Analytical Study of Mode Coupling in Hybrid Inflation

Analytical Study of Mode Coupling in Hybrid Inflation

Laurence Perreault Levasseur, Guillaume Laporte and Robert Brandenberger Department of Physics, McGill University, Montréal, QC, H3A 2T8, Canada
Abstract

We provide an analytical study of the coupling of short and long wavelength fluctuation modes during the initial phase of reheating in two field models like hybrid inflation. In these models, there is - at linear order in perturbation theory - an instability in the entropy modes of cosmological perturbations which, if not cut off, could lead to curvature fluctuations which exceed the current observational values. Here, we demonstrate that the back-reaction of short wavelength fluctuations is too weak to lead to a truncation of the instability for the long wavelength modes on time scales comparable to the typical instability time scale of the long wavelength entropy modes. Hence, unless there are other mechanisms which truncate the instability, then in models such as hybrid inflation the curvature perturbations produced during reheating on scales of current observational interest may be very important.

pacs:
98.80.Cq

I Introduction

The inflationary scenario Guth () is the current paradigm of early universe cosmology. It addresses several conceptual problems which Standard Big Bang cosmology was unable to solve, and - perhaps more importantly - provides a causal mechanism to generate the primordial cosmological fluctuations which lead to the large-scale structure and cosmic microwave background anisotropies which are currently observed ChibMukh (). Reheating after the period of inflation is a key aspect of inflationary cosmology (see the recent review of Reheatingrev ()). During reheating the energy is transformed from the inflaton, the scalar matter field which is responsible for providing the inflationary expansion of space, to the matter we see today. Without reheating, the inflationary universe would leave behind a large universe empty of any matter, in obvious contradiction with observations. Reheating requires some coupling between the inflaton and regular matter.

The initial studies of reheating were based on first order perturbation theory applied to the inflaton decay initial (). This approach, however, fails to take into account the coherent nature of the inflaton field. At the end of the period of inflation, the inflaton begins coherent homogeneous oscillations about the ground state of its potential. In TB () (see also DK (); STB ()) it was realized that this gives rise to a potential parametric instability in any matter field which couples to the inflaton. This instability can take various forms depending on the nature of the coupling between the inflaton and the matter field. Rather generically, the process leads to a rapid energy transfer from the inflaton to the matter fields 111Rapid meaning on a time scale less than the Hubble time at the end of the period of inflation.. In a wide class of models, the resonance is of “broad resonance” KLS1 () type and affects all fluctuations with wavenumbers smaller than a characteristic mass set by the particle physics Lagrangian. In some cases - in particular in the hybrid inflation model which we study in this paper - there is a tachyonic instability in the matter sector at the end of inflation, in which case the reheating instability is of “tachyonic resonance” tachyonic () type and hence extremely efficient.

The inflaton field obviously couples to gravity, and therefore, as first conjectured in BKM1 (), it is possible that its oscillations will lead to a parametric resonance instability of the cosmological fluctuation modes. Due to the exponential growth of the causal horizon during the inflationary phase with constant Hubble radius, even super-Hubble modes can be causally affected by this instability FB1 (). This will inherently cause a long wavelength curvature mode to develop which potentially exhibits a tachyonic resonance. These large scale modes are of prime importance, since they correspond to scales observable today. If an inflationary model fails to control this possible instability of the curvature mode, the observed smallness of the magnitude of curvature perturbations today might rule out this model.

If matter consists of a single scalar field, then - as studied in FB1 () and Afshordi () - there is no parametric instability of super-Hubble scale modes. However, if entropy fluctuations are present, then an instability of entropy modes can occur which will lead to rapid growth of the curvature fluctuations. For this instability to be effective, the entropy mode cannot have been exponentially redshifted during the period of inflation. Models which satisfy the conditions for parametric instability of the metric fluctuations were first discussed in BaVi () and FB2 ().

If the duration of the parametric instability of long wavelength curvature fluctuations were long, then the curvature fluctuations induced by the resonant entropy modes could become larger than the primordial curvature perturbations and - in fact - larger than the observed amplitude of the inhomogeneities. Thus, one might potentially be able to use the parametric resonant instability of the entropy modes of the cosmological fluctuations to rule out large classes of multi-field inflationary models.

One class of inflationary models which at linear order in perturbation theory leads to an instability of the entropy fluctuations is hybrid inflation hybrid (). In this scenario, two scalar fields and , are involved in the inflationary process, the slowly rolling inflaton field and the “waterfall” field . The key point of this class of models is that while is slowly rolling during inflation, the energy density of the Universe is dominated by the potential of . Over the past few years, a lot of interest has been devoted to these models, mainly since they provide a framework for the realization of inflation in the context of both supersymmetry Rachel () and string theory (see stringinflrev () for reviews and Tye () for an original reference). Among the promising approaches stand the D-brane / antibrane inflation models (e.g. the D3/D7 brane inflation model Keshav () and the KKLMMT model KKLMMT ()). Though these models may resolve some of the conceptual problems from which simple single scalar-field driven inflation models suffer, the large number of light moduli fields present in string compactifications can give rise to entropy fluctuation modes, which could in turn enter a parametric resonance phase at early stages of reheating BaVi (); FB2 ().

Hybrid inflation models are characterised by symmetry breaking along the direction: The symmetry is spontaneously broken for field values smaller than a critical value . Inflation takes place while is slowly rolling at field values . Once the inflaton crosses the critical value, the effective potential for develops a tachyonic instability which triggers the rapid rolling of towards one of the ground states of its potential. This causes inflation to end.

Reheating in hybrid inflation models on a fixed Friedmann background cosmology (i.e. no cosmological perturbations) was studied in detail in Felder1 (); Felder2 () (see also hybridpreh ()). By means of numerical simulations it was observed that within a very short time, non-linearities on a length scale given by the mass of the waterfall field develop and dominate the subsequent stages of the reheating process. It is important to study how large-scale metric perturbations evolve during reheating in these models. In previous work, linear evolution of the entropy and curvature modes was studied BDD (); BFL () 222Other work on fluctuations beyond linear analysis in hybrid inflation model focused on non-Gaussianities FIN (); Barnaby (), and it was shown that it is possible that an important entropy perturbation mode develops. In these works it was assumed that non-linear effects, e.g. the back-reaction of small-scale fluctuations, would unlikely have a dominant effect on large-scale fluctuation. Due to the large range of scales (the small wavelength modes we are interested in have wavelengths comparable to the inverse Hubble radius at the end of inflation whereas the wavelengths of modes we are interested in for current cosmological observations are of the order of , assuming that the scale of inflation is about ) numerical studies do not have the dynamical range to study this question. Instead, an analytical understanding of the dynamics is required.

Our work demonstrates, by means of an analytical analysis, that the back-reaction of short wavelength modes with random phases is too weak to truncate the instability of the long wavelength entropy modes. We have studied the effects of short wavelength fluctuations both on the background inflaton and on long wavelength modes of the inflaton and waterfall fields. Most importantly, we have shown that the back-reaction on large scale fluctuations of the waterfall field is too weak to shut off the resonance of this entropy mode one the relevant time scale of the instability. This implies that hybrid inflation models of the type analyzed in this paper may suffer from a potential “entropy mode problem”, unless there are other ways of truncating the resonance (e.g. the exponential decrease in the initial value of the fluctuations modes during the period of inflation - see e.g. BFL () for a discussion) 333There is another way to test these models observationally: hybrid inflation models typically produce topological defects such as cosmic strings. These strings, if stable on cosmological scales as they are in field theory models of hybrid inflation, lead to a scaling solution of strings at all late times (see e.g. CSrevs () for reviews on cosmic strings and structure formation). The strings, in turn, produce line discontinuities in cosmic microwave temperature anisotropy maps KS () which can be searched for in observational maps using edge detection algorithms such as the Canny algorithm, as recently studied in Canny ()..

Our study of the back-reaction of fluctuations in hybrid inflation models is not the first analytical study. For an analytical study of the back-reaction effect of fluctuations on the background using very different techniques than the ones we use see Katrin (). For a study of the effects of metric fluctuations on the observable local expansion rate of the universe due to fluctuations see Ghazal ().

The structure of this paper is as follows. In Section 2, we review the class of hybrid inflation models which are studied here, and introduce the perturbative approach employed to study non-linear interactions that arise during preheating. In Section 3, we solve the background theory, and in Section 4 we review the evolution of quantum fluctuations to linear order. In Section 5, we analyze the generation of long wavelength second order perturbation modes sourced by shorter wavelength first order fluctuations (summing over the contribution from all frequencies and assuming random phases), for both the inflaton and the waterfall field. This is the leading order back-reaction effect between short and long wavelength modes. We also study the back-reaction of the fluctuations on the background. We find that small-scale first-order modes with random phases have a contribution on the evolution of large-scale modes which does not start to dominate until long after the instability of the first order long wavelength modes has had a chance to develop. Hence, in spite of their large phase space, the non-linear effects of the short wavelength perturbations do not become dominant for the evolution of long wavelength modes during the early stages of preheating.

Ii The Model and Perturbative Approach

We focus on the category of hybrid inflation models with Lagrangian density for matter given by:

The equations of motion form a coupled system of partial differential equations

(2)
(3)

The Hubble parameter at time is given by:

(4)

where is the energy density, and is given by:

(5)

where the potential is given in the second line of (II). The turnover value of at which develops a tachyonic instability will then be

(6)

To study this model we work in discrete Fourier space (discrete because of a finite volume cutoff) and expand to second order about a homogeneous and isotropic cosmological background. The expansion parameter is the amplitude of the linear fluctuations. Our goal is to study how first order perturbations of high modes (with wavelengths comparable to the Hubble length at the end of inflation or to the wavelength associated with the mass of the waterfall field, whichever is smaller) influence the low modes (modes which affect cosmological observations today which correspond to a scale of roughly at the end of inflation) at second order. We want to estimate the time interval it will take before this back-reaction effect becomes dominant, and we also want to see whether back-reaction effects on modes with wavelength of the order decreases with the wavelength.

The expansion of the fields to second order in is:

(7)
(8)

Since we will eventually be evaluating mode sums numerically, it is useful to work with real Fourier modes. Hence, we can expand the first and second order field perturbations as follows:

Here, is the size of the finite one-dimensional box inside of which we are performing the discrete Fourier expansion. Physically, we need to take larger (but not much larger) than the largest scale of the problem we study; hence we fix it to be . This effectively imposes a cutoff on the largest scale studied. For clarity, we only write explicitly the one-dimensional expansions, but generalisation to three-dimensional Fourier series is straightforward and will not modify in a crucial way the form of the obtained solutions, unless otherwise mentioned.

Iii Zeroth Order Expansion

Inserting the above ansatz into the equations of motion of the system and expanding to zeroth order in , (2) and (3) reduce to

(9)
(10)

For values of smaller than , there is an instability of the background solution for . Because of this instability, the field grows fast on the scale of a Hubble expansion time. Hence, we expect the Hubble damping term to be negligible, and thus we can set . This amounts to setting to a constant (which we pick to be 1) and The linear equation for then becomes that of a harmonic oscillator and has the solution

(11)

where and are constants depending on the initial conditions on and its derivative. Since in our case, we are interrested in the end of the inflationary era, we want and we also want to be initially slowly rolling, i.e. . Thus, we set to zero and obtain:

(12)
(13)

Iv First Order Expansion

iv.1 Equations

Now, going back to the system (2) and (3) and inserting the ansatz (7) and (8), we expand and keep terms of first order in :

(14)

Inserting the explicit form of the first order perturbations, we make use of the orthogonality relations for trigonometric functions to convert (14) and (IV.1) to discrete Fourier space.

Doing so, we obtain the following set of differential equations for the first order correction to each Fourier mode:

(16)
(17)

Replacing the wavenumber by its vectorial expression yields, without any other modification, the generalisation of the 1+1-dimensional equations to the 3+1-dimensional setting.

iv.2 Solutions

Again setting , the first order equations become that of harmonic oscillators, and so can be solved easily:

(18)

Let us now have a closer look at the characteristic parameter values. We have in mind a hybrid inflation model stemming from string scale physics. Hence, the value of will be taken to be in Planck units. We choose coupling constants and , so that (again in Planck units). We will consider the range (in Planck units). For a value of at the upper end of this interval, the hybrid inflation model will result in cosmological fluctuations of the observed order of magnitude (see e.g. MFB () for a review of the theory of cosmological perturbations and RHBrev2 () for an introductory overview). We explore a range of masses in order to study how the strength of our back-reaction effect depends on the model parameters.

Moreover, we are obviously interested in modes whose wavelength is larger than and larger than the Hubble radius. For such modes . Therefore, (18) can be approximated by:

(19)

which describes stable harmonic oscillation. No instability is manifest in the field.

On the other hand, the first order equations become:

(20)

The equation can be reduced to the Mathieu equation by performing the transformation and using the identity :

(21)

where the prime refers to a derivation with respect to . Defining:

(22)
(23)

we indeed recover the canonical form of the Mathieu equation (see e.g. Mathieu ())

(24)

The paramter is called the “Floquet exponent”, and we will call the “square frequency”.

For the parameter values we are using, the value of is much larger than . Hence, we are in the parameter region of either “tachyonic resonance” (if the tachyonic term in the expression for dominates over the third term, or “broad parametric resonance” if the third term dominates (the first term in is negligible for the modes we are interested in). In either case, all of the infrared modes which we study here will experience an exponential instability with a growth rate characterized by the Floquet exponent. To first order in perturbation theory, every mode evolves in an independent way and there is no interaction between different modes. In particular, a mode that is not initially excited will not grow to first order at any later time (obviously, we expect quantum vacuum fluctuations on all scales to seed the instability). Inserting the expression (6) for , we find that for all values of of interest to us

(independently of the value of ); which means that is always negative. Hence, we conclude that all modes we are interested in undergo tachyonic parametric resonance.

The solution to the second order differential equation (24) can be written in terms of two linearly independent solutions, the so-called Mathieu functions and :

(25)

where the C’s and are the coefficients.

Note that an important property of the solution to (24) for any choice of the parameters is the existence of an exponential instability of parametric resonance type for certain ranges of the parameter value . For values of falling within the instability range, increa-

Figure 1: Analytical solution to (24) in red, in green, and in yellow, all on a logarithmic scale, as a function of ; for and for the reference parameters cited in the text. The function is a better approximation to the MathieuC function, and it is an upper bound for .

ses exponentially. In the cases of tachyonic and broad parametric resonance

(26)

for some constant depending on . In the case of broad parametric resonance KLS1 (), the parameter is of the order (but slightly smaller) than 1.

We can find an approximate solution of (24) valid over the first half of the period of under the assumption that the initial value is (this is a normalization) and that the initial velocity of the mode vanishes (which, as discussed in the next subsection, is a good assumption for the modes we are interested in). Our approximate solution in fact gives an upper bound on the value of the mode function (which is within a factor two of the exact solution) and is given by

(27)

This result is found by approximating the equation of motion as

(28)

and imposing the initial conditions mentioned above. Indeed, if we set

(29)

we obtain

But from our choice of parameters, and . So we have at least from to (in which region ).

iv.3 Initial Conditions

Initial conditions on the first order fluctuation modes are given by the quantum fluctuations at the end of inflation. These modes begin on sub-Hubble scales at the beginning of the inflationary phase in their quantum vacuum state. As reviewed e.g. in MFB (); RHBrev2 (), the fluctuations freeze out and undergo squeezing once the wavelength exits the Hubble radius. The squeezing implies that the velocity of the mode functions will redshift. For purely notational simplicity we have chosen to excite only the and modes and to set the , modes to zero. This implies that we are taking correlated phases for the first order modes. At the end of the calculation we will restore the randomness of the phases and comment on the effect that this has on the strength of the back-reaction. Because of the squeezing of the super-Hubble modes discussed above, we start and with zero velocity, that is to say, we set , while and are determined by the Bunch-Davies state.

When numerically computing mode sums later on in this article, it is important to know the initial amplitude of the mode functions in discrete momentum space. In continuous Fourier space, these amplitudes are given by (in spatial dimensions):

(30)

where we are using the Fourier decomposition in the form

(31)

where is the spatial volume. We want to match this set of initial conditions with our discrete Fourier series:

(32)

We know and , and thus want to relate to in terms of . To do so, we make use of the identity (recalling that )

(33)

in the continuum limit in dimensions. In this limit, the discrete expansion (32) needs to converge to (31), i.e.:

where the last step expresses our requirement of convergence. Hence, for the initial values of the discrete Fourier modes we find the relation:

(35)

However, even though initial velocities of all first order modes must be zero due to freezing outside the Hubble radius, their relative phases must, in general, be random. Thus, the initial conditions must include these random phases. When measuring a first order mode, the expectation value of its amplitude in absolute value needs to be taken, which will divide the obtained amplitude by 2, while for further calculations, keeping the phase general as a random variable will be required.

Bringing everything together, we thus write the solution for the first order and modes:

(36)
(37)

V Second Order Expansion

v.1 Equations

Going back to the system (2) and (3) and again inserting the ansatz (7) and (8), we expand and now keep terms of second order in . We obtain

(38)

and

(39)

Inserting the explicit form of the first and second order perturbations, we make use of the orthogonality relations for trigonometric functions to convert (38) and (39) to discrete Fourier space. However, this time the process is slightly non-trivial due to the presence of the interaction terms at this order in perturbation theory which give rise to mode mixing. Indeed, the last terms in equations (38) and (39) describe how the growth of first order perturbations will source second order pertubations. They involve products of modes, which requires the use of trigonometric identities to split these terms in a way that allows the use of the canonical orthogonality conditions for sines and cosines. After some algebra, this yelds the following set of differential equations for the second order correction to each Fourier mode. In one spatial dimension (recalling that by definition, and because these modes are part of the background) we obtain the following results:

For , the equations describing the back-reaction of the fluctuation modes on the perturbations themselves take the form

(40)
(41)
(42)
(43)

while for , that is, for the back-reaction on the background fields, we have:

(44)
(45)

Note that the phases cancel out in the back-reaction on the inflaton field (in Eq. 44), but not in any of the other equations.

The physics which these equations describe is the following: Since the system initially has no second order perturbations, it is the interaction of two first order modes whose wavenumbers add up to that will source fluctuations of wavenumber at second order. The second order perturbation at wavenumber is affected by all first order modes. Hence, even though the effect of each individual first order mode is of the order , the large phase space of modes which contribute can lead to a large back-reaction effect 444Similarly in spirit, the large phase space of linear perturbation modes can lead to a large back-reaction effect of linear cosmological fluctuations on the background metric, an effect studied in Abramo () and reviewed in RHBrev1 ()..

The above equations can luckily be reduced slightly. Since we have chosen not to excite the and modes, no second order sinusoidal fluctuations will arise, that is, for all , and so there is no need to consider equations (40) and (42). Moreover, can again be set to zero; and every term involving or in the interaction sum acting as a source in each equation can be set to zero. Also, as discussed above, the terms linear in the fields having as coefficient in equations (40) through (43) are negligible compared to the mass term of the fields, and thus can be dropped. However, recall that this conclusion was reached by imposing a cutoff equal to the Hubble radius on the smallest scale excited to first order. Consequently, all sums involving interactions of first order modes acting as source terms for the second order perturbation modes can be performed up to .

Generalizing these equations from the 1+1-dimensional case to the higher-dimensional case this time is a bit more involved than simply replacing the wavenumber by its vectorial expression . In fact, the simple replacement works for every terms except for the interaction sum in each equation, which needs to be modified as follows:

(46)
(47)
(48)
(49)

for the , , and equations, respectively, in the case of spatial dimensions. In particular, for , the sum for the equation (41) can be rewritten as:

(50)

Similarly, for the equation (43):