this has led to more general dark energy or quintessence models:
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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?…”
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