# Tag Archives: Dark energy

## Does Dark Energy Vary with Time?

Einstein introduced the concept of dark energy 100 years ago.

The Concordance Lambda-Cold Dark Matter cosmology appears to fit observations of the cosmic microwave background and other cosmological observations including surveys of large-scale galaxy grouping exceedingly well.

In this model, Lambda is shorthand for the dark energy in the universe. It was introduced as the greek letter Λ into the equations of general relativity, by Albert Einstein, as an unvarying cosmological constant.

Measurements of Λ indicate that dark energy accounts for about 70% of the total energy content of the universe. The remainder is found in dark matter and ordinary matter, and about 5/6 of that is in the form of dark matter.

Alternative models of gravity, with extra gravity in very low acceleration environments, may replace apparent dark matter with this extra gravity, perhaps due to interaction between dark energy and ordinary matter.

The key point about dark energy is that while it has a positive energy, it rather strangely has a negative pressure. In the tensor equations of general relativity the pressure terms act as a negative gravity, driving an accelerated expansion of the universe.

In fact our universe is headed toward a state of doubling in scale in each dimension every 11 or 12 billion years. In the next trillion years we are looking at 80 or 90 such repeated doublings.

That assumes that dark energy is constant per volume over time, with a value equivalent to two proton – antiproton pair annihilations per cubic meter (4 GeV / m³).

But is it?

The Dark Energy Survey results seem to say so. This experiment looked at 26 million galaxies for the clustering patterns, and also gravitational lensing (Einstein taught us that mass bends light paths).

They determined the parameter w for dark energy and found it to be consistent with -1.0 as expected for the cosmological constant model of unvarying dark energy. See this blog for details:

https://darkmatterdarkenergy.com/2017/08/10/dark-energy-survey-first-results-canonical-cosmology-supported/

The pressure – energy density relation is:

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

The parameter w elucidates the relation between the energy density given by ρ and the pressure P. This is called the equation of state. Matter and radiation have w >= 0. In order to have dark energy with a negative pressure dominating, then w should be < -1/3. And w = -1 gives us the cosmological constant form.

Image credit: www.scholarpedia.org

Cosmologists seek to determine w, and whether it varies over time scales of billions of years.

The Concordance model is not very well tested at high redshifts with z > 1 (corresponding to epochs of the universe less than half the current age) other than with the cosmic microwave background data. Recently two Italian researchers, Risaliti and Lusso have examined datasets of high-redshift quasars to investigate whether the Concordance model fits.

Typically supernovae are employed for the redshift-distance relation, and cosmological models are tested against the observed relationship, known as the Hubble diagram. The authors use X-ray and ultraviolet fluxes of quasars to extend the diagram to high redshifts (greater distances, earlier epochs), and calibrate observed quasar luminosities with the supernovae data sets.

Their analysis drew from a sample of 1600 quasars with redshifts up to 5 and including a new sample of 30 high redshift z ~ 3 quasars, observed with the European XMM-Newton satellite.

They claim a 4 standard deviation variance for z > 2, a reasonably high significance.

Models with a varying w include quintessence models, with time-varying scalar fields. If w decreases below -1, it is known as phantom energy. Their results are suggestive of a value of w < -1, corresponding to a dark or phantom energy increasing with time.

For convenience cosmologists introduce a second parameter for possible evolution in w, writing as:

w = w0 + wa*(1-a)   ,where a, the scale factor equals 1/(1+z) and a = 1 for present day.

The best fit results for their analysis are with w0 = -1.4 and wa ~ 1, but these results have large errors, as shown in Figure 4 above, from their paper. Their results are within the red (2 standard deviation, or σ) and orange (3σ) contours. The outer 3σ contours almost touch the cosmological constant point that has w0 = -1 and wa = 0.

These are intriguing results that require further investigation. They are antithetical to quintessence models, and apparently in tension with a simple cosmological constant.

The researchers plan on further analysis in future work by including Baryon Acoustic Oscillation (large scale galaxy clustering) measurements at z > 2.

References

https://darkmatterdarkenergy.com/2017/08/10/dark-energy-survey-first-results-canonical-cosmology-supported/ – Results from Dark Energy Survey of galaxies

Risaliti, G. and Lusso, E. 2018 Cosmological constraints from the Hubble diagram of quasars at high redshifts https://arxiv.org/abs/1811.02590

## 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

## 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.

## 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: darkenergysurvey.org

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

References

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

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

## Dark Energy and the Cosmological 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.

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

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

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

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

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

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

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

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

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

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

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

$\rho_m \approx 1/3$

$\rho_{de} \approx 2/3$

and thus

$p_{de} \approx - 2$

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

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

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

Image: NASA/STScI, public domain

## 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” https://arxiv.org/pdf/1607.08797.pdf . They simulate structure growth while estimating the local values of expansion parameter in many regions as their model evolves.

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

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

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

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

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

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.

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

## 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/

## 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

 Redshift Dark Energy percent Matter percent 0 68 32 0.25 52 48 0.5 39 61 0.75 28 72 1.0 21 79 1.5 12 88

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.

References:

1. Morandi, M. Sun arXiv:1601.03741v3 [astro-ph.CO] 4 Feb 2016, “Probing dark energy via galaxy cluster outskirts”
2. http://chandra.harvard.edu/photo/2016/clusters/