Post on 21-Dec-2015
Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy
Cosmological ConstantCosmological Constant- Einstein -- Einstein -
Constant Constant λλ compatible with all symmetries compatible with all symmetries Constant Constant λλ compatible with all compatible with all
observationsobservations No time variation in contribution to energy No time variation in contribution to energy
densitydensity
Why so small ? Why so small ? λλ/M/M44 = 10 = 10-120-120
Why important just today ?Why important just today ?
Cosmological mass scalesCosmological mass scales Energy densityEnergy density
ρρ ~ ( 2.4×10 ~ ( 2.4×10 -3-3 eV )eV )- 4- 4
Reduced Planck Reduced Planck massmass
M=2.44M=2.44×10×101818GeVGeV Newton’s constantNewton’s constant
GGNN=(8=(8ππM²)M²)
Only ratios of mass scales are observable !Only ratios of mass scales are observable !
homogeneous dark energy: homogeneous dark energy: ρρhh/M/M44 = 7 · = 7 · 10ˉ¹²¹10ˉ¹²¹
matter: matter: ρρmm/M/M4= 3 · 10ˉ¹²¹= 3 · 10ˉ¹²¹
Cosm. Const | Quintessence static | dynamical
QuintessenceQuintessenceDynamical dark Dynamical dark
energy ,energy ,
generated by scalargenerated by scalar
field field (cosmon)(cosmon)
C.Wetterich,Nucl.Phys.B302(1988)668, C.Wetterich,Nucl.Phys.B302(1988)668, 24.9.87 24.9.87P.J.E.Peebles,B.Ratra,ApJ.Lett.325(1988)LP.J.E.Peebles,B.Ratra,ApJ.Lett.325(1988)L17, 20.10.8717, 20.10.87
CosmonCosmon Scalar field changes its value even Scalar field changes its value even
in the in the present present cosmological epochcosmological epoch Potential und kinetic energy of Potential und kinetic energy of
cosmon contribute to the energy cosmon contribute to the energy density of the Universedensity of the Universe
Time - variable dark energy : Time - variable dark energy : ρρhh(t) decreases with time !(t) decreases with time !
V(V(φφ) =M) =M44 exp( - exp( - αφαφ/M )/M )
two key features two key features for realistic cosmologyfor realistic cosmology
1 ) Exponential cosmon potential and 1 ) Exponential cosmon potential and
scaling solutionscaling solution
V(V(φφ) =M) =M44 exp( - exp( - αφαφ/M ) /M )
V(V(φφ →→ ∞ ) → 0 ! ∞ ) → 0 !
2 ) Stop of cosmon evolution by 2 ) Stop of cosmon evolution by
cosmological triggercosmological trigger
e.g. growing neutrino quintessencee.g. growing neutrino quintessence
Evolution of cosmon fieldEvolution of cosmon field
Field equationsField equations
Potential V(Potential V(φφ) determines details of the ) determines details of the modelmodel
V(V(φφ) =M) =M4 4 exp( - exp( - αφαφ/M )/M )
for increasing for increasing φφ the potential the potential decreases towards zero !decreases towards zero !
exponential potentialexponential potentialconstant fraction in dark constant fraction in dark
energyenergy
can explain order of can explain order of magnitude magnitude
of dark energy !of dark energy !
ΩΩh h = 3(4)/= 3(4)/αα22
Asymptotic solutionAsymptotic solution
explain V( explain V( φφ → ∞ ) = 0 ! → ∞ ) = 0 !
effective field equations should effective field equations should have generic solution of this typehave generic solution of this type
setting : quantum effective action , setting : quantum effective action , all quantum fluctuations included:all quantum fluctuations included:investigate generic forminvestigate generic form
Higher dimensional Higher dimensional dilatation symmetrydilatation symmetry
all stable quasi-static solutions of higher all stable quasi-static solutions of higher dimensional field equations , which admit dimensional field equations , which admit a finite four-dimensional gravitational a finite four-dimensional gravitational constant and non-zero value for the constant and non-zero value for the dilaton , have V=0dilaton , have V=0
for arbitrary values of effective couplings for arbitrary values of effective couplings within a certain range :within a certain range :
higher dimensional dilatation symmetry higher dimensional dilatation symmetry implies vanishing cosmological constant implies vanishing cosmological constant
self-tuning mechanismself-tuning mechanism
Cosmic runawayCosmic runaway large class of cosmological solutions which never large class of cosmological solutions which never
reach a static state : runaway solutionsreach a static state : runaway solutions
some characteristic scale some characteristic scale χχ changes with time changes with time
effective dimensionless couplings flow with effective dimensionless couplings flow with χχ ( similar to renormalization group )( similar to renormalization group )
couplings either diverge or reach fixed pointcouplings either diverge or reach fixed point
for fixed point : for fixed point : exact dilatation symmetry exact dilatation symmetry of full of full quantum field equations and corresponding quantum field equations and corresponding quantum effective actionquantum effective action
approach to fixed pointapproach to fixed point
dilatation symmetry not yet realizeddilatation symmetry not yet realized dilatation anomalydilatation anomaly effective potential effective potential V(φ)V(φ) exponential potential reflects exponential potential reflects
anomalous dimension for vicinity of anomalous dimension for vicinity of fixed pointfixed point
V(V(φφ) =M) =M4 4 exp( - exp( - αφαφ/M )/M )
cosmic runaway and the cosmic runaway and the problem of time varying problem of time varying
constantsconstants It is not difficult to obtain quintessence It is not difficult to obtain quintessence
potentials from higher dimensional ( or potentials from higher dimensional ( or string ? ) theoriesstring ? ) theories
Exponential form rather generic Exponential form rather generic
( after Weyl scaling)( after Weyl scaling) Potential goes to zero for Potential goes to zero for φφ →→ ∞∞ But most models show too strong time But most models show too strong time
dependence of constants !dependence of constants !
higher dimensional higher dimensional dilatation symmetrydilatation symmetry
generic class of solutions with generic class of solutions with
vanishing effective four-dimensional vanishing effective four-dimensional cosmological constantcosmological constant
andand
constant effective dimensionless couplingsconstant effective dimensionless couplings
graviton and dilatongraviton and dilaton
dilatation symmetric effective actiondilatation symmetric effective action
simple examplesimple example
in general : in general : many dimensionless many dimensionless parameters characterize effective parameters characterize effective actionaction
dilatation dilatation transformationstransformations
is invariantis invariant
flat phaseflat phase
generic existence of solutions of generic existence of solutions of
higher dimensional field equations higher dimensional field equations withwith
effective effective four –dimensional gravityfour –dimensional gravity andand
vanishing cosmological constantvanishing cosmological constant
torus solutiontorus solutionexample : example : Minkowski space Minkowski space xx D-dimensional torus D-dimensional torus ξξ = const = const solves higher dimensional field equationssolves higher dimensional field equations extremum of effective actionextremum of effective action
finite four- dimensional gauge couplingsfinite four- dimensional gauge couplings dilatation symmetry spontaneously brokendilatation symmetry spontaneously broken
generically many more solutions in flat phase !generically many more solutions in flat phase !
warpingwarping
most general metric with maximal most general metric with maximal four – dimensional symmetryfour – dimensional symmetry
general form of quasi – static solutionsgeneral form of quasi – static solutions( non-zero or zero cosmological constant )( non-zero or zero cosmological constant )
effective four – dimensional effective four – dimensional actionaction
flat phase : flat phase : extrema of Wextrema of Win higher dimensions , those exist generically !in higher dimensions , those exist generically !
extrema of Wextrema of W
provide large class of solutions with provide large class of solutions with vanishing four – dimensional constantvanishing four – dimensional constant
dilatation transformationdilatation transformation
extremum of W must occur for W=0 !extremum of W must occur for W=0 ! effective cosmological constant is effective cosmological constant is
given by Wgiven by W
extremum of W must occur extremum of W must occur for W = 0for W = 0
for any given solution : rescaled metric and for any given solution : rescaled metric and dilaton is again a solutiondilaton is again a solution
for rescaled solution :for rescaled solution :
useuse
extremum condition :extremum condition :
extremum of W is extremum of W is extremum of effective extremum of effective
actionaction
effective four – dimensional effective four – dimensional cosmological constant cosmological constant
vanishes for extrema of Wvanishes for extrema of W
expand effective 4 – d - actionexpand effective 4 – d - action
in derivatives :in derivatives :
4 - d - field 4 - d - field equationequation
Quasi-static solutionsQuasi-static solutions
for arbitrary parameters of dilatation for arbitrary parameters of dilatation symmetric effective action :symmetric effective action :
large classes of solutions with Wlarge classes of solutions with Wext ext = 0 = 0 are explicitly known ( are explicitly known ( flat phaseflat phase ) )
example : example : Minkowski space Minkowski space xx D- D-dimensional torusdimensional torus
only for certain parameter regions : only for certain parameter regions : further solutions with Wfurther solutions with Wext ext ≠ 0 exist : ( ≠ 0 exist : ( non-flat phasenon-flat phase ) )
sufficient condition for sufficient condition for vanishing cosmological vanishing cosmological
constantconstant
extremum of W extremum of W existsexists
effective four – dimensional effective four – dimensional theorytheory
characteristic length characteristic length scalesscales
ll :scale of internal space :scale of internal space
ξξ : dilaton scale: dilaton scale
effective Planck masseffective Planck mass
dimensionless , dimensionless , depends on internal geometry ,depends on internal geometry ,from expansion of F in Rfrom expansion of F in R
effective potentialeffective potential
canonical scalar fieldscanonical scalar fields
consider field configurations with rescaledconsider field configurations with rescaledinternal length scale and dilaton valueinternal length scale and dilaton value
potential and effective Planck mass depend on scalar fieldspotential and effective Planck mass depend on scalar fields
phase diagramphase diagram
stable solutionsstable solutions
phase structure of phase structure of solutionssolutions
solutions in flat phase exist for arbitrary solutions in flat phase exist for arbitrary values of effective parameters of higher values of effective parameters of higher dimensional effective actiondimensional effective action
question : how “big” is flat phase question : how “big” is flat phase ( which internal geometries and warpings ( which internal geometries and warpings
are possible beyond torus solutions )are possible beyond torus solutions )
solutions in non-flat phase only exist for solutions in non-flat phase only exist for restricted parameter rangesrestricted parameter ranges
self tuningself tuning
for all solutions in flat phase : for all solutions in flat phase :
self tuning of cosmological constant self tuning of cosmological constant to zero !to zero !
self tuningself tuning
for simplicity : no contribution of F to for simplicity : no contribution of F to VV
assume Q depends on parameter assume Q depends on parameter αα , , which characterizes internal which characterizes internal geometry:geometry:
tuning required : tuning required : andand
self tuning in higher self tuning in higher dimensionsdimensions
Q depends on higher dimensionalQ depends on higher dimensional
extremum condition extremum condition amounts to field equationsamounts to field equations
typical solutions depend on integration constants typical solutions depend on integration constants γγ
solutions obeying boundary condition exist :solutions obeying boundary condition exist :
fieldsfields
self tuning in higher self tuning in higher dimensionsdimensions
involves infinitely many degrees of freedom !involves infinitely many degrees of freedom !
for arbitrary parameters in effective action : for arbitrary parameters in effective action : flat phase solutions are flat phase solutions are presentpresent
extrema of W existextrema of W exist
for flat 4-d-space : W is functional for flat 4-d-space : W is functional of internal geometry, independent of x of internal geometry, independent of x
solve field equations for internal metric and solve field equations for internal metric and σσ and and ξξ
Dark energyDark energy
if cosmic runaway solution has not yet if cosmic runaway solution has not yet reached fixed point :reached fixed point :
dilatation symmetry of field equations dilatation symmetry of field equations
not yet exactnot yet exact
“ “ dilatation anomaly “dilatation anomaly “
non-vanishing effective potential V in non-vanishing effective potential V in reduced four –dimensional theoryreduced four –dimensional theory
Time dependent Dark Time dependent Dark Energy :Energy :
QuintessenceQuintessence 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 V : potential energy of scalar field or cosmological constantconstant
V/MV/M4 4 is observableis observable
Imagine the Planck mass M increases …Imagine the Planck mass M increases …
Cosmon and Cosmon and fundamental mass scalefundamental mass scale
Assume all mass parameters are Assume all mass parameters are proportional to scalar field proportional to scalar field χχ (GUTs, (GUTs, superstrings,…)superstrings,…)
MMpp~ ~ χχ , m , mprotonproton~ ~ χχ , , ΛΛQCDQCD~ ~ χχ , M , MWW~ ~ χχ ,… ,…
χχ may evolve with time : may evolve with time : cosmoncosmon mmnn/M : ( almost ) constant - /M : ( almost ) constant - observationobservation !!
Only ratios of mass scales are Only ratios of mass scales are observableobservable
theory without explicit theory without explicit mass scalemass scale
Lagrange density:Lagrange density:
recall :recall :
realistic theoryrealistic theory
χχ has no gauge interactions has no gauge interactions χχ is effective scalar field after is effective scalar field after
“integrating out” all other scalar “integrating out” all other scalar fieldsfields
four dimensional four dimensional dilatation symmetrydilatation symmetry
Lagrange density:Lagrange density:
Dilatation symmetry forDilatation symmetry for
Conformal symmetry for Conformal symmetry for δδ=0=0 Asymptotic flat phase solution : Asymptotic flat phase solution : λλ = 0 = 0
Asymptotically vanishing Asymptotically vanishing effective “cosmological effective “cosmological
constant”constant” Effective cosmological constant ~ V/MEffective cosmological constant ~ V/M4 4
dilatation anomaly : dilatation anomaly : λλ ~ ( ~ (χχ//μμ) ) –A–A
V ~ (V ~ (χχ//μμ) ) –A –A χχ4 4 V/MV/M4 4 ~(~(χχ//μμ) ) –A –A
M = M = χχ
It is sufficient that V increases less It is sufficient that V increases less fast than fast than χχ4 4 !!
CosmologyCosmology
Cosmology : Cosmology : χχ increases with time ! increases with time !
( due to coupling of ( due to coupling of χχ to curvature scalar to curvature scalar ))
for large for large χχ the ratio V/M the ratio V/M4 4 decreases to decreases to zerozero
Effective cosmological constant vanishes Effective cosmological constant vanishes asymptotically for large t !asymptotically for large t !
Weyl scalingWeyl scaling
Weyl scaling : gWeyl scaling : gμνμν→→ (M/ (M/χχ))2 2 ggμνμν , ,
φφ/M = ln (/M = ln (χχ 44/V(/V(χχ))))
Exponential potential : V = MExponential potential : V = M44 exp(- exp(-φφ/M)/M)
No additional constant !No additional constant !
quantum fluctuations quantum fluctuations and and
dilatation anomalydilatation anomaly
Dilatation anomalyDilatation anomaly
Quantum fluctuations responsible both Quantum fluctuations responsible both forfor
fixed point and dilatation anomaly close fixed point and dilatation anomaly close to fxed pointto fxed point
Running couplings:Running couplings: hypothesishypothesis
Renormalization scale Renormalization scale μμ : ( momentum : ( momentum scale )scale )
λλ~(~(χχ//μμ) ) –A–A
Asymptotic behavior of Asymptotic behavior of effective potentialeffective potential
λλ ~ ( ~ (χχ//μμ) ) –A–A
V ~ (V ~ (χχ//μμ) ) –A –A χχ44
VV ~ ~ χχ 4–A 4–A
crucial : behavior for large χ !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 Dilatation anomaly and quantum fluctuationsquantum fluctuations
Computation of running couplings Computation of running couplings ( beta functions ) needs unified theory !( beta functions ) needs unified theory !
Dominant contribution from modes Dominant contribution from modes with momenta ~with momenta ~χχ ! !
No prejudice on “natural value “ of No prejudice on “natural value “ of anomalous dimension should be anomalous dimension should be inferred from tiny contributions at inferred from tiny contributions at QCD- momentum scale !QCD- momentum scale !
quantum fluctuations and quantum fluctuations and naturalnessnaturalness
Jordan- and Einstein frame completely Jordan- and Einstein frame completely equivalent on level of effective action equivalent on level of effective action and field equations ( and field equations ( afterafter computation computation of quantum fluctuations ! )of quantum fluctuations ! )
Treatment of quantum fluctuations Treatment of quantum fluctuations depends on frame : Jacobian for depends on frame : Jacobian for variable transformation in functional variable transformation in functional integralintegral
What is natural in one frame may look What is natural in one frame may look unnatural in another frameunnatural in another frame
quantum fluctuations quantum fluctuations and framesand frames
Einstein frame : quantum fluctuations Einstein frame : quantum fluctuations make zero cosmological constant look make zero cosmological constant look unnaturalunnatural
Jordan frame : quantum fluctuations Jordan frame : quantum fluctuations are at the origin of dilatation anomaly;are at the origin of dilatation anomaly;
may be key ingredient for may be key ingredient for solutionsolution of of cosmological constant problem !cosmological constant problem !
fixed points and fluctuation fixed points and fluctuation contributions of individual contributions of individual
componentscomponents
If running couplings influenced by fixed points:If running couplings influenced by fixed points:individual fluctuation contribution can be huge individual fluctuation contribution can be huge
overestimate !overestimate !
here : fixed point at vanishing quartic coupling and here : fixed point at vanishing quartic coupling and anomalous dimension anomalous dimension VV ~ ~ χχ 4–A 4–A
it makes no sense to use naïve scaling argument to it makes no sense to use naïve scaling argument to infer individual contribution infer individual contribution VV ~ h ~ h χχ 4 4
conclusionsconclusions
naturalness of cosmological constant naturalness of cosmological constant and cosmon potential should be and cosmon potential should be discussed in the light of dilatation discussed in the light of dilatation symmetry and its anomaliessymmetry and its anomalies
Jordan frameJordan frame higher dimensional settinghigher dimensional setting four dimensional Einstein frame and four dimensional Einstein frame and
naïve estimate of individual naïve estimate of individual contributions can be very misleading !contributions can be very misleading !
conclusionsconclusions
cosmic runaway towards fixed point cosmic runaway towards fixed point maymay
solve the cosmological constant solve the cosmological constant problemproblem
andand
account for dynamical Dark Energyaccount for dynamical Dark Energy
EnEndd
A few referencesA few references
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P.J.E.Peebles,B.Ratra , Astrophys.J.Lett.325,L17(1988) , received 20.10.1987P.J.E.Peebles,B.Ratra , Astrophys.J.Lett.325,L17(1988) , received 20.10.1987
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