MSc Physics
Theoretical Physics
Master Thesis
On the Naturalness of Multi-field Axion Inflation
by
Adri Duivenvoorden
6141463
July 2015
60 ECTS
Research carried out between May 2014 and July 2015
Supervisor:
Dr. Jan Pieter van der Schaar
Examiner:
Dr. Marieke Postma
A detection of primordial B-mode polarisation in the cosmic microwave background radiation would imply the need for a radiatively stable description of large-field inflation. In this thesis, the N-flation model, which is claimed to provide this by utilising the collective field displacement of a large number (N ) of sub-Planckian axions, is studied from a bottom-up approach. Two expected corrections to the model will be taken into account: first, the non-minimal coupling of axion fields to the Ricci scalar. It is demonstrated that this correction does not lead to the eta problem, even in the presence of N axion fields, but does lead to additional entropy perturbations not suppressed in the long-wavelength limit. The second correction is the inclusion of the renormalisation of the Planck mass that one generally expects in a theory involving a large number of scalar species. The resulting weakening of the gravitational coupling is shown to spoil the mechanism of N-flation at parametrically high N . However, the effect only becomes relevant when considering N ∼ O(106).
This number lies a factor O(102) beyond the maximum number of scalar species the effective
theory is able to describe. This limit is the result of the lower scale of quantum gravity due to the presence of N distinct particle species. Additionally, it is argued that this new scale will introduce higher order operators in the effective field theory that spoil the radiative stability of N-flation.
Acknowledgements
I would like to express my gratitude to my supervisor dr. Jan Pieter van der Schaar for his encouragement and guidance throughout this project. I would also like to thank my examiner dr. Marieke Postma, especially for her help during the final stages of the project. I want to thank my fellow student Lars Aalsma for numerous enlightening discussions and finally, dr. Sera Markoff for her advice and for introducing me to cosmology in the first place.
0.1 Nederlandse samenvatting . . . iii
0.2 Conventions/notation . . . v
1 Introduction 1 1.1 Big Bang puzzles . . . 2
1.1.1 Inflationairy solution . . . 4
1.2 Introduction to inflation . . . 5
1.2.1 De Sitter space . . . 7
1.2.2 Effective field theory of inflation . . . 14
1.3 Slow-roll inflation . . . 22
1.3.1 Inflating with a scalar field . . . 22
1.3.2 Slow-roll approximation . . . 26
1.3.3 Power spectra . . . 27
2 Polarisation of the CMB radiation 32 2.1 Approach based on tensor spherical harmonics . . . 34
2.1.1 Small angle approximation . . . 34
2.1.2 Polarisation on the celestial sphere . . . 36
2.1.3 Scalar and tensor perturbations in the CMB . . . 39
2.2 Approach based on spin ±2 spherical harmonics . . . 42
3 Naturalness in inflation 46 3.1 The eta problem . . . 47
3.2 Natural inflation . . . 48
CONTENTS
4 Multi-field axion inflation 55
4.1 N-flation . . . 57
4.2 Corrections to N-flation . . . 61
4.2.1 Non-minimal coupling . . . 61
4.2.2 Renormalisation of the Planck mass . . . 64
4.3 Corrected N-flation model . . . 72
4.3.1 Background dynamics for general F (φ, R) . . . 73
4.3.2 Single-field description of corrected N-flation . . . 74
5 Effects on the curvature perturbation 81 5.1 Multi-field dynamics for general F (φI, R) . . . . 82
5.1.1 Background dynamics . . . 83
5.1.2 First order equations . . . 83
5.2 Curvature perturbation at superhorizon scales . . . 84
5.2.1 N-flation . . . 85
5.2.2 Corrected N-flation . . . 87
6 Discussion and conclusions 90 6.1 Discussions and future work . . . 90
6.2 Conclusions . . . 94
A Gravity in flat FLRW universes 97
B Most general EFT action for inflation 98 C Logarithmic corrections to the Planck mass 100 D Cosmological perturbation theory 102
0.1
Nederlandse samenvatting
Deze thesis gaat over kosmische inflatie: een beschrijving van het universum in de eerste fractie van een seconde na de oerknal. Inflatie kan worden gezien als een toevoeging op de algemeen bekende oerknaltheorie, waarin het universum uit een enorm heet en klein punt ontstaat en in 13.8 miljard jaar uitzet tot het gigantische universum waarin we nu leven.
Wanneer de allereerste momenten buiten beschouwing worden gelaten, is de evolutie van het universum na de oerknal verrassend goed te beschrijven met de huidige natuurkunde. Er kleeft echter een smet aan het oerknalmodel: de ontwikkeling van het universum hangt sterk af van de condities tijdens de oerknal. Wanneer we het oerknalmodel vragen het huidige universum te produceren, vereist dat exceptioneel nauwkeurig afgestelde begincondities voor alle, net ontstane, materie. Is de snelheid van de collectieve beweging van deeltjes net iets te laag, dan stort het universum vrijwel onmiddellijk weer in onder zijn eigen gewicht. Bij een snelheid die een fractie hoger ligt expandeert het universum juist enorm zodat er na verloop van tijd slechts koude, lege ruimte overblijft. Slechts bij een zeer precies afgestelde initiële snelheid expandeert het universum tot het huidige. Er is echter geen mechanisme dat de materie vertelt welke snelheid de juiste is. Sterker nog, zou een dergelijk mechanisme vlak na de oerknal wel aanwezig zijn, dan zou het de informatie over de juiste snelheid sneller dan het licht moeten verschaffen aan de wegspoedende materie. Door het toevoegen van inflatie kan dit probleem opgelost worden.
De inflatie-theorie postuleert dat een minuscuul stukje ruimte in de eerste fractie na de oerknal in zeer korte tijd enorm uitzet. Typisch zo’n 1026 keer in slechts 10−12 seconde in de modellen die behandeld worden in deze thesis. Na deze korte periode mag het oerknalmodel het roer weer overnemen. Het resulterende zichtbare universum komt nu voort uit dit klein stukje ruimte waarin de snelheden van de deeltjes op een natuurlijke wijze op elkaar zijn afgestemd en waarin, door de enorme expansie, hun collectieve snelheid vanzelf naar de correcte waarde is gedreven, onafhankelijk van beginwaardes. Door de toevoeging van een periode van inflatie, is het huidige universum dus een logisch resultaat van de kosmische evolutie.
De inflatie-theorie maakt bovendien enkele concrete voorspellingen over fluctuaties in de tem-peratuur van de kosmische microgolf-achtergrondstraling: overgebleven warmtestraling van het hete universum relatief kort na inflatie. De temperatuur van de kosmische achtergrondstraling is gemeten en de fluctuaties erin komen overeen met de voorspellingen. Eén voorspelling van de inflatie-theorie is nog niet waargenomen: een bepaald soort polarisatie van de lichtdeeltjes van de achtergrondstraling. De detectie van deze zogenaamde B-mode-polarisatie zou de kers op de taart van de inflatie-theorie zijn. Derhalve zijn er op dit moment meerdere onderzoeksgroepen
CONTENTS
op zoek naar de B-modes. Een hoopgevende, maar niet significante, detectie is reeds gedaan in het afgelopen jaar. De kans op een detectie in de aankomende jaren wordt daarom als aanzienlijk geacht. De thesis gaat uit van de implicaties van een dergelijke waarneming.
Kosmische inflatie is zelf ook niet helemaal onafhankelijk van de condities tijdens de oerknal. Tijdens inflatie is het bijvoorbeeld een vereiste dat de energiedichtheid van het vacuüm positief en vrijwel contant blijft. Alleen nauwkeurig afgestelde begincondities kunnen hier voor zorgen, want door kwantummechanisme correcties aan zwaartekracht1 zal de energiedichtheid typisch juist niet constant blijven. Daarom is de voorgaande eis schijnbaar erg onnatuurlijk (zeker wanneer er een grote hoeveelheid B-mode-polarisatie gevonden wordt). Wanneer waarschijnlijkere begincondities worden gebruikt, kan de energiedichtheid niet lang genoeg (bijna) constant blijven in veel modellen van inflatie.
Om deze relatief lange tijdsduur2 toch te kunnen overbruggen zonder dat kwantumcorrecties roet in het eten gooien, gebruiken de multi-axion modellen uit de titel van deze thesis de (bijna) constante energiedichtheid van de combinatie van een groot aantal axionvelden3. Er wordt gedacht
dat de correcties door kwantumzwaartekracht op de energiedichtheid van de individuele axion-velden verwaarloosbaar klein zijn. Slechts één van deze axionaxion-velden is echter niet voldoende als mechanisme voor inflatie, daarom wordt een groot aantal gebruikt. De resulterende energiedicht-heid is voldoende constant in de tijd; inflatie kan dus plaatsvinden. Uiteraard is het afstemmen van de begincondities nu verruild voor een hoog aantal vereiste axionvelden. Dit wordt echter als minder onnatuurlijk gezien omdat men denkt dat axionvelden vlak na de oerknal toch al in groten getale aanwezig zijn.
In deze thesis wordt gedemonstreerd dat de aanwezigheid van een hoog aantal axionvelden een minder gunstig effect tot gevolg heeft. Er wordt beargumenteerd dat de energieschaal waarop kwantumzwaartekracht een rol gaat spelen omlaag gaat naarmate er meer axionvelden worden toegevoegd aan de theorie. Hierdoor worden de individuele axionvelden toch sterk gecorrigeerd. De resulterende, hevig fluctuerende energiedichtheid vereist opnieuw onwaarschijnlijk precies afge-stemde begincondities, wil inflatie alsnog plaatsvinden. Multi-axion modellen lossen dit probleem van inflatie dus niet op.
1Bij kwantummechanica denkt men aan atomaire of kleinere lengteschalen, terwijl zwaartekracht daar
nor-maal niet mee wordt geassocieerd. Tijdens inflatie zijn de schalen echter zo klein en de energieën zo hoog, dat kwantummechanische correcties aan Einstein’s theorie van zwaartekracht niet te verwaarlozen zijn.
2Een lange tijdsduur is natuurlijk relatief, gezien de voorgaande opmerking dat inflatie slechts een fractie van
een seconde duurt.
3Een axionveld kan beschouwd worden als een, in elk deel van de ruimte aanwezige, energie. Het veld is vrijwel
0.2
Conventions/notation
Throughout this work the speed of light and the reduced Planck constant will be set to one:
c = ~ = 1. The reduced Planck mass: Mpl = (8πG)−1/2 = 2.435 · 1018 GeV, will be kept explicit
and will be denoted simply by the Planck mass. Unless stated otherwise, it will be simultaneously used as the scale of quantum gravity and as the inverse coupling constant of gravity. Our metric signature is (− + ++). Greek indices will denote dimensions in spacetime running from 0 to 3. Lower-case Latin indices will usually denote dimensions in spacetime running from 1 to 3. Upper-case Latin indices are reserved to indicate dimensions in field space. Partial derivates are denoted by commas, i.e. φ,i. Covariant derivates are denoted by semi-colons, i.e. Yµ;i. Conformal
time is denoted by τ . Derivates with respect to time are represented by overdots, derivatives with respect to conformal time with primes.
Chapter 1
Introduction
The choice of subject for this thesis was motivated by the claimed measurement of the
tensor-to-scalar ratio r (r = 0.2) by the BICEP2 experiment [1]. The more pessimistic result from the
follow-up analysis by BICEP2/Keck Array and the Planck satellite [2], that adjusted the claim of a detection to an upper limit on the value of r (r < 0.12), did not make us abandon our research of tensor producing inflationary models. Motivated by the Lyth bound [3], which states that for
r ≥ 0.01, inflation must be described by a large-field model, it is argued that a better theoretical
understanding of large-field inflation is needed in any case.
Since the discovery of the cosmic microwave background (CMB), cosmology has come a long way as a high-precision branch of physics. Arguably, this is mostly due to the inflationary paradigm: a falsifiable model of the early universe. It uses the thoroughly tested theory of General Relativity to solve the problems with the old, hot Big Bang model and uses quantum field theory in curved spacetime to explain the density perturbation seeding the structure observed in the present day universe. However, just like the models predating it, inflation itself depends on initial conditions. It will be demonstrated that, especially, the large-field models motivated by any future measurement of r, are sensitive to these initial conditions.
We argue that at this point in time, due to the vast amount of inflation models that seem to fit the observational data, additional model building is not the way to go. An arguably more constructive approach lies in critically checking the viability, or naturalness, of existing models. To study large-field inflation models in the most general way possible, the choice is made to work in the context of Wilsonian effective field theory (EFT). It is explained how in this approach, one is naturally led to multi-field axion models as candidates for large-field inflation.
In multi-field axion models of inflation the inflaton is described by a combination of several axions: pseudo-Nambu-Goldstone bosons that obey a discrete shift symmetry: φ → φ + 2πf . It
will be demonstrated how this shift symmetry keeps the axions relatively stable under radiative corrections. The focus of this thesis will lie on the N-flation model, which seems to be most general multi-axion inflation model. Two corrections to this model will be introduced: a non-minimal coupling of the axion fields to the Ricci scalar and the renormalisation of the Planck mass due to the presence of a large number of axions. Additionally, it will be demonstrated how the lowering of the scale of quantum gravity in the presence of a large number of particle species will pose the biggest obstacle for the naturalness of N-flation, effectively spoiling the model.
The organisation of this thesis is as follows. In the remainder of this chapter the inflationary solution is introduced in a very general, model independent way. Additionally the concepts of naturalness and the EFT are introduced. Finally, the essential jargon and concepts of slow-roll inflation are explained. In chapter 2 it is explained how experiments might be able to measure the tensor-to-scalar ratio by looking at polarisation in the CMB. In chapter 3 the concept of naturalness is adapted for the use in models of inflation. Furthermore, the axion is introduced. N-flation and the correction to it are discussed in chapter 4. In chapter 5 some additional effects of the non-minimal coupling are discussed. Finally, in chapter 6 the results will be discussed and conclusions will be made.
1.1
Big Bang puzzles
Before the inflationary paradigm took over the cosmological standard model, the leading model was the hot Big Bang model. Despite it being a great change from the models describing a static universe that predated it, it is not without its faults. Its main problem lies in a severe reliance on initial conditions. Without resorting to the fine-tuning of these parameters, the model cannot explain the observed flatness, homogeneity and structure in the present-day universe. Inflation, although not being without faults itself, provides a single explanation for the appearance of our universe. On top of that, it also provides testable predictions, some of which have already been confirmed to be true. Starting with the hot Big Bang model, we will uncover its weaknesses and introduce the solution inflation provides.
On large scales the universe is thought to be homogeneous and isotropic. The most general metric describing such a spacetime is the Friedmann-Lemaître-Robertson-Walker (FLRW) metric. The line element is given by:
ds2 = −dt2+ a2(t) dr 2 1 − kr2 + r 2(dθ2 + sin2 θdφ2) ! . (1.1)
1.1. BIG BANG PUZZLES
This metric can be understood as describing a series of three-dimensional homogeneous and isotropic hypersurfaces that only depend on time. The dimension of time adds the hypersurfaces together to form a four-dimensional spacetime. The spatial hypersurfaces are allowed to be curved. The parameter k denotes whether the curvature is positive (k = 1), zero (k = 0) or negative (k = −1). The spatial part of the line element is given in comoving spherical coordinates. These coordinates will remain fixed as the universe is expanding ( ˙a > 0) or contracting ( ˙a < 0). Here
a(t) is denoted as the scale factor. From the scale factor the Hubble parameter, parametrising
the expansion or shrinking rate of the universe, can be obtained:
H(t) = ˙a
a. (1.2)
In the hot Big Bang model the universe is thought to emerge from an extremely hot and dense state. The universe contains both relativistic particles, e.g. photons and heavier ones that will become non-relativistic at less extreme temperatures. In the beginning, the energy density of the universe is believed to be dominated by radiation. The corresponding expression for the scale factor is proportional to t1/2. Therefore the universe expands. This expansion will stretch the wavelengths of the photons, lowering their energy. Therefore, after a period of time, the photons stop to be the dominating source of energy in the universe. Non-relativistic matter takes over at that point. Such a stage in an universe will have a corresponding scale factor proportional to t2/3.
Consequently, the universe will keep on expanding, speeding up a little in the process. Shortly after this point, the photons in the universe will start to move freely. The universe is expanding such that the scattering rate of photons and matter goes to zero. From that point onward, (almost) all photons will travel freely: the cosmic microwave background (CMB) is created.
An important measure in an expanding universe is the comoving Hubble radius:
(aH)−1. (1.3)
Since the Hubble parameter has dimensions of inverse time (or inverse length because c = 1), the comoving Hubble radius describes a distance evolving with time. It can be shown that points separated by more than (a(ti)Hi)−1are not in causal contact at that specific time (ti); the universe is expanding too rapidly. A light ray emitted from one of the points will not make it to the other. The points are out of causal contact and will not be able to influence each other.
In the hot Big bang model, where the scale factor starts as a ∝ t1/2 and evolves into a ∝ t2/3,
it can be shown that the comoving Hubble radius is always increasing. Therefore more and more points become causally connected as time increases, but points not in causal contact yet, have
never been. This will lead to, arguably, the two largest problems in the hot Big Bang model: the
flatness and horizon problem.
To understand the flatness problem, one should first recall the generality of the FLRW metric depicted in (1.1). Only requiring isotropy and homogeneity at large scales seemingly allows for curvature. Observations indicate that the universe today is well described by a flat, k = 0, FLRW metric. When the flatness of the universe is parametrised as a deviation from the perfectly flat case (Ω(a) = 1), it can be shown that:
|1 − Ω(a)| ∝ (aH)−2. (1.4) Since (aH)−2decreases as one goes back in time, a flat universe now implies an even flatter universe in the past. In the hot Big Bang model, the flatness currently observed implies a deviation from a perfectly flat universe of |1 − Ω(aPL)| ≤ O(10−61) when the age of the universe was one Planck
time: ∼ 5 · 10−44seconds after the Big Bang. Realising that the model does not predict any initial value for |1 − Ω|, it is strange that the initial value is so close to zero [4].
The ever-increasing comoving Hubble distance also indicates that scales that were outside causal contact at the time of the formation of the CMB must have been out of causal contact since the Big Bang. The horizon problem arises, when one realises that we observe a (nearly) homogeneous CMB and universe. How did these disconnected regions exchanged information about their temperature and density? The homogeneity observed in the CMB (and universe) can only be explained in the hot Big Bang model by introducing fine-tuned velocities of all particles created in the Big Bang across causally-disconnected regions of space.
1.1.1
Inflationairy solution
Due to the way the flatness and horizon problem were stated in the previous section, a possible solution to them seems evident: just after the Big Bang, one should postulate a period in which the comoving Hubble radius decreases with time:
d dt (aH) −1 < 0. (1.5) Or equivalently: −¨a (aH)2 < 0, (1.6)
so that it becomes clear that inflation describes an expanding universe that is accelerating, since from the above it is implied that:
d2a
1.2. INTRODUCTION TO INFLATION
Rewriting the second time derivative of the scale factor as a function of the Hubble parameter yields: ¨ a a = H 2(1 − ), (1.7) where: ≡ − ˙ H H2 (1.8)
quantifies the amount of acceleration. For < 1 the universe will undergo accelerated expansion and thus inflate. From (1.4) it can be seen how a decreasing comoving Hubble radius will flatten the spatial hypersurfaces in the FLRW metric such that the initial flatness of the universe can be explained by just having inflation last long enough. The horizon problem is solved by the realisation that during inflation the comoving Hubble radius increases as one goes back in time. Therefore, all seemingly causally-disconnected regions in the CMB (and present day universe), may originate from a single causal region if inflation lasts long enough.
1.2
Introduction to inflation
We have assumed an inflationary phase of the universe as a solution for the problems described in section 1.1. Inflation was defined as a period with a decreasing comoving horizon (aH)−1, which implies an accelerating universe with ¨a/a = H2(1 − ). Since must be small compared to one,
the scale factor is approximately
a(t) ≈ eHt,
up to corrections proportional to .
Formally, the most elegant way to describe a universe with the above properties is by con-sidering a de Sitter universe where the time-invariance of H is slightly broken. A pure de Sitter universe is the most symmetric vacuum solution to the Einstein equations that obeys the above requirements (and thus does not need any additional degrees of freedom on top of the ones from standard general relativity). The only catch is that de Sitter requires a Hubble constant exactly constant with time. This is in contrast to the inflationary solution described in section 1.1.1, which does not have such stringent requirements: it also works with a small, but nonzero, time derivative of H. However, since it is generally easier to break symmetries than to introduce them, we still choose to start with the symmetric de Sitter case and work from there.
Such an approach can be understood as a bottom-up approach. This procedure is most useful when one is unaware of the fundamental theory in the UV describing a phenomenon. In the case of inflation, one might argue that the fundamental processes are described by string theory or some other renormalizable theory of quantum gravity, since the energy scales involved are relatively close the quantum gravity scale (Mpl). String theory is (at the moment) not able to
paint a complete picture of the inflationary mechanism, so the more conservative approach is the bottom-up one. Here one considers all possible processes consistent with the known low-energy theory. One should also take into account any symmetries assumed to be relevant, to forbid some of the possible processes. Next an energy cutoff is introduced, which suppresses UV-dependent processes. Now one raises the cutoff above the inflationary energy scale to have a consistent theory where all uncertainties are parametrised by the cutoff. This theory is then called an effective field theory1 or EFT.
The biggest risk one faces while using the bottom-up approach is that, although everything is consistent with the low-energy theory, one takes the cutoff up to a regime where degrees of freedom from the fundamental UV theory become essential. For example, a symmetry of the UV theory might be spontaneously broken just below the scale of inflation. This symmetry potentially influences the inflationary mechanism, but is not taken into account by the low energy theory that is promoted to the EFT around inflation.
Starting from a known theory of quantum gravity working at Planck scale energies and de-scribing a low-energy process like inflation, dubbed the top-down approach, also arrives at an EFT that describes inflation. Now all relevant degrees of freedom should be known. Why one would want to transform a complete theory in the UV to an EFT at a lower energy scale will become clear later on when an EFT action is constructed. As it will turn out, from the EFT action it is much more straightforward to determine the relevant operators at a certain energy scale than it is from the full theory.
We will start with a quick mathematical overview of the background space-time in a de Sitter universe. Next we will show how, by breaking the time-invariance of this background, one arrives at a model-independent, bottom-up, description of single-field inflation.
1.2. INTRODUCTION TO INFLATION
1.2.1
De Sitter space
De Sitter space in d = 4: dS4, is the maximally symmetric spacetime in d = 4 with positive
curvature2. Since it is maximally symmetric, it must have constant 4-curvature which is homogen-eous and isotropic. Maximally symmetric spacetimes have the maximum number of independent Killing vectors. The number of vectors only depends on the dimensionality d: there exist 12d(d + 1)
Killing vectors for a maximally symmetric space. For de Sitter space we thus expect an isometry group with 10 generators [5]. This claim can be examined by embedding the space into a higher dimensional space; dS4 can be described as a hyperboloid embedded in a d = 5 Minkowski space
[6]:
−X2
0 + X12+ X22+ X32+ X42 = l2, (1.9)
where l is the de Sitter radius, which we will redefine as the Hubble distance, so:
l2 ≡ 1
H2,
where H denotes the Hubble constant. Such an embedding is an extrinsic embedding where the curvature of the spacetime is represented as motion on a curved hypersurface. See figure 1.1. In this embedding, one can check that the isometry group of dS4 is SO(4, 1). This group has 10
generators: 4 boosts and 6 rotations, which matches our previous statement.
One can also choose to describe dS4in four-dimensional space instead of embedding it into
five-dimensional Minkowski space. Now the information about the curvature of the space is encoded in a non-trivial metric. This is denoted as an intrinsic embedding. We will investigate two intrinsic embeddings: one described by planar coordinates and one described by static coordinates.
Planar coordinates With the following coordinate transformations:
X0 = 1 H sinh(Ht) + H 2e Htxix i, −∞ < t < ∞ Xi = xieHt, −∞ < xi < ∞ i = 1, 2, 3 X4 = 1 H cosh(Ht) − H 2 e Ht xixi, the de Sitter metric becomes:
ds2 = −dt2 + e2Htdxidxi. (1.10) 2The other maximally symmetric spacetimes in d = 4 are Minkowski and anti-de Sitter space for no and
Figure 1.1: The embedding of de Sitter space into 5-dimensional Minkowski spacetime. The vertical direction represents the time coordinate x0. In a cylindrical coordinate system with the
height given by X0, the radial direction is given by R = qX2
1 + X22+ X32+ X42. Every point in
the figure represents a unit 3-sphere. The dotted circle represents the most narrow point of the hyperboloid. At this point R = 1/H. Figure taken from [6].
It is clear how de Sitter, with these coordinates, is just a special case of the flat FLWR metric with
a(t) = exp Ht (see appendix A). However, this description of de Sitter is not entirely equivalent
to the one in (1.9). By adding X0 and X4, it becomes clear that the Planar coordinates only
describe half of the de Sitter manifold:
X0+ X4 =
1
He
Ht,
which must be equal or larger than zero. This can be understood as a description of the half of the hyperboloid in figure 1.1 where X0+ X4 ≥ 0. By looking at the metric (1.10), it is clear that
∂t is not a Killing vector. Contrary, to Minkowski space, which is symmetric under translations in time, there exists no global timelike Killing vector in de Sitter space [6]. Nevertheless, de Sitter space is still maximally symmetric and must, just like Minkowski space, be homogeneous.
1.2. INTRODUCTION TO INFLATION
Where in Minkowski space homogeneity is reached by an invariance under translations in space and time, homogeneity in de Sitter space is reached by a combination of translations and proper conformal transformations. These two transformations together form a rotation which is one of the generators we mentioned before. See [7] for a thorough treatment of this point. The end result is the same: de Sitter space is homogeneous (and isotropic), as should be the case for a maximally symmetric space.
The absence of time-translational invariance and the invariance under proper conformal trans-formations can be illustrated by rewriting (1.10) in conformal time: dτ = dt/a. Using a(t) = exp Ht, we obtain: τ = Z t 0 e−Ht0dt0 (1.11) = −1 He −Ht (1.12) and a(t) = − 1 Hτ (t).
The line element then becomes3:
ds2 = a(τ )2−dτ2+ dxidx i
, a(τ ) = − 1
Hτ. (1.13)
We see that the metric in conformal coordinates is also not invariant under translations of τ . However, the invariance under proper conformal transformations has become apparent in these coordinates. Consider:
τ → τ0 = γτ
xi → x0i = γxi
One can check that this conformal rescaling of the coordinates keeps the line element invariant. The Minkowski metric is obviously not invariant under this rescaling. This invariance is not respected if ˙H 6= 0.
We will now have an intermezzo to discuss how the conformal line element provides an in-terpretation of the inflationary solution to the horizon problem [4]. Start with the notion that in the standard hot Big Bang cosmological model, the Big Bang is placed at τ = 0, since a(τ ) 3Note that this expression for a(τ ) is only valid for τ < 0. For positive τ we need to use −a(τ ) as scale factor.
must vanish at that point. The reason for this is that during the radiation and matter dominated periods of the universe the scale factor evolves proportional to τ and τ2 respectively. Thus,
ex-trapolating backwards in conformal time leads to the conclusion that, at τ = 0, the scale factor must vanish, i.e. the Big Bang. As mentioned before, having a radiation and matter dominated stage immediately after the Big Bang leads to the horizon problem. In figure 1.2 the problem is visualised. Simply put, the homogeneity observed in the CMB is due to regions that during recombination were not all in causal contact since they were not given enough conformal time.
Figure 1.2: Conformal diagram of Big Bang cosmology. When considering conformal time, null geodesics are at 45◦ angles to τ . Thus, just as if one is dealing with Minkowski space, one may draw light cones at 45◦ angles. At recombination (τrec) regions of the universe are not necessary
in causal contact. Figure taken from [4].
If one considers inflation described by a de Sitter universe, the conformal history of the universe is expanded to include negative values for the conformal time τ . This can be seen by looking at the scale factor of de Sitter in Planar coordinates (1.13) and noting that this expression is valid for negative τ . In de Sitter, the Big Bang singularity (a(τ ) → 0) is forced back to the infinite conformal past (τ → −∞), which due to (1.12) also corresponds to t → −∞. Consequently, regions of the universe during recombination have had plenty of time to be in causal contact in the conformal past, so the horizon problem is solved.
1.2. INTRODUCTION TO INFLATION
an equally big problem. However, recall from (1.12) that τ → 0 corresponds to t → ∞. So in de Sitter the point τ = 0 is the end of the line. The universe will never get to the radiation dominated phase. Evidently, here one encounters the point were the disadvantages of describing inflation as a pure de Sitter phase become clear. A pure de Sitter phase will extend indefinitely into the past and future and leave no room for any deviation from this course. This is the reason that one speaks of a quasi-de Sitter phase of the universe, i.e. a period in the evolution of the universe in which the universe is approximately de Sitter. The deviation from pure de Sitter is obtained by allowing for a small nonzero time derivative of the Hubble constant. This will allow us to smoothly connect the inflationary phase with a ∼ −(Hτ )−1 for τ < 0 to the radiation dominated phase with a ∼ τ , where τ > 0. This way the problem at τ = 0 is solved. See figure 1.3 for a visualisation of the inflationary solution to the horizon problem.
Static coordinates We now return to our treatment of pure de Sitter space. We have seen that instead of invoking a global timelike Killing vector, the homogeneity in de Sitter is reached by spatial translations and rotations. So de Sitter space can still claim to be one of the three possible maximally symmetric spacetimes in four dimensions. Unfortunately, the lack of a global invariance under time diffeomorphisms, prevents us from globally breaking it weakly to obtain a quasi-de Sitter phase by introducing a small, nonzero ˙H. Why this is worrisome will become clear
when we get the effective field theory description of inflation in section 1.2.2. First, we will show how in a certain part of de Sitter space there does exist a timelike killing vector.
Under the following coordinate transformations, the embedding of de Sitter space is changed from the extrinsic one (1.9) to the intrinsic embedding in static coordinates [6]:
X0 = s 1 H2 − r 2sinh(Ht), X1 = s 1 H2 − r 2cosh(Ht), X2 = r sin(θ) cos(φ), X3 = r sin(θ) sin(φ), X4 = r cos(θ).
Note that X0+ X1 > 0, so, similarly to the embedding in planar coordinates, we only describe
half of the de Sitter manifold. The line element in the static coordinates is given by:
ds2 = (1 − H2r2)dt2− dr
2
1 − H2r2 − r 2
Figure 1.3: Conformal diagram of inflationary cosmology. At recombination (τrec) all regions of
the universe have been in causal contact. The scale factor describing the flat FLRW metric is approximately −(Hτ )−1 during ∞ < τ < 0 and smoothly transforms to a(τ ) ∝ τ around the point τ = 0. At this point inflation ends. Figure taken from [4].
Observe that the line element now is evidently invariant under temporal translations, i.e. ∂t is a timelike Killing vector. Meanwhile, the spatial homogeneity is sacrificed. The line element should remind one of the Schwarzschild metric. Indeed there exist a coordinate singularity at a radius
r = H−1. Thus, an observer located at r = 0, is surrounded by a horizon at a distance of H−1. It can be shown that at the horizon ∂t vanishes and that outside this horizon it becomes a spacelike
1.2. INTRODUCTION TO INFLATION
vector [6]. This validates our previous statement about the non-existence of a global timelike Killing vector in de Sitter space. In the light of our search for a translational invariant part of the de Sitter manifold, we are not satisfied with the description using static coordinates. This is because, despite the existence of a timelike Killing vector in this patch (inside the horizon) of the de Sitter manifold, there is no spatial translational invariance.
Invariance under time translations
We have seen that to find translational invariance in de Sitter, one must only look at a patch of the manifold since there exist no global translational invariance. Furthermore, we can conclude that the description using static coordinates does not suffice. We must make one more concession. We will not look for timelike Killing vectors by considering coordinates of which the metric is independent. Instead we must look at the seemingly more contrived Killing vectors. Remember that de Sitter must have ten independent Killing vectors.
We return to the Planar coordinates with the line element given by (1.10):
ds2 = −dt2 + e2Htdxidxi. Now consider the “dilation" isometry4 of this metric:
t → t0 = t + ∆t, ~x → ~x0 = e−H∆t~x. (1.15) If we treat ∆t as an infinitesimal parameter we may expand the exponential to first order. If we then define the Killing vector K by an infinitesimal isometry, i.e. xµ → xµ+ Kµ∆t, we see that
Kµ is given by:
Kµ = (1, −H~x) . (1.16)
To check if we are truly dealing with a Killing vector, one may plug Kµ into Killing’s equation: ∇(µKν) = 0 [5] and check whether it is satisfied (it is). The norm of Kµ is given by:
gµνKµKν = −1 + e2HtH2|~x|2,
so obviously the vector is timelike for |~x| = 0. The vector will remain timelike up to the point
where it hits a horizon given by: |~x| = e−HtH−1. Note that we can rewrite this horizon as (aH)−1. 4A real dilation cannot be a isometry, since it, by definition, does not preserve distances. In this case, the
dilation is only performed on the spatial part of the metric, such that, when combined with a translation in time, the line element remains unchanged.
So, as one stays inside the comoving Hubble radius (1.3), the Killing vector is timelike and thus de Sitter space can be treated as stationary. Inside the horizon there must exist a coordinate system in which de Sitter has unbroken time translations. This subtlety is largely ignored by authors that use a broken time translational isometry of de Sitter to construct a quasi-de Sitter phase. The authors of [8] came to the same conclusion as was reached here.
1.2.2
Effective field theory of inflation
Effective field theory
The effective field theory of inflation is relatively complicated due to the involvement of gravity. Since the reasoning behind the effective field theory approach builds on general statements in quantum field theory, it is useful to introduce EFT in a simpler case where gravity is not con-sidered. One can then later generalise to a situation which includes gravity. Readers familiar with Wilsonian effective field theories may skip to the next section were we break the dilation isometry in the planar patch of de Sitter space to obtain inflation.
An effective field theory, or more precisely: a Wilsonian effective field theory can be reached both from a top-down as from a bottom-up perspective. In both approaches the regime of validity of the EFT must be specified by introducing an energy cutoff Λ. Particles with masses or momenta above this cutoff are considered to be irrelevant for the validity of the effective theory below the cutoff. Consequently, they will not directly appear in the effective action. However, the contributions of these high energy parameters to the low energy (E ≤ Λ) theory are not completely thrown away. This is because the EFT action, unlike the complete UV theory, will generally contain an infinite number of operators. At low energies only a small number of low-order operators will be significant. However, at energies approaching the cutoff, the higher-order operators will start to dominate. The contribution of the UV theory is thus parametrised by the energy cutoff. Therefore, at any energy scale below Λ, one can estimate the significance of the lost UV information [9].
In the top-down approach one has knowledge of the full theory in the UV, thus, the action appearing in the path integral contains all information about the theory. Consider for example a renormalised scalar theory in 3 + 1 Minkowski space [10]. The path integral is then given by:
Z(J ) =
Z
Dφ eiRd4x[L+J φ]
1.2. INTRODUCTION TO INFLATION
Performing a Wick rotation by defining Euclidean time5: τE ≡ it, yields:
Z(J ) =
Z
Dφ e−Rd3x dτ
E[LE+J φ].
Next, one switches to Euclidean momentum space by using:
φ(x) = Z d4k (2π)4e ikx e φ(k),
and introduces the cutoff |k| > Λ. Performing the path integral for all φ(k) with |k| > Λ ande
setting J (k) = 0 for |k| > Λ yields the Wilsonian effective action:e
e−Seff(φ) =
Z
Dφ|k|>ΛeSE(φ),
where the tildes were dropped. One can now use the effective action in a path integral that only considers φ(k) with |k| < Λ.
The bottom-up approach arrives at a similar EFT. However, since one does not know the full UV theory in this case, the method discussed above is of no use to obtain an EFT. Instead one makes assumptions about the symmetries of the theory that must be present at the energies at which the EFT is valid. Then one constructs the effective action by simply writing down every possible operator allowed by the symmetries.
As an example, consider a scalar in Minkowski space that is assumed to be symmetric under
φ → −φ. The EFT action constructed from the bottom-up perspective will look like [9]: Seff[φ] = Z d4x " Ll[φ] + ∞ X i ci Oi[φ] Λδi−4 # , (1.17)
where the action is split in a renormalisable part Ll and a sum of all possible non-renormalisable operators of dimension δi. The dimensionless coefficients ci are the Wilson coefficients. In some cases no renormalisable terms in the action will be present. However, the EFT action consisting of the sum of non-renormalisable will still provide a valid description of the theory below the cutoff. Note that even for a completely renormalised UV theory, the EFT will have a sum of non-renormalisable operators. Since one is bound by the cutoff, the theory remains well behaved. The sum of non-renormalisable terms in (1.17) is split up into terms depending on the
dimen-5Switching to Euclidean time will assure that all terms in the action will be positive, independently of the field
sionality of the operators O[φ], e.g.
c0Λ4, c2Λ2φ2 : relevant operators (O[φ] : d < 4)
c4φ4 : marginal operators (O[φ] : d = 4) ∞
X
i=6
ci Oi[φ]
Λδi−4 : irrelevant operators (O[φ] : d > 4).
(1.18)
Relevant operators will dominate the dynamics in the IR, i.e. for field values below Λ. In the UV: when field values around or above the cutoff are considered, the sum of irrelevant operators will become dominant. Marginal operators contribute equally in the IR and UV. More operators than the ones depicted above are possible, e.g. derivates of φ. Again the dimensionality of the operator will make it fit in one of the three categories.
Naturalness
In the EFT procedure one introduces the cutoff Λ and the unknown Wilson coefficients. To get an estimate of these parameters, one invokes the notion of naturalness. Both top-down and bottom-up naturalness exist. Since in the following only the bottom-bottom-up approach will be used, we restrict ourselves to discussing bottom-up naturalness here [9].
In a typical bottom-up scenario, one knows the renormalisable part of the action: Ll. Fur-thermore, one also assumes certain symmetries to be present. The Wilson coefficients and cutoff are unknown. One may estimate the size of the Wilson coefficients by calculating loop correction to the parameters of Ll. Parameters are then said to be natural if their measured values are larger than the loop corrections. Raising the cutoff will generally increase the contribution of the loop corrections. Typically, the cutoff can be raised up to a point where loop corrections start to dominate. Just below this point one would expect extra degrees of freedom from the full UV theory to become important in explaining the measured values of the parameters.
For example, by looking at (1.18), it can be seen that the renormalisable mass term in Ll will be corrected by the c2Λ2φ2 term. Even if c2 is taken to be much smaller than one, at some value
for the cutoff, the correction c2Λ2 will inevitably become equal to m2 in Ll. At that point the value of m2 stops being natural.
It can be seen that for Wilson coefficients of order unity, both the scalar mass and the zeroth order part of the potential will go up to the cutoff. The only way to naturally counteract this effect is by invoking a symmetry that suppresses the Wilson coefficients. In essence, both the hierarchy problem associated with the low Higgs mass as the small cosmological constant observed today are due to this effect. In both cases on needs new physics, i.e. new symmetries, at relatively low
1.2. INTRODUCTION TO INFLATION
scales. Without these, the parameters will scale with the cutoff and will far exceed their measured values.
In chapter 3, it will become clear that, in the case of inflation, the scalar mass will be pushed up at least to the Hubble scale when it is not sufficiently protected by symmetries. With such a large inflaton mass, inflation will not last long enough to produce the required 60 e-foldings.
Inflation as spontaneous symmetry breaking
As was mentioned in section 1.2.1, a de Sitter spacetime is not a good description of the early universe. This is because de Sitter space does not allow for a transition to the subsequent stages of the universe. Somehow, space must have a “clock” that tells it when it is time to stop expanding (quasi-)exponentially. From the extrinsic embedding of de Sitter in five-dimensional Minkowski space (1.9), we have seen that the only parameter describing de Sitter is H. In pure de Sitter H is constant in space and time, but for our purposes it is the appropriate (and only) candidate to promote to the cosmological clock. Since we still want to keep most of the exponential expansion de Sitter provides, we only let H depend on time weakly. We thus obtain a quasi-de Sitter phase by introducing a small, but non-zero, time derivative of the Hubble parameter.
As we argued in section 1.2.1, de Sitter space has no global invariance under translations in time, for our purpose we can treat it as if it does. More specially, when using the planar coordinate system, it can be shown that inside the comoving Hubble radius: (aH)−1, there exist at least one timelike and one spacelike Killing vector. However, if we now introduce ˙H 6= 0, it can be checked
that the dilation symmetry (1.15) is broken. Consequently, the timelike Killing vector that was found also vanishes. Effectively, time translational invariance is broken and we only consider the invariance under spatial diffeomorphisms.
In a theory which is only invariant under spatial diffeomorphisms there exists a preferred slicing of spacetime: ˜t(xµ). This introduces a gauge freedom, which we will fix by setting ˜t = t. This
choice is called unitary gauge [8, 11]. It is shown in appendix B that the effective action depicted below is the most general action in unitary gauge with terms constructed from the metric:
S = Z d4x√−g " 1 2M 2 PlR − c(t)g 00− Λ(t) +1 2M2(t) 4(δg00)2+ 1 6M3(t) 4(δg00)3−M¯1(t)3 2 (δg 00)δKµ µ− ¯ M2(t)2 2 δ(K µ µ) 2− M¯3(t)2 2 δK µ νδK ν µ+ . . . # . (1.19)
Note that the perturbed metric is defined around the flat FLRW metric (gµν = gµν
(0)+ δg
µν). K µν
represents the extrinsic curvature, which is a quantity constructed from the induced metric on surfaces of constant ˜t: hµν ≡ gµν+ nµnν. The action (1.19) is an expansion in perturbations and derivative of the metric. From section 1.2.2 we know that terms involving derivatives of the metric are mostly irrelevant in the IR. Furthermore, perturbations of the metric are taken to be small with respect to the unperturbed FLRW metric. As a result, the first three terms of the action will dominate the dynamics [11]. We will show that these three terms encapsulate a large class of single-field inflation models. This is not to say that the full action has no other use. In fact, the framework has proven to be very suitable for describing non-gaussianities in the CMB [12], Dark Energy [8] and stable models that violate the null energy condition [13]. The method has also been generalised for multi-field inflation models [14]. Since the focus of this work lies somewhere else, we will only use the EFT of inflation as a model-independent, bottom-up, introduction to inflation.
We concentrate on the first three terms of (1.19):
S = Z d4x√−g 1 2M 2 PlR − c(t)g 00− Λ(t). (1.20)
These terms determine the unperturbed background evolution of the spacetime [11]. We can use the equations of motion to fix the parameters Λ(t) and c(t). The Einstein-Hilbert term will result in the usual Einstein equations in vacuum. The second and third term will contribute to the right-hand side of Einstein’s equations: Tµν/Mpl2. The energy-momentum tensor: Tµν, can be found by varying the background action:
Tµν = − 2 √ −g δSb δgµν, where Sb is given by:
Sb = Z d4x√−gh−Λ(t) − c(t)g00i = Z d4x√−gh−Λ(t) − c(t)δ0 µδ 0 νg µνi .
Varying Sb leads to:
δSb = Z d4xh−δ√−g Λ(t) − δ√−g c(t)g00−√−g c(t)δ0 µδ 0 νδg µνi .
The first two terms can be rewritten using: δ√−g = 1 2 √ −ggµνδgµν [5]. Leading to: δSb = Z d4x√−g 1 2gµνΛ(t) + 1 2gµνc(t)g 00− c(t)δ0 µδ 0 ν δgµν,
1.2. INTRODUCTION TO INFLATION
which yields:
Tµν = −gµν
Λ(t) + c(t)g00+ 2c(t)δ0µδ0ν.
With this expression for the energy-momentum tensor obtained, one can now derive the Friedmann equations, by taking the 00 component and the trace of the Einstein equations. This will fix Λ(t) and c(t). Note that for the FLRW metric one attains: G00 = 3H2(t) and Gµµ= −12H2(t) − 6 ˙H(t) (See appendix A). By plugging these expressions into the Einstein equations (Gµν = MPl−2Tµν), one obtains: 3H2(t) = 1 M2 Pl h −g00 Λ(t) + c(t)g00+ 2c(t)δ00δ00i = 1 M2 Pl [Λ(t) + c(t)] and −12H2(t) − 6 ˙H(t) = 1 M2 Pl h −δµµ(Λ(t) + c(t)) + 2c(t)g00i = 1 M2 Pl [− (2Λ(t) + 2c(t)) + 4c(t) − 2Λ(t)] = 1 M2 Pl h −6H2(t)M2 pl+ 2c(t) − 2Λ(t) i ⇒ H2(t) + ˙H(t) = −1 3M2 Pl [2c(t) − Λ(t)]
for the 00 component and the trace respectively. Solving for c(t) and Λ(t) yields:
c(t) = −MPl2H(t),˙ Λ(t) = MPl2 h3H2(t) + ˙H(t)i. (1.21)
Appearance of the inflaton
At this point it might seem unclear how the description above will lead to a useful model of inflation. Admittedly, the procedure up to this point has not yielded anything extraordinary. Nevertheless, all required ingredients are already present. To make the theory more accessible, one should note that the actions (1.19, 1.20) describe three degrees of freedom. The two ordinary graviton helicities are joined by a scalar mode. The metric has gained a longitudinal mode due to our gauge choice (unitary gauge). This third degree of freedom can be made explicit by performing the so called Stückelberg trick. When performed, a Nambu-Goldstone boson associated with the spontaneous symmetry breaking (caused by letting ˙H 6= 0) will appear. This boson will be
One starts by performing a broken time diffeomorphism: t → ˜t = t + ξ0(x), ~x → ˜~x = ~x and
then substituting ξ0(x(˜x)) → −˜π(˜x). Under the first transformation the 00 metric transforms as:
g00(x) → ˜g00(˜x(x)) = ∂ ˜x 0(x) ∂xµ ∂ ˜x0(x) ∂xν g µν(x),
while the determinant of the metric transforms as:
g → ˜g(˜x(x)) ∂ ˜x(x) ∂x
!2
.
Changing the integration variable to ˜x will cancel the factor |∂x/∂ ˜x| coming from q−˜g(˜x) such
that we obtain the following expression for the action (1.20):
S = Z d4x√−g 1 2M 2 PlR − Λ(t + π) + c(t + π)h(1 + ˙π)2g00+ 2(1 + ˙π)(∂iπ)g0i+ (∂iπ)(∂jπ)gij i . (1.22)
Note that we have already substituted in π(x) and have dropped the tildes. One can now check that we have restored full diffeomorphism invariance by demanding that π transforms as π(x) →
π(x) − ξ0(x) under the time-diffeomorphism t → t + ξ0(x).
The authors of [11] introduce the Stückelberg trick in the context of a non-Abelean gauge theory where a certain gauge transformation leaves the action changed. The Stückelberg trick is then employed to make the theory invariant under the gauge transformation at the expense of introducing a Goldstone boson for every generator of the gauge group. Unitary gauge then corresponds to setting all Goldstones to zero. The action will then return to its starting point (which is not gauge invariant). So one has two choices: working with a non invariant theory or working with a gauge invariant theory that contains extra particle species.
The case of the non-Abelean gauge theory can be directly translated to the case of a broken time-diffeomorphism. After all, general relativity is perfectly well described by a gauge theory with an invariance under diffeomorphisms as gauge symmetry. We indeed see that the theory becomes invariant after performing the Stückelberg trick and returns to its variant form after going to unitary gauge by setting π(x) = 0.
From the diffeomorphism-invariant action (1.22), it can be seen that the kinetic term of the Goldstone boson is given by:
2c(t + π)∂µπ∂νπgµν = −Mpl2H∂˙ µπ∂νπgµν,
where we have plugged in (1.21). The canonically normalised Goldstone boson is therefore given by: πc ≡ 2MPlH˙1/2π. Note that the kinetic term of the metric is given by the
Einstein-Hilbert term: M2
1.2. INTRODUCTION TO INFLATION
δgc00 ≡ δg00/M
pl. At leading order in derivatives and perturbations, the coupling between the
Goldstone boson: π and the metric in (1.22) is given by:
−2M2 plH ˙πδg˙
00 = − ˙H1/2˙π
cδgc00.
If we now return to the similar case of the non-Abelean gauge theory, we see that a canonical mixing term like to one above must be proportional to the mass of the longitudinal mode of the gauge boson [11]. Translating this back to the above result, leads to the conclusion that the mass of the longitudinal mode of the metric is given by ˙H1/2. This mode is removed from the metric by the introduction of the Goldstone boson. Therefore, one would expect the Goldstone particle to obtain a small mass proportional to ˙H, due to the direct mixing with gravity. Indeed this is
what is found when a more thorough analysis is performed using the ADM formalism [15]. The mass term of the Goldstone becomes equal to −3 ˙Hπ2, so it obtains a small mass: m2
π = 3 ˙H. Conceptually, this result can be understood by recalling the starting point of this section, where we introduced a small ˙H to break the invariance under time diffeomorphisms of de Sitter
space. After the Stückelberg trick, this broken invariance was restored, but with a cost: the appearance of a (canonically normalised) Goldstone boson (the inflaton). It therefore seems very natural to have the Goldstone mass term proportional to the symmetry breaking parameter ˙H.
We will see in the following that inflation can be described equally well by postulating the inflaton field and requiring that its potential is flat up to corrections proportional to ˙H. So there we again
see how having an inflaton mass of ˙H1/2 is required by the theory.
The true advantage of the description in terms of a Goldstone boson is that the high-energy behaviour of the theory can be treated in a controlled way. If one compares the number of derivatives in the kinetic term of the canonical Goldstone with the ones in the term describing the coupling between the Goldstone and metric perturbations, it becomes clear that the kinetic term will dominate in the UV. Therefore, one would expect the theory to break up into two separate theories in the UV. Indeed, this is the case. If one considers high enough energies, the
decoupling limit is reached. Consequently, a description of the correlation functions describing
the scalar fluctuations during inflation in terms of only πc is valid up to fractional corrections of order H2/M2
pl and ˙H/H2 [12]. The decoupling limit is obtained by probing the theory at
energies that are very large compared to the scale describing the Goldstone-gravity coupling: ˙
H1/2. Equivalently, one can also take the limits M
pl→ ∞ and ˙H → 0, while keeping Mpl2H fixed.˙
description. The action becomes: S = Z d4x√−g " 1 2M 2 PlR − Λ(t) − ˙Λπ + (c(t) + ˙cπ) −(1 + ˙π) 2+ (∂iπ)2 a2 !# ,
where we have expanded c(t) and Λ(t) to first order. The terms linear in derivatives of π can be integrated by parts. Take note here that the prefactor √−g = a(t)3, so it only depends on time.
Doing so yields: Sπ = Z d4x√−g " 1 2M 2 PlR − Λ(t) − c(t) ˙π 2 +(∂iπ)2 a2 ! −Λ + ˙c + 6H(t)c(t)˙ π # .
where one can check that the tadpole terms will vanish when Λ(t) and c(t) (1.21) are plugged in:
S = Z d4x√−g " 1 2M 2 PlR − M 2 Pl h 3H2(t) + ˙H(t)i+ Mpl2H(t)˙ ˙π2− (∂iπ) 2 a2 !# = Z d4x√−g " 1 2M 2 PlR − MPl2 h 3H2(t) + ˙H(t)i+1 2 π˙c 2− (∂iπc)2 a2 !# .
This action will provide the starting point for the minimally-coupled, single-field slow-roll models we will consider in the next section. Note that no mass term is present: the Goldstone is massless in the decoupling limit. However, recall that we have seen that π must have a small mass proportional to ˙H1/2.
1.3
Slow-roll inflation
In the previous section it was shown how introducing a small nonzero ˙H would break some of the
isometries in de Sitter space. Using the Stückelberg trick, the weakly broken isometries could then be restored, but at the cost of introducing a scalar particle with a mass given by the symmetry breaking parameter ˙H. In this section we will start by postulating this scalar and show how the
classical background dynamics of inflation follow from this naturally. Additionally, some very brief remarks about how inflation predicts nearly scale-invariant power spectra in the CMB are made.
1.3.1
Inflating with a scalar field
The simplest models of inflation involve a single scalar field minimally-coupled to gravity6, with
an action described by:
S = Z d4x√−g " M2 pl 2 R − 1 2∇φ · ∇φ − V (φ) # . (1.23)
1.3. SLOW-ROLL INFLATION
Expanding V (φ) to second order and assuming φ = −φ, the equation of motion corresponding to the scalar sector is the Klein-Gordon equation as one would expect. However, since the scalar is minimally coupled to gravity, the curvature of the spacetime must be taken into account.
0 = − m2φ = ∇µ∇νφ − m2φ. Generalising to arbitrary V (φ): 0 = −1 2∇µ(−g µν∂ νφ − gµν∂µφ δνµ) − V,φ = ∂µ∂µφ + Γνµλ∂ λφ − V ,φ.
The scalar field can be split into a part describing a homogeneous background ϕ(t) and an in-homogeneous perturbation δφ(xµ). Using the connection of the FLRW metric (see appendix A), the equation of motion for ϕ(t) becomes:
0 = g00∂0∂0ϕ + 3Hg00∂0ϕ − V,ϕ
= ¨ϕ + 3H ˙ϕ + V,ϕ. (1.24) Together with the Einstein equations for the unperturbed FLRW metric, the equation of motion for
ϕ(t) describes the evolution of the universe during inflation. It can be seen that the scalar field is
described by a damped harmonic oscillator, where the expansion rate of the universe: H, provides the friction for the oscillator and where V,ϕ provides the restoring force. By expanding V (ϕ) to second order, it can be understood how it is the mass of the inflaton that is mostly responsible for this force. Intuitively, this behaviour of the scalar field can be apprehended by recalling that a scalar in Minkowski space is described by a simple harmonic oscillator (the solution to the Klein-Gordon equation). However, because the scalar now resides in an expanding (or contracting) space, the harmonic motion is influenced. In an expanding space this can be understood as a friction due to the Hubble expansion. As a result, the scalar field dynamics are described as a scalar (the inflaton) slowly rolling down its potential.
Just as was done in the previous section, varying the action with respect to the inverse metric will result in Einstein’s equations. The Einstein-Hilbert terms will yield the Einstein’s equations in vacuum. The energy-momentum tensor is found by varying the scalar part of the action:
Tµν = − 2 √ −g δSφ δgµν = ∂µφ∂νφ − gµν 1 2∂ ρφ∂ ρφ + V (φ) .
It is now assumed that the scalar field can be described as a combination of a time-dependent back-ground field: ϕ(t), and a small, spacetime-dependent perturbation δφ(xµ). Since the background-value of the field is thought to dominate the evolution of the universe, the perturbations are neglected at this point.
Using φ = ϕ(t), it can be seen that the energy-momentum tensor is described by a perfect fluid (see appendix D.2): Tµν = diag(ρ, a2p, a2p, a2p). The energy density: ρ and pressure: p of the cosmic fluid, are thus given by:
ρ = 1 2 ˙ φ2+ V (φ), (1.25) p = 1 2 ˙ φ2− V (φ). (1.26)
The two Friedmann equations, describing the evolution of the universe in the presence of the nonzero, time-dependent energy-momentum tensor are obtained by by taking the 00 component and the trace of the Einstein equations (see appendix A for Rµν and R). In terms of ρ and p, the Friedmann equations become:
3H2 = ρ M2 pl , (1.27) and ¨ a a = − (ρ + 3p) 6M2 pl . (1.28)
Using the expressions found for ρ and p, the equation can be rewritten in terms of the scalar field:
3H2 = 1 M2 pl " ˙ ϕ2 2 + V (ϕ) # , (1.29) and ¨ a a = − 1 3M2 pl ˙ ϕ2− V (ϕ). (1.30)
The second Friedmann equation or acceleration equation (1.28) can be rewritten using the first Friedmann equation (1.27): ¨ a a = ρ 3M2 pl " 1 −3 2 1 + p ρ !# = H2 1 − 3 2(1 + w) , (1.31)
1.3. SLOW-ROLL INFLATION
where w is the equation of state parameter:
w ≡ p ρ.
It now becomes clear how (1.31), quantifies the deviation from the de Sitter limit: ¨a/a = H2. It
can also be seen how accelerated expansion of the universe will occur as long as the second term in the square brackets is smaller than one (or equivalently, as long as −1 ≤ w < −1/3). We denote this term with the letter :
= 3
2(1 + w). (1.32)
As it turns out, this term is equal to the parameter that was defined in (1.8). This parameter, called the first slow-roll parameter quantifies the expansion of the universe. For < 1 the universe will undergo accelerated expansion as was shown above.
By plugging (1.32) back into (1.31) and substituting: ¨a/a = ˙H + H2, it becomes clear how
(1.32) indeed can be described as:
= −H˙
H2. (1.33)
can also be expressed in terms of ϕ and H by plugging in (1.25) and (1.26): = 1 2M2 pl ˙ ϕ2 H2. (1.34)
During inflation ( < 1), the universe is very close to a de Sitter universe. Consequently, the scale factor grows quasi-exponentially with time: a(t) ∝ exp (Ht), where H is almost constant with time. If we denote the starting time of the inflationary phase with ti and let it end at tend, the universe will have expanded by a factor
a(tend)
a(ti) = e
H(tf−ti),
between ti and tend. The number of e-foldings inflation lasts is thus defined as:
Ne ≡ log
a(tend)
a(ti)
.
Taking into account the small time-dependence of H is done by taking the integral of H when determining Ne:
Ne=
Z tend
ti
It can be shown that to solve the problems mentioned in section 1.1, around 60 e-foldings of inflation are required [4].
Note that the expression for in (1.33) can also be interpreted as the rate of change of H with respect to the number of e-foldings dNe= Hdt since:
= −d log H dNe
. (1.36)
As a result, the existence of a second slow-roll parameter η, whose smallness ensures that stays small sufficiently long to have enough e-foldings, can be understood. This parameter is consequently defined as:
η ≡ −1 2 d log(H2) dNe = − 1 2 d dNe = −1 2 d log ˙H dNe = −1 2 ¨ H ˙ HH (1.37)
One should note that for inflation to occur only has to be smaller than one. However, to reach the required number of e-foldings (Ne ∼ 60) one must additionally demand |η| 1. For large values of |η|, will grow too fast, as can be seen by noting that the first expression for η in (1.37) implies:
= Ne=0e
2 η Ne.
On that account, the number of e-foldings inflation will last, will be:
Ne,f = −
log Ne=0
2η .
So, for an initial value of 0.01 for and 1 for η, this will only result in one e-folding. Note that here it is assumed that η does not vary with Ne.
Just like was done for the first, the second slow-roll parameter can also be expressed in terms of ϕ and H. This is simply done by plugging in (1.34) into (1.37):
η = − ϕ¨
H ˙ϕ. (1.38)
1.3.2
Slow-roll approximation
With inflation requiring both slow-roll parameters to be smaller than one, it can be understood that during inflation it is implied that ˙ϕ2 V (ϕ) and | ¨ϕ| |3H ˙ϕ|, |V
,ϕ|). The equation of motion for ϕ and the first Friedmann equation can thus be approximated as:
3H ˙ϕ ≈ V,ϕ, (1.39) 3H2 ≈ 1
M2 pl
