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
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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