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

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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

I The accelerating universe

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

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

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:

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

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”

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

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

SNIa as standard candles (II)

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

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

The cosmological parameters The universe is accelerating !

II The cosmological constant problem

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

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

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 …

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

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:

III Quintessence

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

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

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

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

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

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)

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

Transition era But the kinetic energy still redshifts as

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

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!

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

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

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

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 !

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

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)

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.

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