Gauge field production in SUGRA inflation: local non-Gaussianity and primordial black holes
When inflation is driven by a pseudo-scalar field coupled to vectors as , this coupling may lead to a copious production of gauge quanta, which in turns induces non-Gaussian and non-scale invariant corrections to curvature perturbations. We point out that this mechanism is generically at work in a broad class of inflationary models in supergravity hence providing them with a rich set of observational predictions. When the gauge fields are massless, significant effects on CMB scales emerge only for relatively large . We show that in this regime, the curvature perturbations produced at the last stages of inflation have a relatively large amplitude that is of the order of the upper bound set by the possible production of primordial black holes by non-Gaussian perturbations. On the other hand, within the supergravity framework described in our paper, the gauge fields can often acquire a mass through a coupling to additional light scalar fields. Perturbations of these fields modulate the duration of inflation, which serves as a source for non-Gaussian perturbations of the metric. In this regime, the bounds from primordial black holes are parametrically satisfied and non-Gaussianity of the local type can be generated at the observationally interesting level .
In a recent series of papers (1); (2); (3) a new broad class of models of chaotic inflation in supergravity has been developed. These models generalize the simplest model of this type proposed long ago in (4); see also (20); (5); (6); (7); (8); (9); (10); (11); (12); (13); (14); (15); (16); (17); (18); (19) for a partial list of other closely related publications.
where is a real holomorphic function such that . Any function which can be represented by Taylor series with real coefficients has this property. The Kähler potential can be chosen to have functional form
In this case, the Kähler potential does not depend on . Under certain conditions on the Kähler potential, inflation occurs along the direction , and the field plays the role of the inflaton field with the potential
All scalar fields have canonical kinetic terms along the inflationary trajectory .
This class of models can be further extended (11); (3) to incorporate a KKLT-type construction with strong moduli stabilization (21), which may have interesting phenomenological consequences and may provide a simple solution of the cosmological moduli and gravitino problems (22); (23).
The generality of the functional form of the inflationary potential allows one to describe any combination of the parameters and . Thus, this rather simple class of models may describe any set of observational data which can be expressed in terms of these two parameters by an appropriate choice of the function in the superpotential. Meanwhile the choice of the Kähler potential controls masses of the fields orthogonal to the inflationary trajectory (1); (2); (3). Reheating in this scenario requires considering the scalar-vector coupling (3); (24). If not only the inflaton but some other scalar field has a mass much smaller than during inflation, one may use it as a curvaton field (25) for generation of non-Gaussian perturbations in this class of models (26).
In this paper, we will study an alternative formulation of this class of models, with the Kähler potential
The simplest version of models of that type, with the Kähler potential was first proposed (4). In this class of models, the Kähler potential does not depend on , which plays the role of the inflaton field with the potential
The difference appears when one notices that in the model (4) the inflaton field is a pseudo scalar, which can have a coupling to vector fields
where and is a dimensionful constant. This coupling is expected to be present since it is compatible with all the symmetries, including a shift-symmetry in .
The study of the phenomenological effects of such a coupling during inflation has received a lot of attention lately (27); (28); (29); (30); (31); (32); (33). In particular, it has been shown in (30); (31) that, if the constant is large enough, such a coupling can lead to a copious production of gauge fields due to the time dependence of . Through their inverse decay into inflaton perturbations, these gauge fields yield an additional contribution to the scalar power spectrum which is both non-Gaussian and violates scale invariance. In this way it is possible to obtain non-Gaussian and non-scale invariant effects that can be observed by the Planck satellite and has not yet been ruled out yet by WMAP, although the parameter space corresponding to such a signal is relatively small (33). In addition, gauge fields source tensor modes and lead to a stochastic gravity wave signal that could be detected at interferometers such as Advanced LIGO or Virgo (34); (32) (see also (35)).
Since the new class of inflationary models in supergravity needs a coupling between the inflaton and gauge fields to have successful reheating, we have to consistently take into account the violations of Gaussianity and scale invariance induced by the inverse decay mechanism. This is the topic of section II.
A potential threat in this model is the overproduction of primordial black holes. As we will see in section III, at very small scales, far beyond what is observable by the CMB, the produced gauge quanta largely increase the curvature power spectrum. At some point, various forms of backreaction stops this growth, but by then the power spectrum has reached . At such high values, a statistical fluctuation might locally increase the density so that primordial black holes are formed. In this way the non-detection of primordial black holes puts an observational upper bound on the power spectrum (36); (37); (38); (39); (40); (41), which we discuss in section IV. Our estimates for the late-time power spectrum land a factor of six above this bound (compare e.g. (33) with (39)). Since we expect our estimate to be reliable up to factors of order one, we cannot definitively claim that the inverse decay mechanism and its interesting phenomenology is incompatible with current data, but our result on production of primordial black holes highlights a clear tension.
In section V we describe an alternative mechanism of generation of non-Gaussian perturbations, proposed in (33). This mechanism requires introduction of a light charged field with mass , where is the Hubble constant during inflation. Inflationary perturbations of this field generate a slightly inhomogeneous distribution of a classical scalar field . This field induces the vector field mass due to Higgs effect.
As a result, the vector field mass takes different values, controlled by fluctuations of the field . In the parts of the universe where the value of the vector field mass is small, the vector field fluctuations are easily produced since the gauge mass quenches the tachyonic instability. This in turns leads to a longer stage of inflation because of the additional friction generated by the gauge fields. Meanwhile in the parts of the universe where the fluctuations of the light scalar field make this field large, the vector field mass becomes larger and inflation is shorter due to the lack of backreaction. As a result, fluctuations of the light scalar field lead to fluctuations of the total number of e-foldings , i.e. to adiabatic perturbations of metric. We will show that this effect may generate significant primordial local non-Gaussianity. Also, in the regime of parameters relevant for this scenario the primordial black hole bounds are satisfied parametrically.
To implement this mechanism in our supergravity-based inflationary scenario, one should find a way to guarantee smallness of the mass of the field during inflation. We will describe a model where the mass squared of this field during inflation is equal to , where can be made small by a proper choice of the Kähler potential.
In section VI we study the evolution of the light field during inflation in our scenario, which is similar to the evolution of the curvaton field in (26), so for simplicity we will continue calling this field the curvaton. One can use the results of (26) for the description of its evolution. However, in the original model of (26), just as in any other curvaton model (25), adiabatic perturbations of metric are generated by perturbations of the field after a complicated sequence of reheating, expansion of the universe, and the subsequent decay of the curvaton field. In our scenario, adiabatic perturbations are produced due to the modulation of the duration of inflation by the perturbations of the field . As we will demonstrate, this mechanism can easily produce local non-Gaussianity in the potentially interesting range from to , even if the coupling constant is not very large.
Finally, in section VII, we find that typical values of the coupling constant considered in this work lead to a relatively high perturbative reheating temperature . This should be read as a lower limit, since the copious non-perturbative production of gauge fields already during inflation could lead to and even higher reheating temperature. This could lead to the cosmological gravitino problem (42), but in the class of models with strong moduli stabilization and gravitino mass TeV this problem does not appear (23).
Ii CMB scales: violations of Gaussianity and scale invariance
Consider a pseudo-scalar inflaton with a potential suitable for inflation. The symmetries of the theory allow for a coupling to some gauge sector. This coupling is essential for reheating in the supergravity models we discussed in section I. We will therefore consider the following bosonic part of the action
Since all relevant effects arise from the couplings above we can safely neglect the gravitational interaction between perturbations and work with an unperturbed FLRW metric
where , and are computed from .
Now let us look at the action expanded around and , i.e. . Organizing the result at various orders in and one finds
where again the classical background fields and solve (7) and (8). Notice that there is a “tadpole” for due to the fact that at the background level we solved an inhomogeneous equation for but just a homogeneous one for . From this term one also sees that will source , hence it will modify the constraint. The equations of motion in Coulomb gauge are
The solution for the constraint equation for is
This is a good approximation as long as (or equivalently ) is not too large (a more quantitative condition is given in (29)), which is the regime we will discuss in this section. Note from (8) and (14) that in this approximation there is no way to tell apart from . In the next section we will see that, since grows with time, towards the end of inflation this description in not valid anymore, and one has to take backreaction into account.
Solving the equation of motion (8) one finds a tachyonic enhancement of the gauge fields. For the growing mode of one of the two polarizations of the gauge field we get
Here denotes the Whittaker function, and is defined as
As we see, the relation between the coupling constant and the value of 60 e-foldings before the end of inflation is model dependent, but for our model there is an approximate relation which is valid for the parameters that we are going to explore:
For the new coupling therefore leads to generation of perturbations of the vector fields around horizon scales. The produced gauge fields then change the dynamics of and . The cosmological homogeneous Klein-Gordon equation and the Friedmann equation get extra contributions from the gauge fields and can now be written as
They are computed as
After renormalization, one can reduce the integration interval to the region , which is where the enhancement in the (derivative of the) gauge field takes place.
From the homogeneous Klein-Gordon equation (15) one reads off that the influence of the produced gauge fields on the homogeneous dynamics of and can be safely neglected as long as
Of these two conditions the first one is always the most stringent. When it stops to hold, backreaction on the homogeneous evolution becomes important and the evolution of and will be slowed down, which makes inflation lasts longer. We will see in the next section that backreaction on the inhomogeneous equation for happens even earlier. In this section we focus on the regime in which all of these effects are negligible, which e.g. for a quadratic potential corresponds roughly to . This is appropriate for the description of CMB scales.
Now we move to the power spectrum. The copiously generated gauge fields may, by inverse decay, produce additional perturbations of the inflaton field , proportional to the square of the vector field perturbations. As was shown in (30); (31), this can be described (up to backreaction effects to be described in the next section) by using (15). The inclusion of the source term leads to an extra contribution to the power spectrum of the curvature perturbation on uniform density hypersurfaces , which has been computed in (30); (31) (we present a quick estimate in Appendix B)
is the amplitude of the vacuum inflationary perturbations as in standard slow-roll inflaton. WMAP (43) has measured , where is the pivot scale that we will take to correspond with e-foldings before the end of inflation. The second term in (24) violates both scale invariance (and Gaussianity as we will see below), since it comes schematically from , i.e. the square of a Gaussian which grows with time as in (16).
where another function defined in (30); (31), which can be computed numerically (see (96) for a useful approximation). The amount of non-Gaussianity, therefore, depends exponentially on . Between and it grows from to and most of the growth takes place in a small interval around .
The analysis of (33) showed that the bounds coming from the power spectrum (especially from WMAP plus ACT, because of the violation of scale invariance) and from the bispectrum (from WMAP) are compatible, with the former being typically slightly more stringent. Specifying a confidence region in requires assuming some prior for this parameter. The physically best motivated prior is log-flat in reflecting the fact the scale of the dimension five coupling could be anywhere (with strong indications that it should be below the Planck scale (44)). In this case at CL one finds . A flat prior on leads to .
Iii Very small scales: strong backreaction
In this section we want to estimate the power spectrum and bispectrum towards the end of inflation, i.e. on scales that are too small to be observed in the CMB. The only observational handle available in this regime is the non-detection of primordial black holes, which puts an upper bound on the power spectrum (36); (37); (38); (39); (40); (41).
To make these estimates it is essential to recognize that many of the formulae described in the previous section and given in the literature about inverse decay are valid only in the regime in which backreaction on the inhomogeneous equation for is small (see (29)). As we show in the following, the scales relevant for the production of primordial black holes leave the horizon when backreaction is large. The authors of (45) did not account for backreaction and therefore their conclusion that gauge field production during inflation leads to black hole production might be premature.
For concreteness, we will consider a quadratic potential , with the mass chosen such that at the pivot scale (that we take to correspond with ) we get .
Let us first look at the dynamics of and . As we already discussed, when enough gauge field quanta have been produced, the conditions in (23) stop to hold (the inequality for is violated first) and and are slowed down. As a result, inflation lasts longer. Let us check this. The behavior of , and as functions of (the number of e-folds left to the end of inflation) follows from simultaneously solving (17), (19) and (20). In Figures 1 and 2 we have plotted the solutions for and , with and without backreaction taken into account. For , the effect of backreaction becomes around .
Now let us consider perturbations. Of course they will be affected by the backreaction on the homogeneous dynamics and that we described above, but there is more. Let us consider (10)-(12). In the last section we solved for in a homogeneous background and used that result (16) to compute the source term in the equation for perturbations. But as and grow larger toward the end of inflation (both of them grow as ) the source in the right-hand side of (10) can not be neglected anymore. If we were able to solve this equation, we would find that now depends on the perturbation . Expanding , which is the source term in (11), in powers of we would find several new terms including additional friction and a modified speed of sound. In (28); (32) it was proposed how to estimate these effects in the regime of strong backreaction by just considering the additional friction term . The equation of motion for the perturbation becomes
with the additional friction term
Here the new term in is caused by the dependence of on (via its dependence on ). The behavior of has been plotted in Figure 3. It is always positive
The new source of backreaction can be neglected as long as
Note (from comparison with (23)) that the factor of makes that backreaction on the power spectrum will become significant before backreaction on and does. For we find that backreaction becomes of order 10% () at .
which leads to the power spectrum
This estimate turns out to be particularly good in the regime in which we can check it, i.e. when when the backreaction is negligible and we can compare with (24) (see appendix B). This gives us confidence to use it also in the strong backreaction regime. It is easy to see that when backreaction becomes large, the second term in (28) dominates, and we end up with
The estimate (31) for the power spectrum has been plotted in Figure 4 together with the formula (24), valid only when backreaction is negligible. Indeed, in the regime of strong backreaction the power spectrum asymptotes the estimate in (32). At the end of inflation we have (for ), which gives
Iv Bounds from primordial black holes
Now let us try to compare this with the existing bounds on the power spectrum coming from the non-detection of primordial black holes. These will form if at horizon re-entry (i.e. smoothing on scales of order ) we have , with denoting the critical value leading to black hole formation. If one assumes that follows a Gaussian distribution (with ) one can express the probability of having in terms of the variance by analyzing the Gaussian probability distribution function. This probability corresponds to the fraction of space that can collapse to form horizon-sized black holes. Hawking evaporation and present day gravitational effects constrain this fraction . Typically one finds in the range , with the strongest bounds coming from CMB anisotropies (37) (spectral distortion and photodissociation of deuterium lead to a bound , as for example in (36)). Setting gives for the upper bound on the power spectrum (40)
Here the lower bound corresponds to and the upper bound to .
However, in our case does not follow a Gaussian distribution. Instead we have (see Appendix B)
The stochastic properties of the vector field are close to those in a free theory, i.e. it has Gaussian perturbations around . As a consequence we can write
with a Gaussian distributed field. This model was studied in (40) and we follow that derivation (see also (38); (39)). The probability distribution function of follows from setting , and takes the form
with . For a given value of we can again infer . Setting (and ) we have which gives
where is the complementary error function. Taking again in the range one gets a tighter upper bound on the power spectrum than in the Gaussian case:
Now let us estimate what value of is relevant for our investigation.
At the end of inflation, the total mass concentrated in the volume associated with perturbations leaving the horizon e-foldings before the end of inflation with the Hubble constant can be estimated by
where we reinserted the reduced Planck mass , which was set to one in the rest of the paper, and is calculated at the end of inflation. In order to study the subsequent evolution of matter in the comoving volume corresponding to this part of the universe, one should distinguish between two specific possibilities depending on the dynamics of reheating after inflation, discussed in section VII.
If reheating is not very efficient, then the universe for a long time remains dominated by scalar field oscillations, with the average equation of state . In this case, the total mass in the comoving volume does not change, and therefore at the moment when the black hole forms, its mass is equal to evaluated in (40). For the parameters of our model, this gives an estimate (see appendix E for details)
On the other hand, if reheating is efficient, then the post-inflationary universe is populated by ultra relativistic particles and the energy density in comoving volume scales inversely proportional to the expansion of the universe. In this case, the black hole mass can be estimated as (see appendix E)
In our estimates of the black hole production we will assume the latter possibility, though in general one may have a sequence of the first and the second regime. The final conclusion will only mildly depend on the choice between these two possibilities.
Now, the bounds on in terms of the would-be black hole mass were given in (36) and updated in (37). Here we follow the result in (37).
Our estimate (33) violates this bound for all by a factor of about six. Since we have made some approximations both in deriving the late time power spectrum and in deriving its observational upper bound, our estimate could well be off by some order one factor and therefore we can not draw a definitive conclusion. It is clear though that the parameter values giving rise to an observable but not yet ruled out violation of scale invariance and non-Gaussianity in the CMB-window produce a late time power spectrum that comes at least very close to the primordial black hole bound. A more precise computation is needed to establish whether this bound is actually violated or not.
However, if such a computation revealed that primordial black holes do indeed constrain these models, that would yield a much stronger bound on as the ones coming from non-Gaussianity and the violation of scale invariance. Since we have seen that the power spectrum has a late-time asymptotic of , this problem persists on a wide range of values for .
For all values of , our estimate for the power spectrum sharply increases before the end of inflation, the closer to the end the smaller is. However, if we disregard black hole bounds for g, which rely on uncertain model dependent assumptions, there are no black hole bounds for . From figure 5 we then see that we get
for the bound on at CMB scales from primordial black hole production. In terms of the coupling constant , this bound implies the constraint
This bound is derived using (42), i.e. radiation domination right after the end of inflation. This assumption fixes the expansion history of the universe and therefore specifies , for the corresponding to CMB scales (see appendix E for a derivation). This is required for consistency but changes the numerics very little. Therefore in all other sections we still use .
For the matter domination regime, the black hole masses would be greater, for a given , see (41), and therefore we would have a slightly stronger constraint on and . We find which corresponds to . Instead of concentrating on it, we will now investigate the model where non-Gaussian perturbations may be generated for much smaller and , without leading to the primordial black hole problem.
V Local non-Gaussianity from heavy vector fields
Now let us turn to a scenario, described in (33), in which the produced gauge fields are massive. The production of gauge quanta decreases with the mass of the gauge fields: for all production is killed. In this scenario, the gauge fields get their mass via the Higgs mechanism. Fluctuations in the Higgs field lead to fluctuations in , which in turn generate fluctuations in the amount of produced gauge quanta, and therefore in the amount of extra friction in the dynamics of and . In the end, one has perturbations in , namely the number of extra e-folds of inflation due to gauge field production. This leads to a non-Gaussian signal in the CMB of the local type (33). Using the formalism one finds
Here is the increase of the duration of inflation for the case where the vector fields are massless, is the Higgs-like field responsible for the mass of the gauge field, is its charge, and we assumed a quadratic inflaton potential, so that .
For a complete description we refer the reader to the original reference (33), section 7. Here we only want to stress that this scenario can also work for . Then it will surely satisfy the bounds from primordial black holes.
Note that the classical field , which gives the vector field mass , can be produced either due to the tachyonic mass of the field at , as in the standard Higgs model, or due to accumulation of long wavelength inflationary perturbations of the field . In both cases, the mechanism of (33) requires that the mass of the field during inflation should be smaller than the Hubble constant. As a result, even if one assumes that the field has the standard Higgs potential, the value of the field during inflation does not correspond to the position of the minimum of the potential. Instead of that, the field takes different values in different exponentially large parts of the universe. The value of in this scenario will depend on a typical local value of the field , which can be determined by the stochastic approach to investigation of curvaton fluctuations (26).
For simplicity, and to make a clear link to the investigation performed in (26), we will call the light field the curvaton, but one should remember that the mechanism of conversion of perturbations of the curvaton field to adiabatic perturbations is different, involving a complicated dynamical processes during reheating. In our case, fluctuations of the field lead to fluctuations during inflation, and thus to a direct production of adiabatic perturbations of metric.
This scenario can work only if we have a charged scalar field with mass much smaller than . At the first glance, one could achieve it by assuming that the relatively light field plays the role of the Higgs field. However, the superpotential would break gauge invariance unless we assume that the field is also charged. This would be inconsistent with the postulated functional form of the Kähler potential.
Fortunately, one can easily do it. Just like in the simplest supersymmetric version of the abelian scalar electrodynamics, one should consider the charged field without any superpotential associated with it. In the global SUSY limit, the simplest version of this theory with vanishing FI coefficient would contain D-term potential , but it would not induce any mass of the field .
However, in supergravity, the radial component of the scalar field does acquire mass, depending on the choice of the Kähler potential. (The complex phase of the field is eliminated due to the Higgs effect.) We will consider the following addition to the Kähler potential of our model:
Terms of similar functional form were included in many versions of our inflationary scenario for stabilization of the inflaton trajectory. One can easily find that the resulting mass squared of the field during inflation is given by
Thus in the absence of the term the field would be too heavy, but by considering models with one can have a consistent theory of a light charged scalar field with mass squared with , as required.
Of course, this requires fine-tuning, but this is just a price which one should be prepared to pay for the description of non-Gaussian inflationary perturbations. We will study observational consequences of this model in the next section.
Vi Stochastic approach
In this section we want to find out how fluctuations in the curvaton field lead to a variable gauge field mass, and therefore to a non-Gaussian signal in the CMB. We will begin our study with investigation of the behavior of the distribution of the fluctuations in , following (26). During inflation, the long-wavelength distribution of this field generated at the early stages of inflation behaves as a nearly homogeneous classical field, which satisfies the equation
which can be also written as
However, each time interval new fluctuations of the scalar field
are generated, with an average amplitude squared
The wavelength of these fluctuations is rapidly stretched by inflation. This effect increases the average value of the squared of the classical field in a process similar to the Brownian motion. As a result, the square of the field at any given point with an account taken of inflationary fluctuations changes, in average, with the speed which differs from the predictions of the classical equation of motion by :
Using , one can rewrite this equation as
By solving this equation with different boundary conditions, one can find the most probable value of the locally homogeneous field .
Now we will consider the case when the mass of the curvaton field is given by
with . This corresponds to the total potential
We assume that , and therefore one can estimate . In this case, Eq. (52) becomes
This equation has a family of different solutions,
where is a constant which could be either positive or negative, depending on initial conditions. During inflation these solutions converge to a simple attractor solution
We are interested in using this formula to estimate the size of non-Gaussianity, which is produced by the conversion of perturbations in into curvature perturbations when the backreaction from gauge fields on the homogeneous evolution becomes substantial, i.e. close to the end of inflation. Hence we should take in (57). For , this solution approaches a constant during the last stages of inflation. Note that this a posteriori justifies the assumption that , as long as .
To give a particular numerical example, we will use (45) for the case . A numerical analysis shows that in this case , and therefore
at the end of inflation with .
All our approximations should work fine if the mass of the vector field is much smaller than , which leads to a constraint .
Consider for example and , which corresponds to the very end of inflation. (We should stress that it would not be consistent to take much larger than in Planck units since that is its value when curvature perturbations are generated in our scenario. Moreover, the main contribution to is given by the last part of the inflationary trajectory where .) In this case
To have non-Gaussian perturbations with one should take .
Vii Gauge field production in SUGRA inflation: reheating
We have found that the coupling needed for reheating in (the pseudo scalar variant of) the new class of SUGRA inflation models proposed in (1); (2); (3) can as well yield an observable non-Gaussian signal. It only remains to be seen what the effects of the typically needed values for are for the reheating in the combined model.
In (3) the reheating temperature for the decay of a scalar inflaton field to two photons due to the interaction was estimated as
A similar estimate is valid in our case. One may also represent it in an equivalent way using relation , and an expression for the slow-roll parameter
As long as one can describe reheating as a particle by particle decay, reheating in inflationary models of this type does not depend much on whether the inflaton field is a scalar or a pseudo scalar. In both types of models, one may consider interactions with , which results in reheating temperature GeV. This solves the cosmological gravitino problem for gravitino in the typical mass range TeV.
However, for , which is required for production of non-Gaussianity in the models based on the pseudo scalar inflaton, an estimate described above gives GeV. It is good for the theory of leptogenesis, but it could be bad from the point of view of the gravitino problem. Moreover, for an entirely different mechanism of reheating is operating. At the end of inflation, when the time-dependent parameter grows and becomes large, a significant fraction of the energy of the inflaton field gradually becomes converted to the energy of the vector field (see Figure 6). This is a very efficient mechanism, which may lead to a very rapid thermalization of energy in the hidden sector. This may exacerbate the gravitino problem. Fortunately, this problem does not appear for superheavy gravitino with mass TeV. Such gravitinos appear in many versions of the models of mini-split supersymmetry, which became quite popular during the last few years, see (23); (48) and references therein.
The new class of chaotic inflation models in supergravity needs a gauge-gauge-inflaton coupling for reheating. The inclusion of this coupling can produce gauge fields and can provide a Planck-observable but not yet ruled out non-Gaussian signal in the CMB.
In this article we have studied two possible realizations of this scenario. Taking the parameter () produces a large amount of gauge quanta, that by inverse decay give rise to an equilateral non-Gaussianity in the CMB, as studied in (30); (31). However, we have estimated that towards the end of inflation the power spectrum grows so much that the model may be ruled out because it overproduces primordial black holes. As our order-one estimate lands within a factor of six from the critical black hole bound on the power spectrum (with the non-Gaussian nature of the signal taken into account), we need a more precise computation to draw a definitive conclusion.
In the second scenario, where the produced gauge fields are massive due to the Higgs effect in presence of a light curvaton-type field, one can take a smaller value for , of order 0.5 - 1, corresponding to from 8 to 15. Then the model is free of black hole trouble. In this case, fluctuations in the curvaton field modulate the duration of inflation and can give rise to adiabatic non-Gaussian perturbations of the local type with . For smaller values of , we return to the standard chaotic inflation scenario with Gaussian adiabatic perturbations.
A.L. is supported by NSF grant PHY-0756174. E. P. is supported in part by the Department of Energy grant DE-FG02-91ER-40671. S. M. wants to thank the Stanford Institute for Theoretical Physics for great hospitality during his two month’s visit in which this work was initiated, and Marieke Postma for many valuable discussions.
Appendix A Variance of
The variance of is defined as
Here we use the polarization tensor conventions given in (31):
which are normalized via . Given our conventions (see footnote II) we are dealing with here.
The angular integral gives , i.e. a third of the whole sphere. The integrals over the modulus are similar to the one in (31) and are computed in the same way
Putting things together one finds
Appendix B Power spectrum estimate
In (30); (31) the power spectrum (24) has been obtained by the Green’s function method. In (32) a quick estimate was introduced to compute the power spectrum in the case of large backreaction (). Here we want to review and further explore this estimate, showing how it leads to (31) and also how, in the case of negligible backreaction, it approximates the precise result (24) within a factor of two.
The full equation of motion for the perturbation is (in real space)
Near horizon crossing we can estimate . Since we have, near horizon crossing, , the first term cancels the third one. The second term can be approximated as . The last term on the left hand side is just a slow-roll correction and can be discarded. This directly gives
and therefore we have
For the position space two point function of we immediately get
with the variance computed in the previous subsection.
To compare the position space power spectrum with the momentum space power spectrum we use
This gives the result (31):