Modified Gravity and Higgs Inflation

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University of Amsterdam

National Institute for Subatomic Physics

NIKHEF

AND

Institute of Physics

IoP - UvA

Master Thesis

Modified Gravity and Higgs Inflation

Author:

Shahzeb Kamal

Supervisor:

Dr. Marieke Postma

Examiner:

Prof. Dr. J. P. van der Schaar

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Abstract

The standard cosmological model derived from General Relativity lacks an explana-tion for a few fundamental cosmological problems, such as the horizon and flatness problem. Both problems are solved by inflation, a period of rapid expansion in the early universe. The non-uniqueness of the Einstein-Hilbert action motivates one to generalize General Relativity while maintaining the second order derivative structure of the equations of motion [1]. One such generalization is the Einstein-Gauss-Bonnet gravity. A scalar field coupled with the Einstein-Gauss-Bonnet gravity allows the previously excluded λφ4 _{potential to be in the experimental limits for inflation}

[2].

The need for a scalar field in inflationary models urges us to think about the Higgs boson, since it is the only scalar particle found experimentally and also has a quar-tic potential with λ ∼ 0.13 [3]. Inflation can be sourced by the Higgs boson if it is coupled non-minimally to the Ricci scalar with a (negative) large coupling constant [17]. The existence of a conformal transformation in such models ensures that the amplitude of scalar perturbations can be enhanced or reduced, by adjusting this new coupling constant, and it can always be made to fit the data. Standard Higgs inflation raises the question if there are other couplings to gravity possible that allow the Standard Model Higgs field to be the inflaton.

We studied the slow roll inflationary models with a coupling between the scalar field and the Gauss-Bonnet term. Inflationary predictions for a generic power law potential and coupling are obtained analytically. We showed that the previously excluded φ4 _{potential can now fit the data with different Gauss-Bonnet couplings.}

We further investigated the possibility of the Higgs boson to be the inflaton in this model. The results show that the corrections to the inflationary observables are of order O(1) from a standard potential driven inflation without a non-minimal cou-pling in a slow roll regime. Therefore, this excludes the Higgs boson to be an inflaton coupled with the Gauss-Bonnet term through a non-minimal coupling constant ξ be-cause it does not give the correct fractional density perturbations δρ_ρ of the CMBR data. However, it has been argued recently in [25] that we can get correct results for the inflationary observables with the SM Higgs coupling if one generalizes the theory from four to five dimensions, and choose an appropriate scale for the new dimension.

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Dedication

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Acknowledgement

I want to thank,

• Erasmus Mundus scholarship program for promoting and funding non-EU stu-dents for higher education in Europe.

• Stan Bentvelsen for offering me a student position at UvA/NIKHEF.

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1 FLRW Cosmology

1.1 General Relativity

1.1.1 The Equivalence principle

The Special Theory of Relativity describes the transformations between inertial frames1 _{moving with a constant relative velocity. It does not deal with non-inertial}

frames2_{. So, can we incorporate gravity into special relativity such that it obeys the}

relativistic transformations? Einstein dealt with the problem in his General Theory of Relativity, published in 1916 [4], based on the Principle of Equivalence.

The Equivalence Principle says that, In a small confined space-time, no experiment

can be performed to distinguish between a uniform gravitational field and an equiv-alent acceleration. In other words, the inertial mass and the gravitational mass of any object are equal and in small enough regions of space-time, the laws of physics reduce to those of special relativity. Moreover, studying the bending of light in an

accelerating frame of reference under the principle of equivalence tells us that it is not only mass that is affected by the gravitational field but all forms of energy and momentum.

1.1.2 Einstein equations from Einstein-Hilbert action

We start with the Einstein-Hilbert action,

SEH =

1 2k

Z

d4x√−gR (1.1)

where R is the Ricci scalar. Our dynamical variable is the metric tensor gµν_{. We}

vary the action with respect to gµν_{, and set k = 8πG = 1}

δSEH = Z d4xδ(√−gR) = Z d4x(√−gδR + (δ√−g)R) (1.2) and use, δ√−g = −1 2 √ −ggµνδgµν (1.3) and, δR = δ(gµνRµν) = Rµνδgµν+ gµνδRµν = Rµνδgµν+ ∇σ(gµνδΓσµν − g µσ_δΓρ ρµ) (1.4)

1_{Non-accelerating frame of reference.}

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For the variation of R, we use the variation of the connection, δΓλ_µν = δgλρgραΓαµν+ 1 2g λρ (∂µδgνρ+ ∂νδgµρ− ∂ρδgµν) = 1 2g λρ_(∇ µδgνρ+ ∇νδgµρ− ∇ρδgµν), (1.5)

by using δgµν = −gµαgνβδgαβ we simplify to,

δΓλ_µν = −1

2(gνα∇µδg

αλ_{+ g}

µα∇νδgαλ− gµαgνβ∇λgαβ) (1.6)

Plug back the variation of the connection in 1.4 to get

δR = Rµνδgµν− ∇µ∇νδgµν+ gµν∇2δgµν (1.7)

The variation of the Ricci scalar is further simplified because the last two terms in 1.7 cancel: Z d4x√−g(−∇µ∇νδgµν+ gµν∇2δgµν) = Z d4x√−gδgµν_(−∇ µ∇ν + gµν∇2δ) = 0 (1.8) So that, δR = Rµνδgµν (1.9)

plugging it back into equation 1.2 gives the field equations

δS = Z d4x√−g(−1 2gµνR + Rµν)δg µν _{= 0,} _(1.10) which imply, (Rµν − 1 2gµνR) = 0 (1.11)

These are the Einstein field equations in vacuum. The Einstein-Hilbert action de-scribed in equation 1.1 is the most simple3 _{action that gives the Einstein field}

equa-tions 1.11. However, this action is not fundamental. We will see in chapter 4 that we can get equation 1.11 from infinite many actions. Einstein equations are unique i.e. they are the only second order4 _{equations we can get from a generic action}

containing up to second order derivatives of the metric.

For a complete description, we have to add the matter action to see how the space-time dynamics are affected by the presence of matter and energy.

3_{Linear in Ricci scalar.}

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1.1.3 Einstein equations with a matter source

In the presence of matter the action of gravity becomes,

S = SEH + SM, (1.12)

where SM is the matter action and its variation with respect to the metric tensor

gives the stress energy-momentum tensor, −2√1

−g

δSM

δgµν = Tµν. (1.13)

By varying the total action, we get the following equation (Rµν−

1

2gµνR) = Tµν. (1.14)

Restoring constants, we get

(Rµν−

1

2gµνR) = 8πGTµν. (1.15)

These are the gravitational field equations in the presence of matter and energy.

1.2 Cosmological principle

Most of the models in cosmology are based on the assumption that the universe looks the same everywhere on large scales. Of course, we do not see it in our local galactic scale where the density variations are high. But, on the largest scales we average over the local density fluctuations and the universe looks the same every-where in all directions.

The Cosmological Principle states that the universe is homogeneous and isotropic

on large scales. Homogeneity means that the metric is the same everywhere. This

is only possible if we consider the largest scales. On the other hand, isotropic means that the universe looks the same in all directions. Cosmic microwave background radiation (CMBR) data supports the concept of isotropy. WMAP and COBE satel-lites confirm that the deviations in the CMBR in comparison with a perfect isotropic universe are extremely small (O ∼ 10−5) [5].

On the large scales the weak nuclear force and the strong nuclear force do not play a role. And since the universe is electrically neutral, the electromagnetic force is also unimportant. Hence, the dynamics of the universe on large scales is governed by gravitational interactions, described by the Einstein equations:

(Rµν−

1

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1.3 Expanding Universe

We will now solve the Einstein equations for our universe. For that we need the explicit form of the metric tensor. Here we assume that the cosmological princi-ple holds i.e. the universe is homogeneous and isotropic. From observations we see that the universe is expanding. Such an expanding universe is described by the Friedmann-Lemaître-Robertson-Walker metric (FLRW), and the line element in spherical coordinates is given by,

ds2 = −dt2+ a2(t)

_dr2

1 − kr2 + r

2_(dθ2_{+ sinθ}2_dφ2₎_, _(1.17)

where k = 0, +1, −1 is the curvature and a(t) is the scale factor that describes the expansion of the universe. For a flat, homogeneous and isotropic solution the metric reduces to,

ds2 = −dt2+ a2(t)(dx2+ dy2 + dz2). (1.18) To solve the Einstein equations with this metric we have to find all the Christoffel symbols, the non-zero components of the Ricci tensor and the Ricci scalar. The calculation is done in appendix A.1.

The Christoffel symbol is defined as, Γσ_µν = 1

2g

ρσ_(∂

µgσν+ ∂νgµσ+ ∂σgµν) (1.19)

The non-zero components of The Ricci tensor are,

R00= − 3ä a , R11= aä + 2 ˙a2, R22= aä + 2 ˙a2, R33= aä + 2 ˙a2 (1.20)

The Ricci then scalar becomes,

R = gµνRµν = g00R00 + g11R11+ g22R22+ g33R33 = 6[¨a a + ˙a a 2 ]. (1.21)

1.3.1 Universe as a perfect fluid

For an isotropic and homogeneous universe, we can describe the matter and energy content in the universe by a perfect fluid. The energy momentum tensor for a perfect fluid is defined as,

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Tµν = (ρ + P )uνuν+ P gµν (1.23)

where ρ and P are the energy density and pressure of the perfect fluid and uµ _{is the}

four velocity of the fluid. We choose the rest frame of the fluid where uµ _is

uµ = (0, 1, 1, 1) (1.24)

and the energy-momentum tensor of the fluid becomes diagonal,

Tµν =      ρ 0 0 0 0 P 0 0 0 0 P 0 0 0 0 P      (1.25)

Now we solve the Einstein equations (including a cosmological constant Λ) with the FLRW metric,

Rµν −

1

2gµνR + Λgµν = 8πTµν (1.26) Contracting with gµν_{, (where the trace of the metric tensor = 4)}

R = 4Λ − 8πT (1.27)

Putting it back into 1.26 yields,

Rµν = Λgµν + 8π(Tµν− 1 2gµνT ). (1.28) The µ = ν = 0 component is R00= Λg00+ 8π(T00− 1 2g00T ) (1.29) where, T = gµνTµν = −ρ + 3P. (1.30)

With this T equation 1.29 becomes, ¨ a a = − 4 3π(ρ + 3P ) + Λ 3. (1.31)

This is Friedmann’s first equation. To get the second equation we plug equation 1.31 in equation 1.28, ˙a a 2 = 8 3πρ + Λ 3 (1.32)

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This is Friedmann’s second equation. These equations describe the dynamics of a flat expanding universe. For Λ = 0 the equations reduces to,

¨ a a = − 4 3π(ρ + 3P ), ˙a a 2 = 8 3πρ (1.33)

We now introduce the Hubble parameter H and the equation of state parameter ω as,

H = ˙a

a, (1.34)

P = ωρ. (1.35)

For matter ω = 0, for radiation ω = 1₃ and for vacuum ω = −1. Figure 1.1 shows the scaling of energy density ρ, scale factor a(t) and Hubble parameter H with respect to time in different eras of the universe.

For radiation dominated universe ρR ∝ a−4 while, the scale factor scales as a ∝ t

1 2,

For matter dominated universe ρM ∝ a−3 and, the scale factor scales as a ∝ t

2 3,

For vacuum dominated universe ρΛ ∝ a0 and, the scale factor scales as a ∝ eHt

(where H = ˙a

a).

Figure 1.1: Scaling of ρ, ω, a(t) and H in different eras

1.4 Problems with Cosmology

The expanding solution of the universe described in 1.3 is a good description of the universe history and is the basis of most cosmological models which stood up in agreement with the experiments. However, there are certain problems with this model which cannot be explained. These are the flatness problem and the horizon

problem.

1.4.1 Flatness problem

We define a density parameter Ω = _ρρ

c, where the critical density ρc =

3H2

8πG. The

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expansion rate at that time. The value of ρc can be calculated for today from the

present value of H today and the density parameter Ω is close to 1. The dependence of Ω on the scale factor a is,

Ω − 1

Ω =

3k

8πGρa2 (1.36)

If energy density is dominated by matter then ρ ∝ _a13, and Ω − 1 Ω ∝ a2_. _(1.37)

If we go backwards in time the scale factor a will approach zero and Ω will approach 1, and going forward in time, Ω deviates from 1 (but is not exactly 1).

H2 = 8πGρ

3 (1.38)

To know the value of Ω at earlier time, we need to know when the transition from radiation domination to matter domination occurred. Which is very hard so we assume that the transition occurred at time tT at which _ρ_matterρrad = 1. And since

ρrad ∝ _a14 and ρm ∝ _a13 ρrad(t) ρm(t) = ρrad(t0) ρm(t0) a(t0) a(t), (1.39)

where t0 is the present time. Using a(t) ∝ t

2

3 during the matter dominated era we

get, ρrad(t) ρm(t) = 10−4 t0 t !2/3 (1.40) we take t0= 13.7 billion years then the above equation gives the transition time

tT = 1.3 × 104 years. Today Ω0 is between 0.1 and 2, thus,

Ω0− 1 Ω0 < 10 (1.41)

and hence during the matter dominated era, we have

Ω − 1 Ω = t t0 !2/3 Ω0− 1 Ω0 ! . (1.42)

and at the time of transition, this ratio is four orders of magnitude smaller, which implies, ΩT − 1 ΩT = 10−4 Ω0− 1 Ω0 ! (1.43)

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and in radiation dominated era, this ratio scales directly with t. At t = 1 in Planck units, Ωn− 1 Ωn = 2 × 10−16 Ω0− 1 Ω0 ! (1.44) which gives, Ω − 1 _t=1sec < 10−60 (1.45)

which is extremely small. This is a big puzzle, why the density parameter is so fine-tuned at the start to give us the density that we have today. There is no explanation in the standard cosmological model.

1.4.2 Horizon problem

The Horizon problem arises from the observed uniformity in the CMBR. The tem-perature of this radiation is isotropic in all directions in the universe to better than 1 part in 104. But, at the time of emission the regions in opposite directions in the sky were well out of causal contact from each other. Since nothing can travel faster than light, no physical process could have brought these two regions in (thermal) equilibrium. To be quantative, we define a Hubble radius as,

RH ≡

1

H (1.46)

The Hubble radius is a measure of causality i.e. if the particles are separated away with a distance more than RH, then they are causally disconnected. All the

in-formation that comes to us from the far edges of the universe travels at the speed of light. So, we can actually calculate the distance light traveled in a certain time period by integrating the action for a light path, ds2 _{= 0,}

l(t) = Z t 0 c a(t0₎dt 0 (1.47) Since the universe is expanding that means the distances are also expanding. It sounds strange but it is true that the space-time around us is expanding, we do not feel it because we are surrounded by high density e.g. Earth, Sun etc. Thus the physical distance can be written as,

lphy(t) = a(t) Z t 0 c a(t0₎dt 0 (1.48) Also called particle horizon. If two observers are separated with a distance more than the particle horizon it means that they were never in causal connection. Suppose

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universe went through radiation and matter dominated eras only, then we can find the particle horizon and Hubble radius from fig 1.1. We find that during matter dominated lphy ∼ 2RH whereas in radiation dominated era RH ∼ lphy. Now here

comes the puzzle, if the universe went through only these eras then particle horizon is equal (at the most, twice) the Hubble radius. With tls = 380000 years, the

time after big bang at which the CMB photons last scattered and t0 = 13.8 billion

years, the current age of the universe, we find that with redshift z ∼ 1100, the causally connected patch at the surface of last scattering to be extended at an angle Θ ∼ (1 + z)tls

t0 ∼ 1

0_{. In total there are 4π steradians∼ 40000 degrees on the sky.}

Which gives 40000 causally disconnected regions of sky when we look at the CMB. Then how it is possible that the temperature fluctuations of CMBR are of the order

δT

T ∼ 10

−5_{? In simple words, how can be the temperature of CMBR the same for}

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2 Inflation

Accroding to general relativity and the observational evidence of expanding universe, the universe started with a Big Bang and went through a certain evolution. The Big Bang itself is a singularity and we cannot predict the behavior of the universe at that point in space-time. The reason is that the entire universe was so small

O(lp = 1.6 × 10−35m) that quantum effects dominated the universe. Hence, close to

the Planck scale we need a quantum theory of gravity. However, general relativity becomes important at energies below the Planck scale with Mp = 2.4 × 1018 GeV.

The standard cosmological model (Λ − CDM ) described in 1.3 is in principle a good description of the evolution of universe. However, the cosmological puzzles i.e. the horizon problem and the flatness problem lack explanation within the standard model.

2.1 Inflation as a solution to cosmological puzzles

The first solution to the cosmological puzzles was proposed by Alan Guth in 1981[6]. A new era of rapid exponential expansion was added to the history of universe before the radiation dominated era. The idea is simple, that the universe grows rapidly to many orders of magnitude. Let us now look again at the problems described in 1.4 under the light of inflation.

2.1.1 Flatness problem revisited

With a non-zero curvature k 6= 0 and the FLRW metric the Friedmann’s first equa-tion can be written as,

H2 = 8πGN

3 ρ −

k

a2. (2.1)

We define a critical density5 _ρ

c as, ρc= 3H2 0 8πGN . (2.2)

Now we can rewrite equation 2.1 as, Ωtotal− Ωk =

ρ ρc

− k

H2_a2 = 1, (2.3)

where, H0 is the Hubble parameter today (73 ± 3 kms−1Mpc−1) and Ωk for today

is given by the measurements i.e. Ωk = _Hk2_a2 = 0.01 ± 0.02. We can see that Ωk is

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close to zero, that means that the energy density in our universe currently is close to the critical density. We want to probe the Friedmann equation at earlier times to see how the curvature term k behaves. From figure 1.1 we find that during the radiation and matter dominated eras, the scale factor evolves as Ha ∝ t−12 and

Ha ∝ t−13 respectively. Hence, we conclude that the curvature term k 2

H2_a2 was very

small in the early universe, which translates into the fine tuning problem. It cannot be a coincidence that the energy density of the early universe was extremely close to the critical density such that after all the evolution, we are left with a negligible curvature term.

During the inflationary phase, the scale factor a ∝ eH0t _{and the curvature term now}

evolve as _H2k_a2 ∝ e

−2H0t_{. This equation tells us that at earlier times the curvature}

term was much bigger such that the energy density could have been far different from the critical density and we do not face a fine tuning problem. To solve tht flatness problem 1.4.1 we need to have a minimum amount of inflation which can be parameterized by the number of e-folds N (t) as,

N (t) = ln a(t) ai ! = Z t 0 H(y)dy (2.4)

where ai is the scale factor at time t = 0. This equation is a measure of the amount

of inflation and in one e-fold the scale factor increases by a factor of e. To solves the flatness problem 1.4.1 with Ωk ∼ 1 at some point before inflation, N (t) = 70 is

required.

2.1.2 Horizon problem revisited

During Inflation the scale factor increases exponentially in time (a ∝ eH0t_{). The}

particle horizon during that phase is,

lphysical(t) = (eH0t− 1)

1

H0

. (2.5)

The horizon also increases exponentially with time while the Hubble radius RH = _H1₀

is constant with H0 = _a˙a = constant. Thus the particle horizon can grow to many

orders of magnitude larger that the Hubble radius and hence the particles cannot communicate now. Thermal equilibrium6 was reached before the inflation and then the patches of universe got causally disconnected in space-time due to inflation.

2.2 Dynamics of Inflation

Inflation is based on the idea that the universe undergoes an accelerated expansion (¨a > 0). This condition can be rewritten as,

 ≡ −

˙

H

H2 < 1. (2.6)

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From equation 1.33, we see that this condition further translates into,

ρ + 3P < 0. (2.7)

We need some sort of a source with negative pressure (P < −1₃ρ). This is possible

with a scalar field. Alan Guth used the idea of spontaneous symmetry breaking [6]. Initially the scalar field is in the minimum with lowest value at φ = 0 and as temperature drops, the second minimum (false vacuum) can take the value which is lower that the value at the true minimum. Thus the scalar field moves from the initial symmetric vacuum to the false non-symmetric vacuum at (φ 6= 0). The scalar field is pulled out of the false vacuum to the true vacuum state, and by doing so, it experiences negative pressure. Guth considered first order phase transition by introducing a barrier between the true and false vacuum. The problem with this model is that the bubbles of the true vacuum state cannot take over the exponential expansion of the universe, so the bubbles never unite and universe never reach the true vacuum. Albrecht and Steinhardt [7] and Linde [8] proposed another infla-tionary model where they assumed that the universe undergo a second order phase transition i.e. no bubbles are formed and the universe will move to the true vacuum state. The condition required for this to happen is that the potential needs to be flat and the field must slowly roll down the potential well. The slow roll inflationary model is the most widely studied and we will explicitly study the dynamics of this scenario.

2.2.1 Slow-roll regime

The simplest inflation model involve a single scalar field φ which depends on time only. Dynamics of this scalar field with the usual Einstein-Hilbert term is governed by the following action.

S = Z d4x√−gh1 2R + Lφ i (2.8) where, Lφ= − 1 2g µν_∂ µφ∂νφ − V (φ). (2.9)

With gµν _{the inverse metric tensor and V (φ) the potential, the equation of motion}

of the scalar field or the Euler-Lagrange equation is,

∂µ(δ( √ −g)L δ(∂µ_φ) ) − δ(√−g)L δφ = 1 √ −g∂ν( √ −ggµν_∂ µφ) − V,φ = 0. (2.10)

For FLRW metric with line element

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and

√

−g = a(t)3_, _(2.12)

the equations of motion reduce further to, 1

a(t)3∂ν(a(t) 3_gµν_∂

µφ) − V,φ = 0. (2.13)

Since φ depends on time only i.e. ∂ν = ∂0 = −∂t we are left with,

¨ φ + 3˙a a ˙ φ + V,φ = 0 (2.14) Slow-roll condition 1: ¨φ 3H ˙φ

For a flat and slow time varying potential, we assume ¨φ 3H ˙φ to get our first

simplified inflation equation from 2.14, 3˙a

a

˙

φ + V,φ = 0. (2.15)

To solve the system we need one more equation which we will get from the Friedmann equations 1.33. For that we need the explicit form of the energy momentum tensor,

Tµν = √2 −g

δ(√−g)L

δgµν

= ∂µφ∂νφ + gµνL. (2.16)

the µ = ν = 0 component is ρ(t) = 1₂φ˙2 _{+ V (φ(t)) and µ = ν = i component is}

P = 1₂φ˙2_{− V (φ(t)). Now by the slow roll condition 2.2.1 and by equation 1.36, the}

equation of state parameter becomes,

ω = P ρ = 1 2φ˙ 2_{− V (φ(t))} 1 2φ˙2 + V (φ(t)) ≈ −1. (2.17)

The condition ¨φ 3H ˙φ can be written in terms of potential by defining a slow roll

parameter η ≡ M2

P

|V00_|

V 1

Slow-roll condition 2: 1₂φ˙2 _{V (φ(t))}

At this point we assume another condition for flatness of the potential i.e. 1 2φ˙

2

V (φ(t)). Hence ω ∼ −1 which will indeed result in accelerated expansion.

Fried-mann equation 1.33 then becomes,

H2 =˙a

a

2

= 8πG

3 V (φ(t)) (2.18)

The condition 1₂φ˙2 _{V (φ(t)) can be written in terms of potential by defining a}

slow roll parameter ≡ MP2

2

V0

V 1. To visualize the accelerated inflation we will

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2.2.2 Case study with a Potential V (φ) = 1 2(

m

~) 2_φ2_(t)

The constraint equation 1.33 with this specific form of the potential is,

H2 = 8πG 3 1 2 m ~ 2 φ2(t) (2.19) which imply H = ± s 8πG 6 m ~φ(t). (2.20) We plug in equation 2.15 and solve for φ(t) to get,

φ(t) = φ0∓ m 3~ s 6 8πGt, (2.21)

where, φ0 = is an integration constant. We plug back this solution in 2.20 to get an

equation for the scale factor a(t),

H = ± s 8πG 6 m ~ [φ0− m2 3~2t] (2.22)

With positive values of H and H = ˙a

a the above equation can be written as,

˙a a = s 8πG 6 m ~ h φ0− m2 3~2 i t (2.23)

To solve for a(t), we assume an exponential solution as follows, let

a(t) = exp[−(k + λt)2] (2.24)

and,

˙a(t) = (−2kλ − 2λ2t)a(t) (2.25)

where k and λ are constants. We compare the assumed solutions with equation 2.23 to find the constants,

λ = m ~ √ 6 (2.26) and k = s 8πG 2 φ0. (2.27)

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Under the slow roll conditions 2.2.1 and 2.2.1, we define a slow roll parameter as,  ≡ 1 2φ˙ 2_(t) V (φ(t)) = 1 18H2_(t) V (φ(t))2 ,φ V (φ(t)) = 1 (6)8πG _{V (φ(t))} ,φ V (φ(t)) 2  1, (2.28)

and with the quadratic potential 2.2.2 it becomes,

 = 1 (6)8πG (m ~) 2_φ2_(t) 1 4 m2 ~2φ 4_(t) = 1 12πG 1 φ2_(t)  1. (2.29)

From the second slow roll condition φ(t)¨

3H(t) ˙φ(t)  1, we define another parameter η as,

¨ φ(t) 3H(t) ˙φ(t) = −1 9H2(∂ 2 φV + 3 ˙H) = −1 24πG ∂_φ2V (φ(t)) V (φ(t)) + 9 = η + 9  1, (2.30) where, H2 = 8πG₃ V (φ(t)), η = _24πG−1 ∂ 2 φV (φ(t)) V (φ(t)) and − ˙H(t) H2_(t) = 1 3 from equation 2.29.

With the quadratic potential 2.2.2 η becomes,

η = −1

12πG 1

φ2_(t) = −. (2.31)

Since 1, we neglect ₉, and by plugging the inflaton field solution 2.21 in η 1, we get s 1 12πG  φ0− m 3~ s 6 8πGt (2.32) which implies, t  3~ m s 8πG 6 φ0− ~ m √ 6 (2.33)

Only when less time has elapsed than this limit, the inflation will continue. The second term is negligible because φ(t) 1 by 2.29, which imply,

t  3~ m

s

8πG

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Inflation will only happen when this condition is satisfied and will end when the condition breaks which can be related to the slow roll parameters as (tend) = 1 [or

η(tend) = 1, if earlier gives explicit definition of tend]. With φ0 = φ(tend) we define

the time at which inflation will stop as,

tend≡ 3~

m

s

8πG

6 φ0 (2.35)

and use this value of t to rewrite a(t) in equation 2.24 as,

−(− √ 8πG 2 φ0+ m ~ √ 6t) 2 _{= −}8πG 4 φ 2 0+ s 8πG 6 m ~ φot − m2_t2 6~2 (2.36)

here m_6~2t22 is negligible so,

−(− √ 8πG 2 φ0+ m ~ √ 6t) 2 _{= −}8πG 4 φ 2 0+ s 8πG 6 m ~ φot (2.37)

Finally the scale factor becomes,

a(t) = e[−8πG4 φ20+ √ 8πG 6 m ~φ0t] (2.38)

which means a rapid exponential expansion!7_{. During Inflation φ field dominates}

and we imposed the slow evolution of φ field, effects are similar to dark energy8. We also assumed that the Kinetic energy density stays small for a long time so the inflation does not ends soon. The inflation is controlled by and η, where measures the slope of V (which should be shallow) and η measures the rate of change for slope of V.

7_{The scale factor grows exponentially.} 8_{Dark energy also has negative pressure.}

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Figure 2.1: A scalar field with an initial value slowly rolls down the potential well

We explicitly saw how the scalar field dynamics with a quadratic potential can lead us to an exponentially growing scale factor under the slow roll assumptions. We can do the same computation with different forms of potentials e.g. quartic potential or exponential potential etc. All these different potentials gives different predictions for the inflationary observables.

2.3 Experimental constraints on Inflation

Inflation, as we saw earlier is a hypothetical era in the early history of universe that was put forward to solve the cosmological puzzles. However, it not only solves those puzzles, but also predict new physical phenomena e.g. quantum gravity [9]. The idea is that during inflation, quantum fluctuations are generated in the scalar field and the metric. Some of these perturbations are frozen and the accelerated expansion amplifies the vacuum fluctuations. To constrain the inflationary models, we have to analyze the scalar metric perturbations and their imprint on the CMBR. The tool for this is the perturbative general relativity, see chapter 7.

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A free scalar field has an equation of motion, −φ = ¨φ + 3˙a

aφ −

∇2

a2φ = 0. (2.39)

Assume that the second term with H is small as compared to the spatial derivative, then we are left with a plane wave equation. We see that for sub-Hubble wavelengths (aH k), the field φk obeys harmonic oscillator equation in Fourier space and

fluctuates. On the other hand, if the damping term with H is larger than the spatial derivative, we can safely neglect the third term. In that case ˙φ ∝ Hφ and H2 k2

a2.

These wavelengths are called the super-Hubble wavelengths. Such a solution φk

is a constant plus a damping term. Now during inflation, there is an increase in the physical length scale due to the expansion, but the Hubble radius remains unchanged. Stated in other words, the sub-Hubble wavelengths can become super-Hubble during expansion. Amplitude of these fluctuations scales inversely with the physical length scale, 1

l. The case is different for the super-Hubble modes, where

the amplitude is amplified during inflation. These amplified scalar perturbations of the metric forms potential wells and seeds the large scale structure. The amplitude can be measured using the two-point function for the perturbations as,

hφkφk0i =

1

k3Ps(k)2π

3_δ3_{(k − k}0

), (2.40)

where Psis the power spectrum of scalar perturbations, a detailed calculation is done

in chapter 7. We evaluate the power spectrum right at the time when the quantum fluctuation becomes super-Hubble, also called Hubble-crossing i.e. k = aH. We then get, Ps= ∆2R k k0 ns−1 , (2.41) where ∆2_R= _24πV2_M4 p _k=k 0

is the amplitude of scalar perturbations at some reference wavelength k0, ns is the spectral index, V is the inflationary potential and is the

slow roll parameter.

CMBR measures the power spectrum. The inhomogeneity from the quantum fluc-tuations leave an imprint on the spectrum of CMBR. Furthermore we find a direct relation between the amplitude of scalar perturbations and the temperature as [16],

∆2_R∼δρ

ρ

2

(2.42) where ρ and δρ are the density and its perturbations, which are related to tem-perature as ρ ∝ T4_{. By comparing the fractional temperature perturbation in the}

CMBR [10], we find

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This number is measured by WMAP[10] with a pivot wavelength k0 = 0.002Mpc−1.

By equation 2.42 we find that for a quartic potential, δρ_ρ ∝ √λ, where λ is the

coupling constant in the potential. The amplitude puts a constraint on λ to be of the order O(10−14_{) for successful inflation.}

Furthermore, there are more parameters that constrains the inflationary potential. Similar to scalar perturbations, there is another quantity called the tensor perturba-tions which are generated in the metric and result in gravitational waves. A similar calculation constrains the power spectrum of tensor perturbations, PT ∝ knT (see

chapter 7). Inflation predicts small corrections to ns from unity which are of the

order of slow roll parameters and for tensor perturbations, it predicts nT = −2.

There is one more observable called the tensor to scalar ratio r found by taking the ratio of both the power spectrums. This ratio is approximately scale invariant and one finds the so called, consistency relation,

r = −8nT. (2.44)

We will see later that this relation breaks if we modify the structure of gravity e.g. Gauss-Bonnet gravity. Fig 2.2 shows the WMAP constraints [10] on a quartic and quadratic inflationary potentials with 1 and 2 σ confidence contours of ns and r.

Fig 2.3 shows the latest results of constraints on inflationary models from Planck [11].

Figure 2.2: WMAP constraints on spectral index ns and tensor to scalar ratio r at

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Figure 2.3: PLANCK constraints on ns and r in different inflationary models.

As we will see that Higgs potential is effectively of the form of λφ4 during inflation and we see that its predictions are well outside the WMAP measurements. The self-coupling of the Higgs should be of the order 10−14, which is in disagreement with the Standard Model Higgs boson (λ ∼ 0.1). However, we can add an extra interaction between the Higgs and gravity coupled by a non-minimal term to allow Higgs to be the inflaton.

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3 Non-minimal Higgs inflation

3.1 Non-minimal inflation

Non-minimal inflation corresponds to certain class of inflationary models in which we have an additional term in the inflation action. This term couples the scalar field to the Ricci scalar through a coupling constant. The action for such a theory can be written as, S = Z d4x√−gM 2 p 2 R − 1 2gµν(∂ µ_φ∂ν_{φ) −} 1 2ξφ 2_{R − V (φ)} (3.1) where, R is the Ricci scalar, φ is the inflaton field and ξ is a coupling constant that couples the scalar field with the Ricci scalar. If ξ 6= 0 then the coupling is said to be non-minimal, otherwise it is called minimal. This theory is similar to the Starobinsky’s R2 _{inflation [12] because it of the equivalence of non-minimal}

inflationary models and Brans-Dicke theories [13] (we will see it in chapter 4). Non-minimal inflationary models can be used to solve the graceful exit problem because

1 2ξφ

2_{R term can be interpreted as a correction to the Planck mass which is now time}

dependent and it can be showed that, a time dependent Planck mass effectively slows down the inflation by changing the potential from exponential to a power law [14]. Non-minimal inflation with a quadratic and a quartic potential is discussed in [15] where, it is showed that inflation is possible with a relatively small coupling constant

ξ ∼ 0.001. On the observational side, the power spectrum for the non-minimal case

is derived with a quartic potential in [16]. It was found that in case of a minimal coupling the amplitude of density perturbations scales as ∆2 ∼ δρ_ρ ∼ √λ for a λφ4

potential. whereas in case of a non-minimal coupling ∆2 ∼ qλ

ξ2. That means for

the standard inflation (ξ = 0) with a λφ4 _{potential, to get the successful inflation}

we need λ ≈ 10−14. However this condition is loosened by the non-minimal coupling because now λ can be as large as we want it to be by choosing appropriate ξ. This fact allows us to introduce Higgs boson as an inflation because the SM Higgs boson has a self-coupling λ ∼ 0.1, which is fine if it is coupled to gravity non-minimally and the non-minimal coupling constant is (negative) large. Higgs boson as an inflaton was first discussed by Shaposhnikov and Berzukov [17]. They used the trick of conformal transformation to write down an effective flat potential in Einstein frame which then lead to inflation. First we will discuss the dynamics of scalar field coupled to gravity and see how it is different from the standard case and then we will choose ξ such that the SM Higgs boson can be a possible candidate for inflation.

Let first now see explicitly how the scalar field inflation works with non-minimal coupling. The action is,

S = Z d4x√−gM 2 p 2 R − 1 2gµν(∂ µ φ∂νφ) − 1 2ξφ 2 R − V (φ) (3.2)

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In order to find the gravitational field equations, we vary the action with respect to the metric to find,

δS = Z d4x√−ghM 2 p 2 − 1 2ξφ 2i (Rµν− 1 2Rgµν) − 1 2ξ(−∇µ∇ν + gµνg ρσ_∇ ρ∇σ)φ2 − 1 2(∂µφ∂νφ) + 1 2gµν h1 2g ρσ_(∂ ρφ∂σφ)V(φ) i δgµν, (3.3) by imposing δS = 0, we get the field equations as,

0 = hM 2 p 2 − 1 2ξφ 2i (Rµν− 1 2Rgµν) − 1 2ξ(−∇µ∇ν + gµνg ρσ_∇ ρ∇σ)φ2 − 1 2(∂µφ∂νφ) + 1 2gµν h1 2g ρσ_(∂ ρφ∂σφ)V(φ) i . (3.4) futhermore by using (∇µ∇ν − gµνgρσ∇ρ∇σ)φ2 = 2φ(−∇µ∇ν+ gµνgρσ∇ρ∇σ)φ + 2[∂µφ∂νφ − gµνgρσ∂ρφ∂σφ], (3.5) we simplify to, (M_p2− ξφ2)Rµν− 1 2Rgµν = (1 − 2ξ)(∂µφ∂νφ) − gµν h1 2− 2ξ i gσρ∂ρφ∂σφ − gµνV (φ) − 2ξφ(−∇µ∇ν + gµνgρσ∇ρ∇σ)φ. (3.6)

Left hand side is the Einstein’s tensor multiplied by the effective Planck mass and right hand side corresponds to the energy momentum tensor for the scalar field. In case of the homogeneous and isotropic FLRW universe with φ = φ(t) and by using φ = − ¨φ − 3H ˙φ +_a12∂_i2φ and ∇µ∇νφ = ∂µ∂νφ − Γαµν∂αφ, the µ = ν = 0 component

of equation 3.6 gives us the constraint equation

H2 = 1 3 1 (M2 p − ξφ2) 1 2 ˙ φ2+ V (φ) + 6ξHφ ˙φ. (3.7) Similarly we find the µ = ν = i components of Einstein tensor gives us the energy constraint equation ˙ H = 1 (M2 p − ξφ2) h (−1 2 + ξ) ˙φ 2_{+ ξφ ¨}_{ξ − ξHφ ˙}_φi_. (3.8) Lastly, we can vary the action with respect to φ to get the equation of motion for field φ i.e.

¨

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where prime represents the derivative with respect to the field φ. A useful thing to do here is to represent R = 6( ˙H + 2H2_{) in terms of φ and ˙}_{φ by using 3.7 and 3.8.}

After doing so, we arrive at the final form of equation 3.9 ¨ φ + 3H ˙φ = (1 − 6ξ)ξφ M2 P − ξ(1 − 6ξ)φ2 φ2 + 1 M2 P − ξ(1 − 6ξ)φ2 h − 4ξφV (φ) − (M2 P − ξφ 2_)V0_(φ)i (3.10)

We note here that one can reduce this equation to a field equation for field φ in a flat space by a conformal transformation of the scalar field by scaling it with a(t). More importantly, there is a special value for ξ that is 1₆, for this value the Ricci scalar vanishes and and equation of motion for the scalar field takes a trivial form, see appendix A.2. Physically R = 0 means that the conformally coupled scalar field behaves as it is in a flat space-time and it does not feel the expansion. We now solve the equations of motion under slow roll conditions.

3.1.1 Slow roll conditions

So far we have three equations i.e. the constraint equation for H2 _{3.7, dynamical}

equation for ˙H 3.8 and the equation of motion for the scalar field φ 3.10. We write

these three equations again to keep track,

H2 = 1 3 1 (M2 p − ξφ2) 1 2 ˙ φ2+ V (φ) + 6ξHφ ˙φ, (3.11) ˙ H = 1 (M2 p − ξφ2) h (−1 2 + ξ) ˙φ 2_{+ ξφ ¨}_{ξ − ξHφ ˙}_φi , _(3.12) and ¨ φ + 3H ˙φ + ξRφ + V0(φ) = 0. (3.13) We now impose the standard slow roll conditions 2.2.1 and 2.2.1 which says that the potential energy dominates over kinetic energy and the field rolls down slowly to the potential well. Mathematically,

¨ φ 3H ˙φ, 1 2 ˙ φ2  V (φ). (3.14)

Let us introduce a quartic potential (because we are interested in Higgs potential) and we also assume a large negative non-minimal coupling −1 ξ,

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Under slow roll conditions 3.14 and large negative coupling, the background equa-tions 3.11 and 3.13 reduce further to,

˙ φ = − M 2 Pλφ 3Hξ(1 − 6ξ) (3.16) H2 = −λφ 2 12ξ => H = s − λ 12ξφ (3.17)

from here we solve for φ(t) to get,

φ(t) = φi− 4M2 P q − λ 12ξ (1 − 6ξ) t (3.18)

where φi is the initial value of the field. We also assume that φf  φi. Similarly we

solve equation 3.17 for a(t) to obtain,

a(t) = aie Hit− 2M 2_PH2_i (1−6ξ)φ2 i t2 (3.19) where ai is the initial scale factor at time t = 0. We can see that the scale factor

grows exponentially during inflation but this is not the only thing we want, we also want to have enough amount of inflation to solve the horizon and flatness problem and that can be obtained by the e-fold condition. We calculate the e-folds as,

N = Z tf ti Hdt = Z φ_f φi H ˙ φdφ = −(1 − 6ξ) 8M2 P [φ2_f − φ2 i]. (3.20)

We suppose that φf = 0 at the end of inflation. Now to solve the horizon and

flatness problem we need N ≥ 70. Hence, 70 ≤ −6ξ 8 h φ2_i M2 P i (3.21) which means that we can loosen the condition on φi more by choosing appropriate

ξ. For ξ = −103_{, the scalar field φ gets an initial value of φ}

i ≥ 0.3Mp. Hence we can

conclude that non-minimal inflationary models also solve the cosmological problems. Our motivation to put forward these models is of course to study Higgs inflation and see what are the bounds on ξ [18]. The above calculation is in Jordan frame. We will now do the computation in Einstein frame by a conformal transformation to see how it effectively transforms the inflationary potential which, later on shows up in the calculation of the amplitude of scalar perturbations and hence can be made fit to the data by choosing appropriate non-minimal coupling.

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3.2 Higgs Inflation

3.2.1 Motivation

The idea of getting inflation from Higgs boson was put forward by Shaposhnikov and Bezrukov [17]. Their motivation and in principle, a general motivation for doing so is because Higgs is the only known scalar particle in the Standard Model. If SM Higgs boson can provide successful inflation with the help of a non-minimal coupling, then we only have one free parameter ξ, which is economical. Otherwise, since inflation occurs due to the negative pressure generated by the scalar field during transition from false vacuum to the true vacuum, we have to experimentally pin down a new scalar particle in order to answer the questions. Furthermore, Higgs inflation results in the production of SM particles (fermions and bosons) during the post inflationary reheating and we do not need a new physical phenomena to get the SM particles from a hypothetical inflaton field.

3.2.2 Dynamics of Higgs Inflation

Higgs inflation is actually an inflationary model with a non-minimal coupling same as we discussed earlier. However, the calculations are involved because we have to go to the Einstein frame through a conformal transformation to extract the physics. Let us see how this work.

Standard Model Higgs boson is a doublet that transforms under SU (2) gauge group9_.

H = φ

0

φ†

!

(3.22) where φ0_{and φ}†_{are complex fields. H becomes a real scalar field in the unitary gauge}

with a vacuum expectation value hφi = v and then the Higgs potential becomes the

Mexican hat potential.

H = φ √ 2 0 ! (3.23) and, V (φ) = 1 4λ(φ 2_{− v}2₎2 _(3.24)

We already saw in the previous section that the initial value of the field is of order (O(MP)) 10, whereas v is much smaller. Hence we can safely ignore it. Then we

are left with a rather simple Higgs potential which is a quartic potential with a coupling constant λ. For the Higgs boson of mass 125 GeV that is found in LHC

9_{Special Unitary Group of 2 × 2 matrices}

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[3] have the self-coupling λ ∼ 0.1. However, for a scalar field inflation without a non-minimal coupling (Linde’s model [3]), the coupling constant of the scalar field should be λ ∼ 10−14 to get the correct amplitude of density perturbations. The new thing about the non-minimal models is that we can add a new parameter i.e. coupling between Higgs and the Ricci scalar and tune the new coupling constant such that inflation happens with the SM Higgs boson. The action for such a theory after the unitary gauge fixing is,

S = Z d4x√−ghM 2 p 2 − 1 2ξφ 2i R − 1 2gµν(∂ µ_φ∂ν_{φ) −} λ 4φ 4 (3.25) To get the observational constraints on inflationary potential we have to compute the power spectrum and the amplitude of the scalar perturbations. Recent work has been done to find out the observables in the non-minimal case while staying in Jordan frame [17]. However, it is easier to compute the power spectrum if one uses the Einstein frame. It is a matter of choice here because physical observable are frame independent. Conformal transformation removes the coupling between the scalar field and gravity. As we will see later that in case of Gauss-Bonner modification to GR, it is not possible to go to Einstein frame because of squared curvature terms in the action. Here we pass to the Einstein frame through a conformal transformation, a detailed calculation of the conformal transformations is done in the appendix A.2. We write our action as,

S = Z d4x√−gM 2 p 2 Ω 2_{R −} 1 2gµν(∂ µ_φ∂ν_{φ) −} λ 4φ 4 (3.26) where we absorbed the interaction term in Ω as,

Ω2 = M 2 P − ξφ2 M2 P . (3.27)

This action simplifies further by doing a conformal transformation to the new metric ¯

gµν = Ω2gµν. Using this new metric with metric compatibility i.e. ¯∇βg¯δγ = 0 and

the identity ¯∇αΩ = ∂αΩ (since Ω is a function of scalar quantities), our action

simplify to, S = Z d4x√−¯gM 2 p 2 ¯ R −h(M 2 P + 6ξ2φ2 M2 PΩ4 i1 2g¯µν(∂ µ φ∂νφ) − Ω−4λ 4φ 4 _(3.28)

We see that this looks like a standard inflation action but with a complicated canon-ical term for the scalar field. We can re-define this field to make the kinetic term canonical as, −1 2g¯µν(∂ µ_ψ(φ)∂ν_{ψ(φ)) = −}1 2 dψ dφ 2 ¯ gµν∂µφ∂νφ (3.29)

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with, dψ dφ = v u u t M2 PΩ2+ 6ξ2φ2 M2 PΩ4 . (3.30)

With this re-scaling of the scalar field our action becomes canonical in the field ψ,

S = Z d4x√−¯gM 2 p 2 R −¯ 1 2g¯µν(∂ µ_ψ∂ν_{ψ) − Ω}−4λ 4φ(ψ) 4 (3.31)

Large field approximation

In the large field approximation we take the limit,

φ  v u u t M2 P −ξ (3.32)

with this limit one can find that Ω2 _{∼ −}ξφ2 M2

P

, and we solve the differential equation 3.30 for ψ to get, ψ = q 6M2 P ln h √ −ξφ MP i (3.33) Large field approximation holds when φ > 0.1MP and for a typical value for ξ = 104,

we have, φ(ψ) ∼ MP ξ e ψ √ 6MP (3.34)

The large field approximation translates into ψ q6M2

P for the new field. Similarly

under the approximation, the effective Higs potential takes the form,

V (ψ) = λM 4 P 4ξ2 ₁ 1 + e h −2ψ √ 6MP i 2 (3.35)

Large field limit is also called the chaotic inflation and in this case the potential above is the effective Higgs potential for inflation.

Small field Approximation

In this case we assume that φ 

r

M2

P

−ξ. With ξ = −10

4 _{this approximation holds if}

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transformation is trivial i.e. Ω = 1, which means that ψ = φ. Therefore the inflationary potential is still the Mexican hat potential for field ψ,

V (ψ) = λ

4(ψ

2_{− v}2₎2_, _(3.36)

which is simply a quartic potential since v is much smaller than Planck scale.

3.3 Experimental constraints on Higgs Inflation

Following the calculation done in [16] for the power spectrum in a non-minimal theory, one finds that the fractional density perturbations depends inversely on ξ,

δρ ρ ∝

s

λ

ξ2. (3.37)

CMBR gives us a number for the fractional density perturbation which is δρ_ρ = 10−4. To agree with the experimental result, one finds that in order for SM Higgs boson (with λ ∼ 0.1) to be the inflaton, ξ should be ξ ∼ −104_.

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Figure 3.1: WMAP constraints on ns and r in different inflationary models at a

reference wavelength k = 0.002Mpc−1. λφ4 _{inflation potential is excluded with 2σ.}

With non-minimal Higgs inflation ξφ2R with SM Higgs potential (green box), ns

and r are inside the 2σ contour.

The story uptil here is that we successfully got inflation by allowing a non-minimal interaction between Higgs and gravity with just one free parameter ξ. Stan-dard Higgs inflation discussed in 3.2 raises the question if there are other couplings to gravity possible that allow the Standard Model Higgs field to be the inflaton. We noted that the Einstein-Hilbert action is not unique, and not completely fixed by some postulates or experimental measurements, as discussed in chapter 4. This further motivates to generalize the fundamental action for gravity, and study its consequences. One special topologically invariant generalization of general relativ-ity is Gauss-Bonnet gravrelativ-ity discussed in 4.3. This theory is special because the action contains fourth order derivatives of the metric, but the equations of motion are nevertheless the same as Einstein equations. This is because the Gauss- Bonnet term is a boundary term in 4 dimensions. However, the Gauss-Bonnet term becomes non-trivial if it is coupled to a scalar field, and thus forms a scalar-tensor theory. Let us first study the non-uniqueness of Einstein-Hilbert action and its possible gen-eralizations i.e. either we will replace the Einstein-Hilbert term with something else or we will add corrections to it. We will see that first of all, inflation is possible

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with changing the structure of gravity e.g. Starobinsky’s case [12]. Secondly we will do the non-minimal inflation with a more generalized theory of gravity which is Gauss-Bonnet gravity and see if Higgs inflation works with that theory. We now start with the modifications of gravity in the next chapter.

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4 Modified Gravity

4.1 Motivation

Many modifications to General Relativity (GR) have been proposed since the de-velopment of Einstein’s GR. Most of them do not fit with the observations and are mathematically complex. There are many reasons to change the structure of gravi-tational theories. Firstly, GR cannot be quantized, hence, it is an effective classical theory. Secondly, there is no argument that forces the Einstein-Hilbert action to be the fundamental action of gravity. In other words, the Einstein-Hilbert action is not the only one which will result in field equations which are second order in derivatives of the metric. However, this is only true for metric theories, i.e. the only independent field in the action is the metric tensor. We can write many actions and still get the same Einstein equations by varying it. So why not explore the whole parameter space? Or is the Einstein-Hilbert action fundamental?

The non-uniqueness of the Einstein-Hilbert action can be proved mathematically using Lovelock’s Theorem.

Lovelock’s Theorem and non-uniqueness of the Einstein-Hilbert action

We can write a generic action for the gravity that depends on the metric and its derivatives up to second order

S =

Z

d4x√−g(gµν, ∂αgµν, ∂β∂αgµν) (4.1)

By varying with respect to the metric we get,

δS = Z d4x√−gδgµνYµν[L] (4.2) where, Yµν[L] = d dxα − d dxβ h ∂L ∂(∂β∂αgµν i + ∂L ∂(∂αgµν) − ∂L ∂(gµν) (4.3) The equations of motion are given by,

Yµν[L] = 0 (4.4)

According to the Lovelock’s theorem [1], the Einstein equations are the only field equations that can be obtained from such a generic action that contain at the second-order derivatives of the metric tensor.

0 = Yµν[L] = √ −g nRµν− R 2g µν_{+ Λg}µν _(4.5)

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where, n and Λ are constants. This theorem is very important because it not only says that the only field equations we will get are second order in derivatives of the metric but also that those equations are actually Einstein equations. Hence the Einstein-Hilbert action is not unique.

SEH =

1 2k

Z

d4x√−gR (4.6)

This is the simplest action (linear in R) that we can write to ge the Einstein equa-tions. But, in principle any action which has the form

S =

Z

d4x√−gL (4.7)

with, L = nR − 2Λ + bG + dI where, b and d are constants and G = R2− 4Rµν_R µν+

Rµνρσ_R

µνρσ is called the Gauss-Bonnet term will result in equations of motion which

are second order in derivatives of the metric. The last term I is an integral over space-time called the index and is equal to, I = µνρσRαβ_µνRαβρσ. These terms do not

contribute in the field equations because,

Yµν[G] = 0 (4.8)

and,

Yµν[I] = 0 (4.9)

The first is true only in four dimensions or less, the second holds universally in any dimension. These terms are topologically invariant which means that by choosing appropriate constants when integrating over space-time, they cancel each other for a given manifold.

So, to get field equations which are not the Einstein equations we should either allow higher than second order derivatives of the metric in the action and/or take into account additional field(s) that are the dynamical variables in the action and/or work in higher (or lower) space-time dimensions and/or write down a completely new description of the gravitational interaction. However, in this thesis we will only focus on higher order derivatives of the metric tensor and adding new dynamical fields in the action.

4.2 f (R) Gravity

The simplest extension to general relativity is to promote the Ricci scalar in the Einstein-Hilbert action to a function of R i.e. R → f (R). This theory has been studied extensivley and further discussions and implications can be found in [19]. Observational implications for such a theory are discussed in [20]. Our goal here is to study f (R) gravity and see how it modifies the structure of gravity and the background equations i.e. the Friedmann equations. We can then replace f (R) with any function of R e.g. R2 _{or R + βR}2 _{to study the inflationary dynamics under the}

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4.2.1 Gravitational field equations

The action of f (R) theory is,

S = 1

2k

Z

d4x√−gf (R) + Sm (4.10)

where k = 8πG and Sm is the matter action. We vary this action with respect to

the inverse metric to get the field equations, 0 = δS = 1 2kδ Z d4x√−gf (R) + δSm = 1 2k Z d4x√−g[−gµν 2 f (R)δg µν_{+ f} ,RδR] + Z d4xδ √ −gLm δgµν δg µν (4.11)

where f,R = df (R)_dR . For a generic field A we have,

AδR = ARµνδgµν + A(gµν − ∇µ∇ν)δgµν (4.12)

We have to calculate AδR. The first term is the usual Ricci scalar while for the second term we use,

A(gµν − ∇µ∇ν)δgµν = AδR − ARµνδgµν = δgµν((gµν − ∇µ∇ν)A) (4.13) Therefore, Z d4x√−gA(gµν − ∇µ∇ν)δgµν = Z d4x√−gδgµν_((g µν − ∇µ∇ν)A) (4.14) and finally, Z d3x√−gAδR = Z d3x√−gA(Rµν + gµν − ∇µ∇ν)δgµν = Z d3x√−gδgµν_(R µν+ gµν − ∇µ∇ν)A (4.15)

Now we set A = f,R and plug back in 4.11 to get the gravitational field equations,

0 = 1 2k Z d4x√−g[−gµν 2 f (R)δg µν + δgµν(Rµν + gµν − ∇µ∇ν)f,R] + Z d4xδ √ −gLm δgµν δg µν (4.16)

We identify the last term as the stress energy tensor for the matter field defined as,

Tµν = − 2 √ −g δSm δgµν = − 2 √ −g δ(√−gLm) δgµν (4.17)

(40)

Hence, 0 = 1 2k Z d4x√−gδgµν_[−gµν 2 f (R) + (Rµν+ gµν − ∇µ∇ν)f,R] − Tµν 2 (4.18)

which implies the following field equations [19], −gµν

2 f (R) + (Rµν + gµν − ∇µ∇ν)f,R= kTµν (4.19) It is important to note that when f (R) = R, we are back to General Relativity 1.16.

4.2.2 Modified Friedmann equations in f (R) theory

For a flat (k = 0) universe, the FLRW metric is,

ds2 = −N2dt2+ a2(t)(dx2+ dy2+ dz2) (4.20) Where, a(t) is the scale factor and N is a gauge that we put equal to 1 at the end [19]. Since we are varying the action with respect to gµν which has components,

g00 = −1 and gij = a2δij, where

√

−g = a3_{, we can write down the action and its}

variation in terms of components of metric tensor and its variation respectively as,

S = 1 2k Z d4x√−g[f (R) + Lm] = 1 2k hZ d4xa3f (R(a)) + Z d4x√−gLm i (4.21)

The variation then becomes,

δS = Z d4xh3a2f δa + a3f,RδR − ka3Tµν δgµν δa δa i = 0 (4.22)

where the stress energy tensor can be written as,

δ√−gLm δa = δ(√−gLm) δgµν δgµν δa = −√2 −g δ(√−gLm) δgµν ( −a3 2 ) δgµν δa = −a 3 2 Tµν δgµν δa (4.23) δ√−gLm δa = − 2 √ −g δ(√−gLm) δgµν ( −a3 2 ) δgµν δa = − a3 2 Tµν δgµν δa (4.24)

Modified Gravity and Higgs Inflation

University of Amsterdam

National Institute for Subatomic Physics

NIKHEF

Institute of Physics

IoP - UvA

Master Thesis

Modified Gravity and Higgs Inflation

Author:

Shahzeb Kamal

Supervisor:

Dr. Marieke Postma

Examiner:

Prof. Dr. J. P. van der Schaar

Abstract

Dedication

Contents

Acknowledgement

1

FLRW Cosmology

1.1

General Relativity

1.2

Cosmological principle

1.3

Expanding Universe

1.4

Problems with Cosmology

2

Inflation

2.1

Inflation as a solution to cosmological puzzles

2.2

Dynamics of Inflation

2.3

Experimental constraints on Inflation

3

Non-minimal Higgs inflation

3.1

Non-minimal inflation

3.2

Higgs Inflation

3.3

Experimental constraints on Higgs Inflation

4

Modified Gravity

4.1

Motivation

4.2

f (R) Gravity