Can observations look Can observations look back back

to the beginning of to the beginning of inflation ?inflation ?

Primordial fluctuationsPrimordial fluctuations inflaton field : inflaton field : χχ primordial fluctuations of inflaton become primordial fluctuations of inflaton become

observable in cosmic microwave observable in cosmic microwave backgroundbackground

Does inflation allow us to Does inflation allow us to observe properties of observe properties of quantum vacuum ?quantum vacuum ?

at which time ?at which time ?

Which primordial epoch do Which primordial epoch do we see ?we see ?

Does the information stored in Does the information stored in primordial fluctuations concernprimordial fluctuations concern

beginning of inflationbeginning of inflationor or horizon crossinghorizon crossing

Initial conditions for Initial conditions for correlation functionscorrelation functions

set at beginning of inflationset at beginning of inflation

Is memory kept at the time of Is memory kept at the time of horizon crossinghorizon crossing

or do we find universal correlations , or do we find universal correlations , uniquely determined by properties of uniquely determined by properties of inflaton potential ?inflaton potential ?

effective action for effective action for interacting inflaton coupled interacting inflaton coupled

to gravity,to gravity,derivative expansion with derivative expansion with

up to two derivativesup to two derivatives

correlation function for correlation function for different initial conditionsdifferent initial conditions

u = k u = k ηη

no loss of memoryno loss of memory at horizon crossingat horizon crossing( u = -1 )( u = -1 )

separate initial condition separate initial condition for every k-modefor every k-mode

amplitude not fixedamplitude not fixed

memory of initial memory of initial spectrumspectrum

scalar correlation scalar correlation functionfunction

correlation function fromcorrelation function fromquantum effective actionquantum effective action

no operators neededno operators needed no explicit construction of vacuum stateno explicit construction of vacuum state no distinction between quantum and no distinction between quantum and

classical fluctuations classical fluctuations one relevant quantity : correlation one relevant quantity : correlation

functionfunction

quantum theory as functional quantum theory as functional integralintegral

background geometry given by vierbein or metricbackground geometry given by vierbein or metric

homogeneous and isotropic cosmologyhomogeneous and isotropic cosmology

M:M: E:E:

quantum effective actionquantum effective action

exact field equationexact field equation

exact propagator equationexact propagator equation

correlation function and correlation function and quantum effective actionquantum effective action

propagator equation propagator equation for interacting scalar field infor interacting scalar field inhomogeneous and isotropic homogeneous and isotropic

cosmologycosmology

general solution and general solution and mode functionsmode functions

Bunch – Davies vacuumBunch – Davies vacuum

αα = 1 , = 1 , ζζ = 0 for all k = 0 for all k

initial conditionsinitial conditions

ββ2 2 + + γγ2 2 = 4 = 4 ζζ ζζ**

mixed state : positive probabilities pmixed state : positive probabilities p ii

absence of loss of memoryabsence of loss of memoryof initial correlationsof initial correlations

amplitude for each k-mode is amplitude for each k-mode is free integration constant !free integration constant !

memory of initial memory of initial spectrumspectrum

αα = 1 + A = 1 + App

example :example :

spectralspectralindex : index :

predictivity of inflation ?predictivity of inflation ? initial spectrum at beginning of inflation : initial spectrum at beginning of inflation :

gets only gets only processedprocessed by inflaton potential by inflaton potential small tilt in initial spectrum is small tilt in initial spectrum is not not

distinguishable distinguishable from small scale violation from small scale violation due to inflaton potentialdue to inflaton potential

long duration of inflation long duration of inflation before horizon before horizon crossing of observable modes : one sees crossing of observable modes : one sees UV-tail of initial spectrum . If flat , UV-tail of initial spectrum . If flat , predictivity retained !predictivity retained !

loss of memory beyond loss of memory beyond approximation of derivative approximation of derivative

expansion of effective expansion of effective action ?action ?

seems likely for long enough duration of seems likely for long enough duration of inflation before horizon crossing of inflation before horizon crossing of observable modesobservable modes

sufficient that modes inside horizon reach sufficient that modes inside horizon reach Lorentz invariant correlation for flat spaceLorentz invariant correlation for flat space

not yet shownnot yet shown equilibration time unknownequilibration time unknown

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

endend

propagator equation inpropagator equation inhomogeneous and isotropic homogeneous and isotropic

cosmologycosmology

ηη > > ηη’ : G’ : G> > = G= Gs s + G+ Gaa

evolution equation forevolution equation forequal time correlation equal time correlation

functionfunction

massless scalar in massless scalar in de Sitter space :de Sitter space :

u = k u = k ηη