Addressing cosmic inflation by extension of standard model.

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

2. Classical Dynamics of Inflation
The Horizon Problem
Slow-Roll Inflation
Quantum fluctuations
Canonical Quantization
Vacuum Fluctuations
Minimal Coupling
The Standard Model Higgs Boson As The Inflaton
The concrete Higgs portal inflation model

3. 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. 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:

6. 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!

8. 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 μ)
Negative pressure
1+3w<0

10. 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

11. 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:

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

14. 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,

15. Variation of S(2) yields the Mukhanov-Sasaki equation for the
mode functions𝑓𝑘.
for each Fourier mode

16. Canonical Quantization
The momentum conjugate to f is
We then promote the felids f(𝜏, x) and π(𝜏,x) to quantum operators
𝑓(𝜏, 𝑥) and π(𝜏, x)
The generalization of the mode expansion is

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

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

19.  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

20. Few more parameters for inflation
𝑤ℎ𝑒𝑟𝑒 𝑛 𝑠, 𝑛 𝑡 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑠𝑐𝑎𝑙𝑎𝑟 𝑎𝑛𝑑 𝑠𝑝𝑒𝑐𝑡𝑟𝑎𝑙 𝑖𝑛𝑑𝑒𝑥 𝑎𝑛𝑑 𝑟 is
the tensor to scalar ratio.

21. Allowed range of 𝑤ℎ𝑒𝑟𝑒 𝑛 𝑠, and 𝑟 for a successful inflation
𝑟
( wmap data 2012)

22. 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?

23. According to the WMAP 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.

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

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

26. The concrete Higgs portal inflation model
A single complex scalar field χ is introduced. Here we assume a
nonminimal coupling of the scalar field χ to the gravity in order to
explain the cosmological inflation problem.
we define the complex scalar field under the new U(1)
symmetry as χ = 𝑟(𝑥)𝑒2𝑖𝛼(𝑥)

27. ∈
The values of 𝑟 ~0.0032 𝑎𝑛𝑑 𝑛 𝑠 ~0.9603.
With a value of λ ~10−2 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝜀 < 4.7 ∗ 104

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.

29. In this work I will be going through a few scalar extensions to the
SM and attempt to elevate the unitary problem from the theory.
We have seen that only SM Higgs does not prove to be a good
inflaton Therefore the future plan in this project is to explore the
scalar extension of SM and find which ones are good.
In particular we will start with various Higgs portal scalar
extended models and check their viability as good inflationary
models, finding out the values of various inflationary parameters
and comparing with the most recent WMAP data

30. [1] F. Bezrukov, M. Shaposhnikov, The Standard Model Higgs boson as the inflaton,
Physics Letters B 659 (2008) 703–706, (DOI: 10.1016/j.physletb.2007.11).
[2] G.F. Giudice, H.M. Lee, Starobinsky-like inflation from induced gravity, Physics
Letters B 733 (2014) 58–62, (DOI: 10.1016/j.physle).
[3] M. Atkins, X. Calmet, Remarks on Higgs Inflation, Physics Letters B 697 (2011)
37–4, (DOI: 10.1016/j.physletb.2011.01.028).
[4] D. Baumann, “TASI Lectures on Inflation", arXiv:0907.5424v1[hep-th].
[5] Fa Peng Huan, Chong Sheng Li, Ding Yu Shao, Jian Wang , Phenomenology of an
Extended Higgs Portal Inflation Model after Planck 2013 , Eur. Phys. J. C 74 (2014)
2990, (DOI: 10.1140/epjc/s10052-014-2990-4)
[6] V. Mukhanov ,Physical foundation of cosmology (2005), isbn-13 978-0-521-
56398-7 ,Cambridge University Press, New York.

32. Ordinary matter satisfies the SEC: 1+3w > 0
The comoving Hubble radius grows and the
comoving horizon gets its largest
contribution from late times
∆𝜏 ∝ 𝑎
1+3𝑤
2 − 𝑎𝑖
1+3𝑤
2
0 = 𝜏𝑖

33. 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