Galileons

Philippe Brax

IPhT Saclay

Axions and the Low Energy Frontier

Comment le sait-on? Mesure de distances!

Dark Energy: anything which leads to the late time acceleration of the expansion of the Universe !

Would like to describe it using some kind of effective field theory at low energy…

GRAVITY ACTS ON ALL SCALES

Looking for extensions of General Relativity valid from small to large scales.

A fruitful approach consists in classifying all possible extensions of GR which are not problematic (ghosts, instabilities, ill-posed initial condition problems, causality issues …) within a minimalistic setting.

In general, these theories require a fine-tuning of the cosmological constant but have unexpected fieldtheoretic properties which go beyond the usual framework of field theory: irrelevant operators dominate, UV completions may not be required (classicalisation)… The most general form of these theories is knownin some cases (Horndeski for one scalar) but one must go into the details of the models to make themwork from the solar system (or the laboratory) to large cosmological scales.

What we have learnt in the last ten years is that such extensions involve light scalar fields, why?:

Massive gravitons have a scalar part

Their interaction with matter generates fifth forces which would have been seen in the laboratory and the solar system.

5 = 2+ 2 + 1

Deviations from Newton’s law are parametrised by:

For large range forces with large λ, the tightest constraint on the coupling β comes from the Cassini probe measuring the Shapiro effect (time delay): Bertotti et al. (2004)

β β

A B

VIOLATION OF THE STRONG EQUIVALENCE PRINCIPLE

C

Tight on the perihelion advance of the moon too (see later)

Lunar ranging experiment

What we have learnt in the last ten years is that such extensions involve light scalar fields, why?:

How do we hide light scalar fields ?

SCREENING

In fact, around a background configuration and in the presence of matter, the Lagrangian of suchextensions can be linearised and the main screening mechanisms can be schematically distinguished :

The Vainshtein mechanism reduces the coupling by increasing Z. The K-mouflage mechanism has the same effect … while the Damour-Polyakov mechanism suppresses β and the chameleon property increases m.

The Vainshtein mechanisms can be easily analysed:

Effective Newtonian potential:

For theories with second order equations of motion:

VainshteinK-mouflage

Cosmological choice

Vainshtein

Newtonian gravity retrieved when the curvature is large enough:

On large cosmological scales, this tells us that overdensities such as galaxy clusters are screened :

On small scales (solar system, galaxies) screening only occurs within the Vainshtein radius:

Let us focus the motion of astrophysical objects in such theories:Trajectories due to gravity + scalar

Unlike chameleons et al., K-mouflage and Vainshtein do not affect the charge Q:

outside the Vainshtein-Kmouflage radius

Outside the Vainshtein-Kmouflage radius, the object feels the scalar force

Chameleons:

The Galileon models are motivated by the requirement that the effective scalar field theory is invariant under a symmetry which prevents the existence of a potential term generalising a cosmological constant. The acceleration of the Universe would have to be self-generated dynamically:

This is a Galilean invariance of the action. The scalar Lagrangian involves only a small number of building blocks:

where X is the kinetic term of the scalar field. The scalar field φ is screened by the Vainshtein mechanism.

The coupling functions are not arbitrary but specified by 4 real coefficients (in the Galileon case)

The simplest example: the cubic Galileon. The scale Λ is the strong coupling scale of the model (see later for its relevance in QFT). These models have a cosmological interest when:

In this case, one can generate a time evolution of the scalar field such that the dynamics of the Universe mimics a fluid with an equation of state close to -1 in the recent past of the Universe. The Universe is self-accelerating.

A simple example: the CUBIC GALILEON

Well inside the Vainshtein radius, Newtonian gravity is restored. Well outside gravity is modified.

The Vainshtein radius is very large for stars, typically 0.1 kpc for the sun, and a mass for the graviton of the order of the Hubble rate .

m graviton mass

What is specific about this model?

In fact, Galileons are simply a subclass of the Horndeski models which are the most general scalar field theories with second order equations of motion, here the quartic Horndeski action:

Equation of state -1 now

Another possible effect of modified gravity is… to change the speed of gravitons compared to the speed of light.

Severe constraints must be taken into account:

If the gravity waves travel slower than the speed of light, relativistic cosmic rays may emit gravitons by the Cerenkov effect. This would suppress them heavily unless:

If the gravity waves travel faster than the speed of light, this would affect the period of binary pulsars unless:

Modified gravity models can have effects on the speed of gravity waves when the term Einstein-Hilbert term is modified in the significant way:

Typically in astrophysical situations, we are interested in the emission of spherical waves in a time-dependent cosmological background. In this context the wave equation for gravitons takes the form:

The speed of gravity wave is “screened”, i.e. hardly modified, if the terms in can be neglected. Could it be that the “Vainshtein mechanism” plays a role here too and screens the speed of gravity waves to a level such that the tight bounds from the Cerenkov effect and the timing of pulsars could be satisfied?

Inside the Vainshtein radius of quartic Galileons, IF spatial gradients are larger than time derivatives:

Where the gradient is essentially constant… and very small… implying that the speed of gravity waves would be screened…

Unfortunately, the time derivatives are smaller than the gradients when:

This is violated for masses of around one solar mass! And gravity waves are not screened.

For Galileon models with an equation of state of -1 now, the speed of gravity waves is way too large… Can be cured using only cubic Galileons… but they have an equation of state -1 only when c2 <0 (Minkowski unstable)… A more favourable situation can be obtained with subdominant “cubic” Galileons:

Can we really have gravity waves faster than the speed of light at the percent level?

If gravitons go faster than the speed of light, they can decay into two photons.

For astrophysical sources, the decay time is much longer than the age of the Universe.

Nothing prevents a graviton from emitting two photons in a Cerenkov way, this entails a loss of energy per unit time:

The typical time-scale for the decay of the energy by Cerenkov emission is much larger than the age of the Universe…

The time delay between photons-neutrinos and gravitons could be a smoking gun for modified gravity models… Indeed gravitational waves are expected to arrive earlier than photons-neutrinos by:

The time delay at emission between neutrinos and gravitons for supernovae like SN1987A is around 1/1000 s… When the speed of gravitons deviate from the speed of light at the percent level, the time difference between seeing photons and detecting gravity for SN1987A could have been 1700 years… For sources 1kpc away from us, this would reduce to 30 years and finally if gravity waves and neutrinos were both detected from supernovae 1 kpc away, we could put a bound on the difference of speed (due to the emission time

difference at the 1/1000 s) at the

The recent detection of gravitational waves does not give a bound on their speed.

On the other hand, combining with the Gamma-ray burst detected by the Fermi experiment, if the emission is from the same source:

In this case, cosmological Galileons would be cubic with a subdominant effect on the acceleration of the Universe.

So it seems that cosmological Galileons with a stable Minkowski limit cannot lead to the acceleration of the Universe on their own. They have a speed of gravitation waves essentially equal to one.

The situation is actually very reminiscent of the axion! Motivated by the QCD ϴ angle, then by Dark Matter… eventually one opens up the parameter space to look for other types of axions with different masses and couplings.

So if the Galileons do not serve as sole progenitor of the acceleration of the Universe, what would they do? They would “simply” modify gravity… Where would this have effects?

Planetary motion Lab tests of gravity Casimir effect

Screening happens inside the Vainshtein radius. Still the tiny deviations depends on r and can lead to an anomalous perihelion in the earth-moon system.

This forces us to consider cosmological Galileons for couplings of matter of order one

The Galileon does not shield in a planar situation but is blanket-shielded by the earth. In the experimental environment of the earth, the effective coupling of the Galileon is reduced.

vanishes

The earth is screened too in fact:

Linear field with coupling:

The eotwash experiment constrains the torque by non gravitational forces between two rotating plates. This gives a tight constraint on the coupling to matter on earth.

Forthcoming experiments like CANNEX will measure the plate-plate Casimir effect. The sensitivity must be high at around a few μm of distance to compete with the Eotwash bound.

So far we have treated the Galileons classically. Galileons have special field theory properties even at the quantum level when including corrections:

The Galileon action is not renormalised

The finite corrections to the Galileon action involve higher order derivatives, and they can be neglected at large enough scale.

How come?

In a given background, expanding the Galileon Lagrangian in perturbation theory leads to a mass and higher order operators depending on the cut-off:

The mass itself is a function of the ratio:

This implies that the quantum corrections appear in the form

These corrections are negligible in the regime:

On earth, this is around 1 cm ! Lab experiments just about in the trustworthy regime.

One can also enlarge the type of couplings of Galileons to matter!

Matter couples to a metric which can differ from the Einstein metric involved in the Einstein-Hilbert term with a constant Newton constant:

Bekenstein (1992) showed that causality and the weak equivalence principle restricts the form of the auxiliary metric:

What is the physics associated with the disformal coupling B(φ,X) ?

Expanding the effect of the disformal coupling to leading order, we obtain a direct coupling with the energy-momentum tensor:

Gravity tests involving static objects are not affected by the disformal coupling.

The only interaction that the disformal coupling induces is a quantum effect. The disformal coupling to the energy momentum tensor is a higher order operator involving two scalars, they can form a loop and lead to a loop induced interaction. In static situations, especially at short distances, this is the leading effect of the disformal coupling.

For matter made out of fermions, the interaction vertex is given by:

This leads to a one loop interaction between two particles, proportional to their masses:

The calculation and the result is similar to the relativistic Van der Waals interaction between polarised molecules.

The Eot-wash experiment constrains the extra interaction compared to Newton’s law down to a distance of 55 μm:

The Casimir force between a plate and a sphere can also be affected. Best tested at the 5% level for d=0.5μm with a sphere of radius R=11.3 cm and a plate of width a=0.5 cm:

One must go to shorter distance scales to feel the full effect of the disformal interaction. The first obvious place is atomic physics where new interactions displace the energy levels of atoms.

One of the most sensitive tests is the 1s-2s gap for hydrogen at the one billionth of eV level:

The disformal interaction could also play a role in the proton radius puzzle… One can test it to even lower scales in neutron scattering experiments between one neutron and neutral gases like Argon where one measures the scattering asymmetry between the scattered and back scattered neutrons with an angle of 45 degrees. The bound is weaker than the atomic bound. Then there are astrophysical effects (burning of stars)…. and even particle physics consequences.

Where do Galileons come from?

They can be obtained in the low energy decoupling limit of bigravity

Taking the limit

One gets Galileon-like theories are low energy!

Conclusions

Light scalar fields could be what remains from massive gravity, string theory …. They need to be screenedin the local environment otherwise tight bounds would be violated.

This prompts one to study the screening mechanism from a bottom-up approach irrespective of their UV completion, if ever needed.

For conformally coupled scalars, there are only three main mechanisms: Vainshtein, K-mouflage and chameleon+Damour-Polyakov.

Galileons play a special role as they are not renormalized.

Their phenomenology is rich and depends on the type of coupling to matter.