# Cosmic Inflation: Background dynamics,
Quantum fluctuations and Reheating

Swagat S. Mishra
(Email<EMAIL_ADDRESS>
###### Abstract

Cosmic inflation is a transient period of rapid accelerated expansion of space
which has been hypothesized to have taken place prior to the hot Big Bang
phase of the universe. It is the leading paradigm of the very early universe
that provides natural initial conditions for the hot Big Bang phase, both at
the background and at the perturbation level. In fact, quantum vacuum
fluctuations during inflation generate both scalar and tensor type primordial
perturbations that are correlated over super-Hubble scales. The scalar
fluctuations, upon their Hubble-entry, lead to the temperature and density
inhomogeneities in the primordial plasma which eventually grow via
gravitational instability to form the large-scale structure of the universe.
Furthermore, the inflationary tensor fluctuations constitute a background of
stochastic Gravitational Waves (GWs) upon their Hubble-entry in the post-
inflationary universe. The correlation functions of inflationary fluctuations
constitute the primary observational tool to probe the high energy physics of
the early universe.

Furthermore, in the inflationary paradigm all constituents of the universe at
the subsequent radiation-dominated epoch were created during a process known
as reheating, which is supposed to have taken place after the end of
inflation. In the simplest inflationary scenario, the energy density of the
universe was concentrated in a slowly evolving scalar field, called the
inflaton. After the end of inflation, the (almost) homogeneous inflaton
condensate started to oscillate coherently around the minimum of its effective
potential. The oscillating inflaton condensate then decayed and transferred
all of its energy to the quanta of other fields that are coupled to the
inflaton. Subsequently, the universe became thermalized at some high
temperature, marking the commencement of the hot Big Bang phase. Since
reheating is the intermediate stage between inflation and the radiation-
dominated hot Big Bang phase, it is associated with a number of important
aspects of the hot Big Bang cosmology.

In these lecture notes, we provide a pedagogical introduction to some aspects
of the inflationary cosmology including the background scalar field dynamics,
generation of primordial seed perturbations via quantum fluctuations during
inflation, and the process of reheating after inflation in the single-field
inflationary paradigm.

Figure 1:  [NASA/ESA, WMAP, PLANCK teams, Max Planck Institute for
Astrophysics, Auguste Rodin.]

Introductory Lecture Notes

Declaration

These are introductory lecture notes primarily based on a set of lectures I
delivered as a supplementary course in Cosmology for the PhD coursework at
IUCAA (India) in Jan/Feb 2024. Presentation of some of the topics originated
from my lectures for the Master students I supervised in between 2017-23. The
lectures are primarily intended for PhD students who have had introductory
courses on general relativity, quantum field theory, particle physics and
cosmology. I have made an attempt to compose a range of topics that are
somewhat complementary to many of the existing (excellent) lecture notes in
the literature and consequently, a number of key aspects of the inflationary
cosmology have not been included in the present version. The approach here is
relatively pedagogical and various concepts are presented from a personal
perspective. Some topics are covered quite comprehensively, and hence I hope
these notes will also be useful to researchers in the field. The present
version may quite possibly contain typos and errors. I would be grateful to
receive comments, suggestions, and typo corrections via email.

“Ages on ages, before any eyes could see

year after year, thunderously pounding the shore as now.

For whom? For what? On a dead planet

with no life to entertain.

Never at rest tortured by energy

wasted prodigiously by the sun, poured into space.

A mite makes the sea roar.

Deep in the sea all molecules repeat

the patterns of one another till complex new ones are formed.

They make others like themselves and a new dance starts.

Growing in size and complexity living things,

masses of atoms DNA, protein,

dancing a pattern ever more intricate.

Out of the cradle onto dry land,

here it is standing:

atoms with consciousness; matter with curiosity.

Stands at the sea, wonders at wondering:

I, a universe of atoms –

An atom in the universe.”

– Richard P. Feynman

These lecture notes are dedicated to all the small town and rural

school children aspiring to accomplish something of great

value in their lives, despite the circumstances.

## Acknowledgements

To begin with, I thank Surhud More for the primary incentive for these
lectures. I am indebted to Yuri Shtanov, Edmund Copeland and Rafid Mahbub for
spending their valuable time in reading the notes, and for providing
insightful comments and suggestions on the preliminary version, which have
greatly improved the quality of presentation in the current version. Special
thanks to many of my mentors, colleagues and students, including Varun Sahni,
Edmund Copeland, Yuri Shtanov, Alexei Starobinsky, L. Sriramkumar, Sanil
Unnikrishnan, Parth Bhargava, Alexey Toporensky, David Stefanyszyn, Anne
Green, Paul Saffin, Antonio Padilla, Oliver Gould, Rafid Mahbub, Sanjit Mitra,
Kandaswami Subramanian, Tarun Souradeep, Mohammad Sami, Prasant Samantray,
Shabbir Shaikh, H. V. Ragavendra and Bhavana Bhat, for various discussions and
exchanges on the topic over the years.

The cover page image has been designed by Siddharth Bhatt, compiling a number
of figures (credit: NASA/ESA, WMAP, PLANCK teams, Max Planck Institute for
Astrophysics, Auguste Rodin) using GIMP. All the figures in the main text have
been primarily generated by the author, at times with crucial assistance from
students and colleagues. In particular, I thank Sanket Dave, Mohammed Shafi,
and Siddharth Bhatt for their help in generating some of the figures.

I am supported by a STFC Consolidated Grant [No. ST/T000732/1] as a
postdoctoral Research Fellow at the University of Nottingham (UK). I am
grateful to IUCAA (India) for their hospitality and to Ranjeev Misra for his
kind support. I further thank the accommodating staff members of the Chef’s
Way cafe, where a significant portion of these lecture notes were compiled.

For the purpose of open access, the author has applied a CC BY public
copyright licence to any Author Accepted Manuscript version arising.

Data Availability Statement: This work is entirely theoretical and has no
associated data.

###### Contents

  1. 1 Introduction
  2. 2 Standard cosmological model: a brief review
  3. 3 Fine-tuning of initial conditions for the hot Big Bang
    1. 3.1 High degree of spatial flatness
    2. 3.2 Horizon problem
    3. 3.3 Super-Hubble correlation of primordial perturbations
  4. 4 The inflationary hypothesis
    1. 4.1 Addressing the flatness problem
    2. 4.2 Addressing the horizon problem
    3. 4.3 Generating super-Hubble fluctuations
  5. 5 Scalar field dynamics of inflation
    1. 5.1 Background dynamics during inflation
    2. 5.2 Quantum fluctuations during Inflation
  6. 6 Correlators of inflationary fluctuations
    1. 6.1 Statistics of vacuum quantum fluctuations
    2. 6.2 Scalar quantum fluctuations during inflation
    3. 6.3 Tensor quantum fluctuations during inflation
    4. 6.4 Beyond linear perturbation theory: primordial non-Gaussianity
    5. 6.5 Post-inflationary evolution of primordial fluctuations
      1. 6.5.1 Evolution of scalar fluctuations after inflation
      2. 6.5.2 Evolution of tensor fluctuations after inflation
  7. 7 Observational constraints on single field inflation
    1. 7.1 Implications of observational constraints
    2. 7.2 Dynamics of popular inflationary models
      1. 7.2.1 Monomial potentials
      2. 7.2.2 Asymptotically flat potentials
  8. 8 Post-inflationary inflaton dynamics and reheating
    1. 8.1 Coherently oscillating scalar field
      1. 8.1.1 Time-averaged equation of state
      2. 8.1.2 Period of an anharmonic oscillator
    2. 8.2 External coupling and decay of the inflaton condensate
      1. 8.2.1 Perturbative reheating
      2. 8.2.2 Non-perturbative reheating
  9. 9 Discussions
  10. 10 Supplementary sources
  11. A Hubble parameter in the early universe
  12. B Kinematics of reheating and the duration of inflation
  13. C Inflaton-clock dynamics: Hamilton-Jacobi formalism
  14. D Mukhanov-Sasaki equation for scalar fluctuations
    1. D.1 Comoving gauge
    2. D.2 Spatially flat gauge
  15. E Analytical solution of the Mukhanov-Sasaki equation
    1. E.1 In terms of Hankel functions
    2. E.2 In terms of Bessel functions
  16. F Inflationary fluctuations as a massive scalar in pure dS

## Units, Notation and Convention

• Einstein’s Gravity (GR) with metric signature $(-,+,+,+)$ • Natural Units
$\hbar,c=1$, reduced Planck mass $m_{p}=\frac{1}{\sqrt{8\pi G}}$ • Hubble
parameter $H$, Hubble radius $R_{H}$, comoving Hubble radius $r_{h}$, physical
size of causal horizon $d_{H}$, comoving horizon $r_{H}$ • $3$D spatial
vectors are denoted by an overhead arrow mark, e.g. the position and momentum
vectors in the comoving coordinates are denoted as $\vec{x}$, $\vec{k}$,
respectively. • Inflaton field $\varphi(t,\vec{x})$, homogeneous background
field $\phi(t)$ such that
$\varphi(t,\vec{x})=\phi(t)+\delta\varphi(t,\vec{x})$ • Fourier transformation
of a scalar field $\Psi(t,\vec{x})=\int\frac{{\rm
d}^{3}\vec{k}}{(2\pi)^{3}}\leavevmode\nobreak\ \Psi_{k}(t)\leavevmode\nobreak\
e^{i\,\vec{k}.\vec{x}}\quad\Rightarrow\quad\Psi_{k}(t)=\int\,{\rm
d}^{3}\vec{x}\leavevmode\nobreak\ \Psi(t,\vec{x})\leavevmode\nobreak\
e^{-i\,\vec{k}.\vec{x}}$ • Power spectrum of vacuum quantum fluctuations of a
field $\boxed{{\cal P}_{\Psi}(k)\,=\,\frac{k^{3}}{2\pi^{2}}\,|\Psi_{k}|^{2}}$
• Variance $\boxed{\sigma_{\Psi}^{2}=\int_{-\infty}^{\infty}\,{\rm
d}\Psi\leavevmode\nobreak\ \Psi^{2}\leavevmode\nobreak\
P[\Psi]\equiv\int_{k_{\rm min}}^{k_{\rm max}}\frac{{\rm
d}k}{k}\leavevmode\nobreak\ {\cal P}_{\Psi}(k)}$ • Scalar (comoving curvature)
fluctuations during inflation $\zeta$, scalar power spectrum ${\cal
P}_{\zeta}(k)$ • Tensor (transverse and traceless) fluctuations $h_{ij}$,
tensor power spectrum ${\cal P}_{T}(k)$ • Conformal time ${\rm
d}\tau=\frac{{\rm d}t}{a(t)}$ • Derivative w.r.t cosmic time
$\frac{\partial\Psi}{\partial t}\equiv\dot{\Psi}$ and w.r.t conformal time
$\frac{\partial\Psi}{\partial\tau}\equiv\Psi^{\prime}$ • Derivatives w.r.t
$\phi$ are denoted as $\frac{{\rm d}V(\phi)}{{\rm d}\phi}\equiv V_{,\phi}$,
$\frac{{\rm d}^{2}V(\phi)}{{\rm d}\phi^{2}}\equiv V_{,\phi\phi}$

##  1 Introduction

Remarkable theoretical and observational advancements in cosmology over the
past five decades have led to the emergence of the standard paradigm of the
universe, popularly known as the Big Bang Theory, within which the spatially
flat ${\rm\Lambda}$CDM model is often used as the fiducial/concordance model.
This standard paradigm of cosmology describes the evolution of the universe
reasonably well, starting from about 1 sec after the Big Bang, all the way
until the present epoch.

However, the success of the concordance model relies upon the existence of
hitherto unknown matter and energy constituents in the universe, in
particular, a non-baryonic dark matter component that reinforces the formation
of the large scale structure (LSS) of the universe in the matter-dominated
(MD) epoch, as well as a negative pressure dark energy component responsible
for the observed accelerated expansion of space closer to the present epoch.
In the concordance model, the dark matter is often assumed to be cold and
pressureless on cosmological scales, and dark energy is assumed to be the
cosmological constant $\Lambda$. The nature of dark matter and dark energy are
two of the greatest puzzles confronting the standard cosmological paradigm in
the $21^{\rm st}$ century.

On the other hand, the success of the standard paradigm also heavily relies
upon the specific primordial initial conditions at the beginning of the hot
Big Bang phase. At the background level, these initial conditions take the
form of (i) extreme spatial flatness of the universe throughout its evolution
history and (ii) high degree of spatial homogeneity and isotropy over large
length scales in the early universe. In particular, the average temperature of
the Cosmic Microwave Background (CMB) seems to be uniform up to a high degree
of precision across all sky. Since the standard Big Bang theory assumes that
the universe was born in the state of being radiation dominated (RD) with
decelerated expansion, both of the aforementioned initial conditions are
highly non-trivial (unnatural) and appear to be extremely fine tuned. More
importantly, we have also established the existence of tiny primordial
curvature fluctuations in the early universe (via temperature and polarisation
fluctuations in the CMB over a range of angular scales) which seed the
formation of the large-scale structure in the late universe via gravitational
instability. The latest CMB observations by the Planck collaboration [1, 2]
indicate that the initial curvature fluctuations were highly Gaussian, nearly
scale-invariant and predominantly adiabatic in nature. Consequently, these
specific properties of the primordial fluctuations, combined with the extreme
spatial flatness, homogeneity and isotropy strongly point towards the
existence of a non-standard (neither radiation nor matter dominated) transient
epoch in the very early universe prior to the commencement of the hot Big Bang
phase. In an expanding universe, assuming Einstein’s theory of Gravity to hold
at early times, these primordial initial conditions can be explained only if
this transient period exhibited rapid accelerated expansion of space, known as
‘cosmic inflation’.

The inflationary paradigm [3, 4, 5, 6, 7, 8, 9] has emerged as the leading
scenario for describing the very early universe and for setting natural
initial conditions for the hot Big Bang phase. One of the key predictions of
the inflationary scenario is the unavoidable quantum-mechanical production of
primordial (scalar) curvature perturbations on super-Hubble scales [10, 11,
12, 13]. In fact, the inflationary predictions for the statistical properties
of these scalar quantum fluctuations (made in the early 1980s) match extremely
well with the latest precision CMB observations by the Planck collaboration
(data released in 2018) over a range of length scales, which consolidates
inflation as a feasible scenario of the pre-hot Big Bang universe [14, 1, 2,
15, 16]. In its simplest realization, inflation is usually assumed to be
sourced by a single canonical scalar field $\varphi$ with a shallow self-
interaction potential $V(\varphi)$ which is minimally coupled to gravity [8,
9, 17]. At early times, when the inflaton is higher up in its potential, the
universe inflates nearly exponentially. While, towards the end of inflation,
the inflaton begins to roll rapidly down its potential before exhibiting
coherent oscillations around its minimum.

Another crucial prediction of inflation is the generation of tensor
perturbations via quantum fluctuations of the transverse and traceless part of
the metric tensor during the accelerated expansion, which constitute a
stochastic background of relic gravitational waves (GWs) [18, 19, 20]. Scalar
and tensor perturbations generated during inflation create distinctive
imprints on the primordial CMB radiation which can be used to deduce the
scalar spectral index $n_{{}_{S}}$ and the tensor-to-scalar ratio $r$ – two of
the most important observables which can be used to rule out competing
inflationary models [21, 17, 1, 2].

It is well known that the inflationary GW power spectrum at large scales
provides us with important information about the nature of an inflaton field
due to its direct relation to the inflaton potential [19, 22]. Of greater
importance is the fact that their spectrum, $\Omega_{{}_{\rm GW}}(f)$ (defined
as the fractional energy density of GWs per logarithm interval of their
frequency $f$ at the present epoch) and the spectral index $n_{{}_{\rm
GW}}={\rm d}\ln{\Omega_{{}_{\rm GW}}}/{\rm d}\ln{f}$ at sufficiently small
scales can serve as key probes to the high energy physics of the post-
inflationary primordial epoch. The primordial spectrum of relic gravitational
radiation at small scales is exceedingly sensitive to the post-inflationary
primordial equation of state (EoS), $w$. In fact, the GW spectrum has
distinctly different properties for stiff/soft equations of state. For a stiff
EoS, $w>1/3$, the GW spectrum shows a blue tilt: $n_{{}_{\rm GW}}>0$, that
increases the GW amplitude on small scales. A softer EoS, $w<1/3$, on the
other hand, leads to a red tilt, whereas the radiation EoS, $w=1/3$, results
in a flat spectrum with $n_{{}_{\rm GW}}\simeq 0$.

Another key aspect of inflationary cosmology is the epoch of reheating during
which the energy stored in the inflaton is transferred into the radiative
degrees of freedom of the hot Big Bang phase. The post-inflationary history of
the universe, prior to the commencement of Big Bang Nucleosynthesis (BBN),
remains observationally inaccessible at present, despite a profusion of
theoretical progress [23, 24, 25, 26, 27, 28, 29, 30, 31] in this direction.
It is expected, however, that the post-inflationary universe in between the
end of inflation and the beginning of the radiation domination phase passed
through a series of physical epochs [32], each of which can be characterized
by an EoS, $w_{i}$. Potential relics from this primordial epoch in the form of
primordial black holes, oscillons and gravitational waves will provide us with
key information about the dynamics of reheating in the upcoming decades.

In the following, we will provide a pedagogical introduction to the
inflationary dynamics. After a brief review of the standard cosmological
paradigm in Sec. 2, we will provide a discussion of the initial conditions for
the early universe in Sec. 3 and demonstrate that, in the standard radiation-
dominated universe, the initial conditions need to be highly fine-tuned. Then
we will discuss the proposal of the inflationary scenario in the pre-hot Big
Bang epoch to address the background initial conditions in Sec. 4. Sec. 5 is
dedicated to the background scalar field dynamics during inflation. In Sec. 6,
we will then turn our attention to the most important predictions of the
inflationary hypothesis: the generation of scalar and tensor fluctuations that
are correlated over super-Hubble scales. Sec. 7 is devoted to a discussion on
the latest observational constraints on inflation, while the post-inflationary
reheating dynamics is introduced in Sec. 8. We will provide a discussion of
some of the key aspects of inflationary cosmology that have not been
considered in detail in these notes in Sec. 9 and conclude by providing
references to supplementary material on inflation in Sec. 10.

Various appendices provide more technical as well as supplementary information
on the subject. Apps. A and B discuss the thermal history and post-
inflationary kinematics of the universe, respectively. App. C provides a
discussion on the Hamilton-Jacobi formalism, with an application to the power-
law inflation. Apps. D, E and F are dedicated to the analytical treatment of
fluctuations in the inflationary and de Sitter spacetimes.

##  2 Standard cosmological model: a brief review

As discussed in the introductory section, the observable universe appears to
be spatially flat, homogeneous, and isotropic on large cosmological scales
throughout its probed history, as inferred from the LSS and CMB observations
as well as BBN constraints. The space-time metric of a nearly homogeneous and
isotropic universe exhibits (approximate) spatial translation and rotation
isometries and hence, is represented by the Friedmann-Lemaître-Robertson-
Walker (FLRW) metric which takes the form

${\rm d}s^{2}=-{\rm d}t^{2}+a^{2}(t)\,\left[\frac{{\rm
d}r^{2}}{1-K\,r^{2}}+r^{2}\,{\rm d}\theta^{2}+r^{2}\,\sin^{2}{\theta}\,{\rm
d}\phi^{2}\right]\,;$ (1)

where $K=0,\,\pm 1$ characterizes the uniform curvature of constant-time
spatial hypersurface111The definition of $K$ used here might be different from
other lecture notes and books in the literature, e.g. some sources use
$\kappa=K/a_{0}^{2}$ where $a_{0}$ being the scalar factor at the present
epoch.. $K=0$ corresponds to a spatially flat $\mathbb{R}^{(3)}$ universe,
$K=1$ (positive spatial curvature) corresponds to a spatially closed
$\mathbb{S}^{(3)}$ universe and $K=-1$ (negative spatial curvature)
corresponds to a spatially open hyperbolic $\mathbb{H}^{(3)}$ universe, see
Refs. [24, 33, 34, 35]. Specializing to the spatially flat FLRW metric for
which $K=0$, we get

${\rm d}s^{2}=-{\rm d}t^{2}+a^{2}(t)\,\delta_{ij}\,{\rm d}x^{i}{\rm
d}x^{j}\,.$ (2)

The rate of spatial expansion is described by the Hubble parameter which is
defined as

$H=\frac{\dot{a}}{a}\,.$ (3)

Expansion of the universe results in redshifting of the wavelength (or
decreasing of the momentum along the geodesics) of massless particles such as
photons. The redshift is related to the scale factor via

$1+z=\frac{a_{0}}{a}\,,$ (4)

where $a_{0}$ is the present day scale factor for which $z=0$. We will not
discuss the kinematics of FLRW space-time and refer the interested readers to
standard textbooks on the subject, for example Ref. [35].

Dynamics of space-time, sufficiently below the Planck scale, is governed by
the Einstein’s Field Equations222We will not discuss modified or extended
theories of gravity in these lectures. Interested readers should see Ref. [36]
and references therein.

$G_{\mu\nu}\equiv
R_{\mu\nu}-\frac{R}{2}\,g_{\mu\nu}=\frac{1}{m_{p}^{2}}\,T_{\mu\nu}\,.$ (5)

For a homogeneous and isotropic distribution of matter-energy constituents
described by the energy-momentum tensor
$T^{\mu}_{\;\>\;\nu}\equiv\mathrm{diag}\left(-\rho,\,p,\,p,\,p\right)$, and
the metric represented by the FLRW line element given in Eq. (1), the time-
time and space-space components of Einstein’s field equations take the form

$\displaystyle H^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{3{m_{p}^{2}}}\,\rho-\frac{K}{a^{2}}\,,$ (6)
$\displaystyle\frac{\ddot{a}}{a}$ $\displaystyle\equiv$
$\displaystyle\dot{H}+H^{2}=-\frac{1}{6{m_{p}^{2}}}\,(\rho+3\,p)\,.$ (7)

$\rho$ and $p$ are energy density and pressure respectively. Eqs. (6) and (7)
are called the Friedmann equations. Eq. (7) determines whether the expansion
of the universe decelerates or accelerates, given the energy density and
pressure of the constituents. Energy-momentum conservation leads to

$\dot{\rho}+3\,H\,\left(\rho+p\right)=0\,,$ (8)

which can also be derived by combining Eqs. (6) and (7). In cosmology, we will
often work with perfect fluids for which the pressure and density will be
related via an equation of state (EoS) of the form

$p=w\,\rho\,,$ (9)

where the equation of state parameter $w\in\left[-1,\,1\right]$. Consequently,
we can write Eq. (8) as

$\dot{\rho}+3\,H\,\left(1+w\right)\rho=0\,,$ (10)

whose general solution for constant $w$ can be written as

$\rho(a)=\rho_{1}\,\left(\frac{a}{a_{1}}\right)^{-3(1+w)}\,,$ (11)

where $\rho_{1}$ is the energy density at some reference epoch $a_{1}$ which
is usually taken to be the scale factor of the present epoch. Incorporating
the above expression into Eq. (6), we obtain

$a(t)=a_{1}\,\left(\frac{t}{t_{1}}\right)^{\frac{2}{3(1+w)}}\,;\quad
H(a)=H_{1}\,\left(\frac{a}{a_{1}}\right)^{-3(1+w)/2}\,.$ (12)

Note that the EoS parameter for radiation (relativistic constituents), matter
(non-relativistic constituents), and a cosmological constant are given by
$w_{r}=\frac{1}{3}$, $w_{m}\simeq 0$, and $w_{\rm DE}=-1$ respectively, which
leads to the following expressions for the energy density

$\rho(a)\propto\begin{cases}1/a^{4}\,;&{\rm RD}\\\ 1/a^{3}\,;&{\rm MD}\,;\\\
{\rm const.}\,&{\rm CCD}\,,\end{cases}$ (13)

and the following expressions for scale factor in a spatially flat universe

$a(t)\propto\begin{cases}t^{1/2}\,;&{\rm RD}\\\ t^{2/3}\,;&{\rm MD}\,;\\\
e^{H\,t}\,&{\rm CCD}\,.\end{cases}$ (14)

A pure cosmological constant-dominated (CCD) universe is called the ‘de
Sitter’ (dS) universe for which the Hubble parameter $H$ is constant and the
scale factor expands exponentially. In general, the universe can be filled
with a number of different constituents at a given time (which is the case for
our own universe). We assume that different constituents with densities
$\rho_{i}$, and EoS parameters $w_{i}$, do not interact with each other (apart
from gravity). Then the Friedmann Eqs. (6), (7) and energy momentum
conservation Eq. (8) take the form

$\displaystyle H^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{3{m_{p}^{2}}}\,\sum_{i}\,\rho_{i}-\frac{K}{a^{2}}\,,$
(15) $\displaystyle\frac{\ddot{a}}{a}$ $\displaystyle\equiv$
$\displaystyle\dot{H}+H^{2}=-\frac{1}{6{m_{p}^{2}}}\,\sum_{i}\,\left(1+3\,w_{i}\right)\,\rho_{i}\,,$
(16) $\displaystyle\dot{\rho_{i}}$ $\displaystyle=$
$\displaystyle-3\,H\,\left(1+w_{i}\right)\rho_{i}\,,$ (17)

Figure 2: The evolution of the density of radiation, matter and dark energy in
the context of the flat $\Lambda$CDM concordance cosmology. Note that, while
the density of the matter and radiative degrees of freedom (represented by the
solid blue and green curves respectively) evolve with time, the density of the
cosmological constant $\Lambda$ (represented by the solid red curve) remains
constant.

A range of cosmological and astrophysical observations relevant to the
standard cosmological paradigm indicate that our universe was radiation
dominated at early times during the hot Big Bang plasma phase. It made a
transition to the matter dominated epoch around a redshift $z\simeq 3400$,
while continuing to exhibit a decelerated expansion throughout most of its
expansion history. However, more recently around the redshift $z\sim 0.7$ the
expansion of the universe started accelerating closer to the present epoch. A
cosmological constant $\Lambda$ with $w_{\rm DE}=-1$ explains most
cosmological observations very well333However, note that we are currently
confronted with the Hubble tension conundrum [14, 1, 37, 38, 39, 39, 40, 41,
42, 43] and a modification to the standard scenario might have some bearing
upon the resolution of this tension.. All the stable constituents of the
Standard Model of particle physics account for the radiation and baryon
fraction of the energy budget of our universe. While pressureless cold dark
matter (CDM) and a small positive cosmological constant ($\Lambda$) as the
dark energy constitute the dominant energy budget of the universe at the
present epoch. This is often referred to as the standard or concordance model
of cosmology, fittingly known as the flat $\Lambda$CDM model.

It is convenient to introduce dimensionless density parameters $\Omega_{i}$
which are defined as

$\Omega_{i}=\frac{\rho_{i}}{\rho_{c}}\,;\quad{\rm
with}\quad\rho_{c}=3\,m_{p}^{2}\,H^{2}$ (18)

being the critical energy density. Hence the Hubble parameter for the flat
${\rm\Lambda}$CDM model can be written as

$H=H_{0}\,\sqrt{\Omega_{0r}\left(a_{0}/a\right)^{4}+\Omega_{0m}\left(a_{0}/a\right)^{3}+\Omega_{0\Lambda}}=H_{0}\,\sqrt{\Omega_{0r}\left(1+z\right)^{4}+\Omega_{0m}\left(1+z\right)^{3}+\Omega_{0\Lambda}}\,,$
(19)

where $\Omega_{0i}$ is the dimensionless density parameter of a constituent at
the present epoch444In many textbooks and research papers, including Ref. [1],
$\Omega_{0i}$ is often denoted by simply $\Omega_{i}$. However, in these
lecture notes, we make a distinction between $\Omega_{0i}$ and $\Omega_{i}$,
where the former represents the density parameter at the present epoch, while
the latter refers to the same at any general epoch.. Note that, in a spatially
flat universe,

$\sum_{i}\Omega_{i}(a)=1$

at all cosmological epochs. The latest CMB observations reported in Ref. [1]
suggest that

$\Omega_{0r}\simeq 9.1\times 10^{-5}\,;\leavevmode\nobreak\ \Omega_{0m}\simeq
0.315\,;\leavevmode\nobreak\ \Omega_{0\Lambda}\simeq 0.685\,.$ (20)

The expression for the Hubble parameter in Eq. (19) is accurate until about
$z\simeq 10^{9}$. At higher redshifts deep inside the radiation dominated
epoch the effective number of relativistic degrees of freedom changes,
implying there will be corrections to the expression for the Hubble parameter
(see App. A). (This is also true for the Standard Model neutrinos, some of
which are known to be massive, and hence, represent part of the non-
relativistic matter at the present epoch, although they were relativistic
(radiation-like) in the early universe.)

In short, the broad outline of our knowledge of modern cosmology can be
succinctly summarised as follows:

  * •

The universe is roughly homogeneous and isotropic on large cosmological length
scales ($>100\leavevmode\nobreak\ {\rm Mpc}$) and exhibits negligible spatial
curvature throughout its (known/probed) expansion history.

  * •

The universe has been expanding for the last 13.8 billion years, at least from
the time when it was about $1$ second old until the present epoch. The metric
of the universe, at the background level, is well approximated by the flat
Friedmann-Lemaître-Robertson-Walker (FLRW) line element given in Eq. (2).

  * •

Observations of the Cosmic Microwave Background (CMB) radiation indicate that
prior to the epoch when our universe was about $370,000$ years old,
corresponding to a redshift of $z\simeq 1100$, it was in a thermal hot dense
plasma state.

  * •

The early universe was thermal and radiation dominated, historically termed as
the hot Big Bang phase, which made a transition to the matter dominated epoch
when it was around $50,000$ years old, at a redshift $z_{\rm eq}\simeq 3400$.
The expansion of the universe was decelerating during both the aforementioned
epochs.

  * •

More recently, the universe has entered into a phase of accelerated expansion
around the time when it was about $8$ billion years old.

  * •

In addition to baryons, photons and neutrinos, our universe also contains a
gravitationally clustering non-relativistic substance known as dark matter
(DM) which is pressureless (and hence ‘cold’) on large extra-galactic scales.
Observations suggest that DM is non-baryonic, in the sense that it does not
interact with the Standard Model constituents at low energy (apart from
gravity, of course). DM plays a dominant role in facilitating the formation of
large-scale structure in the universe [44, 45, 46, 47].

  * •

Additionally, the cosmic recipe also includes a gravitationally non-clustering
negative pressure substance termed as dark energy (DE) in order to fuel the
late time accelerated expansion [48, 49, 50, 47, 51, 52]. The present day
equation of state of the dark energy is close to that of a cosmological
constant $\Lambda$ [53, 54, 55, 56, 57, 58, 59, 60, 61, 47, 62]. The
concordance cosmological model is consequently termed as the flat $\Lambda$CDM
model, or more colloquially as the ‘Big Bang Theory’. Evolution of the energy
density of radiation, matter and dark energy are shown in Fig. 2.

  * •

Starting with an initial spectrum of almost scale-invariant, adiabatic,
Gaussian density fluctuations [1, 2] that are correlated on super-Hubble
scales, the standard cosmological paradigm successfully describes the growth
and formation of structure in our universe, at least on large length scales.

  * •

The nature of these special initial conditions for the concordance cosmology,
in conjunction with CMB and LSS observations, strongly point towards a
transient period of rapid accelerated expansion of space, known as Cosmic
Inflation, which is supposed to have happened in the very early universe [3,
4, 5, 6, 7, 8, 9] prior to the commencement of the radiative hot Big Bang
phase.

Note that there is a classical singularity in the standard Big Bang theory
[63] in the context of classical general relativity where $a=0$ and the
spacetime curvature diverges. We use this classical singularity to set our
initial time $t=0$. However, it is crucial to mention that going backwards in
time, the curvature of the spacetime of the universe will approach Planckian
values before hitting the singularity and hence we expect substantial
modifications to Einstein’s theory due to quantum gravity effects which are
supposed to remove the classical singularity. Hence the existence of the
classical singularity must not to be taken literally. For all practical
purposes, we will assume that the hot Big Bang phase began at some high
temperature below the Planck scale (and much above $1\leavevmode\nobreak\ {\rm
MeV}$ in order to ensure successful BBN), namely

$1\leavevmode\nobreak\ {\rm MeV}\,\ll\,T_{i}^{{\rm
HBB}}\,\leq\,10^{19}\leavevmode\nobreak\ {\rm GeV}\,.$

While the standard Big Bang theory has been remarkably successful in
describing the evolution of the universe [1] both at the background and at the
perturbation level, its success relies heavily upon the following initial
conditions at the beginning of the hot Big Bang phase:

  1. 1.

High degree of spatial flatness: As mentioned before, the background metric of
the universe is consistent with being spatially flat throughout the history of
the universe. In the next section, we will quantify this statement and
demonstrate that this leads to a high degree of fine-tuning of the initial
expansion rate of the hot Big Bang universe, usually known as the flatness
problem.

  2. 2.

Extreme homogeneity of the initial hypersurface: The success of BBN as well as
the precision observations of the CMB indicate that the early universe was
spatially homogeneous and isotropic over large length scales. For example, the
temperature of the CMB is almost uniform all across the sky. In the next
section, we will demonstrate that the aforementioned fact leads to the so-
called horizon problem.

  3. 3.

Super-Hubble correlation of seed perturbations: One of the greatest
discoveries in modern cosmology is the fact that the large scale structure
(LSS) of the universe is formed due to the gravitational collapse of tiny
initial density fluctuations in the early universe. In fact the statistical
properties of these primordial seed fluctuations have been measured very
accurately by CMB and LSS observations and they have three key properties [1,
64, 2].

     * •

Primordial fluctuations are predominantly adiabatic, which indicates that
initial fluctuations in different constituents in the hot Big Bang phase were
almost the same. This further points to the fact that the seed fluctuations in
the early universe emerged out of the perturbations of a single field, namely
the comoving curvature fluctuations $\zeta(t,\vec{x})$ which is a gauge
invariant quantity (i.e. it remains invariant under the gauge transformation
from one local frame to another, that connects different ways of slicing and
threading spacetime into a homogeneous background part and perturbations
around that background.)

     * •

The power spectrum of $\zeta$ is nearly scale-invariant, namely

${\cal
P}_{\zeta}=A_{S}\left(\frac{k}{k_{*}}\right)^{n_{{}_{S}}-1}\,;\leavevmode\nobreak\
k_{*}=0.05\,{\rm Mpc}^{-1}\,$

with

$A_{S}\simeq 2\times 10^{-9}\,,\leavevmode\nobreak\
n_{{}_{S}}-1\simeq-0.035\leavevmode\nobreak\ \leavevmode\nobreak\ ({\rm
small\leavevmode\nobreak\ {\color[rgb]{1,0,0}\textbf{red\leavevmode\nobreak\
tilt}}})\,.$

     * •

The primordial seed fluctuations are approximately Gaussian with probability
distribution function given by

$P[\zeta]={\cal
B}\exp\left[\frac{-\zeta^{2}}{2\sigma_{\zeta}^{2}}\left(1+f_{\rm
NL}\,\zeta+...\right)\right]\,,$

where ${\cal B}$ is a normalisation factor. The variance $\sigma_{\zeta}^{2}$
is given by

$\sigma_{\zeta}^{2}=\int_{-\infty}^{\infty}\,{\rm d}\zeta\leavevmode\nobreak\
\zeta^{2}\leavevmode\nobreak\ P[\zeta]\equiv\int_{k_{\rm min}}^{k_{\rm
max}}\frac{{\rm d}k}{k}\leavevmode\nobreak\ {\cal P}_{\zeta}(k)\simeq
10^{-9}\,.$

Latest CMB and LSS observations are consistent with $f_{\rm NL}=0$ which
implies that we have not found any deviation from Gaussianity so far.

     * •

The temperature power spectrum of the CMB on large angular scales indicates
that the primordial fluctuations exhibit non-vanishing statistical
correlations on super-Hubble scales.

These specific characteristics of the seed fluctuations demand a physical
mechanism to explain their origin. As we will discuss later in these notes,
quantum fluctuations during the accelerated expansion in the inflationary
scenario provide a natural mechanism to explain these observed properties of
the primordial fluctuations.

##  3 Fine-tuning of initial conditions for the hot Big Bang

###  3.1 High degree of spatial flatness

The fine-tuning of the initial expansion rate of the universe for the hot Big
Bang phase can be stated in terms of the famous flatness problem. The first
Friedmann Eq. (6) for a spatially curved universe can be written as

$|\Omega_{\rm
tot}(a)-1|\equiv|\Omega_{K}(a)|=\frac{|K|}{a^{2}\,H^{2}}=(aH)^{-2}\,,$ (21)

where $\Omega_{K}$ corresponds to the dimensionless density parameter
associated with the spatial curvature. From Eq. (12), for a single component
dominated universe, we have

$\left(aH\right)^{-1}=\left(a_{i}H_{i}\right)^{-1}\,\left(\frac{a}{a_{i}}\right)^{\frac{1+3w}{2}}\,.$
(22)

Eq. (21) can then be described more generally for a single component dominated
universe as

$|\Omega_{\rm tot}(a)-1|=|\Omega_{\rm
tot}(a_{i})-1|\,\left(\frac{a}{a_{i}}\right)^{1+3w}\leavevmode\nobreak\
\propto\leavevmode\nobreak\ a^{1+3w}\,,$ (23)

which implies

$|\Omega_{\rm tot}(a)-1|\propto\begin{cases}a^{2}\,;&{\rm RD}\,,\\\ a\,;&{\rm
MD}\,.\end{cases}$ (24)

This shows that $\Omega_{\rm tot}(a)$ continues to diverge away from $1$ in
the radiation and matter-dominated epochs. Given the present day bound on
$|\Omega_{\rm tot}(a_{0})-1|$ (or equivalently, on $|\Omega_{K}(a_{0})|)$ from
the latest CMB observations [14],

$|\Omega_{K}(a_{0})|\leq 10^{-2}\,,$

combined with the fact that $|\Omega_{K}(a)|$ has been growing ever since the
commencement of the hot Big Bang phase, it seems that the early universe was
spatially flat up to an unnaturally high degree of precision. In order to
quantify this, let us compute the bound on $|\Omega_{K}(a_{0})|$ at the
beginning of Big Bang Nucleosynthesis, which corresponds to $z_{{}_{\rm
BBN}}\simeq 4\times 10^{9}$.

$|\Omega_{K}(a_{0})|\leq
10^{-2}\Rightarrow\frac{|\Omega_{K}(a_{0})|}{|\Omega_{K}(a_{\rm
BBN})|}\,|\Omega_{K}(a_{\rm BBN})|\leq 10^{-2}\,,$

which leads to

$|\Omega_{K}(a_{\rm BBN})|\leq 6\times 10^{-18}\,.$ (25)

Further back in time, at the grand unification (GUT) scale ($z_{{}_{\rm
GUT}}\simeq 10^{29}$), the bound on the curvature parameter becomes

$|\Omega_{K}(a_{\rm GUT})|\leq 10^{-56}\,.$ (26)

Our universe during most of its expansion history has been dominated by
radiation and matter555We have ignored the existence of dark energy in the
above analysis. Since the universe began to accelerate around $z\simeq 0.7$,
the scale factor at that epoch is given by $a_{\rm DE}\simeq a_{0}/1.7$. So
the universe spends less than an e-fold of expansion being accelerated closer
to the present epoch and hence, the effect of dark energy can be be safely
ignored while describing the flatness problem. The same applies to the horizon
problem as well., hence the spatial curvature according to equation (23)
should be extremely high today, even though it was small at early times. But
observations suggest that the spatial curvature at the present epoch is still
very small which implies that the spatial curvature in very early times must
have been extremely small. In fact, the simple calculation that we carried out
above demonstrates that $|\Omega_{\rm tot}|$ must have been close to $1$ up to
several decimal places of accuracy at the beginning of the hot Big Bang phase
in the very early universe. Another way to state the fine-tuned spatial
flatness is that the initial density $\rho_{i}^{\rm tot}$ of all matter-energy
components of the hot Big Bang phase must have been close to the critical
density $\rho_{c}=3m_{p}^{2}H^{2}$ up to several decimal places. This also
implies that the initial expansion speed $\dot{a}_{i}$ must be highly fine-
tuned in order for $|\Omega_{K}|$ to remain much smaller than unity at the
present epoch. Such extreme fine-tuning of the initial expansion speed is
known as the flatness problem and suggests an early epoch of expansion prior
to the hot Big Bang phase which can dynamically drive $\Omega_{K}$ towards an
extremely small value by the time the hot Big Bang phase commences.

Note that if $K=0$ exactly, then there is no flatness problem classically.
However, $K=0$ corresponds to a spatially flat universe which is either
infinite in extent or has a non-trivial topology. Additionally, if the
universe began around the Planck scale, then large quantum gravitational
fluctuations might alter the geometry/topology of spatial hypersurfaces.
Furthermore, in quantum cosmology, quantum creation of the universe usually
renders it to be either positively curved [65, 66, 67] or negatively curved
[68, 69].

###  3.2 Horizon problem

CMB observations indicate that the temperature of the early universe, closer
to recombination, is almost uniform across diametrically opposite points in
the sky (with very tiny variations in the temperature of CMB of the order
$\Delta T/T\simeq 10^{-5}$). This suggests that the early universe was
incredibly uniform over large distance scales. However, this observation leads
to a potential problem in the standard hot Big Bang theory, which originates
from the fact that light could have travelled only a finite distance from the
beginning of the hot Big Bang phase until the emission of CMB photons around
the epoch of recombination. Additionally, the spatial hypersurface at the
epoch of BBN is also supposed to be highly uniform and hence the same
arguments may be extrapolated to even earlier epochs. To illustrate this point
explicitly, let us define the physical size of the causal horizon in
cosmology.

The physical size of the causal horizon (also called the particle horizon) at
any epoch $a(t)$ is defined as the maximum distance that any signal could have
travelled from the beginning of the universe at $t=t_{i}$ until that epoch,
and is given by

$d_{H}(t)=a(t)\,r_{H}(t)\,.$ (27)

Where the comoving causal horizon $r_{H}(t)$ can be computed by noting that
signals propagate radially in an FLRW universe, so we have ${\rm d}\theta={\rm
d}\phi=0$. Additionally, for the propagation of light (or any other massless
particle), ${\rm d}s^{2}=0$ in Eq. (2), which leads to

${\rm d}r=\frac{{\rm d}t}{a(t)}\,,$

and hence the expression for the physical size of the causal horizon becomes

$d_{H}(t)\equiv a(t)\,\int_{t_{i}}^{t}\,\frac{{\rm
d}\tilde{t}}{a(\tilde{t})}=a\,\int_{a_{i}}^{a}\,\frac{{\rm
d}\tilde{a}}{\tilde{a}^{2}\,H(\tilde{a})}=(1+z)\,\int_{z}^{z_{i}}\,\frac{{\rm
d}\tilde{z}}{H(\tilde{z})}\,.$ (28)

Note that Eq. (28) can also be written as

$d_{H}(a)=a\,\int_{a_{i}}^{a}\,{\rm
d}\,\ln(\tilde{a})\,\left(\tilde{a}H\right)^{-1}\,.$ (29)

For a single component dominated decelerating universe with EoS parameter
$w>-1/3$, and assuming $t_{i}=0=a_{i}$, we can carry out the integration in
Eq. (29) to obtain

$d_{H}(a)=d_{H}^{i}\,\left(\frac{a}{a_{i}}\right)^{\frac{3(1+w)}{2}}\,,$ (30)

where

$d_{H}^{i}=\left(\frac{2}{1+3w}\right)\,\frac{1}{H_{i}}\,.$ (31)

Since physical length scales grow proportionally to the scale factor in an
expanding universe, $L(a)\propto a$, we have

$\frac{d_{H}(a)}{L(a)}\propto a^{\frac{1+3w}{2}}\,,$ (32)

which shows that the physical size of the causal horizon grows faster than
length scales in a decelerating universe, for which $w>-1/3$. To be precise,
the causal horizon grows as

$d_{H}(a)\propto\begin{cases}a^{2}\,;&{\rm RD}\,,\\\ a^{3/2}\,;&{\rm
MD}\,.\end{cases}$ (33)

Having derived these crucial properties of the causal horizon in a universe
with decelerated expansion, let us compute the angular size subtended by the
horizon at some epoch $z$ in the sky with an observer at the present epoch at
$z=0$, which is defined as

$\Delta\theta(z)\equiv\frac{d_{H}(z)}{d_{A}(z)}\,$ (34)

with

$d_{H}(z)=\frac{1}{a_{0}\,H_{0}}\int_{z}^{\infty}\frac{{\rm
d}\tilde{z}}{\sqrt{\Omega_{0m}\,(1+\tilde{z})^{3}+\Omega_{0r}\,(1+\tilde{z})^{4}+\Omega_{0\Lambda}}}\,,$
(35)

and the angular diameter distance is given by

$d_{A}(z)=\frac{1}{a_{0}\,H_{0}}\int_{0}^{z}\frac{{\rm
d}\tilde{z}}{\sqrt{\Omega_{0m}\,(1+\tilde{z})^{3}+\Omega_{0r}\,(1+\tilde{z})^{4}+\Omega_{0\Lambda}}}\,.$
(36)

To be concrete, let us compute the angular size subtended by the horizon at
recombination with respect to an observer at the present epoch –

$\Delta\theta(z_{\rm CMB})=\frac{d_{H}(z_{\rm CMB})}{d_{A}(z_{\rm
CMB})}=\frac{\int_{z_{\rm CMB}}^{\infty}{\rm d}\tilde{z}\leavevmode\nobreak\
\left[\Omega_{0m}\,(1+\tilde{z})^{3}+\Omega_{0r}\,(1+\tilde{z})^{4}+\Omega_{0\Lambda}\right]^{-1/2}}{\int_{0}^{z_{\rm
CMB}}{\rm d}\tilde{z}\leavevmode\nobreak\
\left[\Omega_{0m}\,(1+\tilde{z})^{3}+\Omega_{0r}\,(1+\tilde{z})^{4}+\Omega_{0\Lambda}\right]^{-1/2}}\simeq
1.16^{\circ}.$ (37)

This implies that the regions in our CMB sky separated by no more than about
$1^{\circ}$ angle were in causal contact, hence they could have been in
thermal equilibrium at the time of recombination, thereby allowing them to
have the same temperature. However, as stressed before, the entire CMB sky
seems to have the same average temperature. This is known as the horizon
problem. Additionally, since the hot Big Bang phase is assumed to be very
uniform also at epochs earlier than recombination, the horizon problem becomes
much more severe as we extrapolate further back into the past. For example, we
can compute the angular size of the horizon at any given epoch in the early
universe using Eq. (34), with the result plotted in Fig. 3.

Figure 3: Angular size of the causal horizon, given by Eq. (34), is plotted as
a function of redshift for the standard ${\rm\Lambda}$CDM universe.

Another intuitive way to understand the Horizon problem is to look at the
conformal time elapsed between the beginning of the universe, until a given
epoch, defined as,

$\Delta\tau=\int_{0}^{t}\,\frac{{\rm
d}\tilde{t}}{a(\tilde{t})}=\int_{0}^{a}\,\frac{{\rm
d}\tilde{a}}{\tilde{a}^{2}\,H(\tilde{a})}\,\,$ (38)

which is analogous to the comoving causal horizon $r_{H}=\frac{d_{H}}{a}$, and
hence gives us the separation between comoving coordinates that have been in
causal contact since the beginning of the universe. The advantage of using
conformal time in eliciting the horizon problem is the following. The FLRW
metric given in Eq. (2) can be written in terms of conformal time as

${\rm d}s^{2}=a^{2}(\tau)\left[-{\rm d}\tau^{2}+\,\delta_{ij}\,{\rm
d}x^{i}{\rm d}x^{j}\right]\,,$ (39)

which is conformally equivalent to the metric of the Minkowski space-time.
Hence, the causal structure of FLRW space-time in terms of the conformal time
is equivalent to that of the Minkowski space-time. So we can draw the usual
space-time diagrams of Special Relativity with light cones subtending
$45^{\circ}$ angle with the space and time axes. Hence, for diametrically
opposite points in the CMB (which are at a fixed comoving distance from us) to
be in causal contact with each other, the conformal time elapsed between the
beginning of the universe and recombination should be greater than the
conformal time elapsed between recombination and today. However, from Eq.
(38), we have

$\Delta\tau(a)\propto\begin{cases}a\,;&{\rm RD}\,,\\\ a^{1/2}\,;&{\rm
MD}\,,\end{cases}$ (40)

which indicates that the conformal time grows with scale factor in the
standard Big Bang theory and hence its size would be extremely small at the
time of recombination, namely,

$\Delta\tau\left(a_{\rm BB}\to a_{\rm CMB}\right)\,\ll\,\Delta\tau\left(a_{\rm
CMB}\to a_{0}\right)\,.$

Hence, diametrically opposite points in the CMB sky could not have been in
causal contact since the birth of the universe, leading to the Horizon
problem. This is illustrated in Fig. 4.

Figure 4: The conformal causal diagram of the universe in the standard
cosmology where the universe began in the hot Big Bang phase being radiation
dominated. This figure schematically illustrates the fact that diametrically
opposite points in the CMB sky could not have been in causal contact by the
time of photon decoupling due to the lack of enough conformal time between the
classical Big Bang singularity and the epoch of recombination.

So we conclude that, in the standard hot Big Bang theory, the early universe
seems to be comprised of an unnaturally large number of causally disconnected
regions, each possessing the same average temperature and density. This again
demands for a physical mechanism prior to the standard hot Big Bang phase
which can bring the entire sky into causal contact by the commencement of the
hot Big Bang phase.

###  3.3 Super-Hubble correlation of primordial perturbations

Towards the end of Sec. 2, we mentioned that the primordial seed fluctuations
of the comoving curvature perturbation $\zeta$ are nearly adiabatic, Gaussian
and almost scale-invariant. However, more interestingly, these fluctuations
have a non-vanishing two-point auto-correlation (power spectrum or variance)
at angular scales that are larger than the angular size of the Hubble radius
in the CMB sky. This can be easily seen from the low-$l$ (large scale) angular
power spectrum of temperature fluctuations in the CMB sky (see Fig. 1 of Ref.
[2]). The physical size of the Hubble radius is defined as

$R_{H}=\frac{1}{H}=\frac{1}{H_{i}}\,\left(\frac{a}{a_{i}}\right)^{\frac{3(1+w)}{2}}\,,$
(41)

while the size of the comoving Hubble radius is given by

$r_{h}=(aH)^{-1}=(a_{i}H_{i})^{-1}\,\left(\frac{a}{a_{i}}\right)^{\frac{1+3w}{2}}\,.$
(42)

Comparing Eq. (31) with Eq. (41), we find the relation between the causal
horizon and Hubble radius in a decelerating single component dominated
universe to be

$d_{H}=\left(\frac{2}{1+3w}\right)\,R_{H}.$ (43)

Thus, a super-Hubble correlation in a decelerating universe also implies a
super-Horizon correlation of primordial fluctuations. This naturally demands
for a physical explanation. At this point, it is important to remind the
reader of the crucial distinction between causal horizon and Hubble radius.
Size of the causal horizon is determined by integrating over the expansion
history, as can be seen from Eq. (28). Consequently, if the distance between
two points in space at a given epoch is larger than the physical size of the
causal horizon, then these two points were never in causal contact since the
beginning of the universe. While, if two points are separated by a distance
greater than the physical size of the Hubble radius at some epoch, then these
points are not in causal contact at that epoch. However, this does not
necessarily imply they were never in causal contact in the past.

Hence the resolution of the conundrum of super-Hubble primordial correlations
involves the existence of an epoch prior to the hot Big Bang phase where the
Hubble radius evolves differently as compared to the causal horizon. In fact,
as we will see, in an accelerating universe the natural spacetime dynamics
leads to length scales constantly being stretched outside the Hubble radius
without violating causality.

##  4 The inflationary hypothesis

In this section, we will stress that a sufficiently long epoch of accelerated
expansion of space prior to the hot Big Bang phase naturally addresses all of
the above problems and provides appropriate initial conditions for the hot Big
Bang phase. Hence, the hot Big Bang phase in the standard Big Bang theory is
not the beginning of our universe, but rather the end of an earlier epoch of
accelerated expansion. In fact, this is the key message that these lecture
notes are meant to convey.

The resolution of the problems of fine tuning of initial conditions emerges
from the key observation that all three initial condition problems of the hot
Big Bang stem from one common dynamics of the decelerating universe, that is
the fact that the comoving Hubble radius $r_{h}=(aH)^{-1}$ increases with
time, as can be seen from Eqs. (21), (29) and (42). Consequently, we can
address all of them by assuming a sufficiently long epoch of expansion history
where the comoving Hubble radius decreases with time, namely

$\frac{{\rm d}}{{\rm d}t}\left(aH\right)^{-1}<0\leavevmode\nobreak\
\Rightarrow\leavevmode\nobreak\ 1+\frac{\dot{H}}{H^{2}}>0\leavevmode\nobreak\
\Rightarrow\leavevmode\nobreak\ \ddot{a}>0,$ (44)

which is equivalent to an accelerating or inflating universe. Let us
explicitly demonstrate how inflation addresses the flatness and horizon
problems, and generates fluctuations that are correlated over super-Hubble
scales.

###  4.1 Addressing the flatness problem

The initial flatness problem of the hot Big Bang phase discussed in Sec. 3.1
can be naturally solved provided if there was a long enough period of
inflation prior to the radiation domination, as discussed above. In
particular, for a perfect fluid with EoS $w$ to drive inflation, we need

$\frac{\ddot{a}}{a}>0\quad\Rightarrow\quad 1\,+\,3\,w<0\quad\Rightarrow\quad
w<-1/3\,,$

leading to

$|\Omega_{\rm tot}(a)-1|=|\Omega_{\rm
tot}(a_{i})-1|\,\left(\frac{a}{a_{i}}\right)^{-|1+3w|}\leavevmode\nobreak\
\propto\leavevmode\nobreak\ a^{-|1+3w|}\,.$ (45)

Hence, during inflation, $|\Omega_{\rm tot}-1|$ or equivalently $|\Omega_{K}|$
decreases with time which results in $\Omega_{\rm tot}(a)$ being driven
towards unity, as long as inflation lasts long enough. In particular, for near
exponential inflation sourced by an effective cosmological constant
$w\simeq-1$, we have

$|\Omega_{\rm tot}(a_{\rm end})-1|=|\Omega_{\rm tot}(a_{i})-1|\,e^{-2\,\Delta
N}\,,$ (46)

where $\Delta N=\ln\left(a_{\rm end}/a_{i}\right)$ is the total number of
e-folds of accelerated expansion during inflation. Assuming inflation began at
some higher energy scale closer to the GUT scale, with the curvature density
being of the same order as the total density i.e. of order unity,

$|\Omega_{K}(a_{i})|=|\Omega_{\rm tot}(a_{i})-1|\approx\mathcal{O}(1)\,,$

Eq. (46) leads to

$|\Omega_{\rm tot}(a_{\rm end})-1|\approx\mathcal{O}(1)\times e^{-2\,\Delta
N}\,.$

We see that $|\Omega_{\rm tot}(a_{\rm end})-1|\leq 10^{-56}$ for $\Delta N\geq
64.47$. This demonstrates that starting from an order unity curvature term at
the GUT scale, about $65$ e-folds of inflation is enough to result in
$\Omega_{0K}\leq 10^{-2}$. In other words, if the hot Big Bang is not the
beginning of our universe, but rather is the end of a sufficiently long enough
period of inflation, then we expect the spatial curvature to be negligible at
the present epoch.

###  4.2 Addressing the horizon problem

During the rapidly accelerated expansion of space, for which
$\frac{1+3w}{2}<0$, the conformal time $\tau$ can be obtained by integrating
Eq. (38) to be

$\tau=\tau_{1}+\frac{1}{a_{1}H_{1}}\,\left|\frac{2}{1+3w}\right|\,\left[1-\left(\frac{a_{1}}{a}\right)^{|1+3w|/2}\right]\,,$
(47)

where $a_{1}$ is the scale factor at some reference epoch $\tau_{1}$. Eq. (47)
shows that the conformal time is negative and diverges at early times,
$\tau\to-\infty$ as $a\to 0$. Thus, accelerated expansion yields a substantial
amount of conformal time during the early history of the universe prior to
$\tau=\tau_{1}$, which in turn brings together a large region of comoving
space into causal contact going back in the past.

In order to see this more explicitly, let us specialise to the case of
exponential inflation, for which

$\tau=\tau_{1}+\frac{1}{a_{1}H}-\frac{1}{aH}\,.$

Taking $\tau_{1}=-1/(a_{1}H)$, we derive the expression for the conformal time
in de Sitter spacetime to be

$\boxed{\tau=-\frac{1}{aH}}\,,$ (48)

which tells us that the conformal time is large and negative at early times
during inflation, while it approaches zero towards the end of inflation,
namely,

$\tau\,:\,-\infty\,\longrightarrow\,0$

during inflation. Hence a sufficient amount of exponential expansion in the
early universe results in bringing the entire CMB sky in causal contact, as
shown schematically in Fig. 5.

Figure 5: The conformal causal diagram of the universe in the presence of an
early stage of exponential inflation before the commencement of the hot Big
Bang phase. From this figure, it is clear that diametrically opposite points
in the CMB sky could have easily been in causal contact if there was a
sufficiently long period of inflation prior to the hot Big Bang phase, which
facilitates for a substantial amount of conformal time to have elapsed in the
past light cone.

Alternatively, the physical size of the causal horizon $d_{H}$ in an
accelerated universe becomes much larger than the Hubble radius. In fact at
the end of inflation, we have

$d_{H}\sim e^{\Delta N}\,R_{H}\,,$

where $\Delta N$ is the total number of e-folds of expansion of space during
inflation. Hence the horizon size at any epoch in the post inflationary
universe is much larger than the Hubble radius, which naturally resolves the
horizon problem discussed in Sec. 3.2.

###  4.3 Generating super-Hubble fluctuations

During the accelerated expansion of space, the comoving Hubble radius,
$r_{h}$, starts shrinking with time, i.e.

$r_{h}\propto a^{-\frac{|1+3w|}{2}}\,$

as can be seen from from Eq. (42). Since comoving length scales corresponding
to the fluctuations remain fixed with time, a shrinking comoving Hubble radius
causes length scales, that were initially shorter than the Hubble radius
(known as sub-Hubble modes), to become larger than the Hubble radius (super-
Hubble) at late times. Thus the rapid accelerated expansion of space during
inflation dynamically stretches the wavelength of (initial) sub-Hubble
fluctuations to large super-Hubble scales at late times. A long enough period
of inflation ensures that by the end of inflation, all of the observable
length scales in the CMB sky, which started out being sub-Hubble, would have
become super-Hubble. On super-Hubble scales the comoving curvature
fluctuations $\zeta$ remain frozen/constant, as we will see in Sec. 6.2.

Figure 6: This figure schematically illustrates the evolution of the comoving
Hubble radius $(aH)^{-1}$ with scale factor, both plotted on a logarithmic
scale (we have assumed $H$ to be almost constant during inflation).
$(aH)^{-1}$ decreases which causes physical scales to exit the Hubble radius
during inflation. As $(aH)^{-1}$ rises during the post-inflationary epochs,
physical scales begin to re-enter the Hubble radius. The CMB pivot scale, as
used by the Planck mission, is set at $k_{*}=0.05\,{\rm Mpc}^{-1}$ and has
been depicted by the dashed blue line.

Inflation thus provides a causal mechanism to generate correlated primordial
fluctuations on super-Hubble scales. The final state of inflationary
fluctuations then acts as the initial seed fluctuations for the hot Big Bang
phase. The transition epoch between the end of inflation and the beginning of
the hot Big Bang phase is called reheating (as mentioned before) which can
last up to a few e-folds of (decelerated) expansion. Evolution of the Hubble
radius during and after inflation is illustrated schematically in Fig. 6 using
a comoving scale, and in Fig. 7 using the physical scale, assuming exponential
expansion of space during inflation. The blue coloured lines indicate
fluctuations of different wavelengths, while the blue stars indicate the
Hubble-exit of the CMB pivot scale during inflation and, later, its Hubble-
entry after the end of inflation. Figs. 6 and 7 demonstrate that long
wavelength fluctuations, which made their Hubble-exit at early times during
inflation, re-entered the Hubble radius at late times closer to the present
epoch. On the other hand, shorter wavelength fluctuations became super-Hubble
towards the end of inflation, and subsequently made their Hubble re-entry
quite early after inflation had ended.

As the comoving Hubble radius began to increase (because of the decelerated
expansion of space after inflation) and one by one, these frozen fluctuations
became sub-Hubble again, they began evolving with time. Eventually, the
fluctuations were amplified by gravitational instability to form the LSS of
the universe. Note that several sources in the literature extensively refer to
Hubble-exit and Hubble-entry as ‘horizon exit’ and ‘horizon entry’. However,
in these lecture notes, we have made an attempt to avoid such phrases and
stick to the more accurate phrases: Hubble-exit and Hubble-entry.

Figure 7: Evolution of the physical Hubble radius $1/H$ is shown with scale
factor (both plotted in logarithmic scale). The physical Hubble radius $1/H$
stays almost constant during inflation which causes physical scales to exit
the Hubble radius during inflation. $1/H$ rises in the post-inflationary epoch
and the physical scales begin to re-enter the Hubble radius.

Hubble-exit and entry can be understood intuitively as follows: during
inflation, accelerated expansion rapidly stretched the wavelengths of
fluctuations. Eventually the frequencies of oscillations of these modes fell
below the rate of expansion of space. Or equivalently, we say the modes became
super-Hubble. Similarly after the end of inflation, the rate of expansion of
space fell rapidly. Hence fluctuations which were super-Hubble can eventually
began to oscillate once the rate of expansion fell below their frequencies of
oscillations. Or equivalently, we say the fluctuation modes became sub-Hubble
after their Hubble re-entry.

In general, a variety of functional forms of the scale factor $a(t)$ which
supports accelerated expansion of space, i.e. $\ddot{a}>0$, can address the
initial conditions problem of the hot Big Bang theory, as long as inflation
lasts long enough. However, from the latest CMB observations we know that if
inflation happened, then the accelerated expansion of space must have been
nearly exponential, $a(t)\propto e^{Ht}$, where $H$ is almost constant during
inflation. Hence we will primarily focus on the quasi-de Sitter expansion of
space during inflation. Typically a period of quasi-de Sitter inflation
lasting for at least 60–70 e-folds of expansion is enough to address the
initial conditions for the hot Big Bang phase, as discussed in App. B. The
comoving Hubble radius plotted in Fig. 6 is particularly convenient to refer
to because of the symmetry between its fall during inflation and rise in the
radiation dominated epoch. In fact, Fig. 6 demonstrates that in order to
ensure that the largest observable length scales are sub-Hubble during
inflation, the universe must spend approximately same number of e-folds of
inflationary evolution as it does in the post-inflationary epoch.

In Einstein’s gravity, an accelerated expansion of space can be sourced by a
negative pressure substance satisfying $w<-1/3$, as can be seen from Eq. (16).
A natural candidate to drive inflation is a singlet scalar field $\varphi$
with self-interaction potential $V(\varphi)$ possessing a low enough effective
mass $m_{\rm eff}^{2}\equiv\frac{{\rm d}^{2}V(\varphi)}{{\rm
d}\varphi^{2}}\lesssim H^{2}$. Since by now the curious reader might have
become bored with preceding long and detailed discussions on inflationary
kinematics, let us move forward to delve into the rich physics of single field
inflationary dynamics without further delay.

##  5 Scalar field dynamics of inflation

In these lecture notes, we focus on the simplest inflationary scenario sourced
by a single scalar field $\varphi$, known as the inflaton field, with a self-
interaction potential $V(\varphi)$ which is minimally coupled to gravity. The
full system of Gravity + Scalar field is described by the action

$S[\varphi]=\int{\rm d}^{4}x\,\sqrt{-g}\;{\cal L}(X,\varphi)\,,$ (49)

where the Lagrangian density ${\cal L}(X,\varphi)$ is a function of the field
$\varphi$ and the kinetic term

$X\equiv\frac{1}{2}\,\partial_{\mu}\varphi\;\partial^{\mu}\varphi=\frac{1}{2}\,\partial_{\mu}\varphi\;\partial_{\nu}\varphi\,g^{\mu\nu}\,.$
(50)

Varying Eq. (49) with respect to $\varphi$ results in the Euler-Lagrange field
equation for the evolution of the scalar field, given by

$\frac{\partial{\cal
L}}{\partial\varphi}-\left(\frac{1}{\sqrt{-g}}\right)\partial_{\mu}\left(\sqrt{-g}\frac{\partial{\cal
L}}{\partial\left(\partial_{\mu}\varphi\right)}\right)=0\,.$ (51)

The energy-momentum tensor of the scalar field associated with the space-time
translational invariance of the action in Eq. (49) is given by

$T^{\mu\nu}=\left(\frac{\partial{\cal L}}{\partial
X}\right)\,\left(\partial^{\mu}\varphi\;\partial^{\nu}\varphi\right)-g^{\mu\nu}\,{\cal
L}\leavevmode\nobreak\ .$ (52)

In Sec. 5.2, we will study linear perturbation theory during inflation. For
that purpose, we will split666The splitting is made explicit in Sec. 5.2. the
system into time-dependent uniform background and space (and time) dependent
perturbations. We start with the discussion of the background scalar field
$\phi(t)$ and space-time dynamics, described by the FLRW metric defined below.

###  5.1 Background dynamics during inflation

Specializing to a spatially flat FLRW universe and a homogeneous part
$\phi(t)$ of the scalar field $\varphi(t,\vec{x})$, one finds

${\rm d}s^{2}=-{\rm d}t^{2}+a^{2}(t)\;\left[{\rm d}x^{2}+{\rm d}y^{2}+{\rm
d}z^{2}\right]\,,$ (53)
$T^{\mu}_{\;\>\;\nu}\equiv\mathrm{diag}\left(-\rho_{{}_{\phi}},\,p_{{}_{\phi}}\,\,p_{{}_{\phi}},\,p_{{}_{\phi}}\right)\,,$
(54)

where the energy density $\rho_{{}_{\phi}}$, and pressure $p_{{}_{\phi}}$, of
the homogeneous inflaton field are given by

$\displaystyle\rho_{{}_{\phi}}$ $\displaystyle=$
$\displaystyle\left(\frac{\partial{\cal L}}{\partial X}\right)\,(2\,X)-{\cal
L}\,,$ (55) $\displaystyle p_{{}_{\phi}}$ $\displaystyle=$ $\displaystyle{\cal
L},$ (56)

with $X=-\frac{1}{2}\dot{\phi}^{2}$. Evolution of the scale factor $a(t)$ is
governed by the Friedmann equations

$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}\equiv
H^{2}=\frac{1}{3m_{p}^{2}}\,\rho_{{}_{\phi}},$ (57)
$\displaystyle\frac{\ddot{a}}{a}\equiv\dot{H}+H^{2}=-\frac{1}{6m_{p}^{2}}\,\left(\rho_{{}_{\phi}}+3\,p_{{}_{\phi}}\right),$
(58)

where $H\equiv\dot{a}/a$ is the Hubble parameter and $\rho_{{}_{\phi}}$
satisfies the conservation equation

${\dot{\rho}_{{}_{\phi}}}=-3\,H\left(\rho_{{}_{\phi}}+p_{{}_{\phi}}\right)\leavevmode\nobreak\
.$ (59)

In the standard inflationary paradigm, inflation is sourced by a scalar field
$\phi$ with a potential $V(\phi)$ (see Fig. 8) which is minimally coupled to
gravity. For a canonical scalar field, the Lagrangian density takes the form

${\cal L}(X,\phi)=-X-V(\phi)\,.$ (60)

Substituting Eq. (60) into Eq. (55) and Eq. (56), the expressions for energy
density and pressure become

$\displaystyle\rho_{{}_{\phi}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}{\dot{\phi}}^{2}+\;V(\phi)\,,$ $\displaystyle
p_{{}_{\phi}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}{\dot{\phi}}^{2}-\;V(\phi)\,,\leavevmode\nobreak\
\leavevmode\nobreak\ $ (61)

consequently the two Friedmann Eqs. (57), (58) and the equation of motion Eq.
(59) become

$\displaystyle
H^{2}=\frac{1}{3m_{p}^{2}}\,\rho_{\phi}=\frac{1}{3m_{p}^{2}}\left[\frac{1}{2}{\dot{\phi}}^{2}+V(\phi)\right],$
(62)
$\displaystyle\dot{H}\equiv\frac{\ddot{a}}{a}-H^{2}=-\frac{1}{2m_{p}^{2}}\,\dot{\phi}^{2}\,,$
(63) $\displaystyle{\ddot{\phi}}+3\,H{\dot{\phi}}+V_{,\phi}=0\,.$ (64)

The evolution of various physical quantities during inflation is usually
described with respect to the number of e-folds of expansion which is given by
$N=\ln(a/a_{i})$, where $a_{i}$ is some arbitrary epoch at very early times
during inflation. It is informative to write and solve the Friedmann equations
in terms of the number of e-folds as follows:

$\displaystyle H^{2}$ $\displaystyle=$
$\displaystyle\frac{V(\phi)}{3\,m_{p}^{2}-\frac{1}{2}\left(\frac{{\rm
d}\phi}{{\rm d}N}\right)^{2}}\,;$ (65) $\displaystyle\frac{{\rm d}H}{{\rm
d}N}$ $\displaystyle=$ $\displaystyle-H\,\frac{1}{2m_{p}^{2}}\left(\frac{{\rm
d}\phi}{{\rm d}N}\right)^{2}\,;$ (66) $\displaystyle\frac{{\rm
d}^{2}\phi}{{\rm d}N^{2}}$ $\displaystyle=$ $\displaystyle-\frac{{\rm
d}\phi}{{\rm d}N}\left(3+\frac{1}{H}\,\frac{{\rm d}H}{{\rm
d}N}\right)-\frac{1}{H^{2}}\,V_{,\phi}\,.$ (67)

A better physical quantity to depict any epoch with scale factor $a$ during
the inflationary expansion is the number of e-folds before the end of
inflation which is defined as

$N_{e}(a)=\ln\left(\frac{a_{e}}{a}\right)=\int_{t}^{t_{e}}H(t)\,{\rm d}t,$
(68)

where $H(t)$ is the Hubble parameter during inflation, and $a_{e}$ denotes the
scale factor at the end of inflation, hence $N_{e}=0$ corresponds to the end
of inflation. Typically a period of quasi-de Sitter (exponential)
inflation777For a detailed account on the geometry of de Sitter spacetime
including its cosmological significance, see Refs. [70, 71, 72]. lasting for
at least 60–70 e-folds is required in order to address the problems of the
standard hot Big Bang model discussed in Sec. 3. We denote $N_{*}$ as the
number of e-folds (before the end of inflation) when the CMB pivot scale

$k_{*}=(aH)_{*}=0.05\leavevmode\nobreak\ \rm Mpc^{-1}$ (69)

left the comoving Hubble radius during inflation. Typically
$N_{*}\in[50,\,60]$ depending upon the details of reheating after inflation
(see Ref. [73] and App. B).

Figure 8: This figure schematically depicts a prototypical inflaton potential
(solid green curve). The ‘CMB window’ represents field values corresponding to
the Hubble-exit epochs of scales
$k\in\left[0.0005,0.5\right]\leavevmode\nobreak\ {\rm Mpc}^{-1}$ that are
observationally accessible with the latest CMB missions.

The quasi-de Sitter like phase corresponds to the inflaton field rolling
slowly down the potential $V(\phi)$. For a variety of functional forms of the
inflaton potential $V(\phi)$, there exists a slow-roll regime of inflation,
ensured by the presence of the Hubble friction term [74, 33, 9, 75, 76, 77] in
Eq. (64). This slow-roll regime can be conveniently characterised by the first
two kinematic Hubble slow-roll parameters $\epsilon_{H}$, $\eta_{H}$, defined
by [78, 9]

$\displaystyle\epsilon_{H}$ $\displaystyle=$
$\displaystyle-\frac{\dot{H}}{H^{2}}=-\frac{{\rm d}\,\ln{H}}{{\rm
d}N}=\frac{1}{2m_{p}^{2}}\,\frac{\dot{\phi}^{2}}{H^{2}}\,;$ (70)
$\displaystyle\eta_{H}$ $\displaystyle=$
$\displaystyle-\frac{\ddot{\phi}}{H\dot{\phi}}=\epsilon_{H}-\frac{1}{2\epsilon_{H}}\,\frac{{\rm
d}\epsilon_{H}}{{\rm d}N}\,,$ (71)

where the slow-roll regime of inflation corresponds to

$\epsilon_{H},\leavevmode\nobreak\ |\eta_{H}|\ll 1\,.$ (72)

Note that $\epsilon_{H}\geq 0$, while $\eta_{H}$ may be negative or positive
depending on whether $\epsilon_{H}$ is increasing or decreasing with time.
Using the definition of the Hubble parameter, $H=\dot{a}/a$, we have
$\ddot{a}/a=\dot{H}+H^{2}=H^{2}(1+\dot{H}/H^{2})$. From the expression for
$\epsilon_{H}$ in Eq. (70), it is easy to see that

$\frac{\ddot{a}}{a}=H^{2}\,\big{(}1-\epsilon_{H}\big{)}\,,$ (73)

which implies that the universe accelerates, ${\ddot{a}}>0$, when
$\epsilon_{H}<1$. Using equation (62), the expression for $\epsilon_{H}$ in
Eq. (70) reduces to $\epsilon_{H}\simeq\frac{3}{2}\frac{\dot{\phi}^{2}}{V}$
when ${\dot{\phi}}^{2}\ll V$. In fact, under the slow-roll conditions in Eq.
(72), the Friedmann equations given by Eqs. (62) and (64) take the form

$\displaystyle H^{2}$ $\displaystyle\simeq$
$\displaystyle\frac{V(\phi)}{3\,m_{p}^{2}}\,;$ (74) $\displaystyle\dot{\phi}$
$\displaystyle\simeq$ $\displaystyle-\frac{V_{,\phi}}{3\,H}\,.$ (75)

The last expression is a consequence of $|\eta_{H}|\ll 1$, which ensures that
the inflaton speed does not change too rapidly, resulting in a long enough
period of accelerated expansion before the end of inflation. The classical
slow-roll dynamics corresponding to Eq. (75) during inflation is similar to
the slow terminal motion of an object where the attractive force and the
restraining drag/friction force are almost balanced with each other. The slow-
roll conditions in Eqs. (74) and (75) can be combined to obtain the slow-roll
trajectory described by

$\boxed{\dot{\phi}\simeq-\frac{m_{p}}{\sqrt{3}}\,\frac{V_{,\phi}}{\sqrt{V(\phi)}}}\,.$
(76)

It is well known that the slow-roll trajectory is actually a local attractor
for a number of different models of inflation, see Refs. [74, 33, 9, 75, 76,
77]. The slow-roll regime of inflation is described more systematically in
terms of the Hubble flow parameters $\epsilon_{n}$ defined by

$\epsilon_{n+1}=\frac{{\rm d}\ln\,\epsilon_{n}}{{\rm d}N}\,;\quad{\rm
with}\quad\epsilon_{1}=\epsilon_{H}\,.$ (77)

Accordingly, from Eq. (71), the second Hubble flow parameter $\epsilon_{2}$ is
related888Note that many references use the symbol $\eta$ to represent
$\epsilon_{2}$. to $\epsilon_{H}$ and $\eta_{H}$ via

$\epsilon_{2}=2\left(\epsilon_{H}-\eta_{H}\right)\,.$ (78)

Apart from the aforementioned kinematic slow-roll parameters, the slow-roll
regime is also often characterised by the dynamical potential slow-roll
parameters [9], defined by

$\displaystyle\epsilon_{{}_{V}}$ $\displaystyle=$
$\displaystyle\frac{m_{p}^{2}}{2}\left(\frac{V_{,\phi}}{V}\right)^{2}\,;$ (79)
$\displaystyle\eta_{{}_{V}}$ $\displaystyle=$ $\displaystyle
m_{p}^{2}\,\left(\frac{V_{,\phi\phi}}{V}\right)\,.$ (80)

For small values of these parameters $\epsilon_{H}\ll 1,\,\eta_{H}\ll 1$, from
Eqs. (70) and (75) one finds $\epsilon_{H}\simeq
V_{,\phi}^{2}/(18\,m_{p}^{2}\,H^{4})$, which, using Eq. (74) becomes

$\epsilon_{H}\simeq\frac{m_{p}^{2}}{2}\left(\frac{V_{,\phi}}{V}\right)^{2}=\epsilon_{{}_{V}}\,.$

Similarly, taking a time derivative of Eq. (75) and incorporating it into Eq.
(74), we get

$\eta_{H}=-\frac{\ddot{\phi}}{H\dot{\phi}}\simeq\eta_{V}-\epsilon_{V}\,.$

The end of inflation is marked by

$\epsilon_{H}(\phi_{e})=1\simeq\epsilon_{{}_{V}}(\phi_{e})\,.$

Furthermore, under the slow-roll approximations, since
$\epsilon_{H}\simeq\epsilon_{{}_{V}}$, the Hubble-exit of the CMB pivot scale
$k_{*}$ is denoted by the number of e-folds before the end of inflation
$N_{*}$, which can be obtained from Eq. (68) to be

$\boxed{N_{*}=\int_{t_{*}}^{t_{e}}{\rm d}t\,H(t)\leavevmode\nobreak\
=\leavevmode\nobreak\
\frac{1}{m_{p}}\,\int_{\phi_{e}}^{\phi_{*}}\,\frac{d\phi}{\sqrt{2\,\epsilon_{H}}}\leavevmode\nobreak\
\simeq\leavevmode\nobreak\
\frac{1}{m_{p}}\,\int_{\phi_{e}}^{\phi_{*}}\,\frac{d\phi}{\sqrt{2\,\epsilon_{{}_{V}}(\phi)}}}\,.$
(81)

The above formula is quite useful in computing the value of $\phi_{*}$ for a
given potential that supports slow roll, as we will see in Sec. 7.2.

Before proceeding further, we remind the reader of the distinction between the
quasi-de Sitter (qdS) and slow-roll (SR) approximations.

  * •

Quasi-de Sitter inflation corresponds to the condition $\epsilon_{H}\ll
1\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\
\frac{1}{2}\,\dot{\phi}^{2}\ll V(\phi)\,.$

  * •

Slow-roll inflation corresponds to both $\epsilon_{H},\,|\eta_{H}|\ll 1\,.$

Note that one can deviate from the slow-roll regime by having $|\eta_{H}|\geq
1$ while still maintaining the qdS expansion by keeping $\epsilon_{H}\ll 1$,
which is exactly what happens during the so-called ultra slow-roll (USR)
inflation [79, 80, 81, 82, 83]. This distinction will not be important for the
rest of these lecture notes in the present version. Under either of the
aforementioned assumptions, the conformal time, $\tau$, is given by Eq. (48)
to be $\tau\simeq-1/(aH)$.

Figure 9: The evolution of the slow-roll parameters $\epsilon_{H}$ and
$\eta_{H}$ is shown as a function of the number of e-folds before the end of
inflation $N_{e}$ for Starobinsky potential (218). From this plot, it is easy
to notice that at early times when $N_{e}\gg 1$, the slow-roll conditions are
satisfied i.e $\epsilon_{H},\,|\eta_{H}|\ll 1$. However, the slow-roll
conditions are violated towards the end of inflation (marked by $N_{e}=0$ and
$\epsilon_{H}=1$).

During inflation, since the inflaton evolves monotonically, one can describe
the evolution of the background dynamics by re-writing Eqs. (62)–(64) such
that $\phi$ behaves as a time variable. Inflationary dynamics in terms of a
$\phi$-clock is known as the Hamilton-Jacobi formalism [78] of inflation,
discussed in App. C.

We conclude that a scalar field with a sufficiently shallow potential
$V(\phi)$ with $\epsilon_{{}_{V}}\ll 1$ leads to nearly exponential (qdS)
expansion of the background space-time during inflation, which successfully
addresses the fine tuning of initial conditions for the background evolution,
such as the flatness and the horizon problems, discussed in Secs. 3.1 and 3.2.
As inflation progresses, since the friction coefficient $H$ falls with time
(although slowly) the slow-roll parameter $\epsilon_{H}$ (also
$\epsilon_{{}_{V}}$) increases and eventually becomes greater than unity, thus
terminating the accelerated expansion of space. Towards the end of inflation,
slow-roll conditions are violated (see Fig. 9). After the end of inflation,
the inflaton field oscillates around the minimum of the potential.
Subsequently, the oscillating inflaton condensate decays into matter and
radiation, via a process called reheating, leading to the commencement of the
hot Big Bang phase. We will return to the topic of reheating in Sec. 8.

Next, we move on to discuss how quantum fluctuations during inflation provides
a natural mechanism for generating the initial seeds of structure formation in
the universe.

###  5.2 Quantum fluctuations during Inflation

Our system is a canonical scalar field minimally coupled to gravity whose
action is given by

$S[g_{\mu\nu},\varphi]=\int\,{\rm
d}^{4}x\,\sqrt{-g}\,\left(\,\frac{m_{p}^{2}}{2}\,R-\frac{1}{2}\,\partial_{\mu}\varphi\,\partial_{\nu}\varphi\,g^{\mu\nu}-V(\varphi)\right).$
(82)

In perturbation theory, we split the metric and inflaton field into their
corresponding homogeneous background pieces and fluctuations, namely

$g_{\mu\nu}(t,\vec{x})=\bar{g}_{\mu\nu}(t)+\delta
g_{\mu\nu}(t,\vec{x})\,;\quad\varphi(t,\vec{x})=\phi(t)+\delta\varphi(t,\vec{x})\,.$

Note that the perturbed metric $\delta g_{\mu\nu}$ has 10 degrees of freedom,
out of which only two are independent, while the rest are fixed by gauge
freedom, and the Hamiltonian and momentum constraints. The perturbed line
element in the Arnowitt-Deser-Misner (ADM) formalism [84, 85] can be written
as

${\rm d}s^{2}=-\alpha^{2}\,{\rm d}t^{2}+\gamma_{ij}\left({\rm
d}x^{i}+\beta^{i}\,{\rm d}t\right)\left({\rm d}x^{j}+\beta^{j}\,{\rm
d}t\right)\,,$ (83)

where $\alpha=1+\delta\alpha$ and $\beta^{i}$ are the lapse and shift
functions, while $\gamma_{ij}$ are the dynamical metric perturbations.

In these lecture notes, we will skip the detailed discussion of relativistic
cosmological perturbation theory (which is an extremely important topic) and
direct the interested readers to Refs. [86, 87, 9, 88, 89, 90, 35]. For the
purpose of pedagogy and simplicity, we work in the comoving gauge999See App. D
for a brief discussion on the perturbed metric in both comoving and spatially
flat gauge choices. defined by

$\delta\varphi(t,\vec{x})=0\,;\quad\gamma_{ij}(t,\vec{x})=a^{2}\bigl{[}\bigl{(}1+2\,\zeta(t,\vec{x})\bigr{)}\,\delta_{ij}\,+\,h_{ij}(t,\vec{x})\bigr{]}\,.$
(84)

In particular, two gauge-invariant light fields are guaranteed to exist in the
single field inflationary scenario, which are

  1. 1.

Scalar-type comoving curvature perturbations $\zeta(t,\vec{x})\,;$

  2. 2.

Tensor-type transverse and traceless perturbations $h_{ij}(t,\vec{x})\,.$

The scalar fluctuations would eventually induce density and temperature
perturbations in the hot Big Bang phase, and subsequently the large scale
structure of the universe, while the tensor fluctuations propagate as
gravitational waves (GWs) at late times and constitute a stochastic
cosmological background of GWs.

The perturbed action in the comoving gauge can be expressed as

$S[g_{\mu\nu},\varphi]=S_{B}[\bar{g}_{\mu\nu},\phi]+S^{(2)}[\zeta]+S^{(2)}[h_{ij}]+S_{\rm
int}[\zeta,h_{ij}]\,,$ (85)

where $S_{B}[\bar{g}_{\mu\nu},\phi]$ is the background action, while
$S^{(2)}[\zeta],\,S^{(2)}[h_{ij}]$ are the quadratic actions for the scalar
and tensor fluctuations (which results in linear field equations for the
fluctuations). The term $S_{\rm int}[\zeta,h_{ij}]$ is the action for
fluctuations beyond the linear order which we do not discuss in these lecture
notes and direct the interested readers to Refs. [9, 85, 91].

The background action $S_{B}[\bar{g}_{\mu\nu},\phi]$ leads to the familiar
Friedmann Equations

$H^{2}=\left(\frac{\dot{a}}{a}\right)^{2}=\left(\frac{1}{3m_{p}^{2}}\right)\rho_{{}_{\phi}},$
$\dot{H}=-\frac{1}{2m_{p}^{2}}\dot{\phi}^{2}\leavevmode\nobreak\ ,$

which result in an accelerated expansion of space for a suitable potential
$V(\phi)$ at the background level. Under slow-roll condition $\epsilon_{H}\ll
1$, we get quasi-dS expansion $a(t)\sim e^{Ht}$.

The quadratic action for the (linear) scalar-type fluctuations in the comoving
gauge is

$S^{(2)}[\zeta]=\frac{1}{2}\int{\rm d}\tau\,{\rm
d}^{3}\vec{x}\leavevmode\nobreak\
\left(am_{p}\sqrt{2\epsilon_{H}}\right)^{2}\,\left[(\zeta^{\prime})^{2}-(\partial_{i}\zeta)^{2}\right]\,,$

and for the (linear) tensor-type perturbations is

$S^{(2)}[h_{ij}]=\frac{1}{2}\int{\rm d}\tau\,{\rm
d}^{3}\vec{x}\leavevmode\nobreak\
\,\left(\frac{a\,m_{p}}{2}\right)^{2}\left[(h^{\prime}_{ij})^{2}-(\partial_{l}h_{ij})^{2}\leavevmode\nobreak\
.\right]$

At linear order in perturbation theory, the scalars, vectors and tensors are
decoupled (the so-called SVT decomposition theorem), and hence their evolution
during inflation can be studied separately, as discussed in Ref. [9]. We will
begin with scalar fluctuations and then move on to study tensor
fluctuations101010It is important to note that vector perturbations are not
created during inflation and any pre-existing vector perturbations decay
rapidly due to the exponential expansion during inflation [9]. Hence, we do
not discuss vector perturbations in these notes.. In the following, we
describe the evolution of perturbations during inflation generated by quantum
fluctuations at linear order in perturbation theory. Note that the
fluctuations are categorized into scalars, vectors and tensors depending on
their transformation properties under the rotation group ${\rm SO}(3)$ on the
three dimensional spatial hypersurfaces.

In the following section, we will delve into the study of scalar and tensor
fluctuations during inflation after providing a brief introduction to the
statistics of quantum vacuum fluctuations.

##  6 Correlators of inflationary fluctuations

The primary observables in cosmology are the late time cosmological
correlators [91] which are in general the $N$-point correlation functions of
different physical quantities, such as fluctuations in temperature and density
contrast. For primordial fluctuations, which are the seeds of the late-time
correlators, we will primarily be interested in the comoving curvature
fluctuations $\zeta(t,\vec{x})$ and the transverse and traceless tensor
fluctuations $h_{ij}(t,\vec{x})$.

As discussed in Ref. [91], an accurate measurement of the late time
correlators of observables ${\cal O}_{i}(t,\vec{x})$ (in the form of CMB
temperature and polarisation anisotropies as well as large scale clustering of
matter in the universe) along with the evolution equations and matter-energy
constituents, would eventually enable us to determine the $N$-point
correlators of the initial fluctuations, denoted by $\Psi_{i}(t_{\rm
ini},\vec{x})$ at the beginning of the hot Big Bang phase, and consequently at
the end of inflation hypersurface. Schematically, this is represented as

$\boxed{\langle\,\hat{\bm{\Psi}}_{1}(t,\vec{x})\,\hat{\bm{\Psi}}_{2}(t,\vec{x})\,\hat{\bm{\Psi}}_{3}(t,\vec{x})\,...\,\rangle_{{}_{\rm
ini}}\leavevmode\nobreak\ \bm{\longleftarrow}\leavevmode\nobreak\
\langle\,\bm{{\cal O}}_{1}(t,\vec{x})\,\bm{{\cal O}_{2}}(t,\vec{x})\,\bm{{\cal
O}_{3}}(t,\vec{x})...\rangle_{{}_{\rm late}}}$

###  6.1 Statistics of vacuum quantum fluctuations

In this subsection, we provide a succinct discussion on the statistics of
cosmological fluctuations at the end of inflation with a particular focus on
the 2-point auto-correlator and the power spectrum. For a detailed account on
the subject, we direct the readers to Refs. [92, 91] and the lecture notes.

We begin with the standard Fourier transform convention used throughout these
notes, namely

$\displaystyle\Psi(t,\vec{x})$ $\displaystyle=$ $\displaystyle\int\frac{{\rm
d}^{3}\vec{k}}{(2\pi)^{3}}\leavevmode\nobreak\
\Psi_{\vec{k}}(t)\leavevmode\nobreak\ e^{i\,\vec{k}.\vec{x}}\quad\,;$ (86)
$\displaystyle\quad\Psi_{\vec{k}}(t)$ $\displaystyle=$
$\displaystyle\int\,{\rm d}^{3}\vec{x}\leavevmode\nobreak\
\Psi(t,\vec{x})\leavevmode\nobreak\ e^{-i\,\vec{k}.\vec{x}}\,,$ (87)

where $\Psi_{k}$ is the Fourier transformation of $\Psi$. If $\Psi(t,\vec{x})$
fluctuations are drawn from a statistical ensemble with a probability
distribution function (PDF) $P[\Psi]$, then the variance of the fluctuations
is defined as

$\sigma_{\Psi}^{2}(t)\equiv\int\,{\rm
d}\Psi\,P[\Psi]\,\Psi^{2}=\langle\,\Psi(t,\vec{x})\,\Psi(t,\vec{x})\,\rangle\,,$
(88)

where $\langle...\rangle$ denotes statistical ensemble average with respect to
the PDF $P[\Psi]$. The power spectrum can be written using the Fourier
decomposition given in Eq. (86) as

$\sigma_{\Psi}^{2}(t)=\int\frac{{\rm
d}^{3}\vec{k}_{1}}{(2\pi)^{3}}\,\int\frac{{\rm
d}^{3}\vec{k}_{2}}{(2\pi)^{3}}\,\langle\,\Psi_{\vec{k}_{1}}(t)\,\Psi_{\vec{k}_{2}}(t)\,\rangle\,e^{i\,\left(\vec{k}_{1}+\vec{k}_{2}\right).\vec{x}}.$
(89)

The 2-point correlator in the momentum space is defined as

$\langle\,\Psi_{\vec{k}_{1}}(t)\,\Psi_{\vec{k}_{2}}(t)\,\rangle\equiv\frac{2\pi^{2}}{k^{3}}\,{\cal
P}_{\Psi}(t,\vec{k}_{1})\,\,(2\pi)^{3}\,\delta_{D}^{(3)}\left(\vec{k}_{1}+\vec{k}_{1}\right)\,,$
(90)

where ${\cal P}_{\Psi}(t,\vec{k}_{1})$ is called the power spectrum of
fluctuations $\Psi$. Assuming the background spacetime to be homogeneous and
isotropic, so that the power spectrum only depends on $k=|\vec{k}_{1}|$, we
get

$\displaystyle\langle\,\Psi_{\vec{k}_{1}}(t)\,\Psi_{\vec{k}_{2}}(t)\,\rangle$
$\displaystyle\equiv$ $\displaystyle\frac{2\pi^{2}}{k^{3}}\,{\cal
P}_{\Psi}(t,k)\,(2\pi)^{3}\,\delta_{D}^{(3)}\left(\vec{k}_{1}+\vec{k}_{1}\right)\,;$
(91) $\displaystyle\sigma_{\Psi}^{2}(t)$ $\displaystyle=$
$\displaystyle\int{\rm d}\ln k\,{\cal P}_{\Psi}(t,k)\,,$ (92)

which shows that the power spectrum ${\cal P}_{\Psi}(t,k)$ is the change in
the variance per logarithm interval of $k$. Similarly, in order to study the
higher-point statistics, we need to compute the skewness and the kurtosis; or
equivalently, in Fourier space, the bispectrum
$\langle\,\Psi_{\vec{k}_{1}}\Psi_{\vec{k}_{2}}\Psi_{\vec{k}_{3}}\,\rangle$ and
the trispectrum
$\langle\,\Psi_{\vec{k}_{1}}\Psi_{\vec{k}_{2}}\Psi_{\vec{k}_{3}}\,\Psi_{\vec{k}_{4}}\,\rangle$
respectively. In the present version of these lecture notes, we primarily
focus on the 2-point correlators or the power spectra of scalar and tensor
fluctuations. However, we provide a brief discussion on higher-point
correlators in Sec. 6.4.

In order to characterize linear quantum fluctuations during inflation, we
begin with the quantum field (in the Heisenberg picture)

$\hat{\Psi}(t,\vec{x})=\int\,\frac{{\rm
d}^{3}\vec{k}}{(2\pi)^{3}}\,\hat{\Psi}_{\vec{k}}(t)\,e^{i\vec{k}.\vec{x}}=\int\,\frac{{\rm
d}^{3}\vec{k}}{(2\pi)^{3}}\,\left[f_{k}(t)\,e^{i\vec{k}.\vec{x}}\,\hat{a}_{\vec{k}}+f^{*}_{k}(t)\,e^{-i\vec{k}.\vec{x}}\,\hat{a}_{\vec{k}}^{\dagger}\right]\,,$
(93)

with

$\hat{\Psi}_{\vec{k}}(t)=f_{k}(t)\,\hat{a}_{\vec{k}}+f^{*}_{k}(t)\,\hat{a}_{-\vec{k}}^{\dagger}\,,$
(94)

where $\hat{a}_{\vec{k}},\,\hat{a}_{\vec{k}}^{\dagger}$ are the creation and
annihilation operators respectively, satisfying the usual commutation relation

$\left[\hat{a}_{\vec{k}_{1}},\,\hat{a}_{\vec{k}_{2}}^{\dagger}\right]=(2\pi)^{3}\,\delta_{\rm
D}^{3}\left(\vec{k}_{1}-\vec{k}_{2}\right)\,,$ (95)

and the vacuum state $|0\rangle$ is defined by

$\hat{a}_{\vec{k}}\,|0\rangle=0\,.$ (96)

Similarly, the mode functions $f_{k}(t)$ satisfy the field equations for
$\hat{\Psi}(t,\vec{x})$ and hence, they only depend upon $k=|\vec{k}|$. These
mode functions are canonically normalized to

$f_{k}\,\dot{f}_{k}^{\ast}-\dot{f}_{k}\,f_{k}^{\ast}=i\,\hbar\,,$ (97)

so that the quantum field $\hat{\Psi}$ and its conjugate momentum
$\hat{\Pi}_{\Psi}=\dot{\hat{\Psi}}$ satisfy the usual canonical commutation
relation

$\left[\hat{\Psi}(t,\vec{x}_{1}),\,\hat{\Pi}_{\Psi}(t,\vec{x}_{2})\right]\,=\,i\,\hbar\leavevmode\nobreak\
\delta_{D}^{(3)}\left(\vec{x}_{1}-\vec{x}_{2}\right)\,.$ (98)

It is important to stress that the splitting of $\hat{\Psi}_{\vec{k}}(t)$ in
terms of $\hat{a}_{\vec{k}}$ and $\hat{a}_{\vec{k}_{2}}^{\dagger}$ in Eq. (94)
is not unique and there exists a family of mode functions $f_{k}$ satisfying
Eqs. (96) and (97), as discussed in Ref. [93]. Hence, in contrast to the
Minkowski spacetime fluctuations, there is no unique vacuum state $|0\rangle$
for the inflationary fluctuations at this stage.

For vacuum quantum fluctuations, using Eqs. (94) and (96), we get

$\langle\,0\,|\,\hat{\Psi}_{\vec{k}_{1}}(t)\,\hat{\Psi}_{\vec{k}_{2}}(t)\,|\,0\,\rangle\,=\langle\,0\,|\,f_{\vec{k}_{1}}f_{\vec{k}_{2}}^{\ast}\,\hat{a}_{\vec{k}_{1}}\hat{a}_{-\vec{k}_{2}}^{\dagger}\,|\,0\,\rangle\,,$

which, using Eq. (95), yields

$\boxed{\langle\,\Psi_{\vec{k}_{1}}(t)\,\Psi_{\vec{k}_{2}}(t)\,\rangle\equiv\langle\,0\,|\,\hat{\Psi}_{\vec{k}_{1}}(t)\,\hat{\Psi}_{\vec{k}_{2}}(t)\,|\,0\,\rangle=|f_{k}(t)|^{2}\,(2\pi)^{3}\,\delta_{D}^{(3)}\left(\vec{k}_{1}+\vec{k}_{1}\right)}\,.$
(99)

Using Eq. (91), we obtain the expression for the power spectrum of vacuum
quantum fluctuations to be

$\boxed{{\cal P}_{\Psi}(t,k)=\frac{k^{3}}{2\pi^{2}}\,|f_{k}(t)|^{2}}\,.$ (100)

We stress that Eq. (91) is the general definition of the power spectrum, while
the definition provided by Eq. (100) is only valid for vacuum quantum
fluctuations. The appearance of the term
$\delta_{D}^{(3)}(\vec{k}_{1}+\vec{k}_{1})$ in Eq. (99) is a consequence of
translational invariance (homogeneity) of the background space, since the
linear correlators respect the background isometries.

Sometimes the cosmological power spectrum is defined as

$\tilde{{\cal P}}_{\Psi}(t,k)=\frac{2\pi^{2}}{k^{3}}\,{\cal P}_{\Psi}(t,k)\,,$
(101)

where $\tilde{{\cal P}}_{\Psi}(t,k)$ is referred to as the dimensionful power
spectrum, while ${\cal P}_{\Psi}(t,k)$ is referred to as the dimensionless
power spectrum. In these notes, we will use Eq. (91) as the definition of
power spectrum, rather than Eq. (101), in order to avoid confusion. However,
it is important to keep this distinction in mind111111The dimensionful power
spectrum $\tilde{{\cal P}}_{\Psi}(t,k)$ is usually denoted as ${\cal
P}_{\Psi}(t,k)$, while the dimensionless power spectrum ${\cal P}_{\Psi}(t,k)$
is denoted as $\Delta_{\Psi}^{2}(t,k)$. However, we will stick to the
notations we defined above throughout these lecture notes..

###  6.2 Scalar quantum fluctuations during inflation

The effective action for the inflationary scalar fluctuations $\zeta$ in the
comoving gauge $\delta\varphi=0$ at linear order in perturbation theory
(hence, the quadratic action) is given by [85, 91] (see App. D)

$\boxed{\,S^{(2)}[\zeta(\tau,\vec{x})]=\frac{1}{2}\int{\rm d}\tau\,{\rm
d}^{3}\vec{x}\leavevmode\nobreak\ \leavevmode\nobreak\
z^{2}\left[\,(\zeta^{\prime})^{2}-(\partial_{i}\zeta)^{2}\,\right]\,}\,,$
(102)

where the ‘pump term’ $z$, and its derivatives are given by

$\displaystyle z$ $\displaystyle=$ $\displaystyle
am_{p}\sqrt{2\epsilon_{H}}\,;$ (103) $\displaystyle\frac{z^{\prime}}{z}$
$\displaystyle=$ $\displaystyle aH\,\left(1+\epsilon_{H}-\eta_{H}\right)\,;$
(104) $\displaystyle\frac{z^{\prime\prime}}{z}$ $\displaystyle=$
$\displaystyle\left(aH\right)^{2}\left[2+2\,\epsilon_{H}-3\,\eta_{H}+2\,\epsilon_{H}^{2}+\eta_{H}^{2}-3\,\epsilon_{H}\,\eta_{H}-\frac{1}{aH}\,\eta_{H}^{\prime}\right]\,.$
(105)

We stress that the above expressions are exact at linear order in perturbation
theory and have not been truncated at quadratic order in slow-roll parameters
$\epsilon_{H},\,\eta_{H}$. Upon change of variable

$v\equiv z\,{\zeta}\,,$ (106)

the scalar action (102) takes the form

$\boxed{\,S^{(2)}[v]=\frac{1}{2}\,\int\,{\rm d}\tau\,{\rm
d}^{3}\vec{x}\,\left[\left(v^{\prime}\right)^{2}-\left(\partial_{i}v\right)^{2}+\frac{z^{\prime\prime}}{z}v^{2}\right]\,}\,,$
(107)

The variable $v$, which itself is a scalar field like $\zeta$, is called the
Mukhanov-Sasaki variable121212Note that the original Mukhanov-Sasaki variable,
as defined in Refs. [94, 95], is ${\cal
Q}(t,\vec{x})=\frac{1}{a(t)}\,v(t,\vec{x})\,.$ in the literature [94, 95]. Eq.
(107) represents the action of a canonical massive scalar field in Minkowski
spacetime. The field equation for $v$ is given by

$v^{\prime\prime}\,+\,\left(-\nabla^{2}\,+\,{\cal M}_{\rm
eff}^{2}\right)v=0\,,$ (108)

where the time-dependent effective (tachyonic) mass term is given by [96]

$\displaystyle\boxed{{\cal M}_{\rm
eff}^{2}(\tau)\equiv-\frac{z^{\prime\prime}}{z}=-\left(aH\right)^{2}\left[2+2\,\epsilon_{H}-3\,\eta_{H}+2\,\epsilon_{H}^{2}+\eta_{H}^{2}-3\,\epsilon_{H}\,\eta_{H}-\frac{1}{aH}\,\eta_{H}^{\prime}\right]}\,.$
(109)

The Fourier modes $v_{k}$ satisfy the Mukhanov-Sasaki equation, given by

$\boxed{v^{\prime\prime}_{k}\,+\,\left(k^{2}+{\cal M}_{\rm
eff}^{2}\right)v_{k}=0}\,.$ (110)

Specializing to slow-roll inflation,
$\epsilon_{H},\,|\eta_{H}|,\,|\eta_{H}^{\prime}|\ll 1$, the above equation in
the quasi-de Sitter limit from Eq. (48) reduces to

$v^{\prime\prime}_{k}\,+\,\left(\,k^{2}-2a^{2}H^{2}\,\right)v_{k}=0\leavevmode\nobreak\
\Rightarrow\leavevmode\nobreak\
\boxed{v^{\prime\prime}_{k}\,+\,\left(\,k^{2}-\frac{2}{\tau^{2}}\,\right)v_{k}=0}\,,$
(111)

whose general solution is given by131313In Eq. (111), we have dropped the
slow-roll parameters in our analysis. If we keep the slow-roll parameters,
then the general solution can be written in terms of Bessel or Hankel
functions of order $\nu$ which is given in terms of the slow-roll parameters
as shown in App. E and Ref. [96]. For qdS approximation, where we drop all the
slow-roll parameters, $\nu=3/2$ and the solution gets reduced to the simple
form given in Eq. (112).

$v_{k}(\tau)=C_{1}\frac{1}{\sqrt{2k}}\left(1-\frac{i}{k\tau}\right)e^{-ik\tau}+C_{2}\frac{1}{\sqrt{2k}}\left(1+\frac{i}{k\tau}\right)e^{+ik\tau}\,.$
(112)

The above expression will be crucial in our computation of the power spectrum
of inflationary fluctuations below. Note that our primary goal is to compute
the power spectrum of vacuum quantum fluctuations of $\zeta$ at late times
(when all the observable modes $\zeta_{k}$ are super-Hubble), which is defined
by

${\cal P}_{\zeta}\equiv\frac{k^{3}}{2\pi^{2}}\,|{\zeta}_{k}|^{2}\Big{|}_{k\ll
aH}=\frac{k^{3}}{2\pi^{2}}\,\frac{|v_{k}|^{2}}{z^{2}}\Big{|}_{k\ll aH}\,.$
(113)

Hence, before quantitatively computing the power spectrum, let us understand
the behaviour of each mode $\zeta_{k}$ in the super-Hubble regime. For this
purpose, we write the following Euler-Lagrange field equation for $\zeta$
corresponding to the quadratic action (102):

$\zeta^{\prime\prime}+2\left(\frac{z^{\prime}}{z}\right){\zeta}^{\prime}-\nabla^{2}\zeta=0\,,$

which leads to the equation for the mode functions $\zeta_{k}$ to be

$\boxed{{\zeta}^{\prime\prime}_{k}+2\left(\frac{z^{\prime}}{z}\right){\zeta}^{\prime}_{k}+k^{2}{\zeta}_{k}=0}\,.$
(114)

At sufficiently late times when a mode is super-Hubble, namely $k\ll aH$ or
equivalently, $-k\tau\ll 1$, we have

${\zeta}^{\prime\prime}_{k}+2\left(\frac{z^{\prime}}{z}\right){\zeta}^{\prime}_{k}\simeq
0\,.$

From Eq. (104), we know that $z^{\prime}/z=aH(1+\epsilon_{H}-\eta_{H})$.
Specializing to slow-roll approximations and dropping
$\epsilon_{H},\,\eta_{H}$, we have $z^{\prime}/z\simeq aH=-1/\tau$, leading to

${\zeta}^{\prime\prime}_{k}\simeq\frac{2}{\tau}\,{\zeta}^{\prime}_{k}\,,$

whose solution is given by

$\zeta_{k}\simeq A_{k}+B_{k}\,\tau^{2}=A_{k}+\frac{B_{k}}{(aH)^{2}}\,,$ (115)

with $A_{k},B_{k}$ being two integration constants. This shows that on super-
Hubble scales, $\zeta_{k}$ comprises of a constant mode and a decaying mode.
Note that at late times, since $\tau\to 0$, the decaying mode can be ignored,
which leads to the conclusion that $\zeta_{k}$ remains constant or frozen
outside the Hubble radius. This is demonstrated in Fig. 10 for the Starobinsky
potential given in Eq. (218).

Figure 10: Evolution of the scalar power spectrum
$\frac{k^{3}}{2\pi^{2}}|\zeta_{k}|^{2}$ is plotted by numerically solving Eq.
(114) for a mode exiting the Hubble radius at about 60 e-folds before the end
of inflation (for the Starobinsky potential given in Eq. 218). At early times
when the mode is sub-Hubble, i.e. $k\gg aH$, the power decreases as ${\cal
P}_{\zeta}\sim(aH)^{-2}$ as expected. After the Hubble-exit, the power freezes
to a constant in the super-Hubble regime when $k\ll aH$. We note down its
value after the mode-freezing as the super-Hubble scale power corresponding to
that mode. Repeating the procedure for a range of scales $k$ yields us the
power spectrum of scalar fluctuations. The same numerical analysis can be
carried out for tensor fluctuations.

The conservation of $\zeta_{k}$ for adiabatic perturbations on super-Hubble
scales is one of the most important consequences of the rapidly accelerated
expansion during inflation. This allows us to translate/extrapolate the power
spectra of inflationary fluctuations (at large cosmological scales) at the end
of inflation through the unknown reheating history of the universe (discussed
in Sec. 8), all the way until these fluctuations enter the Hubble radius
closer to matter-radiation equality and recombination epochs. Thus, it enables
inflation to make firm predictions about the late time structure in the
universe. It is important to stress that the conservation of $\zeta_{k}$ for
adiabatic perturbations on super-Hubble scales is also valid beyond Einstein’s
gravity and potentially beyond linear perturbation theory, as demonstrated in
Refs. [97, 98].

The aforementioned classical dynamics of $\zeta_{k}$ provides us with
intuitive insights into the qualitative properties of super-Hubble scalar
fluctuations during inflation. In order to quantitatively compute the power
spectrum, we need to consider vacuum quantum fluctuations during inflation,
for which we will treat the fields $\hat{\zeta}$ and $\hat{v}$ as quantum
fields. Since the action for $\hat{v}$ in Eq. (107) is in the canonically
normalized form, we will apply the method of canonical quantization discussed
in Sec. 6.1 to quantize it. Working in the Heisenberg picture, we expand
$\hat{v}$ in terms of creation and annihilation operators as follows:

$\hat{v}(\tau,\vec{x})=\int\,\frac{{\rm
d}^{3}\vec{k}}{(2\pi)^{3}}\,\hat{v}_{k}(\tau)\,e^{i\vec{k}.\vec{x}}=\int\,\frac{{\rm
d}^{3}\vec{k}}{(2\pi)^{3}}\,\left[v_{k}(\tau)\,e^{i\vec{k}.\vec{x}}\,\hat{a}_{\vec{k}}+v^{*}_{k}(\tau)\,e^{-i\vec{k}.\vec{x}}\,\hat{a}_{\vec{k}}^{\dagger}\right]\,,$
(116)

where the Fourier modes $\hat{v}_{k}$ are those of the one-dimensional time
dependent quantum harmonic oscillators, that can be written as

$\hat{v}_{k}(\tau)=v_{k}(\tau)\,\hat{a}_{\vec{k}}+v^{*}_{k}(\tau)\,\hat{a}_{-\vec{k}}^{\dagger}\,,$
(117)

with $\hat{a}_{\vec{k}},\,\hat{a}_{\vec{k}}^{\dagger}$ being the annihilation
and creation operators respectively, satisfying the usual commutation
relations

$\left[\hat{a}_{\vec{k}_{1}},\,\hat{a}_{\vec{k}_{2}}^{\dagger}\right]=(2\pi)^{3}\,\delta_{\rm
D}^{3}\left(\vec{k}_{1}-\vec{k}_{2}\right)\,.$ (118)

The mode functions $v_{k}(\tau)$ satisfy the Mukhanov-Sasaki Eq. (110), namely

$v^{\prime\prime}_{k}\,+\,\left(k^{2}-\frac{z^{\prime\prime}}{z}\right)v_{k}=0\,,$
(119)

and hence they only depend on the magnitude $k=|\vec{k}|$ of the comoving
frequency.

Given a mode $k$, at sufficiently early times when it is deep in the sub-
Hubble regime i.e $k\gg aH$, we can assume $v_{k}$ to be the fluctuations of a
massless field in Minkowski spacetime. Consequently, we impose the Bunch-
Davies vacuum initial condition [99]

$\boxed{v_{k}\big{|}_{-k\tau\gg 1}=\frac{1}{\sqrt{2k}}\,e^{-ik\tau}}\,,$ (120)

on each cosmologically observable mode at sufficiently early times during
inflation. At this stage, it is important to point out that we will be using
the results from quantum field theory(QFT) at two important places, namely:
(i) in the definition of power spectrum given in Eq. (113), as discussed in
Sec. 6.1, and (ii) in choosing the initial quantum vacuum which is
characterised by the Bunch-Davies mode functions given in Eq. (120).

As discussed before, during inflation the comoving Hubble radius falls,
causing modes to become super-Hubble i.e $k\ll aH$ and Eq. (110) dictates that
$|v_{k}|\propto z$ and hence $\zeta_{k}$ approaches a constant value. By
solving the Mukhanov-Sasaki equation we can estimate the dimensionless
primordial power spectrum of $\zeta$ using Eq. (113). Imposing standard
normalisation from QFT given in Eq. (97) and the Bunch-Davies initial
conditions (120) on the general solution given in Eq. (112), we get

  1. 1.

Normalisation $\Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\
|C_{1}|^{2}-|C_{2}|^{2}=1$ (where $\hbar=1$) ;

  2. 2.

Bunch-Davies condition $\Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\
v_{k}(\tau)\longrightarrow\frac{1}{\sqrt{2k}}e^{-ik\tau}$ for $k|\tau|\gg
1\,,$

leading to

$C_{1}=1,\leavevmode\nobreak\ C_{2}=0\,,$

which then fixes the vacuum state $|0\rangle$, and yields the expression for
the quasi-dS mode functions of the Mukhanov-Sasaki variable to be

$\boxed{v_{k}(\tau)=\frac{1}{\sqrt{2k}}\left(1-\frac{i}{k\tau}\right)e^{-i\,k\tau}}\,,$
(121)

with

$\Rightarrow\left|v_{k}\right|^{2}=\frac{1}{2k}\,\frac{\left(1+k^{2}\tau^{2}\right)}{k^{2}\tau^{2}}\,.$

Incorporating the Bunch-Davies initial condition imposed mode functions from
Eq. (121), we obtain the scalar power spectrum, defined in Eq. ( 113), to be

${\cal
P}_{\zeta}(k)=\frac{1}{8\pi^{2}}\left(\frac{H}{m_{p}}\right)^{2}\frac{1}{\epsilon_{H}}\left[1+\left(\frac{k}{aH}\right)^{2}\right]\,,$
(122)

which on super-Hubble scales, $k\ll aH$, is given by the (famous) expression

$\boxed{{\cal P}_{\zeta}(k)\Big{|}_{k\ll
aH}=\frac{1}{8\pi^{2}}\left(\frac{H}{m_{p}}\right)^{2}\frac{1}{\epsilon_{H}}}\leavevmode\nobreak\
,$ (123)

which shows that there is finite power on super-Hubble scales which depends on
the Hubble parameter $H$ and the first slow-roll parameter $\epsilon_{H}$
during inflation. The scalar and tensor power spectra determined using the
slow-roll approximations have been plotted in Fig. 11 for the Starobinsky
potential given in Eq. (218). From Eqs. (122) and (123), it is easy to notice
that the frozen value of the scalar power spectrum at late times $k\ll aH$ is
half of its value at the Hubble-crossing $k=aH$, as can be seen in Fig. 10.
However, since $H^{2}/\epsilon_{H}$ is almost constant during slow-roll, this
implies,

$\frac{H^{2}}{\epsilon_{H}}\bigg{|}_{k=aH}\,\simeq\,\frac{H^{2}}{\epsilon_{H}}\bigg{|}_{k\ll
aH}\,,$

we can use the corresponding values of $H$ and $\epsilon_{H}$ at the Hubble
crossing of a particular mode to compute the power spectrum, as long as we are
using Eq. (123), instead of Eq. (122). This is known as the Hubble-crossing
formalism [100, 79].

Figure 11: The power spectra of scalar and tensor quantum fluctuations
(computed using the slow-roll formulae given by Eqs. (123) and (162)
respectively) are shown for comoving modes exiting the Hubble radius at
different number of e-folds $N_{e}$ before the end of inflation for the
Starobinsky potential in Eq. (218). The CMB window (shown as the grey shaded
region) corresponds to comoving modes in the range $k_{\rm
CMB}\in[0.0005,0.5]\leavevmode\nobreak\ {\rm Mpc}^{-1}$ that are being probed
by the current CMB missions. Fluctuations over larger scales (shown as the red
shaded region) are outside the observable universe at present.

In order to compare the inflationary predictions with CMB observations, it is
instructive to write the scalar power spectrum given in Eq. (123) in the form
of a power-law as

${\cal P}_{\zeta}(k)=A_{S}\left(\frac{k}{k_{*}}\right)^{n_{{}_{S}}-1}\,,$
(124)

where, $A_{S}$ is the amplitude of scalar fluctuations at the pivot scale
$k=k_{*}$. The scalar spectral index $n_{{}_{S}}-1$ is defined as

$n_{{}_{S}}-1=\frac{{\rm d}\ln{\cal P}_{\zeta}}{{\rm d}\ln k}=\frac{{\rm
d}\ln{\cal P}_{\zeta}}{{\rm d}N}\,\frac{{\rm d}N}{{\rm d}\ln k}\,,$ (125)

which is to be computed at the Hubble-crossing $k=aH$ of a given mode. Hence
we get

$\frac{{\rm d}\ln k}{{\rm d}N}=1+\frac{{\rm d}\ln H}{{\rm
d}N}=1-\epsilon_{H}\,,$

leading to

$\Rightarrow\frac{{\rm d}N}{{\rm d}\ln k}=\frac{1}{1-\epsilon_{H}}\simeq
1+\epsilon_{H}+{\cal O}\left(\epsilon_{H}^{2}\right)\,.$ (126)

Similarly, from Eq. (123), we get

$\frac{{\rm d}\ln{\cal P}_{\zeta}}{{\rm d}N}=2\,\frac{{\rm d}\ln H}{{\rm
d}N}-\frac{{\rm d}\ln\epsilon_{H}}{{\rm d}N}\,,$

which, using Eq. (71) becomes

$\Rightarrow\frac{{\rm d}\ln{\cal P}_{\zeta}}{{\rm
d}N}=2\,\eta_{H}-4\,\epsilon_{H}\,.$ (127)

Incorporating Eqs. (126) and (127) into Eq. (125) we obtain the expression for
the scalar spectral index $n_{{}_{S}}-1$ to be

$\boxed{n_{{}_{S}}-1=2\,\eta_{H}-4\,\epsilon_{H}}\,.$ (128)

Since $\epsilon_{H},\,|\eta_{H}|\ll 1$ during slow-roll inflation, Eq. (128)
shows that $n_{{}_{S}}-1\simeq 0$, implying the power spectrum is almost
scale-invariant with a tiny spectral tilt. It is possible that in some
specific case, the inflationary dynamics yields $\eta_{H}=2\epsilon_{H}$,
leading to an exact scale-invariant/Harrison-Zeldovich power spectrum with
$n_{{}_{S}}=1$. However, this will require a particular fined tuned functional
form of the potential $V(\phi)$, as shown in Ref. [101].

This is in sharp contrast to the quantum fluctuations of a massless free
scalar field in Minkowski space for which the mode functions are given by

$\boxed{v_{k}^{M}(t)=\frac{1}{\sqrt{2k}}e^{-ikt}}\,.$ (129)

Hence the power spectrum is given by

$\boxed{{\cal P}_{M}(k)=\left(\frac{k}{2\pi}\right)^{2}}\leavevmode\nobreak\
,$ (130)

which has a prominent blue tilt ${\cal
P}_{M}(k)\propto\left(\frac{k}{2\pi}\right)^{2}$ on all scales, because of
which the power is strongly suppressed on large scales and has negligible
macroscopic consequence.

The key difference between inflationary fluctuations and fluctuations in
Minkowski spacetime is the presence of the tachyonic mass term in Eq. (110),
which forces the mode functions $v_{k}$ to grow linearly with scale factor,
thus making sure that $\zeta_{k}$ is conserved on super-Hubble scales. In fact
the growth/instability of $v_{k}$ can be understood from the fact that Eq.
(110) describes an inverted oscillator at late times, when $k^{2}\tau^{2}\ll
1$, featuring a maximum at the centre [102] as shown in Fig. 12. Accordingly,
$\zeta_{k}$ behaves like an over-damped oscillator whose damping term
increases quickly/non-adiabatically as the mode becomes super-Hubble. To see
this qualitatively, let us consider the effective frequency/mass term of the
Mukhanov-Sasaki variable, namely

$\displaystyle\Omega_{v}^{2}(k,\tau)$ $\displaystyle=$ $\displaystyle
k^{2}+{\cal M}_{\rm eff}^{2}(\tau)=k^{2}-\frac{2}{\tau^{2}}\,;$ (131)
$\displaystyle\Omega_{v}^{{}^{\prime}}(k,\tau)$ $\displaystyle=$
$\displaystyle\frac{2}{\tau^{3}\,\Omega_{v}}\,,$ (132)

for which, one obtains

$\bigg{|}\frac{\Omega^{\prime}}{\Omega^{2}}\bigg{|}=\bigg{|}\frac{2}{\left(k^{2}\tau^{2}-2\right)^{3/2}}\bigg{|}\,.$
(133)

The above expression demonstrates that
$\big{|}\frac{\Omega^{\prime}}{\Omega^{2}}\big{|}\geq 1$ when $-k\tau\leq 2$,
indicating violation of adiabaticity, as the mode approaches the Hubble
radius, generating large excitations of the $\hat{v}$ quanta. The effective
mass term of the $\hat{v}$ field becomes tachyonic after Hubble crossing.
Hence, the initial Bunch-Davies type vacuum fluctuations appear to be in a
highly excited state from the point of view of the late time Hamiltonian when
the modes are super-Hubble. In the following, we will explicitly quantify the
above qualitative arguments. Since this excited state corresponds to large
occupation number of particles, the super-Hubble quantum fluctuations can be
considered as classical in this regard [103, 102]. The issue of quantum-to-
classical transition is rather complex and is a topic of intense investigation
(see Refs. [104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116,
117, 118]).

Figure 12: The time-dependent effective potential and the ground state wave
function are plotted in green and purple respectively for a harmonic
oscillator with a time dependent frequency as a representative of the
Mukhanov-Sasaki mode functions $v_{k}$, as described by Eq. (119).

The quadratic action for $\hat{\zeta}$ in Eq. (102) can be written in terms of
the Fourier modes by proceeding as follows. Incorporating the Fourier
decomposition of $\hat{\zeta}$

$\hat{\zeta}(\tau,\vec{x})=\int\frac{{\rm
d}^{3}\vec{k}}{(2\pi)^{3}}\leavevmode\nobreak\
\hat{\zeta}_{\vec{k}}(\tau)\leavevmode\nobreak\ e^{i\,\vec{k}.\vec{x}}$

into the action in Eq. (102), we get

$S^{(2)}[\hat{\zeta}(\tau,\vec{x})]=\frac{1}{2}\int{\rm
d}\tau\,z^{2}\int\frac{{\rm d}^{3}\vec{k}_{1}}{(2\pi)^{3}}\int\frac{{\rm
d}^{3}\vec{k}_{2}}{(2\pi)^{3}}\int{\rm
d}^{3}\vec{x}\,e^{i\left(\vec{k}_{1}+\vec{k}_{2}\right).\vec{x}}\left[\hat{\zeta}^{\prime}_{\vec{k}_{1}}\hat{\zeta}^{\prime}_{\vec{k}_{2}}-\left(i\,\vec{k}_{1}\right).\left(i\,\vec{k}_{2}\right)\hat{\zeta}_{\vec{k}_{1}}\hat{\zeta}_{\vec{k}_{2}}\right]\,.$

Using the definition of Dirac delta function

$\delta_{D}^{(3)}(\vec{k}_{1}+\vec{k}_{2})=\frac{1}{(2\pi)^{3}}\,\int{\rm
d}^{3}\vec{x}\,e^{i\left(\vec{k}_{1}+\vec{k}_{2}\right).\vec{x}}\,,$ (134)

we obtain the expression for the quadratic action of $\hat{\zeta}$ to be

$S^{(2)}[\hat{\zeta}]=\frac{1}{2}\int{\rm d}\tau\int\frac{{\rm
d}^{3}\vec{k}}{(2\pi)^{3}}\,z^{2}\left[\left(\hat{\zeta}_{k}^{\prime}\right)^{2}-k^{2}\,\hat{\zeta}_{k}^{2}\right]\,.$
(135)

Similarly the quadratic action for the Mukhanov-Sasaki variable $\hat{v}$ from
Eq. (107) becomes

$S^{(2)}[\hat{v}]=\frac{1}{2}\int{\rm d}\tau\int\frac{{\rm
d}^{3}\vec{k}}{(2\pi)^{3}}\left[\left(\hat{v}_{k}^{\prime}\right)^{2}-\left(k^{2}+{\cal
M}_{\rm eff}^{2}\right)\hat{v}_{k}^{2}\right]\,.$ (136)

From the above actions written in terms of Fourier modes, we can compute the
vacuum expectation values of any relevant physical quantity associated with
the quantum field $\hat{v}$, since we have fixed the vacuum $|0\rangle$ w.r.t
to the Bunch-Davies condition imposed mode functions given in Eq. (121), re-
written as

$\displaystyle v_{k}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2k}}\left(1-\frac{i}{k\tau}\right)e^{-i\,k\tau}\,;$
(137) $\displaystyle\zeta_{k}(\tau)$ $\displaystyle=$
$\displaystyle\frac{-\tau}{\sqrt{2k}}\left(\frac{H}{m_{p}}\right)\,\frac{1}{\sqrt{2\epsilon_{H}}}\left(1-\frac{i}{k\tau}\right)e^{-i\,k\tau}\,.$
(138)

The conjugate momentum of the Mukhanov-Sasaki field is
$\hat{\Pi}_{v}=\hat{v}^{\prime}$, while its Lagrangian density is given by

${\cal
L}_{v}^{(2)}=\frac{1}{2}\left[\left(\hat{v}^{\prime}\right)^{2}-\left(\partial_{i}\hat{v}\right)^{2}-{\cal
M}_{\rm eff}^{2}\hat{v}^{2}\right]\,.$

Hence the Hamiltonian corresponding to the Mukhanov-Sasaki field is

$\hat{H}_{v}^{(2)}\equiv\int{\rm
d}^{3}\vec{x}\,\left(\hat{v}\,\hat{\Pi}_{v}-{\cal
L}^{(2)}\right)=\frac{1}{2}\int{\rm
d}^{3}\vec{x}\,\left[\hat{\Pi}_{v}^{2}+\left(\partial_{i}\hat{v}\right)^{2}+{\cal
M}_{\rm eff}^{2}\hat{v}^{2}\right]\,,$ (139)

which takes the following form in terms of Fourier modes $\hat{v}_{k}$:

$\hat{H}_{v}^{(2)}=\frac{1}{2}\int\frac{{\rm
d}^{3}\vec{k}}{(2\pi)^{3}}\left[\left(\hat{v}_{k}^{\prime}\right)^{2}+\left(k^{2}+{\cal
M}_{\rm eff}^{2}\right)\hat{v}_{k}^{2}\right]\,.$ (140)

Expanding the quantum field $\hat{v}$ and its momentum $\hat{\Pi}_{v}$ in
terms of creation and annihilation operators, as given in Eq. (116), we obtain

$\displaystyle\hat{v}_{k}(\tau)$ $\displaystyle=$ $\displaystyle
v_{k}(\tau)\,\hat{a}_{\vec{k}}+v_{k}^{\ast}(\tau)\,\hat{a}_{-\vec{k}}^{\dagger}\,;$
(141) $\displaystyle\hat{\Pi}_{v}(\tau)$ $\displaystyle=$ $\displaystyle
v^{\prime}_{k}(\tau)\,\hat{a}_{\vec{k}}+v_{k}^{\prime\ast}(\tau)\,\hat{a}_{-\vec{k}}^{\dagger}\,.$
(142)

Using the above Eqs. (141), (142) in Eq. (140), we obtain

$\displaystyle\hat{H}_{v}^{(2)}=\frac{1}{2}\int\frac{{\rm
d}^{3}\vec{k}}{(2\pi)^{3}}\left[\left(\left(v_{k}^{\prime}\right)^{2}+\Omega_{k}^{2}v_{k}^{2}\right)\hat{a}_{\vec{k}}\hat{a}_{\vec{k}}+\left(\left(v_{k}^{\prime\ast}\right)^{2}+\Omega_{k}^{2}v_{k}^{\ast\,2}\right)\hat{a}_{-\vec{k}}^{\dagger}\hat{a}_{-\vec{k}}^{\dagger}\right.$
$\displaystyle\left.+\left(|v_{k}^{\prime}|^{2}+\Omega_{k}^{2}|v_{k}|^{2}\right)\left(\hat{a}_{\vec{k}}\hat{a}_{-\vec{k}}^{\dagger}+\hat{a}_{-\vec{k}}^{\dagger}\hat{a}_{\vec{k}}\right)\right],$

where the time dependent frequency is given by

$\Omega_{k}^{2}=k^{2}+{\cal M}_{\rm
eff}^{2}(\tau)=k^{2}-\frac{2}{\tau^{2}}\,.$

Using the commutation relation from Eq. (118), we get

$\hat{a}_{\vec{k}}\hat{a}_{-\vec{k}}^{\dagger}=(2\pi)^{3}\,\delta_{D}^{(3)}(0)+\hat{a}_{-\vec{k}}^{\dagger}\hat{a}_{\vec{k}}\,,$

which leads to

$\displaystyle\hat{H}_{v}^{(2)}=\frac{1}{2}\int\frac{{\rm
d}^{3}\vec{k}}{(2\pi)^{3}}\left[\left(\left(v_{k}^{\prime}\right)^{2}+\Omega_{k}^{2}v_{k}^{2}\right)\hat{a}_{\vec{k}}\hat{a}_{\vec{k}}\,+\,\left(\left(v_{k}^{\prime\ast}\right)^{2}+\Omega_{k}^{2}v_{k}^{\ast\,2}\right)\hat{a}_{-\vec{k}}^{\dagger}\hat{a}_{-\vec{k}}^{\dagger}\right.$
$\displaystyle\left.+\left(|v_{k}^{\prime}|^{2}+\Omega_{k}^{2}|v_{k}|^{2}\right)\left(2\hat{a}_{-\vec{k}}^{\dagger}\hat{a}_{\vec{k}}+(2\pi)^{3}\delta_{D}^{(3)}(0)\right)\right]\,.$
(143)

The energy of the quantum field is given by the vacuum expectation value of
the Hamiltonian, namely,

$E_{v}\equiv\langle\,0\,|\,\hat{H}_{v}^{(2)}\,|\,0\,\rangle=\frac{1}{2}\,(2\pi)^{3}\,\delta_{D}^{(3)}(0)\,\int\frac{{\rm
d}^{3}\vec{k}}{(2\pi)^{3}}\,\left[\,|v_{k}^{\prime}|^{2}+\Omega_{k}^{2}|v_{k}|^{2}\,\right]\,.$
(144)

The appearance of the Dirac delta function indicates that formally the total
vacuum energy of the quantum field is divergent, as expected. A better
physical quantity to compute is the vacuum energy density, $\rho_{v}$. Noting
that

$(2\pi)^{3}\delta^{(3)}(0)=\lim_{|\vec{k}|\to 0}\,\int\,{\rm
d}^{3}\vec{x}\,e^{i\,\vec{k}.\vec{x}}=\int\,{\rm d}^{3}\vec{x}\,,$

the vacuum energy density can be obtained from Eq. (144) to be

$\rho_{v}=\frac{1}{2}\,\int\frac{{\rm
d}^{3}\vec{k}}{(2\pi)^{3}}\,\left[\,|v_{k}^{\prime}|^{2}+\Omega_{k}^{2}\,|v_{k}|^{2}\,\right]\,.$
(145)

At this stage, it is important to obtain simplified expressions for $v_{k}$
and $v_{k}^{\prime}$, starting from Eq. (137), as

$\displaystyle v_{k}(\tau)$
$\displaystyle=\frac{1}{\sqrt{2k}}\left(1-\frac{i}{k\tau}\right)e^{-i\,k\tau}\leavevmode\nobreak\
\Rightarrow\leavevmode\nobreak\
|v_{k}(\tau)|^{2}=\frac{1}{2k}\left(1+\frac{1}{k^{2}\tau^{2}}\right)\,;$ (146)
$\displaystyle v_{k}^{\prime}(\tau)$
$\displaystyle=\frac{1}{\sqrt{2k}}\,\frac{1}{\tau}\,\left[-1+i\left(\frac{1}{k\tau}-k\tau\right)\right]e^{-i\,k\tau}\leavevmode\nobreak\
\Rightarrow\leavevmode\nobreak\
|v_{k}^{\prime}(\tau)|^{2}=\frac{k}{2}\left(1-\frac{1}{k^{2}\tau^{2}}+\frac{1}{k^{4}\tau^{4}}\right)\,,$
(147)

which can be used to compute the integrand of Eq. (145) to be

$|v_{k}^{\prime}|^{2}+\Omega_{k}^{2}|v_{k}|^{2}\simeq|v_{k}^{\prime}|^{2}+\left(k^{2}-\frac{2}{\tau^{2}}\right)|v_{k}|^{2}=\frac{k}{2}\left(2-\frac{2}{k^{2}\tau^{2}}-\frac{1}{k^{4}\tau^{4}}\right)\,.$
(148)

It is instructive to obtain the following sub-Hubble and super-Hubble limits
of the above expression:

$\displaystyle|v_{k}^{\prime}|^{2}+\Omega_{k}^{2}|v_{k}|^{2}\bigg{|}_{-k\tau\to\infty}$
$\displaystyle=$ $\displaystyle k\,;$ (149)
$\displaystyle|v_{k}^{\prime}|^{2}+\Omega_{k}^{2}|v_{k}|^{2}\bigg{|}_{-k\tau\to
0}$ $\displaystyle=$
$\displaystyle-\frac{k}{2}\left(\frac{1}{k^{4}\tau^{4}}\right)\,.$ (150)

Incorporating the expression in Eq. (150) into Eq. (145), we obtain the energy
density of the super-Hubble (IR) part of the $\hat{v}$ field to be

$\displaystyle\rho_{v}\big{|}_{\rm IR}(\tau)=-\frac{1}{2}\int_{k_{\rm
IR}}^{-1/\tau}\frac{{\rm d}^{3}\vec{k}}{(2\pi)^{3}}\leavevmode\nobreak\
\frac{k}{2}\,\left(\frac{1}{k^{4}\tau^{4}}\right)=-\frac{1}{4\pi^{2}}\frac{1}{\tau^{4}}\int_{k_{\rm
IR}}^{-1/\tau}{\rm d}\,\left(\ln
k\right)=-\frac{1}{4\pi^{2}}\frac{1}{\tau^{4}}\ln\left(\frac{-1}{k_{\rm
IR}\tau}\right)\,,$ (151)

where $k_{\rm IR}$ is a cut-off introduced to regulate the IR divergence
(since inflation may have a beginning [119], $k_{\rm IR}$ corresponds to the
comoving mode making its Hubble-exit at the beginning of inflation). Eq. (151)
demonstrates that the energy density of the super-Hubble part of $\hat{v}$
diverges at late times, $-\tau\to 0$, which can be attributed to the fact that
the effective mass term is tachyonic, as described previously in Fig. 12 with
the analogy of an inverted oscillator. (Note that the energy density
corresponding to the sub-Hubble (UV) part of the $\hat{v}$ field in Eq. (149)
leads to the usual UV divergence associated with quantum fluctuations in
Minkowski spacetime.)

Similarly we compute the expectation value of the occupation number density of
particles of a given mode $k$ to be

$\langle\,0\,|\,\hat{n}_{v}(k)\,|\,0\,\rangle\equiv\frac{1}{2}\,\Big{|}\frac{|v_{k}^{\prime}|^{2}+\Omega_{k}^{2}|v_{k}|^{2}}{k}\Big{|}_{-k\tau\to
0}-\frac{1}{2}\simeq\frac{1}{4}\,\left(\frac{-1}{k\tau}\right)^{4}\to\infty,$
(152)

which diverges, indicating abundant excitations of $\hat{v}$ quanta, as
discussed before. The corresponding energy density and particle number density
for the $\hat{\zeta}$ field can be obtained by
$\hat{\zeta}=\hat{v}/(am_{p}\sqrt{2\epsilon_{H}})$. Accordingly, the finite
and frozen power spectrum of $\hat{\zeta}$ on super-Hubble scales can be
justified in this Heisenberg picture. Note that similar arguments are also
applicable for the tensor-type fluctuations discussed in Sec. 6.3. See App. F
for a discussion on the quantum fluctuations of a massive scalar field in de
Sitter spacetime and their relation to the inflationary scalar fluctuations.

Before concluding the treatment of scalar fluctuations, let us stress that the
computation of the inflationary scalar power spectrum given in Eq. (123) is
one of the key achievements of modern theoretical cosmology. This is certainly
one of the most important applications of quantum field theory to curved
spacetime in physics, generating the seeds for the large-scale cosmological
structure in the universe141414In April 2014 (following the announcement of
B-mode polarization signal by BICEP2 in Ref. [120]), Juan Maldacena gave a set
of 4 lectures on the computation of inflationary quantum fluctuations at the
International Centre for Theoretical Physics (ICTP), Trieste in Italy. At the
beginning of the first lecture, he stressed the importance of learning these
computations for a theoretical physicist in the form of a funny allegory (here
is the link to the first lecture). I was fortunate to be present in the
classroom as a young student who was inspired by those lectures and later
taught myself the calculations over the period of a few years (and I still
continue to learn new ways of computing and understanding them).. These
results were first obtained in the early 1980s around the time of the famous
Nuffield conference organised by Stephen Hawking and colleagues (see Ref.
[121]) and were published in Refs. [10, 11, 12, 13]. For a historical account
of the subject, see the popular Ref. [122].

###  6.3 Tensor quantum fluctuations during inflation

The gauge-invariant tensor fluctuations $h_{ij}(\tau,\vec{x})$ during
inflation are described by the quadratic action [85, 91]

$S^{(2)}[h_{ij}]=\frac{1}{2}\int{\rm d}\tau\leavevmode\nobreak\ {\rm
d}^{3}\vec{x}\leavevmode\nobreak\
\left(\frac{am_{p}}{2}\right)^{2}\left[h_{ij}^{\prime
2}-(\partial_{l}h_{ij})^{2}\right]\,,$ (153)

where the transverse and traceless tensor field $h_{ij}(\tau,\vec{x})$
satisfies

$\partial^{i}\,h_{ij}=0\leavevmode\nobreak\ \leavevmode\nobreak\
\text{(transverse)}\,;\quad h_{i}^{i}=0\leavevmode\nobreak\
\leavevmode\nobreak\ \text{(traceless)}\,,$ (154)

and can be decomposed into its two orthogonal polarization components,

$\displaystyle h_{ij}(\tau,\vec{x})$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{bmatrix}h_{+}&h_{\times}&0\\\
h_{\times}&-h_{+}&0\\\
0&0&0\end{bmatrix}=\frac{1}{\sqrt{2}}\begin{bmatrix}1&0&0\\\ 0&-1&0\\\
0&0&0\end{bmatrix}h_{+}(\tau,\vec{x})+\frac{1}{\sqrt{2}}\begin{bmatrix}0&1&0\\\
1&0&0\\\ 0&0&0\end{bmatrix}h_{\times}(\tau,\vec{x})$ (155)
$\displaystyle=\frac{1}{\sqrt{2}}\leavevmode\nobreak\
\epsilon^{+}_{ij}\,h_{+}(\tau,\vec{x})+\frac{1}{\sqrt{2}}\leavevmode\nobreak\
\epsilon^{\times}_{ij}\,h_{\times}(\tau,\vec{x})\,,$ (156)

with
$\epsilon^{\lambda}_{ij}\epsilon^{ij\,\lambda^{\prime}}=2\delta^{\lambda\lambda^{\prime}};\leavevmode\nobreak\
\lambda=\\{+,\times\\}$. The action can be rewritten as

$S^{(2)}[h_{+},\,h_{\times}]=\frac{1}{2}\int{\rm d}\tau\leavevmode\nobreak\
{\rm d}^{3}\vec{x}\leavevmode\nobreak\
\left(\frac{am_{p}}{2}\right)^{2}\,\sum_{\lambda=+,\times}\,\left[({h_{\lambda}}^{\prime})^{2}-(\partial_{l}h_{\lambda})^{2}\right]\,.$
(157)

We define the two Mukhanov-Sasaki variables for the tensor perturbations to be

$v_{\lambda}=\left(\frac{am_{p}}{2}\right)h_{\lambda}\,,$ (158)

in terms of which the action becomes

$\boxed{\,S^{(2)}[v_{+},\,v_{\times}]=\frac{1}{2}\int{\rm
d}\tau\leavevmode\nobreak\ {\rm d}^{3}\vec{x}\leavevmode\nobreak\
\,\sum_{\lambda=+,\times}\,\left[({v_{\lambda}}^{\prime})^{2}-(\partial_{i}v_{\lambda})^{2}+\frac{a^{\prime\prime}}{a}\,{v_{\lambda}}^{2}\right]\,}\,,$
(159)

which is equivalent to the action of two massless scalar fields in de Sitter
spacetime, as can be inferred by comparing this with Eq. (107). The full power
spectrum of tensor fluctuations is defined by

$\displaystyle{\cal P}_{T}(k)$
$\displaystyle=\frac{k^{3}}{2\pi^{2}}\left(\absolutevalue{h_{+}}^{2}+\absolutevalue{h_{\times}}^{2}\right)=\frac{k^{3}}{2\pi^{2}}\left(\frac{2}{am_{p}}\right)^{2}\left(\absolutevalue{v_{+}}^{2}+\absolutevalue{v_{\times}}^{2}\right)\,.$
(160)

Borrowing the result from Eq. (121) in Sec. (6.2), we obtain

${\cal
P}_{T}(k)=\frac{2}{\pi^{2}}\left(\frac{H}{m_{p}}\right)^{2}\left(1+k^{2}\tau^{2}\right)\,,$
(161)

which on super-Hubble scales $|k\tau|\to 0$ takes the form

$\boxed{{\cal P}_{T}(k)\Big{|}_{k\ll
aH}=\frac{2}{\pi^{2}}\left(\frac{H}{m_{p}}\right)^{2}}\,.$ (162)

The tensor power spectrum only depends upon the Hubble parameter during
inflation, and hence encodes direct information about the energy scale of
inflation. In contrast, the scalar power spectrum in Eq. (123) depends both on
the Hubble parameter $H$ and the first slow-roll parameter $\epsilon_{H}$, as
stressed before. The tensor power spectrum151515Some papers in the literature
use a different normalisation for the tensor field, instead of using the
canonically normalised fields $\\{v_{+},\,v_{\times}\\}$. While the field
equations are the same for both choices of normalisation, the action will
differ by additional multiplicative factors, therefore the power spectra will
also differ. For this reason, we recommend using the canonically normalised
field variables, in terms of which the tensor action in Eq. (157) takes the
form of two massless fields in de Sitter spacetime. in de Sitter spacetime was
originally computed by Starobinsky in his pioneering work [18] in 1979, a year
before the inception of the inflationary hypothesis.

Similar to the scalar power spectrum in Eq. (124), we can also write the
tensor power spectrum as a power law, that is,

${\cal P}_{T}(k)=A_{T}\left(\frac{k}{k_{*}}\right)^{n_{{}_{T}}}$ (163)

where $A_{T}$ is the amplitude of the tensor modes at the pivot scale $k_{*}$
which is chosen to be $k_{*}=0.05\,\rm Mpc^{-1}$ and $n_{{}_{T}}$ is the
tensor spectral index. An important quantity is the tensor-to-scalar ratio
which is defined as,

$r=\frac{A_{T}}{A_{S}}=\frac{{\cal P}_{T}(k_{*})}{{\cal P}_{\zeta}(k_{*})}.$
(164)

Using Eqs. (123) and (162), we have

$r=16\,\epsilon_{H}$ (165) $n_{{}_{T}}=-2\,\epsilon_{H}$ (166)

Thus, a consistency relation can be established between the tensor spectral
index and the tensor-to-scalar ratio as,

$r=-8\,n_{{}_{T}}\,,$ (167)

which can be used as a smoking gun test of the single field slow-roll
inflationary paradigm.

By this stage, a careful reader must have noticed that the scalar spectral
index is denoted as $n_{{}_{S}}-1$, while the tensor spectral index is denoted
as $n_{{}_{T}}$. Such a difference in the notation is an unfortunate
historical artifact. Since scale-invariance is an approximate symmetry during
inflation, we should have denoted the scalar spectral index by $n_{{}_{S}}$,
where $|n_{{}_{S}}|\ll 1$ is a small number. In fact, we had originally
intended to use this improved notation throughout these lecture notes.
However, this might have created potential confusion for the reader since most
other sources use $n_{{}_{S}}-1$. For this reason161616And for the reason that
it may be notationally convenient for observers to present their measurements
as $n_{{}_{S}}\simeq 0.965$ (which is an order 1 number), rather than as
$n_{{}_{S}}\simeq-0.035$ (which is a small number), as pointed out to the
author by Daniel Baumann., we have (reluctantly) reverted to use the standard
$n_{{}_{S}}-1$ notation for the scalar spectral index. Furthermore, notice
that in Fig. 14, we labelled the $X$-axis as scalar spectral index, although
we plot $n_{{}_{S}}$, rather than $n_{{}_{S}}-1$. Again, we have reluctantly
given in to such a convention in order to be consistent with Fig. 8 of Ref.
[2].

Before moving on to study the dynamics of different popular inflationary
models, it is important to remind the readers that we are working in natural
units, where $\hbar,\,c=1$. Hence they do not appear in the expressions for
the power spectra of quantum fluctuations in Eqs. (123) and (162). Putting
$\hbar$ and $c$ back into the expressions for power spectra, we obtain

$\displaystyle{\cal P}_{\zeta}(k)\Big{|}_{k\ll aH}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{2}}\,\left(\frac{\hbar}{c^{5}}\right)\,\left(\frac{H}{m_{p}}\right)^{2}\,\frac{1}{\epsilon_{H}}\,,$
(168) $\displaystyle{\cal P}_{T}(k)\Big{|}_{k\ll aH}$ $\displaystyle=$
$\displaystyle\frac{2}{\pi^{2}}\,\left(\frac{\hbar}{c^{5}}\right)\,\left(\frac{H}{m_{p}}\right)^{2}\,.$
(169)

###  6.4 Beyond linear perturbation theory: primordial non-Gaussianity

In Secs. 6.2 and 6.3, we computed the power spectra corresponding to the two-
point auto-correlator of the scalar and tensor fluctuations at linear order in
perturbation theory, using the quadratic actions given by Eqs. (102) and
(153). At linear order, the scalar and tensor fluctuations behave like free
(non-interacting) quantum fields and hence their wave functions are Gaussian
[123]. Therefore, the computation of their skewness coming from the three-
point (and higher odd-point) correlators using the quadratic action yields