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Physical Cosmology 23/5/2016

Alessandro Melchiorri

alessandro.melchiorri@roma1.infn.it slides can be found here:

oberon.roma1.infn.it/alessandro/cosmo2016

(2)

Inflation I

Paradoxes of the FRW model:

Flatness:

( ) ⎩ ⎨ ⎧

= Ω

2 2 2/3

1 t

t a

H

t k Radiation dominated era Matter dominated era

( )

2 2

(

0

)

2

0

1 1

1 − Ω = − Ω

a H

t H

In order to have today we need: 1 − Ω

0

≤ 0 . 2 10

4

2

1 − Ω

rm

≤ ×

10

14

3

1 − Ω

nuc

≤ ×

10

60

1

1 − Ω

Planck

≤ ×

At matter-radiation equality At nucleosynthesis

At Planck epoch a

Planck

≈ 2 × 10

32

s

t

Planck

≈ 5 × 10

44

(3)

Inflation II

• Horizons problem. Regions that are not causally connected at recombination

show the same temperature. Why ?

( ) ( )

o ls

A

ls

Hor

rad

Mpc Mpc z

d

t d

Hor

0 . 03 2

13 4 .

0 ≈ ≈

= θ

We have about 20000 not causally connected regions. Why they have

a similar temperature ?

(4)

Inflation III

• Monopole problem. GUT predict that the GUT phase transition creates point-like topological defects that act as magnetic monopoles. The rest energy of the magnetic monopole is

predicted to be m

M

E

GUT

10

12

TeV

( )

3 82

3

36 10

2 ) 1

10

( ≈ ≈ ≈ m

s ct t

n

GUT GUT

M

3 94

2

10 Tev

) (

)

( t

GUT

= n

M

t

GUT

m

M

cm

ρ

M

3 104

4

10 Tev

)

( t

GUT

≈ σ T

GUT

m

ρ

γ

) 10

( )

10

( t

16

s t

16

s

M

> =

= ρ

γ

ρ

The universe should be matter

dominated already in the early universe.

This is impossible because, for example, of BBN constraints !!

Let’s suppose one monopole par horizon.

(5)

Inflation IV

• The solution is to suppose a period of accelerated expansion called inflation in the early universe.

• Let’s model inflation as a cosmological constant acting from t

i

to t

f

( )

( )

( ) t t

t t

t

t t

t t

e a

e a

t t

a t

a

f

f i

i

f t

t H i

t t H i

i i

f i

i i

<

<

⎪ ⎩

⎪ ⎨

=

2 / ) 1

(

) (

2 / 1

/ /

Costant

= Λ

=

i

H

i

(6)

Inflation V

• We can define as number of e-foldings the number N from :

( )

( )

N i

f

e

t a

t

a = N = H

i

( t

f

t

i

)

( t t ) s

s t

H N

i f

GUT i

34

1 36

1

10

10 100

=

3 2 105

Tev/m 8 10

3 ≈

G

H

i

i

π

ρ

2 3

0

0 . 004 Tev/m 8

7 3 .

0 ≈

Λ

G H ρ π

Can’t be the cosmological

constant we see today since:

(7)

Inflation VI

( )

H t

i

e

i

a H

t k

2 2 2

1 − Ω = − ∝

( ) t

f

= e

N

Ω ( ) t

i

e Ω ( ) t

i

Ω ( ) t

i

Ω

1

1 10

1

1

2 200 87

The flatness problem is solved since for an exponential inflation:

(8)

Inflation VII

• The horizon problem is also solved since for an exponential expansion:

( ) ( )

i

t

i i

i i

Hor

c t

t t a c dt a

t d

i

=

=

0

2 /

1

2

/

( )

0

( / )

1/2

exp [ ( )

1/2

] ( 2 +

1

)

⎜ ⎜

+ −

= ∫ ∫

N i i

t

t i i i

t

i i

N i f

Hor

e c t H

t t

H a

dt t

t a c dt

e a t

d

f

i i

( )

( ) t m pc d ( ) t Mpc

d

m t

d

ls Hor

f Hor

i Hor

43 16

28

10 8

. 0 10

2

10 6

=

×

×

(9)

Inflation VIII

( ) t a d ( ) t a ( s ) Mpc Mpc m

d

p f

=

f p 0

≈ 10

34

1 . 4 × 10

4

≈ 3 × 10

23

≈ 0 . 9

( ) t e d ( ) t m

d

p i

=

N p f

≈ 10

44

( ) t m

d

Hor i

≈ 6 × 10

28

We can look at the problem in a different way, if

we go back in time the size of our universe just after inflation was:

This means that before inflation our entire universe was contained in a region of:

Well inside the horizon at that time:

(10)

Inflation IX

• Also the Monopole problem is solved provided that inflation takes place after GUT.

• …today:

( ) t = e

3

n ( ) t e

300

n ( ) t 10

49

m

3

15 pc

3

n

M f N M i M i

( ) t 0 ≈ 10 61 Mpc 3

n M

(11)

Inflation as a Scalar Field

Let’s start with the Lagrangian density of the inflaton field:

where is the potential of the field.

The field time evolution is obtained by minimizing the action:

For a scalar field, the effect of space-time curvature is manifested via the metric in the kinetic term and the determinant of the metric in the integration measure.

This case is called the minimal coupling to gravity.

It is also possible to include in the Lagrangian density terms which couple the scalar field to quantities built from the derivatives from the metric (but we don’t investigate

non-minimal couplings).

(12)

Inflation as a Scalar Field

Minimization of the action leads to the Euler–Lagrange equation:

From this we get the field equation:

Where we have defined

In the case of flat spacetime (Minkowski space), we have:

(13)

Inflation as a Scalar Field

Considering now the case for a free field the potential is simply given by:

We have and the above equation is simply the Klein-Gordon equation for a free particle.

For a FRW metric we have:

where . If, during inflation, the field is homogeneous, we have

and the field equation for a FRW metric is :

Expansion adds a friction term:

(14)

Inflation as a Scalar Field

The Lagrangian density also gives us the energy-momentum tensor

so we have

For the FRW metric, the energy density and pressure measured by an observed comoving with the FRW metric are:

Again, we take the field to be homogeneous so:

(15)

Inflation as a Scalar Field

Let’s consider the field equation of state:

If the kinetic term dominates, w =1; if the potential term dominates, w =−1.

We have also the useful combinations:

Moreover if we consider the continuity equation:

Using the above expressions for density and pressure we get again the field equation:

(16)

Inflation as a Scalar Field

Considering the second equation of

We see that the condition for inflation:

is satisfied if:

If the field is ϕ initially far from the minimum of V (ϕ).

The potential then pulls ϕ towards the minimum. See Figure.

If the potential has a suitable (sufficiently flat) shape, the friction term soon makes ϕ

small enough to satisfy the condition, even if it was not satisfied initially.

(17)

Slow roll conditions (Ryden)

The equation which governs the rate of change of φ is:

the expansion of the universe provides a “Hubble friction” term which slows the transition of the inflaton field to a value which will minimize the potential V.

Just as a skydiver reaches terminal velocity when the downward

force of gravity is balanced by the upward force of air resistance, so

the inflaton field can reach “terminal velocity” when

(18)

Slow roll conditions (Ryden)

If the inflaton field has reached this terminal velocity, then the requirement that

necessary if the inflaton field is to play the role of a cosmo-

logical constant, translates into

(19)

Slow roll conditions (Ryden)

If the universe is undergoing exponential inflation driven by the potential energy of the inflaton field, this means that the Hubble parameter is

which can also be written as

(20)

Slow roll conditions (Ryden)

If the slope of the inflaton’s potential is sufficiently shallow, satisfying this

equation, and if the amplitude of the potential is sufficiently large to dominate the energy density of the universe, then the inflaton field is capable of giving rise to exponential expansion.

As a concrete example of a potential V (φ) which can give rise to inflation, consider the potential shown in Figure. The global

minimum in the potential occurs when the value of the inflaton field is φ = φ0.

Suppose, however, that the inflaton field

starts at φ ≈ 0, where the potential is V (φ) ≈ V0. If

on the “plateau” where V ≈ V

0

, then while φ is slowly rolling toward φ

0

, the

inflaton field contributes an energy density ε

φ

≈ V

0

≈ constant to the universe.

(21)

Models of Inflation I

A scalar field model of inflation consists of 1. a potential V (ϕ) and

2. a way of ending inflation and reheating .

There are two ways of ending inflation:

1. Slow-roll approximation no more valid, as ϕ approaches the minimum of the potential. Inflation ends when either slow roll parameters are of order unity.

2. Extra physics intervenes to end inflation. An example of this is hybrid inflation.

In this case, there are two scalar fields, and inflation ends while the slow-roll approximation is valid for the inflation because of its coupling to the other field.

(22)

Models of Inflation II

Inflation models can be divided into two classes:

1. small-field inflation, Δϕ < MPl in the slow-roll section 2. large-field inflation, Δϕ > MPl in the slow-roll section

Here Δϕ is the range in which ϕ varies during (the relevant part of) inflation (or the distance from the potential minimum). See Figure for typical shapes.

(23)

Models of Inflation III

Examples for small field inflation:

Higgs-like potential:

That can be approximated in the following expansion:

Coleman-Weinberg:

(24)

Models of Inflation IV

Examples for large field inflation Chaotic Inflation:

Natural inflation:

(25)

The Cosmic Microwave Background

Discovered By Penzias and Wilson in 1965.

It is an image of the universe at the time of recombination (near

baryon-photons decoupling), when the universe was just a few thousand years old (z~1000).

The CMB frequency spectrum

is a perfect blackbody at T=2.73 K:

this is an outstanding confirmation

of the hot big bang model.

(26)

Evidence of CMB removed from the article following the suggestion from the PhD supervisor.

Emile Le Roux Phd Thesis (1957) Denisse, Lequeux, Le Roux scientific article (1957)

Earlier and «unknown»

detections of the CMB

Emile probably lost the nobel prize.

(27)

Discovery of the CMB by Penzias and Wilson 1964

1978

(28)

PENZIAS AND WILSON, APJ, 1965, PAG. 419

DICKE ET AL, APJ, 1965, PAG. 415

(29)

The CMB frequency spectrum has been measured with

high precision by the COBE satellite in 1990 (Mather et al.)

(30)

In 1988 Matsumoto et al. claimed a 8% difference from a blackbody

spectrum. This created a lot of interest at that time. The difference was

due to a systematic.

Gush et al. measured the CMB

spectrum in 1990 few months after COBE. They found consistency

with COBE and a perfect black body CMB spectrum.

Some historical remark

(31)

CMB distortions

Bremsstrahlung and free-free cancel any distortion to CMB spectrum from redshifts .

Release of energy at redshifts

produces µ-type distortions.

Compton scattering with

hot gas in clusters produces y-type distortions.

Free free emission from reionization can also

create distortions.

COBE

Limits:

(32)

CMB and foregrounds

(33)

Uniform...

Dipole...

Galaxy (z=0)

The Microwave Sky

COBE

Imprint left by primordial

tiny density inhomogeneities

(z~1000)..

(34)
(35)

Fluctuation generator

Fluctuation amplifier

Dens Hot Sm oot e

h

Ra Cool

refi ed

Clum py

(36)

CMB map from COBE

1992

(37)

CMB map from WMAP

2003

(38)

CMB map from Planck

2015

(39)

In the past 20 years we measured the CMB sky with improved sensitivity, frequency range and angular resolution.

7 degree 0.3 degree 0.08 degree

(40)

Planck maps for 9 frequencies

Main foreground:

Synchrotron

Main

foreground:

Dust

In practice: we see only dust for channels above 353 Ghz

(41)

( ) Δ ( ) =+ ( )

Δ

2 1

2

1

( 2 1 )

2

1 γ γ

γ π

γ C P

T T T

T

The CMB Angular Power Spectrum

R.m.s. of has power per decade in l:

Δ T / T l + ( l 1 ) C

l

/ 2 π

( ) ( ) ( )

+

+

=

Δ l l C d l

l C T

T l

l

rms l ln

2 1 4

1 / 2 2

π π

(42)

Doroshkevich, A. G.; Zel'Dovich, Ya. B.; Syunyaev, R. A.

Soviet Astronomy, Vol. 22, p.523, 1978

A Brief History of the CMB Anisotropies Angular Spectrum

(Theoretical predictions)

(43)

Wilson, M. L.; Silk, J., Astrophysical Journal, Part 1, vol. 243, Jan. 1, 1981, p. 14-25.

1981

A Brief History of the CMB Anisotropies Angular Spectrum

(Theoretical predictions)

(44)

Bond, J. R.; Efstathiou, G.; Royal Astronomical Society, Monthly Notices (ISSN 0035-8711), vol. 226, June 1, 1987, p. 655-687, 1987

A Brief History of the CMB Anisotropies Angular Spectrum

(Theoretical predictions)

(45)

Chung-Pei Ma, Edmund Bertschinger,  Astrophys.J. 455 (1995) 7-25

A Brief History of the CMB Anisotropies Angular Spectrum

(Theoretical predictions)

(46)

Hu, Wayne; Scott, Douglas; Sugiyama, Naoshi; White, Martin.

Physical Review D, Volume 52, Issue 10, 15 November 1995, pp.5498-5515

A Brief History of the CMB Anisotropies Angular Spectrum

(Theoretical predictions)

(47)

A Brief History of the CMB Anisotropies Angular Spectrum

(Experimental Data)

(48)

In 1995 Big Bang Model was nearly dead…

(49)

Collection of CMB anisotropy data from C. Lineweaver et al., 1996

A Brief History of the CMB Anisotropies Angular Spectrum

(Experimental Data)

(50)

Test flight: P. Mauskopf et al, ApJ letters 536, L59, (2000), astro-ph/9911444 A. Melchiorri et al, ApJ letters 536, L63, (2000),astro-ph/9911445 LDB-B00: P. de Bernardis et al, Nature 404 (2000) 955-969, asro-ph/0004404

A. E. Lange et al, PRD D63 (2001) 042001, astro-ph/0005004

Maxima: A. Jaffe et al, Phys. Rev. Lett, (2001) 86, 3475-3479, astro-ph/0007333 LDB-B01: B. Netterfield et al, ApJ in press, astro-ph/0104460

P. de Bernardis et al, ApJ in press, astro-ph/0105296.

BOOMERanG Experiment

(51)
(52)
(53)

Boomerang’s Track


(1 lap in 10.6 days...)

(54)

Blue and Red


are two separate


analyses of same data

Beam

uncertainty
 (correlated)

1.8% of the sky Boomerang Power Spectrum

20% calibration uncertainty

(correlated)

Error bars

10% correlated

11 days of

observations

4 channels

at 150GHz

(55)

CMB anisotropies pre-WMAP (January 2003)

(56)

Wilkinson Microwave Anisotropy Probe

(57)
(58)

Best fit model

cosmic variance

Temperature

Temperature- polarization

1 deg

85% of sky

Spergel et al, 2003

(59)

BOOM03 Flight

Launched: January 6, 2003

! Polarization sensitive receivers 145/245/345 GHz

! Flight January 2003

! 195 hours (11.7 days) of data fsky= 1.8%

! First results published in July 2005

Masi et al. astro-ph/0507509

Jones et al. astro-ph/0507494

Piacentini et al. astro-ph/0507507

Montroy et al. astro-ph/0507514

MacTavish et al. astro-ph/0507503

(60)

Feet

(61)
(62)
(63)

[MacTavish et al. 2005]

(64)

A Brief History of the CMB Anisotropies Angular Spectrum

(2012)

(65)

Planck 2015

(66)

CMB observations

@May 2016

(67)

Temperature Angular spectrum varies with Ω

tot

, Ω

b

, Ω

c

, Λ, τ, h & n

s

etc, etc, etc ....

We can measure cosmological parameters with CMB !

(68)
(69)
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(72)

How to get a measurement of cosmological parameters

DATA

Fiducial cosmological model:

bh2 , Ωmh2 , h , ns , τ, Σmν )

PARAMETER

ESTIMATES

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