Renormalization Group Equations and
Predictions for New Higgs Inflation
by
Melvin van den Bout (10545069)
Master Project Theoretical Physics and Astronomysize 60 ECTS
Supervisor:
Dr. Marieke Postma
Second assessor: Dr. Jan Pieter van der Schaar
in the
FNWI-IoP
Institute for Theoretical Physics Amsterdam Cosmology Group
Nikhef Theory Group
Abstract
The main theme in this thesis is renormalization group equations(RGE) and predictions for new Higgs inflation(NHI). Besides this, we study the RGE and predictions for Higgs inflation(HI) as well. HI is defined with a non-minimal coupling with the Ricci scalar, whereas NHI is defined with a non-minimal coupling with the Einstein tensor. The main difference is that in HI the quantum corrections approximately do not enter in the predictions for the spectral index ns
and tensor-to-scalar ratio r, whereas for NHI the predictions are sensitive to the running of the couplings. Furthermore, we study the relevant background knowledge needed for (N)HI, such as cosmological inflation, modified gravity, Einstein/Jordan frame transformations, non-canonical inflation and the covariant multifield formalism.
New Higgs Inflation
We study original NHI and three variations to the model where the gauge bosons and fermions are non-minimally coupled with the Einstein tensor. We calculate the one-loop RGE for NHI in the Standard Model. Quantum perturbations are studied with a covariant multifield formalism, with which we calculate the one-loop corrections for the self-energy of the propagators. From this we extract the relevant counterterms and hence the beta functions. We separately study quantum corrections in two field regimes as an effective field theory. We parameterize our ig-norance of an ultraviolet completion at the boundary of the two field regimes with threshold corrections. We conclude that the gauge boson mass is heavy for original NHI in the large field regime and is integrated out. Furthermore, we show that the inflationary predictions for NHI are sensitive to the running of the standard model parameters. We apply the RGE to the predictions for NHI and model the effects of quantum corrections numerically in a {ns, r}-plot.
Ultimately, we provide a link between collider observable at the the large hadron collider and cosmic microwave background radiation observables. Explicitly, the predictions for ns and r,
depend on the Higgs and Yukawa top mass at the boundary of the field regimes, up to ignorance due to threshold corrections.
In this thesis we provide a link between two important theories in physics with different energy scales. The first theory is cosmological inflation, which is often referred to as the “bang” of the big bang. Cosmology is defined as the study of the origin and evolution of the universe, where the universe refers to all of space and time. This universe as a whole is governed by gravitational interactions, and can hence be studied with the the theory of general relativity. Inflation is a period of rapid exponential expansion of the early universe. The energy scale of inflation is just below the Planck scale Mpl∼ 1 × 1019GeV, which means that we can use perturbation theory
to describe the interaction of gravity. The quantum fluctuations during inflation left an imprint in the universe as the cosmic microwave background radiation (CMB). The CMB originates from 380.000 years after the big bang, whereas our universe is approximately 13.8 billions years old. The properties of the CMB are detectable, and thus have information available about the very early universe. The CMB provides a link between inflationary theories and experiment, such that the theory becomes empirical.
The second theory, the standard model(SM) of particle physics, describes particle processes at a quantum level. This theory has been confirmed by collider experiments such as the Large Hadron Collider(LHC) at CERN, which operate at ∼ 1 × 103 GeV. Within the SM, we have the Higgs mechanism, which gives mass to the other particles. The corresponding Higgs particle is the only experimentally observed particle with a quantum scalar field. The SM describes three out of the four forces of the universe, the electromagnetic, weak and strong force. The fourth force, gravity, is neglectable in LHC experiments, and is not incorporated in the SM. The energy scale at which gravity becomes important is the Planck scale Mpl, but this energy scale is far
beyond the reach of modern collider experiments.
Inflation can be described by a quantum scalar field with certain properties. The idea is that instead of building an inflation model with new fields and particles, we extend the Higgs mech-anism such that it can provide a period of (slow roll) inflation in the early universe at high energies. We incorporate inflation in the SM and preserve the ”mexican hat” potential at lower energies. We calculate quantum corrections to the theory and conclude that two possible mod-els of Higgs inflation can work under certain conditions. Ultimately, we provide a link between collider observables at the LHC and CMB radiation observables.
Acknowledgements
I would like to express my sincere gratitude to my supervisor Dr. Marieke Postma for guiding me throughout this project. I feel very fortunate to have learned from you. I am grateful for the opportunity to work on this project and to work in collaboration with you and Jacopo Fumagalli. Jacopo, thank you as well for pleasant collaboration. Dr. Jan Pieter van der Schaar, thank you for being second assessor. Sander Mooij, thanks for looking over the presentation and for providing personal notes for Higgs Inflation. Furthermore, I would like to thank my fellow students for the help and fun throughout the last five years.
On a more personal level, I would like to thanks my parents, Bas & Anja, for financial support and dedication for my well being. To my brothers; Bastian, thanks you for your enthusiasm for physics, which has probably made me choose for physics in the first place; Martijn, thank you for the opportunity to work for you during my study. To my lovely wife, Matthea; the last shall be first. Your presence and our marriage is such an enrichment to my life. Without you, without love, I am nothing.
Abstract i
Popular Scientific Abstract ii
Acknowledgements iii
Physical Constants vii
1 Introduction 1
2 Gravity & Cosmology 3
2.1 Dynamics of the Universe . . . 4
2.2 Single Scalar Field Inflation . . . 5
2.3 Connection to Observations . . . 8
2.4 Case Study: φ4 Inflation . . . 10
3 Gravity & Effective Field Theory 12 3.1 Effective Field Theory . . . 12
3.1.1 Cutoff scale . . . 13
3.1.2 Perturbative Quantum gravity . . . 13
3.2 Modified Gravity . . . 14
4 Covariant Multifield Inflation in the Einstein Frame 16 4.1 Conformal Transformation: Single Field . . . 17
4.2 Conformal Transformation: Multiple Fields . . . 19
4.3 Non-canonical Multi-field Inflation . . . 20
4.4 Covariant Multi-field Formalism. . . 22
5 Higgs Inflation: Classical Analysis 24 5.1 Higgs Inflation in the Einstein Frame . . . 24
5.2 Predictions for Higgs Inflation . . . 25
5.3 Higgs & Starobinsky Inflation . . . 28
6 Higgs Inflation: Quantum Analysis 30 6.1 Higgs Inflation as an Effective Field Theory . . . 30
Contents v
6.2 U(1) Abelian Higgs Inflation Model. . . 32
6.3 Interaction Lagrangian in Covariant Multifield Formalism . . . 36
6.4 Feynman Rules & Mass . . . 39
6.5 Renormalization . . . 42
6.5.1 Counterterms . . . 42
6.5.2 Coleman-Weinberg Potential . . . 43
6.6 One-loop Corrections for Higgs & Goldstone Propagator . . . 44
6.7 Beta Functions for U(1) Higgs Inflation Model . . . 45
6.8 Extension to SU (3)c⊗ SU (2)L⊗ U (1)Y Higgs Inflation . . . 48
6.9 Ultraviolet Sensitivity of Higgs Inflation . . . 49
7 New Higgs Inflation 51 7.1 New Higgs Inflation for Standard Model in the Einstein Frame . . . 51
7.2 Predictions: Classical Analysis . . . 55
7.3 Predictions: Quantum Analysis . . . 56
7.4 U(1) Abelian New Higgs Inflation Model . . . 58
7.5 Case A: Quantum Analysis . . . 61
7.5.1 Full Theory in Covariant Formalism . . . 61
7.5.2 Covariant Kinetic Terms. . . 62
7.5.3 Renormalization . . . 64
7.5.3.1 Counterterms. . . 64
7.5.3.2 Coleman-Weinberg Potential . . . 65
7.5.4 One-loop Corrections to the Propagators . . . 65
7.5.5 Counterterms, Beta Functions and Anomalous Dimension . . . 66
7.5.6 Results: RGE for U(1) Model and Standard Model . . . 69
7.5.7 Running of Couplings . . . 70
7.6 Case B: Quantum Analysis . . . 74
7.6.1 Covariant Multifield Expansion . . . 75
7.6.2 Interactions for the Self-energy . . . 76
7.6.3 Counterterms & Beta Functions . . . 77
7.6.4 Results & Running for Standard Model . . . 78
7.7 Case C & D . . . 81
7.8 Discussion . . . 82
8 Summary 84
A Higgs Inflation: Couplings in Covariant Multifield Formalism 86
B New Higgs Inflation Case A: Couplings in Covariant Multifield Formalism 88
C New Higgs Inflation Case B: Couplings in Covariant Multifield Formalism 90
D New Higgs Inflation Case B: Gauge Derivative Loop Diagrams 92
E Dirac Fermions; Weyl Basis and Conventions 93
G One-loop Corrections for Golstone Self-energy 100
Physical Constants
Natural units c = ~ = kB = 1
Reduced Planck mass Mpl−2 = 8πG = 2.4 × 1018GeV2 Metric signature = (−1, 1, 1, 1)
Space-time index = µ,ν = 0,1,2,3 Space indexes = i,j,k = 1,2,3
Field space indexes = I,J,K,.. = 1,2,3,.. N
Introduction
The first theories of cosmology were mythological in kind and date back to 3000BC. Cosmology was rationalized around 300-400BC by the ancient Greeks, such as Plato and Aristotle, and cosmology became more philosophical in kind. Copernicus placed the sun and planets in the right order with his heliocentrism model in 1543. And Kepler was able to describe the planetary system in a mathematical manner in 1619 with his three laws of planetary motion. But since Newton discovered the three laws of gravity in 1687, we could actually studying the universe at larger distances scales in a mathematical manner. The next revolution in cosmology was the discovery of general relativity(GR) by Einstein in 1915. Einstein’s theory of gravity replaced not only Newtonian physics at larger distances scales, but also improved on the calculation in the relativistic limit. Another revolution in cosmology was the discovery by Edwin Hubble in 1929 that the universe is expanding [1]. It did not take long for scientist to realize that if we rewind the expansion of the universe and go back in time, we would get a hot and dense place and even-tually a singularity. This is the begin of the big bang theory, with as foundation GR. Nowadays, cosmology is defined as the study of the origin and evolution of the universe. This universe is broader than the observable universe, and includes all of space and time, which might be infinite. Modern cosmology is centered around two theories. The first, general relativity, describes the interaction of gravity in a curved space-time due to energy and matter sources, and can be applied at the largest distance scale. The second, quantum field theory(QFT), is a theory for the very small, and is a framework to study the quantum behaviour of particle processes. For example, the standard model is a QFT which describes particles processes of the three other forces of nature, namely the electromagnetic, weak and strong force. Gravity is neglected in the SM since quantum gravity is non-renormalizable. Collider experiments, such as the Large Hadron Collider(LHC), verified the SM up to energies scales of ∼ 1 × 103GeV. Gravity is ne-glectable at this energy scale. The energy scale at which gravity becomes important in an QFT is the Planck scale. But this energy scale is far beyond our reach in collider experiments such as the LHC.
Physical Constants 2
The description of the universe on large distances scales requires both GR and QFT. It turns out that the small temperature fluctuations of the cosmic microwave background(CMB) can be explained by quantum fluctuations during inflation. The intrinsic details of these temper-ature fluctuations are described by a power spectrum, which can be measured by theory and experiment. This link between theory and observation is crucial to make the theory empirical. Moreover, it provides an opening to test physics at the energy scale of inflation. This energy scale is probably much higher than the energies which can be reached in the near future at the LHC. Cosmology is not only interesting to satisfy our desire of curiosity about the universe, but also gives an opportunity to probe the energy scale of inflation.
In this thesis we calculate the predictions for two models of inflation: ”Higgs inflation(HI)” and ”new Higgs inflation(NHI)”. These two models differ in their coupling to gravity. The prospects of including inflation in the SM, as these models do, is that it might provide a link between observables at the LHC and CMB. Furthermore, with the Higgs scalar field as the in-flaton field, we minimize the setup. Moreover, the Higgs scalar field is the only experimentally discovered scalar field, as the Higgs particle has been observed at the LHC. The ultimate goal of this thesis will be to understand how the parameters in the model change with energy, which is a pure quantum effect, and how this affects the predictions for inflation.
We start this thesis by an introduction to gravity and cosmology in chapter 2. We will then go on to the study of inflation for modified gravity in the Einstein frame in chapter 3and 4. This gives a sufficient basis to study (new) Higgs inflation classically. We extend on this by studying the full renormalization group equations(RGE) and running for HI in chapter 6. This same method for HI will be applied to NHI to find new results in section7. For HI and NHI we cal-culate the one-loop renormalization group equations(RGE) in the SM. Quantum perturbations are studied with a covariant multifield formalism, with which we calculate the one-loop correc-tions for the self-energy of the propagators. From this we extract the relevant counterterms and hence the beta functions. We expand the theory in the asymptotic regions as an effective field theory(EFT). We parameterize our ignorance of an ultraviolet(UV) completion with threshold corrections at the boundary of the two field regimes.
Gravity & Cosmology
Cosmology studies the evolution and dynamics of the universe at the largest distance scale. It turns out that gravity is the dominant force at the largest distance scale, but it is not a priori clear why the other three forces of nature can often be neglected. The weak and strong force only have a limited spatial reach, to a relatively small distance scale, and can at the largest distance scales be neglected. The electromagnetic force however, does have an infinite range, and does contribute at the largest distance scales. But the universe in total is neutrally charged and we can also neglect the electromagnetic force when studying the universe at the largest distance scale. The range of gravity is infinite and we can approximate gravity to be most relevant force at the largest distance scale.
The second ingredient for a theory of cosmology is the expansion of the universe. Much of modern cosmology is centered about the expansion of the universe. In 1929, Edwin Hubble discovered that the universe is expanding [1]. It did not take long for scientist to realize that if we rewind the expansion of the universe and go back in time, we would get a hot and dense place and eventually a singularity. This is the central idea behind the big bang.
In this chapter we will study the dynamics and evolution of a universe with only gravity and a scalar field. We will rewind time and study an early period of accelerated expansion of the universe. We will show how this period, known as inflation, can be explained with a scalar field. Lastly, we give a overview of how early quantum fluctuations in the scalar field can explain the inhomogeneities in the cosmic microwave background radiation.
In my previous thesis on eternal inflation [2], I have already written some introduction to cosmology. I will repeat the important calculation here and focus on slow roll inflation and the inflation parameters. Details for this chapter are taken from the TASI Lectures on Inflation 2009 [3].
Physical Constants 4
2.1
Dynamics of the Universe
We can describe the interaction of gravity in the universe by a geometrical approach with the use of general relativity(GR). For this we need to describe the curvature of the manifold, which we can describe by defining the space-time metric gµν. For our universe, a 3+1 space+time
dimensional metric. There is good evidence that our universe on the largest scales is isotropic and homogeneous [4]. Such a metric is known as the FRWL metric, for which the line element and metric for a flat universe are given by
ds2 = gµνdxµdxν = −dt2+ a(t)2dx2+ dy2+ dz2 gµν = diag n −1, a (t)2, a (t)2, a (t)2 o , (2.1) where a(t) is the scale factor and characterizes how the universe expands trough time. Because we have good evidence that the universe is expanding, we must distinguish between the comoving distance and physical distance. The scale factor at current time is normalized sucht that a(t = t0) = a0 = 1 and the scale factor at the singularity of the universe is given by a(t = 0) = 0.
The physical distance between two points is given by dphysical= a(t)d, where d is the coordinate
distance. From the scale factor we can define the Hubble parameter H which describes the expansion rate of the universe as
H = ˙a
a (2.2)
where the dot means a derivative with respect to time. The age of the universe is given by the Hubble time t0 ∼ H−1. The observable universe is characterized by the comoving Hubble
radius 1/ (aH). The relevant GR tensors for this FRLW universe are given by: Γ0ij = a ˙aδij R00= −3 ¨ a a R = 6 a2¨aa + ˙a 2 Γij0 = ˙a aδ i j Rij = ¨aa + 2 ˙a2 δij √ −g = a3
where R is the Ricci tensor, R the Ricci scalar and Γ the christoffel symbols. We will apply these tensors and scalars to the Einstein equation to obtain a differential equation for the universe. The Einstein equations describe the interaction of gravity in a curved space-time due to energy and matter sources. For our purpose it is useful to consider the Lagrangian formulation of GR, since we will extend on this in later chapters. The Einstein-Hilbert action contains the Einstein field equations in vacuum and is given by
SEH = Z d4x√−g 1 2M 2 plR − Λ + Lmatter , (2.3) where Λ is the cosmological constant. The field variable is the space-time metric gµν. Varying
Rµν−
1
2Rgµν+ Λgµν = 1
Mpl2Tµν, (2.4) where Tµν is then energy momentum tensor. It is common in GR to model the energy
distri-bution as a fluid with density ρ and pressure p. We will assume a perfect fluid, for which the energy momentum tensor is given in the rest-frame by
Tνµ= diag {−ρ, p, p, p} . (2.5) We would like the energy to be conserved and not ”leak” away, which can be defined as
∇µTµν = 0, (2.6) such that we can find the continuity equation as
˙
ρ = −3˙a
a(ρ + p) . (2.7) This relation between the energy density, pressure and scale factor can be solved if we assume a linear equation of state as p = ωρ. The continuity equation is for a linear equation of state given by
ρ ∝ a−3(1+ω). (2.8) We can consider different types our sources in the universe such as matter, radiation or vacuum to solve for the density. The density of states parameter ω is respectively given by 0, 1/3, −1 for the different sources.
Going back to the Einstein equation, we can write out the µν = 00 and µν = ij components to find the Friedmann equations for the expanding FRLW universe as
˙a a 2 = 1 3M2 pl ρ +Λ 3 & ¨ a a = − 1 3M2 pl (ρ + 3p) +Λ 3. (2.9) The Friedman equations describe the dynamics of the universe with different components of energy when the density is generalized to ρ → P
iρi. When we solve these equations for a
universe without sources, ρ = p = 0, we find in vacuum a De-Sitter space as a (t) ∝ e √
Λ/3t. We
will study that inflation can be described by a quasi De-Sitter space.
2.2
Single Scalar Field Inflation
The use of scalar fields is very prominent in theoretical physics and has applications in most branches. A scalar field associates a scalar to every point in space. We will extend the Einstein-Hilbert action and include a scalar field with an associated potential. The idea is that this
Physical Constants 6 (single) scalar field can drive inflation. Inflation was initially developed to solve the horizon problem, flatness problem and monopole problem. Nowadays, the inflation paradigm can ex-plain several phenomena of the universe, such as the inhomogeneities in the cosmic microwave background radiation(CMB). But to solve these problems we have to postulate a positive expan-sion of the relative early universe, or equivalently, a decreasing comoving horizon. This period of exponential expansion is known as cosmological inflation. We can quantify the postulates of inflation as ¨ a > 0 ←→ ∂t 1 aH < 0. (2.10) We will see that a single scalar field, minimally coupled to gravity, can satisfy this assumption under a special condition. The starting point to study inflation will be the action, minimally coupled to gravity with a canonical kinetic term and potential, given by
Sinflation= Z d4x√−g 1 2M 2 plR − 1 2∂µφ∂ µφ − V (φ) . (2.11) Since the universe is spatially homogeneous and isotropic, we can assume that the scalar field φ will only be a function of time and not of space; φ = φ (t). The equation of motion for a scalar field can be found by varying the action with as dynamical variable the scalar field φ. We find that the dynamics of the scalar field are given by
¨ φ + 3H ˙φ | {z } friction − 1 a2∇ 2φ | {z } omit +V0= 0, (2.12) where ’ denotes a derivative with respect to the field φ and a dot denotes a derivative with respect to time t. Other useful information about the action is the energy density and pressure, which are encoded in the energy-momentum tensor. The general relation for this is given by
Tµν = √2 −g δ (√−gLinf) δgµν = ∂µφ∂νφ − gµν 1 2∂σφ∂ σφ + V . (2.13)
If we use the same energy-momentum tensor as in equation2.5for the rest frame, we find that ρ = 1 2 ˙ φ2+ V p = 1 2 ˙ φ2− V w = p ρ = 1 2φ˙2− V 1 2φ˙2+ V .
If we substitute the energy density ρ and pressure p in the Friedman equation, or equivalently, vary the action with respect to a dynamical metric gµν, we find that
H2= 1 3Mpl2 1 2 ˙ φ2+ V ˙ H = −1 2 ˙ φ2 M2 pl . (2.14)
Focusing on the requirements for inflation, as given in equation 2.10, we can relate it to the scalar field. For a shrinking Hubble sphere, we find that
∂t 1 aH = −˙aH + a ˙H (aH)2 = − 1 a 1 + ˙ H H2 ! = −1 a 1 −3 2(1 + ω) , (2.15) where in the last step we used the Friedman equation. Remember that we need to have a shrinking Hubble sphere; ∂t(1/aH) < 0 for inflation. This translate directly into ω < −13.
Examining the density of state parameter for our action, we conclude that we find ω ∼ −1 when 12φ˙2 V . If the kinetic energy is small compared to the potential, then the scalar field will slowly roll down the hill of the potential. This is known as slow roll inflation. For a FRLW universe with only a density of state parameter of ω ∼ −1, we have found a quasi exponential expanding universe; a (t) ∝ exp (Ht). We define the number of folds as dN = Hdt. Furthermore, to solve the flatness and horizon problem, we need at least some N? e-folds of
inflation to flatten the universe enough. The value of N? is constraint by measurement of the
geometry of the universe and depends on the details of reheating, but is often approximated as N? ∼ 60. To reach this N?, we have to find an approximation such that inflation persists long
enough. This we can do by requiring | ¨φ| |3H ˙φ|, |V0|, which can be expressed as V ≡ − ˙ H H2 ≈ Mpl2 2 V0 V 2 ηV ≡ − ¨ φ Hφ = M 2 pl V00 V , (2.16)
where we need V, |ηV| 1 for slow roll inflation. Inflation ends when V ≈ 1. Furthermore, a
relation between N and the scalar field can be found by Ncmb= Z af ai dln (a) = Z tf ti H (t) dt = Z φ? φf 1 √ 2V dφ Mpl ≈ 60. (2.17) During the slow roll regime we can approximate the Friedman equation and equation of motion by H2 ≈ V 3M2 pl 3H ˙φ ≈ −V0. (2.18)
Physical Constants 8
2.3
Connection to Observations
In the previous section we have studied how a minimally coupled scalar field to gravity can drive inflation under certain conditions. However, we did not impose any restriction to the potential. In this section we will present a method to relate slow roll inflation to observations of the CMB, such that the theory becomes predictive and not all potentials are allowed anymore. Cosmic Microwave Background Radiation (CMB)
In order to understand the CMB, we will have to go back to the universe when it was a hot and dense plasma of free electrons and nuclei. When the universe was approximate 380.000 years old, it cooled down enough such that electrons and nuclei combined to form neutral atoms. Shortly after, the temperature of the free electrons dropped below the ionization energy of the hydrogen atom. The low energy photos began to travel freely through the universe without any interactions, and they still do. But due to the expansion of the universe, and hence red-shift, these photons cooled down to approximately 2.725K. These photons from decoupling are the photons we refer to as the CMB. The CMB is an imprint of the very early universe and we will see that is gives us an opportunity to relate theoretical inflationary models to observations.
Figure 2.1: Map from the CMB with tiny temperature fluctuations. Image from: Planck Collaboration 2013.
Quantum Fluctuations
It turns out that the temperature of the CMB is not perfectly smooth[4]. If we measure the tem-perature of the CMB very precise in every direction, we find that there are tiny fluctuations in the temperature of order δT /T ≈ 10−5K, see figure2.1. The idea is that these small fluctuations
in the CMB, can be understood by quantum fluctuation in the scalar field before inflation. Sup-pose that we expand the scalar field around a homogeneous background as φ = ¯φ (t) + δφ (x, t), then these fluctuations will cause inflation locally to end at a slightly earlier or later time. Since fluctuations will happen everywhere in the scalar field, inflation will end at different times for different parts of the universe. In the process of particle creation, this will cause relative density fluctuations, which eventually cause temperature fluctuations in the CMB. This process gives a opening to relate the physics of the very small, quantum theory, to the physics of the very large, general relativity. The CMB gives hope to test physics at a higher energy scale compared to the LHC. Moreover, we need the fluctuations for the formation of galaxies.
We will not do the exact calculations for the quantum perturbations in the scalar field and metric, but briefly describe the method and give the results. The procedure for this calcula-tion is to do linear perturbacalcula-tion theory around a homogeneous background field for all func-tions of space-time. We will then refer to scalar perturbafunc-tions as perturbafunc-tions in the scalar field; φ = ¯φ (t) + δφ (x, t), and tensor fluctuations are due to a perturbations in the metric; gµν = ¯gµν(t) + δgµν(x, t). The relevant information for these fluctuations is encoded by
cor-relation functions. We can describe the fluctuations statistically in Fourier space by a power spectrum. For a Fourier decomposition in k modes, we relate the power spectrum and correlation function via f (t, x) = Z d3k (2π)3/2e ik·xf k(t) f∗ k1fk2 = δ 3(k 1− k2) 2π2 k3 Pf(k) . (2.19)
We define the dimensionless power spectrum as 2π2∆ (k)2 ≡ k3P (k), and parameterize the
scale dependence as ∆2s(k) = As(k∗) k k∗ ns(k∗)−1 ∆2t = At(k∗) k k∗ nt(k∗) , (2.20)
where k∗ is a reference scale, Ai the amplitude and ni the spectral index. When we canonically
quantize the theory, we can relate the quantum vacuum expectation value of the scalar field with the primordial power spectrum. In this calculation, one finds that the amplitude is approximate constant in the superhorizon limit. The k modes will be evaluated at horizon crossing k = aH. This is the amplitude we measure in the CMB. To connect theory with observations, one can expand the physical CMB temperature fluctuations in a spherical decomposition. In the end this will connect theory with observations and give constraints for the theory side. We will refer to quantities evaluated at horizon crossing, k = aH, with a ?. The result of this calculation
Physical Constants 10 gives Ps(k) = H2 2k3 H2 ˙ φ2 ? −→ ∆2s ≈ 1 24π2 V Mpl4 1 V ? Pt(k) = 4 k3 H2 Mpl2 ? −→ ∆2t ≈ 2 3π2 V Mpl4 ? (2.21)
The scalar dependence and tensor to scalar ratio are respectively given by ns− 1 = dln∆2 s dlnk = 2ηV ?− 6V ? nt= dln∆2t dlnk = −2V ? r = ∆ 2 t ∆2 s = 16V ?. (2.22)
Constraints for the spectral index and tensor-to-scalar ratio can by found in the 2015 analysis from the Planck collaberation [5]. If one desires to describe our universe, the prediction of the inflation model should match with the following constraints on inflation:
V V ? = (0.027Mpl)4 ns= 0.968 ± 0.006 r < 0.11. (2.23)
2.4
Case Study: φ
4Inflation
As a simple case study we will consider a quartic potential, given by V = λ
4φ
4. (2.24)
The parameters for this potential are given by V = 8
Mpl2 φ2 =
3
2ηV (2.25) such that we find for the end of inflation (V ' 1) φ2end = 8Mpl2. This can be used to find φ? in
terms of the number of e-fold as N? ' φ2? 8Mpl2 = 1 V ? = 2 3ηV ? . (2.26)
We can then approximate the inflation parameters in terms of N? as ns= 1 − 7ηV ?' 1 − 7 · 2 3N? N?=60 ≈ 0.92 r ' 16 N? N?=60 ≈ 0.27. (2.27) We see that both of these parameters are not in agreement with the latest constraints on inflation from the Planck Collaboration[5]. We conclude that the φ4 model does not work for our universe. We have chosen this quartic potential to make the connection with Higgs inflation in this thesis. This calculation justifies to some extend why Higgs inflation is non-minimally coupled to gravity, and not minimally coupled as this model. As a bonus, we can find the value of λ? from the scalar power spectrum constraint:
λ?= (0.027)4 16N3 ? N?=60 ≈ 1.6 × 10−13. (2.28)
Chapter 3
Gravity & Effective Field Theory
This intermediate-chapter will give background knowledge needed for the study of RGE for (N)HI. We start by studying the basics for effective field theory(EFT), this will be the basis for our study of quantum perturbations. Besides this, we will study theories of gravity in EFT and study how these can be generalized. These theories of modified gravity, are the roots for (N)HI.
3.1
Effective Field Theory
Usually it is convenient to describe nature at a particular length scale. This is why we have different disciplines in the sciences. Within physics, we often find that the dynamics at a large length scale do not depend on the details of what happen at small length scales. For a quantum system this translates directly into: low energy physics does not (always) depend on the details of the high energy physics. Within the framework of EFT, we can focus on a particular energy scale for a theory such that it is valid up to an energy scale Λ. For our purposes we want the cutoff scale Λ to be around the Planck scale Mpl. Because this is the regime to which we trust
the perturbative quantum fluctuations. The EFT can then only contain light particles φ with mass mφ< Λ. The heavy particles Φ with mass mΨ> Λ should be integrated out. In this thesis
we focus on the ”bottom up” approach for EFT. This approach includes the appropriate degrees of freedom to describe physical phenomena occurring at a chosen energy scale, while ignoring degrees of freedom at higher energies. EFT manifests itself in the Lagrangian formulation and distinguishes the operators by their mass dimension. In general we can write a Lagrangian with all types of operators as
Seff[φ] = Z d4x " LIR[φ] + ∞ X i ci Oδi[φ] Λδi−4 # (3.1) where LIRis the renormalizable low energy (infrared) Lagrangian and Oδi are the operators with
mass dimension δi which are allowed by the symmetries of the theory. The Wilson coefficients
ci are dimensionless and Λ is the cutoff scale for the effective field theory. Systematically, the
operators Oδi can be distinguished by their mass dimension δi
δi < 4 Relevant operators Important for low energies(IR).
δi > 4 Irrelevant operators Important for high energies(UV).
δi = 4 Marginal operators Can be important for all energies.
The relevant and sometimes marginal operators are then usually contained in the renormalizable IR Lagrangian. The systematic approach to construct an EFT is to locate all the relevant and marginal operators which are allowed by symmetry with masses mφ< Λ.
3.1.1 Cutoff scale
An effective field theory is valid up to some energy scale Λ. Above this energy scale the theory can not be trusted anymore and new degrees of freedom will be relevant. New terms in the Lagrangian should be accounted for above the cutoff scale. The cutoff scale can be determined by identifying the energy scale for which tree level unitary is violated. Since the scattering amplitude for N particles, MN, is directly related to the probability of particular scattering
process, we can require MN . 1. This translates directly into the statement that unitarity
in tree level processes is violated when the energy of the amplitude MN for N particles scales
higher then [6]
MN ∝ E4−N. (3.2)
The receipt to determine the cutoff scale is to calculate all N particle amplitudes which violate unitarity and find the energy ΛN. Then the cutoff is defined as Λ = minNΛN. Are more
simplistic approach is to find all terms in the Langragian and use power counting to read off the cutoff scale Λ from the interaction terms. It turn out that the cutoff scale can be field dependent, as we will see is the case for HI.
3.1.2 Perturbative Quantum gravity
Gravity is essential for cosmology, so an EFT for cosmology must include gravitational degrees of freedom. For gravity, this low-energy degrees of freedom is the space-time metric gµν. First,
we will show that we can interpret pure gravity as an EFT. We can do this by expanding the metric around a fixed flat background as
gµν → ˆgµν = ηµν+ κhµν
gµν → gˆµν = ηµν− κhµν+ κ2hµλhνλ+ ...
(3.3) such that we can write the Ricci scalar and volume element as [7]
ˆ
R = κhhλλ− ∂µ∂νhµν
i
Physical Constants 14 p−ˆg =√−η 1 +κ 2h α α− κ2 4 h αβh αβ + κ2 8 (h α α)2+ O(h3) , (3.5) where κ2= Mpl−2. Schematically, we will then find for the Einstein-Hilbert term [7,8]
SEH= Z d4xp−ˆg " 1 2(∂h) 2+ 1 Mpl h (∂h)2+ 1 Mpl2h 2(∂h)2+ ... # , (3.6) where h = hαα. Comparing the non-renormalizable terms with the effective field theory descrip-tion as in 3.1, we conclude that the cutoff for pure gravity is Λ = Mpl.
3.2
Modified Gravity
In chapter 2 we found the Einstein equations in the Lagrangian formalism using the Einstein-Hilbert action (2.3). However, we did not justify the origin of this action and why we only considered a term proportional to the Ricci scalar R. In the context of GR, first derivatives of the metric vanish ∇αgµν = 0[9], therefore we need a scalar for the Lagrangian build from second
deviates of the metric or higher. Hilbert concluded that the simplest scalar for this is the Ricci scalar R, which contains second derivatives of the metric. But we can construct more scalar quantities using the Ricci scalar R. We can expand the effective Lagrangian for pure gravity in order of increasing derivatives ∂i as
Sgravity = Z d4x√−g " MΛ4 −M 2 pl 2 R + c1RµνR µν+ c 2R2+ c3RµνλρRµνλρ+ c4R + 1 M2 c5R 3+ ... + ... # , (3.7)
where the constants ci are dimensionless, MΛ is the renormalized value of the cosmological
constant and M the mass scale of the effective theory [8,7,10,11]. If we note that terms with total derivatives in the Lagrangian will vanish at the boundary due to Stokes’ theorem, then we can use equation 4.7to set c4 = 0. Furthermore, we can use Gauss-Bonnet theorem for the
Euler characteristic on an manifold M which is given by χ(M) = 1 32π2 Z M d4x√−ghRµνλρRµνλρ− 4RµνRµν+ R2 i (3.8) such that the integrand is locally a total derivative. We can always set c3 = 0 if we make the
shift {c1, c2, c3} → {c1− c3, c2+ 4c3, 0}. The cosmological constant MΛ is extremely small
com-pared to Mpl and can be neglected. The remaining higher-curvature terms can only be neglected
if the physical curvature is small. As we approach the Planck scale these higher-curvature terms can contribute to the overall action. For example, for one loop quantization of pure gravity one will find counterterms proportianal to R2 & RµνRµν [12].
For cosmological inflation we will eventually construct a QFT coupled to gravity as an EFT, which is in general given by
Seff[g, φ] = Sg+ Seff[φ] + Z d4x√−g "∞ X i ci Oδi[g, φ] Λδi−4 # , (3.9) where Seff is3.1, Sg is3.7and Oδi[g, φ] are operators with mass dimension δi constructed from
curvature invariants (e.g. R) and φ. For example, when the energy is low compared to the cutoff Λ, the only relevant coupling term is [8]
Z
d4x√−gξφ2R (3.10) where ξ is a dimensionless coupling constant. We will use this type of interaction, with a large ξ for HI. When such a coupling is present in the Lagrangian, we refer to it a non-minimal coupling. Generalizations of the Einstein-Hilbert term with non-minimal coupling or extra R terms are known as f (R) and f (φ) R gravity. When the Lagrangian is of the form of the Einstein-Hilbert action we refer to it as minimal coupling.
Chapter 4
Covariant Multifield Inflation in the
Einstein Frame
In this chapter we will explore extensions of the usual minimal coupling to gravity, and study how this will change inflation. We will present the tools to study such theories in the Einstein frame. Furthermore, a formalism to study quantum perturbations in a covariant manner, for non-canonical inflation, will be presented.
Transformation between Jordan & Einstein Frame
The minimal coupling with gravity can be extended in the form of f (R) or f (φ) R gravity. These non-minimal coupled theories to gravity are defined in the Jordan frame, whereas our understanding of GR, QFT & inflation, is defined in the Einstein Frame. Fortunately, it turns out that we can always switch between the Jordan frame and Einstein frame by a re-scaling and/of transformation of the metric. It has been shown by Postma et al, that the physics in both frames is equivalent [13]. We can write the re-scaling and/of transformation of the metric in the most general form as
ˆ
gµν = A (φ, X) gµν+ B (φ, X) ∇µφ∇νφ, (4.1)
where X = − (∇µφ)2/2 [14]. When B = 0, we refer to this as a conformal transformation.
When A and B are non-zero, we refer to this as a disformal transformation. For example, we can find the inverse metric via gµσgσν = ˜gµσ˜gσν = δνµ and identify possible requirements for a
valid transformation, such as avoiding divergences.
In this chapter we will study that for one field we can always do a conformal rescaling/transfor-mation and end up with a canonical action in the Einstein frame. For multiple fields however, we will end up with a non-canonical action in the Einstein frame. For these non-canonical models, we will study how the inflation parameters change. Furthermore, in order to study quantum
perturbations for non-canonical inflation in a covaraint manner, we need a new formalism which we be presented in the last section.
4.1
Conformal Transformation: Single Field
In chapter 2 we studied the Einstein-Hilbert action (2.3) with matter scalar fields minimally coupled to gravity. We extended this by couplings gravity non-minimal. In this section we will consider a general Lagrangian for f (φ) R gravity and transform the action from the Jordan frame to the Einstein frame. In general we can write a non-minimally coupled action for one scalar fields in D dimensions as
SJordan = Z dDx√−g f (φ)R − 1 2g µν∇ µφ∇νφ − V (φ) . (4.2) Minimal coupling in D=(3+1) corresponds to f (φ) = 12M2
pl. If the term multiplying the Ricci
scalar is a function of the fields then we are in the Jordan frame. When the action takes the form of the usual Einstein-Hilbert action we are in the Einstein frame. If f (φ) is positive definite and more complicated then a constant we can always do a conformal transformation to go from the Jordan frame to the Einstein Frame. Each frame could potentially simplify the mathematics and manifest the physics. In this section we will transform from the Jordan frame to the Einstein frame in D dimensions for a general f (φ), based on the work of [15, 16, 17]. A conformal transformation is a transformation and resaling of the metric and is given by ˆgµν = Ω2(x)gµν,
such that the line element changes to
ds2= Ω(x)2dt2− Ω(x)2a(t)2dxidxj. (4.3)
Since we transform and rescale the metric, the space-time is different in each frame. The inverse metric and volume element are given by
ˆ gµνgˆµν = 1 → gˆµν = 1 Ω2(x)g µν (4.4) p−ˆg = ΩD(x)p−g µν. (4.5)
The Christoffel symbols and Ricci scalar are dependent of the metric, Γ [gµν] , R [gµν], and need transformation as well. The transformed Ricci scalar is given by
ˆ R = 1 Ω2 R − 2(D − 1) Ω Ω − (D − 1)(D − 4) 1 Ω2∇ µΩ∇ µΩ , (4.6) where Ω = gµν∇µ∇νΩ = √1 −g∂µ√−gg µν∂ νΩ . (4.7)
Physical Constants 18 The first term in equation4.2, can then be written as
SR= Z dDx√−g [f (φ)R] = Z dDx √ −ˆg ΩD f (φ) Ω2R +ˆ 2(D − 1) Ω Ω + (D − 1)(D − 4) 1 Ω2g µν∇ νΩ∇µΩ . (4.8)
Comparing the first term from the above equation, with a general Einstein-Hilbert term, will then give us a relation for the transformation to the Einstein frame as
f (φ) ΩD−2R =ˆ M(D)D−2 2 ˆ R → ΩD−2= 2f (φ) M(D)D−2, (4.9) where the reduced Planck mass in D dimensions is defined as M(D)D−2 = (8πGD)−1 and GD is
Newton’s constant in D dimensions. In D=4 we obtain the reduced Planck mass as used in this thesis M42 = Mpl2. Using equation 4.9& 4.7, we notice that
f (φ)Ω(x) = 1 2M D−2 (D) Ω D−2(x) gµν∇µ∇νΩ , (4.10) which can be used to integrate the second term in equation4.8 by parts1. This gives
Z dDxp−ˆg2(D − 1) ΩD+1 f (φ)Ω = Z dDxp−ˆg2(D − 1) ΩD+1 1 2M D−2 (D) Ω D−2 ˆ gµν∇ˆµ∇ˆνΩ = − Z dDxp−ˆg(D − 1)(D − 3)M(D)D−2 1 Ω2gˆ µν∇ˆ νΩ ˆ∇µΩ. (4.11)
If we then use equation 4.9to rewrite the third term in 4.8, we can use that (D − 1)(D − 3) − 1 2(D − 1)(D − 4) = 1 2(D − 1)(D − 2), (4.12) such that SR= Z dDxp−ˆgM D−2 (D) 2 ˆ R − (D − 1)(D − 2) 1 Ω2gˆ µν∇ˆ νΩ ˆ∇µΩ . (4.13) The full action in the Einstein frame can now be written as
SEinstein = Z dDxp−ˆg MD−2 (D) 2 R −ˆ M(D)D−2 2 (D − 1)(D − 2) 1 Ω2ˆg µν∇ˆ νΩ ˆ∇µΩ − 1 2ΩD−2ˆg µν∇ˆ µφ ˆ∇νφ − V (φ) ΩD = Z dDxp−ˆg MD−2 (D) 2 ˆ R − (D − 1) 2(D − 2)M D−2 D 1 f2gˆ µν∇ˆ νf ˆ∇µf − 1 4fM D−2 (D) gˆ µν∇ˆ µφ ˆ∇νφ − V (φ) ΩD (4.14) 1
Note that the transformation does not affect the space-time co¨ordinates xµ and the derivative only acts on scalar fields so that ˆ∇µ= ∇µ= ˆ∂µ= ∂µ
where we have used that ˆ ∇µf (φ(x)) = −M D−2 (D) 2 ∇ˆµΩ D−2(x) = − M(D)D−2 2 (D − 2)Ω D−3∇ˆ µΩ → ∇ˆµΩ = Ω (D − 2)f ˆ ∇µf. (4.15)
Usually, studying a model in normalized canonical form will simplify things. This is because most of the literature for GR and QFT is in this normalized form. The action for equation4.14
is not in canonical form. We will therefore bring the action in the form of 12gµν∂µφ∂µφ. For one
field one can do a field redefinition, φ → χ(φ), such that the action can be brought in canonical form in the Einstein frame. For the action to be in canonical form we define χ (φ) such that
1 2ˆg µν∇ˆ µχ ˆ∇νχ = M(D)D−2 4f gˆ µν ˆ ∇µφ ˆ∇νφ +2(D − 1) (D − 2) 1 f∇ˆµf ˆ∇νf . (4.16) This will give a differential equation for the new χ (φ) field as
dχ dφ = s M(D)D−2 2f (φ)2 s f (φ) +2(D − 1) (D − 2) df (φ) dφ 2 . (4.17) The result is that we can write the action in the Einstein frame in a canonical form as
SEinstein = Z dDxp−ˆg MD−2 (D) 2 R − 1 2gˆ µν∇ˆ µχ ˆ∇νχ − V (χ(φ)) ΩD (4.18) The benefits of studying a model in the Einstein frame is that we generally have a better understanding of the quantities in this frame. Furthermore, all the inflation tools are defined in the Einstein frame which make it easier to study a inflationary model.
4.2
Conformal Transformation: Multiple Fields
In this section we will extend the conformal tranformation of a single field to multiple fields and conclude that this is more problematic. For this we consider the following action in the Jordan frame, with N scalar field φI, with I = 1, ..., N , as
SJordan = Z dDx√−g " f (φ1, ..., φN)R −1 2δIJg µν∇ µφI∇νφJ− V (φ1, ..., φN) # . (4.19)
Physical Constants 20 This action can be brought in the Einstein frame with the same techniques as for single field conformal transformation. This gives in the Einstein frame
SEinstein = Z dDxp−ˆg " MD−2 (D) 2 ˆ R −1 2 (D − 1) (D − 2) M(D)D−2 f2 ˆg µν∇ˆ µf ˆ∇νf −M D−2 (D) 4f δijgˆ µν∇ˆ µφi∇ˆνφj− V ΩD # . (4.20)
Using the chain rule for the N field dependence of f = f (φ1, ..., φN) we can write ˆ
∇µf = ˆ∇µφI∂f
∂φI (4.21)
such that we can write the multiple field action in the Einstein frame as SEinstein= Z dDxp−ˆg " MD−2 (D) 2 ˆ R − 1 2GIJˆg µν∇ˆ µφI∇ˆνφJ− V ΩD # (4.22) where we have written the action in the Einstein frame in term of the field space metric GIJ,
which is given by GIJ = M D2 (D) 2f " δIJ + 2(D − 1) (D − 2) M(D)D−2 f ∂f ∂φI ∂f ∂φJ # . (4.23) The metric GIJ can be interpreted as the metric for the field space. We can apply the usual
GR machinery for the field space metric GIJ(φK) on a manifold. For the case of a single scalar
field dependence, we found a transformation which brought the Einstein action in canonical form, GIJ → ˆGIJ = δIJ. This is only possible when the associated Riemann tensor vanishes,
RIJ KL = 0. It has been shown by Kaiser that this is not possible for N ≥ 2 [15].
We conclude that for multi-field inflation with a non-minimal coupling we can at best transform the action to the Einstein frame with non-canonical kinetic term. We better develop some more understanding of non-canonical kinetic term for inflation. This is what we will do in the next sections.
4.3
Non-canonical Multi-field Inflation
In this section we will study a general non-canonical action for mutli-field inflation in the Einstein frame. The goal will be to understand how slow roll inflation is manifest compared to a minimally coupling single field inflation model. We found the paper from Mori et al. and Elliston et al. helpful for the general analysis [18,19]. We will study
S = Z d4x√−g" M 2 pl 2 R − 1 2g µνG IJ∂µφI∂νφJ − V (φI) # , (4.24)
where gµν is the usual space-time metric, GIJ is the field space metric and determines the
structure of the kinetic terms. When the field space metric is constant and diagonal we retrieve a canonical multi-field action, GIJ = δIJ. The energy momentum tensor for this action is given
by Tµν = GIJ∂µφI∂νφJ − gµν 1 2GIJg αβ∂ αφI∂βφJ+ V φI , (4.25) such that we find for the pressure and energy density in the rest frame
ρ = 1 2GIJφ˙ Iφ˙J+ V φI p = 1 2GIJφ˙ Iφ˙J− V φI . (4.26) When we substitute the pressure and density in the Friedman equations, we find
H2 = 1 3Mpl2 1 2GIJφ˙ Iφ˙J+ V φI ˙ H = −1 2 1 M2 pl h GIJφ˙Iφ˙J i . (4.27)
Furthermore, we can vary the action with respect to the fields φI, such that we can find the equation of motion
Dtφ˙I+ 3H ˙φI+ GIJV,J = 0, (4.28)
where V,J = ∂V /∂φJ. We define the covariant derivative for the field space metric, for any
vector XI in the field space, as
DJXI = ∂JXI+ ΓIJ KXK
DtXI = ˙φJDJXI = ˙XI+ ΓIJ KXJφ˙K.
(4.29) We are ready to generalize our requirements for inflation to obtain the relevant formulas for slow-roll inflation. We require 12GIJφ˙Iφ˙J V φI and |D
tφ˙I| |3H ˙φI|, |GIJV,J| such that we
can approximate the e.o.m and Friedman equations as
3H ˙φI ≈ −GIJV,J & 3Mpl2H2≈ V φI . (4.30)
With these approximations, we can express the inflation parameters as [18,19]
V = 1 2Mpl2 GIJφ˙Iφ˙J H2 = Mpl2 2 GIJV ,IV,J V2 ηV = 2V + 2 GIJφ˙IDtφ˙J HGKLφ˙Kφ˙L = 4V − Mpl4 V DKV,JGKLV,LGJ MV,M V3 . (4.31)
Physical Constants 22
4.4
Covariant Multi-field Formalism
In later chapters we will renormalize (N)HI as an EFT, meaning we have to study perturbations for the quantum fields around a homogeneous background field. The usual method to do such an expansion for a scalar field φI is φI(t, x) = φI0(t) + δφI(t, x). The problem is that for a finite length, the field displacement δφI, does not lie in the tangent space for φI and hence there does not exists a tensorial transformation law. The perturbation δφI is coordinate dependent in field space, so the perturbation is not covariant.
We will adopt the covariant multi-field formalism developed by Gong & Tanaka [20]. We expand all scalar fields φI around a homogeneous background field φI(t, x) = φI0(t) + δφI(t, x). Now we need to find a method to describe the displacement δφI with a tangent-space vector. We will do this by uniquely connecting the points φI(t, x) and φI0(t) by a geodesics for which we label the path by λ. We let the geodesic run from φI(λ = 0) = φI0 to φI(λ = 1). Furthermore, we denote the velocity of the starting point by dφI
0/dλ = QI. Note that φI is not a vector
in the field space manifold, but derivatives ∂µφI do behave as proper vectors. The vector QI
will represent the field fluctuation in a covariant manner. See figure 4.1 for a more insightful representation.
Figure 4.1: Paramatrization for a geodesics which run from λ = 0 to λ = 1. Figure from [20]
The geodesic equation for φI(λ) is given by Dλ2φI = d 2φ2 dλ2 + Γ I J K dφJ dλ dφK dλ = 0, (4.32) where for clarity we used:
Dλ= D dλ, φ I λ=0= φ I 0, DλφI λ=0= dφI dλ λ=0 = QI, (4.33)
where D is the covariant derivative in the λ-direction. Now we can do a Taylor series for φI(λ = = 1) and find (using the geodesic equation)
δφI≡ dφ I dλ λ=0+ 1 2! d2φI dλ2 λ=0+ 1 3! d3φI dλ3 λ=0+ ... = QI− 1 2!Γ I J KQJQK+ 1 3! Γ I LMΓMJ K− ΓIJ K,L QJQKQL+ ... (4.34) where the christoffel symbols have to be evaluated at the background. This expansion is now written in terms of QI vectors, which do lie in the tangent space of φI0. So when ones write everything in terms of this expansion, and hence in terms of QI vectors, the perturbation will transform as an vector. We define the covariant derivative for the field space metric, for any vector XI in the field space, as
DJXI = ∂JXI+ ΓIJ KXK
DtXI = ˙φJDJXI = ˙XI+ ΓIJ KXJφ˙K,
(4.35) where a dot means a derivative with respect to t. Note that these definitions match with equa-tion 4.29
We can also consider a perturbation of a field space scalar function f φI. This expansion will for our purposes be easier to work with. Note that for this field space scalar function, all field space indexes should be contracted. We assume that there are no space-time derivatives in this scalar function for following the derivation. If we use that a ordinary derivative of a field space scalar is identical to a covariant one, we can expand f φI as
f φI = f φI λ=0+ D 1 λf φI λ=0+ 1 2!D 2 λf φI λ=0+ 1 3!D 3 λf φI λ=0+ ... = f λ=0+ f;IQ I+ f ;IJ 1 2!Q IQJ + f ;IJ K 1 3!Q IQJQK+ .., (4.36) where we used the geodesic equation in the second step. The covariant derivatives should be evaluated on the background. A semicolon denotes a covariant derivative for the field space. Mass:
To first order in covariant perturbations, the mass matrix is given by [21] MIJ = m2I
J = G IK(D
Chapter 5
Higgs Inflation: Classical Analysis
In this chapter we will study how the Higgs potential with a non-minimal coupled action to gravity, can give a period of slow roll inflation. We will use the tools presented in this thesis so far, to transform the action for HI to the Einstein frame, such that we can analyze the model. We will study the model classically and conclude that the predictions fall within the constraints from the Planck Collaboration 2015. The idea is to identify the Higgs field as the inflaton field, such that for high energies we can construct a slow roll regime. In the low energy limit, we would like to keep the Higgs mechanism in the SM intact. This model was initially developed by Bezrukov et al. [22, 23]. This inflation model does not introduce any new scalar field and hence associated particles. This is very economical, since the Higgs field and particle is the only experimental verified scalar field. One drawback for HI is the unnatural large parameter ξ. Nevertheless, the predictions for HI fall right in the middle spot for the constraints on inflation. We will also conclude that HI is equivalent to Starobinsky R2 inflation. The following resources were helpful for the study of this chapter [24,25,26]
5.1
Higgs Inflation in the Einstein Frame
In section2.4, we calculated the inflation parameters for a φ4 potential, minimally coupled with gravity to the scalar field, but concluded that the scalar-to-tensor ratio and spectral index do not fall within the observational constraints from the Planck 2015 results. In this section we will include a non-minimal coupling with the scalar field to gravity. Eventually, the idea is to identify the scalar field as the Higgs boson field. The action for the HI model in the SM is given by SJordan = Z d4x√−g" M 2 pl+ ξΦ†Φ 2 R + LSM # , (5.1) where Φ is the Higgs doublet, ξ is a dimensionless constant, M is a mass parameter and LSM
is the standard model Lagrangian. The Higgs potential is included in the SM Lagrangian. The model is defined for large ξ. We assume the chaotic inflation scenario for which the field initially starts at order of the Planck scale. For the conformal transformation to the Einstein frame we
will use the calculations from section 4.1and identify f (φ) = M 2 pl+ ξφ2 2 (5.2) Ω2 = 2f (φ) M2 pl = M 2 pl+ ξφ2 M2 pl = 1 + ξφ 2 M2 pl . (5.3)
Using the above simplifications, we can write the action in the Jordan frame as SJordan = Z d4x√−g" M 2 pl 2 Ω 2R − λΦ†Φ2 # , (5.4) where we have explicitly written down the Higgs potential in the action, and neglected further interactions from LSM. As discussed earlier, it is convenient to work in the Einstein frame. The
action in the Jordan frame can be conformally transformed to the Einstein frame such that SE= Z d4x√−g" M 2 pl 2 R − 1 Ω2 (∂µΦ) † (∂µΦ) − 3ξ 2 M2 plΩ4 ∂µ Φ†Φ ∂µ Φ†Φ −λ Φ †Φ2 Ω4 # (5.5) . In the most general Higgs doublet with 4 degrees of freedom, √2ΦT = {χ1+ iχ2, χ3 + iχ4},
we find that the space field metric will be of the form −1 2Gij∂χi∂χj = − 1 2 " δij Ω2 + 6ξ2 Mpl2Ω4χiχj # ∂χi∂χj. (5.6)
We find for this space field metric Gij, that the curvature on field space is non-zero; R [Gij] 6= 0.
Referring back to section4.1, we concluded that for this type of space field, we can only retrieve canonical kinetic terms for a doublet with only one background field. This is what we will study in the next section to find the predictions for the inflation parameters for this model. However, in order to include HI with quantum perturbations in the SM, we are forced to include the Goldstone bosons in the Higgs doublet. For the full SM with HI, we are required to work with a non-canonical action in the Einstein frame, which will be the focus for the next chapter when we try to renormalize HI in the SM as an EFT.
5.2
Predictions for Higgs Inflation
With the action for HI ready in the Einstein frame, we can employ the inflation tools to find the predictions for this model. The most straightforward method to study the predictions for HI is to choose a Higgs doublet with only one background field. This is the only method to end up with a canonical action in the Einstein frame. Fortunately, we can use the unitary gauge, for which the Higgs doublet reduces to one field. On the background, the doublet is then given
Physical Constants 26 by Φ = √1 2 0 φ !
such that the expectation value is given byΦ†Φ = φ2/2. The degrees of freedom reduce from four to one, with respect to the general gauge from the previous section, and the Higgs field φ is the only degree of freedom left. The Lagrangian in the Jordan frame is given by
SJordan = Z d4x√−g" M 2 pl+ ξφ2 2 R − 1 2∂µφ∂ µφ −λ 4φ 4 # . (5.7) Using the machinery of section 4.1we can bring the action into canonical form in the Einstein frame as SEinstein = Z d4xp−ˆg" M 2 pl 2 ˆ R − 1 2∂ µχ∂ µχ − λ 4Ω4φ 4 # (5.8) where we used for the field redefinition equation 4.17such that
dχ dφ = s M2 pl 2f (φ)2 s f (φ) + 3 df (φ) dφ 2 = s 1 Ω2 + 6ξ2φ2 Ω4M2 pl = s Ω2+ 6ξ2φ2/M2 pl Ω4 . (5.9)
The general solution for this differential equation for large ξ is
χ (φ) = Mpl √ 6 sinh−1 √ 6ξφ Mpl ! − sinh−1 √ 6ξφ Mpl r 1 +Mξφ22 pl . (5.10)
We will explore these equations in two relevant limits for which the equations simplify. In the small field limit φ < Mpl/ξ we have that χ = φ and Ω2 ≈ 1. For this limit the Higgs potential
does not change so that the Higgs mechanism is still intact for low field values. In the regime of the large field limit, φ > Mpl/
√
ξ, we expect a slow roll regime. In this regime we have that Ω2≈ ξφ 2 M2 pl , (5.11) such that dχ dφ = s M2 pl ξφ2 + 6M2 pl φ2 = Mpl φ r 1 ξ + 6 ≈ Mpl √ 6 φ −→ φ = M√pl ξ exp χ √ 6Mpl ! , (5.12)
where we normalized the differential equation for the regime where we expect it to hold. We can use this to write to potential in the Einstein frame as
V (χ) = λM 4 pl 4ξ2 1 − e− 2χ √ 6Mpl 2 . (5.13) This potential will be exponentially flat which will give us the flat plateau for slow roll inflation. We can apply this potential to the inflation analysis from section2.2and calculate the inflation parameters to lowest order of 1/ξ. The derivatives of the potential are given by1
dV (χ) dχ ≈ 4V (χ) √ 6Mpl e− 2χ √ 6Mpl 1 + e− 2χ √ 6Mpl = √4V (χ) 6Mp " M2 pl ξφ2 + Mpl4 ξ2φ4 # ≈ 4V (χ)√ 6 Mpl ξφ2 (5.14) d2V (χ) dχ2 = 4 √ 6Mpl dV (χ) dχ " M2 pl ξφ2 + Mpl4 ξ2φ4 # −√ 4 6Mpl dφ dχ " 2M2 pl ξφ3 + 4Mpl4 ξ2φ5 # ≈ −4V (χ) 3ξφ2 . (5.15)
The number of e-folds N? can now be found as
N? = Z φ? φend dχ dφ 2 dφ M2 pl V dV (φ) dφ ' 3ξ 4M2 pl φ2 ?− φ2end . (5.16)
Furthermore, using that inflation ends when V ≈ 1, we can express the inflation parameters in
terms of N?, and we find that
φend= 4 3 14 M√pl ξ (5.17) φ? ' r 4N? 3 M√?pl ξ . (5.18) V ≈ 4 3 Mpl4 ξ2φ4 φ=φ? ' 12 (4N?)2 (5.19) ηV ≈ −4M2 pl 3ξφ2 φ=φ? ' −4 4N? . (5.20)
The predictions for HI are given by
ns ' 1 − 2 N? N?=60 = 0.967 (5.21) r ' 12 N2 ? N?=60 = 0.003. (5.22) 1
The large field limit for the field χ translates to χ √6Mpl such that we can use a approximation for the geometric serie 1/(1 − x) ≈ 1 + x.
Physical Constants 28 The predicted values for r and ns are well inside the constraints on inflation [5]. We can also
use the constraint on the scalar power spectrum to solve for ξ. Remeber that the scalar power spectrum is defined as [8,4] ∆2s= As k k∗ ns−1 = 2.142 · 10−9± 0.049 (5.23) such that we can express the inflation normalization conditions for N? = 60 as
V (φ?) V = 24π2∆2s(k?)Mpl4 ' 5.244 · 10 −7M4 pl ' 1186λM 4 pl ξ2 . (5.24) Then we find ξ = 47557 √ λ. (5.25)
Energy scale & cutoff
Using equation 5.17 & 5.18, we can estimate the energy scale of the field during inflation as φ ' Mpl/
√
ξ. Applying this to the Friedmann equation in the slow roll regime yields a Hubble scale of H ' r λ 12 Mpl ξ . (5.26)
We can use this to express the energy scale of inflation as Einflation= V1/4 = (3H2Mpl2)1/4= λ 4 14 M√pl ξ ' 10 −1M√pl ξ ' 10 −3 Mpl. (5.27)
We approximated that λ in the running is of order O(10−1) and is positive at the energy scale of inflation [27]. With this approximation we obtain from equation 5.25 that ξ ' 104. This energy scale is just below the cutoff scale for HI, which are given by [28]
Λ ∼ Mpl ξ , φ, Mpl √ ξ , (5.28)
for respectively the following fields regimes: small field: φ < Mpl ξ , mid field: Mpl ξ < φ < M√pl ξ, large field: M√pl ξ < φ. (5.29)
5.3
Higgs & Starobinsky Inflation
In this section we will show that the non-minimal coupling in HI is equivalent to Starobinsky R2 inflation in the large ξ limit. This can be shown if we integrate out the Higgs field. If we
start with HI in the Jordan frame for a one field Higgs doublet, we find SJordan= Z d4x√−g" M 2 pl+ ξφ2 2 R − λ 4φ 4 # . (5.30) If we apply the Euler-Lagrange equation to this action we find that
ξφR − λφ3 = 0 −→ φ2= ξR
λ . (5.31) Plugging this back in the action, and using M2= 3ξλ2, we find
SJordan= Z d4x√−g" M 2 pl 2 R + ξ2 4λR 2 # . (5.32) this is idential to Starobinsky inflation in the ξ 1 limit, which is given by [29]
SStarobinsky= Z d4x√−g Mpl2R + 1 6M2R 2 . (5.33) In the Planck 2015 analysis[5], only Starobinsky inflation is included due to the similarities with the non-minimal coupling model.
Chapter 6
Higgs Inflation: Quantum Analysis
In the previous sections we have simplified the HI model with a Higgs doublet with only one field, in order to quickly calculate the classical inflation predictions. But eventually, it is desir-able to include HI in the standard model(SM) and study the full consequences of the theory. Furthermore, we would like to include quantum corrections to obtain the renormalization group equations(RGE). We will study how the predictions for HI change when quantum effects are included, and conclude that to first order in V, the predictions do approximately not change.
The predictions for the classical tree level analysis are sufficient for first order quantum correc-tions.
We will divide HI in three effective field theories. Our goal is to obtain the full RGE for each of the three regimes. First, we will do the calculation for a simplified model, HI in a U(1) Abelian model coupled to a left and right handed fermion. For the U(1) Higgs model we will calculate the one-loop corrections for the Higgs and Goldstone propagator, using the multifield covariant formalism. From these quantum corrections, we can calculate the beta functions for ξ and λ in the renormalization group. In the last section we will then extend the beta functions to the full standard model SU (3)c⊗ SU (2)L⊗ U (1)Y. Our approach is based upon the work of
M. Postma et al. [25]. In this chapter we set Mpl2 = 1.
6.1
Higgs Inflation as an Effective Field Theory
The SM is described by the symmetry group SU (3)c⊗ SU (2)L⊗ U (1)Y, respectively the strong
interaction, weak interaction and hypercharge. The Higgs mechanism manifest itself when the symmetry of the SU (2)L⊗ U (1)Y group is spontaneously broken. In this chapter we will study
Higgs inflation in the full standard mode. Higgs Inflation in the Jordan Frame
We define Higgs inflation in the Jordan frame with a non-minimal coupling as 30
LHI+SMJordan =p−gJ " 1 2 1 + 2ξΦ†ΦRgJ − λ Φ†Φ −v 2 2 2 −1 4 f a µν 2 −1 4 F a µν 2 −1 4B 2 µν+ X i ¯ ψJi i /D ψJ i −ydQ¯L· ΦdR+ yuu¯R(iσ)2Φ†QL+ h.c. # (6.1)
with ψiJ = {QL, uR, dR}, where QL= (u d)TL and u/d are the up/down quark. fµνa , Fµνa &Bµν
are respectively the SU (3)c, SU (2)L & U (1)Y field strengths. Φ is the Higgs SU (2)L doublet
with three Goldstone bosons and a Higgs field, and is given by Φ = √1 2 χ1+ iχ2 φ0+ φ + iχ3 ! . (6.2)
The covariant derivatives are given by
DµΦ = ∂µ− ig2Aaµτa− iYφg1Bµ Φ,
DµQL= ∂µ− ig3fµata− ig2Aaµτa− iYQg1Bµ QL,
DµuR= ∂µ− ig3fµata− iYug1Bµ uR
(6.3)
where the hypercharges Yφ, YQ & Yu are respectively given by 1/2, 1/6 & 1/3. Futhermore, we
choose ta= σ/2.
Higgs Inflation in the Einstein Frame
We will transform the Lagrangian in the Jordan frame to the Einstein frame with a conformal transformation, defined as gµνE = Ω2gµνJ and use
Ω2=1 + 2ξΦ†Φ. (6.4) When we rescale the fermions with ψE = ψJ/Ω3/2, we get canonical fermion terms and find that the action for HI in the SM is given for the Einstein frame by
LHI+SM Einstein = √ −g " 1 2R − 1 Ω2(DµΦ) † (DµΦ) −3ξ 2 Ω4∂µ Φ†Φ∂µΦ†Φ− λ Ω4 Φ†Φ −v 2 2 2 −1 4 f a µν 2 −1 4 F a µν 2 −1 4B 2 µν+ X i ¯ ψiE i /D ψE i − 1 Ω ydQ¯L· ΦdR+ yuu¯R(iσ)2Φ†QL+ h.c. # . (6.5)
Physical Constants 32 The quadratic action for the gauge and Goldstone fields, is in the SM simply three times the U(1) action [25]. This is why we will not study the SM action explicitly, but study a simpler U(1) model. We can do this because the U(1) model can be easily generalized to the above SM action. We will elaborate on this in more detail in section 6.8.
Three Field Regimes
Our approach to renormalize HI is to divide the model into three regimes, and construct an EFT for each of the regimes. Each of the three regimes defines a specific field space. The small field regime is defined as; ξφ0 1. The mid field regime is defined as; 1/ξ < φ0 < 1/
√ ξ. The large field regime is defined as; 1/ ξφ20 1. The large field regime will be the slow roll regime. We will use s = {1, 0, 0} in equations as an boolean to distinguish between {small, mid, large}. The focus will ultimately be on the large field regime, because there RGE are the most rele-vant for the UV sensitivity of HI inflation. We will use the following expansion parameters; for the small field regime δs = ξφ0 1, and for the large field regime δl = 1/ ξφ20
1. For the mid field regime the expansion is a bit more tricky because we rescaled the condition 1/ξ < φ0 < 1/
√
ξ with ξ → δm−2ξ and φ0 → δm3/2φ0, such that we will work with the expansion
parameters ξφ20 ∝ δm and 1/ (ξφ0)2 ∝ δm.
The three field regimes can always be patched together at the boundaries to model our ig-norance of a UV completion, this has been done by Postma et al. [28]. The focus for this thesis will be on the retrieval of the RGE.
6.2
U(1) Abelian Higgs Inflation Model
In the previous section we have transformed the Lagrangian for HI to the Einstein frame and ended up with a non-canonical term. In this section we will downgrade the model to a U(1) Abelian model for a complex Higgs field, including a left- and right handed Weyl fermion. We will then work out all the terms such that we can identify the interaction terms. Our starting point will be the following Lagrangian in the Einstein Frame:
LEinstein√ −g = 1 2R − 1 Ω2 (DµΦ) † (DµΦ) −3ξ 2 Ω4∂µ Φ†Φ∂µΦ†Φ−λ Φ †Φ2 Ω4 −1 4FµνF µν−y ¯ΨLΦΨR+ ¯ΨRΦ ∗Ψ L Ω + i ¯Ψ /DΨ − m ¯ΨΨ − G2 2ξG − ¯c∂G ∂αc, (6.6)
