Largescale anomalies in the cosmic microwave background
as signatures of nonGaussianity
Abstract
We derive a general expression for the probability of observing deviations from statistical isotropy in the cosmic microwave background (CMB) if the primordial fluctuations are nonGaussian and extend to superhorizon scales. The primary motivation is to properly characterize the monopole and dipole modulations of the primordial power spectrum that are generated by the coupling between superhorizon and subhorizon perturbations. Unlike previous proposals for generating the hemispherical power asymmetry, we do not assume that the power asymmetry results from a single large superhorizon mode. Instead, we extrapolate the observed power spectrum to superhorizon scales and compute the power asymmetry that would result from a specific realization of nonGaussian perturbations on scales larger than the observable universe. Our study encompasses many of the scenarios that have been put forward as possible explanations for the CMB hemispherical power asymmetry. We confirm our analytic predictions for the probability of a given power asymmetry by comparing them to numerical realizations of CMB maps. We find that nonlocal models of nonGaussianity and scaledependent local nonGaussianity produce scaledependent modulations of the power spectrum, thereby potentially producing both a monopolar and a dipolar power modulation on large scales. We then provide simple examples of finding the posterior distributions for the parameters of the bispectrum from the observed monopole and dipole modulations.
I Introduction
It is very tempting to try to use largescale features of the primordial fluctuations, such as the hemispherical power asymmetry, as a clue toward primordial physics. Such signals are both intriguing (maybe they say something about the beginning of inflation) and statistically unfortunate (i.e., not all that unlikely to be a feature of a particular realization of Gaussian, isotropic fluctuations). Many studies of the observed power asymmetry and its statistical significance have been reported using the WMAP data Eriksen et al. (2004); Hoftuft et al. (2009); Bennett et al. (2011) and the Planck data Ade et al. (2014a); Flender and Hotchkiss (2013); Akrami et al. (2014); Quartin and Notari (2015); Ade et al. (2015a); Aiola et al. (2015). A variety of possible explanations for this asymmetry have been discussed (see e.g. Dai et al. (2013)): several of the most intriguing ideas use superhorizon fluctuations to generate the asymmetry, either by using nonGaussianity to couple them to observable perturbations Erickcek et al. (2008a); Erickcek et al. (2009); Schmidt and Hui (2013); Lyth (2013); Kanno et al. (2013); Wang and Mazumdar (2013); D’Amico et al. (2013); Kohri et al. (2014); Liddle and Cortês (2013); McDonald (2013a, b); Mazumdar and Wang (2013); Namjoo et al. (2013); Abolhasani et al. (2014); Assadullahi et al. (2015); Firouzjahi et al. (2014); Jazayeri et al. (2014); McDonald (2014); Namjoo et al. (2014); Lyth (2015); Agullo (2015); Byrnes and Tarrant (2015); Kenton et al. (2015); Kobayashi et al. (2015) or by postulating some different primordial physics that precedes the usual slowroll inflation Donoghue et al. (2009); Liu et al. (2013, 2014); Ashoorioon and Koivisto (2015). Alternatively, one can postulate scenarios that are fundamentally anisotropic on the largest scales Moss et al. (2011); Cai et al. (2014); Chang et al. (2015); Chang and Wang (2013); Kothari et al. (2015).
It is quite general that if the primordial fluctuations are nonGaussian, the likelihood of observing statistical anisotropies in our cosmic microwave background (CMB) changes. Although one might expect that isotropy and Gaussianity are independent criteria for the statistics of the primordial fluctuations, this distinction is not actually clear when we only have access to a finite volume of the universe Ferreira and Magueijo (1997); Lewis (2011); Pearson et al. (2012). If the primordial fluctuations are nonGaussian, the observed largescale “discrepancies” from the simplest isotropic, power law, Gaussian fluctuations need not be a signal of a special scale, time, or feature during the primordial (inflationary) era. Instead, they may simply be a consequence of cosmic variance in a universe larger than the volume we currently observe, filled with nonGaussian fluctuations.
In this paper, we present a single framework to calculate the distribution of expected deviations from isotropy in our observed sky from any model with nonGaussian primordial fluctuations. This framework incorporates most successful proposals for generating the power asymmetry, even some that were not originally formulated as nonGaussian models. The reason is that if the assumption of statistical isotropy is maintained, any explanation of the temperature power asymmetry can be modeled by assuming a fluctuation in a long wavelength modulating field that couples to some cosmological parameter relevant for determining the CMB power spectrum (the fundamental constants, the scalar spectral index and the inflaton decay rate, for example Moss et al. (2011); Dai et al. (2013); Cai et al. (2014); McDonald (2013b, 2014); Zarei (2015)). Since the asymmetry is well modeled by a (small) range of shorter scales all coupling to a longer scale this is, by definition, a sort of nonGaussianity.
Using our analytic framework together with numerical realizations, we show that isotropy violations in our observable universe are more likely in models where subhorizon modes are coupled to longer wavelength modes. In some models they are so much more likely that the extent to which our observed sky is isotropic can be interpreted as a constraint on nonGaussianity.
In addition, we find that scale dependence of such modulations is a generic outcome of nonGaussianity beyond the standard local ansatz. This result is important because a major hurdle in model building for the power anomalies has been their scale dependence, as the observations suggest a very sharp decrease in the power asymmetry amplitude at smaller scales Flender and Hotchkiss (2013); Hirata (2009); Adhikari (2015). Scale dependence of the type observed occurs whenever modes closer to the present Hubble scale couple more strongly to superhorizon modes than very short wavelength modes do. This can be the case even for scaleinvariant bispectra. For example, since equilateraltype nonGaussianity peaks when the momenta configurations are nearly equal, the modulation from an equilateraltype bispectrum is biggest at large scales and quickly dies off at smaller CMB scales.
Further, we will see that any nonGaussianity used to explain the scaledependent power asymmetry will also produce a scaledependent modulation of the powerspectrum monopole. This means that the observed low power on large scales and the dipole power asymmetry can both provide evidence in favor of nonGaussian fluctuations. To quantify the significance of these signals, we perform a parameter estimation of the nonGaussian amplitude and shape using both monopole and dipole modulations, and estimate Bayesian evidences against the Gaussian model.
Finally, we will show that scaledependent nonGaussianity eliminates the need for enhanced inhomogeneity on superhorizon scales to generate the observed asymmetry. The most common method for introducing a dipolar power modulation is to postulate the existence of a largeamplitude superhorizon fluctuation in a spectator field during inflation that then alters the power spectrum on smaller scales via localtype nonGaussianity. (A multiplefield model of inflation, or one that otherwise breaks the usual consistency relation, is required because superhorizon perturbations in the standard inflaton field cannot generate an asymmetry Mirbabayi and Zaldarriaga (2015).) Dubbed the ErickcekKamionkowskiCarroll (EKC) mechanism Erickcek et al. (2008a), this approach has been expanded upon and refined several times since its inception Erickcek et al. (2009); Abolhasani et al. (2014); D’Amico et al. (2013); Kanno et al. (2013); Kohri et al. (2014); Liddle and Cortês (2013); Lyth (2013); Mazumdar and Wang (2013); McDonald (2013a); Namjoo et al. (2013); Wang and Mazumdar (2013); Assadullahi et al. (2015); Firouzjahi et al. (2014); Jazayeri et al. (2014); Lyth (2015); McDonald (2014); Namjoo et al. (2014); Agullo (2015); Byrnes and Tarrant (2015); Kenton et al. (2015); Kobayashi et al. (2015). In these analyses, WMAP and Planck bounds on localtype nonGaussianity forced the amplitude of the superhorizon perturbation to be much larger than predicted by an extrapolation of the observed primordial power spectrum to larger scales. Although possible origins for this largeamplitude fluctuation have been proposed, such as a supercurvature perturbation in an open universe Liddle and Cortês (2013), a bounce prior to inflation Agullo (2015), a nonvacuum initial state Firouzjahi et al. (2014), topological defects Kohri et al. (2014); Jazayeri et al. (2014), or deviations from slowroll inflation Mazumdar and Wang (2013); McDonald (2013a), it is largely taken as an ad hoc addition to the inflationary landscape. The scale dependence of the asymmetry often requires an additional elaboration to the theory, either in the form of scaledependent nonGaussianity Erickcek et al. (2009); Lyth (2013); Kohri et al. (2014); Firouzjahi et al. (2014); Lyth (2015); Byrnes and Tarrant (2015); Agullo (2015) or isocurvature fluctuations Erickcek et al. (2009); McDonald (2013a); Assadullahi et al. (2015).
Dropping the idea of a special large superhorizon fluctuation and instead starting with scaledependent nonGaussianity changes this picture. Importantly, the constraints on the amplitude of the nonGaussianity on large scales is rather weak; the Planck bound of at 68% CL Ade et al. (2015b) assumes a scaleinvariant bispectrum. To generate the observed power asymmetry, we only need nonGaussianity on the scales that are asymmetric. For example, if we consider CMB multipoles up to , the WMAP 5 year data give Smith et al. (2009). (This constraint has not appreciably changed since WMAP5: see also Figure 11 of Ade et al. (2015b) for the most recent plot of Planck’s dependent constraints.) Many arbitrary choices for in the range (and different binning schemes to study the scale dependence) can be found in the literature for computing the largescale power asymmetry amplitude and its significance Eriksen et al. (2004); Flender and Hotchkiss (2013); Ade et al. (2014a, 2015a). We will use as our fiducial value to compute the asymmetry for our numerical tests and statistical analysis later. This also allows us to use the WMAP5 largescale bispectrum constraints computed with .
The weaker constraint on implies that it is possible to generate the observed asymmetry without enhancing the amplitude of superhorizon fluctuations, as was noted by Schmidt and Hui (2013), which considered the powerspectrum modulations generated by anisotropic bispectra. We note, however, that the dipolar bispectrum considered there does not generate a power asymmetry because the symmetry of the power spectrum forbids any modulation to the power spectrum from bispectra that depend on odd powers of the angle between the long and short modes; this point was clarified in Schmidt et al. (2015). In this work, we will focus on isotropic, scaledependent nonGaussianity (although our framework can easily be extended to include fundamentally anisotropic models), and we will show that this is sufficient to generate the observed asymmetry without enhancing the amplitude of superhorizon fluctuations.
The plan of the paper is as follows. In the next section, we use the usual local ansatz to demonstrate the validity of our analytic calculations of the statistical anisotropies expected in nonGaussian scenarios and to illustrate several of the key conceptual points relating nonGaussianity and anisotropies. We also show how our framework encompasses the EKC mechanism and demonstrate that exotic superhorizon perturbations are not required to generate the observed power asymmetry. In Section III, we test our analytic calculations for monopole and dipole power modulations using numerical realizations of CMB maps. We also introduce and discuss our parameter estimation and model comparison methods. In Section IV, we investigate nonGaussianity beyond the local ansatz and in particular, consider a representative model that generates features that are fully consistent with constraints on the isotropic power spectrum and bispectrum. We discuss and summarize important aspects of our work and conclude in Section V. The appendixes contain technical details.
Ii Illustrating the connection between nonGaussianity and isotropy
We assume that at some early time (after reheating but prior to the release of the cosmic microwave background radiation) a large volume of the Universe () contains adiabatic fluctuations described by isotropic but nonGaussian statistics. To compare predictions of this model with observations, we are interested in the statistics of the fluctuations in smaller volumes, , that correspond in size to our presently observable Hubble volume. We will first consider the usual local model with constant for simplicity; we will present more general results in a later section.
ii.1 The local model
Suppose the Bardeen potential is a nonGaussian field described by the local model:
(1) 
where is a Gaussian random field. When the large volume is only weakly nonGaussian, the power spectrum observed in our sky, , will be related to the mean power spectrum in the large volume, , by
where the radial integration for is confined to when the CMB spectrum is the quantity of interest. Our conventions for the power spectrum are stated in Appendix A, and Appendix B provides the derivation of this equation. Any particular model for generating the fluctuations in volume should provide a wellmotivated lower bound on the integral (e.g., from the duration of inflation). However, not all shifts to local statistics are sensitive to the full range of the integral; for localtype nonGaussianity, as we will show later, only the monopole receives contributions from all superHubble modes.
The power spectrum and amplitude of nonGaussianity, , appearing on the righthand side of Eq.(LABEL:eq:modulatedpowerLocal) are those defined in the large volume. However, looking ahead to the result for the dipole modulation from the local ansatz, Eq.(15), and will be shifted to the observed values. In particular, the local nonGaussianity that generates the observed power asymmetry is only the portion that violates Maldacena’s consistency relation Maldacena (2003) and is zero in singleclock inflation Mirbabayi and Zaldarriaga (2015). Since the observed values are ultimately the relevant quantities in the analysis, we will not increase the complexity of the notation to distinguish the large and small volume parameters, except in the appendixes.
The field appearing in the integral is no longer stochastic but consists of the particular realization of the field that makes up the background of a particular Hubble volume. The power spectrum in Eq.(LABEL:eq:modulatedpowerLocal) can depend on position within because an individual realization (local value) of the fluctuations can be nonzero. If we consider the average statistics in the large volume (equivalent to averaging over all regions of size ), then the term proportional to in Eq.(LABEL:eq:modulatedpowerLocal) above averages to zero since . In that case we recover the isotropic power spectrum of . Finally, keep in mind that Eq.(LABEL:eq:modulatedpowerLocal) is still more general than our actual CMB sky: it provides the statistics from which to draw realizations of our observed modes. Any single sky realization will still be subject to the usual cosmic variance that affects the values of small modes and that can generate a power asymmetry even for volumes where the term proportional to is zero.
It is also interesting to consider a twofield extension of Eq.(1):
(3) 
where both and are Gaussian random fields and are uncorrelated. In this case, the power spectrum observed in our sky, , is
(4) 
where is the mean power spectrum in the large volume and is the fraction of power in the field. We will only consider cases with weak nonGaussianity ( and ). The amplitude of the localtype bispectrum for this weakly nonGaussian twofield model is given by . Therefore, the inhomogeneous power spectrum in terms of the observed and the fraction of power is
(5) 
A scaledependent power fraction is a natural way to generate a scaledependent power asymmetry, as can be seen from the inhomogeneous term in Eq.(4); if decreases for large values of , the modulation of the power spectrum will decrease as well. Such mixedperturbation models also have other potentially observable consequences: affects the tensortoscalar ratio and contributes to largescale stochasticity in the power spectra of galaxies Tseliakhovich et al. (2010).
ii.2 Effect on the CMB sky
The imprint of the inhomogeneous power spectrum given by Eq.(LABEL:eq:modulatedpowerLocal) on the CMB can be described in terms of a multipole expansion:
(6) 
where is a spherical harmonic, and is the direction of observation on the last scattering surface. To find the expansion coefficients we make use of the plane wave expansion
(7) 
where is a spherical Bessel function of the first kind, and specifies the position of the observed fluctuation: for the CMB, is the comoving distance to the last scattering surface. Eq.(LABEL:eq:modulatedpowerLocal) then implies that
(8) 
The quantity has a fixed value in any single volume , but when averaged over all small volumes in , . The expected covariance, on the other hand, is nonzero:
(10) 
where in the last line, we have defined the dimensionless power spectrum as . We have again used the subscript to indicate the ensemble average is over the values of in the full volume . Note that both the individual values of and their variance depend on the size of the small volume through the upper limit of integration in Eqs. (8) and (II.2). While the mean statistics in the large volume cannot depend on the scale for the small volume, the variance of the statistics observed in subvolumes generically does. It is now straightforward to study the monopole and dipole contributions from nonGaussian cosmic variance to the modulated component of the power spectrum in a small volume.
In the case of the twofield extension, using Eq.(5) in the definition of the modulation moments Eq.(6) gives
(11)  
That is, for the same amplitude of nonGaussianity observed in the field, the variance of the nonGaussian modulations increases by a factor of compared to the single source () local model.
ii.2.1 Monopole modulation ()
The powerspectrum amplitude shift, , in the parametrization of Eq.(6) is:
(12) 
where can be either positive or negative, but has a lower bound . From the discussion above and Eq.(II.2), it is clear that is Gaussian distributed with zero mean and variance given by:
(13) 
Therefore, the distribution of the monopole power modulation amplitude also follows a normal distribution, for small values of , (), with zero mean and standard deviation:
(14) 
The expression for is sensitive to the infrared limit of the integral. That is, all superHubble modes can contribute. Interesting aspects of cosmic variance arising from this term, including effects on the observed nonGaussianity in small volumes have been subjects of investigation in Linde and Mukhanov (2006); Nelson and Shandera (2013); LoVerde et al. (2013); Nurmi et al. (2013); LoVerde (2014); Bramante et al. (2013); Baytaş et al. (2015). In particular, the observed value of is, in general, shifted from the mean value in the large volume.
For a constant , the effect of the monopole modulation is to change the powerspectrum amplitude on all scales and therefore is not observationally distinguishable from the “bare” value of the powerspectrum amplitude. For scaledependent nonGaussianity, there is a scaledependent power modulation, which can generically be interpreted as shifting the spectral index in the small volume away from the mean value in the large volume. In cases where the amplitude of nonGaussianity is small (and consistent with zero) at small scales (large ), the powerspectrum amplitude from those scales can be taken as , and then one can look for monopole modulation at large scales for which the nonGaussianity constraints are not as strong. The largescale power suppression anomaly Ade et al. (2014b); Contaldi et al. (2003) is exactly such a situation. We will return to this point in more detail in Section IV.
ii.2.2 Dipole modulation ()
The dipole modulation of the power spectrum in the parametrization of Eq.(6) is given by:
(15) 
Since we are interested in the dipole modulation of the observed power spectrum in the CMB sky, the above equation should be obtained from Eq.(LABEL:eq:modulatedpowerLocal) by absorbing the (unobservable) monopole shift to the observed power spectrum. Then, on the righthand side of Eq.(15), is the observed isotropic power spectrum and is the observed amplitude of local nonGausianity within our Hubble volume. See Eq. (67) and the discussion there for details. (Appendix C contains the corresponding expression in terms of bipolar spherical harmonics.) The coefficients are Gaussian distributed with zero mean and a variance
If we pick a direction in which to measure the dipole modulation such that
(16) 
where , then the contribution to the dipole from the nonGaussianity is , which is normally distributed with mean zero and standard deviation:
(17) 
In the twofield model Eq.(3), using Eq.(11), the standard deviation gets modified:
(18) 
This shows that for (e.g. a mixed inflatoncurvaton model), it is easier to generate the hemispherical power asymmetry with a small value of . However, there is a minimal value of that can generate a power asymmetry of a given amplitude: the requirement of weak nonGaussianity in the nonGaussian field [ in Eq.(5)] demands that .
The above discussion of the distribution of the dipole asymmetry assumes that we measure in a fixed direction . However, we have no a priori choice of direction in most situations. This is especially true when considering a power asymmetry that is generated by the random realization of superhorizon perturbations as opposed to a single exotic perturbation mode. Therefore, observations of dipole power modulations are necessarily reported using the amplitude of dipole modulation in the direction of the maximum modulation. To obtain that amplitude, we can consider any three orthonormal directions () on the CMB sky and measure the corresponding three dipole modulation amplitudes () in the three corresponding orthonormal directions for each sky. The amplitude of modulation for the CMB sky (simulated or observed) is then given by . Clearly then, follows the distribution with three degrees of freedom (also known as the Maxwell distribution). In the Section III.1, we will directly test the distributions and parameters obtained in this section using numerical realizations of CMB maps.
ii.2.3 Higher multipole modulations
The anisotropic modulation of the power spectrum is expected to continue to higher multipoles in the presence of nonGaussianity. However, as shown in Figure 1, the expected value of the modulation gets smaller quickly for higher multipoles . The corresponding expected variance of higher multipole modulations for Gaussian CMB maps, however, is only weakly dependent on . See, for example, Figure 2(d) of Akrami et al. (2014). There is no evidence for modulation at higher order multipoles in the Planck temperature anisotropies data (see Figure 34 of Ade et al. (2014a)). In statistical analysis of the kind we discuss later in Section III.4, it may, nevertheless, be useful to add higher multipole modulations (at least the quadrupole ) at large scales as it may provide increased evidence for or against nonGaussian mode coupling. An approximate constraint on may be obtained from the result for the Fourier space quadrupole modulation constraint in Kim and Komatsu (2013). There are two possible scenarios: (i) the expected amplitude of the modulation is larger than that from cosmic variance in the Gaussian case; in this scenario, the lack of observation of such a modulation in the data will disfavor the nonGaussian model that is used to explain the monopole and dipole modulations. (ii) The expected amplitude of the modulation is within the cosmic variance from the Gaussian case, in which case the data are not discriminatory for or against the nonGaussian model.
ii.3 Connection to prior work
Before proceeding, we pause to connect Eq.(15) to the EKC mechanism to better see how scaledependent nonGaussianity eliminates the need for enhanced perturbations on superhorizon scales. In the EKC mechanism, a single superhorizon perturbation mode in a spectator field during inflation is responsible for generating the asymmetry; the original proposal used a curvaton field Erickcek et al. (2008a), but later work extended the mechanism to any source of nonGaussian curvature fluctuations Lyth (2013); Kobayashi et al. (2015). For example, consider a field that generates a curvature perturbation . A superhorizon (SH) sinusoidal fluctuation in ,
(19) 
will generate a dipolar power asymmetry in the curvature power spectrum Kobayashi et al. (2015):
(20) 
to first order in . In terms of the Bardeen potential, , the power asymmetry is
(21) 
where . The Fourier transform of the superhorizon fluctuation given by Eq. (19) is
(22) 
Inserting this expression into Eq.(8) for implies that
(23) 
With this expression for , Eq. (15) matches Eq. (21) to first order in . Therefore, we see that the EKC mechanism can be described by our framework.
For a single superhorizon mode, Eq. (21) implies that the nonGaussian contribution to the dipole is
(24) 
where is the variation of across the surface of last scatter. Since , . The RMS amplitude of values given by extrapolating the observed value of to larger scales is
(25) 
where Ade et al. (2015c). It follows that the amplitude of the superhorizon mode is bounded from below as
(26) 
If (and is entirely due to the nonGaussianity) and , then , which implies that the superhorizon mode must be at least a 10 fluctuation. This is why Eq.(19) was not originally considered to be part of the inflationary power spectrum, but rather a remnant of preinflationary inhomogeneity or a domainwalllike feature in the curvaton field Erickcek et al. (2008a). It was then necessary to consider the imprint this enhanced superhorizon mode would leave on largescale temperature anisotropies in the CMB through the GrishchukZel’dovich (GZ) effect Grishchuk and Zeldovich (1978). Although the curvature perturbation generated by Eq.(19) does not generate an observable dipolar anisotropy in the CMB Turner (1991); Erickcek et al. (2008b); Zibin and Scott (2008), it does contribute to the quadrupole and octupole moments, and observations of these multipoles severely constrain models that employ the EKC mechanism Erickcek et al. (2008b, a).
However, if we relax our upper bound on to 270 or 500, Eq.(26) indicates that a or fluctuation, respectively, could generate an asymmetry with . The odds of generating the observed asymmetry are also improved by accounting for the fact that there are three spatial dimensions, which provide three independent opportunities for a largeamplitude fluctuation. We will see in Section III.4 that considering the combined contributions of several superhorizon modes and accounting for the red tilt of the primordial power spectrum further increases the probability of generating the observed asymmetry, to the point that the value for increases to greater than 0.05 for . Thus, if the perturbations on large scales are sufficiently nonGaussian, there is no need to invoke enhanced superhorizon perturbations to generate the observed power asymmetry.
In the absence of an enhancement of the superhorizon power spectrum, the variance of the quadrupole moments and octupole moments in the CMB will not be altered. Consequently, we do not expect significant constraints on such models from the GZ effect. We note though that the specific realization of modes outside our subvolume will still source quadrupole and octupole anisotropies in the CMB. For realizations that generate a large power asymmetry, the GZ contribution to these anisotropies would likely be larger than expected from theoretical predictions of and and aligned with the power asymmetry. However, this effect may be difficult to disentangle from the monopole power modulation described in Section II.2.1, and we leave a detailed analysis of this observational signature to future work.
Furthermore, the ratio for a given value of and can be significantly reduced if we consider mixed Gaussian and nonGaussian perturbations. Using the mixed perturbation scenario introduced in Eq.(3), (with ),
(27) 
In the last inequality, we employ the fact that is required for the nonGaussianity in the field to be weak enough to make the contribution to negligible. In this case, could be sourced by a 1 fluctuation in the field if . The possibility of using a mixed curvatoninflaton model to generate a scaledependent asymmetry using a single large superhorizon perturbation was explored in Erickcek et al. (2009).
Iii Statistical Anisotropy in the CMB Power Spectrum
iii.1 Numerical tests
We now present numerical tests of our analytic expressions for the dipole power modulation in the case of local nonGaussianity. We will work in the SachsWolfe (SW) regime: we only consider
(28) 
Therefore, for local nonGaussianity, Eq.(1), the temperature fluctuation is given by:
(29) 
We generate 10000 simulated Gaussian SW CMB skies using
(30) 
for ; the primordial power spectrum is given by
(31) 
with and (from Planck TT,TE,TE+lowP column in Table 3. of Ade et al. (2015c)), and as the pivot scale.
Then it is easy to generate nonGaussian SachsWolfe CMB temperature maps using Eq.(29) for a constant . Unlike most CMB analyses, we will keep the dipole variance term . A nonzero is used to model the dipolar anisotropy in density fluctuations on the scale of the observable universe (from the perspective of the large volume ). However, note that the we use is not what we would measure for the CMB dipole, even if we assume that the dominant contribution to the measurement of the dipole from our local motion Aghanim et al. (2014) has been subtracted out. This is because, for adiabatic fluctuations, the leadingorder contribution to the observed CMB dipole from superhorizon perturbations exactly cancels the Doppler dipole generated by the superhorizon perturbations Erickcek et al. (2008b); Zibin and Scott (2008).
It is convenient to set the monopole to zero for the purpose of studying dipole modulations; otherwise, the cosmic variance power asymmetry (i.e. the contribution that is not due to local nonGaussianity) will be different for the weakly nonGaussian realization compared to the Gaussian realization from which it is generated. Therefore, in this section, we use numerical realizations with nonzero values only when testing the monopole modulation formula. The expression for is infrared divergent, so we assume an infrared cutoff ; the same cutoff scale is used to compute the expected amount of monopole power modulation . Numerically,
(32) 
The cutoff can be related to the number of superhorizon efolds of inflation (if interpreted as such) as: . The above integral gets most of its contribution from , and therefore can be well approximated by LoVerde et al. (2013):
(33) 
for , and for . Additional details about our numerical results can be found in Appendix D.
iii.2 Monopole modulation ()
The normally distributed monopole shift amplitude for a local nonGaussian model is given by Eq.(14), and can be written in terms of using Eq.(32) as:
(34) 
The necessary infrared cutoff has already been set by the value of in Eq.(33) to compute . Although it is not possible to observe monopole modulations for a constant local , we can test the expected modulations assuming a value of . The probability distribution of the shift for any is
(35) 
where the variance has contributions from the Gaussian realization and the nonGaussian coupling to the realization of long wavelength modes: . For our numerical tests, is the variance of measured in Gaussian CMB maps. The quantity we measure for from each realization of CMB maps is:
(36) 
where are the input angular powerspectrum values used to obtain the set of numerical CMB maps, and is the angular power spectrum of a particular realization of that set of CMB maps, and . In Figure 2, we plot the distribution of , using Eq.(35), for , along with the distribution obtained from the numerically generated SachsWolfe CMB maps.
iii.3 Dipole modulation ()
A dipole modulation of the power spectrum defined as in Eq.(16) generates a hemispherical power asymmetry with the same amplitude . Therefore, we will look at the quantity:
(37) 
with and , where and refer to two hemispheres in some direction . We will consider s in three orthonormal directions , , and on the sky. Each in a particular direction is normally distributed with zero mean. The variance can be measured from the numerical realizations of Gaussian SachsWolfe CMB maps and depends on the CMB multipoles used in Eq.(37) and the value of the s. For nonGaussian maps, the distribution of the s have an increased variance given by: , where is given by Eq.(17). The power asymmetry dipole amplitude for each CMB sky is then . The probability distribution function (pdf) of is the distribution (or the Maxwell distribution):
(38) 
where . Figure 3 shows that the distribution of asymmetry amplitudes obtained from the CMB realizations agree extremely well with the distribution given above. Note that only is measured from the numerical maps; is directly computed for a value of using Eq.(17).
iii.4 Statistical analysis
In this section, we present examples of how we can perform a statistical analysis using the results from the previous section for the distributions of power modulation on the SachsWolfe CMB sky. While direct comparison of the amplitudes obtained in our SachsWolfe CMB realizations with the reported values of power asymmetry is not possible, we can make a connection between our simpler case and the asymmetry in the observed CMB sky by using the value of the asymmetry. For a given measurement of and the normalized pdf for , [which in our model depends on , see Eq.(38)], the value is simply given by ; i.e. it gives the probability that the observed value of the asymmetry amplitude is greater than some threshold value . We find that an asymmetry amplitude of is approximately , i.e. a value of with respect to the distribution of obtained in our Gaussian SachsWolfe CMB maps. This is approximately equal to some of the more recent reports for the significance of the hemispherical power asymmetry Akrami et al. (2014); Ade et al. (2015d). Therefore, we will use as the value of the asymmetry when making connections with the observations of the anomaly.
When we have a measurement of the power asymmetry amplitude , we can write the likelihood for as , whose expression is given in Eq.(38). From this likelihood, we can infer the posterior distribution for given a measurement of . We can interpret the statistics in different ways:

We can use any power asymmetry as a signal of local nonGaussianity. Using only the largescale CMB multipoles (), for a given value of , we can obtain the posterior distribution for (averaged) for the corresponding range of scales. In Figure 4, we plot the posterior for a few values of the asymmetry , assuming a uniform prior on .

We can combine the largescale bispectrum constraints on with the constraints from the power asymmetry . For this, we use a rough estimate of for of (estimated from Figure 2 of Smith et al. (2009)). We assume that the posterior from WMAP is a normal distribution. However, since the power asymmetry is only sensitive to the magnitude of and not the sign, we use the folded normal distribution (given ):
Then, we multiply the above pdf with to get the combined likelihood from which we can get the posterior for after normalizing. We show the combined posterior distribution of for a few power asymmetry amplitudes in Figure 5.

Although the power asymmetry data alone () prefers as the most likely value (or when the bispectrum constraints are also applied), the probability of an asymmetry increases whenever . In Figure 6 we quantify how the probability of an observed dipole modulation changes as increases. For example, the value of for changes to for (which is within the window of the largescale bispectrum constraint). In other words, even an amplitude of nonGaussianity well below renders the observed asymmetry less “anomalous.”
iii.5 Bayesian evidence
The previous section demonstrated that the power asymmetry data can be used to constrain nonGaussian models, and that the amplitude of the observed asymmetry is less “anomalous” when nonGaussianity is included. However, we also need to ask whether the data are such that the nonGaussian model is preferred over the Gaussian.
To compare the posterior odds for different models , given the data , we compute the Bayes factor
(39) 
where the factors in the numerator and denominator are the model likelihoods for models 1 and 2 respectively. In the simplest comparison, we take as the only parameter of the models. The data we consider include the measured amplitude of the power asymmetry and the CMB constraint on the amplitude of the local bispectrum on large angular scales. For an introduction to Bayesian statistical methods applied to cosmology, see for example Trotta (2008).
The nonGaussian model reduces to the isotropic Gaussian model for (while the probability of the power asymmetry remains nonzero). In that case the evaluation of the Bayes factor can be simplified and a direct Bayesian model comparison can be done using the SavageDickey density ratio (SDDR) Trotta (2007, 2008). The SDDR is given by,
(40) 
where is the more complex model (nonGaussian in our case) that reduces to the simpler model (Gaussian) when the set of parameters goes to (). Here, is the posterior for (plotted in Figure 5) and represents the prior for the parameter in the complex model . Our current case only has one parameter () and one datum (the dipolar asymmetry ). While there may be other interesting possibilities to consider for the prior probability of , we illustrate the calculation of above using the constraint on the parameter from largescale bispectrum measurements as reported by the WMAP and Planck missions as the prior.
In Table 1, we list SDDR for a few values of . For the prior, we have used the folded normal distribution for which is a rough estimate of for the largest scale i.e. up to from Smith et al. (2009). Note that the only value from the prior pdf that is used to compute the SDDR is , so the above consideration from largescale constraints is the same as using a uniform prior for in the range . If the prior range is expanded, then the magnitude of increases thereby reducing the evidence for nonzero . For (whose value roughly corresponds to observed ), the strength of evidence for a nonzero is between weak and moderate in the empirical (Jefferys’) scale Jeffreys (1961) quoted, for example, in Table 1 of Trotta (2008).
value  SDDR ()  

0.02  0.5511  1.1362  0.128 
0.04  0.0381  0.6174  0.482 
0.05  0.0043  0.3211  1.136 
0.055  0.001  0.2012  1.603 
0.06  0.0003  0.1123  2.186 
The results in Table I show that the data we have used, at least in this simple analysis, show no more than a weak preference for the nonGaussian model. A more thorough analysis is unlikely to change this conclusion very much: in Gordon and Trotta (2007), the authors use earlier studies of the power asymmetries in the WMAP data to put the best possible Bayesian evidence of corresponding to odds (, weak support). The method to compute the maximum possible Bayesian evidence is based on Bayesian calibrated values Sellke et al. (2001). The value used from the data analysis of the 3year WMAP maps was Eriksen et al. (2007). If one instead used , which is approximately the level of significance from various more recent analyses of Planck and WMAP temperature anisotropy maps, the best possible Bayesian evidence in favor of the anisotropic model becomes corresponding to the odds (). A value of 0.0003 (about ) is necessary to obtain a best possible , which implies strong evidence Gordon and Trotta (2007).
Iv Beyond the local ansatz
The previous section discussed in detail the effect of localtype nonGaussianity with constant that couples a gradient (induced by superhorizon modes) across the CMB sky to the observable modes. We demonstrated that a dipolar asymmetry is expected in models with localtype nonGaussianity. Of course, local nonGaussianity as the source of the asymmetry is only compatible with the data if we restrict ourselves to the largest scales. Both the amplitude of the asymmetry and the amplitude of nonGaussianity must sharply decrease on smaller scales. The question then is whether there is a different model of nonGaussianity that is consistent with all observational constraints and generates the observed asymmetry in detail. If so, does the current data favor this model over the isotropic, Gaussian assumption? Could future data ever favor such a model?
To address some of these questions, we will first demonstrate that scaledependent modulations are a generic feature of nonGaussian models other than the local model. We will then construct a scenario that is more likely to be preferred by the data by considering scaledependent local nonGaussianity. Finally, we will provide examples of evaluating the Bayesian evidence for this scenario. Although a model of nonGaussianity beyond the local ansatz may add more parameters, if the model has other consequences in the data we might hope to find more evidence for it. This is particularly true if the model has measurable effects on smaller scales, where the usual cosmic variance for Gaussian models is smaller.
iv.1 Power asymmetry from general bispectra
We can easily extend the inhomogeneous power spectrum calculation in the presence of local nonGaussianity to other bispectrum shapes. For example, consider that a Fourier mode of the Bardeen potential is given by Baytaş et al. (2015):
(41) 
where as before is a Gaussian field. The kernel can be chosen to generate any desired bispectrum and the dots represent terms higher order in powers of (which generate treelevel point correlations). Considering only the generic quadratic term, the power spectrum in subvolumes can be computed as in the case of the local bispectrum (see Appendix B), and we get :
(42) 
From the form of the above equation, one can see that a dependent power modulation is a feature of nonlocal nonGaussianity i.e. the dependence of the kernel is carried by the modulated component of the power spectrum in the small volume. The kernels for local, equilateral and orthogonal bispectrum templates are Baytaş et al. (2015):
(43) 
If one uses the kernel for equilateral or orthogonaltype nonGaussianities, then the monopole shifts are not infrared divergent. However, the magnitudes of modulation (both monopole and dipole) are smaller compared to the local case i.e. a very large amplitude of or is necessary for the effect to be interesting. For example, we plot the expected modulation amplitude for local, equilateral and orthogonaltype nonGaussianities in Figure 7. In Figure 8, we illustrate that for local, orthogonal and equilateral bispectra, the power asymmetry is generated by perturbation modes that lie just outside the horizon. The quasisingle field model Chen and Wang (2010) may also be interesting to consider: it has a scaleindependent bispectrum with a kernel that varies between the local and equilateral cases depending on the mass of an additional scalar field coupled to the inflaton.
In general then, if the power asymmetry is coming from mode coupling, the fact that the observed asymmetry falls off on small scales implies that shorter scales are more weakly coupled to superhorizon modes than larger scales are. This is possible with either a scaleindependent bispectrum (as the equilateral and orthogonal cases above demonstrate) or with a scaledependent bispectrum.
iv.2 Generating a scaledependent power asymmetry
To match the observed scale dependence of the power asymmetry anomaly, the strength of coupling of subhorizon modes to the long wavelength background must be scale dependent. The relevant scale dependence in this context can be fully parametrized by introducing two bispectral indices that capture the scale dependence in our observable volume and a more general coupling strength to the long wavelength modes:
Here turns off any power asymmetries on shorter scales. The parameter enhances the sensitivity of the model to infrared modes (as used in Schmidt and Hui (2013); Agullo (2015)). In the case , the dipole asymmetry would be infrared divergent in a universe with a scaleinvariant or redtilt power spectrum. Notice that scaleinvariant bispectra always have , so any scaleinvariant bispectrum that increases IR sensitivity also increases the expected asymmetry on smaller scales. Finally, although we have used to label the coefficient above, a similar expression can be derived from higherorder correlation functions (e.g., to capture the effects of Kenton et al. (2015)). Previous discussions of the power asymmetry from scaledependent nonGaussianity include Erickcek et al. (2009); Lyth (2013); Kohri et al. (2014); Lyth (2015); Firouzjahi et al. (2014); Byrnes and Tarrant (2015); Agullo (2015).
In principle, additional data could eventually constrain all of the parameters introduced above (or at least their values on subhorizon scales). However, here we will consider only one additional measurement (the largescale power suppression) and so we will restrict our attention to the case with just one additional parameter. We take a localshape bispectrum with an amplitude that depends on the scale of the short wavelength mode as
(45) 
In terms of the parameters in Eq.(IV.2), and .
In addition to a scaledependent power asymmetry and a scaledependent bispectrum amplitude, a scaledependent local nonGaussianity also generates a scaledependent modulation of the powerspectrum amplitude. This is the monopole power modulation () discussed in Section II.2 (a similar point was made in Lyth (2015)). When localtype nonGaussianity has a scaleindependent amplitude, the power spectrum amplitude is modulated similarly at all scales, thereby making the effect unobservable. However, the scaledependent case is more interesting as it makes the power modulation scale dependent and therefore an observable effect. This can be easily seen by generalizing Eq.(12) for the case of :
(46) 
As previously discussed, is normally distributed with zero mean and the variance requires a cutoff to limit contributions from arbitrarily large modes, which we have earlier parametrized as the number of superhorizon efolds of inflation. While the above formula is only valid for small modulations , Figure 2 shows that the formula is quite accurate at least up to . There is an additional subtlety because, in the presence of a monopole shift, the observed bispectrum on large scales will not be exactly of the form in Eq.(45). However, the difference is small for small .
Consider a simple example of scaledependent local nonGaussianity given by , and . In multipole space, one can approximate and . These numbers are chosen to facilitate comparison with the scaledependent modulation model results in Aiola et al. (2015) (see Table I therein). In Figure 9, we plot , (the expected amplitude of the power asymmetry in a particular direction due to ), and (the expected shift in the amplitude of the power spectrum due to ), as a function of the multipole number . The purpose of the figure is to illustrate how scaledependent local nonGaussianity can produce more than one signature in the CMB. Therefore, we have not included the effect of the Gaussian cosmic variance, which would add variance to both the power asymmetry and monopole modulation amplitudes; this becomes important when performing parameter estimation of and . While such a full parameter estimation analysis is beyond the scope of this work, we illustrate in the next section that, in a simplified context, adding the constraint can be useful in some situations.
Notice that the scaledependent nonGaussian model of Eq.(45) also generates asymmetries in the spectral index:
(47) 
where is the spectral index of the Gaussian field and