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

2. 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.

3. 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)? …

4. 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.

5. 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.

6. 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.

7. 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!

8. 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…

9. 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).

10. 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.

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

12. Simplified model Solve for. Set: Tree-level: 2 coupled, non-selfinteracting, minimally coupled fields. Markkanen, AT: 2012

13. Scalar field equation of motion Given background (dS, mat. dom., rad. dom., …): Markkanen, AT: 2012

14. Quantum corrected Friedmann eqs. Self-consistently solving for the scale factor: Markkanen, AT: 2012

15. 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?

16. 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.