Cosmic variance in inflation with two light scalars
We examine the squeezed limit of the bispectrum when a light scalar with arbitrary non-derivative self-interactions is coupled to the inflaton. We find that when the hidden sector scalar is sufficiently light (), the coupling between long and short wavelength modes from the series of higher order correlation functions (from arbitrary order contact diagrams) causes the statistics of the fluctuations to vary in sub-volumes. This means that observations of primordial non-Gaussianity cannot be used to uniquely reconstruct the potential of the hidden field. However, the local bispectrum induced by mode-coupling from these diagrams always has the same squeezed limit, so the field’s locally determined mass is not affected by this cosmic variance.
1]Béatrice Bonga, 1]Suddhasattwa Brahma, 1]Anne-Sylvie Deutsch 1,2]and Sarah Shandera Prepared for submission to JCAP
Cosmic variance in inflation with two light scalars
Institute for Gravitation and the Cosmos & Physics Department, The Pennsylvania State University, University Park, PA 16802, USA
Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5 Canada
- 1 Introduction
- 2 Non-Gaussianity from quasi-single field inflation with interactions
- 3 Cosmic variance from super-horizon modes
- 4 Discussion
- A Evaluating new diagrams from the quartic interaction
- B Treating the exchange diagrams
- C The quasi-single field correlations when
The statistics of the primordial fluctuations beyond the power spectrum contain information about the spectrum and interactions of light particles present during inflation. Although the Planck satellite bounds on non-Gaussianity are excellent ( ), they do not yet cross even the highest theoretically interesting (and roughly shape-independent ) threshold to rule out . Future data will help probe the remaining parameter space, and either stronger constraints or detection of non-Gaussianity would provide an important clue about physics at the inflationary scale.
When the primordial fluctuations are entirely or partly sourced by a light field other than the inflaton, the correlation functions can have the interesting property that locally measured statistics depend on the realization of long wavelength modes. For example, the “local” bispectrum generated in curvaton scenarios [3, 4, 5, 6, 7, 8] correlates the amplitude of short wavelength fluctuations with the amplitudes of much longer wavelength modes. This means that in the presence of local ansatz non-Gaussianity, the power spectrum measured in sub-volumes will vary depending on how much the background fluctuations deviate from the mean [9, 10, 11, 12]. While the bispectrum can cause the power spectrum to vary in sub-volumes, the trispectrum can generate shifts to the bispectrum and the power spectrum in biased sub-volumes (that is, sub-volumes whose long wavelength background is not the mean). More generally, long wavelength modes in the -point function can shift any lower order correlation function [9, 13]. Since nearly all sub-volumes will have a long wavelength background that is not zero, statistics measured in any small region are likely to be biased compared to the mean statistics of the larger volume.
This correlation between local statistics and the long wavelength background can be used to detect non-Gaussianity (e.g., through the non-Gaussian halo bias ) when applied to sub-volumes where at least some long wavelength modes are observable. But it may also be relevant to the conclusions we can draw about inflationary physics: our entire observable universe is almost certainly a sub-volume of a larger, unobservable, space. If the perturbations we observe turn out to have any form of long-short mode coupling we must assume there is a ‘super cosmic variance’ uncertainty in comparing observations in our Hubble volume with the mean predictions of inflation models with more than the minimum number of e-folds [15, 9, 10, 11, 16, 12, 17, 13, 18]. In that case, there is not necessarily a one-to-one map between properties of the correlation functions we measure and parameters in an inflationary Lagrangian. In this paper we will explore an example where some, but not all, of the parameters of the Lagrangian are obscured by cosmic variance.
Mathematically, not all non-Gaussian fields have statistics that differ in sub-volumes. In inflationary models the influence of long wavelength fluctuations on locally observed correlations depends on how many degrees of freedom source the background expansion and the observed fluctuations. Single-clock inflation has no significant coupling between modes of very different wavelengths [19, 20, 21, 22, 23]: on a suitably defined spatial slice, the statistics in any sub-volume are the statistics of the mean regardless of the level of non-Gaussianity or the amplitudes of long wavelength fluctuations. In contrast, the family of non-Gaussian fields built from arbitrary local functions of a Gaussian field couples short wavelength modes to all long wavelength modes. As a result, the amplitude of fluctuations and the amplitude of non-Gaussianity () can vary by large factors in sub-volumes [9, 10, 11, 18]. In fact, regardless of whether the mean statistics are weakly or strongly non-Gaussian, sub-volumes that are sufficiently biased (that is, whose long wavelength background is sufficiently far from the mean) all have the statistics of a weakly non-Gaussian local ansatz. However, some properties of the correlation functions are the same in all volumes. In particular, the squeezed limit scaling of the bispectrum is always (very nearly) that of the local template [9, 11]. The preservation of the shape of the bispectrum is not a generic feature of non-Gaussianity; it is easy to construct examples of non-Gaussian fields where the shape of the bispectrum (even in the squeezed limit) changes significantly in biased sub-volumes . Multi-field inflation models can produce a wide range of correlation functions, so it is worthwhile to understand more generally which properties of any potentially observable primordial non-Gaussianity are independent of the long wavelength background and which are not.
An interesting example that naturally interpolates between the single-clock and curvaton (local ansatz) mode coupling is quasi-single field inflation [25, 26], where an additional scalar field is coupled to the inflaton during inflation. This field does not contribute to the background expansion and its self-interactions are not restricted by the approximate shift symmetry that the inflaton is subject to. Observational evidence for this ‘hidden sector’ field would be found in the non-Gaussianity it indirectly sources in the adiabatic mode. Previous studies have shown that when the hidden sector field has a cubic self-interaction, the degree of long-short coupling in the observed bispectrum is determined by the mass of the second field (with the strongest coupling coming from a massless field). The coupling between modes of very different wavelengths is captured by the squeezed limit of the bispectrum, where one momenta corresponds to a much longer wavelength than the others (e.g, ). Measuring the dependence on the long wavelength mode () in the squeezed limit of the bispectrum would reveal the mass of this spectator field [26, 27, 28, 29] (see Eq.(2.15) below). The sensitivity of squeezed limits to the mass and spin of fields in more general multi-field scenarios have been discussed in [30, 31, 32].
In this paper we investigate how robust the quasi-single field bispectrum shape is to cosmic variance when higher order correlations are included in the model. When the hidden sector field is sufficiently light, any additional correlations may bias the statistics observed in sub-volumes. As a simple first case, we consider the correlations generated by contact diagrams from additional (i.e., quartic and beyond) non-derivative self-interactions of the hidden sector fluctuations. By computing the power spectrum and bispectrum in sub-volumes with non-zero long wavelength background modes we will show that (similarly to the local model), any non-derivative self-interaction of the spectator field leads to the same pattern of correlation functions in sufficiently biased sub-volumes. In particular, non-zero long wavelength fluctuations induce a tree-level bispectrum locally even if the mean theory does not contain one. Furthermore, the squeezed limit scaling of the bispectrum is the same in all sub-volumes, while the local amplitude of fluctuations and amplitude of non-Gaussianity are subject to cosmic variance. So, although the diagrams we consider here generate cosmic variance that obscures the details of a light hidden sector field’s potential, that cosmic variance does not affect the measurement of the field’s mass from squeezed limit of the locally observed bispectrum.
In the next section we review the quasi-single field scenario and compute some properties of arbitrary order correlation functions from additional self-interactions of the hidden sector scalar. In Section 3 we compute the statistics observed in sub-volumes. We discuss the results in Section 4. The appendices contain some details of the calculations.
2 Non-Gaussianity from quasi-single field inflation with interactions
Quasi-single field inflation  involves two fields: the inflaton and a scalar , coupled to the inflaton, whose energy density does not significantly contribute to the background expansion. The Lagrangian for the coupled perturbations, and respectively, is
Here is related (at first order) to the curvature mode by
where describes the evolution of the Hubble parameter (the dot denotes a cosmological time derivative). We have explicitly written out all quadratic terms in Eq.(2.1), so the potential starts at cubic order. Since is not the inflaton field, its interactions are not restricted by an approximate shift symmetry. In general the mass of the fluctuation and the potential will depend on the original potential for , expanded about some constant background vacuum expectation value , as well as the terms coupling and .
The coupling between the fields allows the curvature perturbation to inherit non-Gaussianity from the self-interactions. Here we are only interested in the weak mixing case, that is, , since this results in coupling between long and short modes. (In contrast, in the strong mixing case, the two fields effectively behave as a single degree of freedom with a modified speed of sound with no significant coupling between modes of different wavelengths .) Although the quasi-single field model was first introduced with a very specific potential for both the adiabatic and isocurvature perturbations, with the inflaton field following a turning trajectory with a constant radius of curvature , the form of the transfer vertex is generic  in the sense that it comes from the leading order allowed interaction between a (nearly) shift symmetric inflaton and a (spectator) second scalar. Quasi-single field inflation non-Gaussianity was studied in detail with a cubic self-interaction for in [26, 25]. The relevant diagrams for computing the contribution of the spectator field to the observed correlation functions for that case are diagrammatically depicted in Fig. 1. The transfer vertex will allow to contribute to the power spectrum as well as to generate a three-point function for from the interaction.
Since the self-interactions of the inflaton mode will have no significant long-short coupling, in this paper we are only interested in computing correlations coming from self-interaction vertices of . The spectator scalar may well have additional self-interactions beyond the cubic term. Here we will consider
In the next section we compute the correlations generated by these interactions in the limits appropriate for determining the effects of long-short mode coupling on -point functions in sub-volumes.
2.1 Quantization and in-in formalism
To derive the late time correlation functions from the inflationary scenario above we use the in-in formalism [35, 19], where one computes expectation values of fields at a given time. For example, if is a product of field operators whose correlation function we want to evaluate at conformal time , we compute
where is the interaction Hamiltonian in the interaction picture, are the time ordering and anti-time ordering operators. Similarly to the procedure for computing scattering amplitudes, the expression in Eq.(2.4) can be evaluated in perturbation theory by expanding the exponentials, and gathering same-order terms together. One way of arranging those terms together, convenient for our purposes here, is the commutator-form:
The standard prescription should be applied to the lower limits of the time integrals to project onto the Bunch Davies vacuum at early time.
The perturbation fields are quantized in the usual way, with Fourier components in the interaction picture given by:
Here and obey the usual commutation relations. Assuming a quasi-de Sitter background, the mode function for satisfying the equations of motion of the quadratic Lagrangian Eq.(2.1) (with ) is
Note that the order of the Hankel function of the first kind, , is determined by the parameter , which depends on the mass of the spectator field. In the limit of scale-invariant background evolution, this is:
The mode function for the inflaton fluctuations are:
and can be obtained – up to a phase factor – from the mode functions for by taking the massless limit (). We will frequently need the late time approximation for the mode functions, appropriate for when the modes are well outside the Hubble radius. The inflaton mode function in this limit is just , while for the spectator scalar it is
2.2 The -point functions from self-interactions of the hidden sector field
In this section we apply the formalism above to compute the correlation functions of the adiabatic mode at late times. That is, we use the -field contact interactions in the interaction Hamiltonian to evaluate for when all the modes are well outside the inflationary Hubble radius.
We will assume homogeneous and isotropic fluctuations, so that the -point functions depend on independent momenta. For example, the power spectrum , bispectrum and trispectrum are defined from the two-, three- and four-point functions, respectively, in the following way:
For higher order correlations we define
Evaluating the exact higher order correlation functions in quasi-single field with additional interaction terms requires lengthy calculations because of vertices transferring power between and ( integrals and commutators in Eq.(2.5)). However, for our purposes only the degree of long-short coupling in soft limits is required, which simplifies the calculation. In this section we review the previously calculated quasi-single field bispectrum and then use simple arguments (well understood for the bispectrum in e.g., ) to determine the relevant features of higher order correlation functions. Additional details justifying these arguments for the trispectrum can be found in Appendix A.
The full bispectrum from the cubic interaction only, , was calculated in , who also gave an approximate form that accurately captures the behavior in the squeezed limit:
where a scale-invariant power spectrum has been assumed to define . The function is a Bessel function of the second kind (also known as a Neumann function) and numerically fitting the shape ansatz above to the exact result determined the numerical parameter . In terms of the parameters in the Lagrangian, is
chosen to match the normalization of the local ansatz in the configuration. Note that we have departed from the original definitions by moving a factor of into since it will simplify our convention for generic non-Gaussian fields in Eq.(3.2). The function is positive and monotonic (order 1 for and grows rapidly to as ). It is plotted in .111There it is called . The factor is due to the hidden sector cubic coupling, while the factors of come from the three transfer vertices that convert the perturbations in the hidden field back to perturbations of the inflaton. Note that although , the factor of is large and is in the allowed parameter space.
Our interest here is in the so-called squeezed limit of the bispectrum, where one of the momenta is much smaller than the other two. Denoting the long and short wavelength modes and expanding Eq.(2.13) in this limit gives
where we have used . Notice that when the fluctuation of the hidden sector field is massless (), the above expression recovers the local type bispectrum. When is massive () this bispectrum is less divergent as and so has a weaker long-short mode coupling than the local ansatz. If a non-zero bispectrum of the quasi-single field type is detected, measuring its scaling with in the squeezed limit would amount to measuring the mass of the hidden sector fluctuation (although note that even with a detection, the observational precision on this number is likely to be quite poor for the near future).
Adding the interaction term proportional to introduces a new trispectrum to the original quasi-single field scenario. Rather than performing a complete in-in derivation of the exact trispectrum, we will use dimensional analysis in a few simple limits to derive the amplitude and momentum dependence of the trispectrum in squeezed configurations (analogous to the arguments presented in  for the bispectrum). From these estimations we derive an ansatz for the trispectrum, convenient for our calculations. The amplitude of the trispectrum was previously discussed in . Some details of the in-in result can be found in Appendix A.
The four-point correlation function from the quartic interaction vertex is depicted in Fig. 2 and the late time () result can schematically be expressed as:
Notice that the integrals associated with the mixing term (inside the first set of parenthesis on the second line above) each depend only on a single momenta and are dimensionless (once fields are written in terms of mode functions, Eq.(2.6)). So, these integrals should contribute no ratios of momenta and can be approximated by . The momentum dependence of the trispectrum can be extracted from the remaining terms. The multiplicative factor on the first line of Eq. (2.16) is just related to the power spectrum of :
The remaining integral over the self-interaction, the last parenthesis of Eq.(2.16), depends on all momenta. In general, it is clear from the oscillatory form of the mode functions for that contributions from modes deep in the UV will be suppressed. So, in momentum configurations where there is a largest momenta, the dominant contribution to the integral will come when that mode finally crosses the horizon, . Here we are primarily interested in the bispectrum induced in sub-volumes when one mode of the trispectrum is unobservable (e.g., is super-Hubble). Furthermore, we are interested in the squeezed limit of that bispectrum so the momentum configuration is . (This momentum configuration can also be used to work out the correction to the power spectrum in biased sub-volumes when both and are super-Hubble.) In that case the integral above is dominated by , and the result is
where in the last line we have labelled the two (not necessarily equal) long modes , and the short mode .
Putting all the pieces together, and converting to , gives the trispectrum in the configuration
By analogy with the ansatz for the bispectrum, we can guess an approximate form for the trispectrum that is useful because it is symmetric in all momenta:
where is some numerical constant and .
This ansatz also has the correct scaling in the case, as well as when . When one momenta is very soft and the others are in an equilateral configuration the integral in Eq.(2.2.2) can be re-evaluated and the final result agrees with expanding Eq.(2.20):
This limit, together with the previous one, shows that when the trispectrum from the interaction has the same limits as the usual local type trispectrum. The numerical coefficients in Eq.(2.20) could be chosen to match the normalization to that shape. (We have not checked how well the ansatz in Eq.(2.20) works in more general configurations.)
There is also a trispectrum from the cubic interaction alone, coming from a diagram with two cubic interactions connected by a line. There will be many such exchange diagrams at higher orders. These are very interesting, but we wish to focus here on the effects of the contact terms from the series of interaction terms in Eq.(2.3), which in some ways mimics the series expansion of the local ansatz. Appendix B discusses the 4-point exchange diagram from the cubic interaction in more detail, and we will comment later on incorporating it in the expression for the full non-Gaussian field, but otherwise leave a complete discussion of these diagrams for future work.
2.2.3 Higher order correlation functions
In the previous two subsections 2.2.1 and 2.2.2 we studied various squeezed limit(s) of the bispectrum and the trispectrum. Here, we will generalize those results to arbitrarily high point functions generated by self-interaction terms in the hidden sector perturbation field. This will be needed in order to study the contribution from higher order correlation functions on lower order ones through mode-mode coupling.
In particular, we consider the -point function arising from a contact diagram with an interaction term
and transfer vertices. This can easily be estimated using similar arguments as in the previous subsections. The schematic expression of the -point function is given by:
Again the product of all the integrals associated with the transfer vertex can be approximated by . If we consider a configuration with long modes (which can be either super-Hubble or long modes within a sub-volume) and short modes assumed to be all of equal length , the pre-factor can be written as
The remaining integral will be suppressed except when the most UV mode is nearly at the Hubble scale, :
Putting everything together gives the general expression for the -point correlation function with modes taken to be long:
This behavior can be captured by a template similar to the lower order expressions
where is a numerical coefficient that can be chosen to help fit the exact result.
3 Cosmic variance from super-horizon modes
Once the post-inflationary correlation functions have been determined, it is a purely mathematical exercise to compute the statistics in spatial sub-volumes at a fixed time. For this purpose, we introduce in Section 3.1 a formalism to build up the non-Gaussian field from its correlation functions. We introduce a split between long and short modes to derive the non-Gaussian field observed in sub-volumes. Of course, since in the quasi-single field case we also know the dynamical model generating the fluctuations, we could just as well do the whole calculation within the in-in formalism. We do an example in-in calculation in Section 3.3 to confirm that the two methods agree. In addition, Section A.2 in the Appendix contains an in-in calculation that demonstrates aspects of the dynamical calculation that are distinct from the purely statistical effects of sub-sampling.
3.1 Late-time correlation functions and superhorizon modes
In order to provide a framework for our calculations, we first establish our notation for generic non-Gaussian fields in the post-inflationary universe. If the correlation functions of the scalar metric fluctuation are specified on a spatial slice at some early time (but after reheating and any other era that could have transferred isocurvature modes into the adiabatic mode) it is straightforward to determine the distribution of correlation functions observed in sub-volumes.
The non-Gaussian mode can be expressed as a sum of terms that are local or non-local functionals of Gaussian random fields :
In Fourier space, this series is
where is just proportional to the Gaussian field222 In the absence of any mode-coupling effects the coefficient of the linear term, , can just be absorbed into the variance of the Gaussian field . However, in what follows we would like Eq.(3.2) to apply in cases where the amplitude of fluctuations can differ in sub-volumes. The notation for that case is clearer if we allow for the possibility and momentum-dependent.. The higher order terms are convolutions of Gaussian fields. For example,
The kernels are symmetric in the first momenta and are chosen to reproduce the tree level -point function. The structure of the subtracted expectation values ensures that has mean zero and that the non-linear terms only contribute to the connected parts of the correlations. The coefficients and are numbers which can only be unambiguously defined when the kernels are scale-invariant. In that case, the kernels can be normalized so that and agree with, eg, the usual coefficients of the local templates333This works for kernels with non-vanishing equilateral limits, which is true of those we consider in this paper. (although notice that to keep the notation uncluttered we have not separated out the usual factors of used since , etc. are most often defined in the matter era Bardeen potential).
The effective non-Gaussian field that gives the statistics observed in a sub-volume can be found by considering Eq.(3.1) restricted to a spatial region of linear size . This field is approximately the same as that obtained from the simpler procedure of considering Eq.(3.2) with some modes having momenta smaller than a cut-off . We define as with long wave-length modes. For example,
Then the observed field will be
The numerical pre-factors account for the fact that the integrals in the are symmetric in the , so that equivalent contributions come from choosing any of the momenta (not just the last of the ) to be the long-wavelength modes.
The first line in the equation above is the linear field observed in the sub-volume, so to compute the observed non-Gaussian correlations in terms of the observed power spectrum, the should be re-expressed in terms of this field. That shift can be absorbed into a re-definition of the kernels . In other words, an observer in the sub-volume sees statistics generated by
The functional is defined by
with an effective kernel depending on the higher order functionals containing the proper number of long modes
where we have denoted , , etc. Higher order are defined similarly:
In the next subsections we work out the kernels , and higher order ones when the inflaton has the quasi-single field coupling to an additional light scalar with arbitrary non-derivative self-interactions. Because our goal is to understand how long wavelength fluctuations affect the power spectrum and squeezed limit of the bispectrum in biased sub-volumes, we will not need the exact expressions for every momentum configuration. One should keep in mind that the results below are not valid away from the squeezed limits.
3.2 Variance of scalar spectral index and amplitude of power spectrum
Since we begin by considering the field in the entire inflationary volume, . Each higher order involves integrals over momentum, a delta-function for momentum conservation, and a kernel. We choose the two first kernels and to reproduce the squeezed limits of the bispectrum and trispectrum derived in the previous section444The non-Gaussian field built with these kernels is still only approximately that of the full quasi-single field model. The quadratic kernel will generate a contribution to the trispectrum that is not present in the model, which we will ignore because it is suppressed by two factors of (and could be explicitly canceled by adding an appropriate piece to ). In addition, quasi-single field contains contributions to the correlations from exchange diagrams such as the trispectrum piece discussed in Appendix B. This ansatz for captures part, but not all of that diagram. Similarly, our higher order kernels will not capture all the contributions from all exchange diagrams. However, here we want to focus on the effects of the contact diagrams from new interactions so we leave the full discussion of exchange diagrams for future work.:
In order to determine the statistics observed in sub-volumes, we split the Fourier expansion up into “short” modes contained within a sub-volume and “long” modes with wavelengths larger than the size of the sub-volume. The non-linear terms , , etc then contribute to lower order terms in the expansion for a “short” mode when one or more of the integrated momenta are very long wavelength.
For example, when one of the momenta has a long wavelength, the term will contribute a shift to the linear piece of the field observed in a sub-volume, as written in Eq.(3.9). Using the limit of the kernel from Eq.(3.13), the linear term in the expansion of the short wavelength mode is shifted to
collects the terms coming from the expansion of the Neumann function in Eq. (2.13) as well as the normalization factors defined to recover the local shape ansatz in the equilateral limit. This expressions derived above are only correct for a sufficiently squeezed configuration of the bispectrum, , and so in particular one should not naively extrapolate the expressions above for for . However, since the only significant cosmic variance comes from the squeezed limit, the precise form of will not change the results we quote below. The value of is about for , and increases slowly (so that , for example).
In writing Eq.(3.2) also have defined the cumulative long wavelength background – a constant for any particular sub-volume – as
An infrared cut-off, , is only needed for a sufficiently light field. The long wavelength background is a Gaussian field, assumed to be constant over patches of size , with mean zero and variance
Notice that can only be substantially larger than when is very light. Although the derivation of the bispectrum assumed a scale-invariant power spectrum, we can straightforwardly generalize this expression to allow for . For light fields, this is
Confirmation that this is the correct generalization if we repeat the calculation of the quasi-single field bispectrum including leading order slow-roll corrections can be found in Appendix C.
From Eq.(3.2) and Eq.(3.19) we can calculate how locally observed statistics vary about the mean of the large volume. The locally observed power spectrum, for example, can be shifted in both amplitude and scale-dependence from the power spectrum of the global volume (which is also the mean power spectrum, irrespective of sub-volume size):
where the subscript indicates that the power is the Gaussian power only (in the large volume) and the dots include inhomogeneous terms from coupling to long wavelength gradients as well as non-Gaussian corrections. (See Appendix B of  for a careful derivation of this expression.) Keep in mind that depends on .
Performing the integral in Eq.(3.19) gives a better sense of how large the variations due to long wavelength modes can be:
where is the number of e-folds of inflation before the mode exited the horizon (the “extra” e-folds if is roughly the largest observable scale). Since , the variance Eq. (3.22) is small unless these two conditions are satisfied:
There is sufficient power in long wavelength modes, meaning , and
There is sufficient coupling between long and short wavelength modes, which for quasi-single field inflation is equivalent to the requirement that the non-Gaussian field coupled to the inflaton is sufficiently light, . Notice that the value of fixed by observation essentially determines how heavy the field can be before long wavelength modes are irrelevant.
For a sufficiently light field and a sufficiently red power spectrum, this result diverges as the size of the large volume goes to infinity (i.e. ). However, it is reasonable to restrict ourselves to volumes where can be defined as a small fluctuation. Imposing constrains , for example, if is set to the Planck value and on all scales.
In general, the probability that locally observed quantities differ significantly from the global mean depends dramatically on the amplitude and shape of all correlations present in the non-Gaussian statistics in the large volume [9, 10, 11]. However, to demonstrate how important cosmic variance can be even in a very simple case, consider the effect of the quadratic term () only, with a very conservative restriction to weak non-Gaussianity in the large volume. That is, we require that the contribution to the power spectrum in the large volume is small (), which requires . This can be enforced even for if . Note that this restriction simplifies the calculations, but is otherwise not required: the level of non-Gaussianity in the large volume can differ significantly from that in sub-volumes, especially when terms beyond quadratic order are allowed.