Physical Cosmology 11/5/2017
Alessandro Melchiorri
alessandro.melchiorri@roma1.infn.it slides can be found here:
oberon.roma1.infn.it/alessandro/cosmo2017
Dark Matter Candidates
If dark matter (neutrinos, for example) decouple from the primordial plasma when they are relativistic, we usually have a current energy density of:
where g* describe the number of relativistic degrees of freedom at the time of decoupling (13.75 in case of neutrinos).
If we want a value around 0.12 on the left side, the mass
of these particles should be below 15 eV (Hot Dark Matter).
As we will see, this is problematic for structure formation.
Relativistic Thermal Freeze-out
Dark Matter Candidates
On the contrary, if dark matter is non relativistic at decoupling, it will have a suppressed numerical density given by:
Non-relativistic Thermal
Freeze-out In a radiation dominated epoch we have:
The energy density today is therefore given by
Dark Matter Candidates
Thus we find the crucial result that
with only logarithmic dependence on mass since m/Tfr is roughly constant.
This is a very interesting result in that a stable particle at the weak interaction scale of several hundred GeV would give the proper relic density to be dark matter. Any Weakly Interacting Massive Particle (WIMP) might thus be a compelling dark
matter candidate.
Taking the simple scaling that we find the approximate relic density
Direct searches
Up to now, DM direct detection (by nuclear recoil with some material) has not been successful.
We have only the signal from DAMA (since almost 20 years).
Indirect Searches
Several authors have reported, since 2009, the detection of a gamma- ray signal in FERMI data, originating from the inner few degrees
around the Galactic Center. The spectrum and the morphology are
compatible with those expected from annihilating DM particles, and in particular they are best fit by 30-50 GeV DM particles distributed
according to a NFW profile. Unknown foregrounds ? Better foreground modelling can fully explain the signal without DM.
Indirect Searches
(not a WIMP but could be a sterile neutrino)
Indirect Searches
Astro-H (Hitomi)
The spacecraft was launched on 17 February 2016….
Positron fraction in galactic cosmic rays measures
by Pamela, Fermi and AMS-02 satellites. The high energy rise of the spectrum could be dark matter or just due to pulsars.
Antiprotons/protons from AMS-02 seems higher than what predicted from foregrounds models. Dark matter ?
just propaganda…better foregrounds modelling can explain the data without dark matter…
Structure Formation
Until now we have assumed an homogeneous and isotropic universe.
The scale factor, the Friedmann equations, the energy densities they depend just on time.
But this is just an approximation, the observed universe is never perfectly homogeneous and isotropic !
Structure Formation
Galaxy distribution
from the Sloan Digital Sky Survey:
We see clusters,
filaments, structure.
How can we reconcile this with isotropy and homogeneity ?
How did this structure form ?
What we can learn from it ?
Structure Formation
Let’s start from the most simple case: a static universe ! We consider a sphere a radius R.
Outside the sphere we have everywhere a density , constant with time.
Inside the sphere we have a density
with defined as the density
constrast and that varies with time.
How does this overdensity evolves with time ?
Structure Formation
If the density excess δ is uniform within the sphere, then the gravitational acceleration at the sphere’s surface, due to the excess mass, will be
Thus, a mass excess (δ > 0) will cause the sphere to collapse inward ( < 0).
Structure Formation
Conservation of mass tells us that the mass of the sphere,
remains constant during the collapse.
Thus, we can write another relation between R(t) and δ(t) which must hold true during the collapse:
where
Structure Formation
Let us now assume that the density contrast is small:
we can retain values at linear order for δ(t) and neglect higher powers:
Taking the second time derivative yields :
Thus, mass conservation tells us that
Structure Formation
The most general solution of this equation has the form
where the dynamical time for collapse is
Structure formation
If we consider just the growing solution (the initial universe is homogeneous):
In a static universe fluctuations grow exponentially with a dynamic timescale:
i.e. denser the medium faster is the collapse.
Pressure
The density of the air around you is ρ≈ 1kgm−3, yielding a dynamical time for collapse of tdyn ≈ 9 hours.
What keeps small density perturbations in the air from
undergoing a runaway collapse over the course of a few days?
The answer, of course, is pressure.
A non-relativistic gas has an equation-of-state parameter
where T is the temperature of the gas and μ is the mean mass per gas particle. Thus, the pressure of a ideal gas will never totally vanish, but will only approach zero in the limit that the temperature approaches absolute zero.
Pressure
If the pressure is nonzero, the attempted collapse will be
countered by a steepening of the pressure gradient within the perturbation. The steepening of the pressure gradient,
however, doesn’t occur instantaneously. Any change in
pressure travels at the sound speed.Thus, the time it takes for the pressure gradient to build up in a region of radius R will be
where cs is the local sound speed. In a medium with equation- of-state parameter w > 0, the sound speed is
Pressure
Time scale for collapse Time scale for pressure We have collapse only if:
Or, if the size of the perturbation is larger than the Jeans lenght:
Structure Formation
Cold dark matter, always non relativistic:
=0
practically any over density will collapse.
Pressure is negligible.
Photons, or relativistic particles:
=1/3
photons are free streaming. In an expanding universe the Jeans length is close to the horizon size !
We don’t have structure.
Structure Formation
Some energy component has w that depends on time.
For example: baryons.
For z>1100 baryons are tightly coupled to CMB photons.
They make a relativistic plasma with w=1/3.
For z<1100 baryons are decoupled. They have masses
>0.5 MeV and are non relativistic. They have w=0.
Before z=1100 thus we have:
Structure Formation
But before and after decoupling the baryon sound speed changes a lot:
Before:
After:
The Jeans length collapses after decoupling, we have:
Perturbations in the baryon density, from supercluster scales down the the size of the smallest dwarf galaxies, couldn’t grow in amplitude until the time of photon decoupling.
Structure Formation in Expanding Universe
Suppose you are in a universe filled with pressureless matter which has mass density ρ ̄(t). As the universe expands, the density decreases at the rate ρ ̄(t) ∝ a(t). Within a spherical region of radius R, a small amount of matter is added, or
removed, so that the density within the sphere is
In performing a Newtonian analysis of this problem, we are
implicitly assuming that the radius R is small compared to the Hubble distance and large compared to the Jeans length.
Structure Formation in Expanding Universe
The total gravitational acceleration at the surface of the sphere will be
The equation of motion for a point at the surface of the sphere can then be written in the form
Structure Formation in Expanding Universe
Mass conservation tells us that the mass inside the sphere,remains constant as the sphere expands.
and, since we have
Structure Formation in Expanding Universe
Taking two time derivatives of
We get:
Combining with:
Structure Formation in Expanding Universe
We get:
This is the second Friedmann equation.
These two terms cancel
Structure Formation in Expanding Universe
We have:
or, finally:
The expansion of the Universe enters here !
Structure Formation in Expanding Universe
A key point is that the matter component that is collapsing could be different from the dominant energy component.
Introducing the time dependent density parameter:
We can write:
Structure Formation in Expanding Universe
Let us first consider the radiation here. During this epoch:
The equation can be therefore written as:
with solution:
during radiation dominated epoch the perturbations in matter grow only logarithmically !!!
Structure Formation in Expanding Universe
If we move to matter dominated we have:
Structure Formation in Expanding Universe
The equation is therefore:
The solution is a power law:
During matter dominated, matter fluctuations grow with a power of 2/3 (less than in a static universe).
We discard the 1/t solution since we expect a more homogeneous
Universe in the past
Structure Formation in Expanding Universe
Finally, in an epoch dominated by a cosmological constant:
The equation is:
And the solution:
If Lambda dominates the expansion, then the fluctuations in the matter component remain constant.
They are, in practice, frozen.
Summary
If we consider perturbations in a pressureless matter component (Jeans length always zero) their growth depend on which kind of energy component is
dominating the expansion.
We have substantial growth only
here !
Moving to Fourier Space
In order to move to a more physical description let us consider an expansion in Fourier modes of the density contrast field:
Each Fourier mode is given by:
Each Fourier component is a complex number, which can be written in the form:
It is possible to show that assuming a linear perturbation theory, i.e. :
And following a more formal approach, one gets, in the Newtonian regime, for a pressure less fluid:
Moving to Fourier Space
Moving to Fourier Space
This is the same equation we got for the density contrast of a sphere of radius R, but now it applies to each
Fourier mode of a generic density contrast !
The fact that we assumed linear perturbation theory implies that the evolution of each Fourier mode is
independent from the other, i.e. we don’t have
density contrasts of different modes in the equation and their time evolution does not mix.
Pressure term
The previous equation holds for a pressure-less fluid (w=0). If we consider pressure, it is possible to show that the equation modifies to:
New term due to fluid pressure.
k is in comoving coordinates, so at each wavenumber k corresponds a physical scale at time t of:
From the above equation we identify the Jeans wavelength:
Each Fourier mode will therefore evolve with time in a different way if the corresponding k is larger or
smaller than the Jeans wavelength.
If
We can neglect this.
and we have the growth as discussed for a fluid with w=0.
Pressure term
If, on the contrary, we are below the Jeans lenght:
We neglect the first term and the solution is given by a more complicate oscillation term damped in time.
Pressure term
Horizon Scale
Given a time t, the quantity:
provides a causal horizon, i.e. particles that are a distances larger than it are not causally connected.
What happens if I consider a perturbation on scales larger than the horizon scale ?
We can treat them only using general relativity.
As we will see, the solution is also gauge-dependent.
Horizon Scale Horizon Scale
Assuming a synchronous gauge, it is possible to show that perturbations on scales larger than the horizon, i.e. such that at a given time t have:
they always grow, as:
Summary (single fluid)
We have therefore two important scales for structure formation: the horizon scale and the Jeans scale.
For cold dark matter (w=0) what is important is the
horizon scale at equivalence. Perturbations that enter the horizon before the epoch of equivalence are damped
respect to perturbations that enter the horizon later.
For CDM the Jeans scale is always zero.
For baryons, the Jeans length is approximately the
horizon scale until decoupling. The crucial scale is the horizon scale at decoupling. After decoupling baryons have w=0 (approximately).
Perturbations that enter the horizon before decoupling (z=1100) are strongly damped respect to perturbations that enter the horizon later.
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
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
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.
