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