The TheoryObservation connection lecture 2 perturbations

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The Theory/Observation connectionlecture
2perturbations

• Will Percival
• The University of Portsmouth

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Lecture outline

• Describing perturbations
• correlation function
• power spectrum
• Perturbations from Inflation
• The evolution on perturbations before matter
  dominated epoch
• Effects from matter-radiation equality
• Effects from baryonic material
• evolution during matter domination
• linear growth
• linear vs non-linear structure (introduction)

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Perturbation statistics correlation function
overdensity field
definition of correlation function
from statistical homogeneity
from statistical isotropy
can estimate correlation function using galaxy
(DD) and random (RR) pair counts at separations r
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Perturbation statistics power spectrum
definition of power spectrum
power spectrum is the Fourier analogue of the
correlation function
sometimes written in dimensionless form
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The importance of 2-pt statistics
Because the central limit theorem implies that a
density distribution is asymptotically Gaussian
in the limit where the density results from the
average of many independent processes and a
Gaussian is completely characterised by its mean
(overdensity0) and variance (given by either the
correlation function or the power spectrum)
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Do 2-pt statistics tell us everything?
Credit Alex Szalay
Same 2pt, different 3pt
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Correlation function vs Power Spectrum
The power spectrum and correlation function
contain the same information accurate
measurement of each give the same constraints on
cosmological models.
Both power spectrum and correlation function can
be measured relatively easily (and with amazing
complexity)
The power spectrum has the advantage that
different modes are uncorrelated (as a
consequence of statistical homogeneity).
Models tend to focus on the power spectrum, so it
is common for observations to do the same ...
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Phases of (linear) perturbation evolution
Inflation
Matter/Dark energy domination
Transfer function
Non-linear (next lecture)
linear
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Why is there structure?

• Inflation (a period of rapid growth of the early
• Universe driven by a scalar field) was postulated
  to solve some serious problems with standard
  cosmology
• why do causally disconnected regions appear to
  have the same properties? they were connected
  in the past (discussed in last lecture)
• why is the energy density of the Universe close
  to critical density? driven there by inflation
• what are the seeds of present-day structure? -
  Quantum fluctuations in the matter density are
  increased to significant levels

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Driving inflation
Every elementary particle (e.g. electron,
neutrino, quarks, photons) is associated with a
field. Simplest fields are scalar (e.g. Higgs
field), and a similar field ?(x,t) could drive
inflation.
Energy-momentum tensor for ?(x,t)
For homogeneous (part of) ?(x,t)
Acceleration requires field with
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Driving inflation
Acceleration requires field with more potential
energy than kinetic energy
Energy density is constant, so Einsteins
equation gives that the evolution of a is
Gives exponential growth
Problems only way to escape growth is quantum
tunneling, but has been shown not to work.
Instead, think of particle slowly rolling down
potential. Close to, but not perfectly stationary.
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Inflation perturbations
Perturbations in the FRW metric
Scalar potentials usually give rise to so can
interchange these
Spatial distribution of fluctuations can be
written as a function of the power spectrum
During exponential growth, there is no preferred
scale, so (Gaussian) quantum fluctuations give
rise to fluctuations in the metric with
with n1
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Inflation perturbations
Poisson equation translates between perturbations
in the gravitational potential and the overdensity
In Fourier space
So matter power spectrum has the form
Because quantum fluctuations are Gaussian
distributed, so are resulting matter
fluctuations, which form a Gaussian Random Field
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Matter P(k) depends on inflation
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Phases of perturbation evolution
Inflation
Matter/Dark energy domination
Transfer function
linear
Non-linear
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Jeans length

• After inflation, the evolution of density
  fluctuations depends on the scale and composition
  of the matter (CDM, baryons, neutrinos, etc.)
• An important scale is the Jeans Length which is
  the scale of fluctuation where pressure support
  equals gravitational collapse,
• where cs is the sound speed of the matter, and ?
  is the density of matter.

Fma for perturbation growth
depends on Jeans scale
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Transfer function evolution
in radiation dominated Universe, pressure support
means that small perturbations cannot collapse
(large Jeans scale). Jeans scale changes with
time, leading to smooth turn-over of matter power
spectrum. Cut-off dependent on matter density
times the Hubble parameter Wmh.
large scale perturbation
gravitational potential k3/2F
small scale
scale factor of Universe a
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The power spectrum turn-over
Can give a measurement of the matter density from
galaxy surveys
It is hard to disentangle this shape change from
galaxy bias and non-linear effects
varying the matter density times the Hubble
constant
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The effect of neutrinos
The existence of massive neutrinos can also
introduce a suppression of T(k) on small scales
relative to their Jeans length. Degenerate with
the suppression caused by radiation epoch.
Position depend on neutrino-mass equality scale.
???k???????????P(k)
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Configuration space description
Wm0.3, Wv0.7, h0.7, Wb/Wm0.16
position-space description Bashinsky
Bertschinger astro-ph/0012153
astro-ph/02022153 plots by Dan Eisenstein
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Configuration space description
Wm0.3, Wv0.7, h0.7, Wb/Wm0.16
position-space description Bashinsky
Bertschinger astro-ph/0012153
astro-ph/02022153 plots by Dan Eisenstein
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Configuration space description
Wm0.3, Wv0.7, h0.7, Wb/Wm0.16
position-space description Bashinsky
Bertschinger astro-ph/0012153
astro-ph/02022153 plots by Dan Eisenstein
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Configuration space description
Wm0.3, Wv0.7, h0.7, Wb/Wm0.16
position-space description Bashinsky
Bertschinger astro-ph/0012153
astro-ph/02022153 plots by Dan Eisenstein
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Configuration space description
Wm0.3, Wv0.7, h0.7, Wb/Wm0.16
position-space description Bashinsky
Bertschinger astro-ph/0012153
astro-ph/02022153 plots by Dan Eisenstein
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Configuration space description
Wm0.3, Wv0.7, h0.7, Wb/Wm0.16
position-space description Bashinsky
Bertschinger astro-ph/0012153
astro-ph/02022153 plots by Dan Eisenstein
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Configuration space description
Wm0.3, Wv0.7, h0.7, Wb/Wm0.16
position-space description Bashinsky
Bertschinger astro-ph/0012153
astro-ph/02022153 plots by Dan Eisenstein
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Baryon Oscillations in the power spectrum
Wavelength of baryonic acoustic oscillations is
determined by the comoving sound horizon at
recombination
At early times can ignore dark energy, so
comoving sound horizon is given by
varying the baryon fraction
Sound speed cs
Gives the comoving sound horizon 110h-1Mpc, and
BAO wavelength 0.06hMpc-1
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The matter power spectrum
Overall shape of matter power spectrum is given
by
Current best estimate of the galaxy power
spectrum from SDSS no sign of turn-over yet
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Link with CMB
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Link with N-body simulations
LCDM
OCDM
SCDM
tCDM
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Phases of perturbation evolution
Inflation
Matter/Dark energy domination
Transfer function
linear
Non-linear
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Spherical perturbation leading to linear growth
cosmology equation
homogeneous dark energy means that this term
depends on scale factor of background perfectly
clustering dark energy replace a with ap
Consider homogeneous spherical perturbation
evolution is same as mini-universe
Overdense perturbation Radius ap
Background Radius a
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Spherical perturbation leading to linear growth
cosmology equation
definition of d to first order in perturbation
radius (linear approximation) gives
can also be derived using the Jeans equation
only has this form if the dark energy does not
cluster derivation of equation relies on
cancellation in dark energy terms in perturbation
and background
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Linear growth EdS model
For flat matter dominated model, this has
solution
Remember that the gravitational potential and the
overdensity are related by Poissons equation
Then the potential is constant there is a
delicate balance between structure growth and
expansion
Not true if dark energy or neutrinos
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Linear growth general models
For general models, denote linear growth
parameter (solution to this differential
equation)
For lambda models, can use the approximation of
Carroll, Press Turner (1992)
For general dark energy models, need to solve the
differential equation numerically
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Linear growth factor
Present day linear growth factor relative to EdS
value
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Linear vs Non-linear behaviour
z0
non-linear evolution
z1
z2
z3
linear growth
z4
z0
z5
z1
large scale power is lost as fluctuations move to
smaller scales
z2
z3
z4
z5
P(k) calculated from Smith et al. 2003, MNRAS,
341,1311 fitting formulae
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Further reading

• Dodelson, Modern Cosmology, Academic Press
• Peacock, Cosmological Physics, Cambridge
  University Press
• Liddle Lyth, Cosmological Inflation and
  Large-Scale Structure, Cambridge University
  Press
• Eisenstein et al. 2006, astro-ph/0604361
  (configuration space description of perturbation
  evolution)
• Percival 2005, astro-ph/0508156 (linear growth
  in general dark energy models)