# Tag Archives: MOND

## We don’t Need no Stinkin’ Dark Matter

### Extra Acceleration

You’ve heard of dark matter, right? Some sort of exotic particle that lurks in the outskirts of galaxies.

Maybe you know the story of elusive dark matter. The first apparent home for dark matter was in clusters of galaxies, as Fritz Zwicky postulated for the Coma Cluster in the 1930s, due to the excessive galaxy random motions that he measured.

There have been eight decades of discovery and measurement of the gravitational anomalies that dark matter is said to cause, and eight decades of notable failure to directly find any very faint ordinary matter, black holes, or exotic particle matter in sufficient quantities to explain the magnitude of the observed anomalies.

If dark matter is actually real and composed of particles or primordial black holes then there is five times as much mass per unit volume on average in that form as there is in the form of ordinary matter. Ordinary matter is principally in the form of protons and neutrons, primarily as hydrogen and helium atoms and ions.

Why do we call it dark? It gives off no light. Ordinary matter gives off light, it radiates. What else gives off no light? A gravitational field stronger than predicted by existing laws.

Gravitational anomalies are seen in the outer regions of galaxies by examining galaxy rotation curves, which flatten out unexpectedly with distance from the galactic center.  They are seen in galaxy groups and clusters from measuring galaxy velocity dispersions, from X-ray observations of intracluster gas, and from gravitational lensing measurements. A dark matter component is also deduced at the cosmic scale from the power spectrum of the cosmic microwave background spatial variations.

The excessive velocities due to extra acceleration are either caused by dark matter or by some departure of gravity over and above the predictions of general relativity.

Actually at high accelerations general relativity is the required model but at low accelerations Newtonian dynamics is an accurate approximation. The discrepancies arise only at very low accelerations. These excess velocities, X-ray emission, and lensing are observed only at very low accelerations, so we are basically talking about an alternative of extra gravity which is over and above the 1/r² law for Newtonian dynamics.

### Alternatives to General Relativity and Newtonian Dynamics

There are multiple proposed laws for modifying gravity at very low accelerations. To match observations the effect should start to kick in for accelerations less than c * H, where H is the Hubble expansion parameter and its inverse is nearly equal to the present age of the universe.

That is only around 1 part in 14 million expressed in units of centimeters per second per second. This is not something typically measurable in Earth-bound laboratories; scientists have trouble pinning down the value of the gravitational constant G to within 1 part in 10,000.

This is a rather profound coincidence, suggesting that there is something fundamental at play in the nature of gravity itself, not necessarily a rather arbitrary creation of exotic dark matter particle in the very early universe. It suggests instead that there is an additional component of gravity tied in some way to the age and state of our universe.

Do you think of general relativity as the last word on gravity? From an Occam’s razor point of view it is actually simpler to think about modifying the laws of gravity in very low acceleration environments, than to postulate an exotic never-seen-in-the-lab dark matter particle. And we already know that general relativity is incomplete, since it is not a quantum theory.

The emergent gravity concept neatly solves the quantum issue by saying gravity is not fundamental in the way that electromagnetism and the nuclear forces are. Rather it is described as an emergent property of a system due to quantum entanglement of fields and particles. In this view, the fabric of space also arises from this entanglement. Gravity is a statistical property of the system, the entropy (in thermodynamic terms) of entanglement at each point.

### Dark Energy

Now we have a necessary aside on dark energy. Do you know that dark energy is on firmer standing now than dark matter? And do you know that dark energy is just described by additional energy and pressure components in the stress-energy tensor, fully described within general relativity?

We know that dark energy dominates over dark matter in the canonical cosmological model (Lambda-Cold Dark Matter) for the universe. The canonical model has about 2/3 dark energy and the solution for the universe’s expansion approximates a de Sitter model in general relativity with an exponential ‘runaway’ expansion.

### Dark Gravity

As we discuss this no dark matter alternative, we refer to it as dark gravity, or dark acceleration. Regardless of the nature of dark matter and dark gravity, the combination of ordinary gravity and dark gravity is still insufficient to halt the expansion of the universe. In this view, the dark gravity is due to ordinary matter, there is just more of it (gravity) than we expect, again only for the very low c * H or lower acceleration environments.

Some of the proposed laws for modified gravity are:

1. MOND – Modified Newtonian Dynamics, from Milgrom
2. Emergent gravity, from Verlinde
3. Metric skew tensor gravity (MSTG), from Moffat (and also the more recent variant scalar-tensor-vector gravity (STVG), sometimes called MOG (Modified gravity)

Think of the dark gravity as an additional term in the equations, beyond the gravity we are familiar with. Each of the models adds an additional term to Newtonian gravity, that only becomes significant for accelerations less than c*H. The details vary between the proposed alternatives. All do a good job of matching galaxy rotation curves for spiral galaxies and the Tully-Fisher relation that can be used for analyzing elliptical galaxies.

Things are trickier in clusters of galaxies, which are observed for galaxy velocity dispersions, X-ray emission of intracluster gas, and gravitational lensing. The MOND model appears to come up short by a factor of about two in explaining the total dark gravity implied.

Emergent gravity and modified gravity theories including MSTG claim to be able to match the observations in clusters.

### Clusters of Galaxies

Most galaxies are found in groups and clusters.

Clusters and groups form from the collapse of overdense regions of hydrogen and helium gas in the early universe. Collapsing under its own gravity, such a region will heat up via frictional processes and cooler sub-regions will collapse further to form galaxies within the cluster.

Rich clusters have hundreds, even thousands of galaxies, and their gravitational potential is so high that the gas is heated to millions of degrees via friction and shock waves and gives off X-rays. The X-ray emission from clusters has been actively studied since the 1970s, via satellite experiments.

What is found is that most matter is in the form of intracluster gas, not galaxies. Some of this is left over primordial gas that never formed galaxies and some is gas that was once in a galaxy but expelled via energetic processes, especially supernovae.

Observations indicate that around 90% of (ordinary) matter is in the form of intracluster gas, and only around 10% within the galaxies in the form of stars or interstellar gas and dust. Thus modeling the mass profile of a cluster is best done by looking at how the X-ray emission falls off as one moves away from the center of a cluster.

In their 2005 paper, Brownstein and Moffat compiled X-ray emission profiles and fit gas mass profiles with radius and temperature profiles for 106 galaxy clusters. They aggregated data from a sample of 106 clusters and find that an MSTG model can reproduce the X-ray emission with a mass profile that does not require dark matter.

The figure below shows the average profile of cumulative mass interior to a given radius. The mass is in units of solar masses and runs into the hundreds of trillions. The average radius extends to over 1000 Kiloparsecs or over 1 Megaparsec (a parsec is 3.26 light-years).

The bottom line is that emergent gravity and MSTG both claim to have explanatory power without any dark matter for observations of galaxy rotation curves, gravitation lensing in clusters (Brower et al. 2016), and cluster mass profiles deduced from the X-ray emission from hot gas.

Figure 2 from J.R. Brownstein and J.W. Moffat (2005), “Galaxy Cluster Masses without Non-Baryonic Dark Matter”. Shown is cumulative mass required as a function of radius. The red curve is the average of X-ray observations from a sample of 106 clusters. The black curve is the authors’ model assuming MSTG, a good match. The cyan curve is the MOND model, the blue curve is a Newtonian model, and both require dark matter. The point is that the authors can match observations with much less matter and there is no need to postulate additional exotic dark matter.

What we would very much like to see is a better explanation of the cosmic microwave background density perturbation spectrum for the cosmic scale, for either of these dark gravity models. The STVG variant of MSTG claims to address those observations as well, without the need for dark matter.

In future posts we may look at that issue and also the so called ‘silver bullet’ that dark matter proponents often promote, the Bullet Cluster, that consists of two galaxy clusters colliding and a claimed separation of dark matter and gas.

#### References

Brower, M. et al. 2016, “First test of Verlinde’s theory of Emergent Gravity using Weak Gravitational Lensing Measurements” https://arxiv.org/abs/1612.03034v2

Brownstein, J. and Moffat, J. 2005, “Galaxy Cluster Masses without Non-baryonic Dark Matter”, https://arxiv.org/abs/astro-ph/0507222

Perrenod, S. 1977 “The Evolution of Cluster X-ray Sources” http://adsabs.harvard.edu/abs/1978ApJ…226..566P, thesis.

https://darkmatterdarkenergy.com/2018/09/19/matter-and-energy-tell-spacetime-how-to-be-dark-gravity/

https://darkmatterdarkenergy.com/2016/12/30/emergent-gravity-verlindes-proposal/

https://darkmatterdarkenergy.com/2016/12/09/modified-newtonian-dynamics-is-there-something-to-it/

## Matter and Energy Tell Spacetime How to Be: Dark Gravity

Is gravity fundamental or emergent? Electromagnetism is one example of a fundamental force. Thermodynamics is an example of emergent, statistical behavior.

Newton saw gravity as a mysterious force acting at a distance between two objects, obeying the well-known inverse square law, and occurring in a spacetime that was inflexible, and had a single frame of reference.

Einstein looked into the nature of space and time and realized they are flexible. Yet general relativity is still a classical theory, without quantum behavior. And it presupposes a continuous fabric for space.

As John Wheeler said, “spacetime tells matter how to move; matter tells spacetime how to curve”. Now Wheeler full well knew that not just matter, but also energy, curves spacetime.

A modest suggestion: invert Wheeler’s sentence. And then generalize it. Matter, and energy, tells spacetime how to be.

Which is more fundamental? Matter or spacetime?

Quantum theories of gravity seek to couple the known quantum fields with gravity, and it is expected that at the extremely small Planck scales, time and space both lose their continuous nature.

In physics, space and time are typically assumed as continuous backdrops.

But what if space is not fundamental at all? What if time is not fundamental? It is not difficult to conceive of time as merely an ordering of events. But space and time are to some extent interchangeable, as Einstein showed with special relativity.

So what about space? Is it just us placing rulers between objects, between masses?

Particle physicists are increasingly coming to the view that space, and time, are emergent. Not fundamental.

If emergent, from what? The concept is that particles, and quantum fields, for that matter, are entangled with one another. Their microscopic quantum states are correlated. The phenomenon of quantum entanglement has been studied in the laboratory and is well proven.

Chinese scientists have even, just last year, demonstrated quantum entanglement of photons over a satellite uplink with a total path exceeding 1200 kilometers.

Quantum entanglement thus becomes the thread Nature uses to stitch together the fabric of space. And as the degree of quantum entanglement changes the local curvature of the fabric changes. As the curvature changes, matter follows different paths. And that is gravity in action.

Newton’s laws are an approximation of general relativity for the case of small accelerations. But if space is not a continuous fabric and results from quantum entanglement, then for very small accelerations (in a sub-Newtonian range) both Newton dynamics and general relativity may be incomplete.

The connection between gravity and thermodynamics has been around for four 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. This area law entropy approach can be used to derive general relativity as Ted Jacobson did in 1995.

But it may be that the supposed area law component is insufficient; according to Erik Verlinde’s new emergent gravity hypothesis, there is also a volume law component for entropy, that must be considered due to dark energy and when accelerations are very low.

We have had hints about this incomplete description of gravity in the velocity measurements made at the outskirts of galaxies during the past eight decades. Higher velocities than expected are seen, reflecting higher acceleration of stars and gas than Newton (or Einstein) would predict. We can call this dark gravity.

Now this dark gravity could be due to dark matter. Or it could just be modified gravity, with extra gravity over what we expected.

It has been understood since the work of Mordehai Milgrom in the 1980s that the excess velocities that are observed are better correlated with extra acceleration than with distance from the galactic center.

Stacey McGaugh and collaborators have demonstrated a very tight correlation between the observed accelerations and the expected Newtonian acceleration, as I discussed in a prior blog here. The extra acceleration kicks in below a few times $10^{-10}$ meters per second per second (m/s²).

This is suspiciously close to the speed of light divided by the age of the universe! Which is about $7 \cdot 10^{-10}$ m/s².

Why should that be? The mass/energy density (both mass and energy contribute to gravity) of the universe is dominated today by dark energy.

The canonical cosmological model has 70% dark energy, 25% dark matter, and 5% ordinary matter. In fact if there is no dark matter, just dark gravity, or dark acceleration, then it could be more like a 95% and 5% split between dark energy and (ordinary) matter components.

A homogeneous universe composed only of dark energy in general relativity is known as a de  Sitter (dS) universe. Our universe is, at present, basically a dS universe ‘salted’ with matter.

Then one needs to ask how does gravity behave in dark energy influenced domains? Now unlike ordinary matter, dark energy is highly uniformly distributed on the largest scales. It is driving an accelerated expansion of the universe (the fabric of spacetime!) and dragging the ordinary matter along with it.

But where the density of ordinary matter is high, dark energy is evacuated. An ironic thought, since dark energy is considered to be vacuum energy. But where there is lots of matter, the vacuum is pushed aside.

That general concept was what Erik Verlinde used to derive an extra acceleration formula in 2016. He modeled an emergent, entropic gravity due to ordinary matter and also due to the interplay between dark energy and ordinary matter.  He modeled the dark energy as responding like an elastic medium when it is displaced within the vicinity of matter. Using this analogy with elasticity, he derived an extra acceleration as proportional to the square root of the product of the usual Newtonian acceleration and a term related to the speed of light divided by the universe’s age. This leads to a 1/r force law for the extra component since Newtonian acceleration goes as 1/r².

## $g _D = sqrt {(a_0 \cdot g_B / 6 )}$

Verlinde’s dark gravity depends on the square root of the product of a characteristic acceleration a0 and ordinary Newtonian (baryonic) gravity, gB

The idea is that the elastic, dark energy medium, relaxes over a cosmological timescales. Matter displaces energy and entropy from this medium, and there is a back reaction of the dark energy on matter that is expressed as a volume law entropy. Verlinde is able to show that this interplay between the matter and dark energy leads precisely to the characteristic acceleration is $a_0 / 6 = c \cdot H / 6$, where H is the Hubble expansion parameter and is equal to one over the age of the universe for a dS universe. This turns out be the right value of just over $10^{-10}$ m/s² that matches observations.

In our solar system, and indeed in the central regions of galaxies, we see gravity as the interplay of ordinary matter and other ordinary matter. We are not used to this other dance.

Domains of gravity

 Acceleration Domain Gravity vis-a-vis Newtonian formula Examples High (GM/R ~ c²) Einstein, general relativity Higher Black holes, neutron stars Normal Newtonian dynamics 1/r² Solar system, Sun orbit in Milky Way Very low (< c/ age of U.) Dark Gravity Higher, additional 1/r term Outer edges of galaxies, dwarf galaxies, clusters of galaxies

The table above summarizes three domains for gravity: general relativity, Newtonian, and dark gravity, the latter arising at very low accelerations. We are always calculating gravity incorrectly! Usually, such as in our solar system, it matters not at all. For example at the Earth’s surface gravity is 11 orders of magnitude greater than the very low acceleration domain where the extra term kicks in.

Recently, Alexander Peach, a Teaching Fellow in physics at Durham University, has taken a different angle based on Verlinde’s original, and much simpler, exposition of his emergent gravity theory in his 2010 paper. He derives an equivalent result to Verlinde’s in a way which I believe is easier to understand. He assumes that holography (the assumption that all of the entropy can be calculated as area law entropy on a spherical screen surrounding the mass) breaks down at a certain length scale. To mimic the effect of dark energy in Verlinde’s new hypothesis, Peach adds a volume law contribution to entropy which competes with the holographic area law at this certain length scale. And he ends up with the same result, an extra 1/r entropic force that should be added for correctness in very low acceleration domains.

In figure 2 (above) from Peach’s paper he discusses a test particle located beyond a critical radius $r_c$ for which volume law entropy must also be considered. Well within $r_c$  (shown in b) the dark energy is fully displaced by the attracting mass located at the origin and the area law entropy calculation is accurate (indicated by the shaded surface). Beyond $r_c$ the dark energy effect is important, the holographic screen approximation breaks down, and the volume entropy must be included in the contribution to the emergent gravitational force (shown in c). It is this volume entropy that provides an additional 1/r term for the gravitational force.

Peach makes the assumption that the bulk and boundary systems are in thermal equilibrium. The bulk is the source of volume entropy. In his thought experiment he models a single bit of information corresponding to the test particle being one Compton wavelength away from the screen, just as Verlinde initially did in his description of emergent Newtonian gravity in 2010. The Compton wavelength is equal to the wavelength a photon would have if its energy were equal to the rest mass energy of the test particle. It quantifies the limitation in measuring the position of a particle.

Then the change in boundary (screen) entropy can be related to the small displacement of the particle. Assuming thermal equilibrium and equipartition within each system and adopting the first law of thermodynamics, the extra entropic force can be determined as equal to the Newtonian formula, but replacing one of the r terms in the denominator by $r_c$ .

To understand $r_c$ , for a given system, it is the radius at which the extra gravity is equal to the Newtonian calculation, in other words, gravity is just twice as strong as would be expected at that location. In turn, this traces back to the fact that, by definition, it is the length scale beyond which the volume law term overwhelms the holographic area law.

It is thus the distance at which the Newtonian gravity alone drops to about $1.2 \cdot 10^{-10}$ m/s², i.e. $c \cdot H / 6$, for a given system.

So Peach and Verlinde use two different methods but with consistent assumptions to model a dark gravity term which follows a 1/r force law. And this kicks in at around $10^{-10}$ m/s².

The ingredients introduced by Peach’s setup may be sufficient to derive a covariant theory, which would entail a modified version of general relativity that introduces new fields, which could have novel interactions with ordinary matter. This could add more detail to the story of covariant emergent gravity already considered by Hossenfelder (2017), and allow for further phenomenological testing of emergent dark gravity. Currently, it is not clear what the extra degrees of freedom in the covariant version of Peach’s model should look like. It may be that Verlinde’s introduction of elastic variables is the only sensible option, or it could be one of several consistent choices.

With Peach’s work, physicists have taken another step in understanding and modeling dark gravity in a fashion that obviates the need for dark matter to explain our universe

We close with another of John Wheeler’s sayings:

“The only thing harder to understand than a law of statistical origin would be a law that is not of statistical origin, for then there would be no way for it—or its progenitor principles—to come into being. On the other hand, when we view each of the laws of physics—and no laws are more magnificent in scope or better tested—as at bottom statistical in character, then we are at last able to forego the idea of a law that endures from everlasting to everlasting. “

It is a pleasure to thank Alexander Peach for his comments on, and contributions to, this article.

References:

https://darkmatterdarkenergy.com/2018/08/02/dark-acceleration-the-acceleration-discrepancy/ blog “Dark Acceleration: The Acceleration Discrepancy”

https://arxiv.org/abs/gr-qc/9504004 “Thermodynamics of Spacetime: The Einstein Equation of State” 1995, Ted Jacobson

https://darkmatterdarkenergy.com/2017/07/13/dark-energy-and-the-comological-constant/ blog “Dark Energy and the Cosmological Constant”

https://darkmatterdarkenergy.com/2016/12/30/emergent-gravity-verlindes-proposal/ blog “Emergent Gravity: Verlinde’s Proposal”

https://arxiv.org/pdf/1806.10195.pdf “Emergent Dark Gravity from (Non) Holographic Screens” 2018, Alexander Peach

https://arxiv.org/pdf/1703.01415.pdf “A Covariant Version of Verlinde’s Emergent Gravity” Sabine Hossenfelder

## 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, 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

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: 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.

If 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.

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/ sqrt(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).

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.

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

## 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.

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.

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

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.

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

## Dark Gravity: Is Gravity Thermodynamic?

This is the first in a series of articles on ‘dark gravity’ that look at emergent gravity and modifications to general relativity. In my book Dark Matter, Dark Energy, Dark Gravity I explained that I had picked Dark Gravity to be part of the title because of the serious limitations in our understanding of gravity. It is not like the other 3 forces; we have no well accepted quantum description of gravity. And it is some 33 orders of magnitude weaker than those other forces.
I noted that:

The big question here is ~ why is gravity so relatively weak, as compared to the other 3 forces of nature? These 3 forces are the electromagnetic force, the strong nuclear force, and the weak nuclear force. Gravity is different ~ it has a dark or hidden side. It may very well operate in extra dimensions… http://amzn.to/2gKwErb

My major regret with the book is that I was not aware of, and did not include a summary of, Erik Verlinde’s work on emergent gravity. In emergent gravity, gravity is not one of the fundamental forces at all.

Erik Verlinde is a leading string theorist in the Netherlands who in 2009 proposed that gravity is an emergent phenomenon, resulting from the thermodynamic entropy of the microstates of quantum fields.

In 2009, Verlinde showed that the laws of gravity may be derived by assuming a form of the holographic principle and the laws of thermodynamics. This may imply that gravity is not a true fundamental force of nature (like e.g. electromagnetism), but instead is a consequence of the universe striving to maximize entropy. – Wikipedia article “Erik Verlinde”

This year, Verlinde extended this work from an unrealistic anti-de Sitter model of the universe to a more realistic de Sitter model. Our runaway universe is approaching a dark energy dominated deSitter solution.

He proposes that general relativity is modified at large scales in a way that mimics the phenomena that have generally been attributed to dark matter. This is in line with MOND, or Modified Newtonian Dynamics. MOND is a long standing proposal from Mordehai Milgrom, who argues that there is no dark matter, rather that gravity is stronger at large distances than predicted by general relativity and Newton’s laws.

In a recent article on cosmology and the nature of gravity Dr.Thanu Padmanabhan lays out 6 issues with the canonical Lambda-CDM cosmology based on general relativity and a homogeneous, isotropic, expanding universe. Observations are highly supportive of such a canonical model, with a very early inflation phase and with 1/3 of the mass-energy content in dark energy and 2/3 in matter, mostly dark matter.

And yet,

1. The equation of state (pressure vs. density) of the early universe is indeterminate in principle, as well as in practice.

2. The history of the universe can be modeled based on just 3 energy density parameters: i) density during inflation, ii) density at radiation – matter equilibrium, and iii) dark energy density at late epochs. Both the first and last are dark energy driven inflationary de Sitter solutions, apparently unconnected, and one very rapid, and one very long lived. (No mention of dark matter density here).

3. One can construct a formula for the information content at the cosmic horizon from these 3 densities, and the value works out to be 4π to high accuracy.

4. There is an absolute reference frame, for which the cosmic microwave background is isotropic. There is an absolute reference scale for time, given by the temperature of the cosmic microwave background.

5. There is an arrow of time, indicated by the expansion of the universe and by the cooling of the cosmic microwave background.

6. The universe has, rather uniquely for physical systems, made a transition from quantum behavior to classical behavior.

“The evolution of spacetime itself can be described in a purely thermodynamic language in terms of suitably defined degrees of freedom in the bulk and boundary of a 3-volume.”

Now in fluid mechanics one observes:

“First, if we probe the fluid at scales comparable to the mean free path, you need to take into account the discreteness of molecules etc., and the fluid description breaks down. Second, a fluid simply might not have reached local thermodynamic equilibrium at the scales (which can be large compared to the mean free path) we are interested in.”

Now it is well known that general relativity as a classical theory must break down at very small scales (very high energies). But also with such a thermodynamic view of spacetime and gravity, one must consider the possibility that the universe has not reached a statistical equilibrium at the largest scales.

One could have reached equilibrium at macroscopic scales much less than the Hubble distance scale c/H (14 billion light-years, H is the Hubble parameter) but not yet reached it at the Hubble scale. In such a case the standard equations of gravity (general relativity) would apply only for the equilibrium region and for accelerations greater than the characteristic Hubble acceleration scale of  $c \cdot H$ (2 centimeters per second / year).

This lack of statistical equilibrium implies the universe could behave similarly to non-equilibrium thermodynamics behavior observed in the laboratory.

The information content of the expanding universe reflects that of the quantum state before inflation, and this result is 4π in natural units by information theoretic arguments similar to those used to derive the entropy of a black hole.

The black hole entropy is  $S = A / (4 \cdot Lp^2)$ where A is the area of the black hole using the Schwarzschild radius formula and Lp is the Planck length, $G \hbar / c^3$ , where G is the gravitational constant, $\hbar$ is Planck’s constant.

This beautiful Bekenstein-Hawking entropy formula connects thermodynamics, the quantum world  and gravity.

This same value of the universe’s entropy can also be used to determine the number of e-foldings during inflation to be 6 π² or 59, consistent with the minimum value to enforce a sufficiently homogeneous universe at the epoch of the cosmic microwave background.

If inflation occurs at a reasonable ~ $10^{15}$ GeV, one can derive the observed value of the cosmological constant (dark energy) from the information content value as well, argues Dr. Padmanhaban.

This provides a connection between the two dark energy driven de Sitter phases, inflation and the present day runaway universe.

The table below summarizes the 4 major phases of the universe’s history, including the matter dominated phase, which may or may not have included dark matter. Erik Verlinde in his new work, and Milgrom for over 3 decades, question the need for dark matter.

Epoch  /  Dominated  /   Ends at  /   a-t scaling  /   Size at end

Inflation /  Inflaton (dark energy) / $10^{-32}$ seconds / $e^{Ht}$ (de Sitter) / 10 cm

Radiation / Radiation / 40,000 years / $\sqrt t$ /  10 million light-years

Matter / Matter (baryons) Dark matter? /  9 billion light-years / $t^{2/3}$ /  > 100 billion light-years

Runaway /  Dark energy (Cosmological constant) /  “Infinity” /  $e^{Ht}$ (de Sitter) / “Infinite”

In the next article I will review the status of MOND – Modified Newtonian Dynamics, from the phenomenology and observational evidence.

References

E. Verlinde. “On the Origin of Gravity and the Laws of Newton”. JHEP. 2011 (04): 29 http://arXiv.org/abs/1001.0785

T. Padmanabhan, 2016. “Do We Really Understand the Cosmos?” http://arxiv.org/abs/1611.03505v1

S. Perrenod, 2011. Dark Matter, Dark Energy, Dark Gravity 2011  http://amzn.to/2gKwErb

S. Carroll and G. Remmen, 2016, http://www.preposterousuniverse.com/blog/2016/02/08/guest-post-grant-remmen-on-entropic-gravity/