Monday, July 20, 2020

The Peloton Universe


The Peloton Universe: speculations on gravitational drafting as an explanation for the dark matter problem and cosmic de-coupling as an explanation for dark energy

 WARNING:  Highly speculative and far-from-complete post based on a highly superficial understanding of cosmology. Most of the ideas presented here are probably wrong, but perhaps may prove of passing interest to some. If nothing else, some of the Figures may be interesting to review and consider. This is also a work in progress which I put on hold in 2018 and since I likely will not get a chance to take up again in any serious way for a while, I post it for whatever it may be worth.

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 Summary:  Based on analogous terrestrial-scale processes, I propose a form of gravitational drafting in which galactic-scale masses moving through space create a drafting region or wake in which following masses increase their velocities. This provides an alternative explanation for certain observations currently explained by dark matter. Similarly, a drafting or "peloton model" provides an alternative explanation for certain observations currently explained by dark energy.

I.    The dark matter problem

Five main observations are cited as evidence for dark matter. For a summary of these, see: https://medium.com/starts-with-a-bang/five-reasons-we-think-dark-matter-exists-a122bd606ba8.

I focus my proposed framework on three of these five: the unexpectedly high rotational velocity of matter on the galaxy peripheries, galaxy cluster collisions, and the cosmic microwave background (CMB). I look briefly at the other two.

 1.      The high velocities of matter on galaxy perimeters

Rubin (1983) and others observed that baryonic (visible, ordinary) matter observed at galactic peripheries moves at higher velocities than predicted by standard models; this observation is known as the flattening of rotation velocity curves (Sanders, 2014; among others), as shown in Figure 1.



Figure 1. Extended rotation curve of M33. Top curve shows actual (flattening) observed mass velocities, and the lower curve shows the expected velocities based on standard Newtonian equations (from Stefania deluca https://commons.wikimedia.org/wiki/File:M33_rotation_curve_HI.gif)

This increase in expected velocity has been explained by the presence of an invisible halo of matter around the galactic periphery, apparently first called dark matter by Zwicky in 1933 (Debono and Smoot, 2016). Gravitational lensing in the regions of galactic cluster and gaseous collisions is further evidence for dark matter (Crowe et al., 2006).  

Despite many active searches, the particles that should comprise dark matter have yet to be found (Sanders, 2014; physicsworld.com, July 2016). Until these particles are found, alternative explanations for the existing evidence for dark matter are reasonable. These include Milgrom’s modified Newtonian dynamics (“MOND”) (e.g. Milgrom, 2014; for a historical perspective and discussion of other proposed modifications to Newtonian dynamics, see Sanders, 2014), and more recently the “emergent gravity” theory of Verlinde (2017).

There are several criticisms of MOND (DeBono and Smoot, 2016). Two major ones are its failure to predict inhomogeneity and structure in the universe, which is well accounted for by a dark matter model (Dodelson, 2011); and the apparent passage of dark matter through baryonic matter during high velocity galaxy cluster collisions, such as has been proposed to occur for the Bullet Cluster (Clowe et al., 2006). See this presentation for an overview of how baryonic acoustic oscillations and gravitational lensing lead to the dark matter hypothesis: http://slideplayer.com/slide/6878195/



 II.   A novel conceptual framework

I propose that a “cosmological drafting” model to reconcile these problems.  I do not propose a mathematical model that can account for my proposed cosmological drafting framework and I confess my own limitations in mathematics are such that I am largely incapable of producing a mathematical model to support my proposal. Nonetheless, what follows are some ideas which perhaps brighter minds than mine will find interesting enough to test against existing data and mathematical models. 

1.      Gravitational drafting

In this conception, “energy saving” is synonymous with an effective increase in the gravitational force that causes matter to travel at higher velocities than it would in the absence of this energy-saving effect. The proposed effect is analogous to aerodynamic and hydrodynamic drafting in which masses in motion reduce energy expenditure, and effectively increase their velocities when trailing in optimum positions relative to leading bodies, I refer to this as a “gravitational drafting” effect generated in the trailing space behind matter in motion at critical velocities. 

 On the other hand, the effect is unlike aerodynamic and hydrodynamic drafting because there seems to be no drag force in space that slows the leading matter. But the proposal remains that space is reconfigured in the trailing regions behind matter moving through empty space, such that a wake region is generated.

1.1   Gravitational drafting and dark matter

In this conception, dark matter need not be present to account for certain cosmological observations, as prevailing theory demands.  Rather, the implication is that gravitational drafting is a property of space that emerges in a region of space that trails behind matter as it moves through space at critical velocities or energetic intensities, and is not a property of matter per se. I propose that cosmological observations otherwise explained by dark matter may be explained as gravitational drafting.  I suggest this drafting effect may be indicated by an alternative interpretation of available gravitational lensing images, which I explore in this post.

1.2       Gravitational drafting and dark energy

The second general suggestion, consistent with the first, is that dark energy, insofar as it is currently required to explain the observed rapid expansion of the universe about 5 billion years ago, may be explained by a phenomenon in which leading and following (drafting) baryonic matter, on a cosmic scale, de-coupled its drafting interactivity at a critical density in the expansion history of the universe. In this conception, such a “cosmic de-coupling event” caused an epoch of accelerated expansion between regions of matter that were previously coupled by gravitational drafting.  In this model, drafting matter actually decelerates in relation to the leading matter, but the observed effect is an accelerating expansion of the distance between de-coupled matter.


1.3       Proposition: the noted increase in galactic velocities is due to a velocity equalization effect when matter trails other matter at comparatively high velocity inside gravitational wake regions

I propose that the motion of mass through space creates a kind of gravitational trailing wake that acts to magnify the effective force of gravity, thus equalizing the velocities of trailing matter within the wake region and permitting trailing matter to travel at approximately equal speeds to leading massive objects.  The proposed phenomenon is like aerodynamic or hydrodynamic drafting in which drag is reduced in trailing wake regions behind bodies in motion, commonly observed in earth-scale systems.

The closest published theory to my proposition that I have found in the literature is that of Furlanettor and Loeb (2001), who proposed the existence of “hydrodynamic” gravitational wakes with a cone-like structure. In their paper, the authors describe the wake structure in gaseous matter (a “collisional medium”) and dark matter (a “collisionless medium”) as matter moves through space, but they do not appear to suggest that wakes emerge as a property or structure of empty space.  

My proposition goes a step farther to suggest that the movement of matter through space generates a gravitational wake in the structure of the space itself behind the moving object (hereafter “space wake” or “drafting wake”). Trailing behind the baryonic (ordinary) matter in its direction of motion, the space wake creates an effective increase in gravitational force on matter that follows within the boundaries of the drafting (conical) wake, analogous to aerodynamic and hydrodynamic drafting regions that trail objects moving through fluid mediums.

This effective increase in gravitational force accounts for the flattened spiral galaxy rotational curves. The proposed presence of space wakes and the increase in velocities facilitated by gravitational drafting seems to be consistent with Milgrom’s MOND formulation, which effectively extends the force of gravity over greater distances in low gravitational acceleration circumstances, such as found at the peripheries of spiral galaxies (Sanders, 2014; physicsworld.com July 2016). In turn this increases the mean velocities of masses influenced by gravity.
         
I suggest this is equivalent to a drafting or shielding effect, which slows the deceleration rate for a following mass and can also generate a suction force (e.g. Wang et al, 2014, in the context of drafting spheres). In the absence of the shielding mass ahead, the trailing mass would be exposed to higher effective-drag forces and therefore slower speeds or a higher energetic requirement to sustain the given speed; in other words, the speed of the following mass is sustained for longer at lower energetic output. 

Thus, the proposed drafting wake effect, I suggest, is consistent with MOND at least in the range of low accelerations where MOND works, and does not require dark matter to explain the observed flattened rotational curves. Of course, the big questions are therefore how and why should a moving object through empty space generate a drafting wake region, and why should that region behave as though it were a massive object that increases gravity. This seems to imply that the shape of space ahead of a moving mass is different from its shape in the trailing wake region, and that somehow in a way for which I make no suggestions, there is greater effective "drag" ahead of a moving mass, and a corresponding wake region behind it. Perhaps it is at this simple juncture that my ideas fail miserably.

Further, why should a high density interior region of the galaxy not exhibit the drafting effect; i.e. since the quantity of matter in the interior accurately accounts for the observed mass velocities without requiring either dark matter or modified gravity equations. For this, however, I do make a proposition. 


1.4    Enter Verlinde

Erik Verlinde’s recent (2017) paper “Emergent Gravity and the Dark Universe” may shed some light on these questions.  I do not possess the theoretical or analytical tools to offer any reasonable evaluation of his work to determine its consistency with a gravitational drafting effect.  However, I can reasonably ascertain certain general aspects of Verlinde’s paper, including that he applies his theory to an idealized “spherically symmetrical” (p.38, arXiv version) dark matter equivalent displacement region of space; and that for his theoretical framework, the topology or shape of this region is not important.  This suggests that it may be possible to deform Verlinde’s spherical region into the conical drafting region I propose, and that this would be one means of testing both his theoretical framework and the validity of my own propositions.

It is also reasonably clear that Verlinde’s theory involves a volume of baryonic matter space displacement such that the displaced region resides in a kind of “pressureless fluid” (p. 44); and that such a displaced region contains a gravitational force equivalent.  In my highly simplistic understanding, this is just what a drafting region is: a region of volume displacement which permits an increase, or equalization, of the velocities of bodies that follow in the drafting region.  Verlinde says (p. 15):

“The fact that matter causes a displacement of the dark energy medium implies that the medium also carries a reaction force on the matter. That magnitude of this elastic force is determined in terms of a residual elastic strain and stress.  We propose that this force leads to the excess gravity that is currently attributed to dark matter.”

The elastic force seems to be analogous to the “suction” force of drafting, and/or the reduction in drag force that comprises drafting. Of course, with drafting, a leading body provides a shielding effect from drag, which is reduced for the following mass. Verlinde does not speak of drag forces in space, but it seems a reasonable hypothesis that the motion of matter through space perturbs and displaces space conically in a way that generates increased equivalent gravity.


1.5   Drafting at the terrestrial scale

To illustrate my proposition, let’s look a little more closely at certain basic principles of aerodynamic and hydrodynamic drafting. 

  a.     Aerodynamic context

First, the energy saving effect of drafting increases with the size of the leading mass. For example, one cyclist drafting another at 40 km/h saves about 26%, in a no-ambient wind condition; one cyclist drafting two others in single files saves about 27%, while a cyclist in a group of eight, saves about 39%; a cyclist drafting a large box-truck saves about 62%, as shown in Figure 2 (Hagberg and McCole, 1990).  Clearly a small object drafting a much larger one saves more energy.




Figure 2.  Energy saving among cyclists in different configurations (from Burke, 1995, Figure 10.3; as adapted from Hagberg and McCole, 1990).


 Secondly, the magnitude of drafting diminishes as function of distance (Olds, 1998). See Figure 3, with reference to Olds (1998) equation 1 and adapting his Figure 1. This models the diminishing drafting benefit as function of distance between cyclists up to about 3 m when the drafting quantity approaches zero.


Figure 3. Reduction of drafting benefit as a function of distance between cyclists (correction factor from Olds (1998) eq. (1), and corresponds to increasing drag). Rapidly accelerating loss of drafting benefit between about 1.5 m and 3.0 m, corresponds to increased power requirements to maintain the same speed due to increasing drag force as wheel space increases. Adapted from Olds (1998), Fig 1. 


A third important principle of drafting is that its magnitude increases with velocity; for example, below about 16 km/h, there is no drafting benefit for following cyclists (Kyle, 1979).  At cosmic scales, this suggests there may be a threshold velocity above which the drafting effect becomes observable. The function may be derivable by a simple analysis of the rotational curves, as shown in Figure 1. By simplified approximation, according to Figure 1, the threshold mass velocity at which the gravitational drafting effect emerges appears to be about 40 km/s, and the effect increases as a function of mass velocity thereafter.  Indeed, ultra-compact dwarf galaxies with velocity dispersion between 24 to 37 km/s, have been shown not to contain dark matter (Scarpa, 2006; Drinkwater et al., 2003); this suggests the gravitational drafting effect emerges at mass velocities greater than 37 km/s.

If a simple velocity function and threshold is insufficient, I suggest that in addition to mass velocities, the drafting component should be considered along with the density of the mass distribution, and the relative magnitudes (and possibly their volume) of the masses that are coupled by gravitational drafting. Thus, I suggest an equation that models the relationship between these factors will describe matter velocities in high density regions in the interior regions of the galaxies (Scarpa, 2006), and scales proportionately to decreasing matter densities toward galaxy peripheries, such that the drafting effect is small in high density regions (and/or below the velocity threshold), and large in low density or small acceleration regions. To avoid the criticism that I am constructing some arbitrary function without grounding it in physical principles, I will endeavor to show why in principle this relationship can work.

         b.     Hydrodynamic context

In the hydrodynamic context, Hoerner (1965) describes the drafting effect:

“Shielding Effect. In case of two bodies placed one behind the other, the drag of the second one is usually smaller than in free flow, because of reduced dynamic pressure within the wake of the first body. In the example presented in figure 1, the drag of the second disk is even negative, up to a distance of more than 2 diameters, evidently because of suction behind the first plate. As the distance between the two disks is increased, the drag of the second one gradually approaches the value known under free-flow conditions (Cd = l .17). This type of shielding effect can have some consequences in motorcar racing, where a competitor may run for a while within the wake of another car ahead of him.”

Similar to the reduction of drafting effect in the aerodynamic context shown in Figure 3, Hoerner (1965, p. 8-1, Fig 1) indicates that the hydrodynamic drafting benefit diminishes with the distance between the lead and drafting cylinder, and becomes zero at a distance of about 7 diameters, when for the following cylinder Cd = 1.17.  Igarashi (1981) reported the negative drag or suction force to be Cd = -0.65 up to the noted ~2 diameter range, while the reduced drag for such a downstream cylinder beyond the suction zone to be Cd = 0.45.  

Like the increased energy saving for a cyclist drafting several cyclists or a large box-truck, a similar phenomenon is shown among “drafting, kissing, and tumbling” spheres in a liquid medium (Wang et al., 2014; Liao et al, 2015).  When a small sphere and a larger sphere, initially separated by some given distance, are dropped simultaneously under gravity in liquid, the small sphere will catch-up to the large sphere and travel at about the same velocity as the larger sphere. In this configuration, the smaller sphere will travel faster in the drafting region of the larger sphere than it -- the smaller sphere -- would if dropped with a sphere of the same size, or if the initial starting configuration was reversed (i.e. the large sphere followed the small), (Liao et al. 2015), as shown in Figure 4 (from Liao et al., 2015).



Figure 4. From Liao et al., 2015. In Setup-C, the small sphere falls faster if it trails a larger sphere.


 Liao et al. (2015) confirm the finding of Wang et al. (2014), who describe the process (at p. 31) as indicated in Figure 4 above, and as shown in Figure 5, below:

“The reason for this behavior is that there is a strong suction effect on the smaller (trailing) particle due to the low pressure in the wake of the larger (leading) one, which causes the upper particle to fall more rapidly than the lower one…As shown in Figure 16, the wake flow behind the larger particle extends beyond the smaller one, indicating that the smaller particle is strongly influenced by the wake of the larger one, and the smaller experiences less drag and thus it sediments at higher velocity than the larger one.”



Figure 5.  Trailing wake behind larger and smaller spheres (from Wang et. 2014, Fig. 16).


 Extrapolating from Figure 5, we can see that a greater large/small sphere ratio could permit several small spheres to fill the wake space behind it.  Thus, a high density region of spheres reduces the wake space. I will later expand on the importance of this in the context of dark matter.

Note that the spheres and wake regions in Figure 5 are flattened in two dimensions.  Since the drafting events occur in a fluid medium, we would expect the wake to be three-dimensional: an elliptical cone.



Figure 6. Elliptical cone (S. Cheng; retrieved online, July 18, 2017).



1.6    Galactic cluster collisions

The second major source of evidence for dark matter is high velocity galaxy cluster collisions (Markevitch, 2004; Clowe, 2006).

Keeping certain basic principles of aerodynamic and hydrodynamic drafting in mind, let’s look at some of the images of galactic collisions, and their stated dark matter regions. 

The Bullet Cluster:



Figure 7.  Bullet cluster collision (from Chandra Harvard online).

As indicated by Figure 7, under the currently leading paradigm, dark matter (blue) has passed through the gaseous ordinary matter (pink) so that the dark matter has shifted to a location in front of the ordinary matter, in the direction of motion (i.e. blue portions in the image have passed from central regions through the pink to the exterior regions).  

Here is a video that illustrates the prevailing dark matter hypothesis for the bullet cluster collision: https://www.youtube.com/watch?v=eC5LwjsgI4I

This is said to be possible because the velocity of the gaseous matter is reduced by a drag phenomenon, thus allowing the frictionless dark matter to pass through to the ordinary matter so that it resides ahead (to the exterior, in Figure 7) of the baryonic matter.

1.7    The Bullet Cluster viewed under the “peloton model"

Is there at least one other possible interpretation of this image?  What if the dark matter is not matter at all, but is instead a kind of gravitational drafting wake phenomenon, such as observed in pelotons and many other biological and non-biological collectives (for a overview of energy saving mechanisms in nature, see our a review Trenchard and Perc, 2016)?  Such a model does not require the presence of dark matter, but instead is equivalent to an increased gravitational force that trails the ordinary matter as it moves at enormous velocities through space.

In this model, like the aerodynamic and hydrodynamic drafting effects described above, the motion of ordinary matter through space creates a kind of low-pressure fluid wake effect in which the matter that follows in the wake region can, within a certain critical distance, sustain equal speed to the matter ahead. The effect diminishes with distance, just as we would expect from standard Newtonian inverse square law for gravitational effect, and in turn Milgrom’s modified version thereof (Milgrom, 2014; which I will discuss more further).  So we should observe strong gravitational drafting closest to the leading mass, and a threshold distance at which the following matter lags outside the gravitational drafting, thus de-coupling from the leading mass, and advancing at its own intrinsic velocity.

I suggest that this de-coupling effect also has implications for the presence of dark energy, which I will discuss more in paragraphs that follow.  I suggest that this drafting effect, because it has the same effect as gravity in that it permits matter to travel through the drafting region faster than it would in an un-coupled state, distorts space in much the same way as ordinary matter, and therefore can be observed by gravitational lensing.

The analogous circumstance may be present in star and galaxy clusters. The standard Newtonian equation shows that gravitational force diminishes as the square of the distance.  A drafting effect might simply modify the gravitational effect as a function of the nearness of other visible matter such that matter creates an additional "gravitational wake" in a region behind the moving mass (i.e. on the opposite side of the mass to its direction of motion in space).  This predicts therefore that there is an increased effective gravitational force in the region behind the star, star cluster, or galaxy cluster.  On the side of the star that faces the direction of movement, the prediction is that the standard Newtonian or Einsteinian gravitational forces exist. The magnitude of the "wake" diminishes with distance to some unknown threshold distance, beyond which there is no drafting effect. 

A two-dimensional drafting wake trails in an approximate triangle from the leading object, as shown in Figure 3, so it is reasonable to expect a three-dimensional wake in cosmic space to be an elliptical cone, as shown in Figure 4. In a rotating spiral galaxy, for example, the hypothesis here is that passing stars will capture following stars in its conical wake and, in effect, pull them along at velocities that are faster than expected by the standard Newtonian equation.

Therefore, applying what I refer to as a peloton model with approximate elliptical gravitational drafting cones, I suggest the images of the Bullet Cluster collision can be mapped like these:

                                A.

                                B.
  
Figure 8A. Left cluster wake cone.  Apparent dark matter (blue) trails leading matter in a conical, expanding drafting wake, as ordinary matter moves approximately to the right. B. Right cluster wake cone. Apparent dark matter trails in conical wake behind ordinary matter which moves approximately down left and toward the viewer (all cone drawings by HT). Whether the nose of the cone is peaked or flat is difficult to ascertain from the image 


Figure 9.  Intersecting wake cones during collision.


 In this interpretation, we can see regions where we would expect to see more blue imaging if the drafting wakes were uniform. This is a weakness of the proposed model, but perhaps there are discrepancies in the data, effects of other stellar bodies, or space gaps in which there is no ordinary matter around which gravitational lensing might be observed, among other things.  Regardless, I suggest that the high density blue areas shown to the right of the nose of the left cone, and to the left of the right cone, are misleadingly small, and that the gravitational wake region is better represented as conical, as indicated.

Under this model, no discrepancy arises by the intersection of the ordinary matter on the non-matter drafting region behind each cluster, because the matter can fill the wake region without any difficulties in principle.  Also, we should observe gravitational lensing in the entire conical wake region, even those parts that are dominated by gaseous or other matter, and this does appear to be the case.  If this is not the case, then this would be a clear weakness of the model.


1.8    El Gordo collisions     
                   



 Figure 10. El Gordo showing apparent dark matter regions in blue, from Chandra’s x-ray observatory (https://chandra.harvard.edu/photo/2014/elgordo/).


Note the Chandra Harvard X-ray site http://chandra.harvard.edu/photo/2014/elgordo/ affirms the prevailing hypothesis that dark matter passes through the ordinary matter:

“As with the Bullet Cluster, there is evidence that normal matter, mainly composed of hot, X-ray bright gas, has been wrenched apart from the dark matter in El Gordo. The hot gas in each cluster was slowed down by the collision, but the dark matter was not.”

Now, looking again at the El Gordo collision, and consider my alternative hypothesis that apparent dark matter trails in a cone behind the ordinary matter:
                                A.
     
                               B.

 Figure 11A. Left cluster cone moving approximately from left to right.  Nose of cone indicates high density region of ordinary matter. Apparent dark matter “drafting zone” encompasses enormous region of space in a three-dimensional cone, but trails behind the high density ordinary matter, and does not pass through the ordinary matter as weakly interacting particles. B. Right side cluster cone moving from right to left.  Apparent dark matter, or what I refer to as a drafting zone or region, encompasses an enormous volume of space, apparently expanding outward toward the viewer.



Figure 12.  Intersecting right and left cones of the El Gordo collision. If the right cluster (yellow) contains more mass and/or expands into space toward the viewer, it makes sense the cone would either be larger or appear to be larger. Similarly, the nose of the cone would be smaller because it is farther away, on the far side of the combined mass.
1.9    MACS J0025 




Figure 13.  MACS J005 (from Chandra Harvard, online)

In the MACS collision, the matter from the colliding galaxies has been described as having “pooled in the middle” http://spacetelescope.org/images/heic0818a/ as also indicated by the red and green contour map shown in their Figure 9 (Harvey et al. 2015). 

I have not found a reference that shows where the highest densities of the gaseous and galactic matter occur for the two colliding masses.  So, in approximating the drafting cones, there is more guesswork for MACS J0025 than for Bullet Cluster and El Gordo. But if we view the collision between matter fronts as one gaseous/galactic mass going roughly into the page away from the viewer, and one approximately coming towards the viewer, we can reasonably infer the overlapping conical regions as follows:




                                 A.                

   
                               B.

Figure 14A. Left cone, expanding away from view, meaning gaseous/galactic mass approaches from farther out. B. Right cone, apparently expanding toward us, with direction or motion into the page. Perspectives permit reversed approaches.

                              
Figure 15.  Intersecting left and right cones for the MACS J0025 collision. Collision appear to be nearly head-on but into and away from the given view. This results in overlapping cones. Thus, it appears that the left drafting cone (green) is expanding away from view, while the larger visible mass region is closer to us, on this side of the image; the right cone is expanding toward us, with the smaller visible matter region is on the far side of the image.  It is however, not obvious what regions comprise the matter for each drafting cone.


At present, I have not looked closely at images of the more complex Abell and “Trainwreck” collisions (or others) and their dark matter regions to ascertain where the wake cones might be.


1.9      Some thoughts on Markevitch’s dark matter area derivations

The Markevitch et al (2004) and Clowe et al. (2004) papers and their descriptions of the gravitational lensing effects around the Bullet Clusters are the basis for one of the strongest pieces of empirical evidence in favor of the existence of dark matter. For instance, Sean Carroll is emphatic that the Bullet Cluster proves the presence of dark matter: https://www.youtube.com/watch?v=4uogQiH5Yx4.

Far be it for me to be in any position to criticize the approach taken by Markevitch et al. (2004), but there appear to be aspects of their approach that one might reasonably question.  With respect to their weak lensing measurements they say they use a King mass profile to fit the lensing signal to determine the total masses of the subcluster and the main galaxy clusters.  A King mass profile appears relatively straightforward, involving three main paramaters: 1. Central brightness; 2 Core radius, and 3. Tidal radius, the point at which brightness vanishes: http://www.astro.caltech.edu/~aam/science/thesis/total/node20.html

The authors define the main “cluster” as comprising the region to the left of the bullet-shaped cluster, and bullet “subcluster” as the obvious region to the right. They state:

“The subcluster mass signal is detected to r ≃ 150 - 200 kpc from the mass (or galaxy) peak, beyond which the subcluster may be tidally stripped (C04) ... For the sake of modeling, we will adopt a King profile with rc = 70 kpc and ρ0 = 1.3×10-24 gcm-3, truncated at rtr = 150 kpc, which adequately describes the lensing data.”

Figure 1 from the Crowe et al. (2004) paper indicates the 200 kpc white bar (copied below):


Figure 16. Image from Crowe et al. (2006) indicating the white bar is 200 kpc at the distance of the cluster.

The authors appear to arbitrarily constrain the cross-sectional area of what they believe is a dark matter region. This is because the authors appear to have truncated the tidal radius at 150 kpc (as quoted above) when there apparently exists a wider lensing area -- apparent from the faint area of lensing at the top of top right-of-center region in Figure 11, and more easily seen in Figures 2 and 3, and a view by the naked eye of lensing phenomena. 
I confess my analysis here is weak, and needs to be considered a lot more carefully. However, my point is that perhaps the data could be re-analyzed in the context of a drafting wake model. 

1.10    The peloton model appears to be consistent with Milgrom’s modified Newtonian dynamics (MOND)

Under Milgrom’s MOND, dark matter is not a required explanation for the increased velocity of matter observed at galactic extremities (Milgrom, 2014; also see historical review by Sanders, 2014).  Thus, below a certain threshold gravitational acceleration threshold, found largely in lower density galactic regions (Famey, 2005; Scapra, 2006), a modified Newtonian gravitational equation produces predictions that accord well with empirical data (see Sanders, 2014).


1.11      How does this fit a peloton model?

 First, consider some images:


                                         A.
                                        B.
                                        C.

Figure 16A. Compact, high density, peloton, at comparatively low speed. Drafting regions are completely filled, except for peripheral riders. B.  Single file, low density, peloton. All riders are riding at or near maximal sustainable outputs and so the speed of this peloton is expected to be considerably higher than in Figure A. Partial drafting regions to the sides of each cyclist are not filled, but at these outputs, riders must ride in optimal drafting position or be dropped.  C. Mixed compact-single file peloton (image credits in references).


Is the high density interior region of galaxies akin to that of Figure A, in which the total drafting area is generally filled (except on peripheries) and in which there are high acceleration gradients between cyclists because average speeds are slower? Similarly, is the galactic exterior or periphery akin to the high energy output phase shown in B, in which the acceleration gradients between cyclists are small because the speeds are higher, but where drafting cones are only partially filled?  Is an entire galaxy something akin to what we see in C, where there is a high density, lower speed, interior region; and a low density, higher speed, exterior?  The analogy suggests that the passing or acceleration gradient, which is greater in the interior regions of the peloton and smaller at peripheries, is important; i.e. the proposal is that these different acceleration gradients according to density and relative energetic outputs may offer some explanatory power for the MOND equations which do not hold in high density interior galactic regions where the proposed drafting cones are mostly filled, but does apply in peripheral galactic regions where there are greater regions or volumes of unfilled drafting cones. 


III.   Dark energy: cosmic de-coupling

It has also been proposed that the universe has been undergoing an accelerated rate of expansion, since about 5 billion years ago.  This expansion has been attributed to dark energy comprising some energetic principles that have not yet been empirically established.

Thus, when the distance between the two cylinders increases, a critical point is reached when the drag for the downstream cylinder is the same as for the upstream one.  The effect is the same for cyclists, and can be illustrated by a simple plot, based on Olds’ (1998) equation (1).  The magnitude of drafting benefit diminishes to zero as the distance between cyclists increases, up to about 3 m. If a cyclist keeps his front wheel within a meter of the rear wheel of the cyclist ahead, there is negligible loss in drafting benefit.  After 2 m, the reduction in benefit accelerates up to 3 m, when drag force for the following rider is the same as for the leader (assuming the same velocity and other drag parameters), as shown in Figure 11.

If the following cyclist is weaker than the leading cyclist, and can only sustain the speed of the leader by riding in the optimal drafting zone up to 1m, and if she happens to drift backward to a wheel spacing between 2m and 3m, her speed will fall rapidly even at the same power applied in the optimal drafting zone, and wheel spacing will increase suddenly. This is because the rapid onset of increase drag between her and the cyclist ahead will cause her to decelerate. This rapid deceleration is a de-coupling event in which the following cyclist falls outside the drafting zone of the lead cyclist.

Consider cyclist A whose maximum sustainable power (MSP) is 300W; and cyclist B whose MSP is 400W.  By exploiting the drafting zone behind cyclist B, A may sustain the speed of B at ~43 kmh by drafting, but A is otherwise capable of sustaining only about 38 kmh when cycling in isolation on his own.  If the wheel spacing between A and B approaches 3m (or drifts laterally beyond the optimal drafting angle), his speed will suddenly decelerate to ~38 kmh even while sustaining 300W (using: http://www.cyclingpowerlab.com/PowerSpeedScenarios.aspx). This deceleration upon de-coupling results in a rapid and accelerated expansion of the distance between them.

I postulate that an analogous cosmic de-coupling effect may be responsible for the dark energy accelerating expansion of the universe.  The universe expanded at a constant rate until about 9 billion years ago. The accelerated expansion of the universe that began about 5 billion years ago need not be accounted for by dark energy, but is merely the expected effect of de-coupling when the distance between cosmic matter in the universe generally reached a critical quantity.  

Olds (1998) provides a similar equation in the context of drafting cyclists, in which the drafting magnitude diminishes with distance between the rear wheel of the leading cyclist and the front wheel of the following (drafting) cyclist up to 3m, when the drafting effect is zero and the drag force is the same for the front rider and the following cyclist.

This model predicts several cosmological dynamics summarized as follows:

         
Predicted dynamic
Evidence
There is greater “drafting” or energy saving capacity in higher density clusters (1), because the drafting cones are additive and filled.  


Tchieu, A.A., Crowdy, D. and Leonard, A., 2010. Fluid-structure interaction of two bodies in an inviscid fluid. Physics of Fluids, 22(10), p.107101.
High density galactic clusters exhibit higher density dark matter than lower density clusters of roughly equal mass, as explained:


and:

https://physics.aps.org/articles/v9/9 based on Miyatake, H., More, S., Takada, M., Spergel, D.N., Mandelbaum, R., Rykoff, E.S. and Rozo, E., 2016. Evidence of halo assembly bias in massive clusters. Physical review letters, 116(4), p.041301.
As the corollary of the above, drafting capacity diminishes as cluster density falls.

When drafting bodies de-couple there is an accelerated increase in distance between them, which may appear as the accelerated expansion of space between drafting bodies.

For cyclists see:

Olds, T., 1998. The mathematics of breaking away and chasing in cycling. European journal of applied physiology and occupational physiology, 77(6), pp.492-497.

For the equivalent hydrodynamic effect between a downstream drafting cylinder, see:
The threshold point that occurred ~5 billion years ago, when the expansion of the universe began to accelerate, may be the de-coupling point when following ordinary matter exceeded the boundary of the gravitational wake or energy saving region. This dynamic could therefore negate the need for dark energy.  Also, the de-coupling period or region might show a gradual acceleration in expansion as the coupling “elastic” approaches the final and complete point of de-coupling.  It would depend on what part of the expansion phase we are in and able to observe. If the universe is currently still in the “snapped-elastic” phase of cosmic expansion, we should observe a high rate of accelerated expansion. At some point, however, the “wake model” predicts stabilization of the expansion rate.
Once the “cosmic de-coupling event” (CDE) occurred, apparent expansion accelerates rapidly for a period, and then the rate of accelerated expansion decays to zero (expansion stabilizes) and cosmic bodies (and clusters) then proceed at their own intrinsic velocities and expansion at a stable rate continues. 

So, the prediction is that the rate of accelerating expansion will not be the same, depending on how far into the past you look. Thus at different periods , you may find a) a rapid accelerating expansion phase; b) a falling rate of accelerating expansion phase c) a stable expansion phase. All phases comprise CDE which ends in a rate of stable expansion.

The duration of CDE should be calculable. Applying the aero and hydrodynamic analogies, CDE should occur over a comparatively short duration. Under this model we may predict that nearby cosmic bodies exhibit more stable expansion rates, whereas ones observable in time approaching 5 billion years in the past should show higher accelerated expansion.
Even if there was not a generalized CDE, this model could account for differences in measured rates of expansion in different parts of the universe (Racz, et al., 2017; see also: Tian, S. et al., 2017.) 


Nielsen, J.T., Guffanti, A. and Sarkar, S., 2016. Marginal evidence for cosmic acceleration from Type Ia supernovae. Scientific reports, 6, p.35596.


Further evidence would be delineation of differential cosmic expansion rates as a function of how far back in time the expansion is measured.
There should be some detectable gravitational wake exhibited by smaller mass objects, like the sun or planets, resulting in increased velocity.
Anomalous, periodic velocities have been observed in satellite flybys at certain latitudes (Anderson et al., 2008). Perhaps these can be partly attributable to a small-effect earth drafting wake region, or one combined with a solar wake region.
There are optimal drafting positions in which velocity of following mass is greater than in other positions. By analogy, for cyclists, if there is no side wind, the optimal zone is directly behind the rider in front and as close to the rider as possible, with some increased energy saving among a group up to eight cyclists; for fish the zone is at angles to the leading fish since turbulence directly behind reduces energy saving, as it is for birds. 
Unknown, although existing data may indicate whether magnitude of the proposed gravitational wake diminishes with distance.
The drafting cone varies in size according to the mass of the matter it trails, with some relationship to the volume of the matter, and not just its mass.
Kim et al. (2016) have indicated that dark matter likely oscillates around the centre of the visible mass for billions of years. This is consistent with a trailing cone where there is no separation between dark matter and visible matter, but will trail behind the visible matter in the direction of motion. Any change in trajectory, particularly an elliptical trajectory, will move the position of the trailing cone, and it will change size depending on velocity and whether the matter clumps with other matter. Under the prevailing view matter is weakly interacting with visible matter, but can move though visible matter, which suggests the possibility that it can separate from visible matter.




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Images
http://cdn.velonews.com/wp-content/uploads/2010/10/PELOTON-1.jpg




http://d1y1lc6vjc8q07.cloudfront.net/wp-content/uploads/2014/10/0276-300x400.png




http://www.astro.caltech.edu/~aam/science/thesis/total/node20.html (King profile factors) brightness, core radius, tidal radius

http://wwwmpa.mpa-garching.mpg.de/galform/virgo/millennium/

https://www.space.com/31926-galaxy-clusters-dark-matter-complicated-relationship.html
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