The Inflationary Universe The Inflationary Universe

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The Inflationary Universe The Inflationary Universe

Sabino Matarrese

Dipartimento di Fisica “G. Galilei”

Università degli Studi di Padova ITALY

email: [email protected]

Sabino

Sabino Matarrese Matarrese

Dipartimento di

Dipartimento di Fisica Fisica “G. “ G. Galilei Galilei” ” Universit

Università à degli degli Studi Studi di Padova di Padova ITALY

ITALY email:

email: [email protected] [email protected]

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XIX Canary Islands Winter School XIX Canary Islands Winter School

Outline Outline

1. 1. First lecture: First lecture:

Kinematical properties of Inflation Kinematical properties of Inflation

Scalar field dynamics and Inflationary models Scalar field dynamics and Inflationary models

1. 1. Second lecture: Second lecture:

Perturbation theory in a quasi- Perturbation theory in a quasi -de Sitter stage de Sitter stage

Classical evolution of scalar and tensor modes Classical evolution of scalar and tensor modes

2. 2. Third lecture: Third lecture:

Generation of scalar and tensor modes from quantum vacuum oscillations Generation of scalar and tensor modes from quantum vacuum oscill ations

Power- Power -spectrum of scalar and tensor modes and slow spectrum of scalar and tensor modes and slow- -roll parameters roll parameters

3. 3. Fourth lecture: Fourth lecture:

Beyond linear perturbations Beyond linear perturbations

Beyond the power- Beyond the power -spectrum: higher spectrum: higher- -order statistics and primordial non order statistics and primordial non - - Gaussianity

Gaussianity

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1.1 Kinematical properties of Inflation

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From Einstein to

From Einstein to Friedmann Friedmann

Step 1

Step 1 : write the line- : write the line -element given the symmetry of space element given the symmetry of space- -time time Step 2

Step 2 : compute the Christoffel : compute the Christoffel symbols symbols Step 3

Step 3 : compute the Ricci tensor : compute the Ricci tensor Step 4

Step 4 : choose the stress : choose the stress -energy tensor consistently with space - energy tensor consistently with space -time - time symmetries

symmetries

0 2 8 1

;

=

−

≡

b ab

ab ab

T

GT R

g R

G π ^{geometry ⇔} ^⇔ ^⇔ ^{⇔ matter}

invariance under coordinate transformations

equations of motion

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Friedmann

Friedmann vs. Newton vs. Newton

 

 



 +

−

=

−

 =



 





 

 



 +

−

=

2 2 2 2

2

3 3 8

3 3

4 c p a

a

a c G

a a

c p G

a a

ρ ρ

ρ κ π

π ρ

&

F = ma

E = const.

M = const.

2 2

2 c dt a ( t ) d l

ds = − Robertson-Walker metric

scale factor

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The Big Bang crisis The Big Bang crisis

i. i. Horizon problem: does our Universe Horizon problem: does our Universe belong to

belong to … … a set of measure zero? a set of measure zero?

ii. ii. Flatness problem: do we need to fine Flatness problem: do we need to fine - - tune the initial conditions of our

tune the initial conditions of our Universe?

Universe?

iii. iii. Cosmic fluctuation problem: how did Cosmic fluctuation problem: how did perturbations come from?

perturbations come from?

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Horizon problem Horizon problem

Kinney 2003

The comoving scale of causal correlation

r

_H

(t) = 1 / ^a(t)H(t)

grows with time

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Evolution of the comoving Hubble radius

To solve the horizon problem one needs a period when r

^H

decreases with time, i.e. a period of accelerated expansion:

a > 0 ^..

i.e. the comoving scale

of causal correlation

r

_H

(t) decays with time

during inflation

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Solution of the horizon Solution of the horizon

problem (I) problem (I)

About 60 e-folds of inflation suffice to solve the horizon and flatness

problems. Inflation usually lasts much much longer.

Kinney 2003

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Physical scales & horizon crossings

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Solution of the horizon Solution of the horizon

problem (II) problem (II)

The horizon problem is solved if a region that was causally conn

The horizon problem is solved if a region that was causally conn ected ected at the beginning of inflation,

at the beginning of inflation, t t

_i_i

, whose typical size is d , whose typical size is d

_H_H

(t (t

_i_i

) = a(t ) = a(t

_i_i

) ) r r

_H_H

(t (t

_i_i

) after inflating by a factor ) after inflating by a factor

is able to contain the present Hubble radius scaled back to the

is able to contain the present Hubble radius scaled back to the end of end of inflation

inflation t t

_f_f

: :

r r

_H_H

(t (t

_i_i

) ≥ ) ≥r r

_H_H

(t (t

₀₀

) ) This is possible only if

This is possible only if r r

_H_H

(t) decreases with time during inflation: (t ) decreases with time during inflation:

for a suitable time

for a suitable time- -interval interval

exp

inf

) ( exp

) ( / )

( t a t dt H t N

a

Z

^f

i

t i t

f

= ≡

≡ ∫

0 )

( 0

)

( t < ⇔ a t >

r & _H & &

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Evolution of the density parameter:

the flatness problem

The density parameter decreases with time if the Universe expansion is decelerated. One needs a fine-tuning of ~ 60 orders of magnitude (!) at the Planck time in order to allow for a density

parameter of order unity today! A period of

accelerated expansion

automatically solves the

problem.

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“ “ Minimal inflation Minimal inflation ” ” : Z= : Z= Z Z _min _min

Take Z=

Take Z= Z Z _min _min such that r such that r _H _H (t (t ₀ ₀ )= )= r r _H _H (t (t _i _i ). From the definition ). From the definition of Z one finds

of Z one finds

which, for inflation final temperature not far from the which, for inflation final temperature not far from the Planck energy (T

Planck energy (T _Planck _Planck ~10 ~10 ¹⁹ ¹⁹ GeV), and for a nearly de GeV), and for a nearly de Sitter equation of state

Sitter equation of state w w _inf _inf = = - - 1, leads to a minimum 1, leads to a minimum number of inflation e

number of inflation e- - folds folds

N N _inf _inf ~ 60 ~ 60

With such a choice With such a choice Ω Ω ₀ ₀ = = Ω Ω _i _i which automatically solves which automatically solves the flatness problem. More in general

the flatness problem. More in general

( ( Ω Ω ₀ ₀ ^- ^- ¹ ¹ - - 1)/( 1)/( Ω Ω _i _i ^-1 ^- ¹ - - 1)=(Z/Z 1)=(Z/Z _min _min ) ) ^- ^- ^|1+3w| ^|1+3w|

( )

inf inf inf

| 3 1

| / 30 2

min

10 T / T

^inf

w p / ρ

Z ≈

_f _Planck ⁺ ^w

=

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Kinematics of inflation

The accelerated expansion can be realized by many different types of scale factor time- dependence, originating from different equations of state (w=p/ρ) during inflation. For slow-roll inflation this in turn comes from a choice of the inflaton potential V(ϕ).

.

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1.2 1.2 Scalar field dynamics and Inflationary Scalar field dynamics and Inflationary models

models

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YES NO NO!!

YES

The role of vacuum energy The role of vacuum energy

To realize a physical system with negative To realize a physical system with negative isotropic pressure (

isotropic pressure (“ “tension tension” ”) one needs to ) one needs to make use of the properties of the vacuum make use of the properties of the vacuum state in QFT. A well

state in QFT. A well- -known example being known example being the Casimir the Casimir effect effect in QED. in QED.

2

3 0 p 1 c

a & & > ⇔ < − ρ

Which kind of quantum field expectation value is allowed, i.e. is consistent with the observed large-scale homogeneity & isotropy?

fermion condensate

0 0

3 , 2 , 1 , 0 0

0 |

|

〉 ≠

〈

〉 ≠

〈

=

〉 ≠

〈

〉 ≠

= 〈

〉

〈

ψ ψ ψ

φ φ

a A

vac vac

a

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The The Casimir Casimir effect effect

In 1948, the Dutch physicist Hendrick Casimir

proposed to test for the presence of vacuum energy, which in QFT takes the form of particles constantly forming and disappearing on a tiny scale. Normally, the vacuum is filled with particles of almost any wavelength. He argued that if two thin uncharged metal plates are placed very close together, longer wavelengths would be excluded. The extra waves outside the plates generate a force that tends to push them together: the closer the plates, the stronger the attraction. Lamoreux (1996) measured the Casimir effect finding very good agreement with theory. The first measurement with plane parallel geometry is due to Bressi et al. (2002).

Casimir force: F ~ A / d

⁴

(A = area of the plates, d = distance between them)

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A brief history of the A brief history of the inflationary model (I) inflationary model (I)

1979 – 1979 – 1980 1980 A. A. Starobinski Starobinski shows that a de Sitter stage in the early shows that a de Sitter stage in the early Universe is driven by trace anomalies of quantum fields in an ex

Universe is driven by trace anomalies of quantum fields in an external ternal gravitational field; this phase is later terminated by the gener

gravitational field; this phase is later terminated by the generation of ation of scalar field fluctuations (

scalar field fluctuations (“ “scalarons scalarons” ”) ) a stochastic background of a stochastic background of gravity

gravity- -waves should be the observable relic of this early phase waves should be the observable relic of this early phase

1981 1981 A. Guth A. Guth shows that “ shows that “Inflation Inflation” ” (i.e. a quasi- (i.e. a quasi -de Sitter expansion de Sitter expansion phase in the early Universe) is caused by a first

phase in the early Universe) is caused by a first - - order phase transition order phase transition (“ ( “Old Inflation Old Inflation” ”); this phase solves the ); this phase solves the “ “monopole overproduction monopole overproduction

problem

problem” ” of phase transitions at high temperature of phase transitions at high temperature the horizon and the horizon and flatness problems find an automatic solution. However, the phase

flatness problems find an automatic solution. However, the phase transition never actually gets to an end (

transition never actually gets to an end (“ “graceful exit problem graceful exit problem” ”). Similar ). Similar idea were contained in two almost simultaneous papers (

idea were contained in two almost simultaneous papers (Suto Suto 1981; 1981;

Kazanas

Kazanas 1981). 1981).

1982 1982 A dynamical symmetry- A dynamical symmetry -breaking mechanism is invoked to avoid the breaking mechanism is invoked to avoid the graceful exit problem of

graceful exit problem of Guth Guth ’s ’ s model in two independent analyses model in two independent analyses (Albrecht & Steinhardt 1982;

(Albrecht & Steinhardt 1982; Linde Linde 1982). The “ 1982). The “New Inflation New Inflation” ” model is model is based on the slow

based on the slow- -rolling of a scalar field along an almost flat potential. rolling of a scalar field along an almost flat potential.

This scalar field is initially associated to the Higgs sector of

This scalar field is initially associated to the Higgs sector of GUT GUT

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A brief history of the A brief history of the inflationary model (II) inflationary model (II)

1982 – 1982 – 1983 1983 Many independent groups (Guth Many independent groups ( Guth & Pi; Starobinski & Pi; Starobinski; Hawking; ; Hawking;

Bardeen

Bardeen, Steinhardt & Turner) show that during slow , Steinhardt & Turner) show that during slow- -rolling inflation rolling inflation scalar perturbations are created by quantum vacuum oscillations

scalar perturbations are created by quantum vacuum oscillations of the of the scalar field, leading to density fluctuations

scalar field, leading to density fluctuations δρ/ρ δρ/ρ ~ ~ λ λ

^1/2^1/2

where where λ is the self λ is the self - - coupling constant of the scalar field. Consistency with the obse

coupling constant of the scalar field. Consistency with the observed rved isotropy of the CMB constrains

isotropy of the CMB constrains λ to be much less than λ to be much less than 10 10

⁻⁴⁻⁴^.^.

This leads to This leads to two classes of problems: the thermal initial conditions problem

two classes of problems: the thermal initial conditions problem and the and the nature of the scalar field which needs to be very weakly coupled

nature of the scalar field which needs to be very weakly coupled with the with the rest of the world (it must be a

rest of the world (it must be a “ “singlet singlet” ”) )

1983 1983 A. Linde A. Linde proposes a new class of models, called “ proposes a new class of models, called “Chaotic Inflation Chaotic Inflation” ”, , where thermal initial conditions (i.e. a

where thermal initial conditions (i.e. a metastable metastable state) are replaced by state) are replaced by an unstable initial scalar

an unstable initial scalar- -field state motivated by Heisenberg uncertainty field state motivated by Heisenberg uncertainty relation near the Planck scale.

relation near the Planck scale.

1983 1983 Particle- Particle -physics theorists argue that physics theorists argue that Supersymmetry Supersymmetry might be the might be the natural environment for such a weakly coupled scalar field, whic

natural environment for such a weakly coupled scalar field, which can be h can be easily added as a novel scalar sector to the theory: the

easily added as a novel scalar sector to the theory: the “ “Inflaton Inflaton” ”. .

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A brief history of the A brief history of the inflationary model (III) inflationary model (III)

1984 1984 – – 1985 1985 Abbott & Wise show that the actual condition for inflation is Abbott & Wise show that the actual condition for inflation is merely an accelerated phase; the de Sitter solution being just t

merely an accelerated phase; the de Sitter solution being just the simplest he simplest realization of this

realization of this “ “Generalized Inflation Generalized Inflation ”. ” . Lucchin Lucchin & Matarrese show that & Matarrese show that

“Power “ Power- -law Inflation law Inflation” ” is the exact solution of Einstein’ is the exact solution of Einstein ’s equations if the s equations if the inflaton

inflaton has an exponential potential. Scale- has an exponential potential. Scale -free but generally (red free but generally (red- -)tilted )tilted density

density- -fluctuation and gravitational fluctuation and gravitational- -wave power wave power- -spectra are obtained. spectra are obtained.

1986 1986 A. Linde A. Linde notices that chaotic inflation is the most probable state of notices that chaotic inflation is the most probable state of the Universe. Only a tiny fraction of the inflated Universe ends

the Universe. Only a tiny fraction of the inflated Universe ends the the accelerated phase undergoing reheating, thus leading to a post

accelerated phase undergoing reheating, thus leading to a post- -inflation inflation phase resembling the observed Universe. The vast majority of the

phase resembling the observed Universe. The vast majority of the Universe volume undergoes

Universe volume undergoes “ “Eternal Inflation Eternal Inflation” ”. This necessarily calls for . This necessarily calls for the use of the

the use of the “ “Anthropic Anthropic Principle” Principle ” in cosmology. in cosmology.

1989 1989 La & Steinhardt analyze a new class of two- La & Steinhardt analyze a new class of two -field models, which can field models, which can easily accommodate for old inflation while solving the graceful

easily accommodate for old inflation while solving the graceful exit exit problem of

problem of Guth Guth ’s ’ s original model. One of the two fields is a Jordan- original model. One of the two fields is a Jordan -Brans Brans- - Dicke

Dicke (JBD) scalar field characteristic of scalar (JBD) scalar field characteristic of scalar - - tensor theories of gravity. tensor theories of gravity.

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A brief history of the A brief history of the inflationary model (IV) inflationary model (IV)

1992 1992 The discovery of intrinsic CMB temperature anisotropies by the COBE The discovery of intrinsic CMB temperature anisotropies by the COBE satellite boosts the research on the observational consequences

satellite boosts the research on the observational consequences of of inflation. The

inflation. The inflaton inflaton potential reconstruction problem appears affordable. potential reconstruction problem appears affordable.

1994 1994 Hybrid inflation models are proposed by various groups (e.g. Hybrid inflation models are proposed by various groups (e.g.

Copeland et al. 1994;

Copeland et al. 1994; … …). Among the relevant aspects of these models is ). Among the relevant aspects of these models is the possibility of a

the possibility of a “ “blue blue ” ” tilt of the scalar perturbations. tilt of the scalar perturbations.

1996 1996 Motivated by studies of the LSS of the Universe, according to which Motivated by studies of the LSS of the Universe, according to w hich Ω Ω = 0.2 = 0.2 - - 0.4, various groups show that, contrary to previous 0.4, various groups show that, contrary to previous

expectations, inflation does not necessarily lead to

expectations, inflation does not necessarily lead to Ω Ω = 1, but an open = 1, but an open Universe is also a viable possibility without fine

Universe is also a viable possibility without fine- -tuning of the initial tuning of the initial conditions.

conditions.

1999 1999 The detection of the first Doppler peak in the CMB anisotropies The detection of the first Doppler peak in the CMB anisotropies by by the the BOOMERanG BOOMERanG and and MAXIMA MAXIMA collaborations gives strong support to the collaborations gives strong support to the inflationary prediction of a flat (

inflationary prediction of a flat ( Ω Ω = 1) Universe. = 1) Universe.

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A brief history of the A brief history of the inflationary model (V) inflationary model (V)

2001 2001 Alternative scenarios for the generation of scalar perturbation Alternative scenarios for the generation of scalar perturbation are proposed (e.g. the

are proposed (e.g. the curvaton), which do not necessarily rely on curvaton ), which do not necessarily rely on inflaton

inflaton perturbations (Moroi perturbations ( Moroi & Takahashi 2001, 2002; Enqvist & Takahashi 2001, 2002; Enqvist & &

Sloth 2002;

Sloth 2002; Lyth Lyth & Wands 2002; the original idea dating back to & Wands 2002; the original idea dating back to Mollerach

Mollerach 1990). 1990).

2003 2003 WMAP yields spectacular support to all the most important WMAP yields spectacular support to all the most important predictions of inflation: flat Universe, adiabatic and nearly sc

predictions of inflation: flat Universe, adiabatic and nearly scale ale- - invariant density perturbations, T

invariant density perturbations, T- -E cross E cross- -correlation, correlation, … …! Only the ! Only the inflation generated gravitational

inflation generated gravitational- -wave background is yet wave background is yet undetected.

undetected.

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Scalar

Scalar - - field dynamics field dynamics

The action of the (minimally coupled) scalar (inflaton) field reads:

Scalar field eq. of motion: Klein-Gordon equation:

Scalar-field stress-energy tensor:

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Background evolution Background evolution

Split the scalar field as a (classical) background + a (quantum) fluctuation

Dropping the fluctuation we obtain

If the potential energy dominates over the kinetic term er obtain a quasi-de Sitter evolution

a(t) ≈ exp(Ht)

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Inflaton

dynamics (I)

The vacuum expectation value of the inflaton scalar field

behaves as a perfect fluid, but,

unlike standard fluids, it can

have negative pressure.

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R i φ

( V φ)

φ φ φ

bubble nucleation inflation

reheating (

V φ)

(a) (b)

New vs. old inflation dynamics

Different models of inflation derive from different potential and different initial conditions. Old inflation (Guth 1981) assumes thermal initial conditions (which are very difficult to achieve). Chaotic inflation (Linde 1983) is based on the application of the uncertainty principle at Planck energies.

Chaotic inflation Old inflation

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Inflaton

dynamics (II)

If the potential V is very flat the inflaton scalar field ϕ undergoes slow-roll (i.e.

friction-dominated) dynamics (Albrecht & Steinhardt 1982;

Linde 1982). This is common

to most inflation models.

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Quantifying slow

Quantifying slow - - roll dynamics roll dynamics

Using the approximate Friedmann and

Klein-Gordon eqs.:

Define two (first-order)

slow-roll parameters, which need to be << 1 for successful slow-roll inflation to occur

Inflation demands that

 also useful

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Classify Inflationary Models

height

width

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The Universe

thermal history ^IN _F

L A T IO N

The inflationary era had to occur before baryogenesis took place.

The minimum inflation

energy scale can be as

low as 10 ² GeV

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The modern view of the Universe

Observable Universe Observable Universe

Inflation Inflation

Inhomogeneous on small Inhomogeneous on small

scales scales

Almost homogeneous &

isotropic on the horizon isotropic on the horizon scale

scale

Very inhomogeneous on Very inhomogeneous on

super

super -horizon scales - horizon scales

{ {

“… for the practical purposes of describing the observable part of our

Universe one can still speak about the big bang, just as one can still use

Newtonian gravity theory to describe the Solar system with very high

precision.” (A. Linde 1995)

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Self Self - - reproducing Inflationary reproducing Inflationary Universe (I)

Universe (I) A, Linde ~ 1989

«Inflationary theory describes the very early stages of the « Inflationary theory describes the very early stages of the evolution of the Universe, and its structure at extremely large evolution of the Universe, and its structure at extremely large distances from us. For many years, cosmologists believed that distances from us. For many years, cosmologists believed that the Universe from the very beginning looked like an

the Universe from the very beginning looked like an expanding ball of fire. This explosive beginning of the expanding ball of fire. This explosive beginning of the Universe was called the

Universe was called the big bang big bang. In the end of the 70's a . In the end of the 70's a different scenario of the evolution of the Universe was

different scenario of the evolution of the Universe was

proposed. According to this scenario, the early universe came proposed. According to this scenario, the early universe came through the stage of

through the stage of inflation inflation , exponentially rapid expansion , exponentially rapid expansion in a kind of unstable vacuum

in a kind of unstable vacuum- -like state (a state with large like state (a state with large energy density, but without elementary particles). Vacuum energy density, but without elementary particles). Vacuum- - like state in inflationary theory usually is associated with a like state in inflationary theory usually is associated with a scalar field, which is often called

scalar field, which is often called “ “ the the inflaton inflaton field.'' The stage field.'' The stage of inflation can be very short, but the universe within this tim of inflation can be very short, but the universe within this time e becomes exponentially large.

becomes exponentially large. » » (A. Linde (A. Linde) )

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Self Self - - reproducing Inflationary reproducing Inflationary Universe (II)

Universe (II)

«Initially, inflation was considered as an intermediate stage of « Initially, inflation was considered as an intermediate stage of the the evolution of the universe, which was necessary to solve many

evolution of the universe, which was necessary to solve many cosmological problems. At the end of inflation the scalar field cosmological problems. At the end of inflation the scalar field decayed, the universe became hot, and its subsequent evolution decayed, the universe became hot, and its subsequent evolution could be described by the standard big bang theory. Thus,

could be described by the standard big bang theory. Thus,

inflation was a part of the big bang theory. Gradually, however, inflation was a part of the big bang theory. Gradually, however, the big bang theory became a part of inflationary cosmology.

the big bang theory became a part of inflationary cosmology.

Recent versions of inflationary theory assert that instead of be Recent versions of inflationary theory assert that instead of being ing a single, expanding ball of fire described by the big bang theor a single, expanding ball of fire described by the big bang theory, y, the universe looks like a huge growing fractal. It consists of m

the universe looks like a huge growing fractal. It consists of many any inflating balls that produce new balls, which in turn produce mo inflating balls that produce new balls, which in turn produce more re new balls,

new balls, ad infinitum ad infinitum . Therefore the evolution of the universe . Therefore the evolution of the universe has no end and may have no beginning. After inflation the

has no end and may have no beginning. After inflation the universe becomes divided into different exponentially large universe becomes divided into different exponentially large

domains inside which properties of elementary particles and even domains inside which properties of elementary particles and even dimension of space

dimension of space- -time may be different. Thus, the new time may be different. Thus, the new

cosmological theory leads to a considerable modification of the cosmological theory leads to a considerable modification of the standard point of view on the structure and evolution of the standard point of view on the structure and evolution of the universe and on our own place in the world.

universe and on our own place in the world.» » (A. Linde (A. Linde) )

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Self Self - - reproducing Inflationary reproducing Inflationary Universe (III)

Universe (III)

«Domains of the inflationary universe with sufficient energy density

permanently produce new inflating domains due to stochastic generation of long-wavelength perturbations. The Universe evolution in the inflationary scenario has no end and may have no beginning. These processes occur in the very early Universe, at densities just below the Planck one. Classically, the field value should decrease, but quantum perturbations lead to

exponentially large domains

containing values of the scalar field much bigger than its initial value. In particular, the volume of regions of the Universe corresponding to peaks is much bigger than that of regions where the scalar field rolled to its minimum energy density.» (A. Linde)

from: A. Linde

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2.1 2.1 Perturbation theory in a quasi Perturbation theory in a quasi - - de Sitter de Sitter stage

stage

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Physical meaning of

Physical meaning of inflaton inflaton fluctuations fluctuations

)) x (t- ,t)~

x (

t x t

x e

W W HW

- W

} W{δ

(t) t

x δ

e

V H

δ e

V H

δ

,t) x ( δ (t) ,t)

x ( t

e V H

e a(t)

Ht Ht

Ht

r r

&

r

&

r

&

r r

&

(

yields in

order first

to which,

) ( ) (

~ ) , ( reads solution

general most

the times large

at Hence,

times).

(large

0 3

to obeys ,

Wronskian the

Moreover, equation.

same the

satify

and ) , ( hence ,

negligible soon

becomes and

time with

decays term

The

"

3 equation the

to obeys

while

"

3 equation Gordon

- Klein perturbed

the satisfies quantity

The

.

then solution, s

homogeneou a

be ) ( let and

' 3

factor -

scale with

time space

Sitter de

in evolving φ

field scalar a

consider s

let' (1982), Pi

&

Guth Following

3 0 2

2

2 2 2 2

δτ ϕ φ

δτ

ϕ δτ δφ

ϕ φ

ϕ δφ

φ

ϕ ϕ

φ δφ

φ δ φ

δ

φ

φ ϕ

φ φ

φ

−

→

⇒ =

=

∇

−

= +

∇ +

−

≈ +

+

=

∇ +

−

= +

=

−

The effect of

The effect of inflaton inflaton fluctuations is to produce a space fluctuations is to produce a space- -dependent time dependent time- -delay delay δτ δ τ in the in the evolution of the homogeneous mode

evolution of the homogeneous mode ϕ ϕ . We would then expect density fluctuations . We would then expect density fluctuations

δρ δ ρ /ρ / ρ ~ Hδ ~ H δτ τ . .

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Perturbing geometry Perturbing geometry

To allow for deviations from homogeneous and isotropic universe perturb the FRW line-element (scalar perturbations only)

Equivalently, write the covariant metric tensor as:

whose inverse reads:

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Perturbed LHS of Einstein

Perturbed LHS of Einstein ’ ’ s s eqs eqs . .

Expanding to first order in the metric perturbation the Christoffel symbols

first and the Ricci tensor next one arrives at:

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Perturbed RHS of Einstein

Perturbed RHS of Einstein ’ ’ s s eqs eqs . .

& Klein

& Klein - - Gordon Gordon eq eq . .

Next consider the scalar field perturbation and look at its first-order effects on the stress-energy tensor and scalar field equation of motion

φ(x,τ) = φ(τ) + δφ(x,τ)

Klein-Gordon eq.

Stress-energy tensor

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Gauge transformations Gauge transformations

from: Riotto (2002)

A gauge is a map between points of the physical (perturbed) space-time and points of the background.

A gauge transformation is a change of such a map.

This can be mimicked by a coordinate transformation

Its effect on a tensor quantity Q is given by the Lie derivative of the quantity along the infinitesimal four-vector defining the coordinate transformation

E.g., for a scalar f (e.g. the density) one gets:

vector scalars

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Gauge transformations Gauge transformations

& gauge invariance

Under a gauge transformation our

“scalar” perturbations transform as follows:

Beware of pure-gauge modes and of gauge artifacts (“tenacious myths”)!

Two ways out: choose a gauge (but take care of residual gauge ambiguities)

Define gauge-invariant quantities (e.g.

by linearly combining perturbations)

Bardeen’s g.-i. potentials

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The The comoving comoving curvature curvature perturbation

perturbation

The curvature of hypersurfaces at constant (conformal) time is given by

However, this quantity is not gauge-invariant . One can define a g.i. quantity which is related to the previous one on comoving hypersurfaces.

Indeed, the quantity

is the comoving curvature perturbation: it is gauge-invariant by construction and it represents the gravitational potential on comoving hypersurfaces where δφ=0.

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Useful gauge

Useful gauge - - invariant variables invariant variables

The gauge-invariant (by construction) quantity

represents the curvature perturbation on spatial slices of uniform energy density.

For the scalar field fluctuation, one can analogously define the g.-i. variable.

Alternatively, the so-called gauge-invariant Sasaki-Mukhanov variable

represents the inflaton perturbation on spatially flat gauges.

Hence: the inflaton perturbation and the curvature perturbation are related via a gauge- transformation!

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2.2 Classical evolution of scalar and tensor 2.2 Classical evolution of scalar and tensor

modes

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Scalar mode equation of motion Scalar mode equation of motion

in a quasi

in a quasi - - de Sitter stage (I) de Sitter stage (I)

the 0-0 component (energy constraint) and the i-i components give

In the longitudinal gauge (B=E=0), using Einstein’s eqs., the i≠j components yield

the 0-i components (momentum constraints) yield

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Combining the previous eqs. One obtains an equation only for (e.g.) the gravitational potential ψ

On super-horizon scales, using the background equation of motion and the definition of slow-roll parameters one can show that the gravitational potential is nearly constant, which upon

replacement in the momentum constrain, gives

Scalar mode equation of motion Scalar mode equation of motion

in a quasi

in a quasi - - de Sitter stage (II) de Sitter stage (II)

This result can be used in the perturbed Klein-Gordon equation to obtain (still on super-horizon scales)

Rescaling the scalar field variable as δχ_k= δφ_k/a we obtain

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Gauge

Gauge - - invariant derivation invariant derivation

One can write the perturbed Einstein’s equations directly in terms of gauge-invariant variables:

The spatial traceless components give immediately D=0, i.e. Φ=Ψ, which (using background eqs.) implies

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-

Gauge

Gauge - - invariant derivation invariant derivation

Combining the previous eqs. one can obtain an equation for the gauge-invariant variable

= - aQ

which reads

The remaining eqs. give

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Classical evolution of perturbations Classical evolution of perturbations

Before discussing the evaluation of the scalar-field fluctuations, let’s discuss how this information can be transferred to the post-inflationary evolution.

We need a gauge-invariant quantity that changes smoothly when the Universe changes its equation of state

(inflaton radiation matter dark energy domination).

The quantity

ζ remains (approximately) constant outside the horizon, as long as

non-adiabatic pressure terms appear (e.g. isocurvature perturbations)

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Primordial gravitational waves (I) Primordial gravitational waves (I)

GWs are tensor perturbations of the metric. Restricting ourselves to a flat FRW background (and disregarding scalar and vector modes)

ds

²

=a

²

(τ)[ - dτ

²

+ (δ

_ij

+ h

_ij

(x,τ)) dx

ⁱ

dx

^j

]

where h

_ij

are tensor modes which have the following properties

h

_ij

= h

_ji

(symmetric)

h

ⁱ_i

= 0 (traceless) h

ⁱ_j,i

= 0 (transverse) and satisfy the equation of motion

ij ij

ij

ij h h S

a

h + a ' ' −∇ ² =

2 "

Possible source term. It vanishes in linear theory and for a perfect fluid.

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Primordial gravitational waves (II) Primordial gravitational waves (II)

GWs GWs have only (9 have only ( 9 6- 6 -1 1- -3= 3=) ) two two independent degrees of freedom, independent degrees of freedom , corresponding to the two polarization states of the graviton

corresponding to the two polarization states of the graviton

behaviour behaviour: :

k « k « aH aH (outside the horizon) (outside the horizon) φ φ ≈ ≈ const + decaying mode const + decaying mode

k » k » aH aH (inside the horizon) (inside the horizon) φ φ e e

^±ik^±^ikτ^τ

/a /a (gravitational wave; it freely (gravitational wave; it freely streams, experi

streams, experiencing encing redshift redshift and and dilution, just

dilution, just like a free photon) like a free photon)

0 ' '

2 "

) ( ) , ) (

2 ) (

, (

2

3 3

= +

+

= ∫

^•

φ φ

φ

ε τ π φ

τ a k a

k k

k e x d

h

_ij ⁱ^k ^x _ij

v r

r

^r ^r

polarization tensor

free massless, minimally

coupled scalar field

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Frequency (Hz)

10 10 10 1 10 10

Spectral density (Ω h )

-10

-15 -5 5 10

COBE

10 10 10 10 10 10

-16 -14 -12 -10 -8 -6

LIGO I

LIGO II/

VIRGO LISA

Pulsar timing

0.9K graviton blackbody radiation

Extended inflation transition First-order

EW-scale transition Cosmic strings

Global strings

Slow-roll inflation - upper bound

Chaotic inflation Power law inflation 10

10 10 10 10 10

-16 -14 -12 -10 -8 -6

g2 Battye& Shellard1996

The search for primordial The search for primordial

gravitational waves (I)

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3.1 Generation of scalar and tensor modes 3.1 Generation of scalar and tensor modes

from quantum vacuum oscillations

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Generation of cosmological seeds

Particle creation in either strong (Hawking 1972) or rapidly varying

(Parker 1969)

gravitational fields

Schrödinger (1939): “an alarming phenomenon”.

In QED the analogous effect in a strong electric field is known as “Klein paradox”

Kinney 2003

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Solving for the scalar mode Solving for the scalar mode

For constant ν (which is consistent with being at first order in the slow-roll approximation the scalar field eq. of motion is solved by

and its complex conjugate. On super-horizon scales one has

Reminding the definition of the curvature g.i. perturbation

we finally find

With “scalar spectral index”

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Quantization in a FRW background Quantization in a FRW background

Consider a free massive scalar field σ

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3.2 3.2 Power Power - - spectrum of scalar and tensor spectrum of scalar and tensor modes and slow

modes and slow - - roll parameters roll parameters

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Slow Slow - - roll parameters and roll parameters and cosmological observables cosmological observables

Scalar (comoving curvature) perturbation power-spectrum

Tensor (gravity-wave)

perturbation power-spectrum

2 2

2

= M

_P

V ' V ' ' ' / V ξ

24

2

16 2

ln /

2 6

1 ε εη

ξ η ε

− +

−

= +

−

=

−

k d dn n

R R

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Classify Inflationary Models

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“ “ Generic Generic ” ” predictions of single field predictions of single field slow slow - - roll models roll models vs. WMAP 3yr data vs. WMAP 3yr data

from: Spergel et al. 2006

Tensor to scalar ratio Tensor to scalar ratio

< 0.28 (+ SDSS

< 0.28 (+ SDSS data)

data)

< 0.55

< 0.55 r

r

Scalar spectral index Scalar spectral index 0.951

0.951 ±± 0.0220.022 n

n_s_s

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Gravity waves: the

Gravity waves: the “ “ smoking gun smoking gun ” ” of inflation

of inflation

The spectra P _R (k) and P

_T

(k) provide the contact between theory and observations. The WMAP dataset allows to extract an upper bound, r<1.28 (95%) ( Peiris et al.

2003; Kinney et al. 2004 ), or ε<0.08. This limit provides un upper boound on the energy scale of inflation

V V

^1/4

^1/4 < 3.8 x 10 < 3.8 x 10

¹⁶

¹⁶ GeV GeV

A positive detection of the B-mode in CMB polarization, and therefore an indirect evidence of gravitational waves from inflation, once foregrounds due to gravitational

The Inflationary Universe The Inflationary Universe