Category Archives: Dark Energy

Unified Physics including Dark Matter and Dark Energy

Dark matter keeps escaping direct detection, whether it might be in the form of WIMPs, or primordial black holes, or axions. Perhaps it is a phantom and general relativity is inaccurate for very low accelerations. Or perhaps we need a new framework for particle physics other than what the Standard Model and supersymmetry provide.

We are pleased to present a guest post from Dr. Thomas J. Buckholtz. He introduces us to a theoretical framework referred to as CUSP, that results in four dozen sets of elementary particles. Only one of these sets is ordinary matter, and the framework appears to reproduce the known fundamental particles. CUSP posits ensembles that we call dark matter and dark energy. In particular, it results in the approximate 5:1 ratio observed for the density of dark matter relative to ordinary matter at the scales of galaxies and clusters of galaxies. (If interested, after reading this post, you can read more at his blog linked to his name just below).

Thomas J. Buckholtz

My research suggests descriptions for dark matter, dark energy, and other phenomena. The work suggests explanations for ratios of dark matter density to ordinary matter density and for other observations. I would like to thank Stephen Perrenod for providing this opportunity to discuss the work. I use the term CUSP – concepts uniting some physics – to refer to the work. (A book, Some Physics United: With Predictions and Models for Much, provides details.)

CUSP suggests that the universe includes 48 sets of elementary-particle Standard Model elementary particles and composite particles. (Known composite particles include the proton and neutron.) The sets are essentially (for purposes of this blog) identical. I call each instance an ensemble. Each ensemble includes its own photon, Higgs boson, electron, proton, and so forth. Elementary particle masses do not vary by ensemble. (Weak interaction handedness might vary by ensemble.)

One ensemble correlates with ordinary matter, 5 ensembles correlate with dark matter, and 42 ensembles contribute to dark energy densities. CUSP suggests interactions via which people might be able to detect directly (as opposed to infer indirectly) dark matter ensemble elementary particles or composite particles. (One such interaction theoretically correlates directly with Larmor precession but not as directly with charge or nominal magnetic dipole moment. I welcome the prospect that people will estimate when, if not now, experimental techniques might have adequate sensitivity to make such detections.)


This explanation may describe (much of) dark matter and explain (at least approximately some) ratios of dark matter density to ordinary matter density. You may be curious as to how I arrive at suggestions CUSP makes. (In addition, there are some subtleties.)

Historically regarding astrophysics, the progression ‘motion to forces to objects’ pertains. For example, Kepler’s work replaced epicycles with ellipses before Newton suggested gravity. CUSP takes a somewhat reverse path. CUSP models elementary particles and forces before considering motion. The work regarding particles and forces matches known elementary particles and forces and extrapolates to predict other elementary particles and forces. (In case you are curious, the mathematics basis features solutions to equations featuring isotropic pairs of isotropic quantum harmonic oscillators.)

I (in effect) add motion by extending CUSP to embrace symmetries associated with special relativity. In traditional physics, each of conservation of angular momentum, conservation of momentum, and boost correlates with a spatial symmetry correlating with the mathematics group SU(2). (If you would like to learn more, search online for “conservation law symmetry,” “Noether’s theorem,” “special unitary group,” and “Poincare group.”) CUSP modeling principles point to a need to add to temporal symmetry and, thereby, to extend a symmetry correlating with conservation of energy to correlate with the group SU(7). The number of generators of a group SU(n) is n2−1. SU(7) has 48 generators. CUSP suggests that each SU(7) generator correlates with a unique ensemble. (In case you are curious, the number 48 pertains also for modeling based on either Newtonian physics or general relativity.)

CUSP math suggests that the universe includes 8 (not 1 and not 48) instances of traditional gravity. Each instance of gravity interacts with 6 ensembles.

The ensemble correlating with people (and with all things people see) connects, via our instance of gravity, with 5 other ensembles. CUSP proposes a definitive concept – stuff made from any of those 5 ensembles – for (much of) dark matter and explains (approximately) ratios of dark matter density to ordinary matter density for the universe and for galaxy clusters. (Let me not herein do more than allude to other inferably dark matter based on CUSP-predicted ordinary matter ensemble composite particles; to observations that suggest that, for some galaxies, the dark matter to ordinary matter ratio is about 4 to 1, not 5 to 1; and other related phenomena with which CUSP seems to comport.)

CUSP suggests that interactions between dark matter plus ordinary matter and the seven peer combinations, each comprised of 1 instance of gravity and 6 ensembles, is non-zero but small. Inferred ratios of density of dark energy to density of dark matter plus ordinary matter ‘grow’ from zero for observations pertaining to somewhat after the big bang to 2+ for observations pertaining to approximately now. CUSP comports with such ‘growth.’ (In case you are curious, CUSP provides a nearly completely separate explanation for dark energy forces that govern the rate of expansion of the universe.)

Relationships between ensembles are reciprocal. For each of two different ensembles, the second ensemble is either part of the first ensemble’s dark matter or part of the first ensemble’s dark energy. Look around you. See what you see. Assuming that non-ordinary-matter ensembles include adequately physics-savvy beings, you are looking at someone else’s dark matter and yet someone else’s dark energy stuff. Assuming these aspects of CUSP comport with nature, people might say that dark matter and dark-energy stuff are, in effect, quite familiar.

Copyright © 2018 Thomas J. Buckholtz



Primordial Black Holes and Dark Matter

Based on observed gravitational interactions in galactic halos (galaxy rotation curves) and in group and clusters, there appears to be 5 times as much dark matter as ordinary matter in the universe. The alternative is no dark matter, but more gravity than expected at low accelerations, as discussed in this post on emergent gravity.

The main candidates for dark matter are exotic, undiscovered particles such as WIMPs (weakly interacting massive particles) and axions. Experiments attempting direct detection for these have repeatedly come up short.

The non-particle alternative category is MACHOs (massive compact halo objects) composed of ordinary matter.  Planets, dwarf stars and neutron stars have been ruled out by various observational signatures. The one ordinary matter possibility that has remained viable is that of black holes, and in particular black holes with much less than the mass of the Sun.

The only known possibility for such low mass black holes is that of primordial black holes (PBHs) formed in the earliest moments of the Big Bang.

Gravitational microlensing, or microlensing for short, seeks to detect PBHs by their general relativistic gravitational effect on starlight. MACHO and EROS were experiments to monitor stars in the Large Magellanic Cloud. These were able to place limits on the abundance of PBHs with masses from about one hundred millionth of a the Sun’s mass up to 10 solar masses. PBHs from that mass range are not able to explain the total amount of dark matter determined from gravitational interactions.

LIGO has recently detected several merging black holes in the tens of solar mass range. However the frequency of LIGO detections appears too low by two orders of magnitude to explain the amount of gravitationally detected dark matter. PBHs in this mass range are also constrained by cosmic microwave background observations.

Extremely low mass PBHs, below 10 billion tons, cannot survive until the present epoch of the universe. This is due to Hawking radiation. Black holes evaporate due to their quantum nature. Solar mass black holes have an extremely long lifetime against evaporation. But very low mass black holes will evaporate in billions of years or much sooner, depending on mass.

The remaining mass window for possible PBH, in sufficient amount to explain dark matter, is from about 10 trillion ton objects up to those with ten millionths of the Sun’s mass.


Figure 5 from H. Niikura et al. “Microlensing constraints on primordial black holes with the Subaru/HSC Andromeda observation”,  

Here f is the fraction of dark matter which can be explained by PBHs. The red shaded area is excluded by the authors observations and analysis of Andromeda Galaxy data. This rules out masses above 100 trillion tons and below a hundred thousandth of the Sun’s mass. (Solar mass units used above and grams are used below).


Now, a team of Japanese astronomers have used the Subaru telescope on the Big Island of Hawaii (operated by Japan’s national observatory) to determine constraints on PBHs by observing millions of stars in the Andromeda Galaxy.

The idea is that a candidate PBH would pass in front of the line of sight to the star, acting as a lens, and magnifying the light from the star in question for a relatively brief period of time. The astronomers looked for stars exhibiting variability in their light intensity.

With only a single nights’ data they made repeated short exposures and were able to pick out over 15,000 stars in Andromeda exhibiting such variable light intensity. However, among these possible candidates, only a single one turned out to fit the characteristics expected for a PBH detection.

If PBHs in this mass range were sufficiently abundant to explain dark matter, then one would have expected of order one thousand events, and they saw nothing like this number. In summary, with 95% confidence, they are able to rule out PBHs as the main source of dark matter for the mass range from 100 trillion tons up to one hundred thousandth of the Sun’s mass.

The window for primordial black holes as the explanation for dark matter appears to be closing.





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.


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

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.


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, “Dark Energy Survey Year 1 Results: Cosmological Constraints from Galaxy Clustering and Weak Lensing”, 2017, T. Abbott et al., Wikipedia article on weak gravitational lensing discusses galaxy-galaxy lensing and cosmic shear

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.



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

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.


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” . 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

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 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.


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.






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.


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).


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


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

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…

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.


E. Verlinde. “On the Origin of Gravity and the Laws of Newton”. JHEP. 2011 (04): 29

T. Padmanabhan, 2016. “Do We Really Understand the Cosmos?”

S. Perrenod, 2011.

S. Perrenod, 2011. Dark Matter, Dark Energy, Dark Gravity 2011

S. Carroll and G. Remmen, 2016,

Galaxy Clusters Probe Dark Energy

Rich (large) clusters of galaxies are significant celestial X-ray sources. In fact, large clusters of galaxies typically contain around 10 times as much mass in the form of very hot gas as is contained in their constituent galaxies.

Moreover, the dark matter content of clusters is even greater than the gas content; typically it amounts to 80% to 90% of the cluster mass. In fact, the first detection of dark matter’s gravitational effects was made by Fritz Zwicky in the 1930s. His measurements indicated that the galaxies were moving around much faster than expected from the known galaxy masses within the cluster.


Image credit: X-ray: NASA/CXC/Univ. of Alabama/A. Morandi et al; Optical: SDSS, NASA/STScI (X-ray emission is shown in purple)

The dark matter’s gravitational field controls the evolution of a cluster. As a cluster forms via gravitational collapse, ordinary matter falling into the strong gravitational field interacts via frictional processes and shocks and thermalizes at a high temperature in the range of 10 to 100 million degrees (Kelvins). The gas is so hot, that it emits X-rays due to thermal bremsstrahlung.

Recently, Drs. Morandi and Sun at the University of Alabama have implemented a new test of dark energy using the observed X-ray emission profiles of clusters of galaxies. Since clusters are dominated by the infall of primordial gas (ordinary matter) into dark matter dominated gravitational wells, then X-ray emission profiles – especially in the outer regions of clusters – are expected to be similar, after correcting for temperature variations and the redshift distance. Their analysis also considers variation in gas fraction with redshift; this is found to be minimal.

Because of the self similar nature of the X-ray emission profiles, X-ray clusters of galaxies can serve as cosmological probes, a type of ‘standard candle’. In particular, they can be used to probe dark energy, and to look at the possibility of the variation of the strength of dark energy over multi-billion year cosmological time scales.

The reason this works is that cluster development and mass growth, and corresponding temperature increase due to stronger gravitational potential wells, are essentially a tradeoff of dark matter and dark energy. While dark matter causes a cluster to grow, dark energy inhibits further growth.

This varies with the redshift of a cluster, since dark energy is constant per unit volume as the universe expands, but dark matter was denser in the past in proportion to (1 + z)^3, where z is the cluster redshift. In the early universe, dark matter thus dominated, as it had a much higher density, but in the last several billion years, dark energy has come to dominate and impede further growth of clusters.

The table below shows the percentage of the mass-energy of the universe which is in the form of dark energy and in the form of matter (both dark and ordinary) at a given redshift, assuming constant dark energy per unit volume. This is based on the best estimate from Planck of 68% of the total mass-energy density due to dark energy at present (z = 0). Higher redshift means looking farther back in time. At z = 0.5, around 5 billion years ago, matter still dominated over dark energy, but by around z = 0.3 the two are about equal and since then (for smaller z) dark energy has dominated. It is only since after the Sun and Earth formed that the universe has entered the current dark energy dominated era.

Table: Total Matter & Dark Energy Percentages vs. z 


Dark Energy percent

Matter percent



















The authors analyzed data from a large sample consisting of 320 clusters of galaxies observed with the Chandra X-ray Observatory. The clusters ranged in redshifts from 0.056 up to 1.24 (almost 9 billion years ago), and all of the selected clusters had temperatures measured to be equal to or greater than 3 keV (above 35 million Kelvins). For such hot clusters, non-gravitational astrophysical effects, are expected to be small.

Their analysis evaluated the equation of state parameter, w, of dark energy. If dark energy adheres to the simplest model, that of the cosmological constant (Λ) found in the equations of general relativity, then w = -1 is expected.

The equation of state governs the relationship between pressure and energy density; dark energy is observed to have a negative pressure, for which w < 0, unlike for matter.

Their resulting value for the equation of state parameter is

w = -1.02 +/- 0.058,

equal to -1 within the statistical errors.

The results from combining three other experiments, namely

  1. Planck satellite cosmic microwave background (CMB) measurements
  2. WMAP satellite CMB polarization measurements
  3. optical observations of Type 1a supernovae

yield a value

w = -1.09 +/- 0.19,

also consistent with a cosmological constant. And combining both the X-ray cluster results with the CMB and optical results yields a tight constraint of

w = -1.01 +/- 0.03.

Thus a simple cosmological constant explanation for dark energy appears to be a sufficient explanation to within a few percent accuracy.

The authors were also able to constrain the evolution in w and find, for a model with

w(z) = w(0) + wa * z / (1 + z), that the evolution parameter is zero within statistical errors:

wa = -0.12 +/- 0.4.

This is a powerful test of dark energy’s existence, equation of state, and evolution, using hundreds of X-ray clusters of galaxies. There is no evidence for evolution in dark energy with redshift back to around z = 1, and a simple cosmological constant model is supported by the data from this technique as well as from other methods.


  1. Morandi, M. Sun arXiv:1601.03741v3 [astro-ph.CO] 4 Feb 2016, “Probing dark energy via galaxy cluster outskirts”

Gravitational Waves and Dark Matter, Dark Energy

What does the discovery of gravitational waves imply about dark matter and dark energy?

The first detection of gravitational waves results from a pair of merging black holes, and is yet another magnificent confirmation of the theory of general relativity. Einstein’s theory of general relativity has passed every test thrown at it during the last 100 years.

While the existence of gravitational waves was fully expected to be confirmed, the discovery took several decades and represents a technological tour de force. Detected at the two LIGO sites, one in Louisiana and one in Washington State, the main event lasted only 0.2 seconds, and was seen as a change of length in the “arms” of the detector (laser interferometers) of only one part in a thousand billion billion.

LIGO signal 2

The LIGO detection of gravitational waves. The blue curve is from the Louisiana site and the red curve from the Washington state site. The two curves are shifted by 7 milliseconds to account for the speed-of-light delay between the two sites. Note that most of the power in the signal occurs within less than 0.2 seconds. The strain is a measure of proportional change in length of the detector arm and is less than 1 part in 10²¹.

Nevertheless, this is the most energetic event ever seen by mankind. The merger of two large black holes totaling over 60 times the Sun’s mass resulted in the conversion of 3 solar masses of material into gravitational wave energy. Imagine, there were 3 Suns worth of matter obliterated in the blink of an eye. During this brief period, the generated power was greater than that from the light of all of the stars of all of the galaxies in our known universe.

What the discovery of gravitational waves has to say about dark matter and dark energy is essentially that it further confirms their existence.

Although there is as of now no direct detection of dark matter, we infer the existence of dark matter by using the equations of general relativity (GR), in a number of cases, including:

  1. Gravitational lensing – Typically, a foreground cluster of galaxies distorts and magnifies the image of a background galaxy. GR is used to calculate the bending and magnification, primarily caused by the dark matter in the foreground cluster.
  2. Cosmic microwave background radiation (CMBR) – The CMBR has spatial fluctuation peaks (harmonics) and the first peak tells us about ordinary matter and the third peak about the density of dark matter. A GR-based cosmological model is used to determine the dark matter average density.

Dark matter is also inferred from the way in which galaxies rotate and from the velocities of galaxies within galaxy clusters, but general relativity is not needed to calculate the dark matter densities in such cases. However, results from these methods are consistent with results from the methods listed above.

In the case of dark energy, it turns out to be a parameter in the equations of general relativity as first formulated by Einstein. The parameter, lambda, (Λ) is known as the cosmological constant, and represents the minimum energy of the vacuum. For many years astronomers and cosmologists thought it might take the value of zero. However in 1998 multiple teams confirmed that the value is positive and not zero, and it turns out that dark energy has more than twice the energy content of dark matter. Its non-zero value is actually another stunning success for general relativity.

Thus the detection of gravitational waves indirectly provides further support for the canonical cosmological model ΛCDM, with both dark matter and dark energy, and fully consistent with general relativity.

References – ScienceMag article

B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett. 116, 061102 – Published 11 February 2016 –

NEW BOOK just released:

S. Perrenod, 2016, 72 Beautiful Galaxies (especially designed for iPad, iOS; ages 12 and up)