Conventionally, the 2-point correlation function of a random variable is regarded as a classical
object, related to the power spectrum
via the relation
When we look at in terms of a quantum field in momentum space, we need to reinterpret the
average
as the expectation value of the 2-point function over a determined quantum state. This
raises several issues that are usually ignored in a classical analysis. For instance, the value of the
expectation value depends in the algebra of the annihilation and creation operators that compose the field
operator. Any non-trivial algebra such as a non-commutative one, leads to non-trivial power spectra. Also,
the quantum expectation value depends on the state of the field, and different choices can lead to radically
different results.
Suppose that represents a perturbation propagating on an inflationary background. Upon
quantization, we have
In addition to UV divergences, there are infrared (IR) ones in long-range correlations. Usually, one
tames these by putting the universe in a box and cutting off super-horizon correlations. However, several
authors have recently proposed more sensible IR regulating techniques, see e.g., [394, 523]. Very natural
ways to obtain IR finite results are to take into account the presence of tiny spatial curvature or a
pre-inflationary phase which alters the initial conditions [479, 523]. In principle these regularizations will
leave an imprint in the large-scale structure data, in the case that regularization scale is not too far
beyond the present horizon scale. If this pre-inflationary phase is characterized by modified field
theory, such as modified dispersion relations or lower dimensional effective gravity, the scalar
and tensor power spectra show a modification whose magnitude is model-dependent, see e.g.,
[770].
The two-point function of a scalar field is constructed from basic quantum field theory, according to a set of rules determined in the context of relativistic quantum mechanics. In particular, the usual commutation rules between position and momentum are promoted to commutation rules between the field and its canonical conjugate. A modification of the fundamental quantum mechanical commutation rules can easily be generalized to field theory. The most popular case is represented by non-commutative geometry, which implies that coordinate operators do not commute, i.e.,
where One can construct models where the inflationary expansion of the universe is driven by
non-commutative effects, as in [24, 769]. In this kind of models, there is no need for an inflaton field and
non-commutativity modifies the equation of state in the radiation-dominated universe in a
way that it generates a quasi-exponential expansion. The initial conditions are thermal and
not determined by a quantum vacuum. For the model proposed in [24], the predictions for
the power spectra have been worked out in [517]. Here, Brandenberger and Koh find that the
spectrum of fluctuations is nearly scale invariant, and shows a small red tilt, the magnitude of
which is different from what is obtained in a usual inflationary model with the same expansion
rate.
On the other hand, non-commutativity could introduce corrections to standard inflation. Such a,
perhaps less radical approach, consists in assuming the usual inflaton-driven background, where scalar and
tensor perturbations propagate with a Bunch and Davies vacuum as initial condition, but are subjected to
non-commutativity at short distance. It turns out that the power spectrum is modified according to (see
e.g., [522], and references therein)
Furthermore, it is interesting that the violation of isotropy can also violate parity. This could provide
what seems a quite unique property of possible signatures in the CMB and large-scale structure. However,
there is also an ambiguity with the predictions of the simplest models, which is related to interpretations of
non-commuting quantum observables at the classical limit. This is evident from the fact that one
has to consider an effectively imaginary in the above formula (4.5.4
). Reality of physical
observables requires the odd parity part of the spectrum (4.5.4
) to be imaginary. The appearance of
this imaginary parameter
into the theory may signal the unitary violation that has been
reported in theories of time-space non-commutativity. It is known that the Seiberg–Witten
map to string theory applies only for space-space non-commutativity [810]. Nevertheless, the
phenomenological consequence that the primordial fluctuations can distinguish handedness,
seems in principle a physically perfectly plausible – though speculative – possibility, and what
ultimately renders it very interesting is that we can test by cosmological observations. Thus, while
lacking the completely consistent and unique non-commutative field theory, we can parametrize
the ambiguity by a phenomenological parameter whose correct value is left to be determined
observationally. The parameter
can be introduced [522] to quantify the relative
amplitude of odd and even contributions in such a way that
, where
.
The implications of the anisotropic power spectra, such as (4.5.4), for the large-scale structure
measurements, is discussed below in Section 4.5.3. Here we proceed to analyse some consequences of the
non-commutativity relation (4.5.3
) to the higher order correlations of cosmological perturbations. We find
that they can violate both isotropy and parity symmetry of the FRW background. In particular, the latter
effect persists also in the case
. This case corresponds to the prescription in [18] and in the
remainder of this subsection we restrict to this case for simplicity. Thus, even when we choose this
special prescription where the power spectrum is even, higher order correlations will violate
parity. This realizes the possibility of an odd bispectrum that was recently contemplated upon
in [489].
More precisely, the functions defined in Eq. (3.3.4
) for the three-point function of the curvature
perturbation can be shown to have the form
We now focus on this part in the following only and set all components of equal to zero. This gives
Moreover, it is worth noting that the result (4.5.10) depends on the wave vectors
and
and
hence on the shape of the momentum space triangle. This is in contrast with the commutative case, where
the scale dependence is given by the same result (4.5.7
)for all shape preserving variations,
,
regardless of triangle shape. This allows, in principle, to distinguish between the contributions arising from
the non-commutative properties of the theory and from the standard classical inflationary physics or
gravitational clustering.
To recapitulate, parity violations in the statistics of large-scale structures would be a smoking gun signature of timespace non-commutativity at work during inflation. Moreover, purely spatial non-commutativity predicts peculiar features in the higher order correlations of the perturbations, and in particular these can be most efficiently detected by combining information of the scale- and shape-dependence of non-Gaussianity. As discussed earlier in this document, this information is extractable from the Euclid data.
Besides the non-commutative effects seen in the previous section, anisotropy can be generated by the
presence of anisotropic fields at inflation. Such could be spinors, vectors or higher order forms which modify
the properties of fluctuations in a direction-dependent way, either directly through perturbation dynamics
or by causing the background to inflate slightly anisotropically. The most common alternative is vector
fields (see Section 4.5.2.1). Whereas a canonical scalar field easily inflates the universe if suitable initial
conditions are chosen, it turns out that it much less straightforward to construct vector field alternatives. In
particular, one must maintain a sufficient level of isotropy of the universe, achieve slow roll and keep
perturbations stable. Approaches to deal with the anisotropy have been based on a “triad” of
three identical vectors aligned with the three axis [59], a large number of randomly oriented
fields averaging to isotropy [401], time-like [521] or sub-dominant [313] fields. There are many
variations of inflationary scenarios involving vector fields, and in several cases the predictions of the
primordial spectra of perturbations have been worked out in detail, see e.g., [946]. The generic
prediction is that the primordial perturbation spectra become statistically anisotropic, see e.g.,
[7
].
Anisotropy could be also regarded simply as a trace of the initial conditions set before inflation. One then assumes that inflation has lasted just about the 60 e-folds so that the largest observable scales were not yet smoothed out, or isotropized, by the early inflationary expansion [734]. Such a scenario can also be linked to various speculative ideas of pre-inflationary physics such as gravitational tunnelling into an anisotropic universe, see e.g., [9].
Also in this case the interest in such possibilities has been stimulated by several anomalies observed in
the temperature WMAP maps, see [254] for a recent review (some of them were also present in the COBE
maps). Their statistical evidence is quite robust w.r.t. the increase of the signal-to-noise ratio over the years
of the WMAP mission and to independent tests by the international scientific community, although the a
posteriori choice of statistics could make their interpretation difficult, see [122]. Apart from those already
mentioned in Section 4.3.1, these anomalies include an alignment between the harmonic quadrupole and
octupole modes in the temperature anisotropies [288], an asymmetric distribution of CMB power
between two hemispheres, or dipole asymmetry [350], the lack of power of the temperature
two-point correlation function on large angular scales (
), asymmetries in the even vs. odd
multipoles of the CMB power spectra (parity symmetry breaking), both at large [503, 409
]
and intermediate angular scales [122
]. Some of the anomalies could be connected among each
other, e.g., the CMB parity breaking has been recently linked to the lack of large-scale power
[628, 253, 504].
Various inflationary models populated by vector fields can be described with a Lagrangian of the following form
whereA general prediction from all these scenarios is that the power spectrum of primordial perturbations can be written as
where However, beyond the various concrete realizations, the expression (4.5.12), first introduced in [7],
provides a robust and useful way to study observable consequences of a preferred direction during
inflation and also a practical template for comparison with observations (see below). Usually the
amplitude
is set to a constant
. A generalization of the above parametrization is
, where
are spherical harmonics with only even
multipoles
[746
]. Interestingly enough, inflationary models with vector fields can also generate
higher-order correlators, such as bispetrum and trispectrum, which display anisotropic features as well (e.g.,
[973, 492, 92, 91]).
The alignment of low CMB multipoles and the hemispherical power asymmetry observed in the CMB
anisotropies can find an explanation in some models where the primordial gravitational perturbation is the
result of fluctuations within our Hubble volume, modulated by super-horizon fluctuations. The primordial
gravitational perturbation can thus be thought of as a product of two fields and
([333],
and references therein)
Groeneboom and Eriksen [405], using WMAP5 year data (up to multipoles
), claimed a detection
of a quadrupolar power spectrum of the form of Eq. (4.5.12
) at more than
(
) with
preferred direction
. Subsequently this result has been put under further
check. [423] confirmed this effect at high statistical significance, pointing out however that beam
asymmetries could be a strong contaminant (see also [424]). The importance of this systematic effect
is somewhat debated: [404
], including polarization and beam asymmetries analysis excluded
that the latter can be responsible for the observed effect. Their claim is a
detection with
. However, the preferred direction shifted much closer to the ecliptic poles, which
is probably an indication that some unknown systematic is involved and must be corrected
in order to obtain true constraints on any primordial modulation. Foregrounds and noise are
disfavored as possible systematic effects [122, 405
]. Thus the cause of this kind of asymmetry is
not definitely known. Planck should be able to detect a power quadrupole as small as 2% (at
) [746
, 405
, 404]. It is of course desirable to test this (and other anisotropic effects) with other
techniques.
What about large-scale structure surveys? Up to now there are just a few analyses testing anisotropies in large-scale structure surveys, but all of them have been crucial, indicating that large-scale structure surveys such as Euclid offer a promising avenue to constrain these features.
Hirata [437] used high-redshift quasars from the Sloan Digital Sky Survey to rule out the simplest
version of dipole modulation of the primordial power spectrum. In comparison the Planck mission using the
CMB hemispherical asymmetry would only marginally distinguish it from the standard case [348]. The
constraints obtained by high-redshift quasars require an amplitude for the dipole modulation 6 times
smaller than the one required by CMB. This would disfavor the simple curvaton spatial gradient scenario
[346] proposed to generate this dipole modulation. Only a curvaton scenario with a non-negligible fraction
of isocurvature perturbations at late times could avoid this constraint from current high-redshift quasars
[345].
Pullen and Hirata [745] considered a sample of photometric luminous red galaxies from the SDSS survey
to assess the quadrupole anisotropy in the primordial power spectrum of the type described by Eq. (4.5.12
).
The sample is divided into eight redshift slices (from
up to
), and within each slice
the galaxy angular power spectrum is analysed. They also accounted for possible systematic
effects (such as a modulation of the signal and noise due to a slow variation of the photometric
calibration errors across the survey) and redshift-space distortion effects. In this case [745
]
Assuming the same preferred direction singled out by [405], they derive a constraint on the anisotropy
amplitude (
), thus finding no evidence for anisotropy. Marginalizing over
with a uniform prior they find
at 95% C.L. These results could confirm
that the signal seen in CMB data is of systematic nature. However, it must be stressed that
CMB and LSS analyses probe different scales, and in general the amplitude of the anisotropy
is scale dependent
, as in the model proposed in [345]. An estimate for what an
experiment like Euclid can achieve is to consider how the uncertainty in
scale in terms of
number of modes measured and the number of redshift slices. Following the arguments of [745],
the uncertainty will scale roughly as
, where
is the maximum multipole
at which the galaxy angular power spectrum is probed, and
is the number of redshift
slices. Considering that the redshift survey of Euclid will cover redshifts
, there is
an increase by a factor of 3 in distance of the survey and hence a factor 3 increase in
(
, see the expression for the selection function after Eq. (4.5.15
)). Taking
the effective number of redshift slices is also increased of a factor of
(
, with
the radial width of the survey). Therefore, one could expect
that for a mission like Euclid one can achieve an uncertainty (at
)
or
, for a fixed anisotropy axis or marginalizing over
, respectively. This will be
competitive with Planck measurements and highly complementary to it [702, 409]. Notice that
these constraints apply to an analysis of the galaxy angular power spectrum. An analysis of the
3-dimensional power spectrum
could improve the sensitivity further. In this case the
uncertainty would scale as
, where
is the number of independent Fourier
modes.
http://www.livingreviews.org/lrr-2013-6 |
Living Rev. Relativity 16, (2013), 6
![]() This work is licensed under a Creative Commons License. E-mail us: |