Modiﬁcations of Gravity vs. Dark Matter/Energy

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Modifications of Gravity vs. Dark Matter/Energy

Daniel Grumiller

Massachusetts Institute of Technology Marie-Curie Fellowship MC-OIF 021421

Finnish-Japanese Workshop on Particle Cosmology Helsinki, March 2007

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Outline

1 Gravity

Problems with General Relativity

2 Galactic rotation curves Statement of the problem Dark matter?

Alternatives

3 General relativistic description Motivation

Axially symmetric stationary solutions Toy model for a galaxy

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Outline

1 Gravity

Alternatives

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Theoretical problems

“Standard folklore”

GR = beautiful and experimentally established only “a few details” are missing

No comprehensive theory of quantum gravity Several experimental “anomalies”

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Theoretical problems

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Theoretical problems

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Experimental anomalies

Dark Energy Dark Matter Pioneer anomaly other anomalies?

Note intriguing coincidences:

a_Λ

|{z}

Dark Energy

≈ a_MOND

| {z }

Dark Matter

≈ a_P

|{z}

Pioneer anomaly

≈ a_H

|{z}

Hubble

≈10⁻⁹−10⁻¹⁰m/s²

in natural units:a≈10⁻⁶¹−10⁻⁶²

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Experimental anomalies

a_Λ

|{z}

Dark Energy

≈ a_MOND

| {z }

Dark Matter

≈ a_P

|{z}

Pioneer anomaly

≈ a_H

|{z}

Hubble

≈10⁻⁹−10⁻¹⁰m/s²

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Experimental anomalies

a_Λ

|{z}

Dark Energy

≈ a_MOND

| {z }

Dark Matter

≈ a_P

|{z}

Pioneer anomaly

≈ a_H

|{z}

Hubble

≈10⁻⁹−10⁻¹⁰m/s²

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Experimental anomalies

a_Λ

|{z}

Dark Energy

≈ a_MOND

| {z }

Dark Matter

≈ a_P

|{z}

Pioneer anomaly

≈ a_H

|{z}

Hubble

≈10⁻⁹−10⁻¹⁰m/s²

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Experimental anomalies

a_Λ

|{z}

Dark Energy

≈ a_MOND

| {z }

Dark Matter

≈ a_P

|{z}

Pioneer anomaly

≈ a_H

|{z}

Hubble

≈10⁻⁹−10⁻¹⁰m/s²

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Outline

1 Gravity

Alternatives

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Some experimental data of galactic rotation curves

Rotation curve of our Galaxy

Inconsistent with Newton!

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Sketch of experimental data

A typical non-Keplerian galactic rotation curve

c Wikipedia

Curve A: Newtonian prediction Curve B: Observed velocity profile

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Newtonian calculation

v²

|{z}

kinetic

∝ R ρdV

r

| {z }

potential

∝ M r

Regime withρ=const.: v ∝r Regime with M =const.: v ∝1/√

r

Conclusion

Flat rotation curves not described well by Newton

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Outline

1 Gravity

Alternatives

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Saving Newton

Postulate existence of Dark Matter

Fit Dark Matter density as to “explain” rotation curves Note: other hints for Dark Matter, e.g. gravitational lensing!

Exciting prospect for near future

Dark Matter might be discovered next year at LHC!

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Outline

1 Gravity

Alternatives

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Example: MOND

Modified Newton Dynamics

~F =m~a·µ(a/a_MOND) with

µ(x)→1 for x 1 µ(x)→x for x 1

critical acceleration: a_MOND≈10⁻¹⁰m/s²

Phenomenologically rather successful, but no deeper theoretical understanding

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Main issues

Two important questions:

Do we have the correct theory of gravity?

Are we applying it correctly?

Possible answers:

Regarding 1: e.g. MOND, IR modifications of GR, Yukawa-type corrections, ...

Regarding 2: Newtonian limit justified?

Consider the second point

Study appropriate General Relativistic (exact) solutions

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Main issues

Possible answers:

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Main issues

Possible answers:

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Main issues

Possible answers:

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Outline

1 Gravity

Alternatives

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Validity of Newtonian approximation

Locally: can use Newton almost everywhere in galaxy (except for central galactic Black Hole region and near jet axis)

But: GR is a non-linear theory!

Not granted that Newton is applicable globally Thus, use GR instead of Newton!

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The Cooperstock-Tieu attempt

Ansatz:

Axial symmetry Reflection symmetry Stationarity

Pressurless perfect fluid sources Corotating perfect fluid

Weak field limit

Claim of Cooperstock-Tieu:

GR differs essentially from Newton However, technically incorrect!

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The Cooperstock-Tieu attempt

Ansatz:

Weak field limit

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The Cooperstock-Tieu attempt

Ansatz:

Weak field limit

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The Cooperstock-Tieu attempt

Ansatz:

Weak field limit

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The Cooperstock-Tieu attempt

Ansatz:

Weak field limit

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The Cooperstock-Tieu attempt

Ansatz:

Weak field limit

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The Cooperstock-Tieu attempt

Ansatz:

Weak field limit

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Outline

1 Gravity

Alternatives

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Assumptions

H. Balasin and D. Grumiller,astro-ph/0602519

Drop weak field limit, but keep rest:

Axial symmetry→spacelike Killing

Reflection symmetry (around galactic plane) Stationarity→timelike Killingξ^a= (∂t)^a

Pressurless perfect fluid sources→Tâb =ρuâu^b Corotating perfect fluid→uâ= (u⁰(r,z),0,0,0)

Consequently, line element can be brought into adapted form:

ds²=−(dt−Ndφ)²+r²dφ²+exp(ν)(dr²+dz²) functionsρ,N, ν depend solely on r,z

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Einstein equations

rν_z+N_rN_z =0, (1)

2rν_r +N_r²−N_z²=0, (2) ν_rr +ν_zz+ 1

2r²

N_r²+N_z²

=0, (3)

N_rr− 1

rN_r +N_zz =0, (4)

1 r²

N_r²+N_z²

=κρe^ν. (5)

Important notes:

(4) is a linear PDE for N!

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Einstein equations

rν_z+N_rN_z =0, (1)

2rν_r +N_r²−N_z²=0, (2) ν_rr +ν_zz+ 1

2r²

N_r²+N_z²

=0, (3)

N_rr− 1

rN_r +N_zz =0, (4)

1 r²

N_r²+N_z²

=κρe^ν. (5)

Important notes:

For known N (1)-(3) yieldν (integration constant!) For known N, ν(5) establishesρ

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Einstein equations

rν_z+N_rN_z =0, (1)

2rν_r +N_r²−N_z²=0, (2) ν_rr +ν_zz+ 1

2r²

N_r²+N_z²

=0, (3)

N_rr− 1

rN_r +N_zz =0, (4)

1 r²

N_r²+N_z²

=κρe^ν. (5)

Important notes:

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Einstein equations

rν_z+N_rN_z =0, (1)

2rν_r +N_r²−N_z²=0, (2) ν_rr +ν_zz+ 1

2r²

N_r²+N_z²

=0, (3)

N_rr− 1

rN_r +N_zz =0, (4)

1 r²

N_r²+N_z²

=κρe^ν. (5)

Important notes:

For known N (1)-(3) yieldν (integration constant!) For known N, ν(5) establishesρ

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Inverse problem

Predicting mass density from velocity profile

Physical meaning of N:

V(r,z) = N(r,z) r

V is 3-velocity as seen by asymptotic observer at rest with respect to center of galaxy

“Inverse problem”:

Take V as experimental input (rotation curve) Calculate mass densityρ

Compareρwith experimental data

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Inverse problem

Predicting mass density from velocity profile

Physical meaning of N:

V(r,z) = N(r,z) r

V is 3-velocity as seen by asymptotic observer at rest with respect to center of galaxy

“Inverse problem”:

Take V as experimental input (rotation curve) Calculate mass densityρ

Compareρwith experimental data Compare with Newtonian prediction forρ

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Boundary conditions and asymptotics

Solution mathematically trivial (separation Ansatz) Physical input:

Modes bounded for|z| → ∞ Modes bounded for r → ∞

No exoticδ-sources in galactic plane!

General solution for r >0:

N(r,z) =r²

∞

Z

0

dxC(x)X

±

((z±x)²+r²)^−3/2

Remaining task:

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Boundary conditions and asymptotics

Solution mathematically trivial (separation Ansatz) Physical input:

Modes bounded for|z| → ∞ Modes bounded for r → ∞

No exoticδ-sources in galactic plane!

General solution for r >0:

N(r,z) =r²

∞

Z

0

dxC(x)X

±

((z±x)²+r²)^−3/2

Remaining task:

Choose spectral density C(x)appropriately!

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Outline

1 Gravity

Alternatives

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Choice of spectral density

Take simple 3-parameter family of C(x):

Parameter V₀: flat region velocity (about 200km/s) Parameter r₀: bulge radius (about 1kpc)

Parameter R: choose about 100kpc With these choices V(r,0)looks as follows:

5 10 15 20

0.2 0.4 0.6 0.8

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Asymptotic non-flatness

The necessity of jets

Crucial observation:

Solutions (necessarily) singular close to axis r =0 for|z| ≥r₀

0 200 400 600 800 1000

-1000 -500 0 500 1000

r z

0 2 4 6 8 10

-4 -2 0 2 4

r z

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Prediction for mass density

Mass density

ρ= β

|{z}

int.const.

N_r²+N_z² r²

on large scales:

0 200

400 600

800 1000-1000

-500 0

500 1000 0

2.5·10^-9 5·10^-9 7.5·10^-9 1·10^-8

0 200

400 600

800 1000

Integratingρover “white” region yields≈10¹¹ solar masses

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Comparison with Newton

For any given velocity profile V Newton predicts:

ρNewton∝ V²+2rVV⁰ r² Ratio of GR/Newton:

ρ

ρ_Newton = ˜β

1+ r²(V⁰)² V²+2rVV⁰

Assume a spectral density which yields a linear regime V ∝r and a flat regime V =const.

Important remaining task:

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Comparison with Newton

For any given velocity profile V Newton predicts:

ρNewton∝ V²+2rVV⁰ r² Ratio of GR/Newton:

ρ

ρ_Newton = ˜β

1+ r²(V⁰)² V²+2rVV⁰

Assume a spectral density which yields a linear regime V ∝r and a flat regime V =const.

Important remaining task:

Have to fix free constantβ!˜

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Difference to Newton

Fixβ˜such that

Newton and GR coincide in the linear regime V ∝r

This yieldsβ˜=3/4.

Calculate the ratio in the flat regime V =const.

ρ ρNewton

flat

= ˜β= 3 4

Conclusion

Newton predicts 133% of the mass density as compared to GR

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Difference to Newton

Fixβ˜such that

ρ ρNewton

flat

= ˜β= 3 4 Conclusion

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Difference to Newton

Fixβ˜such that

ρ ρNewton

flat

= ˜β= 3 4 Conclusion

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Summary

Problem of flat galactic rotation curves May be solved by Dark Matter

GR attempts to avoid Dark Matter have failed GR predicts slightly less Dark Matter than Newton

Outlook:

Improve the toy model Perform patching

Understand better role of jets and central galactic black holes for global galactic dynamics

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Summary

Outlook:

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Summary

Outlook:

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Summary

Outlook:

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Summary

Outlook:

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Literature I

C. Lammerzahl, O. Preuss and H. Dittus, “Is the physics within the Solar system really understood?,”

gr-qc/0604052.

F. I. Cooperstock and S. Tieu,astro-ph/0507619.

M. Korzynski,astro-ph/0508377; D. Vogt and P. S. Letelier,astro-ph/0510750.

H. Balasin and D. Grumiller,astro-ph/0602519.