Can observations look back to the beginning of inflation ?

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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 kmodefor every kmode
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 kmode is amplitude for each kmode 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 UVtail of initial spectrum . If flat , UVtail 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