Dark Matter Nature

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Dark Matter Nature

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Title: Dark Matter Nature

1
Dark Matter Nature

Roberto Decarli

First evidences of Dark Matter in Universe
Baryonic or non baryonic?
Thermal or non thermal?
Hot, Warm, Cold or other?
Axions or LSP?
Looking for Dark Matter Particles
Collisional or non collisional?
A skeptical point of view

Observational Cosmology A.Y. 2004-2005
2
First evidences of Dark Matter - 1

At the beginning of the century, Oort studied the
motion of stars in a radius of 500 pc from the
Sun. Neglecting systematic rotation of stars
around galactic center, he assumed hydrostatic
equilibrium in the direction normal to galactic
disk (z).
r?m(L)F(L)dL0,1 M pc-3
gr- ?z (rs2)
r(z)r(0) exp-z/h
The i-th star speed may be assumed to be vi
3(vi)heliocentric, and
s2 S (vi - ltvigt)2/N
If equilibrium is guaranteed, Gauss theorem
assures ?z g 4pGrm
leading to rm 2,8 r (but theres a strong
dependence on the number of stars considered).

r stellar density m(L) stellar mass F(L)
luminosity function L luminosity g acceleration
along z h characteristic thickness of galactic
disk s speed dispersion N number of stars rm
matter density ?z partial derivation along z
M solar mass G Newton gravitational constant
3
First evidences of Dark Matter - 2

Lenticular and spiral galaxies present important
angular momenta, responsible of the disk-like
structure. If star rotation around galactic
center had been keplerian, the whole galaxy mass
being considered as a point, star orbital speed
would be
Fmvr2/rG GmMG/rG2 vrvGMG/rG
Indeed, a typical rotational curve appears
constant at large rG.
If vr const, we must have MG(rG) ? rG or
rm(rG) ? rG-2which can be obtained referring
to an isothermal sphere model. Thus, there must
be a matter halo surrounding the galaxy.

MG galaxy mass m stellar mass rG star distance
from galactic center vr orbital speed
4
First evidences of Dark Matter - 3

Virial Theorem may be applied to galaxy clusters
(since 1933 when Zwicky studied gravitational
bounding for Coma cluster). The hypothesis is
that the cluster is an isolated system. Pressure
effects are neglected.
2K U S mivi2 Gmimj/rij 0
Dynamical mass must be confronted to the mass
associated to luminosity. A big galaxy such as MW
typically has M/L 10 M/L. Even with some
assumption on intracluster materials (linked to
x-ray bremsstrahlung emission), we hardly reach
dynamical value of M/L 180 M/L.
Considerations about dynamical masses in galaxy
clusters may also start from SZ effect studies
and from cluster gravitational lensing.

K kinetic energy U potential energy
modulus rij distance between i-th and j-th
galaxies L Sun luminosity
5
Baryonic or non baryonic? - 1

Dynamical mass in the universe gives Wm 0,3. Can
this matter be made of atoms?
Baryonic dark matter consists of

Planets, rocks, brown dwarves, protostars
Gas and dust
Black holes
Wm density parameter due to matter

Experiments Macho and Eros tested the presence of
jupiters and brown dwarves in LMC using
gravitational lensing. Total density due to lt 0,5
M objects cannot explain more than 20-50 of
galactic halo.

6
Baryonic or non baryonic? - 2

If large amounts of neutral gas were diffused in
the universe, there would be strong absorption of
the radiation from quasars. If the gas were
ionized, we would see important effects both on x
and microwave backgounds. None of these features
are observed.
Low masses black holes would evaporate because of
Hawking radiation, with strong gamma emission. BH
with masses in the order of solar mass would
explain Macho observations, but cannot fit with
density requests. Super massive black holes
cannot explain dark matter effects on small
galaxies scale.
Most of all, non baryonic dark matter is needed
according to measures of residual Helium,
Deuterium and Lythium from Big Bang
Nucleosynthesis Wblt 0,04 and from CMB fits
Wm0,280,04 Wb 0,03-0,04 Wtot1,00,1.

Wb density parameter due to baryonic
matter Wtot total density parameter
7
Thermal or non thermal?

Since the most important component of dark matter
is non baryonic, we must consider what kind of
particles it is and how it is distributed.
If a particle was at first in thermal equilibrium
with the other components, it is said to be a
thermal component. Some models consider particles
which have never been in thermal equilibrium with
the other components these particles are called
non-thermal components.
Neutrinos and supersymmetric particles are
examples of thermal candidates for dark matter.
Axions are the most important non-thermal
candidate for dark matter.
Thermal components follow thermal distributions
as long as they are coupled with other
components we cannot say anything about
non-thermal component distribution.

8
Is thermal decoupling necessary?

Consider a thermal component, such as neutrinos.
When in equilibrium with the remains of the
Universe, events such as n e n e
frequently happen.
tcoll(nsvn)-1?T-a th ?T-2 for radiation
dominated universe
n?T3, s ? T2 and vn c a 5
So tcoll/th ? lcoll/lh? T-3 , that is, as
temperature decreases, the mean free path grows
until it exceeds the causal horizon at
TdT(thtcoll).
Thermal components have vth? T1/2 (Boltzmann).
Theorical reasons say that no event can be
characterized by a cross-section temperature
dependence equal or minor than T-1, so thermal
decoupling is ineludible.

tcoll time between two events th universe
life-time n neutrino numerical density s
neutrino cross-section vn neutrino speed c
light speed T temperature lcoll collisional
mean free path lh horizon scale vth thermal
component speed
9
Comoving entropy
For physical/comoving coordinates and RW metric,
see Appendix A

During decoupling, comoving entropy SgT3a3 is
conserved, if the transition is adiabatic
(pressure has explicit dependence only from T)
dU r dV -p dV T dS dS/dVs (rp)/T
F U TS s ?p/?T
S (d/dt) sa3 T-1a3 p a3(rp) T-2 T 0

.
.
.
.
S comoving entropy total derivation in t gi
statistical weight of i-th component a universe
growth factor in Robertson-Walker metric r
energy density p pressure s entropy per
volume a0,T0 nowadays values for a,T Nbosons,
Nfermions number of spin states for bosons and
fermions.
.
.
.
.
p (?p/?T)TsT
(d/dt)ra3 -3pa2a (from Tijj0)

Keeping in mind neutrino example, we have
Sin/Sfin(gnTn3a3)/(gnT0n3a03)(ggeTge3a3)/(ggT0
g3a03)
Since initial thermal equilibrium is accepted,
TnTge, which leads to T scaling
T0n(gge/gg)1/3 T0g 0,71 T0g
From density considerations in the phase space
gi Ni bosons 7/8 Ni fermions gge 27/2
11/2
gg 2

10
Decoupling temperature
For a brief explanation of H and its role in the
expanding universe, see Appendix B

A thermal component follows Boltzmanns law
(generalized for expanding backgrounds)
ni 3Hni ltsannvigtni2 Y
At equilibrium Y ltsannvigtneq2
In comoving coordinates we have
a neq-1 (dni/da) -(th/tcoll)(ni/neq)2-1
where th(a/a)-1 and tcoll(ltsannvigtneq)-1
For thtcoll, nineq, while
for thtcoll, ni const n(td)
Decoupling epoch is defined so that
th(td)tcoll(td)

.
.
H Hubble parameter sann cross-section for
annihilation events Y source term td
decoupling epoch
11
Hot, warm, cold dark matter

A particle is relativistic as long as kbT(t)
mc2.
The epoch in which this inequality is violated
is the derelativization time (tr).
If trtd, the particle is still relativistic when
decoupling happens. We call this kind of matter
hot. Neutrinos are hot particle. Today their
speed is near c.
If trlttd, the particle is already
non-relativistic when decoupling happens. We call
this kind of matter cold.
If a particle derelativizes after decoupling, but
nowadays has low speed, we call it warm matter.

kb Boltzmanns constant
12
Phase space population

Let f(x, p, t) be the distribution function of
the particle considered. We can express it in
covariant formalism as
f(xh, pj) d(pkpk m2c4)
So numerical and energy densities result
n(x,t)?d3p f(x, p, t)
r(x,t)?d3p E(p) f(x, p, t)
Hypothesis of universe homogeneity lets us drop
position dependence isotropic assumption reduces
momenta dependence to p dependence.
From spin-statistic theorem, relativistic
particles have
f(p)(2p)-3 ep/T 1-1
where is used for fermions, - for bosons.
In this case, n(T) z(3) p-2 g T3
For non-relativistic particles
f(p)(2p)-1 exp(p2/2m -mc2)/kbT

x particle position in 3D space p particle
momentum in 3D momentum space xh particle
tetraposition pj particle tetramomentum E(p)
particle energy as a function of its
momentum z(3) Riemanns function 1,2 g
statistical weight Nbos 3/4 Nferm
13
HDM and WDM

HDM and WDM models use relativistic particles.
For HDM
nn z(3) p-2 3/2 Nn Tn3 z(3) p-2 6/11 Nn Tg3
nn S mn 3/11 ng S mn rcr Wn h2 Wn
h2S mn /93 eV
Three neutrino families with 5 eV masses would
explain a value of Wn near 0,3.
The greater masses are considered, the earlier
decoupling happens. g must take in account a
larger number of particles, that means more
degrees of freedom. g maximum value doesnt
exceed 100. The relation obtained is now
Wm h2S m/1650 eV
which leads to a maximum mass of 240 eV.

Known from CMB observations
Known from flatness studies
Nn number of neutrino families mn neutrino
mass nn photon numerical density rcr critical
density Wn density parameter for neutrino
component h Hubble parameter normalized to
H0100 km/s/Mpc
14
Neutrino mass measurements

From double beta decays, beta spectra and
oscillations one can make considerations on
neutrino mass. No way has been found to detect
directly cosmological neutrinos, yet.
At the time, we know electronic neutrino to have
mn elt 5 eV, Dm2n m-e lt 6 eV2 tauonic neutrino
has greater mass.
Neutrino oscillations were proposed by Bruno
Pontecorvo. Studies were made through
SuperKamiokande and SNO experiments, using Solar
neutrinos. At the time, Opera experiment is on
set. It will study the decay of muonic to
electronic neutrinos shot from Cern to Gran Sasso
laboratories.
Neutrino mass measurements succeed in guarantee
energy density due to dark matter.

mn x mass of the neutrino belonging to x-th
family
15
HDM the main component of DM?

Since neutrino cross-section is negligible, we
can consider HDM component as a collisionless
fluid. So, Liouvilles theorem is guaranteed
distribution function f is conserved.
Consider a maximum of phase space density for
relativistic particles
(dN /Vd3p)max (g (2ph)-3 / eE/T 1)max ½
g (2ph)-3
If nowadays we assume a Maxwellian distribution
dN /Vd3p r m-4(2ps)-3/2 exp(-v2/2s2) r
m-4(2ps)-3/2
Equaling the two equations we have the
Tremaine-Gunn Limit m4 r s -3
Neutrinos can explain DM effects in galaxy
clusters, but not in dwarf galaxies!!! This
effect is known as free-streaming neutrinos flow
out of potential wells, so they cannot be the
most important component of DM.

As E tends to 0
As E tends to 0
m particle mass N number of particles V
three-dimensional volume h Plancks constant s
speed dispersion E energy
16
What about WDM?
For supersymmetric particles, see Supersymmetric
particles For transfer function, see Appendix C

As we said, heavier particles decouple earlier.
That means, their speed may be quite low, today.
These particles, called WDM, have masses lt240 eV.
Such a mass resist free-streaming until galaxy
halo scales. WDM candidates may be right-handed
neutrinos or the supersymmetric particle
gravitino.
Perturbation spectra studies show that WDM models
imply a reduction in small galaxy-sized objects
formation. At the beginning, this feature granted
quite a fortune for WDM models, because of an
observed deficiency of small galaxies.
Unluckly, this effect was not so important as WDM
models show. WDM hardly matches galaxy and
cluster formation.

17
Cold Dark Matter

In order to prevent free-streaming, we can
increase the mass of DM particles in our models.
tr exceeds td, so particles are non-relativistic
while decoupling.
Boltzmanns distributions leads to
n(T) g (mCDMkbT/2p)3/2 exp-mCDMc2/kbT
which leads to
Wmh2 mCDM -2
In order to have Wmh2 0,1 1, mCDM must be in
the order of some GeV. These particles must be
non baryonic, stable, or we would see gamma ray
emissions as they decay they also must be
extremely weakly interactive thats why they are
called WIMPs (Weakly Interactive Massive
Particles).
Theorists proposed the WIMPs to be supersymmetric
particles called neutralinos. This particle
should have a mass around 10 GeV.

18
Supersymmetric particles

Supersymmetries were firstly thought as an
extension of symmetry groups finalized to
envelope a unified theory for all interactions
(GUT and TOE).
In order to explain the great difference of
energy scales from TGUT (1015 GeV) to electroweak
energy (80 GeV), it was considered a possibility
that fermions could be vectors of interactions,
such as bosons W, Z0, photons and gravitons.
This can be made considering fermionic
counterparts to known bosons, and bosonic
counterparts to fermions
Q bosongt fermiongt Q fermiongt bosongt
While coupled with ordinary matter in the
primordial universe, supersymmetric particles
should not decay in ordinary particles after the
Soft Break of SUSY (Super Symmetries, T10-100
TeV). So all supersymmetric particles will decay
into the lightest state (LSP, lightest
supersymmetric particle), called Neutralino.
Probably, were dealing with photon counterpart,
called photino.

19
Direct detection of WIMPs

Direct WIMPs observations are based on scattering
on proper targets. Scattering would produce an
exponential distribution for recoiling energy,
depending on mass the hope is to detect a
contribution from a low background in the
differential energy spectrum distribution
observed.
If interaction is spin-dependent, the number of
useful targets falls, and measurement difficulty
increases.
External factors such as gamma rays and
radioactivity may produce electron recoiling and
perturbations in spectra. Extremely low signals
force the threshold of detectability to be very
low.
Three direct approaches are used
1) based on accurately radioactivity background
measurements. Genius experiment will use a ton of
Germanium.
2) based on scintillators (mainly NaI). This is
the best way for spin-dependent interactions, but
threshold is quite high.
3) based on simultaneous scintillation and
ionization detection in liquid xenon (CDMS and
CRESST experiments).

20
Indirect detection of WIMPs

Indirect measurements are based on the hypothesis
that dark matter is made of particles and
antiparticles, too. So, annihilations may be
produced, and we should be able to detect them
through gamma emission.
Measures in galactic halo do not offer the needed
precision because of the error in modeling WIMPs
distribution.
Another method is based on detection of
high-energy neutrinos coming from Earth or Sun
center. The idea is that some WIMPs fall into the
core of astronomical objects and annihilate each
others, producing neutrinos. Detections can be
made through large detectors (1000 m2) such as
SuperKamiokande, Baksan or Macro. Even if
constrains may be introduced for WIMPs mass and
cross-section, a new generation of experiments is
needed to reach direct measurements sensitivity.

21
Axions

While non-thermal components do not follow any
temperature-dependent distribution, axions are
often referred to as one of the most believable
candidates for cold dark matter.
Such as WIMPs, axions were introduced for
non-cosmological reasons (linked to strong parity
violation in this case).
Axions are the product of the breaking of a new
symmetry, which takes name from Peccei and Quinn
(1977). The mechanism is analogous to the one
introduced by Higgs for the mass problem of W
and Z0. A potential well has its minimum as the
field f is null as temperature decrease, the
minimum is taken to non-null values. In axion
case, the second state minimum isnt degenerate
that state corresponds to a new particle. The
energy gap between this minimum and the V(0) is
used to give the particle a mass, of the order of
some tens of meV (thats why its a non-thermal
component).

22
Detecting axions

Axion detection experiments are based on
resonances with magnetic fields, which may
produce faint microwave radiation in a cavity. A
differential radio-receiver is able to amplify
the signal and to make the signal detectable.

Experiments are set in Livermore, MIT, Florida
and Chicago. Heres the scheme of the receiver
used in Lawrence Livermore Laboratory.

No signal has been detected yet. In other words,
if 2,25-3,25 meV axions exist, and our knowledge
of their physics is deep enough, their energy
density in galactic halo cannot exceed 0,45 GeV
cm-3.

23
Mixed models

If we assume a CDM model, with a Zeldovich
primordial fluctuation spectrum, and we normalize
to Cobe data, we obtain a universe model with
less clusters than observed.
In order to avoid this problem, composite models
in which both cold and hot dark matter play
important roles in cosmology may be considered
CHDM, MDM are examples.
Composite models are promising, because they can
explain a wide range of phenomena. On the other
hand, they count an increasing number of
parameters, which weakens the solidity of the
physics beyond them.

24
Collisional Dark Matter

Even if extremely low interactive, dark matter
particles may have a finite cross-section. This
idea can be useful to prevent singularity
behavior of dark matter distribution in potential
wells.
The effect of considering a collisional fluid is
that distributions are flattened, even if
collision rate doesnt exceed few events in a
Hubble time. Small gravitationally bound
substructures are suppressed (see Yoshida et al.,
2000).
On the other hand, if a constant cross-section is
assumed, such a collisional fluid cannot explain
dark matter effects on the dwarf galaxy scale
here collisional time is nearly 50 times the
universe age. Collisional effects would be
important only with a wider cross-section, but it
hurts with cluster dynamics (all clusters would
appear spherical). It may be assumed that
cross-section is energy-dependent, so that
scattering is less effective when particle speed
is high enough. A better understanding of dark
matter physics is needed.

25
A skeptical point of view

No direct observation of dark matter succeeded in
its goal. We only have indirect observations,
fundamentally based on the lack of matter for the
observed gravitational interactions.
A question arises Are we looking in the right
direction? Could a better understanding of
gravity avoid the hypothesis of a non detectable
matter?
This approach leads to the development of
cosmologies which do not need dark matter, and
dark energy is taken account only as topological
characteristic of space-time curvature. An
example of this theory may be found in Moffat
(2005). Einsteins General Relativity theory (GR)
is extended to Non-symmetric Gravity Theory (NGT)
whose simplification is the Metric-Skew-Tensor
Gravity (MSTG). Details exceed the aim of this
work the key idea is that G isnt treated as a
constant. Accordingly, constant rotational curves
(at large radii) are obtained. Values for
gravitational lensing and speed dispersion in
galaxy clusters consistent with observation are
expected.

26
MOND -1

The most important theory which doesnt consider
dark matter is MOND (MOdified Newtonian Dynamic),
developed by Bekenstein Milgrom.
MOND consists in a modification of Newtonian
theory, in which, instead of Poissons equation,
potential f follows the law
?m(?f/a0) ?f 4pGr
In Newtonian view, if f is assumed to solve
Poissons equation, a mass density r (4pG)-1 ?
2f is deduced, and a dark contribution appears as
rD r r.
Studies are on run in order to evaluate MOND
success in describing dark matter role in dwarf
galaxies and in galactic clusters (see Knebe
Gibson, 2004, Milgrom, 2001, Read Moore, 2005
for examples).

m(x) x for x 1 m(x) 1 for x 1 r density
distribution a0 acceleration constant of MOND
1,2 x 108 cm/s2
27
MOND -2

Knebe Gibson results for N-body simulations are
shown. MOND model is in the middle row.
At z5 MOND model shows few density peaks
(lighter spots), that means MOND predictions are
affected by an extremely fast evolution.

28
Appendix A

Physical or Comoving Coordinates?
Since universe is expanding, as proved by Hubble
observations, we may consider a coordinate system
in which space coordinates do not depend on time
(that is, unity length do not take part to
universal expansion). We call these coordinates
comoving
Dl (t) a(t) Dx
Robertson-Walker metric.
In general relativity tetradimensional space
geometry is defined by its metric, which produce
the line element ds
ds2 c2dt2 dl 2
Because of universe expansion, we can express
time dependence of spatial terms in comoving way
ds2 c2dt2 a(t) (dx12 dx22 dx32)
Because of universe curvature, we must consider a
correction vanishing as x tends to zero. Assuming
isotropy
ds2 c2dt2 a(t) (1-kr 2)-1dr 2 r 2dW 2
which is called Friedmann-Robertson-Walker
metric.

Robertson-Walker metric
dl length element
29
Appendix B

Friedmann equations.
From Einsteins general relativity theory, we
have
ds2 gij dxidxj Rij 8pGc-4(Tij ½ gij T)
Ricci tensor may also be written as a function of
Christoffels symbols
Rij ?a Gija - ?i Gaja GijaGabb GibaGajb
Gabc ½ gck (?a gkb ?b gka - ?k gab)
while energy-momentum tensor may be written as
Tij (rp) uiuj p gij
which, in homogenous, isotropic comoving case,
becomes
T00 r Tii -p Tij 0 for i?j
Through some algebra, we are now able to deduce
Friedmanns equations
H2 8pGr/3 ka-2
H2q 4pG/3 (r3p)
Friedmanns equations are not independent to
energy conservation substituting Tijj0 into
one of the equations leads to the other.

gij metric element Rij Ricci tensor Tij
energy-momentum tensor T trace of Tij ui
tetraspeed i-th component q deceleration
parameter
30
Appendix C

Transfer function.
Friedmanns equations explain how universe
expansion depends on its energy density. In
primordial universe, density fluctuations were
produced by quantistic effects.
Continuity equation, Gauss theorem and
Boltzmanns generalized law rule the evolution of
a density fluctuation.
dfin(k) T(k) din(k)
Pfin(k) dfin(k)2 T(k)2 Pin(k)
Initial power spectrum is thought to be a power
law in k, P(k)? kn, with n1 (Zeldovich
spectrum). It may be proved that fluctuations
scale as k-2 after entering the causal horizon,
so final power spectrum grows as k for
fluctuations outside causal horizon, and falls as
k-3 for fluctuations inside causal horizon.

dfin final density fluctuation dr/r din initial
density fluctuation T(k) transfer function k
wave number of the fluctuation P(k) fluctuation
power spectrum
31
References