Essay on variable constants

Space-time Invariance & the Nature of Physics - David Crosby
1 Introduction
1.1 Symmetry and the Laws of Physics
1.1.1 The Equivalence Principle
Abstract
The laws of physics, particularly cosmology make extremely important assumptions
as to the nature of reality. The issue and history of the invariance of the
physical laws is discussed and the most up to date experimental measurements presented
together with some of the latest theoretical developments on the issue. It is
concluded that to date only a single valid non-zero result for the possible variation
of a fundamental constant, namely the fine-structure constant, has been measured.
The principle of Occam’s razor, “non sunt multiplicanda entia praeter necessitatem”, still
has great relevance to modern physics, it states that entities (such as assumptions or theories)
should not be multiplied by more than necessity, or more colloquially, the simplest
explanation is usually best. The notions embodied within this statement are at the heart
of the quest to constantly reduce and unify the laws of physics, hopefully towards some sort
of Theory of Everything (sic.). An example of this is the relationship between symmetry
and the traditional conservation laws (energy, linear & angular momentum). Established
by the famous mathematician Emmy Nöther in 1918, what was to become known as
Nöther’s theorem unified the notions of basic physical symmetries and the conservation
laws so dear to the physicist’s heart on a firm mathematical basis [1].
Nöther’s work showed that every conservation law associated with a system arising from
a variation principle comes from a symmetry property. For example, the Euler-Lagrange
equations of dynamics conform to such a formulation. Consequently, Nöther was able
to show that the conservation of linear momentum arises from the linear translational
invariance of the laws of physics (the symmetry that space is homogeneous), the conservation
of angular momentum arises from rotational invariance (space is isotropic) and the
conservation of energy from invariance of the laws of physics with respect to an arbitrary
translation in time [2]. Furthermore, Nöther showed that, within her framework, the
statements of symmetry or conservation law were both necessary and complete meaning
that one need only to invoke one (it doesn’t matter which) to imply the other, illustrating
somewhat of a philosophical resonance to Occam’s razor.
Modern field theories in physics are grounded upon the notions that the most fundamental
physical laws correspond to some invariance, which in turn is equivalent to a collection of
changes forming a symmetry group. This issue is summarized rather succinctly by Steven
Weinberg [3] with the aphorism “symmetries imply conservation laws”.
In 1907, Einstein first published a proposal of the equivalence principle [4], which states
that there are no means, by experiments confined to infinitesimally small regions of spacetime,
to distinguish one local Lorentz frame in one region of space-time from any other
local Lorentz frame in the same or any other region. Immediate consequences of this
principle include the prediction that gravitational fields should cause the path traced by
a ray of light to curve, from symmetry with the observation of a curved path expected
if one were inside an appropriately accelerating box. The powerful symmetry implied by
the equivalence principle allows the generalization of all the special relativistic laws of
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Figure 1: Emmy Amalie Nöther
1.2 Variable Constants
physics to the curved space-time geometry implied by general relativity, with breakdown
only occurring at a singularity (where the curvature of space-time becomes infinite).
The modern definition of Einstein’s equivalence principle (EEP) may be broken down
into three basic assumptions. The weak equivalence principle (WEP) requires that any
test body with a given initial velocity and position in space-time follow a trajectory in
a given gravitational field independent of the internal structure or composition of that
body. Local position invariance (LPI) requires that the outcome of non gravitational test
experiments performed within a local, freely falling frame in a background gravitational
field are independent of the frame’s location, and local Lorentz invariance (LLI), requires
independence with respect to the frame’s velocity. The modern interpretation of the EEP
is embodied by the combination of WEP, LPI and LLI [5].
The symmetries used by Nöther and Einstein form the basis of the Cosmological Principle,
first named and formalized by Milne in 1933 [6], and the basis of virtually all modern work
in cosmology. It states that “the Universe as seen by any observer at any place or time is
homogeneous and isotropic”. Bondi [7] extended this principle with an additional postulate
of temporal equivalence, leading to what is known as the Perfect Cosmological Principle.
Defining the objective goals of physics is a matter of some opinion, however, the grand
aim of cosmology, to explain the origin and evolution of the Universe, has to be one of
the loftiest and most important. Thus by association, the assumptions upon which that
cosmology is based must also be important.
The subject of physics is an empirical beast demanding that any and all assumptions invoked
should be tested and verified experimentally to the best of our capability. Clearly,
assumptions about the nature of physical reality with implications as important as the
conservation laws of energy, linear and angular momentum, the theory of general relativity
and the basis of cosmology demand the very highest degree of verification given the ramifications
should they prove to be false. Thus one should seek to measure the symmetry or
invariance of the laws of physics in both space and time. One must now determine a basis
upon which such verifications can be made. It is reasonable to expect that any variations
will initially express themselves in the most simple way conceivable, to which end, one
may attempt to determine to what degree the fundamental constants of physics truly are
invariant in space-time. Excluding grand contrivances, bizarre and radical variations of
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2 Numerology
the laws of physics would almost certainly have an impact on any extracted constants.
2.1 The Importance of Being Dimensionless
The equations of physics are permeated with constants that relate one quantity to another.
Most of these constants, such as the speed of light in a vacuum or the charge of the electron
are dimensional quantities, representing the dimensionless ratio of the quantity to some
unit standard. For example, the second is defined as a number of oscillations of a hyperfine
state in a Caesium atom, the number being the dimensionless ratio and the unit standard
being the time for a single hyperfine oscillation. By redefining the unit of time using
some other standard, values such as the speed of light will be changed. Of course, this
change in a constant (in this case the speed of light) does not imply any change in the
laws of physics, merely a different dimensional scale upon which those laws are measured.
This raises the question as to whether it is appropriate to look for genuine changes in
the laws of physics based on variations of dimensional constants? One cannot be sure
of the equivalence of the standard units in different reference frames moving relative to
eachotheroratdifferent points in space-time. By postulating such a correspondence of
standard units necessitates that statements of the constancy of dimensional constants will
be circular [8]; any experimental evidence for a varying dimensional constant can always
be absorbed into a redefinition of units. However, if one were to measure a sufficient
number of dimensional constants to span the dimensional basis space (e.g. mass, length,
time and electric charge) then no such arbitrary absorption of changes is possible, but this
is tantamount to measuring an appropriate dimensionless constant, which would seem the
more elegant option.
By appropriate cancellation of the dimensional factors of various dimensional constants
one obtains a pure number, a dimensionless ratio independent of the set of basis units
used. For example, the fine-structure constant, α = e 2 / (}c) (cgs units)= e 2 / (4πε 0 }c)
(SI units), remains the same (approximately 137 −1 ), although differences between the
definition of the units of electron charge, e, requiresdifferent expressions to yield the
equivalent dimensionless value.
Variations in dimensional constants are not meaningless, they are merely difficult to separate
from the definitions upon which they are based. This caveat presents the experimenter
with the challenge of making measurements of the appropriate dimensionless quantity involving
a dimensional constant of interest. Thus, one can consider models and theories
involving the variation of any constant, but must reconcile implied results with experimental
observations of the correspondingly involved dimensionless constants.
The values of dimensionless constants, being fixed by the physical world, are in some
sense more fundamental than their dimensional brethren. The current status quo of modern
physics necessitates that these constants be measured, even the highly successful
standard model of particle physics relies on a set of parameters requiring experimental
determination. A more deep understanding would be apparent if one were able to derive
such values apriori. Ideas such as this are very much at the heart of certain quests for
so called Theories of Everything, although one should exercise great caution in the use
of such a phrase in any technical sense. A popular treatise of this subject has been conducted
by Barrow [9] and a more technical discussion exercised by Albrecht [10]. Aside
from a discussion verging on metaphysics with regard to what constitutes a fundamental
theory it is certainly reasonable to suppose that being able to determine the cosmological
dynamics, if any, of the fundamental constants could help signpost the way towards new
theories.
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2.1.1 Candidate Constants
The array of dimensionless constants, particularly those amenable to measurements over
meaningful epochs with reference to the Cosmological Principle, is relatively limited, the
following summarizes the quantities of most significance with regard to the measurements
conducted to date (cgs units assumed where appropriate).
α = e2
}c ' 7.3 × 10−3 , the fine-structure constant.
µ = m e
m p
' 5.4 × 10 −4 , the electron-to-proton mass ratio.
β = gm2 p c
}
' 9 × 10 −6 , where g is Fermi’s constant of the weak interactions.
3
γ = Gm2 p
}c
' 5 × 10 −39 , where G is the constant of gravitation.
ε = Gρ ' 2 × 10 −3 , where ρ is the mean density of mass in the Universe.
H0
2
The necessities of experimental design also result in other ”over-expressed” dimensionless
quantities such as ¡ ¢
α 2 g p m e /m p ,wheregp is the gyromagnetic ratio of the proton
(this particular measurement corresponds to a comparison between the 21 cm hyperfine
transition of hydrogen and an optical resonance transition [11]). Of the above constants
α is particularly accessible from an experimental point of view, due to the prevalence of
obvious electromagnetic phenomena, it is the measurement of the secular variation of this
quantity that has received the most emphasis.
2.2 Dirac’s Large Number Hypothesis
2.2.1 Numerical Coincidences
In 1857 Kirchoff noticed the proximity of the ratio of electric and magnetic charges,
measured as 3.107×10 10 cm s −1 by Weber & Kolorausch the previous year, to the speed of
light [8]. This strange numerical ”coincidence” sufficiently impressed James Clerk Maxwell
to help prompt the development of his theory of electromagnetism, which, of course, was to
demonstrate that the coincidence is exact. This early success of numerological reasoning
prompted further investigations. In 1905 Planck noticed a possible relationship between
the constant he had introduced, the electric charge and the speed of light, i.e. that
h ∼ e 2 /c. However, it wasn’t until 1916 that Sommerfeld, in constructing his relativistic
modification to the Bohr theory of the atom, brought the three quantities together as the
fine-structure constant, α. Defined originally as the ratio of the electrostatic energy of
repulsion between two elementary charges e separatedbyasingleComptonwavelength
(the characteristic length scale at which relativistic interactions become important) to the
rest energy of a single charge
α = 2πe2 / (h/mc)
mc 2
= e2
}c . (1)
Sommerfeld clearly realised that this quantity had fundamental significance in the new
physics of the era, “e is the representative of the electron theory, h fittingly represents the
quantum theory, and c comes from the theory of relativity” [12].
2.2.2 Eddington’s Fundamental Theory
Arthur Stanley Eddington, a brilliant relativity theorist, felt that the dimensionless constants
of physics could be derived purely from mathematical arguments alone [13]. He
indulged in some of the most extreme examples of numerological reasoning in trying to
construct his ‘Fundamental Theory’ and α −1 , with its close proximity to the value 137
a number that is both integral and prime, received particular attention. Eddington’s
approach to deriving these values was, to say the least, somewhat novel and resulted in
considerable derision. For example, he once argued that α −1 is equal to the number of
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terms in a symmetric 16-dimensional tensor, 136. Later unity was added to this value to
align it better with the experimental value of 137.036 [8], such arbitrary changes did little
to enhance the credibility of the work. Eddington, despite the sceptical reaction, worked
assiduously on his combinatorial derivations throughout a considerable period of his life,
generating a vast array of results that still lack a coherent basis. Whilst much of his
reasoning was incoherent, it is clear that Eddington realised that many of the problems
presented by cosmology would require radically different approaches to deliver meaningful
answers, and this would seem a more fitting tribute to the man than simple taunts.
In 1923 Eddington began to ponder the discrepancy between the typical values of the
dimensionless constants. Considering the constants above: γ −1 and δ −1 have values of
approximately 10 40 , whilst α −1 , β −1 and ε −1 have values of approximately 10 3 . Eddington
did not believe that very large pure numbers should occur naturally as fundamental constants
of some purely mathematical infrastructure. Instead he proposed a line of Machian
reasoning, that one could remove the difficulty in accounting for these values if one should
associate with them the total number of particles in the world, a number presumably
decided by pure accident. Consequential to this reasoning Eddington evaluated the ’Eddington
number’, the total number of particles in the observable Universe N ∼ 10 79 .This
number is necessarily integral and was of great interest to Eddington, who felt that, in
principle, one could calculate its value exactly.
2.2.3 The Large Number Hypothesis
The unnatural appearance of very large dimensionless constants was also to cause Dirac
some consternation, culminating in a letter to Nature in 1937 in which he expounded what
was to become Dirac’s Large Number Hypothesis (LNH) [14]. Dirac suggested that only
the relatively small dimensionless constants, such as α −1 , β −1 and ε −1 held fundamental
significance. The much larger numbers were instead linked to parameters associated with
the dimensions of the universe, and hypothesized to be related to each other by factors of
order unity. The original letter to Nature cites one such possible relationship,
N 1 =
t 0
e 2 /m e c 3 ∼ age of universe
1040 =
atomic light-crossing time , (2)
N 2 =
e 2
∼ 2.3 × 10 39 electric force between proton & electron
=
Gm N m e gravitational force between proton & electron . (3)
The LNH demands that these two quantities are approximately equal, therefore for an
evolving universe, in which t 0 is not infinite, one can see that N 2 must somehow be
a function of time. Furthermore, Dirac supposed that these numbers be linked to the
Eddington number in the fashion N 1 ≈ N 2 ≈ N 1 2 , thereby implying a cosmology involving
the continuous creation of matter. Dirac suggested identifying temporal dependence with
the gravitational constant, i.e. that G ∝ t −1
0 . Whilst Dirac himself did not consider the
possible secular variation of α it was his ideas that mobilized the subject; “the constancy
of the fundamental physical constants should be checked in an experiment”. Note that
it was not concerns of symmetry or conservation laws that led Dirac to wonder upon
the constancy of the constants, rather it was a mathematical intuition that it would be
clumsy and inelegant of Nature to create very large pure fundamental numbers. This line
of reasoning was typical of the idiosyncratic P. A. M. Dirac, who sought mathematical
elegance above all else in his pursuit of new theories [15, 16]. Indeed, one suspects that
the reasoning behind the LNH was so radical that if it were proposed by a physicist of
lesser stature the subject would not have commanded such a forum.
Given the assumption that the Universe is approximately 10 billion years old one can
exclude the variation originally proposed by Dirac, that α, β, ε are constant and δ,
γ ∼ t −1 , with relative ease. For example, Teller [17] considered the effect varying G
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would have had on evolution of the solar system. In a varying G cosmology the radius of
the Earth’s orbit about the Sun, R orb , would have been much smaller and the luminosity
of the sun, due to the increased internal pressure resulting from a larger G leadingtoa
greater internal fusion rate, much higher
L¯ ∝ G 7 ; R orb ∝ G −1 . (4)
Where L¯ denotes the luminosity of the Sun. (4) implies a change in temperature at the
surfaceoftheEarth,T ⊕ of the form
2.2.4 Anthropic Principles
T ⊕ ∼
µ
L¯
R 2 orb
1/4
∝ G 9/4 ∝ t −9/4 , (5)
implying a very hot Earth. Teller’s calculation in 1948, assumed a Universe three billion
years old, and that life began 500 million years ago (the pre-Cambrian era). With these
parameters he was able to show that the variation implied by (5) would lead to boiling
oceans, prohibiting the development of life. Dicke [18] was able to show that revised data
for both the age of the Universe (approximately 12 billion years old) and the beginnings
of life on Earth (3.4 billion years ago) maintained cogency for the argument, a prediction
that life could not have formed when it did. Other arguments regarding the evolution of
satellites and stellar models also prohibit Dirac’s variation. Dirac returned to the LNH in
his later years [19, 20], however, the results are only of historic interest now.
Dirac’s large number hypothesis prompted the development of several purely phenomenological
cosmological models utilising numerological reasoning and the LNH to moot possible
variation of dimensionless constants. The Brans-Dicke hypothesis [21] suggested that
α, β, ε remain constant, and γ ∼ t −r , δ ∼ t −1 ,thisdiffers from Dirac’s hypothesis in that
r is a small number of the order 0.05. Accordingly, one expects the gravitational forces to
decrease with time by a fraction between 10 −12 and 10 −11 per year. Gamow [22] made the
first suggestion that α might vary with time. Gamow supposed that the natural quantity
to discuss was not the quantum mechanical constants α and γ but rather the ratio
γ/α = Gm 2 p/e 2 =7× 10 −37 (6)
of the gravitational attraction to the electrostatic repulsion between two protons. AccordingtotheLNH(γ/α)
shouldvaryast −1 , Gamow insisted that γ remain constant
leadingtothemodel: β, γ, ε remain constant, α ∼ t, δ ∼ t −1 . This predicts a secular
variation for the fine-structure constant of ( ˙α/α) ' +5×10 −11 year −1 . Finally, Teller [17]
proposed a model wherein β, ε remain constant, α ∼ (ln t) −1 , γ ∼ t −1 , δ ∼ t −1 , leading
to a predicted variation of ( ˙α/α) ' −5 × 10 −13 year −1 . An excellent review summarising
these models may be found in [23], however, it should be noted that modern observations
can now exclude all of these hypotheses by at least three orders of magnitude.
Dicke [18] in objecting to Dirac’s LNH employed a method of argument that has subsequently
become known as the Weak Anthropic Principle (WAP). Dicke argues that the
famous large number coincidences are not random but conditioned by biological factors.
Using simple models of stellar evolution, one may derive a characteristic minimum timescale
necessary for the eventual production of creatures (physicists) capable of pondering
such big questions (a few billion years). Dicke also invokes Mach’s Principle, the philosophical
supposition that one may only measure acceleration relative to distant stars.
Although it should be noted that there are many interpretations of how Mach’s principle
might fit in with the context of physics, almost as many as there are physicists arguing
about Mach’s principle. The Machian assumption is employed to argue for a minimum
mass required to obtain the gravitational coupling constant observed. Subsequently, it is
shown that the need for observers to exist is sufficient to obviate the conspiracy of the
LNH and any special relationships attached therein.
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The Anthropic Principle was subsequently expanded and formalised by Carter [24], leading
to the Final Anthropic Principle (FAP) of Barrow & Tipler [8]: Intelligent informationprocessing
must come into existence in the Universe, and, once it comes into existence, it
will never die out. This is essentially a summary of what is to the best of our knowledge the
greatest most complicated technical accomplishment of the Universe, namely the evolution
of intelligent, sentient beings (the majority of mankind if one excludes children’s TV
presenters). No anthropomorphic conceit is intended, it is merely a statement of fact that
the conditions of the Universe required to allow the evolution of complex multi-cellular
life forms are much more stringent the range of parameters permitted for the formation
of, for example, a neutron star. Indeed, it is also not unreasonable to state that the
human brain is the most complicated physical structure that we are presently aware of.
Consequently, from an Anthropic and empirical point of view, the necessity for a set
of physical laws suitable to produce a properly functioning human brain is the supreme
determining factor for the nature of the actual laws of physics, regardless of how hopeless
our present knowledge of the brain and nervous system may be, from a fundamental point
of view.
An example of the application of the Anthropic Principle with regard to the constancy
ofphysicsisthenucleosynthesisof 12 C [25]. The carbon level at 7.65 MeV lies only 0.3
MeV higher than the sum of masses of three α-particles, resulting in resonantly enhanced
cross-section of the reaction 3α → 12 C. The 8 Be nucleus is unstable, prohibiting significant
production of carbon via two body α+ 8 Be collisions. Without the 7.65 MeV resonance
the three-body formation rate would not be fast enough in comparison with the competing
reaction α+ 12 C→ 16 O and consequently there would not be enough carbon to create life
(carbon being required since only it supports a rich enough chemistry to facilitate life).
This process depends on the strong interaction and to a smaller extent the electromagnetic
force. Consequently one may develop a crude test for the variation of α based on the
maximum permitted variation that will not overtly disturb the carbon nucleosynthesis,
[9] suggests a maximum variation of about 5% for (∆α/α), corresponding to an epoch of
order 10 10 yr ( ˙α/α ≤ 4 × 10 −12 /yr).
The dependence of our existence on so many ”coincidences” (see [8]) have resulted in the
Anthropic Principle have become strongly tied with the many worlds interpretations of
quantum mechanics and some of the consequences of inflationary theories of the big bang
model. Both of these answer can the question of why we have such a special Universe by
allowing an infinite number of possible Universes to exist, for example, the currently more
favourable inflation theories, allow the inflationary field to be seeded many times, creating
many alternative Universes, within which the laws of physics need not be the same [26],
and between which a grand form of Darwinian natural selection operates (i.e. Universes
leading to closed spaces rapidly collapse upon themselves and blink out of existence).
Indeed Barrow [8] sees nothing wrong and no conflict with Occam’s razor by invoking the
notion that all logically possible realities must exist. However, since at present there is no
means by which these ideas might be verified by experiment these discussions are more
suited to the domain of metaphysics and philosophy.
3 MeasuringtheConstancyoftheConstants
The measurements employed to search for variation of the constants can be broadly
classified into three regimes, local/laboratory tests, geophysical tests and astrophysical/cosmological
tests. It should be noted that all of the results quoted, unless stated
otherwise, assume a monotonic variation for the constant in question, the limiting variations
make no account for the possibility of oscillatory behaviour, which if it existed
would require careful and numerous measurements to render tractable. Unfortunately,
the assumption of monotonicity is an inevitable necessity of having one’s measurements
limited by the availability of appropriate test systems, prohibiting the arbitrary selection
of measured data points. Most of the measurements are associated with the fine-structure
constant thus it is first prudent to more properly clarify the nature of this number.
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3.1 The fine-structure variable?
3.2 Local Measurements
3.2.1 Atomic Clocks
The modern interpretation of the fine-structure constant, within the framework of Quantum
Electrodynamics, defines α as the coupling constant for the electromagnetic force.
To the limits of current knowledge, any electromagnetic phenomena may be described
in terms of powers of α [27]. Quantum electrodynamics also sheds light (so to speak)
onthetruenatureofα; the conventional value of approximately 137 −1 actually represents
the low energy (or long range) non-relativistic limit. In QED an electron can emit
virtual photons, which can subsequently emit virtual electron-positron pairs (e + , e − ), resulting
in the ”bare” electron charge becoming screened by the polarisation arising from
the attraction of the virtual positrons and repulsion of the virtual electrons. Thus, the
conventional value of α, being proportional to e 2 ,maybedefined as the square of an
effective charge screened by vacuum polarization and seen from an infinite distance [28].
Processes occurring at shorter distances corresponding to higher energies can result in
partial penetration of the vacuum polarization charge screen, resulting in a larger value
of α. For example, at an energy of about 91 GeV, corresponding to the mass of the Z
boson, α (m Z ) is approximately 129 −1 [25]. Given this modern context, there would seem
to be nothing particularly special about the value 137 −1 in any numerological sense.
The well understood variation of α with energy is an entirely separate issue to any cosmological
variations of the zero-energy asymptotic value, and it is this value that is of
interest for most of the tests employed as a measure of the constancy of the physical laws.
An energy scale of 100 GeV corresponds to a distance scale of 10 −18 m, and is far beyond
the domain of the processes involved in the measurements for constancy of α. Atamore
modest energy scale of 1 keV, typical of the ionised atomic processes employed in some
astrophysical measurements of atomic spectra, the change in α from the zero energy limit
(∆α (1 keV) = α (1 keV)−α (0)) is predicted to be less than 1 part in 10 14 [29], given that
the best measurements of α is accurate to 4ppb [30], especially since these measurements
occur at an even lower energy scale (an energy scale of 10 eV gives rise to a deviation of
less than 1 in 10 18 ). Thus, the variation of α with energy is not an important consideration
for the processes involved.
The fine structure constant has a value assigned by the 1986 CODATA least squares adjustment
produced a value for α of 7.29735308(33) × 10 −3 , a relative uncertainty of 0.045
ppm [31]. However, the most precise measurement, using the anomalous magnetic moment
of the electron and positron, which measure the resonant interaction of an electron
with the electromagnetic modes of the surrounding microwave cavity, obtain a precision
an order of magnitude better than this [30]. Figure 2, illustrates the results from several
different systems for measuring α (Mhfs-muon hyperfine structure, h/m n -atom interferometer,
acJ&γ p -A.C. Josephson Effect, q.Hall-quantum hall effect, a e -Electron anomalous
magnetic moment) the systematic deviations are believed at present to be attributable to
untraced experimental differences and not indicative of new physics.
By making single observations of α in a given experimental system limits one to measuring
α stability to the precision of the experiment (constant systematics may be neglected), in
all cases a result worse that 1 part in 10 9 . However, a method based on the comparison
of different atomic maser systems (essentially atomic clocks) has been developed that can
achieve a much better test. The work of Prestage et al. [32] compares the hyperfine
transition of hydrogen with a hyperfine transition in Hg + . To the lowest order in α and
m e /m p the hydrogen hfs is the splitting used as the clock transition in the H maser,
a s = 8 3 α2 g p (m e /m p )R ∞ c,whereR ∞ c is the Rydberg constant in frequency units. The
theory for the hyperfine splitting in alkali atoms, A alkali , is less well developed than that of
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Figure 2: Values of the fine-structure constant determined by several means.
3.2.2 Atom interferometers
hydrogen, however the authors determined theoretical predictions for the clock transition
frequencies to the 1% level, this is sufficient to allow for a very accurate test.
ThetransitioninHg + exhibits relativistic corrections and a Casimir correction factor,
which for an S 1/2 state electron is given by F rel (αZ) =3 £ λ ¡ 4λ 2 − 1 ¢¤ −1
,whereλ =
h
1 − (αZ) 2i 1/2
,thusFrel is a strong function of α for high Z nuclei, for Hg F rel =2.26. A
time variation in α induces a change in the frequency of an H maser relative to the Hg +
maser:
µ
d
dt ln Aalkali
= α d µ 1
a hydrogen dα ln (F dα
rel) . (7)
α dt
Assuming that all other quantities remain constant (maximal variation in α). A measurement
over 140 days was made, establishing a limit of 2.1(0.8) × 10 −16 /day ( ˙α/α ≤
3.7 × 10 −14 /yr) between the two long term stable clocks [32].
Atomic clocks have also been used to verify the local position invariance (LPI) principle
embodied in the Einstein equivalence principle [33]. In this case a ”null” gravitational
redshift experiment is conducted, the rate of a magnesium frequency standard is compared
with that of a caesium reference clock, searching for a dependence on the solar gravitational
potential during a period of 430 days (since the Earth’s orbital motion is an ellipse it
follows that the Earth will move between a maximum and minimum gravitational potential
relative to the sun). The measured variation enables the establishment of a relative
frequency variation of 7 × 10 −4 .
A new approach to the determination of α utilizes the atomic recoil from the absorption
of photons [34]. Basically speaking, two pulses are applied orthogonal to an atom moving
along the +z direction, the atom absorbs a π pulse of laser beam propagating in the
+x direction tuned to resonance, conservation of momentum dictates that the atom’s
velocity changed by }k/m, where m is the mass of the atom. The atom is then deexcited
by another π pulse, this time propagating in the opposite direction, and the
laser beam stimulates the release of a photon of momentum −}k. The excitation and deexcitation
frequencies are separated by ∆ω =2}k 2 /m, thusmeasurementsof∆ω and the
photon wavelength determine }/m. Atom interferometry improves on this by exploiting
the wave-like nature of matter, replacing each π pulse with two π/2 pulses facilitates
coherent superposition and interference (assuming the apparatus is capable of maintaining
a long enough coherence length) between beam paths. The formation a double-atom
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interferometer aligned along the z-axis and the resulting Ramsey spectroscopy interference
patterns can be used to determine 2}k 2 /m, given by the displacement between the sets
of Ramsey fringes [30]. Subsequently, a relation to the fine-structure constant may be
constructed by writing α as [35]:
µ µ µ 2R∞ m h
α =
(8)
c m e m
where the last term is measured by atom interferometry and R ∞ is the Rydberg constant,
determined independently by laser resonance spectroscopy of the 1S − 2S transition in
hydrogen [36]. The current experimental resolution reported results in ∆α/α ∼ 10 −8 for
two hours of integration time, this is competitive with the other methods for measuring
α although this is not yet suitable for measuring the variation of α (as yet, differential
schemes have not been developed). Significant improvements are expected with the introduction
of a Bose condensate source (moving towards atom laser interferometry) and
using improved laser pulse sequences to increase the momentum separation. Atom interferometers
are also capable of extremely sensitive gravitational measurements, which could
eventually have an impact on verifications of the equivalence principle. The most recent
result reported by Peters et al. [37] achieves an absolute certainty for the measurement
of the acceleration of gravity of ∆g/g ≈ 3 × 10 −9 .
3.3 Geophysical Measurements
3.3.1 The Oklo Phenomenom
A long-lived nucleus with a decay-rate λ dependent on α canbeusedinconjunctionwith
the abundance of the nucleus in nature to measure the variation of α over the period since
the creation of that nucleus[38]. Dyson [23]demonstrates how the radioactive half-life of
alpha and beta decay. Given evidence that the 238 U decay rate has remained within 20%
of its present value over the last 2×10 9 years one may infer a limit of ˙α/α ≤ 2 × 10 −13 /yr.
In 1972 scientists at the French Atomic Energy Commision made the discovery that spontaneous
chain reactions had occured within the Oklo uranium deposits in the distant past.
Oklo is located in Gabon, West Africa and it was during routine isotopic abundance analysis
of a natural uranium sample that a very small, but statistically significant shift in
the isotope ratio of 235 U was measured. It was found to be 0.7171 ± 0.0007% compared to
the range considered normal for natural uranium of 0.7202 ± 0.0006% [39]. Consequential
to this measurement a whole series of abnormally low results were obtained, including
an incredibly low content of 0.2% and 0.4% not being untypical. A hypothesis of a prehistoric
natural nuclear reactor was proposed and rapidly gained convincing support in
theformofwellcorrelated 235 U fission fragments from and isotope abundances being
detected in the ore. For example, the Oklo site exhibited highly anomalous abundance
of 40.0% Samarium-147, against the world average of 15.0%[40]. Closer inspection of
the site (including ore samples at a resolution of 5cm in some places) revealed it to be
truly remarkable, including features of natural neutron moderation from the surrounding
groundwater water and carbon strata.
Extrapolation of the radioactive isotopes and decay paths backwards in time enabled
simulation of when the reactor would had been operational. The result was approximately
1.8 billion years ago (obtained using the U/Pb method [41]), with criticality being achieved
1.84 ± 0.07 × 10 9 years ago, and an operational lifespan of 2.29 ± 0.7 × 10 5 years [42]. The
total energy produced by the reactor is estimated at 1.5 × 10 4 MW year, giving a mean
power output of a rather meagre 25 kW.
The relevance of the Oklo phenomenon to determining constancy of physics was first
calculated by Shlyakhter [43] in 1976. The method relies on an anomalous resonance for
thermal neutron capture in 149 Sm which has a very high cross-section of 42000 barns
10

compared to the more usual ∼100 barns for this region of the periodic table [40]. Usually
the single neutron resonances lie at energies of ∼ few eVs with widths of ∼ 100 meV.
However,inthecaseof 149 Sm the resonance lies at 98 meV and has a width of 63 meV,
giving rise to an enormous enhancement of the thermal neutron capture cross-section.
Modelling a neutron temperature of around 1000 K (approx. 80 meV) and combining
the 147 Sm and 148 Sm relative abundances with the 149 Sm abundance leads to a limit
for the position of the neutron resonance to within 10 meV of its present energy. The
resonance occurs in an optical potential ∼ 50 MeV deep, implying that the single neutron
coupling constant to the nucleus has changed by less than one part in 5×10 9 over the
past ∼ 2 × 10 9 years [43]. Translating this as the variation of the coupling constant for
the strong interaction gives rise to a maximal varying rate of | ˙α strong /α strong | ≤ 10 −19 /yr,
assuming all other quantities remain the same. If the entire variation is attributed to a
change in the fine-structure constant, which represents at most ∼ 5% of the interaction
one abtains a bound of the secular variation of α : | ˙α/α| ≤ 5 × 10 −17 /yr [44]. Similarly,
the weak interaction yields the result ¯ ˙β/β
¯ ≤ 10 −12 /yr. One cannot rule out a paradigm
in which the coupling constants conspire to vary in such a way so as to cancel each other
out, although it seems unlikely. This caveat could be resolved if a similar analysis could
be conducted on some other isotopic abundance, where one would expect the ratio of the
strong, electromagnetic and weak effects to be different, unfortunately, as yet no such
opportunity has presented itself.
3.4 Astrophysical Measurements
Astrophysical measurements offer the possibility of being able to measure the constancy
of physics over a very large range in space-time. However, one cannot, as yet, ”poke
a neutron star”, so to speak, meaning that the astrophysicist must employ ”detective”
skills to extract a desired measurement. Viking lander data, stellar evolution, Anthropic
arguments, molecular spectra, primordial nucleosynthesis and even the cosmic microwave
background along with the more conventional methods of atomic absorption and emission
spectroscopy, make up the menagerie of methods employed to determine constancy of
physical laws on an astrophysical basis.
3.4.1 The expansion of the universe
In 1929 Edwin Hubble [45] published results demonstrating a linear redshift to distance
relation for distant extra-galactic nebulae,
v = H 0 d. (9)
Where v is the recessional velocity of an object at distance d, and is calculated on the
assumption that the redshift observed for spectral features measured in the distant object
arise due to the Doppler effect:
z = λ obs − λ rest
λ rest
= ∆λ
λ rest
;define redshift parameter z, (10)
The relationship between the observed wavelength λ obs and the rest (or laboratory) wavelength
λ rest , assuming radial motion is,
λ obs = λ rest
s
1+v/c
1 − v/c , (11)
Hence, one may write the redshift in terms of the recessional velocity,
s
1+v/c
z =
1 − v/c − 1 (12)
11

Figure 3: The three basic scenarios arising from the Friedmann model.
The constant of proportionality H 0 is the Hubble constant and conventionally has units of
km s −1 Mpc −1 ,thedifficulty of measuring astronomical distances accurately means that
there is still considerable uncertainty in this value, the most recent measurement from the
Hubble Space Telescope program [46] is 70 km s −1 Mpc −1 and is accurate to about 10%.
(9) is now referred to as the Hubble law. The observations of Edwin Hubble suggested
an expanding Universe with all distant galaxies moving away from each other (nearby
galaxies have to contend with local gravitational interactions which can overcome the bulk
expansion motion), providing observational evidence in support of a class of cosmological
models derived by Friedmann from Einstein’s field equations of general relativity [47].
Friedmann-type models still represent the best candidates for describing the evolution of
our Universe. Given a reasonable expanding (flat) Universe model 1/H 0 is approximately
the age of the Universe [48].
The model derived by Friedmann describes the dynamics of a variable R, the radius of
the Universe:
Ṙ 2 = C R + 1 3 ΛR2 − k, (13)
where C = 8 3 πR3 ρ is the total mass of the Universe multiplied by two (ρ being the mean
mass density), Λ is the cosmological constant (see below) and k = −1, 0, +1, a description
of the different cases resulting from this model can be found in [49]. Crudely speaking
all the Friedmann models describe three kinds of behaviour, i) a universe which expands
indefinitely, ii) a universe that reaches a point of maximum expansion and then collapses
upon itself and iii) a flat universe of critical density (expressed by the normalised parameter
Ω = 1) where the expansion of the universe converges on some asymptotic maximum.
There is much cosmological evidence to suggest that our own Universe conforms closely
(perhaps exactly according to some inflationary models) to a flattened space of Ω =1,
although the observable matter only amounts to Ω =0.3 at most [50]. Note that more
properly one should write Ω m + Ω Λ ,whereΩ m is the mean mass density and Ω Λ the
cosmological constant. A flat geometry conforms to Ω m + Ω Λ =1.
An interesting diversion to consider is the length scale over which the expansion of the
Universe governed. The Friedmann model above only describes the gross behaviour of the
size of the Universe, it says nothing about the expansion of the contents. The point is of
12

3.4.2 Astronomical spectroscopy
somerelevancetotheissueofthevariationofconstants,sincetheinterpretationofthe
expanding universe dictates that space itself is given to expand. If this were to happen
on, say, an atomic scale one would expect the energy spectrum to change, a result which
might be mis-interpreted as some other phenomenon. The first solution was proposed by
Einstein and Straus [51], who showed that non-expanding ’vacuoles’ could exist around
a massive body such as a star inside a surrounding, expanding space-time. However, it
has recently been shown that this model relies critically the assumption of spherically
symmetric mass distribution within a vacuole [52, 53], which is somewhat unreasonable
given the distribution of matter observed in reality.
An alternative speculative proposal has recently been published [54], wherein a simple
model of the hydrogen atom is considered in a flat, but expanding, space-time. The
author then considered the change in the orbit radius of a classical hydrogen atom as the
surrounding universe expands. The lack of a quantum theory of gravity prohibits a full
quantum formulation, instead the calculation was performed classically by formulating the
Maxwell equations in the background space-time and solving for the first Bohr orbit. The
problem is relevant since the electromagnetic interaction is responsible for holding together
the structure of extended bodies (such as ourselves). The calculation compared how the
radius of electron’s trajectory changes after a single orbit around the proton compared
withthedistanceaneutralparticlewouldmoveinthesametimeinterval(giventhe
expanding geometry). Using the size of the hydrogen atom and the current age of the
Universe, the author was able to show that the change in radius of the electron orbit is
10 −67 times smaller than the change in position of the neutral particle, a reassuring result
although from a rather speculative formulation.
It is clear that measuring the redshift to a distant body together with the application
of the Hubble law facilitates a some measurement of the laws of physics at a different
point both in space and time. Indeed, the power of the best optical and radio telescopes
available is such that one can observe very distant faint objects over an epoch to about
5% the present age of the Universe.
The energy levels of a hydrogenic atom, neglecting the effects of QED and recoil are given
by the exact expressions:
E n,j = m e c 2 f (n, j) (14a)
f (n, j) =
" ¡ ¢ # − 1
Zα
2 2
1+
(n − β) 2 (14b)
q ¡j
β = j + 1 2 − ¢
+
1 2 2
2 − (Zα)
(14c)
where n is the principle quantum number and j the total orbital angular momentum.
In principle, a measurement of an atomic transition between a pair of known statesE n,j →
E n 0 ,j0, along with knowledge of the appropriate parameters, should be sufficient to obtain
a measurement of the fine-structure constant, including QED and recoil effects merely
complicates the expression to which α is matched, but makes no qualitative difference.
However, astrophysical observations are accompanied by some red shift due to the expansion
of the universe, this will cause an artificially deviant value of α to arise if not
accounted for properly. For small Z the following expansion is reasonable, it is the nonlinear
terms in (Zα) 2 that can be exploited to yield higher order terms may be exploited
to yield measurements of α independent of any overall bulk redshift
13

µ
f (n, j) = 1− (Zα)2
2n 2 − (Zα)4 1
2n 3 j + 1 2
Ã
− (Zα)6
8n 3
1
¡
j +
1
2
3
¢ 3
+
n ¡ j + 1 2
− 3
4n
¢ 2
+ 5
2n 3 − 6
n ¡ ¢
2 j + 1 2
!
+ ... (15)
To simultaneously measure a red shift and a spectral line one requires two identified
spectral lines, however, just as a redshift can be interpreted as a variation of one’s constants
the opposite is also true. It would be inappropriate to explicitly calculate the redshift for
an observed line in a system for use with an alternative line. Savedoff [55] suggested
the possibility of using differential measurements of redshifted lines to construct a proper
framework capable of yielding an explicit presentation of desired quantities. Following the
framework of Tubbs [11] one may define the frame (x ∗ ,t ∗ )atlinesourceandtheframe
(0,t 0 ) at the point of measurement, thus,
(v a /v b ) 0
=(v a /v b ) ∗
(16)
is an equivalent statement of the assumption that the photon is infinite in time, or any
effects (such as tired life) involve only a linear relationship between distance and photon
energy. Comparing this ratio with the equivalent laboratory standard (v a /v b ) lab
leads
to some measurement implicitly comparing the values of various constants. Depending
on the line system in question and any assumptions made regarding which constants are
under consideration for maximal variation determines the explicit form of the constant
measurement made. If v a corresponds to a transition between fine-structure states and v b
arises from a resonance transition then the measurement is approximately
¡
(v a /v b ) ¢ 0 α
2
∗
≈
(v a /v b ) lab
(α 2 , (17)
) lab
since it is sufficient given the smallness of α variation (less than 1 part in 10 4 )toconsider
(15) to (Zα) 4 only as this is the dominant nonlinear (in (Zα) 2 ) term. Alternately, if v a
corresponds to the 21cm transition, so beloved of radio astronomers, between the hyperfine
states in the ground level of hydrogen instead [11]
¡
α 2 g p m e /m p
¢
(v a /v b ) 0
≡ 1+z opt
∗
≈
(v a /v b ) lab
1+z 21 (α 2 . (18)
g p m e /m p ) lab
where z opt and z 21 are the measured redshifts of the optical and 21cm line respectively.
Ideal astrophysical sources to conduct these tests should present a suitable selection of
candidate spectral lines and be observable over a large range of distances to ensure a
test spanning a wide range in space-time. Early observations by Savedoff [55] made use
of doublets in N II and Ne III and the hydrogen 21 cm line measured in Cygnus A, an
extra-galactic nebulae. Assuming the redshift to be due to relative motion enabled the
following restrictions ¡
α
2 ¢ Cyg. A
(α 2 ) lab
=1.0036 ± 0.0032, (19)
from the optical observations, and using the 21cm line
(g p m e /m p ) Cyg. A
(g p m e /m p ) lab
=0.9967 ± 0.0032. (20)
Given that Cygnus A is 3 × 10 8 light years away one may derive the limits | ˙α/α| ≤
6 × 10 −12 /yr or |ẋ/x| ≤ 1 × 10 −12 /yr where x ≡ α 2 g p m e /m p .
14

Figure 4: HIRES spectra of the QSO APM 08279 + S255
3.4.3 Quasar Absorption Line Systems
Quasar absorption line system observations that make up the bulk of measurements.
Absorbing gas in front of QSOs, provides ideal objects for astrophysical examination of
the invariance of dimensionless constants, due to the apparently high intrinsic brightness
of quasars and the ready accessibility of these objects at medium redshift, consequently
providing a selection of sources spanning a wide range of space-time, although the relative
paucity of high redshift (z & 3) objects limits measurement to the earlier parts of the
history of the universe. Gamow [22] originally suggested that the redshift of distant
quasars could be due to a variation in α. However, observations of the fine-structure
splittings of O III and Ne III emissions of the QSO’s 3C 47 and 3C 147 by Bahcall
and Salpeter [56] were able to rule out any bulk variations of α capable of making a
significant contribution to the redshift. Subsequently Bahcall and Schmidt [57] made
QSO observations to a redshift of about 0.2, corresponding to approximately 2×10 9 years
ago producing a limit of | ˙α/α| ≤ 5 × 10 −13 /yr, which is valid over a timescale comparable
to the Oklo observations.
Modern instruments, such as the HIRES echelle spectrograph fittedtotheKeck-Itenmeter
telescope [58], have greatly extended both the span of observations and the stringency
of the resulting tests. Figure 4 shows the spectra obtained with HIRES of the typical
QSO absorption line systems used in these analyses [59].
Onemaymakeadditionaluseofhydrogenfurthertothe21cmline,thepresenceof
molecular hydrogen makes the electronic, vibrational and rotational energies of the H 2
molecule available for analysis. These levels exhibit different dependencies on the electronto-proton
mass ratio (µ ≡ m e /m p ) which for the purposes of the measurements required
may be adequately expressed in the Born-Oppenheimer approximation, as used by Cowie
and Songalia [60]
E = E el + µ 1/2 E vib + µE rot . (21)
The energy shift in any vibration-rotation transition j in the Lyman series therefore has
15

the form
∆E j = a el + b j µ 1/2 + c j µ (22)
=⇒ (∆E j − ∆E i ) ∼ b ji µ 1/2 + c ji µ, (23)
where i is a similar transition. To the lowest order, a change in µ results in a change in
∆E j − ∆E i such that
δµ
µ ≈ δυ µ 2∆Ei
, (24)
c ∆E i − ∆E j
where δυ is the mean offset compared to the laboratory value of the energy difference
between the two sets of lines when represented as a velocity difference. Potekhin et al. [61]
have obtained the strongest observational constraint to date for the direct measurement
of the variation of µ, | ˙µ/µ| < 1.5 × 10 −14 /yr. This used the Cerro-Tololo Inter-American
Observatory (CTIO) 4m telescope and was based on an observation of the QSO PKS 0528-
250 at redshift z =2.811, corresponding to ∼ 10 10 years ago (the exact figure depends
upon the model used for the expansion of the Universe, or rather to which model our
Universe conforms). Cowie and Songalia [60] have made various observations with HIRES
& Keck. Measurement of alkaline Si IV doublets in 4 high redshift QSOs consistent
with the formalism of (17) yielded | ˙α/α| ≤ 1 × 10 −14 /yr, it is interesting to note that
the precision of this is experiment currently hampered by the accuracy of the laboratory
standard. Use of the hyperfine 21cm/optical resonance method (18) of Q1331 + 170,
in which the redshift of the hyperfine absorption is known to very high precision (z 21 =
1.77642±2×10 −5 ) combined with a recent observation of C 0 absorption and fine-structure
(resulting in z opt =1.77644 ± 2 × 10 −5 )provide
δx
x =7× 10−6 ± 1.1 × 10 −5 . (25)
Which corresponds to a 95% confidence range of
µ µ dx 1
=(−2.2, 4.2) × 10 −15 /yr. (26)
dz x
Given that x ≡ α 2 g p (m e /m p ) one may derive the implied maximal variations of the
component dimensionless constants, giving | ˙µ/µ| < 4.2 × 10 −15 /yr and | ˙α/α| ≤ 2.1 ×
10 −15 /yr.
Most recently, observations by Webb et al. [62] have pointed to the tantalising possibility
of a statistically significant variation of α in an intermediate epoch between z ' 1and
z ' 1.6, this does not conflict with the Oklo result which is consistent with a redshift of
z ' 0.15. This work is notable in that it considers a large number of observations over a
range of redshifts z ' 0.6to1.6, facilitating the possibility of accounting for nonmonotonic
variations of α. The observations also made use of a new inter-line comparison method
[63, 64]. Under this formalism the energy equation for a transition from the ground state
within a particular multiplet, observed at some redshift z, isgivenby
" µαz # 2 µ 2 µ 4
E = E 0 + Q 1 Z 2 − 1 + K 1 (LS) Z 2 αz
+ K 2 (LS) 2 Z 4 αz
. (27)
α 0 α 0 α 0
Here E 0 and Q 1 describe the position of the configuration centre, K1 and K2 describe the
level splitting within one configuration, L is the total orbital angular momentum, S is the
total electron spin. α 0 is the laboratory value of α and α z is the astrophysical value of
thesourceatredshiftz, this expression may be re-arranged to give
" µαz #
" 2 µαz # 4
E z = E z=0 +[Q 1 + K 1 (LS)] Z 2 − 1 + K 2 (LS) 2 Z 4 − 1 . (28)
α 0 α 0
16

Figure 5: QSO illuminated absorption lines observed by Webb et al.
The method makes use of two alkali doublets in two different species, one of considerably
higher Z than the other, measured against each other. The relativistic corrections will
be much larger in the higher Z species, leading to much larger induced changes in the
fine-structure with changes in α. The relative shifts are argued by the author’s to be
substantially greater when using a lighter Z species to ”anchor” against the larger higher
Z shifts, than observations of a single doublet alone. The observations measured the
Mg 7II 2796Å/2803Å doublet compared with up to five Fe II transitions (2344, 2374,
2383, 2587, 2600 Å), the experiment was conducted on HIRES at Keck. Figure 6 shows
the results obtained from this survey. The results taken over the whole redshift range
(0.6

Figure 6: Results of the fine-structure constant survey by Webb et al.
3.4.4 High z Measurements
The astrophysical data measure variation over an epoch to approximately 5% of the present
age of the Universe, however, since the observations are of large evolved structures places
a limit upon the ultimate epoch over which one may test the constancy of the constants.
This is particularly distressing given that it is during the phases of the very early universe
that one expects to observe the largest possible variations in the constants.
By modelling primordial nucleosynthesis for various constant variation regimes and comparison
with the observed primordial abundances one may determine a limit on the size
of constant variation applicable just a few minutes after the big bang. Such an analysis
has been conducted by Kolb et al. [65] requiring the yield of primordial 4 He to be
within acceptable limits. The derived result gives | ˙α/α| ≤ 1.5 × 10 −14 h −1 /yr, where h is
a dimensionless parameter of order unity to represent the propagated uncertainty of the
Hubble constant in the derived figure (in principle all of the quotes for variation limits
should be accompanied by this parameter). Primordial nucleosynthesis represents the
most primeval measurement yet devised to test the variation of the constants.
The cosmic microwave background is also due to be recruited into providing cosmological
limitations for the invariance of physics. Kaplinghat et al. [66] argue that any time variationinthefine-structure
constant will alter the ionization history of the universe, which
in turn will change the pattern of cosmic microwave background fluctuations, expected
to be dominated by the affected change in the redshift of recombination due to a shift in
the binding energy of hydrogen. This represents a measurement equivalent to a redshift
of z ∼ 1000, the recombination era some 10 5 years after the big bang [40]. It is proposed
that the forthcoming MAP (Microwave Anisotropy explorer from NASA, scheduled for
launch in 2000) and PLANCK (named after Max Planck, from ESA, due for launch in
2007) satellite experiments could reach sensitivities of |∆α/α| ∼ 10 −2 − 10 −3 .
18

4 Cosmological Implications
4.1 The Accelerating Universe
The cosmological constant, Λ, was introduced by Einstein as a mathematically admissible
parameter in the gravitational field equations. The Newtonian formulation of gravity represents
the weak-field, non-relativistic asymptote of general relativity where the curvature
of space is essentially zero. The introduction of the cosmological constant results in a
modified Poisson’s equation for the Newtonian gravitational potential, ϕ [49],
5 2 ϕ + Λ =4πGρ. (32)
Solution of this equation leads to a central harmonic potential involving the cosmological
constant Λ, a positive value implies a repulsive central potential and a negative value is
correspondingly attractive. Thus, if one places a mass at the centre of the coordinate system
and invoke a test mass at a position r¯i, one will observe a linear term in the resulting
force acting on that body, Force ∝ − ¡ 1/ri 2 + Λr i¢
(r¯i/r i ). The cosmological constant
therefore determines the basic property of empty space, or more appropriately, the basic
property of the vacuum. Since general relativity relates the curvature of space time to the
energy density (e.g. mass energy) contained therein, an extremely important nexus may
exist between the quantum vacuum energy density and the cosmological constant (e.g.
Λ ≡ 8πGρ vac ). In fact the vacuum is the subject of considerable speculative theoretical
work, the Soviet physicist A. D. Sakharov [67] in 1968 suggested that gravity was not a
fundamental force, but rather an induced effect associated with the zero-point fluctuations
of the vacuum, in a similar fashion to the Casimir and van-der-Waals forces. Such work
has been extended by Puthoff [68], although the lack of development would seem to suggest
a theoretical dead-end for this avenue to reconcile gravity with quantum mechanics.
However, given that vacuum energy and zero-point fluctuations are real phenomena, as
confirmed by Casimir force experiments [69], and given that relativity associates gravitation
with energy density it would seem most reasonable to expect some fundamental link
to exist between the two, although possibly not within the current theoretical framework
and most probably not leading to the fantastic ”infinite energy horizons” of free-energy
evangelists [70, 71].
The equations of general relativity naturally give rise to a dynamical formulation for the
behaviour of space-time [72]. This conflicted with the prevailing philosophical view of
the early 20 th century which favoured a static everlasting Universe. Einstein envisaged Λ
assuming some critical value to exactly cancel the dynamical behaviour (for example, a
collapsing universe could be balanced by a positive, and therefore repulsive, Λ). However,
in 1930 Eddington showed that such a solution could not satisfy a stable equilibrium for
a static universe as the inhomogeneous distribution of mass in the universe would lead
to either expansion or collapse [73]. Prior to this Friedmann had produced his set of
cosmological models describing an expanding Universe (embodied by (13)) and Hubble
had produced his observational evidence to back this up. As the support for a static
universe collapsed so did the need for a cosmological constant, and it was confined to the
dustbin. Einstein was to describe the introduction of the cosmological constant as ”his
greatest blunder”, among other things he missed out on using his theory to make one of
the greatest predictions of all time, the expanding universe! An excellent description of
the epic of the cosmological constant may be found in [74], a less detailed online colloquial
reference is [75].
Many observations, such as distant supernovae, galactic density and clustering with respect
to distance, and gravitational lensing events all seemed to confirm the very great
flatness of space-time, and thus point to a minute cosmological constant. ”Flatness” implies
that the bulk geometry (i.e. not in the locality of massive bodies) of space-time is
Euclidean, by making predictions for the density of various events such as those mentioned
based upon a homogeneous Universe projected to earlier redshifts one may construct a test
19

for such a flatness. For example, a positive cosmological constant, causing accelerating
expansion, would lead to a relative paucity of events.
One of the most exciting and controversial discoveries of 1998 was the evidence that the
expansion of the universe may be accelerating [76, 50], the two teams made observations
oftheapparentbrightnessofhighredshifttypeIasupernovae(SNeIa). SNeIaare
believed to be excellent ”standard candles” [48], that is the physics of these events is
such that their intrinsic brightness does not vary much. The results of the observations
by the supernovae team found them to be unexpectedly dim, which may be interpreted
as evidence for a positive cosmological constant. Without a cosmological constant the
observed light curves were as much as 9 standard deviations away from the expected
values, a value that reduced significantly upon inclusion of the extra parameter (exactly
how much is difficult to say without other independent constraints as the measurement
used is only capable of restricting models based upon (13) to a set of values). Recently
Zehavi and Dekel [77] have produced evidence for a positive constant from an independent
constraint; a set of simulated models based upon deviations of galaxy velocities from a
smooth universal expansion yielding a lower bound to the mass density (since the lower
the mass density the smoother the universe and smaller the galactic motion defects are
expected to be). After decades of null measurements there finally exists the possibility of
a non-zero value for Λ.
The inclusion of a discussion on the cosmological constant is not included here to lend
moral support to a quest currently bogged down by null results, or to lend credence to the
daydreams of the more outlandish propulsion scientists hoping one day to build ”warpdrives”
[78] (an extremely exciting possibility unfortunately still stuck firmly in the realm
of science fiction). Rather, the issue of a possible non-zero value for the cosmological constant
has some relevance to the issue of the variation of the constants. Possible alternative
explanations for the anomalously faint supernovae data include a novel dust model, capable
of attenuation without the spectral reddening associated with normal dust [79] or
possible bias in the SNe Ia calibration data due to a bias introduced by the nonlinear
response of film (used for the early observations and still part of the SNe Ia calibration set
due to the relative paucity of SNe Ia events), as compared with the excellent linearity of
CCDs [80], leading to a skewed dataset of SNe Ia models, although the authors claim that
some corrections to account for this have been made. However, it is the work of Albrecht
& Magueijo [81] that proposes time varying speed of light as a solution to the problem.
A higher value of c in the early Universe could reproduce the anomalous faintness of the
distant Supernovae by making them appear closer than they really are (a higher value
of c would entail a smaller redshift for the same Hubble recessional velocity). Indeed,
Albrecht goes further to suggest a varying speed of light model as being capable of solving
both the flatness problem (that Ω = 1) and the horizon problem (that the early universe
is extremely smooth and homogeneous) which are the two key observations for which the
theory of inflation was proposed. Barrow & Magueijo make a similar proposition [82] in
addition to suggesting instead a variation of the electronic charge e. Both of these variations
will be expressed in α and therefore could, in principal, be measured. As yet neither
of the authors have presented quantitative predictions which can be compared against
the measured variation rates. An earlier model proposed by [83] suggested a variation
of the form c ≈ t −1/3 , however this theory simultaneously required both G and Planck’s
constant, h to follow convenient gauge relations, and the results of which can now be
excluded by modern observations.
The work associated with the cosmological constant represents an exciting area where
experiment is perhaps leading the theory, with most of the models being proposed being
phenomenological in form and thus not meshed with the minutiae of the main body of
physics. However, there also exists unification theories which approach the subject from a
much more fundamental point of view. Principally these theories are higher dimensional
theories of the Kaluza-Klein type (for example, [84]), or superstring theories (for example,
[85]). Both kinds of theory are relatively generic and allow for enormous latitude in the
20

4.2 Superstring theories
4.3 Kaluza-Klein Theories
variations expressed within them.
Superstring theories, considered a real candidate able to unify gravity with all the other
interactions at the Planck energy scale (E Planck ≡ p }c 5 /G ' 1.2 × 10 19 GeV), the
superstring theory reduces to classical general relativity at lower energies. However, in
addition to the normal tensorial graviton, all versions of the theory predict the existence
of a scalar dilaton partner [27]. As a result all the coupling constants and masses of
elementary particles being dependent upon the dilaton scalar field, φ and thus rendered
space and time dependent, thus yielding a small, but non-zero, observable consequences
including variations of the fine-structure constant and other gauge coupling constants.
˙α
α v kH 0 (φ − φ m ) 2 , (33)
where k is the main parameter determining the efficiency of the cosmological relaxation
of the dilaton field φ towardsit’sextremevalueφ m . Since equation (33) depends on
cosmological evolution of the dilaton field (the dynamics of which have not yet been
determined) it is possible that non-monotonous and non-homogeneous variations may be
expressed into α.
Damour & Polyakov [85] attempt to derive a numerical prediction for variations of α
and deviations from the equivalence principle, resulting in the expression connecting the
variation of the fundamental constants:
Ġ
G = Θk ˙α α ' 3332.5k ˙α α , (34)
where Θ =2 ¡ ln ¡ Λ s /mc 2¢¢ 2
is a numerical coefficient, Λs ' 5 × 10 17 GeV is a string
mass scale & m =1.661 × 10 −24 g is the atomic mass unit. G and α oscillate for k ≥ 1,
whereas for k ¿ 1 leads to monotonic evolution. For k ≥ 1 models a measured bound
on Ġ/G gives a bound on ˙α/α or vice versa, thus the result of Webb et al. implies
¯
that ¯Ġ/G¯¯¯ ¿ 1 × 10 −10 /yr. Forthcoming satellite experiments that should measure
gravitational acceleration, a, to an accuracy of ∆α/α ∼ 10 −17 , should enable a reversal
of this test and instead place a much more stringent limit on α (within this superstring
framework), or combining independent observations of Ġ/G and ˙α/α to impose some limit
on k. Indeed, this is perhaps a specific place where one can identify the analysis of the
variation of constants as being able to have a direct impact upon restricting and guiding
a new theory of physics.
Like string theories, Kaluza-Klein theories invoke a higher number of dimensions than
the familiar three spatial plus one time. The extra dimensions are perceived to be compactified
into some small manifold so that they do not express themselves strongly until
one reaches the Planck scale. Variation of the coupling constants in the 3+1 dimensional
world is dependent upon the evolution of the additional dimensions within which
the associated true fundamental couple constant. One may write this variation more
explicitly by invoking a geometric mean scale factor R (t, x) (whereitisbelievedthat
R now ∼ l Planck ≡ p }G/c 3 =1.6 × 10 −33 cm), of the cosmological evolution of the additional
D spatial dimensions [86], thus for Kaluza-Klein theories the fine, α, weak,α w ,
and strong, α s , gauge coupling constants observed in four dimensional space-time would
evolve as:
α ∝ α w ∝ α s ∝ R −2 . (35)
Similarly for the popular ten-dimensional superstring theories:
α ∝ α w ∝ α s ∝ R −D , (36)
21

assuming that each of the forces is expressed in all D additional dimensions. Thus enabling
the possibility of placing some restrictions on R, which could consequently be worked into
the cosmology of a given theory. The primordial nucleosynthesis of Kolb et al. [65] places
a restriction that the size of the extra dimensions in Kaluza-Klein models must have
been within 1% of their current value(∆R/R < 0.01), whilst those for ten dimensional
superstring models must have been within 0.5% of their current value (∆R/R < 0.005).
5 Summary
The recent observations of possible variation of the fine-structure constant, combined with
the re-ignited interest in the cosmological constant has added fresh impetus to the issue
of the improving the measurements of the variation of constants and hence improving the
rigidity of the important Cosmological Principle. The current most stringent restrictions
placed on the constants for various regimes are summarised below.
Source Restriction Regime
H-Maser/Hg + -Maser [32] | ˙α/α| ≤ 3.7 × 10 −14 /yr local, ˜140days
Oklo [44] | ˙α/α| ≤ 5 × 10 −17 /yr 1.8 × 10 9 yr, z ' .15
Q1331+170, 21cm [60] | ˙α/α| ≤ 2.1 × 10 −15 /yr z =1.8
Q1331+170, 21cm [60] | ˙µ/µ| < 4.2 × 10 −15 /yr z =1.8
Multiple QSO [62] | ˙α/α| ≤ (−2.2 ± 5.1) × 10 −16 /yr 0.6

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