Quantum fields for Cosmology

1. Quantum fields for Cosmology Anders Tranberg Universityof Stavanger In collaborationwith Tommi Markkanen (Helsinki) JCAP 1211 (2012) 027/arXiv: 1207:2179 arXiv: 1303.0180 CPPP, Helsinki 4.-7. June 2013

2. Precision Cosmology • Unprecedentedprecision in observationsrequiresimprovedprecision in theoreticalpredictions and computations. Planck 2013! • Standard dynamics: • Inflation from classicallyslow-rolling homogeneousfield. • CMB from free, lightscalarfield modes in deSitterspacevacuum, freezing in semi-instantaneously at horizoncrossing. • New observables: • Non-gaussianity (bi-spectrum, tri-spectrum, spikes, …). • Scaledependencebeyondpowerlaw (spectralindex, running, runningofrunning…). • Efoldswithprecision +/- 10. • But: Inflaton is an interactingquantumfield.

3. Corrections? Dynamics -> Value at horizoncrossing? Interactingvacuumstate? Dynamics -> End ofinflation -> valueof H(k)? Interactions -> high-order nontrivial correlators? Freeze-in afterhorizoncrossing? Reheatingdynamics -> H(k)? …

4. Whatwe all know, butrarelystate. • The ”inflaton” is reallythemean-field (1-point function) of a quantumdegreeoffreedom (fundamental scalarfield, composite order parameter, …). • The ”potential” V is reallythequantumeffectivepotential, computed to some order in someexpansion. • Degreeoffreedomdisplaced from potential minimum -> inflation.

5. Effectivepotential • 1) Lowenergyeffective action; integrateoutdegreesoffreedomabovesomeenergyscale -> effectiveinteractions for low-energydegreesoffreedom. • Ex. (Fermi theory <-> Electroweakinteractions, Standard Model <-> MSSM, …). • Still quantuminteractionsoflow-energydegreesoffreedom. • 2) Quantum effective action; integrateout all degreesoffreedomexceptthemeanfield/order parameter. • No more ”quantum” interactions. Treat as ”classical” dynamics in effectivepotential.

6. Classical, classical and classical • Trulyclassicaltheory: no h-bar, noquantumfluctuations • Classicalequationsof motion • Toasters, macroscopicmagneticfields, gravity, cosmicstrings • Classicallimit.

7. Classical, classical and classical • Classicalapproximation: • In a squeezedstate (largeoccupationnumbers), dynamicsareclassical-like. • Still need to average over ensemble representingthe initial state! • CMB-prescription: Replace ensemble average by average over the sky. Standby for Arttu’s talk! Starobinsky, Mukhanov, Garcia-Bellido, Grigoriev, Shaposhnikov, Tkachev, Smit, Serreau, Aarts, AT, Rajantie, Linde, Kofman, Hindmarsh, Felder, Saffin, Berges, Borsanyi, …

8. Classical, classical and classical • Quantum effectivepotential: • Meanfieldevolutionfollows as ”classical” equationof motion from effectivepotential. • Meanfield~ ”theclassicalfield” (dangerous!) • Trulyclassical = trivial limit ofquantumeffectivepotential. • Computeeffective action: • Pickfavourite (renormalizable) tree-level action. • Compute diagrams untilyou run outofgraduate students. • Renormalize relative to somevacuum. • 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…

9. Quantum effective action in FRW • Example: One-loop 1PI effective action oftwocoupledscalarfieldsand metric. Treatmetric as classicalfield(nogravitational loops).

10. Issues • Vacuum? • Identifyingdivergences -> anyvacuumcorrect to 4 derivatives (order H^4) is ok! • Useadiabaticvacuum? • Computing effective action? • Expansion in diagrams, and probably in gradients (adiabatic, Schwinger-deWitt, …). • Computeclose to whereyouneed it? • Renormalization? • Divergencesaregone. Applyrenormalizationconditions to fix parameters. • At whichscale? • To whichvalues? • Onlycounterterms for invariant operators.

11. Quantum effective action in FRW • Most general case: Markkanen, AT: 2012

12. Simplifiedmodel • Solve for . • Set: • Tree-level: 2 coupled, non-selfinteracting, minimallycoupledfields. Markkanen, AT: 2012

13. Scalarfieldequationof motion • Given background (dS, mat. dom., rad. dom., …): Markkanen, AT: 2012

14. Quantum correctedFriedmanneqs. • Self-consistentlysolving for thescalefactor: Markkanen, AT: 2012

15. More issues • Infrared problems for masslessfields? • Becauseweuse ”perturbative” propagators, withmean-fieldinsertions. • Interactingtheory -> dynamicalmass. • End ofinflation? • Nonperturbativebehaviour (reheating, preheating, defects…). • Thermalization, imaginaryself-energies. • Needself-consistent, dynamicalpropagatorequation -> 2PI effective action. Calzetta, Hu, Cornwall, Jackiw, Tomboulis, … • Serreau 2011: 2PI-resummation to LO -> alwaysnon-zero mass in dS. (alsoBoyanovsky, deVega, Holman, Sloth, Riotto, Parentani, Garbrecht, Prokopec…) • LO is still Gaussian! NLO AT 2008 • Need a spacelatticeand a finitenumberof modes; all eventuallyredshiftintothe IR. Problem. AT 2008 • How to renormalizeconsistently?

16. Conclusions • ModernCosmologicalobservationsareprecise to 10 (5?) e-folds. • Detectionof non-gaussianity is imminent (…maybe…). • For precisioncomputations, weneed to thinkoftheinflaton/curvaton as quantumfields. • Simple! Computetheeffectivepotential, and do as usual…maybewithout SR. • Useful! Onlyallowsrenormalizableinteractions-> restrictive (buteffectivetheories…). • Easy? Well…thetechniquesexist: • 1PI for massive fieldswithperturbativelysmallexcitations • 2PI for anyfieldswith non-perturbativelylargeexcitations. • -> alsoclassical-statisticalapproximation for verylargeexcitations.