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Dark Energy- a cosmic mystery. Dunkle Energie – Ein kosmisches Raetsel. Quintessence. C.Wetterich. A.Hebecker,M.Doran,M.Lilley,J.Schwindt, C.M ü ller,G.Sch ä fer,E.Thommes, R.Caldwell,M.Bartelmann. What is our universe made of ?. fire , air, water, soil !. quintessence !.

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Dark Energy-

a cosmic mystery

Dunkle Energie –

Ein kosmisches Raetsel







what is our universe made of
What is our universe made of ?

fire , air,

water, soil !

quintessence !

dark energy dominates the universe
Dark Energy dominates the Universe

Energy - density in the Universe


Matter + Dark Energy

30 % + 70 %


Matter : Everything that clumps

Abell 2255 Cluster

~300 Mpc

critical density
critical density
  • ρc =3 H² M²

critical energy density of the universe

( M : reduced Planck-mass , H : Hubble parameter )

  • Ωb=ρb/ρc

fraction in baryons

energy density in baryons over critical energy density

dark matter
Dark Matter

Most matter is dark !

So far tested only through gravity

Every local mass concentration gravitational potential

Orbits and velocities of stars and galaxies measurement of gravitational potential

and therefore of local matter distribution


Ωm= 0.3

gravitational lens , HST

spatially flat universe
spatially flat universe

Ωtot = 1

  • theory (inflationary universe )

Ωtot =1.0000……….x

  • observation ( WMAP )

Ωtot =1.02 (0.02)

dark energy
Dark Energy

Ωm + X = 1

Ωm : 30%

Ωh : 70% Dark Energy

h : homogenous , often ΩΛ instead of Ωh

Dark Energy density isthe same at every point of space “ homogeneous “ No force –“ In what direction should it draw ? “
two important predictions
Two important predictions

The expansion of the Universe

accelerates today !

Structure formation : One primordial

fluctuation- spectrum

composition of the universe
Composition of the Universe

Ωb = 0.045 visible clumping

Ωdm= 0.22 invisibleclumping

Ωh = 0.73 invisiblehomogeneous


Dark Energy-

a cosmic mystery

Dunkle Energie –

Ein kosmisches Raetsel


What is Dark Energy ?

Cosmological Constant


Quintessence ?

cosmological constant einstein
Cosmological Constant- Einstein -
  • Constant λ compatible with all symmetries
  • No time variation in contribution to energy density
  • Why so small ? λ/M4 = 10-120
  • Why important just today ?

Cosm. Const. | Quintessence

static | dynamical

cosmological mass scales
Energy density

ρ ~ ( 2.4×10 -3 eV )- 4

Reduced Planck mass


Newton’s constant


Cosmological mass scales

Only ratios of mass scales are observable !

homogeneous dark energy: ρh/M4 = 6.5 10ˉ¹²¹

matter: ρm/M4= 3.5 10ˉ¹²¹

time evolution
Time evolution

tˉ² matter dominated universe

tˉ3/2 radiation dominated universe

  • ρm/M4 ~ aˉ³ ~
  • ρr/M4 ~ aˉ4~ t -2radiation dominated universe

Huge age small ratio

Same explanation for small dark energy?


Dynamical dark energy ,

generated by scalar field


C.Wetterich,Nucl.Phys.B302(1988)668, 24.9.87

P.J.E.Peebles,B.Ratra,ApJ.Lett.325(1988)L17, 20.10.87

prediction homogeneous dark energy influences recent cosmology of same order as dark matter
Prediction : homogeneous dark energyinfluences recent cosmology- of same order as dark matter -

Original models do not fit the present observations

…. modifications


Cosmon – Field φ(x,y,z,t)

similar to electric field , but no direction ( scalar field )

Homogeneous und isotropic Universe : φ(x,y,z,t)=φ(t)

Potential und kinetic energy of the cosmon -field

contribute to a dynamical energy density of the Universe !

fundamental interactions
“Fundamental” Interactions

Strong, electromagnetic, weak


On astronomical

length scales:






evolution of cosmon field
Evolution of cosmon field

Field equations

Potential V(φ) determines details of the model

e.g. V(φ) =M4 exp( - φ/M )

for increasing φ the potential decreases towards zero !

cosmological equations
Cosmological equations

cosmon +matter


  • Scalar field changes its value even in the presentcosmological epoch
  • Potential und kinetic energy of cosmon contribute to the energy density of the Universe
  • Time - variable dark energy :

ρh(t) decreases with time !

  • Tiny mass
  • mc ~ H
  • New long - range interaction
dynamics of quintessence
Dynamics of quintessence
  • Cosmonj: scalar singlet field
  • Lagrange density L = V + ½ k(φ)¶j ¶j

(units: reduced Planck mass M=1)

  • Potential : V=exp[-j]
  • “Natural initial value” in Planck era j=0
  • today: j=276
cosmon mass changes with time
cosmon mass changes with time !

for standard kinetic term

  • mc2 = V”

for standard exponential potential , k = const.

  • mc2 = V”/ k2 = V/( k2 M2 )

= 3 Ωh (1 - wh ) H2 /( 2 k2 )

cosmic attractors
Cosmic Attractors

Solutions independent of initial conditions

typically V~t -2

φ ~ ln ( t )

Ωh ~ const.

details depend on V(φ)

or kinetic term

early cosmology

equation of state
Equation of state

p=T-V pressure kinetic energy

ρ=T+V energy density

Equation of state

Depends on specific evolution of the scalar field

negative pressure
Negative pressure
  • w < 0 Ωh increases (with decreasing z )
  • w < -1/3 expansion of the Universe is


  • w = -1 cosmological constant

late universe with

small radiation component :

small early and large present dark energy
small early and large presentdark energy

fraction in dark energy has substantially increased since end of structure formation

expansion of universe accelerates in present epoch

quintessence becomes important today1
Quintessence becomes important “today”

No reason why w should

be constant in time !

time dependence of dark energy
Time dependence of dark energy

cosmological constant : Ωh ~ t² ~ (1+z)-3


early dark energy
Early dark energy

A few percent in the early Universe

Not possible for a cosmological constant

a few percent early dark energy
A few percent Early Dark Energy

If linear power spectrum fixed today ( σ8 ) :

More Structure at high z !



How to distinguish Q from Λ ?

A) Measurement Ωh(z) H(z)

i) Ωh(z) at the time of

structure formation , CMB - emission

or nucleosynthesis

ii) equation of state wh(today) > -1

B) Time variation of fundamental “constants”

C) Apparent violation of equivalence principle

quintessence and time variation of fundamental constants
Quintessence and time variation of fundamental constants

Generic prediction


Strong, electromagnetic, weak


C.Wetterich , Nucl.Phys.B302,645(1988)



time varying constants
Time varying constants
  • It is not difficult to obtain quintessence potentials from higher dimensional or string theories
  • Exponential form rather generic

( after Weyl scaling)

  • But most models show too strong time dependence of constants !
are fundamental constants time dependent
Are fundamental “constants”time dependent ?

Fine structure constant α (electric charge)

Ratio nucleon mass to Planck mass


Quintessence and Time dependence of “fundamental constants”

  • Fine structure constant depends on value of

cosmon field : α(φ)

(similar in standard model: couplings depend on value of Higgs scalar field)

  • Time evolution of φ

Time evolution of α


standard model of electroweak interactions higgs mechanism
Standard – Model of electroweak interactions :Higgs - mechanism
  • The masses of all fermions and gauge bosons are proportional to the ( vacuum expectation ) value of a scalar field φH ( Higgs scalar )
  • For electron, quarks , W- and Z- bosons :

melectron = helectron * φHetc.

restoration of symmetry at high temperature in the early universe
Restoration of symmetryat high temperature in the early Universe

high T :

less order

more symmetry



Low T


<φH>=φ0≠ 0

High T




Quintessence :Couplings are still varying now !Strong bounds on the variation of couplings -interesting perspectives for observation !


Abundancies of


light elements





if present 2-sigma deviation of He –abundance

from CMB/nucleosynthesis prediction would be

confirmed :

Δα/α ( z=1010 ) = -1.0 10-3 GUT 1

Δα/α ( z=1010 ) = -2.7 10-4 GUT 2



Time variation of coupling constants

must be tiny –

would be of very high significance !

Possible signal for Quintessence



Everything is flowing

  • Ωh = 0.7
  • Q/Λ : dynamical und static dark energy

will be distinguishable

  • Q : time varying fundamental coupling “constants”

violation of equivalence principle

fundamental mass scale
Fundamental mass scale
  • Fixed parameter or dynamical scale ?
  • Dynamical scale Field
  • Dynamical scale compared to what ?

momentum versus mass

( or other parameter with dimension )

cosmon and fundamental mass scale
Cosmon and fundamental mass scale
  • Assume all mass parameters are proportional to scalar field χ (GUTs, superstrings,…)
  • Mp~ χ , mproton~ χ , ΛQCD~ χ , MW~ χ ,…
  • χ may evolve with time : cosmon
  • mn/M : ( almost ) constant - observation!

Only ratios of mass scales are observable


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…)

dilatation symmetry
Dilatation symmetry
  • Lagrange density:
  • Dilatation symmetry for
  • Conformal symmetry for δ=0
dilatation anomaly
Dilatation anomaly
  • Quantum fluctuations responsible for

dilatation anomaly

  • Running couplings:
  • V~χ4-A , Mp(χ )~ χ
  • V/Mp4 ~ χ-A : decreases for increasing χ
  • E>0 : crossover quintessence

Cosmology : χ increases with time !

( due to coupling of χ to curvature scalar )

for large χ the ratio V/M4 decreases to zero

Effective cosmological constant vanishes asymptotically for large t !

weyl scaling
Weyl scaling

Weyl scaling : gμν→ (M/χ)2 gμν ,

φ/M = ln (χ4/V(χ))

Exponential potential : V = M4 exp(-φ/M)

No additional constant !

realistic cosmology
Realistic cosmology

Hypothesis on running couplings

yields realistic cosmology

for suitable values of A , E , φc


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 ?


A few references

C.Wetterich , Nucl.Phys.B302,668(1988) , received 24.9.1987

P.J.E.Peebles,B.Ratra , Astrophys.J.Lett.325,L17(1988) , received 20.10.1987

B.Ratra,P.J.E.Peebles , Phys.Rev.D37,3406(1988) , received 16.2.1988

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)

growth of density fluctuations
Growth of density fluctuations
  • Matter dominated universe with constantΩh :
  • Dark energy slows down structure formation

Ωh < 10% during structure formation

  • Substantial increase of Ωh(t) since structure has formed!

negative wh

  • Question “why now” is back ( in mild form )


quintessence models
  • Kinetic function k(φ) : parameterizes the

details of the model - “kinetial”

    • k(φ) = k=const. Exponential Q.
    • k(φ ) = exp ((φ – φ1)/α) Inverse power law Q.
    • k²(φ )= “1/(2E(φc – φ))” Crossover Q.
  • possible naturalness criterion:

k(φ=0)/ k(φtoday) : not tiny or huge !

- else: explanation needed -

sn and equation of state
SN and equation of state

Riess et al. 2004


Cosmon mediates new long-range interaction

Range : size of the Universe – horizon

Strength : weaker than gravity

photon electrodynamics

graviton gravity

cosmon cosmodynamics

Small correction to Newton’s law

violation of equivalence principle
Violation of equivalence principle

Different couplings of cosmon to proton and neutron

Differential acceleration

“Violation of equivalence principle”





only apparent : new “fifth force” !

differential acceleration
Differential acceleration η

For unified theories ( GUT ) :


Q : time dependence of other parameters


Link between time variation of α

and violation of equivalence principle

typically : η = 10-14

if time variation of α near Oklo upper bound

to be tested by MICROSCOPE

variation of fine structure constant as function of redshift
Variation of fine structure constant as function of redshift

Three independent data sets from Keck/HIRES

Δα/α = - 0.54 (12) 10-5

Murphy,Webb,Flammbaum, june 2003


Δα/α = - 0.06 (6) 10-5

Srianand,Chand,Petitjean,Aracil, feb.2004

z ≈ 2