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“Varying Constants” workshop Leiden May 2009. Fundamental Scalar Fields. Outline. 1-Higgs Field: symmetry breaking, inflation and dark energy (and possibly the first scalar field to be detected) 2-Moduli Fields (string inspired models)

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“Varying Constants” workshop

Leiden May 2009

Fundamental Scalar Fields

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1-Higgs Field: symmetry breaking, inflation and dark energy(and possibly the first scalar field to be detected)

2-Moduli Fields(string inspired models)

3- f(R) theories and ChameleonFields(modifications of gravity)

4-Quantum Difficulties(The mass and cosmological constant hierarchy problems)

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The Higgs scalar

Scalar fields are particularly useful as they can have a vacuum expectation value which preserves Lorentz invariance.

Vector and Fermion vevs would break Lorentz invariance.

Scalar vevs can break gauge symmetry:

The Higgs field

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Symmetry Breaking

  • The Higgs field has a Mexican hat potential:

  • The vev at the minimum is given by:

  • In general the Higgs field is coupled to matter fermions and gives a mass to the quarks and the leptons

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Higgs Inflation

  • What if the Higgs coupled to gravity as well as to fermions?

  • The only gauge invariant and dimension four operator is:

  • Going to the Einstein frame where Newton’s constant is constant via

  • This is constant and drives inflation for large values of H

Time dependent Newton’s constant

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Higgs Dark Energy

  • Who says that the Higgs vev is not a field χ?

  • This field may also couple to gravity:

  • Initially, the Higgs field is driven towards the minimum of the valley and inflation happens

Quantum corrections

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Exponential Higgs Quintessence

  • Along the Higgs valley, the Einstein frame potential is not flat:

  • Normalising the field,

    the potential becomes exponentially decreasing:

  • Such a runaway potential leads to the acceleration of the universe as χ goes to large values.

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Nasty Beasts: the Moduli

KKLT scenario

Beyond the standard model physics leads to the existence of myriads of scalar fields: the moduli

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A few examples

The dilaton is a moduli as string perturbation theory does not fix its value:

Dilaton gauge coupling constant

Cosmological variation of the dilaton would lead to a time variation of the fine structure constant

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A few examples

Moduli can have a direct geometric origin, e.g. shape moduli (“volume” )

ShapeNewton’s constant

This would lead to a time variation of masses and the possibility of a variation of the electron to proton mass ratio

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Moduli Problems?

No massless scalar particles have ever been observed. Their masslessness would cause many problems. Need to generate a potential for the moduli.

Moduli can be useful to generate both inflation and darkenergy (flat potentials).

If the potential is too steep, then need to find a minimum where the field is stabilised with a finite mass.

How massive?? (in the stable case)

Which potential?? ( inflation or dark energy)

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How Massive?

Long lived moduli couple gravitationally with ordinary matter, leading to the presence of a new Yukawa interaction:

Unless the coupling is dynamically reduced (chameleons), fifth force experiments imply that the mass of moduli must be

corresponding to a range lower than a millimetre.

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Long Lived Moduli

Gravitational problems are only present for long-lived moduli.

If decay rate:

smaller than the Hubble rate now, moduli have not decayed, i.e.

moduli behave like matter in the radiation era. Its energy density would dominate now unless

Low mass moduli with gravitational couplings are not compatible with cosmology.

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Short Lived Moduli

Moduli can be very massive and may have decayed well before the present day.

The decay must happen before Big Bang Nucleosynthesis as it releases an enormous amount of entropy

The reheat temperature due to the decay is greater than 1 MeV (BBN) provided:

very heavy moduli are favoured.

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Runaway Moduli?

If stable then moduli must be very massive- TeV range.

Another attractive possibility: runaway behaviour. Possibility of generating inflation and/or dark energy

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The w sieve

Energy density and pressure:

Runaway fields can be classified according to

very fast roll

slow roll (inflation)

gentle roll (quintessence)

strong gravitational constraints

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Which Potential?

A very interesting class of moduli appear as they parameterise flat directions in supersymmetric theory. Necessitates non-perturbative arguments like in ordinary QCD.

Prototype non-perturbative results occur in SUSY QCD

The low energy physics is well described by the physics of mesons (condensates of quarks and antiquarks)

The flat direction is lifted by the superpotential:

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Ratra- Peebles Quintessence

During matter era, moduli with potential

converge to an attractor

whose equation of state is

If α not too large, can generate w close to -1.

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Modular Inflation

Models where moduli play the role of the inflaton

Uses the the potential in the imaginary part of T (whose real part is the radius of compactification) to generate inflation.

The racetrack model uses exponential potentials:

The inflaton starts from a saddle point and rolls down towards the Minkowski minimum.

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Effective Potential

  • All the racetrack potentials are described by an effective potential around the saddle point

  • The spectral index is a function of the η parameter at the saddle point only.

  • The error is of the order of 1 per mil explaining the robustness of the spectral index in racetrack inflationary models.

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f(R) gravity

  • General Relativity well-tested down to scales 0.1 mm and up to subgalactic scales.

  • Could it be that dark energy reflects a complete misunderstanding of gravity?

  • The simplest modification is f(R) gravity (and ghost free):

  • The function f(R) must be close to R, so f(R)= R+ h(R), h<< R in the solar system.

  • f(R) gravity addresses the dark energy issue for certain choices of h(R).

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f(R) vs Scalar-Tensor Theories

f(R) totally equivalent to aneffective field theory with gravity and scalars

The potential V is directly related to f(R).

Same problems as dark energy: coincidence problem, cosmological constant value etc…

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A Few Examples

A large class of models is such that h(R) C for large curvatures. This mimics a cosmological constant for large value of

Another class of models leads to a quintessence like behaviour:

Ratra-Peebles ! n=-(p+1)/p

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Gravitational Tests

Scalar-tensor theories and f(R) theories suffer from the potential presence of a fifth force mediated by the scalar field.


Non-existent if the scalar field has a mass greater than :

If not, strong bound from Cassini experiments on the gravitational coupling:

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Dark Energy Effective Theory

Effective field theories with gravity and scalars

deviation from Newton’s law

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Variations of Constants

  • Scalar tensor theories predict a variation of the fundamental particle masses:

  • The gauge coupling constants are field-independent.

  • The mass of the proton is essentially a pure gluon condensation effect in Quantum ChromoDynamics, hence constant too.

  • The electron to proton ratio is field dependent:

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The Chameleon Mechanism

When coupled to matter, scalar fields have a matter dependent effective potential

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Chameleon field: field with a matter dependent mass

A way to reconcile gravity tests and cosmology:

  • Nearly massless field on cosmological scales

Massive field in the laboratory

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Chameleon Cosmology

Electron kick during BBN

Late time acceleration

Possibility of variation of constants

Lurking cosmological constant

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Loop Corrections

Quantum fluctuations destabilise all the previous results

Cosmological constant problem

Hierarchy problem (Higgs mass)

Large contributions due to scalars

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Higgs Corrections

  • At the very least there is a quantum correction due to the Higgs field

  • The standard model of particle physics is not valid up to the Planck scale. Perturbativity of the Higgs coupling λ implies that the cu-off scale is larger than 1 TeV.

  • The correction due to the mass of Higgs

  • is given by:

  • The vacuum energy is shifted to a large value. The mass of the Higgs field is also shifted to large values.

  • The mass hierarchy problem can be solved using supersymmetry. There is no known solution to the vacuum energy hierarchy problem. This is the cosmological constant problem.

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  • Scalar fields are too easy to obtain! Especially in string inspired models.

  • They can easily lead to inflation and dark energy, possibly leading to the variation of constants.

  • Unfortunately, they are associated with quantum problems. The LHC results on the Higgs sector may shed light on this issue, hinting at a solution of the mass hierarchy problem. This would be of tremendous relevance to both inflation and dark energy model building.