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 is 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 modify ing th e ri gh t erm T , the ma tter Quintessence, K-essence, Tachyons, Chaplygin gas, coupled models with dark matter, etc. (-1<w<1 <w < -1/3) Either modifying the left term G, the gravity/geometry f(R) gravity models , Tensor -scalar models, brane worlds , massive gravity, inhomogeneities, etc. (w < -1 possible) With sometimes overlap between the two types of solution The various possibilities A cosmological constant, with w =-1, standard CDM But also anyy, new fluid, or scalar field , with p=w, and w < -1/3 The simplest is the scalar field , with potential V(), pressure and energy given by V() 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 dynamica l systems, w hose evo lution 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, but 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 Could t his 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!