Cosmological constant, or quintessence?

Transcription

1 Chaire Galaxies et Cosmologie Cosmological constant, or quintessence? Françoise Combes

2 Outline Formulation of the problem Scalar field (cosmon or quintessence) with Equation of state P=w w < -1/3 Example of cosmon, coupling with neutrinos The chameleons The Galileons Principal models of quintessence, and their limits K-essence Chaplygin gas The Tachyons Conclusion

3 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 = - c 2 The vacuum «gravity» is negative For a gas, N ~1/V decreases The pressure pushes the piston

4 How to explain this value zero pt = GeV 4, if the cut-off is at Planck scale = GeV 4 if the cut-off is at the electroweak interaction One observes ~( ev) 4, ie i.e GeV 4 Problems of Fine tuning Temporal coincidence One must explain the passage From dominant matter Dominant energy at z=0.5

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)

6 Two solutions The dark energy problem can be solved in 2 ways Either you add g in the right hand or you subtract it on the left hand of Einstein equation G 8 GT Either modifying i the righ term T, the matter Quintessence, K-essence, Tachyons, Chaplygin gas, coupled models with dark matter, etc. (-1<w< < -1/3) Either modifying the left term G, the gravity/geometry f(r) gravity models, Tensor-scalar scalar models, brane worlds, massive gravity, inhomogeneities, etc. (w < -1 possible) With sometimes overlap between the two types of solution

7 The various possibilities A cosmological constant, with w =-1, standard CDM But also any 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 conservation equation is which yieds the Klein-Gordon equation (relativistic Schrödinger equation)

8 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 = w 0 + w 1 z w = w 0 + w 1 (a-1)

9 Opinion of Lev Landau on cosmologistes Often in error, Never in doubt! Proceed by elimination Static Universe, or H 0 = 500km/s Cosmic strings, topological defects Phase transition, baryogenesis Magnetic monopoles Hot dark matter Unstable particles Extra-dimensions,

10 When in doubt, add a scalar field! Classical physics: a scalar field like a potential, a temperature (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 photon does not interact with the Higgs field, and has a zero mass The mass of all fermions is due to their interaction with the Higgs field V( ) The Higgs has a mass by a break of symmetry

11 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 c s = p/ = speed of light c The fluid oscillates, and does not collapse, Its own pressure resists gravity Perturbations of Klein Gordon s equation in Fourier space At scall scale (large k = 2 ), the term k 2 dominates Matter The perturbation oscillates around zero, as an acoustic wave

12 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

13 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 For the quintessence, a small mass is required m ~10-33 ev ~10-60 M Pl V( ) Slow roll V( )

14 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 5 th force, and variation of constants

15 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- Goldstone, with a sinusoidal V( ) potential and by nature a small mass and weak interactions (the Nambu-Goldstone boson mass=0 spin=0, in condensed matter, fluids, phonons) Carroll 2009

16 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 interesting constraints Galaverni et al 2016

17 Problem of the mass Suppose that all mass parameters are proportionnal to the scalar (quantum fields, superstrings, unification ) M p ~, m proton ~, Λ QCD ~, M W ~, may evolve with time: cosmon (t) m p /M : almost constant: constraints from observations at 10-7! Only mass ratios are observable Dark Energy 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

18 Cosmon and neutrinos: As the transition radiation-matter occurs at z= 4000 (equivalence) because -4-3 rad a, and m a, one can think of another transition, with a fluide whose mass varies with time m(a) a 3 Suxh that its density g a -3(1- ), selected so that the transition is z=0.5 A possibility could be the neutrinos (Amendola, Baldi, Wetterich ) 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» (as in dynamical systems, whose evolution is irreversible, Cf Theory of chaos)

19 Temporal evolution Radiation dominates a(t) varies as t 1/2 Aftger equivalence, a(t) varies as t 2/3 ρ m ~ a -3 t -2 universe dominated by matter t -3/2 radiation era radiation matter energy time ρ rad ~ a -4 t -2 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?

20 Neutrinos coupled to cosmon In the case where is large (~4), neutrinos would be massive today, but of negligible ibl mass at the epoch of structure formation cannot prevent the formation of galaxies The coupling of cosmon with baryons << gravity with dark matter << gravity (constraint from equivalence principle, 5th force, etc.) The coupling of cosmon with neutrinos > gravity The cosmon mass is negligible mc 2 ħh 0 or kg ~10-60 M pl Very long range interaction H= 2/3 (1+ )/t

21 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.

22 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 = 005eV 0.05 Either inversed m1~m2 >> m3, m1 = 0.05 ev, or degenerate m1~m2~m3 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 Could this be due to gravity? No, since then m = 10-5 ev

23 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

24 Why neutrinos? Mass scales : Dark energy density : ρ ~ ( 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) Th li f t i ith th ld b The coupling of neutrinos with the cosmon could be stronger than with gravity

25 The growth of neutrino s mass leads to the transition of dark energy to a quasi 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)

26 m 0 = 2.3 ev Evolution of the model a = 1/(1+z) CDM DE CDM rad baryons neutrinos neutrinos DE baryons z=10 3 z=0 Energy of neutrinos totalt Non-rel Mass of neutrinos Cosmon+neutrino cosmons Amendola et al (2008)

27 Evolution of densities ~ a -3 matter ~ a -4 photons T 0 = 13.5 Gyr t -2 t -2 The quintessence becomes equal to matter only recently quintessence It then becomes asymptotically constant 27

28 Formation of structures with neutrinos Simulation of the coupling cosmon+neutrino, until z=1 CDM Velocity increasing N-body simulation Baldi et al 2011

29 Neutrinos at larger scale CDM Velocity increases Baldi et al 2011

30 Formation of structures with neutrinos The coupling cosmon-neutrino is equivalent to a force which attract neutrinos, then forming clumps These particles behave like CDM + dark energy Ayaita et al 2013

31 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 Ayaita et al 2012 Radius (Mpc)

32 Chameleons for dark energy Particles with a variable mass, varying as a function of environnement. Mass eff 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 of scalar fields V( ) in the theory Departure from standard model Translated into a force F = Gmm /r 2 (1 + e -mr ) Khoury & Weltman 2004 V eff ( ) = V( ) + m A( )

33 Chameleon potential : coupling The coupling with matter changes considerably V( ) The chameleon camouflages! Total Field alone coupling coupling Low density High density

34 Chameleon principle: screening System with spherical symmetry: Khoury and Weltman (2004) The force due to chameleons is only that of a shell (r) ( 6/2)lnF où V ( ) (RF f )/2F 2 B A Inside and outside of the system Minima and maxima at Q A Q B V, ( A) Qe A 0, V, ( B ) Qe B 0 If the field has a large mass, the system a thin shell Parameter of the shell r c B A 1 c : gravitational r c 6Q c potential il at the surface

35 Constraints in the solar system The constant of effective gravitation G eff = G(1+ r c /r c ) Constraints provided by experiments on the Casimir force r c B The measures in the solar system yield r c 6 c V( ) eff Faible-densite (mass~0) For the Sun ( c 10 6 ), (massif) B Haute-densite. B. This is satisfied if 1 /(3 f, 2. RR is large in the region M R H 0 2 )

36 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 vacuum 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 The chameleons do not cross: too massive, no tunnelling effect Their reflections are incoherent with those of photons

37 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 Experiment e with an aluminium u sphere e in which the best vacuum is made A beam of cesium atoms is separed 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!

38 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 = GeV M Already around g Tomorow well below

39 Simulations of chameleons Oyaizu, Lima, Hu (2008) No chameleon effect

40 Scalar fields Goon et al (2014) Theories of Galileons The Vainshtein screening mechanism suppress the 5th force within a certain radius around massive sources, where the regime becomes es highly non-linear (discontinuity ywhen m-grav 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 phenomenon is described by equations which remain of second order, which implies the absence of ghosts and intabilities

41 Galileons and Goldstone modes After a symmetry break, the energy is transferred in oscillations of low energy, the Goldstone modes Linked to the Goldstone bosons (m=0, spin =0) There exists also a screening effect with Galileons Analogus to the Vainshtein mechanism th L i i b ilt t ti f th G lil t the Lagrangian is built to satisfy the Galilean symmetry In a flat universe

42 The galileons not favored by observations Galileons: Galileon Late-time tracking w -1.0 f(r) Galileons, only the solutions late-time tracker are allowed Future Present Galileon Tracker Solutions simple tracker ruled out z

43 The proto-type of quintessence A typical model, where all main characteristics are discussed V( ) Two free parameters: 0 M: energy scale n: power index p = ½ (d /dt) 2 V( ) must be negative, thus d /dt must be weak V( ) slowly variable

44 Models of quintessence Quintessence, where w depends mainly on potential (a) Freezing models (b) Thawing models.. w decreases until -1 w increases from -1 eg e.g. PNGB Pseudo-boson (Nambu- Goldstone) k-essence, where w depends mainly on kinetic energy Typically, the evolution of w is similar to models (b) of thawing

45 Freezing -- Thawing Allows to distinguish 2 types of quintessence whatever the model V( ) adopted as a function of w, w Models begin above, but end below the limit after z=1 P/ = Caldwell & Linder 2005 (1) The Hubble damping ( ) would have frozen the system out of equilibrium, (w=-1) and today is thawing (2) System moving at the start then slows down and freezes w -1

46 Thawing Typical examples Freezing V( ) = V 1 (1+ cos /a) V( ) = M 5 / approx Numerical «tracker» solution e.g. PNGB Pseudo-boson (Nambu-Goldstone) Mass of field very low m 2 =V <H~10-33 ev = 2x10-69 kg Tsujikawa 2013 z=0 z=0 Type of the supersymmetric theory SU(Nc) Nc colors Nf flavours Binetruy (1999)

47 Limits of the quintessence Several models represented In order to separate them with observations, the resolution required in dw/dlna is of the order of (1+w)!! V ~ n, n=1,2,4 For <M P Then the models should have 1+w > (thawing) 1+w > 0.01 (freezing) Limits of quintessence Caldwell & Linder 2005 V ~ n, n <0 There exist «tracker» solutions

48 K-essence Quintessence: 5th element, after the baryons, CDM, photons & neutrinos the quintessence corresponds to a field k-essence, kinetic-essence The negative pressure of dark energy would be due to the non-linear term of the field kinetic energy The cosmon field has an evolution following that of the underlying back-ground Can change the evolution of the various era: radiation, matter, etc. W= today radiation matter DE Armendariz-Picon et al

49 K-essence (following) According to the back-ground, these solutions are called «tracker» -- during 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

50 The principle of tracker Very large range of initial conditions, which lead to the same final state = radiation CDM baryons Tracker solution= Attractor without fixed point Zlatev et al 1999

51 Quintessence Tracker CDM + cosmological constante Zlatev et al 1999

52 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( today ) : non extreme -unless, special case to explain rad mat Armendariz-Picon et al

53 Comparison: Theory and Observations Theory Observations, for Freezing models have problems Thawing models are still compatible with data

54 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 :

55 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)

56 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 (not only adiabatic) b ) Other extensions: viscosity Holographic model Has a supersymmetric generalisation

57 Other values of P(k) k Large scales Small scales small, too similar to CDM, > 3 would be a superluminic solution (Moschella, 2008)

58 Tachyonic models v/c = pc/ (p 2 c 2 +m 2 c 4 ) 1/2 Tachyons negative mass m For a scalar field M 2 = 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

59 Large 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»

60 Dark Force

61 Conclusion A cosmological constant is still possible But the expected value of the vacuum energy is 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) freezing, w -1, with a tracker solution Correspond to supersymmetric theories, with symmetry breaking (1) A pseudo-nambu-goldstone boson, or super-gravity theory Powerful experiments required to discriminate