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The Consistency Equation in Single Field Infaltion Models

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					  The Consistency Equation in
 Single Field Inflationary Models
               astro-ph/0603016




           Andrew R. Liddle
             Marina Cortês
           Astronomy Centre
          University of Sussex


May 06
                                    Inflation
                                    25 years
                                         Starobinsky,1980
                                            Guth,1981
                                             Sato,1981
                                    Albrecht & Steinhardt, 1982
                                      Hawking & Moss, 1982
                                            Linde 1983




•period of accelerated expansion in the early universe
•proposed to explain problems of the standard Hot Big Bang Model:horizon, flatness, monopole
•Most interesting feature: means to explain appearance of structure
•Quantum Cosmology: we are observing the effects of quantum fluctuations in the microwave sky
& in the Dark-Matter Power Spectrum
•DIY: scalar field:
                    -dynamical expansion has to be switched off
                    -self interacting field ->potential energy with small slope: field rolling down the hill with
                     friction
      Perturbations:simplest form
•   Scale invariant equal on all scales: no divergences on small scales or large
    scales

•   Adiabatic: all particle species are perturbed in fixed ratio: perturbing the
    photons perturbs the baryons
•   Gaussian random noise

•   When light encompasses the wave the wave starts to oscillate
        Slow Roll approximation
• Kinetic energy has to be small and must remain
  small for some time
• Einstein Field Equation and Klein Gordon:



 2 8  1  2         2 8
 H  2    V ( )  H  3m 2 V ( )
       3mPl  2            Pl
  3H  V ( )
                     3H  V ( )
                           
                      
Fluctuations: lowest order
expressed in terms of Hubble parameter: makes computations easier
•Friedmann equation translates one into the other




                                        4 H2
                             AS (k ) 
                                       5mPl H 
                                         2
V H                                                  k  aH

                                           H
                                           2
                             AT (k ) 
                                       5  mPl        k  aH
Infinite Hierarchy of Slow Roll paramaters:
derivatives of the Potential must be small in an order-by-order expansion




                                     mPl  H  
                                      2            2

                                     
                                     4  H 
                                     mPl H 
                                      2
                                  
                                     4 H
                                     mPl  H H  
                                      2                12

                                     2 
                                     4  H 
                                                            13
                                     mPl  H 2 H ( 4) 
                                      2
                                       H2         
                                     4               
                             Next order
                               (Stewart & Lyth 1993)
                     (measuring the errors in slow roll predictions)


•                                          
    Compare the slow roll approximation 3H  V ( )

     with the exact expression           
                                    3H  V ( )


               
                
• The error is
               H    



•   Spectra will pick up errors of order            and    
•Corrections in      and      :




                                              2
 AS (k ) 
            4
                1  (2C  1)  C  H
              2
           5mPl                       H          k  aH


 AT (k )      1  (C  1) 
                  2            H
           5                  mPl   k  aH
                    Next order
                      (Stewart & Lyth 1993)
            (measuring the errors in slow roll predictions)



•   No assumptions beyond linear perturbation theory
    (drops slow roll) =>Next order in dynamics not perturbations




•   Ananda, Clarkson and Wands, ICG:
        - next order in perturbations
        - extra effect: mixing of scalars and tensors
         (to appear)
                   d ln AS2        d ln AT2
          nS  1             nT 
                    d ln k          d ln k


                       
nS  1  4  2   (8C  8) 2  (6  10C )  2C 2   
               
   nT  2   (6  4C ) 2  (4  4C )    
 dnS
        8 2  10  2 2 
d ln k
        (40  32C ) 3  (60  62C ) 2          
                                                  3
        (12  20C )  (8  14C )  2C  2C 
                         2               2   2
                                                   
 dnT
        4 2  4 
d ln k
        (28  16C ) 3  (40  28C ) 2 
                                          
        (8  8C )  (4  4C )        
                     2               2
                                          
   Consistency Equation
               Smoking gun of inflation

                                           2
               A
       nT  2                             T
                                           2
               A                           S

• Connects the two spectra in a way unique to inflation
• Correlates the relative amplitude of the two spectra with
  the slope of the tensor spectrum
•Single field

•Multifield:
      - isocurvature modes
testing the consistency equation
                  (prospects for detecting tensor modes)

•   CMB and space based experiments
•   Planck higher sensitivity to polarization




                                                Figures computed by A. Challinor: The Planck Consortia
                                                   Clover &
        Planck
                                                    QUIET
• Should be able to map out the           • Ground based
  electric polarization power
   spectrum ClE on all scales up          • Large arrays of polarization
   to and beyond the global                 sensitive detectors (necessary
   maximum of l~1000                        to detect the wavelength of
                                            primordial gravitational waves)


• Will at best only be able to            • distant future (NASA funding)
  detect tensor modes if r is
   greater than a few percent
   (already a considerable achievement)
                              structure
                                                  2
                                A
                         A  2
                             2
                             S
                                                  T
                                nT
•   Correlates the relative amplitude of the two spectra with the slope of the tensor
    spectrum

•   Tensors are one order higher

•   Differential equation for the tensors

•   Integral equation for the scalars

                     => tensors     fully specify the physical situation
                                                  2
                                                AT
                                         A  2
                                            2
                                            S
                                                nT
                                   Taylor expand both sides:

                                        d   2 nT 
                                                A2
                                                         d 2   2 nT 
                                                                   A2
        2                2   2
                                                   k 1  T 2 k  2
                                         T  ln                    ln
      dA     k 1 d A        k       A
AS 
 2      S
           ln          ln 2   2
                             S
                               2
                                                                   T
                                                                    2
                                                                              
     d ln k k0 2 d ln k     k0      nT   d ln k      k0 2 d ln k           k0

            Order of approximation is maintained in differentation and integration:


                Infinite Hierarchy of Consistency
                             Equations
                                                 d   2 AT 
                                                           2
                                                            (i )
                                    d (i ) AS
                                            2
                                                      nT 
                                                                                   i  0,1,...
                                               
                                   d ln k (i )    d ln k (i )

                                    In terms of spectral indices
                                   d (i 1) (nS  1)    d (i 1) nT   d (i ) ln nT
                                                                                    i  1,2,...
                                     d ln k (i 1)     d ln k (i 1) d ln k (i )
                      Consistency Equation Hierarchy
                                    Next Order in Slow Roll
   – First Consistency Equation                     – Second Consistency Equation
                      Next Order                                      Next Order


                   AT  AT
                    2      2
                                       
            nT  2 2 1  2  (nS  1)
                   AS  AS             
                                                                                   n                     dnS 
                                                          nT nT  (nS  1)  nT  T (nT  (nS  1)) 
                                                   dnT
                                                  d ln k                           2                    d ln k 
                                                                                                                
    Separate tensors and scalars

                                     1 
                                2
     AS 1  (nS  1)  2
                              AT
                                    1  2 nT 
      2

                              nT             




Tensors no longer specify the situation!
First Consistency Equation Lowest Order
                      2
                     AT
              nT  2 2
                     AS

First Consistency Equation Next Order


                     2
                    AT    AT 2
                                           
          nT  2    2   1  2  (1  nS )
                    AS    AS              



Second Consistency Equation Lowest Order


                      nT nT  (nS  1)
               dnT
              d ln k


 Second Consistency Equation Next Order


                                   n                      dnS 
          nT nT  (nS  1)  nT  T nT  (nS  1)  
   dnT
  d ln k                           2                     d ln k 
                                                                 
    The approximate Consistency Equation
                     (models with running)
           Chung, Shiu & Trodden astro-ph/0305193
             Chung & Romano astro-ph/0508411


• Propose similarity of two scales
• Suppose n-1 changes sign at some scale
• This involves a large or at least non-
  negligible running
• Running cannot be sustained for long
  (must achieve 60 e-folds)=> flat bump in
  the potential
    Two Scales:

              nS changes sign: spectrum changes from blue to red




                  k 1: nS  1  0
              Flat bump in the potential



                             V 
                                              2

                  k2 :   ~    0  nT  nS  1
                             V 
•But: Consistency equation exhausts connections
       between spectra!
              (nS  1)  nT         nS  1
N                                            No inflationary input
          dnT d ln k  dnS d ln k dnS d ln k


denominator is one order higher in slow roll

                      N        will be large


                           1
                     N ~ O 
                            
    Enforcing the relation between scalars and tensors turns the expression into
                                                 
                                 N                      at scale k1
                                            4 2
                                             2




                        In terms of observables
                                             nT
                              N  
                                        n  dnS d ln k
                                         2
                                         T




        Comparing with the non-inflationary expression at scale k1
                                         (nS  1)  nT
                            N  
                                     dnT d ln k  dnS d ln k


                  We see that their proposal is to confirm that

                                 dnT
                                        nT
                                          2

                                d ln k
Which is the second consistency equation at scale k1!

    So the proposed new consistency equation introduces no novelty
              on the relation between scalars and tensors
                           Constant Running
                               Lidsey & Tavakol


                                                        1
 A2
          (nS  1)      2
                             2               nS  1  2  ~
  S
   exp                           erf               c
 A2
  T     2 dnS d ln k   dnS d ln k 
                                          2 dnS d ln k 
                                                         

  Relation advertised as independent of the potential
  Could be used to test for inflation




However this is not a consistency equation:
-does not include the tensors
-as it is it seeks out to measure constant c which
  is the same as measuring the running
-typical inflation models predict some deviation from constant running
=> as such it is not testing for a signature of inflation
                         Conclusions
•   We presented expressions for the spectral indices and their running to
    lowest and next order in Slow-Roll



•   We showed that the consistency equation can be differentiated to present a
    complete description between the spectra in inflationary models: Infinite
    Hierarchy of Consistency Equations



•   As this exhausts the connection between the spectra in inflationary models
    any other relations between these must derive from one of the consistency
    equations or from combinations of more of them
                     prospects
• Study the relation between scalars and
  tensors to next order
• Consistency equation for Multi-field
 (the standard situation is an inequality)

• Next Order Corrections in perturbation
  theory (relevant?)

				
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posted:4/9/2010
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