Quantum fields for Cosmology

Anders TranbergUniversity of Stavanger

In collaboration withTommi Markkanen (Helsinki)

JCAP 1211 (2012) 027 /arXiv: 1207:2179arXiv: 1303.0180

CPPP, Helsinki 4.-7. June 2013

Precision Cosmology• Unprecedented precision in observations requires improved precision in

theoretical predictions and computations. Planck 2013!• Standard dynamics:

– Inflation from classically slow-rolling homogeneous field.– CMB from free, light scalar field modes in deSitter space vacuum, freezing in semi-

instantaneously at horizon crossing.

• New observables:– Non-gaussianity (bi-spectrum, tri-spectrum, spikes, …).– Scale dependence beyond power law (spectral index, running, running of running…).– Efolds with precision +/- 10.

• But: Inflaton is an interacting quantum field.

Corrections?

Dynamics -> End of inflation -> value of H(k)?

Dynamics -> Value at horizon crossing?

Interacting vacuum state?

Interactions -> high-order nontrivial correlators?Freeze-in after horizon crossing?Reheating dynamics -> H(k)?…

What we all know, but rarely state.• The ”inflaton” is really the mean-field (1-point function) of a quantum

degree of freedom (fundamental scalar field, composite order parameter, …).

• The ”potential” V is really the quantum effective potential, computed to some order in some expansion.

• Degree of freedom displaced from potential minimum -> inflation.

Effective potential• 1) Low energy effective action; integrate out degrees of freedom above

some energy scale -> effective interactions for low-energy degrees of freedom.– Ex. (Fermi theory <-> Electroweak interactions, Standard Model <-> MSSM, …).– Still quantum interactions of low-energy degrees of freedom.

• 2) Quantum effective action; integrate out all degrees of freedom except the mean field/order parameter.– No more ”quantum” interactions. Treat as ”classical” dynamics in effective potential.

Classical, classical and classical• Truly classical theory: no h-bar, no quantum fluctuations

– Classical equations of motion– Toasters, macroscopic magnetic fields, gravity, cosmic strings

• Classical limit.

Classical, classical and classical• Classical approximation:

– In a squeezed state (large occupation numbers), dynamics are classical-like.

• Still need to average over ensemble representing the initial state!• CMB-prescription: Replace ensemble average by average over the sky.

Starobinsky, Mukhanov, Garcia-Bellido, Grigoriev, Shaposhnikov, Tkachev, Smit, Serreau, Aarts,AT, Rajantie, Linde, Kofman, Hindmarsh, Felder, Saffin, Berges, Borsanyi, …

Standby for Arttu’s talk!

Classical, classical and classical• Quantum effective potential:

– Mean field evolution follows as ”classical” equation of motion from effective potential.– Mean field ~ ”the classical field” (dangerous!) – Truly classical = trivial limit of quantum effective potential.

• Compute effective action:– Pick favourite (renormalizable) tree-level action.– Compute diagrams until you run out of graduate students.– Renormalize relative to some vacuum.– In real-time (in-in, CTP, Schwinger-Keldysh, …).

Parker, Toms, Birrell, Davies, deWItt, Lyth, Shore, Shaposhnikov, Bezrukov, Barvinsky, Bilandzic, Prokopec, Kirsten, Elizalde, Enqvist, Lerner, Taanila, AT, Markkanen, Garbrecht, Postma…

Quantum effective action in FRW• Example: One-loop 1PI effective action of two coupled scalar fields and

metric. Treat metric as classical field (no gravitational loops).

Issues• Vacuum?

– Identifying divergences -> any vacuum correct to 4 derivatives (order H^4) is ok!– Use adiabatic vacuum?

• Computing effective action?– Expansion in diagrams, and probably in gradients (adiabatic, Schwinger-deWitt, …).– Compute close to where you need it?

• Renormalization?– Divergences are gone. Apply renormalization conditions to fix parameters.– At which scale?– To which values?– Only counterterms for invariant operators.

Quantum effective action in FRW• Most general case:

Markkanen, AT: 2012

Simplified model• Solve for .• Set:• Tree-level: 2 coupled, non-selfinteracting, minimally coupled fields.

Markkanen, AT: 2012

Scalar field equation of motion• Given background (dS, mat. dom., rad. dom., …):

Markkanen, AT: 2012

Quantum corrected Friedmann eqs.

• Self-consistently solving for the scale factor:

Markkanen, AT: 2012

More issues• Infrared problems for massless fields?

– Because we use ”perturbative” propagators, with mean-field insertions.– Interacting theory -> dynamical mass.

• End of inflation?– Nonperturbative behaviour (reheating, preheating, defects…).– Thermalization, imaginary self-energies.

• Need self-consistent, dynamical propagator equation -> 2PI effective action. Calzetta, Hu, Cornwall, Jackiw, Tomboulis, …

• Serreau 2011: 2PI-resummation to LO -> always non-zero mass in dS. (also Boyanovsky, deVega, Holman, Sloth, Riotto, Parentani, Garbrecht, Prokopec…)

• LO is still Gaussian! NLO AT 2008

• Need a space lattice and a finite number of modes; all eventually redshift into the IR. Problem. AT 2008

• How to renormalize consistently?

Conclusions– Modern Cosmological observations are precise to 10 (5?) e-folds.– Detection of non-gaussianity is imminent (…maybe…).

• For precision computations, we need to think of the inflaton/curvaton as quantum fields.– Simple! Compute the effective potential, and do as usual…maybe without SR.– Useful! Only allows renormalizable interactions -> restrictive (but effective theories…).– Easy? Well…the techniques exist:

• 1PI for massive fields with perturbatively small excitations• 2PI for any fields with non-perturbatively large excitations.• -> also classical-statistical approximation for very large excitations.