Density profile of dark matter

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(1)Density Profile of Dark Matter. Liz Maria Arcila Osejo Thesis presented as a requirement for Undergraduate Physicist Director Dr. Marek Nowakowski. UNIVERSIDAD DE LOS ANDES SCIENCE FACULTY PHYSICS DEPARTMENT BOGOTA D.C. MAY 2008.

(2) To my mom, Paola and Camilo.. 1.

(3) Introduction There are certain phenomena in our universe that does not seem to fit into what we already know of the universe. What happens then? Assuming that observations were made carefully enough, the only thing left to do is to question the theory or the universe. Dark matter proposes a solution to one problem of this type. The rotation curves for galaxies are in discrepancy to what is expected by Newton’s legacy, instead of declining at large radii, these rotation curves flatten after certain radius, leaving us with the uncertainty of whether Newton is wrong or whether there is something out there unable to give us some type of proof of its existency except for these rotation curves. Several physicists have tried to solve this puzzle over a long period of time. The main stream believes in the existence of dark matter, a new form of matter, not discovered yet, and one that we cannot see because it does not emit light although it does interact gravitationally. So if we consider this new matter, and take it into account within the rotation curves, everything seems to fit into place, and naturally, the rotation curves flattens after the luminous part of the galaxy. After a while of its proposal, dark matter has also helped astronomers to explain other phenomena, such as the acceleration in the expansion of the universe and the formation of galaxies in the early and homogeneous universe. Nevertheless, there are certain physicists who believe that this solution is rather ideal, and therefore propose a different solution; to assume that Newton was right but only for large accelerations. This new theory is known as MOND (Modified Newtonian Dynamics) and was proposed in the early 800 s. Even though MOND doesn’t have the same amount of followers as dark matter, it has not been ruled out, and it has also gained certain strength after a new theory was proposed to reconcile MOND and General Relativity. This thesis paper, was developed to investigate this topic, as a summary of several efforts made by many physicists in these two areas. In the first chapter we analyze the existence of dark matter, first proposed by Fritz Zwicky, assume its existence as a fact and explain certain studies made in this area that may evidence its existence. Then, we explain what this matter could be made of.. 2.

(4) In the second chapter we developed a brief introduction into the cosmological term, which will be included in chapter three as part of the virial theorem. Subsequently, chapter four develops a summary of a density profile for dark matter by Navarro, Frenk and White, and an application of the virial theorem to this profile. Finally, chapter five presents the work developed by Mordehai Milgrom, the founder of the theory called MOND.. 3.

(5) Contents Introduction 1. 2. Dark Matter 1.1 An introduction to Dark Matter . . . . . . . . . . . 1.1.1 The missing mass problem . . . . . . . . . 1.2 Early Type Galaxies and Their Dark Matter Content 1.3 Dark Matter Candidates . . . . . . . . . . . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 7 7 7 9 9. 2. The Cosmological Constant 12 2.1 The Cosmological Constant and General Relativity . . . . . . . . . . . . . . 12 2.2 The Cosmological Constant and the universe . . . . . . . . . . . . . . . . . . 13 2.3 The Friedmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14. 3. The Virial Theorem 3.1 General Results . . . . . . . . . . 3.2 The Tensor Virial Theorem . . . . 3.3 The Scalar Virial Theorem . . . . 3.4 Continuous Distribution of Matter 3.5 Particular Example . . . . . . . .. 4. 5. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 16 16 17 19 20 22. NFW Density Profile 4.0.1 Numerical Experiments . . . . . . . . . . . . . . . . . . . . 4.0.2 Effects of Numerical Limitations . . . . . . . . . . . . . . . . 4.0.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Numerical Results of the NFW profile . . . . . . . . . . . . . . . . . 4.1.1 Considering the system as a non uniform distribution of matter 4.2 Dark Matter according to Navarro et al. and the Virial Theorem . . . . 4.3 Results for an exponential distribution of matter . . . . . . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 24 25 25 26 32 32 33 37. Modified Newtonian Dynamics (MOND) 5.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 MOND as modified gravity . . . . . . . . . . . . . . . . . . . . . . . . . . .. 41 41 42 44. . . . . .. 4. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . ..

(6) 5.4. MOND as a modified inertia . . . . . . . . . . . . . . . . . . . . . . . . . . 47. Conclusions. 48. Bibliography. 50. Appendixes. 52. 5.

(7) List of Figures 1.1 1.2. The rotation curve of a spiral galaxy, Image taken from An introduction to Cosmology,Jayant Vishnu Narlikar [3] . . . . . . . . . . . . . . . . . . . . . 8 Mass To Light Ratios in Early Type Galaxies, Image taken from the article Mass-To-Light Ratios in early type galaxies and the dark matter Content [1] . 10. 2.1. Vacuum pressure, image taken from [14] . . . . . . . . . . . . . . . . . . . . 13. 4.1 4.2 4.3 4.4. 4.8. Density profiles found by NFW [10] . . . . . . . . . . . . . . . . . . . . . . Time evolution of halos of different mass, Navarro et al. image taken from [11] Circular Velocity Profiles from Navarro et al. image taken from [10] . . . . . Circular Velocity Profiles of only two halos from Navarro et al. Image taken from [10] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Shapes of the Rotation Curves of Disk Galaxies found by NFW in their paper [10] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concentration as a function of the mass, by Navarro et al. Image taken from [10] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concentration c as a function of the mass of the halo, red points considered for the linear regression. Image taken from [10] . . . . . . . . . . . . . . . . Numerical Parameters by Navarro et al. Table taken from [10] . . . . . . . .. 5.1 5.2. Mass discrepancy against the typical size, image taken from [17] . . . . . . . 44 Mass discrepancy against the system acceleration, image taken from [17] . . 45. 4.5 4.6 4.7. 6. 26 28 29 30 31 32 34 35.

(8) Chapter 1 Dark Matter 1.1 1.1.1. An introduction to Dark Matter The missing mass problem. Galaxies are to cosmology as atoms are to atomic physics, and it was here precisely where some discrepancies began to arise. When scientists tried to determined the total mass that could be found in the universe one suitable technique could be to determine the mass contained in each galaxy. Since it could be complicated to add up an infinite number of galaxies, only an estimate of the percentage of mass could be determined through this technique. The mass-to-light ratio (M/L) is a widely used technique in astronomy, which relates the mass and luminosity of any given luminous system, and it is usually given in solar units. When studying the M/L ratio in elliptical and spiral galaxies it was found that the contribution of its inner parts was roughly the same, but when studying the contribution of the outer parts of the galaxies, these regions of low luminosity and low density showed mass-to-light ratios much larger than those found for the inner parts of the galaxies. One possible explanation for this discrepancy is that in these regions of low density there are predominantly present low mass objects like dwarfs or planetary sized-objects or high mass objects like black holes and neutron stars. It could also be that these regions contained some form of non-baryonic matter[3]. One way to determine the amount of mass contained in a galaxy is through its rotation curve. If we consider the galaxy as disc-shaped, the rotation velocity v at a distance r from the center of the disc according to a gravitational force F is given by: v2 =F (1.1) r Now, continuing with this equation, according to a circular Keplerian orbit, the velocity at some point from the center of the galaxy will be: M (r). 7.

(9) r. F M (r) (1.2) r Where M (r) is the mass at the r radius from the center of the galaxy. This is taken from the fact that we use the gravitational force as if most of the mass were concentrated at the center of the galaxy (as seen in image [1.1], which is a good approximation according to the light distribution in spiral galaxies). If we consider a point A of the galaxy (which represents the extent of visible matter in the galaxy) and assume that all of the mass is observable we would observe that M (r) = constant beyond A, and the velocity at this extent would behave as: v=. v∝. 1 1. r2. (1.3). Figure 1.1: The rotation curve of a spiral galaxy, Image taken from An introduction to Cosmology,Jayant Vishnu Narlikar [3] Nevertheless as observed in figure [1.1] at even two or three times beyond A, approximately at B, v is more or less constant. A possible explanation for this kind of behavior is to consider some type of non luminous mass at the outer part of the galaxy in order maintain this constancy in the velocity, this is why this problem is addressed as the “Missing mass problem”. Another explanation is that the law of gravitation no longer holds for the galactic distant range. This possibility will be discussed in Chapter 5.. 8.

(10) The mass to light ratio is also useful to evaluate the mass required to stop a system such as a pair of galaxies, group or cluster from flying apart. This ratio is found to be about 100M/L for galaxy pair and 300M/L for groups and clusters of galaxies. According to this inferred ratio, and what we actually observe, over 95% of the measured mass consists of dark matter. The first scientist who found this inconsistencies was Fritz Zwicky, who in 1933, estimated the total amount of mass in seven galaxies of the Coma Cluster. He found out that each galaxy had a radial velocity that differed from that of the cluster about 700km/s RMS dispersion. Using this estimate in the dispersion he considered it as a measure of the kinetic energy per unit mass. Then he measured the total mass of the cluster with an estimate of the radius using the virial theorem. Comparing his analysis with the observations, he found out that there was actually 400 times more mass than expected [4]. Zwicky proposed the existence of some form of matter that can’t be directly observed.. 1.2. Early Type Galaxies and Their Dark Matter Content. Capaccioli, Napolitano and Arnaboldi[1] made a study of the behavior of M/L distribution on Early Type Galaxies. After analyzing a number of possible methods to determine the M/L ratios in a sample of 14 galaxies, they determined that the Planetary Nebulae technique was the most appropiate to obtain the M/L∗ for the inner estimate and the M/Lh for the outermost region. The graphics obtained are shown in figure [1.2], where the inner and outer parts of the galaxies are connected by lines to indicate the behavior of the ratio. The right handed graphic was used to homogenize the results. In this graph all of the ratios were shifted so that, arbitrarily, one could obtain M/L∗ = 6 and analyze from a better perspective the results. This distribution classifies galaxies in three types according to their gradient: first, the type of galaxy for which the dark matter distribution is not necessary (nevertheless it doesn’t have to be excluded), second, a type of galaxy for which the dark matter distribution is necessary and third a type of galaxy with steep gradients, possibly the result of wrong assumptions on the equilibrium conditions for those galaxies that are considered as interacting systems.. 1.3. Dark Matter Candidates. There have been several proposes as to what dark matter might be, which have been split in two main categories: baryonic and non-baryonic matter, the latter subdivided into hot and cold dark matter. According to Jayant V. Narlikar [3], the candidates for baryonic matter as non luminous 9.

(11) Figure 1.2: Mass To Light Ratios in Early Type Galaxies, Image taken from the article MassTo-Light Ratios in early type galaxies and the dark matter Content [1] matter are brown dwarfs, which are stars with masses too low to ignitiate (M ≤ 0.08M ), massive black holes, neutral and ionized gas in the form of hydrogen and small solid bodies like comets, asteroids and dust grains. The latter two account only to a small proportion of mass contribution. All these are classified as MACHOS (Massive Astrophysical Compact Halo Objects). Gravitational microlensing is the technique used to detect MACHOS such as brown dwarfs and other sub-stellar mass objects. It is based on the phenomenom of bending of light when it passes near a massive object due to the warping of space around this object. This technique is capable to detect non-luminous massive objects of masses from 0.01 to 5 times the solar mass. The non-baryonic matter candidates are divided in two categories: Cold Dark Matter and Hot Dark Matter. Cold dark matter is known as WIMPS (Weakly Interacting Massive Particles) which are massive particles (≈ 100GeV ) that move at non relativistic velocities. Their name is given by the fact that they interact through gravity and weak interaction with baryonic matter, leptons or light, but they cannot be observed electromagnetically. Even though there have been postulated a great number of WIMPS such as the photinos or axions, these have never been detected. Hot dark matter candidates are light weight particles that move close to the speed of light. One 10.

(12) candidate is the neutrino, considering its rest mass is between one million and one thousandth the mass of the electron.. 11.

(13) Chapter 2 The Cosmological Constant Initially, the cosmological constant was proposed by Einstein as a way to fulfill the requirements of the time in which the universe was meant to be static. This new term in his equation could balance the attractive force generated by gravity. Later, Hubble discovered that galaxies are receding from us, and hence he inferred that the universe was expanding. Nowadays, this term which was believed to be a mistake, is used again to explain density and pressure related with empty space. Quantum Mechanics predicts that there is some type of energy associated with empty space, called the zero-point energy (defined as the lowest energy of a system, also called vacuum energy). It predicts that since in empty space there are particles and antiparticles which only exist for a short period of time before annihilating, they give the vacuum some type of potential energy since all types of energy and matter should gravitate [14].. 2.1. The Cosmological Constant and General Relativity. The Robertson-Walker metric is a suitable equation in order to explain the shape of spacetime in a homogenous and isotropic universe, see equation [2.1]. ds2 = c2 dt2 − α2 (t) [dx2 + f (x)2 dθ2 + sin2 θdφ2 ]. (2.1). α(t) is the scale factor, it measures the expansion rate of the universe [15] and the term in the square brackets is the square of the co-moving distance dl2 . A co-moving distance expands with the universe, so that the real distance ~r and the co-moving coordinate are related by equation [2.2]. The co-moving distance between two points as the universe expands remains constant, while the real distance does change with time. ~r = α(t)~l. (2.2). Einstein’s field equations describe mathematically how the shape of space time is curved by 12.

(14) the presence of matter and energy, therefore these equations also describe the behavior of the metric [2.3] 8πGN 1 Rjk − Rgij − Λgij = Tij (2.3) 2 c4 Here, gij represents the metric or the distance function, Rij is the Ricci Tensor, R is the trace of the Ricci tensor (which could be understood as the radius of curvature of space-time), and Tij is the stress-energy tensor (this tensor describes the distribution of matter and energy). As stated before, the cosmological constant can be associated with a vacuum energy density given by equation [2.4]: Λ = 8πGN ρvac c2. (2.4). The pressure (which also contributes to the gravitational potential), associated with this vacuum energy density is [2.5] Pvac = −ρvac c2. (2.5). With the vacuum density given by ρvac , the energy necessary to move the piston (see image [2.1]) by a volume dV is given by ρvac c2 dV so that this is exactly the amount of pressure that the vacuum does on the piston Pvac = −ρvac c2. Figure 2.1: Vacuum pressure, image taken from [14] Thus there is a pressure caused by the cosmological constant which is not like the pressure of matter or radiation, since this is a negative pressure which creates a repulsive gravitational force, which instead of attracting acts to expand the universe. This cosmological constant is really a constant over time since the work P dV done by the vacuum provides the necessary energy in order to fill this new volume with the same density.. 2.2. The Cosmological Constant and the universe. It is believed that after the Big Bang the universe passed through a period of exponential expansion called the cosmic inflation caused by the negative pressure of vacuum density energy. Due of this period of inflation, the universe smoothed out making it geometrically flat, 13.

(15) which means that Ωtotal ≈ 1. Astronomers calculate that the amount of matter and energy in the universe today, accounts to only 30% of the necessary amount in order to explain a flat universe. Then, the cosmological constant is a way to explain the rest. Also, there is the “age problem” in which calculations for an open universe give a certain age of the universe that is much less to what is calculated for the oldest stars. This could be fixed if we consider a positive cosmological constant in a flat universe.. 2.3. The Friedmann equation. The Friedmann equation is an equation of the time evolution of the cosmic scale factor or size of the universe. This equation is shown in equation [2.6], where ρ is the mass density, k gives the curvature of the universe (if k < 0 then the spatial geometry of the universe is hyperbolic, if k = 0 then the universe is flat and if k > 0 then the universe is spherical), Λ is the cosmological constant and c is the speed of light. 2 8πGN k Λ 1 dα 2 = ρ− 2 + (2.6) H ≡ α dt 3 α 3 Originally, Friedmann wrote this equation without the term of the cosmological constant. Today we could define the contributions to Ωtotal given by matter, the cosmological constant and the curvature constant (the radiation contribution is not taken into account because is of negligible size today): Ωm0 ≡. 8πGN ρm0 3H02. ΩΛ0 ≡. Λ 3H02. Ωk0 ≡ −. k H02. So that: Ωtotal0 ≡ Ωm0 + ΩΛ0 + Ωk0 = 1 It is also possible to define: Ωtotal0 ≡ Ωm0 + ΩΛ0 = 1 − Ωk0 If Ωm0 + ΩΛ0 < 1 then the universe is an open universe, if on the other side Ωm0 + ΩΛ0 = 1 then the universe is flat and if Ωm0 + ΩΛ0 > 1 then the universe is a closed universe. Now, from the Friedmann equation: α̈ 4πGN =− α 3. 3P Λ ρ+ 2 + c 3. (2.7). And with a positive acceleration of the universe α̈ > 0 then Λ 6= 0. An approximation to the understanding of an accelerated universe is given by the Newtonian Limit. Considering the Schwarzschild-de Sitter metric for an object of mass M and spherical symmetry, [20] 14.

(16) ds2 = −eν(r) dt2 + e−ν(r) dr2 + r2 dθ2 + r2 sin2 θdφ2 With:. ν(r). g00 = −e. 2rs r2 =− 1− − 2 r 3rΛ rs = GN M. 1 rΛ = √ Λ Equation [2.8] shows the connection between the gravitational potential and the zero-zero component of the metric:. g00 ≈ − (1 + 2φ). (2.8). Then; φ (r) = −. rs 1 r2 − 2 r 6 rΛ. (2.9). The last term in equation [2.9] represents a repulsive external force, so that every two points in space go apart because of this cosmological constant. This is the part of the cosmological expansion that survives to the newtonian limit. This term (the Λ term) will be taken into account later using the virial theorem for dark matter distribution.. 15.

(17) Chapter 3 The Virial Theorem 3.1. General Results. Considering a force F~ (~x) over a unit of mass in the position ~x generated by a gravitational attraction over a mass distribution ρ(~x), then Newton’s Gravitational Law is stated as: m1 m2 F~ = −G 2 rˆ12 r Now, we consider a force obtained by adding small contributions: δ F~ = −G. ~x − x~0 δm(x~0 ) 0 3 ~ |~x − x |. (3.1). (3.2). i.e. F~ (~x) = −G. ~x − x~0 ρ(x~0 )d3 x0 |~x − x~0 |3. (3.3). ρ(x~0 ) 3 0 1 d x − Λ|~x|2 0 ~ 6 |~x − x |. (3.4). Z. Now, we define a gravitational potential: Z Φ(~x) = −G. The force in terms of this gravitational potential is given by: F~α = −mα ∇x Φ(~x) F~ α = mα G. Z. d x ρ(x~0 )∇x 3 0. . . 1 |~x − x~0 |. (3.5) 1 + mα Λ∇x (|~x|2 ) 6. (3.6). Where we know that ∇x. 1 |~x − x~0 |. =− 16. (~x − x~0 ) |~x − x~0 |3. (3.7).

(18) ∇x (|~x|2 ) = 2~x. (3.8). Now the force may be written as: F~α = −mα G. Z. d 3 x0. ρ(x~0 )(~x − x~0 ) 1 + mα Λ~xα 0 3 ~ 3 |~x − x |. (3.9). By definition we know that: F~ α = mα~ẍα. (3.10). So, we may conclude that: Z. ~α. ẍ = −G. ρ(x~0 )(~x − x~0 ) 1 α + Λ~x 3 |~x − x~0 |3. (3.11). Gmβ (xβj − xαj ) 1 α + Λxj 3 |~xβ − x~α |3. (3.12). d3 x 0. If we have a discrete distribution of matter: ẍαj = Σβ=1,β6=α. 3.2. The Tensor Virial Theorem. By definition of the moment of inertia, we have: Ijk =. N X. mα xαj xαk. (3.13). α=1. Where α = 1, 2, .., N is the number of particles, and j, k = 1, 2, 3 are spatial coordinates. If we now take the time derivative of equation [3.13]: N. dIjk X mα ẋαj xαk + xαj ẋαk = dt α=1. (3.14). And equation [3.13] second time derivative: N. d2 Ijk X α α α α α α α α ẍ + ẋ ẋ + x = m ẍ x + ẋ ẋ α j k k j k j k j dt2 α=1. (3.15). N. d2 Ijk X α α α α α α = m ẍ x + 2 ẋ ẋ + x ẍ α j k j k j k dt2 α=1 Using the result obtained in equation [3.12] and replacing: 17. (3.16).

(19) 2. N X. N X. N X. Gmβ xβj − xαj. d Ijk =2 mα ẋαj ẋαk + mα xαk dt2 |~xβ − ~xα |3 α=1 α=1 β=1,β6=α β α N N N N − x Gm x X X X β k k 1 1 X α α α + Λ + Λ mα xk xj + m α xj mα xαk xαj β −~ α |3 3 α=1 3 |~ x x α=1 α=1 β=1,β6=α. (3.17). (3.18). By definition we have the tensor kinetic energy: N. Kjk. 1X ≡ mα ẋαj ẋαk 2 α=1. (3.19). If we substitute the definition of the tensor kinetic energy, in equation [3.17]: N o X 2 Gmα mβ n β d2 Ijk β α α α α − x x = 4K + ΛI + x + x − x (3.20) jk jk j k k xj j k 3 2 α β dt 3 |~ x − ~ x | α,β=1,α6=β. Now we define the potential energy tensor for a system of particles: N xαj xβk − xαk X Wjk ≡ G mα mβ |~xα − ~xβ |3 α,β=1,α6=β. (3.21). Now if we interchange the “dummy” indices α and β and add this result to what we already have:. 2Wjk = G. N X α,β=1,α6=β. mα mβ. xαj xβk − xαk |~xα − ~xβ |3. +G. N X. mβ mα. α,β=1,α6=β. xβj xαk − xβk |~xβ − ~xα |3. (3.22). So; . N X. 1 Wjk = − G mα mβ 2 α,β=1,α6=β. xαj − xβj. xαk − xβk. |~xα − ~xβ |3. (3.23). From equation [3.23] we may conclude that Wjk = Wkj and using this result we found the Tensor Virial Theorem [3.24]: d2 Ijk 2 = 4Kjk + 2Wjk + ΛIjk 2 dt 3 18. (3.24).

(20) 3.3. The Scalar Virial Theorem. If we now take the trace of the tensors: N. 3. 1 XX mα ẋαi ẋαi K = T race(Kjk ) = 2 α=1 i=1. (3.25). N. 1X K= mα vα2 ≡ Kinetic Energy 2 α=1. N X. 1 W = T race(Wjk ) = − G 2 α,β=1,α6=β. 3 X. mα mβ. j=1. xαj − xβj. (3.26). 2. |~xα − ~xβ |3. N X 1 mα mβ W =− G ≡ Potential Energy 2 α,β=1,α6=β |~xα − ~xβ |. I = T race(Ijk ) =. N X 3 X. mα xαi xαi. =. α=1 i=1. N X. mα |~xα |2. (3.27). (3.28). (3.29). α=1. Then: d2 I 2 = 4K + 2W + ΛI 2 dt 3 If we consider a system in a stable state: d2 I =0 dt2 2 4K + 2W + ΛI = 0 3 Now we have obtained the Scalar Virial Theorem: 1 2K + W + ΛI = 0 3. 19. (3.30). (3.31) (3.32). (3.33).

(21) 3.4. Continuous Distribution of Matter. If we now consider a continuous distribution of matter its moment of inertia is given by equation [3.34]: Z Ijk = ρxj xk d3 x (3.34) For the kinetic energy we have: Z 1 ρvj vk d3 x (3.35) Kjk = 2 If we now consider that the mean value vj vk may be separated in two parts: one contribution is due to the streaming motion, and the other due to the fact that at a certain point not all of the stars have the same velocity σij2 , where: σij2 ≡ (vi − vi ) (vj − vj ) = vi vj − v̄i v¯j. (3.36). Then, the Kinetic Energy may be written as: 1 Kjk = Tjk + Πjk 2. (3.37). Where:. Tjk. 1 = 2. Z Z. Πjk =. ρv¯j v¯k d3 x. (3.38). 2 3 ρσjk dx. (3.39). In the same manner for the potential energy: Wjk. 1 =− G 2. Z Z. x0 − xj (x0 − xk ) j k ρ (~x) ρ x~0 d3 xd3 x0 3 x~0 − ~x. (3.40). For the scalar virial theorem for a continuous distribution of matter we need the trace of the tensors: Z 1 T race (Kjk ) = K = ρ (~x) v¯2 d3 x 2 Z 1 T race (Wjk ) = W = ρ (~x) Φ (~x) d3 x 2 Z T race (Ijk ) = I = ρ (~x) |~x|2 d3 x. 20. (3.41) (3.42) (3.43).

(22) For the Trace of the potential energy tensor, it is necessary to remember the definition of the gravitational potential: Z ρ x~0 0 Φ (~x) = −G d3 x (3.44) ~x − x~0 If we now take as an example a sphere with radius R and a constant density ρ, with an average velocity for each particle taken as v̄ 2 , by definition its potential energy is given by: Z W ≡−. ρ (~x). 3 X j=1. xj. ∂Φ 3 dx ∂xj. (3.45). Also, as stated before: Z W =−. 3 X. ∂Φ 3 1 ρ (~x) xj d x= ∂xj 2 j=1. Z. ρ (~x) Φ (~x) d3 x. (3.46). Which may be written as: Z W =−. ρ (~x) ~x · ∇Φd3 x. (3.47). Also, we know that: dΦ GM (r) r̂ = r̂ dr r2 For a sphere of radius R and constant density ρ: ∇Φ =. 4 M (r) = πρr3 3. (3.48). (3.49). Using the definition for a solid angle: Z. 2π. π. Z. Ω≡. sinφdθdφ 0. (3.50). 0. We find: 4 W = −G πρ2 3. Z. Z. R. dΩ 0. r4 dr = −. 16π 2 2 5 Gρ R 15. In the same manner for the kinetic energy and the moment of inertia: Z 1 4 3 1 3 2 K= ρv d x = πR ρ v 2 2 2 3 Z 4π 5 I = ρ (~x) |~x|2 d3 x = ρR 5 21. (3.51). (3.52) (3.53).

(23) Now, using these results in the scalar virial theorem for a steady state: 1 2K + W + ΛI = 0 3. (3.54). 4 16π 2 2 5 1 4 0 = πR3 ρv 2 − Gρ R + Λ πρR5 3 15 3 5. (3.55). 1 4 v 2 + R2 Λ = πGρR2 5 5. (3.56). ρ=. 5 Λ 2+ v 4πGR2 4πG. (3.57). ρ=. 5 v 2 + 2ρvac 4πGR2. (3.58). Then:. By definition: Λ (3.59) 8πG The term ρvac introduced in equation [3.59] was introduced as a modification to Einstein’s equations [4], and it is equivalent to the addition of a fluid with a density ρvac , and a pressure [3.60] ρvac =. P =−. Λc2 8πG. (3.60). Λ is known as the Cosmological Constant and has units of (time)−2 . If we don’t consider the term due to the cosmological term, then the density of mass is less than the one obtained using it.. 3.5. Particular Example. If we now consider a density profile of the form: −r. ρ(r) = ρc e r0. (3.61). −r 4 M (r) = πρc r3 e r0 3. (3.62). Then the mass is given by:. 22.

(24) So that: −r 4 ∇φ = πρc GN re r0 3 Then the potential energy is given by: Z R Z 2π Z π −2r 4 2 4 r0 πρc GN r e dr dφ W =− sinθdθ 0 3 0 0. (3.63). (3.64). Solving: 16π 2 ρ2c GN W =− 3. . 4 3 5 1 −2R 3 2 2 3 4 r0 3r0 + 6r0 R + 6r0 R + 4r0 R + 2R r − r0 e 4 0 4. For the kinetic energy: 1 K = v2 2. R. Z. ρc e. −r r0. Z. 2. π. Z. 2π. dφ. sinθdθ. r dr. (3.65). 0. 0. 0. n o −R K = 2πv 2 ρc 2r03 − r0 e r0 2r02 + 2r0 R + R2. (3.66). For the moment of inertia: R. Z. 2. I=. r ρc e. −r r0. Z. 2. r dr. π. Z sinθdθ. 0. 0. 2π. dφ. (3.67). 0. Which gives: n o −R I = 4πρc 24r05 − r0 e r0 24r04 + 24r03 R + 12r02 R2 + 4r0 R3 + R4. (3.68). Then, the virial theorem for these system is:. 8πr03 ρc e. −R r0. . −v 2. 2. +4πρc r0 R e. −R r0. 2. + πρc GN R e. −R r0. − 2ΛR. 2. . +8πρc r02 Re. −R r0. −R 2 2 2 r0 2 2 −v + πρc GN R e − ΛR 3 3. −R −R −R 2 1 2 r0 2 2 −v − πρc GN R e + ΛR + 8πρc r04 Re r0 πρc GN e r0 − 3Λ 3 3 −R. +4πρc r05 e r0. . −R πρc GN e r0 − 8Λ + 4πρc r03 2v 2 − πρc GN r02 + 8Λr02 = 0. We will come back to the example of matter distribution given by (3.61) in section 4.3.. 23.

(25) Chapter 4 NFW Density Profile Julio Navarro, Carlos Frenk, and Simon White (NFW) developed through a series of investigations a density profile based on N-body simulations in the standard cold dark matter1 cosmogony [10]. According to the theory of galaxy formation, galaxies formed because an increase of the primordial fluctuations2 which attracted gravitationally gas and dark matter to more dense areas. As the universe began to cool down, dark matter clusters began to condensate, and with them, the gas that surrounded this lumps. According to the investigation carried by Navarro et al., N-body simulations are necessary to study the evolution of these dark matter halos. Nevertheless, since more massive halos are better resolved because they have a larger number of particles, halos are first modeled in N-body simulations for large periodic boxes, and then re-simulated at a higher resolution. The regions outside this high-resolution box were filled with thousands of particles in order to have into account tidal effects at a larger scale [11]. According to this density profile, each virialized halo was modeled as an isothermal sphere characterized by the velocity dispersion and the core radius. The velocity dispersion is also known as the circular velocity [equation 4.1] and it is assumed that this velocity is proportional to the characteristic velocity of the galaxy in which we could find the halo. Nevertheless, it has been found that there is no apparent relation between the luminosity of binary galaxies and the relative velocity of its component, which would be the case if more massive halos could be associated with brighter halos and faster rotating disks.. Vc2 =. GM r. 1. (4.1). Assumption which implies that most of the matter in the universe is in the form of cold dark matter and that systems grow hierarchically. 2 Changes in density occurred during the early universe. These primordial fluctuations began as quantum fluctuations and without them the universe would probably be homogeneous. 24.

(26) There is no firm evidence that could prove the existence of the core, a central region in the dark halo where the density is approximately constant. There have been advocated large radii for these cores in order to account for the contribution of the luminous part in the rotation curves of disk galaxies, but the effects of gravitational lensing (giant arcs) of background galaxies produced by clusters of galaxies are only explained if a small radius is considered for the cluster core. All these inconsistencies led Navarro, Frenk and White to consider dark matter halos described by the model of Hernquist (1990) as shown in equation [4.2], as a gently changing logarithmic slope rather than an isothermal sphere. NFW proposed their own density profile in which they included rs called a scale radius [4.3]. ρ (r) ∝. 1 . r 1+ ρ (r) ∝. r rs. 3. (4.2). 2. (4.3). 1 . r 1+. r rs. As an observation, NFW pointed out that the rotation velocity for the luminous radius of the galaxy is constant. For less luminous galaxies their rotation curves tend to be rising while for more luminous galaxies their rotation curves tend to be declining. Then, if the disk is rotating rapidly the curve tends to decline, but if the disk is rotating slowly the curve tends to rise. In image [4.5] is observed that the disk’s M/L ratio increases with mass and that in bright galaxies the contribution of the halo is less important.. 4.0.1. Numerical Experiments. During the simulation there were considered 19 systems of different mass ranging from big clusters to dwarf galaxies. For each system there was defined a virial radius denoted as r200 . This radius was determined by enclosing a sphere of mean overdensity of 200, using the center as the center of mass of the clumps. Then, there were selected the systems covering a mass range of ( 3 ∗ 1011 < M200 /M < 3 ∗ 1015 ). These mass ranges were selected so that there were halos grouped into circular velocities of 100km/s, 250km/s and > 450km/s.. 4.0.2. Effects of Numerical Limitations. There are a number of Numerical Limitations that can influence the density profile considered for these halos; in their work they considered two main limitations: the initial redshift and the gravitational softening. During their investigation it was found that at radii larger than the gravitational softening and with a large number of particles (since they have a large number of particles, more massive systems are better resolved), these numerical factors do not affect the structure of the halos when chosen carefully. 25.

(27) 4.0.3. Results. The results found by Navarro, Frenk and White for the Density Profiles are shown in image [4.1]. In this image are the density profiles of four dark halos, with different mass (ranging from the lighter, with a mass of 3 ∗ 1011 to the heavier with a mass 3 ∗ 1015 ). In the image the mass decreases from right to left, and the smooth curves represent the fit using the model found by NFW [10]. This image reflects the fact that the NFW profile suits well for all of the halos regarding the difference in mass:. Figure 4.1: Density profiles found by NFW [10] δc ρ(r) = r ρcrit ( rs )(1 +. r 2 ) rs. (4.4). In equation [4.4] rs is called a characteristic radius: r200 c ρcrit is the critical density for which the spatial geometry would be flat: rs =. 3H (t)2 ρcrit = 8πG Where H (t) is the Hubble’s constant and δc and c are two dimensionless parameters. The. 26.

(28) with an uncertainty of 15%3 . The current value for the Hubble’s constant is H0 = 77 km/s mpc dimensionless parameter δc could be understood considering that low mass halos are denser than high mass halos, then δc has a higher value for low mass halos. As a reference, δc is called the characteristic overdensity of the halo, c its concentration and rs the scale radius. The critical overdensity is defined as: 200 c3 3 [ln(1 + c) −. δc =. (4.5). c ] (1+c). This definition could be obtained using the definition of the mass at the virial radius M (r200 ) Z 2π Z π Z r200 2 dφ Senθdθ ρ (r) r dr M (r200 ) = 0. 0. 0. Which gives: M (r200 ) =. 4πρcrit δc rs3. . −r200 − Ln[rs ] + Ln[rs + r200 ] rs + r200. (4.6). Also according to the paper, δc and c, are linked by the fact that the mean density is equal to two hundred times the critical density, which is: ρ (r200 ) = 200ρcrit Then, the mass of the halo at this radius is: 4 3 M200 = (200ρcrit ) πr200 3 Since these two definitions of the mass at r200 should be the same: 4 3 −r200 3 − Ln[rs ] + Ln[rs + r200 ] = 200ρcrit πr200 4πρcrit δc rs rs + r200 3. (4.7). If we solve for δc : 200 δc = 3. . r200 rs. 3 . −1 −r200 − Ln[rs ] + Ln[rs + r200 ] rs + r200. (4.8). Using the fact that c = rr200 , we obtain for δc what we already had in equation[4.5]. Now it s remains to fix c which we will discuss in detail below. An interesting mass dependence in the halos could be understood considering that low mass halos are denser, since they have a higher redshift (which means that they formed earlier, when the mean density of the universe was higher)[10]. 3. NASA’s Chandra X-ray Observatory value for the Hubble http://spaceflightnow/news/n0608/08hubbleconstant/, recovered on April 4, 2008. 27. parameter,. 2006,.

(29) Time Evolution Figure [4.2] shows the assembly of three different halos of different mass carried out by Navarro et al. Time increases downward, while the third image accounts for the halo with more mass. From these simulations, it could be stated that less massive halos assembly earlier than more massive ones.. Figure 4.2: Time evolution of halos of different mass, Navarro et al. image taken from [11]. Circular Velocity Profiles Navarro et al. noticed that the information contained in the circular velocity profiles was roughly the same for 19 systems of different mass (see figure [4.3]), since the density is similar for all of them. The information in these profiles was taken from the fact that: Fg =. GN M m v2 = m r2 r. 28.

(30) r v=. GN M r. Figure 4.3: Circular Velocity Profiles from Navarro et al. image taken from [10] As a conclusion seen from figure [4.4], where the upper profile corresponds to the lower mass system and the bottom profile to the largest mass system, a conclusion is that the maximum circular velocity of each halo Vmax is larger than the velocity at the virial radius V200 . In fact, larger systems have velocities that continue to rise to a larger fraction of the virial radius than what is observed for smaller systems. All these conclusions lead to the fact that low mass systems are more concentrated [10]. Rotation Curves The shapes of the rotation curves found by NFW [10] are shown in image [4.5] where the solid lines represent the galaxy rotation curve (disk + halo), the dashed lines represent the velocity of the adiabatically compressed halo and the dotted lines represent the slopes of observed rotation curves near the optical radius ropt (The optical radius is defined as the radius that encircles 83% of the total luminosity of the galaxy in the B-band). It is also considered that the halo will adjust according to the formation of the disk to the mass and radius of the disk, and that during this process of formation the disk formed slowly while the halo compressed adiabatically.. 29.

(31) Figure 4.4: Circular Velocity Profiles of only two halos from Navarro et al. Image taken from [10] Mass dependence As seen before, low mass systems are more centrally concentrate, which could be explained because low mass systems collapse at a higher redshift, when the mean density of the universe was higher. Figure [4.6] relates the concentration c with the mass of the halo at the virial radius, M200 , normalized at the nonlinear mass M∗ ≈ 3.3 ∗ 1013 M . In this figure is seen that large halos are less concentrated than smaller ones. In the model proposed by the CDM cosmogony, the density at the time when the halo has already collapsed, its a constant multiple of the density of the universe at the time it formed. Then, it is necessary to state a formal definition of the time of formation of the halo in order to understand the relation between the halo mass and its characteristic density. Since the halos are constantly changing gaining more mass through mergers, one definition states the time of formation of the halo, as the first time when half of its total mass was in its progenitors, each of these with individual masses that exceeded some fraction f of the final mass M of the halo [10]. Lacey & Cole [1993] stated that the probability that a halo with mass M identified at z0 was 30.

(32) Figure 4.5: The Shapes of the Rotation Curves of Disk Galaxies found by NFW in their paper [10] part of a progenitor with mass of at least f M earlier at a z was: ( ) δ0 (z − z0 ) P (> f M, z|M, z0 ) = erf c p 2 [∆20 (f M ) − ∆20 (M )]. (4.9). Where erf c is the complementary error function [13] defined as: Z ∞ 2 2 e−t dt Erf c(z) = √ π z With special values for erf c given by; erf c(−∞) = 2 erf c(0) = 1 erf c(∞) = 0 And ∆20 (M ) is the variance of the power linear spectrum at z = 0. From equation [4.9] it is possible to find the formation redshift, zf orm (M, f, z0 ) by setting the probability equal to 12 . Then the mean density of the universe at that redshift gives us the characteristic overdensity: δc (M, f, z0 ) = C(f )[1 + zf orm (M, f,0 )]3 31. (4.10).

(33) Figure 4.6: Concentration as a function of the mass, by Navarro et al. Image taken from [10] The shapes in image [4.6] show that the prediction between the mass and density given by equation [4.10] improves for smaller values of f .. 4.1 4.1.1. Numerical Results of the NFW profile Considering the system as a non uniform distribution of matter. If we now use the NFW profile for a system with a non uniform distribution of matter, this assumption is a better approximation to the problem of the dark matter halo. For the mass distribution: Z r r0 0 (4.11) M (r) = 4πρcrit δc rs 2 dr 0 r 0 1 + rs −r 3 = 4πρcrit δc rs − Ln (rs ) + Ln (rs + r) (4.12) rs + r Then: G ∇φ = 2 4πρcrit δc rs3 r. . −r − ln (rs ) + ln (rs + r) rs + r 32. (4.13).

(34) With this result we may obtain the potential energy:. W = −16π 2 ρ2crit δc2 GN rs4. i h   −rs −R (2rs + R) + 2rs (rs + R) Ln[ rsr+R ]  s. 2. 2 (rs + R). . (4.14). . For the moment of inertia; I=. 4.2. 4πρcrit δc rs3. . R (−6rs2 − 3rs R + R2 ) + 3rs2 ln 2 (rs + R). . rs + R rs. (4.15). Dark Matter according to Navarro et al. and the Virial Theorem. For the Kinetic Energy it is known that: K≥0 And using the fact that: 1 2K + W + ΛI = 0 3. (4.16). Then 1 0 ≤ 2K = −W − ΛI 3 1 0 ≤ −W − ΛI 3. (4.17) (4.18). Since for our system the Potential Energy is negative: 1 0 ≤ W − ΛI 3 1 W ≥ ΛI 3 By definition, ρvac =. Λ 8πGN. And that: ρvac = 0.7ρcrit 33. (4.19) (4.20).

(35) Then the inequality: W ≥. 0.7 3. 8πGN ρcrit I. (4.21). In order to check if this inequality holds for the model proposed by Navarro et al., the previous results are used to obtain a value for W and I and to find a relation that correlates c with the virial mass. It is proposed a linear regression based on the data in red of image [4.7], for which a table is extracted [4.1] with the closest values (chosen by hand) to model a straight line:. Figure 4.7: Concentration c as a function of the mass of the halo, red points considered for the linear regression. Image taken from [10]. ≈ Log. . M200 M∗. . -0.1 -1.0 -1.2 -1.9 0.4 1.5. ≈ Log c 1.05 1.15 1.18 1.25 0.98 0.81. Table 4.1: Data taken from image [4.7]. 34.

(36) Using this data, the result given by the linear regression is given by equation [4.22] M200 + 1.0205 Log c = −0.128 Log M∗. (4.22). Navarro et al. used certain numerical parameters to proceed with their simulations; these parameters are shown in figure [4.8]. Based on these parameters, three values for the mass are taken in order to corroborate the virial inequality.. Figure 4.8: Numerical Parameters by Navarro et al. Table taken from [10] The first mass selected in order to work with the virial inequality was the third numerical parameter found in the table: M200 = 0.414 ∗ 1012 M So that: Log c = −0.128 Log. 0.414 ∗ 1012 M 3.3 ∗ 1013 M. Log c = 1.2641 c = 18.3609. With this result and using equation [4.5] δc could be found: 35. .

(37) δc = 204802.77. (4.23). Now, r200 can be found using: M200 = 200 ρcrit Using a value for ρcrit = 1.118 ∗ 10−26 found:. Kg , m3. 4π 3 r 3 200. . and knowing that M. (4.24) = 1.989 ∗ 1030 Kg we. r200 = 4.44 ∗ 1021 m Now, this result obtained for r200 is used to find rs (given by definition in the paper): r200 = 2.421 ∗ 1020 (4.25) c Using these results we obtain the potential energy W and the moment of inertia I, using equations [4.14] and [4.15] respectively: rs =. W = −2.298 ∗ 1052 J. (4.26). I = 3.336 ∗ 1084 Kg m2. (4.27). And the inequality yields: 2.298 ∗ 1052 J ≥. 1 (8π) (GN )(0.7)ρcrit (3.336 ∗ 1084 Kg m2 ) 3. 2.298 ∗ 1052 J ≥ 1.458 ∗ 1049 J Since the inequality seems to hold for small masses, the same calculation is done for bigger masses, first for M200 = 28.15 ∗ 1012 M Which yields the result:. 1.469 ∗ 1055 J ≥ 2.02 ∗ 1052 J Now, for M200 = 1109.9 ∗ 1012 M : 5.81 ∗ 1057 J ≥ 9.92 ∗ 1054 J It is found that this inequality does not hold only for large and unrealistic values of M200 .. 36.

(38) 4.3. Results for an exponential distribution of matter. Using the results discussed in the previous section, now we study how an exponential density profile would behave according to the virial theorem: 1 2K − W + ΛI = 0 3 1 0 ≤ W − ΛI 3 1 0 ≥ ΛI − W 3 Knowing that: 8πGN ρvac 3. Λ= We obtain:. 8πGN W ρvac ≤ 3 I 3W ρvac ≤ 8πGN I If we define: W =. GN W̃ 2. ρvac ≤. (4.28). 3W̃ 16πI. By definition: Z |W̃ | =. ρ(r) Z. I=. Φ 3 dr Gn. r2 ρ(r)d3 r. (4.29) (4.30). If we consider a constant density, ρ and an arbitrary geometry: ρvac ≤ ρ. 3W̃ 0 16πI 0. So that: ρvac 3W̃ 0 ≤ ρ 16πI 0 Where, by definition, 37. (4.31).

(39) W̃ 0. Z. Φ 3 dr GN ρ. = V. I0 =. Z. r2 d3 r. Now, for a spherical geometry: R. Z. ρ(r)M (r)rdr. W = 4πGN 0. Z. ρ(r)r∇Φd3 r. W =− Where ∇Φ =. GN M (r) r̂ r2. Then R. Z W = 4πGN. ρ(r)M (r)rdr 0. Using the definition for W̃ ; R. Z W̃ = 8π. ρ(r)M (r)rdr 0. Where r. Z. r2 ρ(r)dr. M (r) = 4π 0. And for the moment of inertia: R. Z. r4 ρ(r)dr. I = 4π 0. Now, we take the exponential density profile of the form: −r. ρ(r) = ρc e r0 Where: ρ(0) = ρc −R. ρb = ρc e r0 For the moment of inertia: 38. (4.32).

(40) R. Z. −r. r4 e r0 dr. I = 4πρc 0 r , r0. Using the substitution x = Ln[ ρρcb ] = rR0 gives: I=. 4πρc r05. integrating by parts, and using the fact that. ρb ρc. −R. = e r0 , and that,. ρb ρc 2 ρc 3 ρc 4 ρc − 24 + 24Ln + 12Ln + 4Ln + Ln + 24 ρc ρb ρb ρb ρb (4.33). For the mass: Z M (r) = 4πρc. r. 0. −r 0. r 2 e r0 dr0. (4.34). 0 r0 , r0. so that, r0 dx = dr0 , then gives: " ) ( 2 # −r r r − +2 M (r) = 4πρc r03 e r0 −2 − 2 r0 r0. Using the substitution x =. (4.35). Now for W̃ ; (. R. Z. 4πρc r03. W̃ = 8π. −e. −r r0. ". 2+2. 0. r r0. . +. r r0. 2 #. ). −r. + 2 ρc e r0 rdr. (4.36). Which gives: ( W̃ = 32π 2 ρ2c r05. −2. . ρb ρc. . −2. ρb ρc. . 2 2 ) ρc ρc 11 ρb 11 ρb Ln + + Ln ρb 8 ρc 4 ρc ρb . ( ) 2 2 ρ ρ 7 ρ 1 ρ 5 b c b c +32π 2 ρ2c r05 Ln2 + Ln3 + 4 ρc ρb 2 ρc ρb 8 So far for the inequality we have: ρvac ≤. 3W̃ 16πI. Replacing and solving: ρc 2 −1 ≤ A ∗B ρvac 3 Where, in equation [4.37] A and B are two functions given by:. 39. (4.37).

(41) A = −2. 2 2 2 ρb ρb ρc 11 ρb 11 ρb ρc 7 ρb 2 ρc −2 Ln + + Ln + Ln ρc ρc ρb 8 ρc 4 ρc ρb 4 ρc ρb. 2 1 ρb 5 3 ρc + Ln + 2 ρc ρb 8 ρc ρb 2 ρc 3 ρc 4 ρc 24 + 24Ln + 12Ln + 4Ln + Ln B=− ρc ρb ρb ρb ρb. (4.38). Given a commonly used exponential density profile and the two values, the central density ρc and the boundary density ρb , we can check via equations [4.37] and [4.38] if the body is virialized or not (then equation [4.38] is satisfied). This is the usefulness of equations [4.37] and [4.38] derived in this thesis.. 40.

(42) Chapter 5 Modified Newtonian Dynamics (MOND) Modified Newtonian Dynamics or MOND is an alternative proposal in order to explain the flat rotation curves observed in galaxies, such as those shown in Fig [1.1]. MOND was proposed by Mordehai Milgrom, an Israeli physicist at the department of Condensed Matter Physics at the Weizmann Institute as an alternative to dark matter.. 5.1. History. Mordehai Milgrom began his investigations studying ultracompact neutron stars in binary star systems. After a while, during a sabbatical period in Princeton in 1979 Milgrom became interested in the galactic rotation curves that had intrigued physicists over a while. During the 0 70, physicists had used Newton dynamics in order to extract a measure of the total mass in a galaxy. This type of rotation curves work perfectly well for the solar system, just as expected by Newton’s law [5.1]. Nevertheless, this same procedure on galaxies show rotation curves, that instead of decreasing with radius tends to stay flat after a certain point. M GN (5.1) r2 At this point, physicists had three options to explain this behavior: the first one was to question the data, the second one to question the theory, or the third one, to think of the possibility of some type of non-luminous matter. Milgrom decided to use the second one. aN =. At the beginning, Milgrom’s ideas were not received warmly and it was not until 1983, that his three original papers were published in the Astrophysical Journal. Milgrom knew that his new theory did not explained anything from its principles, it was just a mere adjustment in order to explain these flat rotation curves, and while he continued to work on his theory, dark matter had gained many followers and explained much more than just these rotation curves. Something necessary in order to make MOND a serious alternative to dark matter was to reconcile it with The General Theory of Relativity. This was achieved about four years ago 41.

(43) by Milgrom’s colleague, Jacob Bekenstein, who called this new theory TeVeS (an acronym for tensor, vector and scalar). TeVeS could also be used to explain gravitational lensing.. 5.2. Basic Ideas. Starting from the knowledge that the rotation curves worked well for systems like our solar system, Milgrom established the possible properties differing from the solar system to a galaxy in order to modify the original equations. Galactic systems and planetary systems differ by their masses, sizes and angular momentum, as a first attempt one may consider to modify the distance dependence on gravity [19], making it not as strong as proposed by Newton (r−2 ). But in 1980 Milgrom proposed the acceleration as the parameter taken into account for modification, which at large distances (galactic systems) is much smaller. Then according to MOND, the standard Newtonian dynamics work for accelerations that are much larger than a new constant with dimensions of acceleration called a0 , where a0 is of the order 10−8 m s−2 . Milgrom assumed that when the acceleration was large enough compared with a0 : a = aN =. M GN for a >> a0 r2. (5.2). But for the other limit: a2 M GN = for a << a0 (5.3) a0 r2 This change only works when the acceleration falls below one 10-billionth of a meter per second every second [16]. With an interpolating function: a MG µ (5.4) a= 2 a0 r Where function µ satisfies: µ(x) ≈ 1 when x >> 1 µ(x) ≈ x. when x << 1 This means that for the Newtonian limit, when aa0 >> 1, therefore a >> a0 , we obtain the original Newtonian equation, while in the MOND limit, when aa0 << 1, therefore a << a0 , then we obtain the new modified dynamics as shown in equation [3.11].. 42.

(44) The function µ is not defined specifically but several possible forms have been postulated for it: 1. µ(x) = x(1 + x2 )− 2. µ(x) =. x 1+x. µ(x) = 1 − e−x Since the main basis of MOND is to explain the flat rotation curves, at large radius (small accelerations) MOND describes the orbital speed around a mass M as: p v = 4 M Ga0. Where we used: √ M G N a0 v2 a= = r r This means that the velocity rotation is independent of the radius. Then, when a system presents a smaller acceleration it must also show a larger mass discrepancy. Figure [5.1] shows the mass discrepancy for various systems, where the dependent axis shows the relation between the dynamical mass (the mass that is predicted from the motion of the system) and the observed mass, versus the typical system size given in Kiloparsecs shown in the independent axis. As it can be seen from figure [5.1], the mass discrepancy does not depend on the actual size of the system, which could be easily seen by the fact that the small dwarf spheroidals show large discrepancies, while galaxy clusters show a small discrepancy. On the other hand, figure [5.2] plots the mass discrepancy against the relation aa0 as predicted by MOND. Systems with the larger mass discrepancy are those for which a0 >> a, which refers to those on the right side of the plot, when aa0 >> 1. Nevertheless, there is one system, the cores of X-ray clusters, that seem to violate this mass discrepancy according to the acceleration. Within the X-ray clusters, the discrepancy in the core still behaves as predicted by MOND, while as we increase in the radius away from the core, the discrepancy decreases and eventually disapepars. The only explanation for this that would not contradict MOND is the 43.

(45) Figure 5.1: Mass discrepancy against the typical size, image taken from [17] presence of some type of not detected baryonic matter. In the form of dim stars or warm gas. At the non-relativistic level, MOND could be understood either as a modified gravity or as a modified inertia. These two are explained in the next sections.. 5.3. MOND as modified gravity. The first proposed attempt was to modify the dependence of gravity and distance, still making the gravitational force proportional to the masses involved, but the decline not as strong as the r−2 law at large distances. In the interpretation of MOND as a modified gravity, the equation that establishes the gravitational field caused by some sort of mass distribution changes from what we originally know. And the gravitational potential φ determined by the mass distribution ρ from a modification of Poisson’s equation is given by: h i ~ = 4πGρ ~ · µ ∇φ ~ /a0 ∇φ ∇ (5.5) Which we originally know to be the simple Poisson equation:. 44.

(46) Figure 5.2: Mass discrepancy against the system acceleration, image taken from [17]. ∇2 φ = 4πGN ρ. (5.6). In MOND as a modified gravity, only those systems that are governed by pure gravity are affected. Also, the acceleration of a test particle depends only on the position of the particle on the field [18]. Then, working with the new modification for the gravitational potential and having the equation of motion: ~ ~r¨ = −∇φ. (5.7). For a spherical symmetry and establishing µ(x) as: µ(x) =. x 1+x. Then, from equation [5.5] we get: ~ · ∇ 0. Where in equation [5.8], φ is. . 0 φ 0 φ r̂ = 4πρ(r)GN a0 + φ0. dφ . dr. 45. (5.8).

(47) Now solving: 0 1 d 2 φ 0 r φ = 4πGN ρ(r) r2 dr a0 + φ0 0 d 2 φ 0 r φ = 4πr2 ρ(r)GN dr a0 + φ0 Z 0 φ 0 2 r φ = 4πr2 ρ(r)GN dr a0 + φ0. (5.9) (5.10) (5.11). Which could be written as: 0. φ2 M (r)GN 0 = a0 + φ r2. Because: Z M (r) = 4π 0. φ 2 = GN. 0. φ 2 − 2GN. 0. φ 2 − 2GN. drr2 ρ(r). M (r) 0 a + φ 0 r2. M (r) 0 M (r) φ = G a0 N 2r2 r2. 2 M (r) 0 G2N M 2 (r) M (r) 2 M (r) φ + = G a + G N 0 N 2r2 4r4 r2 4r4. 2 2 M (r) M (r) 0 2 M (r) φ − GN = G a + G N 0 N 2r2 r2 4r4. M (r) φ − GN =± 2r2 0. r GN. 46. 2 M (r) 2 M (r) a + G 0 N r2 4r4. (5.12).

(48) 0. If φ > GN M2r(r) 2 then; r 2 M (r) M (r) 2 M (r) G φ − GN = ± a + G N 0 N 2r2 r2 4r4 0. (5.13). 0. And if φ < GN M2r(r) 2 then; M (r) φ − GN = (−) ± 2r2 0. r GN. 2 M (r) 2 M (r) a + G 0 N r2 4r4. (5.14). Given ρ, one can calculate M (r) and φ. Nevertheless if it were true that the discrepancies increased with the size of the system, then the dwarf spheroidal galaxies would show much more smaller discrepancies than those presented in some large galaxies, which is not the case.. 5.4. MOND as a modified inertia. MOND as a modified inertia, is understood as a modification of the equation of motion while the force fields do not change, as seen before in MOND as a modified gravity. In this theory, the standard kinetic action is replaced by one a little bit more complicated, which could be written in the form: Am S [~r(t), a0 ]. (5.15). In equation [5.15], Am depends on the mass of the body, and S depends only on the trajectory of the body and the parameter a0 . What we expect from this new function is that in the limit a0 → 0, then it should behave like what we originally know from Newtonian Dynamics, while for the limit when a → ∞ the MOND behavior holds. As we can see in modified inertia, the modification depends on the details of the trajectory of the particle, which means, that unlike modified gravity, modified inertia not only depends on the position of the particle but also on its trajectory.. 47.

(49) Conclusions This thesis work began as an attempt to understand the flat rotation curves problem. It followed the main stream current of physicists who believed in the existence of some type of non-luminous matter in the universe, called dark matter, and studied a density profile proposed by Navarro et al. which seems to adjust well to observations. This density profile was the result of N-body simulations for large periodic boxes. This new profile is in great agreement to what is observed in the flat rotation curves, not to mention that the inclusion of dark matter gives astronomers a powerful explanation of how galaxies formed in the first place. This density profile was studied under the scope of the virial theorem with a small modification, the cosmological constant. And we see that it is in great agreement with an inequality derived from the virial theorem for systems with masses corresponding to those observed in the universe. Even though this thesis work was based on the density profile developed by Navarro et al., it was necessary not to leave behind dark matter’s competition, MOND. This theory was also created as an explanation for the rotation curves observed in galaxies. It is an ad hoc theory; this is why it fits perfectly well with most of the systems for which this discrepancy was observed. Originally, MOND had two proposals, the first one considered MOND as a modification of the way gravity was affected by distance, and the second one proposed MOND as a modification in inertia. The former was still proportional to the involved masses in the system but considered that the decline with the distance was not as strong as it was originally proposed, for the latter the field does not change but rather the equation of motion suffered an alteration. MOND is expected to fit well in every system, since this is the purpose of the theory, nevertheless some discrepancies have been found in the cores of X-ray clusters. The reason why these two theories are still competing is because they both have flaws and neither of them have the ultimate proof. On one side we have some type of matter that seems 48.

(50) to explain many things in the universe, but that has never been detected (this is the main reason for skepticism) and on the other side there is a theory that was created to fix the problem, but that doesn’t seem to fit into any other aspect of our universe (because of its implications) and it still has a system for which it doesn’t work perfectly well. Finally, we believe that more work is necessary in order to conclude which of the two theories is the best. It is necessary to concentrate on the flaws of both of them in order that one can succeed, or a third one emerge. For now, it seems that dark matter has the lead on the race.. 49.

(51) Bibliography [1] M.Capaccioli, N.R.Napolitano, M.Arnaboldi,Mass-to-Light ratios in Early-Type Galaxies and the Dark Matter content, Capodimonte Astronomical Obs, Italy, Dept of Physical Sciences, Italy, Pino Torinese Astronomical Obs, 2002. [2] Michael Rowan-Robinson, Cosmology, Blackett Laboratory, Imperial College, London, Third Edition, 1996. [3] Jayant Vishnu Narlikar, An Introduction to Cosmology, Inter University Centre for Astronomy and Astrophysics, Pune India, Third Edition, 2002. [4] James Binney and Scott Tremaine,Galactic Dynamics,Princeton University Press, Princeton, New Jersey, 1987. [5] National Center for Supercomputing Applications, More the eye, University of Illinois, recovered on March http://archive.ncsa.uiuc.edu/Cyberia/Cosmos/MoreMeetsEye.html.. that 20,. meets 2008,. [6] Tom Ritchey, About Fritz Zwicky, Swedish Morphological Society, recovered on March 20, 2008, http://www.swemorph.com/zwicky.html. [7] Martin White, Dark Matter, Berkeley Astronomy Department, University of California, recovered on March 31, 2008, http://astro.berkeley.edu/ mwhite/darkmatter/essay.html. [8] Martin White, Dark Matter, Berkeley Astronomy Department, University of California, recovered on March 31, 2008, http://astro.berkeley.edu/ mwhite/darkmatter/dm.html. [9] Chris Miller, Cosmic Hide and Seek: the Search for the Missing Mass, recovered on April 5, 2008, http://www.eclipse.net/cmmiller/DM/. [10] J. Navarro, C.Frenk, S.White, The Structure of Cold Dark Matter Halos, The Astrophysical Journal, 462:563-575, 1996 May 10. [11] J. Navarro, C. Frenk, S. White A Universal Density Profile From Hierarchichal Clustering, The Astrophysical Journal, 490:493-508, 1997 December 1. [12] Marek Nowakowski, The Consistent Newtonian Limit of Einstein’s Gravity with a Cosmological Constant, Departamento de Fisica, Universidad de los Andes, International Journal of Modern Physics, Vol 10, 2000. 50.

(52) [13] Weisstein, Eric W. , “Erfc”. From MathWorld-Wolfram, recovered April 14, 2008, http://mathworld.wolfram.com/Erfc.html. [14] Eli Michael, Andrew Hamilton The Cosmoλogical Constant, Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, http://supercolorado.edu/ michaele/Lambda/lambda.html, recovered on April 8, 2008. [15] Carlos Felipe Uribe Muoz, Analysis of the Anisotropies in the Microwave Background Radiation, Universidad de los Andes, Facultad de Ciencias, Departamento de Fisica, 2007. [16] Adam Frank, Gravity’s Gadfly, Discover Magazine-New York, Vol 27, Num 8, Pages 32-39, Buena Vista Magazines, USA, 2006. [17] Mordehai Milgrom, The Modified Dynamics-A Status Review, Department of Condensed-Matter Physics, Weizmann Institute, Rehovot, Israel, astro-ph/9810302v1, October 2008. [18] Mordehai Milgrom, Does MOND follow from the CDM paradigm?, Institute of Astronomy, Cambridge, UK, astro-ph/0110362v1, October 2001. [19] Mordehai Milgrom, MOND a pedagogical review, Department of Condensed Matter Physics, Weizmann Institute, Rehovot, Israel, astro-ph?0112069v1, December 2001. [20] M. Nowakowski, I. Arraut, Living with Λ, Departamento de Fsica, Universidad de los Andes, Bogot, Colombia, April 11,2008.. 51.

(53) Appendixes The Boltzmann Collisionless Equation Despite the former development of the virial theorem, there is another way to arrive at this result departing from the Boltzmann Collisionless Equation (This result is just a summary of the original result presented at [4]): Consider a potential Φ(~x, t) and a number of stars under the influence of this potential. A complete description of this system at any time t may be given by specifying the number of stars through a density function [5.16] with positions at a small volume d3 x centered about ~x and with velocities in a small range d3 v centered about ~v . f (~x, ~v , t) d3 xd3 v. (5.16). If we now consider a phase space in which the stars move along their orbits, we may find the coordinates in this phase space as a 6 dimension vector given by: w ~ ≡ (w1 , w2 , ..., w6 ) ≡ (~x, ~v ). (5.17). The velocity of this flux of stars may be written as: ẇ = (ẋ, v̇) = (~v , −∇Φ). (5.18). The stars described by ẇ is that in the absence of encounters, the stars travel softly in space, rather than jumping from one point in the phase space into another. Analogous to the density of an ordinary flux, the density of stars f (w, ~ t) satisfies the continuity equation: 6. ∂f X ∂ (f ẇα ) + =0 ∂t α=1 ∂wα. (5.19). If we now integrate equation [5.19] in some volume of the phase space the first term describes the rate at which stars increase inside this volume, and an application of the divergence theorem to the second term determines the rate at which stars leave this volume. Now the flux described by ẇ:. 52.

(54) 6 X ∂ ẇα. 3 X ∂vi. ∂ v̇i = + ∂w ∂x ∂vi α i α=1 i=1 6 3 X ∂ ẇα X ∂ ∂Φ = =0 − ∂w ∂v ∂x α i i α=1 i=1. (5.20) (5.21). ∂vi This result was obtained thanks to the fact that ∂x = 0 because vi and xi are independent i coordinates of the phase space and because, ∇Φ has no dependence on the velocity. If we use the equation at [5.21] to simplify [5.19] we may obtain the Boltzmann Collisionless Equation: 6. ∂f ∂f X + ẇα =0 ∂t α=1 ∂wα. (5.22). 3 ∂Φ ∂f ∂f X ∂f + vi − =0 ∂t ∂x ∂x ∂v i i i i=1. (5.23). And in vector notation: ∂f ∂f + ~v · ∇f − ∇Φ · =0 (5.24) ∂t ∂~v Since f (w, ~ t) is a seven variables distribution function, it could be quite difficult to find a solution the Boltzmann Collisionless Equation, nevertheless if we integrate equation [5.23] over all possible velocities: Z Z Z ∂f 3 ∂f 3 ∂Φ ∂f 3 d v + vi d v− d v=0 (5.25) ∂t ∂xi ∂xi ∂vi ∂ This range of velocities does not depend on time, so ∂t may be taken out of the integral, in the same manner, vi does not depend on xi . Finally, if we consider that f (~x, ~v , t) = 0 for a v big enough (which is to say that there aren’t stars that move infinitely fast). If we define the spatial density of the stars as ν (~x) and the media stellar velocity ~v¯ (~x) as:. Z. ν = f d3 v Z 1 v̄i ≡ f vi d3 v ν. (5.26) (5.27). Then: ∂ν ∂ (ν v̄i ) + =0 ∂t ∂xi 53. (5.28).

(55) We could identify equation [5.28] as a continuity equation, when compared to the continuity equation [5.29]: ∂ρ + ∇ · (ρ~v ) = 0 (5.29) ∂t If we multiply equation [5.28] by vj and integrate over all velocities: Z Z Z ∂f 3 ∂Φ ∂f 3 ∂ 3 f vj d v + vi vj d v− vj d v=0 (5.30) ∂t ∂xi ∂xi ∂vi For the last term: Z Z Z ∂f 3 ∂vj 3 vj d v=− f d v = − δij f d3 v = −δij ν (5.31) ∂vi ∂vi All these because the term in equation [5.31] could be transformed by the divergence theorem, and using the fact that f = 0 for large v. Now, equation [5.30]: ∂ (ν v̄j ) ∂ (νvi vj ) ∂Φ + +ν =0 ∂t ∂xi ∂xj. (5.32). where: Z 1 vi vj f d3 v vi vj ≡ ν Equation [5.32] is known as the Jeans Equation. If we now use this equation and multiply it by xk and integrate over all possible positions, we find a simple tensor equation that relates the global properties of the galaxy and the mean-square stream velocity. Z Z Z ∂ (ρv̄j ) 3 ∂ (ρvi vj ) 3 ∂Φ 3 d x = − xk d x − ρxk dx (5.33) xk ∂t ∂xi ∂xj In equation [5.33] we related ν with the mass density ρ. The second term in this equation is by definition the potential energy tensor: Z ∂Φ 3 Wjk ≡ − ρ (~x) xj dx (5.34) ∂xk The second term can be written as, assuming that ρ vanishes for large r: Z Z ∂ (ρvi vj ) 3 xk d x = − δki ρvi vj d3 x = −2Kkj (5.35) ∂xi Where we used the kinetic energy tensor definition: Z 1 Kjk ≡ ρvj vk d3 x (5.36) 2 54.

(56) Using the result of equation [3.36] we may divide K into the random and organized contributions: 1 Kjk = Tjk + Πjk 2. (5.37). Where: Tjk. 1 ≡ 2. Z Z. Πjk ≡. ρv̄j v̄k d3 x. (5.38). 2 3 ρσjk dx. (5.39). ∂ out of the integral since xk does not depend on time, and For equation [5.33] we may take ∂t finally we may take an average to obtain: Z 1d ρ (xk v̄j + xj v̄k ) d3 x = 2Tjk + Πjk + Wjk (5.40) 2 dt If we remember the definition for the moment of inertia tensor I: Z Ijk = ρxj xk d3 x (5.41). Taking the derivative with respect to time: Z 1 ∂ρ 1 ∂Ijk = xj xk d3 x (5.42) 2 ∂t 2 ∂t With equation [5.28] we may write: Z Z 1 1 ∂ (ρv̄i ) 3 xj xk d x = − ρv̄i (xk δji + xj δki ) d3 x (5.43) 2 ∂xi 2 The equality in equation [5.43] is obtained by the divergence theorem. If we substitute equation [5.43] into equation [5.42]: Z 1 dIjk 1 = ρ (v̄j xk + v̄k xj ) d3 x (5.44) 2 dt 2 Combining equation [5.40] and equation [5.44] we obtain the tensor virial theorem: 1 d2 Ijk = 2Tjk + Πjk + Wjk (5.45) 2 dt2 For many applications we assume a system at a steady state, for which I is independent of time. Now to obtain the scalar virial theorem: 1 T race (T ) + T race (Π) ≡ K 2 T race (W ) = W 55. (5.46) (5.47).

(57) For a steady state system, we obtain the scalar virial theorem: 2K + W = 0. (5.48). The Boltzmann Collisionless System In the previous appendix we introduced the virial theorem through a series of mathematical descriptions based on the Boltzmann Collisionless equation. This appendix pretends to demonstrate why this system of stars could be considered as a Collisionless System [4]. If we consider a number of particles moving in a straight line, its mean free path is given by: 1 (5.49) nσ Where n is the number density and σ is the cross section. Using a rough estimate in which we approximate all the stars to have the same radius as the Sun R , then its cross section is given by: λ=. σ = π (2R )2. (5.50). If we spread 1011 stars over a radius of 10Kpc and thickness 1Kpc, then the number density is n = 0.3pc−3 , and the mean free path is λ = 5 ∗ 1014 pc. With this mean free path we may calculate the time between collisions as λv where v is the random velocity of the stars at a specific place. , then the time between collisions is t = 1019 yr which If we consider this velocity to be 40 Km s is 109 more time than the age of the galaxy.. 56.

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