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GRAVITATIONAL BACKREACTION IN SPACETIMES WITH CONSTANT DECELERATION

˚ 1˚. GRAVITATIONAL BACKREACTION IN SPACETIMES WITH CONSTANT DECELERATION. Tomislav Prokopec, ITP & Spinoza Institute, Utrecht University. Based on : Tomas Janssen & Tomislav Prokopec, arXiv:0906.0666 & 0707.3919 [gr-qc] Tomas Janssen, Shun-Pei Miao & Tomislav Prokopec, Richard Woodard,

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GRAVITATIONAL BACKREACTION IN SPACETIMES WITH CONSTANT DECELERATION

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  1. ˚ 1˚ GRAVITATIONAL BACKREACTION IN SPACETIMES WITH CONSTANT DECELERATION Tomislav Prokopec, ITP & Spinoza Institute, Utrecht University Based on: Tomas Janssen & Tomislav Prokopec, arXiv:0906.0666 &0707.3919 [gr-qc] Tomas Janssen, Shun-Pei Miao & Tomislav Prokopec, Richard Woodard, arXiv: 0808.2449 [gr-qc], Class. Quant. Grav. 25: 245013 (2008); 0904.1151 [gr-qc] JCAP (2009) Tomas Janssen & Tomislav Prokopec, arXiv:0807.0477 (2008) Jurjen F. Koksma & Tomislav Prokopec, 0901.4674 [gr-qc] Class. Quant. Grav. 26: 125003 (2009) Bielefeld, Sep 23 2009

  2. ˚ 2˚ WHAT IS (QUANTUM) BACKREACTION? Einstein’s Equations background gravitational fields & corresp. (quantum) fluctuations background matter fields & corresponding (quantum) fluctuations ◘ Classical Equations: are not correct in presence of strong backreaction from (quantum) fluctuations ◘ Quantum Einstein Equations: ( physical state) includes (conserved) contribution from graviton fluctuations includes (conserved) contribution from quantum matter fluct’s NB: Since gravitons couple to matter, it is better to write:

  3. ˚ 3˚ THE PROBLEM(S) WITH BACKREACTION Quantum Einstein’s Equations has to be determined by solving dynamical equations for matter and graviton matter perturbations in the expanding Universe setting. Hopelessly hard! ◘ Statements: Dark energy can be (perhaps) explained by the backreaction of small scale gravitational + matter perturbations onto the background space time ♦ Hard to (dis-)prove. Naïve argument against: grav. potential is small: ♦ Maybe too naïve, because of secular (growing) terms generated by perts HERE: I will discuss the simplest (!?) possible TOY MODEL: - a homogeneous universe with constant deceleration parameter q=ε-1, ε=-(dH/dt)/H² - a massless dynamical scalar  but gravity is non-dynamical Q1: What is `toy’ about this model?; Q2: Why is it interesting to study anyway?

  4. ˚ 4˚ A FEW WORDS ON THE CCP PROBLEM Quantum corrections in scalar QED, scalar theories (^4) are positive Quantum corrections from integrating fermions (QED, yukawa) are negative FERMIONS + YUKAWA SCALARS + VECTORS/GRAVITONS RECALL: Shun-Pei M. $1000000 Q: Can solving for Veff self consistently with the Friedmann equation stop the Universe from collapsing into a negative energy (`anti-de Sitter’) universe? JFK+TP, in progress The Universe can also be stabilised by adding a sufficiently many vectors and scalars NB: An effective potential of the form Veff ~ -(^4)ln(²/H²) would solve the CCP problem (the dynamics of  would drive 0). But, how to get it? Tomislav Prokopec, gr-qc/0603088

  5. ˚ 5˚ BACKGROUND SPACE TIME LINE ELEMENT (METRIC TENSOR): FRIEDMANN (FLRW) EQUATIONS (=0): ● for power law expansion the scale factor reads:

  6. ˚ 6˚ SCALAR 1 LOOP STRESS ENERGY ♦CENTRAL Q: Under what conditions can the backreaction from the one loop fluctuations become so large to change evolution of the (background) Universe ◘ QUANTUM STRESS ENERGY TRACE:   i(x;x): scalar propagator at coincidence  : conformal coupling ◘ QUANTUM ENERGY DENSITY & PRESSURE NB: This specifies q and pq up to a term that scales as radiation, 1/a^4

  7. ˚ 7˚ SCALAR THEORY MASSLESS SCALAR FIELD ACTION (V0, R plays role of a `mass’) What is ?  SCALAR EOM  Field quantisation (V0):  PROPAGATOR EQUATION

  8. ˚ 8˚ SCALAR PROPAGATOR IN FLRW SPACES in D dimensions Janssen & Prokopec 2009, 2007 Janssen, Miao & Prokopec 2008 MMC SCALAR FIELD PROPAGATOR (V’’=0, ε=const) ► EOM ► IR unregulated ( SPACE) PROPAGATOR (ε=const) l= geodesic distance in de Sitter space ► IN D=4 ► NB: =1/2 for a conformally coupled scalar,=1/6

  9. ˚ 9˚ MATCHING • We match radiation era (ε=2, =1/2) onto a constant εhomogeneous FLRW Universe  RECALL: observed inhomogeneities /~1/10000 @ cosmological scales (0 ε 3: COVERS ALL KNOWN CASES IN THE HISTORY OF THE UNIVERSE) ε = 3/2: MATTER ε = 3: KINATION ε = 1: CURVATURE RADIATION ε = constant ε << 1: INFLATION ε=2 Another regularisation: the Universe in a finite comoving box L>RH with periodic boundaries MATCHING Tsamis, Woodard, 1994 Janssen, Miao, Prokopec, Woodard, 2009 ► NB: ε=const. space inherits a finite IR from radiation era ► NB2: ε=const. space keeps memory of transition: a LOCAL function of time

  10. ˚10˚ IR SINGULARITY IN DE SITTER SPACE Scalar field spectrum Pφ in de Sitter (=3/2, ε=0) Source of scalar cosmological perturbations  IR log SINGULAR  UV quadratically SINGULAR

  11. ˚11˚ SCALAR THEORY: IR SINGULARITIES the IR singularity of a coincident propagator:  BD vacuum is IR singular (in D=4) for 0 ≤ ε≤ 3/2, when =0  large quantum backreaction expected When 0 the coincident propagator is IR singular in the shaded regions:

  12. ˚12˚ SCALAR THEORY: IR OF BD VACUUM R>0 (ε<2) R=6(2-ε)H²<0 (ε>2)

  13. ˚13˚ QUANTUM & CLASSICAL ENERGY SCALING ◘ SCALINGS: ◘ SCALING: ◘ IF : wq<wb ◘ TYPICALLY: Q: What is the self-consistent evolution for t>tcr, when qb ? Q2: Can q play the role of dark energy?

  14. ˚14˚ ♦ AFTER A LOT OF WORK .. we obtain q & pq , i.e. how they scale with scale factor a

  15. ˚15˚ CLASSICAL vs. QUANTUM DYNAMICS ◘ QUANTUM & CLASSICAL ENERGY DENSITY SCALING: wq vs wb Wq  wb Wb= -1+(2/3)ε NB: We expect that for ε<1 the graviton undergoes the same scaling:  forε>1 we expect the =0 scaling:

  16. ˚16˚ SCALAR QUANTUM EOS PARAMETER : >0 ◘ QUANTUM ENERGY DENSITY & PRESSURE: >0 (meff²>0) Wq, =0.1 Wb =1 =1 ε=εcr NB: For small positive : wq>wb ε; for large >1/6, wq<wb if ε>εcr>2

  17. ˚17˚ SCALAR QUANTUM EOS PARAMETER: <0 ◘ QUANTUM ENERGY DENSITY & PRESSURE: >0 (meff²>0) Wq, =-0.1 Wb =-0.5 NB: For negative : wq<wb ε< εcr; 1<εcr<2

  18. ˚18˚ THE (,ε) REGIONS WHERE wq<wb ◘ QUANTUM ENERGY DENSITY & PRESSURE SHADED REGIONS: wq<wb: after some time, q will become dominant over b

  19. ˚19˚ CAN QUANTUM FLUCTUATIONS BE DARK ENERGY? ◘ SIMPLE ESTIMATE: IMAGINE that quantum fluctuations generated at matter-radiation equality (z~3200) are responsible for dark energy  Ups! It does not work! A much earlier transition is needed!  But, from: we have learned that typically a large time delay occurs between the transition and q~b  NB: It does not work for radiation ε=2 Q: What is the self-consistent evolution for t>tcr, when qb ? Q2: Can q play the role of dark energy?

  20. ˚20˚ SUMMARY AND DISCUSSION The quantum backreaction from massless scalars in ε=const spaces can become large at 1 loop, provided conformal coupling <0 (ε<2). OPEN QUESTIONS: What about other IR regularisations: (scalar) mass, positive curvature, finite box What about the backreaction from scalars/gravitons at higher loop order, non-constant ε FLRW spaces, inhomogeneous spaces, .. ? The backreaction from fermions is large, and distabilises the Universe, driving it to a negative energy Universe: can that be stabilised? Koksma & Prokopec 2009 ► What is the effect of dε/dt 0 (mode mixing)? Janssen & Prokopec 2009 (?) ► is the backreaction gauge dependent (for gravitons)? (Exact gauge?) Miao & Woodard 2009 (?)

  21. ˚21˚ LAGRANGIAN FOR PERTURBATIONS Graviton: lagrangian to second order inhmn ► 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 (Woodard,Tsamis): ● upon a suitable rotation tensor, vector and 2 scalar fields decouple on shell

  22. ˚22˚ GRAVITON PROPAGATOR IN FLRW SPACES Janssen, Miao & Prokopec 2008 Janssen & Prokopec 2009 EOM (symbolic) ► VECTOR DOFs: ► GHOST DOFs: GRAVITON PROPAGATORS

  23. ˚23˚ GRAVITON PROPAGATORS ► SCALAR AND TENSOR DOFs (G=3x3 operator matrix):

  24. ˚24˚ GRAVITON 1 LOOP EFFECTIVE ACTION ►EFFECTIVE ACTION: ☀ When renormalized, one gets the one loop effective action: Janssen, Miao & Prokopec 2009 ► 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:

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