Monte Carlo Markov Chain Analysis ofCMB + LSS experimental data

within theEffective Field Theory of Inflation

Claudio Destriclaudio.destri@mib.infn.it

Dipartimento di Fisica – Universita Milano–Bicocca

C.D., H.J. de Vega, N. Sanchez, astro-ph/0703417

Paris 17/05/2007 – p. 1/21

Outline of the talk

• Monte Carlo Markov Chains and cosmological parameters

• The COSMOMC program

• Current CMB and LSS data and ΛCDM model

• Effective Field Theory of Inflation

• Trinomial new inflation

• Trinomial chaotic inflation

• Summary and conclusions

Paris 17/05/2007 – p. 2/21

Outline of the talk

• Monte Carlo Markov Chains and cosmological parameters

• The COSMOMC program

• Current CMB and LSS data and ΛCDM model

• Effective Field Theory of Inflation

• Trinomial new inflation

• Trinomial chaotic inflation

• Summary and conclusions

Paris 17/05/2007 – p. 2/21

Outline of the talk

• Monte Carlo Markov Chains and cosmological parameters

• The COSMOMC program

• Current CMB and LSS data and ΛCDM model

• Effective Field Theory of Inflation

• Trinomial new inflation

• Trinomial chaotic inflation

• Summary and conclusions

Paris 17/05/2007 – p. 2/21

Outline of the talk

• Monte Carlo Markov Chains and cosmological parameters

• The COSMOMC program

• Current CMB and LSS data and ΛCDM model

• Effective Field Theory of Inflation

• Trinomial new inflation

• Trinomial chaotic inflation

• Summary and conclusions

Paris 17/05/2007 – p. 2/21

Outline of the talk

• Monte Carlo Markov Chains and cosmological parameters

• The COSMOMC program

• Current CMB and LSS data and ΛCDM model

• Effective Field Theory of Inflation

• Trinomial new inflation

• Trinomial chaotic inflation

• Summary and conclusions

Paris 17/05/2007 – p. 2/21

Outline of the talk

• Monte Carlo Markov Chains and cosmological parameters

• The COSMOMC program

• Current CMB and LSS data and ΛCDM model

• Effective Field Theory of Inflation

• Trinomial new inflation

• Trinomial chaotic inflation

• Summary and conclusions

Paris 17/05/2007 – p. 2/21

Outline of the talk

• Monte Carlo Markov Chains and cosmological parameters

• The COSMOMC program

• Current CMB and LSS data and ΛCDM model

• Effective Field Theory of Inflation

• Trinomial new inflation

• Trinomial chaotic inflation

• Summary and conclusions

Paris 17/05/2007 – p. 2/21

MCMC and cosmological parametersExperimental data:CMB, LSS, SN, Lensing, Ly−α , . . .

←→ Cosmological model:Ωb, Ωc, H0, τ , As, ns, r, . . .

Paris 17/05/2007 – p. 3/21

MCMC and cosmological parametersExperimental data:CMB, LSS, SN, Lensing, Ly−α , . . .

←→ Cosmological model:Ωb, Ωc, H0, τ , As, ns, r, . . .

From probability distribution of data to likelihood over model parameters:

Pr(

Xdata

)

−→ L(

θ1, θ2, . . . , θn|Xdata

)

Gaussian case:

L = e−χ2/2 , χ2 =∑

data

[

X(θ)model −Xdata

]2, θ = θ1, θ2, . . . , θn

Paris 17/05/2007 – p. 3/21

MCMC and cosmological parametersExperimental data:CMB, LSS, SN, Lensing, Ly−α , . . .

←→ Cosmological model:Ωb, Ωc, H0, τ , As, ns, r, . . .

From probability distribution of data to likelihood over model parameters:

Pr(

Xdata

)

−→ L(

θ1, θ2, . . . , θn|Xdata

)

Gaussian case:

L = e−χ2/2 , χ2 =∑

data

[

X(θ)model −Xdata

]2, θ = θ1, θ2, . . . , θn

MCMC are sets of parameters distributed according to L, produced with anacceptance/rejection one-step algorithm (e.g. Metropolis)

W (θn+1, θn) = g(θn+1, θn) min

1,L(θn+1) g(θn+1, θn)

L(θn) g(θn, θn+1)

Paris 17/05/2007 – p. 3/21

MCMC and cosmological parametersExperimental data:CMB, LSS, SN, Lensing, Ly−α , . . .

←→ Cosmological model:Ωb, Ωc, H0, τ , As, ns, r, . . .

From probability distribution of data to likelihood over model parameters:

Pr(

Xdata

)

−→ L(

θ1, θ2, . . . , θn|Xdata

)

Gaussian case:

L = e−χ2/2 , χ2 =∑

data

[

X(θ)model −Xdata

]2, θ = θ1, θ2, . . . , θn

MCMC are sets of parameters distributed according to L, produced with anacceptance/rejection one-step algorithm (e.g. Metropolis)

W (θn+1, θn) = g(θn+1, θn) min

1,L(θn+1) g(θn+1, θn)

L(θn) g(θn, θn+1)

L includes COSMIC VARIANCE (in primordial fluctuations, ` . 300)

Paris 17/05/2007 – p. 3/21

WMAP 3-years (http://lambda.gsfc.nasa.gov)

Paris 17/05/2007 – p. 4/21

WMAP 3-years (http://lambda.gsfc.nasa.gov)

Spergel et al. 2006

param 100Ωbh2 Ωmh2 H0 τ ns σ8

best fit 2.22 0.127 73.2 0.091 0.954 0.756

∆ 0.073 0.008 3.2 0.03 0.016 0.049

Paris 17/05/2007 – p. 4/21

The COSMOMC program• http://cosmologist.info/cosmomc/

• Developped and mantained by Antony Lewis (Cambridge)

• Publicly available F90 open source code

• Comes with CMB and LSS data (except WMAP)

• Interfaces directly with the WMAP likelihood software

• Theoretical calculations with CAMB (derived from CMBFAST)

• Quite accurate, easy to use, fairly well documented

• Comes with tool for analyzing results and useful MATLAB scripts

• Not very fast, but runs well on clusters

Paris 17/05/2007 – p. 5/21

The COSMOMC program• http://cosmologist.info/cosmomc/

• Developped and mantained by Antony Lewis (Cambridge)

• Publicly available F90 open source code

• Comes with CMB and LSS data (except WMAP)

• Interfaces directly with the WMAP likelihood software

• Theoretical calculations with CAMB (derived from CMBFAST)

• Quite accurate, easy to use, fairly well documented

• Comes with tool for analyzing results and useful MATLAB scripts

• Not very fast, but runs well on clusters

Paris 17/05/2007 – p. 5/21

The COSMOMC program• http://cosmologist.info/cosmomc/

• Developped and mantained by Antony Lewis (Cambridge)

• Publicly available F90 open source code

• Comes with CMB and LSS data (except WMAP)

• Interfaces directly with the WMAP likelihood software

• Theoretical calculations with CAMB (derived from CMBFAST)

• Quite accurate, easy to use, fairly well documented

• Comes with tool for analyzing results and useful MATLAB scripts

• Not very fast, but runs well on clusters

Paris 17/05/2007 – p. 5/21

The COSMOMC program• http://cosmologist.info/cosmomc/

• Developped and mantained by Antony Lewis (Cambridge)

• Publicly available F90 open source code

• Comes with CMB and LSS data (except WMAP)

• Interfaces directly with the WMAP likelihood software

• Theoretical calculations with CAMB (derived from CMBFAST)

• Quite accurate, easy to use, fairly well documented

• Comes with tool for analyzing results and useful MATLAB scripts

• Not very fast, but runs well on clusters

Paris 17/05/2007 – p. 5/21

The COSMOMC program• http://cosmologist.info/cosmomc/

• Developped and mantained by Antony Lewis (Cambridge)

• Publicly available F90 open source code

• Comes with CMB and LSS data (except WMAP)

• Interfaces directly with the WMAP likelihood software

• Theoretical calculations with CAMB (derived from CMBFAST)

• Quite accurate, easy to use, fairly well documented

• Comes with tool for analyzing results and useful MATLAB scripts

• Not very fast, but runs well on clusters

Paris 17/05/2007 – p. 5/21

The COSMOMC program• http://cosmologist.info/cosmomc/

• Developped and mantained by Antony Lewis (Cambridge)

• Publicly available F90 open source code

• Comes with CMB and LSS data (except WMAP)

• Interfaces directly with the WMAP likelihood software

• Theoretical calculations with CAMB (derived from CMBFAST)

• Quite accurate, easy to use, fairly well documented

• Comes with tool for analyzing results and useful MATLAB scripts

• Not very fast, but runs well on clusters

Paris 17/05/2007 – p. 5/21

The COSMOMC program• http://cosmologist.info/cosmomc/

• Developped and mantained by Antony Lewis (Cambridge)

• Publicly available F90 open source code

• Comes with CMB and LSS data (except WMAP)

• Interfaces directly with the WMAP likelihood software

• Theoretical calculations with CAMB (derived from CMBFAST)

• Quite accurate, easy to use, fairly well documented

• Comes with tool for analyzing results and useful MATLAB scripts

• Not very fast, but runs well on clusters

Paris 17/05/2007 – p. 5/21

The COSMOMC program• http://cosmologist.info/cosmomc/

• Developped and mantained by Antony Lewis (Cambridge)

• Publicly available F90 open source code

• Comes with CMB and LSS data (except WMAP)

• Interfaces directly with the WMAP likelihood software

• Theoretical calculations with CAMB (derived from CMBFAST)

• Quite accurate, easy to use, fairly well documented

• Comes with tool for analyzing results and useful MATLAB scripts

• Not very fast, but runs well on clusters

Paris 17/05/2007 – p. 5/21

The COSMOMC program• http://cosmologist.info/cosmomc/

• Developped and mantained by Antony Lewis (Cambridge)

• Publicly available F90 open source code

• Comes with CMB and LSS data (except WMAP)

• Interfaces directly with the WMAP likelihood software

• Theoretical calculations with CAMB (derived from CMBFAST)

• Quite accurate, easy to use, fairly well documented

• Comes with tool for analyzing results and useful MATLAB scripts

• Not very fast, but runs well on clusters

Paris 17/05/2007 – p. 5/21

CMB, LSS and the FlatΛCDM+r modelDatasets: WMAP3, ACBAR, CBI2, BOOMERANG03, SDSSMCMC parameters: ωb, ωc, τ , Θ (slow), As, ns, r (fast)Context: Ων = 0, . . . ; standard priors, no SZ, no lensing, linear mpk, . . .

Paris 17/05/2007 – p. 6/21

CMB, LSS and the FlatΛCDM+r modelDatasets: WMAP3, ACBAR, CBI2, BOOMERANG03, SDSSMCMC parameters: ωb, ωc, τ , Θ (slow), As, ns, r (fast)Context: Ων = 0, . . . ; standard priors, no SZ, no lensing, linear mpk, . . .

0.021 0.023 0.025Ωb h2 0.09 0.1 0.11 0.12

Ωc h2 1.03 1.035 1.04 1.045 1.05

θ

0.05 0.1 0.15τ

0.9 0.95 1 1.05ns

2.9 3 3.1 3.2log[1010 As]

0 0.1 0.2 0.3r

0.7 0.75 0.8ΩΛ

13.2 13.4 13.6 13.8 14Age/GYr

0.7 0.75 0.8 0.85σ8

5 10 15zre

70 75 80H0

param best fit

100Ωbh2 2.232

Ωmh2 0.128

τ 0.956

H0 73.03

σ8 0.771

log[1010As] 0.303

ns 0.956

r 0.016

− log(L) 2714.038Paris 17/05/2007 – p. 6/21

The marginalized ns− r probabilityCL: 12%, 27%, 45%, 68% and 95%

ns

r

0.9 0.92 0.94 0.96 0.98 1 1.02 1.040

0.05

0.1

0.15

0.2

0.25

0.3

Paris 17/05/2007 – p. 7/21

EFT of inflation (à la Boyanowski–De Vega–Sanchez)Single scalar field φ + quantum fluctuations

H2 =ρ

3 M2Pl

=8πG

3

[

1

2φ2 + V (φ)

]

(Friedmann)

φ + 3 Hφ + V ′(φ) = 0 (condensate dynamics)

units: = 1, c = 1, MPl = 2.4× 1018 GeV

Paris 17/05/2007 – p. 8/21

EFT of inflation (à la Boyanowski–De Vega–Sanchez)Single scalar field φ + quantum fluctuations

H2 =ρ

3 M2Pl

=8πG

3

[

1

2φ2 + V (φ)

]

(Friedmann)

φ + 3 Hφ + V ′(φ) = 0 (condensate dynamics)

units: = 1, c = 1, MPl = 2.4× 1018 GeVSlow roll: M = energy scale of inflation ∼MGUT ∼ 1016 GeV is fixed bythe amplitude (measured in CMB!) of adiabatic scalar perturbations at somecosmologically-relevant reference scale kpivot

V (φ) = N M4 w(χ) , χ =φ√

N MPl

, w(χ) ∼ O(1) , N & 50

whereN = − 1

M2Pl

∫ φend

φexit

dφV (φ)

V ′(φ)= −N

∫ χend

χexit

dχw(χ)

w′(χ)

is the number of efolds from kpivot horizon exit to the end of inflation

Paris 17/05/2007 – p. 8/21

EFT of inflation (à la Boyanowski–De Vega–Sanchez)Single scalar field φ + quantum fluctuations

H2 =ρ

3 M2Pl

=8πG

3

[

1

2φ2 + V (φ)

]

(Friedmann)

φ + 3 Hφ + V ′(φ) = 0 (condensate dynamics)

units: = 1, c = 1, MPl = 2.4× 1018 GeVSlow roll: M = energy scale of inflation ∼MGUT ∼ 1016 GeV is fixed bythe amplitude (measured in CMB!) of adiabatic scalar perturbations at somecosmologically-relevant reference scale kpivot

V (φ) = N M4 w(χ) , χ =φ√

N MPl

, w(χ) ∼ O(1) , N & 50

whereN = − 1

M2Pl

∫ φend

φexit

dφV (φ)

V ′(φ)= −N

∫ χend

χexit

dχw(χ)

w′(χ)

is the number of efolds from kpivot horizon exit to the end of inflation

Wide separation of scales←→ EFT + slow-roll

Paris 17/05/2007 – p. 8/21

EFT of inflation (à la Boyanowski–De Vega–Sanchez)Single scalar field φ + quantum fluctuations

H2 =ρ

3 M2Pl

=8πG

3

[

1

2φ2 + V (φ)

]

(Friedmann)

φ + 3 Hφ + V ′(φ) = 0 (condensate dynamics)

units: = 1, c = 1, MPl = 2.4× 1018 GeVSlow roll: M = energy scale of inflation ∼MGUT ∼ 1016 GeV is fixed bythe amplitude (measured in CMB!) of adiabatic scalar perturbations at somecosmologically-relevant reference scale kpivot

V (φ) = N M4 w(χ) , χ =φ√

N MPl

, w(χ) ∼ O(1) , N & 50

whereN = − 1

M2Pl

∫ φend

φexit

dφV (φ)

V ′(φ)= −N

∫ χend

χexit

dχw(χ)

w′(χ)

is the number of efolds from kpivot horizon exit to the end of inflation

Understanding start and end of inflation −→ N

Paris 17/05/2007 – p. 8/21

H M MPl −→quantum loops = double expansion in H2

M2

Pl

and 1N

slow roll = expansion in 1N

graviton corrections suppressed by H2

M2

Pl

Paris 17/05/2007 – p. 9/21

H M MPl −→quantum loops = double expansion in H2

M2

Pl

and 1N

slow roll = expansion in 1N

graviton corrections suppressed by H2

M2

Pl

See–saw–like inflaton mass and slow-roll Hubble parameter

m =M2

MPl∼ 2.45× 1013 GeV , H =

√N mH ∼ 1014 GeV

Paris 17/05/2007 – p. 9/21

H M MPl −→quantum loops = double expansion in H2

M2

Pl

and 1N

slow roll = expansion in 1N

graviton corrections suppressed by H2

M2

Pl

See–saw–like inflaton mass and slow-roll Hubble parameter

m =M2

MPl∼ 2.45× 1013 GeV , H =

√N mH ∼ 1014 GeV

H2 =1

3

[

1

2 N

(

dχ

dτ

)2

+ w(χ)

]

, τ =t M2

MPl

√N

Paris 17/05/2007 – p. 9/21

H M MPl −→quantum loops = double expansion in H2

M2

Pl

and 1N

slow roll = expansion in 1N

graviton corrections suppressed by H2

M2

Pl

See–saw–like inflaton mass and slow-roll Hubble parameter

m =M2

MPl∼ 2.45× 1013 GeV , H =

√N mH ∼ 1014 GeV

H2 =1

3

[

1

2 N

(

dχ

dτ

)2

+ w(χ)

]

, τ =t M2

MPl

√N

To first order, with χ ≡ χexit for brevity:

ns = 1− 1

N

3

[

w′(χ)

w(χ)

]2

− 2w′′(χ)

w(χ)

,dns

d ln k= O

(

1

N2

)

r =8

N

[

w′(χ)

w(χ)

]2

, |∆(S)ad |2 =

N2

12π2

(

M

MPl

)4w3(χ)

w′2(χ)

Paris 17/05/2007 – p. 9/21

H M MPl −→quantum loops = double expansion in H2

M2

Pl

and 1N

slow roll = expansion in 1N

graviton corrections suppressed by H2

M2

Pl

See–saw–like inflaton mass and slow-roll Hubble parameter

m =M2

MPl∼ 2.45× 1013 GeV , H =

√N mH ∼ 1014 GeV

H2 =1

3

[

1

2 N

(

dχ

dτ

)2

+ w(χ)

]

, τ =t M2

MPl

√N

To first order, with χ ≡ χexit for brevity:

ns = 1− 1

N

3

[

w′(χ)

w(χ)

]2

− 2w′′(χ)

w(χ)

,dns

d ln k= O

(

1

N2

)

r =8

N

[

w′(χ)

w(χ)

]2

, |∆(S)ad |2 =

N2

12π2

(

M

MPl

)4w3(χ)

w′2(χ)

Trinomial realization: V (φ) = V0 ± 12m2φ2 − 1

3mgφ3 + 14λφ4

Paris 17/05/2007 – p. 9/21

Trinomial new (= small field) inflation

w(χ) = −1

2χ2 − 1

3h

√

y

2χ3 +

1

32yχ4 +

2

yF (h)

Paris 17/05/2007 – p. 10/21

Trinomial new (= small field) inflation

w(χ) = −1

2χ2 − 1

3h

√

y

2χ3 +

1

32yχ4 +

2

yF (h)

g = −h

√

y

2 N

(

M

MPl

)2

, λ =y

8 N

(

M

MPl

)4

,2

yF (h) =

V0

NM4

Paris 17/05/2007 – p. 10/21

Trinomial new (= small field) inflation

w(χ) = −1

2χ2 − 1

3h

√

y

2χ3 +

1

32yχ4 +

2

yF (h)

g = −h

√

y

2 N

(

M

MPl

)2

, λ =y

8 N

(

M

MPl

)4

,2

yF (h) =

V0

NM4

∆ =√

h2 + 1 , h ≥ 0 −→ absolute minimum = χend =

√

8

y(∆+h) > 0

F (h) =8

3h4 + 4h2 + 1 +

8

3h∆3 −→ w(χend) = w′(χend) = 0

Paris 17/05/2007 – p. 10/21

Trinomial new (= small field) inflation

w(χ) = −1

2χ2 − 1

3h

√

y

2χ3 +

1

32yχ4 +

2

yF (h)

g = −h

√

y

2 N

(

M

MPl

)2

, λ =y

8 N

(

M

MPl

)4

,2

yF (h) =

V0

NM4

∆ =√

h2 + 1 , h ≥ 0 −→ absolute minimum = χend =

√

8

y(∆+h) > 0

F (h) =8

3h4 + 4h2 + 1 +

8

3h∆3 −→ w(χend) = w′(χend) = 0

fixing N −→

y = z − 2h2 − 1− 2h∆ +4

3h(h + ∆−

√z)

+16

3h(∆ + h)∆2 log

[

1

2

(

1 +

√z − h

∆

)]

− 2F (h) log[√

z(∆− h)]

where z ≡ y8χ2

exit and z1 = 1− z(∆+h)2 acts as normalized effective coupling.

Paris 17/05/2007 – p. 10/21

dz

dy< 0 , 0 ≤ z ≤ (∆ + h)2 , 0 ≤ y <∞

Paris 17/05/2007 – p. 11/21

dz

dy< 0 , 0 ≤ z ≤ (∆ + h)2 , 0 ≤ y <∞

dz1

dy> 0 , 0 ≤ z1 ≤ 1 , 0 ≤ y <∞

Paris 17/05/2007 – p. 11/21

dz

dy< 0 , 0 ≤ z ≤ (∆ + h)2 , 0 ≤ y <∞

dz1

dy> 0 , 0 ≤ z1 ≤ 1 , 0 ≤ y <∞

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

100

z1

y1/2

h = 0

h = 20

h = 100

Paris 17/05/2007 – p. 11/21

ns = 1 −6y

N

z(z + 2h√

z − 1)2ˆ

F (h) − 2z + 8

3hz3/2 + z2

˜

2+

y

N

3z + 4h√

z − 1

F (h) − 2z + 8

3hz3/2 + z2

r =16y

N

z(z + 2h√

z − 1)2ˆ

F (h) − 2z + 8

3hz3/2 + z2

˜

2

Paris 17/05/2007 – p. 12/21

ns = 1 −6y

N

z(z + 2h√

z − 1)2ˆ

F (h) − 2z + 8

3hz3/2 + z2

˜

2+

y

N

3z + 4h√

z − 1

F (h) − 2z + 8

3hz3/2 + z2

r =16y

N

z(z + 2h√

z − 1)2ˆ

F (h) − 2z + 8

3hz3/2 + z2

˜

2

0.88 0.89 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.970

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

ns

r

z1 −−> 1

z1 −−> 0

h = 0

h = 20

N = 50

h = 0h = 0.5h = 1h = 5h = 20

Paris 17/05/2007 – p. 12/21

ns = 1 −6y

N

z(z + 2h√

z − 1)2ˆ

F (h) − 2z + 8

3hz3/2 + z2

˜

2+

y

N

3z + 4h√

z − 1

F (h) − 2z + 8

3hz3/2 + z2

r =16y

N

z(z + 2h√

z − 1)2ˆ

F (h) − 2z + 8

3hz3/2 + z2

˜

2

ns

r

N = 50

0.9 0.92 0.94 0.96 0.98 1 1.02 1.040

0.05

0.1

0.15

0.2

0.25

0.3

Paris 17/05/2007 – p. 12/21

CMB, LSS and trinomial new inflationDatasets: WMAP3, ACBAR, CBI2, BOOMERANG03, SDSSMCMC parameters: ωb, ωc, τ , Θ (slow), As, z1, h (fast)Context: Ων = 0, . . . ; standard priors, no SZ, no lensing, linear mpk, . . .

Paris 17/05/2007 – p. 13/21

CMB, LSS and trinomial new inflationDatasets: WMAP3, ACBAR, CBI2, BOOMERANG03, SDSSMCMC parameters: ωb, ωc, τ , Θ (slow), As, z1, h (fast)Context: Ων = 0, . . . ; standard priors, no SZ, no lensing, linear mpk, . . .

0.021 0.022 0.023Ωb h2 0.095 0.1 0.105 0.11 0.115

Ωc h2 1.035 1.04 1.045

θ

0.05 0.1 0.15τ

0.2 0.4 0.6 0.8

(1−z1)1/2

2.9 3 3.1log[1010 As]

0.2 0.4 0.6 0.8

h/(1+h)

0.7 0.75 0.8ΩΛ

13.4 13.6 13.8 14Age/GYr

0.7 0.75 0.8 0.85σ8

5 10 15zre

68 70 72 74 76 78H0

param best fit

100Ωbh2 2.22

Ωmh2 0.128

τ 0.847

H0 72.9

σ8 0.767

log[1010As] 0.302

z1 0.885

h 0.266

− log(L) 2714.153Paris 17/05/2007 – p. 13/21

FlatΛCDM+r vs. trinomial new inflation

param best fit 1 best fit 2

100Ωbh2 2.232 2.22

Ωmh2 0.128 0.128

τ 0.956 0.847

H0 73.03 72.9

σ8 0.771 0.767

log[1010As] 0.303 0.302

ns 0.956 0.9557

r 0.016 0.054

− log(L) 2714.038 2714.153

Paris 17/05/2007 – p. 14/21

The lower bound on r

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z1

0.94 0.945 0.95 0.955 0.96 0.9650

0.2

0.4

0.6

0.8

1

ns

0 0.05 0.1 0.150

0.2

0.4

0.6

0.8

1

r

0 5 10 15 200

0.2

0.4

0.6

0.8

1

y

Paris 17/05/2007 – p. 15/21

The lower bound on r

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z1

0.94 0.945 0.95 0.955 0.96 0.9650

0.2

0.4

0.6

0.8

1

ns

0 0.05 0.1 0.150

0.2

0.4

0.6

0.8

1

r

0 5 10 15 200

0.2

0.4

0.6

0.8

1

y

CL: r > 0.016 (95%) , r > 0.049 (68%)ML: r = 0.054 , ML: y = 1.2 , MV: r = 0.077 , MV: y = 2.03

Paris 17/05/2007 – p. 15/21

Trinomial chaotic (= large field) inflation

w(χ) =1

2χ2 +

1

3h

√

y

2χ3 +

1

32yχ4 , −1 < h ≤ 0

Paris 17/05/2007 – p. 16/21

Trinomial chaotic (= large field) inflation

w(χ) =1

2χ2 +

1

3h

√

y

2χ3 +

1

32yχ4 , −1 < h ≤ 0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(y/8)1/2 χ

(y/8

) w(χ

)

h = 0

h = −0.5

h = −0.8

h = −1

Paris 17/05/2007 – p. 16/21

Trinomial chaotic (= large field) inflation

w(χ) =1

2χ2 +

1

3h

√

y

2χ3 +

1

32yχ4 , −1 < h ≤ 0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(y/8)1/2 χ

(y/8

) w(χ

)

h = 0

h = −0.5

h = −0.8

h = −1

fixing N −→y = z +

4

3h√

z +

„

1 −4

3h2

«

log`

1 + 2h√

z + z´

−4h

3∆

„

5

2− 2h2

« »

arctan

„

h +√

z

∆

«

− arctan

„

h

∆

«–

where z ≡ y8χ2

exit acts as effective coupling, 0 ≤ y <∞, 0 ≤ z <∞

Paris 17/05/2007 – p. 16/21

ns = 1− y

2Nz

[

3(1 + 2h

√z + z)

2

(

1 + 43h√

z + 12z)2 −

1 + 4h√

z + 3z

1 + 43h√

z + 12 z

]

r =4 y

Nz

(1 + 2h√

z + z)2

(

1 + 43h√

z + 12z)2

Paris 17/05/2007 – p. 17/21

ns = 1− y

2Nz

[

3(1 + 2h

√z + z)

2

(

1 + 43h√

z + 12z)2 −

1 + 4h√

z + 3z

1 + 43h√

z + 12 z

]

r =4 y

Nz

(1 + 2h√

z + z)2

(

1 + 43h√

z + 12z)2

ns

r

h = −0.999h = −0.99

h = −0.95

h = −0.9

h = −0.85h = −0.8

h = −0.5h = 0

h = 0.99

h = 0 h = 20

N = 50φ2

φ4

0.9 0.92 0.94 0.96 0.98 1 1.02 1.040

0.05

0.1

0.15

0.2

0.25

0.3

Paris 17/05/2007 – p. 17/21

Some limiting cases

Paris 17/05/2007 – p. 18/21

Some limiting casesThe flat limit h→ −1+ and then z → 1− :

ns → 1− 4

N, r → 0 , y ∼ 1

1−√z→∞ , |∆(S)

ad |2 ∼(

N M2

MPl

2

y

)2

M√

y is the true scale of inflation and the theory becomes massless, m→ 0.

Paris 17/05/2007 – p. 18/21

Some limiting casesThe flat limit h→ −1+ and then z → 1− :

ns → 1− 4

N, r → 0 , y ∼ 1

1−√z→∞ , |∆(S)

ad |2 ∼(

N M2

MPl

2

y

)2

M√

y is the true scale of inflation and the theory becomes massless, m→ 0.

The flat limit h→ −1+ when z > 1 is meaningless since ns − 1 and r

diverge.

Paris 17/05/2007 – p. 18/21

Some limiting casesThe flat limit h→ −1+ and then z → 1− :

ns → 1− 4

N, r → 0 , y ∼ 1

1−√z→∞ , |∆(S)

ad |2 ∼(

N M2

MPl

2

y

)2

M√

y is the true scale of inflation and the theory becomes massless, m→ 0.

The flat limit h→ −1+ when z > 1 is meaningless since ns − 1 and r

diverge.Harrison–Zeldovich limit z = 1, h→ −1

ns → 1 , r → 0 y ∼ (h+1)−1/2 →∞ , |∆(S)ad |2 ∼

(

M(h + 1)−1/4

MPl

)4

Paris 17/05/2007 – p. 18/21

Some limiting casesThe flat limit h→ −1+ and then z → 1− :

ns → 1− 4

N, r → 0 , y ∼ 1

1−√z→∞ , |∆(S)

ad |2 ∼(

N M2

MPl

2

y

)2

M√

y is the true scale of inflation and the theory becomes massless, m→ 0.

The flat limit h→ −1+ when z > 1 is meaningless since ns − 1 and r

diverge.Harrison–Zeldovich limit z = 1, h→ −1

ns → 1 , r → 0 y ∼ (h+1)−1/2 →∞ , |∆(S)ad |2 ∼

(

M(h + 1)−1/4

MPl

)4

But, even if M ≡M(h + 1)−1/4 is kept fixed:

m ∼ M2

MPl(h + 1)1/2 , g ∼ M

2

M2Pl

(h + 1)1/4 , λ ∼ M4

M4Pl

(h + 1)1/2 ,

all vanish yielding a scale–invariant massless free–field.

Paris 17/05/2007 – p. 18/21

The marginalized z − h probabilityCL: 12%, 27%, 45%, 68% and 95%

z

h

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

−0.95

−0.9

−0.85

−0.8

−0.75

−0.7

Paris 17/05/2007 – p. 19/21

The marginalized z − h probabilityCL: 12%, 27%, 45%, 68% and 95%

z

h

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

−0.95

−0.9

−0.85

−0.8

−0.75

−0.7

Paris 17/05/2007 – p. 19/21

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

z0 5 10 15 20

0

0.2

0.4

0.6

0.8

1

y

0.9 0.95 1 1.05 1.10

0.2

0.4

0.6

0.8

1

ns0 0.1 0.2 0.3 0.4

0

0.2

0.4

0.6

0.8

1

r

ML: y = 4.2 , MV: y = 3.1

Paris 17/05/2007 – p. 20/21

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

z0 5 10 15 20

0

0.2

0.4

0.6

0.8

1

y

0.9 0.95 1 1.05 1.10

0.2

0.4

0.6

0.8

1

ns0 0.1 0.2 0.3 0.4

0

0.2

0.4

0.6

0.8

1

r

ML: y = 4.2 , MV: y = 3.1

Paris 17/05/2007 – p. 20/21

Summary and conclusions• Effective FT of inflation is not just phenomenology

• N plays a distingushed role

• Trinomial potentials are adequate

• Trinomial new inflation is- generically adequate (also when fully symmetric)- moderately coupled (Ginzburg–Landau safe)- provides a lower bound for r

• Trinomial chaotic inflation is- confined to a corner of parameter space- more strongly coupled- in tension with the Ginzburg–Landau picture

• Other potentials should be similarly studied (not reconstructed )

Paris 17/05/2007 – p. 21/21

Summary and conclusions• Effective FT of inflation is not just phenomenology

• N plays a distingushed role

• Trinomial potentials are adequate

• Trinomial new inflation is- generically adequate (also when fully symmetric)- moderately coupled (Ginzburg–Landau safe)- provides a lower bound for r

• Trinomial chaotic inflation is- confined to a corner of parameter space- more strongly coupled- in tension with the Ginzburg–Landau picture

• Other potentials should be similarly studied (not reconstructed )

Paris 17/05/2007 – p. 21/21

Summary and conclusions• Effective FT of inflation is not just phenomenology

• N plays a distingushed role

• Trinomial potentials are adequate

• Trinomial new inflation is- generically adequate (also when fully symmetric)- moderately coupled (Ginzburg–Landau safe)- provides a lower bound for r

• Trinomial chaotic inflation is- confined to a corner of parameter space- more strongly coupled- in tension with the Ginzburg–Landau picture

• Other potentials should be similarly studied (not reconstructed )

Paris 17/05/2007 – p. 21/21

Summary and conclusions• Effective FT of inflation is not just phenomenology

• N plays a distingushed role

• Trinomial potentials are adequate

• Trinomial new inflation is- generically adequate (also when fully symmetric)- moderately coupled (Ginzburg–Landau safe)- provides a lower bound for r

• Trinomial chaotic inflation is- confined to a corner of parameter space- more strongly coupled- in tension with the Ginzburg–Landau picture

• Other potentials should be similarly studied (not reconstructed )

Paris 17/05/2007 – p. 21/21

Summary and conclusions• Effective FT of inflation is not just phenomenology

• N plays a distingushed role

• Trinomial potentials are adequate

• Trinomial new inflation is- generically adequate (also when fully symmetric)- moderately coupled (Ginzburg–Landau safe)- provides a lower bound for r

• Trinomial chaotic inflation is- confined to a corner of parameter space- more strongly coupled- in tension with the Ginzburg–Landau picture

• Other potentials should be similarly studied (not reconstructed )

Paris 17/05/2007 – p. 21/21

Summary and conclusions• Effective FT of inflation is not just phenomenology

• N plays a distingushed role

• Trinomial potentials are adequate

• Trinomial new inflation is- generically adequate (also when fully symmetric)- moderately coupled (Ginzburg–Landau safe)- provides a lower bound for r

• Trinomial chaotic inflation is- confined to a corner of parameter space- more strongly coupled- in tension with the Ginzburg–Landau picture

• Other potentials should be similarly studied (not reconstructed )

Paris 17/05/2007 – p. 21/21

Summary and conclusions• Effective FT of inflation is not just phenomenology

• N plays a distingushed role

• Trinomial potentials are adequate

• Trinomial new inflation is- generically adequate (also when fully symmetric)- moderately coupled (Ginzburg–Landau safe)- provides a lower bound for r

• Trinomial chaotic inflation is- confined to a corner of parameter space- more strongly coupled- in tension with the Ginzburg–Landau picture

• Other potentials should be similarly studied (not reconstructed )

Paris 17/05/2007 – p. 21/21

Summary and conclusions• Effective FT of inflation is not just phenomenology

• N plays a distingushed role

• Trinomial potentials are adequate

• Trinomial new inflation is- generically adequate (also when fully symmetric)- moderately coupled (Ginzburg–Landau safe)- provides a lower bound for r

• Trinomial chaotic inflation is- confined to a corner of parameter space- more strongly coupled- in tension with the Ginzburg–Landau picture

• Other potentials should be similarly studied (not reconstructed )

Paris 17/05/2007 – p. 21/21

Summary and conclusions• Effective FT of inflation is not just phenomenology

• N plays a distingushed role

• Trinomial potentials are adequate

• Trinomial new inflation is- generically adequate (also when fully symmetric)- moderately coupled (Ginzburg–Landau safe)- provides a lower bound for r

• Trinomial chaotic inflation is- confined to a corner of parameter space- more strongly coupled- in tension with the Ginzburg–Landau picture

• Other potentials should be similarly studied (not reconstructed )

Paris 17/05/2007 – p. 21/21