3.1 Inflationary observables in the \delta N formalism

Abstract

Supersymmetric hybrid inflation is an exquisite framework to connect inflationary cosmology to particle physics at the scale of grand unification. Ending in a phase transition associated with spontaneous symmetry breaking, it can naturally explain the generation of entropy, matter and dark matter. Coupling F-term hybrid inflation to soft supersymmetry breaking distorts the rotational invariance in the complex inflaton plane—an important fact, which has been neglected in all previous studies. Based on the formalism, we analyze the cosmological perturbations for the first time in the full two-field model, also taking into account the fast-roll dynamics at and after the end of inflation. As a consequence of the two-field nature of hybrid inflation, the predictions for the primordial fluctuations depend not only on the parameters of the Lagrangian, but are eventually fixed by the choice of the inflationary trajectory. Recognizing hybrid inflation as a two-field model resolves two shortcomings often times attributed to it: The fine-tuning problem of the initial conditions is greatly relaxed and a spectral index in accordance with the PLANCK data can be achieved in a large part of the parameter space without the aid of supergravity corrections. Our analysis can be easily generalized to other (including large-field) scenarios of inflation in which soft supersymmetry breaking transforms an initially single-field model into a multi-field model.

April 2014 DESY 14-005

SISSA 16/2014/FISI

IPMU 14-0083


Hybrid Inflation in the Complex Plane

W. Buchmüller, V. Domcke, K. Kamada, K. Schmitz

[3mm] a Deutsches Elektronen-Synchrotron DESY, 22607 Hamburg, Germany

b SISSA/INFN, 34100 Trieste, Italy

c Institut de Théorie des Phénomènes Physiques, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland

d Kavli IPMU (WPI), TODIAS, University of Tokyo, Kashiwa 277-8583, Japan


.

1 Introduction

Supersymmetric hybrid inflation is a promising framework for describing the very early universe. Not only does it account for a phase of accelerated expansion; it also provides a detailed picture of the subsequent transition to the radiation dominated phase. Different versions are F-term [2, 1], D-term [3, 4] and P-term [5] inflation, with supersymmetry during the inflationary phase being broken by an F-term, a D-term or a mixture of both, respectively.

Hybrid inflation is very attractive for a number of reasons. It can be naturally embedded into grand unification, and the GUT scale yields the correct order of magnitude for the amplitude of the primordial scalar fluctuations [2]. Moreover, supergravity corrections are typically small, since during inflation the value of the inflaton field is , i.e. much smaller than the Planck scale. Hybrid inflation ends by tachyonic preheating, a rapid ‘waterfall’ phase transition in the course of which a global or local symmetry is spontaneously broken [6]. Pre- and reheating have recently been studied in detail for the case where this symmetry is , the difference between baryon and lepton number. The decays of heavy Higgs bosons and heavy Majorana neutrinos can naturally explain the primordial entropy, the observed baryon asymmetry and the dark matter abundance [7, 8, 9].1 Finally, inflation, preheating and the formation of cosmic strings are all accompanied by the generation of gravitational waves that can be probed with forthcoming gravitational wave detectors [12, 13, 14, 15, 16].

The supersymmetric extension of the Standard Model with local symmetry is described by the superpotential

(1)

The first term is precisely the superpotential of F-term hybrid inflation, with the singlet superfield containing the inflaton and the waterfall superfields and containing the Higgs field responsible for breaking at the scale . The next two terms involve the singlet superfields whose fermionic components represent the charge conjugates of the three generations of right-handed neutrinos. These two terms endow the singlet neutrinos with a Majorana mass term and a Yukawa coupling to the MSSM Higgs and lepton doublets, denoted here by and in notation. and are coupling constants.

In a universe with an (almost) vanishing cosmological constant, F-term supersymmetry breaking leads to a constant term in the superpotential,

(2)

where is the vacuum gravitino mass at low energies and a model-dependent parameter. In the Polonyi model, one has  [17]. For definiteness, we choose in the following. We assume that the supersymmetry breaking field is located in its minimum and that its dynamics can be neglected during inflation. Together with the non-vanishing F-term of the inflaton field during inflation, , this constant term in the superpotential induces a term linear in the real part of the inflaton field in the scalar potential [18],

(3)

The real and the imaginary part of the inflaton field are thus governed by different equations of motion, requiring an analysis of the inflationary dynamics in the complex inflaton plane. As a consequence, all of the inflationary observables are sensitive to the choice of the inflationary trajectory. In this sense, the measured values of these quantities do not point to a particular Lagrangian or specific values of the fundamental model parameters. To large extent, they are the outcome of a random selection among different initial conditions which has no deeper meaning within the model itself. We emphasize that these conclusions apply in general to every inflationary model in which inflation is driven by one or several large F-terms. In the presence of soft supersymmetry breaking, these F-terms will always couple to the constant in the superpotential and thus induce linear terms in the scalar potential of exactly the same form as in Eq. (3). The analysis in this paper can hence be easily generalized to other models of inflation, in particular also to models of the large-field type.

Taking the two-field nature of hybrid inflation into account, we find that the initial conditions problem of hybrid inflation is significantly relaxed and we can obtain successful inflation in accordance with the PLANCK data [19] without running into problems due to cosmic strings [20]. First results of this two-field analysis were presented in Ref. [21]. Non-supersymmetric multi-field hybrid inflation, commonly referred to as ‘multi-brid’ inflation, has been studied in Refs. [22, 23]. The model investigated here differs from multi-brid inflation in two regards: (i) we embed inflation into a realistic model of particle physics and (ii) we study inflation in the context of softly broken supersymmetry. Furthermore, we note that, during the final stages of preparing this paper, evidence for a potentially primordial B-mode signal in the polarization of the cosmic microwave background (CMB) radiation was announced by the BICEP2 Collaboration [24]. In App. B, we discuss the implications of this very recent development on F-term hybrid inflation.

Our discussion is organized as follows. In Sec. 2, we analyze the connection between and the spectral index analytically for inflation along the real axis. In Sec. 3, we then turn to the generic situation of arbitrary inflationary trajectories in the complex plane. We perform a full numerical scan of the parameter space, based on a customized version of the formalism, in order to determine the inflationary observables and again reconstruct our results analytically. Sec. 4 demonstrates how these results relax the initial conditions problem of F-term hybrid inflation and Sec. 5 is dedicated to an investigation of the allowed range for the gravitino mass. Finally, we conclude in Sec. 6. As a supplement, we derive in App. A simple analytical expressions that allow to estimate the scalar amplitude as well as the scalar spectral tilt in general multi-field models of inflation in the limit of negligible effects due to isocurvature perturbations.

2 Hybrid inflation on the real axis

2.1 Successes and shortcomings

The potential energy of the complex inflaton field , determined by the superpotential given in Eqs. (1) and (2), receives contributions from the classical energy density of the false vacuum [1], from quantum corrections [2], from supergravity corrections [25] and from soft supersymmetry breaking [18],2

(4)
(5)
(6)
(7)
(8)

where denotes the reduced Planck mass. During inflation, the energy density of the Universe is dominated by the false vacuum contribution , while the inflaton dynamics are governed by the field-dependent terms and/or . Inflation ends when the waterfall field becomes tachyonically unstable at . The scalar potential determines the predictions for the amplitude and the spectral tilt of the scalar power spectrum as well as the amplitude of the local bispectrum. These should be compared to the recent measurements by the PLANCK satellite [20, 26],

(9)

In the following, we shall consider Yukawa couplings , comparable to Standard Model Yukawa couplings, and . In this case, supergravity corrections are negligible, cf. Ref. [18].3 Most analyses also neglect the linear term in Eq. (8), which arises due to soft supersymmetry breaking. For small values of and sufficiently large gravitino masses, this term is however important and can even dominate the inflaton potential [18].

Hybrid inflation with a linear term has been analyzed in detail in Ref. [27]. The authors focused on initial conditions along the real axis with , to avoid fine-tuning of the initial conditions.4 The linear term induces a local minimum at large field values in the inflaton potential and for the inflaton may get trapped in this false minimum, preventing successful inflation if the initial conditions are chosen unfittingly. For , successful inflation is difficult to achieve but possible for carefully chosen parameter values. The observed spectral index can be obtained by resorting to a non-minimal Kähler potential [28].

Figure 1: Contour lines in the plane along which inflation on the real axis succeeds in reproducing the observed value of ; the red and blue contours correspond to and , respectively. The gravitino mass is varied over four different values, , , and  TeV, and consistency with the observed value of the scalar spectral index at  C.L. is indicated by the green contour segments. The region to the right of the thick light-blue line is excluded due to the non-observation of cosmic strings.

Recently, it has been observed that for inflation along the real axis with the observed spectral index can be obtained for a canonical Kähler potential [29, 30] in the hill-top regime of hybrid inflation [31], if one allows for severe fine-tuning of the initial conditions. Furthermore, the current bound on the tension of cosmic strings [20] is naturally satisfied in this case,5

(10)

where is Newton’s constant and is the string tension [15]. These are interesting results despite the fine-tuning problem of initial conditions. In both cases, and , the inflaton phase remains unchanged during inflation, so that at the end of inflation the final phase either corresponds to or to . In Fig. 1, we compare the constraints on the parameters and imposed by the normalization of the scalar power spectrum for these two situations. In doing so, we also vary the gravitino mass and determine the parameter combinations for which the scalar spectral index falls into the range around the measured best-fit value. The results shown in Fig. 1 are based on the numerical analysis described in Sec. 3. We observe that, while the case (blue contours) is almost excluded by the cosmic string constraint, this constraint is automatically satisfied in most of the parameter space for the case (red contours). To sum up, we find that hybrid inflation on the positive real axis is able to reproduce the scalar spectral index for a canonical Kähler potential and is in less severe tension with the non-observation of cosmic strings. At the same time, hybrid inflation on the negative real axis has the virtue that it does not require the initial position of the inflaton to be finely tuned.

2.2 Understanding the hill-top regime

In this section, our goal is to analytically reconstruct our results for and depicted in Fig. 1 for the case of hybrid inflation on the real axis in the the hill-top regime () based on a canonical Kähler potential. This analysis will prove to be a useful preparation for our general investigation of hybrid inflation in the complex plane in Sec. 3. As the amplitude of the local bispectrum is slow-roll suppressed in the single-field case, we do not study it in this section; for a discussion of in the general two-field scenario, cf. Sec. 3.2.

The inflaton field is a complex scalar, , and the relevant variables are its real and imaginary parts normalized to the symmetry breaking scale , and . During the inflationary phase, the inflaton potential is flat in global supersymmetry at tree-level. The one-loop quantum and tree-level supergravity corrections only depend on , the absolute value of the inflaton field. Supersymmetry breaking generates an additional term linear in , such that one obtains for the scalar potential

(11)

where we have neglected the quartic supergravity term and with the one-loop function

(12)

Here, is nothing but the Coleman-Weinberg potential, , cf. Eq. (6). We choose the sign convention such that . For , i.e. , inflation can take place, ending in a waterfall transition at .6 In the slow-roll regime, the equations of motion for the two real inflation fields and as well as the Friedmann equation for the Hubble parameter read,

(13)

As vastly dominates the potential energy for all times during inflation, we shall approximate by in the following for the purposes of our analytical calculations.

The number of -folds between a critical point , at which inflation ends, and an arbitrary point in the complex plane are given by a line integral along the inflationary trajectory,

(14)

As explained in App. A, in general multi-field models of inflation, the scalar amplitude and the scalar spectral tilt are approximately given by the simple single-field-like expressions

(15)

if (and only if) isocurvature modes during inflation do not give a significant contribution to the scalar power spectrum. and are the slow-roll parameters along the inflationary trajectory,

(16)
(17)

with the inflaton ‘flavor’ indices , and all running over and . In the following, we shall use these expression to obtain simple analytical estimates for and . Hence, in order to make connection between our predictions and the measured values for the inflationary observables, we need to evaluate and in Eqs. (16) and (17) -folds before the end of inflation, when the CMB pivot scale exits the Hubble horizon.

In this section, we shall restrict ourselves to inflation along the real axis. Since

(18)

with , the real axis with is a indeed a stable solution of the slow-roll equations. In  direction, one has

(19)

If the constant term can be neglected, one obtains the standard form of hybrid inflation. In this case, -folds correspond in field space to a point , where , which leads to the spectral index

(20)

This value is disfavoured by the recent PLANCK data. It deviates from the measured central value by about .

Figure 2: Scalar potential for inflation along the real axis in the complex inflaton field space after adding a constant term to the superpotential. Slow-roll inflation is possible for both and . Here, we have chosen parameter values  GeV, and  TeV.

For sufficiently large values of , an interesting new regime opens up for field values very close to the critical point [30]. This is apparent from Fig. 2, where the potential is displayed for representative values of , and . Note that the first derivative of the loop-induced potential is always positive,

(21)

As a consequence, for initial conditions , cancellations between the gradients of the linear term and the one-loop potential can lead to extreme slow roll. The second derivative of the loop potential is always negative,

(22)

and diverges for . This allows small values of , if is sufficiently close to the critical point. For the example shown in Fig. 2, the point of 50 -folds is . Note that successful inflation requires carefully chosen initial conditions. The inflaton rolls in the direction of the critical point only if . We will come back to the problems related to the necessary tuning of the initial conditions in more detail in Sec. 4. Also for initial values , the linear term significantly modifies the loop-induced potential, but qualitatively the picture does not change.

Let us now consider the hill-top regime quantitatively. Close to the critical point, i.e. for , one has for the first and the second derivative of the one-loop function

(23)
(24)

The value of is determined by, cf. Eq. (14),

(25)

and using Eq. (23) one obtains

(26)

where the parameter measures the relative importance of the two contributions to the slope of the potential in Eq. (11),

(27)

Consistency (i.e. the inflaton rolling towards the critical line) requires , which yields an upper bound on the gravitino mass, cf. also the discussion in Sec. 5,

(28)

Clearly, tuning and , one can move very close to the critical point. This enhances the amplitude of the scalar fluctuations,

(29)

From Eqs. (15), (17) and (24) one obtains for the spectral index

(30)

Finally, eliminating by means of Eq. (29), one obtains a relation between the spectral index and the amplitude of scalar fluctuations, which is independent of the gravitino mass,

(31)

Note that this relation is very different from standard hybrid inflation, where and are determined by and , respectively, and where the dependence on is very weak.

Figure 3: Constraints on the model parameters of hybrid inflation, , and , imposed by the measured values for the inflationary observables and the cosmic string bound (light-blue curve) for . For each () pair, the gravitino mass is adjusted, as indicated by the grey contour lines in the left panel, such that the scalar amplitude comes out right. In the region labeled , our prediction for is always larger than the observed value , even if is set to . Along the solid black lines, the best-fit value for the scalar spectral index is reproduced, with the green band indicating the corresponding confidence interval. All of the black and grey contour curves in both panels are the result of our full numerical calculation, cf. Sec. 3. The red and blue curves in the right panel are by contrast based on our (semi-)analytical results for in the small- and large- regime, respectively, cf. Eqs. (31) and (32). The initial field values are indicated by the grey dot-dashed contour lines.

For larger couplings , the gradient of the one-loop potential increases and a longer path in field space is needed to obtain -folds. To achieve this for GUT-scale field values, i.e. , a larger gravitino mass is needed to reduce the gradient of the total potential. A rough estimate for the spectral index can be obtained by using for the second derivative of the potential the approximation for large field values, , which yields

(32)

This expression agrees with Eq. (31) up to an factor. Note that a numerical determination of is needed in order to obtain quantitative result for .

The domain of successful inflation in the plane reproducing the measured amplitude of the scalar fluctuations and the spectral index is displayed in Fig. 3. The left panel shows the result of a numerical analysis. Since the real axis is merely a special case of all possible trajectories in the complex plane, these results were obtained using the two-field method described in Sec. 3.1. For each pair, the measured amplitude of the primordial fluctuations is used to fix the gravitino mass, cf. the grey contour lines. In the green band, the spectral index lies in the range , cf. Eq. (9). In the right panel, the numerical results are compared with the analytical estimates. The small- approximation in Eq. (31) works approximately up to whereas Eq. (32), after inserting numerical values for , provides a rough estimate for . The four parameter points discussed in Ref. [30] correspond to , i.e. they require a rather strong fine-tuning of the initial position of the inflaton field.

3 Hybrid inflation in the complex plane

So far, we have considered inflation for . Due to the linear term in the inflaton potential, a new interesting hilltop region has emerged, which allows for a small spectral index consistent with observation. This improvement in is only achieved, however, at the price of a considerable fine-tuning of the initial position of the inflaton field on the real axis.

The situation changes dramatically once we take into account the fact that, also due to the linear term in the inflaton potential, F-term hybrid inflation is a two-field model of inflation: As we have demonstrated in Sec. 2, the potential depends in fact differently on the real and the imaginary part of the inflaton field and not only on its absolute value . The rotational invariance in the complex plane is thus broken, which is why, depending on the initial value of the inflaton phase, , the inflaton may actually traverse the field space along complicated trajectories that strongly deviate in shape from the simple trajectories on the real axis.7 In order to obtain a complete picture of hybrid inflation, it is therefore important to extend our analysis from the previous section to the general case of inflation in the complex plane. To do so, we will first introduce our formalism, by means of which we are able to calculate predictions for the inflationary observables in the case of multi-field inflation. Then, we will apply this formalism to hybrid inflation in the complex plane and present our numerical results. After that, we will finally demonstrate how our numerical findings can be roughly reconstructed based on analytical expressions.

Figure 4: Two-field dynamics of the complex inflaton in field space. The solid green lines represent possible inflationary trajectories in the scalar potential (dot-dashed orange contour lines.) Lines of constant are marked by dashed blue contours, with the beginning and end of inflation ( and , respectively) marked by thicker contours. Along the light-blue trajectory, the measured values of and are reproduced. The model parameters are again set to  GeV, and  TeV, cf. Fig. 2.
Figure 5: Effect of a large constant term in the superpotential, , on the inflationary trajectories in the complex plane. Colour code, labels and parameter values as in Fig. 4, but now with  TeV.

3.1 Inflationary observables in the formalism

The analytical estimates presented in the previous section were mostly based on an effective single-field approximation. However, in order to fully capture the two-field nature of hybrid inflation, we have to go beyond this approximation and perform a numerical analysis of the inflationary dynamics in the complex plane. In doing so, we shall employ an extended version of the so-called ‘backward method’ developed by Yokoyama et al. [35, 36] in the context of the formalism [37, 38, 39, 40, 41, 42].

The essence of the formalism is that it identifies the curvature perturbation  on uniform energy density hypersurfaces as the fluctuation in the number of -folds which is induced by the fluctuation of the inflaton in field space, , around its homogeneous background value,8

(33)

In calculating , one is free to either specify a boundary condition at early or at late times and then either evolve forward or backward in time. Obviously, the backward method described by Yokoyama et al. [35, 36] pursues the latter approach, cf. also the geometrical analysis presented in Ref. [43]. The former approach is implemented in the ‘forward method’ developed by the authors of Refs. [44, 45]. Either way, it is important to notice that the formalism in its standard formulation, cf. Eq. (34), comes with intrinsic limitations. For instance, the possible interference between different modes at the time of Hubble exit is usually neglected and all perturbations are instead taken to be uncorrelated and Gaussian. Likewise, the universe is assumed to eventually reach the adiabatic limit with no isocurvature modes remaining at late times. Finally, the decaying modes in the curvature perturbation spectrum cannot be accounted for by the formalism. More advanced computational techniques to overcome this latter problem have recently been proposed in the literature [46]. But, as we do not have to deal with any, say, temporal violation of the slow-roll conditions, these decaying modes are negligible in our case just as in any other ‘standard scenario’ of slow-roll inflation. We therefore do not have to resort to a more sophisticated method and can safely stick to the formalism. Similarly, as the two slow-roll parameters and are always small except during the last few -folds of inflation, we will make use of the slow-roll approximation for most of the inflationary period. At the same time, the smallness of and also guarantees that the ‘relaxed slow-roll conditions’ stated in Ref. [35] are satisfied for most times. This justifies why Yokoyama et al.’s backward method is applicable to our inflationary model in its slow-roll formulation.

In the formalism, the inflationary observables , and are all determined by the derivatives of the function w.r.t. to the various directions in field space,

(34)

where and are the first and second partial derivatives of in the sense of a function on field space and with a prime denoting differentiation w.r.t. to in the sense of a time coordinate. For an arbitrary number of canonically normalized real inflaton fields , we have9

(35)

As we shall not consider the possibility of a non-canonical Kähler potential, we have assumed canonical kinetic terms for all scalar fields in writing down Eq. (34).10

In order to obtain predictions for , and which can be compared with observations, all quantities on the right-hand sides of the relations in Eq. (34) need to be evaluated at . As for and , the traditional way to do this, followed by many authors in the literature, is to directly calculate as function on field space by solving the equations of motion for the scalar fields and then to take the partial derivatives of the such obtained expression for . This ‘brute force’ approach is, however, prone to numerical imprecisions and in particular not suited for comparing results from different authors. Every author has to come up with his own numerical procedure to compute and its derivatives, which impedes the comparability of independent studies. By contrast, the backward method by Yokoyama et al. is an elegant and standardizable means of computing the derivatives and directly as the solutions of simple first-order differential equations, rendering the intermediate step of calculating the function first obsolete. Let us now outline how we adapt this method to the scenario of hybrid inflation in the complex plane.

It is convenient to divide the evolution of the inflaton field in field space into three stages: (i) the phase of slow-roll inflation at early times, (ii) the phase of fast-roll inflation shortly before the instability in the scalar potential is reached, and (iii) preheating in the course of the waterfall transition at the end of inflation. In order to quantify the time at which the transition between the slow-roll and the fast-roll stages takes place, we generalize the slow-roll parameters and in Eqs. (16) and (17) to the case of multi-field inflation [37],

(36)

The physical difference between these two sets of slow-roll parameters is the following: While and parametrize the gradient and the curvature of the scalar potential in the direction of the trajectory, cf. Eq. (73) in App. A, and quantify the total gradient and the total curvature of the scalar potential at the momentary location of the inflaton. Since in the slow-roll approximation the inflaton happens to roll in the direction of the potential gradient, coincides with . The parameter , however, is proportional to the Frobenius norm of the Hessian matrix of the scalar potential, , and thus receives contributions from the directions in field space perpendicular to the trajectory, which are not contained in . In this sense, is only a good approximation to , as long as the contributions from isocurvature perturbations to are negligible, i.e. as long as the inflationary dynamics in the complex plane are effectively very similar to the dynamics of ordinary single-field inflation. Slow-roll inflation is now characterized by both generalized slow-roll parameters being at most of . As for all times during inflation, the end of slow-roll inflation is marked by the time when . The radial inflaton component at this time, , can be readily estimated making use of the second derivative of the one-loop potential in the limit of a large field excursion during inflation, . To good approximation, we have11

(37)

As long as , the slow-roll approximation is valid and the evolution of and is governed by the slow-roll equations,

(38)

In order to solve these equations, we specify boundary conditions for them at the end of slow-roll inflation, and , where is nothing but the free parameter labeling the different possible trajectories in field space, which we introduced in Sec. 2.1. At this point, it is worth emphasizing that technically is not defined as the inflaton phase at the onset of the waterfall transition, but as the phase at the end of slow roll. If we were to define as the inflaton phase at the end of fast roll, it would no longer suffice to parametrize the set of inflationary trajectories; in addition to , one would also have to know the final inflaton velocity in order to fully characterize a particular trajectory. For small values of , this distinction between the different possibilities to define is of course irrelevant, since . In the large- regime, the inflaton phase might however drastically change during the stage of fast roll, in which case it is important to precisely define what is meant by .

In Eq. (38), we have omitted the interaction between the inflaton and the waterfall field. This reflects the fact that we assume the waterfall field to be stabilized at its origin throughout the entire inflationary phase. Of course, unknown Planck-scale physics could result in the waterfall field having a large initial field value and/or a large initial velocity. But as long as we focus on the field dynamics around the GUT scale, it is natural to assume that the waterfall field has rolled down to its origin before the onset of the last -folds due to is inflaton-induced GUT-scale mass. Guided by this expectation, we restrict ourselves to the study of slow-roll inflation in the so-called ‘inflationary valley’, in which the waterfall field vanishes. An extension of our analysis incorporating arbitrary initial field values and velocities for the inflaton as well as for the waterfall field is left for future work,12 cf. also our discussion in Sec. 4.

For given values of , , and , the slow-roll equations in Eq. (38) have unique solutions, which describe the time evolution of the homogeneous background fields and . At the same time, the slow-roll equations for the fluctuations and together with the relation  [37, 40] may be used to derive the following slow-roll transport equations for the partial derivatives  [35] and  [44],

(39)

Here, and are functions of and its partial derivatives evaluated along the inflationary trajectory, and ,

(40)

According to Yokoyama et al.’s backward formalism, we specify the initial conditions for the differential equations in Eq. (39) at the end of slow-roll inflation, when . In Cartesian coordinates, the hypersurface in field space on which this condition is satisfied is given by

(41)

Often it is assumed that at the end of slow-roll inflation the universe has already reached the adiabatic limit, which is equivalent to taking the energy density or equivalently the Hubble rate on this hypersurface to be constant, . This renders Yokoyama et al.’s method insensitive to the further evolution of the inflaton field at times after . As a consequence of this assumption, the conversion of isocurvature into curvature perturbations during the final stages of inflation as well as after inflation is neglected, which may however have important effects in some cases such as, for instance, multi-brid inflation [22, 23, 49]. To remedy this shortcoming of the backward method in its original formulation, we explicitly take into account the variation of the function on the hypersurface. Let us denote by , such that all of the four following conditions are equivalent to each other,

(42)

After some algebra along the lines of Refs. [35, 36], we then find the initial values of and at time ,

(43)

where all quantities on the right-hand sides of these two relations are to be evaluated at and with being defined as

(44)

Our result for is identical to the one derived in Ref. [43], which represents the first analysis properly taking care of the fact that is in general actually not a constant. By contrast, our expression for has not been derived before. It represents a straightforward generalization of the initial conditions for stated in Refs. [35, 36, 44, 45] to the case of non-constant . As we will see shortly, the universe reaches the adiabatic limit in the course of the preheating process. This allows us to fix the origin of the time axis, , at some appropriate time during preheating and distinguish between two contributions to the function : the number of -folds elapsing during the final fast-roll stage of inflation, , as well as the number of -folds elapsing during preheating, ,

(45)

In this sense, our improved treatment of the initial conditions for and now also includes the evolution of curvature and isocurvature modes during fast-roll inflation as well as preheating.

For a given slow-roll trajectory hitting the hypersurface for some inflaton phase , we compute by solving the full equations of motion for the two inflaton fields between the point and the instability in the scalar potential.13 These equations are of second order and thus require us to specify the initial velocities of the inflaton fields on the hypersurface, and . The unique choice for these initial conditions ensuring consistency with our treatment of the slow-roll regime obviously corresponds to the expressions in Eq. (38) evaluated at and it is precisely these velocities that we use in computing . Nonetheless, we observe that our results for are rather sensitive to the values we choose for and . This sensitivity becomes weaker once we lower , the critical value of dividing the fast-roll from the slow-roll regime. On the other hand, going to a smaller value of also reduces the portion of the inflationary evolution during which the transport equations in Eq. (39) are to be employed, the simplicity of which motivated us to base our analysis on Yokoyama et al.’s backward method in the first place. It is therefore also under the impression of these observations that, seeking a compromise between too large and too small , we set to an intermediate value such as rather than to or .

In order to compute , we solve the full second-order equations of motion for the two inflaton fields and as well as for the waterfall field from the onset of the phase transition up to the time when the Hubble rate has dropped to some fraction of its initial value and the universe has reached the adiabatic limit. Here, our numerical calculations indicate that a fraction of is enough, so as to obtain a sufficient convergence of all inflationary trajectories. Moreover, we note that, as is a classically stable solution, it is necessary to introduce a small artificial shift of the field at the beginning of preheating, so as to allow the waterfall field to reach the true vacuum. The two parameters and are physically meaningless and just serve as auxiliary quantities in our numerical analysis. Their values must therefore be chosen such that our results for remain invariant under small variation of these parameters.

Our procedure to determine captures of course only the classical dynamics of the waterfall transition and misses potentially important non-perturbative quantum effects. A treatment of preheating at the quantum level however requires numerical lattice simulations, which goes beyond the scope of this paper—and which is actually also not necessary for our purposes. As we are able to demonstrate numerically, and its derivatives never have any significant effect on our predictions for , , and , if solely computed based on classical dynamics. Barring the unlikely possibility that quantum effects yield a substantial enhancement of , the evolution of the inflaton during the waterfall transition is thus completely negligible from the viewpoint of inflationary physics. Because of this, we will simply discard the contribution from preheating to the function in the following and approximate it by its fast-roll contribution, . This also automatically entails that we do not need to consider the evolution of the waterfall field any further. As we focus on hybrid inflation in the inflationary valley, we can simply set to at all times.

In conclusion, we summarize that, for given values of the parameters , , and , we have to perform four steps in order to compute our predictions for the observables , and . (i) First, we determine by solving the second-order equations of motion for and from the hypersurface to the instability in the scalar potential. Here, we specify the initial velocities of and such that they are consistent with Eq. (38) evaluated on the hypersurface. (ii) Subsequently, we solve the slow-roll equations for and in Eq. (38) starting on the hypersurface and then going backward in time up to the point when the CMB scales leave the Hubble horizon, i.e., in terms of the number of -folds, from up to . (iii) With the slow-roll solutions for and at hand, we are able to solve the transport equations for the partial derivatives and in Eq. (39) in the interval . In doing so, we employ the initial conditions for and at the time in Eq. (43). (iv) The derivatives and evaluated at time eventually allows us to calculate the inflationary observables according to Eq. (34).

3.2 Phase dependence of the inflationary observables

Inflationary trajectories in the complex plane

As a first application of the above developed formalism, we are now able to study the dynamics of the inflaton field in the complex plane. In order to find all viable inflationary trajectories in the complex inflaton field space, we impose two conditions: (i) on the hypersurface, the slope of the scalar potential in the radial direction must be positive14 and (ii) the fast-roll motion during the last stages of inflation must end on the instability in the scalar potential,

(46)

Together, these two requirements are sufficient to ensure that the inflaton does not become trapped in the local minimum on the positive real axis. For vanishing or small gravitino mass, they are always trivially fulfilled and can take any value between and . However, once the slope of the linear term begins to exceed the slope of the one-loop potential, the range of allowed values becomes more and more restricted, until eventually only phases remain viable. This effect is illustrated in Figs. 4 and 5, which respectively show the set of possible inflationary trajectories for an intermediate as well as for a large value of the gravitino mass, while and are set to identical values in both plots. Note that for Fig. 4 we have chosen the same parameter values as for Fig. 2, which renders this figure the continuation of Fig. 2 from the real axis to the complex plane. Both Fig. 4 and Fig. 5 demonstrate how the linear term distorts the rotational invariance of the scalar potential by adding a constant slope in the direction of the real inflaton component . As for Fig. 5, the situation is however more extreme in consequence of the enhanced gravitino mass compared to Fig. 4. Inflation on the positive real axis is, for instance, no longer possible for such a large gravitino mass; instead, has at least to be slightly larger than . Moreover, as an important consequence of our ability to determine all inflationary trajectories, we are now in the position to identify the region in field space which may provide viable initial conditions for inflation. In fact, this region is nothing but the fraction of field space traversed by all inflationary trajectories for . We will return to the issue of initial conditions for inflation in Sec. 4.

Inflationary observables for individual parameter points

Figure 6: Amplitude and spectral index of the primordial scalar power spectrum as functions of the inflaton phase at the end of slow-roll inflation, , for , and .

In the next step, as we now know the trajectories along which the inflaton can move across field space, we are able to compute the inflationary observables for given values of , and and study their dependence on . In the limit of very small gravitino masses, when the slope of the inflaton potential is dominated by the one-loop potential, this dependence becomes increasingly negligible and , and as functions of approach constant values. On the other hand, for very large values of , all viable trajectories start out at a similar initial inflaton phase and run mostly in parallel to the real axis, cf. Fig. 5. Due to this similarity between the different viable trajectories, the dependence of the inflationary observables on is again rather weak for the most part. There is however one crucial exception: In the large- regime, is bounded from below, and once approaches , the scalar and the bispectrum amplitudes, and , begin to rapidly increase. This is due to the fact that for the inflaton trajectory hits the instability in the scalar potential at a very shallow angle, so that initial isocurvature perturbations induce large shifts in , and hence large curvature perturbations, at late times. But at any rate, the most interesting case is the one of intermediate gravitino masses, when the gradients of the one-loop potential and the linear term are of comparable size and the inflationary observables strongly depend on . An example for such a situation is given in Figs. 6 and 7, in which we show , and as functions of for the same parameter values that we also used for Figs. 2 and 4. Now it becomes evident that for these parameter values and a final phase of the observed values for and can be nicely reproduced, while safely stays within the experimental bounds.

An important lesson which we learn from Figs. 6 and 7 is that the Lagrangian parameters, , and , and hence the functional form of the scalar potential do not fix the inflationary observables at all. Under a variation of the inflationary trajectory, , and vary over significant ranges, in which the observed values are not singled out in any way. We therefore conclude that the values for the inflationary observables realized in our universe do not point to a particular Lagrangian, but rather seem to be a mere consequence of an arbitrary selection among different possible trajectories. This is a very characteristic feature of hybrid inflation in the complex plane, which distinguishes it from other popular inflation models. In  inflation [50] or chaotic inflation [51], for instance, the shape of the scalar potential is the key player behind the predictions for the inflationary observables. As we now see, the philosophical attitude in hybrid inflation is certainly a different one: Here, the main virtue of inflation are mainly its qualitative aspects—the fact that it solves the initial conditions problems of big bang cosmology, explains the origin of the primordial density perturbations and is consistent with a compelling model of particle physics at very high energies. Its quantitative outcome is the mere result of a selection process that has no deeper meaning within the model itself.

Figure 7: Amplitude of the local bispectrum (black curve) and naive single-field slow-roll estimate for this quantity (grey curve), cf. Eq. (50), as functions of the inflaton phase at the end of slow-roll inflation, , for , and .

Amplitude and spectral tilt of the scalar power spectrum

In the third step of our numerical investigation, we perform a calculation of the inflationary observables, as we just did it for one parameter point, for all values of , , and of interest. In this scan of the parameter space, we shall cover the following parameter ranges,

(47)

The ranges for and are chosen such that on the one hand, for values of not much smaller than typical Standard Model Yukawa couplings, the measured value of the scalar amplitude can be reproduced and that on the other hand the bound on the cosmic string tension in Eq. (10) is obeyed in most cases. At the same time, the range covers all values of gravitino masses which are commonly assumed in supersymmetric models of electroweak symmetry breaking. As our results will confirm, the such defined parameter space contains all the phenomenologically interesting parameter regimes for hybrid inflation.

Figure 8: Viable parameter space for hybrid inflation in the complex plane for two different values of the final inflaton phase . Color code and labels as in the left panel of Fig. 3. In contrast to Fig. 3, we now also find regions in parameter space where our predictions for the scalar amplitude are always smaller than the observed value . In these regions, it is impossible to increase by going to larger gravitino masses as this would cause the inflaton to miss the instability in the scalar potential and reach the local minimum on the positive real axis, cf. our discussion of Eq. (46).

Let us first focus on and , the two observables related to the scalar power spectrum, before we then comment on , the amplitude of the local bispectrum. Both and depend on all three Lagrangian parameters , and as well as on the choice among the different inflationary trajectories, which we label by . As has been measured very precisely by the various CMB satellite experiments, cf. Eq. (9), we are able to eliminate one free parameter, say, the gravitino mass, by requiring that our prediction for must always coincide with the observed best-fit value for the scalar amplitude, ,

(48)

This renders all remaining inflationary observables functions of , and only. Next, we demand that our prediction for must fall into the range around the measured best-fit value for the scalar spectral index, ,

(49)

which provides us with  C.L. exclusion contours in the plane for every individual value of . As examples of such exclusion contours, we show the viable region in the plane for and in Fig. 8. These two plots generalize the left panel of Fig. 3 from hybrid inflation on the real axis to the full two-field scenario.

By comparing our parameter constraints in the two-field case with the results obtained in Sec. 2.2, we are able to identify the similarities and differences between hybrid inflation on the real axis and hybrid inflation in the complex plane. These observations belong to the most important results of our analysis. First of all, we note that for small but nonzero and fixed , we always find two pairs of values such that and are successfully