|
This paper discusses the idea of using a
Lemaitre-Tolman (L-T) solution to model the universe. The authors fit the two
free functions in these models by matching observables with their corresponding
values in a $\Lambda$CDM FRW cosmology. This is achieved using a method devised
by Mustapha et al. The main claim of the paper is then that the space-time that
results should be described as having a ``giant hump'' in the density profile,
rather than a ``giant void''.
The mathematics in the paper all appears to be
correct (it is largely reproduced from the previous work of Mustapha et al), and
I can see no errors in the manuscript of this kind. The interpretation of the
mathematical results, however, seems less clear, and there are some issues that
need to be addressed and clarified. As the main claims of the paper involve the
interpretation of mathematical results, this seems to be an important issue. I
will outline my concerns below.
My one principle problem with this paper is the highly gauge dependent nature of
the quantities that are being discussed. The authors wish to describe their
mathematical results as a local over-density in the energy density on a
space-like hypersurface of constant time. They try to make clear at numerous
places in the text that this is different to a ``giant void'' (i.e. an
under-density in the energy density). However, these statements are gauge
dependent, and it seems to me that one could very easily describe the same
space-time as either a ``void'' or a ``hump'', simply by choosing different
gauges. This ambiguity is not discussed in the manuscript at all, and appears to
me to be of central importance to the claims they are making. As the authors
state in the manuscript, neither an under-density nor an over-density on a
space-like hypersurface is directly observable, and so I see no reason to favor
one description over the other.
Also, the authors make much of the description ``giant void'', which has been
used in numerous previous papers to describe the L-T models that have resulted
from various considerations. They appear to consider it an inaccurate
description of L-T models that can reproduce $\Lambda$CDM observations. However,
the previous authors who have used the description ``giant void'' have often
worked in a different gauge to the one that is used by the present authors. That
two different analyses lead to two different energy density profiles, in two
different gauges, is not particularly surprising. To make a meaningful
comparison the authors should transform their results into the gauge choice made
by the authors who use the description ``giant void'' (or vice versa).
What is more, any attempt at a preferred description of a quantity on a surface
of ``t=now'' requires a reasonable definition of ``now''. The most sensible
description would appear to me to be given by insisting that the interval along
a curve of a comoving fluid element should be given by a set amount, otherwise
the universe appears to have a different `age' at different spatial locations.
(The authors raise an objection in their text to describing $t-t_B$ as the age
of the Universe, due to the neglect of radiation. In terms of the L-T model
itself, however, it seems like an appropriate description). This corresponds to
a particular gauge in which $t_B=$constant, which is the case in many of the
papers who use the description ``void''. The present authors use a different
gauge, and so their hypersurface ``t=now'' has fluid elements of different ages.
Comparing the energy density in different spatial regions of the Universe with
different ages will naturally give different results to comparing regions when
they have the same age. For example, one could consider an FRW model in a gauge
in which space-like hypersurfaces have non-constant density. Voids and humps
would then appear on hypersurfaces of constant time, and whether a region is
described as a void or a hump would be sensitive to the exact choice of gauge.
This does not make either the description ``void'' or ``hump'' more appropriate
for any given spatial region, and in the more sensible constant bang time gauge
there is none of either.
I consider the issues above to be very important for the manuscript under
consideration.
I also have a number of other more minor points relating to interpretation and
presentation. In the text there are numerous issues linked to those I have
outlined above, but I will not push these points any further. Instead, I will
try and outline the other concerns that I have, in the order they appear in the
article (not necessarily in order of importance).
In the introduction the authors say ``We do not claim we are living at or near
the centre of any spherically symmetric universe.'' The rest of the paper then
goes on to describe a global L-T space-time. Such models appear to put the
observer in a very special place, contrary to the statements made in this
paragraph.
Later in the introduction the authors write ``Why, then, has this information
been overlooked and several researchers have been led astray by the frequent
claims of a giant void being implied by an L--T model? We suspect the reason
might be twofold.'' They then write about half a page on the alleged existence
of the Hubble bubble and suggest this is why previous authors have found a
``giant void'' when considering L-T models. I think this is quite unlikely. Most
of the papers that describe giant voids are the results of fitting to data
(often with a priori constraints on the initial data, as the authors rightly
point out). However, I very much doubt these initial constraints were
constructed with the intention of making a giant void. They appear to be done
for simplicity. If the data had suggested a giant hump after these initial
simplifications I expect it would have been reported by most (if not all) of
these studies. These comments therefore seem inappropriate to me.
I do not see the need for Fig.1. The L-T model in the figure is not described
anywhere that I can see, and showing that an anonymous model gives different
results to $\Lambda$CDM seems unnecessary (it doesn't seem to add anything
beyond the statements made in the text).
The authors make clear they are trying to reproduce $\Lambda$CDM observables in
an L-T model with no $\Lambda$. This is not the same thing as reconstructing the
L-T functions from observable data, however, even if $\Lambda$CDM is consistent
with all observations. This is hinted at by the authors in various places, but
not made very clear. This seems especially important as the observables the
authors consider have not yet been observed out to the redshifts they consider.
Number counts, for example, as far as I am aware are only generally considered
reliable out to redshifts considerably less than 1. Looking at Fig. 10 out to
such distances gives a very different picture to considering much larger
(unobserved) distances: In fact, it would appear to suggest a local void rather
than a hump.
In the results sections the authors implement a special procedure to deal with
the apparent horizon. I have no problems with this procedure, and see why it is
necessary. However, it breaks up the flow of the paper a little and I would
suggest to the authors that they consider putting it in an appendix instead
(this is very much an issue of style only, and I only mention it as a suggestion
to improve the flow of the text).
Figures 2 and 3 seem unnecessary. The same quantities, with different arguments,
seem to appear later in Figures 5 and 6. I don't see why they need to be
displayed twice. There are also some oddities with these figures. The x-axis is
labelled with a coordinate distance in units of proper distance. The `scale
factor' is now a function of r, so shouldn't the proper distance be a non-linear
function of r? Also, the quantities being plotted could be more usefully
displayed as -2 E/r2 and M/r3, as they would then have straightforward
interpretations in terms of FRW quantities (which is what many readers will be
interested in). Removing the r2 and r3 factors would probably make the plots
more useful also, as at present they look very much like simply r2 and r3 only.
(I, for one, would also be interested to see if the spatial curvature at the
centre of symmetry is higher or lower than the asymptotic value. Such a
rescaling of variables may make this more apparent). These comments can be
applied to relevant later plots as well.
The authors write just before the start of Section 3.2 the following: ``Finally,
as seen from Fig. 10, the current density profile does not exhibit a giant void
shape. Instead, it suggests that the universe smoothed out around us with
respect to directions is overdense in our vicinity up to Gpc-scales.'' It seems
worth pointing out that at present observables do not extend as far as the
authors have plotted these quantities. In fact, if the graph were restricted to
the region covered by current observation it would show a void rather than a
hump. The existence of the giant hump that the authors describe does not,
therefore, seem to be a consequence of current observations, but rather of their
expectation that future observations out to larger redshifts will follow the
$\Lambda$CDM prediction. This seems worth making explicit.
In the Discussion section the authors write: ``As we said earlier in this paper,
the belief that an L--T model fitted to supernova Ia observations necessarily
implies the existence of a giant void with us at the centre was created by
needlessly, arbitrarily and artificially limiting the generality of the model.''
I do not consider these choices to be needless or arbitrary, although they do
limit generality. It seems to me that the previous work that is being described
by these words often has a considerably different aim to the present paper: They
are attempting something closer to hypothesis testing (in a Bayesian sense,
often). To try and compare a model with arbitrarily many free parameters (i.e.
free function) to a model with one constant would results in the L-T models
being dismissed as highly improbable. By parameterizing the functions in some
way the number of constants to be fitted is then reduced to a workable number,
making the proposed (less general) model much more favorable. In this sense,
limiting generality is a very sensible, if restrictive, thing to do. I
understand the ambiguity in this kind of reasoning, but certainly do not
consider it needless (for the goals mentioned above). These comments could be
applied to earlier discussion, too.
Figure 20 shows a giant void model and the authors' giant hump model. I can't
see what model the giant void is referring to though. It would be helpful to
mention this in the caption and the text (if it is not already, and I can't see
it). It's probably also worth mentioning that (I expect) the giant void model
was fitted to data at low z, and not out to z=4. This would make the
(unobserved) difference with the giant hump model more understandable.
The authors go on to write: ``Thus, while dealing with an L--T (or any
inhomogeneous) model, one must forget all Robertson--Walker inspired prejudices
and expectations.'' This seems a bit too much, as FRW cosmology is very useful
for understanding lots of aspects of more general cosmologies. I would suggest
``be cautious when applying'', rather than ``forget all''.
Shortly afterwards the authors write: ``This putative opposition can then give
rise to the expectation that more, and more detailed, observations will be able
to tell us which one to reject. In truth, there is no opposition.'' This may not
be strictly true. Observations of the kSZ effect, and the growth of linear
structure may give effects in L-T models that cannot be easily reproduced by
FRW. For the kSZ effect see
arXiv:0807.1326 Title: Looking the void in the eyes - the kSZ effect in LTB
models Authors: Juan Garcia-Bellido, Troels Haugboelle
And for linear structure see
arXiv:0903.5040 Title: Perturbation Theory in Lemaitre-Tolman-Bondi Cosmology
Authors: Chris Clarkson, Timothy Clifton, Sean February.
These effects seem worth mentioning.
Later the authors write: ``Thus, if the Friedmann models, $\Lambda$CDM among
them, are considered good enough for cosmology, then the L--T models can only be
better''. This is true in terms of reproducing some observations, but for
hypothesis testing they could be disfavoured (see comments above on hypothesis
testing).
Shortly afterwards the authors then write: ``the question that should be asked
is `what limitations on the arbitrary functions in the model do our observations
impose?' rather than `which model better describes a given situation: a
homogeneous one of the FLRW family, or an inhomogeneous one?'''. In terms of
trying to reproduce $\Lambda$CDM observations with an L-T model this may be true
(although see comments above). But in terms of trying to fit the two models to
observations it seems perfectly plausible to ask which model better describes
them: The L-T model will probably usually fit better, but it is perfectly
possible that $\Lambda$CDM could in the future be shown to be inconsistent with
observations, or more favorable, in a Bayesian sense.
Finally, the last paragraph seems contrary to the rest of the paper. The paper
up until this point has been about reconstructing the L-T functions by matching
observables to their expected values in $\Lambda$CDM. If this is the programme,
then there is only going to be one result -- the model the authors have found.
In this case there seems no point comparing to any other model, as it will not
reproduce $\Lambda$CDM as well. Presuming that future observations are in
keeping with FRW, the giant hump model will fit better. Of course, future
observations may turn out to be in conflict with $\Lambda$CDM, but this does not
seem to be what this paragraph is considering.
The list of references also seems to be missing a few recent publications that
should probably be included for completeness. These are:
arXiv:0807.1326 Title: Looking the void in the eyes - the kSZ effect in LTB
models Authors: Juan Garcia-Bellido, Troels Haugboelle
arXiv:0903.5040 Title: Perturbation Theory in Lemaitre-Tolman-Bondi Cosmology
Authors: Chris Clarkson, Timothy Clifton, Sean February.
arXiv:0810.4939 Title: The radial BAO scale and Cosmic Shear, a new observable
for Inhomogeneous Cosmologies Authors: Juan Garcia-Bellido, Troels Haugboelle
arXiv:0909.1479 Title: Rendering Dark Energy Void Authors: Sean February, Julien
Larena, Mathew Smith, Chris Clarkson
arXiv:0807.1443 Title: Living in a Void: Testing the Copernican Principle with
Distant Supernovae Authors: Timothy Clifton, Pedro G. Ferreira, Kate Land
arXiv:0809.3761 Title: Can we avoid dark energy? Authors: J. P. Zibin, A. Moss,
D. Scott
arXiv:0902.1313 Title: What the small angle CMB really tells us about the
curvature of the Universe Authors: Timothy Clifton, Pedro G. Ferreira, Joe Zuntz
The last of these seems particularly relevant, as it also finds that an
inhomogeneous bang time is necessary to fit to cosmological observables.
If the authors were to cut back on some of the Figures, and reduce the
discussion in the introduction, I expect the paper could be reduced in size by
as much as 25\% without losing any scientific content.
|