Scale Dependent Local Non-Gaussianity from Loops
We analyze multi-field inflationary systems which yield strongly scale dependent non-Gaussianity with a shape that is very close to the local shape. As in usual multi-field models, the non-Gaussianity arises from the non-linear transfer of scalar field fluctuations to curvature perturbations. Here we consider models in which higher order terms (loops) dominate over the lowest order source of non-linearity. The magnitude of non-Gaussianity depends on an infrared cutoff which is determined by our observational probes measuring non-Gaussianity. In our models, the running is positive and large () on CMB scales. The magnitude of the bispectrum is maximally of order , and grows on small scales. This can lead to interesting signals for large scale structure.
With the advent of precise cosmological data, it is now possible to constrain models of inflation by the measured magnitude and scale-dependence of correlated temperature perturbations in the cosmic microwave background (CMB) and from tracking density perturbations in dark matter from measuring the Large Scale Structure (LSS) of our universe. In these observations, it is found that the primordial perturbations coming from inflation are Gaussian to a remarkable accuracy, in agreement with the predictions of most single field models of inflation.
Non-Gaussianity (NG) can be quantified by the magnitude of the bispectrum denoted (this is usually quoted at the equilateral point in momentum space where all three momenta are equal). For most slow-roll models, is smaller than 1 [1, 2]. By comparison, the most recent constraints from WMAP5  data are for the local shape and for the equilateral shape . The Planck satellite is expected to improve the bounds to . There are also a large number of running and upcoming experiments probing LSS scales (such as LSST, DES, SDSS, etc.) and they may allow us to eventually probe non-Gaussianity on smaller scales.
In this note, we shall consider multi-field models with a large bispectrum (three-point correlation function) that is strongly scale dependent111There has been much recent work in calculating the bispectrum and trispectrum in multi-field inflation, for some recent references see [6, 7, 8, 9, 10, 11, 12].. The running is positive (or blue which means that the NG grows as increases) and can be achieved while keeping the power spectrum nearly scale invariant. It arises from loops (or higher order terms in the local ansatz) and the shape of the bispectrum is very well approximated by the local shape multiplied by a logarithm. We provide a consistent setup where the 1-loop effect dominates the bispectrum while giving a subdominant contribution to the power spectrum, and where higher loop contributions can be neglected. Since the running is positive, we can engineer a set-up where the curvature perturbation on CMB scales are extremely Gaussian while having a detectable NG on LSS scales.
Running NG has already been considered in the context of DBI inflation [13, 14]. This model can have a strong NG signal due to a small and varying sound speed for the inflaton fluctuations . The amplitude of the 3-pt can strongly run with scale if the sound speed varies but the running of the sound speed is exactly cancelled by the quickly varying Hubble constant along the trajectory. This is the key point of this type of model where the potential is steep but the inflaton moves slowly because of a speed limit. This causes the power spectrum to be scale invariant while the bispectrum can run wildly [16, 17].
The prospect of detecting large NG with large scale structure data has spurred much activity recently. LoVerde et al  have examined the possibility of using cluster counts and the galaxy bispectrum to constrain running . It was also realized in [19, 20], that NG of the local shape can induce a scale dependence of the galaxy/halo bias (see also [21, 22, 23, 24, 25, 26]). This effect can be easily found in the data and it results in a competitive bound on NG with local shape . At the time of this writing, there exists no significant experimental bound on the running of NG with scale. Recently, Sefusatti et al  argued that Planck could bound , the running of non-Gaussianity, with a precision for a local (equilateral) shape of non-Gaussianity.
In our models, we find NG with a (nearly) local shape with a scale dependence such that the NG signal grows on small scales. The magnitude of the bispectrum grows with with a model independent running of at CMB scale and on LSS scale. The strongest constraint on the magnitude of NG arises from . We find that can be achieved in principle. We also calculate the trispectrum , which also runs. Before getting into the details, we summarize the basic idea and results.
2 Scale Dependence from Loops
Local shape NG can be obtained in multi-field models of inflation, where each field is Gaussian but a non-linear relation between the inflaton perturbations and curvature perturbations induces NG. The original definition of the local ansatz for the curvature perturbation was done in real space 
where is the Gaussian piece of the curvature perturbation. in this formula is by definition scale invariant. In momentum space, the above ansatz leads to the following bispectrum
where denotes a convolution, is the power spectrum (which is assumed to be scale invariant, for simplicity) and defines the local shape. Many multi-field models (such as curvatons [29, 30]) have local scale invariant NG of this type. The NG can also be scale dependent even if the shape is nearly local; for example, this is expected to happen when the NG is generated throughout the whole trajectory as opposed to simply at some fixed later time, such as in curvaton models. A particular model with this feature was considered by Byrnes et al [31, 32], where the scale-dependence arises from the dependence of on the (time-dependent) slow-roll and Hubble parameters. In their case, the NG decreases on small scales.
This higher order contribution to the bispectrum has a structure similar from a loop contribution as it involves an integral over internal momenta. The integral converges in the UV but contains IR divergences if the power spectrum is nearly scale invariant. One can ‘regulate’ this divergence by introducing an IR cutoff in momenta 222These loops have been called c-loops . They must not be confused with q-loops, or loops coming from the expansion of the quantum evolution operator prior to horizon crossing . There has been much discussion recently on the physical significance of the IR divergences in loop calculation in inflation. For c-loops, this IR cutoff is physical and depends on the observational probe and on how we measure the zero mode of curvature perturbations. We will justify this point of view in more detail in Sec. (3.1).. Doing so, the shape of this term is close to local up to a log [33, 36]
If this term dominates the bispectrum, we will have a scale dependence with a running of order . As we will show later, the cutoff is well approximated by the size of the universe today such that around CMB scale and . The NG grows with scale becoming more important for smaller wavelength. Needless to say this is the interesting case as it gives rise to a stronger signal for LSS.
Recently, Cogollo et al  and Rodriguez et al  have argued that loops can dominate in a particular 2-brid model. While their idea is very similar to what we propose, their particular model suffers from a problem pointed out in . One of the fields that is assumed to follow a smooth classical trajectory is actually dominated by its quantum fluctuations, undermining part of their analysis.
As we will show, the field that gives rise to NG in our model is also dominated by its quantum fluctuations. But this field plays no role in the inflationary trajectory and there is no inconsistency. We consider multi-field models of hybrid inflation where the inflationary trajectory is dictated by a single field but the surface of reheating (determined by when an extra waterfall/tachyon field starts condensing) fluctuates due to two fields [39, 40] (as originally envisioned by [41, 42] – see also [43, 44, 45] for similar models).
In section 3, we describe the detailed set-up for the model, and describe the infra-red momentum cutoff. In section 4 we compute the power spectrum, and in section 5 we compute the bispectrum and trispectrum. We conclude in section 6 with a discussion of these results.
3 Multi-Field Model
A simple way to move beyond single field slow-roll and generate NG is to have multiple fields. This type of model can quickly become very complicated and in order to simply illustrate the main physical effect of interest (namely large scale dependent NG from loops), we will consider a very simplified set-up. More general models and in-depth analysis of the model we present is left for future work. Consider a model of hybrid inflation with two real light scalar fields ( and ) and a waterfall field which ends inflation when it becomes tachyonic and condenses. In this paper, we will consider a rather general action, a more detailed and worked example is given in Appendix A. The action is (we follow the notation of ):
The only coupling between and are through the tachyon which acts as a mediator or messenger. The form of is taken to be
The function interpolates from large and positive values (in Hubble units) during inflation to negative values after the system crosses a critical line in field space. Therefore during inflation, has a large positive mass, its vev is driven to zero and its potential vanishes. Because of its large mass, this field will not fluctuate and it can be integrated out of the theory. In this model, inflation ends suddenly when the mass of the tachyon vanishes, which occurs on a line in field space parameterized by
where the index “” denotes the value of the fields at the end of inflation. During the inflationary phase, and have no direct coupling. To simplify further, we assume that and we refer to as the inflaton from now on. The Hubble scale is then approximately given by
and is a “hidden” field during inflation which fluctuates but without much impact on the total energy density of the Universe. Nevertheless, its quantum fluctuations are still important as they will be felt as ripples on the surface of reheating. Indeed, at different point in space, the (slightly) different value of will mean different critical value for the inflaton resulting in more or less inflation in these different regions. This correlates directly in curvature perturbations (See Fig. (1))
Since the quantum perturbations of mainly affect the surface of reheating, this system is well amenable to analysis through the (or separable universe) formalism . The idea is that the curvature perturbation on large scales is simply given by the perturbation in the number of efolds for each trajectories
where the curvature perturbation is given by fluctuations of the scale factor and the difference in number of efolds is from a initial flat hypersurface to a uniform energy density final hypersurface. This formula does not take into account possible interactions between the various fields inside the horizon (on small scale) and it is only valid after horizon crossing where the evolution of the curvature perturbation is classical333The formalism will not account correctly for multi-field effects for modes inside the horizon. In our case, because the fields are uncoupled during inflation, we can solve for and are horizon exit independently and follow the subsequent evolution of with the formalism..
The surface where inflation ends Eq. (7) is not a uniform energy density hypersurface and a correction term must be included as discussed in [47, 44]. The correction term is very small in the hybrid scenario where the potential is very flat and it will be dropped in what follows. The number of efolds is given by . For the case where the classical trajectory is determined by a single field , one has
where the critical value of depends on the value of the field at the end of inflation (we dropped the subscript and unless otherwise specified444The field is evolving stochastically and the value of the field at the end of inflation is the sum of all fluctuations created for each mode as they exit the horizon.) and refers to horizon crossing for a given mode. By varying and then with
where we denote the zero mode of by , that is (for notational simplicity, the bar is omitted in any derivative subscript). We get at second order (using )
where denotes derivatives with respect to . This can be reproduced using the formula of Vernizzi and Wands , for the case albeit they implicitly assume that all fields obey their equation of motion which is not true here for the field . It is simple to show that where the slow-roll parameters are
The terms with involve derivatives of slow-roll parameters and will therefore be suppressed. To simplify the formula and the analysis we will consider the case where the slow-roll parameter at horizon crossing and at the end are equal, . This is not true in many models and we will discuss at the end how that would affect our results. We thus drop all subscript referring to the time of evaluation. The mean of Eq. (12) is non-zero and as it is we will generate a one-pt function. To ensure that the mean is zero we can subtract a constant piece (keeping only the leading terms)
which is of the form Eq. (1).
Note that this series terminates if
the function is such that and higher derivatives are small.
and higher derivative contributions are small.
In this type of model, the function could be anything and in the case where the quadratic piece in will dominate over the linear piece (in ) which ensures that the loop contribution to the bispectrum will dominate
as we advocated earlier. In order for the power spectrum to be nearly scale invariant we will still need the piece to be the dominant contribution to the power spectrum. There is no contradiction since the linear perturbation in does not contribute to the bispectrum (or gives a very small slow-roll suppressed contribution). Furthermore, in the case where the higher derivatives of are suppressed, the higher loop contribution can be neglected, ensuring a consistent truncation.
Another important point is that for the loop to dominate, the zero mode of at the end of inflation ( which is the mean averaged over the size of the universe at the end of inflation) must be smaller then the 1- deviation value of the perturbation around the mean. Taking the quantum perturbation to be of order , we must have . This is better seen in a specific model such as the one presented in Appendix A. There, we use a model where such that and and the series truncate. In such models it is clear that the quadratic term dominate over the linear piece when
It is then clear that the field has to behave stochastically and is in no way following a classical equation of motion. The fact that has essentially no effect on the inflationary dynamics prior to the reheating tells us that the stochastic behavior is unimportant during inflation. The value of , being stochastic, could have any value and it is therefore a free parameter. Before going into more details of the calculation, we need to discuss the choice of IR cutoff in the loop calculation.
3.1 The IR cutoff
There has been much discussion in the literature about the choice of cutoff that should be used in loop calculations. For the calculation of quantum loops in the in-in formalism (prior to horizon exit), the correlations function of scalars appear to be sensitive to this choice of cutoff, and there is no clear understanding of how this cutoff should be set. But for the c-loops which we consider in this paper, the situation is considerably simpler and there is a natural choice of cutoff . We will define the observed zero modes of the fields as
where is the largest scale over which we have measured the fields. The perturbations of the fields are then defined as .
When computing correlators of , we are actually interested in the correlations functions of the perturbations e.g. . From the definition of the perturbations, we see that the effect of subtracting the zero mode is to remove all Fourier modes with momentum . Hence for , and zero otherwise. Similarly, we find
The effect is to include a cutoff on any momentum integral. Due to the cutoff, the correlation functions will have an explicit dependence on . This can be traced back directly to the fact that we are calculating correlation functions of perturbations like , which have a direct dependence on through . In this formalism, it is clear that all the dependence on comes from the variation in the zero mode as a function of as was discussed in more details in  (see also ).
To summarize, there is a natural cutoff determined by the biggest scale on which we are able to measure the background zero mode of curvature. This is maximally the size of the universe today . This coincides with the lowest perturbations that are possible to observe now. Since there are about 5 efolds between when the lowest observable wavenumber leaves the horizon and when CMB scales leave the horizon, we have . LSS are about two orders of magnitude greater than CMB scales, giving .
4 The Power Spectrum
We will first consider the two-point function . For the scalar fields, we have
and we consider a model where these expectation values are approximately constant and where is scale invariant (independent of ). We will also assume that any intrinsic 3-pt functions are negligible, and for odd. In Fourier space the curvature perturbation is given by (from Eq. (14))
The “tree-level” contribution to the power spectrum arises from linear terms in the expansion of , and it is easily seen to give
However, there is also a “one-loop” contribution which arises from the non-linear terms in the expansion which leads to
where the factor of is from the combinatorics. For a scale invariant power spectra , the integral is approximately
where we use as the upper limit because for the denominator goes as with , and the integrand drops rapidly. The integrand has two simple poles which give logarithmic divergences. We regulate these by putting an IR cutoff on the integral. Hence for this example, we get
This contribution will depend on the IR limit of the momentum integration. This limit is given by the size of the observable universe today, as we discussed in Sec. (3.1). Modes of longer wavelength are already summed in the background value of the field. We thus find
Combining these terms yields
We have defined the power spectrum for curvature with the superscript . The spectral index is
Note that the log contribution is positive (blue) and if this is the only contribution, we cannot match to the currently observed value of . For now, we simply impose that the log contribution contribute no more than a percent correction to
which in turn implies that the non-linear contribution to the 2-point function must be subleading if .
5 Higher Point Functions
We now compute the 3-point function . Again, we find that this correlation function can easily be computed by expanding in terms of and . Since and are Gaussian fields, the only non-trivial contributions will come from non-linearities in the expansion. As in the case of the 2-point function, there is a natural separation into “tree-level” and “loop” contributions . The contribution which is of lowest order in is
The next term in the expansion is
Now the loop integral involves two different momenta
Diagrammatically this is equivalent to a triangular loop of scalars (see Fig. (2)).
We note that near the poles at , we get logarithmic divergences which are cut off by the IR scale . This logarithmic dependence breaks scale invariance. So our shape is a function of three variables which we choose to simply be the norm of all three vectors . An estimate of the shape can be obtained by simply evaluating the integral around each poles, cutting off the momentum integration in the infrared at scale . So for example, the integrand
has a pole around , and the integrand falls off rapidly when becomes of the same order as or . Hence we can approximate the integral around that pole as
The same thing can be done for the other poles and for the various permutations. There are also points in parameter space where the integrand has a pole of order 4. These poles occur in the squeezed limit where and hence . This shows that the bispectrum diverges in the squeezed limit, as is usual for the local shape. In principle, we can only measure to a resolution and the bispectrum, while large, is finite and of order in this limit. Hence the stronger poles that we have neglected are only important in the squeezed limit and they give contributions of the same order as the log terms in that limit. The full approximative shape is
The term is only relevant for scales smaller than . For larger the shape is very well approximated by
We show numerically in Appendix B that this is a good approximation. In Figure (3), we plotted the shape given by Eq. (36) in term of the usual variable and . When the bispectrum is scale invariant, is fixed to 1 (arbitrarily) but here we plotted the shape for different value of . As the figure clearly shows, the graph is very close to local and the magnitude grows as increases.
At the equilateral point , the loop contribution to the bispectrum simplifies to
where the first term is the tree-level contribution, and the second term is the one-loop contribution.
In the case of the two-point function, experimental bounds on the spectral index required the loop-contribution to be subleading. But there is no such requirement for the bispectrum. The loop contribution will dominate if
In this limit we have
where we have utilized the bound and the normalization from COBE data. We thus find, in this scenario, that one can easily generate local non-Gaussianity which is not ruled out by WMAP5 and can potentially be probed at Planck. Note that the magnitude of the non-Gaussianity increases logarithmically with momentum, suggesting that non-Gaussianity can have an important impact on the formation of structure at smaller scales. If we define the running of at the equilateral point
one gets in the loop dominated limit
In the limit where non-linearities dominate, the running of is thus independent of , and .
As in the case of the 3-point function, the only non-vanishing contributions will arise from the non-linear dependence of on , so we can ignore fluctuations. To simplify notation, we define
where and . The last term ensures that we only keep the connected part of every diagrams. The tree level contribution (the term of lowest order in ) is
where we have used that and the shape is given by
with the notation . The magnitude of the trispectrum is usually given by two numbers ( and ) corresponding to two distinct shapes:
The lowest order contribution thus corresponds to and . The 1-loop contribution comes from the following term
The integral over momentum is difficult in general, so we will only estimate its value at the equilateral point
We see that the trispectrum is dominated by the non-linear contributions in largely the same regime as the bispectrum. Given the bound from , the maximum value for in this loop dominated regime is
Interestingly, the bound from WMAP5 on this parameter is while Planck is expected to improve this bound up to .
We have studied a simple class of models in which non-Gaussianity is dominantly produced by higher-order non-linearities in the transfer of fluctuations from the fundamental scalars to the curvature. These higher-order non-linear order contributions are often referred to in the literature as “c-loops”, and can dominate the lowest order “tree-level” contribution in the limit where , where and parameterize the non-linear transfer of fluctuations. In particular, can be achieved in these models.
We have also found in these models that the magnitude of non-Gaussianity is scale dependent, with at CMB scales and at LSS scale. Interestingly, the non-Gaussianity of the bispectrum is stronger at smaller scales, where it can potentially be observed by large scale structure experiments. The shape of our NG signal is very nearly local. Moreover, this class of models yields a non-trivial trispectrum (parameterized by ) that also runs.
A number of open issues remain. In our model, we have assumed that the slow-roll parameter is constant throughout inflation. This was necessary in order to have an observable effect from the end of inflation, but it requires tuning and it leads to a very flat power spectrum. It would be interesting to either relax this assumption in our scenarios or to look at a completely different set-up where the NG is not generated at the end of inflation. We expect that we can relax this assumption since we could have a case where is very small on CMB scales but grows to be detectable on LSS scales. We note though that D-term inflation with a Coleman-Weinberg potential (as illustrated in Appendix A) has a natural regime with the required flat potential, . From an effective field theory point of view (and from string theory models such as [40, 53]), the real tuning is in keeping all other allowed terms (such as a mass term for ) subdominant to the Coleman-Weinberg potential.
We have also assumed that the fundamental scalars ( and ) are Gaussian, and that all non-Gaussianity is induced by the non-linear transfer of fluctuations to the curvature. Non-trivial NG can also arise from non standard kinetic terms, or a steep potential for (which unlike the inflaton does not have to satisfy slow-roll conditions). Loop corrections then have a richer structure although the basic idea remains the same. Of particular interest are models like DBI inflation where the spectral index is nearly one and entropy modes being converted to curvature at the end of inflation can also be observable . This scenario has been analyzed recently in  based on methods developed in [56, 57] (see also ) and a mixture of equilateral and local NG has been found. It would be interesting to consider the regime where the loop dominate in this kind of models.
Acknowledgments.We are particularly thankful to Bhaskar Dutta for early collaboration on this project. We are grateful to Niayesh Afshordi, Sarah Shandera, Martin Sloth, Xerxes Tata and Andrew Tolley for useful discussions. L.L. would like to thank the organizers of the workshop on Effective Field Theory of Inflation at the Perimeter Institute and of the Phenomenology workshop at Cooks Branch Conservancy where part of this work was presented. L.L. would also like thank the KITP and the Aspen Institute for their hospitality. LL is supported in part by NSF Grant No. PHY–0505757. AR is supported in part by NSF Grant No. PHY–0653656. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada though Industry Canada and by the province of Ontario through the Ministry of Research & Innovation.
Appendix A A Specific Model
The discussion in the text is very general and the ultimate goal of having dominant loop contribution in the bispectrum inducing a large running may be achievable in a variety of ways. Here we we illustrate the necessary ingredients with a specific model (based on ). Take an inflationary potential
where , has mass dimension 2 and and are dimensionless couplings. The reader will recognize this as the Coleman-Weinberg potential. There is a regime in parameter space where the inflaton does not move very much with
and the slow-roll parameter is also nearly constant555By integrating the EoM of motion of , in the limit , one can check that is a good approximation as long as where is the number of efolds between horizon crossing and the surface of reheating.
The and power spectrum are simply given by and they will remain approximately constant until the end of inflation if and are much smaller than 1. Now consider a simple potential for
This potential drives to but the field will fluctuate and acquire some stochastic value which in general will be non-zero (although small). The tachyon potential is of the form Eq. (6) with the surface of reheating defined by
We choose a model such that . Since the function is quadratic in both fields, the transfer function is simply
and while . Note that and