Quintessence and the Accelerating Universe

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Quintessence and the Accelerating Universe

• Jérôme Martin

Institut dAstrophysique de Paris
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Bibliography
1) The case for a positive cosmological Lambda
term, V. Sahni and A. Starobinsky,
astro-ph/9904398.
2) Cosmological constant vs. quintessence, P.
Binétruy, hep-ph/0005037.
3) The cosmological constant and dark energy,
P. Peebles and B. Ratra, astro-ph/0207347.
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/9807002.
7) P. Brax and J. Martin, Phys. Lett 468B, 40
(1999).
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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
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I
The accelerating universe
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The luminosity distance (I)
The flux received from the source is
is the distance to the source
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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
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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
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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
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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
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The acceleration parameter
Equation giving the acceleration of the scale
factor
Friedmann equation
Matter
Radiation
Critical energy density
Vacuum energy
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SNIa as standard candles (I)
The luminosity distance is
absolute luminosity
where
apparent luminosity
Clearly, the main difficulty lies in the
measurement of the absolute luminosity
the width of the light curve is linked to the
absolute luminosity
SNIa
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SNIa as standard candles (II)
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The Hubble diagram
Hubble diagram luminosity distance (standard
candles) vs. redshift in a FLRW Universe
The universe is accelerating
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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
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The cosmological parameters
The universe is accelerating !
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II
The cosmological constant problem
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The cosmological constant (I)
Bare cosmological constant
Contribution from the vacuum
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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
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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
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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
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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
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III
Quintessence
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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
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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
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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
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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
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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
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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)
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Kinetic era
The potential energy becomes constant even if
the kinetic one still dominates!
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Transition era
But the kinetic energy still redshifts as
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Potential era
The sound velocity has to change
The potential energy still dominates
The potential era cannot last forever
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The attractor (I)
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!
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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
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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
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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
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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 !
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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
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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)
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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.
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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