THE GRACEFUL EXIT FROM INFLATION AND DARK ENERGY By Tomislav Prokopec Publications: Tomas Janssen and T. Prokopec, arXiv: ; Tomas Janssen, Shun-Pei Miao & T. Prokopec, in preparation. Nikhef, Amsterdam, 18 Dec 2007 ˚ 1˚

The cosmological constant problem ˚ 2˚ Vacuum fluctuates and thereby contributes to the stress-energy tensor of the vacuum (Casimir 1948): 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: 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 ●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: 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 Λ ₁, which after tunnelling reduces to Λ ₀  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 Λ₀Λ₀ Λ1Λ1 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 Λ eff = 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

Scalar backreaction in FLRW spaces The quantum Friedmann equations from 1 loop scalar field fluctuations: ● When solved for the expansion rate H (with Λ=0), one gets: ˚ 7˚ NB1: Λ eff (probably) does not drop fast enough to explain dark energy  At late times t  (today), H drops as 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 calculation Our approximation is valid when -d  /dt<<  H [  =(dH/dt)/H ² ]: ˚ 8˚ NB: The condition -d  /dt <<  H is met (uniformly) when w<-1/3 Classical (de Sitter space) attractor Quantum (Minkowski space) attractor w=  /p=0

Gaviton backreaction in FLRW spaces The quantum Friedmann equations from 1 loop graviton fluctuations: ● When solved for the expansion rate H (with Λ=0), one gets: ˚ 9˚  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  Λ/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 quantum classical

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 (Λ eff ) causes acceleration -> Perlmutter; Riess 1998 Evidence: distant supernovae appear fainter than they would in a decelerating Universe, implying accelerated expansion Λ causes a (tiny) repulsive force which increases with distance: must be measured at cosmological distances

Dark energy and cosmological constant Dark energy has the characteristics of a cosmological constant Λ eff, yet its origin is not known ˚11˚ But why is Λ eff 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 discussion 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. ˚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) 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