# Presentation: 416027

## -vac perturbation theory

- David Lowe
- Brown University
- with K. Goldstein, hep-th/0208167
- with K. Goldstein, hep-th/0302050
- with G.L. Alberghi and K. Goldstein, astro-ph/0307413
- with K. Goldstein, hep-th/0308135

## Plan

- Present a simple class of initial states for inflation that encode effects of trans-Planckian physics
- -vacua
- Potentially observable consequences on CMBR and cosmic ray observations
- Develop perturbation theory around -vacua
- Either coupling to gravity gives problems due to nonlocality
- Or perturbative expansion fails due to IR divergent pinch singularites
- Cutoff -vacua still possible if right strongly coupled and/or non-local, novel physics happens at UV cutoff

- David Lowe, Brown University

## vacua

- Chernikov, Tagirov (1968) Mottola, Allen (1985) showed one-complex parameter family of vacua invariant under all symmetries of de Sitter group
- Any initial state should asymptote to one of these at late time if inflation sufficiently long-lived
- Unknown trans-Planckian physics can be encoded in nontrivial value for 
- Parameterization of new physics (Starobinsky,Tkachev, Danielsson, Goldstein,Lowe)
- Previous studies of inflation always start with Bunch-Davies vacuum state (infinity)
- Rationale: free propagator matches with expected flat space limit as points coincide (Hadamard condition)
- Purely mathematical restriction, since this asymptotic limit not attainable in a cutoff effective field theory

- David Lowe, Brown University

## What happens if we give up this restriction to Hadamard states?
General vacuum
Non-equilibrium states
Unruh detector has non-thermal response
Non-Hadamard
Interplay between curvature and short-distance physics
Possible interesting consequences for ultra-high energy cosmic ray production and CMBR spectrum

- General vacuum
- Non-equilibrium states
- Unruh detector has non-thermal response
- Non-Hadamard
- Interplay between curvature and short-distance physics
- Possible interesting consequences for ultra-high energy cosmic ray production and CMBR spectrum

- David Lowe, Brown University

## Choose conformal metric

Make mode expansion of field

’s are Hankel functions of the second kind
Euclidean vacuum, aka Bunch-Davies

- Make mode expansion of field
- ’s are Hankel functions of the second kind
- Euclidean vacuum, aka Bunch-Davies

- Basic Setup

- David Lowe, Brown University

## General class obtained by taking linear combinations of + and – frequency modes

MA showed these vacua give de Sitter invariant two-point functions
local commutator
Wightman function has peculiar anti-podal singularities 

Much studied in context of dS/CFT
	Bousso, Maloney and Strominger
	marginal coupling in CFT

- MA showed these vacua give de Sitter invariant two-point functions
- local commutator
- Wightman function has peculiar anti-podal singularities
- Much studied in context of dS/CFT
- 	Bousso, Maloney and Strominger
- 	marginal coupling in CFT

- David Lowe, Brown University

## Consequences for CMBR

- Reheating, transition of vacuum energy to zero
- Power-law expansion begins, modes begin to re-enter the horizon
- vacuum generates huge back-reaction (k-independent)

- David Lowe, Brown University

## Cutoff -vacuum

- Work with proper wavenumber cutoff Mcut on set of “nice” timeslices (c.f. Corley and Jacobson)
- Assume  trans-Planckian physics above cutoff scale Mcut, places modes in an -vacuum by the time they expand out to cutoff scale

- David Lowe, Brown University

## Back-reaction problem solved

	Easy to satisfy

Correction to CMBR spectrum

Contribution largely unconstrained

- 	Easy to satisfy
- Correction to CMBR spectrum
- Contribution largely unconstrained

- David Lowe, Brown University

## Expect trans-planckian physics to tie to Mcut
E.g. Danielsson, 
	Easther, et al.
Back-reaction bound becomes

Main conclusion: if new physics places modes in an vacuum below some UV cutoff, there are no obvious problems, and observable effects are produced
Danielsson; Goldstein,Lowe, : rate of thermalization to Bunch-Davies too slow
See also (Starobinsky,Tkachev; Alberghi, Goldstein, Lowe) for studies of ultra-high energy cosmic ray production

- E.g. Danielsson,
- 	Easther, et al.
- Back-reaction bound becomes
- Main conclusion: if new physics places modes in an vacuum below some UV cutoff, there are no obvious problems, and observable effects are produced
- Danielsson; Goldstein,Lowe, : rate of thermalization to Bunch-Davies too slow
- See also (Starobinsky,Tkachev; Alberghi, Goldstein, Lowe) for studies of ultra-high energy cosmic ray production

- David Lowe, Brown University

## Can local perturbative physics place modes in an -vacuum?

- Define vacuum in an interacting theory
- Basic assumptions:
- Theory must be de Sitter invariant
- Suitably defined two-point correlators should reduce to MA result
- Try to send cutoff to infinity and look for problems...

- David Lowe, Brown University

**Notes:**
Renormalization of vacuum energy and back-reaction
Violates Wilsonian decoupling?
Non-local interactions
No known way to quantize with interactions? Does this mean no thermalization?
Thermalization time – dynamics just not there- consider al=0 and prop. Of high freq mode

## Two approaches

- Do everything in real time formalism
- Can think of -vacuum as squeezed state on Bunch-Davies vacuum
- Real time ordered propagators match those of MA (Einhorn,Larsen; Collins,Holman,Martin)
- Start with theory on S4 then continue to real-time
- Imaginary time propagators match continuation of MA
- Analyticity leads to new form of real-time ordering (Goldstein, Lowe)

- David Lowe, Brown University

## Real-time approach

- Can represent -vac as a squeezed state on asymptotic future or past infinity
- In position space, time-ordered propagator contains a linear combination of terms with differing i prescriptions

- David Lowe, Brown University

## Pinch singularities

- The mixed iterms give pinch singularities in loops (Einhorn, Larsen)
- Note: not a UV divergence, but an IR divergence arising when points are null separated
- Can be regulated
- Finite time cutoff
- Define resummed propagator
- Perturbation theory difficult, c.f. non-equilibrium statistical field theory
- IR divergence is a signal of breaking of dS symmetry
- This approach with cutoffs (and subsequent slow decay to Bunch-Davies) still viable phenomenologically (Danielsson; Collin and Martin) but requires fine-tuning of initial state

- David Lowe, Brown University

## Imaginary time approach

- Follow similar approach, but for imaginary time, then continue back at end (Goldstein, Lowe)
- In general this approach computes  expectation values with respect to a density matrix.
- Truncation of real time contour gives transition amplitudes
- Pinch singularities cannot arise in Euclidean signature
- All divergences take form of UV divergences that can be regulated with local counter-terms

- David Lowe, Brown University

## Analytic continuation back to real time

- Can think of field as a bi-local linear combination of Bunch-Davies vacuum fields:
- Real-time propagator has new time-ordering prescription

- x

- y

- x

- y

- x

- y

- x

- y

- David Lowe, Brown University

## Path integral

- Can represent transition amplitudes with a path integral with action
- Only sign of non-locality in kinetic term

- David Lowe, Brown University

## In fixed de Sitter background
Commutators still local due to local interactions
Set of finite renormalizable transition amplitudes
However always need to couple to gravity at end
Stress energy tensor non-local
Fluctuations in geometry allow for observable violations of causality
Expect strong quantum gravity effects to kick in as in chronology protection conjecture (Hawking)
Conclude imaginary time approach cannot be consistently coupled to gravity at non-perturbative level

- Commutators still local due to local interactions
- Set of finite renormalizable transition amplitudes
- However always need to couple to gravity at end
- Stress energy tensor non-local
- Fluctuations in geometry allow for observable violations of causality
- Expect strong quantum gravity effects to kick in as in chronology protection conjecture (Hawking)
- Conclude imaginary time approach cannot be consistently coupled to gravity at non-perturbative level

- David Lowe, Brown University

## Exotic solutions

- Place graviton in -vacuum
- End up with extremely non-local interactions
- Orbifold by antipodal map (Parikh et al.)
- Solves non-locality problem
- No longer time orientable, conventional global quantization techniques fail
- Some other non-local and/or non-perturbative solution

- David Lowe, Brown University

## Conclusions

- Constructed a useful parameterization of trans-Planckian effects in slow-roll inflation
- No go theorem for producing an -vacuum via local, perturbative means
- Have gained a more physical understanding of uniqueness of Bunch-Davies vacuum and Hadamard condition
- Exotic effects and/or fine tuning required to “pump up” the -vacua
- If UV “pump” can be explained, decay to Bunch-Davies can be very slow
- Potentially interesting effects in the CMBR spectrum and production of ultra-high energy cosmic rays

- David Lowe, Brown University