Incompatible GravityIncompatible Gravity

Pouria KhalajPhysics Department

Sharif University of TechnologyMay 11, 2010

OutlineTerminologyIntroductionThe ProblemSolutions:

Modified Newtonian Dynamics (MOND)Dark MatterCompound TheoriesModified Gravity (!)

SummaryProposal

Γ= 𝑀𝐿 (Γ⊙ = 1)

TerminologyMass to Light Ratio: Surface Brightness:

HSB Galaxies: High-Surface Brightness GalaxiesLSB Galaxies: Low-Surface Brightness GalaxiesDynamical Mass: Derived from spectroscopy.Luminous Mass: Derived form photometry.

Σ= 𝐿𝐴 ൬Σ= 𝐿𝜋𝑟2൰

IntroductionIn the 17th century Sir Isaac Newton could

successfully give a unified description of the motion for celestial and terrestrial objects in one consistent theory. Newton’s universal law of gravitation and its three laws of motion was enough for us to go to the Moon. However precise measurements showed some significant deviations from what is predicted by the theory. (Precession of the planetary orbits (esp. Mercury), light deflection, time dilation and gravitational red shift.)

Albert Einstein published Theory of General Relativity in 1915. In contrast to Newtonian gravity, GR was able to solve the above problems. However one big problem still remains unsolved.

The Problem

• In 1937 the Swiss astronomer Zwicky measured the velocity dispersion of the Coma cluster. Astronomers got familiar with the idea that the universe is filled by dark matter.

• In 1978, flat rotation curves of galaxies posed a serious problem in astronomy.(Using Doppler effect, we can draw rotation curves of spiral galaxies. By doing so we’ll reach unexpected results. Instead of having a descending curve we have a flat curve. It indicates that gravitation must be an stronger than expected. )

Flat Rotation Curves

Observed Curve (gives dynamical mass)

Expected Curve (from M/L ratio)

Examples

Galaxy ClustersFrom Virial Theorem we know:

By applying Virial Theorem to galaxy clusters we have:

𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝐸𝑛𝑒𝑟𝑔𝑦+ 2× 𝐾𝑖𝑛𝑒𝑡𝑖𝑐 𝐸𝑛𝑒𝑟𝑔𝑦= 0

< 𝑈> +2 < 𝑇>= 0

𝛼𝐺𝑁𝑀2𝑅 = 2× 12× 𝑀× 𝑣2 → 𝑣 = ඨ𝛼𝐺𝑁𝑀𝑅

The ProblemIn both cases we conclude that the dynamical

mass is not equal to the luminous mass. There are two possible explanations:

1- A kind of unseen matter exists in the galaxy.

2-We are facing a failure of the law of gravitation.

MOND

Dark Matter

MONDMOND theory was proposed by M.Milgrom

in 1983. In this theory the known Newtonian acceleration is replaced by a modified acceleration so that in the case of gravity, it has a 1/r dependency at large distances from the center of the spiral galaxies. Thus it can explain flat rotation curves.

𝐹= 𝑚𝑎𝜇൬𝑎𝑎0൰ 𝑎0 = 1.2× 10−10 𝑚𝑠2 𝜇൬𝑎𝑎0൰𝒂 = −∇Φ𝑁

MOND

𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 = 𝑎0 𝑎 ≫𝑎0 →𝑎𝑁= 𝑎 →𝜇= 1

𝑎 ≪𝑎0 →𝑎 ∝ 1𝑟 →𝜇= 𝑎𝑎0

𝑎0 = 1.2± 0.27× 10−10 𝑚𝑠−2

MOND

What is the general form of µ? Bekenstein & Milgrom (1984)

There are also other forms:Famaey & Binney (2005), Bekenstein (2004), and Zhao et al.

(2006).

𝑎𝑁= 𝐺𝑀𝑟2 , 𝑎𝑁= 𝑎𝜇൬𝑎𝑎0൰ 𝑎<𝑎0ሳልልሰ 𝐺𝑀𝑟2 = 𝑎2𝑎0

𝑎 = 𝑣2𝑟 →𝐺𝑀𝑟2 = 𝑣4𝑟2𝑎0 → 𝑣 = ඥ𝐺𝑀𝑎04 = 𝑐𝑜𝑛𝑠𝑡𝑛𝑎𝑡

𝜇ሺ𝑥ሻ= 𝑥ξ1+ 𝑥2 → 𝑎 = 𝑎𝑁ቌ12+ 12ඨ1+൬2𝑎0𝑎𝑁൰

2ቍ

12

ExampleIf we model the Galactic disk as a simple,

infinitesimally thin Kuzmin disk (Kuzmin 1956), the corresponding Newtonian potential (Binney & Tremaine 1987) is given by:

Dark MatterDoes not interact via electromagnetic

force.Does interact via gravitational force.Can not be seen by its radiation. (It has no

radiation at all)Comes into three types:

- Cold Dark Matter (CDM): Non-relativistic- Warm Dark Matter (WDM): Relativistic - Hot Dark Matter (HDM): Ultra-Relativistic

Dark Matter or Missing Mass? Nucleosynthesis Theory constraint on baryonic

matter:

Primordial black holes (CDM)Neutrinos (HDM,CDM)MACHOs: Massive Compact Halo Objects

(CDM)Brown Dwarfs (CDM)Unseen gas and dust (CDM)Non-baryonic dark matter (WIMP)

0.016 < Ω𝐵ℎ2 < 0.024

Examples (CDM halo Models)Power-law potential

Logarithmic potential

MOND versus DMIssues for DM:• Does the stuff we call Dark Matter really exist?• Can a dark matter based theory explain the

MONDian phenomenology observed in rotation curves?

• Why should the mass discrepancy only appear at a particular acceleration scale?

Issues for MOND:• Can a modified force law explain all

observations?• Can a satisfactory theory encompassing both

General Relativity and MOND be found?

Curve Fitting (MOND +1)In contrast to DM, MOND gives very little

flexibility in curve fitting which is good!!

MOND has one free parameter which is M/L. Sometimes M/L becomes irrelevant.

DM has three free parameters which are M/L, the halo core radius and the halo asymptotic velocity.

Example (NGC6503)

Solid curve: Three-parameter dark-halo itDashed line: Stellar componentDotted line: Gas componentDash-dotted line: Dark halo component

Fake Curve Fitting

Data form UGC128 and NGC2403

Magellanic Stream

Tully-Fisher Relation (MOND +1)T-F relation (spiral galaxies) in MOND:

T-F relation in Newtonian gravity:

which is wrong! Since we have HSBs and

LSBs.

𝐿∝ 𝑣4

𝑣4 = 𝐺𝑀𝑎0 ∝ 𝑀𝐿𝐿∝ Γ𝐿

𝑣4 = 𝐺2𝑀2𝑟2 ∝ 𝑀2Σ𝐿2 𝐿∝ Γ2Σ𝐿 → Σ= 𝑐𝑜𝑛𝑠𝑡

HSBs and LSBs

(left) T-F relation (right) F-J relation

Bullet Cluster (DM +1)

Bullet Cluster (DM +1)

Gravitational Lensing (DM +1)MOND with some approximation can explain

gravitational lensing. Modified MOND is not acceptable!

Compound Theories and Modified GravityThere are some situations that one needs

to assume that both dark matter and MOND theory are true. In this case we can ask what is the benefit of MOND at all when we still need some dark matter?

Is Dark Matter another epi-cycle?How about dark energy?Instead of MOND, one may use other forms

of equations for gravity. Asymmetric gravity for example.

SummaryObservational

TestDark Matter MOND

LSBG Tully-Fisher Relation

?? +

M/L-Surface Brightness Relation

X +

Stellar Mass to Light Ratios

NP +

Mass Surface Densities

X +

Local M/L NP +

Galaxy Clusters ? ??

Gravitational Lensing

? ??

CMB Geometry + NT

+ = prediction confirmedX = prediction falsified*? = uncertain but promising?? = uncertain but not promisingNP = no predictionNT = no test

Proposal

One big question of mine:How Cosmic Rays can be used as a tool to

determine whether MOND or Dark Matter is true?

ReferencesHoekstra, H., Franx, M., Kuijken, K., 1999,

arXiv:astro-ph/9911106v1Brownstein, J. R., arXiv:0908.0040v1Bekenstein, J. D. 2010, arXiv:1001.3876v1Haghi, H., Rahvar, S., Hassani-Zanooz, A. 2006,

ApJ, 652, 354Liddle, Andrew, An Introduction to Modern

Cosmology, WILEY, 2003Haghi, H., Hassani-Zanooz, A., Rahvar, S.,

Magellanic stream: a possible tool for studying dark halo model

www.wikipedia.org

Any questions?

Thank you for your time.