The Theory/Observation connection lecture 5 the theory behind (selected) observations of structure formation

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The Theory/Observation connectionlecture 5the
theory behind (selected) observationsof
structure formation

• Will Percival
• The University of Portsmouth

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Lecture outline

• Dark Energy and structure formation
• peculiar velocities
• redshift space distortions
• cluster counts
• weak lensing
• ISW
• Combined constraints
• parameters
• the MCMC method
• results (brief)

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Structure growth depends on dark energy

• A faster expansion rate makes is harder for
  objects to collapse
• changes linear growth rate
• to get the same level of structure at present
  day, objects need to form earlier (on average)
• for the same amplitude of fluctuations in the
  past, there will be less structure today with
  dark energy
• If perturbations can exist in the dark energy,
  then these can affect structure growth
• for quintessence, on large scales where sound
  speed unimportant
• scale dependent linear growth rate (Ma et al
  1999)
• On small scales, dark energy can lead to changes
  in non-linear structure growth
• spherical collapse, turn-around does not
  necessarily mean collapse

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Peculiar velocities
All of structure growth happens because of
peculiar velocities
Initially distribution of matter is approximately
homogeneous (? is small)
Final distribution is clustered
Time
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Linear peculiar velocities
Consider galaxy with true spatial position
x(t)a(t)r(t), then differentiating twice and
splitting the acceleration d2x/dt2g0g into
expansion (g0)and peculiar (g) components, gives
that the peculiar velocity u(t) defined by
a(t)u(t)dx/dt satisfies
The peculiar gravitational acceleration is
So, for linearly evolving potential, u and g are
in same direction
In conformal units, the continuity and Poisson
equations are
Look for solutions of the continuity and Poisson
equations of the form uF(a)g
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Linear peculiar velocities
Solution is given by
where
Zeldovich approximation mass simply propagates
along straight lines given by these vectors
The continuity equation can be rewritten
So the power spectrum of each component of u is
given by
k-1 factor shows that velocities come from
larger-scale perturbations than density field
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Peculiar velocity observations
So peculiar velocities constrain f.can we measure
these directly?
Obviously, can only hope to measure radial
component of peculiar velocities
To do this, we need the redshift, and an
independent measure of the distance (e.g. if
galaxy lies on fundamental plane). Can then
attempt to reconstruct the matter power spectrum
The 1/k term means that the velocity field probes
large scales, but does directly test the matter
field. However, current constraints are poor in
comparison with those provided by other
cosmological observations
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Redshift-space distortions
We measure galaxy redshifts, and infer the
distances from these. There are systematic
distortions in the distances obtained because of
the peculiar velocities of galaxies.
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Large-scale redshift-space distortions
In linear theory, the peculiar velocity of a
galaxy lies in the same direction as its motion.
For a linear displacement field x, the velocity
field is
Displacement along wavevector k is
Line-of-sight
The displacement is directly proportional to the
overdensity observed (on large scales)
Kaiser 1987, MNRAS 227, 1
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Redshift space distortions
At large distances (distant observer
approximation), redshift-space distortions affect
the power spectrum through
Large-scale Kaiser distortion. Can measure this
to constrain ?
On small scales, galaxies lose all knowledge of
initial position. If pairwise velocity dispersion
has an exponential distribution (superposition of
Gaussians), then we get this damping term for the
power spectrum.
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Redshift space distortion observations
Therefore we usually quote ?(s) as the
redshift-space correlation function, and ?(r)
as the real-space correlation function. We can
compute the correlation function ??rp, ?),
including galaxy pair directions
Fingers of God
Expected
Infall around clusters
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Cluster cosmology

• Largest objects in Universe
• 10141015Msun
• Discovery of dark matter
• (Zwicky 1933)
• Can be used to measure
• halo profiles
• Cosmological test based on hypothesis that
  clusters form a fair sample of the Universe
  (White Frenk 1991)

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Cluster cosmology

• Cluster X-ray temperature and profile give
• total mass of system
• X-ray gas mass

Can therefore calculate
If we know s and b, where
We can measure
Allen et al., 2007, MNRAS, astro-ph/0706.0033
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Cluster cosmology
Saw in lecture 3 that the Press-Schechter mass
function has an exponential tail to high mass
Number of high mass objects at high redshift is
therefore extremely sensitive to cosmology
Problem is defining and measuring mass.
Determining whether halos are relaxed or not
Borgani, 2006, astro-ph/0605575
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Cluster observations

• Short-term
• Weak-lensing mass estimates used to constrain
  mass-luminosity relations
• Need to link N-bosy simulation theory to
  observations - will we ever be able to solve
  this?
• Longer term
• Large ground based surveys will find large
  numbers of clusters in optical
• PanSTARRS, DES
• SZ cluster searches

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Weak-lensing
General relativity Curvature of spacetime
locally modified by mass condensation
Deflection of light, magnification, image
multiplication, distortion of objects directly
depend on the amount of matter. Gravitational
lensing effect is achromatic (photons follow
geodesics regardless their energy)
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Weak-lensing

• Assumptions
• weak field limit v2/c2ltlt1
• stationary field tdyn/tcrossltlt1
• thin lens approximation Llens/Lbenchltlt1
• transparent lens
• small deflection angle

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Weak-lensing
The bend angle depends on the gravitational
potential through
So the lens equation can be written in terms of a
lensing potential
The lensing will produce a first order mapping
distortion (Jacobian of the lens mapping)
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Weak-lensing
We can write the Jacobian of the lens mapping as
In terms of the convergence
And shear

• represents an isotropic magnification. It
  transforms a circle into a larger / smaller
  circle
• ? Represents an anisotropic magnification. It
  transforms a circle into an ellipse with axes

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Weak-lensing
Galaxy ellipticities provide a direct measurement
of the shear field (in the weak lensing limit)
Need an expression relating the lensing field to
the matter field, which will be an integral over
galaxy distances ?
The weight function, which depends on the galaxy
distribution is
The shear power spectra are related to the
convergence power spectrum by
As expected, from a measurement of the
convergence power spectrum we can constrain the
matter power spectrum (mainly amplitude) and
geometry
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Weak-lensing observations

• Short-term
• CFHT-LS finished
• 5 constraints on ?8 from quasi-linear power
  spectrum amplitude. Split into large-scale and
  small-scale modes.
• Theory develops
• improvements in systematics - intrinsic
  alignments, power spectrum models
• Longer term
• Large ground based surveys
• PanSTARRS, DES
• Large space based surveys
• DUNE, JDEM
• Will measure ?8 at a series of redshifts,
  constraining linear growth rate
• Will push to larger scales, where we have to
  make smaller non-linear corrections

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Integrated Sachs-Wolfe effect
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Integrated Sachs-Wolfe effect

• line-of-sight effect due to evolution of the
  potential in the intervening structure between
  the CMB and us
• affects the CMB power spectrum (different
  lecture)

• can also be measured by cross-correlation
  between large-scale structure and the CMB
• detection shows that the potential evolves and
  we do not have this balance between linear
  structure growth and expansion
• need either curvature or dark energy

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Now quickly look at combining observations
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Model parameters (describing LSS CMB)
content of the Universe total energy density
Wtot (1?) matter density Wm baryon
density Wb neutrino density Wn (0?) Neutrino
species fn dark energy eqn of state w(a) (-1?)
or w0,w1
perturbations after inflation scalar spectral
index ns (1?) normalisation s8 running
a dns/dk (0?) tensor spectral index nt
(0?) tensor/scalar ratio r (0?)
evolution to present day Hubble
parameter h Optical depth to CMB t
parameters usually marginalised and
ignored galaxy bias model b(k)
(cst?) or b,Q CMB beam error B CMB calibration
error C
Assume Gaussian, adiabatic fluctuations
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WMAP3 parameters used
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Multi-parameter fits to multiple data sets

• Given WMAP3 data, other data are used to break
  CMB degeneracies and understand dark energy
• Main problem is keeping a handle on what is
  being constrained and why
• difficult to allow for systematics
• you have to believe all of the data!
• Have two sets of parameters
• those you fix (part of the prior)
• those you vary
• Need to define a prior
• what set of models
• what prior assumptions to make on them (usual to
  use uniform priors on physically motivated
  variables)
• Most analyses use the Monte-Carlo Markov-Chain
  technique

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Markov-Chain Monte-Carlo method
MCMC method maps the likelihood surface by
building a chain of parameter values whose
density at any location is proportional to the
likelihood at that location p(x)
an example chain starting at x1 A.) accept x2 B.)
reject x3 C.) accept x4
given a chain at parameter x, and a candidate for
the next step x, then x is accepted with
probability

• p(x) gt p(x)
• p(x)/p(x) otherwise

-ln(p(x))
A
B
C
x
x1 x2 x4 x3
for any symmetric proposal distribution q(xx)
q(xx), then an infinite number of steps leads
to a chain in which the density of samples is
proportional to p(x).
CHAIN x1, x2, x2, x4, ...
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MCMC problems jump sizes
q(x) too broad chain lacks mobility as
all candidates are unlikely
q(x) too narrow chain only moves slowly
to sample all of parameter space
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MCMC problems burn in
Chain takes some time to reach a point where the
initial position chosen has no influence on the
statistics of the chain (very dependent on the
proposal distribution q(x) )
Approx. end of burn-in
Approx. end of burn-in
2 chains jump size adjusted to be large
initially, then reduce as chain grows
2 chains jump size too large for too long, so
chain takes time to find high likelihood region
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MCMC problems convergence
How do we know when the chain has sampled the
likelihood surface sufficiently well, that the
mean std deviation for each parameter are well
constrained?
Gelman Rubin (1992) convergence test Given M
chains (or sections of chain) of length N, Let
W be the average variance calculated from
individual chains, and B be the variance in the
mean recovered from the M chains. Define Then
R is the ratio of two estimates of the variance.
The numerator is unbiased if the chains fully
sample the target, otherwise it is an
overestimate. The denominator is an underestimate
if the chains have not converged. Test set a
limit Rlt1.1
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Resulting constraints
From Tegmark et al (2006)
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Supernovae BAO constraints
SNe
WMAP-3
6-7 measure of ltwgt
SNe
BAO
BAO
SNLSBAO (No flatness)
SNLS BAO simple WMAP Flat
(relaxing flatness error in ltwgt goes from 0.065
to 0.115)
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Further reading

• Redshift-space distortions
• Kaiser (1987), MNRAS, 227, 1
• Cluster Cosmology
• review by Borgani (2006), astro-ph/0605575
• talk by Allen, SLAC lecture notes, available
  online at
• http//www-conf.slac.stanford.edu/ssi/2007/lateReg
  /program.htm
• Weak lensing
• chapter 10 of Dodelson modern cosmology,
  Academic Press
• Combined constraints (for example)
• Sanchez et al. (2005), astro-ph/0507538
• Tegmark et al. (2006), astro-ph/0608632
• Spergel et al. (2007), ApJSS, 170, 3777