Tag Archives: X-ray clusters

Modified Gravity

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.


Brower, M. et al. 2016, “First test of Verlinde’s theory of Emergent Gravity using Weak Gravitational Lensing Measurements”

Brownstein, J. and Moffat, J. 2005, “Galaxy Cluster Masses without Non-baryonic Dark Matter”,

Perrenod, S. 1977 “The Evolution of Cluster X-ray Sources”…226..566P, thesis.


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”

Dark Matter: Made of Sterile Neutrinos?


Composite image of the Bullet Group showing galaxies, hot gas (shown in pink) and dark matter (indicated in blue). Credit: ESA / XMM-Newton / F. Gastaldello (INAF/IASF, Milano, Italy) / CFHTLS 

What’s more elusive than a neutrino? Why a sterile neutrino, of course. In the Standard Model of particle physics there are 3 types of “regular” neutrinos. The ghost-like neutrinos are electrically neutral particles with 1/2 integer spins and very small masses. Neutrinos are produced in weak interactions, for example when a neutron decays to a proton and an electron. The 3 types are paired with the electron and its heavier cousins, and are known as electron neutrinos, muon neutrinos, and tau neutrinos (νe, νμ, ντ).

A postulated extension to the Standard Model would allow a new type of neutrino, known as a sterile neutrino. “Sterile” refers to the fact that this hypothetical particle would not feel the standard weak interaction, but would couple to regular neutrino oscillations (neutrinos oscillate among the 3 types, and until this was realized there was consternation around the low number of solar neutrinos detected). Sterile neutrinos are more ghostly than regular neutrinos! The sterile neutrino would be a neutral particle, like other neutrino types, and would be a fermion, with spin 1/2. The number of types, and the respective masses, of sterile neutrinos (assuming they exist) is unknown. Since they are electrically neutral and do not feel the standard weak interaction they are very difficult to detect. But the fact that they are very hard to detect is just what makes them candidates for dark matter, since they still interact gravitationally due to their mass.

What about regular neutrinos as the source of dark matter? The problem is that their masses are too low, less than 1/3 of an eV (electron-Volt) total for the three types. They are thus “too hot” (speeds and velocity dispersions too high, being relativistic) to explain the observed properties of galaxy formation and clumping into groups and clusters. The dark matter should be “cold” or non-relativistic, or at least no more than “warm”, to correctly reproduce the pattern of galaxy groups, filaments, and clusters we observe in our Universe.

Constraints can be placed on the minimum mass for a sterile neutrino to be a good dark matter candidate. Observations of the cosmic microwave background and of hydrogen Lyman-alpha emission in quasar spectra have been used to set a lower bound of 2 keV for the sterile neutrino’s mass, if it is the predominant component of dark matter. A sterile neutrino with this mass or larger is expected to have a decay channel into a photon with half of the rest-mass energy and a regular (active) neutrino with half the energy.

A recent suggestion is that an X-ray emission feature seen at 3.56 keV (kilo-electron Volts) from galaxy clusters is a result of the decay of sterile neutrinos into photons with that energy plus active (regular) neutrinos with similar energy. This X-ray emission line has been seen in a data set from the XMM-Newton satellite that stacks results from 73 clusters of galaxies together. The line was detected in 2 different instruments with around 4 or 5 standard deviations significance, so the existence of the line itself is on a rather strong footing. However, it is necessary to prove that the line is not from an atomic transition from argon or some other element. The researchers argue that an argon line should be much, much weaker than the feature that is detected.

In addition, a second team of researchers, also using XMM-Newton data have claimed detection of lines at the same 3.56 keV energy in the Perseus cluster of galaxies as well as our neighbor, the Andromeda galaxy.

There are no expected atomic transition lines at this energy, so the dark matter decay possibility has been suggested by both teams. An argon line around 3.62 KeV is a possible influence on the signal, but is expected to be very much weaker. Confirmation of these XMM-Newton results are required from other experiments in order to gain more confidence in the reality of the 3.56 keV feature, regardless of its cause, and to eliminate with certainty the possibility of an atomic transition origin. Analysis of stacked galaxy cluster data is currently underway for two other X-ray satellite missions, Chandra and Suzaku. In addition, the astrophysics community eagerly awaits the upcoming Astro-H mission, a Japanese X-ray astronomy satellite planned for launch in 2015. It should be able to not only confirm the 3.56 keV X-ray line (if indeed real), but also detect it within our own Milky Way galaxy.

Thus the hypothesis is for dark matter composed primarily of sterile neutrinos of a little over 7.1 keV in mass (in E = mc^2 terms), and that the sterile neutrino has a decay channel to an X-ray photon and regular neutrino. Each decay product would have an energy of about 3.56 keV. Such a 7 keV sterile neutrino is plausible with respect to the known density of dark matter and various cosmological and particle physics constraints. If the dark matter is primarily due to this sterile neutrino, then it falls into the “warm” dark matter domain, intermediate between “cold” dark matter due to very heavy particles, or “hot” dark matter due to very light particles.

The abundance of dwarf satellite galaxies found in the Milky Way’s neighborhood is lower than predicted from cold dark matter models. Warm dark matter could solve this problem. As Dr. Abazajian puts in in his recent paper “Resonantly Produced 7 keV Sterile Neutrino Dark Matter Models and the Properties of Milky Way Satellites”

the parameters necessary in these models to produce the full dark matter density fulfill previously determined requirements to successfully match the Milky Way galaxy’s total satellite abundance, the satellites’ radial distribution, and their mass density profile..