renormalized stress tensor for trans-planckian cosmology francisco diego mazzitelli universidad de...
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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
• Modified dispersion relations: adiabatic renormalization
• Examples and related works
•Conclusions
D. Lopez Nacir, C. Simeone and FDM, PRD 2005
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
The Semiclassical Einstein Equations: usual dispersion relation
Up to fourth adiabatic order
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
In Robertson Walker spacetimes the procedure above is equivalent to the so called adiabatic subtraction:
usual dispersion relation
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)
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
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
Assumption: “trans- Planckian physics” may change theusual dispersion relation
+ higher powers of k2
Higher spatial derivatives inthe lagrangian
SCALAR FIELD WITH MODIFIED DISPERSION RELATIONLemoine et al 2002
=
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
+….
Components of the stress tensor in terms of Wk
NO DIVERGENCES AT FOURTH ADIABATIC ORDER (power counting)
Zeroth adiabatic order
after integration by parts….
Zeroth adiabatic order:
The divergence can be absorbed into a redefinition of in the SEE:
can be rewritten as
Second adiabatic order – minimal coupling
Second adiabatic order – additional terms for nonminimal coupling
After integration by parts and “some” algebra:
where
Non-minimal coupling
<T00> is proportional to G00
<T11> is proportional to G11
Summarizing:
Renormalized SEE:
No need for higher derivative terms if wk ~ k or higher4
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
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
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)
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