Physical Cosmology 1/5/2016
Alessandro Melchiorri
alessandro.melchiorri@roma1.infn.it slides can be found here:
oberon.roma1.infn.it/alessandro/cosmo2016
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 ?
Indirect Searches
(not a WIMP but could be a sterile neutrino)
Indirect Searches
Astro-H (Hitomi)
The spacecraft was launched on 17 February 2016….
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 !
