scalar fields : welcome to wonderland -...

Scalar fields : welcome to wonderland

Pierre Binétruy, APC, Paris Diderot

Zuoz, 15 July 2008

The early days with an artillery officer

LISA = LHC in the sky

The bonnes à tout faire of cosmology

Leftovers from unification

« Mr Galileo was correct in his findings »

Collisions in the cosmos

Higgs particle discovered at CERN

A Higgs event in CMS

Higgs particle discovered at CERN

When (if) the Higgs particle is discovered…

the first fundamental scalar field will be discovered.

This will confirm a picture of our world that has now developped much beyond the Standard Model.

A story that started 80 years ago…

… with SchrödingerE = P2/2m

E ↔ i h ∂/∂t ∇

P ↔ - i h ∇ → →

→

ih ψ = - Δψ∂

∂t 2mh2

(1926)

… with SchrödingerE = P2/2m

E ↔ i h ∂/∂t ∇

P ↔ - i h ∇ → →

→

ih ψ = - Δψ∂

∂t 2mh2

trying to get a relativistic equivalent of his famous equation

E2 = P2c2 + m2c4 ( + ) ψ = 0 m2c2

h2

≡ - ∇2∂2

∂t2

(1926)

(1926)

→

• Classical Schrödinger equation

Probabilistic interpretation:

ρ = ψ* ψ positive definite

J = - (ψ* ∇ ψ - ψ ∇ ψ* )

∂ρ /∂t + ∇. J = 0 → →

→→

→ ih2m

• Classical Schrödinger equation

Probabilistic interpretation

ρ = ψ* ψ positive definite

J = - (ψ* ∇ ψ - ψ ∇ ψ* )

∂ρ /∂t + ∇. J = 0 → →

→→

→ ih2m

• Relativistic Schrödinger equation

J = - (ψ* ∇ ψ - ψ ∇ ψ* )

ρ = - (ψ* Δ ψ - ψ Δ ψ* ) not positive∂ρ /∂t + ∇. J = 0

→ → →

→→2mc2

2m

ih

ih

This led Dirac to propose his equation...


In any case, the relativistic Schrödinger equation could not describe the electron because it did not include the spin in the description.

( + ) ψ = 0 m2c2

h2Klein-Gordon equation

describes a spinless particle i.e. A SCALAR FIELD


In any case, the relativistic Schrödinger equation could not describe the electron because it did not include the spin in the description.

( + ) ψ = 0 m2c2

h2Klein-Gordon equation

describes a spinless particle i.e. A SCALAR FIELD

N.B. Q = e ∫ρ d3x = e ∑k (nk- - nk

+)nk

- ≡ a†k ak nk

+ ≡ b†k bk

But does this equation describes a fundamental field?

Yes, if the Higgs is discovered at LHC or rather

φ = - V’(φ)

V(φ) scalar potential :

V(φ)

h = c = 1

Higgs production at LHC

Can we probe the electroweak phase transition?

Is it second order i.e. smooth?

or first order?

Difficult to tell at LHC !

T

• in the Standard Model, requires mh < 72 GeV (ruled out)• MSSM requires too light a stop but generic in NMSSM• possible to recover a strong 1st order transition by including H6 terms in SM potential• other symmetries than SU(2)xU(1) at the Terascale (→ baryogenesis)

Pros and cons for a 1st order phase transition at the Terascale:

If the transition is first order, nucleation of true vacuum bubblesinside the false vacuum

Collision of bubbles and turbulence → production of gravitational waves

A quick tutorial on graviton (i.e. gravitational wave) production

Ω = ρ-1 dρ/dlogf

If gravitons were in thermal equilibrium in the primordial universe

γ

g

A quick tutorial on graviton (i.e. gravitational wave) production

Ω = ρ-1 dρ/dlogf

If gravitons were in thermal equilibrium in the primordial universe

γ

g

A quick tutorial on graviton (i.e. gravitational wave) productionare not

When do gravitons decouple?

Interaction rate Γ~ GN2 T5 ~ ----T5

MPl4

Expansion rate H ~ ----

---- ~ ----

T2

T3

MPl

MPl3

Γ

H

Gravitons decouple at the Planck era : fossile radiation

(radiation dominated era)

Gravitons of frequency f* produced at temperature T* are observed at a redshifted frequency

f = 1.65 10-7 Hz --- ( ----- ) ( ---- )ε1 T*

1GeV

g*

100

1/6

At production λ* = ε H*-1 (or f* = H*/ ε)

Horizon lengthWavelength

LF band0.1 mHz - 1 Hz

Gravitational wave detection

VIRGO

for ε=1ΩGW = --- --------

d ρGWd logf

ρc

1 , ρc = 3H0/(8πGN)

for ε=1ΩGW = --- --------

d ρGWd logf

ρc

1 , ρc = 3H0/(8πGN)

Gravitons produced at the electroweak phase transition should be observed in the LISA window.

α = ---------Efalse vac

aT*4

radiation energy at transition

Two basic parameters todiscuss the dynamics:

β= time variation of bubble nucleation rate

β-1 > 10-3 H-1

duration of phase transition

(ε ~ πH/β)

α = ---------Efalse vac

aT*4

radiation energy at transitionh0

2 ΩGW

f in mHz

turbulence bubble collision

Nicolisgr-qc/0303084

Two basic parameters : β= time variation of bubble nucleation rate

β-1 ~ 10-2 H-1

duration of phase transition

V(φ) = λ(φ*φ-v02)2/2

Cosmic stringµ mass per unit length :

φ = v0 einθ

v0

φ = 0µ ~ v0

2

value of GNµ

v0 ~ 1014 GeV

v0 ~ 1012 GeV

LISA position

Earth-LISA distance50 million kms

gij = Ψ4 δij

Pretorius 2005

Re Ψ4 r in z=0 plane

energy

LHC

intensity

100 GeV1 GeV

SU(3)

U(1)

SU(2)

strong int.

e.m.int.

weak Int. UNIFICATION?

The bottom-up road to unification

Unification

energy

LHC

intensity

100 GeV1 GeV

SU(3)

U(1)

SU(2) FUNDAMENTALTHEORY

top-down viewpoint

The natural low energy theory is an abelian gauge theory:electrodynamics (i.e. the quantum theory of electromagnetism) …

energy

LHC

intensity

100 GeV1 GeV

SU(3)

U(1)

SU(2)

mass of scalar field

top-down viewpoint

FUNDAMENTALTHEORY

The natural low energy theory is an abelian gauge theory:electrodynamics (i.e. the quantum theory of electromagnetism) …without a fundamental scalar field!

Λ

∫Λ

d4kk2 - m2

~ Λ2

Fermions are protected by chiral symmetry e.g. uL and uR transform differently.

energy

LHC

intensity

100 GeV1 GeV

SU(3)

U(1)

SU(2) Mass of scalar field

Symmetry can protect the mass of a scalar field: e.g. SUPERSYMMETRY

FUNDAMENTALTHEORY

Then proliferation of scalar degrees of freedom: one for each fermionic degree

Inflation scenario proposed first in the context of the phasetransition associated with grand unification (Guth, 81)

Fluctuations in CMB predicted at the level observed by the COBE satellite : V0 = ε1/4 6.7 1016 GeV

ε slowroll parameter :2ε=(MPV’/V)2 « 1

Cosmology

Scalar fields are the « bonnes à tout faire » of cosmology

Scalar fields easily provide a diffuse background

Speed of sound cs2 = δp / δρ

In most models, cs2 ~ 1, i.e. the pressure of the scalar field

resists gravitational clustering :

scalar field dark energy does not cluster

An illustrative example : dark energy

Models for accelerating the expansion of the Universe

Extended gravity

L = f (R) Brane models

(DGP model)Dark energy

Quintessence

K-essenceRatra-Peebles Exp.

PGB String inspired Brane models

Chaplygin gas Tachyon

Models for accelerating the expansion of the Universe

Extended gravity

L = f (R) Brane models

(DGP model)Dark energy

Quintessence

K-essenceRatra-Peebles Exp.

PGB String inspired Brane models

Chaplygin gas Tachyon

All scalar fields!

Example of quintessence :

V

ϕ

Acceleration of expansion if w = pϕ/ρϕ = < -1/3

if ϕ2 < V(ϕ)

ϕ2/2- V(ϕ)

ϕ2/2+ V(ϕ).

.

.

V

ϕ

Masse m2 ~ V’’ ~ V/mP2 ~ H0

2 ~ (10-33 eV/c2)2

Quintessence thus mediates a very long range force: H0-1 ~ 1026 m

→ Very weakly coupled to ordinary matter → invisible at LHC

mP

Dark energy scalar field might still be coupled:

• dark matter• neutrinos

Looking for standard candles

Gamma ray bursts

Determine the luminosity through a relation between the collimation corrected energy Eγ and the peak energy

Supernovae of type Ia

mB = 5 log(H0dL) + M - 5 log H0 + 25

Coalescence of supermassive black holes

Inspiral phase

Key parameter : chirp mass M = (m1 m2)3/5

(m1 + m2)1/5(z) (1+z

)

Inspiral phase


(m1 + m2)1/5

Amplitude of the gravitational wave:

h(t) = F (angles) cos Φ(t) M(z)5/3 f(t)2/3

dL

Luminosity distance

frequency f(t) = dΦ/2πdt

(z) (1+z)

Inspiral phase


(m1 + m2)1/5

Amplitude of the gravitational wave:

h(t) = F (angles) cos Φ(t) M(z)5/3 f(t)2/3

dL

Luminosity distance poorly known in the case of LISA

Δθ~ 10 arcmin 1 HzSNR fGW

(z) (1+z)

z = 1 , m1 = 105 M, m2 = 6.105 M

δθ (arcminutes)

δdL/dL

3°

5%

Holz & Hughes

Using the electromagnetic counterpart

Allows both a measure of the direction and of the redshift

Holz and HughesδdL/dL

0.5%

Dalal et al. astro-ph/0603275

3000 supernovae

100 SMBH sources

Because the scalar fields that we have considered are singlets, they might appear in the coupling of matter to the spacetime metric

In Einstein frame, the matter action would read :

Smatter = S ( ψmatter, A(φ) gµν)

our scalar field!

Alternate theory of gravity: scalar-tensor theory

Equivalence principle :

mi a = mg g ⇒ a = g

inertial mass mi = gravitational mass mg

Universality of free fall

Test of equivalence principle : version 0

Astronaut David Scott from Apollo 15

Astronaut David Scott from Apollo 15

« That proves that Mr Galileo was correct in his findings! »


ZARM tower in Bremen


Buzz Aldrin

lunar retroreflector from Apollo 11

Lunar Laser Ranging

The Apache Point Observatory Lunar Laser-ranging Operation (Apollo)

distance Earth-Moon known to 50 ps/1 cm level test of equivalence principle at a 10-13 level

Microscope satellite


Test of equivalence principle at 10-15 level

• Fundamental scalar fields are everywhere in a theorist world.

• Do they also exist in the real world?

• If the Higgs does not exist as a fundamental scalar field, arethe other scalar fields part of our world? If not, what plays their role?

Conclusion

Sweet dreams …

Sweet dreams …

…with the scalar fields!

scalar fields : welcome to wonderland -...

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