Lecture 16: The Nature of Dark Matter
Summary: Empirical evidence for dark ma5er A: Dynamics mgrav >> mvisible (three different cases) •  Galaxy rota<on curves: radial separa<on of baryons and dark maAer •  Cluster galaxy dynamics and hydrosta<c support: well-­‐mixed baryons and dark maAer •  Bullet cluster (see later): non-­‐radial separa<on of baryons and dark maAer B: P(k) of density fluctua<ons in the Universe and forma<on of structure •  As expected from primordial n ~ 1 with the effects of “stagspansion” but without signature of Silk damping eradica<ng fluctua<ons on scales < 1013 Msun. •  Beau<ful CMB Cl requires underlying DM to be consistent with today’s structure (“baryons fall in to underlying dark maAer fluctua<ons”) C: Primordial nucleosythesis gives baryon density << observed maAer density D: Aspects of galaxy forma<on •  Galaxies exist (c.f. Silk damping) •  Baryonic densi<es and sizes NB: Quite independent physics and lines of argument.
see how any kind of “modified gravity” would work.
Very hard to
The problem of Dark MaAer is that: •  There are no candidate DM par<cles within the Standard Model of Par<cle Physics. •  There are very many candidates outside the Standard Model, e.g. •  axions (10-­‐5 eV) •  massive sterile neutrinos •  super-­‐symmetric WIMPs (100 GeV) •  primordial black holes (very massive) Standard CDM assumes:
“C”: No free-streaming damping (particles either massive or non-thermalized)
“D”: Completely non-interacting
To what extent are these
true, and what constraints
can we apply from
astrophysical
observations?
Astrophysical “problem areas” (1)
The slope of the (stellar) mass function of
galaxies is flatter than expected
•  Strong dependence of efficiency of starformation as f(mass)?
•  Warm DM component leads to
suppression of small scale fluctuations?
Astrophysical “problem areas” (2)
The slope of the inner profile in dwarf
galaxies is flatter than r-1 expected, i.e.
“core” rather than “cusp”.
•  Interactions between DM particles?
•  (Gravitational) interactions between
baryonic and DM components?
•  Warm DM component?
What Dark Matter is not
Normal dark baryons (planets, gas at some unfavourable temperatures etc):
Won’t work globally from primordial nucleosynthesis: ΩB ~ 0.15 Ωm
Within our halo, could the halo be made of Massive Compact objects (MACHOs):
•  Increasingly unlikely from lensing searches.
Mass constraint from absence of freestreaming damping (for thermalized particles)
−2
M fs ~ 3×1015 m30eV
M sun
−1
d fs ~ 40 m30eV
comoving Mpc
Observations conservatively
suggest λfs < 3 Mpc,
implying Mfs < 3×1013 Msun
mDM > 300 eV
Note: If particles formed
“cold” and never thermalized
then this limit will not apply.
E.g. “axion” 10-5 eV.
The Tremaine-Gunn (1979) constraint for fermions in haloes
Fermions obey the Pauli Exclusion Principle, and there is therefore a
maximum density in phase-space.
Per unit (spatial) volume the number of fermions cannot exceed
3
4π pmax
N ≤ 2g
3 h3
In a self-gravitating halo of mass M and radius R, we know that the typical
velocities σ will be given by the virial condition
GM
σ ~
R
2
and no particle can exceed the escape velocity
So there is a maximum momentum
vesc ~ 2σ ~
pmax
2GM
R
2GM
~
mf
R
If we suppose that the mass is dominated by this fermion
species, so
3
4π pmax
N ≤ 2g
3 h3
4π 3
M~
R Nm f
3
We can then trivially re-arrange to get a constraint on the mass of the fermion
3
"
%
9
π

m 4f ≥ $
'
2
# 8 2g & Gσ R
1.5
mf ≥
eV
1/4
2
(σ 1000km/s RMpc )
So, cluster of galaxies with σ ~ 1000 kms-1 and R ~ 1 Mpc yields m > 1.5 eV
Extreme dwarf galaxies with σ ~ 50 kms-1 and R ~ 1 kpc yield m > 100 eV
Constraints weaken as Nf1/4 if there are multiple species, but still rule out light
neutrinos
The Bullet cluster: 1E 0657-56
• 
One of the most luminous X-ray
clusters at z = 0.296
• 
High resolution X-ray image
reveals main cluster and smaller
bow shocked feature “the bullet”
• 
Optical galaxies and the peaks in
mass map from weak lensing are
displaced from the centers of the
X-ray emission
A small 7 × 1013 Msun sub-cluster has
fallen through a larger 2 × 1015 cluster.
Velocity indicated (from X-ray shock
Mach number) of 4500 ± 1000 kms-1
(which is high but not crazy)
X-ray gas
Mass distribution
Significance
•  Non-radial separation of dark matter and the (dominant) baryonic mass (in
the hot gas) effectively rules out modifications of Newtonian gravity as
explanation of dark matter
•  Separation of dark matter and gas gives direct constraints on the DM-DM
cross-section (c.f. terrestrial experiments give only DM-B cross-section)
The Bullet cluster (cont)
Consider that the number density in the sub-cluster, of linear dimension ~ l, is
n. This corresponds to a projected surface mass density Σ = nlm. The
observed Σ (from lensing measurements) is around 0.2 gm cm-2
l
n
The chance of a given DM particle in the main cluster having an interaction as
it streams through the volume of the sub-cluster is given by “optical depth” τ
σ
τ = nσ l = Σ
m
Unless τ < 1, DM gas will behave as interacting fluid, like the baryonic gas.
This already implies σ
< 5 cm 2g-1
m
Aside: Why do we get the
constraint on σ/m?
A more stringent constraint on τ comes from considering mass loss from the subcluster. The M/L comparing the DM to the luminous galaxies in the sub-cluster is
within about 10% of that of the main cluster (and clusters generally), suggesting no
more than about 20% of the DM particles have been stripped off as a result of
collisions during the transit.
Consider single elastic collision in rest-frame of subcluster v1=v0 cos α1
Sub-cluster loses DM particle if both v1 and v2 > vesc
from sub-cluster.
α1
v0 ~ 4500 kms-1, vesc ~ 1200 kms-1
v
sin α1 > esc
v0
v
and cos α1 > esc
v0
α2
α1 + α 2 =
v0
v2=v0 sin α1
1/2
2 &
# vesc
vesc
⇒
< sin α1 < %1− 2 (
v0
v0 '
$
Expect α to be uniformly distributed in 3-d between 0 and π/2: P(α)dα=2π sinα dα
Fraction of collisions leading to escape probability per collision χ is high:
2
vesc
χ ~ 1− 2 2 ~ 0.86
v0
π
2
Now constraint becomes
σ
χτ = Σχ < 0.2
m
σ
< 1 cm 2g-1
m
Less direct arguments along the same lines give similar
constraints:
e.g. “cold” low-σ DM particles in low mass sub-haloes
embedded in “hot” high-σ clusters would be heated by
conduction (i.e. particle-particle interactions) if σ/m were not
low. Sub-haloes would then “evaporate”, i.e. lose mass and
disappear.
Is σ/m < 0.1 cm2g-1 an interesting constraint?
•  It is very much higher than
terrestrial limits for DMbaryon interactions
(equivalent to about 106 pb
for 100 GeV particles)
•  It does largely rule out DMDM interactions as
explanations for e.g.
eliminating cores in DM
haloes which need of order
1-10 cm2g-1
Indirect searches for a decay or annihilation signal from DM
Can we detect gamma-rays from DM annihilation, c.f. 511 keV line from e+eannihilation in the Galaxy.
Has to be indirect since DM
cannot directly interact with
photons. Could be
•  via pions
•  via UHE e+e- etc
Decay signal proportional to nDM,
i.e. ρDM ΓmDM-1
Annihilation signal proportional to nDM2,
i.e. ρDM2 σmDM-2
So, inner profiles of haloes important.
Indirect searches for a decay or annihilation signal from DM
•  Fermi LAT imager in space
20 MeV -300 GeV range
•  Ground-based Cerenkov
telescopes 100 GeV – 100
TeV range
Could conceivably measure a
signal, but no convincing
detections so far
Significant problem: Gammarays will be indirect. There are
other ways to produce TeV
particles through acceleration,
especially in high density areas
(centers of galaxies, clusters etc)
that could produce DM signal
Weakly interacting particles
1. Weakly interacting light particles: e.g. three light neutrino species
Recall from Nucleosynthesis discussion that these are still relativistic when the
weak interactions freeze-out, so their number density is fixed relative to the
photon density, and thus the present-day density (and thus contribution to Ωmh2)
today is set only by the sum of the rest masses, plus small adjustments for
subsequent production of photons due to e+e- annihilation.
∑m
ν ,i
Ωm,ν =
i
93.5eV
To have Ωm ~ 0.3 implies <m> ~ 10 eV, which is already in serious trouble with
Tremaine-Gunn and free-streaming constraints (and anyway is more massive
than current limits on the light neutrino masses).
While light neutrinos with non-zero mass may be a detectable contributor to Ωm,
they cannot provide Ωm ~ 0.3
Weakly interacting particles (cont.)
2. Weakly interacting massive neutrinos
Consider more massive neutrino-like particles that become non-relativistic
earlier when they are still interacting. As with e.g. e+e-, their number
densities will drop as they annihilate.
dn
= −n 2 β
dt
Annihilation is two body, so rate is proportional to
−1
physical density n2, and the annihilation timescale will be
τ ann ∝ ( β n )
proportional to n-1
The physical density n will be given by the comoving density times a-3. The a
at which the particles become non-relativistic will be proportional to m-1.
The comoving density will drop through
annihilations until the annihilation time τann
equals the Hubble time. Hubble time is
proportional to a2 (radiation-dominated) and
thus to m-2
τ ann ∝ ( β nc m
τ H ∝ m −2
3 −1
)
Weakly interacting particles (cont.)
Resulting comoving
density of particles
The comoving density of
the particles (and antiparticles) at freeze-out
will scale as
1
nf ∝
mβ
What is interaction
strength β?
τ ann ∝ ( β nc m
3 −1
)
τ H ∝ m −2
Completely non-­‐interac<ng before freeze-­‐out Neutral leptons Charged leptons hadrons Weakly interacting particles (cont.)
For neutrinos expect** β
scales as m2
So, freeze-out comoving
number density should scale
as m-3 and present-day density
should scale as m-2.
Around 10GeV expect to get
ΩM ~ 0.3
But known neutrinos are
not this massive!
1
nf ∝
mβ
The WIMP Miracle
WIMP = Weakly interacting massive particle
For Supersymmetric WIMPs
expect** β scales as m-2
So, freeze-out comoving number
density should scale as m1 and
present-day density should scale
as m2.
Around 100 GeV expect to get
ΩM ~ 0.3
This is known as the “WIMP
Miracle” because 100 GeV is the
Fermi scale of electroweak
unification
1
nf ∝
mβ
Is there some connection with Baryogenesis itself?
Strange co-incidence: ΩB ~ ΩCDM
And today at least:
~ ΩΛ
At face value these have come from completely different Physics:
•  Baryogenesis and ε asymmetry (complete annihilation)
•  WIMP miracle (if relevant ??) involving partial annihilation
•  Vacuum energy, scalar field etc
Is this coincidence telling us something?
Or is it anthropic? Arguably there would be no Life as we know it:
•  if ΩΒ too low to allow cooling to form galaxies or if ΩCDM too low to
prevent Silk eradication of density fluctuations on galaxy scales (i.e.
need to have 10-3 > ΩΒ/ ΩM < 10 ?)
•  if Λ dominated at z > 100 (i.e. ΩΛ > 106 ΩM)
Key points
Dark matter exists and is non-baryonic and there are many independent lines
of evidence for this.
Simple CDM provides an extremely successful cosmogony.
Possible areas to watch are:
• 
inner profiles of haloes
• 
number of low mass haloes
but these are precisely where we need to worry about baryon physics.
Constraints on the mass of the DM particle come from
free-streaming damping (unless born cold?)
phase-space density in dwarf galaxy haloes (for fermions)
DM-DM cross-section constrained from astrophysical data (e.g. Bullet
cluster), but these are much higher than the DM-baryon cross-section
constraints from terrestrial searches.
We can predict density of particles knowing their interactions. e.g. “WIMP
miracle”.