# Galaxy rotation curve and dark matter

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I am reading "The Essential Cosmic Perspective" by Jeffrey O. Bennett, Megan O. Donahue, Nicholas Schneider, Mark Voit. In Chapter 14, it is stated that an evidence of the presence of dark matter in our galaxy is that the rotational curve does not match that obtained from calculation. In the calculation, the following formula is used $$M_r=frac{rv^2}{G}$$ where $M_r$ is the mass enclosed inside a radius $r$ from the galactic center.

My understanding is that the above equation is a result of the shell theorem. But we know that shell theorem applies to spherically symmetric mass distribution only, and most galaxies are disks. So why can we still do this?

We can't. That is an over simplification only used in elementary treatments simply to provide the the flavour of the argument. If you see it done somewhere in the refereed literature, it is probably incorrect. Of course it may be true that the mass is almost spherically symmetric, especially if it is dominated by a spherically symmetric dark matter component, but that the visible light is not. Or, it may be true that at large distances from even an asymmetric mass distribution, a Keplerian potential is a good approximation for the orbits of distant objects (e.g. satellite galaxies to the Milky Way). A measured galaxy rotation curve makes no such assumption, it is merely a measurement of rotation speed as a function of radius. It is only the interpretation and modelling that needs to deal with the mass distribution.

The real situation is much more complex. See for example http://ned.ipac.caltech.edu/level5/March01/Battaner/revision.html

However, even if one assumed all the mass was concentrated into a disk-like shape, the only way you can get flat rotation curves is to assume that the mass in the disk does not "follow the light" - that the mass-to-luminosity ratio increases vastly with radius - which is essentially still saying that you have "dark matter", just in a disk.

1. Your claim that sphericity is assumed is incorrect when referred to hard scientific evidence.
2. Even if that was assumed, the resulting error in the rotation speed is small for spheroidal galaxies and smallish for flat galaxies, provided they are centrally concentrated (which their stellar distribution is).

Scientists are, of course, well aware of the errors emanating from this (and other) assumptions (in particular if the situation is as simple as here) and care is taken (if not by every individual scientist, then certainly by the concensus reached by the scientific community) that these errors have no bearing on the scientific conclusions. Your quote from a undergraduate-level textbook proves nothing in this respect.

## Further Clues to Dark Matter in Galaxy Rotation Curves

Title: How the Self-Interacting Dark Matter Model Explains the Diverse Rotation Curves
Authors: A. Kamada, M. Kaplinghat, A. B. Pace, H.-B. Yu
First Author’s Institution: Department of Physics and Astronomy, University of California, Riverside, CA
Status: Submitted to the arXiv

Spiral galaxies spin in captivating, celestial displays—a subtle showcase of the conservation of angular momentum. The speeds of the gas and stars within tell yet another story. Most stars in a spiral live in a central “bulge,” which dictates that the stars and gas within the surrounding disk orbit the bulge ever more slowly the further they are from the center of the galaxy. But what we’ve observed is that no matter their distance, the stars and gas appear to orbit at the same—if not increasing—velocity away from bulge. This implies that spiral galaxies must be tiny gems embedded in a vast ball of invisible, or “dark,” matter. The presence of overwhelming amounts of dark matter in the universe was later confirmed by merging galaxy clusters, and the idea that giant unseen spheres of matter, what we now call “dark matter halos,” encapsulate all galaxies became an everyday fact.

Dark matter in its most simplistic rendition, cold dark matter (CDM), is a subatomic particle that only obeys the force of gravity (unlike the more familiar baryonic matter we’re made out of, which obeys four fundamental forces). CDM has had remarkable success in explaining observations across cosmic time and across cosmic scales, but has met problems at the scales of galaxies and smaller, what astronomers call the “small scale” problems of CDM. Galaxies often have constant density “cores” rather than steeply rising, or “cuspy” densities in their central parts. The Milky Way has too few satellite galaxies, and they aren’t as massive as CDM predicted they’d be.

These small scale problems hatched the idea that the dark matter particle might obey another force: one that causes dark matter particles to interact with other dark matter particles. This form of dark matter, self-interacting dark matter (SIDM), can undergo collisions that redistribute angular momentum between dark matter particles. The dense central regions of halos, where collisions are frequent, lose their cuspiness as dark matter particles “thermalize” (much like a familiar ideal gas) and flatten into “isothermal,” or constant density cores. The smallest satellites are destroyed, and larger ones become more susceptible to mass loss.

But can SIDM explain the velocities of stars and gas in spiral galaxies—the first clue as to the existence of dark matter? Rotation curves, the velocity profile of a spiral galaxy, are typically flat or increasing with SIDM as with CDM (see top panel of Fig. 1). However, galaxies with halos of the same mass can have rotation curves that look very different. Some are incredibly flat—pointing to SIDM-like constant density cores—while others rise more steeply, as in CDM. The authors of today’s paper set out to determine how this diversity can be produced with SIDM.

Figure 1. Rotation curves of halos with different concentrations and baryons. At top is shown the rotation curves of a halo with CDM (gray) vs. SIDM (blue). The shaded regions denote the range of curves produced by the observed variation in halo concentration. The SIDM curves rise less steeply than the CDM curves. At bottom, a stellar disk was included. The scale length of the disk, Rd, describes how centrally concentrated the disk is. Smaller disks are more concentrated, causing the rotation curve to become steeper.

The authors look into two natural ways to produce diversity. Halos form by merging with and accreting other halos, and differences between halos’ formation histories manifest as differences in how centrally concentrated the halos end up being. More concentrated halos cause the rotation curves of galaxies to become steeper. The authors find that including the observed variation in concentration can account for some, but not all, of the range in rotation curves.

Which brings us back to a matter of a more familiar sort: baryons. Baryons, the stars and gas that make up a large part of the disk and bulge of a galaxy, dominate in the central parts of some galaxies more than others. This further increases the central concentration of the total matter (dark + baryonic), which allows rotation curves to be even steeper, increasing the range of curves permissible by SIDM to ranges that have been observed (see bottom panel of Fig. 1).

To further show that SIDM halos can explain observed rotation curves if differences in concentration and baryon dominance are included, the authors fit SIDM rotation curves to those observed. They selected a sample of spiral galaxies with observed rotation curves that exemplify their diversity. Two such galaxies are shown in Fig. 2. All galaxies have observed curves (shown as black points) that asymptote to the same velocity, but become flat quickly (NGC 6503, left) or slowly (UGC 128, right). The SIDM fit (red) with baryons produced remarkably good fits to the rotation curves of galaxies.

Figure 2. Fits to the observed rotation profiles of two galaxies with SIDM and baryons. The observed profile is shown in black, while the fit is shown in red. The contributions from gas and stars in pink, and the SIDM halo in blue (solid). While both galaxies asymptote to the same velocity at large radii, NGC 6503 (left) sharply rises, becoming flat at

5 kpc, while UGC 128 (right) increases only slowly, becoming flat at

20 kpc. The dashed blue line denotes the contribution CDM would have had, and the blue asterisks denote the contribution SIDM would have had without baryons.

This is an important piece of work that demonstrates that SIDM may be able to reproduce many of the small scale observations of the universe. Further work awaits to determine if more detailed modeling that includes, for instance, not just the mass of the baryons but also their physics—heating and feedback from supernovae and winds, among others—can change their ability to explain the curves.

Featured image: The spiral galaxy NGC 6503, one of the galaxies studied in today’s paper (the galaxy’s rotation curve is plotted in Figure 2). Image credit: NASA, ESA, D. Calzetti, H. Ford, and the Hubble Heritage Team.

## Spiral Galaxy Rotation Curve Builder

The rotation curves of spiral galaxies are the standard way to introduce students to the evidence for dark matter. I wanted a good publicly available tool which lets students adjust the amounts of dark and luminous matter and see for themselves what happens to the rotation curve, so I suggested this as a project for Bethany Baldwin-Pulcini and Steven Hyatt in winter quarter 2014. They built a tool which should be useful to many astronomy students.

A common misconception is that dark matter = black hole. This interactive tool helps students realize that the supermassive black hole at the center of a typical spiral galaxy cannot account for the observed rotation curve, both because of the shape of the curve and the overall amount of mass.

 Click on the image to start.

Details: the model for the mass in stars follows an exponential density distribution, as does the light from the disk in a typical disk galaxy. The students integrated over this disk to find the enclosed stellar mass at any given radius. The slider controls the normalization only (in other words, the mass-to-light ratio). The dark matter is modeled as having density proportional to 1/(a 2 +r 2 ) (where a is some constant) so that it is proportional to r -2 at large r (where it provides the flat rotation curve after being integrated over a sphere) but tends toward constant density at small r (to avoid the rotation curve being flat all the way in to an infinite-density center). The slider again controls the normalization only. Back to main Interactive Figures for Astrophysics page.

## Galaxy rotation curve and dark matter - Astronomy

At large distances from the galactic centre the gravitational potential should be that produced by a central point mass and, in the absence of forces other than gravitation, it should be expected that GM / R 2 = / R ( G , universal gravitation constant M , galactic mass R , galactocentric radius , rotation velocity), therefore R -1/2 , which is called, for obvious reasons, the Keplerian rotation curve. This Keplerian decline was not observed, but rather, flat rotation curves with =cte were obtained. Apparently, this has the direct implication that M R , thus depending on the quality of the telescope used. The "Dark Matter" (DM) hypothesis interprets this result in the sense that the Keplerian regime holds at much greater distances than those at which we obtain observations. There should be great quantities of dark matter extending far beyond the visible matter in a more or less spherically symmetric DM halo. If its distribution is spherically symmetric, the mass interior to a sphere of radius R would be M ( R ) R , so that we obtain a first rough model of DM density distribution: = (1/4 R 2 ) dM / dR = /4 GR 2 , i.e. R -2 , for distances far beyond the visible radius. This model is obviously over simplified, as we will see, but it coincides with the so called "nonsingular isothermal" profile

(with and R 0 being constants), one of the most frequently types of halos.

The interpretation of rotation curves of spiral galaxies as evidence of DM halos was probably first proposed by Freeman (1970) who noticed that the expected Keplerian decline was not present in NGC 300 and M33, and considered an undetected mass, with a different distribution for the visible mass. The observation of flat rotation curves was later confirmed and the DM hypothesis reinforced by successive studies. Rubin, Ford and Thonnard (1980) and Bosma (1978, 1981a, b) carried out an extensive study, after which the existence of DM in spiral galaxies was widely accepted. Van Albada et al. (1985) analyzed the rotation of NGC 3198 and the distribution of its hypothetical DM, concluding that this galaxy has a dark halo, in agreement with the paper by Ostriker and Peebles (1987) about the stability of disks. The rotation of spirals was soon considered the most solid proof for the existence of DM in the Universe, particularly important when it was later believed that = 1. Other decisive papers were produced by Begeman (1987) and Broeils (1992).

The initial conclusion could be schematized by considering the rotation curve to be high and flat. If it is high, the dark halo should be very massive if it is flat, the dark halo should be very large. Indeed, the flatness of the rotation curves was explained "too" well, because if the disk and halo had such different distributions, very careful matching was required between the falling disk rotation curve and the rising halo one. The curve was "too" flat there was a "disk-halo conspiracy" (Bahcall and Casertano, 1985, van Albada and Sancisi, 1986).

The only explanation offered for this "conspiracy" is the adiabatic compression of the halo material when the disk was formed (Barnes 1987, Blumenthal et al. 1986) (which is commented later) although Bosma (1998) gave a list of galaxies for which this mechanism is not fully operational. The disk-halo conspiracy is a problem that remains to be satisfactorily solved. The problem is not why curves are flat (not all are flat) but why the transition from disk to halo domination is so smooth.

Different procedures have been used to obtain dark matter distribution: stellar distribution is determined from photometric observations and must then be complemented with CO and HI observations (with a correction factor to include the He mass) mainly for late spirals, in order to assess the gas profiles. These data determine the densities of bulge, disk and gas in the disk, or rather their contribution to the rotation velocity through the so called "circular velocity", V c ( R ), which would coincide with the true rotation velocity , if the component were dominant in the galaxy. The rotation curve, ( R ), is determined mainly with 21 cm maps. The addition of the different visible components does not, in general, coincide with ( R ), from which the existence of a DM halo is deduced.

Then, to obtain its distribution, there are several different techniques. One of the most widely used is the "maximum disk hypothesis" (see for instance, Begeman, Broeils and Sanders, 1991). Here, the mass to light, M/L, ratio is fixed for the different visible components, with values higher than about 10 being difficult to assign to a stellar population. Then, the innermost regions are adjusted so that the disk is able to produce the observed rotation curve without a halo. The disk M/L obtained is then kept constant at all radii and the circular velocity of the halo is then obtained for higher radii. Another possibility is the so called "best fit" technique. In this case, it is necessary to adopt a halo profile defined with several adjustable parameters. Most decompositions have adopted the isothermal spherical profile. At present, it might be profitable to investigate the alternative NFW profile, as this has a higher theoretical justification (we will come back to this point in the section devoted to theoretical models). The problem with the best fit procedure is that the halo distribution function must be known although, in part, this is precisely what we want to obtain.

The maximum disk technique was introduced by van Albada and Sancisi (1986). There are some psychological aspects to their introduction: "dark matter is a daring assumption the intention is therefore to make the halo as small as possible, at least in the traditional optical best known innermost regions and reserve the exotic physics for the outer radio observable regions. It can therefore be found as noticeable that "maximum disk" fits are reasonable and do not very much differ from other fits. This gives us a first information: the inner parts do not require large amounts of dark matter". This conclusion was "a priori" not obvious. At present, it is considered that the amounts of dark and visible matter in the optical disks are similar, with not so much DM being needed as in the outermost disks.

The basic initial description consisting of an innermost region in which ( R ) increased linearly followed by a constant in the outer region was soon considered too simple. Casertano and van Gorkom (1991) found galaxies with declining rotation curves and analyzed current observations to show that bright compact galaxies have slightly declining rotation curves and that rising curves are predominant in low-luminosity galaxies (see also Broeils, 1992). This latter fact indicated that low-luminosity galaxies are more DM rich and that, in general, there is an increase in the dark matter fraction with decreasing luminosity (Persic and Salucci, 1988, 1990). Nearly all rotation curves belonging to the different types of spirals can be described by means of a single function, the so called "Universal Rotation Curve" (Persic, Salucci and Stel, 1996 Salucci and Persic, 1997) which is a successful fit of galactic astronomy that will be commented later.

## Acceleration relation found among spiral and irregular galaxies challenges current understanding of dark matter

In spiral galaxies such as NGC 6946, researchers found that a 1-to-1 relationship between the distribution of stars plus gas and the acceleration caused by gravity exists.

In the late 1970s, astronomers Vera Rubin and Albert Bosma independently found that spiral galaxies rotate at a nearly constant speed: the velocity of stars and gas inside a galaxy does not decrease with radius, as one would expect from Newton's laws and the distribution of visible matter, but remains approximately constant. Such 'flat rotation curves' are generally attributed to invisible, dark matter surrounding galaxies and providing additional gravitational attraction.

Now a team led by Case Western Reserve University researchers has found a significant new relationship in spiral and irregular galaxies: the acceleration observed in rotation curves tightly correlates with the gravitational acceleration expected from the visible mass only.

"If you measure the distribution of star light, you know the rotation curve, and vice versa," said Stacy McGaugh, chair of the Department of Astronomy at Case Western Reserve and lead author of the research.

The finding is consistent among 153 spiral and irregular galaxies, ranging from giant to dwarf, those with massive central bulges or none at all. It is also consistent among those galaxies comprised of mostly stars or mostly gas.

In a paper accepted for publication by the journal Physical Review Letters and posted on the preprint website arXiv, McGaugh and co-authors Federico Lelli, an astronomy postdoctoral scholar at Case Western Reserve, and James M. Schombert, astronomy professor at the University of Oregon, argue that the relation they've found is tantamount to a new natural law.

An astrophysicist who reviewed the study said the findings may lead to a new understanding of internal dynamics of galaxies.

"Galaxy rotation curves have traditionally been explained via an ad hoc hypothesis: that galaxies are surrounded by dark matter," said David Merritt, professor of physics and astronomy at the Rochester Institute of Technology, who was not involved in the research. "The relation discovered by McGaugh et al. is a serious, and possibly fatal, challenge to this hypothesis, since it shows that rotation curves are precisely determined by the distribution of the normal matter alone. Nothing in the standard cosmological model predicts this, and it is almost impossible to imagine how that model could be modified to explain it, without discarding the dark matter hypothesis completely."

McGaugh and Schombert have been working on this research for a decade and with Lelli the last three years. Near-infrared images collected by NASA's Spitzer Space Telescope during the last five years allowed them to establish the relation and that it persists for all 153 galaxies.

The key is that near-infrared light emitted by stars is far more reliable than optical-light for converting light to mass, Lelli said.

The researchers plotted the radial acceleration observed in rotation curves published by a host of astronomers over the last 30 years against the acceleration predicted from the observed distribution of ordinary matter now in the Spitzer Photometry & Accurate Rotation Curves database McGaugh's team created. The two measurements showed a single, extremely tight correlation, even when dark matter is supposed to dominate the gravity.

"There is no intrinsic scatter, which is how far the data differ on average from the mean when plotted on a graph," McGaugh said. "What little scatter is found is consistent with stellar mass-to-light ratios that vary a little from galaxy to galaxy."

Lelli compared the relation to a long-used natural law. "It's like Kepler's third law for the solar system: if you measure the distance of each planet from the sun, you get the orbital period, or vice versa" he said. "Here we have something similar for galaxies, with about 3,000 data points."

"In our case, we find a relation between what you see in normal matter in galaxies and what you get in their gravity," McGaugh said. "This is important because it is telling us something fundamental about how galaxies work."

Arthur Kosowsky, professor of physics and astronomy at the University of Pittsburgh, was not involved but reviewed the research.

"The standard model of cosmology is remarkably successful at explaining just about everything we observe in the universe," Kosowsky said. "But if there is a single observation which keeps me awake at night worrying that we might have something essentially wrong, this is it."

He said McGaugh and collaborators have steadily refined the spiral galaxy scaling relation for years and called this latest work a significant advance, reducing uncertainty in the mass in normal matter by exploiting infrared observations.

"The result is a scaling relation in the data with no adjustable parameters," Kosowky said. "Throughout the history of physics, unexplained regularities in data have often pointed the way towards new discoveries."

McGaugh and his team are not pressing any theoretical interpretation of their empirical relation at this point.

"The natural inference is that this law stems from a universal force such as a modification of gravity like MOND, the hypothesis of Modified Newtonian Dynamics proposed by Israeli physicist Moti Milgrom. But it could also be something in the nature of dark matter like the superfluid dark matter proposed by Justin Khoury," McGaugh said. "Most importantly, whatever theory you want to build has to reproduce this."

## The Galaxy Rotation Problem

Physicists are painfully aware of the fact that spiral galaxies are spinning faster than they should be, given the amount of matter — and therefore, gravity — they contain. At the speed that some of them are spinning, current theory says that the stars, planets, dust, and other matter should be flung off into space. Because they are not, physicists have hypothesized that “dark matter” we cannot see or otherwise detect is causing the extra gravitational pull, keeping these galaxies together. This matter is said to account for about 25 percent of the universe, but Verlinde believes that there may be another answer that can account for the deviations between the expected and observed rotation curves.“What is observed is that the deviations that we see in the rotational curves of galaxies, which is just derived by looking at the matter that we see, always seems to occur at one particular acceleration,” he says.

That particular acceleration happens to play an important role in the relationship between a galaxy’s distance and the speed with which it’s moving away from our own, which is governed by the expansion of the universe, known as Hubble’s Law. A 2017 paper by Alexandre Chaloum Elbeze in the Journal of Modern Physics outlines how the expansion rate of the universe, or H0, is linked through a new parameter, which he calls E0, is linked to the rotation curves of galaxies measured by astronomers.

Verlinde believes that this is an indication that he is on to something.

“That fact kind of hints that it has something to do with the Hubble expansion [of the universe], which at present is due to the presence of dark energy,” he says.

The Hubble constant describes the observed accelerating expansion of the universe. This acceleration is unexplained, but has been attributed to “dark energy,” which Verlinde says can be used to explain away the idea of dark matter.

“Dark energy is quite an important part of my theory,” Verlinde says. “I don’t do away with everything that’s called ‘dark,’ I just explain what is what we now call ‘dark matter’ by thinking about what the influence of dark energy would be, and that [dark energy] actually gives the same effect.”

It should be noted that Verlinde is tackling the problem with dark matter from a specific point of view as a string theorist and is working to fit it into that perspective. Mark Van Raamsdonk, a professor of physics at the University of British Columbia, says that this method should be approached with caution.

“This possibility is intriguing, but as far as I’m aware, it’s not based on a precise model that is mathematically consistent,” Van Raamsdonk says. “Rather, he’s using his intuition to piece together a set of ideas and provide a story for how things might work. He is a very accomplished physicist, so I think his intuition is worth paying attention to.”

## Galaxy rotation study rules out modified gravity, or does it?

Can modified Newtonian dynamics (MOND) explain the curious behaviour of rotating galaxies? Two research groups have independently studied the dynamics of large numbers of galaxies to test MOND and have reached different conclusions. MOND is an alternative to dark matter — a hypothetical substance that is thought to affect the rotation of galaxies via its gravitational pull – and the conflicting studies could help solve problems with our current understanding of galaxy dynamics.

At first glance, Newtonian gravity appears to fail spectacularly when used to calculate the dynamics of galaxy rotation. The problem is that stars far from the galactic centre rotate much faster than predicted and should be flung away from the galaxy. The conventional explanation is that enormous quantities of cold dark matter (CDM) provides additional gravitational glue that binds the galaxies together. This, however, gives physicists the task of explaining the nature of dark matter – which despite its apparent abundance, has never been detected directly.

A minority of physicists, however, take the opposite approach and call for a revision of Newton’s laws. Extraordinary as this suggestion sounds, it does offer potential solutions to some otherwise troubling problems in galactic dynamics.

### Curious correlations

Despite being a pillar of the Standard Model of cosmology, CDM does not offer a complete explanation for the observed dynamics of galaxies. In 2016, for example, Federico Lelli of Case Western Reserve University in the US and colleagues studied a sample of 175 galaxies. They looked at the rotation rate at different distances from the centre of each galaxy. They calculated that the radial acceleration at an arbitrary point in each galaxy is correlated with the amount of visible matter attracting it – but the relationship does not match that predicted by Newtonian dynamics.

The CDM model explains this discrepancy by assuming the visible matter is attracted by dark matter as well as other visible matter. However, dark matter could be found in different quantities and different places in different galaxies, so this relationship should have quite a lot of scatter. A mathematically predictable deviation from the predictions of Newtonian dynamics is hard to explain under the CDM model.

MOND, however, proposes that, at very large radii and small accelerations, gravity decays with distance more slowly than Newton’s inverse square law. This removes the need for dark matter, providing a clear explanation for the tight non-Newtonian correlation between visible matter and radial acceleration.

### Universal scale

In one of the new studies, Davi Rodrigues of Federal University of Espírito Santo in Brazil and colleagues examine 193 disk galaxies (most of which had previously been studied by Lelli) to see whether there is a fundamental acceleration scale. This would be a universal scale factor relating the predictions of Newtonian dynamics and MOND.

“We do a full Bayesian [statistical] analysis in order to find the error bars of this radial acceleration relation for each galaxy,” explains Rodrigues. Having done this, the researchers conclude that there is no scale factor that is not ruled out at a statistical significance of at least 10σ – which means that it is extremely unlikely that the finding is a result of statistical fluctuations in the data.

The researchers therefore rule out any fundamental theory that extends MOND without amending its underlying dynamics. Instead, they suggest that the apparent correlation between visible matter and galactic dynamics could arise from hypothetical complex interactions between visible matter and dark matter.

Working independently, Lelli and colleagues address the same question using different statistical techniques. The researchers fit the radial acceleration relation to data from their set of 175 galaxies. They calculate a value for the scatter in the data that is much lower than the value arrived at by Rodrigues.

### Plane uncertainty

Lelli’s group argue that the other study has ignored the uncertainty in the plane of inclination of disk galaxies relative to the angle of observation – which is an additional source of error in their calculations. Furthermore, says Lelli – now at the European Southern Observatory in Germany – his team found that allowing the scale factor to vary from galaxy to galaxy did not improve the fit. Therefore, the researchers suggest, the observed scatter in the data is better explained by observational errors than by an underlying inconsistency between the data and the fundamental acceleration scale predicted by MOND.

James Binney of the University of Oxford notes that the Rodrigues group’s paper does not question Lelli and colleagues’ 2016 conclusion that there is a mathematically predictable relationship within galaxies between the visible matter and the radial acceleration. Whether that relationship can be fitted by a single parameter that applies to all galaxies is, he says, “subsidiary”.

Rogrigues and colleagues describe their work in Nature Astronomy. Lelli’s team will publish its results in Astronomy & Astrophysics and a preprint is available on arXiv.

## 17 Galaxies & Dark Matter

(a) By observing orbital velocities at different distances from the center of a disk galaxy (M64, the “Evil Eye” galaxy, 5 Mpc distant), astronomers can plot a rotation curve for the galaxy.

(b) Rotation curves for some nearby spiral galaxies indicate masses of a few hundred billion times the mass of the Sun.

The corresponding curve for our own Galaxy is marked in red for comparison.

Rotation curvers allow measurement of galaxy masses .

(a) In a binary galaxy system, galaxy masses may be estimated by observing the orbit of one galaxy about the other.

(b) The mass of a galaxy cluster may be estimated by observing the motion of many galaxies in the cluster and estimating how much mass is needed to prevent the cluster from flying apart.

Galaxies need 3-10 times more mass than can be observed to explain their rotation curves.

The discrepancy is even larger in galaxy clusters about 10-100 times.

The total mass needed is more than the sum of the dartk matter associated with each galaxy.

Astronomers speculate that some galaxies may be composed almost entirely of dark matter, emitting virtually no visible light.

(Right) Has the long stream of gas leaving galaxy UGC 10214 been torn out by a close encounter with a dark companion at bottom right?

Or is the companion simply hidden behind, or even in front of, the visible galaxy?

(a) X-ray image of Abell 85, an old, distant cluster of galaxies. The cluster’s X-ray emission is shown in orange. The green graphs display a smooth, peaked intensity profile centered on the cluster, but not associated with individual galaxies.

(b) The contour map of the X rays is superimposed on an optical photo, showing its X rays are strongest around Abell 85’s central supergiant galaxy.

This shows that the space between the galaxies within galaxy clusters is filled with superheated gas .

(c) Superposition of infrared and X-ray radiation from another distant galaxy cluster. The X rays are shown as a fuzzy, bluish cloud of hot gas filling the intracluster spaces among the galaxies.

(d) IR image of the central region, showing the richness of this cluster, which spans about a million parsecs.

It is believed this gas is primordial, dating from the very early days of the Universe.

There is not nearly enough of this gas to be the needed dark matter in galaxy clusters.

## Physicist theorizes that dark matter is a superfluid

Physicists have been striving to understand dark matter—the invisible material that makes up most of the universe—for decades. In the standard cosmological model, this mysterious substance is a cold, thinly dispersed medium composed of weakly interacting particles, which are considered “collisionless” in that they either hardly interact with each other, or don’t interact at all. Additionally in this model, dark matter keeps the outermost galaxies within galaxy clusters from eluding the grasp of gravity and drifting into distant space, or from being torn apart.

However, a new theory proposed by physicist Justin Khoury, a professor in the School of Arts and Sciences, and his former postdoctoral researcher Lasha Berezhiani, now at the Max Planck Institute for Physics, stands to shake up how scientists consider dark matter. Their hypothesis that dark matter is a liquid has the potential to explain cosmological mysteries that have previously eluded researchers. The findings have been published in Journal of Cosmology and Astroparticle Physics.

As a theoretical principle, the standard rendering of dark matter functions well to explain large-scale cosmological observations, such as the gravitational retention of outer galaxies within a galaxy cluster, but issues arise when applied to smaller-scale mysteries in the universe, such as the rotation curves of individual galaxies.

A rotation curve plots the orbital speeds of objects in relation to their distance from a central mass. In a solar system, objects revolve around the concentrated mass at the center of the system, and, based on Kepler’s second law of planetary motion, the speed of rotating objects decreases as they move farther away from the center.

Galaxy rotation curves are unusual. Objects far from the galactic center orbit at a speed independent from their distance from the center, instead of the decreasing speed expected from Kepler’s second law. Because the velocity becomes constant instead of decreasing, this behavior is usually interpreted as evidence for dark matter. Alternatively, physicists sometimes resort to modifying Newton’s law of gravitational force to explain these galactic observations.

Khoury and Berezhiani approach this differently. Their theory suggests that dark matter in galaxies behaves as a special type of fluid, known as a superfluid, which lacks viscosity and flows indefinitely when stirred. In this approach, the excitations of the superfluid result in a new long-range force, on top of the usual gravitational force, in such a way that explains the unexpected behavior seen in galaxies. Meanwhile, on the large scale of galaxy clusters, according to their theory, dark matter behaves as in the conventional model.

Even at the absolute zero temperatures found in space, a superfluid remains a smoothly flowing liquid, and its individual particles behave as a collective entity. For something to reach a superfluid state, along with being exposed to extremely low temperatures, the fluid needs to have densely-packed atoms.

“A superfluid is a very interesting kind of fluid,” says Khoury. “Most substances we find in the laboratory form a solid when cooled to a low enough temperature. But a certain set of substances, such as liquid helium, effectively remain a liquid all the way to absolute zero temperature. The uncertainty principle of quantum mechanics forbids these atoms from arranging into crystalline structure, and instead they remain fluid, but it’s a very peculiar kind of fluid. It flows without viscosity, doesn’t carry any entropy, and has many other strange behaviors.”

For earlier work in this area, Khoury and Berezhiani received the 2017 Buchalter Cosmology Prize, and the researchers have continued exploring the topic. Receiving the prize, Khoury says, was an honor, rewarding innovative new ideas in the field of cosmology that will hopefully fill the gaps in the current understanding of the universe.

While researchers can calculate the density of the total amount of dark matter in the universe, no one has determined the mass of an individual particle. But it is known that to create a superfluid on Earth, scientists squeeze particles together—such as liquid helium particles—to create extremely dense matter, and then rapidly cool the matter to near absolute zero. This bodes well for the superfluid theory, since the gravitational pull of a galaxy creates the squeeze. The leading paradigm of dark matter already suggests it is extremely cold, which indicates dark matter could be a viable superfluid in certain states.

“What we discovered, essentially, was an analogy,” Khoury says. “If you look at the types of theories people utilized to explain these galactic observations using modified gravity, those equations take the same form as those of a superfluid. We realized that if the dark matter was this particular kind of superfluid, then you could explain both sets of observations, the large scale and the small scale.”

The work was supported by the U.S. Department of Energy, the Université de Strasbourg, the National Science Foundation, NASA, and the Charles E. Kaufman Foundation of the Pittsburgh Foundation.

Justin Khoury is a professor in the Department of Physics and Astronomy at the University of Pennsylvania.

Homepage photo: An artistic rendering displays how superfluid vortices might appear within the dark matter halo around individual galaxies. (Image courtesy of Markos Kay for Quanta Magazine).

## An astronomer scratching his head (1979)

At first sight, it seems that there are only two possible solutions for the discrepancy. Either stage 1 is wrong and we missed something important with our mass modeling, or stage 2 is wrong and the application of the fundamental laws of physics is not valid in these cases.

The vast majority of physicists support the first explanation. According to their view, there must be some additional matter in galaxies. This matter cannot be seen (because the visible mass we see is not enough to explain rotation curves) and must be distributed differently from the visible matter (otherwise rotation curves would still be declining). This kind of matter is known as dark matter . Today, dark matter has grown to be necessary in other fields of astrophysics, but back then in the 70's, rotation curves were the only reason for inventing this concept*.

Some physicists, however, support the second explanation. According to their view, the very application of Newton's laws is wrong. These laws were never put to test in galactic scales before. Perhaps the laws are not general enough. Specifically, according to this theory, Newton's second law should take a different form. The proposed generalization is of such a nature that Newton's original law is obtained as an approximation when "normal" accelerations are used. In galaxies, however, the accelerations are so low that the new prediction deviates significantly from the Newtonian prediction (see the MOND original paper ).

Both theories have advantages and disadvantages. The main disadvantage of dark matter is the simple fact that it hasn't been discovered (although scientists have been searching for it for decades). As long as this substance is not found, it must be taken as a weakness of the theory. No matter how deep dark matter is integrated in other fields of physics (i.e. cosmology), it is still a speculation . It is an extra assumption we make on the physical reality. This issue, together with other fundamental problems** have led some astrophysicists to seriously doubt the current paradigm. However, the resources allocated to investigate alternatives are still very limited.

The main disadvantage of the MOND theory is its inconsistency with GR (General Relativity). After all, Newton's laws are an approximation of GR, and if you're going to change Newton's laws you need to change GR in some way or another. This is a whole new story. First, GR is the most successful theory we have to date. Second, it is constructed on basic principles. Changing it in order to meet galactic observations without changing the whole shield is a very difficult goal. These heroic attempts have been made by some physicists in the past, but their overall success is unclear.

In this site we wish to offer a third explanation for the discrepancy. An explanation which does not invoke extra matter or the need of changing Einstein's general relativity. For a comprehensive review of the new approach please visit this page. Our current work is focused only on galaxy rotation curves. However, it's not unlikely that the same core idea will be used for the investigation of other discrepancies (e.g. ellipticals, gravitational lensing). We express our deepest hopes that this work will contribute to our understanding of nature.

* The history of the hidden mass problem is actually more complicated than that. The "desire" for extra mass in cosmology has already existed in the 1970's. An excellent overview can be found here.

** A comprehensive review of the dark matter paradigm and the unproportional credit it receives nowadays can be found in our website.