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Renormalized stress tensor for trans-Planckian cosmology

Renormalized stress tensor for trans-Planckian cosmology. Francisco Diego Mazzitelli Universidad de Buenos Aires Argentina. PLAN OF THE TALK Motivation Semiclassical Einstein equations and renormalization: usual dispersion relation

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Renormalized stress tensor for trans-Planckian cosmology

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  1. Renormalized stress tensor for trans-Planckian cosmology Francisco Diego Mazzitelli Universidad de Buenos Aires Argentina

  2. PLAN OF THE TALK • Motivation • Semiclassical Einstein equations and renormalization: • usual dispersion relation • Modified dispersion relations: adiabatic renormalization • Examples and related works • Conclusions D. Lopez Nacir, C. Simeone and FDM, PRD 2005

  3. MOTIVATIONS • scales of cosmological interest today are sub-planckian at the beginning of inflation potential window to observe Planck-scale physics • (Brandenberger, Martin, Starobinsky, Niemeyer, Parentani....) • quantum gravity suggests modified dispersion relations for quantum • fields at high energies • potential implications: • - signatures in the power spectrum of CMB • - backreaction on the background spacetime metric Aim of this work: handle divergences in the Semiclassical Einstein Equations

  4. The Semiclassical Einstein Equations: usual dispersion relation Up to fourth adiabatic order

  5. This subtraction works for some quantum states of the scalar field: those for which the two-point function reproduces the Hadamard structure. These are the physical states of the theory. The infinities can be absorbed into the gravitational constants in the SEE. Alternative to point-splitting -> dimensional regularization

  6. In Robertson Walker spacetimes the procedure above is equivalent to the so called adiabatic subtraction: usual dispersion relation

  7. Solve the equation of the modes using WKB approximation keeping • up to four derivatives of the metric 2) Insert this solution into the expression for different components of the stress tensor (note dimensional regularization) + …….

  8. 3) Compute the renormalized stress tensor and dress the bare constants Renormalized stress tensor Divergent part, to be absorbed into the bare constants Zeldovich & Starobinsky 1972, Parker, Fulling & Hu 1974, books on QFTCS

  9. For the numerical evaluation, one can take the n->4 limit inside the integral A simpler example: renormalization of Only the zeroth adiabatic order diverges

  10. Assumption: “trans- Planckian physics” may change the usual dispersion relation 2 + higher powers of k Higher spatial derivatives in the lagrangian

  11. = SCALAR FIELD WITH MODIFIED DISPERSION RELATION Lemoine et al 2002

  12. We can solve the equation using WKB approx. for a general dispersion relation The 2j-adiabatic order scales as w +…. 2-2j k Modification to the dispersion relation

  13. Components of the stress tensor in terms of Wk NO DIVERGENCES AT FOURTH ADIABATIC ORDER (power counting)

  14. Zeroth adiabatic order after integration by parts….

  15. The divergence can be absorbed into a redefinition of  in the SEE: can be rewritten as Zeroth adiabatic order:

  16. Second adiabatic order – minimal coupling

  17. Second adiabatic order – additional terms for nonminimal coupling

  18. After integration by parts and “some” algebra: Non-minimal coupling <T00> is proportional to G00 <T11> is proportional to G11 where

  19. Renormalized SEE: 4 No need for higher derivative terms if wk ~ k or higher Summarizing:

  20. Explicit evaluation of regularized integrals for some particular dispersion relations …. …. From this one can read the relation between bare and dressed constants and the RG equations

  21. In the massless limit If m0: more complex expressions in terms of Hypergeometric functions Finite results in the limit n->4: similar to usual QFT in 2+1 dimensions

  22. 6 OK fork and minimal coupling Too many restrictions on the initial state, should coincide with adiabatic vacuum up to order 4 Relation with our approach? Work in progress • Related works: • drop the zero-point energy for each Fourier mode (Brandenberger • & Martin2005) • assume that the Planck scale physics is effectively described by • a non trivial initial quantum state for a field with usual dispersion • relation. Usual renormalization. • (Anderson et al2005) • Ibidem, but considering a general initial state. Additional divergences • are renormalized with an initial-boundary counterterm • (Collins and Holman2006, • Greene et al 2005)

  23. CONCLUSIONS • we have given a prescription to renormalize the stress- tensor in • theories with generalized dispersion relations • the method is based on adiabatic subtraction and dimensional • regularization • although the divergence of the zero-point energy is stronger than • in the usual QFT, higher orders are suppressed and it is enough • to consider the second adiabatic order. For • the second adiabatic order is finite – subtract only zero point energy • the renormalized SEE obtained here should be the starting point • to discuss the backreaction of transplanckian modes on the • background method

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