Classical and Quantum Proper3es of Screened Modified Gravity Philippe Brax IPhT Saclay P.B, C. van de Bruck, A.C. Davis, B. Li, H. Winther, G. Zhao etc and in progress «Quantum vacuum» workshop, LKB December 2012
Outline 1) Dark energy or modified gravity: models and unified descrip3on 2) Field theore3cal problems 3) A possible resolu3on: classicalisa3on In progress
The Big Puzzle
Evidence: The Hubble Diagram The explosion of high red- shiy SN Ia (standard candles): Within General Rela3vity, link to ma]er and dark energy Dark Energy must exist!
The Cosmic Microwave Background Fluctua3ons of the CMB temperature across the sky lead to acous3c peaks and troughs, snapshot of the plasma oscilla3ons at the last sca]ering surface when the universe became transparent The posi3on of the first peak: The universe is spa3ally flat WMAP data
The accelera3on of the Universe could be due to either: ϕ Modified Gravity Dark Energy In both cases, current models use scalar fields. In modified gravity models, this is due to the scalar polarisa3on of a massive graviton. In dark energy, it is by analogy with infla3on. The fact that the scalar field acts on cosmological scales implies that its mass must be large compared to solar system scales.
Dark Energy or Modified Gravity? Field rolling down a runaway poten3al, reaching large values now. If coupled to CDM and/or baryons, this modifies gravity due to the existence of a long range fiyh force whose range is of the size of the observable Universe.
Devia3ons from Newton s law are parametrised by: For fields of zero mass or of the order of the Hubble rate now, the 3ghtest constraint on β comes from the Cassini probe measuring the Shapiro effect (3me delay): The effect of a long range scalar field must be screened to comply with this bound.
Around a background configura3on and in the presence of ma]er, the Lagrangian can be linearised and the main screening mechanisms appear directly: The chameleon mechanism makes the range become smaller in a dense environment by increasing m The Damour- Polyakov mechanism reduces β in a dense environment The Vainshtein mechanism reduces the coupling in a dense environment by increasing Z
The effect of the environment When coupled to ma]er, scalar fields have a ma]er dependent effec3ve poten3al Environment dependent minimum Chameleon=constant coupling The field generated from deep inside is Yukawa suppressed. Only a thin shell radiates outside the body. Hence suppressed scalar contribu3on to the fiyh force.
Symmetron
Dilaton
For small enough bodies, and distances less than the range of the force outside the body, the fiyh force is not screened
Thin shell When objects are big enough/dense enough, the field is screened outside. Inside it is nearly constant apart from inside thin shell whose size is inversely propor3onal to Newton s poten3al at the surface. For a chameleon: The fiyh force is suppressed outside:
ϕ ϕ+ For all chameleon, dilaton, symmetron models where either the poten3al and/or the coupling β is a non- linear func3on of ϕ, the screening criterion is simply:
The family of screened modified gravity models (excluding Vainshtein) can be much more easily analysed using a reconstruc3on procedure. The existence of a minimum for a medium of density ρ allows one to define a mapping between ρ and the value of the field and the value of the poten3al at the minimum. This implicit way of defining V(φ) is very useful as ρ itself can be parameterised by the expansion rate of the Universe This implicit defini3on of the models depends only on the mass m(a) and the coupling β(a) at the minimum.
The non- linear poten3al of the model and the values of the field can be evaluated using: The full non- linear dynamics is reconstructed parametrically using the mass and the coupling func3on as a func3on of redshiy! It works explicitly for chameleons, f(r), dilatons, symmetrons and one can invent new models! As the Universe evolves from pre- BBN to now, the density of ma]er goes from the density of ordinary ma]er (10g/cm3) to cosmological densi3es. The minimum of the effec3ve poten3al experiences all the possible minima from sparse densi3es (now) to high density (pre- BBN).
This implies a reformula3on of the screening condi3on: where the bounds of the integral are such that the densi3es inside and outside the body correspond to the ma]er density at these respec3ve scale factors. Inverse power law chameleons (3/2 <r<3) and large curvature f(r) models (r>3) are described by: Given m(a) and β(a), one can construct a model defined by V(ϕ) and, and then check explicitly if screening occurs.
The loosest screening condi3ons requires that the Milky way is marginally screened, this corresponds to the absence of disrup3on of the galac3c halo dynamics: This implies the crucial bound: Effects of modified gravity can appear at most on the Mpc scale. Similar bound obtained from pulsar- white dwarf system.
Power Spectrum (symmetron models)
The modified gravity models that we have presented so far are only effec3ve field theories. At best, they would describe the physics at low energy ayer BBN when energies are below 1 MeV and all standard model par3cles have decoupled (apart from neutrinos). A minimal requirement for these theories would be to extend them to energies higher than 1 MeV and couple them directly to standard model par3cles. This leads to a severe problem: a viola3on of unitarity This nasty effect can assuaged due to the non- linearity of the theory.
We consider a scalar- tensor theory coupled to an extension of the standard model, valid up to a large cut- off scale: In the Einstein frame, the par3cles couple to a rescaled metric and their effec3ve mass feels the scalar field: We incorporate a cosmological constant in the ma]er sector: We also include a cosmological constant in the scalar field poten3al:
In this model we retrieve the effec3ve poten3al, valid as long as the par3cles cannot be considered as non- rela3vis3c fluids: Below the electron mass, ma]er par3cles can be considered as fluids: As the par3cles decouple, they give a contribu3on to the effec3ve poten3al. Hence the true effec3ve poten3al below the electron mass is the sum of the classical one, involving fluids, and the quantum correc3ons.
Let us now examine the quantum correc3ons: This induces a change of the cosmological constant: Coleman- Weinberg correc3on in the Jordan frame The complete low energy effec3ve poten3al: This large correc3on would prevent the ma]er dependence of the vacuum and therefore screening.
Reformula3on of the cosmological constant problem: At low energy we have therefore: The cosmological constant Λ is what remains ayer this cancella3on. We will concentrate on chameleons from now:
Around the low energy minimum: The effec3ve poten3al can be expanded: or equivalently Perturba3on theory is valid when λ is smaller than 1. Non- renormalisable interac3on suppressed by a cut- off scale
For chameleons we find that: Perturba3on theory is only valid in low density media, below a few g/cm3 The low energy cut off is simply given by the vacuum value of the field, always bigger than Λ but most of the 3me below 1 GeV. This confirms that chameleons are only low energy models, fine below BBN and for gravity tests.
In this low energy regime where perturba3on theory makes sense, one must consider the effect of the loops on the effec3ve poten3al: As long as perturba3on is valid and λ is small, the quantum correc3ons have a small effect: This is not the case at energies above the cut off, even when λ is small!
The non- renormalisable operators in the effec3ve poten3al have a nasty effect: High energy probe High energy chameleon fragmenta3on At high energy above the cut off, high energy chameleons create many soyer chameleons with a probabibility which breaks unitarity.
This type of behaviour is common in field theory: Above the weak scale, the Fermi interac3on needs to be opened up and becomes the electro- weak theory. W This is a UV comple3on of the theory.
High energy probe Crea3on of an extended object What I will argue is that, like in other cases as advocated by Dvali et al., the chameleons defend themselves against high energy probes and create extended objects which behave semi- classically.
In the non- perturba3ve regime above the cut- off, one must use new arguments different from the usual perturba3ve ones. Here we consider the operator valued equa3on of mo3on in the Heisenberg picture, which depends on the operator: We will solve the operator- valued Klein- Gordon equa3on in an averaged sense for a class of states which sa3sfy: which belong to the Fock space of scalar and par3cle excita3ons and are a tensor product of the in- state formed with par3cles and probing the chameleons, and a scalar state which we are looking for. In par3cular, if we probe with massive fermions: The probe corresponds to a ma]er distribu3on:
Quick review of coherent states: Using the crea3on and annihila3on operators one can form the operators: And the associated coherent states: sa3sfying and
The averaged value of the quantum operator in the coherent state is the classical field: For normal ordered operators, the same property implies the factorisa3on: We are reduced to the average of the Klein- Gordon equa3on: which we analyse in the rest frame of the high energy probe:
The typical size of the probe is its Compton wave length and its Newtonian poten3al: At low enough energy, the probe is not screened for: At scales shorter than a typical distance, the state behaves like a free state whose size is much larger than the Compton wave length of the probe. For larger distances, the interac3ons cannot be neglected. outside free inside
At higher energies, the chameleons react in an even stronger way to the probe: the probe becomes screened and the state is a free configura3on of size the Compton wavelength in vacuum. inside outside Screened probe of ever smaller Compton wavelength The chameleon field is free and forms a lump whose size is independent of the probe.
At high energy above the cut off scale, chameleons do not need to have a UV comple3on to restore unitarity, they classicalise: Forma3on of lump whose size is the Compton wavelength of the chameleon much larger than the Compton wavelength of the probe. Analogous to the forma3on of black holes in the TransPlanckian regime.
Summary Screening is an essen3al property of dark energy/modified gravity models Models ill defined at high energy Strong hints that chameleons classicalise