Chaire Galaxies et Cosmologie
Cosmological constant, or quitintessence?
Françoise Combes Outline Formulation of the problem Scalar field ((qcosmon or quintessence ) with Equation of state P=w w < -1/3
Example of cosmon, coupling with neutrinos The chameleons The Galileons
Princippq,al models of quintessence, and their limits K-essence Chaplygin gas The T ach yons
Conclusion Vacuum energy
• The vacuum energy must be a constant, depends on nothing • It is necessary to provide energy to pull the piston, since there is more energy afterwards • If the piston provides energy , negative pressure P = - c2 The vacuum « gravity » is negative
For a gas, N ~1/V decreases The pressure pushes the piston How to explain this value
72 4 zero pt = 10 GVGeV , if th e cut-off i s at Pl anck scal e =2 109 GeV4 if the cut-off is at the electroweak interaction One observes ~(2~ (2×10-3 eV)4, iei.e. 10-47 GeV4 Problems of Fine tuning Temporal coincidence
One must explain the passage From dominant matter Dominant energy at z=0.5 Why now? The problem of the coincidence is the more striking as we speak of derivatives: is zero in the past, increases very recently, (z=0.5, 5Gyr) to become 1 in the future.
a = 1/(1+z) Two solutions
The dark energy problem can be solved in 2 ways
Eith er you add g ihihhdin the right hand or you subibtract it on thlfhe left hand of Einstein equation
G 8GT Either modifyi ng th e ri gh t erm T , the ma tter Quintessence, K-essence, Tachyons, Chaplygin gas, coupled models with dark matter, etc. (-1
The kinetic term must be small, slowly rolling
The conservati on equati on i s which yieds the Klein-Gordon equation
(relativistic Schrödinger equation) Equation of state w=1 The equation of state P=w, writes
According to extreme cases: Dominant kinetic energy w 1 rigid fluid Equipartition w 0 CDM, baryons Dominant potential energy w -1 cosmological constant Negative kinetic energy w < -1 phantom
The equation of state varies with time One can write, in the linear approx (a(t) is the Universe radius)
w = w0 + w1 z w = w0 + w1 (a-1) Opinion of Lev Landau on cosmologistes
Often in error, Never in doubt !
Proceed by elimination
Static Universe, or H0 = 500km/s Cosmic strings, topological defects Phase transition, baryogenesis Magnetic monopoles Hot dark matter Unstable particles Extra-dimensions, … When in doubt, add a scalar field!
Classical physics: a scalar field like a potential, a tempp(gerature (its gradient is a vector field, ,) a force)
Quantum field theory (QFT): Scalar field, of spin 0, the only one in the standard model and the most famous: the Higgs field The interaction with the field gives mass to the particles
The ppgg,hoton does not interact with the Higgs field, and has a zero mass The mass of all fermions is due to thiiheir interacti on wi ihhth the Hi ggs fi fildeld V() The Higgs has a mass by a break of symmetry Characteristics of the scalar field
Action of the scalar field
with the Lagrangian Dark Energy
For such a scalar field, the propagation speed (sound speed) is
cs = p/ = speed of light c The fluid oscillates,,p, and does not collapse, Its own pressure resists gravity
Matter
Perturbations of Klein Gordon’s equation in Fourier space At scall scale (large k = 2 ), the term k 2 dominates
The perturbation oscillates around zero, as an acoustic wave Mass of “cosmon” or quintessence
The excitation of the quintessence field is a particle of quintessence, or cosmon
In quantum field theory, one never sees a particle “alone” but the collective effect of all summed interactions with all possible particles or virtual fields Fine tuning of the mass The effect of these virtual particles is to increase the mass Unless there exists a symmetry to decrease it which is the case for stable known particles
Except the Higgs, which precisely is very massive (broken symmetry) A massive field rolls very rapidly its potential V() For the quintessence, a small mass is required -33 -60 m ~10 eV ~10 MPl
Slow roll V() Problem of interactions
Besides the field should interact with all other particles/fields It is difficult to keep a new field completely isolated
If the field interacts with one of them Interacts with all In particular the particles of the standard model Constraints on the 5th force, and variation of constants Solution: broken symmetry
These two fine tunings, on small masses and interactions can be solved out by a slightly broken symmetry
+ 0 The quintessence would be a pseudo-boson of Nambu- GldGoldstone, w ihith a si nusoid idlal V() potential and by nature a small mass and weak interactions (the Nambu-Goldstone boson mass=0 spin=0, in condensed matter, fluid s, ph onons) Carroll 2009 Observational constraints Interaction coupling the quintessence to electromagnetic fields
This interaction produces a cosmological birefringence: The polarization vectors of E, B fields should rotate during their travel across the quintessence field Rotation independent of frequency
Constraints from CMB WMAP, Planck rotation < 2° Radio-galaxy GRB
Radio-galaxies and GRB iiinteresting constrai nts Galaverni et al 2016 Problem of the mass
• Suppose that all mass parameters are proportionnal to the scalar (quantum fields , superstrings , unification …) Dark Energy • Mp~ , mproton~ , ΛQCD~ , MW~ ,… • may evolve with time: cosmon (t) -7 • mp/M : almost constant: constraints from observations at 10 ! • Only mass ratios are observable
• can correspond to the transition N-Dim to 4D universe (actual) • Or to the quantum anomaly of dilatation (no scale invariance) E >0 quintessence
Let us take and example of quintessence Cosmon and neutrinos: As the transition radiation-matter occurs at z= 4000 (equivalence) -4 -3 because rad a , and m a , one can think of another transition , with a fluide whose mass varies with time m(a) a3
-3(1-) Suxh that its density g a , selected so that the transition is z=0.5 A possibility could be the neutrinos (Amendola, Baldi, Wetterich 2008-13)
In this model, the quintessence of the scalar field « cosmon » , is coupled to neutrinos , and this coupling makes the mass m vary At the transition, the field tends to a constant acceleration
The field is of nature « tracker », i.e. whatever the initial conditions, the evolution is towards and « attractor » (d(as in dynamical systems, wh ose evol ution is irreversibl e, Cf Theory of chaos) radiation Temporal evolution matter Radiation dominates a(t) varies as t1/2 energy Aftger equivalence, a(t) varies as t 2/3
time -3 -2 • ρm ~ a t unididbiverse dominated by matter t-3/2 radiation era
-4 -2 • ρrad ~ a t universe dominated by radiation
The very large ratio matter/radiation today is due to the large age of the Universe Same explanation for dark energy? Neutrinos coupled to cosmon
In the case where is large (~4), neutrinos would be massive today, b ut o f neg ligibl e mass at th e epoch of structure f ormati on cannot prevent the formation of galaxies
The coupling of cosmon with baryons << gravity with dark matter << ggyravity (constraint from equivalence principle, 5th force, etc.) The coupling of cosmon with neutrinos > gravity
2 -68 -60 The cosmon mass is negligible mc ħH0 or 10 kg ~10 Mpl Very long range interaction
H= 2/3 (1+)/t Neutrinos are different
• If the coupling baryons -cosmon is strongly constrained by the tests of equivalence principle and variation of fundamental constants No constraint on the coupling neutrino-cosmon
• In particle physics : the mechanism to acquire mass for the neutrinos differs from that for charged fermions. The mechanism of “seesaw” involves heavy particles whose mass could depend on the value of the cosmon field. Mass of neutrinos
In the standard model of particles, neutrinos have zero mass. How to explain their observed mass ( ~eV) millions times lower than then of leptons? Either normal hierarchy m1 << m2 << m3, with m3 = 005eV0.05 eV Either inversed m1~m2 >> m3, m1 = 0.05 eV, or degenerate m1~m2 ~m3
Mixture Not known wether (Majorana) or Dirac The mass of fermions is created by a break of symmetry (Higgs) , but this does not work for neutrinos -5 Coul d this be due to g rav ity? No, s ince t he n m = 10 eV The “Seesaw”mechanism
Type I: Hypothesis of very massive “sterile” neutrinos
RH (right-handed), non interacting with EW
The masses are the eigen values of a 2x2 matrix, with M, and m M >> m, keeping the product constant (Mm) , implying the seesaw In the majorana case, M ~scale of GUT, for Dirac, only EW
Type II, Higgs triplet charged Leptons
Neutrinos LH, light
Sterile neutrinos RH, heavy Seesaw diagram WhWhy neutri nos?
Mass scales : Dark energy density : ρ ~ ( 2×10-3 eV )-4. Neutrino’s mass : eV or below
Cosmological declic : Neutrinos become non-relativistic rather late in the history of Universe (z~300, or z=0.5 if m variable)
The coup ling of neut ri nos with th e cosmon could b e stronger than with gravity The growth of neutrino’s mass leads to the transition of dark energy to a q uasi constant value As soon as the neutrino becomes non-relativistic, there is the declic This moment has occurred recently, 5 Gyr ago! Yields the right scale for dark energy!
Mass growth of neutrinos
Amendola et al (2008) m = 2.3 eV 0 Evolution of the model a = 1/(1+z)
CDM DE CDM
rad neutrinos DE baryons baryons neutrinos
z=103 z=0 Energy of neutrinos ttltotal
Mass of neutrinos Cosmon+neutrino Non-rel cosmons
Amendola et al (2008) Evolution of densities
T0= 13.5 Gyr ~ a-3 matter’ ~ a-4 photons
t-2 t-2
quintessence The quintessence becomes equal tttlto matter only recentl y
It then becomes asymptotically constant 27 Formation of structures with neutrinos Simul ati on of th e coupli ng cosmon+ neut ri no, until z=1
CDM
Velocity increasing
N-body simulation Baldi et al 2011 Neutrinos at larger scale
CDM
Velocity
increases
Baldi et al 2011 Formation of structures with neutrinos The coupling cosmon-neutrino is equivalent to a force which attract neutrinos, then forming clumps These parti cl es b eh ave lik e CDM + d ark energy
Ayaita et al 2013 back-reaction : T of neutrinos There is a big difference between the background evolution (hypothesis of homogeneity ---) And the real world with cosmon evolution (______)
Pressure
neutrinos
Radius (Mpc) Ayaita et al 2012 Chameleons for dark energy
Particles with a variable mass, varying as a function of environnement. Masseff increases with density It has thus a large mass in the solar system, with a small interaction range (1mm). At large scale, the range is much larger (> kpc) escape the detection as a 5th force
Addition o f sca lar field s V() iththin the theory
Departure from standard model
Translated into a force
F = Gmm’/r / r2 (1 + ’ e-mr)
Khoury & Weltman 2004 Veff() = V() + m A() Chameleon potential : coupling
The coupling with matter changes considerably V() The chameleon camouflages!
Total
Field alone coupling
coupling
Low didensity Hi ihdigh density Chameleon principle: screening Sys tem w ith sph eri cal symmet ry: Khoury and Weltman (2004) The force due to chameleons is (6/2)lnF only that of a shell (r) V() (RF f )/2F 2 où
B A Inside and outside of the system Minima and maxima at
QA QB V, ( A ) Qe A 0, V, ( B ) Qe B 0 If the field has a large mass , the system ≡ a thin shell r : Parameter of the shell c B A 1 c gravitational rc 6Qc potentilial at the surface Constraints in the solar system
The constant of effective gravitation
Geff= G(1+G ( 1 + rc/rc) Constraints provided by experiments on the Casimir force
r The measures in the solar system yield c B 1.15 105 rc 6c 6 For the Sun (),c 10 V()eff Haute-densite 11 (massif) B 6.0 10 B . This is satisfied if M 2 1/(3 f ) . ,RR Faible-densite (0)(mass~0) . is large in the region 2 R H0 Detection tests The formula for the scalar field are similar to that of the axion There exist experiments for detecting the axion A laser beam crosses a zone of strong magnetic field The photons create chameleons. Slowed down, they remain a while in vacuu m chamber. After a certain time photons to detect
Experiment GammeV (FermiLab, Chou et al 2008)
The chameleons do not cross: too massive , no tunnelling effect Their reflections are incoherent with those of photons Burrage et al 2015 Test in a vacuum chamber Laser beam, atoms of cesium, inner vacuum sphere (10cm): atomic interference Since the chameleon effect depends on density , the range of their force is maximum in vacuum Experime nt w ith an alu miniu m sp h ere in w hic h the best vacuum is made
A beam of cesium atoms is seppyared by the laser The laser pulses make the atoms to interfere with each other These atomic interferences should be perturbed by the chameleon’s force negative result
Could still be improved by a factor 10! Upper limits on chameleons The atoms of cesium are not numerous enough to weaken the chameleon’s field
force of the field, M the coupling Mp = 2 1018 GeV
M Already around g Tomorow well below Simulations of chameleons
NhNo chame leon e fftffect
Oyaizu, Lima, Hu (2008) Theories of Galileons Scalar fields Goon et al (2014)
The Vainshtein screening mechanism suppress the 5th force within a certain radius around massive sources, where the regime becomes hig hly no n-linee(dscouywear (discontinuity when m-ggvrav 0)
The galileon theories can be interpreted naturelly as branes traveling in space-times with extra dimensions
Galileons ~ Goldstone modes occuring when some symmetries of the space-time are spontaneously broken The ppyqhenomenon is described by equations which remain of second order, which implies the absence of ghosts and intabilities Galileons and Goldstone modes After a symmetry break, the energy is transferred in oscillations of low energy, the Goldstone modes Link ed to th e G old stone b osons ( m=0 , spi n =0)
There exists also a screening effect with Galileons Analogus to the Vainshtein mechanism
the L agrangi an i s b uilt t o sati sf y th e G alil ean symmet ry In a flat universe The galileons not favored by observations Ga lileons:
0.0
Galileon Late-time -0.50 tracking
f(R) Galileons, only the -1.0 solutions late-time tracker w are allowed
-1.5 Solutions simple tracker Galileon ruled out Tracker -2.0 Future Present
0.1 1 10 1 + z The proto-type of quintessence
A typical model, where all main characteristics are discussed
V() Two free parameters:
M: energy scale n: power index
p = ½ (d/dt)2 − V() 0 must be negative, thus d/dt must be weak V() slowly variable Models of quintessence
Quintessence, where w depends mainly on potential
(a) Freezing models (b) Thawing models . ege.g. PNGB Pseudo-boson (Nambu- GldtGoldstone) . w decreases until -1 w increases from -1
k-essence, where w depends mainly on kinetic energy Typically, the evolution of w is similar to models (b) of thawing Freezing -- Thawing
Allows to distinguish 2 types of quintessence whatever the model V( ) adopted as a function of w, w’
(1) The Hubble damping ( ) would have frozen the system out of equilibrium, Models begin above, but end below the limit after z=1 (w=-1) and today is thawing
P/= (2) System moving at the start then slows down and freezes w -1 Caldwell & Linder 2005 Typical examples Thawing Freezing
V() = V1 (1+ cos /a) V() = M5/
approx
NilNumerical « tktracker » solution
z=0 z=0 e.g. PNGB Pseudo-boson Type of the sup ersy mmetric (Nambu-Goldstone) theory SU(Nc) Mass of field very low Nc colors 2 -33 -69 m =V= V’’ Several models represented V ~n, n=1,2,4 In order to separate them with observations, the resolution required in dw/dlna is of the order of (1 +w) !! For Then the models should have 1+w > 0. 004 (thawing) 1+w > 0.01 (freezing) V ~n, n <0 Limits of quintesse nce Caldwell & Linder 2005 There exist « tracker » solutions K-essence Quintessence: 5th element, after the baryons, CDM, photons & neutrinos the quintessence corresponds to a field k-essence, kinetic-essence The neggpative pressure of dark ener gy would be due to the non-linear term of the field kinetic energy The cosmon field has an evolution W=-077td0.77 today following that of the underlying back-ground radiation Can change the evolution DE of the various era: radiation, matter matter, etc. Armendariz-Picon et al 2000-01 K-essence (following) According to the back -ground, these solutions are called « tracker » -- duringg, the radiation era, the k-essence is not dominant, but its evolution is parallel to that of radiation. The density ratio remains fixed vs radiation (for example 1/100). The ratio close to 1 would be due to some kind of energy equipartition -- during the matter era , where p=0 , the k -essence cannot follow and remains frozen, with constant density -- at the end, the value relaxes towards an asymptotic one corresponding to -1 < w < 0 -- the « tracker » solution is an attractor in the sense that the system tends to the same solution , whatever the initial conditions The principle of tracker Very large range of initial conditions, which lead to the same final state = 070.7 radiation Tracker baryons solution= Attractor without CDM fixed point Zlatev et al 1999 Quintessence Tracker CDM + cosmological constante Zlatev et al 1999 K-essence models • Kinetic function k() : parametrizes the details of the k-essence model – k() = k=const. an exponential potential – k( ) = exp (( 1)/α) inverse power law – k²( )= “1/(2E(c – ))” transition • Criterium of natural character essence k(=0)/ k( ) : non extreme today rad mat -unlilliless, special case to explain Armendariz-Picon et al 2000-01 Comparison: Theory and Observations Observations, for Theory Freezing models have problems Thawing models are still compatible with data Model of the Chaplygin gas Gaz of Chaplygin (1904) Generalized Chaplygin gas Chaplygin Corresponds to an unified model where the dark energy and dark matter are a same component: called also UDM quartessence Past: very large (dark matter) Today: small (dark energy) Continuity equation : Parametrization of Chaplygin With the Friedmann equation One can deduce The equation of state P = w , is past: Future: The value today is One can obtain the Chaplygin gas with an action of d-branes evolving in a space-time (d + 2) Chaplygin and the observations must be small enough, unless the sound speed prevents the structure formation Power spectrum of matter =0, 0.1, 0.2 Besides, there must exist entropy perturbations (t(not onl y adi dibtiabatic) Other extensions: viscosity Holographic model… Has a supersymmetric generalisation Other values of P(k) k Large scales Small scales small, too similar to CDM, > 3 would be a superluminic solution (Moschella, 2008) Tachyonic models v/c = pc/ (p 2c2 +m+ m2c4)1/2 Tachyons negative mass m For a scalar field M2 = V’’() unstable ? Construction of a scalar field, with the same evolution as the Chaplygin gas Certain elaborated models (Gorini et al 2004) of tachyonic fields can explain the acceleration of expansion, and end in a « Big Brake » Big deceleration A. Sen 2005 Larggye variety of behaviours For w < 0, the system crosses acceleration phases, then deceleration phases. Gorini et al 2004 Deceleration Acceleration Deceleration Tachyonic field « Big Brake » Dark Force Conclusion A cosmological constant is still possible But the expected value of the vacuum energy is 60-120 orders of magnitude superior to the observations Better to suppose it equal to zero, and think of a scalar field, a quintessence with a dynamic evolution Two large types of solution: (1) thawing w=-1 at start, then increases <-0.7 (2) fifreezing, w -1, wihith a tracker solilution Correspond to supersymmetric theories, with symmetry breaking (1) A pseudo-Nambu-Goldstone boson, or super-gravity theory Powerful experiments required to discriminate