Scalar fields : welcome to wonderland -...
Transcript of Scalar fields : welcome to wonderland -...
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
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.
… 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
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
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
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
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
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
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
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
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
Inspiral phase
Key parameter : chirp mass M = (m1 m2)3/5
(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
Key parameter : chirp mass M = (m1 m2)3/5
(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)
Using the electromagnetic counterpart
Allows both a measure of the direction and of the redshift
Holz and HughesδdL/dL
0.5%
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
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 : version 4
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