Physical Cosmology 9/5/2016
Alessandro Melchiorri
alessandro.melchiorri@roma1.infn.it slides can be found here:
oberon.roma1.infn.it/alessandro/cosmo2016
Today lecture is also slightly based on:
Cosmological «Circuit»
Generator of Perturbations (Inflation)
Amplifier (Gravity)
Low band pass filter.
Cosmological and
Astrophysical effects
Tend to erase small scale (large k) perturbations
For example, we can consider two modes, one entering the horizon before the matter-radiation equivalence and another one entering after it.
log δ
a log
Perturbation in Red: enters the horizon AFTER the equivalence.
Perturbation in Blue: enters the horizon BEFORE equivalence.
aEQ
a2
k1
k2
) (
/ 2 1 2
2 a c H a
k = −
1
2
k
k >
a1
Evolution for a w=0 (no pressure)
component.
Perturbations with k larger
than
are damped respect to perturbations with k smaller Cold dark
matter (only)
For example, we can consider two modes, one entering the horizon before the decoupling and another one entering after it.
log δ
a log
Perturbation in Red: enters the horizon AFTER the decoupling.
Perturbation in Blue: enters the horizon BEFORE decoupling.
a2
k1
k2
) (
/ 2 1 2
2 a c H a
k = −
1
2
k
k >
a1
Evolution
for the baryon component.
Perturbations with k larger
than
are strongly damped respect to perturbations with k smaller Baryons
(only)
We ar
e below the Jens lenght.
Damping+Oscillations
log δ
a log
Baryon/CDM Perturbation in Red: enters the horizon AFTER the decoupling.
Baryon Perturbation in Blue: enters the horizon BEFORE equivalence.
CDM Perturbation in Green: enters the horizon BEFORE equivalence.
a2
k1
) (
/ 2 1 2
2 a c H a
k = −
a1
CDM
+Baryons
aEQ
The situation is different if we consider a CDM+Baryon case.
Baryons “feel" the CDM gravitational potential.
Baryons,
after decoupling fall in the CDM potential wells.
k2
k2
Power Spectrum
Each Fourier component is a complex number, which can be written in the form
The mean square amplitude of the Fourier components defines the power spectrum :
where the average is taken over all possible orientations of the wavenumber. If δ(⃗r) is isotropic, then no information is lost,
statistically speaking, if we average the power spectrum over all angles and we get an isotropic power spectrum:
Correlation function
Let us consider the autocorrelation function of the density field (usually called the correlation function):
Where the brackets indicates an average over a volume V.
We can write:
and, performing the integral we have:
Correlation function
Since the correlation function is a real number, assuming an isotropic power spectrum we have:
If the density field is gaussian, we have that the value of δ at a randomly selected point is drawn from the Gaussian
probability distribution:
where the standard deviation σ can be computed from the power spectrum:
Summary
In practice, our theory cannot predict the exact value of
in a region of the sky. But if we assume that the initial perturbations are gaussian we can predict the correlation function, the variance of the fluctuations and its the power spectrum P(k).
These are things that we can measure using, for example, galaxy surveys and assuming that galaxies trace the
CDM distribution.
Cosmological «Circuit»
Generator of Perturbations (Inflation)
Amplifier (Gravity)
Low band pass filter.
Cosmological and
Astrophysical effects
Tend to erase small scale (large k) perturbations
Power spectrum for CDM
The analysis we have presented up to now is very qualitative and approximated (just to have an idea…)
The true power spectrum for CDM density fluctuations can be computed by integrating a system of differential equations.
We will see this in better detail in the next lectures.
In any case, we assume a power law as initial power spectrum as:
The motivation of using this type of primordial spectrum comes from inflation (we will discuss this also in a future lecture). We have two free parameters: the amplitude A and the spectral index ns.
Power spectrum for CDM
The first numerical results on the CDM power spectrum appeared around 1980.
Bardeen, Bond, Kaiser and Szalay (BBKS) in 1986 proposed the following fitting function:
where: This is correct fit for a pure CDM model (no baryons or massive neutrinos).
Note the dependence on
that defines the epoch of equivalence.
Horizon at equivalence
The redshift of equivalence is given by:
The Hubble parameter at equivalence is then
The physical size is then:
And, in comoving coordinates:
P(k) for LCDM (from numerical computations).
Spectral index is assumed
n=0.96
Note these oscillations in the
CDM P(k).
Gravitational feedback from
baryons.
The position of the peak is related to size of the
horizon at equivalence, i.e.
to the matter density since radiation is fixed.
Primor dial r
egime
Damping (scales that enter
ed horizon befor
e quality)
If we plot the P(k) in function of h/Mpc, then the dependence is just on
Effect of the Cosmological parameters
Cold dark matter
Effect of the Cosmological parameters
Baryon density
Effect of the Cosmological parameters
Spectral index
CDM vs data (1996)
If we assume a flat universe made just of matter with
we get too much power on small scales
to match observations.
Already in 1996 the best fit was for
i.e. suggesting a low matter density universe.
CDM vs data (2003)
The 2dfGRS provided
the following constraints:
(assuming just CDM, no massive neutrinos)
2 sigma detection of baryons:
Measurements from SDSS
Massive Neutrinos
We saw that massive neutrinos contribute to the matter density when they are non-relativistic as:
However, until they are relativistic, neutrino have
w=1/3, i.e. perturbations in the neutrino component dissipates when they enter the horizon.
The wavelength of the horizon when they start to be non relativistic is given by:
Evolution of the density fields in the synchronous gauge (top panels) and the conformal Newtonian gauge (bottom panels) in the Ων = 0.2 CDM+HDM
model for 3 wavenumbers k= 0.01, 0.1 and 1.0 Mpc−1. In each figure, the five lines represent δc , δb , δγ , δν and δh for the CDM (solid), baryon (dash-
dotted), photon (long-dashed), massless neutrino (dotted), and massive neutrino (short-dashed) components, respectively.
Numerical evolution of density fluctuations
Model with CDM, neutrinos, massive neutrinos, photons, baryons
Focus just on the Top panel
(Syncronous Gauge)
These fluctuations enter the horizon
just after decoupling
CDM, Baryons and MN grow as a^2 and a
Photons and
massless neutrinos oscillates and damp
after entering the horizon
Focus just on the Top panel
(Syncronous Gauge)
CDM grow as a^2, ln aˆ2 and a
Until decoupling baryons oscillates and are damped, they quickly follow CDM
afterwards.
Massive neutrinos
(nearly 1.5 eV in this case)
are semi
relativistic at equality and damped.
They slowly catch CDM afterwards.
These fluctuations enter the horizon just after equality
Photons and massless neutrinos oscillates and damp
after entering the horizon
Focus just on the Top panel
(Syncronous Gauge)
Until decoupling baryons oscillates and are damped, they quickly follow CDM
afterwards.
Massive neutrinos are first
relativistic and strongly damped because of free streaming.
They try to catch CDM afterwards.
These fluctuations enter the horizon well before
equality
Photons and massless neutrinos oscillates and damp
after entering the horizon
CDM grow as a^2, ln aˆ2 and a
small feedback from b and mnu
Baryons vs Massive Neutrinos
If a perturbation in baryons enters the horizon before
decoupling oscillates with a decreasing amplitude because of diffusion damping (photons can go from hotter to
colder regions after several scattering).
After decoupling, baryons fall in the CDM potential well.
If a perturbation in massive neutrinos enter the horizon before the non relativistic regime is strongly damped
because of free streaming (neutrinos are collision-less).
Neutrino are very light and also afterwards they still
suffer from free streaming and practically don’t cluster.
Massive Neutrinos
The growth of the fluctuations is therefore suppressed on all scales below the horizon when the neutrinos become
nonrelativistic
the small scale suppression is given by:
Larger is the neutrino mass, large is the suppression.
Hu et al, arXiv:astro-ph/9712057
eff
Massive Neutrinos
Massive Neutrinos
...but we have degeneracies...
• Lowering the matter density suppresses the power spectrum
• This is virtually degenerate with non-zero neutrino mass
Inclusion of CMB data is important to break degeneracies
Conservative limit from CMB+P(k) measurements:
Mass fluctuations
Given a theoretical model a quantity can be often easily compared with observations is the
variance of fluctuations on a sphere of R Mpc:
where:
Usually one assumes R=8 Mpc hˆ-1, where the linear approximation is valid.
P(k) for LCDM (from numerical computations).
Spectral index is assumed
n=0.96
The
gives the P(k)
amplitude around these scales
Examples from CAMB
http://lambda.gsfc.nasa.gov/toolbox/tb_camb_form.cfm
