“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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Leiden May 2009
Fundamental Scalar Fields
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)
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
Time dependent Newton’s constant
the potential becomes exponentially decreasing:
Beyond the standard model physics leads to the existence of myriads of scalar fields: the moduli
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
Moduli can have a direct geometric origin, e.g. shape moduli (“volume” )
This would lead to a time variation of masses and the possibility of a variation of the electron to proton mass ratio
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)
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.
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.
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.
If stable then moduli must be very massive- TeV range.
Another attractive possibility: runaway behaviour. Possibility of generating inflation and/or dark energy
Energy density and pressure:
Runaway fields can be classified according to
very fast roll
slow roll (inflation)
gentle roll (quintessence)
strong gravitational constraints
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:
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.
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.
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…
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
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:
Effective field theories with gravity and scalars
deviation from Newton’s law
When coupled to matter, scalar fields have a matter dependent effective potential
Chameleon field: field with a matter dependent mass
A way to reconcile gravity tests and cosmology:
Massive field in the laboratory
Electron kick during BBN
Late time acceleration
Possibility of variation of constants
Lurking cosmological constant
Quantum fluctuations destabilise all the previous results
Cosmological constant problem
Hierarchy problem (Higgs mass)
Large contributions due to scalars