[meteorite-list] What If Copernicus Was Wrong?
JoshuaTreeMuseum
joshuatreemuseum at embarqmail.com
Thu Sep 10 15:55:31 EDT 2009
Dark Energy v. The Void: What if
Copernicus was Wrong?
Living in a Void: Testing the Copernican Principle with Distant Supernovae
Timothy Clifton,? Pedro G. Ferreira, and Kate Land
Oxford Astrophysics, Physics, DWB, Keble Road, Oxford, OX1 3RH, UK
A fundamental presupposition of modern cosmology is the Copernican
Principle; that we are not
in a central, or otherwise special region of the Universe. Studies of Type
Ia supernovae, together
with the Copernican Principle, have led to the inference that the Universe
is accelerating in its
expansion. The usual explanation for this is that there must exist a 'Dark
Energy', to drive the
acceleration. Alternatively, it could be the case that the Copernican
Principle is invalid, and that
the data has been interpreted within an inappropriate theoretical
frame-work. If we were to live in
a special place in the Universe, near the centre of a void where the local
matter density is low, then
the supernovae observations could be accounted for without the addition of
dark energy. We show
that the local redshift dependence of the luminosity distance can be used as
a clear discriminant
between these two paradigms. Future surveys of Type Ia supernovae that focus
on a redshift range
of 0.1 ? 0.4 will be ideally suited to test this hypothesis, and hence to
observationally determine
the validity of the Copernican Principle on new scales, as well as probing
the degree to which dark
energy must be considered a necessary ingredient in the Universe.
The concordance model of the Universe combines two
fundamental assumptions. The first is that space-time
is dynamical, obeying Einstein's Equations. The second
is the 'Cosmological Principle', that the Universe is then
homogeneous and isotropic on large scales - a generalisation
of the Copernican Principle that "the Earth is
not in a central, specially favored position" [1]. As a result
of these two assumptions we can use the Freidmann-
Robertson-Walker (FRW) metric to describe the geometry
of the Universe in terms of a single function, the scale
factor a(t), which obeys
H2 =
8G
3
?
k
a2 (1)
where H ? ? a/a is the Hubble rate, is the energy density,
k is the (constant) curvature of space, and overdots
denote time derivatives. The scale factor can then be
determined by observing the 'luminosity distance' of astrophysical
objects. At small z ? a0/a(t)?1 this is given
by
H0DL ? cz +
1
2
(1 ? q0)cz2, (2)
where q ? ?¨aa/ ?a2 is the deceleration rate, and subscript
0 denotes the value of a quantity today. Recent measurements
of (z, DL) using high redshift, Type Ia Supernovae
(SNe) have indicated that q0 < 0, i.e. the Universe
is accelerating in its expansion [2, 3]. Accelerating expansion
is possible in an FRW universe if a fraction of
is in the form of a smoothly distributed and gravitationally
repulsive exotic substance, often referred to as
Dark Energy [4]. The existence of such an unusual substance
is unexpected, and requires previously unimagined
amounts of fine-tuning in order to reproduce the observations.
Nonetheless, dark energy has been incorporated
into the standard cosmological model, known as CDM.
Electronic address: tclifton at astro.ox.ac.uk
An alternative to admitting the existence of dark energy
is to review the postulates that necessitate its introduction.
In particular, it has been proposed that the SNe
observations could be accounted for without dark energy
if our local environment were emptier than the surrounding
Universe, i.e. if we were to live in a void [5, 6, 7].
This explanation for the apparent acceleration does not
invoke any exotic substances, extra dimensions, or modifications
to gravity - but it does require a rejection of the
Copernican Principle. We would be required to live near
the centre of a spherically symmetric under-density, on
a scale of the same order of magnitude as the observable
Universe. Such a situation would have profound consequences
for the interpretation of all cosmological observations,
and would ultimately mean that we could not
infer the properties of the Universe at large from what
we observe locally.
Within the standard inflationary cosmological model
the probability of large, deep voids occurring is extremely
small. However, it can be argued that the centre of a
large underdensity is the most likely place for observers
to find themselves [8]. In this case, finding ourselves in
the centre of a giant void would violate the Copernican
principle, that we are not in a special place, but it may
not violate the Principle ofMediocrity, that we are a 'typical'
set of observers. Regardless of what we consider the
a priori likelihood of such structures to be, we find that
it should be possible for observers at their centre to be
able to observationally distinguish themselves from their
counterparts in FRW universes. Living in a void leads to
a distinctive observational signature that, while broadly
similar to CDM, differs qualitatively in its details. This
gives us a simple test of a fundamental principle of modern
cosmology, as well as allowing us to subject a possible
explanation for the observed acceleration to experimental
scrutiny.
Some efforts have gone into identifying the observational
signatures that could result from living in a void.
The cosmic microwave background (CMB) supplies us
with the tight constraint that we must be within 15 Mpc
of the center of the void [9]. There have also been some
attempts at calculating predictions for CMB anisotropies
and large scale-structure [10, 11, 12], as well as the kinematic
Sunyaev-Zeldovich effect [13].
General Relativity allows a simple description of timedependent,
spherical symmetric universes: the Lema^itre-
Tolman-Bondi (LTB) models [14, 15, 16], whose lineelement
is
ds2 = ?dt2 +
a2
2(t, r)dr2
1 ? k(r)r2 + a2
1(t, r)r2d2, (3)
where a2 = (ra1)', and primes denote r derivatives . The
old FRW scale factor, a, has now been replaced by two
new scale factors, a1 and a2, describing expansion in the
directions tangential and normal to the surfaces of spherical
symmetry. These new scale factors are functions of
time, t, and distance, r, from the centre of symmetry,
and obey a generalization of the usual Friedman equation,
(1), such that
? a1
a12
=
8G
3
~ ?
k(r)
a2
1
. (4)
Here ~ = m(r)/a3
1, and is related to the physical energy
density by = ~+~'ra1/3a2. The two free functions, k(r)
and m(r), correspond to the curvature of space, and the
distribution of gravitating mass in that space. We choose
initial conditions such that the curvature is asymptotically
flat with a negative perturbation near the origin,
and so that the gravitational mass is evenly distributed.
As the space-time evolves the energy density in the vicinity
of the curvature perturbation is then dispersed, and
a void forms. Observations of distant astro-physical objects
in this space-time obey a distance-redshift relation
DL = (1 + z)2rEa1(tE, rE) (5)
where
1 + z = expZ rE
0
( ?a1r)'
?1 ? kr2
dr, (6)
and subscript E denotes the value of a quantity at the
moment the observed photon was emitted. This expression
is modified from equation (2), allowing for the possibility
of apparent acceleration without dark energy.
We find that the form of the void's curvature profile
is of great importance for the observations made by astronomers
at its centre. In Figure 1 we plot some simple
curvature profiles, together with the corresponding distance
moduli as functions of redshift (distance modulus,
dm, is defined as the observable magnitude of an astrophysical
object, minus the magnitude such an object
would have at the same redshift in an empty, homogeneous
Milne universe). It is clear from Figure 1 that
for the void models there is a strong correlation between
k(r) and dm; at low redshifts dm(z) traces the shape
of k(r). Hence, for a generic, smooth void dm starts
off with near zero slope, where it is locally very similar
to a Milne universe, it then increases at intermediate z,
and later drops off like an Einstein-de Sitter universe.
For CDM, we have dm ? ?5
2 q0z at low z, i.e. a
http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.1443v2.pdf
Phil Whitmer
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