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., .
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.,
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 , the predictions for the power spectra have been worked out in . 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., , 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 . 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  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  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 .
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 184.108.40.206). 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 , a large number of randomly oriented fields averaging to isotropy , time-like  or sub-dominant  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., . The generic prediction is that the primordial perturbation spectra become statistically anisotropic, see e.g., .
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 . 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., .
Also in this case the interest in such possibilities has been stimulated by several anomalies observed in the temperature WMAP maps, see  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 . 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 , an asymmetric distribution of CMB power between two hemispheres, or dipole asymmetry , 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 . 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
A general prediction from all these scenarios is that the power spectrum of primordial perturbations can be written as
However, beyond the various concrete realizations, the expression (4.5.12), first introduced in , 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 . 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 (, and references therein)
Groeneboom and Eriksen , 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.  confirmed this effect at high statistical significance, pointing out however that beam asymmetries could be a strong contaminant (see also ). The importance of this systematic effect is somewhat debated: , 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  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 . 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  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 .
Pullen and Hirata  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 
Assuming the same preferred direction singled out by , 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 . 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 , 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.
Living Rev. Relativity 16, (2013), 6
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