Dark Energy in the Early Universe Joel Weller arXiv:gr-qc/0812.2843 University of Sheffield Beyond Part III 2009 Based on work with Carsten van de Bruck, Philippe Brax and Lisa Hall

Contents Introduction Inflation Perturbations Reheating Summary Scalar fields in Cosmology Aims Inflation The Model Quintessence during Inflation. Slow-roll Parameters Perturbations Perturbations of the Q-field The Power Spectrum Reheating Summary

Scalar Fields: Introduction When would we expect to see an accelerated expansion of the universe? Inflation (the very early universe) Dark energy (present day) Introduction

Scalar Fields: Introduction The starting point of most cosmological models is the assumption that the universe is isotropic and homogeneous on large scales. The quantity a(t) is the scale factor, determining the ‘size’ of the universe at a given time. The matter content of the universe is modelled as a perfect fluid with energy density ρ and pressure p, linked by the relationship: Introduction

Scalar Fields: Introduction Solving the Einstein equations gives the Friedmann equations for the evolution of the scale factor (H is the Hubble parameter å/a): To get an accelerating universe one needs the universe to be dominated by a component with w < -1/3. The equation of state for a scalar field, Q, varies with the value of the field. One can achieve acceleration if the field is slowly rolling. Introduction

Scalar Fields: Inflation Inflation is a period of accelerated expansion in the early universe introduced to counter some of the shortcomings of the big bang model. The universe is dominated by a light scalar field, the inflaton, slowly rolling down its potential. Simple models of inflation use a power-law potential. The field rolls down the potential and stops in the minimum, whereupon inflation ceases. At the end of inflation, the temperature of the universe is very low. So, to recover a high entropy hot universe dominated by radiation, the inflaton decays into relativistic particles. This is called reheating. Introduction

Scalar Fields: Quintessence The acceleration of the universe observed in recent times can be explained by a cosmological constant. However, a dynamic solution is more attractive as it allows one to bypass the coincidence problem. Introduction

Scalar Fields: Quintessence The behaviour of the Q-field is determined by the form of the field potential. For acceleration one needs a period of slow-roll i.e. a flat potential. Two common choices are: Introduction

Scalar Fields: Quintessence By choosing the parameters appropriately, one can find tracker solutions that, starting from a wide range of initial conditions, match the energy density of the dominant component of the universe. As the universe expands, the field value decreases at a lesser rate than the dominant component. So quintessence will eventually become the dominant component itself. Solid = quintessence; dashed = matter; dotted = radiation Picture credit: J. Martin, 2008 arXiv:0803.4076 (Inverse Power Law) Introduction

Aims In this work we examine a model in which the quintessence field was present in the early universe as a subdominant field coupled to the inflaton field. Using the mathematical framework of inflation and reheating, we calculate the changes to the normal inflationary scenario due to the way the quintessence field affects the other components of the universe. Dark matter is characterised by the fact that it only interacts gravitationally with ‘normal’ matter and light, but does it interact with dark energy? We assume that, after inflation, the Q-field remains coupled to the decay products of the inflaton field. Thus, the observational constraints on the coupling between dark matter and quintessence in later times can be used to test the constraint the parameters of the model. Introduction

The Model Inflation The action for the model is: The function A(Q) is a measure of the coupling between the Q-field and the inflaton. We assume that it takes the form: Inflation

What happens to Q during Inflation? We can get the following equations of motion for the inflaton and the quintessence field. We choose a (standard) quadratic potential for the inflaton and an exponential potential for Q with lambda positive. The field value of Q increases with time. Inflation

What happens to Q during Inflation? The squared mass of a scalar field is the second derivative of the potential with respect to the field value. We can differentiate the effective potential to get the effective mass. During slow-roll, the kinetic term is negligible and U dominates the energy density of the universe, so the effective mass is large and the field will quickly find its minimum. Inflation

What happens to Q during Inflation? One can understand this behaviour by plotting the effective potential. Veff ~ V + e4βQU(Φ) Inflation

Slow Roll Parameters During inflation, when the fields are slowly rolling and Q is in its minimum, the Friedmann equation takes the form One can also derive expressions for the slow-roll parameters. The condition for inflation is ε, η << 1. Inflation

Cosmological Perturbations A byproduct of Inflation is the Theory of Cosmological Perturbations. During inflation, the universe expands very quickly but the horizon stays constant. This means that quantum fluctuations of the inflaton field are pushed beyond the horizon and remain frozen. After inflation, the horizon grows with the steady expansion of the universe and fluctuations with wavelength smaller than the horizon re-enter the observable universe. This causes a set of scale invariant curvature perturbations that can act as the seeds for large-scale structures. Over time, regions that are denser than average will become denser. Perturbations

Cosmological Perturbations The large effective mass of the Q field means that the perturbations in Q are suppressed. Integrating the perturbations equations numerically confirms this result. The effect of the perturbations of Q can be neglected in our analysis. Perturbations of the inflation Perturbations of the Q-field Perturbations

The Power Spectrum We can calculate the power spectrum of curvature perturbations. The equation for the spectral index takes a similar form to the standard case: Perturbations

The Power Spectrum Using these relations we can see how the values of the inflaton mass (as obtained by normalizing the power spectrum to COBE) and spectral index vary with the parameters λ and β . Perturbations

Reheating When the slow-roll regime comes to an end, the inflaton oscillates around the minimum of its potential and decays to radiation. Following the standard treatment, we add a friction term Γr to the equation of motion. It also acts as a source term for the radiation produced. Reheating

Reheating In the standard situation, the radiation produced by the inflaton quickly reaches a maximum. Meanwhile, the inflaton continues to lose energy. When H ~ Γr the inflaton rapidly decays, leaving a radiation dominated universe. ρ radiation ρ inflaton Reheating

Reheating This is not the case when the Q-field is present. Before the onset of reheating the Q-field represents a fraction of the universe’s energy density. So, although the inflaton field loses energy, the Hubble damping stays relatively large, reducing the efficiency of the reheating process. With the inflaton rapidly decaying, one has to ensure that the energy density of radiation is greater than that of the Q-field to get a radiation-dominated universe. Reheating

Reheating Current constraints set the potential exponent to be |λ| < 0.95. (Bean et al., 2008). Using this, it is not possible for the radiation to become dominant. Reheating

Reheating If one uses a larger value for λ, the potential for the Q-field is steeper. This reduces the damping and allows the radiation produced from the inflaton to dominate the universe. Models of Quintessence with an exponential potential with |λ| < 0.95 are only consistent if the quintessence field does not couple to the inflaton field. Reheating

Summary In this model Quintessence is coupled to the inflaton in the very early universe and to dark matter thereafter. Quintessence is forced into the minimum of the effective potential during inflation and thus cannot affect the cosmological perturbations produced at this time. The equations for the spectral index and other inflationary parameters differ from the standard results for single field inflation. For successful reheating, a large value of λ is required, which is not compatible with quintessence models that have |λ| < 0.95. Summary