# Modified Newtonian Dynamics (MOND) as a Solution to the Cosmic Mass Discrepancy Problem

Steven Paul Keyte[1], Faculty of Science, Monash University

## Abstract

Keywords: Cosmology, dark matter, galaxy dynamics, gravity, gravitational lensing.

## Introduction

The cosmic mass discrepancy problem is one of the most confounding and long-standing problems in astronomy and cosmology. Assuming that our understanding of gravity is complete, the luminous mass in the Universe does not explain the dynamics of galactic systems or the structure of the Universe itself (Famaey and McGaugh, 2013). The earliest evidence for this problem was inferred by measuring the rotational velocities of material in galaxies. According to our understanding of gravity, galaxies are measured to rotate at rates that should cause them to fly apart (Rubin et al., 1982). This observation, combined with the cosmic microwave background, large-scale structure of the Universe and the dynamics and gravitational lensing of galaxy clusters, suggests that either the current theory of gravity is incomplete or an unseen form of mass pervades the Universe on all scales (Dodelson, 2011).

The standard solution for this problem is to infer a significant amount of non-luminous cold dark matter (CDM). This hypothetical matter is termed 'dark' because it apparently does not interact with electromagnetic radiation at any wavelength. While the prevailing cosmological model ΛCDM (where Λ, pronounced lambda, is a cosmological constant associated with dark energy) has achieved significant empirical success, an inability to detect a CDM particle, and a growing number of observational anomalies are becoming problematic (Akerib, 2009; Famaey and McGaugh, 2013). Additionally, the Standard Model of particle physics must be extended in order to accommodate the existence of a particle with the necessary properties of CDM (Easther et al., 2014; Sanders and McGaugh, 2002).

Rather than invoke an unobserved source of mass, Modified Newtonian Dynamics (MOND) proposes an adjustment of Newton's law of gravitation as an alternate solution to the mass discrepancy problem (Milgrom, 1983). While MOND has gained some support, due primarily to its superior ability to predict galaxy rotation curves, it remains an incomplete theory, not yet unified with General Relativity and, as such, cannot currently match the success of ΛCDM in large-scale systems or in cosmology (Dodelson, 2011). At present, MOND is generally viewed as somewhat of a fringe theory, with ΛCDM being preferred by the vast majority of astronomers and cosmologists.

Discovering the underlying physics of the mass discrepancy problem is of central importance to elementary particle physics, observational astronomy, theoretical astrophysics and cosmology. This article discusses the physical basis for inferring CDM and assesses the challenges currently faced by ΛCDM. We review the evidence for MOND as an alternative to CDM, including the relative success of current theories that attempt to reconcile MOND with General Relativity and cosmology. In summary, we evaluate whether an alternative to ΛCDM is even necessary and, hence, whether or not efforts to unify MOND with cosmology are warranted.

## Background

##### Newtonian dynamics and General Relativity

In the seventeenth century Isaac Newton laid the foundations of classical physics. One of Newton's most important findings was his Law of Universal Gravitation, which, stated in its simplest form, is expressed in Equation 1:

$F=\frac{GMm}{r^2}\qquad(1)$

where F is the attractive force between two bodies, G is the universal gravitational constant, M and m are the masses of the respective bodies and r is the distance between those bodies. Using Newton's second law of motion F=ma and the fact that, in circular motion, a=v2/r, the velocity v of a body orbiting a centre of mass can be derived:

$m\frac{v^2}{r}=\frac{GMm}{r^2}$

therefore:

$v=\bigg(\frac{GM}{r}\bigg)^{\frac{1}{2}}\qquad(2)$

where M is now the total mass contained within the body's orbit. A body orbiting in such a manner is said to exhibit Keplerian motion.

In 1915 Albert Einstein showed that Newton's law of gravity was actually a specific case of a more general theory, General Relativity (GR) (Einstein, 1920). While the complexities of GR are beyond the scope of this article, it is sufficient to understand that, rather than being the result of attractive forces between masses, as in Newton's model, gravity is instead a result of the masses causing spacetime to warp (Einstein, 1920). The long-lasting success of Newton's classical physics can be attributed to the fact that its predictions coincide very closely with GR for all commonly experienced phenomena. It only becomes necessary to invoke GR for objects moving at 'relativistic' speeds (speeds approaching the speed of light) and for very high mass phenomena (Einstein, 1920). Newtonian dynamics is said to be the weak field limit of GR.

##### The mass discrepancy problem

The mass discrepancy problem manifests itself through multiple independent phenomena (Famaey and McGaugh, 2012). The first evidence was provided by an analysis of galaxy rotation curves, which plot the rotational velocity of visible matter as a function of galactic radius (Rubin et al. 1982). The flat rotation curve of NGC3198, depicted in Figure 1, is representative of all spiral galaxies for which rotational velocities have been accurately measured (Aryal et al., 2013; Chadwick et al., 2013; de Blok et al., 2001; Ho et al. 2010). In order to have a flat rotation curve while still satisfying Equation 2, a galaxy's mass must increase in proportion to its radius, making the right hand side of the equation approximately constant. This necessitates some form of dark matter, since the mass of luminous material drops off rapidly with increasing radius (Rubin et al. 1982).

Figure 1: Measured rotational velocity (in km s-1) of visible matter, as a function of distance from the galactic centre, for the galaxy NGC3198. If the visible matter is all that is present, then by Equation 2 we expect the velocity of material in a galaxy to drop off as 1/r1/2 once the bulk of the galaxy's mass is contained (Rubin, 1982). We see here that the velocity remains approximately constant as radius increases. (Source: Begeman, 1989)

Mass discrepancies exist on larger scales also, with our local group of galaxies providing a good case in point. The standard cosmological model predicts that the material that was to become our local group of galaxies was initially expanding at the rate predicted by Hubble's Law (Bell et al., 2003: 117). At present, the Milky Way and its nearest spiral neighbour, M31, are approaching each other at around 100 km/s. To have overcome the initial expansion and arrived at their present separation and relative velocities, approximately 80 times more mass is required than is present in these galaxies' stars alone (Watkins et al., 2010: 275). A vast quantity of non-luminous mass is therefore required to explain the local galaxy cluster dynamics (Bell et al., 2003: 118–19, Watkins et al., 2010: 276). Gravitational lensing (i.e. the relativistic effect of massive objects bending light) by galaxy clusters also implies more mass than can be accounted for by the visible matter observed (Clowe et al., 2006; Feix et al., 2008: 313).

Even on the Universe's largest scales, cosmology provides strong motivation for CDM. The thermal radiation left over from the Big Bang, known as the cosmic microwave background (CMB), is observed to be very homogenous (Figure 2) (Komatsu et al., 2011). This implies that the Universe itself was initially homogenous, however today the Universe is inhomogenous. With gravity being the only attractive long-range force able to initiate structure formation, the growth rate of large-scale structure in the early Universe cannot be reconciled with the inferred mass density of baryons (i.e. atoms) in the Universe, unless we assume some other form of mass is present (Llinares et al., 2008: 1778–79) (Figure 3). Additionally, Big Bang Nucleosynthesis, which refers to the production of light nuclei in the early Universe, places a constraint on the number of baryons in the Universe. This not only implies that dark matter exists, but that it must be both non-baryonic and cold. 'Non-baryonic' meaning that it is something other than atoms (or constituents of atoms) and 'cold' meaning that it moves slowly enough to condense and form structure (Komatsu et al., 2011; Llinares et al., 2008: 1778).

Figure 2: Wilkinson Microwave Anisotropy Probe all sky image of the CMB, where colour indicates temperature variation. The CMB was released when atoms first formed and free electrons disappeared. At this point, matter and radiation decoupled, and the Universe became transparent (Hu and Dodelson, 2002). Immediately after the Big Bang, quantum fluctuations caused infinitesimal differences in gravity, initiating density fluctuations that led to temperature anisotropies in the CMB (Hu and Dodelson, 2002). The CMB has a nearly homogeneous temperature of 2.7 K; the largest fluctuations being ~10-5K. In this figure red regions are ~10-5K above the mean CMB temperature and blue regions are ~10-5K below the mean. Though tiny, these anisotropies allow us to infer the composition of the Universe (Source: Bennett et al., 2013). See also Figure 3 for further discussion of the CMB.

Figure 3: The Acoustic Power Spectrum of the CMB, caused by sound waves propagating in the early Universe, shows the angular size of anisotropies in the CMB. The largest fluctuations in the CMB indicate how far a particle could have travelled between the instant of the Big Bang and the instant of decoupling (Komatsu et al., 2011). The angular size of features in the CMB depends on the composition of the Universe and time that has elapsed since decoupling. The time since decoupling is known by measurements of the expansion of the Universe, therefore the composition of the Universe can be inferred. The location of first peak tells us the total mass-energy density of the Universe and the pattern of subsequent peaks, which is caused by the decoupling of baryons and photons, tells us the amount of baryonic matter in the Universe. The ratio of baryonic matter to total matter can be determined from these peaks (Komatsu et al., 2011; Llinares et al., 2008). (Source: Hinshaw et al., 2013)

In relation to the problems discussed above, the ΛCDM paradigm has proved to be an extremely successful one. This is particularly true in large-scale systems and cosmology, where it has been found repeatedly to correspond to observation very favourably (Bennett et al., 2013; Clowe et al., 2006: L109).

## Challenges for dark matter

##### Where are the dark matter particles?

ΛCDM requires the mass density of dark matter to be approximately five times that of baryonic matter, with the preferred candidate particles being Weakly Interacting Massive Particles (WIMPs) (Akerib, 2009: 34c). However, an inability to detect WIMPs is confounding, if indeed they exist in the required quantity (Akerib, 2009). This is not only because of the proposed abundance of WIMPs in the Universe; more significant is the fact that direct searches by particle physicists, which look for particles exhibiting particular parameters (e.g. mass, energy), have now excluded most of the range of parameters that it is thought WIMPs could have (Famaey and McGaugh, 2012: 12; Aprile et al., 2011; 5). Furthermore, particle colliders, such as the Large Hadron Collider (LHC) at CERN, should have produced evidence for WIMPs through mass or momentum losses during collisions, however no such evidence has been identified (Aaltonen et al., 2012a; Aaltonen et al., 2012b).

While the ability of dark matter particles to escape detection remains unsatisfying to some, it is pertinent to note that, by definition, dark matter must be composed of particles that do not interact except through gravitation. Therefore the elusive nature of dark matter particles does not preclude CDM's existence. Kobach (2013: 5) states that, depending on their mass, kinematic information alone may be insufficient to unambiguously identify the presence dark matter particles at the LHC.

##### Galactic dynamics

In galactic dynamics, the problem of galactic disk stability emerges in the CDM paradigm. If galaxies are surrounded by a dark matter halo, dynamical friction between baryonic and dark matter should cause angular momentum to be transferred to the CDM halo, rendering galaxies unstable (Brada and Milgrom, 1999: 590; Tiret and Combes, 2008: 719). Additionally, cosmic N-body computer simulations, which have gained predictive power in recent years, tend to restrict the ways in which CDM can be arranged within a galaxy. This means that ΛCDM predictions, which are based off these simulations, are often difficult to reconcile with observed galactic dynamics (Famaey and McGaugh, 2012: 18–23). Subtly changing some parameters of ΛCDM can sometimes improve the situation somewhat, so an unobserved prediction is not necessarily a failure of the CDM paradigm, however there are numerous observations that are surprising irrespective of the specifics of a particular CDM model (Famaey and McGaugh, 2012: 18-23). These observations can be generalised as empirical relations between the dynamics of galaxies and their baryonic distribution. There is no reason for such relations to exist in systems that are dominated by CDM, whereas they are fully consistent with a modified force law, such as MOND (Famaey and McGaugh, 2012: 18–34). A concise statement of the observed relationship between baryons and galactic dynamics, known as Renzo's rule, states that for any feature in the luminosity (i.e. brightness) profile of a galaxy, there is a corresponding feature in the rotation curve (Famaey and McGaugh, 2012: 31–34; Sancisi, 2004: 237–39).

##### Baryonic Tully-Fisher Relation

The Baryonic Tully-Fisher Relation (BTFR) is an observed relation between the baryonic mass Mb in a spiral galaxy and the rotational velocity Vf in the outer region of the galaxy (McGaugh, 2011; 1). This relation is well described by a power law whereby a log-log plot of Mb vs. Vf is linear, with almost zero scatter (Begum et al., 2008: 138; McGaugh, 2011; 1; Trachternach et al., 2009: 585).

Some debate exists over whether or not the BTFR can fit within ΛCDM. Recent work by Foreman and Scott (2012) suggest that the BTFR is an untrustworthy test of CDM, while Dutton (2012: 3127–28) claims that the basic features of the BTFR are consistent with a ΛCDM-based model. However, some argue that the BTFR is not inherent in any conceivable variation of the CDM paradigm and that the only way to achieve such a relation in ΛCDM is by invoking some elaborate fine-tuning between the detected baryonic mass and the total baryonic mass within a galaxy (McGaugh, 2011: 2–3; Trachternach et al., 2009). The problem with this is that the fraction of baryonic mass detected could conceivably lie anywhere between zero and one, yet the BTFR has negligible scatter (McGaugh, 2011: 3). That every galaxy conceals the precise fraction of baryonic matter to match its observed rotation and ensure the BTFR holds seems like a remarkable coincidence (Famaey and McGaugh, 2012: 20–21).

Nonetheless, the BTFR has been confirmed for all rotating galaxies, irrespective of mass, luminosity and star-to-gas ratio. The problem for ΛCDM is particularly significant in the case of tidal dwarf galaxies, which are formed in collisions of larger galaxies. Such collisions should separate baryonic and dark matter very effectively. In fact it is generally agreed that tidal dwarfs do not contain CDM, however they still obey the BTFR (Gentile et al., 2007: L28).

##### The a0 scale

The BTFR is one of many tight relations that are observed in galactic dynamics. The most curious phenomenon to come out of these relations, and the most confounding in the context of a CDM-dominated Universe, is the appearance of a characteristic acceleration scale. In the BTFR, the relation between Mb and Vf suggests a force law that is consistent with an acceleration scale a0, which acts to modify Newton's law of gravitation (McGaugh, 2011: 3). The Faber-Jackson relation, which is an empirical observation relating the luminosity of an elliptical galaxy to its velocity dispersion, implies this same modified force law by a factor of a0 (McGaugh and Milgrom, 2013). This scaling relation is counterintuitive in ΛCDM, but it appears in so many apparently unrelated galactic phenomena that it can either be viewed as a dramatic coincidence or evidence of an error in the current cosmological model (McGaugh, 2011: 4). Significantly, a0 is of the same order as two other important cosmological constants; the Hubble constant H0 for the expansion of the Universe; and the cosmological constant Λ from the Einstein's equations of GR, which describes the energy density of empty space in the ΛCDM model (Aryal et al., 2013: 1). This fact is expressed in Equations 3 and 4 (where the speed of light c has been implicitly set to unity).

$a_0\sim{H_0}\qquad(3)$

$a^2_0\sim\Lambda\qquad(4)$

As well as being difficult to fit within the ΛCDM model, a possible connection to H0 and Λ adds credence to the suggestion that a0 may have some cosmological significance beyond mere coincidence (Famaey and McGaugh, 2012: 12–1 3).

## Modified Newtonian dynamics

The typical acceleration experienced by a star in a galaxy is around 11 orders of magnitude less than Earth's gravity. In 1983, Mohedrai Milgrom proposed that the mass discrepancy problem is related to a breakdown of Newtonian dynamics at such tiny accelerations. He introduced the concept of MOND, imposing a modification to gravity below an acceleration constant a0 (Milgrom, 1983).According to MOND, in the limit where the gravitational acceleration g >> a0, Newtonian physics holds. However when g << a0, the modification stated by Equation 5 is applied:

$g=\sqrt{g_Na_0}\qquad(5)$

where g is the actual gravitational acceleration and gN is that which is expected under Newtonian gravity (Milgrom, 1983: 366). Returning to the derivation of Equation 2, but using this modified expression for gravity, we have:

$m\frac{v^2}{r}=m\sqrt{\frac{GMa_0}{r^2}}$

and, therefore:

$v=(GMa_0)^{\frac{1}{4}}\qquad(6)$

At very low accelerations, Equation 6 suggests that a galaxy's rotational velocity approaches a constant as the mass M approaches the total galactic mass (given that G, M and a0 are all constants at this limit). Thus, MOND predicts flat rotation curves as v asymptotes to this constant (McGaugh and Milgrom, 2013: 35–37; Milgrom, 1983). Indeed, Equation 6 expresses that the BTFR is inherent in MOND.

A smooth transition across all acceleration scales is achieved by an interpolating function. Rearranging Equation 5 we have:

$g_N=\frac{g^2}{a_0}=\Big(\frac{g}{a_0}\Big)g \qquad(7)$

We can then introduce the interpolating function into Equation 7:

$\mu\Big(\frac{g}{a_0}\Big)g=g_N \qquad(8)$

where

$\mu\Big(\tfrac{g}{a_0}\Big) \rightarrow 1$

if

$\frac{g}{a_0}\gg 1$

and

$\mu\Big(\frac{g}{a_0}\Big) \rightarrow \Big(\frac{g}{a_0}\Big)$

if

$\frac{g}{a_0}\ll 1$

In words, Equation 8 says that if g is large compared to a0, then g equals gN and gravity behaves in a Newtonian manner, whereas if g is small compared to a0 then the modified force law applies (Milgrom, 1983: 366–6 7).

Assuming MOND to be true, one can make predictions about galactic dynamics, which are not predicted by ΛCDM. McGaugh and Milgrom (2013) present a striking example of just how closely MOND predictions often agree with observation in their analysis of velocity dispersion data of 17 nearby dwarf spheroidal galaxies. Predictions in their study were based solely on the luminosities of the galaxies and assumed no CDM. There are many plausible scenarios with CDM that would give rise to dynamics that MOND could not explain, so there is no guarantee that MOND will work in the context of the CDM paradigm (McGaugh and Milgrom, 2013: 27; Milgrom, 1983: 370). Despite this, the observed velocity dispersions of all 17 galaxies are within uncertainty of what MOND models predict. The authors of this study claim that their predictions are in such good agreement with observation that one might easily mistake them for actual observational data (McGaugh and Milgrom, 2013: 27).

While superficially the various predictions of MOND seem unrelated to each other, each is dependent in some way on this characteristic a0 scale. This seems to suggest the existence of some universal force law generated by baryons alone. In light of this, the motivation for studying MOND as an alternative to ΛCDM is not necessarily to replace dark matter, but to explain why galaxies seem to be mimicking such a law and to discover the significance, if any, of a0 to cosmology, as implied by Equations 3 and 4 (Famaey and McGaugh, 2012: 43).

## Relativistic interpretations of MOND

Despite its success in predicting galactic dynamics, MOND must be unified with GR if it is to have any predictive power beyond the scale of individual galaxies (Sanders and McGaugh, 2002: 285–87). A number of relativistic interpretations of MOND currently exist, with the general theme being to invoke new fields with which baryonic matter interacts, that generate the MOND behaviour (Famaey and McGaugh, 2012: 88). A theoretical field was first proposed in 1984 (Bekenstein and Milgrom, 1984). The gradient of the field has the dimensions of acceleration (i.e. a0) and can therefore be used to mediate MOND. Modern relativistic interpretations of MOND include Tensor–Vector–Scalar Gravity (TeVeS), which incorporates multiple fields and is able to predict many observed phenomena on galactic and cosmic scales (Chiu, 2011). TeVeS has been demonstrated to have problems however. In particular, a computation made by Dodelson (2011) showed it to be very hard to reconcile with large-scale structure formation.

To date, a completely satisfactory theory has not been developed. However much progress has been made and one can speculate as to how problems in galaxy clusters and large-scale structure might be addressed by such a theory (Famaey and McGaugh, 2013; 9). For example, if the fields generating MOND behaviour in galaxies are less energy dense than the baryonic one, larger scale phenomena currently attributed to CDM may be produced, but without destroying galactic stability (Tiret and Combes, 2008). This would effectively fix the disk stability problem faced by ΛCDM (Tiret and Combes, 2008: 725–26). The addition of new fields may also be able to explain certain aspects of cosmology and cosmic structure, based on the effectively stronger gravity at very low accelerations (Llinares et al., 2008). Angus and Diaferio (2011) showed numerically that structure formation under a MOND-like driving force would initially be slower than for ΛCDM, but would rapidly speed up. Large-scale structure evolution may in fact be faster with MOND than is predicted by ΛCDM, without the need for any additional unseen mass (Llinares et al., 2008: 1789).

## Challenges for MOND

One of the biggest challenges for MOND lies in the dynamical mass of galaxy clusters. In short, even if one assumes an altered gravitational force law, there is still a mass discrepancy problem on the cluster scale (Clowe et al., 2006: L112; Watkins et al. 2010: 276–77). Weak gravitational lensing by galaxy clusters is also problematic for MOND (Schmidt, 2008). Figure 4 shows a gravitational lensing map of the bullet cluster, which shows that the apparent distribution of mass does not coincide with the distribution of baryonic matter (Clowe et al., 2006). This provides strong evidence for the presence of dark matter and illustrates the problem that weak gravitational lensing causes for MOND (Clowe et al., 2006; Feix et al., 2008). Feix et al. (2008) demonstrated that TeVeS cannot explain the lensing observation in Figure 4 without assuming additional dark mass. As such, weak gravitational lensing is currently unable to be reconciled within a MOND paradigm (Feix et al., 2008: 325). It must, however, be emphasized that, while this observation does imply additional unseen mass, it does not require that the mass be non-baryonic CDM. It may be baryonic matter that escapes detection (Feix et al., 2008).

Figure 4: Gravitational lensing map of the bullet cluster 1E0657-56. The bullet cluster is actually two subclusters colliding at high speed. The hot gas stripped from each subcluster by the collision, which constitutes most of the cluster's baryonic mass, is coloured pink and purple. The green lines represent a contour map of the distribution of mass, as indicated by gravitational lensing. Clowe et al. (2006) infer from this that the gravitational potential does not coincide with the observed baryonic mass. Instead it approximately traces the distribution of galaxies. This suggests most of the mass in the system is unseen (Clowe et al., 2006). (Source: Clowe et al., 2006)

Another significant challenge for current MOND-based theories is predicting the angular power spectrum of the CMB (cf. Figure 3). The angular power spectrum strongly implies a CDM-dominated Universe. However, the predictions of any particular model within a MOND-based cosmology cannot simultaneously match each of the observed acoustic peaks (Komatsu et al., 2011, Skordis et al., 2006: 3–4). MOND therefore appears to be refuted by observational cosmology. Some suggest, based on CMB observations alone, that a fully satisfactory MOND cosmology may never emerge (Spergel et al., 2007: 382).

These challenges make MOND untenable for some. The aforementioned problems for ΛCDM make it unacceptable to others. All things considered, Occam's Razor does appear to be in favour of the prevailing dark matter paradigm. However, in the absence of irrefutable evidence one way or the other, it may come down to the question of whether the introduction of additional fields is in some way more fundamentally satisfactory than the introducing a new state of matter and, if so, why. Indeed, the reality may be that some combination of MOND and ΛCDM is present in the Universe (Ho et al., 2010). This could be interpreted as the presence of both dark fields and dark matter, in an appropriate combination, which can simultaneously bring about the observed phenomena on all scales. Ho et al. (2010) have made such a suggestion, introducing the term MONDian dark matter. In their model, MONDian dark matter would behave as CDM at cluster and cosmological scales, but on galactic scales it would act as MOND (Ho et al., 2010).

## Conclusion

If one considers cosmology and large-scale structure alone, the evidence for ΛCDM appears irrefutable and to consider a model based on MOND seems unwarranted. Conversely, if one considers only the dynamics of individual galaxies, MOND is in such good agreement with observation and ΛCDM has become so problematic, that it appears equally as unreasonable to assume a large component of non-baryonic dark matter. If MOND is to survive as a theory, the missing mass responsible for galaxy cluster dynamics must be discovered. However, if CDM is correct, a legitimate explanation for MOND phenomenology must be found. In short, before we can reject ΛCDM in favour of MOND, we need a simpler theory that, in addition to correcting for the current discrepancies, can reproduce its successes. At present, this seems a long way off. However, the remarkable success of MOND on galactic scales certainly warrants further attempts to embed it within a relativistic theory of gravity.

Just as Einstein's GR revolutionised our understanding of gravity a century ago, it could be that a new revolution in our thinking is necessary to fully understand dynamics on galactic and cosmic scales. Investigating whether or not MOND may lead to this revolution is an exciting and necessary step for the progression of cosmology and physics in general.

## List of figures

Figure 1: Rotation curve of NGC3198. Created by author. Adapted from Begeman (1989).

Figure 2: Planck SMICA cosmic microwave background map. Source: http://lambda.gsfc.nasa.gov/product/map/current/m_images.cfm (2014). Thank you to the WMAP Science Team for making this image public domain.

Figure 3: Acoustic Power Spectrum of the CMB. Source: http://lambda.gsfc.nasa.gov/product/map/current/m_images.cfm (2014). Thank you to the WMAP Science Team for making this image public domain.

Figure 4: Gravitational lensing map of the bullet cluster. Source: Clowe et al. (2006). Public domain under Wikimedia Commons.

## Notes

[1] Steven is currently nearing the end of his Bachelor of Science, with majors in physics and applied mathematics, at Monash University in Melbourne, Australia. He intends to continue with graduate studies in theoretical physics in 2015.

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To cite this paper please use the following details: Keyte, S.P. (2014), 'Modified Newtonian Dynamics (MOND) as a Solution to the Cosmic Mass Discrepancy Problem', Reinvention: an International Journal of Undergraduate Research, Volume 7, Issue 2, http://www.warwick.ac.uk/reinventionjournal/archive/volume7issue2/keyte Date accessed [insert date]. If you cite this article or use it in any teaching or other related activities please let us know by e-mailing us at Reinventionjournal at warwick dot ac dot uk.