Lecture 20 (Revision)

PHY326/426 DARK MATTER!

Exam GuideAll material in the lectures and lecture notes is potentially examinable including that from guest lecturers

The course is slightly changed from previous years - less on axions, more on WIMPs; a bit more on neutrinos; a broader overview of the subject (see later)

You should be aware of and understand the meaning of the equations presented....some you need to know and this is indicated in the notes

Exams are generally around 50-60% qualitative knowledge, the rest quantitative calculationsIf there are particular equations needed that it would be unreasonable for you to know I will give them

Exam GuideIf you donʼt write anything you WILL get zero marks!If you write something

chances are you will get some marks

Itʼs what you know, not what you donʼt know, that is important,

in general

Revision AdviceRead the lecture notes carefully

Do the little questions I put in the notes as a guide

If there is a term you donʼt understand - google it or ask me

Do past exam questions - if you attempt them and show me I will go through the answers with you, BUT:

2006/07 exam: do NOT do Q4:

2007/08 exam: do NOT do Q5 (try parts of it if you wish)

2008/09 exam: do NOT do: Q3(c,d), Q5

If you want to do pre 2006/07 exams ask me...

2009/10 exam: all is ok

2010/11 exam: all is ok

Question 1 compulsory

Exam Format

5 questions [changed from previous exams]

Question 2-5 - answer any two questions from these 4

Each is divided into 4 or 5 RELATED subsections (a, b, c, d, e)

Each has a mixture of description and calculation

PAST EXAMS ARE AVAILABLE ON THE WEB OR FROM THE OFFICE

Example2008/09 exam Q4

Example2009/10 exam Q3

(a) Observation of the velocity rotation curves of galaxies provide a major piece of evidence for the existence of dark matter. With the aid of sketches explain and compare the rotation curves expected for a Keplarian system, such as planets orbiting a star, with that observed for a typical spiral galaxy. How does this lead to the interpretation that the matter in galaxies is mainly dark matter?

ANS:Keplarian: Rotation following Kepler's 3rd law is shown as planet-like or differential rotation. The orbital speeds falls off as you go to greater radii within the Galaxy. This is called a Keplerian rotation curve where v goes as r1/2

Spiral galaxy: like all galaxies we observe a flat rotation curve at large distances from the centre:

The gap between the observed flat rotation curve and the keplarian form that would be appropriate for the visible matter indicates the existence of a dark matter halo, e.g. example above.

(e) Briefly explain why this distribution is so different from that expected for the baryonic matter in galaxies?

ANS:The baryonic matter interacts strongly with itself so tends to lose angular momentum and hence collapse into a disc, whereas the non-interacting particle dark matter remains with a halo structure.

The observable Hubble law

For objects having redshifts of more than about 0.5, use the following second order result (ignoring the dots):

q0 is called the deceleration parameter, which takes account of evolution of the expansion rate.

For very nearby objects

LECTURE 1-2

Remember this

Consider the motion of a galaxy of mass m at the edge of a spherical region. According to Hubble law its velocity is v = Hor and the kinetic energy T = mv2/2The potential energy at the edge of a sphere of mass M is U = -GMm/r

Thus the total energy is: E = T + U = mv2/2-GMm/r

If the mean density of the Universe is ρ the mass is M = (4πr3/3)ρ the value of ρ that gives E = O is called the critical density ρc.

ρc = 3Ho2/8πG

Some Basic Cosmology

Remember this derivation

LECTURE 1-2

How to Quantify Dark MatterBut first we need a way to quantify the amount of DM

astronomers use Mass to Light (luminosity) ratios

solar mass (1.98892×1030 kg)Mo

Lo

ML

η=

i.e. we define the mass to luminosity ratio for a given system using units of Solar mass to light ratio, with M and L being the totals for that system (e.g. Galaxy, cluster, part of galaxy etc).

(For the Sun Mo/Lo = 0.51 g erg-1 s)

In this form we would say ηSun = 1

LECTURE 3-4

Rotation curve of NGC3198

Keplerian expectation

allow for central buldge

as measured

we infer this DARK MATTER halo

LECTURE 3-4

Mass change with radius continuedNow, we know that

Differentiate this with respect to r,

Equate the two expressions for dM/dr.

Rearrange to get an expression for the density at radius r:

So indeed IF the dark matter dominates the density, and IF the halo is spherically symmetric to beyond the gas radius, a 1/r2 distribution of matter fits the ʻflatʼ rotation curve data.

LECTURE 3-4

Applying the modified form of Newtonʼs second law to the gravitational force acting on a star outside of a galaxy of mass M leads us to

which in the low acceleration limit (large r, a ≪ a0) yields

Equating this with the centrifugal acceleration associated with a circular orbit, we arrive at

MOND - Modifying Newtonʼs Laws

!

F =GMmr2

= maµμ = a/a0

!

a =GMa0r

!

GMa0r

=v 2

r" v = (GMa0)

1/ 4

LECTURE 5-6

Zwicky then assumed the galaxies are evenly distributed within a sphere of radius R, and calculates the gravitational potential as

gravitational constant total mass of the cluster

Zwickyʼs value for R was 2 x 106 light years (613 kpc)

!

VGPE

= "3GM

5R

simplify previous equation: indicates the average taken over both time and mass

thus the total mass of the cluster

This equation now only depends on the velocities of each galaxy!

Mivi

2

i

" = M v2

!

M =5R v

2

3G

Virial theoremLECTURE 5-6

Radius of the Einstein

Ring

LECTURE 7-8Gravitational Lensing (Weak, Strong, Micro)

!

A(t) =u2(t) + 2

u(t) u2(t) + 4

!

u(t) =r(t)

b= u

0

2+

t " t0

#

$

% &

'

( )

2

Microlensing Theory?

distance of the deflector (MACHO) from the line of sight.

This magnification A, is given by:

Impact parameter

Massive Object

Star in nearbygalaxy

Earth

LECTURE 8-9Evidence for Dark Baryons

Can Neutrinos be Dark Matter?It is also interesting to ask for a lower limit on Ωh2 which the dominant dark-matter component must obey.

Allowing for a significant baryon fraction indicates that particle dark matter (PDM) should obey Ων > 0.2

!

0.05 " #$ h2" 0.4

So a reasonable range where this dark matter candidate could be all of the nonbaryonic dark matter, therefor neutrinos with mass ABOUT:

Taking h > 0.5 as a lower limit for the expansion rate implies

!

4eV " m# " 40eV

could represent all of the dark matter

LECTURE 8-9

LECTURE 8-9

Li-7

He-3

D

He-4

Data consistent with about 4% of the matter density of the universe being baryonic.

The measured values (circles) line up to suggest a value of

~ 0.04

Big Bang Nucleosynthesis (BBNS)

Hot, Cold and Warm Dark MatterWe can divide the candidates into three types (HDM, CDM and WDM) useful for understanding the influence they have of structure formation in the Universe:The difference reflects how fast the particles were moving at the time they decoupled from the baryonic matter (i.e. when they stopped interacting with it as the Universe cooled). This distinction is important because whether dark matter is in the form of CDM or HDM critically influences how we would expect the Universe to form. Remember which ever it is (CDM or HDM) it dominates the Universe so its bound to affect the structure of the visible bit we can see (galaxies etc.).

CDM HDM

LECTURE 10-11

The New Particle Zoo

(1) Neutralino Particles (2) Kaluza-Klein Particles

(3) Axions

Masses and interaction strengths span many, many orders of magnitude. But independent of cosmology, we expect new particles.

Most Important dark matter candidates

Weakly Interacting

Massive Particles

Supersymmetry theory

Extra Dimension

theory

CP violation theory

LECTURE 10-11

(2) Neutral, so not going to interact with the electrons in nuclei, going to get elastic scatters (not inelastic)

(1) Weakly Interacting, so donʼt interact much despite the number of them

(3) Non-relativistic, so we can use classical kinematics

(4) Slow moving, with mass comparable to that of nuclei, so turns out the energy released is quite low, keV region

How do Neutralinos/WIMPs interact?

The Observed Diff. Energy Spectrum

!

dR

dEOBS

= R0S(E

R)F

2(E

R)I

We will aim to derive this formula....

LECTURE 11-12

WIMP Signal RateLast time, we (finally) finished deriving the rate for WIMPnuclear interactions, assuming spin independence.

The quantities (other than constants) in this formula are:total mass of sensitive material in detector

atomic mass of target in atomic mass unitsWIMP, target nucleus, reduced rest energies

[GeV] width of energy bin used for event rate countdark matter rest energydensity in halo

WIMP nuclear form factor at recoil energy ERWIMP-nucleon cross section at zero recoil energy

virial velocity multiplied by divided by speed of lightevents per GeV of energy bandwidth per second

Example Expected Recoil Spectra

so here is what we might expect in a detector for an example 100 GeV WIMP for different target nuclei

LECTURE 13 (Klinger)SUST - LSP; UED - LKP

LECTURE 14 (Kudryavtsev)Vitaly Kudryavtsev

LECTURE 15 (Daw)Axion Dark Matter?

What is the Axion Mass ?

ACCELERATOR SEARCHES

AXION DARK MATTER OVERCLOSES UNIVERSE

KSVZ DF

SZ RED GIANT STAR BOUNDS

HALO DECAY EXPERIMENTS

SUPERNOVA 1987A

SEARCH WINDOW, 10-6 eV < ma < 10-3 eVLOWER$MASSES MEAN AXIONS MORE ABUNDANT

Axions, CP violation LECTURE 15

Ed Daw

How to Reduce Background - WIMPs(1) Passive shielding - i.e. (a) surround your detector by material to absorb the background gammas and neutrons (b) make detector pure

(3) Active discrimination - identify and reject background events in the detector through measuring some property that identifies them as not due to WIMP interactions

(2) Active shielding - detect the background particle scattering in a “veto” detector around the main detector, reject coinicences

How to Identify a Real WIMP Signal(1) Observe different energy spectra - i.e. use the kinematics of earlier - should get different response from different target nuclei(2) Observe annual modulation - i.e. use the motion of the earth around the Sun to see an annual change in the spectra(3) Observe directionality of the recoils - i.e. use the motion of the earth through the galaxy to see a preferred direction of the recoils

LECTURE 16

What About a Real Signal

• Target dependence (depends on A2, form factor; neutron rejection); Spectral shape (exponential form, but can look like typical background)

Recoil Spectra

• Annual modulation of flux and spectrum (few % effect at threshold)

Annual Modulation

• Recoil direction modulation (large diurnal effect, requires gas target)

Directionality

To summarize, there are three main ideas for a signal:

LECTURE 13-14

LECTURE 17(Walker)Case study of a WIMP detector

Populations in Nuclear Equilibrium

NP

NI

ND

For equilibrium, the number of decays from parent to intermediate in time dt must be equal to the number of decays from intermediate to daughter.Mean number of decays of parent and intermediate intime interval dt is

Therefore in nuclear equilibrium:

In a chain if you wait long enough we can get Nuclear Equilibrium

LECTURE 18

dEdX

Phonons

Ionisation Light

Discplacement

Experiments and TechniquesLECTURE 18

e.g. Light and Charge Technique This combination, again to get us information on the dE/dx, of the interactions, is possible thanks to amazing properties of LIQUID NOBLE GASES - Xenon and Argon:

WIMP

Argon or Xenon

Recoil

Light produced from excited Ar, Xe atoms !

!

e-

e-

e-

Ionizationelectrons

We get both charge and light BUT get proportionaly LESS charge for a nuclear recoil vs. an electron recoil

Results of Supernova Surveys

Dark EnergyLECTURE 19

Why Dark Energy wins at late timesIf you think of dark energy as a source of energy out of the vacuum, the bigger the volume of spacetime, the more dark energy there is. But if spacetime is full of matter, and then expands, this does not increase the amount of matter, so that as time increases, the dark energy content of the Universe increases, but the matter content stays constant.

In terms of densities, the matter density of the Universe drops as it expands, but the vacuum energy content of the Universe maintains a constant energy density, or at least drops less rapidly than the mass energy density.

Conclusions OF the Universe

PHY 326/426Neil Spooner

n.spooner@sheffield.ac.ukoffice: E23 , extension 2-4422

www.shef.ac.uk/physics/teaching/phy323

How to Contact Me

The End