Varying alpha from N-body Simulations

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Draft version September 9, 2010
Preprint typeset using LATEX style emulateapj v. 11/10/09
VARYING ALPHA FROM N -BODY SIMULATIONS
Baojiu Li1,2, David F. Mota3 and John D. Barrow2
1DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
2Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge CB3 0HA, UK and
3Institute of Theoretical Astrophysics, University of Oslo, 0315 Oslo, Norway
Draft version September 9, 2010
ABSTRACT
We have studied the Bekenstein-Sandvik-Barrow-Magueijo (BSBM) model for the spatial and tem-
poral variations of the fine structure constant, α, with the aid of full N -body simulations which
explicitly and self-consistently solve for the scalar field driving the α-evolution . We focus on the
scalar field (or equivalently α) inside the dark matter halos and find that the profile of the scalar field
is essentially independent of the BSBM model parameter. This means that given the density profile of
an isolated halo and the background value of the scalar field, we can accurately determine the scalar
field perturbation in that halo. We also derive an analytic expression for the scalar-field perturbation
using the Navarro-Frenk-White halo profile, and show that it agrees well with numerical results, at
least for isolated halos; for non-isolated halos this prediction differs from numerical result by a (nearly)
constant offset which depends on the environment of the halo.
1. INTRODUCTION
During the past ten years there has been continual in-
terest in the possible time and space variation of fun-
damental constants of Nature. This interest was stimu-
lated by observations of quasar absorption spectra that
are consistent with a slow increase of the fine structure
“constant”, α, over cosmological time scale (Webb et al.
1999, 2001). Experimental and observational efforts to
constrain the level of any possible time variation in fun-
damental constants have a history that pre-dates mod-
ern theories about how they might vary (for overviews
see Barrow (2002, 2005), Uzan (2003) and Olive & Qian
(2004)). Until quite recently all the observational stud-
ies found no evidence for any variations. However, high-
quality data from a number of astronomical observations
have provided evidence that at least two of these con-
stants, the fine structure constant: α = e2/ ~c, and
the electron-proton mass ratio: µ = me/mp might have
varied slightly over cosmological time. Using a data set
of 128 KECK-HIRES quasar absorption systems at red-
shifts 0.5 < z < 3, and a new many-multiplet (MM)
analysis of the line separations between many pairs of
atomic species possessing relativistic corrections to their
fine structure, Webb et al. (1999, 2001) found the ob-
served absorption spectra to be consistent with a shift in
the value of fine structure constant, α, between those
redshifts and the present day, with ∆α/α ≡ α(z) −
α(0)/α(0) = −0.57± 0.10× 10−5. A smaller study of 23
VLT-UVES absorption systems between 0.4 ≤ z ≤ 2.3 by
Chand et al. (2004) and Siranand et al. (2004) initially
found ∆α/α = −0.6±0.6×10−6 by using an approximate
version of the full MM technique. However, the reanaly-
sis of the same data set by Murphy, Webb & Flambaum
(2007, 2008) using the full and unbiased MM method
increased the uncertainties and suggested a revised fig-
ure of ∆α/α = −0.64 ± 0.36 × 10−5 for the same data.
Email address: b.li@damtp.cam.ac.uk
Email address: d.f.mota@astro.uio.no
Email address: j.d.barrow@damtp.cam.ac.uk
These investigations relied on the statistical gain from
large samples of quasar absorption spectra. Most re-
cently, this observational programme has been extended
to both hemispheres of the sky using KECK and VLT sam-
ples of 153 absorption systems by Webb et al. (2010)
and finds evidence consistent with an increase in α the
northern sky but consistent with a slow decrease in α
with time in the south. When combined these overlap-
ping data sets are well fitted by a dipole with ∆α/α0 =
(1.10±0.25)×10−6r cos θ, at measurement position r¯ (rel-
ative to Earth at r¯ = 0) where θ is the angle between the
measurement and the axis of the dipole. These observa-
tions suggest that we should develop an understanding
of the spatial as well as the temporal consequences of
varying constants.
By contrast to these statistical searches for varying α,
probes of the electron-proton mass ration can use sin-
gle objects effectively. Reinhold et al. (2006) have found
a 3.5σ indication of a variation in the electron-proton
mass ratio µ ≡ me/mpr over the last 12Gyrs: ∆µ/µ =
(−24.4 ± 5.9) × 10−6 from H2 absorption in a different
object at z = 2.8. However, Murphy et al. (2008) have
exploited the µ sensitivity of ammonia inversion transi-
tions (Flambaum & Kozlov 2007) compared to rotational
transitions of CO, HCN, and HCO+ in the direction of
the quasar B0218+357 at z = 0.68466 to yield a result
that is consistent with no variation in µ when systematic
errors are more fully accounted for: ∆µ/µ = (+0.74 ±
0.47stat±0.76system)×10−6, corresponding to a time vari-
ation of µ˙/µ = (−1.2 ± 0.8stat ± 1.2system) × 10−16yr−1
in the best-fit ΛCDM cosmology.
Any variation of α and µ today could also be con-
strained by direct laboratory searches. These are per-
formed by comparing clocks based on different atomic
frequency standards over a period of months or years.
Until very recently, the most stringent constraints on
the temporal variation in α arose by combining mea-
surements of the frequencies of Sr (Blatt et al. 2008),
Hg+ (Fortier et al. 2006), Yb+ (Peik et al. 2004), and
H (Fischer et. al. 2004) relative to Caesium: α˙/α =

2(−3.3 ± 3.0) × 10−16 yr−1. Cingoz et al. (2007) also re-
cently reported a less stringent limit of α˙/α = −(2.7 ±
2.6)×10−15 yr−1; however, if the systematics can be fully
understood, an ultimate sensitivity of 10−18 yr−1 is pos-
sible with their method Nguyen et al. (2004). If a lin-
ear variation in α is assumed with cosmic time then the
Webb et al. (1999, 2001) quasar measurements equate to
α˙/α = (6.4±1.4)×10−16 yr−1. If the variation is due to a
light scalar field described by a theory like that of Beken-
stein, Sandvik, Barrow andMagueijo (BSBM from here on)
(Bekenstein 1982; Sandvik, Barrow & Magueijo 2002),
then the rate of change in the constants is exponen-
tially damped during the recent dark-energy-dominated
era of accelerated expansion, and one typically predicts
α˙/α = 1.1 ± 0.3 × 10−16 yr−1 from the Murphy et al.
data, which is not ruled out by the atomic clock con-
straints mentioned above. For comparison, the Oklo nat-
ural reactor constraints, which reflect the need for the
Sm149 + n → Sm150 + γ neutron capture resonance at
97.3meV to have been present 1.8 − 2Gyr (z = 0.15)
ago, as first pointed out by Shlyakhter (1976), are cur-
rently (Fujii et al. 2000) ∆α/α = (−0.8± 1.0)× 10−8 or
(8.8±0.7)×10−8 (because of the double-valued character
of the neutron capture cross-section with reactor temper-
ature) and (Lamoureaux 2004) ∆α/α > 4.5× 10−8 (6σ)
when the non-thermal neutron spectrum is taken into
account. However, there remain significant environmen-
tal uncertainties regarding the reactor’s early history and
the deductions of bounds on constants. The quoted Oklo
constraints on α apply only when all other constants are
held to be fixed. If the quark masses to vary relative to
the QCD scale, the ability of Oklo to constrain variations
in α is greatly reduced (Flambaum 2007).
Recently, Rosenband et al Rosenband et al. (2008)
measured the ratio of aluminium and mercury single-
ion optical clock frequencies, fAl+/fHg+, over a period of
about a year. From these measurements, the linear rate
of change in this ratio was found to be (−5.3 ± 7.9) ×
10−17 yr−1. These measurements provide the strongest
limit yet on any temporal drift in the value of α: α˙/α =
(−1.6 ± 2.3) × 10−17 yr−1. This limit is strong enough
to conflict with theoretical explanations of the change in
α reported by Webb et al. (1999, 2001) in terms of the
slow variation of an effectively massless scalar field, even
allowing for the damping by cosmological acceleration,
unless there is a significant new physical effect that slows
the locally observed effects of changing α on cosmological
scales. Also, one expects inhomogeneous changes in α in
scenarios where variations in α are induced by a heavy
scalar field with a mass (mφ). One would expect varia-
tions in α on cosmological scales to differ from those on
scales below the field’s Jeans length, which is O(1/mφ)
(for a detailed analysis of global-local coupling of vari-
ations in constants see Clifton, Mota & Barrow (2005);
Mota & Barrow (2004a,b); Shaw & Barrow (2006c,a,b);
Olive & Pospelov (2008)).
It has also been noticed that if ‘constants’ such as α
or µ could vary, then in addition to a slow temporal
drift one would also expect to see an annual modulation
in their values. In many varying constant theories, the
Sun perturbs the values of the constants by a factor
roughly proportional to the Sun’s Newtonian gravita-
tional potential (Magueijo, Barrow & Sandvik 2002;
Barrow, Sandvik & Magueijo 2002; Barrow & Mota
2002) (the contribution from the Earth’s gravitational
potential is about 14 times smaller than that of the Sun’s
at the Earth’s surface). Hence the ‘constants’ depend
on the distance from the Sun. Since the Earth’s orbit
around the Sun has a small ellipticity, the distance, r,
between the Earth and Sun fluctuates annually, reaching
a maximum at aphelion around the beginning of July
and a minimum at perihelion in early January. It was
shown that in many varying constant models, the values
of the constants measured here on Earth oscillate in a
similar seasonal manner. Moreover, in many cases, this
seasonal fluctuation is predicted to dominate over any
linear temporal drift (Barrow & Shaw 2007).
In this paper we will study the BSBM model for the
spatial and temporal variations of the fine structure con-
stant α, with the aid of full N -body simulations which
explicitly and self-consistently solve for the scalar field
driving α-evolution. We focus on the trend of the scalar
field (or equivalently, α) inside the dark-matter halos,
and find that the profile of the scalar-field fluctuation
is essentially independent of the BSBM model parameter.
This means that given the density profile of an isolated
halo and the background value of the scalar field, we
can accurately determine the scalar field perturbation in
that halo. We also derive an analytic expression for the
scalar-field perturbation using the Navarro-Frenk-White
(NFW) halo profile, and show that it agrees well with
numerical results, at least for isolated halos. For non-
isolated halos this exact prediction differs from numeri-
cal result by a (nearly) constant offset, depending on the
environment of this halo.
A brief outline of the remaining of this paper is as fol-
lows: in § 2 we list the minimal set of necessary equations
to understand the physics, and briefly describe our algo-
rithm; § 3 displays the main numerical results and we
then discuss and conclude in § 4.
2. EQUATIONS AND ANALYSIS
This section lists the equations which will be used in
the N -body simulations for the BSBM varying-α model
Barrow & Mota (2003); Barrow, Magueijo & Sandvik
(2002); Barrow, Sandvik & Magueijo (2002);
Sandvik, Barrow & Magueijo (2002); Mota & Barrow
(2004b,a).
2.1. The Basic Equations
The Lagrangian density for the BSBM model could be
written as
L= 12
[R
κ −∇
aϕ∇aϕ
]
− Lm − e−2
√
κϕLEM − Lr , (1)
where R is the Ricci scalar, κ = 8πG with G being the
gravitational constant, ϕ is the scalar field; Lm,LEM,Lr
represent respectively the Lagrangian densities for dust,
electromagnetic field (including photons) and other radi-
ation (such as neutrinos). The coupling function between
the scalar field and the electromagnetic field in the BSBM
model is e−2
√
κϕ where √κ is added so that √κϕ ≡ ψ is
dimensionless. In the simplest version of the model there
is no potential for the scalar field.
The dust Lagrangian for a point particle with mass m0

3is
Lm(y)=−
m0√−g δ(y − x0)
√
gabx˙a0 x˙b0, (2)
where y is the general coordinate and x0 is the coordinate
of the centre of the particle. From this equation we derive
the corresponding energy-momentum tensor:
T abm =
m0√−g δ(y − x0)x˙
a
0 x˙b0. (3)
Also, because gabx˙a0x˙b0 ≡ gabuaub = 1, in which ua is the
four-velocity of the dark-matter particle centred at x0,
the Lagrangian can be rewritten as
Lm(y)=−
m0√−g δ(y − x0). (4)
This result will be used below.
Eq. (3) is just the energy-momentum tensor for a single
matter particle. For a fluid consisting of many particles,
the energy-momentum tensor will be
T abm =
1
V
∫
V
d4y√−g m0√−g δ(y − x0)x˙
a
0 x˙b0
= ρCDMuaub, (5)
where V denotes a volume which is microscopically large
but macroscopically small, and we have extended the 3-
dimensional δ function to a 4-dimensional one by adding
a time component. Here, ua is the averaged four-velocity
of the matter fluid.
Using
T ab=− 2√−g
δ (√−gL)
δgab
, (6)
it is straightforward to show that the energy-momentum
tensor for the scalar field is
Tϕab=∇aϕ∇bϕ− 12g
ab∇cϕ∇cϕ. (7)
Therefore the total energy-momentum tensor is
Tab=∇aϕ∇bϕ−
1
2gab∇cϕ∇
cϕ
+Tmab + T rab + e−2
√
κϕTEMab (8)
where Tmab = ρmuaub, T rab is the energy-momentum ten-
sor for radiation fields except photons, and TEMab for pho-
tons. The Einstein equations are
Gab=κTab (9)
in which Gab = Rab− 12gabR is the Einstein tensor. Notethat due to the extra coupling between the scalar field,
ϕ, and the electromagnetic field, the energy-momentum
tensors for either will no longer be separately conserved,
but instead we have
∇bT abEM=2
√
κ
(
gabLEM + T abEM
)
∇bϕ. (10)
However, the total energy-momentum tensor is certainly
conserved.
Meanwhile, the scalar field equation of motion is
�ϕ=−2
√
κe−2
√
κϕLEM, (11)
where � ≡ ∇a∇a. This equation governs the time evo-
lution and spatial configuration of the scalar field.
Eqs. (8, 9, 10, 11) summarize all the physics needed
for the following analysis. However, when making use of
them we should also specify the form of the electromag-
netic matter. For example, if it is a pure electromagnetic
field (photons), then we have LEM = 12
(
E2 −B2
)
= 0
in which E,B stand for the electric and magnetic fields.
Thus from the time component of Eq. (10) we obtain the
(background) evolution equation for photon density as
ρ˙r + 4Hρr=2ψ˙ρr (12)
where remember that ψ = √κϕ.
It then seems that, by the same reason, the right
hand side of Eq. (11) also vanishes, leaving the scalar
field unsourced. This may not be true however, as non-
relativistic matter could also contribute to LEM and thus
TEMab . For example, in baryonic matter LEM ≈ E2/2, and
for neutrons and protons this electromagnetic contribu-
tion to the total mass can be of order 10−4; in supercon-
ducting cosmic strings LEM ≈ −B2/2 where ρEM ≈ B2/2
so that |LEM/ρEM| ∼ 1. In the BSBM model, in order to
simplify the situation, it is assumed that LEM/ρm = ζ
where ζ is a constant with a modulus between 0 and
≈ 1, either positive or negative, and ρm is the density for
non-relativistic matter.
Thus the scalar field equation gets sourced by a term
proportional to ζ:
�ϕ=−2
√
κζe−2
√
κϕρm. (13)
Since the part of LEM which affects the scalar field is a
constituent of the non-relativistic matter and is presum-
ably moving with the matter particles, we could combine
Eq. (10) and the conservation equation for the dust mat-
ter (no including electromagnetic contribution) to write
a new conservation equation for the particle:
∇bT abm+EM=2
√
κ
(
gabLEM + T abEM
)
∇bϕ. (14)
Although we have assumed above that LEM = ζρm, we
still lack a knowledge about T abEM, whose relation to LEM
could be complicated. Here for simplicity we assume that
T abEM = −ζρmuaub. Then it is easy to find that the time
component of this equation reads
ρ˙m+EM + 3Hρm+EM=0 (15)
while the i-th spatial component of it gives the following
(modified) geodesic equation
x¨i0 + Γiabx˙a0x˙b0 =2ζ
√
κ
(
gib − uiub
)
∇bϕ. (16)
where x0 is the coordinate of the centre of a parti-
cle, and the right hand side represents a fifth force on
the particle Li & Zhao (2009, 2010). The assumption
T abEM = −ζρmuaub might seem unappealing, but as we
shall see below, because |ζ| ≪ 1, the fifth force is much
weaker than gravity and thus has negligible effects in the
clustering of matter in any case; ultimately it is only the
BSBM assumption LEM = ζρm that is important in theo-
retical predictions of the spatial and temporal variations
of α, given that α = e2ψ e
2
0
c~ .
2.2. Analytical Approximation

4The scalar field equation of motion, which controls the
dynamics of the scalar field ϕ, is generally complicated,
because it depends nonlinearly on ϕ, which both evolves
in time and fluctuates in space. Fortunately, for the ma-
jority of applications the scalar field potential and cou-
pling function are not nonlinear enough to give the scalar
field a very heavy mass so as to make it fluctuate strongly.
The nice thing about this is that in certain places of
the equations one may then forget the scalar field per-
turbation and simplify these equations accordingly. In
Li & Barrow (2010), for example, it is shown that such
simplification is very good an approximation (see how-
ever Li & Zhao (2009, 2010) for an opposite extreme for
which the scalar field potential is very nonlinear so that
such simplification does not work).
In the BSBM model, there is no scalar field potential
and the coupling function is close to linear if √κ|ϕ| ≪ 1
(which is the case for our interested parameter space
(Barrow, Magueijo & Sandvik 2002)). We therefore ex-
pect the fluctuation of ϕ to be very weak and assume
that √κ|δϕ| ≪ √κ|ϕ| ≪ 1 (which we shall confirm below
using numerical simulations). Under such an assumption
the Poisson equation could be written as
∇2xΦ=4πGa3ρm
[
1 + ζe−2
√
κ(ϕ¯+δϕ)
]
−4πGa3ρ¯m
[
1 + ζe−2
√
κϕ¯
]
≈ 4πGa3 (ρm − ρ¯m) (17)
where ϕ¯ the background value of ϕ and δϕ its perturba-
tion; ρm, ρ¯m are respectively the local and background
matter density; Φ is the gravitational potential and a
the cosmic scale factor; ∇x is the derivative with respect
to the comoving coordinate x. To obtain Eq. (17) we have
used the fact that in the BSBM model |ζe−2ϕ¯| ≪ 1. Note
that Eq. (17) clearly indicates that the gravitational po-
tential is essentially not influenced by the scalar field ϕ.
For the scalar field equation of motion Eq. (13), be-
cause the background part (which has no spatial depen-
dence) can be solved easily, so we subtract that from the
full equation to obtain an equation of motion for δϕ only
(remember that ϕ = ϕ¯+ δϕ). Furthermore, we drop the
time derivative terms of δϕ as they are small compared
with the spatial gradients (i.e., work in the quasi-static
limit). The final equation for δϕ then becomes
∇2x
(
a
√
κδϕ
)
=2ζκ
[
ρme−2
√
κ(ϕ¯+δϕ) − ρ¯me−2
√
κϕ¯
]
a3
≈ 16πGa3ζ (ρm − ρ¯m) (18)
where we have used κ = 8πG and √κ|δϕ| ≪ √κ|ϕ| ≪ 1.
Comparing Eqs. (17, 18), it is evident that the source
terms are the same up to a constant coefficient 4ζ. Con-
sequently, we shall have
a
√
κδϕ(x)≈ 4ζΦ(x). (19)
Note that this equation, together with the geodesic equa-
tion Eq. (16), implies that the magnitude of the fifth force
(force due to exchange of scalar field quanta between par-
ticles) |f | satisfies
|f | ∼ ζ|~∇(a
√
κδϕ)| ∼ 4ζ2|~∇Φ|. (20)
Therefore the ratio between the magnitudes of the fifth
force and gravity is of order ζ2 . 10−12−10−8 ≪ 1. This
Fig. 1.— The time evolution of ϕ¯ for the four models considered
in this work, with ζ = −2 × 10−6 (solid curve), 2 × 10−6 (dotted
curve), −5×10−6 (dashed curve) and 5×10−6 (dash-dotted curve)
respectively. The horizontal axis is the cosmic scale factor a(t) and
the vertical axis plots √κϕ¯ in unit of 10−4.
implies that the fifth force cannot influence the structure
formation significantly.
3. SIMULATIONS AND RESULTS
3.1. ϕ Perturbation vs. Gravitational Potential
To study in details the behaviour of the scalar field and
hence the fine structure constant α in the BSBM model,
we have performed N -body simulations for four different
models, with ζ = ±2× 10−6 and ±5× 10−6 respectively.
The physical parameters we adopt in all simulations are
as follows: the present-day dark-energy fractional energy
density ΩDE = 0.743 and Ωm = ΩCDM + ΩB = 0.257,
H0 = 71.9 km/s/Mpc, ns = 0.963, σ8 = 0.761. These are
in accordance with the concordance cosmological model
preferred by current data sets. Our simulation box has a
size of 64h−1 Mpc, in which h = H0/(100 km/s/Mpc).
In all those simulations, the mass resolution is 1.114 ×
109h−1 M⊙; the particle number is 2563; the domain
grid (i.e., the coarsest grid which covers the whole simu-
lation box) has 1283 equal-sized cubic cells and the finest
refined grids have 16384 cells on each side, corresponding
to a force resolution of ∼ 12h−1 kpc. Detailed descrip-
tion about the N -body simulation technique and code
for the (coupled) scalar field models could be found in
Li & Zhao (2009, 2010) and will not be presented here.
Because the BSBM model (with the parameter ζ con-
strained by data) involves a very weak coupling between
matter and the scalar field ϕ, the presence of the latter
and its coupling have negligible influences on the back-
ground (ΛCDM) cosmology, although the opposite is not
true since there is no scalar field potential and thus the
dynamics of ϕ is controlled entirely by the coupling. In
our simulations, we compute the full background cos-
mology and evolution of ϕ on a predefined time grid us-
ing MAPLE, and interpolate to obtain the corresponding
quantities which are needed in N -body simulations. De-
tails of this procedure can be found in the Appendix C
of Li & Barrow (2010), and Fig. 1 shows the background
evolution of √κϕ for the 4 models considered here. Ob-
viously the condition √κϕ¯≪ 1 is satisfied, justifying the
approximation we used above to derive Eq. (19).

5Fig. 2.— (Color Online) Scatter plot of the scalar field perturbation, a√κδϕ, (vertical axis, in unit of 10−11) vesus gravitational potential
Φ (horizontal axis) in code units. The black solid line is the analytical approximation aδϕ ∝ Φ; each green dot represents the corresponding
result measured in a cell from a particular slice that is randomly selected from the simulation box. The columns are for four models with
ζ = ±2,±5 × 10−6 as shown above each panel, and the rows are for three different output times a = 0.3, 0.5, 1.0, also shown above each
panel. Note that the slopes for the solid lines differ because of the different values of ζ.
One nice thing about the BSBM model is its sim-
plicity, and it turns out that the background evolu-
tion of ϕ (and thus α) in different cosmic epochs
can be well described by some analytical formulae
Barrow, Magueijo & Sandvik (2002). Therefore in the
present study we shall mainly focus on the spatial varia-
tion of ϕ and α (especially in virialized halos).
As mentioned above, because there is no potential for
the scalar field ϕ and because √κϕ ≪ 1 for our choices
of ζ, so the scalar field equation of motion (in the quasi-
static limit) for δϕ and the Poisson equation share the
same source up to a constant coefficient, and therefore
we expect aδϕ ∝ Φ across the whole space. This is what
we have found in Li & Barrow (2010) for a different cou-
pled scalar field model where the scalar field potential is
negligible. Indeed, this could serve as a test of the scalar
field solver in our N -body simulation code.
To check that our code does recover the analytical ap-
proximation, we have plotted in Fig. 2 the comparison of
the scalar field perturbation a√κδϕ and gravitational po-
tential Φ from a slice of the simulation box. As indicated
by this figure, the agreement between the numerical re-
sults (green dots) and analytical approximation (black
solid line) is remarkably good, implying that the numeri-
cal code works well. Therefore to a high precision we can
assume that a√κδϕ ∝ Φ everywhere, a fact which shall
be used below to obtain an analytical expression of δϕ
in dark matter halos.
3.2. Spatial variation of ϕ in Halos
In the standard picture, the galaxies where observers
live generally locate inside the dark matter halos, which
to the simplest approximation are just spherical clusters
of matter with a universal NFW (Navarro et al. 1996)
density profile.
We are certainly interested in the (possible) variation
of α inside the halo we reside in. For example, there
has been a great deal of analytical work on how signifi-
cantly the local value of α could deviate from its cosmo-
logical counterpart (Mota & Barrow 2004b,a; Jacobson
1999; Wetterich 2003; Shaw & Barrow 2006c,a,b). Also,
if the spatial variation of α is strong enough, then it
might have impact on our observation of the spectra for
the stars from our Galaxy and other galaxies.
From our simulation output, it is easy to identify the
dark matter halos (Knebe & Gibson 2004; Li & Barrow
2010). Now what we want to do here is to measure the
quantity √κδϕ as a function of distance R to the halo