Signatures of Trans-Planckian Dissipation in Inflationary Spectra

Transcription

1 Signatures of Trans-Planckian Dissipation in Inflationary Spectra 3. Kosmologietag Bielefeld Julian Adamek ITPA University Würzburg 8. May 2008 Julian Adamek 1 / 18

2 Trans-Planckian Dissipation in Inflationary Spectra Outline Outline Inflationary Cosmology & the Trans-Planckian Question The Dissipative Model Numerical Analysis of Signatures in the Power Spectrum Conclusion Julian Adamek 2 / 18

3 Inflationary Cosmology & the Trans-Planckian Question The Standard Picture During inflation, quantum fluctuations of the vacuum are redshifted across the horizon scale H = t a/a, where they freeze and turn into classical stochastic perturbations. After inflation has ended, these perturbations eventually re-enter the horizon, where they initiate structure formation in the universe. Julian Adamek 3 / 18

4 Inflationary Cosmology & the Trans-Planckian Question Framework Minimally coupled massless scalar field Φ Flat FLRW metric ds 2 = dt 2 + a 2 (t) dx 2 = a 2 (η) ( dη 2 + dx 2) Comoving coordinates (p = comoving three-momentum, p = p, proper momentum P = p/a) Julian Adamek 4 / 18

5 Inflationary Cosmology & the Trans-Planckian Question The Power Spectrum of Primordial Perturbations (Definition) Inflationary perturbations are characterized by their power spectrum (the Fourier transformed two-point function): 1 P p (η) (2π) 3 d 3 x Φ (η,x) Φ (η,0) e ipx = 1 d 3 p [ Φ p (η),φ 2 p (η) ] + Julian Adamek 5 / 18

6 Inflationary Cosmology & the Trans-Planckian Question The Power Spectrum of Primordial Perturbations (Quantization) For a minimally coupled massless scalar field Φ, the mode decomposition a(η) Φ p (η) Φ p (η) = â p ϕ p (η) + â p ϕ p (η) is canonically quantized by imposing [âp,â p ] = δ3 ( p p ), [âp,â p ] = 0. The mode functions ϕ p are solutions of the mode equation [ 2 η + p 2 2 ηa/a ] ϕ p = 0. Julian Adamek 6 / 18

7 Inflationary Cosmology & the Trans-Planckian Question The Power Spectrum of Primordial Perturbations (Bunch-Davies Vacuum) The choice of the vacuum relies on the identification of the positive-frequency mode functions, which is usually done on subhorizon scales (P = p/a ) by matching the inflationary solution to the one of Minkowski space (equivalence principle). In the slow-roll regime of inflation ( t H/H 2 1), the choice of this vacuum (= Bunch-Davies vacuum) implies P p = H2 p 2p 3, i.e. a (nearly) scale invariant power spectrum. H p denotes the Hubble rate taken at the horizon exit of the mode p. Julian Adamek 7 / 18

8 Inflationary Cosmology & the Trans-Planckian Question Conceptional Problems Depending on the total amount of inflation, the structures we observe today may originate from fluctuation scales that were arbitrarily small at the onset of inflation. Trans-Planckian Question: If these scales had been beyond the Planck scale, would we see signatures of trans-planckian physics in the power spectrum? Or from another point of view: The identification of the vacuum in the limit P appears questionable, since we have no certainty about the nature of the vacuum at small scales. Julian Adamek 8 / 18

9 Inflationary Cosmology & the Trans-Planckian Question Phenomenological Approach to UV Effects Nontrivial phenomenology may arise from e.g. Setting initial conditions at UV cutoff ( instantaneous vacuum ) Modified dispersion relation ( ω 2 (p) = p P n ) Λ n +... Dissipative effects? ( ) P n γ(p) = p Λ n +... Julian Adamek 9 / 18

10 The Dissipative Model Action The effective theory is defined by the action S = 1 d 3 p[s Φ (p) + S Ψ (p) + S int (p)], 2 with the free contributions S Φ (p) = dη Φ [ p 2 η ωp 2 (η) ] Φp, [ S Ψ (p) = dt 2 t Ω 2 ] k Ψp,k, dkψ p,k and the interaction part S int (p) = dηg p (η) ) dk ( Φp η Ψ p,k + h.c.. Julian Adamek 10 / 18

11 This yields the two Euler-Lagrange equations [ 2 t + Ω 2 k] Ψp,k = t (g p Φ p ) [ 2 η + ωp 2 (η)] Φ p = g p (η) The Dissipative Model, Ω k πλ k, dk η Ψ p,k. Equations of Motion We solve the first equation Ψ p,k = Ψ 0 p,k dt G Ψ k ( t,t ) t (g p Φp ), and insert this general solution into the second equation. We then use dk t G Ψ ( k t,t ) = δ (t t ) Λ and arrive at a local equation... Julian Adamek 11 / 18

12 The Dissipative Model Effective Equation with Dissipation [ 2 η + 2γ p (η) η + ωp 2 ] (η) + η γ p Φp = g p (η) dk η Ψ 0 p,k with the effective decay rate γ p (η) = g2 p (η) 2Λ ( ) P n = p Λ n +... Julian Adamek 12 / 18

13 The Dissipative Model The Late-Time Power Spectrum The general solution of the effective equation of motion reads Φ p (η) = Φ 0 p }{{ (η) + dη G Φ ( p η,η )( ( g p η ) ) dk η Ψ 0 p,k. } damped As soon as the damped part has dissipated, we can write the power spectrum as 1 P p (η) = 2a 2 dη dη G Φ ( p η,η ) G Φ ( p η,η ) ( N p η,η ), (η) where we have introduced the noise kernel ( N p η,η ) = g p (η ) g p (η ) δ 3 (p p dkdk [ η Ψ 0 p,k ), ]+ η Ψ0 p,k. Julian Adamek 13 / 18

14 The Dissipative Model Summary The late-time power spectrum is entirely governed by the properties (the state) of the Planckian degees of freedom Ψ. For any given state (e.g. ground state), the power spectrum can be calculated from a double integral. In case of scale separation Λ H it is possible to apply some analytic approximations in order to estimate the spectral power in the vacuum. In any other case, the power spectrum can be computed numerically. Julian Adamek 14 / 18

15 Numerical Analysis of the Power Spectrum Input Input for the numerical analysis Inflationary background in terms of a(η) (de Sitter space, power law inflation) Effective decay rate γ p /p in terms of a function (polynomial) of P/Λ State of the Ψ fields (e.g. vacuum) In order to have a well-behaved system it is necessary to assume that the coupling does not grow without bounds in the ultraviolet sector. As a simple realization, we choose (( ) γ p P n ) p = κtanh κ 1. Λ Julian Adamek 15 / 18

16 Numerical Analysis of the Power Spectrum de Sitter Space, Vacuum [ Spectral power Pp in units of H 2 /2p 3 ] κ = 0.1 κ = 0.5 κ = 2 κ = 10 leading order estimate Deviation [ in units of H 2 /2p 3] κ = 0.1 κ = 0.5 κ = 2 κ = 10 leading order estimate Λ [in units of H] Λ [in units of H] Julian Adamek 16 / 18

17 Numerical Analysis of the Power Spectrum Power Law Inflation, ε th/h 2 = 0.2, Vacuum [ in units of Λ 2 ] Spectral power p 3 Pp κ = 0.1 κ = 0.5 κ = 2 κ = Comoving momentum p [in units of Λ] Julian Adamek 17 / 18

18 Conclusion In the vacuum, the power spectrum remains robust if Λ H. The leading order modification scales as (H/Λ) n. The complicated next-to-leading order modifications can (at least partially) be attributed to non-adiabatic transitions. The dissipative setting may also be seen as a framework for mode creation. Julian Adamek 18 / 18

19 Appendix de Sitter Space, Finite Temperature T 10 3 [ in units of H 2 /2p 3 ] Spectral power Pp H 10 1 Λ Environment temperature T [in units of H] Julian Adamek 19 / 18

20 Appendix Non-Adiabaticity and Memory Loss The degree of non-adiabaticity in the evolution of the modes can be characterized by η ω eff ωeff 2, where ω eff denotes the effective mode frequency. The amount of dissipation between any event η and the end of inflation is given by the integrated damping rate ( exp 0 η γ p ( η ) dη ). Julian Adamek 20 / 18

21 Appendix Scale Separation and Non-Adiabatic Transitions Exponential damping term κ = 0.5 κ = Degree of non adiabaticity z pη Julian Adamek 21 / 18