1. Can observations look back to the beginning of inflation ?

2. Primordial fluctuations inflaton field : χ inflaton field : χ primordial fluctuations of inflaton become observable in cosmic microwave background primordial fluctuations of inflaton become observable in cosmic microwave background

3. Does inflation allow us to observe properties of quantum vacuum ? at which time ?

4. Which primordial epoch do we see ? Does the information stored in primordial fluctuations concern beginning of inflation beginning of inflationor horizon crossing horizon crossing

5. Initial conditions for correlation functions set at beginning of inflation Is memory kept at the time of horizon crossing Is memory kept at the time of horizon crossing or do we find universal correlations, uniquely determined by properties of inflaton potential ? or do we find universal correlations, uniquely determined by properties of inflaton potential ?

6. effective action for interacting inflaton coupled to gravity, derivative expansion with up to two derivatives

7. correlation function for different initial conditions correlation function for different initial conditions u = k η no loss of memory at horizon crossing at horizon crossing ( u = -1 )

8. separate initial condition for every k-mode amplitude not fixed

9. memory of initial spectrum

10. scalar correlation function

11. correlation function from quantum effective action no operators needed no operators needed no explicit construction of vacuum state no explicit construction of vacuum state no distinction between quantum and classical fluctuations no distinction between quantum and classical fluctuations one relevant quantity : correlation function one relevant quantity : correlation function

12. quantum theory as functional integral background geometry given by vierbein or metric homogeneous and isotropic cosmology M:E:

13. quantum effective action quantum effective action exact field equation exact propagator equation

14. correlation function and quantum effective action

15. propagator equation for interacting scalar field in homogeneous and isotropic cosmology

16. general solution and mode functions

17. Bunch – Davies vacuum α = 1, ζ = 0 for all k

18. initial conditions β 2 + γ 2 = 4 ζ ζ* mixed state : positive probabilities p i

19. absence of loss of memory of initial correlations amplitude for each k-mode is free integration constant !

20. memory of initial spectrum α = 1 + A p example : spectral index :

21. predictivity of inflation ? initial spectrum at beginning of inflation : gets only processed by inflaton potential initial spectrum at beginning of inflation : gets only processed by inflaton potential small tilt in initial spectrum is not distinguishable from small scale violation due to inflaton potential small tilt in initial spectrum is not distinguishable from small scale violation due to inflaton potential long duration of inflation before horizon crossing of observable modes : one sees UV-tail of initial spectrum. If flat, predictivity retained ! long duration of inflation before horizon crossing of observable modes : one sees UV-tail of initial spectrum. If flat, predictivity retained !

22. loss of memory beyond approximation of derivative expansion of effective action ? seems likely for long enough duration of inflation before horizon crossing of observable modes seems likely for long enough duration of inflation before horizon crossing of observable modes sufficient that modes inside horizon reach Lorentz invariant correlation for flat space sufficient that modes inside horizon reach Lorentz invariant correlation for flat space not yet shown not yet shown equilibration time unknown equilibration time unknown

23. vacuum in cosmology simply the ( averaged ) state of the Universe result of time evolution minimal “particle number” ? depends on definition of particle number as observable

24. end

25. propagator equation in homogeneous and isotropic cosmology η > η’ : G > = G s + G a

26. evolution equation for equal time correlation function massless scalar in de Sitter space : u = k η