GRAVITON BACKREACTION & COSMOLOGICAL CONSTANT ˚ 1˚ GRAVITON BACKREACTION & COSMOLOGICAL CONSTANT Tomislav Prokopec, ITP & Spinoza Institute, Utrecht University Based on: Tomas Janssen & Tomislav Prokopec, arXiv:0707.3919 [gr-qc] (2007) Tomas Janssen, Shun-Pei Miao & Tomislav Prokopec, arXiv:0807.0439 [gr-qc] Tomas Janssen & Tomislav Prokopec, arXiv:0807.0477 (2008) Munchen, Oct 9 2008

THE COSMOLOGICAL CONSTANT PROBLEM ˚ 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: THE COSMOLOGICAL CONSTANT PROBLEM: The expected energy density of the vacuum is about 122 orders of magnitude larger than the observed value: Q: H²Λ/3 is a classical attractor. Does it remain so in quantum theory?

BACKGROUND SPACE TIME ˚ 3˚ LINE ELEMENT (METRIC TENSOR): FRIEDMANN (FLRW) EQUATIONS (=0): ● for power law expansion the scale factor reads:

SCALAR THEORY ˚ 4˚ (MASSLESS) SCALAR FIELD ACTION  SCALAR EOM In momentum space (=0, V=0): Scalar field spectrum Pφ in de Sitter (ε=0) CONTAINS IR SINGULARITY

SCALAR THEORY: SINGULARITIES ˚ 5˚ SCALAR THEORY: SINGULARITIES the IR singularity of a coincident 2 point function: is IR singular for 0 ≤ ε ≤ 3/2  large quantum backreaction expected coincident 2 point function (propagator) in dS limit: ● when e=constant, the 1/e term can be subtracted ● when e=ε(t), but slowly changing in time, s.t. dε/dt<<Hε ● singularities occur when e = 3/2, 4/3, 5/4,.., 1,.. 4/5, 3/4, 2/3, 1/2 & 0 close to matter era: e=3/2+ε: ►we find: implying a secular growth of vacuum fluctuations that can compensate a cosmological term

CLASSICAL ATTRACTOR IN FLRW SPACES ˚ 6˚ CLASSICAL ATTRACTOR IN FLRW SPACES Einstein’s equations in FLRW spaces (0): ► CLASSICAL SOLUTION Classical (de Sitter) attractor Quantum behaviour (?) Q: can quantum vacuum fluctuations change the late time de Sitter attractor behaviour?

SCALAR PROPAGATOR IN FLRW SPACES ˚ 7˚ SCALAR PROPAGATOR IN FLRW SPACES Janssen & Prokopec 2007 Janssen, Miao & Prokopec 2008 SCALAR PROPAGATOR ► EOM ► Ansatz: l = geodesic distance in de Sitter space ► This propagator allows for determination of the quantum backreaction and more generally effects of quantum scalar fields in dynamical FLRW spaces HOPE: THAT THIS SCALAR PROPAGATOR RESUMS THE LOGS OF a:

LAGRANGIAN FOR PERTURBATIONS ˚ 8˚ LAGRANGIAN FOR PERTURBATIONS Graviton: lagrangian to second order in hmn ► PERTURBATIONS ► GRAVITON-SCALAR MIXING ● lagrangian must be diagonalized w.r.t. the scalar fields 00 &  ►GAUGE: graviton propagator in exact gauge is not known. We added a gauge fixing term: ● upon a suitable rotation tensor, vector and 2 scalar fields decouple on shell

GRAVITON PROPAGATOR IN FLRW SPACES ˚ 9˚ GRAVITON PROPAGATOR IN FLRW SPACES Janssen, Miao & Prokopec 2008 EOM (symbolic) ► VECTOR DOFs: ► GHOST DOFs: GRAVITON PROPAGATORS

► SCALAR AND TENSOR DOFs (G=3x3 operator matrix): GRAVITON PROPAGATORS ˚10˚ ► SCALAR AND TENSOR DOFs (G=3x3 operator matrix):

SCALAR 1 LOOP EFFECTIVE ACTION ˚11˚ SCALAR 1 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 can compensate Λ. NB: Can be calculated by knowing the relevant propagator. NB2: Propagators are not known for general spaces; now known for FLRW spaces with constant ε. Janssen, Miao & Prokopec 2008

GRAVITON 1 LOOP EFFECTIVE ACTION ˚12˚ GRAVITON 1 LOOP EFFECTIVE ACTION ►EFFECTIVE ACTION: ☀ When renormalized, one gets the one loop effective action: Janssen, Miao & Prokopec 2008 ► i: renormalization dependent constants ► H0: a Hubble parameter scale ► (z)=dln[(z)]/dz: digamma function ► can be expanded around the poles of (z): ● the poles 0, 1, 2 (dS, curv, rad) are not relevant. NB: Q & pQ can be obtained from the conservation law:

DYNAMICS NEAR THE DE SITTER POLE ˚13˚ ►near the de Sitter pole (ε=0): small quantum effect (A,B: undetermined constants) ● Late time dynamics: asymptotes a nearly-classical de Sitter attractor: Quantum corrected attractor Classical (de Sitter) attractor

DYNAMICS NEAR THE MATTER ERA POLE ˚14˚ ►near the matter era pole (ε=3/2): secular growth and large quantum effects ● Late time dynamics: asymptotes the near-pole classical attractor: Classical branch INSET 1. Quantum branch

DYNAMICS NEAR THE MATTER ERA POLE II ˚15˚ ►Hubble parameter vs time (εp~3/2, <0): ● Late time dynamics: asymptotes the near-pole classical attractor: Quantum corrected attractor Classical (de Sitter) attractor

DYNAMICS NEAR THE 5/4 POLE ˚16˚ ►near the ε=5/4 pole (>0): secular growth and large quantum effects ● Late time dynamics: asymptotes the near-pole classical attractor: Quantum branch INSET Classical branch

DYNAMICS NEAR THE 5/4 POLE II ˚17˚ ►Hubble parameter vs time (εp~5/4, >0): ● Late time dynamics: asymptotes the near-pole classical attractor: Classical (de Sitter) attractor Quantum attractor

DARK ENERGY AND COSMOLOGICAL CONSTANT ˚18˚ 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!? This work suggests that it may be the gravitational backreaction of gravitons (plus matter).

SUMMARY AND DISCUSSION ˚19˚ SUMMARY AND DISCUSSION We considered the quantum backreaction from massless scalar and graviton 1 loop vacuum fluctuations in expanding backgrounds Scalar matter and graviton VACUUM fluctuations in a near de Sitter universe induce a weak quantum backreaction at 1 loop order (also at 2 loops?). 1 loop backreaction can be strong when e = 3/2, 4/3, 5/4,.., 1,.. 4/5, 3/4, 2/3, 1/2 (-2/3≤w≤0) OPEN QUESTIONS: ► we calculated in the approximation ε=(dH/dt)/H²=const. What is the effect of dε/dt  0 (mode mixing)? Janssen & Prokopec 2008 ► is the backreaction gauge dependent? (Exact gauge?) ► what happens at 2 loop? Miao & Woodard 2008, .. ► What is the quantum backreaction of other quantum fields (fermions, photons)? Koksma & Prokopec 2008