosmological distances
The luminosity vs distance relation for distant
Type Ia supernovae reveals the Universe is
expanding at an accelerated pace
DARK ENERGY (?eff) causes acceleration -gt
Evidence distant supernovae appear fainter than
they would in a decelerating Universe, implying
accelerated expansion
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Dark energy and cosmological constant
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Dark energy has the characteristics of a
cosmological constant ?eff, yet its origin is not
known
But why is ?eff so small?
EXPLANATION?
UNKNOWN SYMMETRY?
GRAVITATIONAL BACKREACTION!?
OUR ANALYSIS SHOWS scalar (matter) fields
PERHAPS! (though unlikely)but not the
gravitons! (awaits confirmation from a 2 loop
calculation hard) Tsamis, Woodard, 1995
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Summary and discussion
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We have learned that
The (scalar) matter VACUUM fluctuations in an
accelerating universe induce strong quantum
backreaction at the one loop order gravitons do
not.
These vacuum fluctuations may be the key for
understanding the vacuum structure of
inflationary models, and the origin of dark
energy.
Q How these scalar and graviton vacuum
fluctuations affect the inflationary dynamics?
(in progress with Ante Bilandic, Nikhef
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Measuring vacuum fluctuations
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Physicists measure routinely effects of vacuum
fluctuations in accelerator experiments
E.g. Fine structure constant (strength of em
interactions)
becomes stronger when electrons and photons in
Compton scattering have larger energy
Compton scattering
charge screening of an electron at higher
energies, one sees more of the negative
electric charge