Dark Matter

Evidence ofDark Matter

GautamSharma Evidence of Dark Matter

Gravitational Lensing, Bullet Cluster and CosmologicalMicrowave Background

Gautam Sharma

Harish Chandra Research Institute

19 Feb,2015

Prof. Raj Gandhi


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Gravitational Lensing

First used by Einstein to measure deflection of light by sun, in1919.While the first major evidence was seen in a quasar lensedby a galaxy in 1979.

The idea is that light rays from galaxies residing behind thecluster get bent by the gravitational field of the cluster.


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Deflection of an ultrarelativistic particle

The Schwarszchild metric is

ds2 = −ψ(r)dt2 +1

ψ(r)dr2 + r2dθ2 + r2sin2(θ)dφ2

In the orbit equation of the above metric, by substitutingu = 1/r .We get

d2u

dφ2+ u =

GMm2

L2+

3GM

c2u2

Taking a solution of the form u = b−1cosφ+ f (φ) withf (φ) << 1/b.We have

f ′′ + f ∼=GMm2

L2+

3GM

2c2b2[cos2φ+ 1]

solving the above differential equaton gives

u =1

bcosφ− GM

c2b2cos2φ+

GMm2

L2+

2GM

c2b2


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At u = 0(i .e. r =∞) ignoring the cos2φ terms

−cosφ ∼=GMm2b

L2+

2GM

c2b≡ q

φ = ±[(π/2) + q] and net deflection δφ = 2q

At r =∞,Angular momentum L = bp∞ = bγmv∞.Above relations give

θ = 2q =2GM

bv2∞

(1 +

v2∞c2

)For c →∞ (Newtonian limit), θ = 2GM

bv2∞

For v∞ = c(photons), θ = 4GMbv2

∞


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Weak Lensing,Lens Equation and Einstein radius

We assume that distances to source and lens are very large anddeflection angle is very small (α̃ ≤ 1arc sec).The deflectionresults in two images of the source at different positions.

Figure : Schematic view of the lens geometry


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From the figure we note that, β = ηDS

, θ = ξDL

and hence

θDS = βDS + α̃DLS (1)

From our previous derivation of angle of deflection we have

α̃(ξ) =4GM(ξ)

c21

ξ(2)

where M(ξ) is the mass inside radius ξ.

Using (1) and (2)we get

β = θ − DLS

DSα̃

4GM

c21

ξ(3)


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with α given by

α =DLS

DS

4GM

c21

DL|θ|=

θ2E|θ|2

θ

with θE given by

θ2E =DLS

DLDS

4GM

c2

Now if we define RE = θEDL

RE =

√4GMDLDLS

c2DS

RE is the Einstein’s radius. So we can obtain the mass of thelens, from the above formula if we know RE .


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Galaxies acting as gravitational Lenses

Most spectacular observations have been made with galaxiesacting as gravitational lenses. But it is poorly described assuperposition of point sources. So we need superposition ofpoint masses or we may use a smooth mass density. Forsuperposition our formula is modified as

α̃(ξ) = Σ4Gmi

c2

~ξ − ~ξi|~ξ − ~ξi |2

We introduce a continuous mass distribution dm = Σ(ξ)d2ξwith a 2-dimensional mass density Σ(ξ) =

∫ρ(ξ, z)dz , so that

α̃(ξ) =4G

c2

∫d2~ξ′Σ(~ξ′)

~ξ − ~ξi|~ξ − ~ξi |2


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For symmetric mass distributions it reduces to, α̃ = 4GM(ξ)c2ξ

Using our previous theory we can thus calculate the total massresponsible for deflection.

Figure : The Collision of two galaxy clusters Abell 520 from an X-rayexposure by Chandra (red) and a point by point evaluation of lens ingeffects (blue). The red colour shows the distribution of “normal”matter, blue is the distribution of dark matter derived from lensing.


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Bullet Cluster

Bullet Cluster(1E0657-558) is a unique cluster merger, thatenables direct detection of dark matter, independent ofassumptions regarding the nature of the gravitational force law.

Due to the collision of two clusters, the dissipation less stellarcomponent and the fluid-like X-ray emitting plasma arespatially separated as observed in the map.Galaxies will behaveas collisionless particles but the plasma will experience rampressure.

We assume that in absence of dark matter,the gravitationalmatter will trace the dominant visible matter component,whichis X-ray plasma.But if the dominant matter is dark matter thegravitational field will trace dark matter.

To verify this the gravitational potential of the system wasmapped using gravitational lensing , to determine the dominantpart.


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Figure : This is a composite image of the Bullet Cluster (1E 0657-558)that shows the Xray light in purple, the optical light in white, and the darkmatter map in blue. source: NASA


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It is visible from the figure that the gravitational lensing mapdon’t trace the plasma distribution(the dominant baryonicmass) but rather traces the galaxies.

Figure : This image of the Abell 2218 galaxy cluster shows how amassive cluster can lens the galaxies that are behind it. Clearly seenin this image are multiple stretched galaxies.


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Using the above data, the the ellipticity of the the backgroundgalaxies from their brightness distribution was measured.

The ellipticity of each galaxy is a direct measurement of thereduced shear (stretching), g= γ/(1− κ), where γ is the shear,and κ is the convergence.

In Newtonian gravity, κ is equal to the surface mass density ofthe lens divided by a scaling constant. In modified gravitymodels,κ is no longer linearly related to the surface massdensity but is instead a nonlocal function that scales as themass raised to a power. It is this difference that allows theauthors to compare nonstandard models of gravity withNewtonian.


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Figure : On the left the colour image from the Magellan telescope.On the right is the Chandra Xray image. The green contours in bothimages are the weak lensing convergence map.

From GR, κ ∝ Σ, showing the concentrations of masses.

The peaks of the contours occur both offset from the brightestgalaxy.


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After the lensing contour maps, the masses and locations ofbaryonic matter were measured.

Figure : The masses of the stellar components and the Xray gas weremeasured independent of any gravity or dark matter models.

The amount of mass in the stellar component is much smallerthan the amount of mass in the Xray plasma, by a large factor.Regardless, the centroid of the gravitational well map is alignedwith the stellar components, indicating most of the massshould be there.


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Cosmological Microwave Background

Before the neutral hydrogen was formed, the matter wasdistributed almost uniformly in space.Gravity pulled the normal and dark matter in toward the centerof each fluctuation. While the dark matter continued to moveinward, the normal matter fell in only until the pressure ofphotons pushed it back, causing it to flow outward until thegravitational pressure overcame the photon pressure and thematter began to fall in once more.When the neutral hydrogen formed, areas into which thematter had fallen were hotter than the surroundings. Areasfrom which matter had streamed out, were cooler.This pattern of temperature variations was frozen into thecosmic microwave background when the electrons and protonsformed neutral hydrogen. So a map of the temperaturevariations in the CMB traces out the location and amount ofdifferent types of matter at 390,000 years after the Big Bang.


GautamSharma In the early 1989, NASA’s Cosmic Background Explorer

(COBE) spacecraft used a pair of radio telescopes to measuredifferences among relic photons to one part per million betweentwo points in the sky.

A subsequent spacecraft, the Wilkinson Microwave AnisotropyProbe (WMAP), made an even more precise map. Thisrevealed hot and cold spots about 1.8 degrees in size across thesky that vary in intensity by a few parts per million.

The angular size and the extent of variation indicate that theuniverse contained about five times as much dark matter asnormal matter when the neutral hydrogen formed.


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Figure : Map of the temperature variations in the cosmic microwavebackground measured by the WMAP satellite.

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