Dark Matter NatureRoberto Decarli

Observational Cosmology A.Y. 2004-2005

-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

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).

*=∫m(L)(L)dL≈0,1 M° pc-3

g*=- ∂z (*2)*(z)=*(0) exp[-z/h]

- The i-th star speed may be assumed to be vi = 3(vi)heliocentric, and:

2 = (vi - <vi>)2/N

- If equilibrium is guaranteed, Gauss theorem assures: ∂z g = 4Gm

leading to: m≈ 2,8 * (but there’s a strong dependence on the number of stars considered).

*= stellar densitym(L)= stellar mass(L)= luminosity

functionL= luminosityg= acceleration along zh= characteristic

thickness of galactic disk

= speed dispersionN= number of starsm= matter density∂z= partial derivation

along z M°= solar massG= Newton

gravitational constant

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:

F=mvr2/rG = GmMG/rG

2 vr=√GMG/rG

- Indeed, a typical rotational curve appears constant at large rG.

- If vr ≈ const, we must have: MG(rG) ∝ rG or m(rG) ∝ rG

-2

which can be obtained referring to an isothermal sphere model. Thus, there must be a matter halo surrounding the galaxy.

MG= galaxy massm= stellar massrG= star distance from

galactic centervr= orbital speed

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 = 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

Baryonic or non baryonic? - 1- Dynamical mass in the universe gives m~ 0,3. Can this

matter be made of atoms?- Baryonic dark matter consists of:

m= density parameter due to matter

Gas and dust Planets, rocks, brown dwarves, protostars Black holes

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

Baryonic or non baryonic? - 2

b= density parameter due to baryonic matter

tot= total density parameter

- 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: b< 0,04 and from CMB fits: m=0,28±0,04; b= 0,03-0,04; tot=1,0±0,1.

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.

Is thermal decoupling necessary?- Consider a thermal component, such as neutrinos. When

in equilibrium with the remains of the Universe, events such as: + e± + e± frequently happen.

coll=(nv)-1∝T- th ∝T-2 for radiation dominated universen∝T3, ∝ T2 and v≈c = 5

So coll/th ∝ coll/h∝ T-3 , that is, as temperature decreases, the mean free path grows until it exceeds the causal horizon at Td=T(th=coll).

- 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.

coll= time between two events

th= universe life-timen= neutrino numerical

density= neutrino cross-

sectionv= neutrino speedc= light speedT= temperaturecoll= collisional mean

free pathh= horizon scalevth= thermal

component speed

- Keeping in mind neutrino example, we have:Sin/Sfin=(gT

3a3)/(gT03a0

3)=(g+eT+e3a3)/(gT0

3a03)

Since initial thermal equilibrium is accepted, T=T+e, which leads to T scaling:

T0=(g+e/g)1/3 T0≈ 0,71 T0

- From density considerations in the phase space:gi= Ni bosons + 7/8 Ni fermions g+e= 2+7/2 = 11/2

g= 2

Comoving entropy- During decoupling, comoving entropy S=gT3a3 is

conserved, if the transition is adiabatic (pressure has explicit dependence only from T):dU = dV = -p dV + T dS dS/dV== (+p)/T

F = U – TS = ∂p/∂TS = (d/dt) [a3] = T-1a3 p – a3(+p) T-2 T = 0

S= comoving entropy = total derivation in tgi= statistical weight of

i-th componenta= universe growth

factor in Robertson-Walker metric

= energy densityp= pressure= entropy per volumea0,T0= nowadays values

for a,TNbosons, Nfermions= number

of spin states for bosons and fermions.

. ..

(d/dt)[a3] = -3pa2a (from Tij;j=0)

.p = (∂p/∂T)T=T. ..

For physical/comoving coordinates and RW metric, see Appendix A

.

Decoupling temperature- A thermal component follows Boltzmann’s law

(generalized for expanding backgrounds):ni + 3Hni + <annvi>ni

2 =

At equilibrium: = <annvi>neq2

- In comoving coordinates we have:a neq

-1 (dni/da) = -(th/coll)[(ni/neq)2-1]

where th=(a/a)-1 and coll=(<annvi>neq)-1

- For th»coll, ni≈neq, while:for th«coll, ni ≈ const ≈ n(td)

- Decoupling epoch is defined so that:th(td)=coll(td)

H= Hubble parameterann= cross-section for

annihilation events= source termtd= decoupling epoch

.

.

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

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 tr»td, 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 tr<td, 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= Boltzmann’s constant

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) (pkpk – m2c4)

So numerical and energy densities result:n(x,t)=∫d3p f(x, p, t)

(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 havef(|p|)=(2)-3 [e|p|/T ± 1]-1

where + is used for fermions, - for bosons.

- In this case, n(T) = (3) -2 g* T3

- For non-relativistic particles: f(|p|)=(2)-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

(3)= Riemann’s function ~1,2

g*= statistical weight =Nbos + 3/4 Nferm

HDM and WDM- HDM and WDM models use relativistic particles. For HDM:

n = (3) -2 3/2 N T

3 = (3) -2 6/11 N T

3

n m= 3/11 n m= cr h2 h2=m/93 eV

- Three neutrino families with ~5 eV masses would explain a value of 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 doesn’t exceed 100. The relation obtained is now:

m h2≈m/1650 eVwhich leads to a maximum mass of 240 eV.

N= number of neutrino familiesm= neutrino massn= photon numerical densitycr= critical density= density parameter for neutrino componenth= Hubble parameter normalized to H0=100 km/s/Mpc

Known from CMB observations

Known from flatness studies

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 me< 5 eV, m2

-e < 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.

mx= mass of the neutrino belonging to x-th family

HDM: the main component of DM?- Since neutrino cross-section is negligible, we can consider

HDM component as a collisionless fluid. So, Liouville’s theorem is guaranteed: distribution function f is conserved.

- Consider a maximum of phase space density for relativistic particles:

(dN /Vd3p)max = (g (2h)-3 / [e|E|/T + 1])max= ½ g (2h)-3

If nowadays we assume a Maxwellian distribution:dN /Vd3p = m-4(2)-3/2 exp(-v2/22) = m-4(2)-3/2

- Equaling the two equations we have the “Tremaine-Gunn Limit”: m4 ~ -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. m= particle mass

N= number of particlesV= three-dimensional volumeh= Planck’s constant= speed dispersionE= energy

As E tends to 0

As E tends to 0

What about WDM?- As we said, heavier particles decouple earlier. That means,

their speed may be quite low, today. These particles, called WDM, have masses <240 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.

For supersymmetric particles, see Supersymmetric particles

For transfer function, see Appendix C

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.

- Boltzmann’s distributions leads to: n(T) = g (mCDMkbT/2)3/2 exp[-mCDMc2/kbT]

which leads to:mh2 ~mCDM

- In order to have mh2 ≈ 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: that’s 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.

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 |boson> = |fermion> Q |fermion> = |boson>

- 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, T~10-100 TeV). So all supersymmetric particles will decay into the lightest state (LSP, lightest supersymmetric particle), called Neutralino. Probably, we’re dealing with photon counterpart, called photino.

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).

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.

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 is null; as temperature decrease, the minimum is taken to non-null values. In axion case, the second state minimum isn’t 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 eV (that’s why it’s a non-thermal component).

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. Here’s the scheme of the receiver used in Lawrence Livermore Laboratory.

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

Mixed models- If we assume a CDM model, with a Zel’dovich 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.

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 doesn’t 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.

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). Einstein’s 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 isn’t 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.

MOND -1- The most important theory which doesn’t consider dark

matter is MOND (MOdified Newtonian Dynamic), developed by Bekenstein & Milgrom.

- MOND consists in a modification of Newtonian theory, in which, instead of Poisson’s equation, potential follows the law:

∇[(|∇|/a0) ∇] = 4G*

- In Newtonian view, if is assumed to solve Poisson’s equation, a mass density =(4G)-1 ∇ 2 is deduced, and a dark contribution appears as D = – *.

- 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).

(x) ≈ x for x « 1(x) ≈ 1 for x » 1*= density distributiona0= acceleration constant of MOND = 1,2 x 108 cm/s2

MOND -2- Knebe & Gibson

results for N-body simulations are shown. MOND model is in the middle row.

- At z=5 MOND model shows few density peaks (lighter spots), that means MOND predictions are affected by an extremely fast evolution.

Appendix APhysical 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:

l (t) = a(t) x

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 + dx2

2 + 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 2d2]which is called Friedmann-Robertson-Walker metric.

Robertson-Walker metric

dl= length element

Appendix BFriedmann equations.- From Einstein’s general relativity theory, we have:

ds2 = gij dxidxjRij = 8Gc-4(Tij – ½ gij T) - Ricci tensor may also be written as a function of Christoffel’s

symbols:Rij = ∂a ij

a - ∂i aja + ij

aabb – ib

aajb

abc = ½ gck (∂a gkb + ∂b gka - ∂k gab)

while energy-momentum tensor may be written as:Tij = (+p) uiuj – p gij

which, in homogenous, isotropic comoving case, becomes:T00 = Tii = -p Tij = 0 for i≠j

- Through some algebra, we are now able to deduce Friedmann’s equations:

H2 = 8G/3 – ka-2

H2q = 4G/3 (+3p)

- Friedmann’s equations are not independent to energy conservation: substituting Ti

j;j=0 into one of the equations leads to the other.

gij= metric elementRij= Ricci tensorTij= energy-momentum

tensorT= trace of Tij

ui= tetraspeed i-th component

q= deceleration parameter

Appendix CTransfer function.- Friedmann’s 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 Boltzmann’s generalized law rule the evolution of a density fluctuation.

fin(k) = T(k) in(k)Pfin(k)= |fin(k)|2 = |T(k)|2 Pin(k)

- Initial power spectrum is thought to be a power law in k, P(k)∝ kn, with n≈1 (Zel’dovich 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. fin= final density

fluctuation /in= initial density

fluctuationT(k)= transfer functionk= wave number of the

fluctuationP(k)= fluctuation power

spectrum

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