This has led to more general Dark Energy or Quintessence models:

Evolving scalar field which ‘tracks’ the matter density

Convenient parametrisation: ‘Equation of State’

Can we measure w(z) ?

wP

Matter 0Radiation 1/3Curvature -1/3‘Lambda’ -1Quintessence w(z)

2/1

)1(300 )1(

i

ww

i

izHH

iw

Inflation for astronomers

We have been considering but suppose thatin the past . From the Friedmann equations it would then be very difficult to explain why it is so close to zero today.

00 k0k

Present day ‘closeness’ of matter density to the critical density appears to require an incredible degree of ‘fine tuning’ in the very early Universe.

FLATNESS PROBLEM

How do we explain the isotropy of the CMBR, when opposite sides of the sky were ‘causally disconnected’ when the CMBR photons were emitted?

HORIZON PROBLEM

From Guth (1997)

CMBR

Big Bang

time

space

Our world line

Now

A B

Our past light cone

Solution (first proposed by Alan Guth in 1981) is…

INFLATIONINFLATION

…a period of accelerated expansion in the very early universe.

Small, causallyconnected region

Limit of observable Universe today

INFLATION

Inflationary solution to the Horizon Problem

From Guth (1997)

Inflationary solution to the Flatness Problem

From Guth (1997)

Inflationary solution to the Flatness Problem

Suppose that in the very early Universe:

Suppose there existed

Easy to show that:-

i.e. vacuum energy will dominate as the Universe expands, and drives to zero

0init, k 0init,rad

0init,vac

2

init

vac

R

Rk

4

init

vac

rad

R

R

k

HtRR

Rexp

3

De Sitter solution;exponential growth

CoBE map of temperature across the sky

CMBR fluctuations are the seeds of today’s galaxies

LSS formation is sensitive to the pattern, or power spectrum, of CMBR temperature fluctuations

Basics of large scale structure formation - 1o LSS assembled under by gravitational instability

o Express in terms of density contrast

o Can decompose into Fourier modes

o These evolve independently provided the fluctations are small (linear regime) – evolution depends on parameters of the background model

),( tx

)(),(),( ttxtx

)(

2)( 3

3

kekd

x xki

(at a given epoch)

Basics of large scale structure formation - 2o Density perturbations handled statistically, e.g. via 2-point correlation function

o Assuming statistical homogeneity

o Inflation predicts a primordial spectrum of the form

with n = 1

2

3

3

)(2

)()()( kekd

rxxr rki

)(

)sin(

2)( 2

2

kPkr

krdkkr

Power spectrum; measures strength of clustering on scale, knkkP )(

Harrison-Zel’dovich spectrum

Basics of large scale structure formation - 3o Late time (i.e. today) power spectrum is different; modified by transfer function – describes principally how different wavelengths were affected by radiation pressure before CMBR epoch.

Key points:-

Structure can only grow on scales k smaller than horizon

Scales with small k entered horizon in radiation era; radiation pressure suppresses growth on these

scales

When a given scale entered the horizon depends on the expansion rate, and hence on cosmological parameters.

Transfer function also depends on nature of dark matter

),(),(),( prim2 RkPRkTRkP

Basics of large scale structure formation - 4o Putting all this together: measuring the present day power spectrum of galaxy clustering is a sensitive probe of the cosmological model

BUT are galaxies faithful tracers of the mass distribution?…

CMBR fluctuationso In many ways the CMBR is a ‘cleaner’ probe of the initial power spectrum – perturbations are much smaller!

Decompose temperature fluctuations in spherical harmonics

define angular 2-point correlation function:-

= angular power spectrum

mmm Ya

T

T

,

)(cos)12(

4

1)(

cos21

21

PCT

T

T

TC

Spherical harmonics

Legendre polynomials

mmaC

2

12

1

Adapted from Lineweaver (1997)

The CMBR angular power spectrum is sensitive to many cosmological parameters, which can be estimated by comparing observations with theory

Theoretical curve

But what do all the squiggles mean?…

Max Tegmark (2001)

Early Universe too hot for neutral atoms

Free electrons scattered light (as in a fog)

After ~380,000 years, cool enough for atoms (T ~ 3000K; z ~ 1000); fog clears!

Last Scattering Surface

Wayne Hu (1998)

100

~

2/1

LSS0hor 1000

1

zm

Simplified CMBR Power Spectrum

Adapted from Lineweaver (1997)

Damping

2/1

LSS0hor 1000

1

zm

100

~

Simplified CMBR Power Spectrum

Sachs-Wolfe Effect

Caused by large scale primordial fluctations in gravitational potential on super-horizon scales (inflationary origin?)

Photons at LSS are blue / redshifted as they fall down / climb out of potential hills (hotspots) and valleys (cold spots)

Size of super-horizon SW effect independent of scale

Adapted from Lineweaver (1997)

Simplified CMBR Power Spectrum

2

3

29

25

21

2

5

4n

n

n

n

QC

nkkP )(For

For

20

‘Quadrupole’ 2C

1n)1(5

24 2

Q

C

constant)1( C

Adapted from Lineweaver (1997)

What about sub-horizon scales?…

Universe today is matter dominated

i.e. Matter-radiation equality at z ~ 3500

25rad,0 104 h -1-1

0 Mpckms100hH

4

0

rad,0

rad

)(

)(

zR

Rz

3

0

mat,0

mat

)(

)(

zR

Rz

)1()(

)(

mat

rad zz

z

20

4eq 104.2)1( hz m

What about sub-horizon scales?…(1)

Adapted from Lineweaver (1997)

(2)

(1) Radiation era ends

Baryonic matter begins to collapse into potential wells as they enter the horizon (‘drags along’ photons);

acoustic oscillations on scales smaller than sound horizon

(2) Last Scattering Surface

Baryons and photons decouple; photons carry ‘imprint’ of acoustic oscillations in density, velocity at LSS

Pattern of acoustic peaks, valleys

What about sub-horizon scales?…

(1) Radiation era ends

Baryonic matter begins to collapse into potential wells as they enter the horizon (‘drags along’ photons);

acoustic oscillations on scales smaller than sound horizon

(2) Last Scattering Surface

Baryons and photons decouple; photons carry ‘imprint’ of acoustic oscillations in density, velocity at LSS

Pattern of acoustic peaks, valleys

(1)

Adapted from Lineweaver (1997)

(2)

A

B

C

D

A

B

C

D

Simplified CMBR Power Spectrum

Adapted from Lineweaver (1997)

Damping

2/1

LSS0hor 1000

1

zm

100

~

Beyond

Further anisotropies due to secondary post-LSS effects:

(reionisation, Vishniac, S-Z)

Strongly damped

Can compute CMBR power spectrum using:

CMBFAST

Sensitive to a large number of parameters

1.0

STnnQ

h

T

bm

/

Adapted from Lineweaver (1997)

Each acoustic peak corresponds to a fixed physical scale

We observe peak at a particular angular scale – depends on:-

angular diameter distance to LSS

Position of peaks constrains Omegas, Hubble parameter –

LSS

2/1)1(3

00 )1(

1ang

z

z

dz

i

iwiw

hd

Baryon density constrained by height of peaks

2

crit

hbb

bb

Baryon density constrained by height of peaks

2

crit

hbb

bb

Q. How can we distinguish degenerate models?

A. Combine observations from different sources…

Hubble constant ( )

Hubble Diagram of Distant Supernovae

Large Scale Structure / Galaxy Clustering

Strong and weak gravitational lensing

Cluster abundance / baryon fraction

Abundance of light elements / nucleosynthesis

Age of the oldest star clusters

etc, etc …

Crucial test of systematic errors

2hii

Tegmark et al (1998)

Hubble diagram of distant supernovae

Consider an object of intrinsic luminosity

from which we observe a flux

Define the Luminosity Distance via:-

L

24 Ld

L

Distance required to give observed flux if Universe has a flat geometry

Hubble diagram of distant supernovae

Consider an object of intrinsic luminosity

from which we observe a flux

Define the Luminosity Distance via:-

L

24 Ld

L

Distance required to give observed flux if Universe has a flat geometry

Actual distance depends on true geometry, and expansion history of the Universe

Hubble diagram of distant supernovae

Consider an object of intrinsic luminosity

from which we observe a flux

Define the Luminosity Distance via:-

L

24 Ld

L

Distance required to give observed flux if Universe has a flat geometry

Actual distance depends on true geometry, and expansion history of the Universe

),;()1(),;( ang2

LL mm zdzzdd

Adapted from Schmidt (2002)

25log5)(

Mpc

dMm L

mag

Distance Modulus

Fractional distance change ½(mag change)

e.g.

0.1 mag difference is 5% distance difference

Adapted from Schmidt (2002)

White dwarf star with a massive binary companion. Accretion pushes white dwarf over the Chandrasekhar limit, causing thermonuclear disruption

Type Ia SupernovaType Ia Supernova

Good standard candle because:-

Narrow range of luminosities at maximum lightObservable to very large distances

log z

Model with positive cosmological constant

Model with zero cosmological constant

Models with different matter density

Hubble diagram of distant Type Ia supernovae

Straight line relation nearby

Perlmutter (1998) results

2 competing teams:-

Supernova Cosmology Project (Saul Perlmutter, LBL)

Supernova High-z Project (Brian Schmidt, Mt Stromlo)

Consistent Results

Tegmark et al (1998)

SNIa measure:-

CMBR measures:-

Together, can constrain:-

mq2

10

mk 1

,m

And the answer is?…

Microwave Anisotropy Probe

First year WMAP results published Feb 2003

First year WMAP results published Feb 2003

From Bennett et al (2003)

Accuracy of measurements across first two peaks sufficient to effectively break most degeneracies

From Bennett et al (2003)

From Bennett et al (2003)

From Bennett et al (2003)

Key WMAP results:-

Consistent with flat geometry; nS ~ 1

Excellent agreement of Hubble constant with HST Key project results

Polarisation: large-scale correlation reionisation

anti-correlation super-horizon fluctuations

Reionisation at z ~ 20 age of the first stars;

age of the Universe

Incompatible with warm dark matter

Universe made up of: 73% dark energy22% cold dark matter 5% baryons

Constant Lambda term favoured, but result not conclusive

Can we distinguish a constant term from quintessence?…

Not from current ground-based SN observations (combined with e.g. LSS)

Adapted from Schmidt (2002)

Can we distinguish a constant term from quintessence?…

Not from current ground-based SN observations (combined with e.g. LSS)…

Adapted from Schmidt (2002)

Can we distinguish a constant term from quintessence?…

Not from current ground-based SN observations (combined with e.g. LSS)…

…or from future ground-based observations (even with LSS + CMBR)

Adapted from Schmidt (2002)

Can we distinguish a constant term from quintessence?…

Not from current ground-based SN observations (combined with e.g. LSS)…

…or from future ground-based observations (even with LSS + CMBR)

Adapted from Schmidt (2002)

Can we distinguish a constant term from quintessence?…

Not from current ground-based SN observations (combined with e.g. LSS)…

…or from future ground-based observations (even with LSS + CMBR)

Main goal of the SNAP satellite(launch during next decade?)

Adapted from Schmidt (2002)

““The Concordance Model in The Concordance Model in Cosmology:Cosmology:

Should We Believe It?…”Should We Believe It?…”