Renormalized stress tensor for trans-Planckian cosmology
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Transcript of Renormalized stress tensor for trans-Planckian cosmology
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Renormalized stress tensor for trans-Planckian cosmology
Francisco Diego Mazzitelli
Universidad de Buenos Aires
Argentina
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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
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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
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The Semiclassical Einstein Equations: usual dispersion relation
Up to fourth adiabatic order
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This subtraction works for some quantum statesof the scalar field: those for which the two-point functionreproduces 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
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In Robertson Walker spacetimes the procedure above is equivalent to the so called adiabatic subtraction:
usual dispersion relation
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1) 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)
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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
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A simpler example: renormalization of
Only the zeroth adiabatic order diverges
For the numerical evaluation, one can take the n->4 limit inside the integral
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Assumption: “trans- Planckian physics” may change theusual dispersion relation
+ higher powers of k2
Higher spatial derivatives inthe lagrangian
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SCALAR FIELD WITH MODIFIED DISPERSION RELATIONLemoine et al 2002
=
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Modification to the dispersion relation
2-2jk
The 2j-adiabatic order scales as w
We can solve the equation using WKB approx. for a general dispersion relation
+….
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Components of the stress tensor in terms of Wk
NO DIVERGENCES AT FOURTH ADIABATIC ORDER (power counting)
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Zeroth adiabatic order
after integration by parts….
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Zeroth adiabatic order:
The divergence can be absorbed into a redefinition of in the SEE:
can be rewritten as
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Second adiabatic order – minimal coupling
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Second adiabatic order – additional terms for nonminimal coupling
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After integration by parts and “some” algebra:
where
Non-minimal coupling
<T00> is proportional to G00
<T11> is proportional to G11
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Summarizing:
Renormalized SEE:
No need for higher derivative terms if wk ~ k or higher4
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Explicit evaluation of regularized integrals for some particular dispersion relations
….
….
From this one can read the relation between bare anddressed constants and the RG equations
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In the massless limit
Finite results in the limit n->4: similar to usual QFT in 2+1 dimensions
If m0: more complex expressions in terms of Hypergeometric functions
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Related works:
• drop the zero-point energy for each Fourier mode (Brandenberger & Martin 2005) OK for k and minimal coupling 6
• 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 al 2005) Too many restrictions on the
initial state, should coincide withadiabatic vacuum up to order 4
Relation with our approach?Work in progress
• Ibidem, but considering a general initial state. Additional divergences are renormalized with an initial-boundary counterterm (Collins and Holman 2006, Greene et al 2005)
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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