the graceful exit from inflation and dark energy

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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˚

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˚ 1˚. 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. ˚ 2˚. The cosmological constant problem. - PowerPoint PPT Presentation

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Page 1: THE GRACEFUL EXIT FROM INFLATION AND DARK ENERGY

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˚

Page 2: THE GRACEFUL EXIT FROM INFLATION AND DARK ENERGY

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:

Page 3: THE GRACEFUL EXIT FROM INFLATION AND DARK ENERGY

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)

Page 4: THE GRACEFUL EXIT FROM INFLATION AND DARK ENERGY

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

Page 5: THE GRACEFUL EXIT FROM INFLATION AND DARK ENERGY

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

Page 6: THE GRACEFUL EXIT FROM INFLATION AND DARK ENERGY

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

Page 7: THE GRACEFUL EXIT FROM INFLATION AND DARK ENERGY

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

Page 8: THE GRACEFUL EXIT FROM INFLATION AND DARK ENERGY

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

Page 9: THE GRACEFUL EXIT FROM INFLATION AND DARK ENERGY

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

Page 10: THE GRACEFUL EXIT FROM INFLATION AND DARK ENERGY

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

Page 11: THE GRACEFUL EXIT FROM INFLATION AND DARK ENERGY

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?

Page 12: THE GRACEFUL EXIT FROM INFLATION AND DARK ENERGY

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]

Page 13: THE GRACEFUL EXIT FROM INFLATION AND DARK ENERGY

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