Institute of Cosmology and Gravitation University of ...

Theoretical cosmology

David WandsInstitute of Cosmology and Gravitation

University of Portsmouth

5th Tah Poe School on Cosmology July 2019

Theoretical cosmology

• Physics of the Cosmic Microwave Background

• Cosmological perturbations

• Inflation in the Very Early Universe

slides online at icg.port.ac.uk/tpcosmov-2019

image: Pablo Carlos Budassi


40 light-minutes to Jupiter


4 light-years to nearest star


30,000 light-yearsto galactic centre

Sloan Digital Sky Surveyover one million galaxies

each dot is a system of billions of stars, like our own Milky Way

see www.galaxyzoo.org

2 billion light-years

how far back can we see?can we see the Big Bang?

image: NASA

Before last-scattering: hot electron-proton plasma (scatters photons) After last scattering: cool neutral atoms (transparent)

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45 billion light-yearsto last-scattering surface

Physics of the CMB• Isotropic Cosmic Microwave Background

– isotropic hot big bang cosmology– CMB spectrum

• Anisotropic Cosmic Microwave Background– density and metric perturbations– Sachs-Wolfe effect

• Acoustic oscillations in the CMB– cosmological parameter constraints from CMB physics

• New frontiers– polarisation and gravitational waves– non-linearity, non-Gaussianity and weak-lensing

Cosmic Microwave Background radiation

• relic thermal radiation from the hot big bang

• 3 Kelvin, just three degrees above absolute zero

• Many ground- and balloon-based experiments since– e.g., South Pole Telescope, Atacama

Cosmology Telescope

• discovered in 1965 byArno Penzias and Robert Wilson

CoBE satellite launched by NASA in 1990

2.7 K in all directions© NASA

+/- 3.3 mK Doppler shift due local motion(at 1 million miles per hour)

+/- 18 µK intrinsic anisotropies

COBE launched 1990

© NASA

WMAP launched 2001, final data 2012

Planck launched 2009, first data 2013

© ESA Planck satellite (2015)

small temperature anisotropies, 1 part in 100,000

Structure at the edge of our Universe

Einstein’s theory of gravity: General Relativity

spacetime tells matter how to move

& matter tells spacetime

how to curve

• slice up 4D spacetime into expanding 3D space with uniform matter density and spatial curvature

• Friedmann equation – from Einstein’s energy constraint:

Friedmann’s dynamic cosmology

scale factor a(t)Hubble rate H ≡ !a / a

H 2 =8πG

3ρ +

Λ3−κa2

⇒ 1= Ωm + ΩΛ + Ωκ

H0 = 100h km s�1 Mpc�1

h ⇡ 0.7

Hot big bang dynamics• Spatially flat metric:

• Einstein energy constraint G00 + evolution Gij equations for ℋ ≝ #$/#:

• Energy conservation

ds2 = �c2dt2 + a2�ijdxidxj = a2

⇥�d⌘2 + �ijdx

idxj⇤

5.2.1 Dynamical equations

Background: FRW metric

ds2 = a

2n�d⌘

2 + �ijdxidx

jo. (111)

The Einstein equations give the Friedmann constraint and evolution equation for thebackground (spatially flat, = 0) FRW universe

H2 =

8⇡G

3a2⇢ , (112)

H0 = �

4⇡G

3a2 (⇢+ 3P ) , (113)

and energy conservation gives the continuity equation

⇢0 = �3H (⇢+ P ) , (114)

where ⇢ and P = w⇢ are the total energy density and the total pressure, a prime denotesa derivative with respect to conformal time, ⌘, the scale factor is a, and H ⌘ a

0/a is the

conformal Hubble parameter.

Radiation domination:

P� = ⇢�/3 , ⇢� / a�4

, a / t1/2

/ ⌘ . (115)

Matter domination:

Pm = 0 , ⇢m / a�3

, a / t2/3

/ ⌘2. (116)

Vacuum domination

Pv = �⇢vc2

⇢v = constant , a / eHt

/ 1/(⌘1 � ⌘) (117)

30



ds2 = a

2n�d⌘

2 + �ijdxidx

jo. (111)


H2 =

8⇡G

3a2⇢ , (112)

H0 = �

4⇡G

3a2 (⇢+ 3P ) , (113)


⇢0 = �3H (⇢+ P ) , (114)


0/a is the



P� = ⇢�/3 , ⇢� / a�4

, a / t1/2

/ ⌘ . (115)

Matter domination:

Pm = 0 , ⇢m / a�3

, a / t2/3

/ ⌘2. (116)

Vacuum domination

Pv = �⇢vc2


/ 1/(⌘1 � ⌘) (117)

30



ds2 = a

2n�d⌘

2 + �ijdxidx

jo. (111)


H2 =

8⇡G

3a2⇢ , (112)

H0 = �

4⇡G

3a2 (⇢+ 3P ) , (113)


⇢0 = �3H (⇢+ P ) , (114)


0/a is the



P� = ⇢�/3 , ⇢� / a�4

, a / t1/2

/ ⌘ . (115)

Matter domination:

Pm = 0 , ⇢m / a�3

, a / t2/3

/ ⌘2. (116)

Vacuum domination

Pv = �⇢vc2


/ 1/(⌘1 � ⌘) (117)

30



ds2 = a

2n�d⌘

2 + �ijdxidx

jo. (111)


H2 =

8⇡G

3a2⇢ , (112)

H0 = �

4⇡G

3a2 (⇢+ 3P ) , (113)


⇢0 = �3H (⇢+ P ) , (114)


0/a is the



P� = ⇢�/3 , ⇢� / a�4

, a / t1/2

/ ⌘ . (115)

Matter domination:

Pm = 0 , ⇢m / a�3

, a / t2/3

/ ⌘2. (116)

Vacuum domination

Pv = �⇢vc2


/ 1/(⌘1 � ⌘) (117)

30



ds2 = a

2n�d⌘

2 + �ijdxidx

jo. (111)


H2 =

8⇡G

3a2⇢ , (112)

H0 = �

4⇡G

3a2 (⇢+ 3P ) , (113)


⇢0 = �3H (⇢+ P ) , (114)


0/a is the



Pr = ⇢r/3 , ⇢r / a�4

, a / t1/2

/ ⌘ . (115)

Matter domination:

Pm = 0 , ⇢m / a�3

, a / t2/3

/ ⌘2. (116)

Vacuum domination

Pv = �⇢vc2


/ 1/(⌘1 � ⌘) (117)

Matter-radiation equality

1 + zeq ⌘a0

aeq=

⇢m,0

⇢r,0= 3.4⇥ 103

⌦mh

2

0.14

!

(118)

30

Planck - new standard model of primordial cosmology

© ESA

Cosmological parameters (Ade et al: Planck 2013 results. XVI)

⌦⇤ = 0.686± 0.020

⌦matter = 0.314± 0.020

⌦ = �0.04± 0.05

h = 0.674± 0.014

Precision cosmology from angular power spectrum

background.uchicago.edu/~whu

angular scale indicates flat space geometry, but also depends on nature of energy density in the universe

The isotropic CMB• a brief thermal history

CMB = Black-body spectrum,T0=2.725K

Kom

atsu

201

1

Black-body spectrum => thermal equilibrium

Kom

atsu

201

1Photon energy:

Einstein-Boltzmann distribution:

Number density:

Energy density:

but CMB photons no longer in thermal equilibrium with matter today

f(p) =1

exp(~p/kBT )� 1

E = h⌫ = ~p

n� =2

(2⇡)3

Z4⇡p2f(p) dp ' 2.4

⇡2

✓kBT

~c

◆3

⇢�c2 =

2

(2⇡)3

Z4⇡p2f(p) ~p dp =

⇡2

15~3c3 (kBT )4

Photons in expanding Universe• Line element in spatially-flat FLRW metric:

– distance travelled by light = horizon size = conformal time:

• 4-momentum: !" = $, $ &'(– where photon trajectory:

⌘ =

Zc dt

a

ds2 = �c2dt2 + a2�ijdxidxj = a2

⇥�d⌘2 + �ijdx

idxj⇤

pi = pn̂i

Wikipedia

Einstein-Boltzmann equation in expanding universe• Photon 4-momentum: !" = $, $ &'(

• Geodesic equation:

leads to cosmological redshift:

– preserves Einstein-Boltzmann distribution:

– where temperature redshifts with cosmic expansion:

dPµ

d�+ �µ

⌫⇢P⌫P ⇢ = 0 ) 1

p

dp

d⌘= �1

a

da

d⌘

1 + z ⌘ p

p0=

a0a

f(p) =1

exp(~p/kBT )� 1=

1

exp(~p0/kBT0)� 1

1 + z =T

T0=

a0a

Temperature-time relation:At sufficiently high temperature, !"# ≫ %&', all particles become relativistic. In thermal equilibrium the density

( = *eff-'

30ℏ1&2!"# 3

where geff=effective number of degrees of freedom (g=2 for 2 photon spin states, g=2x7/8 for fermions)

Friedmann equation relates density to Hubble rate

( =34'

8-67=

332-679'

where in radiation-dominated cosmology : ∝ 9</', hence 4 = :̇/: = 1/29

Thus we have a temperature time relation in the Hot Big Bang cosmology

9 =3

32-6730ℏ1&2

*eff-'1

!"# '

or more simply:@

<ABC=<

Deff<MeVGHI

'

for photons (and other relativistic particles)

frequency = (wavelength)-1 = wavenumber = momentum = mass = energy = temperature

by convention they have different units (seconds, metres, etc) which we have to relate using fundamental constants

much easier to use natural units, such that fundamental constants

leaves only one dimensional constant = Newton’s constant, G

natural units:

so only one fundamental unit remains = Planck mass

natural units:

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