University of Amsterdam
Insitute of Physics
MSc Physics
Theoretical Physics
Master Thesis
THE POLYMER UNIVERSE
Resolving Dark Matter From a Heuristic Point of View
by Wieger Steggerda 5821991 July 2014 54 ECTS July 2013 - July 2014 Supervisor:
Prof. Dr. Erik Verlinde
Second Examiner: Dr. Ben Freivogel
Abstract
The dark matter paradigm is successful in explaining the dynamics of galaxies, galaxy clusters and cosmology. However, there is still no clear understanding of the nature of dark matter. Entropic gravity offers a new perspective. Using a heuristic argument, which applies polymer physics to space-time, we are able to combine baryonic matter, dark matter and dark energy to obtain a description of gravity that agrees strongly with observations on galaxies and galaxy clusters. Although there is a lot of work to be done in validating this description, we suggest that polymer physics offers a new and promising point of view on space-time, dark matter and dark energy.
Contents
1 Introduction 5
2 Introducing the Dark Matter Formula 5
3 Observational Evidence 7
3.1 dark matter in spiral galaxies . . . 7
3.2 dark matter in galaxy clusters . . . 9
4 An Anology With Polymer Physics 17 4.1 elastic material . . . 18
4.2 the ideal chain . . . 20
4.3 elasticity in polymers . . . 22
4.3.1 free energy of a polymer melt . . . 23
4.3.2 young’s modulus for a polymer melt . . . 24
4.4 dark matter, dark energy and gravity from a heuristic point of view . . . . 25
4.4.1 dark matter . . . 26
4.4.2 newtonian gravity . . . 26
4.4.3 continued calculation . . . 28
5 Challenges / Discussion 29
A Galaxy Cluster Mass Profiles 30
1
Introduction
Dark matter forms a huge gap in our understanding of physics. Our current models of cosmology claim that only 15.5% of the matter in our universe is made up out of ordinary matter. The rest is an unknown substance which is referred to as dark matter [1]. This enormous discrepancy shows a weak spot in our current understanding of physics. In ”On the Origin of Gravity and the Laws of Newton” Erik Verlinde develops the idea that gravity is – instead of fundamental – entropic in nature. In this thesis we will develop this new perspective, to improve our understanding of gravity and dark matter.
Various attempts to find this missing mass in the form of an exotic particle have been made. We will argue however, that the perspective presented in this thesis, results in a reinterpretation of dark matter as a feature of gravity, rather than an unobserved particle. We are not the first to suggest a reinterpretation of gravity, we are however, very successful in describing gravity for all scales in the visible universe. From the smallest scales where gravity is too small to be measured, to the size of the observable universe.
In the second section of this thesis we introduce a formula from which we can calculate a dark matter distribution, provided we know the radial baryonic matter distribution. In the third section we test the validity of this dark matter formula against observational data from galaxies and galaxy clusters. In the final section we present a scenario in which space-time is subject to polymer physics. From this picture we derive the Newtonian laws of gravity and the dark matter formula. Therefor this picture not only tells us how to interpret dark matter, it also presents a reinterpretation of gravity.
2
Introducing the Dark Matter Formula
In this section we introduce the dark matter formula, an equation that relates baryonic, or ordinary, matter to dark matter. We start with the energy of the gravitational field due to dark matter. The gravitational field’s kinetic energy inside a spherical region with radius R is given by
ED(R) = 1 8πG Z r≤R |∇ΦD|2dV = Z R 0 GMD2(r) 2r2 dr (2.1)
Z R 0 GMD2(r) r2 dr = MB(R)R cH0 2π (2.2)
where MB(R) is the amount of baryonic matter, MD(R) is the amount of dark matter
inside a spherical region with radius R and H0 is the Hubble constant.
In this formula baryonic and dark matter are directly related. This is hard to explain in a paradigm where dark matter manifests itself as particles. Moreover the Hubble constant appears in this formula – why would dark matter have anything to do with cosmology.
We take the derivative of the formula to R as is shown below
GMD2(R) R2 = MB(R) cH0 2π 1 +d log MB d log R (2.3)
The definition of the logarithmic derivative is given by d log fd log x := f (x)x dxdf. It’s value gives the slope in a log-log plot.
By defining the average baryonic- and dark matter densities as ¯ρB(R)4πR3/3 =
MB(R) and ¯ρD(R)4πR3/3 = MD(R), we obtain another representation of the dark
matter formula ¯ ρB(R)ρcrit ¯ ρ2 D(R) = πH0R/c 4 + αB(R) (2.4)
here αB(R) is the logarithmic derivative of the average barionic density ¯ρB(R).
In the following sections we investigate the validity of the dark matter formula using observational data about dark matter inside galaxies and galaxy clusters. Then we provide some theoretical foundation of polymer physics, with which we can heuristically derive the formula.
3
Observational Evidence
3.1 dark matter in spiral galaxies
At the start of the 1930s, astrophysicists noticed an enormous discrepancy between the mass derived from gravitational effects, and the mass derived from the luminosity of stars inside the Milky Way. The speed at which stars revolve around the core of the galaxy is too high to be solely caused by the attraction of the luminous matter. This discrepancy was not only encountered in the Milky Way. As more galaxies were inspected, researchers established that they all carry this extra weight. Apparently, when this is true for all galaxies, most of the matter in the universe is non-luminous and has only gravitational interaction with ordinary matter. This is known as the missing matter problem. The mysterious missing matter has been baptized ’dark matter’.
The overwhelming amount of evidence for the existence of dark matter has fertilized the minds of many scientists bringing forth a wealth of theories and opinions about the manifestation of dark matter [4]. Despite the lack of confirmation received from particle accelerators, the idea that dark matter consists of subatomic particles has found the most support. It seems that observations on the cosmic microwave background are compatible with this paradigm. These observations yield the desired amount of baryonic matter, dark matter and dark energy in the Universe.
There are alternative theories that avoid the need for dark matter. mond (Modi-fied Newtonian Dynamics) adapts the laws of Newton for extremely small accelerations present in the expanses of space far away from huge gravitational wells such as galaxies. mond describes rotational curves of spiral galaxies extremely well [3]. Its popularity is limited however, because the theory is purely phenomenological and fails when used to describe clusters of galaxies.
Take the outer region of a spiral galaxy such that almost all of the ordinary (baryonic) matter is enclosed within a radius R: MB(r) = MB. In Newtonian physics, gravitational
accelerations found in stable orbits are canceled by centripetal ones GMB/r2 = v2/r.
Hence the orbital velocity v falls of as 1/√r. Observed galaxies’ rotational curves in-dicate however, that stars outside the core rotate at velocities much greater than the picture sketched by the Newtonian paradigm. Their rotational curves flatten and become constant. This phenomenon is demonstrated in figure 3.1.
Let’s see if the dark matter formula (2.2) describes the correct rotation curves of galaxies. Again at large distances, where most of the galaxy is enclosed the mass becomes
R
Sp
eed observed
expected
Figure 1: A typical rotation curve of a spiral galaxy. The observed curve flattens and deviates a lot from the Newtonian prediction.
GMD(R)
R2 =
pGMBcH0/2π
R (3.1)
The gravitational acceleration represented in the l.h.s. is canceled by the centripetal acceleration v2/r on the l.h.s. We now deduce that the rotational velocity becomes
v =p4
GMBcH0/2π (3.2)
Equation (3.1) corresponds with the assumptions of the mond paradigm [7]. There-fore the accomplishments of mond can be used to strengthen the validity of the dark matter formula.
The appearance of the Hubble constant is hard to explain in the context of a con-ventional dark matter scenario. Mordehai Milgrom, the initiator of mond, exposed this connection with the constant before [6]. The phenomenological nature of mond however, prohibits a satisfactory explanation of the constants presence as well. The source of its presence is further analyzed in part 4 of this thesis.
Since the dark matter formula describes rotational curves of spiral galaxies well, it is now time to see if it also works on the gravitational structures that mond cannot describe correctly: galaxy clusters.
3.2 dark matter in galaxy clusters
mond is known to precisely describe gravity in spiral galaxies [3]. mond fails however, to describe gravitation in galaxy clusters. To account for this, mond also claims the existence of non-luminous matter. Newtonian physics needs a dark-to-baryonic matter ratio of around nine to explain the gravitational features of the icm, while mond only needs a factor of three or four [9]. In this section we apply the dark matter formula on galaxy clusters and compare our results to those of mond and Newtonian physics.
Galaxy clusters are the biggest gravitationally-bound structures in the Universe. The baryonic mass of these cosmological giants is composed of a collection of galaxies, together with the icm (Intracluster Medium): a hot (1-15 keV) x-ray gas that permeates throughout the expanses of space in between the galaxies. The ionized hydrogen and helium atoms that compose the gas, carry – counter intuitively – the lions share of the baryonic mass: typically about 90%. The thermal light that the icm emits can be observed using x-ray telescopes like the xmm-Newton.
The icm is repulsed by gaseous pressure and attracted gravitationally by the New-tonian mass: the dark and baryonic matter combined. The temperature produced by the pressure therefore informs us about the amount of total mass in the clusters MN. A
large quantity of dark matter, for example, generates high pressures, resulting in high icm temperatures.
In the next paragraphs we use the observed x-rays to derive the amount of dark and luminous matter present in clusters. We use a sample of the 106 brightest galaxy clusters [8] to deduce the mass profiles for the total mass MN(R) and gas mass Mg(R)
of the clusters. With the dark matter formula we can numerically calculate the baryonic mass profile MB(R) that is needed to produces the dynamic mass MN(R).
We start off by measuring the x-ray surface brightness I(r) and the gas temperature kT (r). The temperature is nearly constant in r and I(r) is fit with King’s β-model [5]
I(r) = I0
"
1 + r rc
2#−3β+1/2
The intensity is a 2D projection of the electron number density ne(r), which is given
by the isothermal β-model [5]: a three-fit-parameter description of an isothermal radially symmetric gas sphere
ne(r) = n0 " 1 + r rc 2#−3β/2 (3.3)
The hydrogen/helium ratio of the gas mixture yields a gas density of ρg(r) =
1.17 ne(r)mp, where mp is the proton mass [8]. The β-model therefore provides the
following gas density
ρg(r) = ρ0 " 1 + r rc 2#−3β/2 (3.4)
We can deduce a relation between baryonic mass density and pressure by assuming that the galaxy cluster is in a hydrostatic equilibrium: a state where the pressure is cancelled by the gravitational potential. When the gas is in this relaxed state, the gas density abides the isotropic collisionless Boltzmann equation
dP (r)
dr = −ρg(r) |∇Φ(r)| (3.5)
The pressure is derived from the ideal gas law
P (r) = ρg(r) µmp
kT (r) (3.6)
where µ = 0.609 is the mean atomic weight.
The gravitational acceleration |∇Φ(r)| is determined solely by the Newtonian mass MN(r)
|∇Φ(r)| = GMN(r)
r2 (3.7)
Substituting equations (3.6) and (3.7) into equation (3.5), followed by performing the derivatives, we are left with the total mass
MN(r) = − rkT Gµmp d log ρg(r) d log r + d log kT (r) d log r (3.8)
We discard the right-hand side because the temperature distribution is nearly con-stant; the left-hand side is substituted by the gas density given in equation (3.4)
MN(r) = 3βkT Gµmp r3 r2+ r2 c (3.9)
This formula clearly diverges for increasing r. Galaxy clusters however, do have a finite extent. We claim that the β-model describes the gas density correctly until rout. rout is the radius where the gas density equals 250 times the critical density of
barions: 250 0.05ρcrit. Thereafter the gas density approaches zero. The values for the
fit-parameters β and rc, together with the temperature of the cluster kT are substituted
in (3.9) so that we receive the observed Newtonian mass MN(r).
We take the derivative of the dark matter formula (2.2) and get
GMD(R) R2 = MB(R)(1 + R MB dMB dR ) cH0 2π (3.10)
This identity becomes a differential equation when we substitute MD(R) with MN(R)−
MB(R). We can solve this differential equation numerically for MB(R).
To summarize: From the gas density and temperature we received the pressure. Since the pressure is canceled by the gravitational attraction in a hydrostatic equilibrium, we can calculate the total mass MN(r). The dark matter formula gives a constraint between
MB(r) and MD(r) and thus we are able to calculate MB(r) numerically. The gas mass
profile is derived by integrating the gas density radially: Mg(r) = R 4πr2ρg(r)dr. We
apply this calculation to the sample of 106 galaxy clusters in appendix B. We also included the mond dynamic mass: the baryonic mass that mond predicts from the Newtonian mass profile MN(r). The resulting mass profiles for the Newtonian mass, the
baryonic mass, the gas mass and the mond dynamic mass are shown in appendix A. We highlight the Abell 1367 cluster in figure 3.2.
0.1 0.2 0.5 1 rc R2500 R500 rout 1010 1011 1012 1013 1014 1015 1016
A1367
Figure 2: A log-log plot of the enclosed mass profiles of cluster A1369: the Newtonian mass (blue), our prediction for the baryonic mass (red), mond’s predicted barionic mass (green) and the observed gas mass (gold dashed). On the x-axis R is in units of Mpc, and the mass profiles are given in units of solar masses.
We expect that the baryonic mass is dominated by galaxies for the small radii around rc. For large radii we expect that the gas mass starts to dominate. At rout about 90%
of the cluster’s baryonic mass should be gaseous.
We investigate how the gas fraction fg = Mg/MB correlates with the barionic mass
MB, the three fit parameters ρ0, β, rc, and the temperature kT , at the two radii rc
and rout. We expect that the fraction fg(rc) should be small because the baryons are
dominated by galaxies. We expect fg(rout) to be near 0.9 because the gas mass dominates
the baryonic mass at large distances. The results are given in figures 3.3, 3.4, 3.5 and 3.6. There is a clear correlation between fg(rc) and rc– refer to the third panel in figure
3.4. Because of this correlation we made the data points with large rc brighter in color.
The fraction fg(rc) also correlates with ρ0 because large rc is generally associated with
0.0 0.2 0.4 0.6 0.8 1.0 0 5.0 ´ 1012 1.0 ´ 1013 1.5 ´ 1013 2.0 ´ 1013 2.5 ´ 1013 3.0 ´ 1013
MB
Figure 3: MB against fg(rc) = Mg(rc)/MB(rc), the brightness of the colors depends on
the size of the core radius rc of that cluster.
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Ρ0 0.0 0.2 0.4 0.6 0.8 1.0 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Β 0.0 0.2 0.4 0.6 0.8 1.0 0 100 200 300 400 500 Rc 0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 10 12 kT
Figure 4: ρ0, β, rcand kT against fg(rc) = Mg(rc)/MB(rc), the brightness of the colors
0.0 0.5 1.0 1.5 2.0 2.5 0 5.0 ´ 1013 1.0 ´ 1014 1.5 ´ 1014 2.0 ´ 1014 2.5 ´ 1014 3.0 ´ 1014
MB
Figure 5: MBagainst fg(rout) = Mg(rout)/MB(rout), the brightness of the colors depends
on the size of rc of that cluster.
0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8 1.0 Ρ0 0.0 0.5 1.0 1.5 2.0 2.5 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Β 0.0 0.5 1.0 1.5 2.0 2.5 0 100 200 300 400 500 Rc 0.0 0.5 1.0 1.5 2.0 2.5 0 2 4 6 8 10 12 kT
Figure 6: ρ0, β, rc and kT against fg(rout) = Mg(rout)/MB(rout), the brightness of the
0.02 0.05 0.1 0.2 0.5 1.0 10-6 10-5 10-4 10-3 10-2 0.1 1.0
Figure 7: Mean mass profiles of MN
(blue), our predicted barionic mass (red), mond’s predicted barionic mass (green thin), Mg (gold dashed) and Mc
(blue dashed) normalized on MN(rout)
for the 84 clusters with rc< 247 kpc
0.02 0.05 0.1 0.2 0.5 1.0 10-6 10-5 10-4 10-3 10-2 0.1 1.0
Figure 8: Mean mass profiles of MN
(blue), our predicted barionic mass (red), mond’s predicted barionic mass (green thin), Mg (gold dashed) and Mc
(blue dashed) normalized on MN(rout)
for the 22 clusters with rc< 247 kpc
We distinguish between clusters that have a core radius above 247 kpc and below 247 kpc. There are 22 clusters with rc> 247 kpc and 84 with rc< 247 kpc. For both cases
we create average mass profiles in the following manner. We scale the profiles to units of rout and we normalize over the total mass of the cluster MN(rout). For the gas profile
for example, ¯Mg(x) = (#clusters)−1PclustersMg(x rout)/MN(rout). We also added a
profile that indicates the amount of mass that is needed to complement the gas mass, i.e. ¯Mc(x) = ¯MB(x) − ¯Mg(x). The two graphs of the average mass profiles are given in
figures 7 and 8.
Again, we expect that near the core, a large portion of the baryonic matter consists of stars, while near the boundary at rout the gas should dominate the contents of the
baryonic matter.
In both cases the dark matter formula correctly describes the boundary of the cluster at rout. If the dark matter formula is valid, we would expect that it yields the best
results where our understanding of the baryonic mass is the greatest. Our knowledge of the baryonic matter is best at the boundary of the cluster since we know that almost all of the baryonic matter consists of gas at this radius. Therefore, this result reinforces the validity of the dark matter formula.
(the dashed blue line) contains almost all of the barionic matter near the core, and then, at the boundary of the cluster, obtains a negative slope. This means that the enclosed mass decreases for increasing radii, which is impossible because negative mass doesn’t exist.
The case with rc > 247 kpc doesn’t have the problem with the negative mass and
the profiles also seem to agree with the data. Very near the core it predicts that the gas mass becomes bigger than the barionic mass. This is impossible, it doesn’t pose a big problem however, because the β-model description doesn’t describe the gas mass for small radii.
The 22 clusters with the big core radii have strong agreement with the dark matter formula. The other clusters are more troublesome. There are a number of reasons why this could be the case:
1. The β-model describes clusters with big core radii better;
2. Clusters with small core radii are harder to measure, therefore the β-model fits might be flawed for these clusters;
3. The assumption that the clusters are in a hydrostatic equilibrium is wrong; 4. The β-model systematically fails to describe the gas mass profile.
Interestingly, as mentioned before, in clusters with small core radii, we observe that rc and the baryonic gas fraction fg are correlated. This correlation could give us insight
to a better understanding of, and might lead to a prediction on, the inner mass profiles of the cores of galaxy clusters.
4
An Anology With Polymer Physics
Polymer physics is the physics of macromolecules. This research field has been pioneered by Nobel laureate Paul Flory in the 1930s. The field was revolutionized in the 70s by another Nobel prize winner, Pierre-Gilles de Gennes.
The conventional approach to study polymer melts, which is a concentrated solution of polymers, is to look at a idealized polymers. The ideal chain is an abstraction of a polymer to a random walk and has the same importance in polymer physics, as the
are constrained by other polymers, they behave similarly to individual ideal chains sub-merged in a heat bath. Since the polymers in a melt behave like ideal chains, we can deduce physical properties of polymer melts by studying ideal chains.
Polymer melts have elastic properties. In this section we’ll find out that they have features analogues to gravity, dark matter and dark energy. To examine these parallels, we need to explore some very basic principles of continuum mechanics and polymer physics first. The next sections are dedicated to explain these concepts and they will culminate in the mathematical machinery needed to deduce Newtonian gravity and the dark matter formula from a heuristic perspective.
4.1 elastic material
This section presents a concise summary of the concepts of continuum mechanics which are relevant to the elastic properties of a polymer melt.
There are many different ways to exert stress on a material. Three of them are illustrated in figure 4.1. A material’s resistance to stress depends on the way this stress is exerted. For isotropic, uniaxial and shear stress the stiffness is described by the bulk modulus K, the Young modulus Y and the shear modulus G respectively.
Figure 9: Isotropic-, uniaxial- and shear stress
Consider a bulk of elastic material acted on by external forces. The stress that is caused internally results in a deformation. The strain tensor describes the relative deformation of neighboring points and is defined as
ij =
1
2(∂iuj+ ∂jui) (4.1)
where u(x) describes the translation of a point x due to deformation.
The stress tensor σij describes the internal forces in the material. The stress tensor
ma-terial having normal vector n. The stress vector T(n)i , which is illustrated in figure 4.1, describes the force exerted on a point on an imaginary surface S through the material. It is defined as
Tj(n) = dFj
dS = σijni (4.2)
The component of Tj(n) that is parallel to the surface S is called shear stress. The orthogonal component is called normal stress.
Figure 10: Stress vector
To summarize: strain describes internal deformations, while stress describes internal forces.
The strain tensor ij and stress tensor σij are related through a linear transformation
that is dependent on the type of material. This transformation is called the stiffness tensor cijkl
σij = cijklkl (4.3)
The linear transformation consists of 91 components. Luckily, for isotropic materials, the stiffness tensor is represented by just two independent components. It reduces to,
λ and µ are called the Lam´e parameters. The first parameter λ = K − 2µ/3, where K is the bulk modulus. The second Lam´e parameter is µ = G is the shear modulus1. Young’s modulus can be calculated from the other two moduli: Y = 9Kµ/(3K + µ).
Young’s modulus is a useful tool that describes how much force a material exerts when it stretches along an single axis. It is defined as the stress-strain ratio σ/ for small deformations. When a force F is applied on an orthogonal surface A0 of a material with
length L0. The material gets deformed by a small distance ∆L, we get a stress strain
ratio of Y = σ = F/A0 ∆L/L0 (4.5)
From the amount of work that is done when deforming the material, we can calculate the elastic energy
Ue= Z F d∆L = Z Y A 0∆L L0 d∆L = EA0∆L 2 2L0 (4.6)
The energy density is thus equal to
u = Ue A0L0
= 1 2Y
2 (4.7)
Generally the elastic energy per unit volume is given by
U = 1
2cijklijkl (4.8)
4.2 the ideal chain
In general, a polymer is a collection of connected nodes referred to as monomers, in which the connected monomers lie close to each other. An example of a polymer is a string of dna held together by hydrogen bonds.
1The most common notation for the shear modulus is G, in this document however we use µ, because
In the ideal chain model, this notion of a polymer is presented in a more abstract way as a random walk. The monomers are joint at a fixed length which is the only interaction they adhere to. There are no preferred angles and monomers can pass through each other freely. Surprisingly, polymers that are very stiff like dna can still be accurately described by an ideal chain. In the stiff dna molecule, neighboring monomers are very straight and will have an angle of almost 180◦. However, if we only focus on the monomers which are separated at a large distance of 500˚A, we observe a random walk with no preferred direction at all. These monomers do indeed behave like an ideal chain. The length at which we can view the polymer as an ideal chain is called the Kuhn length. The Kuhn length l and the length of a totally stretched chain L define all physical characteristics of the ideal chain. Figure 4.2 illustrates that, although the dna molecules are stiff, we observe a random walk when we zoom out to the Kuhn length scale.
R
l
Figure 11: Random walk in dna
To calculate the physical properties of the ideal chain, we start by looking at the size of the chain. The end-to-end vector is defined as R = P ri, where ri are the
N vectors between every neighboring monomer pair. Due to spherical symmetry, the mean of the end-to-end vector is hRi = 0. The central limit theorem tells us that the probability density functions of Rx, Ry and Rz are normally distributed with a variance
of σ2Rx = σ2rxN . The variance σ2rx of a random vector with length l can be calculated using a surface integral over a sphere Sl with radius l
σr2x = 1 4πl2 Z Sl x2dA = l 2 3 (4.9)
P (R) = ( 3 2πN l) 3 2 · e −3R2 2N l2
Until thus far this calculation has been static. The dynamic properties emerge when we introduce a temperature T and apply the laws of thermodynamics.
The entropy S(R) cannot be calculated directly because the multiplicity Ω(R) is not defined for continues systems. We can however, claim that the probability density function is proportional to the multiplicity Ω(R) ∝ P (R). With this claim the entropy becomes
S(R) = k log Ω(R) = −3kR
2
2N l2 + cnst. (4.10)
where k is Boltzmann’s constant.
Because there are no interactions between the monomers, the internal energy U only depends on the temperature. To calculate the amount of work available in the ideal chain we need an expression for the free energy. The free energy is defined as
F = U − T S = −3kT R
2
2N l2 + cnst. (4.11)
The force f exerted on a polymer with end-to-end vector R is obtained by taking the gradient of the free energy
f = ∇F = −3kT
N l2R (4.12)
In this equation, Hooke’s law F = −kx can be recognized. The stiffness constant k is proportional to the temperature T , which is characteristic to entropic forces.
4.3 elasticity in polymers
Thus far we have found that a single polymer behaves springlike. In this section we will look at the collective elastic properties of a very large number of loosely cross-linked polymers: a polymer melt. In a polymer melt, the nodes – at which the polymers entangle – form a network of smaller polymers as in figure 4.3.
Figure 12: A network of loosely cross-linked polymers before and after a uniaxial exten-sion
4.3.1 free energy of a polymer melt
We deform the material affinely by stretching along the three different axes
x → λxx y → λyy z → λzz (4.13)
The end-to-end vectors change (Rx, Ry, Rz) → (λxRx, λyRy, λzRz), and thus the free
energy of the polymers adjust to the deformation as well
∆Fpolymer = 3kT 2Ll Rnew 2− R old2 = 3kT 2Ll (λ 2 x− 1)R2x+ (λ2y − 1)R2y+ (λ2z− 1)R2z
The free energy of the entire polymer melt in the volume V is obtained by averaging the free energy over the ideal chains
∆F = V V0 h∆Fpolymeri = V V0 3kT 2Ll (λ 2 x− 1)Rx2 + (λ2y− 1)R2y + (λ2z− 1)R2z
where V0 is the volume occupied by a single polymer. Because R2x
= R2y = R2
z = Ll/3 (4.9) we obtain the change in free energy
∆F = V V0 kT 2 λ 2 x+ λ2y+ λ2z− 3
We have found the free energy by taking the polymer melt and adding the effects of its individual polymers. The free energy of the individual polymers is obtained by making the abstraction to the ideal chain model. The free energy of an ideal chain is derived in the previous section. In the next section we will use this free energy to calculate Young’s modulus for polymer melts.
4.3.2 young’s modulus for a polymer melt
The goal of this section is to calculate the Young’s modulus for polymer melts. The modulus describes the materials resistance against a uniaxial deformation – a deforma-tion caused by a force along a single axis. By applying a deformadeforma-tion along the x-axis the ideal chains in the melt grow, or shrink, and this difference in size changes the free energy of the bulk. The free energy allows us to calculate the amount of force that is applied to facilitate the deformation. From this force we derive the stress and with it we can calculate Young’s modulus.
Consider an extension or compression of a polymer melt along the x-axis: x → λx, or in other words: λx= λ. We can easily relate λ to the strain
= du dx =
d(λxx − x)
dx = λ − 1 (4.14)
The molecules of the rubber are very resistant to compression – more technically: the bulk modulus is approximately 5-6 orders of magnitude larger than Young’s modulus.
This implies that the volume doesn’t change, so Vnew:= λxλyλzV = V . It follows that
λyλz = 1/λ. Considering cylindrical symmetry, λy = λz = 1/
√
λ. From equation (4.11) we deduce the free energy of the polymer melt in the uniaxial extension
∆F = V V0 kT 2 λ 2+ 2/λ − 3 (4.15)
To calculate the strain, we start with a cross-section of the stressed material with area A = LyLz and derive the force f on this surface by taking the derivative of the free
energy: f = dFL = L1
x dF
dλ. Since the volume is given by V = LxLyLz, the stress becomes
σ = 1
LyLz
f = kT 2V0
(2λ − 2/λ2) (4.16)
Because Young’s modulus is the stress-strain ratio for small extensions λ ≈ 1, we can calculate it by taking the linear part of the Taylor-series
Y = dσ d λ=0= dσ dλ λ=1= 3kT V0 (4.17)
Finally we obtained the Young’s modulus for polymer melts. As it turns out, the modulus only depends on the temperature and the volume V0. Surprisingly, the
micro-scopic details of the polymer, like the mass, shape or the strength of the bonds, have no effect on the elasticity. In the next section we will use Young’s modulus for polymer melts, to show that the Newtonian potential and dark matter emerge from a model in which space-time manifests itself in the form of a polymer melt.
4.4 dark matter, dark energy and gravity from a heuristic point
of view
Consider a Universe without mass, only a Hubble constant H0 – a de Sitter space. The
metric for such a space is
horizon is a0 = cH0. We can calculate the (dark) energy density ρcritc2 = 3H 2 0c2
8πG via the
Friedman equations.
Something interesting happens when we place a mass M in the middle of this Uni-verse. We have to combine the metric of the de Sitter space with a Schwartzschild metric
ds2= − 1 − r2H02− 2GM/r dt2+ 1 − r2H02− 2GM/r−1
dr2+ r2dΩ22 (4.19)
We now observe two event horizons, which are located at the roots of f (r) = 1 − r2H02/c2 − 2GM/c2r. One is at the Schwartzschild radius r = 2GM/c2. The other,
previously located at R0, has now come a little bit closer at r = R0− GM/c2.
4.4.1 dark matter
The volume that is removed from the Universe equals ∆V = 4πR20GM/c2. We claim that this volume previously contained polymers that traversed from the cosmological horizon to the central mass. This is depicted in figure 4.4.1. Because the polymers are attached to the cosmological horizon, the amount of polymer’s volume inside a radius R equals R/R0. If we multiply the total amount of removed polymer volume with
this ratio, we end up with the removed polymer volume inside a sphere with radius R: ∆V (R) = ∆V (R0)R/R0 = 4πGM R/cH0. Finally we can calculate amount of dark
energy that is associated with this volume
E(R) = ∆V (R)ρcritc2=
3
2M RcH0 (4.20)
Remarkably, this is equal – neglecting a factor 3π ≈ 9 – to the energy distribution ED(R) which we used to derive the dark matter formula.
4.4.2 newtonian gravity
Because of the removed volume, our space has been deformed radially with a distance u(R) = −∆V (R)/4πR2 = −GM/cH0R. The strain is equal to = dRdu = GM/cH0R2.
We now assume that every polymer contains one degree of freedom: one quantum bit of information. Then, according to the equipartition theorem, the energy contained in
Figure 13: The removed polymers were connected to the mass and the cosmological horizon. The empty space that they leave contain energy from which we derive the dark matter formula.
this polymer equals kT /2. With the volume V0 that is occupied by the polymer, we can
identify the energy density ρcritc2= kT /2V0. Without declaring values for kT and V0, we
are able to derive Young’s modulus for polymer melts: Y = 3kT /2V0 = 6ρcritc2. From
Young’s modulus, together with the strain, we find that the energy density associated with the deformation equals
1 2Y
2 = 9GM2
8πR4 (4.21)
Amazingly this is equal – again neglecting a factor 9 – to the energy density in the Newtonian gravitational field
4.4.3 continued calculation
We used the idea that space is build up out of polymers in both calculations. We used it to find ∆V (R) which led to the acceleration due to dark matter and we found that Young’s modulus for polymers resulted in the Newtonian gravitational acceleration GM/R2.
There is a way to get rid of the factor of nine. If we assume that the displaced space is three times smaller than the one we find from the de Sitter space u = GM/3cH0R we
get the Newtonian potential exactly
1 2Y 2= GM2 8πR4 = |∇φ|2 8πG (4.23)
We check how much energy is removed from the universe when the radius of the cosmological horizon gets GM/3c2 smaller. The total energy in the Universe U = ρcc24πR30/3 = R0c4/2G changes by, ∆U = dU dR0 GM 3c2 = 1 2M c 2 (4.24)
By multiplying with the fraction R/R0 of the polymers within a radius R, we now
find the dark matter formula
ED(R) = M R
cH0
6 (4.25)
Well, neglecting the small factor 2π/6 which might be an error in the dark matter formula that we started with. This slightly different dark matter formula does also agree with the observations.
We not only changed the displacement u in the calculation, we also calculated the amount of energy ∆U received from the volume ∆V differently. Since the de Sitter space gets smaller, the critical density gets a bit bigger: ρcrit = 3c2/8πGR20. Some
of the energy that was in ∆V , needs to be spent on the increased energy density of the universe. We took this into account in this calculation by obtaining the Universe’s change in energy from ∆U = dRdU
5
Challenges / Discussion
We have shown that from polymer physics we are able to heuristically derive Newtonian gravity and a formula that describes dark matter. We tested this formula on spiral galaxies and galaxy clusters. The formula works great on spiral galaxies yielding the same results as the phenomenological mond paradigm.
The dark matter formula also seems to work on galaxy clusters. It averages out to a desired result on the outskirts of the cluster where we have a relatively good understand-ing of the total baryonic mass. However, the dark matter formula has some trouble to describe the baryonic mass found in the central regions of the clusters. The correlation between rc and fg(rc) in clusters may be a clue of a systematic failure and thus may
provide new insights about the internal mass profiles of galaxy clusters.
Measurements on strong and weak gravitational lensing provide another way to find the Newtonian mass MN(r) in clusters. We can use these measurements to enhance the
analyses of section 3.2.
Although we present some very promising results about spiral galaxies and galaxy clusters, there are a number of gravitational structures we did not look at yet. Star clusters, ultra-faint dwarfs, dwarf spheroidal galaxies and elliptical galaxies are examples of objects that the dark matter formula needs to be tested on as well.
In the derivation of the dark matter formula, we use a result from general relativity: the de Sitter metric and the Schwarzschild metric are used to find out how much volume dissipates from the universe after a mass M is placed in the center. We are however, looking for ways to understand gravity without using existing theories of gravity. There-for, we are not content with the use of general relativity to calculate the removed volume ∆V .
We have shown that when we apply polymer physics to space-time, we receive New-tonian gravity and the dark matter formula. It is important to note that we are not claiming that space-time is indeed literally made out of polymers. What we do claim however, is that space-time adheres to polymer physics. The arguments in this thesis are heuristic. Therefor, we can’t deduce a microscopic description of gravity and space-time from them. We can argue however, that the connection with polymer physics narrows our search for this description.
A
Galaxy Cluster Mass Profiles
0.2 0.5 1 rc R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A3376
0.2 0.5 1 rc R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A2065
0.2 0.5 1 rc R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A3395
0.1 0.2 0.5 1 rc R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A3395s
0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A2255
0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A3827
0.1 0.2 0.5 1 2 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A2256
0.1 0.2 0.5 1 2 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A3266
0.1 0.2 0.5 1 2 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A2163
0.1 0.2 0.5 1 2 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A0119
0.1 0.2 0.5 1 rc R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016S405
0.1 0.2 0.5 1 2 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A0399
0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016
ZwCl1215
0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A3530
0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A3888
0.1 0.2 0.5 1 2 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A3695
0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A0576
0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A1800
0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A1367
0.1 0.2 0.5 1 2 rc R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A1736
0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016IIZw108
1010 1011 1012 1013 1014 1015 1016A2634
1010 1011 1012 1013 1014 1015 1016A3822
1010 1011 1012 1013 1014 1015 1016COMA
0.1 0.2 0.5 rc R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016
S636
0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A3921
0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 10163C129
0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016RXJ2344
0.1 0.2 0.5 1 2 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A1644
0.1 0.2 0.5 1 2 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A3627
0.05 0.1 0.2 0.5 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016IIIZw54
0.1 0.2 0.5 1 2 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A2319
0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A3532
107 108 109 1010 1011 1012 1013 1014 1015 1016A1650
107 108 109 1010 1011 1012 1013 1014 1015 1016A3667
107 108 109 1010 1011 1012 1013 1014 1015 1016OPHIUCHU
0.1 0.2 0.5 1 2 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016
TRIANGUL
0.05 0.1 0.2 0.5 1 2 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A3158
0.05 0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A1775
0.05 0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A3560
0.05 0.1 0.2 0.5 1 2 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A0401
0.05 0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A0754
0.05 0.1 0.2 0.5 1 2 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 101615
0.05 0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A3391
0.05 0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016ZwC174
1010 1011 1012 1013 1014 1015 1016A1914
1010 1011 1012 1013 1014 1015 1016A3558
1010 1011 1012 1013 1014 1015 1016A2734
0.05 0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016
A0644
0.05 0.1 0.2 0.5 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A0548w
0.05 0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A2877
0.05 0.1 0.2 0.5 1 2 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A1651
0.05 0.1 0.2 0.5 1 2 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A3571
0.05 0.1 0.2 0.5 1 2 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A1413
0.05 0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A3528n
0.05 0.1 0.2 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016FORNAX
0.05 0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016AWM7
107 108 109 1010 1011 1012 1013 1014 1015 1016A1689
107 108 109 1010 1011 1012 1013 1014 1015 1016A0400
107 108 109 1010 1011 1012 1013 1014 1015 1016A2142
0.05 0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016
A0539
0.05 0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016EXO0422
0.05 0.1 0.2 0.5 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016UGC03957
0.05 0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A2199
0.05 0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016S540
0.05 0.1 0.2 0.5 1 2 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A2244
0.05 0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A2657
0.05 0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A0548
0.05 0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A2589
1010 1011 1012 1013 1014 1015 1016A2063
1010 1011 1012 1013 1014 1015 1016MKW8
1010 1011 1012 1013 1014 1015 1016A3528s
0.02 0.05 0.1 0.2 0.5 1 2 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016
A3562
0.02 0.05 0.1 0.2 0.5 1 2 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A0478
0.02 0.05 0.1 0.2 0.5 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A1060
0.02 0.05 0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A4059
0.02 0.05 0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016HCG94
0.02 0.05 0.1 0.2 0.5 1 2 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A0085
0.02 0.05 0.1 0.2 0.5 1 2 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A2029
0.02 0.05 0.1 0.2 0.5 1 2 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A1795
0.02 0.05 0.1 0.2 0.5 1 2 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016PKS0745
107 108 109 1010 1011 1012 1013 1014 1015 1016A2151w
107 108 109 1010 1011 1012 1013 1014 1015 1016A2204
107 108 109 1010 1011 1012 1013 1014 1015 1016MKW3S
0.02 0.05 0.1 0.2 0.5 1 2 rc R2500R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016
PERSEUS
0.02 0.05 0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A3112
0.02 0.05 0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A4038
0.02 0.05 0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A2597
0.02 0.05 0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016S1101
0.01 0.02 0.05 0.1 0.2 0.5 1 rc R2500R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016HYDRA_A
0.01 0.02 0.05 0.1 0.2 0.5 1 rc R2500R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A0133
0.01 0.02 0.05 0.1 0.2 0.5 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016NGC1550
0.01 0.02 0.05 0.1 0.2 0.5 1 rc R2500 R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A0262
1010 1011 1012 1013 1014 1015 1016A2052
1010 1011 1012 1013 1014 1015 1016A3526
1010 1011 1012 1013 1014 1015 1016A3581
0.01 0.02 0.05 0.1 0.2 0.5 1 rc R2500R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016
2A0335
0.010.02 0.05 0.1 0.2 0.5 1 2 rc R2500R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016A0496
0.01 0.02 0.05 0.1 0.2 rc R2500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016NGC5813
0.01 0.02 0.05 0.1 0.2 rc R2500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016NGC499
0.01 0.02 0.05 0.1 0.2 0.5 rc R2500R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016NGC507
0.01 0.02 0.05 0.1 0.2 rc R2500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016M49
0.010.02 0.05 0.1 0.2 0.5 1 rc R2500R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016MKW4
0.01 0.02 0.05 0.1 0.2 0.5 rc R2500R500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016NGC5044
0.01 0.02 0.05 0.1 rc R2500 rout 107 108 109 1010 1011 1012 1013 1014 1015 1016NGC5846
107 108 109 1010 1011 1012 1013 1014 1015 1016NGC4636
B
Galaxy Cluster Sample
cluster kT ρ0 β rc [keV] [1025g/cm] [kpc] A3376 4.0 0.02 1.054 531.7 A2065 5.5 0.04 1.162 485.9 A3395 5.0 0.02 0.981 473.2 A3395s 5.0 0.03 0.964 425.4 A2255 6.87 0.03 0.797 417.6 A3827 7.08 0.05 0.989 417.6 A2256 6.6 0.05 0.914 413.4 A3266 8.0 0.05 0.796 397.2 A2163 13.29 0.1 0.796 365.5 A0119 5.6 0.03 0.675 352.8 S405 4.21 0.02 0.664 323.2 A0399 7.0 0.04 0.713 316.9 ZwCl1215 5.58 0.05 0.819 303.5 A3530 3.89 0.03 0.773 296.5 A3888 8.84 0.1 0.928 282.4 A3695 5.29 0.04 0.642 281.0 A0576 4.02 0.03 0.825 277.5 A1800 4.02 0.04 0.766 276.1 A1367 3.55 0.03 0.695 269.7 A1736 3.5 0.03 0.542 263.4 IIZw108 3.44 0.03 0.662 257.0 A2634 3.7 0.02 0.64 256.3 A3822 4.9 0.04 0.639 247.2 COMA 8.38 0.06 0.654 242.3 S636 1.18 0.01 0.752 242.3 A3921 5.73 0.07 0.762 231.0 3C129 5.6 0.03 0.601 223.9 RXJ2344 4.73 0.07 0.807 212.0 A1644 4.7 0.04 0.579 211.3 A3627 6.02 0.04 0.555 210.6 IIIZw54 2.16 0.04 0.887 203.5 A2319 8.8 0.1 0.591 200.7 A3532 4.58 0.05 0.653 198.6 A1650 6.7 0.08 0.704 197.9 A3667 7.0 0.07 0.541 196.5 OPHIUCHU 10.26 0.13 0.747 196.5 TRIANGUL 9.6 0.1 0.61 196.5 A3158 5.77 0.08 0.661 189.4 A1775 3.69 0.06 0.673 183.1 A3560 3.16 0.03 0.566 180.3 A0401 8.0 0.11 0.613 173.2 A0754 9.5 0.09 0.698 168.3 15 4.91 0.03 0.444 167.6 A3391 5.4 0.05 0.579 164.8 ZwC174 5.23 0.1 0.717 163.4 A1914 10.53 0.22 0.751 162.7 A3558 5.5 0.09 0.58 157.7 A2734 3.85 0.06 0.624 149.3 A0644 7.9 0.15 0.7 143.0 A0548w 1.2 0.02 0.666 139.4 A2877 3.5 0.03 0.566 133.8 A1651 6.1 0.15 0.643 127.5 cluster kT ρ0 β rc [keV] [1025g/cm] [kpc] A1413 7.32 0.19 0.66 126.1 A3528n 3.4 0.07 0.621 125.4 FORNAX 1.2 0.02 0.804 122.5 AWM7 3.75 0.09 0.671 121.8 A1689 9.23 0.33 0.69 114.8 A0400 2.31 0.04 0.534 108.5 A2142 9.7 0.27 0.591 108.5 A0539 3.24 0.06 0.561 104.2 EXO0422 2.9 0.13 0.722 100.0 UGC03957 2.58 0.09 0.74 100.0 A2199 4.1 0.16 0.655 97.9 S540 2.4 0.08 0.641 91.5 A2244 7.1 0.23 0.607 88.7 A2657 3.7 0.1 0.556 83.8 A0548 3.1 0.05 0.48 83.1 A2589 3.7 0.12 0.596 83.1 A2063 3.68 0.12 0.561 77.5 MKW8 3.29 0.05 0.511 75.4 A3528s 3.15 0.09 0.463 71.1 A3562 5.16 0.11 0.472 69.7 A0478 8.4 0.5 0.613 69.0 A1060 3.24 0.09 0.607 66.2 A4059 4.4 0.2 0.582 63.4 HCG94 3.45 0.11 0.514 60.6 A0085 6.9 0.34 0.532 58.5 A2029 9.1 0.56 0.582 58.5 A1795 7.8 0.5 0.596 54.9 PKS0745 7.21 0.97 0.608 50.0 A2151w 2.4 0.16 0.564 47.9 A2204 7.21 0.99 0.597 47.2 MKW3S 3.7 0.31 0.581 46.5 PERSEUS 6.79 0.63 0.54 45.1 A3112 5.3 0.54 0.576 43.0 A4038 3.15 0.26 0.541 41.5 A2597 4.4 0.71 0.633 40.8 S1101 3.0 0.55 0.639 39.4 HYDRA A 4.3 0.63 0.573 35.2 A0133 3.8 0.42 0.53 31.7 NGC1550 1.43 0.15 0.554 31.7 A0262 2.15 0.16 0.443 29.6 A2052 3.03 0.52 0.526 26.1 A3526 3.68 0.29 0.495 26.1 A3581 1.83 0.31 0.543 24.6 2A0335 3.01 1.07 0.575 23.2 A0496 4.13 0.65 0.484 21.1 NGC5813 0.52 0.18 0.766 17.6 NGC499 0.72 0.2 0.722 16.9 NGC507 1.26 0.23 0.444 13.4 M49 0.95 0.26 0.592 7.7 MKW4 1.71 0.57 0.44 7.7 NGC5044 1.07 0.67 0.524 7.7 NGC5846 0.82 0.47 0.599 4.9References
[1] Planck Collaboration. Planck 2013 results. i. overview of products and scientific results. arXiv preprint arXiv:1303.5062, 2013.
[2] Private communications with E.P. Verlinde, 2013.
[3] B. Famaey and S.S. McGaugh. Modified newtonian dynamics (mond): Observational phenomenology and relativistic extensions. Living Rev. Relativity, 15(10), 2012. [4] D. Hooper G. Bertone and J. Silk. Particle dark matter: Evidence, candidates and
constraints. Physics Reports, 405(5):279–390, 2005.
[5] I.R. King. The structure of star clusters. iii. some simple dynamical models. Astro-nomical Journal, 71:64–75, 1966.
[6] M. Milgrom. A modification of the newtonian dynamics: Implications for galaxies. The Astrophysical Journal, 270:371–383, 1983.
[7] M. Milgrom. Mond – a pedagogical review. 2001.
[8] T.H. Reiprich. Cosmological implications and physical properties of an x-ray flux-limited sample of galaxy clusters. Ph.D. dissertation Max–Planck-Institut f¨ur ex-traterrestrische Physik, Ludwig–Maximilians–Universit¨at M¨unchen, 2001.
[9] RH Sanders. Clusters of galaxies with modified newtonian dynamics. Monthly Notices of the Royal Astronomical Society, 342(3):901–908, 2003.
