MODIFIED GRAVITY THEORIES: ALTERNATIVES TO THE MISSING MASS
AND MISSING ENERGY PROBLEMS
By
MARC EDWARD SOUSSA
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2005
Copyright 2005
by
Marc Edward Soussa
To myf iIK.,I,;/
ACKNOWLEDGMENTS
The order in which I shall make my acknowledgements in no way reflects the
relative importance each represents. There are those who directly and indirectly
impacted my life as a graduate student here, and therefore their influences are
unique in almost every respect.
Professionally, I thank my committee members (James Fry, David Groisser,
Pierre Ramond, Pierre Sikivie, and Richard Woodard) for agreeing to serve and
making the necessary time commitments required. Of those members, I make
special acknowledgments to my advisor, Richard P. Woodard. His dedication,
excellence, and unwavering commitment to his work and his students serve as an
example to which any physicist could aspire. My life and my mind will forever be
influenced by his tutelage, and for that I am eternally grateful.
I also make a special acknowledgment to Pierre Ramond for his constant
and continuing support of my pursuits, and his committed effort to ensuring
that my physics education received a well-rounded balance from many fields.
His ability to find a unique perspective in which to view problems (related and
especially unrelated to his own field of research) is something I will ahv-- -i admire
and work to emulate. I thank James Fry and Pierre Sikivie for drafting letters
of recommendation on my behalf and for being available to my questions and
requests, whether administrative or physics-related.
Of the remaining faculty, I acknowledge Hai-Ping C'!, 1;s, my first advisor,
and who encouraged me to pursue my interests and took interest in my progress
throughout my graduate school career. I thank Mark Meisel who has ahv-- -i been
available, encouraging, and sincerely concerned for my status and progress as a
graduate student.
I thank the entire administrative staff in the physics department. They greatly
eased the burden of paperwork and administrative requirements placed on graduate
students. Of the staff I especially recognize Darlene Lattimer, Susan Rizzo, and
Yvonne Dixon.
I acknowledge the following individuals for their impact my life on a personal
level. I especially thank my parents, to whom I owe a debt of gratitude which could
never be sufficiently expressed here. Vous etes, vous etiez toujours la pour moi.
Sans vous rien est possible. Je vous aime; je vous remercie.
I would like to thank Sudarshan Ananth for being (I can quite confidently -iw)
the only other graduate student with whom I could spend 4 years in an office. Our
countless mature conversations, remarkable analogies, and delicate treatment of
each other and our colleagues will be something I will carry throughout my career
with fond recollection.
I thank Jacqueline, Jessie, Bailey, and family for their support. Some of my
fondest memories and some of my happiest moments in Gainesville are directly
attributable to them.
I thank my friends -past, present, and unnamed.
TABLE OF CONTENTS
ACKNOW LEDGMENTS .............................
page
iv
ABSTRACT .......................
1 INTRODUCTION ..............................
2 DARK MATTER: THE MISSING MASS ....
Introduction .. .. .. .. .. .. .. ..
Dark Matter Taxonomy ...........
Dark Matter Distribution ..........
Problems and Drawbacks to the CDM Halo
Concluding Remarks ............
Models
3 MODIFIED NEWTONIAN MECHANICS ....
3.1 Nonrelativistic Formulation .......
3.1.1 M otivation .............
3.1.2 Action Principle ..........
3.2 Relativistic Formulation .........
3.2.1 M otivation .............
3.2.2 Scalar-tensor Approach ......
3.2.3 Purely Metric Approach .....
3.3 The MOND No-Go Statement for Purely
3.3.1 M otivation .. ...........
3.3.2 The Statement .. .........
3.3.3 A Connection with TeVeS .....
3.3.4 Revisiting the No-Go Statement
Metric
Approaches
4 DARK ENERGY: THE MISSING ENERGY. ............. ..54
4.1 Introduction .................. ........... .. 54
4.2 The Many Faces of Dark Energy .................. .. 55
5 LATE TIME ACCELERATION WITH A MODIFIED EINSTEIN-HILBERT
ACTION .................. ............. .. .. 61
Late-time Acceleration .. ....................
The Gravitational Response .. .................
Remarks on our Calculation and Future Work .. .........
6 CONCLUSIONS ............................. 75
REFERENCES ................................... 80
BIOGRAPHICAL SKETCH ........ ........ .. .......... 86
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
MODIFIED GRAVITY THEORIES: ALTERNATIVES TO THE MISSING MASS
AND MISSING ENERGY PROBLEMS
By
Marc Edward Soussa
May 2005
C'!I ii: Richard P. Woodard
M, 1 ri Department: Physics
Modified theories of gravity are examined and shown to be alternative possibil-
ities to the standard paradigms of dark matter and dark energy in explaining the
currently observed cosmological phenomenology. Special consideration is given to
the relativistic extension of Modified Newtonian Dynamics ( MOND) in supplanting
the need for dark matter. A specific modification of the Einstein-Hilbert action
(whereby an inverse power of the Ricci scalar is added) is shown to serve as an
alternative to dark energy.
CHAPTER 1
INTRODUCTION
The advent of precision ..-1i1, ,i,!-l~ics in the past 20 years has provided cos-
mologists, particle theorists, and general relativists with a healthy volume of data
and measurements with which to work and explain. It has become clear that a
large fraction of the universe's energy content is unknown to us. Indeed, we are
entering an exciting era of astrophysical investigation whereby experiments (past,
present, and future) will guide physicists to understand the fundamental nature of
the universe.
Our thesis considers two problems: namely, what I will term, with no attempts
at originality, the missing mass and the missing energy problem. Each shall be
approached by considering first the predominant or orthodox approach to their
explanations. In the case of the missing mass problem this corresponds to the
concept of dark matter. Respectively, the missing energy problem introduces
a substance dubbed dark energy to describe the late-time acceleration of the
universe. Each of these approaches will be shown to possess advantageous and
seemingly "natural" features which seem to justify their acceptance as the leading
candidates. However, these approaches are far from being satisfactory resolutions of
their targeted problems. Inconsistencies and ambiguities remain. While that is so,
alternative approaches must be vigorously researched.
One of the most important points to underline in this thesis is the fact that
neither dark matter nor dark energy have been detected in a laboratory setting;
they have only been observed in a gravitational context. Direct determination of
dark matter would certainly quell (if not completely put to rest) the notion that
perhaps we do not understand gravity even at the classical level. However, until
strong evidence of either particle dark matter or dark energy is obtained we may
admit the possibility that it is new gravitational physics, not missing substances
which is responsible for our observed universe.
Einstein's equation possesses two sides: the source side and the gravity side.
G, =- 87wGT,,, (1.1)
where T,, is the matter-energy stress tensor and G,, R,, is the Einstein
tensor. Each is obtained by varying the respective actions,
2 6 4 -2 6 [
G,= dX R 4 -gM(OA, ....),
(1.2)
where LM is a matter Lagrangian density. We will use a time-like signature
throughout the thesis. GC can be thought of as the source of gravity due to the
matter components constituting T,. Clearly, we change the behavior of gravity by
altering the sources present in T,,. Indeed, the "dark horses" of modern physics,
dark matter and dark energy, are both appendices to the source side, albeit in
peculiar forms. Gravitational observation can only tell us about the gravity side,
and thus the ability in/, ; n- exists to add terms to the source side to make the
gravity side true. Only by solving the equations of motion of a new particle field
and confirming the solution experimentally can one conclude real existence.
Consider the more general situation,
9, = 8sTGT,, (1.3)
Here, g,, is a not necessarily the Einstein tensor. We will refer to it as the i ,Il i
- ii- .i Obviously, one must make restrictions when formulating the gravity ten-
sor. General covariance reconciles the requirements of stability with local Lorentz
invariance. Higher spin fields are notorious for possessing negative energy degrees
of freedom. However, in a generally covariant theory these degrees of freedom are
either unphysical (e.g., the non-transverse modes of the photon in Lorentz gauge)
or are constrained degrees of freedom (e.g., the Newtonian potential of gravity).
In addition to conforming with general covariance, phenomenological requirements
must be met. The most poignant of these is Newton's law of gravitation. Gravity is
well understood at the solar system scale. Any proposed modification to Einstein's
equation would have to faithfully reproduce Newton's law in this regime. Further,
Einstein's relativity (which has proven to be extremely robust over a broad range
of physical scale) must emerge from any candidate theory under the appropriate
conditions. Gravitational lensing, the bending of light due to matter sources, will
be shown to serve as an important phenomenological constraint. A-l i.|1 r, -ical
data coming from recent experiments such as the Wilkinson Microwave Anisotropy
Probe (WMAP) and Cosmic Microwave Explorer (COBE); and data anticipated in
the future from the Supernova Acceleration Probe (SNAP) and the Planck Satellite
simultaneously furnish us with bizarre challenges now and place more and more
stringent restrictions on models in the future.
The thesis is organized as follows: C'! lpter 2 discusses the missing mass
problem and describes the dark matter approach to its explanation. In C'! Ilpter
3, the alternative proposition of MOND is introduced and shown as a viable
alternative. First, we discuss the nonrelativistic successes of MOND, followed by
a thorough analysis of the relativistic extensions currently under consideration.
Specifically, the scalar-vector-tensor theories Bekenstein, Milgrom, and Sanders
are survn i, d1 in some detail followed by a complete treatment of the author's
contribution to the purely metric approach. Chapter 4 introduces the dark energy
problem discussing some of the more common approaches. C'! Ilpter 5 introduces
a specific alteration to the Einstein-Hilbert action one can make to reproduce the
same effect as dark energy. The author's contribution to the computation of the
force of gravity due to such an alteration is completely treated. It is shown to place
4
severe limitations on this wide class of models. ('! i lter 6 summarizes the results
with some remarks concerning the implications for present and future work.
CHAPTER 2
DARK MATTER: THE MISSING MASS
2.1 Introduction
When the velocities of satellites orbiting spiral galaxies are measured, they be-
have in a quite peculiar fashion. The velocities are of the order 102km/s (therefore
v/c ~ 10-3), and one would naively expect a Newtonian description of their gravi-
tational dynamics. From elementary physics, we ascribe centripetal acceleration to
a particle in a circular orbit outside a matter distribution M(r),
V2(r) GM(r)
,a (2.1)
r r
and thus well outside the mass distribution the .-i-~!,lil ic velocity behaves as,
S2 (2.2)
However, rather than a Keplerian fall off of the .-i-,,!,l')itic velocity, satellites are
observed to .-i-:i .,l')te to a constant velocity far from the central galactic bulge,
o, = constant (2.3)
Such behavior, one can easily imagine, can be described by inserting more
matter into the system in the appropriate distribution. Indeed, this phenomenon
has served as a in1 i, r impetus for invoking the existence of an unknown matter
component (dark matter) that pervades the galactic systems of our universe, and
the universe's entirety itself. As we have not directly witnessed this matter via
the electromagnetic, electroweak, or strong nuclear forces, it has acquired the
"I ,i : nomenclature. Its only interaction that we have observed (if it were a true
component of our universe) is its gravitational interaction with luminous and
nonluminous matter (and of course the cosmological fluid itself).
The application of this idea faces many obstacles from the onset. Immediately,
its origins come into question. That is, one must use cosmological motivations and
evidence to account for the existence of dark matter. Galaxy formation becomes
a critical issue when discussing dark matter since early fluctuations in the C'\ ll
give evidence to the density fluctuations that were present at recombination. These
imprints constrain the possibilities of how much matter energy density can be
involved in galaxy formation.
Many candidates have been proposed, of which a few will be discussed later
-but we may temporarily spotlight the necessity to quantify dark matter's
standing in the particle description of matter. Specifically, does dark matter
consist of the usual suspects (i.e. the Standard Model particles)? Or is dark
matter the implementation of i, .-- ply-r, particles and fields not currently
captured in the Standard Model? Its distribution and -y iili. .I ic" relationship
with the luminous matter in the galaxy and the universe must be identified.
Further, with the recent augmentation of our experimental abilities in measuring
galactic observables, dark matter must be embedded into our galactic systems in
a self-consistent fashion such that the current observables are not voided by the
introduction of a new exotic.
2.2 Dark Matter T.::;..i-n.i-
We will limit the scope of our discussion of dark matter to issues concerning
galactic systems. However, it should be noted that there is an enormous amount
of theoretical and experimental work extant dealing with dark matter's possible
role in cosmology. Dark matter, if existent, occupies far more of the intergalactic
medium than the galactic one. This is evident from recent and quite constraining
data from the WMAP probe [1], which reveals the large discrepancy between dark
and luminous matter critical fractions, QCDM ~ 0.27 and QB ~ 0.03, respectively
(here CDM and B refer to cold dark matter and baryonic matter, respectively,
explained below). Any cosmological component represents some fraction of the
critical mass density (the density for which the universe expands to a critical radius
and freezes) pc = 3H2/87rG,
x -- (2.4)
Pc
where H is the Hubble expansion parameter. The WMAP data confirms the hot
big bang theory followed by a period of inflation giving rise to a flat universe.
In terms of critical fractions, the total fraction is uTotal ~ 1 with a dark energy
component, QA ~ 0.7, which will be discussed later in greater detail. The question
is then whether the :il' energy component (i.e. that which is not coming from
dark energy) is a true matter component or whether there is new physics at the
level of new particles and fields, or fundamental spacetime principles.
In addition, proof that dark matter really does exist must come from its
observed interactions. Many dark matter candidates have emerged through
the years. Roughly, one may divide them into two categories: nonluminous
baryonic matter such as brown dwarfs, black holes, and large planets (\!.ACHOS
-Massively Compact Halo Structures); and weakly interacting particles such
as neutrinos, axions, and neutralinos (WIMPS -Weakly Interacting Massive
Particles) which pervade large portions of the universe. Of these two candidates, it
has been experimentally and phenomenologically determined that if MACHOS do
exist, they constitute very little of the total possible dark matter observed [2].
WIMPS can be further categorized by their cosmological history. Some
particles formed in the big bang are relativistic for some period of time. Depending
on their masses and couplings to other particles, we are able to predict and
observe their transition from the relativistic regime to the nonrelativistic one.
Those that are relativistic at the onset of galaxy formation are classified as hot
dark matter (HDM), whereas those which have dipped into the nonrelativistic
regime are classified as cold dark matter (CDM). A light neutrino (< 20 eV) and a
heavy neutrino (~ 100 GeV) serve as candidates for CDM and HDM, respectively
[3]. Finally, a third type of particle dark matter, the axion [4], arises from the
Peccei-Quinn mechanism to solve the strong CP problem of QCD [5]. The axion
is a particle which exhibits a particular symmetry that ensures (by reaching the
minimum of its potential) that CP-symmetry is not violated in any strong nuclear
interactions. Depending on the values of the axion mass and couplings, it is
possible to account for a large fraction of the dark matter [6].
Further analysis using WMAP data, however, strongly tilts the favor toward a
CDM scenario [7]. Because galactic formation depends upon the nature of its dark
matter constituency [8], CDM has emerged as it is able (unlike HDM) to provide
sufficient clumping on galactic scales we observe tod -i. Therefore, we will survey
the conservative approach to galactic dark matter and regard it as CDM.
Galactic dark matter has been most commonly introduced itself in the
literature and in scientific investigations (both theoretical and phenomenological)
as halos in which one may either view the galaxy embedded in the halo, or the
halo embedded in the galaxy. The amount of dark matter projected is often on the
order of 9 greater than that of ordinary matter. Since the rotation curves of the
inner regions of galaxies are well reproduced by considering only luminous matter,
it must be that the in i i, ily of the dark matter resides outside the central bulge to
ensure the .,-i-',,' .1l' ically constant satellite velocities.
We will further restrict ourselves to rotation curves of spiral galaxies, as they
have been by far the most studied. The problem is to find how the dark matter
distributes itself in and around the luminous matter of the galactic system. It
cannot have a great impact on dynamics in the inner region since luminous matter
accounts for the rotation curves there. It also faces the challenge of reproducing
the rotation curves outside the inner region for a wide range of physical scales
distance, luminous mass, mass-to-light ratios, etc. Thus, dark matter profiles face
the further challenge of being universal (or at least exhibiting universality).
2.3 Dark Matter Distribution
A thorough review of the different halo models is given by [9]. We consider
only two here, which is more than sufficient to capture the most important features
of halos. The simplest CDM distribution which can successfully account for the
rotation curves of spiral galaxies is the isothermal sphere proposed by Gunn and
Gott [10] with mass density,
p(r = o (2.5)
1 + (r/r,)2
Immediately from Equation 2.5 we see there are two parameters which must be
determined, the central density po and the core radius re. Clearly, this profile
gives rise to the observed rotation curves. The circular velocity of a particle in the
isothermal distribution is,
v2(r) 4G dr'r /2 p' (2.6)
r Jo
= 47Gpor2 i arctan (2.7)
In the limit of r > re, Equation 2.7 reduces to,
v4 V4Gpor (2.8)
the .i-,i ill ic velocity required to explain the rotation curves. A possibly un-
fortunate feature of Equation 2.8 is that the velocity never ceases to fall off, and
therefore certainly this scenario can only serve as a first approximation.
Another popular class of formulations is the so-called "universal" N i' 11 o,
Frenk, and White (NFW) profiles [11, 12],
p(v) Kc
p r + /)2 (2.9)
p rc (+r/r.s)2
where pc is the same as in Equation 2.4. There are again two parameters to
determine, as is alv--,v- the case with a CDM profile: a density parameter (in this
case a dimensionless characteristic number 6c); and a length scale (here represented
by the scale radius r,). These profiles have inherited the classification "universal"
for the similarities in the profiles between halos of widely varying mass, which came
as a surprise, in light of the power-spectrum data [12].
Identical to the isothermal halo, we calculate the velocity from the NFW
profile Equation 2.9,
v2 ) 4Gp) 1 (2.10)
r 1 + r,/r
The square velocity has the limiting behaviors,
2 27rGcpcrFr r < Fr ,
v2(r) -- (2.11)
c ln(v/r,) r >r,
These limiting behaviors are clearly superior in the phenomenological sense to the
isothermal halo on account of the quasi-Keplerian fall off at large enough distances.
The nontrivial solution to the transcendental equation,
x(1 + 2x)- (1 + x)2 In(1 + x)= 0 (2.12)
where x = r/r,, gives the ratio of radii necessary to achieve the maximum velocity.
This equation can be numerically solved to give x 2.16. The velocity approx-
imately drops to 0.82vmax and 0.85vmax for x = 0.1x and x = 10x, respectively.
Therefore, it is quite evident that flat rotation curves can be described with the
NFW profile, with the added feature that at .,-i- !,ll)tically large distances, a fall
off is exhibited.
2.4 Problems and Drawbacks to the CDM Halo Models
Although the isothermal halo model enjoys success in reproducing many of
the observed rotation curves of disk galaxies, the assumptions leading to it exposes
its lack of universality. The isothermal model assumes what has been called the
in i::in -d-hI: hypothesis, which assigns to the disk the maximum mass-to-
light ratio consistent with velocity measurements in the inner region [11]. The
maximum-disk hypothesis effectively separates the disk from the halo. Therefore,
satellites in the inner region inherit all but a negligible fraction of their velocities
from the disk alone. This has the effect, of course, of limiting the core density of
the halo. However, inclusion of data from dwarf galaxies shows this assumption to
break down [13]. The rotation curves for these samples were of quite a different
shape and could not be explained using the isothermal plus maximum-disk model.
Further, velocity-dispersion measurements in the normal direction to the disk
showed that the disk only contributed approximately I .to the inner velocity of
satellites [14], thereby nullifying the maximum-disk hypothesis.
The maximum-disk hypothesis attempts to regard rotation curves as functions
of the luminosity alone. This was tendentiously proposed [14] in response to the
observation that in low-luminosity galaxies, rotation curves rise slowly and continue
rising past the optical radius; whereas in higher luminosity galaxies, the curves rise
more sharply to their maximum, leveling off and sometimes even declining past the
optical radius. However, it was discovered [14] that within the subset of observed
galaxies that exhibit similar luminosities, differently shaped curves were measured
(distinguished by the galaxy's surface brightness).
The NFW profiles for galactic halos and X-ray clusters have been studied
extensively using N-body/ ,- i-Lvi ,i.i1 Ll simulations of CDM in a flat, low-density,
cosmological constant-dominated universe [15, 16]. Inserting nonsingular isothermal
halos into N-body simulations shows these structures to poorly fit the data [15].
The NFW profiles by far constitute a more sophisticated approach and are able
to overcome some of the problems of the isothermal model. That said, the NFW
approach currently requires further dynamical input into determining the specific
profiles that can account for the broad range of observed galaxy brightnesses and
masses observed. For example, low surface-brightness galaxies are not as well fit by
NFW profiles [17] where rotation curves rise more sharply in the inner region than
the profile predicts.
Lastly, one of the more disturbing features of CDM halos is their large
parameter range. Although it is by no means an indicator of futility of this
approach, it certainly spurs the advent of more sophisticated models (if not
radically new ideas). By trading off disk mass for halo concentration, one is .l ,-
capable of producing similar velocity curves. In fact, i;1, rotation curve can be fit
by setting the disk's mass-to-light ratio to zero and tuning the halo parameters.
Thus, differentiating among a class of halo models becomes a daunting task that
can only be simplified by injecting new data or new fundamental concepts into the
profile-building process.
This fine-tuning problem comes to light when considering the Tully-Fisher
relation: the observation that a galaxy's luminosity is correlated to its peak
rotation velocity via,
L o v,. (2.13)
The rotation velocity is mostly set by the halo, whereas the infrared luminosity
comes from the visible matter in the galaxy [18]. The fine-tuning that arises from
the symbiotic relationship between the disk and halo must somehow consistently
reproduce Equation 2.13. The observational precision of Equation 2.13, however,
is not to be easily expected from statistical processes involved in galaxy formation.
Therefore, the Tully-Fisher relation remains an issue for halo profiles that clearly
needs to be addressed before any specific profile or halo mechanism is deemed
satisfactory.
2.5 Concluding Remarks
Evidence for dark matter, regardless of the galaxy rotation curves, is quite
extensive. The Standard Cosmology cannot do without it; at least not without a
radical change to fundamental physics. Processes such as Big Bang Nucleosynthesis
(BBN) [19, 20], structure formation [21, 22], and the cosmic microwave background
(C \!11i) [23, 24] all indicate the Lambda CDM (cold dark matter with a cosmo-
logical constant) approach to be the superior scenario. Successes in these areas
cannot be ignored. The conservative approach to favor dark matter as the leading
candidate of the iii,--iig In i-- phenomenon. It can be said that halo models are
just extending past their infancy. Indeed, any ,- i-v ii 1iii l process incorporates
an enormous amount of complexity. Attempting to find universality among galaxies
is a daunting task if dynamical histories have distinct imprints in the rotation
curves. Currently, NFW profiles offer the most universal approach in halo models.
Despite their inability to accurately fit low-surface-brightness galaxies, their success
encourages us to add new galactic dynamics to the model instead of abandoning
it altogether. These profiles are inferred using N-body simulations, and therefore
questions as to the validity of these simulations certainly enter. For example, one
could argue that the limited number of particles employ, .1 prevents an accurate
reproduction of the dynamics; and the singular nature of NFW profiles is certainly
an unattractive feature that does not exist in nature.
The enormous parameter space also leads one to conclude that dark matter's
existence cannot be proved by galaxy rotation curves alone. Even more narrowly,
perhaps neither can any particular halo model (at least not any from the current
arsenal). Along with cosmological evidence and particle searches for dark matter
14
properties, abundances, and composition, galaxy rotation curves will serve as one
of the approaches to investigate its possibility. In the event of a -iil. .iig gun"
observation of dark matter's existence, rotation curves will pl i. an important role
in understanding galaxy formation and dynamical evolution.
CHAPTER 3
MODIFIED NEWTONIAN MECHANICS
MOND was proposed by Milgrom in 1983 [25] as an empirical alternative to
dark matter in explaining the rotation curve phenomena. By altering gravity at
low acceleration scales, one can reproduce the .,-i-,_i,!il.1 ically constant velocities of
satellites outside the central galactic bulge [26]. This chapter will first introduce
the nonrelativistic formulation of Milgrom and Bekenstein. Next, relativistic
extensions of this theory will be discussed: first the scalar-tensor varieties whose
in i ji proponents have been Bekenstein and Sanders, and secondly the purely
metric approach of Soussa and Woodard. We will end with the phenomenological
constraints and implications of each of these relativistic approaches.
3.1 Nonrelativistic Formulation
3.1.1 Motivation
One may formulate MOND by altering Newton's second law to be nonlinear in
the acceleration d,
FNewt = m/ d a where pI(x) = (3.1)
lx V < l.
The function p(x) is constructed to reproduce Newton's 2nd law, F = ma for
accelerations a < ao (corresponding to p > 1); and F ma2 /ao for accelerations
a > ao (corresponding to p < 1). The numerical value of ao has been determined
by fitting to the rotation curves of nine well-measured galaxies [27],
ao (1.20 0.27) x 10-10m s-2 (3.2)
However, when one considers the enormous internal accelerations of galactic and
stellar constituents relative to the center-of-mass acceleration, it becomes preferable
to regard MOND as a modification of the ,ian ;..:Ul..,.,il force at low accelerations 1
(F~ewt f i Vx 1 ,
FMOND = f Newt Newt where f(x)= (3.3)
mao / .
This empirical law is constructed to ensure the ivmptotically constant velocities
observed in the galactic rotation curves. That is, a particle orbiting a mass
M at an acceleration a < .- will follow a trajectory governed by the force
FMOND = m /aGM/r. Setting mv2/r FMOND gives,
/aoGM v2 4
-aoGM = va (3.4)
r r
In the absence of dark matter, a galaxy's luminosity L should be a constant times
its mass where the constant depends on the type of galaxy. Therefore, MOND is
able to automatically reproduce the expectation L ~ v,, which is the observed
Tully-Fisher relation [29].
When the data are analyzed, MOND is shown to be an impressively robust
predictive tool. Using only the measured distributions of gas and stars, and the
fitted mass-to-luminosity ratios for gas and stars, MOND has accurately matched
the data of more than 100 measured galaxies. A review by Sanders and McGaugh
[30] summarizes the data and lists the primary sources. Two significant things
should be noted: first, MOND agrees in detail, even with low-surface-brightness
1 For example, if we were to consider neutral Hydrogen as a classical planetary
system, the proton would be accelerating with a value of about 1019ms-2 about
the atomic barycentre and therefore well above MOND's characteristic acceleration
scale. Milgrom has considered strongly nonlocal nonrelativistic particle actions in
which the onset of MOND might be governed by the center-of-mass acceleration
[28]
galaxies [17, 31]; second, the fitted mass-to-luminosity ratios are not unreasonable
[32].
On the other hand, when MOND is applied to intergalactic scales, or clusters
of galaxies, it has proven as yet to be less successful. Some dark matter must be
invoked to explain the temperature and density profiles at the cores of large galaxy
clusters [33], and data from the Sloan digital sky survey claims that satellites of
isolated galaxies violate MOND when care is taken to exclude interloper galaxies
(defined as dwarfs with large physical distances from the primary galaxies which
are claimed to make the halo mass profile difficult to measure) [34]. This objection,
however, has serious difficulty in being applied to all the rotation curves. Prada et
al. -ii--.- -1 that this is what leads to the systematic effects that fools the observer
into measuring a constant velocity dispersion in a I. v instances [34]. To regard
100 rotation curves -most of which have not been systematically checked for
the presence of this purported interloper phenomenon -as a few instances is
currently an overestimation of the interloper hypothesis. Recently, a paper by
Penarrubia and Benson [35] analyzed the effects of dynamical evolution on the
distribution of dark halo substructures using semi-analytic methods (checked
with the latest N-body simulations). Their goal was to disentangle the effects of
processes acting on the substructures. They conclude that orbital properties of
substructure components are determined a priori by the intergalactic environment
[35] precluding the interloper hypothesis.
3.1.2 Action Principle
Milgrom and Bekenstein were able to obtain the MOND force law from a
nonrelativistic Lagrangian that respected all the symmetry (and hence conserva-
tion) principles we demand of nonrelativistic theories [36]. Considering a general
gravitational potential Q sourced by a mass density p, they proposed the following
Lagrangian,
L =- Jdr p(r#(r) + (87G)-laF (3.5)
where the interpolating function F is related to the function p of Equation 3.1 via,
p(x) F'(2) (3.6)
Assuming that the potential vanishes on the boundary, varying L with respect
to 0 gives the equation of motion,
V. [f(||V |/ao) ] = 47rGp, where f(x) F'(2) (3.7)
Consider an isolated mass M. Using Gauss's law and spherical symmetry we
trivially have,
GM
r 3
/(llVll/ao) GMr. (3.8)
in view of our empirical requirement set forth in Equation 3.3, we see that the
.I-i~,IIl ,'I ic behavior of the acceleration of an object due to an object of mass M
must behave as,
S- aoGM + 0(r-2) (3.9)
F-2
Note the Equation 3.9 applies in all regimes: MOND and Newtonian. Requiring
that,
2 G ,M (3.10)
leads to the trivial solution,
) G--- VoGMln(r/ro) + O(r-) (3.11)
where ro is an arbitrary radius. Solving the field Equation 3.7 can be done by get-
ting the first integral leaving one with an algebraic problem. From the rotational,
space, and time translational symmetry of Equation 3.5 we immediately obtain
the conserved quantities of linear momentum, angular momentum, and energy,
respectively.
The form of the interpolating function is obviously not unique and many dif-
ferent forms may be taken as long as the limiting behavior matches the requirement
Equation 3.3. A typical example is,
11 4
f(x) 2 2 (3.12)
and thus,
F(x) = xx2 + ln(x + V+x). (3.13)
3.2 Relativistic Formulation
3.2.1 Motivation
The purpose of the previous section, and that of the original authors, was to
demonstrate: first, the MOND force law is derivable from an action principle; and
second (and a direct consequence of the first), the action principle would possess
the spacetime symmetries associated with conservation of energy and momentum.
The need to extend MOND into the relativistic domain is easily seen. Esthet-
ically, the nonrelativistic formulation of MOND leaves a theorist almost aching to
extend it to a fully relativistic description. However, and perhaps even fortunately,
it is phenomenology which serves as the greatest impetus. As is, MOND proclaims
itself as an alternative to dark matter, and therefore any phenomena to which the
presence of dark matter has succeeded in explaining must now be equally or better
described by MOND.
If one is interested in cosmology and gravitational lensing, there is no escape
from the requirement of a generally covariant formulation of MOND that includes
at least the usual metric. If MOND is indeed a viable alternative approach, it must
account for the deficiency observed in the general relativity with no dark matter
prediction [37]. Milgrom, Bekenstein, and Sanders have a scalar-tensor approach to
this end [36]
3.2.2 Scalar-tensor Approach
We will not in anyway here attempt to thoroughly consider scalar-tensor theo-
ries. Rather, we will take a more taxonomic approach, and in the process list their
respective strengths and weaknesses from the theoretical perspective. Bekenstein
[18] gives a more exhaustive review of these theories. The main phenomenological
issue is whether the metric encodes the MOND force law or whether it is coming
from a scalar field. All the scalar-tensor approaches possess the latter feature,
and therefore one may immediately see that a new kind of dark matter emerges.
Namely, if these added fields are real then we are again faced with the challenge of
detecting them as any other dark matter candidate.
Aquadratic Lagrangian: AQUAL
Bekenstein and Milgrom first proposed a relativistic formulation of MOND as
an appendix to their principal theme of devising a nonrelativistic potential theory
in [26]. Their original approach was to introduce a dynamical degree of freedom
in the form of a scalar field i in the spirit of scalar-tensor theories. Particles no
longer follow geodesics of the Einstein metric gy,, but rather that of a conformally
related pliy-cal" metric gy= e2g,. MOND physics comes from the scalar field
Lagrangian density,
87GL2 f(L2 (3. 4)
where f is an a priori known function which is constructed so as to reproduce
MOND in the appropriate regimes and L is a constant length. In this theory,
particles follow geodesics of gy. That is, if we parameterize a particle's worldline
to be X"(T), it has the following action,
S -mn dre g(x(r))Y(r) (r) (3.15)
where a dot indicates differentiation with respect to r. To make contact with the
nonrelativistic theory, expand the particle action,
Sm,= m dr(1+ N+ s + +.... (3.16)
identifying KN = -(1 + gtt)/2 as the Newtonian potential, determined by mass
density p via the linearized Einstein equations. If we further restrict ourselves to
the quasi-static case, it is straightforward to recover the MOND equation of motion
Equation 3.7 in the weak field limit, with the acceleration of a particle governed by,
S= -V(4N + ) (3.17)
Phase coupling gravitation: PCG
AQUAL was discovered to possess the debilitating feature that ip could
propagate superluminally [26]. To see this, consider the wave equation for free
propagation of i that follows from Equation 3.14,
[f(L2"py ,'jg) & ,i 0 (3.18)
Here a semicolon represents covariant differentiation with respect to g,,. Now
linearize Equation 3.18 for small perturbations in i) and consider the highest
derivative terms. Following Bekenstein's coordinate prescription [26], it is possible
to find a local Lorentz frame to point in the x-direction, allowing one to expand
Equation 3.18 as,
0= S,tt + (1 + 2j)6Q,,xx + J+ + JQ, + ... (
(3.19)
where = dln f'(y)/dlny and the dots indicate terms with only one derivative. To
determine whether b can propagate acausally, we need only consider the highest
derivative term and their respective coefficients. Since ( > 0, the coefficients in
Equation 3.19 clearly display Jb's ability to violate causality.
Another downfall of AQUAL comes from the conformal nature in which
the field couples to the Einstein and the physical metrics. It cannot influence
gravitational lensing. We simply state this fact here: any conformal transformation
of the form gy, Q-2 9y has no impact on the bending of light. We will
revisit this statement and discuss it at greater length when considering a purely
metric theory. This means galaxies induce gravitational lensing only to the extent
predicted by general relativity without dark matter. This is far too small [38].
In order to prevent the superluminal propagation of the scalar field inherent to
the relativistic AQUAL theory, one can add a second scalar field which couples to
'b to ensure causality. This incarnation of MOND, PCG (Phase Coupled Gravity)
[39], has now for a scalar Lagrangian density,
[, A] -= [g"P(A,,A,, + r- 2A2 ,, ,) + V(A2)] (3.20)
and equation of motion for A,
A'; r-2A,, AV'(A2) 0 (3.21)
Including a point mass M, the equation of motion for b follows from Equation
3.20,
(A2gp,);~ ; ij2e M63() (3.22)
Small values of Ill justifies dropping the first term in Equation 3.20 and allows us
to solve for A in terms of '. Inserting this into Equation 3.22 reproduces the same
type of equation exhibited by AQUAL for '.
For the choice V(A2) -= -'2A6 with c a constant one can show that a
particle acceleration for spherically symmetric solutions behaves as,
GM 92M
a r 2 (3.23)
where K 2-3/2 ( + + 4 ( ) ). Making an appropriate choice of e and thus
K and identifying the critical acceleration scale in terms of our PCG parameters,
ao -, (3.24)
47Ge
will satisfy the rotation curve requirement. Thus PCG is capable of reducing to
the AQUAL behavior and thus the nonrelativistic regime which is responsible
for rotation curves. It also removes the acausal propagation of the scalar field.
Naively, one would assume that since first derivatives of i in Equation 3.21 enter
quadratically that causality ensues. A more thorough analysis by Bekenstein [40]
shows that although this is not sufficient, considering only stable backgrounds
enforces the desired property of causal propagation.
The parameters rl and c are stringently constrained by solar system tests. The
accuracy to which we know the perihelion precession of Mercury proves enough to
marginally rule out PCG (See [18] for a detailed discussion).
Finally, PCG suffers the same problem as AQUAL: the conformal coupling of
the metrics leads to no enhancing of gravitational lensing in the general relativity
with no dark matter hypothesis.
Disformally transformed metrics: Stratified gravitation
The failure of both the AQUAL and PCG theories stem from the conformal
relation between the Einstein and physical metrics. Consider, rather a "disformal"
relation [41],
gi = e-2 C (A + BL2(3 ,, '') ,
(3.25)
where A and B are functions of the invariant g~" ,i, and L is a constant length.
The second term in Equation 3.25 is responsible for the additional light deflection
needed to explain observed galaxy lensing without dark matter. However, if one
demands causality then it was found [42] that the sign of B would have to be such
that the effect of the disformal transformation would be to decrease the amount
of gravitational lensing. One way to overcome this shortcoming is to replace the
second term in Equation 3.25 by a non-dynamical vector field which is purely
time-like [43]. This stratified framework, which chooses
gy = e-2 g9, 2UU, sinh(2Q) (3.26)
g" UU, =-1 (3.27)
is able to successfully describe observed gravitational lensing phenomena, and
satisfies local solar system tests of gravity. However, it is clearly a preferred-frame
theory, and has no a priori principle in which to select the preferred direction.
Further, U, must be of a bizarre form to satisfy Equation 3.27 at all spacetime
points -undoubtedly an unnatural feature.
Tensor-Vector-Scalar Theory: TeVeS
Recently, Bekenstein has built on the successes of Sanders' stratified theory
[18], by formulating a theory incorporating a vector, a scalar, and a tensor field
which work cooperatively to exhibit the desired phenomenology -gravitational
lensing beyond general relativity alone and causal propagation of scalar modes. In
addition, this tensor, vector, scalar (TeVeS) theory is no longer a preferred-frame
theory like the stratified framework theory of Sanders by virtue of the vector field's
dynamics.
The TeVeS theory proposes the following transformation between the physical
and Einstein metric,
gY = e-26g, 2UUl sinh(20) (3.28)
g/""UU = -1 (3.29)
Now, however, the vector follows from the action [18],
Sv = j d g [g3gPu[,]U[3,v] 4(A(x)/K)(g1PUpU, + 1)] (3.30)
Sv (0327G
where A(x) is a Lagrange multiplier enforcing Equation 3.29, and K is a constant of
dimension zero. Further, we employ the notation V~,] = Vo,p V,. 2
The scalar action is,
S, = d4x g 2(gp U-PUv0,,v U + 2f 4F(kGu 2) (3.31)
where F is a function again constructed to reproduce MOND behavior and k
and f are constants of length dimension zero and one, respectively. The gravity
action is the usual Einstein-Hilbert action with metric g,,, but the matter action is
constructed coupling to g,, instead of g,,.
TeVeS's scalar equation of motion is,
[p(kf2,10 ,a] ; kGN(p + 3p)eC-2 (3.32)
where p(y) is defined by the equation,
F(p) F'(p) (3.33)
2
2 It should be noted that Jacobson et al. [44] have considered models such as
Equation 3.30 (Einstein-Aether Theories).
Quantities with tildes are constructed using the physical metric. Equation 3.32 is
exact. To exhibit AQUAL behavior, we take g3P -- 'r3 and e-2~ 1. Further, we
neglect p relative to p. Equation 3.32 can then be approximated,
S- [p(kf2(v)2t) kGp (3.34)
By properly constructing p, Equation 3.34 reproduces the nonrelativistic scheme of
AQUAL. Similarly, one can work out from Equation 3.32 the MOND limit and the
GR limit -in fact, Equation 3.32 is the starting point for most analyses.
Parameterizing the metric as,
.,, ..,dx3 = -e)dt2 + e(() [dQ2 + Q2(d02 + sin2 Odj2)] (3.35)
One can show [18] that gravitational lensing in TeVeS is achieved by,
b f" v'- (' + 4p'
Ap = dx (3.36)
2 J_,
where x J2 b2 is the Cartesian coordinate along the light-ray characterized
by distance p and impact parameter b from the source. Further, Equation 3.36 can
be approximated to leading order via the relation [18],
A = 2b dx (3.37)
where K = + LN. This result is consistent with the GR plus dark matter
prediction.
TeVeS's 1 i ri setback is that (like dark matter) it introduces new parameters
to which one must then examine their origins. Taking appropriate limits of TeVeS's
three parameters k, K allows one to properly go from general relativity to
MOND, etc. At least two (we may assume that at least one of these parameters
are related explicitly to ao which is determined from rotation curve data) added
experiments have to be performed to determine these parameters. We can imagine
that perhaps solar system tests provide one, lensing another, and rotation curves a
third. Thus, like many current problems in current physics, we may recast an old
problem in terms of new unknown parameters.
Another drawback is that if the scalar and vector fields are real entities, they
are in essence a new form of dark matter which need explanation. One can thus
argue that this is simply dark matter in a peculiar guise. Undoubtedly, however,
the TeVeS is an improvement over the previous scalar-tensor approaches, and its
phenomenological success alone makes it a viable and serious candidate explanation
for the current astrophysical data.
3.2.3 Purely Metric Approach
Motivation
One interested in simplicity, viz. by pure degrees of freedom would certainly
want to consider a purely metric extension of MOND, if for no other reason than
theoretical completeness. Further, it is certainly arguable that a purely metric ap-
proach is closer to the "spirit" of general relativity. However, as is usually the case
in physics, one gains simplicity in one facet of a theory only to lose it in another.
We discovered that no local theory can reproduce MOND behavior. Consider the
weak-field expansion of general relativity about a Minkowski background,
S 6-G d6gR ] d4x{h fw-hj+O(h2)} (3.38)
We want MOND corrections to "turn on" at a characteristic gravitational
acceleration. The Ricci scalar, however, vanishes for general relativity outside a
source. So there is no way a putative MOND correction term based upon R can
.!I 'v.--" to turn on. You can get terms which don't vanish by using the Kretchman
invariant, R ,asR"'6; these can indeed tell when to turn on but functions of this
invariant automatically inherit the Ostrogradskian instability [45].
The only local actions which can avoid the higher derivative problem involve
functions of the Ricci scalar alone. The Gauss-Bonnet invariant avoids this
instability but is purely topological. Since the nonrelativistic MOND force law
involves I||V 2, the weak-field expansion must start at cubic order in the action.
Abandoning locality is certainly not an uncommon occurrence in ,'div's
theoretical physics. Effective theories have become more and more commonplace in
regimes where ignorance of fundamental principles dominates. Quantum gravity's
effective action, of course, falls in this category; and although we are unable to
even -z'i what the full effective action is, nothing prevents us from guessing its
form. We chose the simplest class of guesses which would be capable of satisfying
our nonrelativistic constraint Equation 3.7, acting with the inverse covariant
d'Alembertian on the Ricci scalar3 We will refer to this as the small potential,
1 1
p[g] = R where D = --,(/gg"" ,) (3.39)
(We use the convention R, FP P ,P + PFP~ P" .) Embedding
MOND in a nonlocal Lagrangian has the form,
6G R + ca(4C40 2g( P,,v g, (3.40)
where F(x) is an interpolating function whose form for small x controls the onset
of MOND behavior as in the nonrelativistic case considered previously.
Although the field equations are nonlocal, they do not possess additional
graviton solutions in weak-field perturbation theory. To see this, expand the metric
3 This is not an unprecedented approach, but has been utilized in examining the
physics of the post-inflationary universe [46].
about a Minkowski background,
9g, = Tpv + h (3.41)
The Ricci scalar follows easily,
1
R = h" 1 2h, + O(h2), (3.42)
where graviton indices are raised and lowered using the Lorentz metric. Using our
gauge freedom we choose de Donder gauge,
1
h",- 2 h, = 0, (3.43)
to show that the small potential is local in the weak-field limit,
p[ + h] -h +0(h2) (3.44)
2
Since the Lagrangian depends upon the first derivative of p, this theory contains no
higher derivative solutions in weak-field perturbation theory. Further, all solutions
to the source-free Einstein equations are solutions to this theory since R = 0
throughout spacetime, which implies p = 0 as well. Therefore, our modification to
the Einstein-Hilbert action in Equation 3.40 is the change in response to sources,
without adding new weak-field dynamical degrees of freedom -a clear distinction
from the scalar-tensor theories so far discussed.
The class of nonlinear gravity theories we are considering are known to have
a connection to scalar-tensor theories through a conformal rescaling of the metric,
thereby introducing a minimally coupled, massive scalar field [47]. Regardless,
the number of dynamical degrees of freedom remain equivalent. The scalar-tensor
theories of Milgrom, Bekenstein, and Sanders all introduce new degrees of freedom,
and therefore these approaches are truly distinct. Were the small potential y[g] an
independent dynamical variable, rather than a functional of the metric, the purely
gravitational sector of these models would be identical, and thus the matter sector
would serve to distinguish them. Matter couples in the usual way to gy, in our
class of models whereas it couples to p2g in the Bekenstein-Milgrom formalism.
Note in particular that the field equation associated with a truly dynamical scalar,
(pF')" 0 (3.45)
is not generally solved by p = O-1R.
Phenomenological constraints
The two physical processes we will use to guide us in formulating our relativis-
tic extension of MOND are the rotation curves and gravitational lensing. We will
consider circular orbits as an approximation to the typical orbit a satellite of a spi-
ral galaxy follows. The invariant length element in a static, spherically symmetric
geometry can be expressed as,
ds2 gdxdx -B(rdt2 + A(r)dr2 + r2d (3.46)
The worldline of a test particle moving in this geometry may be parameterized by
xt(t) and obeys the geodesic equation,
i"(t) + FM^(x(t))xP(t)kx(t) 0 (3.47)
It is a straightforward exercise to obtain the nonzero connection coefficients from
Equation 3.46,
B' B' A'
Fttr- Irtt B rrr -
tr 2B" r 2A' 2A'
r r 1
Froo A' AF sin2 Or -
S- sincos, cot. (3.48)
Er 1 E06 sin O0cos 0, E0 cotO (3.48)
r 6
Now parameterize the worldline for the case of circular motion,
(xW,X 0,X,) = r, 2, (t) (3.49)
The only nontrivial geodesic equation in Equation 3.47 is,
B' r 2
0B'. (3.50)
2A A c2
The A(r) is thus irrelevant. In circular orbits, the velocity has the relation to
the angular velocity v = r. In the MOND limit v2 approaches the constant
v = /aGM, and thus the MOND limiting form of B(r) must obey,
B'(r) 2 4 (3.51)
For weak-fields we can write,
A(r) 1 + a(r), B(r) 1 + b(r), (3.52)
where la(r)| < 1 and Ib(r)| < 1. A large spiral galaxy has on the order of 1011
solar masses in dust and gas, or M ~ 1041kg. Therefore, such a galaxy would enter
the MOND limit at a radius,
Rgal -M 102m (3.53)
V ao
It is significant that the natural length scale corresponding to the MOND accelera-
tion,
c2
RMOND ~ 1027m, (3.54)
ao
is greater than the Hubble radius. This has the important consequence that,
-- t 1, (3.55)
RnMOND
on galaxy and galactic cluster scales, and therefore powers of r do not necessarily
distinguish 'large' and 'small' terms.
We will now propose a phenomenological Ansatz for the .-i ll..' ic behavior
of the weak-fields in terms of four order one parameters 4
GM [aoGM GM aoGM ( r
a(r) -+ VT b(r)- 62 2 In (3.57)
C 2 C C2 C gal
From Equation 3.51 we see that MOND predicts C2 = 2. General relativity without
dark matter predicts el = C2 = 0 and 61 = -62 = 2. The insertion of an isothermal
dark halo whose density is chosen to reproduce v = aGM would lead to the
general relativity prediction of el = 61 -62 = 2.
To see what the remaining parameters in our Ansatz must be to be consistent
with phenomenology, we consider the angular deflection of light from a mass M in
terms of the turning point Ro,
A = 2 dr (3.58)
JRo r 2 2 B(Ro) 1
nr o B(r) 1
Expanding this expression in powers of the weak-fields and changing variables to
r Ro csc 0 yields,
A0 2 j dr 1 +a(r) 1 b(Ro) b(r) (3.59)
P"r V+ +---- (3.59)
R 2 r 2 2 2 1 (R>o)
= I dO {a(Ro sec 0) csc2 0[b(Ro) b(Ro sec 0)] + .. } (3.60)
4 One may be worried about the logarithmic growth in b(r), but it is of no prac-
tical concern. The change in b(r) from the onset of MOND all the way to the cur-
rent horizon (Rhor ~ 1026m) is
b(Rhor) b(Rgai) ~ -62 x 10-6 + 2 x 10-5 (3.56)
The weak-field regime is therefore applicable throughout the Hubble volume.
Substituting in the Ansatz Equation 3.57 gives,
A0 (6 62)M+ ( +2) + a M ... (3.61)
C2 0L 2V 2 4
Without dark matter, general relativity gives too little deflection at large Ro to
be consistent with the frequency of lensing by galaxies. General relativity with an
isothermal dark halo is consistent with the existing data [37, 38]. Therefore, for
MOND to faithfully adhere to the current observations, it is required to have the
sum 1c + C2 to be positive and of order one.
The field equations
In this section the field equations that would be derived from Equation 3.40
are presented. Here we give a more heuristic approach to this end. To avoid
digression, we simply state here that the one does not get causal field equations by
varying a temporally nonlocal action. Further, if one considers this class of models
in the context of quantum field theory, then issues of nonreal operator eigenvalues
quickly present themselves. We will here derive the field equations without light
of the above concerns. Instead we will use a trick so to speak to obtain causal and
conserved field equations from Equation 3.40. Therefore, one may as well regard
the resulting equations of motion, rather than Equation 3.40 as defining the model.
The method we present reconciles the requirements of causality and conser-
vation. Using retarded boundary conditions, one may easily add corrections to
the field equations to enforce causality. However, it is immensely more difficult to
guess symmetric tensors of any complexity that will combine to have a vanishing
covariant divergence. Of course, field equations derived from any coordinate invari-
ant action will automatically be conserved; however, varying actions which involve
nonlocal operators result in equations at x" which depend upon fields in the future
as well as in the past of x".
The method is simplest to describe by comparing with the correct variation.
Consider an arbitrary functional of the metric, f[g](y). We can write,
S[g] (Y) 1 601 1 1 6R(y)
f[](y) f [g](y) 1 Ry) + (3.62)
f gPW (x) ([ Dy gl(x) EyR(Y) + Dy6gl-(x) f
where we used the fact that '0-1 I(D60)-1.
Varying the covariant d'Alembertian and the Ricci scalar gives,
6J g(y)E(y) a 1 64(X 1 a
6gpW(x) YP ay g/y y- 2g 9 Py17 (3.63)
R(y) [R,{,(y) + DD, g,(y)D]64(x- y) (3.64)
We now define the small potential using the retarded Green's function,
4[g](y) Jd4z Gret(y; Z)R(z). (3.65)
Using the well known property Gret(x; y) = Gadv(Y; X), we have,
d "(x) ) 2 v 2- +9 Pf
6gW 1 11dv 1
Eadv
The trick is to simply replace the advanced Green's function in Equation 3.66 with
the retarded ones,
6 4[gl 1 1
y f () (x 2 R f +_" ") 2- ] 2 f
+ [R,, + DDD, g,,] -f (3.67)
Since conservation depends only upon the differential equations obeyed by the
Green function, it is not affected by this replacement. The source for our gravi-
tational equations of motion is the stress-energy tensor from the variation of the
matter action S,,
2 6S,
T,, =- (3.68)
V Fg 6gl '
Finally, taking 167Gc-4/ g times the variation of our nonlocal action Equation
3.40 -in the sense of our trick Equation 3.67 -and defining the I.' ,/ potential5 ,
K [g] (
(3.69)
gives the following field equations,
8 -Gc-4T, 2[;,,, gjD0] + [1 2+]G,,
+ [g9 P, 'P4,P P,P, il,, + + 7,, '
29gp. (3.70)
2C4
It is worthwhile to explicitly demonstrate conservation as we have not rigorously
derived Equation 3.70. Taking the covariant divergence DV of Equation 3.70 gives,
2[4;,, gv,0N ];"
([ 2}]G,));"
[g 'P [ ;, g~,,2;
-1^):
2RP-, ,
-2R + R ,
-9p,w9 9R%, ,
-^.'^+
which obviously sum to zero. Because 1/H denotes the retarded Green's function,
these equations are causal in the sense that the equations at x" depend only upon
points within the x"'s past light-cone6
5 Note that the large potential would vanish identically had p been a fundamen-
tal scalar as in the scalar-tensor models of Bekenstein and Milgrom [36].
6 This issue of causality should be distinguished from causal propagation. Field
equations for which the highest derivative term is nonlinear can admit superluminal
propagation as in the relativistic model of Bekenstein and Milgrom [36].
(3.71)
(3.72)
(3.73)
(3.74)
(3.75)
The Schwarzchild-MOND solution
Here we work out the small and potentials: Equation 3.39 and Equation 3.69,
respectively, for a spherically symmetric and static metric Equation 3.46. They give
rise to two independent equations of motion deriving from Equation 3.70 for this
geometry.
The generally spherically symmetric and static geometry Equation 3.46,
Equation 3.48 gives rise to the following Ricci scalar,
B" B' A/ B' 2 ( B' 2 1
R + -+ + +' .- (3.76)
AB 2ABB (A B) rA A B r2 A
For this case, and acting upon a function only or r, the covariant d'Alembertian
reduces to,
1 d (r B d (3.77)
T/AB dr A dr
The differential equation which defines the small potential therefore takes the form,
S(3.7)A- --
(3.78)
Assuming the parenthesized terms above vanish at r = 0, we can write,
B (' B B'\
-p' (r) + -( t) -d AA -t))
B r r B 0 AA
(3.79)
The differential equation that defines the large potential is,
.( g/v7 w),. a( g c,/) z=7r (2 (r2 c'.') (3.80)
Assuming again that the parenthesized terms vanish at r = 0 we can write,
-(C4 12
'() seco'(nd' covariant derivatives we shall need are,
a A(re)
The second covariant derivatives we shall need are,
B' A'
ibV/ 4" B )]4/A/)
; tt A 02
2A r 2A
1 1 B' A' ( )
A r 2 B A
(3.81)
Only the diagonal components of the field equations Equation 3.70 are
nontrivial in this geometry. The 00 and 00 components are proportional to
one another, and are identically obtained from the rr and tt equations from
conservation. We have therefore the two independent equations of motion,
8_4GA 4 AG a f A'
8GT -= 24 + + (1 2+) + AF 4' '~ (3.83)
C4B r B 2c4 A
8xrG 4 a2 B'
r 2c'+ G,,( 2) -O', (3.84)
c4 r 2c4 B
and the two from conservation,
TT r3 B'AT d 2 A' B'
2 = Too =Tu + [+ --+ T, (3.85)
sin2(0) 2 2B B dr r A 2BT "
The tt and rr components of the Einstein tensor are,
A Gtt A' + (A ) G,, B' A (3.86)
B rA r 2 rB r2
At this point we begin our perturbative analysis in this geometry, and recall
that we may express A(r) and B(r) in terms of the weak-fields a(r) and b(r),
A(r) = 1 + a(r) (3.87)
B(r) 1 + b(r) (3.88)
In the MOND regime, we have la(r)| < 1 and Ib(r)| < 1. Therefore, to leading
order in the weak-fields Equation 3.79 becomes,
2a
-- -2 b' + . (3.89)
r
Notice that the integrand in Equation 3.79 vanishes exactly in the general relativity
regime when A = B-1. In the MOND regime, the integrand is no longer zero, but
it is second order in the weak fields and we are therefore justified in ignoring this
term altogether for our present analysis.
In the .,-vmptotic regime we can assume that each derivative adds a factor of
1/r. Hence p'(r) goes like 1/r times the small numbers a(r) or b(r). It follows that
KI'/r is much larger in magnitude than 9p''. By similar reasoning we recognize that
K'/r and 4" dominate the other MOND corrections,
1 ", > / A4 B- a (3.90)
r A B C 4
We will assume Trr = 0 in the weak-field limit and allow for a nonzero
A/BTtt = p. Including the first two terms in Equation 3.90 with the general
relativity terms allows us to express Equation 3.83 and Equation 3.84 to leading
order in the weak-fields,
4 a' a 8wG
2l" + '+ + + p(r (3.91)
r r r 2 C4
4 b' a
S+ +... 0. (3.92)
The first of these equations Equation 3.91 can be integrated to give,
4 2a K 167rG
V+ + 6G+ + d/2p( /) (3.93)
2 3 4 3 g
Rgal
where K is the constant of integration. Adding Equation 3.92 and Equation 3.93
cancels the leading MOND corrections,
b' a K 167rG
+ + + -- j dr 2( r) (3.94)
S r2 r3 C43 gal
Notice that Equation 3.94 is independent of the still unknown interpolating
function F. We can therefore make general statements about all models of the type
Equation 3.40. In the absence of dark matter, the mass integral must eventually
stop growing, for which case the left hand side of Equation 3.93 must fall as 1/r3.
To satisfy this situation, b'(r) must go as a constant times 1/r and a(r) must
go like minus this same constant. In terms of our Ansatz of the previous section
Equation 3.57, we have just demonstrated that,
1+ 2 0. (3.95)
We acquire no lensing at leading order -a phenomenologically unacceptable
result.
It is still worthwhile to see if we can find an interpolating function F(x) to
reproduce MOND rotation curves. We will consider a sphere of mass M and radius
R with very low, constant density,
3 Mc2
p(r) = 3 (R r) (3.96)
47R3
If the density is small enough the MOND regime prevails throughout, as in a low
surface brightness galaxy. This means that Equation 3.91 can be integrated all the
way down to r = 0 to give,
a 8rG "
2' + a + G 'dr12p(r) (3.97)
r C4r2 J0
We can also use Equation 3.89 to eliminate b'(r) in -r times Equation 3.92,
4)' + a' + 0 (3.98)
r
Now eliminate a(r) by adding Equation 3.97 and Equation 3.98, and then use
Equation 3.81 to obtain an equation for the small potential,
S'[1 + 6SF'( ) + < dr',2rp(r') (3.99)
For r > R the mass integral is constant,
c4,12 ] 2GM>
P[1 + 6F( ) + .. 2G Vr > R. (3.100)
To get flat rotation curves we determined that MOND requires C2 = 2. We have
just computed explicitly that any model of the type Equation 3.40 must have
e1 = -62 and therefore the weak-field limit Equation 3.89 for the small potential
implies,
Pr --W 6 /a0CM
'()- ... (3.101)
It follows that the constant term within the square brackets of Equation 3.100 must
exactly cancel, and that the next order term must involve one power of 9p'. It is
straightforward to compute our interpolating function,
3
1 x X X2
F'() + O(x) (x) 2 + O(~2) (3.102)
6 108 6 162
The associated weak-fields are,
4GM aoGM
a(r) 32 2 (3.103)
3C2r C4
8 GM aoGM r
b(r) + 2 ln (3.104)
For the general weak-field Ansatz Equation 3.57 of section 2 we have just shown
-261 = 62= -' and = =2 2.
We have a large amount of freedom in enforcing the MOND limit with regard
to choosing the interpolating function F(x). The MOND limit is enforced by
determining only the first two terms in the small x expansion of F(x). Therefore
depending on the level of suppression desired its functional form is far from unique.
Our only requirement is that the MOND corrections must be sufficiently small
when entering the general relativity regime, i.e. x| > 1. For example, we can make
F(x) --- -1 x for large Ix with the following extension,
7'z (x) 2 sgn(x)
() 9 2 (3.105)
22 ( 7
F(x) = + 441n ( + xl .X (3.106)
l+ |.| V 6 /
For |x| > 1 this would typically suppress MOND corrections by some characteristic
length of the system divided by c2/ao 1027 m. If that is not sufficient one can
alv-i-, extend F(x) differently to obtain more suppression.
The FRW-MOND Solution
Even though we just discovered our relativistic formulation is unable to
produce enough lensing, it is still instructive to see what it does for cosmology.
Further, it serves as a potential gauge of how much a general formulation of MOND
changes in passing from a static geometry to the time dependent one of cosmology.
We begin with the usual Friedmann-Robertson-Walker metric for homogeneous and
isotropic cosmologies,
ds2 -C2t2 + a(t)dX d. (3.107)
In this geometry the Ricci scalar is,
c2R =6H + 12H2 where H (3.108)
The small potential is defined by the equation,
(t) -a-3 d (a3 R(t) (3.109)
We define the initial values of po and its first derivative to be zero, in which case the
small potential becomes,
u(t) = dt'a-3(') dt"t3 (t") (6ft") + 12Ht2 ") (3.110)
The large potential is defined by the differential equation,
V(t) = dt'd(t')F C2a (_d (l/) .) (3.111)
If we again assume null initial values the result is,
S(t) u- t/ '(t')o' 2 (w) (3.112)
Jo
The nonzero components of the second covariant derivative are,
);oo = C -2
;i = -C-2H H-i .
For a perfect fluid, the stress-energy tensor is,
TI, = j ,,,,, + (p + p)uu .
Stress-energy conservation implies, P
the Einstein tensor are,
-3H(p + p). The nonzero components of
c2Goo 3H2 (3.115)
c2Gj = -(2H + 3H2)gij (3.116)
The two nontrivial equations of motion from Equation 3.70 in this geometry
are,
2
8rGc-2p -6HW + 3H2(1 24) + aF ,
2c2
8srGc-2p 2$ + 4H( (2H + 3H2)(1 24) A
22
2C2
Conservation tells us only one of these equations is independent.
In the MOND regime we can therefore express Equation 3.117 as,
-H + 3H2(1 _
For cosmology the argument x = -(c/
the same sign as the small potential,
S2 + = 87Gc-2p .
(3.119)
ao)2 is negative so the large potential has
1
4(t) ((t) .
6
(3.120)
In the MOND regime we can therefore express Equation 3.117 as,
S2 + 8rGc-2p .
(3.113)
(3.114)
(3.117)
(3.118)
H + 3H2 _-
3
(3.121)
Of special interest to cosmology is the case of a power-law scale factor,
a(t) = ( + Ht) (3.122)
Here Hi is 1/s times the Hubble parameter at t = 0. Substituting into Equation
5.13 gives the small potential,
(({) -6s ( ) n [1 + i] -(1 3s)-1( + t) 1-3 ] (3.123)
The logarithm term dominates in Equation 3.124 at late times. In this regime, we
can express ip in terms of the Hubble parameter H,
() = -6s( s { ) [1 + Hi] -(1 3s)-1[( + ,t) 1- ] (3.124)
We can therefore write the MOND analog of the Friedman equation for power law
expansion,
3{1 + 2a a2 + 2sa7ln l + Hit] }2(t) + 8- 8Gc-2p(t) (3.125)
where a (2s 1)/(3s 1). For s > 1 the logarithm term serves to gradually slow
the expansion -consistent with the MOND strengthening of the force of gravity in
the weak-field regime.
For the case of radiation domination (s = 1/2 and a = 0) we note that
p(t) = 0, and hence so too K(t) = 0. The equations are therefore those of general
relativity, but with the energy and pressure coming from ordinary matter. This is
of course unacceptable in light of recent observations which show that nonbaryonic
matter must predominate over baryonic matter by about a factor of six [1].
3.3 The MOND No-Go Statement for Purely Metric Approaches
In the previous section we discussed scalar-tensor theories of MOND. These
models have had successes in complying with the phenomenological restrictions
currently available, but to date they have not shown to be completely satisfac-
tory for the reasons already presented. We went on to develop systematically a
relativistic version of MOND using a purely metric approach. The purely metric
class of models suffered from far too little lensing. It is the purpose of this sec-
tion to demonstrate that ,:;, purely metric theory of MOND will suffer the same
phenomenological shortcoming.
3.3.1 Motivation
It was our intention in developing a phenomenologically viable theory of
MOND which would satisfy the requirements of gravitational lensing and still
reproduce the rotation curves. What we discovered, however, was that to leading
order in the weak-fields in our class of models Equation 3.40, the MOND predicts
no additional lensing to the prediction of general relativity. This is inconsistent
without invoking the presence of dark matter. One is immediately tempted to
consider different classes of models which can overcome the lensing I -I For
example, one might replace the covariant d'Alembertian with the conformal one in
defining a small potential,
1 1
cp[g] = R whereO, -R (3.126)
11, 6
However, the distinction between E and El, disappears in the weak-field regime
since R scales as one power of the weak-fields times 1/r2. Therefore, this class of
models would have no hope of having any success in acquiring a nonzero lensing
contribution to leading order in the weak-fields.
Further, any MOND action which only contains the Ricci scalar as the
source upon which the nonlocal operator acts will have no impact during the
radiation phase of the universe -an entirely unacceptable feature. The next most
complicated scalar potential would seem to be,
c4 t
[] 4- (R/"R, (3.127)
Because 92 has roughly two derivatives acting upon two powers of the weak-fields,
one must also change the Lagrangian,
4
2 l= G-4 -[.)] c g (3.128)
For this class of models F2(x) would be linear in the MOND regime.
Instead of embarking on a program to discover a class of theories which are
able to satisfy the lensing requirements, we propose to study the general features
I,';/ purely metric formulation of MOND possesses. This approach is obviously
the more powerful if a definitive result can be obtained (or at least if some firmer
guidelines as to which classes of models can be considered in making MOND
relativistic).
3.3.2 The Statement
We intend to show here that no phenomenologically viable, purely metric
approach of MOND can be constructed within a set of given assumptions.
We assume that the gravitational force is mediated by the metric tensor
g,,(x), and that its source is the usual stress-energy tensor T,,(x). In four space-
time dimensions the metric is determined by the set of ten equations having the
form,
GO[g] 8wGT, (3.129)
where the gravity tensor, g~, is a functional of the metric (for ordinary general
relativity it is simply the Einstein tensor G,). The stress-energy tensor is obtained
as usual by varying the matter action.
We assume that the gravity tensor is covariant and is covariantly conserved,
gP"DpP = 0 (3.130)
At this point we have made no restrictions on the gravity tensor. In particular, we
allow it to involve higher derivatives, and even to be a nonlocal functional of the
metric.
Recall that MOND and Newtonian gravity are distinguishable only for very
small accelerations. These accelerations are expressed as derivatives of the metric.
It is well known that diffeomorphism invariance bestows the freedom to choose a
coordinate frame in which the metric agrees with the Minkowski metric r1, at a
single point. The observed fact that the gradients of the metric are small allow us
to make the metric numerically quite close to rTl, over a large region. Therefore, we
are justified in expanding the gravity tensor in weak-field perturbation theory,
gpw(x) = ,r- v + h,(x) (3.131)
[g] = 9, v[ + h] (3.132)
The crucial observation now in specializing to MOND, is to notice that the
MOND force law Equation 3.4 scales like the square root of the mass,
FMOND (3.133)
r
As a result, at least one component of h., must scale like (/GM (for spherical
distributions this would be the rr component but this does not matter). It is
obvious that the right hand side of Equation 3.129 scales like GM, and therefore
91, [l + h] must have at least one nonzero component whose lowest term is of order
h2
If we further assume gravity to be absolutely stable then not all ten compo-
nents of Q,, [Tl + h] can begin at quadratic order in the weak-field expansion. This
is due to the fact that the dynamical subset of field equations are obtained from
varying the gravitational Hamiltonian. If its variation were quadratic then the
Hamiltonian would be cubic, and this would be inconsistent with stability. The
conclusion therefore is that only a subset of the ten components of Qv,[] + h] can
begin at order h2.
This subset must be distinguished in some covariant fashion. A symmetric,
second rank tensor in four dimensions has two distinguished components: its
covariant derivative and its trace. We can immediately see from Equation 3.130
that conservation occurs at all orders in perturbation theory, and therefore the
covariant divergence cannot be responsible for the required h2 term. We are left
with the trace as the only remaining possibility,
g1"G,[T4 + h] = (h2). (3.134)
Equation 3.134 implies .i~,- ill ic conformal invariance. Although we are able
to reproduce the MOND rotation curves, the corrections to gravitational lensing of
general relativity with no dark matter come in at quadratic order, and are therefore
far too weak [37].
To see this, note that in the weak-field limit, one can perform a local, confor-
mal rescaling of the metric,
gp9(x) Q2(x)g,(x) (3.135)
and completely remove the corrections -the linearized MOND weak-fields are
traceless. It has been known for some time that traceless metric field equations
imply invariance under the conformal transformation in Equation 3.135 [48]. The
full field equations are not traceless, and so neither is the full theory conformally
invariant. This means that the linearized field equations only determine the metric
up to a conformal factor (and a linearized diffeomorphism, but this is irrelevant
for the argument). The conformal part of the metric is fixed by the order h2 term
in the trace of the field equations, and this is how one component hy, contrives to
scale like /-GM.
This is a disaster for the phenomenology of gravitational lensing. Recall that
for a general metric g,, the Lagrange density of electromagnetism is,
1
L = -4FF1Pg/Pg"h-c g (3.136)
where F,, -_ ,A, 0,A,. Equation 3.136 is invariant under the metric rescaling
Equation 3.135, and thus oblivious to MOND corrections to general relativity in
the weak-field limit.
An Example
Here we will illustrate our no-go statement's using our nonlocal, purely metric
model. The equations of motion Equation 3.70 imply the identification,
GpQj[g] = 2[);pt, gD,0] + [1 21]G,
2
+ [g ,p -jP, 9, l + P,9 ,' (3.137)
We have already worked out the leading order terms in the small and large
potentials, and the interpolating function as well. We may leave the large potential
4[g] = (poPF');, in terms of the small potential o, in which case the necessary
relations are,
1
F(x) -- (3.138)
1
4 [g] --1. (3.139)
6
In taking the weak-field limit of 9,, we may neglect any products of R,,, o or
4, such as GA, p'PK,p, and 9,p4,w. Henceforth, the weak-field limit of g, is
contained in the four terms,
1 1
L./, ;/) + RP 1 2 PR (3.140)
3 2 (3.14)
Of course these terms contain higher powers of h.., but more importantly they
contain all the linear pieces. And the terms we have kept are exactly traceless,
gPIg gp G F ( ,, 3;v + R v gvR (3.141)
=O R 0 (3.142)
Note that tracelessness (and hence conformal invariance) is not a feature of the
full field equations. In particular,
2a2
g[", = -60D R[1 2] + 2 +'2 + ', a' F (3.143)
This scales like h2 in the weak-field expansion.
3.3.3 A Connection with TeVeS
We have constructed MOND as a purely classical theory of gravity in the
sense that no attempt at quantization has been made (that said, the class of
models we derived can be thought of as originating from the effective action of
gravity). It should be noted that Bekenstein's TeVeS model can be put in the form
of a nonlocal, purely metric theory as long as the scalar and vector fields are not
directly observed. This is done by integrating out those fields, leaving one with a
nonlocal, purely metric action.
A second and extremely important distinction between the TeVeS theory and
the purely metric theory is the coupling of gravity to matter. The TeVeS theory
possesses a physical metric and an Einstein metric. One simple and direct con-
sequence of this fact is that gravitational and electromagnetic radiation traveling
from distant sources should possess disparate travel times. Recall that in the TeVeS
theory, the physical (gy,) and Einstein metric (g,,) are related via the scalar field Q
and the vector field U,,
gY = e-2gp,, 2UU, sinh(2Q) (3.144)
Suppose we were to observe a -5i,11 1,il," .,-1 ii1r,'!-i I1 event (such as a Super-
nova) from a source such as the Large Magellanic Cloud (LMC)7 Assuming a
Minkowski background (which is a reasonable first approximation) we would expect
gravitational waves to take a time T =D/c to reach us, while ultrarelativistic
neutrinos would take a time,
T eD-2dr (3.145)
c o
ST(1 2Q) (3.146)
In obtaining Equation 3.146 we made two assumptions. First, that the vector
of Equation 3.144 is directed in the direction of the cosmological evolution pa-
rameter, i.e. U, = 60 Secondly, we assume that the value of the scalar field is
approximately constant and has a magnitude Q << 1. This last assumption can
be understood in far greater detail in [18]. However, a reasonable choice would
be Q 10-6. The distance to the LMC is D w 105lyrs; and therefore this value
of 0 would correspond to a delay of AT w 10 min. Consequently, the parameter
space of theories like TeVeS should ostensibly be constrained, and perhaps either be
bolstered or falsified.
7 For example, an observer at the Laser Interferometer Gravitational-Wave Ob-
servatory (LIGO) and an observer at SuperKamiokande (SuperK) would be capable
of detecting the gravitational waves and neutrino flux from the event in the LMC.
3.3.4 Revisiting the No-Go Statement
This section examines the no-go statement in more detail. Specifically we
consider circumventing the gravitational lensing disaster by relaxing the assumption
of gravitational stability.
Let us revisit the assumptions which led to our no-go result:
1. The gravitational force is carried by the metric with its source being the
usual stress-energy tensor.
2. Gravity is described by a covariant theory.
3. The MOND force law can be realized in weak-field perturbation theory.
4. The theory of gravity is absolutely stable.
5. Electromagnetism couples conformally to gravity.
The third and fifth assumptions are the most rigid. The third, if untrue,
would inhibit us from working with any relativistic theory of MOND; if there is no
region for which the MOND force is weak (or at least as weak as the Newtonian
gravitational force), then there is no hope in passing even solar system experiments.
The first assumption may be violated if one makes a distinction between a
1pl -i. I!" and Xi vi'l ,I ,,i i 1" metric. In such a case test particles would follow
geodesics of the former while gravity would behave according to dynamics of the
latter. This is obviously a violation of the strong equivalence principle, but it
is worth noting that to date there has been no conclusive data forbidding this
possibility (See [36] for a detailed consideration). This old idea has been explored
with many more modern theories such as Brans-Dicke [49], Dirac's variable
gravitational constant [50], and string theories [51] to name a few.
The second assumption is easily foregone if one specifies a preferred-frame as
we have seen in a previous section. However, by losing covariance or perturbability
would certainly seem to be counter to the spirit in which we set upon in construct-
ing a purely metric theory. Our fundamental prescription makes strong use of
the strong equivalence principle whereby gravity and matter are described by one
metric, not two.
Relaxing the fourth assumption, that of gravitational stability, is seemingly the
most reasonable choice to overcome the no-go statement. Given the choice between
a stable and unstable theory when examining an observably stable system, the
physicist will alv--, choose the former. However, when phenomenologically driven,
the latter may be the choice of greater utility. When the instability manifests itself
at scales outside or nearly outside the physical scale, or at least in regimes where
perturbative predictions no longer hold, the phenomenologist may cautiously accept
(or at least consider accepting) the unstable solution as a candidate explanation. If
the gravitational stability is on the super-cluster scale or larger, we may consider
the possibility of all ten of the linearized MOND equations vanishing -the trace
component is no longer distinguished.
How does this relaxation affect the no-go statement? Imagine that all of the
linearized MOND weak-fields vanish in the field equations, in which case there
would no longer be a linearized theory -a sufficient bending of light could be
realized. We have already discussed the fact that if the MOND weak-fields begin
at order h2, then the gravitational Hamiltonian begins at cubic order. This signals
an instability, but not necessarily a fatal one. There are two weak-field regimes
the weak-field (or Newtonian) and ultra-weak-field (or deep MOND). In regions
such as the solar system it would be the Newtonian regime which dominates and
thus we experience no deviation from well established physics. At larger scales
(galactic and/or cosmological) we expect the deep MOND regime to enter the
fold. At these scales, the unstable solution would proceed to decay into large
wavelength particles diffusing as the universe expands. The result is that a return
to the Newtonian regime could occur as decay products build a sufficiently large
gravitational potential. The instability would, in essence, turn itself off just as it
becomes too large to become quantitatively reliable. We would no longer have a the
tracelessness of the linearized equations -there would be no linearized theory, and
hence gravitational lensing could be affected by MOND corrections.
CHAPTER 4
DARK ENERGY: THE MISSING ENERGY
4.1 Introduction
One of the greatest surprises to i,-1 !1,vi--icists and particle physicists alike
in the last 20 years is the recent observation from Type Ia Supernovae that
the universe is entering a phase of acceleration. The Standard Cosmology is
characterized by an early period of accelerated expansion (inflation) leading to
a flat universe, a process supported by the large-scale isotropy observed in the
Cosmic Microwave Background (C' \!l) [1]. The matter we see 'ti-i is the result of
gravitational collapse over 13.76 Gyrs after the initial singularity -which in turn
is a manifestation of the density perturbations created by quantum fluctuations
at the end of inflation. The sum of the critical energy fractions is very nearly one,
with its current decomposition consisting of nearly 3:l,' from matter (of which
only approximately !' is ordinary), and over 711' from an unknown source. The
first year's data of WMAP [1] gives us (in terms of the Friedmann equation of
cosmology in the present epoch) the critical fractions,
-tot = Qk + m, + 2r + tx 1.02 0.02 (4.1)
k r =0 (95', CL) (4.2)
m = 0.27 0.04 (4.3)
x = 0.73 0.04. (4.4)
Qx of course represents the source of critical energy responsible for the acceleration
of the universe. This "dI ,1: energy has commanded the attention of a wide variety
of researchers, both theoretical and experimental.
The term dark energy, much like dark matter, is a rather broad encompass-
ment of theoretical ideas -essentially referring to some added component to the
right hand side of the Einstein equation which represents a -,lI-I ,i' which
exerts a negative pressure and therefore induces expansion. Interestingly, Einstein's
greatest blunder, the cosmological constant introduced in his research to ensure
a static universe, has now almost impishly reintroduced itself into the theoretical
arena.
4.2 The Many Faces of Dark Energy
Here we review the fundamental physics behind dark energy. Simple observa-
tional ideas -coupled with theory -allows one to easily understand the nature of
dark energy. We will survey the modern landscape of theoretical ideas with a broad
brush attempting to capture the more important themes and identify the common
properties each must share.
On large scales the universe is homogeneous and isotropic, allowing one to
use the Friedman-Robertson-Walker (FRW) metric to define the invariant length
element in natural units,
ds2 g, (x. idxV = dt2 +a2(t)dix di. (4.5)
Reading off the metric components, assuming the perfect-fluid form (with energy
and pressure densities p(t) and p(t), respectively), and inserting them into the
Einstein equation yields the two independent equations,
3 -) 87Gp, (4.6)
2- = -8rGp. (4.7)
a a
(4.8)
Taking the linear combination of Equations 4.6 and 4.7 gives,
(t) 4G [p(t) + 3p(t)], (4.9)
a(t) 3
then it is clear that in order for the universe to undergo accelerated expansion (i.e.
a > 0), we must have,
p < 0, and pl ~ |p| (4.10)
A useful but unfortunately named quantity is q, the deceleration parameter defined
by,
q aa. (4.11)
Clearly, a2 and a are positive definite, and therefore an accelerating universe
demands q < 0.
Observationally, one can understand late-time expansion using a Hubble plot.
The cosmological redshift, z, can be equivalently defined using the ratios of photon
wavelengths or the ratios of scale factors at different times,
/now 80O
z=- -1 z= -- (4. 12)
Then a(t)
where ao is the value of the scale factor now and time t = 0 is the present and all
values of t > 0 involve the past. Physical distances are determined via the relation,
dphys = a(t)dco-moving (4.13)
Supernovae have the desirable feature of having well-determined luminosities, and
thus are good for distance and velocity measurements. The flux F one measures is
related to the luminosity L thusly:
L
4rd = (4.14)
where dL is the luminosity distance (physical distance) to the supernova. It is
simple to show using Equation 4.12 and the Hubble parameter H = a/a that the
luminosity distance can be calculated with the integral,
dL -(1 + ) d' (4.15)
Jo H(z')
Substituting into Equation 4.15 the expansion of H(z) for small z gives,
dL (=+$i) dz'(1- Hz'+ ... (4.16)
Ho JO z Ho
Using Equation 4.11 and the chain rule allows us to make the identification
H' = (1 + q)Ho. Integrating Equation 4.16 term by term and collecting powers
results in the following power series in z,
dL- + (1 qo)z+.. (4.17)
Ho 2
The first term in Equation 4.17 represents Hubble's law -namely, v = Hod. The
second term is the first deviation of Hubble's law. Therefore, by measuring dL and
z and plotting them one infers the curvature (or deviation from linearity). For
z > 1, this expansion breaks down; in which case numerical integration can be
performed using the energy density one assumes to be present in Equation 4.6 (this
obviously introduces some model-dependent effects).
To determine the evolution of the missing energy component, we define the
parameter w which relates the energy and pressure densities at any given time by
the equation of state,
p =wp (4.18)
Equations 4.9 and 4.18, along with the requirement that the universe accelerate
forces the inequality,
p(l + 3w) < 0. (4.19)
Since p > 0, dark energy must give rise to w < Recently, w has become slightly
more constrained as measurements have improved. In order for the structure
formation we currently observe to exist from the density perturbations indicated
by C\!l l anisotropy measurements, we must have w < -' [52]. Additionally, the
absence of any intragalactic physics due to dark energy leads one to believe that its
distribution be smooth and homogeneous on large-scales.
The Cosmological Constant
The history of the cosmological constant is now so well known it needs little
development. Einstein introduced a constant to his general relativity equations
to balance the collapsing effect that matter alone would exert on the cosmic fluid.
By doing this it imposed what he felt at the time to be the natural state of the
universe -static.
Of course, the observed redshift of distant galaxies quickly did away with
the notion of a static universe; however, the cosmological constant would undergo
a conceptual it-ulution" soon after, when particle theorists were forced to
incorporate the quantum fluctuations of the vacuum which persist in gravity even
after renormalization. For example, consider the Hamiltonian of the quantum
harmonic oscillator with N degrees of freedom in terms of the raising and lowering
operators at and a, respectively:
N t N t 0)
H [a a+a + + (4.20)
i=1 i=1
The transition to field theory takes the number of degrees of freedom to infinity,
H = [at (k) a(k) + h(k ). (4.21)
k
Clearly, the ground state contributes an infinity to Equation 4.21. The usual
practice is to redefine the Hamiltonian by shifting the energy by an infinite
amount as only energy differences are observable quantities. This procedure,
however, cannot be employ, -1 with gravity. Theories like QED, QCD, and the
EW force all possess dimensionless expansion parameters. Thus, one may ahl--iv
find enough counter terms in the renormalization scheme at all energy scales.
The expansion parameter of gravity is Newton's constant GN, which in natural
units has dimensions M-2. Thus, as one increases in energy (i.e. probing the
ultraviolet) it takes more and more counter terms to renormalize to a finite value
-an infinite such counter terms for higher order terms and thus gravity in this
sense is nonperturbatively finite. Admitting our ignorance we may insert an ad hoc
cutoff,
A
HVAC ~ k) A. (4.22)
The cutoff scale is often chosen to be the Planck mass, A Mp ~ 1019GeV,
at which point new physics is needed to make predictions as to how gravity and
spacetime behave. Further, the Casimir effect, which in QED is the force registered
by two neutral, conducting plates as a result of quantum vacuum fluctuations lends
credence that the vacuum p1 i' a definite role at certain scales [53].
Vacuum energy is naturally homogeneous, isotropic, and of course must enter
Einstein's equation covariantly,
TVAC = PVAC / ,, 3" /lg (4.23)
Vacuum energy has the inherent properties that PVAC is uniform throughout
spacetime and that PVAC = -PVAC (i.e. w = -1).
The proposition of a cutoff introduces an awkward problem which we must
face. Currently, the value of the constant can be grossly estimated by ..--iiii:.-
3H2
P PA 10-48GeV4 (4.24)
87G
If we take a cutoff seriously, then a bare cosmological constant would have a value,
PA bare ~ A4 ~ 1076GeV4 (4.25)
Thus, we are forced to account for a discrepancy of 120 orders of magnitude
between the expected and the observed. One may do away with many orders of
magnitude if supersymmetry is included (with a cutoff scale MsUSY ~TeV), or
if the cutoff is not the Planck scale but rather the electroweak scale of 100 GeV;
however, it does not do nearly enough and we are left with essentially the same
questions, if but perhaps in a slightly less embarrassing form.
The observation of a small but non-zero cosmological constant which leads
to the so-called coincidence problem: namely, why has it only recently achieved
relative dominance [54, 55]?
There have been many attempts at understanding these critical problems
[56, 57, 58, 59, 60, 61, 62], none of which can be deemed satisfactory solutions,
else we would certainly have something far more profound to v- about dark
energy. Introducing a homogeneous scalar field which possesses dynamics will work
[63, 64, 65, 66], but one must understand why it is homogeneous [67] and again
why it has achieved dominance now. This approach, named quintessence, works as
a tracker solution, whereby the energy density of the scalar field follows the energy
density of the universe in such a way as to produce late-time acceleration.
Long-range forces have been si:r:.- -i .1 [68] whereby one introduces a charged
scalar field with a long-range, self-interacting force mediated by vector gauge boson.
If the gauge boson mass were to vanish at the minimum of the scalar potential, the
field would be unable to relax to its minimum, and cosmic acceleration could be
achieved [68]. Unlike quintessence, this model predicts an oscillating equation of
state [68] which can ostensibly be observed by high-z Supernovae; and therefore,
this model is distinguishable.
CHAPTER 5
LATE TIME ACCELERATION WITH A MODIFIED EINSTEIN-HILBERT
ACTION
As was the case with dark matter, dark energy pl i,- the role of an added
and hitherto unknown component to the right hand side of the Einstein equation.
Endowing it with the special property that it exerts a negative pressure on the
cosmological fluid provides us with a somewhat natural mechanism with which
to explain late-time acceleration. And just as MOND announces itself as an
alternative to the dark matter hypothesis, modified Einstein-Hilbert gravities
position themselves as alternative candidate explanations.
This chapter illustrates how adding a term proportional to an inverse power
of the Ricci scalar gives rise to an accelerating universe in late-time cosmology, i.e.
post big-bang inflation. We then examine the effect an added inverse Ricci term in
the action has on the resulting force of gravity.
5.1 Late-time Acceleration
Carroll, Duvvuri, Trodden, and Turner proposed a purely gravitational
approach [69]. Late time acceleration is achieved by considering a subset of
nonlinear gravity theories in which a function of the Ricci scalar is added to the
usual Einstein-Hilbert action,
S,] =dx R+f(R), (5.1)
where,
f(R) _-2(p+1)R-p V p > 0 (5.2)
From dimensionality we see that p is an a priori unknown parameter of mass
dimension one. Some connections to braneworlds have been proposed in which
terms with inverse powers of the Ricci scalar are exhibited [70].
For simplicity we will consider the case p = 1 at no loss of qualitative
understanding. We also include the matter action for completeness, in which case
the action is,
S d4 Rd4xM (5.3)
The equations of motion follow directly from Equation 5.28 via the variation,
where TM is the matter energy-momentum tensor. It is quite evident that the limit
p -- 0 in Equation 5.29 takes us back to the usual Einstein equation of motion.
Equation 5.29 can be trivially solved for R if one is considering the constant-
non-zero,
S2
Rvac = v/32 (5.6)
unlike their Minkowski counterpart. Of course, a (negative) positive constant-
curvature solution is precisely (anti) de Sitter space, and we therefore see imme-
diately how our equation of motion Equation 5.29 is capable of providing a purely
gravitational mechanism for explaining cosmological acceleration.
We wish to consider cosmological scenarios. Thus, on the grounds of large-
scale isotropy and homogeneity of the cosmological fluid, we restrict ourselves to
the perfect fluid form of the energy-momentum tensor,
(5.7)
T = (M + PM)UU + PMg1 ,
where U" is the fluid rest-frame four velocity, pM is the energy density of matter
and radiation, PM is the pressure of matter and radiation which is related to the
energy density via the equation of state PM = i,'rr. In a matter dominated
universe, w = 0; and in a radiation dominated universe, w = 1/3.
Homogeneity and isotropy allows also to limit our analysis to metrics of the
Robertson-Walker form,
ds2 = -gdxPdx v
It is straightforward to compute the Ricci
from Equation 5.8,
R a a2)
where H is the Hubble parameter,
dt2 + a2(t)d d .
(5.8)
scalar in terms of the scale factor a(t)
6(H + 2H2) ,
(5.9)
(5.10)
With Equation 5.9 and Equation 5.8, we obtain the two time-time and space-space
equations of motion from Equation 5.29,
31
3 2
2
4
H2 -(2HH + 15H2
12(H + 2H2)3
72( (4H + 9H2 + I2
72(H + 2H2)2 ( R
'H + 2H2 + 6H4)
- 6 P+ 4H
R2 P
8rTGpM (5.11)
-4WGPM,
(5.12)
respectively.
These fourth-order equations are cumbersome and therefore extracting their
cosmological implications not an easy task in their present form. Instead, Carroll
et al. [69] performed a specific conformal transformation on the original degrees of
freedom,
gp = p()gV p exp ( = 1 + )2, (5.13)
r / R2
dt =-pdt a(t) = vpa(t) (5.14)
PM P-2PM PM P-2PM (5.15)
where 0 is a real scalar function on space-time. This transformation has been
extensively treated [47], and involves representing metric degrees of freedom in
terms of a fictitious scalar field. The transformation leads to the following equation
of motion for the transformed expansion parameter,
H2 8- (pG +M), (5.16)
and scalar equation of motion,
/dV(O) (1 3w)
+ 3H' d+ 6( )pM, (5.17)
where a prime denotes differentiation with respect to t. We have introduced the
potential,
V(O) = (5.18)
87G p2
and here we identify the transformed energy density and scalar energy density,
K 47iG
M 3(w) exp -( 3w) (5.19)
Pt = \^ + V( (5.20)
respectively. Carroll et al. [69] considered three qualitatively distinct cases,
assuming an initial value for the scalar field to satisfy,
1
00 <. Mp (5.21)
vl67rG
From Equation 5.18, we see that the potential vanishes when Q -> 0 and Q oo.
The limit Q -> 0 would normally correspond to the Minkowski vacuum, but from
Equation 5.13 it is clear that instead a curvature singularity exists in this limit.
Although Q oo corresponds to R -> 0 and seems like a possible Minkowski
vacuum solution. However, from Equation 5.17 and Equation 5.13 we see that the
solution is oscillatory at .i, -ii:d! l ically large values of Q and therefore unphysical.
When the initial condition, 'o = where 0' is the critical value for which
the scalar field comes to rest at the peak of the potential, the scalar field energy
density becomes constant. Therefore,
H[O = c] = constant (5.22)
This of course is the hallmark of a de Sitter expansion, albeit under unstable
conditions since any perturbations in the scalar field will have it exhibit one of the
alternative qualitative possibilities.
For the scenario 0o < the scalar field never reaches the maximum but rolls
back toward 0 and the universe collapses upon itself. As V 0 and H goes to
a constant, the deceleration parameter and the Ricci scalar, both of which depend
upon H or H', are singular since H ~ V o.
Alternatively, the scalar field can be endowed with 0o > 0' in which case the
scalar field becomes quite large with time and the potential behaves as,
V p-3/2 p2M exp (- 3 (5.23)
If we seek a power law solution for the scale factor,
a t ocp- H (5.24)
then this implies,
Q' ->p ~ t/3 (5.25)
t:
Thus, the scale factors behave as,
a t4/3 (5.26)
a t2/3 (5.27)
It is possible to consider the above situations for the more realistic case of pM / 0,
which was considered in [69]. However the results are no more instructive, and we
therefore direct to the aforementioned article for a more thorough discussion.
To this point we have said nothing of the p parameter. Although the 'o = 'c
scenario is unstable, one may argue that this theory holds phenomenological
relevance. This eternal de Sitter inflation is not too absurd if the decay rate of
the phase is on the order of 7-1 ~ (14 Gyrs)1 the inverse age of the universe.
In terms of a mass scale, this corresponds to p r 10-33eV. Therefore, one can
argue on phenomenological grounds that this theory is worthy of consideration
since it clearly gives rise to late-time acceleration. Of course, p is no better than
a tuned parameter serving the function of giving credence to the above statement.
Nevertheless, it is a viable alternative to the dark energy mechanism, and as such
merits further investigation.
5.2 The Gravitational Response
We have shown in the previous section that with an inverse Ricci scalar term
in the gravity action, it is possible to explain late-time acceleration. However, we
have yet to see what this theory i- about the force of gravity on cosmological and
local scales. That is the task of this section, and we will restrict ourselves to the de
Sitter solution which was discussed to be unstable, but with a slow enough decay
rate to justify its study.
Before we embark on calculating the force of gravity with an inverse Ricci
term added to the gravitational action, we should comment on the inherent
features of such actions. As was apparent from Equation 5.11 and Equation
5.12, our equations of motions of motion are of the higher derivative v ,i I 1 i
(that is, they possess more than two time derivatives on one of the degrees of
freedom). Typically, higher derivatives bring negative energy degrees of freedom;
however, endowing the Lagrangian with nonlinear functions of the Ricci scalar
can sometimes be permitted [71]. This will only give rise to a single, spin zero
higher derivative degree of freedom. But since the lower derivative spin zero is
a constrained, negative energy degree of freedom (the Newtonian potential), its
higher derivative counterpart can occasionally carry positive energy.
There have been several recent articles which examine aspects of this model.
Dick considered the Newtonian limit in perturbation theory about a maximally
symmetric background [72]; while Dolgov and Kawasaki discovered and discussed
an instability in the interior of a matter distribution [73]. However, Nojiri and
Odintsov have shown than R2 can be added to the action without changing the
cosmological solution, and that the coefficient of this term can be chosen to
enormously increase the time constant of the interior instability [74]. Meng and
Wang have explored perturbative corrections to cosmology [75]; and others have
drawn connections with a special class of scalar-tensor theories [76, 77].
What we wish to consider here is the gravitational response to a diffuse matter
source after the epoch of acceleration has set in. The procedure will be to solve
for the perturbed Ricci scalar, whence we determine the gravitational force carried
by the trace of the metric perturbation. We will constrain the matter distribution
to have the property that its rate of gravitational collapse is identical to the rate
of spacetime expansion, thereby fixing the p,, ;/,. 'l1 radius of the distribution
to a constant value. Further, we impose the condition that inside the matter
distribution the density is low enough to justifiably employ a locally de Sitter
background, in which case the Ricci scalar can be solved exactly and remains
constant.
The Calculation
We shall consider a gravitational action parameterized by p > 0,
S[g 16 d4X [R -d 2(p+1) R-] (5.28)
(We employ a space-like metric with Ricci tensor R,~ FP,,P FP, P, + FP,,I -
P,,F' ,.) Functionally varying with respect to the metric and setting it equal to
the matter stress energy tensor leads to the equations of motion,
[1 +p2(p+1)R -(+1)] R [1 2(p+1)R-(+I)] Rg1
+ p 2(p+1)(gO DD)R-(P+) 8 GT,,. (5.29)
D. is the covariant derivative and O (-g)-1/2Q( 2ggP/ ') is the covariant
d'Alembertian.
Although one must really solve all components of the field equations Equa-
tion 5.29 we can get an important part of the gravitational response by simply
taking the trace. We shall also restrict to p = 1 for simplicity. Inside the matter
distribution the trace equation is,
R + 3 + 31/O = S1GgS T1, T. (5.30)
(Note that T is negative.) Normally, one would expect the matter stress energy
to be redshifted by powers of the scale factor in an expanding universe. However,
recall that this matter distribution possesses a rate of gravitational collapse equal
to the rate of universal expansion, and thus T remains constant. Since our matter
source is also diffuse, we may perturb around a locally de Sitter background. For
the interior solution, we are able to solve for R exactly using Equation 5.30 for the
case T is constant and DR = 0,
2 T 12+ .
Rill= 2 1 + (.1
Obtaining de Sitter background obviously selects the negative root. Further, we
concentrate on the situation IT7 < p2
T
Rin = 2 + (5.32)
Outside the matter source we perturb around the de Sitter vacuum solution,
Outside the matter source we perturb around the de Sitter vacuum solution,
Rout = /p2 + 6R.
(5.33)
Substituting Equation 5.33 into Equation 5.30 and expanding to first order in 6R
yields the equation defining the Ricci scalar correction,
DSR(x) + 3p26R(x) = 0.
(5.34)
In our locally de Sitter background the invariant length element is,
dst2- d2 + a2(t)dx di,
(5.35)
with a(t) having the property,
H=
a
constant.
(5.36)
We can relate the Hubble constant H to the parameter p via the vacuum Ricci
scalar,
R = 12H2 3= /2.
(5.37)
Identifying E = a-30 (a3gp7t,), we expand Equation 5.34,
[a2 3HOo + 12H2] 6R(t, x) = 0,
(5.38)
where 02 = -2 + a-2V2. It is evident from Equation 5.38 that the frequency term
has the wrong sign for stability [69]. However, since the decay time is proportional
to 1/H, we may safely ignore this issue.
Seeking a solution of the form 6R = 6R(H11I |.11) allows us to convert Equation
5.38 into an ordinary differential equation,
[(1 d2 ( 2 2)d + 12] R =0, (5.39)
dy y dy +
where y HIIII.T \. To solve this equation we try a series of the form,
00
f uy)= fn+". (5.40)
n=0
Substituting this series into Equation 5.38 yields a solution with a = 0,
0 V + 3 ')P(n + + ) (2y))2n
fo () 4 4 (5.41)
fo() S r(+ 57) (2n+ (541
and a solution with c = -1,
S1 Pn7)( + 1 + v) (2y)2n
iu) E( v-_ 7)P + v57) (2n)!( )
f _1( ) 4 4 4 4 (
Both solutions converge for 0 < y < 1. Both also have a logarithmic singularity
at y = 1, which corresponds to the Hubble radius. We can therefore employ them
quite reliably within the visible universe.
The solution we seek is a linear combination,
6R(y) 1 fo(y) + 2f- l(y), (5.43)
whose coefficients are determined by the requirements that 6R(y) and its first
derivative are continuous at the boundary of the matter distribution. We employ
a spherically symmetric distribution of matter, centered on the comoving origin.
If the matter distribution collapses at the same rate as the expansion of the
universe, its physical radius is a constant we call p. (This means that the comoving
coordinate radius is p/a(t).) If the total mass of the distribution is M we can
identify T as the constant,
87rGM 6GM
T 4 (5.44)
mp3 p
In terms of our variable y = Ha(t) I1.7T, the boundary of the matter distribution is
at yo = Hp. Demanding continuity of the Ricci scalar and its first derivative at yo
gives the following result for the combination coefficients of the exterior solution
Equation 5.43,
3MG foQo) f --o1
01 =3 [fo(yo) _- (of-i(yo) (5.45)
3MG f' l(yo)
02 P f-I(Yo) f' o) ) (5.46)
P3 0 fyo)
where a prime represents the derivative with respect to the argument.
We are now in a position to calculate the gravitational force carried by the
trace of the graviton field. The metric perturbation modifies the invariant length
element as follows,
ds2 -(1 hoo)dt2 + 2a(t)hoidtdxi + a2 t) (6 + hj)dx'dxj. (5.47)
Further defining h = -hoo + hii and imposing the gauge condition,
S 2 h, + 3h (lna),, 0, (5.48)
allows us to express the Ricci scalar in terms of h,
6R = --2hh + 4HOoh,. (5.49)
(Recall that we define 0, = (dt, a-V).) Assuming h h(y) as we did for 6R gives
the equation for the gravitational force carried by h,
[ 1, 26R(y) 2 (5.50)
(y2- 1) + l(,- 2) h'(y)-) (5.50)
dy y H2
The solution to Equation 5.50 is,
h'(y) 2 j- 2)3/ '2 ~ 1/26R (5.51)
At this point it is useful to consider the y values which are relevant. The
Hubble radius corresponds to y = 1, whereas the typical distance between galaxies
corresponds to about y = 10-4, and a typical galaxy radius would be about
y = 10-6. We are therefore quite justified in assuming that yo < 1, and in
specializing to the case of yo < y < 1. Now consider the series expansions,
1 3MG
fo(y) = 1 2y2 + 1 + 0(y6) 3 + O(yo2), (5.52)
5 p
f-i(y) 1 -- 7y2 + t44 + O(6) ,/2 1 G3 + O(). (5.53)
y 3 3p
We see first that |12 1 which means 6R(y) Pifo(y) -and second, that
fo(y) ~ 1 which implies 6R(y) -T/2. This means that the integrand in
Equation 5.51 fails to fall off for y > yo, so the integral continues to grow outside
the boundary of the matter distribution. For small y > yo we have,
2GM
h'(y) = + O(y3). (5.54)
H2 p3
To see that this linear growth is a phenomenological disaster it suffices to
compare Equation 5.54 with the result that would follow for the same matter
distribution, in the same locally de Sitter background, if the theory of gravity had
been general relativity with a positive cosmological constant A = 3H2. In that case
6R(y) = -TO(yo y) and, for y > yo, the integral in Equation 5.51 gives,
h'(y) 1 { arcsin(yo) yo( 2y) y. (5.55)
R 4H2 2(1 2
4GMH
2 + 0(). (5.56)
112 1)
The linear force law Equation 5.54 of modified gravity is stronger by a factor of
( y)3. For the force between two galaxies this factor would be about a million.
5.3 Remarks on our Calculation and Future Work
We have determined the gravitational response to a diffuse matter source in a
locally de Sitter background. Our result is the leading order result in the expansion
variable y, the fractional Hubble distance. Equation 5.54 clearly forces us to
disregard the class of theories considered here Equation 5.28 when compared to GR
with a cosmological constant (for example, the correction to the gravitational force
between the Milky Way and Andromeda increases by six orders of magnitude).
The two assumptions made in our analysis were:
the matter distribution is gravitationally bound,
the matter distribution has a mean stress energy T1 l p2.
The second of these assumptions can be viewed rather flexibly if interested only in
phenomenological implications. Regardless of whether it is satisfied, we still would
expect a linearly growing response far from the source. To see this, recall that
the dominant piece of the solution, fo(y), from equation Equation 5.43 remains
constant and approximately equal to one for many orders of magnitude (for
instance, fo(108-) fo(10-3) W 10-6). Therefore, although the exterior solution
would not be very reliable near the matter source, we can be confident that at
cosmic or even intergalactic scales perturbing about de Sitter becomes appropriate
and a growing solution would still be observed.
This analysis was performed for p = 1, but of course nothing restricts us from
considering arbitrary powers of the inverse Ricci scalar. To no surprise, however,
varying the power only changes the coefficient of the gravitational force leaving
its qualitative behavior alone. The instability found by Dolgov and Kawasaki [73]
and the growing solution calculated in this work seem to preclude all such theories
phenomenologically. The two problems seem to complement one another because
either problem could be avoided by the addition of an R2 term, which would not
alter the cosmological solution [74]. However, avoiding the interior instability seems
to require the R2 term to have a large coefficient, whereas avoiding the exterior
growth requires a smaller value [77].
None of these issues diminishes the importance that should be placed on
considering novel approaches to understanding the dark energy problem. It is the
responsibility of both theorists and experimentalists to construct and constrain
candidate theories, and it is truly an exciting epoch of human investigation for
which we are just beginning to acquire these capabilities. Greater freedom can
be obtained by adding different powers of R. (Note that this generally alters the
cosmological solution.) Although such models seem epicyclic when considered as
modifications of gravity, the same would not be true if they were to arise from
fundamental theory. For example, it can be shown that the braneworld scenario
of Dvali, Gabadadze and Shifman [78] avoids both the interior instability and the
linearly growing force law [79].
CHAPTER 6
CONCLUSIONS
This thesis has examined alternative explanations to the dark matter and
dark energy problems. Each problem has been presented with an alternative that
modifies the Einstein-Hilbert action of gravity in four dimensions with a function of
the Ricci scalar,
1 PX.r
S,[g = SEH[g4 + gf(R) (6.1)
The subsequent phenomenology has been discussed and used to make definitive
statements as to the standing of these theories and prospects for future investiga-
tion.
Dark matter's successes -particularly a A-CDM scenario -leaves many with
the impression that its role in galactic rotation curves is a necessary feature. CDM
is able to explain galaxy formation by providing enough gravitational presence to
ensure luminous matter clumping on the scales we see tod -v. If one takes seriously
the Peccei-Quinn mechanism as a solution to the strong-CP problem of QCD,
then the axion is a real particle and thus a prime candidate for dark matter. Big
bang nucleosynthesis also cannot do without dark matter. Baryonic matter alone
is unable to account for the density required to allow BBN to occur. We clearly
see that dark matter's connection to the entire cosmology of the universe is too
intertwined for its existence to not be taken as a possible reality.
Nevertheless, new gravitational physics which occurs at different scales is
certainly not an impossibility. The evidence stated above is only gravitational
in nature. That is, it only serves to identify dark matter via its gravitational
interactions. Observing a particle gravitationally is an insufficient method of
detection. This only serves to determine the metric in a fixed gauge, whence
one may then construct the Einstein tensor and then define the matter-stress
energy tensor so as to make the Einstein equation true. Therefore, flipping the
solution on its head: there currently exists modifications due to gravity which are
capable of accounting for the observed cosmology; and these modifications can be
interpreted as the presence of matter stress-energy we call dark matter (much like
the organizing principle of perturbation theory in classical general relativity).
The current dark matter profiles which have been studied suffer definite
problems which we have discussed in this work to some length. They are unable to
explain: the Tully-Fisher relation -the proportionality of the absolute luminosity
to the quartic power of the maximum rotational velocity; and Milgrom's law -the
fact that dark matter needs to be evoked when satellites possess an acceleration
a < ao 10-10m/s2. Additionally, their fine-tuning features (by virtue of their
three parameter fitting) all leave one to conclude that the rotation curves alone
an amazingly consistent phenomenon -cannot presently enable one to v- much
about the fundamental nature of dark matter.
MOND is purely gravitational at the nonrelativistic level, and by design is
constructed to satisfy Milgrom's law. Therefore, at the empirical level, it is vastly
superior to dark matter halos. It is at the fundamental level where one properly
di-pl',1i, reservations as to its viability in light of the successes of dark matter
in several key physical processes. The need for a covariant metric formulation
of MOND becomes immediately evident -one which can be directly measured
alongside its dark matter competitor.
MOND's relativistic extension has been treated in this thesis by considering
the two predominant approaches: the scalar-tensor theories of Milgrom, Bekenstein,
and Sanders, and the purely metric approach of Soussa and Woodard. At present,
it can be said conclusively that of the scalar-tensor varieties, the TeVeS theory of
Bekenstein is the most viable candidate. Naturally, all approaches are constructed
to be able to reproduce the nonrelativistic version of MOND. However, the key
issues in extending MOND have been the lack of sufficient gravitational lensing
of light, the acausal propagation of dynamical fields, and inherent ambiguities in
regard to its cosmological impact.
Bekenstein's TeVeS is the first relativistic version which is able to resolve
the first two of these three issues (under appropriate assumptions) and not be a
preferred-frame theory. As it is fully relativistic, one may see what it -Zi- about
cosmology. Presently, no definitive conclusions can be made and is the subject
for future work. TeVeS, however, it not without problems. Namely, the large
parameter space creates ambiguity and it is not overly clear which observables can
set or constrain these parameters. One possibility would be to take advantage of
the disparate travel times that gravitons and neutrinos would possess from distant
.i- i.' .1l, -ical sources. We found after a simple computation that the d.l li in
arrival times of a gravitational wave and a pulse of neutrinos could be on the order
of a few minutes under reasonable assumptions of the parameters. Therefore one
would ostensibly be able to constrain their values (or ratios thereof). Solar system
tests serve as good constraining tools in scalar-tensor theories (e.g. Brans Dicke
gravity), and therefore it is certainly not unreasonable to have a degree of optimism
in the falsifiability inherent to a theory like TeVeS.
The purely metric theory gains the advantage in overall i i, lu IIl~. -"
that is, the purely metric degrees of freedom, if sufficient to describe MOND
in all regimes consistent with .i-1 i !,!1, -i I1 observations, would follow Ockam's
razor. Avoiding philosophical vicissitudes, we shall unabashedly assume the
preferential treatment of such a feature in the purely metric theory. The avoidance
of scalar and vector degrees of freedom results in fewer parameters. What we
discovered, unfortunately from the model builder's perspective, is that, under
conservative assumptions, ,;, purely metric theory will never give enough lensing
due to the conformal invariance of the linearized MOND equations. However, this
result should be viewed in a positive light, as any predictive and ultra-restricting
statement in physics should. We may conclude that the most plausible way to
avoid the lensing disaster in a purely metric formulation of MOND is by foregoing
the notion of gravitational stability, a less pleasant but not unprecedented nor
unfathomable situation.
In similar fashion, we have surveyed the current landscape of the dark energy
problem. Like dark matter, many of the approaches have centered on adding a new
component to the universe such as a constant scalar field, a dynamical scalar field
designed to turn on at the appropriate time, charged scalars that exhibit long-range
forces, etc. By construction, all these models serve their purpose -they give rise
to late-time acceleration in the universe. Each, however, begs the question to their
detection at the level of new particles and fields.
Contrarily, modifications of the Einstein-Hilbert action interact with all
matter and energy, and there signatures are in the evolutions and dynamics of the
universe's constituents. This thesis has considered specifically the modification
of Carroll et al. [69] in which an inverse power of the Ricci scalar is added to the
action. This type of term is shown to give rise to late-time acceleration under
appropriate assumptions.
The work of Carroll et al. [69] did not consider the effect this kind of term
would have on the force of gravity. This work presents this very calculation in a
locally de Sitter background for the case of a diffuse matter source. The result
clearly shows that this type of term can in no way be phenomenologically viable.
The solution possesses a term which grows linearly with distance, and therefore
even physics on the cluster scale completely rules out such a model. Further, the
lack of a Newtonian limit which surfaces as an instability in the inner regions
of matter sources in Dolgov and Kawasaki's work [73] seemingly dooms such a
proposal.
One may consider adding terms proportional to R2, and R3, and/or using the
Palatini formalism to remove the inherent instabilities of only having a 1/R term
in the action. Presently, it does not seem clear at all that one may both remove
the instability found by Dolgov and Kawasaki and the linearly growing solution
discussed here. Adding more and more terms to the action in the epicyclic spirit
seems counter to how we should seek solutions. However, upon doing so a larger
theory or a more fundamental gravitational principle may emerge -we may
find these terms to naturally arise from some larger theory, either as an effective
field theory, or perhaps from a string theory. Measurement, phenomenology, and
consistency are our guides to this end.
Dark energy and dark matter are without question the consensus -the
currently orthodox approach to explaining '11.' of the universe's energy. It is,
undoubtedly, extremely peculiar that we have not directly detected i,;, of this '.'.
-never once. The only means at our disposal to ?- anything empirically, the
sole fashion we may claim to have observed either of these two phenomena, is via
gravity. Simply stated: Einstein's theory works. Therefore, changes to it at any
scale should and will meet resistance from the wealth of data that exists -not
to mention the theoretical challenges which must be overcome. That said, there is
serious reasons to believe that general relativity even at the classical level is unable
to account for all of the observed universe. The processes discussed here have
all been gravitational and their orthodox explanations can all be recast into the
form of a purely gravitational solution. This fact serves not only as an incentive
to search for alternatives, but almost obliges the physicist, in conformity with the
scientific spirit, to allow its possibilities.
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BIOGRAPHICAL SKETCH
Marc Soussa was born in York, Pennsylvania. He received a B.A. from Cornell
University in the field of biochemistry and chemistry. He went on to study high-
energy theory in the Physics Department at the University of Florida under the
supervision of Richard Woodard.