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The Curvaton Scenario in Supersymmetric Theories

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The Curvaton Scenario in Supersymmetric Theories The curvaton scenario in supersymmetric theories Marieke Postma The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34100 Trieste, Italy (November 2002) We analyze the curvaton scenario in the context of supersymmety. Supersymmetric theories contain many scalars, and therefore many curvaton candidates. To obtain a scale invariant perturbation spectrum, the curvaton mass should be small during inflation m H. This can be achieved by ≪ invoking symmetries, which suppress the soft masses and non-renormalizable terms in the poten- tial. Other constraints on the curvaton model come from nucleosynthesis, gravitino overproduction, and thermal evaporation. The curvaton coupling to matter should be very small to satisfy these constraints (e.g. h < 10−8 for typical soft masses m TeV). ∼ ∼ I. INTRODUCTION in the quadratic potential. The curvaton behaves as cold matter; its energy density red shifts as ρ a−3, with a φ ∝ It is now widely believed that the early universe went the scale factor of the universe. After inflaton decay, the through a period of superluminal expansion, known as in- universe becomes radiation dominated, with the energy density in radiation red shifting as ρ a−4. During this flation. In addition to explaining the isotropy and homo- I ∝ geneity of our observable universe, inflation can generate state of mixed matter, i.e., radiation and cold dark mat- the seeds for structure formation. In the usual picture the ter, the isocurvature perturbations are transformed into quantum fluctuations of the slowly rolling inflaton field adiabatic ones. The perturbations grow until the curva- “freeze in” soon after horizon exit, and become essentially ton becomes to dominate the universe, or if this never a classical perturbation which remains constant until the happens, until the curvaton decays. moment of horizon re-entry. The produced perturbations In this paper we reanalyze the curvaton scenario in only depend on the potential during inflation, and there- the context of supersymmetry (SUSY). SUSY theories fore severely constrain possible inflationary models. contain many scalar fields, and therefore many possible Recently there has been a revived interest in the idea curvaton candidates. However, the finite energy density that not the perturbations in the inflaton field are respon- in the early universe breaks supersymmetry, and induces sible for the cosmic microwave background (CMB) fluctu- soft masses of the order of the Hubble constant for all ations, but instead isocurvature fluctuations in another scalar fields [8,9]. Such heavy fields lead to a perturba- scalar field, the “curvaton” field, that is sub-dominant tion spectrum with a large spectral tilt, in contradiction during inflation. After inflation the isocurvature fluctu- with observations. Symmetries can be invoked to pro- ations of the curvaton field are converted into adiabatic tect scalars from large soft masses. In D-term inflation ones. If at decay the contribution of the curvaton to the Hubble induced soft masses are forbidden by gauge in- energy density is considerable, this can lead to perturba- variance [10]. In no-scale gravity models, and generaliza- tions of the observed size. The generation of curvature tions thereof, there is a Heisenberg symmetry that pro- does not depend on the nature of inflation, beyond the tects scalar fields from soft mass terms at tree level [11]. Another possibility is that the curvaton field is a (pseudo) arXiv:hep-ph/0212005v2 24 Jan 2003 requirement that the Hubble constant must be nearly constant. This mechanism of generating perturbations Goldstone boson. was first described years ago [1], but it did not attract Non-renormalizable terms in the potential can lift the much interest until recently [2–6]. flat potential for the large field values needed in the cur- The usual set up is the following † : During inflation vaton scenario. For the curvaton model to work such the curvaton field acquires a large expectation value if it terms should be very small or absent up to some high has a negative mass squared, or if its mass is much smaller order. Here SUSY helps, since the non-renormalization than the Hubble constant. In the post-inflationary epoch theorem assures that operators absent in the superpoten- the Hubble expansion acts as a friction term in the equa- tial are absent at all scales. tions of motion, and the field remains effectively frozen There are several constraints on the curvaton model. at a large field value. This stage ends when the Hub- The produced perturbations should be adiabatic (which ble constant becomes of the order of the curvaton mass, requires that the curvaton decays after inflaton decay), H mφ, at which point the curvaton starts oscillating nearly scale invariant (the curvaton mass should be much ∼ smaller than the Hubble constant during inflation), and of the correct magnitude. For the perturbations to be siz- able, the curvaton energy density needs to be a significant †For an alternative implementation of the curvaton model fraction of the total energy density in the universe. To see [7] avoid gravitino overproduction the reheat temperature 1 should be sufficiently low. Curvaton decay should not of inflation and the moment the potential approaches a destroy the nucleosynthesis predictions. Furthermore, quadratic form by the parameter q: thermal damping and evaporation should be taken into δφ δφ account. = q , (6) φ φ In the next section we discuss these various constraints osc ∗ on the curvaton model in more detail. In section III we where the subscript osc denotes the quantity at the onset consider the possible (soft) mass terms that can arise in of oscillations, when H m. For a quadratic or flat SUSY theories, and analyze the parameter space for the potential the evolution Eqs.∼ (2, 3) for φ and δφ are the various cases. We end with conclusions. same and q = 1. The oscillating curvaton field behaves as non- II. CONSTRAINTS relativistic matter. After inflaton decay, the energy den- sity becomes a mixture of radiation and matter, and isocurvature perturbation are transformed into adiabatic A. Density perturbations ones. This period ends when the curvaton becomes to dominate the energy density or, if that never happens, The density perturbations for the curvaton field have when it decays. been analyzed in [3]. We briefly review their results. The prediction of the curvaton model for the curvature Consider the curvaton with minimal kinetic term and perturbation can be written as (for small ηφ): scalar potential V (φ). We can expand the field in a clas- sical part plus quantum fluctuations 1/2 rdecq H∗ ζ = , (7) P 3π φ∗ φ(x)= φ + δφ(x). (1) where the subscript dec means the corresponding quan- The unperturbed and perturbed curvaton field satisfy re- tity evaluated at the time of φ-decay, i.e., when H Γ . ∼ φ spectively Furthermore, we have defined the parameter r as the ra- tio of energy density in the curvaton field to the total ¨ ˙ φ +3HI φ + Vφ =0 , (2) energy density in the universe r = ρφ/ρ. The COBE data requires ¨ ˙ 2 1/2 δφk +3HI δφk + (k/a) + Vφφ δφk =0 . (3) (COBE) = 4.8 10−5, Pζ × n(COBE) = 0.93 0.13. (8) Here δφk are the Fourier components of δφ. An over- ± dot denotes ∂/∂t, and a subscript φ denotes ∂/∂φ. We Moreover, the non-detection of tensor perturbations sets have made the first order approximation δ(V )= V δφ. φ φφ an upper limit on the energy density during inflation The fluctuations of a generic massive scalar field gen- 14 erated during de Sitter stage are then found to be on H∗ < 10 GeV. (9) superhorizon scales (where the gradient term in Eq. (3) ∼ is negligible) B. Initial conditions 3/2−νφ H∗ k δφk 3 , (4) | |≃ 2k aH∗ The initial field value of the curvaton at the begin- ning of inflation is a free parameter. The only constraint where the subscript denotes the time of horizon exit. is that the energy density in the curvaton field is sub- ∗ 2 2 Further, νφ = (9/4 m/H ). For ηφ = (m/3H ) 1 2 2 − ≪ dominant. The inflaton energy density is ρI HI MP, one has 3/2 νφ ηφ and the spectrum is nearly scale ∼ − ≃ with HI the Hubble constant during inflation. This invariant. To be more precise, the spectral tilt of the means that for masses m2 H2 a typical initial field perturbation is given by ≪ I value will be large φ MP. We consider scalar≫ potentials of the form d ln φ nφ P =2ηφ 2ǫH , (5) ≡ d ln k − 1 λ V (φ)= m2φ2 + φ4+n. (10) 2 M n and it is assumed that ǫ H/H˙ 1. H ≡ ≪ The field remains overdamped until H m when the The non-renormalizable operators, suppressed by the field starts oscillating in the potential.∼ The potential Planck scale or some other ultraviolet cutoff, will gen- 2 2 2 rapidly approaches a quadratic form, after which the frac- erate an effective mass δmHO = ∂ V/∂φ Vφφ >H for tional perturbation δφ/φ remains constant. We param- large initial field values. The curvaton is≡ underdamped, eterize the change in fractional density between the end and decreases exponentially fast. 2 The effective mass squared term during inflation can fractional energy density in the curvaton field increases be positive or negative. The finite energy density in the until a maximum value at curvaton decay: early universe breaks supersymmetry. This leads to soft 2 α mass terms of the form m2 = cH2. The sign of c is ρφ 1 φosc ΓI I rdec = , (13) determined by the K¨ahler potential. ρ ∼ h M 2 m dec P If the effective mass squared is negative and the initial curvature amplitude is large, the curvaton field will ap- with α = 0 for ΓI > m, and α = 1/2 for ΓI < m.
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