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The Theory/Observation connection lecture 2 perturbations. Will Percival The University of Portsmouth. Lecture outline. Describing perturbations correlation function power spectrum Perturbations from Inflation The evolution on perturbations before matter dominated epoch
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The University of Portsmouth
from statistical homogeneity
from statistical isotropy
can estimate correlation function using galaxy (DD) and random (RR) pair counts at separations ~r
power spectrum is the Fourier analogue of the correlation function
sometimes written in
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 (overdensity=0) and variance (given by either the correlation function or the power spectrum)
Credit: Alex Szalay
Same 2pt, different 3pt
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 ...
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:
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
Acceleration requires field with more potential energy than kinetic energy
Energy density is constant, so Einstein’s 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.
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
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
where csis the sound speed of the matter, andis the density of matter.
“F=ma” for perturbation growth
depends on Jeans scale
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 parameterWmh.
gravitational potential k3/2F
scale factor of Universe a
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
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.
“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
Sound speed cs
Gives the comoving sound horizon ~110h-1Mpc, and BAO wavelength 0.06hMpc-1
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
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”
definition of d
to first order in perturbation radius
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
For flat matter dominated model, this has solution
Remember that the gravitational potential and the overdensity are related by Poisson’s equation
Then the potential is constant: there is a delicate balance between structure growth and expansion
Not true if dark energy or neutrinos
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
Present day linear growth
factor relative to EdS value
large scale power
is lost as fluctuations
move to smaller scales
P(k) calculated from Smith et al. 2003, MNRAS, 341,1311 fitting formulae