Modified Newtonian Dynamics (MOND) and the Bullet Cluster (1E

Modiﬁed Newtonian Dynamics (MOND) and the Bullet Cluster (1E 0657-558)

Alan G. Aversa

ABSTRACT

Modified Newtonian Dynamics (MOND) is a theory that modifies Newton’s force law to explain observations that most astronomers interpret as evidence for dark matter. Observations of 1E 0657-558, two colliding galaxy clusters known colloquially as the “Bullet Cluster,” offer the best evidence against MOND and MOND-like theories and some of the best evidence for dark matter. We present herein an overview of the theory of MOND and its success at explaining certain observations as well as the Bullet Cluster and its case against MOND.

Contents

1 Modiﬁed Newtonian Dynamics (MOND) 3

1.1 Observations ...... 3

1.1.1 Galaxy Rotation Curves ...... 3

1.1.2 Tully-Fisher Law ...... 5

1.2 Predictions ...... 7

1.3 Theoretical Formulation ...... 10

2 Bullet Cluster (1E 0657-558) 11

2.1 Observations ...... 11

2.2 Case Against MOND ...... 11

3 Conclusions 12

1. Modiﬁed Newtonian Dynamics (MOND)

Milgrom (1983) was the first to propose a modification of Newtonian dynamics (MOND) in order to explain observations that the scientific community says indicate the presence of a weakly- interacting, massive, non-luminous matter (dark matter). In his paper, he formulated MOND and – 2 – described how it would explain various observational laws. Firstly, we overview its explanation of both the Tully-Fisher law—that a galaxy’s mass M and its its rotational velocity V are related by M ∝ V α, where α ≈ 4—and of the rotation curves of galaxies, that they are asymptotically flat at large radii from their respective centers. Secondly, we show one of MOND’s predictions. Lastly, we introduce a theoretical formulation of MOND.

1.1. Observations

1.1.1. Galaxy Rotation Curves

That galaxies’ rotation curves are flat at large radii was the first piece of observational evidence pointing to the possibility that there must be more mass in a galaxy than its luminous matter shows. √ Rotation curves such as those in Figure 1 show that a Keplerian law, that velocity V ∝ 1/ r, is not possible. Figure 1 shows that high and low surface brightness galaxies have very different rotation curves, something that a modified Newtonian law will have to explain.

One way to modify Newtonian dynamics would be to modify Newton’s gravitational force law such that it matches these rotation curves. To lessen the dependence on distance of the gravitational force for large radii is what Milgrom (1983) did in his original paper:

GM F = f(r/r ), (1) gravity r2 0 where f(x) = 1, x 1, f(x) = x, x 1. Instead of Newton’s second law as F~ = m~a, Milgrom (1983) modiﬁed F~ with the function µ(a), where a is acceleration. Then Newton’s second law in MOND reads:

F~ = m~aµ(a/a0), (2) where µ(x) = 1, x 1, µ(x) = x, x 1. Thus, according to Sanders & McGaugh (2002), one can relate the true gravitational acceleration

~g to the Newtonian gravitation acceleration ~gn by

~gn = ~gµ(|g|/a0). (3)

These aforementioned results will help us understand how MOND can explain the Tully-Fisher Law. – 3 –

Fig. 1.— Galaxy rotation curves of NGC 1560, a high surface brightness galaxy, and NGC 2903, a low surface brightness galaxy, from Sanders & McGaugh (2002). The dotted lines are the Newtonian rotation curves due to the visible components of the galaxy, and the dashed lines are the Newtonian rotation curves due to the neutral hydrogen gaseous component, as measured in the radio at 21 cm. MOND’s predictions (solid lines) fall very nearly on the observed points. – 4 –

1.1.2. Tully-Fisher Law

Figure 2 shows the Tully-Fisher (T-F) relation between Vrot and K-band luminosity. K-band luminosity is the optimal tracer of a galaxy’s underlying stellar masses. Qualitatively, the T-F law makes sense: the more mass you have the more gravity the galaxy has to keep itself together when rotating at faster velocities.

√ 2 From Equation 3 we can see that in the low acceleration regime g = gna0. Setting g = V /r yields 4 V = GMa0, (4) or that M ∝ V 4, the T-F law. Sanders & McGaugh (2002) then convert Equation 4 into the luminosity-velocity relation

log (L) = 4 log (V ) − log (Ga0hM/Li), (5) where hM/Li is the average mass-to-light ratio of the galaxy. From observations of the T-F relation, such as those of the Ursa Major galaxies in Figure 2, and from Equation 5, one can determine the acceleration scale a0 that diﬀerentiates MONDian dynamics from Newtonian dynamics. This −8 −2 acceleration scale a0 is 10 cm s . Although it may seem like numerology, Milgrom realized that a0 ≈ cH0 within a factor of 5-6, thus he speculates that MOND may have applications to cosmology.

1.2. Predictions

Using only the M/L ratio as a free fitting parameter, MOND fits a variety of galaxies’ rotation curves, as shown in Figure 3. Only a small fraction of the galaxies in Sanders & Verheijen (1998)’s Ursa Major sample did not fit a MOND rotation curve.

Turning the problem around, one can assume an acceleration scale a0 and fit a M/L ratio for the Sanders & Verheijen (1998) Ursa Major spirals. Doing so yields Figure 6, which shows M/L ratios versus B − V colors. The B-band M/L ratio versus color plot shows that M/L ratios are higher for redder colors, whereas the K-band M/L ratio versus color does not vary significantly with color. That the B-band M/L ratios should vary with color has been confirmed previously by Bell & de Jong (2001), who used stellar population synthesis models. Therefore, MOND has made an independent prediction.

Although we have only discussed spiral galaxies’ rotation curves, MOND has been applied to structures on many diﬀerent scales, from dwarf spheroidal galaxies to galaxy superclusters (Sanders & McGaugh 2002). – 5 –

Fig. 2.— The Tully-Fisher relation between rotational velocity Vrot and K-band (IR) luminosity, a proxy for mass M, of Ursa Major spiral galaxies (Sanders & McGaugh 2002). – 6 –

Fig. 3.— With only the M/L ratio as a free parameter, MOND can ﬁt very well a variety of galactic rotation curves, such as those of these Ursa Major spiral galaxies at 15.5 Mpc (Sanders & Verheijen 1998). Asterisks represent kinematically disturbed galaxies, and the solid, dashed, and dotted lines are the same as in Figure 1. – 7 –

Fig. 4.— Predicted mass-to-light ratios versus B − V colors of the Ursa Major spiral galaxies of Sanders & Verheijen (1998). Solid lines are results from the Bell & de Jong (2001) stellar population synthesis models. – 8 –

1.3. Theoretical Formulation

The MONDian force (Equation 2) does not obey linear momentum conservation, which is a big theoretical obstacle. It is not MOND’s duty to try to invalidate this law of physics, so theoreticians have been trying to place MOND on solid theoretical grounds. Bekenstein & Milgrom (1984) developed the Bekenstein-Milgrom Theory, a non-relativistic Langrangian theory with the action being Z " 2 # 3 −1 2 ∇ Sf = − d r ρφ + (8πG) a0F 2 , (6) a0 where φ is the scalar potential. The condition for Sf to be stationary is " # |∇φ| ∇ · µ 2 ∇φ = 4πGρ, (7) a0 from which they derive a Poisson equation of the form

n 2D/2−1 o ∇ · (∇φ) ∇φ = αDGρ, (8) where D = 2 corresponds to the normal Newtonian Poisson equation, D = 3 corresponds to the

MOND phenomenology, and αD is a constant related to D. According to Sanders & McGaugh (2002), the Poisson equation is conformally invariant, just like many laws of physics, including Maxwell’s laws of electromagnetism. Thus getting MOND into a Poisson equation like Equation 8 is a start in formulating a theoretical framework for MOND.

Recently, Bekenstein (2004) formulated the Tensor-Vector-Scolar (TeVeS) theory of gravity, in which they eﬀectively invent a theory of general relativity that obeys MOND’s requirements. This is useful when comparing MOND cosmology to that of the Cold Dark Matter (CDM) models.

2. Bullet Cluster (1E 0657-558)

One of the best observational arguments for dark matter and against MOND are the observations of the Bullet Cluster (1E 0657-558). The Bullet Cluster has two components that have passed through each other, leaving hot x-ray gas behind due to ram pressure. By analyzing where the x-ray gas is and what weak gravitational lensing tells about the cluster’s mass distribution, one can rule out MONDian gravity in favor of dark matter.

2.1. Observations

Clowe et al. (2004) observed the interacting galaxy clusters 1E 0657-558 (Bullet Cluster) with the VLT and Chandra. They used I-band VLT images to construct a surface mass density map based upon the cluster’s weak gravitational lensing of background sources. They measured how, – 9 – for example, the ellipticities of the lensed galaxies were distributed and ran an iterative code called the KS93 algorithm to ﬁnd a convergent solution to the cluster’s surface mass density. Chandra X-ray maps allow tracing of hot gas masses. Figure 5 shows these maps with surface mass density contours overlaid.

2.2. Case Against MOND

Looking at Figure 5, one can see that the highest surface mass density is not coincident with the peak x-ray intensity. In a CDM universe, one would expect the x-ray gas, only a small fraction of the dark matter mass, to be spatially offset compared to the peak surface mass density as determined by the weak gravitational lensing. In a MONDian universe, the x-ray gas would be the dominant source of mass in the galaxy cluster, thus one would expect the highest surface mass density to be coincident with the location of peak x-ray emission. This is not the case according to Clowe et al. (2004)’s observations. They admit, however, that truly to rule out MOND, one would need a general-relativistic derivation of MOND, such as TeVeS, with which to compute the weak gravitational lensing. Since this did not exist when Clowe et al. (2004) did their analysis, they could not predict how a photon would travel in a MOND universe, so they assumed standard general relativity to compute how mass and a static gravitational field affects a photon’s propagation.

3. Conclusions

There are two prominent observational facts that a modified theory of Newtonian dynamics needs to explain without invoking dark matter: the Tully-Fisher law and galaxy rotation curves. MOND successfully fulfills these requirements. Besides this, though, MOND would have to explain the Bullet Cluster and cosmology as successfully as the standard ΛCDM theory in order for it to become a more scientifically accepted theory of gravity. Dark matter, although no one has yet detected it unequivocally in the laboratory, still offers a simpler explanation which holds true over many orders of magnitude. Thus the scientific community has chosen to follow Occam’s razor.

REFERENCES

Bekenstein, J., & Milgrom, M. 1984, ApJ, 286, 7

Bekenstein, J. D., & Sanders, R. H. 2006, EAS Publications Series, 20, 225

Bekenstein, J. D. 2004, Phys. Rev. D, 70, 083509

Bell, E. F., & de Jong, R. S. 2001, ApJ, 550, 212

Clowe, D., Gonzalez, A., & Markevitch, M. 2004, ApJ, 604, 596 – 10 –

Milgrom, M. 1983, ApJ, 270, 365

Pointecouteau, E., & Silk, J. 2005, MNRAS, 364, 654

Sahni, V. 2005, The Physics of the Early Universe, 653, 141

Sanders, R. H., & McGaugh, S. S. 2002, ARA&A, 40, 263

Sanders, R. H., & Verheijen, M. A. W. 1998, ApJ, 503, 97

This preprint was prepared with the AAS LATEX macros v5.2. – 11 –

Fig. 5.— (left) VLT I-band image with surface mass density contours overlaid. (right) Chandra x-ray image.