THE GRACEFUL EXIT FROM INFLATION AND DARK ENERGY

1. ˚ 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. ˚ 2˚ The cosmological constant problem 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 Einstein’s 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:

3. ˚ 3˚ Cosmic inflation ●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)

4. ˚ 4˚ Scalar inflationary models Guth 1981, Starobinsky 1980 ●EOM for a classical scalar field (t) in an expanding Universe H = expansion rate, V=scalar potential ● in the slow roll paradigm d²/dt²can be neglected. Take V’=m², then the FRIEDMANN EQUATION: SOLUTION: V() SCALAR FIELD TRAJECTORY 

5. ˚ 5˚ Graceful exit problem Guth 1981, Linde 1982 Λ1 ●Inflation realised in de Sitter space with cosmological term Λ₁, which after tunnelling reduces to Λ₀  0. tunneling Λ₀ THE GRACEFUL EXIT PROBLEM: 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. 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 loopscalar field fluctuations do precisely that!

6. ˚ 6˚ Scalar field one loop effective action 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 Λ. 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

7. ˚ 7˚ Scalar backreaction in FLRW spaces The quantum Friedmann equations from 1 loop scalar field fluctuations: Janssen & Prokopec 2007 ● When solved for the expansion rate H (with Λ=0), one gets:  At late times t (today), H drops as Classical (de Sitter) attractor Quantum behaviour NB1:Λeff (probably) does not drop fast enough to explain dark energy NB2:Minkowski space is the late time attractor (NOT the classical H²=/3)

8. ˚ 8˚ Validity of the backreaction calculation Our approximation is valid when -d/dt<<H [=(dH/dt)/H²]: Quantum (Minkowski space) attractor w=/p=0 Classical (de Sitter space) attractor NB: The condition -d/dt << His met (uniformly) when w<-1/3

9. ˚ 9˚ Gaviton backreaction in FLRW spaces The quantum Friedmann equations from 1 loop graviton fluctuations: Janssen, Miao& Prokopec 2007 ● When solved for the expansion rate H (with Λ=0), one gets: • at early timest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Λ/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

10. ˚10˚ Dark energy and acceleration Perlmutter; Riess 1998 Λ 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 (Λeff) causes acceleration -> Evidence: distant supernovae appear fainter than they would in a decelerating Universe, implyingaccelerated expansion

11. ˚11˚ Dark energy and cosmological constant Dark energy has the characteristics of a cosmological constant Λeff, yet its origin is not known But why is Λeff 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]

12. ˚12˚ Summary and discussion We have learned that: The (scalar) matter VACUUM fluctuations in an accelerating universe induce strongquantum 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 Bilandžić, Nikhef]

13. ˚13˚ Measuring vacuum fluctuations 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