Cosmic variance in inflation with two light scalars
Abstract
We examine the squeezed limit of the bispectrum when a light scalar with arbitrary nonderivative selfinteractions 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 subvolumes. This means that observations of primordial nonGaussianity cannot be used to uniquely reconstruct the potential of the hidden field. However, the local bispectrum induced by modecoupling 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]AnneSylvie 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
Contents
1 Introduction
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 nonGaussianity are excellent ( [1]), they do not yet cross even the highest theoretically interesting (and roughly shapeindependent [2]) threshold to rule out . Future data will help probe the remaining parameter space, and either stronger constraints or detection of nonGaussianity 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 nonGaussianity, the power spectrum measured in subvolumes 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 subvolumes, the trispectrum can generate shifts to the bispectrum and the power spectrum in biased subvolumes (that is, subvolumes 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 subvolumes 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 nonGaussianity (e.g., through the nonGaussian halo bias [14]) when applied to subvolumes 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 subvolume of a larger, unobservable, space. If the perturbations we observe turn out to have any form of longshort 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 efolds [15, 9, 10, 11, 16, 12, 17, 13, 18]. In that case, there is not necessarily a onetoone 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 nonGaussian fields have statistics that differ in subvolumes. 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. Singleclock 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 subvolume are the statistics of the mean regardless of the level of nonGaussianity or the amplitudes of long wavelength fluctuations. In contrast, the family of nonGaussian 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 nonGaussianity () can vary by large factors in subvolumes [9, 10, 11, 18]. In fact, regardless of whether the mean statistics are weakly or strongly nonGaussian, subvolumes that are sufficiently biased (that is, whose long wavelength background is sufficiently far from the mean) all have the statistics of a weakly nonGaussian 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 nonGaussianity; it is easy to construct examples of nonGaussian fields where the shape of the bispectrum (even in the squeezed limit) changes significantly in biased subvolumes [24]. Multifield 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 nonGaussianity are independent of the long wavelength background and which are not.
An interesting example that naturally interpolates between the singleclock and curvaton (local ansatz) mode coupling is quasisingle 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 selfinteractions 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 nonGaussianity it indirectly sources in the adiabatic mode. Previous studies have shown that when the hidden sector field has a cubic selfinteraction, the degree of longshort 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 multifield scenarios have been discussed in [30, 31, 32].
In this paper we investigate how robust the quasisingle 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 subvolumes. As a simple first case, we consider the correlations generated by contact diagrams from additional (i.e., quartic and beyond) nonderivative selfinteractions of the hidden sector fluctuations. By computing the power spectrum and bispectrum in subvolumes with nonzero long wavelength background modes we will show that (similarly to the local model), any nonderivative selfinteraction of the spectator field leads to the same pattern of correlation functions in sufficiently biased subvolumes. In particular, nonzero long wavelength fluctuations induce a treelevel bispectrum locally even if the mean theory does not contain one. Furthermore, the squeezed limit scaling of the bispectrum is the same in all subvolumes, while the local amplitude of fluctuations and amplitude of nonGaussianity 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 quasisingle field scenario and compute some properties of arbitrary order correlation functions from additional selfinteractions of the hidden sector scalar. In Section 3 we compute the statistics observed in subvolumes. We discuss the results in Section 4. The appendices contain some details of the calculations.
2 NonGaussianity from quasisingle field inflation with interactions
Quasisingle field inflation [26] 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
(2.1) 
Here is related (at first order) to the curvature mode by
(2.2) 
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 nonGaussianity from the selfinteractions. 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 [33].) Although the quasisingle 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 [25], the form of the transfer vertex is generic [34] in the sense that it comes from the leading order allowed interaction between a (nearly) shift symmetric inflaton and a (spectator) second scalar. Quasisingle field inflation nonGaussianity was studied in detail with a cubic selfinteraction 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 threepoint function for from the interaction.
Since the selfinteractions of the inflaton mode will have no significant longshort coupling, in this paper we are only interested in computing correlations coming from selfinteraction vertices of . The spectator scalar may well have additional selfinteractions beyond the cubic term. Here we will consider
(2.3) 
In the next section we compute the correlations generated by these interactions in the limits appropriate for determining the effects of longshort mode coupling on point functions in subvolumes.
2.1 Quantization and inin formalism
To derive the late time correlation functions from the inflationary scenario above we use the inin 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
(2.4) 
where is the interaction Hamiltonian in the interaction picture, are the time ordering and antitime 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 sameorder terms together. One way of arranging those terms together, convenient for our purposes here, is the commutatorform:
(2.5) 
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:
(2.6)  
Here and obey the usual commutation relations. Assuming a quaside Sitter background, the mode function for satisfying the equations of motion of the quadratic Lagrangian Eq.(2.1) (with ) is
(2.7) 
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 scaleinvariant background evolution, this is:
(2.8) 
The mode function for the inflaton fluctuations are:
(2.9) 
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
(2.10) 
2.2 The point functions from selfinteractions 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 fourpoint functions, respectively, in the following way:
(2.11)  
For higher order correlations we define
(2.12) 
Evaluating the exact higher order correlation functions in quasisingle 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 longshort coupling in soft limits is required, which simplifies the calculation. In this section we review the previously calculated quasisingle field bispectrum and then use simple arguments (well understood for the bispectrum in e.g., [30]) to determine the relevant features of higher order correlation functions. Additional details justifying these arguments for the trispectrum can be found in Appendix A.
2.2.1 Bispectrum
The full bispectrum from the cubic interaction only, , was calculated in [26], who also gave an approximate form that accurately captures the behavior in the squeezed limit:
(2.13) 
where a scaleinvariant 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 [26]. In terms of the parameters in the Lagrangian, is
(2.14) 
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 nonGaussian fields in Eq.(3.2). The function is positive and monotonic (order 1 for and grows rapidly to as ). It is plotted in [26].^{1}^{1}1There 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 socalled 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
(2.15) 
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 longshort mode coupling than the local ansatz. If a nonzero bispectrum of the quasisingle 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).
2.2.2 Trispectrum
Adding the interaction term proportional to introduces a new trispectrum to the original quasisingle field scenario. Rather than performing a complete inin 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 [30] 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 [25]. Some details of the inin result can be found in Appendix A.
The fourpoint correlation function from the quartic interaction vertex is depicted in Fig. 2 and the late time () result can schematically be expressed as:
(2.16) 
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 :
(2.17) 
The remaining integral over the selfinteraction, 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 subvolumes when one mode of the trispectrum is unobservable (e.g., is superHubble). 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 subvolumes when both and are superHubble.) In that case the integral above is dominated by , and the result is
(2.18) 
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
(2.19) 
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:
(2.20) 
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 reevaluated and the final result agrees with expanding Eq.(2.20):
(2.21) 
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 4point exchange diagram from the cubic interaction in more detail, and we will comment later on incorporating it in the expression for the full nonGaussian 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 selfinteraction 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 modemode coupling.
In particular, we consider the point function arising from a contact diagram with an interaction term
(2.22) 
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:
(2.23) 
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 superHubble or long modes within a subvolume) and short modes assumed to be all of equal length , the prefactor can be written as
(2.24) 
The remaining integral will be suppressed except when the most UV mode is nearly at the Hubble scale, :
(2.25) 
Putting everything together gives the general expression for the point correlation function with modes taken to be long:
(2.26) 
This behavior can be captured by a template similar to the lower order expressions
(2.27) 
where is a numerical coefficient that can be chosen to help fit the exact result.
3 Cosmic variance from superhorizon modes
Once the postinflationary correlation functions have been determined, it is a purely mathematical exercise to compute the statistics in spatial subvolumes at a fixed time. For this purpose, we introduce in Section 3.1 a formalism to build up the nonGaussian field from its correlation functions. We introduce a split between long and short modes to derive the nonGaussian field observed in subvolumes. Of course, since in the quasisingle field case we also know the dynamical model generating the fluctuations, we could just as well do the whole calculation within the inin formalism. We do an example inin calculation in Section 3.3 to confirm that the two methods agree. In addition, Section A.2 in the Appendix contains an inin calculation that demonstrates aspects of the dynamical calculation that are distinct from the purely statistical effects of subsampling.
3.1 Latetime correlation functions and superhorizon modes
In order to provide a framework for our calculations, we first establish our notation for generic nonGaussian fields in the postinflationary 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 subvolumes.
The nonGaussian mode can be expressed as a sum of terms that are local or nonlocal functionals of Gaussian random fields :
(3.1) 
In Fourier space, this series is
(3.2) 
where is just proportional to the Gaussian field^{2}^{2}2 In the absence of any modecoupling 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 subvolumes. The notation for that case is clearer if we allow for the possibility and momentumdependent.. The higher order terms are convolutions of Gaussian fields. For example,
(3.3)  
(3.4) 
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 nonlinear 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 scaleinvariant. In that case, the kernels can be normalized so that and agree with, eg, the usual coefficients of the local templates^{3}^{3}3This works for kernels with nonvanishing 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 nonGaussian field that gives the statistics observed in a subvolume 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 cutoff . We define as with long wavelength modes. For example,
(3.5)  
(3.6) 
Then the observed field will be
(3.7) 
The numerical prefactors 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 longwavelength modes.
The first line in the equation above is the linear field observed in the subvolume, so to compute the observed nonGaussian correlations in terms of the observed power spectrum, the should be reexpressed in terms of this field. That shift can be absorbed into a redefinition of the kernels . In other words, an observer in the subvolume sees statistics generated by
(3.8) 
where
(3.9) 
The functional is defined by
(3.10) 
with an effective kernel depending on the higher order functionals containing the proper number of long modes
(3.11)  
where we have denoted , , etc. Higher order are defined similarly:
(3.12)  
In the next subsections we work out the kernels , and higher order ones when the inflaton has the quasisingle field coupling to an additional light scalar with arbitrary nonderivative selfinteractions. Because our goal is to understand how long wavelength fluctuations affect the power spectrum and squeezed limit of the bispectrum in biased subvolumes, 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
We can now use the late time correlation functions computed in Section 2 to express the nonGaussian perturbation as an expansion in terms of a Gaussian field, Eq.(3.2).
Since we begin by considering the field in the entire inflationary volume, . Each higher order involves integrals over momentum, a deltafunction 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 section^{4}^{4}4The nonGaussian field built with these kernels is still only approximately that of the full quasisingle 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, quasisingle 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.:
(3.13)  
(3.14) 
In order to determine the statistics observed in subvolumes, we split the Fourier expansion up into “short” modes contained within a subvolume and “long” modes with wavelengths larger than the size of the subvolume. The nonlinear 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 subvolume, 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
(3.15) 
Here,
(3.16) 
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, [26], 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 subvolume – as
(3.17) 
An infrared cutoff, , 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
(3.18) 
Notice that can only be substantially larger than when is very light. Although the derivation of the bispectrum assumed a scaleinvariant power spectrum, we can straightforwardly generalize this expression to allow for . For light fields, this is
(3.19) 
where
(3.20) 
Confirmation that this is the correct generalization if we repeat the calculation of the quasisingle field bispectrum including leading order slowroll 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 scaledependence from the power spectrum of the global volume (which is also the mean power spectrum, irrespective of subvolume size):
(3.21) 
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 nonGaussian corrections. (See Appendix B of [36] 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:
(3.22) 
where is the number of efolds of inflation before the mode exited the horizon (the “extra” efolds 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 quasisingle field inflation is equivalent to the requirement that the nonGaussian 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 nonGaussian 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 nonGaussianity 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 nonGaussianity in the large volume can differ significantly from that in subvolumes, especially when terms beyond quadratic order are allowed.