Stochastic Theory of Inflation by theoryman

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									                                                                      ˚1˚
     Stochastic Theory of Inflation



By Tomislav Prokopec (Utrecht University)


                          Based on work of
                            Starobinsky;
                       Starobinsky Yokoyama;
                        Woodard; Morikawa;
                       Prokopec, Rigopoulos
                     Prokopec, Tsamis, Woodard




      Nonperturbative Dynamics in the Early Universe, Madrid, Sep 13 2006
                                                                                                                                              ˚2˚
             Scalar theory in de Sitter space
   Scalar action in de Sitter space a  exp(Ht )  1 / H 

  Sφ   d 4 x  g
                     1
                     2
                        
                        g μν μ φ  φ  m φ2  Rφ2  2V (φ) ,
                                           
                                                                                                        ds 2  g dx dx  dt 2  a 2dx 2
                                                                                                                                                   



 Scalar operator equation
                        2
        2                                                                                                                      
         t  3H  t  2  mφ  R
                              
                                                      φ(x )  V '(φ)  0,
                                                       ˆ           ˆ                         [ φ, Πφ ]  i 3  x  x ' 
                                                                                               ˆ ˆ
                       a                            
can be recast as the Yang-Feldman equation (Woodard’s heuristic derivation)
                 φ(x )  φ0 (x )   d 4 x ' a '3 Gret ( x, x ')V '  φ( x ')  ,
                 ˆ       ˆ                                            ˆ
  Tree level (noninteracting) solution                                         m  φ
                                                                                    
                                                                                         R  0 
                                                                                                        
                                                                                   
                   d 3k ikx                                                                                   H        ik 
                         e u (t , k )a(k )  e u *(t , k )a  (k ) ,                                                            exp  ik / Ha 
                            
      φ0 (x )  
      ˆ                              ˆ         ik  x
                                                          ˆ                                          u (t , k )       1 
                  (2 )3                                                                                          2k 3
                                                                                                                        Ha 

  The MMC scalar retarded Green function
                                                                                                       
                                                       d 3k
                                                                                                           
                                                                                                                   
                     Gret ( x, x ')  i (t  t ')          u (t , k )u *(t ', k )  u *(t , k )u (t ', k ) e ik ( x  x ')
                                                      (2 )3
                                                                   
                                        H2                 H x  x '  H (   ') 
                                                                                                                            
                                                                                                                               
                                           (t  t ')                                       x  x '  (   ')  
                                        4              
                                                                aa ' H x  x '                                                
                                                                                                                               
                                                                                                                                      ˚3˚
  Stochastic scalar theory (Woodard’s derivation)
  Scalar field is split into IR and UV parts
                                                                                                  
         ,                           d 3k
                                                                                                                         
                                                                                     
   ˆ ˆ       ˆ                 φ (x )  
                               ˆ                   ( aH  k ) e u(t , k )a(k )  e u *(t , k )a  (k ) , (  1)
                                                                 ik  x
                                                                           ˆ         ik  x
                                                                                                ˆ
                                           (2 )3


Mode functions u and the Green function can be expanded as
                                           H  1 k  i  k               
                                                            2        3
                                                                                                                  
                               u (t , k )        1              ..                               (x  x  x ' ,     ')
                                            2k 3  2  Ha  3  Ha 
                                                 
                                                                            
                                                                            
                                                                                                                           
                                                                                                                      1 2
                            Gret ( x, x ') 
                                              1
                                             3H
                                                                   1
                                                                          
                                                                                1 
                                                                            a a' 
                                                                                     
                                                 (t  t ') 3  x  x '  3  3   O N 1 3  x  x ' , N 1  
                                                                                        ˆ          
                                                                                                            ˆ
                                                                                                                      4 ( Ha)2

 Here we made use of:                      (|  | x)
                                                                                                4 3   1 1 
                                                                                                                      3
                                                    1                 1
              x       x     x     '  x   2   ''  x   3  ..         x      ..
                                                    2                 6                        3H 3           a a'

  such that the Yang-Feldman equation reduces to (drop φ>):
                                                                                                                             
                      1
                                                                                                                                 
                                                                             3

                        V '  φ ( x)  ,
                                                                                                                 
                                                        (x )   (aH ) d k  ( aH  k ) 1 eik  x a(k )  e  ik  x a  (k )  O( N 1 ) ,
  φ ( x)   ( x) 
  ˆ                           ˆ                 ( x)  φ0
                                                        ˆ                 (2 )3              3
                                                                                                      ˆ                  ˆ             ˆ
                     3H                                                                     2k

 NB: The correction O(1/N) is small only when є<<1!
 NB2: When modes u are truncated at leading order, the field commutes!
                                                                          ˚4˚
                  Scalar field spectrum
Free scalar field spectrum in de Sitter space
                                  dk         sin(k x)         
        0  ( x, ) ( x ', ) 0   P (k , )
          ˆ        ˆ                                    , x  x  x '
                                    k             k x
                            H2       k2  H2       1     
                P (k , ) 
                                1           1       
                            4 2  (aH)2  4 2  4n(k, ) 




                                                           n(k, )  1



                n(k, )  1
                                                                                                                          ˚5˚
Stochastic scalar theory (Morikawa’s derivation)
    Scalar field is split into IR and UV parts      ,
                                               ˆ ˆ      ˆ
                                                              
                d 3k
                                                                               
                                                      
   φ (x )  
   ˆ                  W (k , ) e u (t , k )a(k )  e u *(t , k )a (k ) , e.g. W (k , )   (k   aH )
                                 ik  x
                                            ˆ         ik  x
                                                                 ˆ
               (2 )3

 Use the functional integral technique (in Schwinger-
  Keldysh formalism) to integrate out the UV field:
        Sφ   d 4 x  g  g μν μ φ  φ  m φ2  Rφ2  2V (φ)  ,
                            1                         

                             2
       Sφ[ ]  S [ ]  S [ ]  Sint [ ,  ], S [ ]  S [ ], S [ ]  S [ ]
      Sint [ ,  ]   d4 x  μ a3 g   ν  m  R    higher order,
                                                      



Double time contour path integral (S^+/-:for-/backward contour)
                                                                  
                                                                
    exp(i[ ])   d  d  (   )  D  D exp iS+  iS- 
                                                 


                                i 4
                                   d x d x 'a  (x)R (x, x')a'  (x') -2 d x d x ' (t - t')a  (x)I (x, x')a'  (x')
[ ,  ]  S< ( )  S< ( ) 
                                       4   3                 3            4    4              3                 3 C

                                2
           ,  
                    C       1 
                                  ,
                                 2
                                        
                                                                              
                                                         Pt  W(t) +3HW(t) +2W(t) t
                  d3k                                                         d3k
R (x, x') = Im        0 [ Ptk (x)][ Pt 'k (x')] 0 ,
                                                         I (x, x') = Re         0 [ Ptk (x)][ Pt 'k (x')] 0 ,
                                                                                                       
                 (2 )3                                                      (2 )3
                                                                                                          ˚6˚
                       Morikawa’s derivation (2)
    For MMC scalar field
  exp(iS[  ]) =  D P[ ] exp(iSeff [  ,  ]), Seff [  ,  ] = S[  ] -S[  ] +  d4 x (x)a3  (x)
                                                                                                      


 where ξ is the (classical) stochastic random field, with the correlator
                                                 H3 sin( Har)                              
            (x) (x')   D P[ ]  (x) (x') = 2
                                                 4     Har
                                                                (t - t'),          r = x - x' 
                                                          2
                      Seff [ ,  ]
                               
                                                          
                                            = 0   +3H - 2   V'( ) -   3H  0
                                                                        
                            
                                       0
                                                          a

For a slow roll inflation (d²φ/dt²<< dφ/dt) this reduces to the Starobinsky’s result
                       2
     1               1       1            H ˆ      1 d 1                                        ˆ
 
      V'( ) -             (   )   N-1 
                                                            V'( ) +..   0,                    N-1  4 2
    3H              3H a2    3H             12      3H dt  3H            
                                                                                  ˚7˚
             Starobinsky-Yokoyama solution
     Starobinsky’s Langevin theory
                       1                                 H3
                 
                        V'( ),         (x, t) (x', t)  2 δ(t - t')
                      3H                                   4

   Stochastic (statistical) average of a function(al) F of φ
                            1
                    
                             V'( ),      F[]   d ( , t)F[ ]
                           3H
Where the probability density satisifies the Fokker-Planck equation
                                 1                        H3 
                 t  ( , t)      [V'( )  ( , t)] + 2   ( , t) (*)
                                3H                        8
Assume  ( )  lim  ( , t) exists, s.t.  t  ( )  0, and that H≈ const.
                  t ∞
                    →




                                                   8 2V ( ) 
then the solution of (*) is:  ( )  0 exp                
                                                     3H4 
For quartic potential, V ( )    4 ,
                                  1 4!
                     2   2  4        2  4 
          ( )           4  exp            
                   (1/ 4)  9H          9 H4 
                                                                                                                   ˚8˚
        Woodard Tsamis scalar field correlator
    Asymptotic form for scalar field correlator
                                                    2  4   9H4 
                                                                                       n/2
                                                      n    1
                           2n
                   lim  (x, t)   d  ( ) 
                                               2n
                                                                   
                    t ∞
                    →
                                                      4    2 
                                                        1



    NB: Resummed perturbative result -> nonperturbative in !
    (when 0 the correlator spreads to ∞)
    For a finite time t, perturbative result in ln²(a)
    Fokker-Planck equation implies
                                 n(2n -1)H               n    
                  t  (x, t)2n             (x, t)2n-2      (x, t)2n+2
                                     4π2                   9H
 whose solution is:
                                    n
      2n               H2ln(a)   ln2 (a) 
   (x, t)  (2n -1)!!           F
                                   n          ,           2zF'(z) +nF  nF  n(2n +1)zF  0, F (0) =1
                                                              n       n    n-1          n+1    n
                         4π2   36π2 
                                                       n
                                              H2ln(a)     n(n +1) ln2 (a) n(35n3 +170n2 +225n + 74)  ln2 (a)  2 
                                                                                                                         
This yields
                         
                       (x, t)2n  (2n -1)!!             1                                                2 
                                                                                                                       ..
                                              4π              2     36π               280              36π 
                                                   2                       2
                                                           
                                                                                                                         
                                                                                                                          

NB: in (partial) agreement with perturbative result (UV scalars?)
                                                               ˚9˚
        Scalar stochastic theory:
       An intermediate conclusion
Stochastic theory can be used for a (nonperturbative)
evaluation of leading order (in ln(a)) correlators
                                    
                       2n
                             , T (x, t) , etc.



(A part of) perturbation series can be resummed. It is not clear
yet whether one is guaranteered to get the correct leading log
result. The main obstacle are the loops which contain UV scalar
propagators.
                                                                                              ˚10˚
          Classicality and decoherence
   Classicality
  Required: a (nearly) scale
  invariant spectrum & ε<<1
Recall that kmax=εHa  n=(Ha/2k)²,
        s.t. nmin=1/(4ε²) >> 1

   The infrared theory is characterised by a suppressed
   commutator (ε³ suppression):
           (x, t),  (x', t)   i 2 ( H)3 sin( Har) -( Har)cos( Har)
                     
            ˆ        ˆ
                               2                       ( Hr)3
If ε0, the theory is classical ; if ε∞, one gets canonical commutator.
                                                                          1
The corrections to Starobinsky theory:                              
                                                                    
                                                                         3H
                                                                            V'( ) -   0,
are suppressed either as 1/N~4ε² by ε³
or by slow roll parameters, and thus small
                                                                                     ˚11˚
        Classicality and decoherence (2)
   Decoherence:
UV modes: become IR when k/a = εH, and thus represent a
(stochastic) source current in the IR theory, which induces
fluctuations and dissipation of IR fields and thus decoherence
    Schematically: the reduced density matrix ρ
                ,     D [ , ][ , ], i t [ , ]  H [ , ]

  satisfies   t   i [H,  ]  " jbd  diss[  ]"
                                                       [work in progress with G. Rigopoulos]
 Physical picture: the constant physical
 UV momentum cutoff is imposed by
 the observer (recall the angular
 resolution scale of CMBR bolometers)
 Position space basis: is a natural
 POINTER BASIS in which classicallity
 and decoherence are manifest
                                                                         ˚12˚
    Interactions in the functional picture
 There are PASSIVE and ACTIVE fields (Woodard). Passive fields are
the fields which are not classical in the IR, and the UV of active fields.
  Active fields are the ones which develop classical IR fluctuations.
          Examples are the IR MMC scalars and IR gravitons.

 Prescription (example: scalar QED):
                                    [work in progress with Tsamis & Woodard]

      split active fields       ;
                            ˆ ˆ ˆ
      integrate out passive fields (Aμ)
      integrate out the UV part of active fields   ;

Problem: step 2 and 3 do not decouple!
Potential Problem: integrating UV scalars lead to nonlocal
 interaction terms. (Solution: these terms are ε² suppressed.)
                                          [work in progress with G. Rigopoulos]
                                                                                                     ˚13˚
     Interactions in the functional picture (2)
Since the scalar QED action is quadratic in (passive) gauge fields Aμ, they
 can be integrated out. (Gauge fields are passive since they couple conformally.)
                            1                                                                 
       SφQED   d 4 x  g   g μ g ν F F  g μν (Dμ φ)*D φ  m φ * φ  Rφ * φ  V (φ)  ,
                                          
                                                                     

                            4                                                                 
                                                        Gauge field mass insertions:
    Diagramatically:
                                                                     mA = e2
                                                                      2      *


 1A  iTrln Dμν (x, x';mA ( ))  ,
                            2                  (requires massive gauge propagator Dμν)
         i
  1  Tr ln
         2
           UV        -g( --m -V''( )) ,
                              2
                                           
    Caveat: this integration does not capture loops containing
           both gauge and UV scalar propagators, e.g.


 Two loop diagrams:                                          +
                                                                                    ˚14˚
                                   Scalar QED
Thus, it is impossible to integrate out all
passive fields in an interacting theory

 Hope: Integrating out gauge field may be enough
       to recover the leading log result!
Since ultimately interested in backreaction onto the metric
  tensor, we calculate the stress energy tensor in φQED:
             T (x)    QED
                                T (x) + TΑ (x)

             T (x)  D (x)  *D (x) - g D(x)  *D(x)g  V( ) 
                                                                               
                                    1
             TΑ (x)  FF g  gFF g g
                                            
                                    4

Diagramatically     T (x) 
(up to 2 loops):
                                                                                                                                  ˚15˚
                                                       Scalar QED (2)
Some results:

Stochastic field strength expectation value
                                                                                           D          D   D +1       D +1    
                                                                                            1   2                  
                                                                                H D
                                                                                           2          2  2           2       
              F    (x)F (x) 
                                       QED,stoch
                                                      g g - g g 
                                                                              (4 )D/2         D                1     1     
                                                                                                1                   
                                                                                               2                2     2     
                                                  D-3     e2 *
                                                               2

                                                     2        , D = # space time dimensions
                                                  2         H2

       NB: no agreement yet with perturbative 2 loop result

Stochastic scalar bilinear
                                                                    D               D   D +1       D +1              
                                           D -1       H    D          2  1   2  2    2      2    D -1  
                                                                                                                            
                                                                                                            
 D  (x)  *D  (x)
            
                         QED,stoch
                                      
                                            4
                                                g
                                                    (4 )D/2
                                                                   
                                                                            D                  1     1           D 
                                                                                                                              
                                                                            1                            1
                                                                   
                                                                           2                  2     2            2  


       NB: agreement with the perturbative 2 loop result!
                                                                 ˚16˚
                        Discussion
The main motivation for studying the stochastic limit of interacting
theories in inflation: (HOPE) allows for resummation of perturbative
series (generalisation of RG), which in turn allows to study the
backreaction of matter fields and gravitons onto the background
metric, and eventually understanding the dynamical relaxation of
the cosmological term (CC).
Second motivation: observer induced classicisation and
decoherence of comological perturbations (bolometer window
function) and the position basis as a natural pointer basis in
which the density matrix (first) decoheres.
LIST OF (CURRENT INCOMPLETE) RESULTS:
 stochasticisation of Yukawa theory and of scalar QED in
  de Sitter inflation; calculation of stress-energy tensors

DESIREABLE FUTURE WORK:
 resolving the role of UV active field loops as regards leading logs
 stochasticising gavity: cosmological perturbations and gravitons;
  understanding the question of backreaction and CC relaxation

								
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