Why the cosmological constant goes to zero, and why we see it now

Quintessence C.Wetterich A.Hebecker, M.Doran, M.Lilley, J.Schwindt, C.Müller, G.Schäfer, E.Thommes, R.Caldwell, M.Bartelmann, K.Kharwan, G.Robbers, T.Dent, S.Steffen, L.Amendola, M.Baldi, N.Brouzakis, N.Tetradis, D.Mota, V.Pettorino, T.Krüger, M.Neubert

Dark Energy dominates the Universe Energy - density in the Universe Energy - density in the Universe = Matter + Dark Energy Matter + Dark Energy 25 % + 75 % 25 % + 75 %

Cosmological Constant - Einstein - Constant λ compatible with all symmetries Constant λ compatible with all symmetries Constant λ compatible with all observations Constant λ compatible with all observations No time variation in contribution to energy density No time variation in contribution to energy density Why so small ? λ/M 4 = Why so small ? λ/M 4 = Why important just today ? Why important just today ?

Cosmological mass scales Energy density Energy density ρ ~ ( 2.4×10 -3 eV ) - 4 ρ ~ ( 2.4×10 -3 eV ) - 4 Reduced Planck mass M=2.44×10 18 GeV Newton’s constant G N =(8πM²) Only ratios of mass scales are observable ! homogeneous dark energy: ρ h /M 4 = 7 · 10ˉ¹²¹ homogeneous dark energy: ρ h /M 4 = 7 · 10ˉ¹²¹ matter: ρ m /M= 3 · 10ˉ¹²¹ matter: ρ m /M 4 = 3 · 10ˉ¹²¹

Cosm. Const | Quintessence static | dynamical

Quintessence Dynamical dark energy, generated by scalar field generated by scalar field (cosmon) (cosmon) C.Wetterich,Nucl.Phys.B302(1988)668, P.J.E.Peebles,B.Ratra,ApJ.Lett.325(1988)L17,

Prediction : homogeneous dark energy influences recent cosmology - of same order as dark matter - Original models do not fit the present observations …. modifications

Cosmon Scalar field changes its value even in the present cosmological epoch Scalar field changes its value even in the present cosmological epoch Potential und kinetic energy of cosmon contribute to the energy density of the Universe Potential und kinetic energy of cosmon contribute to the energy density of the Universe Time - variable dark energy : Time - variable dark energy : ρ h (t) decreases with time ! ρ h (t) decreases with time ! V(φ) =M 4 exp( - αφ/M )

two key features 1 ) Exponential cosmon potential and scaling solution scaling solution V(φ) =M 4 exp( - αφ/M ) V(φ) =M 4 exp( - αφ/M ) V(φ → ∞ ) → 0 ! V(φ → ∞ ) → 0 ! 2 ) Stop of cosmon evolution by cosmological trigger cosmological trigger

Evolution of cosmon field Field equations Field equations Potential V(φ) determines details of the model Potential V(φ) determines details of the model V(φ) =M 4 exp( - αφ/M ) V(φ) =M 4 exp( - αφ/M ) for increasing φ the potential decreases towards zero ! for increasing φ the potential decreases towards zero !

Cosmic Attractor Solutions independent of initial conditions V ~ t -2 φ ~ ln ( t ) Ω h ~ const. early cosmology

exponential potential constant fraction in dark energy can explain order of magnitude of dark energy ! Ω h = 3(4)/α 2

realistic quintessence fraction in dark energy has to increase in “recent time” !

Quintessence becomes important “today” No reason why w should be constant in time !

coincidence problem What is responsible for increase of Ω h for z < 6 ? Why now ?

growing neutrino mass triggers transition to almost static dark energy growingneutrinomass

cosmon coupling to neutrinos basic ingredient :

Cosmon coupling to neutrinos can be large ! can be large ! interesting effects for cosmology if neutrino mass is growing interesting effects for cosmology if neutrino mass is growing growing neutrinos can stop the evolution of the cosmon growing neutrinos can stop the evolution of the cosmon transition from early scaling solution to cosmological constant dominated cosmology transition from early scaling solution to cosmological constant dominated cosmology L.Amendola,M.Baldi,… Fardon,Nelson,Weiner

growing neutrinos

crossover due to non –relativistic neutrinos growingneutrinomass

end of matter domination growing mass of neutrinos growing mass of neutrinos at some moment energy density of neutrinos becomes more important than energy density of dark matter at some moment energy density of neutrinos becomes more important than energy density of dark matter end of matter dominated period end of matter dominated period similar to transition from radiation domination to matter domination similar to transition from radiation domination to matter domination this transition happens in the recent past this transition happens in the recent past cosmon plays crucial role cosmon plays crucial role

cosmological selection present value of dark energy density set by cosmological event present value of dark energy density set by cosmological event ( neutrinos become non – relativistic ) ( neutrinos become non – relativistic ) not given by ground state properties ! not given by ground state properties !

connection between dark energy and neutrino properties present equation of state given by neutrino mass ! present dark energy density given by neutrino mass

dark energy fraction determined by neutrino mass constant neutrino - cosmon coupling β variable neutrino - cosmon coupling

varying neutrino – cosmon coupling specific model specific model can naturally explain why neutrino – cosmon coupling is much larger than atom – cosmon coupling can naturally explain why neutrino – cosmon coupling is much larger than atom – cosmon coupling

neutrino mass seesaw and cascademechanism omit generation structure triplet expectation value ~ doublet squared

cascade mechanism triplet expectation value ~ M.Magg, … G.Lazarides, Q.Shafi, …

varying neutrino mass triplet mass depends on cosmon field φ neutrino mass depends on φ ε ≈ -0.05

“singular” neutrino mass triplet mass vanishes for φ → φ t neutrino mass diverges for φ → φ t

strong effective neutrino – cosmon coupling for φ → φ t

crossover from early scaling solution to effective cosmological constant

early scaling solution ( tracker solution ) neutrino mass unimportant in early cosmology

growing neutrinos change cosmon evolution modification of conservation equation for neutrinos

effective stop of cosmon evolution cosmon evolution almost stops once cosmon evolution almost stops once neutrinos get non –relativistic neutrinos get non –relativistic ß gets large ß gets large This always happens for φ → φ t !

effective cosmological trigger for stop of cosmon evolution : neutrinos get non-relativistic this has happened recently ! this has happened recently ! sets scales for dark energy ! sets scales for dark energy !

dark energy fraction determined by neutrino mass constant neutrino - cosmon coupling β variable neutrino - cosmon coupling

cosmon evolution

Hubble parameter as compared to ΛCDM m ν =0.45 eV

Hubble parameter ( z < z c ) only small differencefrom ΛCDM !

Can time evolution of neutrino mass be observed ? Experimental determination of neutrino mass may turn out higher than upper bound in model for cosmological constant Experimental determination of neutrino mass may turn out higher than upper bound in model for cosmological constant ( KATRIN, neutrino-less double beta decay ) ( KATRIN, neutrino-less double beta decay ) GERDA

neutrino fluctuations neutrino structures become nonlinear at z~1 for supercluster scales stable neutrino-cosmon lumps exist N.Brouzakis, N.Tetradis,… D.Mota, G.Robbers, V.Pettorino, …

neutrino fluctuations time when neutrinos become non – relativistic time when neutrinos become non – relativistic sets free streaming scale sets free streaming scale neutrino structures become nonlinear at z~1 for supercluster scales neutrino structures become nonlinear at z~1 for supercluster scales stable neutrino-cosmon lumps exist stable neutrino-cosmon lumps exist N.Brouzakis, N.Tetradis,… D.Mota, G.Robbers, V.Pettorino, …

Conclusions Cosmic event triggers qualitative change in evolution of cosmon Cosmic event triggers qualitative change in evolution of cosmon Cosmon stops changing after neutrinos become non-relativistic Cosmon stops changing after neutrinos become non-relativistic Explains why now Explains why now Cosmological selection Cosmological selection Model can be distinguished from cosmological constant Model can be distinguished from cosmological constant

two key features 1 ) Exponential cosmon potential and scaling solution scaling solution V(φ) =M 4 exp( - αφ/M ) V(φ) =M 4 exp( - αφ/M ) V(φ → ∞ ) → 0 ! V(φ → ∞ ) → 0 ! 2 ) Stop of cosmon evolution by cosmological trigger cosmological trigger

Why goes the cosmological constant to zero ?

Time dependent Dark Energy : Quintessence What changes in time ? What changes in time ? Only dimensionless ratios of mass scales Only dimensionless ratios of mass scales are observable ! are observable ! V : potential energy of scalar field or cosmological constant V : potential energy of scalar field or cosmological constant V/M 4 is observable V/M 4 is observable Imagine the Planck mass M increases … Imagine the Planck mass M increases …

Cosmon and fundamental mass scale Assume all mass parameters are proportional to scalar field χ (GUTs, superstrings,…) Assume all mass parameters are proportional to scalar field χ (GUTs, superstrings,…) M p ~ χ, m proton ~ χ, Λ QCD ~ χ, M W ~ χ,… M p ~ χ, m proton ~ χ, Λ QCD ~ χ, M W ~ χ,… χ may evolve with time : cosmon χ may evolve with time : cosmon m n /M : ( almost ) constant - observation ! m n /M : ( almost ) constant - observation ! Only ratios of mass scales are observable Only ratios of mass scales are observable

Example : Field χ is connected to mass scale of transition from higher dimensional physics to effective four dimensional description

theory without explicit mass scale Lagrange density: Lagrange density:

realistic theory χ has no gauge interactions χ has no gauge interactions χ is effective scalar field after “integrating out” all other scalar fields χ is effective scalar field after “integrating out” all other scalar fields

Dilatation symmetry Lagrange density: Lagrange density: Dilatation symmetry for Dilatation symmetry for Conformal symmetry for δ=0 Conformal symmetry for δ=0

Asymptotically vanishing effective “cosmological constant” Effective cosmological constant ~ V/M 4 Effective cosmological constant ~ V/M 4 λ ~ (χ/μ) –A λ ~ (χ/μ) –A V ~ (χ/μ) –A χ 4 V/M 4 ~(χ/μ) –A V ~ (χ/μ) –A χ 4 V/M 4 ~(χ/μ) –A M = χ M = χ It is sufficient that V increases less fast than χ 4 !

Cosmology Cosmology : χ increases with time ! ( due to coupling of χ to curvature scalar ) for large χ the ratio V/M 4 decreases to zero Effective cosmological constant vanishes asymptotically for large t !

Weyl scaling Weyl scaling : g μν → (M/χ) 2 g μν, φ/M = ln (χ 4 /V(χ)) φ/M = ln (χ 4 /V(χ)) Exponential potential : V = M 4 exp(-φ/M) No additional constant ! No additional constant !

Quintessence from higher dimensions

geometrical runaway and the problem of time varying constants It is not difficult to obtain quintessence potentials from higher dimensional ( or string ? ) theories It is not difficult to obtain quintessence potentials from higher dimensional ( or string ? ) theories Exponential form rather generic Exponential form rather generic ( after Weyl scaling) ( after Weyl scaling) Potential goes to zero for φ → ∞ Potential goes to zero for φ → ∞ But most models show too strong time dependence of constants ! But most models show too strong time dependence of constants !

runaway solutions geometrical runaway geometrical runaway anomalous runaway anomalous runaway geometrical adjustment geometrical adjustment

Quintessence from higher dimensions An instructive example: Einstein – Maxwell theory in six dimensions Einstein – Maxwell theory in six dimensions with J. Schwindt hep-th/

Metric Ansatz with particular metric ( not most general ! ) which is consistent with d=4 homogeneous and isotropic Universe and internal U(1) x Z 2 isometry B ≠ 1 : football shaped internal geometry

Conical singularities deficit angle singularities can be included with energy momentum tensor on brane bulk point of view : describe everything in terms of bulk geometry describe everything in terms of bulk geometry ( not possible for modes on brane without tail in bulk ) ( not possible for modes on brane without tail in bulk )

Exact solution m : monopole number ( integer) cosmology with scalar and potential V :

Asymptotic solution for large t

Naturalness No tuning of parameters or integration constants No tuning of parameters or integration constants Radiation and matter can be implemented Radiation and matter can be implemented Asymptotic solution depends on details of model, e.g. solutions with constant Ω h ≠ 1 Asymptotic solution depends on details of model, e.g. solutions with constant Ω h ≠ 1

geometrical runaway V ~ L D V ~ L D M p 2 ~ L D M p 2 ~ L D V/ M p 4 ~ L - D V/ M p 4 ~ L - D

problem : time variation of fundamental constants relative change order one for z around one gauge coupling goes to zero for large time

primordial abundances for three GUT models T.Dent,S.Stern,… present observations : 1σ He D Li

three GUT models unification scale ~ Planck scale unification scale ~ Planck scale 1) All particle physics scales ~Λ QCD 1) All particle physics scales ~Λ QCD 2) Fermi scale and fermion masses ~ unification scale 2) Fermi scale and fermion masses ~ unification scale 3) Fermi scale varies more rapidly than Λ QCD 3) Fermi scale varies more rapidly than Λ QCD Δα/α ≈ Δα/α ≈ allowed for GUT 1 and 3, larger for GUT 2 Δln(M n /M P ) ≈40 Δα/α ≈ allowed

toy model for asymptotically vanishing dark energy effective d=4 cosmology involves graviton, cosmon and gauge bosons ( radiation ) effective d=4 cosmology involves graviton, cosmon and gauge bosons ( radiation ) asymptotically : gauge bosons are free, cosmon becomes massless asymptotically : gauge bosons are free, cosmon becomes massless dark energy goes to zero without tuning as a consequence of simple geometry dark energy goes to zero without tuning as a consequence of simple geometry example that naïve estimate of size of cosmological constant from quantum fluctuations fails badly example that naïve estimate of size of cosmological constant from quantum fluctuations fails badly can be understood in terms of effective dilatation symmetry can be understood in terms of effective dilatation symmetry

stabilizing the couplings… in toy model : gauge couplings go to zero as volume of internal space increases ways to solve this problem: volume or curvature of internal space is irrelevant for modes on brane volume or curvature of internal space is irrelevant for modes on brane possible stabilization by fixed points in scale free models possible stabilization by fixed points in scale free models

Warped branes model is similar to first co-dimension two model is similar to first co-dimension two warped brane model : C.W. Nucl.Phys.B255,480(1985); warped brane model : C.W. Nucl.Phys.B255,480(1985); see also B253,366(1985) see also B253,366(1985) first realistic warped model first realistic warped model see Rubakov and Shaposhnikov for earlier work ( no stable solutions, infinitely many chiral fermions) see Rubakov and Shaposhnikov for earlier work ( no stable solutions, infinitely many chiral fermions) see Randjbar-Daemi, C.W. for arbitrary dimensions see Randjbar-Daemi, C.W. for arbitrary dimensions

Brane stabilization idea : all masses and couplings of standard model depend only on characteristic scale and geometry of brane all masses and couplings of standard model depend only on characteristic scale and geometry of brane generalized curvature invariant, which is relevant for V, scales with inverse power of characteristic length scale L generalized curvature invariant, which is relevant for V, scales with inverse power of characteristic length scale L L → ∞ while brane scale remains constant L → ∞ while brane scale remains constant analogy with black hole in cosmological background analogy with black hole in cosmological background

scales in gravity gravity admits solutions with very different length or mass scales gravity admits solutions with very different length or mass scales example : black hole in expanding universe example : black hole in expanding universe

quantum fluctuations and dilatation anomaly

Dilatation symmetry Lagrange density: Lagrange density: Dilatation symmetry for Dilatation symmetry for Conformal symmetry for δ=0 Conformal symmetry for δ=0

Dilatation anomaly Quantum fluctuations responsible for Quantum fluctuations responsible for dilatation anomaly dilatation anomaly Running couplings: hypothesis Running couplings: hypothesis Renormalization scale μ : ( momentum scale ) Renormalization scale μ : ( momentum scale ) λ~(χ/μ) –A λ~(χ/μ) –A

Asymptotic behavior of effective potential λ ~ (χ/μ) –A λ ~ (χ/μ) –A V ~ (χ/μ) –A χ 4 V ~ (χ/μ) –A χ 4 V ~ χ 4–A V ~ χ 4–A crucial : behavior for large χ !

Without dilatation – anomaly : V= const. Massless Goldstone boson = dilaton Dilatation – anomaly : V (φ ) Scalar with tiny time dependent mass : cosmon

Dilatation anomaly and quantum fluctuations Computation of running couplings ( beta functions ) needs unified theory ! Computation of running couplings ( beta functions ) needs unified theory ! Dominant contribution from modes with momenta ~χ ! Dominant contribution from modes with momenta ~χ ! No prejudice on “natural value “ of anomalous dimension should be inferred from tiny contributions at QCD- momentum scale ! No prejudice on “natural value “ of anomalous dimension should be inferred from tiny contributions at QCD- momentum scale !

quantum fluctuations and naturalness Jordan- and Einstein frame completely equivalent on level of effective action and field equations ( after computation of quantum fluctuations ! ) Jordan- and Einstein frame completely equivalent on level of effective action and field equations ( after computation of quantum fluctuations ! ) Treatment of quantum fluctuations depends on frame : Jacobian for variable transformation in functional integral Treatment of quantum fluctuations depends on frame : Jacobian for variable transformation in functional integral What is natural in one frame may look unnatural in another frame What is natural in one frame may look unnatural in another frame

quantum fluctuations and frames Einstein frame : quantum fluctuations make zero cosmological constant look unnatural Einstein frame : quantum fluctuations make zero cosmological constant look unnatural Jordan frame : quantum fluctuations are at the origin of dilatation anomaly; Jordan frame : quantum fluctuations are at the origin of dilatation anomaly; may be key ingredient for solution of cosmological constant problem ! may be key ingredient for solution of cosmological constant problem !

fixed points and fluctuation contributions of individual components If running couplings influenced by fixed points: individual fluctuation contribution can be huge overestimate ! here : fixed point at vanishing quartic coupling and anomalous dimension V ~ χ 4–A it makes no sense to use naïve scaling argument to infer individual contribution V ~ h χ 4

conclusions naturalness of cosmological constant and cosmon potential should be discussed in the light of dilatation symmetry and its anomalies naturalness of cosmological constant and cosmon potential should be discussed in the light of dilatation symmetry and its anomalies Jordan frame Jordan frame higher dimensional setting higher dimensional setting four dimensional Einstein frame and naïve estimate of individual contributions can be very misleading ! four dimensional Einstein frame and naïve estimate of individual contributions can be very misleading !

End

How can quintessence be distinguished from a cosmological constant ?

Time dependence of dark energy cosmological constant : Ω h ~ t² ~ (1+z) -3 M.Doran,…

small early and large present dark energy fraction in dark energy has substantially increased since end of structure formation fraction in dark energy has substantially increased since end of structure formation expansion of universe accelerates in present epoch expansion of universe accelerates in present epoch

effects of early dark energy modifies cosmological evolution (CMB) modifies cosmological evolution (CMB) slows down the growth of structure slows down the growth of structure

interpolation of Ω h G.Robbers,M.Doran,…

Summary o Ω h = 0.75 o Q/Λ : dynamical und static dark energy will be distinguishable o growing neutrino mass can explain why now problem o Q : time varying fundamental coupling “constants” violation of equivalence principle violation of equivalence principle

A few references C.Wetterich, Nucl.Phys.B302,668(1988), received P.J.E.Peebles,B.Ratra, Astrophys.J.Lett.325,L17(1988), received B.Ratra,P.J.E.Peebles, Phys.Rev.D37,3406(1988), received J.Frieman,C.T.Hill,A.Stebbins,I.Waga, Phys.Rev.Lett.75,2077(1995) P.Ferreira, M.Joyce, Phys.Rev.Lett.79,4740(1997) C.Wetterich, Astron.Astrophys.301,321(1995) P.Viana, A.Liddle, Phys.Rev.D57,674(1998) E.Copeland,A.Liddle,D.Wands, Phys.Rev.D57,4686(1998) R.Caldwell,R.Dave,P.Steinhardt, Phys.Rev.Lett.80,1582(1998) P.Steinhardt,L.Wang,I.Zlatev, Phys.Rev.Lett.82,896(1999)

Cosmon coupling to atoms Tiny !!! Tiny !!! Substantially weaker than gravity. Substantially weaker than gravity. Non-universal couplings bounded by tests Non-universal couplings bounded by tests of equivalence principle. of equivalence principle. Universal coupling bounded by tests of Brans- Dicke parameter ω in solar system. Universal coupling bounded by tests of Brans- Dicke parameter ω in solar system. Only very small influence on cosmology. Only very small influence on cosmology.

effective cosmological constant linked to neutrino mass realistic value α φ t / M ≈ 276 : needed for neutrinos to become non-relativistic in recent past - as required for observed mass range of neutrino masses φ t / M : essentially determined by present neutrino mass adjustment of one dimensionless parameter in order to obtain for the present time the correct ratio between dark energy and neutrino energy density no fine tuning ! no fine tuning !

effective cosmological constant realistic value for α φ t / M ≈ 276

neutrino fraction remains small m ν = 0.45 eV ΩνΩνΩνΩν z

equation of state present equation of state given by neutrino mass !

oscillating neutrino mass

crossing time from matching between early solution and late solution

approximate late solution variables : approximate smooth solution ( averaged over oscillations )

dark energy fraction

neutrino equation of state

cosmon equation of state

fixed point behaviour : apparent tuning

Growth of density fluctuations Matter dominated universe with constant Ω h : Matter dominated universe with constant Ω h : Dark energy slows down structure formation Dark energy slows down structure formation Ω h < 10% during structure formation Ω h < 10% during structure formation P.Ferreira,M.Joyce

Early quintessence slows down the growth of structure

bounds on Early Dark Energy after WMAP’06 G.Robbers,M.Doran,…

Cluster number relative to ΛCDM Two models with 4% Dark Energy during structure formation Fixed σ 8 ( normalization dependence ! ) Little Early Dark Energy can make large effect ! Non – linear enhancement More clusters at high redshift !Bartelmann,Doran,…