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

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

THE PROBLEM(S) WITH BACKREACTION Quantum Einstein’s Equations ˚ 3˚ ◘ 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: has to be determined by solving dynamical equations for matter and graviton matter perturbations in the expanding Universe setting. Hopelessly hard! ♦ 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?

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

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

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

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

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

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

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

˚11 ˚ When  0 the coincident propagator is IR singular in the shaded regions: 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

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

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

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

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

SCALAR QUANTUM EOS PARAMETER :  >0 ˚16˚ ◘ QUANTUM ENERGY DENSITY & PRESSURE:  >0 (m eff ²>0) W q,  =0.1  =1 WbWb NB: For small positive  : w q >w b  ε; for large  >1/6, w q ε cr >2 ε=ε cr

˚17˚ ◘ QUANTUM ENERGY DENSITY & PRESSURE:  >0 (meff²>0) SCALAR QUANTUM EOS PARAMETER:  <0 W q,  =-0.1  =-0.5 WbWb NB: For negative  : w q <w b  ε< ε cr ; 1<ε cr <2

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

CAN QUANTUM FLUCTUATIONS BE DARK ENERGY? ˚19˚ ◘ SIMPLE ESTIMATE: IMAGINE that quantum fluctuations generated at matter-radiation equality (z~3200) are responsible for dark energy Q: What is the self-consistent evolution for t>t cr, when  q   b ? Q2: Can  q play the role of 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

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

LAGRANGIAN FOR PERTURBATIONS ˚21˚ Graviton: lagrangian to second order in h  ► PERTURBATIONS ►GAUGE: graviton propagator in exact gauge is not known. We added a gauge fixing term (Woodard,Tsamis) : ► GRAVITON-SCALAR MIXING ● lagrangian must be diagonalized w.r.t. the scalar fields  00 &  ● upon a suitable rotation tensor, vector and 2 scalar fields decouple on shell

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

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

GRAVITON 1 LOOP EFFECTIVE ACTION ☀ When renormalized, one gets the one loop effective action: ˚24˚ Janssen, Miao & Prokopec 2009 ►  i : renormalization dependent constants ► H 0 : a Hubble parameter scale ►  (z)=dln[  (z)]/dz : digamma function ► can be expanded around the poles of  (z): ►EFFECTIVE ACTION: ● the poles 0, 1, 2 (dS, curv, rad) are not relevant. NB:  Q & p Q can be obtained from the conservation law: