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THE GRACEFUL EXIT FROM INFLATION AND DARK ENERGY
By Tomislav Prokopec
Publications: Tomas Janssen and T. Prokopec, arXiv:0707.3919;Tomas Janssen, Shun-Pei Miao & T. Prokopec, in preparation.
Nikhef, Amsterdam, 18 Dec 2007
˚ 1˚
The cosmological constant problem
μν μν μν2 4
(vacuum matter)gravitationalgeometry energy momentumcoupling tensor
8 G ˆG (g) g = Tc c
(μ,ν =0,1,2,3)
˚ 2˚
Vacuum fluctuates and thereby contributes to the stress-energy tensor of the vacuum (Casimir 1948):
vac vac geom vacobs 2
8 G(T ) g
c
COSMOLOGICAL CONSTANT PROBLEM: The expected energy density of the vacuum
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 Einstein’s equation this vacuum energy curves space-time such that it induces an accelerated expansion:
4 76 4vac Pl~m ~10 GeV
2 -46 40obs Pl~(H m ) ~10 GeV
is about 122 orders of magnitude larger than the observed value:
Cosmic inflation●a period of accelerated expansion of the primordial
Universe
EVIDENCE for inflation:▪a nearly scale invariant spectrum of cosmological
perturbations
▪gaussianity of CMBR fluctuations
▪a near spatial flatness of the Universe
Temperature fluctuations of CMBR
˚ 3˚
CMBR power spectrum (WMAP
3year, 2006)
Scalar inflationary models
3 ( ) 0,H V
2 22
26 Pl
mH
M
●EOM for a classical scalar field (t) in an expanding Universe
● in the slow roll paradigm d²/dt²can be neglected. Take V’=m², then the FRIEDMANN EQUATION:
,23
0 tmM Pl
,6
)(
PlM
tmH
2 20
0 exp 066 PL
m t m ta a a
M
SOLUTION:
˚ 4˚
Guth 1981, Starobinsky 1980
SCALAR FIELD TRAJECTORY
V()
H = expansion rate, V=scalar potential
Graceful exit problem
●Inflation realised in de Sitter space with cosmological term Λ₁, which after tunnelling reduces to Λ₀ 0.
Upon tunnelling, bubbles form and grow, but INFLATION does not complete: the growth of the false vacuum Λ₁ wins
over that of the true vacuum Λ₀ 0.
tunneling Λ₀
Λ1
THE GRACEFUL EXIT PROBLEM:
The graceful exit problem would be solved if Λ would be (in part) compensated
by quantum effects resulting in a decreasing effective Λeff=Λ(t).
I SHALL ARGUE: The one loop scalar field fluctuations do precisely that!
˚ 5˚
Guth 1981, Linde 1982
Scalar field one loop effective action
When the determinant is evaluated in a FLRW space, it leads to a backreaction that compensates Λ.
ONE LOOP (MASSLESS) SCALAR FIELD EFFECTIVE ACTION:
˚ 6˚
DIAGRAMMATICALLY 1 LOOP(vacuum bubble):
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
[ ] [ ] [ ]
1/ 2
1loop contribution
1[ ] [ ] ln
2[ ]
i iS iS ie D e e S Tr g
Det g
..][ 24 gxdS
Scalar backreaction in FLRW spaces
8 6 72
2 3
3 1100 : 3 8 0
2 2 4N
N
G H H H HH G
H H H
The quantum Friedmann equations from 1 loop scalar field fluctuations:
● When solved for the expansion rate H (with Λ=0), one gets:
˚ 7˚
6 7
300 : 2 8 ( ) 4 0
2N
N
G H H Hii H G p
H H
NB1: Λeff (probably) does not drop fast enough to explain dark energy
1/ 3
13
3 4NGH t t
At late times t (today), H drops as
2 / 3eff
23
( )4NGt t t
NB2: Minkowski space is the late time attractor (NOT the classical H²=/3)
Classical (de Sitter) attractor
Quantum behaviour
Janssen & Prokopec 2007
Validity of the backreaction calculationOur approximation is valid when -d/dt<<H
[=(dH/dt)/H²]:
˚ 8˚
NB: The condition -d/dt << H is met (uniformly) when w<-1/3
H
Classical (de Sitter space) attractor
Quantum (Minkowski space) attractor
w=/p=0
Gaviton backreaction in FLRW spacesThe quantum Friedmann equations from 1 loop graviton
fluctuations:
● When solved for the expansion rate H (with Λ=0), one gets:
0ln2
2183:00
20
222
H
HHHGGH NN
˚ 9˚
00 : 2 8 ( ) 0Nii H G p
p
wtw
H
HHG
w
H N
,
2
)1(31ln
2
)1(211
0
at early times t0 (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. H²Λ/3 is still late time attractor, albeit slightly increased. The scale factor a approaches the de Sitter exponential expansion, albeit it gets slightly reduced (there is a small `delay time’).
Janssen, Miao& Prokopec 2007
20 0.1NG H
quantum
classical
maxH
The luminosity vs distance relation for distant Type Ia supernovae reveals: the Universe is expanding at an accelerated pace:
˚10˚Dark energy and acceleration
DARK ENERGY (Λeff) causes acceleration
->
Perlmutter; Riess 1998
Evidence: distant supernovae appear fainter than they would in a decelerating Universe, implying accelerated expansion
0az( t)= -1
a( t)
Λ causes a (tiny) repulsive force which increases with distance: must be measured at cosmological distances
Dark energy and cosmological constantDark energy has the characteristics of a cosmological constant Λeff, yet its origin is not known
˚11˚
But why is Λeff so small?
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]
EXPLANATION?
Summary and discussionWe 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.
˚12˚
Q: How these scalar and graviton vacuum fluctuations affect the inflationary dynamics? (in progress with Ante Bilandžić, Nikhef]
Physicists measure routinely effects of vacuum fluctuations in accelerator experiments
˚13˚Measuring vacuum fluctuations
E.g. Fine structure constant (strength of em interactions)
2
e
e 1=
4 c 137becomes 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