1. Quintessence from time evolution of fundamental mass scale

2. Quintessence and solution of cosmological constant problem should be related !

3. Ω m + X = 1 Ω m : 25% Ω m : 25% Ω h : 75% Ω h : 75% Dark Energy Dark Energy ?

4. 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 …

5. Fundamental mass scale Unification fixes parameters with dimensions Unification fixes parameters with dimensions Special relativity : c Special relativity : c Quantum theory : h Quantum theory : h Unification with gravity : Unification with gravity : fundamental mass scale fundamental mass scale ( Planck mass, string tension, …) ( Planck mass, string tension, …)

6. Fundamental mass scale Fixed parameter or dynamical scale ? Fixed parameter or dynamical scale ? Dynamical scale Field Dynamical scale Field Dynamical scale compared to what ? Dynamical scale compared to what ? momentum versus mass momentum versus mass ( or other parameter with dimension ) ( or other parameter with dimension )

7. 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

8. Example : Field χ denotes scale of transition from higher dimensional physics to effective four dimensional description in theory without fundamental mass parameter (except for running of dimensionless couplings…)

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

10. 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 E > 0 : crossover Quintessence E > 0 : crossover Quintessence

11. 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 !

12. 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 !

13. 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 ~ (χ/μ) –A χ 4 M = χ M = χ V/M 4 ~(χ/μ) –A V/M 4 ~(χ/μ) –A

14. 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 !

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

16. Crossover Quintessence ( like QCD gauge coupling) ( like QCD gauge coupling) critical χ where δ grows large critical φ where k grows large k²(φ )=δ(χ)/4 k²(φ )= “1/(2E(φ c – φ)/M)” k²(φ )= “1/(2E(φ c – φ)/M)” if  c  ≈ 276/M ( tuning ! ) : if  c  ≈ 276/M ( tuning ! ) : this will be responsible for relative increase of dark energy in present cosmological epoch this will be responsible for relative increase of dark energy in present cosmological epoch

17. Realistic cosmology Hypothesis on running couplings Hypothesis on running couplings yields realistic cosmology yields realistic cosmology for suitable values of A, E, φ c for suitable values of A, E, φ c

18. Quintessence cosmology - models -

19. Dynamics of quintessence Cosmon  : scalar singlet field Cosmon  : scalar singlet field Lagrange density L = V + ½ k(φ)  Lagrange density L = V + ½ k(φ)  (units: reduced Planck mass M=1) (units: reduced Planck mass M=1) Potential : V=exp[-  Potential : V=exp[-  “Natural initial value” in Planck era  “Natural initial value” in Planck era  today:  =276 today:  =276

20. Quintessence models Kinetic function k(φ) : parameterizes the Kinetic function k(φ) : parameterizes the details of the model - “kinetial” details of the model - “kinetial” k(φ) = k=const. Exponential Q. k(φ) = k=const. Exponential Q. k(φ ) = exp ((φ – φ 1 )/α) Inverse power law Q. k(φ ) = exp ((φ – φ 1 )/α) Inverse power law Q. k²(φ )= “1/(2E(φ c – φ))” Crossover Q. k²(φ )= “1/(2E(φ c – φ))” Crossover Q. possible naturalness criterion: possible naturalness criterion: k(φ=0)/ k(φ today ) : not tiny or huge ! k(φ=0)/ k(φ today ) : not tiny or huge ! - else: explanation needed - - else: explanation needed -

21. More models … Phantom energy ( Caldwell ) Phantom energy ( Caldwell ) negative kinetic term ( w < -1 ) negative kinetic term ( w < -1 ) consistent quantum theory ? consistent quantum theory ? K – essence ( Amendariz-Picon, Mukhanov, Steinhardt ) K – essence ( Amendariz-Picon, Mukhanov, Steinhardt ) higher derivative kinetic terms higher derivative kinetic terms why derivative expansion not valid ? why derivative expansion not valid ? Coupling cosmon / (dark ) matter ( C.W., Amendola ) Coupling cosmon / (dark ) matter ( C.W., Amendola ) why substantial coupling to dark matter and not to ordinary matter ? why substantial coupling to dark matter and not to ordinary matter ? Non-minimal coupling to curvature scalar – f(φ) R - Non-minimal coupling to curvature scalar – f(φ) R - can be brought to standard form by Weyl scaling ! can be brought to standard form by Weyl scaling !

22. kinetial Small almost constant k : Small almost constant Ω h Small almost constant Ω h Large k : Cosmon dominated universe ( like inflation ) Cosmon dominated universe ( like inflation )

23. Cosmon Tiny mass Tiny mass m c ~ H m c ~ H New long - range interaction New long - range interaction

24. cosmon mass changes with time ! for standard kinetic term for standard kinetic term m c 2 = V” m c 2 = V” for standard exponential potential, k ≈ const. for standard exponential potential, k ≈ const. m c 2 = V”/ k 2 = V/( k 2 M 2 ) m c 2 = V”/ k 2 = V/( k 2 M 2 ) = 3 Ω h (1 - w h ) H 2 /( 2 k 2 ) = 3 Ω h (1 - w h ) H 2 /( 2 k 2 )

25. Quintessence becomes important “today”

26. Transition to cosmon dominated universe Large value k >> 1 : universe is dominated by scalar field Large value k >> 1 : universe is dominated by scalar field k increases rapidly : evolution of scalar fied essentially stops k increases rapidly : evolution of scalar fied essentially stops Realistic and natural quintessence: Realistic and natural quintessence: k changes from small to large values after structure formation k changes from small to large values after structure formation

27. crossover quintessence k(φ) increase strongly for φ corresponding to present epoch Example (LKT) : exponential quintessence:

28. Why has quintessence become important “now” ?

29. coincidence problem What is responsible for increase of Ω h for z < 10 ?

30. a) Properties of cosmon potential or kinetic term Early quintessence Ω h changes only modestly Ω h changes only modestly w changes in time w changes in timetransition special feature in cosmon potential or kinetic term becomes important “now” special feature in cosmon potential or kinetic term becomes important “now” tuning at ‰ level tuning at ‰ level Late quintessence w close to -1 Ω h negligible in early cosmology needs tiny parameter, similar to cosmological constant

31. attractor solutions Small almost constant k : Small almost constant Ω h Small almost constant Ω h This can explain tiny value of Dark Energy ! This can explain tiny value of Dark Energy ! Large k : Cosmon dominated universe ( like inflation ) Cosmon dominated universe ( like inflation )

32. Transition to cosmon dominated universe Large value k >> 1 : universe is dominated by scalar field Large value k >> 1 : universe is dominated by scalar field k increases rapidly : evolution of scalar fied essentially stops k increases rapidly : evolution of scalar fied essentially stops Realistic and natural quintessence: Realistic and natural quintessence: k changes from small to large values after structure formation k changes from small to large values after structure formation

33. b) Quintessence reacts to some special event in cosmology Onset of Onset of matter dominance matter dominance K- essence K- essence Amendariz-Picon, Mukhanov, Amendariz-Picon, Mukhanov, Steinhardt Steinhardt needs higher derivative needs higher derivative kinetic term kinetic term Appearance of non-linear structure Back-reaction effect needs coupling between Dark Matter and Dark Energy

34. Back-reaction effect Needs large inhomogeneities after structure has been formed Local cosmon field participates in structure

35. End

36. work with J. Schwindt hep-th/0501049 Quintessence from higher dimensions

37. 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) But most models show too strong time dependence of constants ! But most models show too strong time dependence of constants !

38. Quintessence from higher dimensions An instructive example: Einstein – Maxwell theory in six dimensions Einstein – Maxwell theory in six dimensions Warning : not scale - free ! Dilatation anomaly replaced by explicit mass scales.

39. Field equations

40. Energy momentum tensor

41. 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

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

43. Free integration constants M, B, Φ(t=0), (dΦ/dt)(t=0) : continuous m : discrete

44. 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 ( no modes on brane without tail in bulk )

45. Asymptotic solution for large t

46. 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

47. problem : time variation of fundamental constants

48. Dimensional reduction

49. Time dependent gauge coupling

50. ???????????????????????? Why becomes Quintessence dominant in the present cosmological epoch ? Are dark energy and dark matter related ? Can Quintessence be explained in a fundamental unified theory ?

51. Cosmon dark matter ? Can cosmon fluctuations account for dark matter ? Can cosmon fluctuations account for dark matter ? Cosmon can vary in space Cosmon can vary in space