1. Quintessence and the Accelerating Universe
Jérôme Martin
Institut d’Astrophysique de Paris

2. Bibliography
1) “The case for a positive cosmological Lambda term”, V. Sahni and A. Starobinsky, astro-ph/
2) “Cosmological constant vs. quintessence”, P. Binétruy, hep-ph/
3) “The cosmological constant and dark energy”, P. Peebles and B. Ratra,
astro-ph/
4) “The cosmological constant”, S. Weinberg, Rev. Mod. Phys. 61, 1 (1989).
5) B. Ratra and P. Peebles, Phys. Rev. D 37, 3406 (1988).
6) I. Zlatev, L. Wang and P.J. Steinhardt, Phys. Rev. Lett. 82, 896 (1999), astro-ph/
7) P. Brax and J. Martin, Phys. Lett 468B, 40 (1999).

3. Plan I) The accelerating universe:
SNIa, CMB
II) The cosmological constant problem:
Why the cosmological constant is not a satisfactory candidate for dark energy
III) Quintessence
The notion of tracking fields

4. I
The accelerating universe

5. The luminosity distance (I)
The flux received
from the source is
is the distance to the source

6. The luminosity distance (II)
Let us now consider the same physical situation but in a FLRW curved spacetime:
If we define
, then the luminosity distance takes the form: as
For small redshifts, one has
with,
Hubble parameter
Acceleration parameter

7. Equations of motion (I)
The dynamics of the scale factor can be calculated from the Einstein equations:
For a FLRW universe:
pressure
energy density
The equation of state of a cosmological constant is given by:

8. Equations of motion (II)
The Einstein equations can also be re-written as:
with
These equations can be combined to get an expression for the acceleration of the scale factor:
In particular this is the case for a cosmological constant

9. Acceleration: basic mechanism
Relativistic term
A phase of acceleration can be obtained if two basic principles
of general relativity and field theory are combined :
General relativity: “any form of energy weighs”
Field theory: “the pressure can be negative”

10. The acceleration parameter
Equation giving the acceleration of the scale factor
Friedmann equation
Matter
Radiation
Critical energy density
Vacuum energy

11. SNIa as standard candles (I)
The luminosity distance is
where
: absolute luminosity
: apparent luminosity
Clearly, the main difficulty lies in the measurement of the absolute luminosity
SNIa:
the width of the light curve is linked to the absolute luminosity

12. SNIa as standard candles (II)

13. The Hubble diagram
Hubble diagram: luminosity distance (standard candles) vs. redshift in a FLRW Universe:
The universe is accelerating

14. The CMB anisotropy measurements
COBE has shown that there are temperature fluctuations at
the level
The two-point correlation function is
The position of the first peak depends on

15. The cosmological parameters
The universe is accelerating !

16. II
The cosmological constant problem

17. The cosmological constant (I)
Bare cosmological constant
Contribution from the vacuum

18. The cosmological constant (II)
The Einstein equations can be re-written under the following form
The cosmological constant problem is :
“ Answer “: because there is a deep (unknown!) principle such
that the cancellation is exact (SUSY?? …) .
However, the recent measurements of the Hubble diagram indicate

19. The cosmological constant (III)
Maybe super-symmetry can play a crucial role in this unknown principle ?
The SUSY algebra
yields the following relation between the Hamiltonian and the super-symmetry generators
but SUSY has to be broken …

20. The cosmological constant (IV)
Since a cosmological constant has a constant energy density, this means that its initial value was extremely small in comparison with the energy densities of the other form of matter
Coincidence problem, fine-tuning of the initial conditions
Radiation
Matter
orders of magnitude
Cosmological constant

21. The cosmological constant (V)
It is important to realize that the cosmological constant problem
is a “theoretical” problem. So far a cosmological constant is still
compatible with the observations
The vacuum has the correct equation of state:

22. III
Quintessence

23. Quintessence: the main idea (I)
1) One assumes that the cosmological constant vanishes due
to some (so far) unknown principle.
2) The acceleration is due to a new type of fluid with a
negative equation of state which, today, represents 70%
of the matter content of the universe.
This is the fifth component (the others being baryons, cdm, photons and neutrinos) and the
most important one … hence its name
Plato

24. Scalar fields
A simple way to realize the previous program is to consider a scalar field
The stress-energy tensor is defined by:
The conservation of the stress-energy tensor implies

25. Quintessence: the main idea (II)
A scalar field Q can be a candidate for dark energy. Indeed, the time-time and space-space components of the stress-energy tensor are given by:
This is a well-known mechanism in the theory of inflation at very high redshifts. The theoretical surprise is that this kind of exotic matter could dominate at small redshifts, i.e. now.
A generic property of this kind of model is that the equation
of state is now redshift-dependent

26. The proto-typical model
A typical model where all the main properties of quintessence
can be discussed is given by
Two free parameters:
: energy scale
: power index

27. Evolution of the quintessence field
The equations of motion controlling the evolution of the system are (in conformal time):
1) Friedmann equation:
Background: radiation or matter
quintessence
2) Conservation equation for the background :
3) Conservation equation for the quintessence field:
Using the equation of state parameter and the “sound velocity”,
the Klein-Gordon equation can be re-written as

28. Initial conditions 1) The initial conditions are fixed after inflation
2) One assumes that the quintessence field is subdominant
initially.
Equipartition
Quintessence is a test field
The free parameters are chosen to be
(see below)

29. Kinetic era The potential energy becomes constant even if
the kinetic one still dominates!

30. Transition era
But the kinetic energy still redshifts as

31. Potential era The potential era cannot last forever
The sound velocity has to change
The potential energy still dominates
The potential era cannot last forever

32. The attractor (I) The equation of state is negative!
At this point, the kinetic and potential energy become
comparable
If the quintessence field is a test field, then the Klein-Gordon equation with the inverse power –law potential has the solution
Redshifts more slowly than the background and
therefore is going to dominate
The equation of state tracks the background equation of state
The equation of state is negative!

33. The attractor (II) Equivalence between radiation and matter
One can see the change in the quintessence equation of state when the background equation of state evolves

34. The attractor (III)
Let us introduce a new time defined by and define and by
Particular solution
The Klein-Gordon equation, viewed as a dynamical system in the plane , possesses a
critical point and small perturbations around this point, , obey
Solutions to the equation are
The particular solution is an attractor

35. The attractor (IV)
This solution is an attractor and is therefore insensible to the initial conditions
The equation of state obtained is negative as required
Different
initial conditions

36. Consequences for the free parameters
(valid when the quintessence field is about to dominate)
SuperGravity is going to play an important role in the model building problem
In order to have
one must choose
For example
High energy physics !

37. A note of the model building problem
A potential
arises in supersymmetry in the
study of gaugino condensation.
The fact that, at small redshifts, the value of the quintessence
field is the Planck mass means that supergravity must be used
for model building. A model gives
usual term
Sugra correction
At small redshifts, the exponential factor pushes the equation of state towards –1 independently of
. The model predicts

38. Problems with quintessence
The mass of the quintessence field at very small redshift (i.e. now)
The quintessence field must be ultra-light (but this comes “naturally” from the value of M)
This field must therefore be very weakly coupled to matter
(this is bad)

39. Quintessential cosmological perturbations
The main question is: can the quintessence field be clumpy?
and one has to solve the
At the linear level, one writes
perturbed Klein-Gordon equation:
Coupling with the perturbed metric tensor
No growing mode for
NB: there is also an attractor for the perturbed quantities, i.e. the final result does not depend on the initial conditions.

40. Conclusions Quintessence can solve the coincidence and (maybe) the
fine tuning problem: the clue to these problems is the concept
of tracking field.
There are still important open questions: model building,
clustering properties, etc …
A crucial test: the measurement of the equation of state
and of its evolution
SNAP