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Dark Energy Survey First Results: Canonical Cosmology Supported

The Dark Energy Survey (DES) first year results, and a series of papers, were released on August 4, 2017. This is a massive international collaboration with over 60 institutions represented and 200 authors on the paper summarizing initial results. Over 5 years the Dark Energy Survey team plans to survey some 300 million galaxies.

The instrument is the 570-megapixel Dark Energy Camera installed on the Cerro Tololo Inter-American Observatory 4-meter Blanco Telescope.

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Image: DECam imager with CCDs (blue) in place. Credit: darkenergysurvey.org

Over 26 million source galaxy measurements from far, far away are included in these initial results. Typical distances are several billion light-years, up to 9 billion light-years. Also included is a sample of 650,000 luminous red galaxies, lenses for the gravitational lensing, and typically these are foreground elliptical galaxies. These are at redshifts < 0.9 corresponding to up to 7 billion light-years.

They use 3 main methods to make cosmological measurements with the sample:

1. The correlations of galaxy positions (galaxy-galaxy clustering)

2. The gravitational lensing of the large sample of background galaxies by the smaller foreground population (cosmic shear)

3. The gravitational lensing of the luminous red galaxies (galaxy-galaxy lensing)

Combining these three methods provides greater interpretive power, and is very effective in eliminating nuisance parameters and systematic errors. The signals being teased out from the large samples are at only the one to ten parts in a thousand level.

They determine 7 cosmological parameters including the overall mass density (including dark matter), the baryon mass density, the neutrino mass density, the Hubble constant, and the equation of state parameter for dark energy. They also determine the spectral index and characteristic amplitude of density fluctuations.

Their results indicate Ωm of 0.28 to a few percent, indicating that the universe is 28% dark matter and 72% dark energy. They find a dark energy equation of state w = – 0.80 but with error bars such that the result is consistent with either a cosmological constant interpretation of w = -1 or a somewhat softer equation of state.

They compare the DES results with those from the Planck satellite for the cosmic microwave background and find they are statistically significant with each other and with the Λ-Cold Dark MatterΛ model (Λ, or Lambda, stands for the cosmological constant). They also compare to other galaxy correlation measurements known as BAO for Baryon Acoustic Oscillations (very large scale galaxy structure reflecting the characteristic scale of sound waves in the pre-cosmic microwave background plasma) and to Type 1a supernovae data.

This broad agreement with Planck results is a significant finding since the cosmic microwave background is at very early times, redshift z = 1100 and their galaxy sample is at more recent times, after the first five billion years had elapsed, with z < 1.4 and more typically when the universe was roughly ten billion years old.

Upon combining with Planck, BAO, and the supernovae data the best fit is Ωm of 0.30 with an error of less than 0.01, the most precise determination to date. Of this, about 0.25 is ascribed to dark matter and 0.05 to ordinary matter (baryons). And the implied dark energy fraction is 0.70.

Furthermore, the combined result for the equation of state parameter is precisely w = -1.00 with only one percent uncertainty.

The figure below is Figure 9 from the DES paper. The figure indicates, in the leftmost column the measures and error bars for the amplitude of primordial density fluctuations, in the center column the fraction of mass-energy density in matter, and in the right column the equation of state parameter w.

DES.aug17.paper1.fig9.jpeg

The DES year one results for all 3 methods are shown in the first row. The Planck plus BAO plus supernovae combined results are shown in the last row. And the middle row, the fifth row, shows all of the experiments combined, statistically. Note the values of 0.3 and – 1.0 for Ωm and w, respectively, and the extremely small error bars associated with these.

This represents continued strong support for the canonical Λ-Cold Dark Matter cosmology, with unvarying dark energy described by a cosmological constant.

They did not evaluate modifications to general relativity such as Emergent Gravity or MOND with respect to their data, but suggest they will evaluate such a possibility in the future.

References

https://arxiv.org/abs/1708.01530, “Dark Energy Survey Year 1 Results: Cosmological Constraints from Galaxy Clustering and Weak Lensing”, 2017, T. Abbott et al.

https://en.wikipedia.org/wiki/Weak_gravitational_lensing, Wikipedia article on weak gravitational lensing discusses galaxy-galaxy lensing and cosmic shear

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Dark Energy and the Comological Constant

I am seeing a lot of confusion around dark energy and the cosmological constant. What are they? Is gravity always attractive? Or is there such a thing as negative gravity or anti-gravity?

First, what is gravity? Einstein taught us that it is the curvature of space. Or as famous relativist John Wheeler wrote “Matter tells space how to curve, and curved space tells matter how to move”.

Dark Energy has been recognized with the Nobel Prize for Physics, so its reality is accepted. There were two teams racing against one another and they found the same result in 1998: the expansion of the universe is accelerating!

Normally one would have thought it would be slowing down due to the matter within; both ordinary and dark matter would work to slow the expansion. But this is not observed for distant galaxies. One looks at a certain type of supernova that always has a certain mass and thus the same absolute luminosity. So the apparent brightness can be used to determine the luminosity distance. This is compared with the redshift that provides the velocity of recession or velocity-determined distance in accordance with Hubble’s law.

A comparison of the two types of distance measures, particularly for large distances, shows the unexpected acceleration. The most natural explanation is a dark energy component equal to twice the matter component, and that matter component would include any dark matter. Now do not confuse dark energy with dark matter. The latter contributes to gravity in the normal way in proportion to its mass. Like ordinary matter it appears to be non-relativistic and without pressure.

Einstein presaged dark energy when he added the cosmological constant term to his equations of general relativity in 1917. He was trying to build a static universe. It turns out that such a model is unstable, and he later called his insertion of the cosmological constant a blunder. A glorious blunder it was, as we learned eight decades later!

Here is the equation:

G_{ab}+\Lambda g_{ab} = {8\pi G \over c^{4}}T_{ab}

The cosmological constant is represented by the Λ term, and interestingly it is usually written on the left hand side with the metric terms, not on the right hand side with the stress-energy (and pressure and mass) tensor T.

If we move it to the right hand side and express as an energy density, the term looks like this:

\rho  = {\Lambda \over8\pi G }

with \rho  as the vacuum energy density or dark energy, and appearing on the right it also takes a negative sign. So this is a suggestion as to why it is repulsive.

The type of dark energy observed in our current universe can be fit with the simple cosmological constant model and it is found to be positive. So if you move \Lambda to the other side of the equation, it enters negatively.

Now let us look at dark energy more generally. It satisfies an equation of state defined by the relationship of pressure to density, with P as pressure and ρ denoting density:

P = w \cdot \rho \cdot c^2

Matter, whether ordinary or dark, is to first order pressureless for our purposes, quantified by its rest mass, and thus takes w = 0. Radiation it turns out has w = 1/3. The dark energy has a negative w, which is why you have heard the phrase ‘negative pressure’. The simplest case is w = -1, which the cosmological constant, a uniform energy density independent of location and age of the universe. Alternative models of dark energy known as quintessence can have a larger w, but it must be less than -1/3.

275px-EquationofState.gif

Credit: http://www.scholarpedia.org/article/Cosmological_constant

Why less than -1/3? Well the equations of general relativity as a set of nonlinear differential equations are usually notoriously difficult to solve, and do not admit of analytical solutions. But our universe appears to be highly homogeneous and isotropic, so one can use a simple FLRW spherical metric, and in this case one end up with the two Friedmann equations (simplified by setting c =1).

\ddot a/a  = - {4 \pi  G \over 3} ({\rho + 3 p}) + {\Lambda \over 3 }

This is for a (k = 0) flat on large scales universe as observed. Here \ddot a is the acceleration (second time derivative) of the scale factor a. So if \ddot a is positive, the expansion of the universe is speeding up.

The \Lambda term can be rewritten using the dark energy density relation above. Now the equation needs to account for both matter (which is pressureless, whether it is ordinary or dark matter) and dark energy. Again the radiation term is negligible at present, by four orders of magnitude. So we end up with:

\ddot a/a  = - {4 \pi  G \over 3} ({\rho_m + \rho_{de} + 3 p_{de}})

Now the magic here was in the 3 before the p. The pressure gets 3 times the weighting in the stress-energy tensor T. Why, because energy density is just there as a scalar, but pressure must be accounted for in each of the 3 spatial dimensions. And since p for dark energy is negative and equal to the dark energy density (times the square of the speed of light), then

\rho + 3 p is always negative for the dark energy terms, provided w < -1/3. That unusual behavior is why we call it ‘dark energy’.

Overall it is a battle between matter and dark energy density on the one side, and dark energy pressure (being negative and working oppositely to how we ordinarily think of gravity) on the other. The matter contribution gets weaker over time, since as the universe expands the matter becomes less dense by a relative factor of (1=z)^3 , that is the matter was on average denser in the past by the cube of one plus the redshift for that era.

Dark energy eventually wins out, because it, unlike matter does not thin out with the expansion. Every cubic centimeter of space, including newly created space with the expansion has its own dark energy, generally attributed to the vacuum. Due to the quantum uncertainty (Heisenberg) principle, even the vacuum has fields and non zero energy.

Now the actual observations at present for our universe show, in units of the critical density that

\rho_m \approx 1/3

\rho_{de} \approx 2/3

and thus

p_{de} \approx - 2

And the sum of them all is around -1, just coincidentally. Since there is a minus sign in front of the whole thing, the acceleration of the universe is positive. This is all gravity, it is just that some terms take the opposite side. The idea that gravity can only be attractive is not correct.

If we go back in time, say to the epoch when matter still dominated with \rho_m \approx 2/3 and  \rho_{de} \approx 1/3 , then the total including pressure would be 2/3 +1/3 – 1, or 0.

That would be the epoch when the universe changed from decelerating to accelerating, as dark energy came to dominate. With our present cosmological parameters, it corresponds to a redshift of z \approx 0.6, and almost 6 billion years ago.

Image: NASA/STScI, public domain


Yet Another Intermediate Black Hole Merger

Another merger of two intermediate mass black holes has been observed by the LIGO gravitational wave observatories.

There are now three confirmed black hole pair mergers, along with a previously known fourth possible, that lacks sufficient statistical confidence.

These three mergers have all been detected in the past two years and are the only observations ever made of gravitational waves.

They are extremely powerful events. The lastest event is known as GW170104 (gravitational wave discovery of January 4, 2017).

It all happened in the wink of an eye. In a fifth of a second, a black hole of 30 solar masses approximately merged with a black hole of about 20 solar masses. It is estimated that the two orbited around one another six times (!) during that 0.2 seconds of their final existence as independent objects.

The gravitational wave generation was so great that an entire solar mass of gravitational energy was liberated in the form of gravitational waves.

This works out to something like 2 \cdot 10^{47} Joules of energy, released in 0.2 seconds, or an average of 10^{48} Watts during that interval. You know, a Tera Tera Tera Terawatt.

Researchers have now discovered a whole new class of black holes with masses ranging from about 10 solar masses (unmerged) to 60 solar masses (merged). If they keep finding these we might have to give serious consideration to intermediate mass black holes as contributors to dark matter.  See this prior blog for a discussion of primordial black holes as a possible dark matter contributor:

https://darkmatterdarkenergy.com/2016/06/17/primordial-black-holes-as-dark-matter/

IMG_0462

Image credit: LIGO/Caltech/MIT/Sonoma State (Aurore Simonnet)


No Dark Energy?

Dark Energy is the dominant constituent of the universe, accounting for 2/3 of the mass-energy balance at present.

At least that is the canonical concordance cosmology, known as the ΛCDM or Lambda – Cold Dark Matter model. Here Λ is the symbol for the cosmological constant, the simplest, and apparently correct (according to most cosmologists), model for dark energy.

Models of galaxy formation and clustering use N-body simulations run on supercomputers to model the growth of structure (galaxy groups and clusters) in the universe. The cosmological parameters in these models are varied and then the models are compared to observed galaxy catalogs at various redshifts, representing different ages of the universe.

It all works pretty well except that the models assume a fully homogeneous universe on the large scale. While the universe is quite homogeneous for scales above a billion light-years, there is a great deal of filamentary web-like structure at scales above clusters, including superclusters and voids, as you can easily see in this map of our galactic neighborhood.

399px-2MASS_LSS_chart-NEW_Nasa

Galaxies and clusters in our neighborhood. IPAC/Caltech, by Thomas Jarrett“Large Scale Structure in the Local Universe: The 2MASS Galaxy Catalog”, Jarrett, T.H. 2004, PASA, 21, 396

Well why not take that structure into account when doing the modeling? It has long been known that more local inhomogeneities such as those seen here might influence the observational parameters such as the Hubble expansion rate. Thus even at the same epoch, the Hubble parameter could vary from location to location.

Now a team from Hungary and Hawaii have modeled exactly that, in a paper entitled “Concordance cosmology without dark energy” https://arxiv.org/pdf/1607.08797.pdf . They simulate structure growth while estimating the local values of expansion parameter in many regions as their model evolves.

Starting with a completely matter dominated (Einstein – de Sitter) cosmology they find that they can reasonably reproduce the average expansion history of the universe — the scale factor and the Hubble parameter — and do that somewhat better than the Planck -derived canonical cosmology.

Furthermore, they claim that they can explain the tension between the Type Ia supernovae value of the Hubble parameter (around 73 kilometers per second per Megaparsec) and that determined from the Planck satellite observations of the cosmic microwave background radiation (67 km/s/Mpc).

Future surveys of higher resolution should be able to distinguish between their model and ΛCDM, and they also acknowledge that their model needs more work to fully confirm consistency with the cosmic microwave background observations.

Meanwhile I’m not ready to give up on dark energy and the cosmological constant since supernova observations, cosmic microwave background observations and the large scale galactic distribution (labeled BAO in the figure below) collectively give a consistent result of about 70% dark energy and 30% matter. But their work is important, something that has been a nagging issue for quite a while and one looks forward to further developments.

 

Measurements of Dark Energy and Matter content of Universe

Dark Energy and Matter content of Universe


Distant Galaxy Rotation Curves Appear Newtonian

One of the main ways in which dark matter was postulated, primarily in the 1970s, by Vera Rubin (recently deceased) and others, was by looking at the rotation curves for spiral galaxies in their outer regions. Although that was not the first apparent dark matter discovery, which was by Fritz Zwicky from observations of galaxy motion in the Coma cluster of galaxies during the 1930s.

Most investigations of spiral galaxies and star-forming galaxies have been relatively nearby, at low redshift, because of the difficulty in measuring these accurately at high redshift. For what is now a very large sample of hundreds of nearby galaxies, there is a consistent pattern. Galaxy rotation curves flatten out.

M64

M64, image credit: NASA, ESA, and the Hubble Heritage Team (AURA/STScI)

If there were only ordinary matter one would expect the velocities to drop off as one observes the curve far from a galaxy’s center. This is virtually never seen at low redshifts, the rotation curves consistently flatten out. There are only two possible explanations: dark matter, or modification to the law of gravity at very low accelerations (dark gravity).

Dark matter, unseen matter, would case rotational velocities to be higher than otherwise expected. Dark, or modified gravity, additional gravity beyond Newtonian (or general relativity) would do the same.

Now a team of astronomers (Genzel et al. 2017) have measured the rotation curves of six individual galaxies at moderately high redshifts ranging from about 0.9 to 2.4.

Furthermore, as presented in a companion paper, they have stacked a sample of 97 galaxies with redshifts from 0.6 to 2.6  to derive an average high-redshift rotation curve (P. Lang et al. 2017). While individually they cannot produce sufficiently high quality rotation curves, they are able to produce a mean normalized curve for the sample as a whole with sufficiently good statistics.

In both cases the results show rotation curves that fall off with increasing distance from the galaxy center, and in a manner consistent with little or no dark matter contribution (Keplerian or Newtonian style behavior).

In the paper with rotation curves of 6 galaxies they go on to explain their falling rotation curves as due to “first, a large fraction of the massive high-redshift galaxy population was strongly baryon-dominated, with dark matter playing a smaller part than in the local Universe; and second, the large velocity dispersion in high-redshift disks introduces a substantial pressure term that leads to a decrease in rotation velocity with increasing radius.” 

So in essence they are saying that the central regions of galaxies were relatively more dominated in the past by baryons (ordinary matter), and that since they are measuring Hydrogen alpha emission from gas clouds in this study that they must also take into account the turbulent gas cloud behavior, and this is generally seen to be larger at higher redshifts.

Stacy McGaugh, a Modified Newtonian Dynamics (MOND) proponent, criticizes their work saying that their rotation curves just don’t go far enough out from the galaxy centers to be meaningful. But his criticism of their submission of their first paper to Nature (sometimes considered ‘lightweight’ for astronomy research results) is unfounded since the second paper with the sample of 97 galaxies has been sent to the Astrophysical Journal and is highly detailed in its observational analysis.

The father of MOND, Mordehai Milgrom, takes a more pragmatic view in his commentary. Milgrom calculates that the observed accelerations at the edge of these galaxies are several times higher than the value at which rotation curves should flatten. In addition to this criticism he notes that half of the galaxies have low inclinations, which make the observations less certain, and that the velocity dispersion of gas in galaxies that provides pressure support and allows for lower rotational velocities, is difficult to correct for.

As in MOND, in Erik Verlinde’s emergent gravity there is an extra acceleration (only apparent when the ordinary Newtonian acceleration is very low) of order. This spoofs the behavior of dark matter, but there is no dark matter. The extra ‘dark gravity’ is given by:

g _D = sqrt  {(a_0 \cdot g_B / 6 )}

In this equation a0 = c*H, where H is the Hubble parameter and gB is the usual Newtonian acceleration from the ordinary matter (baryons). Fundamentally, though, Verlinde derives this as the interaction between dark energy, which is an elastic, unequilibrated medium, and baryonic matter.

One could consider that this dark gravity effect might be weaker at high redshifts. One possibility is that density of dark energy evolves with time, although at present no such evolution is observed.

Verlinde assumes a dark energy dominated de Sitter model universe for which the cosmological constant is much larger than the matter contribution and approaches unity, Λ = 1 in units of the critical density. Our universe does not yet fully meet that criteria, but has Λ about 0.68, so it is a reasonable approximation.

At redshifts around z = 1 and 2 this approximation would be much less appropriate. We do not yet have a Verlindean cosmology, so it is not clear how to compute the expected dark gravity in such a case, but it may be less than today, or greater than today. Verlinde’s extra acceleration goes as the square root of the Hubble parameter. That was greater in the past and would imply more dark gravity. But  in reality the effect is due to dark energy, so it may go with the one-fourth power  of an unvarying cosmological constant and not change with time (there is a relationship that goes as H² ∝ Λ in the de Sitter model) or change very slowly.

At very large redshifts matter would completely dominate over the dark energy and the dark gravity effect might be of no consequence, unlike today. As usual we await more observations, both at higher redshifts, and further out from the galaxy centers at moderate redshifts.

References:

R. Genzel et al. 2017, “Strongly baryon-dominated disk galaxies at the peak of galaxy formation ten billion years ago”, Nature 543, 397–401, http://www.nature.com/nature/journal/v543/n7645/full/nature21685.html

P. Lang et al. 2017, “Falling outer rotation curves of star-forming galaxies at 0.6 < z < 2.6 probed with KMOS^3D and SINS/ZC-SINF” https://arxiv.org/abs/1703.05491

Stacy McGaugh 2017, https://tritonstation.wordpress.com/2017/03/19/declining-rotation-curves-at-high-redshift/

Mordehai Milgrom 2017, “High redshift rotation curves and MOND” https://arxiv.org/abs/1703.06110v2

Erik Verlinde 2016, “Emergent Gravity and the Dark Universe” https;//arXiv.org/abs/1611.02269v1


Emergent Gravity in the Solar System

In a prior post I outlined Erik Verlinde’s recent proposal for Emergent Gravity that may obviate the need for dark matter.

Emergent gravity is a statistical, thermodynamic phenomenon that emerges from the underlying quantum entanglement of micro states found in dark energy and in ordinary matter. Most of the entropy is in the dark energy, but the presence of ordinary baryonic matter can displace entropy in its neighborhood and the dark energy exerts a restoring force that is an additional contribution to gravity.

Emergent gravity yields both an area entropy term that reproduces general relativity (and Newtonian dynamics) and a volume entropy term that provides extra gravity. The interesting point is that this is coupled to the cosmological parameters, basically the dark energy term which now dominates our de Sitter-like universe and which acts like a cosmological constant Λ.

In a paper that appeared in arxiv.org last month, a trio of astronomers Hees, Famaey and Bertone claim that emergent gravity fails by seven orders of magnitude in the solar system. They look at the advance of the perihelion for six planets out through Saturn and claim that Verlinde’s formula predicts perihelion advances seven orders of magnitude larger than should be seen.

hst_saturn_nicmos

No emergent gravity needed here. Image credit: NASA GSFC

But his formula does not apply in the solar system.

“..the authors claiming that they have ruled out the model by seven orders of magnitude using solar system data. But they seem not to have taken into account that the equation they are using does not apply on solar system scales. Their conclusion, therefore, is invalid.” – Sabine Hossenfelder, theoretical physicist (quantum gravity) Forbes blog 

Why is this the case? Verlinde makes 3 main assumptions: (1) a spherically symmetric, isolated system, (2) a system that is quasi-static, and (3) a de Sitter spacetime. Well, check for (1) and check for (2) in the case of the Solar System. However, the Solar System is manifestly not a dark energy-dominated de Sitter space.

It is overwhelmingly dominated by ordinary matter. In our Milky Way galaxy the average density of ordinary matter is some 45,000 times larger than the dark energy density (which corresponds to only about 4 protons per cubic meter). And in our Solar System it is concentrated in the Sun, but on average out to the orbit of Saturn is a whopping 3.7 \cdot 10^{17} times the dark energy density.

The whole derivation of the Verlinde formula comes from looking at the incremental entropy (contained in the dark energy) that is displaced by ordinary matter. Well with over 17 orders of magnitude more energy density, one can be assured that all of the dark energy entropy was long ago displaced within the Solar System, and one is well outside of the domain of Verlinde’s formula, which only becomes relevant when acceleration drops near to or below  c * H. The Verlinde acceleration parameter takes the value of 1.1 \cdot 10^{-8}  centimeters/second/second for the observed value of the Hubble parameter. The Newtonian acceleration at Saturn is .006 centimeters/second/second or 50,000 times larger.

The conditions where dark energy is being displaced only occur when the gravity has dropped to much smaller values; his approximation is not simply a second order term that can be applied in a domain where dark energy is of no consequence.

There is no entropy left to displace, and thus the Verlinde formula is irrelevant at the orbit of Saturn, or at the orbit of Pluto, for that matter. The authors have not disproven Verlinde’s proposal for emergent gravity.

 

 

 

 

 


Leo II Dwarf Orbits Milky Way: Dark Matter or Emerging Gravity

In a prior blog, “The Curiously Tangential Dwarf Galaxies”, I reported on results from Cautun and Frenk that indicate that a set of 10 dwarf satellite galaxies near the Milky Way with measured proper motions have much more tangential velocity than expected by random. Formally, there is a 5 standard deviation negative velocity anisotropy with over 80% of the kinetic energy in tangential motion.

While in no way definitive, this result appears inconsistent with the canonical cold dark matter assumptions. So one speculation is that the tangential motions are reflective of the theory of emergent gravity, for which dark matter is not required, but for which the gravitational force changes (strengthens) at very low accelerations, of order c \cdot H, where H is the Hubble parameter, and the value at which the force begins to strengthen works out to be accelerations of only less than about 2 centimeters per second per year.

One of the 10 dwarf galaxies in the sample is Leo II. The study of its proper motion has been reported by Piatek, Pryor, and Olszewski. They find that the galactocentric radial and tangential velocity components are 22 and 127 kilometers per second, respectively. While there is a rather large uncertainty in the tangential component, for their measured values some 97% of the kinetic energy is in the tangential motion.

local_group_and_nearest_galaxies

Artist’s rendering of the Local Group of galaxies. This representation is centered on the Milky Way, you can see a large number of dwarf galaxies near the Milky Way and many near the Andromeda Galaxy as well. Leo II is in the swarm around our Milky Way. Image credit: Antonio Ciccolella. This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.

So let’s look at the implications for this dwarf galaxy, assuming that it is in a low-eccentricity, nearly circular orbit about the Milky Way, which seems possible. We can compare calculations for Newtonian gravity with the implications from Verlinde’s emergent gravity framework.

Under the assumption of a near circular orbit, either there is a lot of dark matter in the Milky Way explaining the high tangential orbital velocity of Leo II, or there is excess gravity. So what do the two alternatives look like?

Let’s look at the dark matter case first. The ordinary matter mass of the Milky Way is measured to be 60 billion solar masses, mostly in stars, but considering gas as well. The distance to the Leo II dwarf galaxy is 236 kiloparsecs (770,000 light-years), well beyond the Milky Way’s outer radius.

So to first order, for a roughly spherical Milky Way, including a dark matter halo, we can evaluate what the total mass including dark matter would be required to hold Leo II in a circular orbit. This is determined by equating the centripetal acceleration v²/R to the gravitational acceleration inward GM/R². So the gravitational mass under Newtonian physics required for velocity v at distance R for a circular orbit is M = R v² / G. Using the tangential velocity and the distance measures above yields a required mass of 870 billion solar masses.

This is 14 times larger than the Milky way’s known ordinary matter mass from stars and gas. Now there are some other dwarf galaxies such as the Magellanic Clouds within the sphere of influence, but they are very much smaller, so this estimate of the total mass required is reasonable to first order. The assumption of circularity is a larger uncertainty. But what this says is something like 13 times as much dark matter as ordinary matter would be required.

Now let’s look at the emergent gravity situation. In this case there is no dark matter, but there is extra acceleration over and above the acceleration due to Newtonian gravity.  To be clear, emergent gravity predicts both general relativity and an extra acceleration term. When the acceleration is modest general relativity reduces to Newtonian dynamics. And when it is very low the total acceleration in the emergent gravity model includes both a Newtonian term and an extra term related to the volume entropy contribution.

In other words, gT = gN + gE is the total acceleration, with gN = GM/R² the Newtonian term and gE the extra term in the emergent gravity formulation. The gN term is calculated using the ordinary mass of 60 billion solar masses, and one gets a tiny acceleration of gN = 1.5 \cdot 10^{-11} centimeters / second / second (cm/s/s).

The extra, or emergent gravity, acceleration is given by the formula gE = sqrt (gN \cdot c \cdot H / 6 ), where H is the Hubble parameter (here we use 70 kilometers/second/Megaparsec). The value of c \cdot H / 6 turns out to be 1.1 \cdot 10^{-8} cm/s/s. This is just a third of a centimeter per second per year.

The extra emergent gravity term from Verlinde’s paper is the square root of the product of 1.1 \cdot 10^{-8} and the Newtonian term amounting to 1.5 \cdot 10^{-11} . Thus the extra gravity is 4.1 \cdot 10^{-10} cm/s/s, which is 27 times larger than the Newtonian acceleration. The total gravity is about 28 times that or 4.3 \cdot 10^{-10} cm/s/s. Now a 28 times larger gravitational acceleration leads to tangential orbital velocities over 5 times greater than expected in the Newtonian case.

Setting v²/R = 4.3 \cdot 10^{-10} cm/s/s and using the distance to Leo II results in an orbital velocity of 177 kilometers/second. With the Newtonian gravity and ordinary matter mass of the Milky Way, one would expect only 33 km/s, a velocity over 5 times lower.

Now the observed tangential velocity is 127 km/s, so the calculated number with emergent gravity is a bit high, but there is no guarantee of a circular orbit. Also, Verlinde’s model assumes quasi-static conditions, and this assumption may break down for a dynamically young system. The time to traverse the distance to Leo II using its radial velocity is of order 10 billion years, so the system may not have settled down sufficiently. There could also be tidal effects from neighbors, or possibly from Andromeda.

This is not a clear argument demonstrating that the Leo II dwarf galaxy’s observed tangential velocity is explained by emergent gravity. But it is a plausible alternative explanation, and made here to show how the calculations work out in this sample case.

So the main alternatives are a Milky Way dominated by dark matter and with a mass close to a trillion solar masses, or a Milky Way of ordinary matter only amounting to 60 billion solar masses. But in that latter case, the Milky Way exerts an extra gravitational force due to emergent gravity that only becomes apparent at very small accelerations less than about 10^{-8} cm/s/s.

Future work with the Hubble and future telescopes is expected to determine many more proper motions in the Local Group so that a fuller dynamical picture of the system can be developed. This will help to discriminate between the emergent gravity and dark matter alternatives.

 

 

 

 


Emergent Gravity: Verlinde’s Proposal

In a previous blog entry I give some background around Erik Verlinde’s proposal for an emergent, thermodynamic basis of gravity. Gravity remains mysterious 100 years after Einstein’s introduction of general relativity – because it is so weak relative to the other main forces, and because there is no quantum mechanical description within general relativity, which is a classical theory.

One reason that it may be so weak is because it is not fundamental at all, that it represents a statistical, emergent phenomenon. There has been increasing research into the idea of emergent spacetime and emergent gravity and the most interesting proposal was recently introduced by Erik Verlinde at the University of Amsterdam in a paper “Emergent Gravity and the Dark Universe”.

A lot of work has been done assuming anti-de Sitter (AdS) spaces with negative cosmological constant Λ – just because it is easier to work under that assumption. This year, Verlinde extended this work from the unrealistic AdS model of the universe to a more realistic de Sitter (dS) model. Our runaway universe is approaching a dark energy dominated dS solution with a positive cosmological constant Λ.

The background assumption is that quantum entanglement dictates the structure of spacetime, and its entropy and information content. Quantum states of entangled particles are coherent, observing a property of one, say the spin orientation, tells you about the other particle’s attributes; this has been observed in long distance experiments, with separations exceeding 100 kilometers.

400px-SPDC_figure.pngIf space is defined by the connectivity of quantum entangled particles, then it becomes almost natural to consider gravity as an emergent statistical attribute of the spacetime. After all, we learned from general relativity that “matter tells space how to curve, curved space tells matter how to move” – John Wheeler.

What if entanglement tells space how to curve, and curved space tells matter how to move? What if gravity is due to the entropy of the entanglement? Actually, in Verlinde’s proposal, the entanglement entropy from particles is minor, it’s the entanglement of the vacuum state, of dark energy, that dominates, and by a very large factor.

One analogy is thermodynamics, which allows us to represent the bulk properties of the atmosphere that is nothing but a collection of a very large number of molecules and their micro-states. Verlinde posits that the information and entropy content of space are due to the excitations of the vacuum state, which is manifest as dark energy.

The connection between gravity and thermodynamics has been around for 3 decades, through research on black holes, and from string theory. Jacob Bekenstein and Stephen Hawking determined that a black hole possesses entropy proportional to its area divided by the gravitational constant G. String theory can derive the same formula for quantum entanglement in a vacuum. This is known as the AdS/CFT (conformal field theory) correspondence.

So in the AdS model, gravity is emergent and its strength, the acceleration at a surface, is determined by the mass density on that surface surrounding matter with mass M. This is just the inverse square law of Newton. In the more realistic dS model, the entropy in the volume, or bulk, must also be considered. (This is the Gibbs entropy relevant to excited states, not the Boltzmann entropy of a ground state configuration).

Newtonian dynamics and general relativity can be derived from the surface entropy alone, but do not reflect the volume contribution. The volume contribution adds an additional term to the equations, strengthening gravity over what is expected, and as a result, the existence of dark matter is ‘spoofed’. But there is no dark matter in this view, just stronger gravity than expected.

This is what the proponents of MOND have been saying all along. Mordehai Milgrom observed that galactic rotation curves go flat at a characteristic low acceleration scale of order 2 centimeters per second per year. MOND is phenomenological, it observes a trend in galaxy rotation curves, but it does not have a theoretical foundation.

Verlinde’s proposal is not MOND, but it provides a theoretical basis for behavior along the lines of what MOND states.

Now the volume in question turns out to be of order the Hubble volume, which is defined as c/H, where H is the Hubble parameter denoting the rate at which galaxies expand away from one another. Reminder: Hubble’s law is v = H \cdot d where v is the recession velocity and the d the distance between two galaxies. The lifetime of the universe is approximately 1/H.

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The value of c / H is over 4 billion parsecs (one parsec is 3.26 light-years) so it is in galaxies, clusters of galaxies, and at the largest scales in the universe for which departures from general relativity (GR) would be expected.

Dark energy in the universe takes the form of a cosmological constant Λ, whose value is measured to be 1.2 \cdot 10^{-56} cm^{-2} . Hubble’s parameter is 2.2 \cdot 10^{-18} sec^{-1} . A characteristic acceleration is thus H²/ Λ or 4 \cdot 10^{-8}  cm per sec per sec (cm = centimeters, sec = second).

One can also define a cosmological acceleration scale simply by c \cdot H , the value for this is about 6 \cdot 10^{-8} cm per sec per sec (around 2 cm per sec per year), and is about 15 billion times weaker than Earth’s gravity at its surface! Note that the two estimates are quite similar.

This is no coincidence since we live in an approximately dS universe, with a measured  Λ ~ 0.7 when cast in terms of the critical density for the universe, assuming the canonical ΛCDM cosmology. That’s if there is actually dark matter responsible for 1/4 of the universe’s mass-energy density. Otherwise Λ could be close to 0.95 times the critical density. In a fully dS universe, \Lambda \cdot c^2 = 3 \cdot H^2 , so the two estimates should be equal to within sqrt(3) which is approximately the difference in the two estimates.

So from a string theoretic point of view, excitations of the dark energy field are fundamental. Matter particles are bound states of these excitations, particles move freely and have much lower entropy. Matter creation removes both energy and entropy from the dark energy medium. General relativity describes the response of area law entanglement of the vacuum to matter (but does not take into account volume entanglement).

Verlinde proposes that dark energy (Λ) and the accelerated expansion of the universe are due to the slow rate at which the emergent spacetime thermalizes. The time scale for the dynamics is 1/H and a distance scale of c/H is natural; we are measuring the time scale for thermalization when we measure H. High degeneracy and slow equilibration means the universe is not in a ground state, thus there should be a volume contribution to entropy.

When the surface mass density falls below c \cdot H / (8 \pi \cdot G) things change and Verlinde states the spacetime medium becomes elastic. The effective additional ‘dark’ gravity is proportional to the square root of the ordinary matter (baryon) density and also to the square root of the characteristic acceleration c \cdot H.

This dark gravity additional acceleration satisfies the equation g _D = sqrt  {(a_0 \cdot g_B / 6 )} , where g_B is the usual Newtonian acceleration due to baryons and a_0 = c \cdot H is the dark gravity characteristic acceleration. The total gravity is g = g_B + g_D . For large accelerations this reduces to the usual g_B and for very low accelerations it reduces to sqrt  {(a_0 \cdot g_B / 6 )} .

The value a_0/6 at 1 \cdot 10^{-8} cm per sec per sec derived from first principles by Verlinde is quite close to the MOND value of Milgrom, determined from galactic rotation curve observations, of 1.2 \cdot 10^{-8} cm per sec per sec.

So suppose we are in a region where g_B is only 1 \cdot 10^{-8} cm per sec per sec. Then g_D takes the same value and the gravity is just double what is expected. Since orbital velocities go as the square of the acceleration then the orbital velocity is observed to be sqrt(2) higher than expected.

In terms of gravitational potential, the usual Newtonian potential goes as 1/r, resulting in a 1/r^2 force law, whereas for very low accelerations the potential now goes as log(r) and the resultant force law is 1/r. We emphasize that while the appearance of dark matter is spoofed, there is no dark matter in this scenario, the reality is additional dark gravity due to the volume contribution to the entropy (that is displaced by ordinary baryonic matter).

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Flat to rising rotation curve for the galaxy M33

Dark matter was first proposed by Swiss astronomer Fritz Zwicky when he observed the Coma Cluster and the high velocity dispersions of the constituent galaxies. He suggested the term dark matter (“dunkle materie”). Harold Babcock in 1937 measured the rotation curve for the Andromeda galaxy and it turned out to be flat, also suggestive of dark matter (or dark gravity). Decades later, in the 1970s and 1980s, Vera Rubin (just recently passed away) and others mapped many rotation curves for galaxies and saw the same behavior. She herself preferred the idea of a deviation from general relativity over an explanation based on exotic dark matter particles. One needs about 5 times more matter, or about 5 times more gravity to explain these curves.

Verlinde is also able to derive the Tully-Fisher relation by modeling the entropy displacement of a dS space. The Tully-Fisher relation is the strong observed correlation between galaxy luminosity and angular velocity (or emission line width) for spiral galaxies, L \propto v^4 .  With Newtonian gravity one would expect M \propto v^2 . And since luminosity is essentially proportional to ordinary matter in a galaxy, there is a clear deviation by a ratio of v².

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 Apparent distribution of spoofed dark matter,  for a given ordinary (baryonic) matter distribution

When one moves to the scale of clusters of galaxies, MOND is only partially successful, explaining a portion, coming up shy a factor of 2, but not explaining all of the apparent mass discrepancy. Verlinde’s emergent gravity does better. By modeling a general mass distribution he can gain a factor of 2 to 3 relative to MOND and basically it appears that he can explain the velocity distribution of galaxies in rich clusters without the need to resort to any dark matter whatsoever.

And, impressively, he is able to calculate what the apparent dark matter ratio should be in the universe as a whole. The value is \Omega_D^2 = (4/3) \Omega_B where \Omega_D is the apparent mass-energy fraction in dark matter and \Omega_B is the actual baryon mass density fraction. Both are expressed normalized to the critical density determined from the square of the Hubble parameter, 8 \pi G \rho_c = 3 H^2 .

Plugging in the observed \Omega_B \approx 0.05 one obtains \Omega_D \approx 0.26 , very close to the observed value from the cosmic microwave background observations. The Planck satellite results have the proportions for dark energy, dark matter, ordinary matter as .68, .27, and .05 respectively, assuming the canonical ΛCDM cosmology.

The main approximations Verlinde makes are a fully dS universe and an isolated, static (bound) system with a spherical geometry. He also does not address the issue of galaxy formation from the primordial density perturbations. At first guess, the fact that he can get the right universal \Omega_D suggests this may not be a great problem, but it requires study in detail.

Breaking News!

Margot Brouwer and co-researchers have just published a test of Verlinde’s emergent gravity with gravitational lensing. Using a sample of over 33,000 galaxies they find that general relativity and emergent gravity can provide an equally statistically good description of the observed weak gravitational lensing. However, emergent gravity does it with essentially no free parameters and thus is a more economical model.

“The observed phenomena that are currently attributed to dark matter are the consequence of the emergent nature of gravity and are caused by an elastic response due to the volume law contribution to the entanglement entropy in our universe.” – Erik Verlinde

References

Erik Verlinde 2011 “On the Origin of Gravity and the Laws of Newton” arXiv:1001.0785

Stephen Perrenod, 2013, 2nd edition, “Dark Matter, Dark Energy, Dark Gravity” Amazon, provides the traditional view with ΛCDM  (read Dark Matter chapter with skepticism!)

Erik Verlinde 2016 “Emergent Gravity and the Dark Universe arXiv:1611.02269v1

Margot Brouwer et al. 2016 “First test of Verlinde’s theory of Emergent Gravity using Weak Gravitational Lensing Measurements” arXiv:1612.03034v


The Curiously Tangential Dwarf Galaxies

There are some 50 or so satellite galaxies around the Milky Way, the most famous of which are the Magellanic Clouds. Somewhat incredibly, half of these have been discovered within the last 2 years, since they are small, faint, and have low surface brightness. The image below shows only the well known ‘classical’ satellites. The satellites are categorized primarily as dwarf spheroidals, and most are low in gas content.

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Image credit: Wikipedia, Richard Powell, Creative Commons Attribution-Share Alike 2.5 Generic

“Satellite galaxies that orbit from 1,000 ly (310 pc) of the edge of the disc of the Milky Way Galaxy to the edge of the dark matter halo of the Milky Way at 980×103 ly (300 kpc) from the center of the galaxy, are generally depleted in hydrogen gas compared to those that orbit more distantly. The reason is the dense hot gas halo of the Milky Way, which strips cold gas from the satellites. Satellites beyond that region still retain copious quantities of gas.” – Wikipedia article

In a recent paper “The tangential velocity excess of the Milky Way satellites“, Marius Cautun and Carlos Frenk find that a sample of satellites (drawn from those known for more than a few years) deviates from the predictions of the canonical Λ – Cold Dark Matter (ΛCDMcosmology. (Λ refers to the cosmological constant, or dark energy).

“We estimate the systemic orbital kinematics of the Milky Way classical satellites and compare them with predictions from the Λ cold dark matter (ΛCDM) model derived from a semi-analytical galaxy formation model applied to high resolution cosmological N-body simulations. We find that the Galactic satellite system is atypical of ΛCDM systems. The subset of 10 Galactic satellites with proper motion measurements has a velocity anisotropy, β = −2.2 ± 0.4, that lies in the 2.9% tail of the ΛCDM distribution. Individually, the Milky Way satellites have radial velocities that are lower than expected for their proper motions, with 9 out of the 10 having at most 20% of their orbital kinetic energy invested in radial motion. Such extreme values are expected in only 1.5% of ΛCDM satellites systems. This tangential motion excess is unrelated to the existence of a Galactic ‘disc of satellites’. We present theoretical predictions for larger satellite samples that may become available as more proper motion measurements are obtained.”

Radial velocities are easy, we get those from redshifts. Tangential velocities are much tougher, but can be obtained from relatively nearby objects by measuring their proper motions. That is, how much do their apparent positions change on the sky after many years have passed. It’s all the more tough when your object is not a point object, but a fuzzy galaxy!

For a ‘random’ distribution of velocities in accordance with ΛCDM cosmology, one would expect the two components of tangential velocity to be each roughly equal on average to the radial component, and thus 2/3 of the kinetic energy would be tangential and 1/3 would be radial. But rather than 33% of the kinetic energy being in radial motion, they find that the Galactic satellites have only about 1/2 that amount in radial, and over 80% of their kinetic energy in tangential motion.

Formally, they find a negative velocity anistropy, β, which as it is defined in practice, should be around zero for a ΛCDM distribution. They find that β differs from zero by 5 standard deviations.

One possible explanation is that the dwarf galaxies are mainly at their perigee or apogee points of their orbits. But why should this be the case? Another explanation: “alternatively indicate that the Galactic satellites have orbits that are, on average, closer to circular than is typical in ΛCDM. This would mean that MW halo mass estimates based on satellite orbits (e.g. Barber et al. 2014) are biased low.” Perhaps the Milky Way halo mass estimate is too low. Or, they also posit, without elaborating, do the excess tangential motions “indicate new physics in the dark sector”?

So one speculation is that the tangential motions are reflective of emergent gravity class of theories, for which dark matter is not required, but for which the gravitational force changes (strengthens) at low accelerations, of order c \cdot H, where H is the Hubble parameter, and the value works out to be around 2 centimeters per second per year. And it does this in a way that ‘spoofs’ the existence and gravitational affect of dark matter. This is also what is argued for in Modified Newtonian Dynamics, which is an empirical observation about galaxy light curves.

In the next article of this series we will look at Erik Verlinde’s emergent gravity proposal, which he has just enhanced, and will attempt to explain it as best we can. If you want to prepare yourself for this challenging adventure, first read his 2011 paper, “On the Origin of Gravity and the Laws of Newton”.


Modified Newtonian Dynamics – Is there something to it?

You are constantly accelerating. The Earth’s gravity is pulling you downward at g = 9.8 meters per second per second. It wants to take your velocity up to about 10 meters per second after only the first second of free fall. Normally you don’t fall, because the floor is solid due to electromagnetic forces and also it is electromagnetic forces that give your body structural integrity and power your muscles, resisting the pull of gravity.

You are also accelerating due to the Earth’s spin and its revolution about the Sun.

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International Space Station, image credit: NASA

Our understanding of gravity comes primarily from these large accelerations, such as the Earth’s pull on ourselves and on satellites, the revolution of the Moon about the Earth, and the planetary orbits about the Sun. We also are able to measure the solar system’s velocity of revolution about the galactic center, but with much lower resolution, since the timescale is of order 1/4 billion years for a single revolution with an orbital radius of about 25,000 light-years!

It becomes more difficult to determine if Newtonian dynamics and general relativity still hold for very low accelerations, or at very large distance scales such as the Sun’s orbit about the galactic center and beyond.

Modified Newtonian Dynamics (MOND) was first proposed by Mordehai Milgrom in the early 1980s as an alternative explanation for flat galaxy rotation curves, which are normally attributed to dark matter. At that time the best evidence for dark matter came from spiral galaxy rotation curves, although the need for dark matter (or some deviation from Newton’s laws) was originally seen by Fritz Zwicky in the 1930s while studying clusters of galaxies.

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NGC 3521. Image Credit: ESA/Hubble & NASA and S. Smartt (Queen’s University Belfast); Acknowledgement: Robert Gendler 

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Galaxy Rotation Curve for M33. Public Domain, By Stefania.deluca – Own work,  https://commons.wikimedia.org/w/index.php?curid=34962949

If general relativity is always correct, and Newton’s laws of gravity are correct for non-relativistic, weak gravity conditions, then one expects the orbital velocities of stars in the outer reaches of galaxies to drop in concert with the fall in light from stars and/or radio emission from interstellar gas, reflecting decreasing baryonic matter density. (Baryonic matter is ordinary matter, dominated by protons and neutrons). As seen in the image above for M33, the orbital velocity does not drop, it continues to rise well past the visible edge of the galaxy.

To first order, assuming a roughly spherical distribution of matter, the square of the velocity at a given distance from the center is proportional to the mass interior to that distance divided by the distance (signifying the gravitational potential), thus

   v² ~ G M / r

where G is the gravitational constant, and M is the galactic mass within a spherical volume of radius r. This potential corresponds to the familiar 1/r² dependence of the force of gravity according to Newton’s laws.  In other words, at the outer edge of a galaxy the velocity of stars should fall as the square root of the increasing distance, for Newtonian dynamics.

Instead, for the vast majority of galaxies studied, it doesn’t – it flattens out, or falls off very slowly with increasing distance, or even continues to rise, as for M33 above. The behavior is roughly as if gravity followed an inverse distance law for the force (1/r) in the outer regions, rather than an inverse square law with distance (1/r²).

So either there is more matter at large distances from galactic centers than expected from the light distribution, or the gravitational law is modified somehow such that gravity is stronger than expected. If there is more matter, it gives off little or no light, and is called unseen, or dark, matter.

It must be emphasized that MOND is completely empirical and phenomenological. It is curve fitted to the existing rotational curves, rather successfully, but not based on a theoretical construct for gravity. It has a free parameter for weak acceleration, and for very small accelerations, gravity is stronger than expected. It turns out that this free parameter, a_0 , is of the same order as the ‘Hubble acceleration’ c \cdot H. (The Hubble distance is c / H and is 14 billion light-years; H has units of inverse time and the age of the universe is 1/H to within a few percent).

The Hubble acceleration is approximately .7 nanometers / sec / sec or 2 centimeters / sec / year  (a nanometer is a billionth of a meter, sec = second).

Milgrom’s fit to rotation curves found a best fit at .12 nanometers/sec/sec, or about 1/6 of a_0 . This is very small as compared to the Earth’s gravity, for example. It’s the ratio between 80 years and one second, or about 2.5 billion. So you can imagine how such a variation could have escaped detection for a long time, and would require measurements at the extragalactic scale.

The TeVeS – tensor, vector, scalar theory is a theoretical construct that modifies gravity from general relativity. General relativity is a tensor theory that reduces to Newtonian dynamics for weak gravity. TeVeS has more free parameters than general relativity, but can be constructed in a way that will reproduce galaxy rotation curves and MOND-like behavior.

But MOND, and by implication, TeVeS, have a problem. They work well, surprisingly well, at the galactic scale, but come up short for galaxy clusters and for the very largest extragalactic scales as reflected in the spatial density perturbations of the cosmic microwave background radiation. So MOND as formulated doesn’t actually fully eliminate the requirement for dark matter.

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Horseshoe shaped Einstein Ring

Image credit: ESA/Hubble and NASA

Any alternative to general relativity also must explain gravitational lensing, for which there are a large number of examples. Typically a background galaxy image is distorted and magnified as its light passes through a galaxy cluster, due to the large gravity of the cluster. MOND proponents do claim to reproduce gravitational lensing in a suitable manner.

Our conclusion about MOND is that it raises interesting questions about gravity at large scales and very low accelerations, but it does not eliminate the requirement for dark matter. It is also very ad hoc. TeVeS gravity is less ad hoc, but still fails to reproduce the observations at the scale of galaxy clusters and above.

Nevertheless the rotational curves of spirals and irregulars are correlated with the visible mass only, which is somewhat strange if there really is dark matter dominating the dynamics. Dark matter models for galaxies depend on dark matter being distributed more broadly than ordinary, baryonic, matter.

In the third article of this series we will take a look at Erik Verlinde’s emergent gravity concept, which can reproduce the Tully-Fisher relation and galaxy rotation curves. It also differs from MOND both in terms of being a theory, although incomplete, rather than empiricism, and apparently in being able to more successfully address the dark matter issues at the scale of galaxy clusters.

References

Wikipedia MOND entry: https://en.wikipedia.org/wiki/Modified_Newtonian_dynamics

M. Milgrom 2013, “Testing the MOND Paradigm of Modified Dynamics with Galaxy-Galaxy Gravitational Lensing” https://arxiv.org/abs/1305.3516

R. Reyes et al. 2010, “Confirmation of general relativity on large scales from weak lensing and galaxy velocities” https://arxiv.org/abs/1003.2185

“In rotating galaxies, distribution of normal matter precisely determines gravitational acceleration” https://www.sciencedaily.com/releases/2016/09/160921085052.htm