In addition, cosmic acceleration seems to be a recent phenomenon at least for standard dark-energy models, which gives rise to the coincidence problem. The epoch in which dark energy begins to play a role is close to the epoch in which most of the cosmic structures formed out of the slow linear gravitational growth. We are led to ask again: can the acceleration be caused by strong inhomogeneities rather than by a dark energy component?
Finally, one must notice that in all the standard treatment of dark energy one always assumes a perfectly isotropic expansion. Could it be that some of the properties of acceleration depends critically on this assumption?
In order to investigate these issues, in this section we explore radical deviations from homogeneity and isotropy and see how Euclid can test them.
In recent times, there has been a resurgent interest towards anisotropic cosmologies, classified in terms of
Bianchi solutions to general relativity. This has been mainly motivated by hints of anomalies in the cosmic
microwave background (CMB) distribution observed on the full sky by the WMAP satellite
[288, 930, 268, 349]. While the CMB is very well described as a highly isotropic (in a statistical sense)
Gaussian random field, and the anomalies are a posteriori statistics and therefore their statistical
significance should be corrected at least for the so-called look elsewhere effect (see, e.g., [740, 122
] and
references therein) recent analyses have shown that local deviations from Gaussianity in some directions
(the so called cold spots, see [268]) cannot be excluded at high confidence levels. Furthermore,
the CMB angular power spectrum extracted from the WMAP maps has shown in the past
a quadrupole power lower than expected from the best-fit cosmological model [335]. Several
explanations for this anomaly have been proposed (see, e.g., [905, 243, 297, 203, 408]) including the
fact that the universe is expanding with different velocities along different directions. While
deviations from homogeneity and isotropy are constrained to be very small from cosmological
observations, these usually assume the non-existence of anisotropic sources in the late universe.
Conversely, as suggested in [520, 519, 106, 233, 251], dark energy with anisotropic pressure acts as a
late-time source of anisotropy. Even if one considers no anisotropic pressure fields, small departures
from isotropy cannot be excluded, and it is interesting to devise possible strategies to detect
them.
The effect of assuming an anisotropic cosmological model on the CMB pattern has been studied by [249, 89, 638, 601, 189, 516]. The Bianchi solutions describing the anisotropic line element were treated as small perturbations to a Friedmann–Robertson–Walker (FRW) background. Such early studies did not consider the possible presence of a non-null cosmological constant or dark energy and were upgraded recently by [652, 477].
One difficulty with the anisotropic models that have been shown to fit the large-scale CMB pattern, is that they have to be produced according to very unrealistic choices of the cosmological parameters. For example, the Bianchi VIIh template used in [477] requires an open universe, an hypothesis which is excluded by most cosmological observations. An additional problem is that an inflationary phase – required to explain a number of feature of the cosmological model – isotropizes the universe very efficiently, leaving a residual anisotropy that is negligible for any practical application. These difficulties vanish if an anisotropic expansion takes place only well after the decoupling between matter and radiation, for example at the time of dark energy domination [520, 519, 106, 233, 251].
Bianchi models are described by homogeneous and anisotropic metrics. If anisotropy is slight, the dynamics of any Bianchi model can be decomposed into an isotropic FRW background linearly perturbed to break isotropy; on the other side, homogeneity is maintained with respect to three Killing vector fields.
The geometry of Bianchi models is set up by the structure constants , defined by the commutators
of (these) three Killing fields
:
Type | ![]() |
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|
|
|
|
|
|
|
I | 0 | 0 | 0 | 0 | |
V | ![]() |
0 | 0 | 0 | |
VII![]() |
0 | 0 | 1 | 1 | |
VII![]() |
![]() |
0 | 1 | 1 | |
IX | 0 | 1 | 1 | 1 | |
While deviations from homogeneity and isotropy are constrained to be very small from cosmological
observations, these usually assume the non-existence of anisotropic sources in the late universe. The CMB
provides very tight constraints on Bianchi models at the time of recombination [189, 516, 638] of order of
the quadrupole value, i.e., . Usually, in standard cosmologies with a cosmological constant the
anisotropy parameters scale as the inverse of the comoving volume. This implies an isotropization of the
expansion from the recombination up to the present, leading to the typically derived constraints on the
shear today, namely
. However, this is only true if the anisotropic expansion is not
generated by any anisotropic source arising after decoupling, e.g., vector fields representing anisotropic dark
energy [519].
As suggested in [520, 519, 106, 233, 251], dark energy with anisotropic pressure acts as a late-time source of anisotropy. An additional problem is that an inflationary phase – required to explain a number of feature of the cosmological model – isotropizes the universe very efficiently, leaving a residual anisotropy that is negligible for any practical application. These difficulties vanish if an anisotropic expansion takes place only well after the decoupling between matter and radiation, for example at the time of dark energy domination [520, 519, 106, 233, 251].
For example, the effect of cosmic parallax [749] has been recently proposed as a tool to assess the presence of an anisotropic expansion of the universe. It is essentially the change in angular separation in the sky between far-off sources, due to an anisotropic expansion.
A common parameterization of an anisotropically distributed dark energy component is studied in a class of Bianchi I type, where the line element is
The expansion rates in the three Cartesian directionsThe anisotropic expansion is caused by the anisotropically stressed dark energy fluid whenever its energy density contributes to the global energy budget. If the major contributions to the overall budget come from matter and dark energy, as after recombination, their energy-momentum tensor can be parametrized as:
respectively, whereConsidering the generalized Friedmann equation, the continuity equations for matter and dark energy and no coupling between the two fluids, the derived autonomous system reads [520, 519]:
where Anisotropic distribution of sources in Euclid survey might constrain the anisotropy at present, when the
dark energy density is of order 74%, hence not yet in the final dark energy dominant attractor
phase (4.3.8).
Inhomogeneity is relatively difficult to determine, as observations are typically made on our past light cone,
but some methods exist (e.g., [242, 339, 600]). However, homogeneity may be tested by exploring the
interior of the past light cone by using the fossil record of galaxies to probe along the past world line of a
large number of galaxies [428
]. One can use the average star formation rate at a fixed lookback time as a
diagnostic test for homogeneity. The lookback time has two elements to it – the lookback time of the
emission of the light, plus the time along the past world line. The last of these can be probed using
the integrated stellar spectra of the galaxies, using a code such as vespa [890], and this is
evidently dependent only on atomic and nuclear physics, independent of homogeneity. The
lookback time can also be computed, surprisingly simply, without assuming homogeneity from [428]
Nonlinear inhomogeneous models are traditionally studied either with higher-order perturbation theory or
with -body codes. Both approaches have their limits. A perturbation expansion obviously breaks down
when the perturbations are deeply in the nonlinear regime.
-body codes, on the other hand, are
intrinsically Newtonian and, at the moment, are unable to take into account full relativistic effects.
Nevertheless, these codes can still account for the general relativistic behavior of gravitational collapse
in the case of inhomogeneous large void models, as shown recently in [30
], where the growth
of the void follows the full nonlinear GR solution down to large density contrasts (of order
one).
A possibility to make progress is to proceed with the most extreme simplification: radial symmetry. By assuming that the inhomogeneity is radial (i.e., we are at the center of a large void or halo) the dynamical equations can be solved exactly and one can make definite observable predictions.
It is however clear from the start that these models are highly controversial, since the observer needs to
be located at the center of the void with a tolerance of about few percent of the void scale radius,
see [141, 242], disfavoring the long-held Copernican principle (CP). Notwithstanding this, the idea that we
live near the center of a huge void is attractive for another important reason: a void creates an apparent
acceleration field that could in principle match the supernovae observations [891, 892, 220, 474]. Since we
observe that nearby SN Ia recede faster than the predicted by the Einstein–de Sitter universe, we
could assume that we live in the middle of a huge spherical region which is expanding faster because it is
emptier than the outside. The transition redshift
, i.e., the void edge, should be located
around 0.3 – 0.5, the value at which in the standard interpretation we observe the beginning of
acceleration.
The consistent way to realize such a spherical inhomogeneity has been studied since the 1930s in the
relativistic literature: the Lemaître–Tolman–Bondi (LTB) metric. This is the generalization of a FLRW
metric in which the expansion factor along the radial coordinate is different relative to the surface line
element
. If we assume the inhomogeneous metric (this subsection follows closely
the treatment in [49])
Considering the infinitesimal radial proper length , we can define the radial Hubble
function as
In terms of the two Hubble functions, we find that the Friedmann equations for the pressureless matter
density are given by [28]
If one imposes the additional constraint that the age of the universe is the same for every observer, then
only one free function is left [380]. The same occurs if one chooses
(notice that this is
different from
, which is another possible choice), i.e., if the matter density fraction is
assumed homogeneous today (and only today) [341]. The choice of a homogeneous universe age guarantees
against the existence of diverging inhomogeneities in the past. However, there is no compelling reason to
impose such restrictions.
Eq. (4.3.20) is the classical cycloid equation whose solution for
is given parametrically
by
where is the inhomogeneous “big-bang” time, i.e., the time for which
and
for a point at comoving distance
. This can be put to zero in all generality by a redefinition of
time. The “time” variable
is defined by the relation
As anticipated, since we need to have a faster expansion inside some distance to mimic cosmic
acceleration, we need to impose to our solution the structure of a void. An example of the choice of
) is [380
]
In order to compare the LTB model to observations we need to generalize two familiar concepts: redshift and luminosity distance. The redshift can be calculated through the equation [27]
where The proper area of an infinitesimal surface at is given by
. The
angular diameter distance is the square root of
so that
. Since the
Etherington duality relation
remains valid in inhomogeneous models, we have [530]
Besides matching the SN Ia Hubble diagram, we do not want to spoil the CMB acoustic peaks and we
also need to impose a local density near 0.1 – 0.3, a flat space outside (to fulfil inflationary
predictions), i.e.,
, and finally the observed local Hubble value
. The CMB
requirement can be satisfied by a small value of
, since we know that to compensate for
we
need a small Hubble rate (remember that the CMB essentially constrains
). This fixes
. So we are left with only
and
to be constrained by SN Ia. As anticipated
we expect
to be near
, which in the standard
CDM model gives a distance
. An analysis using SN Ia data [382
] finds that
and
.
Interestingly, a “cold spot” in the CMB sky could be attributed to a void of comparable size
[269, 642].
There are many more constraints one can put on such large inhomogeneities. Matter inside the void moves with respect to CMB photons coming from outside. So the hot intracluster gas will scatter the CMB photons with a large peculiar velocity and this will induce a strong kinematic Sunyaev–Zel’dovich effect [381]. Moreover, secondary photons scattered towards us by reionized matter inside the void should also distort the black-body spectrum due to the fact that the CMB radiation seen from anywhere in the void (except from the center) is anisotropic and therefore at different temperatures [197]. These two constraints require the voids not to exceed 1 or 2 Gpc, depending on the exact modelling and are therefore already in mild conflict with the fit to supernovae.
Moreover, while in the FLRW background the function fixes the comoving distance
up to
a constant curvature (and consequently also the luminosity and angular diameter distances), in the LTB
model the relation between
and
or
can be arbitrary. That is, one can choose the
two spatial free functions to be for instance
and
, from which the line-of-sight values
and
would also be arbitrarily fixed. This shows that the “consistency” FLRW relation
between
and
is violated in the LTB model, and in general in any strongly inhomogeneous
universe.
Further below we discuss how this consistency test can be exploited by Euclid to test for large-scale
inhomogeneities. Recently, there has been an implementation of LTB models in large-scale structure
-body simulations [30], where inhomogeneities grow in the presence of a large-scale void and seen to
follow the predictions of linear perturbation theory.
An interesting class of tests on large-scale inhomogeneities involve probes of the growth of
structure. However, progress in making theoretical predictions has been hampered by the increased
complexity of cosmological perturbation theory in the LTB spacetime, where scalar and tensor
perturbations couple, see for example [241]. Nevertheless, a number of promising tests of large-scale
inhomogeneity using the growth of structure have been proposed. [29] used -body simulations
to modify the Press–Schechter halo mass function, introducing a sensitive dependence on the
background shear. The shear vanishes in spatially-homogeneous models, and so a direct measurement
of this quantity would put stringent constraints on the level of background inhomogeneity,
independent of cosmological model assumptions. Furthermore, recent upper limits from the ACT and
SPT experiments on the linear, all-sky kinematic Sunyaev–Zel’dovich signal at
, a
probe of the peculiar velocity field, appear to put strong constraints on voids [986]. This result
depends sensitively on theoretical uncertainties on the matter power spectrum of the model,
however.
Purely geometric tests involving large-scale structure have been proposed, which neatly side-step the
perturbation theory issue. The Baryon Acoustic Oscillations (BAO) measure a preferred length scale,
, which is a combination of the acoustic length scale,
, set at matter-radiation decoupling, and
projection effects due to the geometry of the universe, characterized by the volume distance,
. In
general, the volume distance in an LTB model will differ significantly from that in the standard model, even
if the two predict the same SN Ia Hubble diagram and CMB power spectrum. Assuming that the
LTB model is almost homogeneous around the decoupling epoch,
may be inferred from
CMB observations, allowing the purely geometric volume distance to be reconstructed from
BAO measurements. It has been shown by [990] that, based on these considerations, recent
BAO measurements effectively rule out giant void models, independent of other observational
constraints.
The tests discussed so far have been derived under the assumption of a homogeneous Big Bang
(equivalent to making a particular choice of the bang time function). Allowing the Big Bang to be
inhomogeneous considerably loosens or invalidates some of the constraints from present data. It has been
shown [187] that giant void models with inhomogeneous bang times can be constructed to fit the SN Ia
data, WMAP small-angle CMB power spectrum, and recent precision measurements of simultaneously.
This is contrary to claims by, e.g., [767], that void models are ruled out by this combination of observables.
However, the predicted kinematic Sunyaev–Zel’dovich signal in such models was found to be severely
incompatible with existing constraints. When taken in combination with other cosmological observables,
this also indicates a strong tension between giant void models and the data, effectively ruling them
out.
In general, we would like to compute directly the impact of the inhomogeneities, without requiring an exact
and highly symmetric solution of Einstein’s equations like FLRW or even LTB. Unfortunately there is no
easy way to approach this problem. One ansatz tries to construct average quantities that follow equations
similar to those of the traditional FLRW model, see e.g., [185, 751, 752, 186]. This approach is often called
backreaction as the presence of the inhomogeneities acts on the background evolution and changes it. In this
framework, it is possible to obtain a set equations, often called Buchert equations, that look surprisingly
similar to the Friedmann equations for the averaged scale factor , with extra contributions:
However, it is not possible to directly link this formalism to observations. A first step can be done by
imposing by hand an effective, average geometry with the help of a template metric that only holds on
average. The probably simplest first choice is to impose on each spatial hypersurface a spatial metric with
constant curvature, by imagining that the inhomogeneities have been smoothed out. But in general the
degrees of freedom of this metric (scale factor and spatial curvature) will not evolve as in the FLRW case,
since the evolution is given by the full, inhomogeneous universe, and we would not expect that the
smoothing of the inhomogeneous universe follows exactly the evolution that we would get for a smooth
(homogeneous) universe. For example, the average curvature could grow over time, due to the collapse of
overdense structure and the growth (in volume) of the voids. Thus, unlike in the FRLW case, the average
curvature in the template metric should be allowed to evolve. This is the case that was studied
in [547].
While the choice of template metric and the Buchert equations complete the set of equations, there are
unfortunately further choices that need to be made. Firstly, although there is an integrability condition
linking the evolution of and
and in addition a consistency requirement that the effective
curvature
in the metric is related to
, we still need to impose an overall evolution by hand as
it was not yet possible to compute this from first principles. Larena assumed a scaling solution
, with
a free exponent. In a dark energy context, this scaling exponent
corresponds to
an effective dark energy with
, but in the backreaction case with the template metric the
geometry is different from the usual dark energy case. A perturbative analysis [569] found
,
but of course this only an indication of the possible behavior as the situation is essentially
non-perturbative.
The second choice concerns the computation of observables. [547] studied distances to supernovae and
the CMB peak position, effectively another distance. The assumption taken was that distances could be
computed within the averaged geometry as if this was the true geometry, by integrating the equation of
radial null geodesics. In other words, the effective metric was taken to be the one that describes distances
correctly. The resulting constraints are shown in Figure 51. We see that the leading perturbative mode
(
) is marginally consistent with the constraints. These contours should be regarded as
an indication of what kind of backreaction is needed if it is to explain the observed distance
data.
One interesting point, and maybe the main point in light of the discussion in the following section, is
that the averaged curvature needs to become necessarily large at late times due to the link between it and
the backreaction term , in order to explain the data. Just as in the case of a huge void, this effective
curvature makes the backreaction scenario testable to some degree with future large surveys like
Euclid.
http://www.livingreviews.org/lrr-2013-6 |
Living Rev. Relativity 16, (2013), 6
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