Cosmological consequences from sub-Planckian inflation

Cosmological consequencesfrom Sub-Planckian inflation

Sayantan Choudhury

Physics and Applied Mathematics UnitIndian Statistical Institute, Kolkata

Date : 31/07/2014Contact: [email protected]

Webpage: http://isical.academia.edu/sayantanchoudhury

PLAN OF THESIS

1. Introduction

2. MSSM inflation from various flat directions

3. Inflation from background supergravity

4. Reheating & Leptogenesis in a supergravityinspired brane inflation

5.Primordial non-Gaussian features from DBIGalileon inflation

6.Improved constraints on primordial graviataionalwaves for sub-Planckian models of inflation

1

PLAN OF THESIS

1. Introduction

2. MSSM inflation from various flat directions ⇐=

3. Inflation from background supergravity

4. Reheating & Leptogenesis in a supergravityinspired brane inflation

5.Primordial non-Gaussian features from DBIGalileon inflation

6.Improved constraints on primordial graviataionalwaves for sub-Planckian models of inflation ⇐=

2

PUBLICATIONS(Thesis is based on the papers marked by the symbol ∗)

1. SC and S. Pal, PRD 85, 043529 (2012)∗.2. SC and S. Pal, NPB 857 (2012) 85∗.3. SC and S. Pal, JCAP 04 (2012) 018∗. ⇐=

4. SC and S. Pal, NPB 874 (2013) 85∗.5. SC and S. Pal, arXiv:1210.4478 ∗.6. SC, A. Mazumdar and S. Pal, JCAP 07 (2013) 041∗. ⇐=

7. SC and A. Mazumdar, NPB 882 (2014) 386∗. ⇐=

8. SC and A. Mazumdar, PLB 733 (2014) 270∗.9. SC and S. SenGupta, JHEP 02 (2013) 136.10. SC, T. Chakraborty and S. Pal, NPB 880 (2014) 155.11. SC and S. SenGupta, arXiv:1306.0492 .12. SC, S. Sadhukhan and S. SenGupta, arXiv:1308.1477 .13. SC and A. Dasgupta, NPB 882 (2014) 195.14. SC and S. SenGupta, arXiv:1311.0730 .

3

PUBLICATIONS15. SC, A. Mazumdar and E. Pukartas,JHEP 04 (2014) 077.16. SC, JHEP 04 (2014) 105.17. SC, PLB 735 (2014) 138 . ⇐=

18. SC and A. Mazumdar, arXiv:1403.5549 .⇐=

19. SC and A. Mazumdar, arXiv:1404.3398 .20. SC, J. Mitra and S. SenGupta, Accepted in JHEP .21. SC, arXiv:1406.7618 . ⇐=

22. SC , J. Mitra and S. SenGupta, To appear soon .23. SC , B. K. Pal, B. Basu and P. Bandyopadhyay To appearsoon .24. SC and S. Panda, To appear soon .25. SC, To appear soon .26. SC and A. Dasgupta, To appear soon .Conference Proceedings:1. SC and S. Pal, J. Phys.: Conf. Series 405 (2012) 012009.

4

Highlights · · ·

⋆Basics of inflation and allied issues....

⋆Observational status....

⋆Low scale saddle point inflation.....

⋆High scale inflection point inflation.....

⋆Anti Lyth bound after Planck and BICEP2.....

⋆Reconstruction of potential (from GR andbraneworld).....

⋆Bottom lines....

⋆Open issues....

⋆Future projects....

5

Basics of inflation and allied issues · · ·

6

•What is Inflation (in Cosmology)?

7

•What is Inflation (in Cosmology)?⇒Evolution of the comoving Hubble radius, (aH)−1 , which

implies the comoving Hubble sphere shrinks during inflationand expands after inflation. Inflation is therefore a mechani smto ‘zoom-in’ on a smooth sub-horizon patch .

7-A

•What is Inflation (in Cosmology)?⇒Evolution of the comoving Hubble radius, (aH)−1 , which

implies the comoving Hubble sphere shrinks during inflationand expands after inflation. Inflation is therefore a mechani smto ‘zoom-in’ on a smooth sub-horizon patch .

7-B

•Inflation (from Friedmann eqn.) ⇒a > 0 ⇐⇒ d

dt (aH)−1 < 0 ⇐⇒ − HH2

(= d lnH

dN

)> 0 ⇐⇒ (ρ+ 3P ) < 0.

where N = ln(aend

a ) =∫ tend

tHdt measures the no. of e-foldings.

8


dt (aH)−1 < 0 ⇐⇒ − HH2

(= d lnH

dN

)> 0 ⇐⇒ (ρ+ 3P ) < 0.

where N = ln(aend

a ) =∫ tend


• Inflation (via Slow-Roll) ⇒1. |φ| << |3Hφ|, V ′

(φ) ⇒ 3Hφ ≈ −V′(φ).

2. φ2 << V (φ) ⇒ H2 ≈ V (φ)3M2

p.

8-A


dt (aH)−1 < 0 ⇐⇒ − HH2

(= d lnH

dN

)> 0 ⇐⇒ (ρ+ 3P ) < 0.

where N = ln(aend

a ) =∫ tend


• Inflation (via Slow-Roll) ⇒1. |φ| << |3Hφ|, V ′

(φ) ⇒ 3Hφ ≈ −V′(φ).

2. φ2 << V (φ) ⇒ H2 ≈ V (φ)3M2

p.

⇒Flatness/Slow-Roll Parameters ⇒ǫV =

M2p

2

(

V′(φ)

V (φ)

)2

, ǫV = M2p

(

V′′(φ)

V (φ)

)

ξ2V = M4p

(V ′(φ)V ′′′(φ)

V 2(φ)

)

, σ3V = M6

p

(V ′2(φ)V ′′′′(φ)

V 3(φ)

)

.

where in slow-roll regime ǫV , |ηV |, |ξ2V |, |σ3V | << 1.

⋆Inflation ends when the SR violated (i.e. ηV (φend) = 1 · · · ).

8-B

•Inflation (and Observables) ⇒

9


⋆Two Point FunctionA. Scalar Scalar : 〈ζ(~k)ζ(~k′)〉 = (2π)3PS(k)δ

3(~k + ~k′)

B. Tensor Tensor :〈hij(~k)hjk(~k′)〉 = (2π)3δikPT (k)δ

3(~k + ~k′)

where

PS(k) = PS(k⋆)(

kk⋆

)nS−1+αS2! ln( k

k⋆)+κS

3! ln2( kk⋆)+···

PT (k) = PT (k⋆)(

kk⋆

)nT+αT2! ln( k

k⋆)+κT

3! ln2( kk⋆)+···

10


⋆Two Point FunctionA. Scalar Scalar : 〈ζ(~k)ζ(~k′)〉 = (2π)3PS(k)δ

3(~k + ~k′)

B. Tensor Tensor :〈hij(~k)hjk(~k′)〉 = (2π)3δikPT (k)δ

3(~k + ~k′)

where

PS(k) = PS(k⋆)(

kk⋆

)nS−1+αS2! ln( k

k⋆)+κS

3! ln2( kk⋆)+···

PT (k) = PT (k⋆)(

kk⋆

)nT+αT2! ln( k

k⋆)+κT

3! ln2( kk⋆)+···

In GR1. Scalar power spectrum : PS(k⋆) =

V24π2ǫV M4

p

2. Tensor power spectrum : PT (k⋆) =2V

3π2M4p

3. Tensor-to-scalar ratio : r(k⋆) =PT (k⋆)PS(k⋆)

= 16ǫV

10-A


4. Scalar spectral tilt : nS(k⋆)− 1 = d lnPS(k)d ln k |⋆ = 2ηV − 6ǫV

5. Tensor spectral tilt : nT (k⋆) =d lnPT (k)

d ln k |⋆ = −2ǫV

6. Running of scalar spectral tilt :αS(k⋆) =

dnS

d ln k |⋆ = 16ηV ǫV − 24ǫ2V − 2ξ2V7. Running of tensor spectral tilt :

αT (k⋆) =dnT

d ln k |⋆ = 4ηV ǫV − 8ǫ2V8. Running of the running of scalar spectral tilt :

κS(k⋆) =d2nS

d ln k2 |⋆ = 192ǫ2V ηV − 192ǫ3V + 2σ3V − 24ǫV ξ

2V

+ 2ηV ξ2V − 32η2V ǫV

9. Running of the running of tensor spectral tilt :κT (k⋆) =

d2nT

d ln k2 |⋆ = 56ηV ǫ2V − 64ǫ3V − 8η2V ǫV − 4ǫV ξ

2V

⋆Note: Slow Roll ≡ Cosmological perturbation.

11


12

•Inflation (and Consistency relations) ⇒

nT = −r

8

(

2− r

8− nS

)

+ · · · , αT =dnT

d ln k=

r

8

(r

8+ nS − 1

)

+ · · · ,

nr =dr

d ln k=

16

9

(

nS − 1 +3r

4

)(

2nS − 2 +3r

8

)

+ · · · ,

κT =d2nT

d ln k2=

2

9

(

nS − 1 +3r

4

)(

2nS − 2 +3r

8

)(r

8+ nS − 1

)

+r

8

[

αS +2

9

(

nS − 1 +3r

4

)(

2nS − 2 +3r

8

)]

+ · · · ,

κr =d2r

d ln k2

=16

9

(

2nS − 2 +3r

8

)

αS +4

3

(

nS − 1 +3r

4

)(

2nS − 2 +3r

8

)

+16

9

(

nS − 1 +3r

4

)

2αS +2

3

(

nS − 1 +3r

4

)(

2nS − 2 +3r

8

)

+ · · · .

13

Observational status · · ·

Sr. Inflationary PLANCK+WP(+BICEP2) PLANCK+WP

No. observables

1 ln(1010PS) 3.089+0.024−0.027 3.089+0.024

−0.027

2 nS 0.9600± 0.0071 0.9603± 0.0073

3 αS −0.022± 0.010 −0.013± 0.009

4 κS 0.020+0.016−0.015 0.020+0.016

−0.015

5 r 0.2+0.07−0.05 < 0.12

6 nT 1.36± 0.83 ?

> −0.76 ?

14

Observational status · · ·

Sr. Inflationary PLANCK+WP(+BICEP2) PLANCK+WP

No. observables

1 ln(1010PS) 3.089+0.024−0.027 3.089+0.024

−0.027

2 nS 0.9600± 0.0071 0.9603± 0.0073

3 αS −0.022± 0.010 −0.013± 0.009

4 κS 0.020+0.016−0.015 0.020+0.016

−0.015

5 r 0.2+0.07−0.05 < 0.12

(r = 0 ruled out at 7σ)

6 nT 1.36± 0.83 (Blue = NBD/bounce..) ?

> −0.76 (Red = Inf + ...) ?

(nT = 0 ruled out at 3σ)

15

•Blue Gravity Waves from BICEP2? ⇒

16

•Constraints from Planck+WP+high L+BICEP2+.....? ⇒

17

•Scale of Inflation and Reheating temperature ⇒

A. V ≤ (1.96× 1016GeV)4 r0.12

B. H ≤ 9.241× 1013√

r⋆0.12 GeV

C. Trh ≤(

30π2g⋆

) 14

(1.96× 1016GeV) 4√

r0.12 .

18

•Scale of Inflation and Reheating temperature ⇒

A. V ≤ (1.96× 1016GeV)4 r0.12

B. H ≤ 9.241× 1013√

r⋆0.12 GeV

C. Trh ≤(

30π2g⋆

) 14

(1.96× 1016GeV) 4√

r0.12 .

•Inflaton (and CMBPOL) ⇒

1. < (aXlm)∗aYl′m′ >= CXY

l δll′ δmm′ where m = −l, ...,+l,X, Y = T,E,B

2. CXYl = 2

π

∫ kH(>k⋆)

kL(=ke)dk k2

INFLATION︷︸︸︷

P (k)

ANISOTROPIES︷︸︸︷

∆Xl(k)∆Y l(k) .where P (k) := PS(k), PT (k)

and∆Xl(k) =

∫ η0

0dη SX(k, η)

︸︷︷︸

SOURCES

PXl(k[η0 − η])︸︷︷︸

PROJECTION

.

18-A

⇒ SC, PLB 735 (2014) 138

21

Low scale saddle point inflation · · ·

⇒ SC and SP, JCAP 04 (2012) 018

Gauge invariant superpotential

W4 ≈ λ4

4MΦ4 ∀QQQL,QuQd,QuLe,uude

22


⇒ SC and SP, JCAP 04 (2012) 018


W4 ≈ λ4


Effective potential

V(φ, θ) = 12m2

φφ2 + λ4A

4Mφ4Cos(4θ + θA) +

λ2

4

M2φ6

⇒λ4 = λ4,0

[

1 + D3 log

(

φ2

µ20

)]

, A =

A0

[

1+D2 log

(

φ2

µ20

)]

[

1+D3 log

(

φ2

µ20

)] , m2φ

= m20

[

1 + D1 log

(

φ2

µ20

)]

mφ soft SUSY breaking mass, Φ = φ exp(iθ) (∈ C)

22-A


⇒ SC and SP, JCAP 04 (2012) 018


W4 ≈ λ4


Effective potential

V(φ, θ) = 12m2

φφ2 + λ4A

4Mφ4Cos(4θ + θA) +

λ2

4

M2φ6

⇒λ4 = λ4,0

[

1 + D3 log

(

φ2

µ20

)]

, A =

A0

[

1+D2 log

(

φ2

µ20

)]

[

1+D3 log

(

φ2

µ20

)] , m2φ

= m20

[

1 + D1 log

(

φ2

µ20

)]

mφ soft SUSY breaking mass, Φ = φ exp(iθ) (∈ C)

Flat direction contentQI1

a = 1√2(Φ, 0)T ,QI2

b = 1√2(Φ, 0)T ,QI3

c = 1√2(Φ, 0)T ,LI4

3 =1√2(0,Φ)T ,dB1

a = Φ√2,uB2

b = Φ√2,uB3

c = Φ√2, e3 = Φ√

2

22-B

Saddle point condition

(1) V′(φ0) = 0 ⇒

φ0 =√

M4λ4(3+D3)

[

A(1 + D2

2

)±

√

A2(1 + D2

2

)2 − 8m2φ(1 +D1)(3 +D3)

] 12

(2) V′′(φ0) = 0 ⇒

A =√

2(3 +D3)G1G2G3mφ(φ0)

(3) V′′′(φ0) = 0 ⇒

D3 = MA0

4λ4,0φ20

(

37+60 log(

φ0µ0

))

D2

(

13 + 12 log(

φ0

µ0

))

−2m2

φ(φ0)D1M

λ4,0A0φ20

+ 6

(

1− 20λ4,0φ20

MA0

)

(4) V′′′′(φ0) < 0

where Gi = Gi(D1, D2, D3) for i=1,2,3.

23

Saddle point condition

(1) V′(φ0) = 0 ⇒

φ0 =√

M4λ4(3+D3)

[

A(1 + D2

2

)±

√

A2(1 + D2

2

)2 − 8m2φ(1 +D1)(3 +D3)

] 12

(2) V′′(φ0) = 0 ⇒

A =√

2(3 +D3)G1G2G3mφ(φ0)

(3) V′′′(φ0) = 0 ⇒

D3 = MA0

4λ4,0φ20

(

37+60 log(

φ0µ0

))

D2

(

13 + 12 log(

φ0

µ0

))

−2m2

φ(φ0)D1M

λ4,0A0φ20

+ 6

(

1− 20λ4,0φ20

MA0

)

(4) V′′′′(φ0) < 0

where Gi = Gi(D1, D2, D3) for i=1,2,3.Inflaton potential

Around the saddle point φ0 V (φ) = C0 + C4(φ− φ0)4

23-A

For C0 = 2.867× 10−36M4, C4 = −1.685× 10−13 and N = 70 weget: Ps(= ∆2

s) = 2.498× 10−9, ns = 0.957, nt = −1.550× 10−30,r = 1.240× 10−29, αs = −0.612× 10−3, κs = 1.749× 10−5.

0.92 0.93 0.94 0.95 0.96 0.97

0.00002

0.00004

0.00006

0.00008

0.0001

ns

Ds

24

0

1000

2000

3000

4000

5000

6000

1 10 100 1000

l(l+1)C

l/2π[µ K

2 ]

l

CMB TT Angular Power Spectrum

WMAP data setBest fit

25

0

1000

2000

3000

4000

5000

6000

1 10 100 1000

l(l+1)C

l/2π[µ K

2 ]

l

CMB TT Angular Power Spectrum

WMAP data setBest fit

Output parametrs from COSMOMC(CAMB) ⇒

t0 zReion Ωm Ωb ΩΛ Ωk ηRec η0

Gyr Mpc Mpc

13.707 10.704 0.2670 0.04 0.7329 0.0 285.10 14345.1

25-A

One loop RG flowD1 ≈ −0.056ζ2, D1

2 ≈ −0.074ζ, D22 ≈ −0.071ζ,

D32 ≈ −0.031ζ, D1

3 = D23 = D3

3 ≈ −0.048− 0.168ζ2 where ζ = m/mφ.

0 5 10 15 20 25

1

H L

0 5 10 15 20 25

2

4

H L

0 5 10 15 20 25

1.0

1.2

1.4

1.6

1.8

H L

0 5 10 15 20 254

6

8

10

12

14

H L

0 5 10 15 20 250.5

1.0

1.5

2.0

2.5

H L

0.0

0.2

0.4

0.6

0.8

H L

26

One loop RG flowD1 ≈ −0.056ζ2, D1

2 ≈ −0.074ζ, D22 ≈ −0.071ζ,

D32 ≈ −0.031ζ, D1

3 = D23 = D3

3 ≈ −0.048− 0.168ζ2 where ζ = m/mφ.

0 5 10 15 20 25

1

2

3

4

5

log10HΜL

A1HΜL

A1HΜ

0L

U

0 5 10 15 20 25

1

2

3

4

5

log10HΜL

A2HΜL

A2HΜ

0L

D

0 5 10 15 20 250.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

log10HΜL

A3HΜL

A3HΜ

0Lã

0 5 10 15 20 254

6

8

10

12

14

log10HΜL

miHΜLH

GevL

0 5 10 15 20 250.5

1.0

1.5

2.0

2.5

log10HΜL

mΦ2@ΜD

mΦ2@Μ

0D

Qu

Qd

0 5 10 15 200.0

0.2

0.4

0.6

0.8

log10HΜLΛΒHΜL

ΛΒHΜ

0L

26-A

High scale inflection point inflation · · ·

⇒ SC, AM and SP, JCAP 07 (2013) 041


W6 ≈ λ6MPL

Φ6 ∀udd,LLe

27


⇒ SC, AM and SP, JCAP 07 (2013) 041


W6 ≈ λ6MPL

Φ6 ∀udd,LLe

High scale potential ( H ≫ mφ)

V (φ) = V0 +cHH2

2 |φ|2 − aHHφ6

6M3PL

+ λ2|φ|10M6

PL

27-A


⇒ SC, AM and SP, JCAP 07 (2013) 041


W6 ≈ λ6MPL

Φ6 ∀udd,LLe


V (φ) = V0 +cHH2

2 |φ|2 − aHHφ6

6M3PL

+ λ2|φ|10M6

PL

Inflection point constraints

(1)Tuning ⇒ a2H

40c2H

= 1− 4δ2

(2)Flatness ⇒ V ′′(φ0) = 0

(3)VEV/IP⇒ φ0 =(√

cH10 HM3

PL

)1/4

27-B


⇒ SC, AM and SP, JCAP 07 (2013) 041


W6 ≈ λ6MPL

Φ6 ∀udd,LLe


V (φ) = V0 +cHH2

2 |φ|2 − aHHφ6

6M3PL

+ λ2|φ|10M6

PL

Inflection point constraints

(1)Tuning ⇒ a2H

40c2H

= 1− 4δ2

(2)Flatness ⇒ V ′′(φ0) = 0

(3)VEV/IP⇒ φ0 =(√

cH10 HM3

PL

)1/4

Inflaton potential Around the inflection point φ0

V (φ) = α + β(φ− φ0) + γ(φ− φ0)3 + κ(φ− φ0)

4

27-C

⇒ α = V (φ0) = V0 +(

415 + 4

3δ2)cHH2φ2

0 +O(δ4),

β = V′(φ0) = 4δ2cHH2φ0 +O(δ4),

γ = V′′′

(φ0)3! = cHH2

φ0

(32− 80δ2

)+O(δ4),

κ = V′′′′

(φ0)4! = cHH2

φ20

(384− 1260δ2

)+O(δ4)

SHAPE OF THE POTENTIAL

0.0 0.2 0.4 0.6 0.8 1.0

4.5´10-9

5.´10-9

5.5´10-9

6.´10-9

6.5´10-9

ΦHin MPLL

VHΦLH

inM

PL

4L

POTENTIAL NEAR INFLECTION

Point of inflection

0.000 0.001 0.002 0.003 0.004 0.005 0.0064.45´10-9

4.5´10-9

4.55´10-9

4.6´10-9

4.65´10-9

4.7´10-9

xHin MPLL

VHxLH

inM

PL

4L

Here x := φ− φ0.

28

⇒ SC, AM and EP, JHEP 04 (2014) 077

29

⇒ SC, AM and EP, JHEP 04 (2014) 077

Kahler corrections

K(1) = φ†φ+ s†s+a

M2p

φ†φs†s+ · · · ,

K(2) = φ†φ+ s†s+b

2Mps†φφ+ h.c.+ · · · ,

K(3) = φ†φ+ s†s+c

4M2p

s†s†φφ+ h.c.+ · · · ,

K(4) = φ†φ+ s†s+d

Mpsφ†φ+ h.c.+ · · · ,

Parameter Space

cH ∼ O(10− 10−6) ,

aH ∼ O(30− 10−3) ,

Ms(= V1/40 ) ∼ O(9.50× 1010 − 1.77× 1016) GeV .

30

cS = 0.02

cS = 1

0.00 0.02 0.04 0.06 0.08 0.10 0.12

0.9996

0.9997

0.9998

0.9999

1.0000

r*

a

a vs r* plot

cS = 0.02

cS = 1

0.00 0.02 0.04 0.06 0.08 0.10 0.12

0.92

0.94

0.96

0.98

1.00

r*

b

b vs r* plot

cS = 1

cS = 0.02

0.00 0.02 0.04 0.06 0.08 0.10 0.120.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

r*

c

c vs r* plot

cS = 0.02

cS = 1

0.00 0.02 0.04 0.06 0.08 0.10 0.120.4

0.5

0.6

0.7

0.8

0.9

1.0

r*

d

d vs r* plot

31

cS = 0.02

cS = 1

0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.991.´10-9

1.5´10-9

2.´10-9

2.5´10-9

3.´10-9

nS

PS

PS vs nS plot

33

Anti Lyth bound after Planck and BICEP2 · · ·

⇒ SC and AM, NPB 882 (2014) 386In this work we provide an accurate bound on generatingprimordial gravitational waves from inflation, where inflat ionoccurs below the Planck scale.

Large tensor to scalar ratio can be generated as required byPLANCK and BICEP2.

38

Anti Lyth bound after Planck and BICEP2 · · ·

⇒ SC and AM, NPB 882 (2014) 386In this work we provide an accurate bound on generatingprimordial gravitational waves from inflation, where inflat ionoccurs below the Planck scale.

Large tensor to scalar ratio can be generated as required byPLANCK and BICEP2.

If inflation has to make connection with the real particlephysics it must be explained within an EFT description whereit can be trustable below the UV cut-off of the scale of gravit y.

We also establish a new closed relationship between VEV,scale of inflation, tensor to scalar ratio in a modelindependent fashion.

38-A

What is “THE LYTH BOUND”?A useful way of normalizing the primordial GW amplitude isthe tensor-to-scalar ratio

r(k⋆) =PT (k⋆)PS(k⋆)

= 16ǫV = 8M2

pl

φ2

H2 = 8M2

pl

(dφ

dN

)2

The value of r(k⋆) determines whether primordial GW aredetectable in future CMB observations.

39

What is “THE LYTH BOUND”?A useful way of normalizing the primordial GW amplitude isthe tensor-to-scalar ratio

r(k⋆) =PT (k⋆)PS(k⋆)

= 16ǫV = 8M2

pl

φ2

H2 = 8M2

pl

(dφ

dN

)2

The value of r(k⋆) determines whether primordial GW aredetectable in future CMB observations.

The total field evolution between the time when CMBfluctuations exited the horizon and the end of inflation

⇒LYTH BOUND

∆φ

Mpl=

∫ N⋆

NenddN

√r(N)8

=∫ k⋆

kend

dkk

√r(k)8

≃ O(1)×√

r(k⋆)0.01

39-A

Detectable large values of the tensor-to- scalar ratio,r(k⋆) > 0.01 and correlates with ∆φ > Mpl.∆φ > Mpl implies preferance to the large-field models ofinflation. Example: String theory.

40

Detectable large values of the tensor-to- scalar ratio,r(k⋆) > 0.01 and correlates with ∆φ > Mpl.∆φ > Mpl implies preferance to the large-field models ofinflation. Example: String theory.This implies the lower bound of the energy scale of inflation

is around the GUT scale (V 1/4 ≥ O(1016 GeV )) but cannotaccomodate feasible αS & κS .For ∆φ > Mpl one can go beyond the upper cut-off of theeffective theory of gravity ΛUV = Mpl but the models becomeUV unprotective.

40-A

Detectable large values of the tensor-to- scalar ratio,r(k⋆) > 0.01 and correlates with ∆φ > Mpl.∆φ > Mpl implies preferance to the large-field models ofinflation. Example: String theory.This implies the lower bound of the energy scale of inflation

is around the GUT scale (V 1/4 ≥ O(1016 GeV )) but cannotaccomodate feasible αS & κS .For ∆φ > Mpl one can go beyond the upper cut-off of theeffective theory of gravity ΛUV = Mpl but the models becomeUV unprotective.Within ∆φ < Mpl it is possible to accomodate feasible αS &κS which can generate large “r” as observed by BICEP2 andPlanck.

⋆NOTE:Higher order SL corrections (HOSLC) ≡ PrecisionCosmology (fits CMBPOL within 2 < l < 2500.)

40-B

In presence of feasible αS+κS+HOSLC, the tensorto scalar ratio “r” can be expressed as:

r = 16ǫV[1−(CE+1)ǫV ]2

[1−(2CE+1)ǫV +CEηV ]2= 8

M2pl

(1−ǫV )2[1−(CE+1)ǫV ]2

[1−(2CE+1)ǫV +CEηV ]2

(dφ

dln k

)2

where CE = 4(ln 2 + γE)− 5.

41

In presence of feasible αS+κS+HOSLC, the tensorto scalar ratio “r” can be expressed as:

r = 16ǫV[1−(CE+1)ǫV ]2

[1−(2CE+1)ǫV +CEηV ]2= 8

M2pl

(1−ǫV )2[1−(CE+1)ǫV ]2

[1−(2CE+1)ǫV +CEηV ]2

(dφ

dln k

)2

where CE = 4(ln 2 + γE)− 5.

To evaluate the total field evolution ∆φ:

∫ k⋆

ke

dkk

√

r(k)8

≈∆φ

Mpl

1 + 1∆φ

[

(2CE − 1)∫ φ⋆

φedφ ǫV − 2CE

∫ φ⋆

φedφ ηV

]

+ ....

where

r(k) = PT (k)PS(k)

= r(k⋆)(

kk⋆

)a+ b2ln( k

k⋆)+ c

6ln2( k

k⋆)+....

with

a = nT − nS + 1, b = (αT − αS) , c = (κT − κS).

41-A

We Taylor expand a generic inflationary potential V (φ)

around the vicinity of VEV φ0 where inflation occurs.

√r(k⋆)

8

∣∣(a2 − b

2 + c2 − 2

) [1− e−∆N

]−(a2 − b

2 + c2 − 1

)∆N e−∆N

+(b4 − c

4

)(∆N)2 e−∆N − c

12 (∆N)3 e−∆N∣∣ ≈ ∑∞

n=0 Gn

(|∆φ|Mp

)n

where Gn =

1 +∞∑

m=0

Am

(φe − φ0

Mp

)m

︸︷︷︸

≪1

∼ 1 for n = 1

<< 1 for n ≥ 2 ,

and Am = Mm+2p

[(CE − 1

2

)Cm − CEDm

].

⋆NOTE: Gn << 1 reqd. for convergence of the Taylor series.

42

Here the slow-roll parameters show non-monotonicbehaviour within the interval ∆N = Ne −N⋆ = ln

(k⋆

ke

)

≈ 17.

0.00 0.02 0.04 0.06 0.08 0.10

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

HΦ-Φ0L in Mp

Ε VHΦL

ΕV HΦL vs HΦ-Φ0L plot

0.00 0.02 0.04 0.06 0.08 0.10

0.005

0.010

0.015

0.020

0.025

0.030

0.035

HΦ-Φ0L in Mp

ÈΗVHΦLÈ

ÈΗV HΦLÈ vs HΦ-Φ0L plot

0.00 0.02 0.04 0.06 0.08 0.10

0.000

0.002

0.004

0.006

0.008

0.010

HΦ-Φ0L in Mp

ÈΞ2 VHΦLÈ

ÈΞ2V HΦLÈ vs HΦ-Φ0L plot

0.00 0.02 0.04 0.06 0.08 0.10

0.002

0.004

0.006

0.008

0.010

0.012

0.014

HΦ-Φ0L in Mp

ÈΣ3 VHΦLÈ

ÈΣ3V HΦLÈ vs HΦ-Φ0L plot

43

10-5 10-4 0.001 0.01 0.1 1

2.0´10-9

1.5´10-9

kHin Mpc-1L

P SHkL

PSHkL vs k plot

10-5 10-4 0.001 0.01 0.10.9

1.

1.1

1.2

1.3

kHin Mpc-1L

n SHkL

nSHkL vs k plot

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

kHin Mpc-1L

ΑSH

kL

ΑSHkL vs k plot

-25 -20 -15 -10 -5 0 5 10

3.9

4.

4.1

4.2

4.3

4.4

lnHkMpc-1L

lnNHkL

lnNHkL vs lnk plot

44

Neglecting the Planck scale suppressed contribution in thelimit ∆φ < Mpl we get relation between tensor-to-scalar ratioand ∆φ:

√r(k⋆)

2

∣∣∣

r(k⋆)16 − ηV (k⋆)

2 − 1−(6CE + 23

3

)ǫ2V (k⋆)−

η2V (k⋆)6 + (CE − 1)

ξ2V (k⋆)2

−(2CE − 8

3

)ηV (k⋆)ǫV (k⋆)− σ3

V (k⋆)2 + · · ·

∣∣∣ ≈ |∆φ|

Mp

45


√r(k⋆)

2

∣∣∣

r(k⋆)16 − ηV (k⋆)

2 − 1−(6CE + 23

3

)ǫ2V (k⋆)−

η2V (k⋆)6 + (CE − 1)

ξ2V (k⋆)2

−(2CE − 8

3

)ηV (k⋆)ǫV (k⋆)− σ3

V (k⋆)2 + · · ·

∣∣∣ ≈ |∆φ|

Mp

SCALE OF INFALTION: V ≤ (1.96× 1016GeV)4 r(k⋆)0.12

45-A


√r(k⋆)

2

∣∣∣

r(k⋆)16 − ηV (k⋆)

2 − 1−(6CE + 23

3

)ǫ2V (k⋆)−

η2V (k⋆)6 + (CE − 1)

ξ2V (k⋆)2

−(2CE − 8

3

)ηV (k⋆)ǫV (k⋆)− σ3

V (k⋆)2 + · · ·

∣∣∣ ≈ |∆φ|

Mp

SCALE OF INFALTION: V ≤ (1.96× 1016GeV)4 r(k⋆)0.12

Finally we get a relation between ∆φ and scale of inflation:

|∆φ|Mpl

≤√V

(2.20×10−2 Mpl)2

∣∣∣

V

(2.78×10−2Mpl)4− ηV (k⋆)

2 − 1

−(6CE + 23

3

)ǫ2V (k⋆)−

η2V (k⋆)6 + · · ·

∣∣∣

45-B

⇒ SC and AM, arXiv:1404.3398

46

⇒ SC and AM, arXiv:1404.3398

N = 70

N = 50

0.90 0.92 0.94 0.96 0.98 1.00-0.040

-0.035

-0.030

-0.025

-0.020

nS

n T

nT vs nS plot

N = 70

N = 50

0.950 0.955 0.960 0.965 0.970-0.0006

-0.0005

-0.0004

-0.0003

-0.0002

-0.0001

0.0000

0.0001

nS

ΑT

ΑT vs nS plot

N = 70N = 50

0.950 0.955 0.960 0.965 0.970

-0.00060

-0.00055

-0.00050

-0.00045

-0.00040

-0.00035

nS

Κ T

ΚT vs nS plot

N = 70

N = 50

0.10 0.15 0.20 0.25 0.30-0.005

0.000

0.005

0.010

0.015

r0.002

n r

nr vs r0.002 plot

47

Reconstruction (in GR) : ⇒ SC and AM, arXiv:1404.3398

V (φ⋆) =32PS(k⋆)r(k⋆)π

2M4p ,

V′(φ⋆) =

32PS(k⋆)r(k⋆)π

2√

r(k⋆)8 M3

p ,

V′′(φ⋆) =

34PS(k⋆)r(k⋆)π

2(

nS(k⋆)− 1 + 3r(k⋆)8

)

M2p ,

V′′′(φ⋆) =

32PS(k⋆)r(k⋆)π

2[√

2r(k⋆)(

nS(k⋆)− 1 + 3r(k⋆)8

)

− 12

(r(k⋆)

8

) 32 − αS(k⋆)

√2

r(k⋆)

]

Mp,

V′′′′(φ⋆) = 12PS(k⋆)π

2

κS(k⋆)2 − 1

2

(r(k⋆)

8

)2 (

nS(k⋆)− 1 + 3r(k⋆)8

)

+ 12(

r(k⋆)8

)3

+ r(k⋆)(

nS(k⋆)− 1 + 3r(k⋆)8

)2

+

[√

2r(k⋆)(

nS(k⋆)− 1 + 3r(k⋆)8

)

− 12

(r(k⋆)

8

) 32 − αS(k⋆)

2r(k⋆)

]

×[√

r(k⋆)8

(

nS(k⋆)− 1 + 3r(k⋆)8

)

− 6(

r(k⋆)8

) 32

]

48

⇒ SC, arXiv:1406.7618

r(k) =

r(k∗)

r(k∗)(

kk∗

)nT (k∗)−nS(k∗)+1

r(k∗)(

kk∗

)nT (k∗)−nS(k∗)+1+αT (k∗)−αS(k∗)

2! ln( kk∗ )

r(k∗)(

kk∗

)nT (k∗)−nS(k∗)+1+αT (k∗)−αS(k∗)

2! ln( kk∗ )+

κT (k∗)−κS(k∗)

3! ln2( kk∗ )

⇓ ⇓ ⇓ ⇓

49

⇒ SC, arXiv:1406.7618

r(k) =

r(k∗)

r(k∗)(

kk∗

)nT (k∗)−nS(k∗)+1

r(k∗)(

kk∗

)nT (k∗)−nS(k∗)+1+αT (k∗)−αS(k∗)

2! ln( kk∗ )

r(k∗)(

kk∗

)nT (k∗)−nS(k∗)+1+αT (k∗)−αS(k∗)

2! ln( kk∗ )+

κT (k∗)−κS(k∗)

3! ln2( kk∗ )

⇓ ⇓ ⇓ ⇓

Field− excursion (in GR) :

∣∣∣∣

∆φ

Mp

∣∣∣∣=

O(2.7− 5.1)

O(2.7− 4.6)

O(0.6− 1.8)

O(0.2− 0.3)

49-A

Inflation from RS braneworld : ⇒ SC, arXiv:1406.7618

5D Model action

SRS =

∫

d5x√

−(5)g

[M3

5

2(5)R− 2Λ5 + Lbulk + (Lbrane − σ) δ(y)

]

50


5D Model action

SRS =

∫

d5x√

−(5)g

[M3

5


]

50-A


5D Model action

SRS =

∫

d5x√

−(5)g

[M3

5


]

51


5D Model action

SRS =

∫

d5x√

−(5)g

[M3

5


]

4D Modified Friedmann Equation within Slow-Roll ⇒CovariantShiromizu-Maeda-Sasaki approach+ Isriel-Junction condi tion

H2 ≈ ρ

3M2p

1+

︷︸︸︷ρ

2σCorrection term

≈ V(φ)

3M2p

(

1+V(φ)

2σ

)

Cut-off scale and brane tension

M35 =

√

4πσ

3Mp σ =

√

− 3

4πM3

5Λ5 > 0

51-A

Reconstruction (in RS) : ⇒ SC, arXiv:1406.7618

V (φ∗) =3√

2π2PS(k∗)r(k∗)M4/3p σ2/3,

V′(φ∗) =

√PS(k∗)σ

24 π r(k∗)Mp,

V′′(φ∗) = 2−4/3(PS(k∗)r(k∗))2/3π4/3

(

nS(k∗)− 1 + r(k∗)4

)

M2/3p σ1/3,

V′′′(φ∗) = 2−5/3(PS(k∗)r(k∗))4/3π5/3

[r(k∗)

3

(

nS(k∗)− 1 + r(k∗)4

)

− 18(

r(k∗)24

)2

− αS(k∗)

]

M1/3p σ1/6,

V′′′′(φ∗) =

V 4(φ∗)

8M6p (V

′ (φ∗))2

[

κS(k∗)2 − 4

(r(k∗)

8

)2 (

nS(k∗)− 1 + r(k∗)4

)

+ 96(

r(k∗)24

)3

+ r(k∗)3

(

nS(k∗)− 1 + r(k∗)4

)2

− 4M4pσ

2(V′(φ∗))

2V′′′

(φ∗)

V 4(φ∗)

(

nS(k∗)− 1− r(k∗)12

)]

52

⇒ SC, arXiv:1406.7618

Brane tension : σ ≤ O(10−9) M4p,

5D Scale : M5 ≤ O(0.04) Mp,

5D Cosmological Constant : Λ5 ≥ −O(10−15) M5p

⇓ ⇓ ⇓ ⇓

53

⇒ SC, arXiv:1406.7618


5D Scale : M5 ≤ O(0.04) Mp,


⇓ ⇓ ⇓ ⇓

Field− excursion (in RS) :

∣∣∣∣

∆φ

Mp

∣∣∣∣=

O(0.24− 0.81)

O(0.24− 0.73)

O(0.05− 0.28)

O(0.02− 0.05)

53-A

⇒ SC, arXiv:1406.7618


5D Scale : M5 ≤ O(0.04) Mp,


⇓ ⇓ ⇓ ⇓

Field− excursion (in RS) :

∣∣∣∣

∆φ

Mp

∣∣∣∣=

O(0.24− 0.81)

O(0.24− 0.73)

O(0.05− 0.28)

O(0.02− 0.05)

⋆⋆⋆

Note:Large (detectable) r+ |∆φ| < Mp (EFT)= running/Beyond GR

(RS)/muiltifield/.....

53-B

BOTTOM LINES

•High scale inflation is more favoured after Planck andBICEP2.

54

BOTTOM LINES


•Large tensor modes can be generated from high scalemodels.

54-A

BOTTOM LINES



• High scale models fitted well with the CMB TT spectra asobserved by Planck within 2 < l < 2500.

54-B

BOTTOM LINES




•Using higher order corrections to slow roll accurateanalytical bound on tensor modes has been established inpresence of considerable amount of running.

54-C

BOTTOM LINES





•Super-Planckian models (String Theory) cannot provideconsiderable amount of running.

54-D

BOTTOM LINES





•Super-Planckian models (String Theory) cannot provideconsiderable amount of running.

•To get large ‘r’+ sub-Planckian field excursionrunning+HOSLC, braneworld, multi field etc are required.

54-E

OPEN ISSUES

•Finding tensor modes from Planck and BICEP2 is notsufficient. Need to figure out the components of B (tensor)modes (=INFLA+PMF+NG+LENSING+...) and to separate all ofthem. This only gives the correct value of “r′′.

55

OPEN ISSUES


•Fitting any models at the pivot k⋆ is not suffiecient enough.Need to check whether they fitted well the CMB(POL) spectrawithin 2 < l < 2500.

55-A

OPEN ISSUES



• Need to find out analytical and statistical tools throughwhich one can rule out several models and break thedegeneracy between the cosmological parameters.

55-B

OPEN ISSUES



• Need to find out analytical and statistical tools throughwhich one can rule out several models and break thedegeneracy between the cosmological parameters.

• If the tensor mode is really blue tilted then need to figure outits possible sources (=NBD VAC/BOUNCE/INFLATION+......?) .

55-C

OPEN ISSUES

•Need to check whether the models really can address thethermal history of early universe (i.e. reheating etc.).

56

OPEN ISSUES

•Need to check whether the models really can address thethermal history of early universe (i.e. reheating etc.).

•The primordial non-Gausiianity is not yet been detected wit hhigh statistical accuracy (i.e. f local

NL = 2.7± 5.8 andτ localNl ≤ 2800). But if it is detected in future CMB experimentsthen it is possible to rule out various models.

56-A

FUTURE PROJECTS

•FERMION LOCALIZATION IN 5D WITHOUT ANY EXTERNALFIELD: SC, J. Mitra and S. SenGupta.

•MAGNETOGENESIS BEYOND GR: SC.

• CONSTRAINTS ON ASSISTED TACHYONIC NFLATION: SCand S. Panda.

57

FUTURE PROJECTS

•FERMION LOCALIZATION IN 5D WITHOUT ANY EXTERNALFIELD: SC, J. Mitra and S. SenGupta.

•MAGNETOGENESIS BEYOND GR: SC.

• CONSTRAINTS ON ASSISTED TACHYONIC NFLATION: SCand S. Panda.

•TORSION INDUCED INFLATION FROM ECKS THEORY OFGRAVITY: SC, B. K. Pal, B. Basu and P. Bandyopadhyay.

•MODEL INDEPENDENT ANALYSIS FOR UNIFYING INFLATIONAND DARK MATTER IN BRANEWORLD: SC and A. Dasgupta.

•ISSUE OF CLASSICALIZATION AND IMPACT ONCONSISTENCY RELATIONS: SC and M. Sami.

57-A

FUTURE PROJECTS

•BOUNCING FROM HIGHER RANK TENSOR FIELDS: ANALTERNATIVE APPROACH TO INFLATION: SC, S. Kar and S.SenGupta.

•WIMPY BARYOGENESIS MIRACLE IN BRANEWORLD: SC andA. Dasgupta.

• EFFECTIVE THEORY OF DARKOLEPTOGENESIS: SC and S.Sadhukhan.

58

FUTURE PROJECTS

•BOUNCING FROM HIGHER RANK TENSOR FIELDS: ANALTERNATIVE APPROACH TO INFLATION: SC, S. Kar and S.SenGupta.

•WIMPY BARYOGENESIS MIRACLE IN BRANEWORLD: SC andA. Dasgupta.

• EFFECTIVE THEORY OF DARKOLEPTOGENESIS: SC and S.Sadhukhan.

•LOOPS AND LEGS I: A COMPLETE FEYNRULE FOR FORCMB LENSING SC.

•LOOPS AND LEGS II: DIAGRAMATICS FOR LARGE SCALESTRUCTURE: SC.

•LOOPS AND LEGS III: BEHAVIOUR OF IR AND UVDIVERGENCES AND SPECTRAL REGULARIZATION: SC.

58-A

SPECIAL THANKS TO .........

⋆Soumitra SenGupta ⋆R. H. Brandenberger

⋆Sayan Kar ⋆Marc Kamionkowski

⋆Sudhakar Panda ⋆Daniel Baumann

⋆Md. Sami ⋆Leonardo Senatore

⋆Anupam Mazumdar ⋆Cliff Burgess

⋆Supratik Pal

⋆Andrew Liddle

⋆David H. Lyth

⋆Claudia-de-Rahm

59

THANKS FOR YOUR TIME.........

60

THANKS FOR YOUR TIME.........

61

BACK UP SLIDES.........

62

BETA FUNCTIONS FOR QQQL,QuQd,QuLe,uude ⇒(a) For Soft mass:µ(m2

φ)QQQL = 18π2

(3m2

Q3+m2

U3+ |A33

U |2) (

λ33U

)2

− 18π2

(3g22 |m2|2 + 4g23 |m3|2

),

µ(m2φ)QuQd = 1

2π2

(m2

Q3+m2

U3

) (λ33U

)2

− 18π2

(32g

22 |m2|2 + 16

3 g23 |m3|2),

µ(m2φ)QuLe = 3

8π2

(m2

Q3+m2

U3

) (λ33U

)2

− 18π2

(32g

22 |m2|2 + 8

3g23 |m3|2

),

µ(m2φ)uude = 1

2π2

(m2

Q3+m2

U3

) (λ33U

)2 − 12π2 g

23 |m3|2,

(b) For Trilinear A- term:

µAaaD = δb3

(λ33U

)2 A33U

8π2

− 14π2

(718g

21 |m1|2 + 3

2g22 |m2|2 + 8

3g23 |m3|2

),

µAabU =

3(1+δa3)A33U

8π2

(λ33U

)2

− 14π2

(1318g

21 |m1|2 + 3

2g22 |m2|2 + 8

3g23 |m3|2

),

µAaaE = − 1

4π2

(32g

21 |m1|2 + 3

2g22 |m2|2

),

63

(c) For Fourth level Yukawa coupling:µλaa

U = 3(1+δa3)8π2

(λ33U

)3 − λaaU

4π2

(1318g

21 +

32g

22 +

83g

23

),

µλabD = δb3

(λ33U

)2 λabD

8π2 − λabD

4π2

(718g

21 +

32g

22 +

83g

23

),

µλaaE = −λaa

E

4π2

(32g

22 +

32g

23

)

(d) Gaugino masses and couplings:

µgi =di

2 g3i , µ

(mi

g2i

)

= 0 ∀ i where

a, b = 1, 2, 3(family), i = 1(U(1)Y), 2(SU(2)L), 3(SU(3)C) andd1 = 11

8π2 ,d2 = 18π2 ,d3 = − 3

8π2 .

SOLUTION ⇒gi(µ) =

gi(µ0)√

1−dig2i (µ0) ln

(

µµ0

)

,mi(µ) = mi(µ0)(

gi(µ)gi(µ0)

)2

,

∆m2φ =

∑3i=1 f

iF∆m2

i , ∆Aabβ = 1

2

∑3i=1(C

iβ)

ab∆mi,

λabβ (µ) = λab

β (µ0)∏3

i=1

(gi(µ0)gi(µ)

)(Ciβ)

ab

with ∆Aβ = Aβ(µ)−Aβ(µ0), ∆mi = mi(µ0)−mi(µ),∆m2

φ = m2φ(µ)−m2

φ(µ0),∆m2i = m2

i (µ0)−m2i (µ),

64

.

CONSTANTS DETERMINED FROM RGE

D1 = − 18π2

∑3i=1 Ji

(mi

mφ0

)2

g2i (µ0), Dβ2 = − 1

4π2

∑3i=1K

βi(

mi

A0

)

g2i (µ0)

where J1 = 0,J2 = 3 and J3 = 4 for i = 1, 2, 3 and

Kβi i = 1(U(1)Y) i = 2(SU(2)L) i = 3(SU(3)C)

β=1(U) 1318

32

83

β=2(D) 718

32

83

β=3(E) 32

32 0

Universal mSUGRA B. C. at GUT scale Constraints on λ0 from proton decay ( p → π0

e+, p → π+νe

etc.) ⇒D1 ≈ −0.056ζ2, D1

2 ≈ −0.074ζ, D22 ≈ −0.071ζ, D3

2 ≈ −0.031ζ,

D13 = D2

3 = D33 ≈ −0.048− 0.168ζ2 where ζ = m/mφ.

65

New constraint between PGW and PBH · · ·

PBH formation by Press-Schechter theory

f(≥ M) = 2γ∫∞Θth

dΘP(Θ;M(kPBH)) = γ erf[

Θth√2ΣΘ(kPBH)

]

66


PBH formation by Press-Schechter theory

f(≥ M) = 2γ∫∞Θth

dΘP(Θ;M(kPBH)) = γ erf[

Θth√2ΣΘ(kPBH)

]

where P(Θ; kPBH) =1√

2πΣΘ(kPBH)exp

(

− Θ2

2Σ2Θ(kPBH)

)

ΣΘ(kPBH) =

√∫∞0

dkk

exp(

− k2

k2PBH

)

PΘ(k)

Θ(k) = 25

(kaH

)2Rc(k)

PΘ(k) =425

(1+w)2

(1+ 35w)

2

(kaH

)4PS(k)

PS(k) = PS(k⋆)(

kk⋆

)nS−1+αS2

ln( kk⋆)+κS

6ln2( k

k⋆)+···

kPBH =√γ

5.54×10−24

(MPBH

1 g

)− 12 ( g⋆

3.36

)− 16 Mpc−1

66-A


ΣΘ(kPBH) = 15

(1+w)√

PS(k⋆)

(1+ 35w)

(k⋆

aH)2 A+BnS(k⋆) + CαS(k⋆)

+DκS(k⋆) + · · · 12

where A = 2B =4k2

PBHk2⋆

[

1 − exp

(

−k2Λ

k2PBH

)]

,

C = 12

ln

(

kΛk⋆

)

+ 14

Γ

[

0,k2L

k2PBH

]

− Γ

[

0,k2Λ

k2PBH

]

,

D =k2Λ

6k2PBH

P FQ

[

(1, 1, 1) ; (2, 2, 2) ;−k2Λ

k2PBH

]

+k2PBH3k2

⋆

ln2(

kΛk⋆

)

+ γE ln

(

kLkΛ

)

.

0 2´1012 4´1012 6´1012 8´1012 1´1013

0.14

0.16

0.18

0.20

MPBHHgmL

fHMPB

HL

67


Regularized SD in terms of the slow-rollparameters along with higher order corrections:

ΣΘ(kPBH) =(1+w)

√

AV⋆ǫV (k⋆)

8√6πM2

pl(1+35w)

1 +

(CE + 2

5

)ηV (k⋆)

−(2CE + 11

5

)ǫV (k⋆)− C

5 ξ2V (k⋆) +

2D5 σ3

V (k⋆) + · · ·

68



ΣΘ(kPBH) =(1+w)

√

AV⋆ǫV (k⋆)

8√6πM2

pl(1+35w)

1 +

(CE + 2

5

)ηV (k⋆)

−(2CE + 11

5

)ǫV (k⋆)− C

5 ξ2V (k⋆) +

2D5 σ3

V (k⋆) + · · ·

Relationship between GW & PBH abundance

ΩGWh2(η0) =(

ΣΘ(kPBH)10−2

)2 ( MPBH1020 g

) 12 ΩPBHh2(η0)

(1.7×1021)γ erf[

Θth√2ΣΘ(kPBH )

]

68-A



ΣΘ(kPBH) =(1+w)

√

AV⋆ǫV (k⋆)

8√6πM2

pl(1+35w)

1 +

(CE + 2

5

)ηV (k⋆)

−(2CE + 11

5

)ǫV (k⋆)− C

5 ξ2V (k⋆) +

2D5 σ3

V (k⋆) + · · ·


ΩGWh2(η0) =(

ΣΘ(kPBH)10−2

)2 ( MPBH1020 g

) 12 ΩPBHh2(η0)

(1.7×1021)γ erf[

Θth√2ΣΘ(kPBH )

]

Next we use the constraints for sub-Planckianinflation.

68-B



ΣΘ(kPBH) =(1+w)

√

AV⋆ǫV (k⋆)

8√6πM2

pl(1+35w)

1 +

(CE + 2

5

)ηV (k⋆)

−(2CE + 11

5

)ǫV (k⋆)− C

5 ξ2V (k⋆) +

2D5 σ3

V (k⋆) + · · ·


ΩGWh2(η0) =(

ΣΘ(kPBH)10−2

)2 ( MPBH1020 g

) 12 ΩPBHh2(η0)

(1.7×1021)γ erf[

Θth√2ΣΘ(kPBH )

]

Next we use the constraints for sub-Planckianinflation.Further using the real root from the closed

constraint relation we express “r′′ in terms of thefield displacement ∆φ.

68-C


Hence we express the regularized SD in terms ofinflationary observables using the sub-Planckianconstraints.

69


Hence we express the regularized SD in terms ofinflationary observables using the sub-Planckianconstraints.Finally we get a colsed cosmic constraint relation

between GW & PBH abundance connected via thethe sub-Planckian inflationary constraints:

ΩGWh2 ≤ 6×10−18

γ

(MPBH1020 g

) 12 O2

PBHΩPBHh2

erf

(

Θth√2OPBH

)

whereOPBH = 5

√A(1+w)(8.17×10−3)2

12√2π(1+ 3

5w)

1 + 2ηV (k⋆)5

+ 1500

(r(k⋆)0.12

)

− 3ǫV (k⋆)−C

5ξ2V (k⋆) +

2D

5σ3V (k⋆) + · · ·

69-A


PLANCK 1Σ

BAND

-0.025 -0.020 -0.015 -0.010 -0.005 0.000 0.005

0.009970

0.009971

0.009972

0.009973

0.009974

0.009975

WPBHh2 vs ΑS plot

70


71

POWER SPECTRUM

PS(k⋆) = [1− (2CE + 1)ǫV + CEηV ]2V

24π2M4PLǫV

PT (k⋆) = [1− (CE + 1)ǫV ]2 2V

3π2M4PL

SPECTRAL TILT

nS − 1 ≈ (2ηV − 6ǫV )− 2CEξ2V +2

3η2V + 2(8CE + 3)ǫ2V

+2ǫV ηV

(

6CE +7

3

)

−4CE(CE+1)ξ2V ǫV +2C2EηV ξ

2V

nT ≈ −2ǫV + 2

(

2CE +5

3

)

ǫV ηV − 2

(

4CE +13

3

)

ǫ2V

72

RUNNING OF SPECTRAL TILTαS ≈

(16ηV ǫV − 24ǫ2V − 2ξ2V

)− 2CE(4ǫV ξ2V − ηV ξ

2V − σ3

V )

+4

3ηV (2ηV ǫV − ξ2V ) + 4(8CE + 3)ǫV (4ǫ

2V − 2ηV ǫV )

− 4CE(CE +1)[ǫV (4ǫV ξ

2V − ηV ξ

2V − σ3

V ) + ξ2V (4ǫ2V − 2ηV ǫV )

]

+ 2C2Eξ

2V (2ηV ǫV − ξ2V ) + 2C2

EηV (4ǫV ξ2V − ηV ξ

2V − σ3

V )

αT ≈ (4ηV ǫV − 8ǫ2V ) + 2

(

2CE +5

3

)[ǫV (2ηV ǫV − ξ2V )

+ ηV (4ǫ2V − 2ηV ǫV )

]− 4

(

4CE +13

3

)

ǫV (4ǫ2V − 2ηV ǫV )

⋆NOTE:HIGHER ORDER SLOW ROLL CORRECTIONS≡PRECISION COSMOLOGY (FITS CMBPOL THROUGHOUT THEMULTIPOLE 2 < l < 2500.)

73

INFLATON POTENTIALAround the saddle point φ0

V (φ) = C0 + C4(φ− φ0)4

where(1) C0 = V (φ0) =

m3φ(φ0)M

6√6λ4

3(1 + D1

2 − D3

6

)

[

1 +D1 log(

φ20

µ20

)]

− 3(1 + D1

2 − D3

6

)2[

1 +D2 log(

φ20

µ20

)]

+(1 + D1

2 − D3

6

)2[

1 +D2 log(

φ20

µ20

)]

,

(2) C4 = 14!V

′′′′(φ0) =

m2φ(φ0)

24√6φ2

0

(1 + D1

2 − D3

6

)

[(360√

6− 12

√6)

+ (684D3 − 50√6D2)

] (1 + D1

2 − D3

6

)

− 2√6D1

(1+D12 −D3

6 )

+(1 + D1

2 − D3

6

) (360D3√

6− 12

√6D2

)

log(

φ20

µ20

)

.

75

Cosmological consequences from sub-Planckian inflation

Documents

Transcript of Cosmological consequences from sub-Planckian inflation