We are addressing the scenario called inflation from the particle physics point of view, Putting more constrained over the minimal higgs inflation by the polynomial modification of potential. My model are agreeing with the recent WMAP data

1. OUSEPH C.J.
1603151012
(Discipline of physics)
Dr. Subhendu Rakshit ( Supervisor)
1

2. Introduction
 Classical Dynamics of Inflation
 Minimally Coupled Model
My work
 Generalized Starobinsky potential and Inflation
 Modification of Inflationary Potential
 Minimal Higgs Inflation
 New constrained over minimal Inflation
Literature survey
 The Standard Model Higgs Boson as the
Inflaton
 Higgs Portal Inflation
 Ameliorating the unitarity issue via the
Singlet assistance
 Inert Doublet as an inflaton
2

3. The Horizon Problem
Why is the CMB so uniform?
• Two points have never been in causal contact
if their past light cones don’t intersect:
World line of the early universe
3

4. Light has travelled a finite distance since the Big Bang:
Particle horizon is give by
∆𝒙 = ∆𝝉 = 𝟎
𝒕 𝒅𝒕
𝒂 𝒕
= 𝒂𝒊
𝒂
𝒂𝑯 −𝟏 d lna
𝒂𝑯 −𝟏
∝ 𝒂
(𝟏+𝟑𝒘)
𝟐
𝒅𝒔 𝟐=𝒂 𝟐 𝒅𝝉 𝟐 − 𝒅𝒙 𝟐
Hubble radius w= 𝒑/𝝆
4

5. A Shrinking Hubble Sphere
𝑑
𝑑𝑡
(𝑎𝐻)−1 < 0
so ,1+3w <0
∆𝝉 =
𝒂𝒊
𝒂
(𝒂𝑯)−𝟏 𝒅 𝒍𝒏𝒂
𝝉𝒊 =
𝟐
𝟏 + 𝟑𝒘
𝒂
𝟏
𝟏+𝟑𝒘 𝒂𝒊 → 𝟎, 𝒘 < −𝟏/𝟑
-ְ∝
There was more time between the singularity
and recombination than we had thought!
5

6. World line of the early universe
6

7. Conditions for Inflation
Accelerated expansion-From the relation
𝑑
𝑑𝑡
(𝑎𝐻)−1= −
𝑎
𝑎2 , 𝑎>0
 Slowly-varying Hubble parameter,
𝐻
𝐻2 < 1 call it as ∈ , −
𝐻
𝐻 𝐻
< 1(call it as η)
Negative pressure
1+3w<0
7

8. Slow-roll parameters
slow-roll parameters in terms of derivatives of the potential
The easiest way to achieve Inflation was to introduce a scalar
field 𝜑, which has to fulfill the following conditions:
8

9. The quantum origin of density perturbations is quite
intuitive:
9

10. Few more parameters for inflation
𝑤ℎ𝑒𝑟𝑒 𝑛 𝑠, 𝑛 𝑡 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑠𝑐𝑎𝑙𝑎𝑟 𝑎𝑛𝑑 𝑠𝑝𝑒𝑐𝑡𝑟𝑎𝑙 𝑖𝑛𝑑𝑒𝑥 𝑎𝑛𝑑 𝑟 is the tensor to
scalar ratio.
𝑛 𝑠= 1 − 6𝜖 + 2 η
𝑛 𝑡 = −2 𝜖
𝑟 = 16𝜖
10
Allowed range of 𝑛 𝑠~0.953 − 0.986
For that of r~10−2

11. In this case the parameter 𝜀 is set to zero and the system
is said to be minimally coupled,
MINIMAL COUPLING
Can this give rise to inflation?
11

12. According to the COBE data the value of
𝑉
𝜖
= (0.027𝑀 𝑝𝑙)4
• For this kind of a value the value of λ ~10−13
Such an
extremely fine-tuned coupling constant seems very
unphysical.
• The value of r≈ 0.26 which is also in conflict with the
observed value of r.
12

13. THE STANDARD MODEL HIGGS BOSON AS THE INFLATON
Non-minimal coupling of gravity with scalar field
𝑆𝐽 = 𝑑𝑥4 −𝑔
𝑀2 𝑅
2
+ ξ 𝐻+ 𝐻𝑅 + 𝐿 𝑆𝑀
conformal transformation from the Jordan frame to the
Einstein frame,
Action in the Einstein frame,
13

14. For Higgs inflation,
𝑟~0.0033 , 𝑛 𝑠~0.97
Could the Higgs Boson be the Inflaton?
While calculating(COBE normalization)
𝑉
𝜖
=
(0.027𝑀 𝑝𝑙)4
we found that ,
The value of ξ~104 this large coupling makes unitarity
violation
14

15. HIGGS PORTAL INFLATION
 A combination of the Higgs and a singlet plays the role of inflaton.
 The Real singlet scalar coupled non-minimally to the curvature 𝑅
Lagrangian in the Jordan frame written as,
𝐿𝐽 = −
𝑀 𝑃𝑙
2
2
𝑅 −
ξℎ
2
ℎ2
𝑅 −
ξ 𝑠
2
𝑠2
𝑅 +
1
2
(𝜕 𝜇ℎ)2
+
1
2
(𝜕 𝜇 𝑠)2
− 𝑉(ℎ, 𝑠)
And 𝑉(ℎ, 𝑠) is given by
15

16. The transformation to the Einstein frame is given by
The potential energy transformed as
𝑉𝐸 =
1
ξℎℎ2 + ξ 𝑠 𝑠2 2 𝑉 ℎ, 𝑠
By the introduction of new variable 𝜑 and 𝜏,
𝜑 =
3
2
log ξℎℎ2
+ ξ 𝑠 𝑠2
, 𝑥 =
ℎ
𝑠
The potential energy transformed as
𝑔 𝜇𝜗 = 𝑔 𝜇𝜗 𝛺 2
; 𝛺 2
= (1 +
ξℎℎ2+ξ 𝑠 𝑠2
𝑀 𝑃𝑙
2 )
𝑈 𝜑 =
𝜆 𝑒𝑓𝑓
4ξℎ
2 1 + exp
−2𝜑
6
−2
Where 𝜆 𝑒𝑓𝑓 =
1
4
4𝜆ℎ 𝜆 𝑠−𝜆ℎ𝑠
2
𝜆 𝑠+𝜆ℎ 𝑥2−𝜆ℎ𝑠 𝑥
16

17. The values of 𝜆 𝑒𝑓𝑓 and Inflation
𝜆 𝑒𝑓𝑓 is the composite of inflation ,
𝜆 𝑒𝑓𝑓= 𝜆ℎ, for Higgs Inflation
𝜆 𝑒𝑓𝑓=𝜆 𝑠/𝑥2
, for singlet Inflation
Using COBE normalization
𝑈
𝜖
= (0.027𝑀 𝑃𝑙)4, we can fix 𝜆 𝑒𝑓𝑓 and
𝜀ℎ.
𝜆 𝑒𝑓𝑓~1 , ξℎ~104
𝜆 𝑒𝑓𝑓
17

18. Inflationary parameters
The slow roll parameter 𝜖 𝑎𝑛𝑑 η are given by
𝜖 =
4
3ξℎ
2
ℎ4
, η =
4
3ξℎℎ2 With ℎ =
1
ξℎ
exp(−
2𝜑
6
)
Initial value of the inflation for a given number of e-folds N is ℎ = 4𝑁 3ξℎ
Using the values of 𝜆 𝑒𝑓𝑓, ξℎ with 60 e-fold, the value of Spectral
Index and tensor 𝑛 𝑠 to scalar ratio 𝑟 calculated as
𝑛 𝑠 = 1 − 6ϵ + 2η ≈ 0.97 𝑎𝑛𝑑 𝑟 = 16 𝜖 ≈0.0033
18

19. Ameliorating the unitarity issue via
the singlet assistance
𝐿𝑗
−𝑔
=
1
2
𝑀2 + ξ 𝜎2 + 2ζ𝐻𝐻† 𝑅 −
1
2
𝜕𝜇 𝜎
2
− 𝐷𝜇 𝐻
2
−
1
4
𝜆 𝜎 𝜎2 − 𝑢2 + 2
𝜆 𝐻𝜎
𝜆 𝜎
𝐻† 𝐻
2
−
𝜆 𝐻 −
𝜆 𝐻𝜎
2
𝜆 𝐻
𝐻† 𝐻 −
𝑣2
2
2
Λ 𝑈𝑉 = 1 + 6𝑟𝜀
𝑀 𝑃𝑙
ξ
We have used the effective quartic coupling introduced due to the addition of the
real singlet and it's mixing with the Higgs doublet, 𝜆 𝐻 −
𝜆 𝐻𝜎
2
𝜆 𝐻
.The Basic idea is to
make the unitarity cut-off large
Where 𝑟 =
ξ 𝑢2
𝑀 𝑃𝑙
2 , the values of 𝑟 runs between zero and one
19

20. We are assuming that the Non- minimal coupling of higgs with gravity is absent in
this case. Under unitarity gauge 𝐻 𝑇=
1
2
(0. ℎ) the lagrangian change as,
𝐿𝑗
−𝑔
=
1
2
ξ 𝜎2 𝑅 −
1
2
(𝜕𝜇 𝜎)2−
1
2
(𝜕𝜇ℎ)2−
1
4
𝜆 𝜎 𝜎2 − 𝑢2 + 2
𝜆 𝐻𝜎
𝜆 𝜎
ℎ2
2
−
𝜆 𝐻 −
𝜆 𝐻𝜎
2
𝜆 𝐻
ℎ2 −
𝑣2
2
2
The mass of the sigma field calculated as 𝑀 𝜎
2 ≈ 𝜆 𝜎
𝑀 𝑃𝑙
2
3ξ2 ,using
COBE normalization mass of the sigma field calculated as 𝑀 𝜎 ≈
1013 𝐺𝑒𝑉
20

21. Inert Doublet as an Inflaton
The action of the Model is given by
𝑆𝐽 = −𝑔 𝑑4 𝑥(−
1
2
𝑀 𝑝𝑙
2
𝑅 − 𝜕𝜇 𝜙1 𝜕 𝜇 𝜙1 − ξ1 𝜙1
2
𝑅 − 𝜕𝜇 𝜙2 𝜕 𝜇 𝜙2
−ξ2 𝜙2
2 𝑅 − 𝑉 𝜙1, 𝜙2
𝑉 𝜙1, 𝜙2 = 𝑚1
2
𝜙1
2 + 𝑚2
2
𝜙2
2 + 𝜆1 𝜙1
4 + 𝜆2 𝜙2
2 + 𝜆3 𝜙1
2 𝜙2
2 +
𝜆4(𝜙1
+
𝜙2) (𝜙2
+
𝜙1)+
𝜆5
2
(𝜙1
+
𝜙2)2+𝐶. 𝐶 ,
where 𝜙1 𝑎𝑛𝑑𝜙2 are given by
𝜙1 =
1
√2
ᵡ
ℎ
, 𝜙2 =
1
√2
𝑞
𝑥𝑒 𝑖𝜃
21

22. By conformal transformation 𝑔 𝜇𝜗 = 𝛺2
𝑔 𝜇𝜗 ,
𝛺2= 1 +
ξ1
𝑀 𝑝𝑙
2 ᵡ2 + ℎ2 +
ξ2
𝑀 𝑝𝑙
2 𝑞2 + 𝑥2
The potential in the Einstein Frame is defined as
𝑉𝐸 =
𝜆2 𝑀 𝑝𝑙
4
4ξ2
2
[1 − exp(−
2
3
)
1
2
𝐴
𝑀 𝑝𝑙
]2 ,
𝐴 = √(
3
2
)𝑀 𝑝𝑙 𝑙𝑛𝛺2
This model required the value of non-minimal coupling ξ2 ~104 in order to
agreement with the density perturbations( Using COBE data ).
In this model Unitarity Problem associated With the Higgs boson is
transferred to Inert sector
22

23. Generalized starobinsky like Potential and
Inflation
Potential of the form 𝐾 1 − exp 𝜑 𝑛 known as
starobinsky potential.
COBE normalization
𝑉
𝜖
= (0.027𝑀 𝑝𝑙)4
for Starobinsky like
potential always gives,
ξ≈ 104
𝜆.
23

24. Modification of Inflationary Potential
Using the conformal transformation Inflationary potential modified as
𝑉 𝜙 = 𝜆
𝑚4−𝑛
𝜙 𝑛
1 +
𝜙
𝛼
𝑛
Where 𝜆, m and n are the free parameters .Where the parameter 𝜆 defined for n=4
Realized as the Higgs Minimal Inflation.
𝛼 associated with the Inflationary scale by Λ= 𝜆𝑚4−𝑛 𝛼 𝑛
24

25. Minimal Higgs inflation
Here the polynomial modification of potential help SM Higgs to drive Inflation.
𝑉 𝜙 = 𝜆
(𝐻† 𝐻 − 𝑣2
)2
1 +
𝐻† 𝐻
𝛼
2
Inflationary parameters with 𝛼~1015 𝐺𝑒𝑣, 𝜆=10−1
𝑛 𝑠= 0.967, 𝑟 = 2 ∗ 10−8 𝑓𝑜𝑟 𝑁 = 50
𝑛 𝑠= 0.972, 𝑟 = 1 ∗ 10−8
𝑓𝑜𝑟 𝑁 = 60
This model successfully explained inflation However, It fail to give the allowed values of
tensor to scalar ratio .
WMAP data constrained the value of 𝑟 as, 𝑟 ≤ 10−2
25

26. New constraints on Minimal Higgs Inflation.
The model
𝑠 = 𝑑4 𝑥 −𝑔
1
2
𝑀 𝑃𝑙
2
𝑅 −
1
2
𝑔 𝜇𝜗 𝜕𝜇ℎ𝜕 𝜗ℎ +
𝜆
4
ℎ4
1 + ℎ4
𝛼4
Here I am expanding the potential as Taylor series
𝑉(𝜙)=𝑉0 𝜙0 + 𝑉1 𝜙0 𝜙 − 𝜙0 +
1
2
𝑉2 𝜙0 𝜙 − 𝜙0
2 About 𝜙0
𝑉0 𝜙0 =
3𝜋2
2
𝑀 𝑃𝑙
4
𝐴 𝑠 𝑟
𝑉1 𝜙0 =
3𝜋2
4 2
𝑀 𝑃𝑙
3
𝐴 𝑠 𝑟
3
2
𝑉2 𝜙0 =
3𝜋2
4
𝑀 𝑃𝑙
2
𝐴 𝑠 𝑟
3
8
+ (𝑛 𝑠 − 1)
26

27. Where 𝐴 𝑠 is nothing but
𝑉
𝜖
.
With 𝑁 number of e-folds the value of ℎ(𝜙0) can be determined as
ℎ(𝜙0) =(24𝑁𝑀 𝑃𝑙
2
𝛼4
)
1
6
The value of 𝑉2 𝜙0 makes the new constrains in the parameter space.
All inflationary parameter fall in the allowed range of WMAP data with 𝛼~1017
𝐺𝑒𝑣 and
𝜆~10−2 .
Inflationary parameter of the Model
𝑛 𝑠= 0.968, 𝑟~10−2 𝑓𝑜𝑟 𝑁 = 50
𝑛 𝑠= 0.982, 𝑟~10−2 𝑓𝑜𝑟 𝑁 = 60
27

28.  Inflation explains why the universe appears flat, homogeneous
and isotropic.
 With a large non-minimal coupling the Higgs boson could drive
inflation which agrees with CMB data.
The Higgs inflation model suffer from unitarity problems.
 By the addition of a Heavy singlet scalar we can ameliorating the unitarity
issue associated with the SM Higgs .
 The inert doublet coupled non-minimally to gravity could act as in
inflaton.
 Generalized Starobinsky potential required high value of non minimal
coupling of gravity in order to agreement with the observed density
perturbation.
 Minimally coupled Higgs can be an inflaton with a modified potential.
28

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30

31. 31

32. The perturbed inflaton field
for the unperturbed FRW metric ,action will be
To get the linearized equation of motion for f(𝜏 x), we need
to expand the action,
32

33.  solution of the Mukhanov-Sasaki equation
The minimum energy mode
function in Minkowski is:
The mode function is given by
33

34. Vacuum Fluctuations
Finally, we can compute the variance of inflaton fluctuations
due to quantum zero-point fluctuations.
We define the power spectrum as,
34

35. Most of the time we will work with the
dimensionless power spectrum.
∆ 𝑠
2(k)=
𝑘3 𝑝 𝑟
(2π)2 =
𝐻4
(2π)2 𝜑2
 power spectra for tensor fluctuation,
∆ 𝑡
2
(𝑘) =
2𝐻2
π2
35

36. Scalar Field Dynamics
𝑠 = 𝑑4
𝑥 −𝑔
𝑀2
2
𝑅 −
1
2
𝑔 𝜇𝜗
𝜕𝜇 𝜑𝜕 𝜗 𝜑 − 𝑣 𝜑
In a flat FRW background, we have
Friedman
Klein-Gordon
𝐻2
=
1
3𝑀2
1
2
𝜑2
+ 𝑉 𝜑
𝜑 + 3𝐻 𝜑 = −𝑉′
Continuity
𝐻 = −
1
2
𝜑2
𝑀2
∈=
1
2
𝜑2
𝑀2 𝐻2<1
𝜇 =
2 𝜑
𝐻 𝜑
<<1
Consider a scalar field minimally coupled to gravity
36

37. The apparent problems of Higgs inflation come from
consideration of Higgs-Higgs scattering.
The Jordan frame metric is expanded about flat spacetime
Then,
The first term give the vertex
And the energy is scaled by Λ =
𝑀 𝑝
𝜀
dimensionless coupling of this vertex is of order
𝜀𝐸
𝑀 𝑝
=
𝐸
Λ
Because of large coupling 𝐸 ≥ Λ
37