1 Introduction



Evolution of vacuum fluctuations generated

during and before inflation

Hajime Aoki, Satoshi Iso and Yasuhiro Sekino

Department of Physics, Saga University, Saga 840-8502, Japan

KEK Theory Center, High Energy Accelerator Research Organization (KEK),

Graduate University for Advanced Studies (SOKENDAI),

Ibaraki 305-0801, Japan


We calculate the time evolution of the expectation value of the energy-momentum tensor for a minimally-coupled massless scalar field in cosmological spacetimes, with an application to dark energy in mind. We first study the evolution from inflation until the present, fixing the Bunch-Davies initial condition. The energy density of a quantum field evolves as in the matter-dominated (MD) period, where and are the Hubble parameters during inflation and at each moment. Its equation of state, , changes from a negative value to in the radiation-dominated period, and from to in the MD period. We then consider possible effects of a Planckian universe, which may have existed before inflation, by assuming there was another inflation with the Hubble parameter . In this case, modes with wavelengths longer than the current horizon radius are mainly amplified, and the energy density of a quantum field grows with time as in the MD period, where and are the scale factors at each time and at present. Hence, if is of the order of the Planck scale , becomes comparable to the critical density at the present time. The contribution to from the long wavelength fluctuations generated before the ordinary inflation has in the free field approximation. We mention a possibility that interactions further amplify the energy density and change the equation of state.

1 Introduction

Our universe is well described by the so-called spatially-flat CDM model. Energy density of our universe is close to the critical density. According to the PLANCK 2013 results [1], only 5.1% of the energy density is attributed to a known form of baryonic matter, while 26.8% is attributed to cold dark matter (weakly interacting non-relativistic matter), and 68.3% to dark energy (or the cosmological constant). Explaining the origin of these unknown ingredients, dark matter and dark energy, is one of the biggest challenges in modern physics.

There are various proposals for dark matter, such as supersymmetric particles, axions, and so on. Dark matter may well be one of these. However, the origin of dark energy is totally unclear. Although its equation of state, , seems like that of vacuum energy of quantum fields, there is no reasonable explanation for its magnitude, with meV=eV. Here, is the (reduced) Planck scale and is the current Hubble parameter. is far smaller than the expected magnitude of the vacuum energy in a theory with an ultra-violet (UV) cutoff, , with any reasonable choice for . If we take to be , is smaller than by more than 120 orders of magnitude. Even if we take to be the supersymmetry breaking scale, electroweak scale, or any other natural scale in high-energy physics, is still much smaller than . This is the cosmological constant problem [2]. It may turn out that the solution to this problem is given by the anthropic principle [3], but attempts at a dynamical explanation of dark energy are undoubtedly important.

In this paper, we study some aspects of vacuum energy of quantum fields in cosmological spacetimes, with an application to the cosmological constant problem in mind. Let us first note that the order of magnitude of the energy density of quantum fields is not necessarily given by the value of the UV cutoff. Expectation values of the energy-momentum tensor should be renormalized by subtracting the cutoff-dependent (divergent) term. There is a well-defined method for regularization and renormalization, which yields finite expectation values for energy-momentum tensors, which are covariant and conserved [4]. The terms to be subtracted are a combination of spacetime curvature tensors. Typical terms in the renormalized expectation value (such as the terms responsible for the Weyl anomaly) are of the order of the background curvature.

In particular, vacuum energy for fluctuations in the de Sitter background with the Hubble parameter is of the order (for massless fields), and has  [5, 6]. Since our present universe is close to de Sitter space, one may wonder if dark energy can be explained as vacuum energy in de Sitter with the current Hubble parameter , but this does not seem to be plausible. Dark energy that we observe, , is much larger than the expected contribution from a single field, .

However, local curvature is not the only dimensionful quantity that affects the renormalized energy-momentum tensor. To compute expectation values of fluctuations, we need to specify the vacuum state. This may depend on global properties of the geometry and the whole history of the universe; thus, different scales might be introduced in the problem.

There is by now strong evidence [7] that there has been a period of inflation in our past. The spacetime during inflation is nearly de Sitter space with the Hubble parameter being much larger than . Inflation should have lasted long enough (e-foldings ) to solve the flatness and horizon problems. It would be reasonable to take the vacuum to be the Bunch-Davies vacuum [5] for de Sitter space with . The Bunch-Davies vacuum is the one obtained by the Euclidean prescription, and reduces to the Minkowski vacuum in the short wavelength limit. Even if different initial states are taken, correlation functions will be attracted to those taken with respect to the Bunch-Davies vacuum, as shown quite generally [8].

Fluctuations of the massless scalar in the de Sitter background (in four spacetime dimensions) is of the order , as is clear from dimensional analysis. Massless fluctuations are frozen (remain constant) outside the Hubble radius (see, e.g., [9]); thus, infra-red (IR) modes could have a large value in the universe after inflation. In fact, these fluctuations are considered to be the origin of the fluctuations of the cosmic microwave background (CMB) that is observed today [1, 7].

The purpose of this paper is to understand how the IR modes of quantum fields could affect the energy density in the present universe. We will mostly consider a massless minimally-coupled scalar field, and study the time evolution of the energy-momentum tensor in detail. Our result will shed light on the effect of almost massless and non-interacting fields, such as axions. Our work is related to the studies of the fluctuations of the graviton or the inflaton [10, 11, 12], but further modifications are necessary since energy-momentum tensors for gravitons have different tensor structures from those for scalars and those for the inflaton have contributions from the classical value of the field. We believe our work serves as a starting point for the study of the time evolution of those fields.1

We will find that the magnitude of the present energy-momentum tensor of a massless field is of the order , with a possible factor logarithmic in the scale factor. The contribution from each momentum mode can be renormalized separately before integration over is performed. The equation of state for the (un-renormalized) contribution from each has : The IR mode has ; for larger , is larger, and the UV limit has , which is the same as radiation. The equation of state may become due to renormalization, but the terms arising from renormalization will be of the order of the background curvature. In the present universe, these will be of the order , and are negligible.2

The value is smaller than dark energy (or total energy density) in our universe, since the scale of inflation is smaller than the Planck scale, , as suggested by the observations of the CMB [7]. But, in this paper, we point out a possibility that the energy density of a quantum field takes a larger value.3 Even though the e-folding of inflation has to be large enough to make the region inside our horizon smooth, it does not have to be infinite. Finiteness of the e-foldings means the long wavelength modes have not been in causal contact during the inflationary era. There is no reason to expect that those modes are in the Bunch-Davies vacuum. It would not be too surprising if there are large fluctuations in the far IR. After all, the point of inflation is to push away inhomogeneities beyond our horizon. In particular, in the context of eternal inflation [17, 18, 19], our universe is generally surrounded by the region with a larger Hubble parameter where there are large fluctuations.

It is difficult to know what happened before inflation. In this paper, as an explicit example, we consider a double inflation model. We assume there was an inflation with the Hubble parameter of the order of the Planck scale (such as Starobinsky’s inflation [20], for example), followed by either a radiation-dominated or curvature-dominated transition period, before the usual inflation with starts. We fix the initial condition of the fields in the Planckian inflation period by taking the Bunch-Davies vacuum for de Sitter with the Hubble , and study the time evolution afterwards. In this case, IR mode is enhanced to .

By computing the evolution of the energy-momentum tensor for the double inflation model, we find the present value of vacuum energy to be of the order of . One may worry that the fluctuations become so large that they contradict the observed value of CMB fluctuations. This will depend on what field we are considering, and needs careful study. In this paper, we argue that it is possible to enhance the modes that have a longer wavelength than the scales that are observed in CMB, leaving shorter wavelength modes essentially unaffected.

The message of this paper is that vacuum energy of the order of the energy density of our present universe may arise, due to the enhancement of the IR fluctuations generated in the very early universe. In our analysis of free fields, we were able to obtain only , which cannot drive acceleration. However, we should mention that the free field approximation is not likely to be valid in the far IR where large fluctuations generated before the ordinary inflation exist. At the end of the paper, we will mention possible directions for future study to take interactions into account.

The paper is organized as follows. We review the basics of quantization of scalar fields in curved spacetimes in Section 2. After specifying the cosmic history of background geometry in Section 3, we obtain the wave function of a massless minimally-coupled scalar field, with the initial condition fixed in the inflationary era in Section 4. Then we calculate the energy-momentum tensor, paying special attention to the contributions from the IR modes in Section 5. We explain the prescription for treating the UV divergence, and mention subtleties associated with a physical interpretation of cutoff depenent terms, and present the renormalized energy-momentum tensor in Section 6. We consider time evolution of the energy-momentum tensor from the inflationary era until the present in Section 7. In Section 8, we consider a double inflation model, and discuss the effects of a period that may have preceded the ordinary inflation. In section 8.2, we summarize time evolution of energy densities generated in the inflation period and in the pre-inflation period. They are shown in Figure 10. Section 9 is devoted to conclusions and discussion. In Appendix A, we consider a double inflation model with a different intermediate stage. In Appendix B, we investigate IR behaviors of the wave functions.

2 Scalar field in curved spacetimes

In this section we briefly review some basics of the scalar field on curved spacetimes (see, for instance, ref. [4]).

A scalar field with a mass and a coupling to the scalar curvature in -dimensional spacetime is described by the action


which gives the equation of motion


The energy momentum tensor is given by


where and . The conformally coupled scalar is described by . In this paper we will study the minimally coupled scalar with .

For background geometries, we consider Robertson-Walker spacetimes, which enjoy homogeneous and isotropic spaces, with the metric


is the scale factor, and and are the conformal time and spacial coordinates. An explicit form of will be specified in Section 3.

For quantizing the field, one expands the field as


where the mode functions with the comoving momentum are the solutions of the equation of motion (2.2), and are chosen to asymptote to positive-frequency modes in the remote past. A vacuum is then defined by . The vacuum , which is an in-state, evolves as increases, and if an adiabatic condition is broken the state gets excited above an adiabatic ground state at each moment, .

In the Robertson-Walker spacetime (2.4), the wave equation (2.2) is written as


where and


The expectation value of the energy momentum tensor in the state is given by


The right-hand side is obtained by inserting (2.5) into (2.3), and represents the differential operator that acts on in (2.3). The subscript ‘un-ren’ indicates that the UV divergence has not been subtracted yet. Regularization and renormalization will be discussed in Section 6. Note that (2.8) is independent of the space coordinates , due to the spacial homogeneity of the Robertson-Walker spacetime.

3 The background geometry

Our universe is well approximated by the Robertson-Walker spacetime (2.4) in four spacetime dimensions, . It experienced the inflation, radiation-dominated (RD), and matter-dominated (MD) periods. We describe the three stages of the cosmic history by the following scale factor :


which is specified by the eight parameters . is the Hubble parameter in the inflation period. We show below that both and must be continuous at the boundaries of the inflation-RD and the RD-MD periods. Then only four of the eight parameters are independent. As the four independent parameters, we will use where and are the present scale factor4 and the Hubble parameter, respectively, and is the red-shift factor at the matter-radiation equality. All the eight parameters are written in terms of them.

The continuity condition of can be easily understood as follows. The scale factor satisfies the Friedmann equation


where and are energy and pressure densities. The second equation demands that should be continuous unless the second term has a singularity. Then, according to the first equation, is continuous and so is as well as .

The continuity conditions between the inflation and RD periods are given by


They give the relation


Similarly, the condition between the RD and MD periods is given by


which lead to


Now we determine . The third equation of (3.1) gives


and we can write and in terms of and as


Also, the same equation gives


and is written in terms of and (hence, ). The other parameters can be similarly solved in terms of the four parameters. Here, we note a relation:


It can be proved straightforwardly from the above equations, and understood from the evolution of the Hubble parameters, and in the RD and MD periods, respectively, which are obtained by and (3.1).

Finally, we estimate the numerical values of various parameters. is determined by (3.12) with the use of the present Hubble:


Note that, as seen from (2.4), definition of has a rescaling ambiguity that can be absorbed into , and only the combination of and has a physical meaning. , and hence , is determined by (3.14) as


where we used


Similarly, , and hence , is determined by (3.15) as


where we have used the constraint from the CMB fluctuations:


As a final comment, we note that the higher derivatives of with respect to are not continuous. Consequently, we will see that the Bogoliubov coefficients have a long UV tail as a function of the momentum . Such a long tail is an artifact of the rapid change of the scale factor and can be removed by smoothing the connections between the stages. It will be discussed in Section 6.

4 Time evolution of wave functions

We now solve the wave equation (2.6) to obtain the wave function in the cosmic history (3.1). In this paper we consider the minimally coupled case, , in four dimensions, . Then the wave equation (2.6) becomes




The relation with (2.7) becomes


Eq. (4.1) is interpreted as the time-independent Schrödinger equation for a one-dimensional quantum system with a potential , by regarding as the spatial position. Figure 1 shows the potential (4.2) for the case. Note that it is discontinuous at the boundaries of the inflation, RD, and MD periods.

Figure 1: Potential for the wave equation (4.1) with . The solid line depicts the potential (4.2), while the dashed line represents the right-hand side of (4.1). to are the boundaries of the inflation, RD, and MD periods. is the present time. Their numerical values are specified at the end of Section 3. In particular, heights of the potential peaks are determined by (3.17) and (3.19). If we normalize the height of the potential at the present as 1, the heights at and are given by and , respectively.

In the massless case, the solutions of (4.1) are given as


in the inflation, RD, and MD periods, respectively, where the wave functions are


The wave function (4.4) contains only the positive frequency mode, which corresponds to taking the Bunch-Davies vacuum in the de Sitter spacetime. The constants , , , and are easily determined by using the junction conditions of the wave functions, i.e., the continuity of and . They become


These constants give the coefficients of the Bogoliubov transformations, and the Bunch-Davies vacuum is interpreted as an excited state on the adiabatic vacuum defined in the RD and MD periods, respectively.5 Let us look at the particle creation in the RD period. We expand in terms of the adiabatic wave functions at each moment in the RD period as


The adiabatic vacuum is defined by . If we start from the Bunch-Davies vacuum defined by , the state evolves into a highly excited state on the adiabatic vacuum . By using the solutions of the wave equation and comparing (2.5) and (4.11), we have


The coefficients satisfy the relation . Furthermore, we can always make real by redefining the phase of . So the coefficients can be parametrized as


The inverse of the transformation (4.12) is generated by using the squeezing operator




The Bunch-Davies vacuum is then written as


Since the squeezing operator is bilinear in the creation operators, the Bunch-Davies vacuum is a collection of excited states with multiple pairs of particles. Concretely, can be expanded as


The number of created particles is calculated as


Hence, it becomes very large for . The same calculation is performed in the MD period, and the number of created particles is given by .

Now, let us investigate the IR, i.e., small-, behavior of the wave functions. When , (4.7) behaves as . Terms of cancel and the next term starts with . By using (4.3) with (3.1), the IR behavior of becomes


in the inflation period. This IR behavior of the wave function is kept until the RD and MD periods. It generally holds that the leading part of the superhorizon modes of a massless field is time-independent, i.e., frozen. It gives the seeds of the CMB fluctuations. (See, for instance, sections 7.3.2, 8.4, and 9.9 of ref. [9].)

Let us confirm the above IR behavior of the wave function. Since and in (4.9) are proportional to in the IR region for , the wave function in the RD period (4.5) seems to behave for . Similarly, the behavior indicates much more violent IR behavior, , for . However, lots of cancellations occur and the IR behavior becomes milder. Indeed, in the RD period, (4.9) is rewritten as


where the terms with and the terms with cancel in the square bracket. Then the wave function (4.5) behaves as


in the IR regions. Note that the terms with cancel each other and the next terms start with . Moreover, by using in (3.6), the leading term in (4.22) gives


where we used (3.1) and (3.5). Hence, the IR behavior (4.20) is shown to be maintained in the RD period as well.

As we show in Appendix B.1, the same IR behavior holds in the MD period.

5 Energy-momentum tensors

For a minimally-coupled massless scalar in four dimensions, i.e., , , and , the energy density and the pressure density ( is not summed over) are given from (2.3) by

Here, the expectation values are taken in the Bunch-Davies vacuum. The superscript ‘un-ren’ means that the UV divergences are not yet subtracted.

We first examine the contributions from the IR modes. By substituting the IR behavior of the wave function (4.20), which is kept until the RD and MD periods, into (5) and (5), one obtains


Since the IR wave function (4.20) is frozen and time-independent, only the spacial derivative terms contribute to and .6 Due to the amplification of the wave function (4.20), and are enhanced to the order of . The leading term in (5.5) and (5.6) gives .

The two-point correlation function of massless fields receives a logarithmic IR growth:


In contrast, the energy and pressure densities, (5.5) and (5.6), are IR finite since they involve derivatives, and the IR divergences are canceled. This is due to the fact that for an exactly massless field, the constant part of the field (which gives rise to the logarithmic divergence) does not have a physical meaning.

However, if we take the massless limit of a massive field, the mass term in the energy-momentum tensor gives a non-vanishing contribution, since . This can be computed by using the exact wave function for massive fields, which is written in terms of Hankel functions in de Sitter space [4, 5], or can be seen as follows. We compute


By introducing a small mass, the potential in the inflation period changes from to , and the IR behavior of the massless wave function (4.20) is modified to


Inserting it into (5.8), we obtain


Here, the integration is performed up to , which corresponds to the horizon scale , since the IR behavior (5.9) is valid below this scale. The result (5.10) is independent of the mass. Hence, the massless limit of a massive theory gives an additional contribution to the expectation value of the energy-momentum tensor. But we should note that, in order to obtain the contribution, we implicitly assumed that there is no other IR cutoff. If there exists a physical IR cutoff, , the logarithmic IR divergence is automatically cured by and no such singular behavior with appears. Such an IR cutoff may be given, e.g., by the initial time of the inflation period. In that case, the expectation value of a massless scalar will be proportional to the physical time interval since the initial time [22, 23, 24, 25], and will not be infinite.7 Then vanishes in the limit and the massless limit of a massive theory gives the same answer as the purely massless theory. In the following analysis of the present paper, we assume an existence of such an IR physical cutoff. A further possible contribution of to the vacuum energy will be investigated in a separate paper.

6 Renormalized energy-momentum tensors

The energy and pressure densities (5.5) and (5.6) are UV divergent, and must be regularized and renormalized. This has been studied extensively in the past (see, for instance, section 6 of ref. [4]), so we will keep the description brief and just present the result.

We first make the integral finite by using one of the regularization prescriptions, such as the dimensional regularization or the covariant point-splitting. We then perform renormalization by subtracting the terms that can be absorbed by redefinitions of coupling constants in the gravity action that have the dimensions four (the cosmological constant), two (the Newton constant inverse), and zero (the coefficients for curvature tensor squared terms). In general, the renormalized energy-momentum tensor takes the form


where stands for the derivative operators in (2.8). is the two-point correlation function obtained in the previous sections. is the subtraction term, which is the two-point function obtained using the first few orders of the DeWitt-Schwinger expansion [30]. This is an expansion around the flat spacetime, and the expansion depends only on the information of local geometry, namely, the subtraction terms are written in terms of curvature tensors.

Another way of obtaining subtraction terms, which has been shown to be equivalent to the DeWitt-Schwinger expansion in various cases, including the Friedmann-Robertson-Walker (FRW) universe, is the adiabatic regularization [31, 32]. In this scheme, is obtained by using the WKB expansion of the mode functions up to the adiabatic order four. With this prescription, the subtraction can be performed at each separately, and we can obtain finite integrals without the need of an explicit regularization. In this sense, the adiabatic regularization is a prescription for subtraction, not a regularization method.

The subtraction term contains an IR divergent term for massless theories; thus, the renormalized energy-momentum tensor should be defined by taking the massless limit of a massive theory. The pieces that remain finite as a result of this procedure are given, e.g., in Eqs. (3.14) and (3.15) of [32] for the general FRW universe. The Weyl anomaly arises from this procedure. However, since these contributions are smaller than the terms that we are mainly interested in, we will ignore these contributions in the analyses of the paper.

Before starting the analysis of the renormalized energy-momentum tensor in our background, we would like to make a side remark about the UV divergent (cutoff-dependent) terms. We emphasize here that a physical interpretation of these terms is a very subtle issue. For example, consider the quartically divergent term in the effective action, which is the leading divergence in four spacetime dimensions,


where is the UV cutoff for the momentum. This term is often interpreted as a contribution from quantum fields to the cosmological constant.

On the other hand, if one computes , the quartically divergent term has , which is the equation of state for radiations, not the cosmological constant. This is indeed expected from the fact that in the limit of high momentum, there is no particle creation, and the field behaves as a collection of radiations. One can also see directly from the coefficients of the terms in the integrand of (6.4) and (LABEL:pRDAB) below.

It is intuitively unclear how these two different equations of state for the quartically divergent terms are consistent with each other. The argument in the literature [32] is based on the dimensional regularization: The quartic divergence in is regularized, then one looks at the contribution at the pole and finds that it is proportional to . Thus, it is removed by renormalizing the cosmological constant.

The above difference of equation of state in the two approaches may be attributed to the fact that the UV cutoff itself depends on the metric. One can introduce a UV cutoff by putting the fields at two points separated by a coordinate time interval (though we expect that the details of the regularization will not affect the conclusion). Then will depend on ; we assume it is of the noncovariant form . By taking this into account,8 the energy-momentum tensor for (6.2) becomes


where the second term comes from the variation of with respect to . The expression (6.3) indeed has when the background metric is diagonal.

In our opinion, it could be misleading to discuss the cosmological constant problem by looking at cutoff-dependent quantities with UV power divergences9 since the subtracted term is not generally covariant. In this paper, we will always consider the renormalized quantities when we discuss physical effects.

6.1 RD period

Let us now evaluate and for the state (4.5) in the RD period. By substituting the wave function (4.5) into (LABEL:rhogenchi) and (LABEL:pgenchi), one obtains


The terms with and represent interference between the positive and negative frequency modes.

We now perform the subtraction of the UV divergences following the adiabatic regularization procedure. Since the potential for the wave equation vanishes in the RD period, the adiabatic wave function, or the WKB wave function, agrees with the plane-wave solution (4.8). The terms to be subtracted are given by the first line in (6.4) and (LABEL:pRDAB) with , . Thus, the renormalized expression is obtained by simply replacing by in (6.4) and (LABEL:pRDAB).

The subtraction term has been obtained for the FRW universe with the general scale factor . See, for instance, eqs. (2.30) and (2.35) in ref. [32] (see also (2.10) in [33]). Substituting the present form of in (3.1), one indeed obtains the above mentioned subtraction term. Apart from that term, there are finite terms of the order (where is the Hubble parameter at each moment), which arise as a result of performing the integration in massive theory, and taking the massless limit in the end. This procedure is necessary since some of the terms in the adiabatic expansion are IR divergent, and the massless theory cannot be studied directly. These terms are important since they give the Weyl anomaly, and also contribute to the vacuum energy of pure de Sitter space. However, at late times, these terms of the order are much smaller than the total contribution that we are studying. The latter is enhanced due to the IR behavior of wave functions, as we have seen in (5.5) and (5.6). Thus, we will ignore this finite contribution of the order in this paper.

We now consider the consequences of our approximation, in which the scale factor (3.1) changes its functional forms instantaneously and its higher derivatives with respect to are not continuously connected at the boundaries of the stages. Due to this approximation, we have non-differentiable sharp peaks in the potential at the boundaries (see Figure 1). In this potential, the reflection coefficient decreases only with a power, , as given in (4.9). Even at high , over-the-barrier scattering by the sharp potential is not strongly suppressed. This is different from the general behavior for the scattering by a smooth potential, where the reflection coefficients fall off exponentially at large . Namely, if wavelengths are smaller than the typical curvature radius at the peak of the potential, over-the-barrier scattering cannot occur. (See, for instance, section 52 of ref. [37].)

The power law tail of for the sharp potential produces problems in the UV behavior of the -integral in (6.4) and (LABEL:pRDAB). The coefficient in the first line of (6.4) and (LABEL:pRDAB) after the UV subtraction behaves as , where the unitarity relation was used. It then follows that the leading term gives rise to a UV logarithmic divergence. A problem also arises from the interference terms. In the IR region of , the coefficients behave as , as given by (4.9), and the term in (LABEL:pRDAB) decreases as while oscillating. The integrand will be seen below in Figure 2. However, in the UV region of , the coefficients behave as and , and the term grows as a linear function of though it oscillates.

These problems are caused by our setting where the scale factors are not sufficiently smoothly connected at the boundaries of the stages. The subtraction term obtained by the DeWitt-Schwinger expansion or adiabatic regularization, which is based on the adiabatic expansion of the background geometry, could not cancel all the UV divergences in such cases. Since modes with infinitesimally small wavelengths are affected by the sharp potential at the boundary, the UV behaviors at later times become dependent on the past history and cannot be controlled only by the local quantities.

We assume that the scale factors in the realistic settings are smoothly connected, so that the reflection coefficients decay quickly when . To take this behavior into account, we will introduce a mask function , which takes a value close to 1 for , and falls off rapidly for . The scattering coefficients have to satisfy the unitarity relation . The simplest way to apply a mask function, , that is consistent with this relation would be to make the following replacements in (4.9):


In the actual analysis, we will take