Initial conditions for Inflation in an FRW Universe
We examine the class of initial conditions which give rise to inflation. Our analysis is carried out for several popular models including: Higgs inflation, Starobinsky inflation, chaotic inflation, axion monodromy inflation and non-canonical inflation. In each case we determine the set of initial conditions which give rise to sufficient inflation, with at least e-foldings. A phase-space analysis has been performed for each of these models and the effect of the initial inflationary energy scale on inflation has been studied numerically. This paper discusses two scenarios of Higgs inflation: (i) the Higgs is coupled to the scalar curvature, (ii) the Higgs Lagrangian contains a non-canonical kinetic term. In both cases we find Higgs inflation to be very robust since it can arise for a large class of initial conditions. One of the central results of our analysis is that, for plateau-like potentials associated with the Higgs and Starobinsky models, inflation can be realized even for initial scalar field values which lie close to the minimum of the potential. This dispels a misconception relating to plateau potentials prevailing in the literature. We also find that inflation in all models is more robust for larger values of the initial energy scale.
a]Swagat S. Mishra, a]Varun Sahni b,c]and Alexey V. Toporensky Prepared for submission to JCAP
Initial conditions for Inflation in an FRW Universe
Inter-University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411 007, India
Sternberg Astronomical Institute, Moscow State University, Universitetsky Prospekt, 13, Moscow 119992, Russia
Kazan Federal University, Kremlevskaya 18, Kazan, 420008, Russia
- 1 Introduction
- 2 Methodology
- 3 Inflation with Power-law Potentials
- 4 Higgs Inflation
- 5 Starobinsky Inflation
- 6 Discussion
- 7 Acknowledgements
- A The values of and for several inflationary models
- B Jordan to Einstein frame transformation for Higgs inflation
- C Derivation of asymptotic forms of the Higgs potential in the Einstein frame
Since its inception in the early 1980’s, the inflationary scenario has emerged as a popular paradigm for describing the physics of the very early universe [1, 2, 3, 4, 5]. A major reason for the success of the inflationary scenario is that, in tandem with explaining many observational features of our universe – including its homogeneity, isotropy and spatial flatness, it can also account for the existence of galaxies, via the mechanism of tiny initial (quantum) fluctuations which are subsequently amplified through gravitational instability [6, 7, 8, 9].
An important issue that needs to be addressed by a successful model of inflation is whether the universe can inflate starting from a sufficiently large class of initial conditions. This issue was affirmatively answered for chaotic inflation in the early papers [10, 11]. Since then the inventory of inflationary models has rapidly increased. In this paper we attempt to generalize the analysis of [10, 11] to other popular inflationary models including Higgs inflation, Starobinsky inflation etc., emphasising the distinction between power law potentials and asymptotically flat ‘plateau-like’ potentials. As we shall show, our results for asymptotically flat potentials do not provide support to the ‘’ raised in 111See  for an analysis of other problems with plateau-like potentials raised in ..
Our paper is organized as follows. We introduce our method of analysis in section 2. Section 3 discusses power law potentials and includes chaotic inflation and monodromy inflation. Section 4 discusses Higgs inflation in the context of both the non-minimal as well as the non-canonical framework222As pointed out in  non-canonical scalars permit the Higgs field to play the role of the inflaton.. Section 5 is devoted to Starobinsky inflation. Our results are presented in section 6.
We work in the units and the reduced Planck mass is assumed to be . The metric signature is . For simplicity we assume that the pre-inflationary patch which resulted in inflation was homogeneous, isotropic and spatially flat. An examination of inflation within a more general cosmological setting can be found in .
The action for a scalar field which couples minimally to gravity has the following general form
where the Lagrangian density is a function of the field and the kinetic term
Varying (2.1) with respect to results in the equation of motion
The energy-momentum tensor associated with the scalar field is
Specializing to a spatially flat FRW universe and a homogeneous scalar field, one gets
where the energy density, , and pressure, , are given by
and . The evolution of the scale factor is governed by the Friedmann equations:
where satisfies the conservation equation
For a canonical scalar field
Noting that one finds , which informs us that the expansion rate is a monotonically decreasing function of time for canonical scalar fields which couple minimally to gravity. The scalar field equation of motion follows from (2.3)
Within the context of inflation, a scalar field rolling down its potential is usually associated with the Hubble slow roll parameters 
and the potential slow-roll parameters 
For small values of these parameters , one finds and . The expression for in (2.17) can be rewritten as which implies that the universe accelerates, , when . For the scalar field models discussed in this paper so that , which reduces to when .
The slow-roll parameters play an important role in determining the spectral index of scalar perturbations, since333Here , where is the power spectrum of scalar curvature perturbations., . Observations indicate  which suggests that on scales associated with the present cosmological horizon. The fact that are required to be rather small might appear to imply that successful inflation can only arise under a very restricted set of initial conditions, namely those for which . This need not necessarily be the case. As originally demonstrated in the context of chaotic inflation [10, 11], a scalar field rolling down a power law potential can arrive at the attractor trajectory from a very wide range of initial conditions. In this paper we shall apply the methods developed in [10, 11, 17] to several inflationary models with power law and plateau-like potentials in order to assess the impact of initial conditions on these models.
In addition to the field equations developed earlier, we shall find it convenient to work with the parameter
which describes the number of inflationary e-foldings since the onset of inflation. For our purpose it will also be instructive to rewrite the Friedman equation (2.14) as
where is the sign of (this definition ensures that and have the same sign). Clearly, holding fixed and varying and , one arrives at a set of initial conditions which satisfy the constraint equation (2.20) defining the boundary of a circle of radius . Adequate inflation is then qualified by the range of initial values of and for which the universe inflates by at least 60 e-foldings, i.e. .
3 Inflation with Power-law Potentials
3.1 Chaotic Inflation
We first consider the potential 
where is assumed, in agreement with observations of the cosmic microwave background [16, 19] (see Appendix A) . The generality of this model is studied by plotting the phase-space diagram ( vs ) and determining the region of initial conditions which gives rise to . Equations (2.15), (2.16), (2.19) have been solved numerically for different initial energy scales . The phase-space diagram corresponding to is shown in figure 1.
To study the effect of different energy scales on inflation, we take different values of () and determine the range of initial values of that lead to adequate inflation with . (The initial value of is conveniently determined from the consistency relation (2.20).) Our results are summarized in figure 3. The solid blue lines correspond to initial values, , which always result in adequate inflation (irrespective of the sign of ). The dashed red lines corresponding to , result in adequate inflation only when points in the direction of increasing (represented by blue arrows). Inadequate inflation is associated with the region . If the initial scalar field value falls within this region then one does not get adequate inflation irrespective of the sign of . This region is shown in figure 3 by the solid red line. The dependence of and on the initial energy scale is given in table 1.
To determine the fraction of initial conditions that do not lead to adequate inflation (we call this ‘the degree of inadequate inflation’), we consider a uniform measure on the distribution of initial conditions for and . These initial conditions are described by a circle of circumference with (in Planck units) which is illustrated in figure 4. The degree of inadequate inflation and marginally adequate inflation (corresponding respectively to and in figure 3) is and , where and are illustrated in figure 4.
The dependence of , and , on the commencement scale of inflation is shown in table 1. We see that the fraction of initial conditions that leads to inadequate inflation, , decreases with an increase in the initial energy scale . This result is also illustrated in figures 13 and 13 where we compare chaotic inflation with monodromy inflation.
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3.2 Monodromy Inflation
where . In this paper our focus will be on two values of , namely . (However our methods are very general and easily carry over to other values of .)
One should note that for the potential (3.2) is not differentiable at the origin. This might lead to problems with reheating since the latter usually occurs during rapid oscillations of around . We circumvent this problem by the following useful generalization of (3.2)
where is an integer (we have taken n=4). In this expression the value of is chosen so that for whereas for . It is well known that inflation ends when the slow-roll parameter in (2.18) grows to unity. Substituting (3.2) in (2.18) and setting one finds which can be used to set a value to , namely . With this value of we proceed with a generality analysis for which will be followed by a similar analysis for . (Note that our main results are quite insensitive to the value of .)
Linear Monodromy Inflation
Consider first the linear potential
where is in agreement with the CMB  (see Appendix A). This potential and its modification via (3.3) is shown in figure 5. We see that for , the modified potential behaves like , and rapidly converges to for .
Initial values of that lead to adequate or inadequate inflation are schematically shown in figure 8. Inadequate inflation arises when the scalar field originates in the region , shown by solid red line. Blue lines represent initial field values , which always result in adequate inflation. Note that is the maximum allowed value of for a given initial energy scale, as determined from the consistency equations (2.14), (2.20). Initial conditions , shown by dashed red lines, lead to adequate inflation only when points in the direction (shown by blue arrows) of increasing . The dependence of and on the initial energy scale is shown in table 2.
The values of and in table 2 have been determined by assuming a uniform distribution of initial values of and on the circular boundary (2.20). We find that and decrease with an increase in , as expected.
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Fractional Monodromy Inflation
Next we consider
where CMB constraints imply  (see Appendix A). The potential (3.5) and its generalization (3.3) are shown in figure 9. As expected, the cusp at in (3.5) is absent in the modified potential (3.3), which is shown by the green line in figure 9.
Initial values of that lead to adequate or inadequate inflation are schematically shown in figure 12. Inadequate inflation arises when the scalar field originates in the region , shown by solid red lines. Blue lines represent initial field values , which always result in adequate inflation. Note that is the maximum allowed value of for a given initial energy scale, as determined from the consistency equations (2.14), (2.20). The initial conditions , shown by dashed red lines, lead to adequate inflation only when points in the direction (shown by blue arrows) of increasing . We refer to this as marginally adequate inflation. The dependence of and on the initial energy scale is shown in Table 3.
As in the case of chaotic inflation, we determine the fraction of initial conditions that do not lead to adequate inflation (the degree of inadequate inflation), by assuming a uniform distribution of initial values of and on the circular boundary (2.14), (2.20) with given by (3.5). Our results are given in table 3. As was the case for quadratic chaotic inflation, we once more find that and decrease with an increase in ; see table 3, figures 13 and 13.
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3.3 Comparison of power law potentials
In this subsection we compare the generality of inflation for the power law family of potentials, , by plotting the fraction of initial conditions that do not lead to adequate inflation ( and ) in figures 13 and 13; also see tables 1-3. These figures demonstrate that the set of initial conditions which give rise to adequate inflation (with ) increases with the energy scale of inflation, . We also find that inflation is sourced by a larger set of initial conditions for the monodromy potential , which is followed by and respectively. Finally we draw attention to the fact that our conclusions remain unchanged if we determine the degree of inflation by a different measure such as and , where is the maximum allowed value of for a given inflationary energy scale.
4 Higgs Inflation
It would undoubtedly be interesting if inflation could be realized within the context of the Standard Model () of particle physics. Since the has only a single scalar degree of freedom, namely the Higgs field, one can ask whether the Higgs field (4.2) can source inflation. Unfortunately the self-interaction coupling of the Higgs field, in (4.2), is far too large to be consistent with the small amplitude of scalar fluctuations observed by the cosmic microwave background .
This situation can however be remedied if either of the following possibilties is realized: (i) the Higgs couples non-minimally to gravity, or (ii) the Higgs field is described by a non-canonical Lagrangian444Another means of reconciling the () potential with observations is through a field derivative coupling with the Einstein tensor of the form . This approach has been discussed in .
Indeed, as first demonstrated in , inflation can be sourced by the Higgs potential if the Higgs field is assumed to couple non-minimally to the Ricci scalar. The resultant inflationary model provides a good fit to observations and has been extensively developed and examined in [23, 24, 25, 26, 27]. A different means of sourcing Inflation through the Higgs field was discussed in  where it was shown that the Higgs potential with a non-canonical kinetic term fits the CMB data very well by accounting for the currently observed values of the scalar spectral index and the tensor-to-scalar ratio . We shall proceed to study Higgs inflation first in the non-minimal framework in section 4.1 followed by the same in the non-canonical framework in section 4.2.
4.1 Initial conditions for Higgs Inflation in the non-minimal framework
Inflation sourced by the Standard Model () Higgs boson was first discussed in . In this model the Higgs non-minimally couples to gravity with a moderate value of the non-minimal coupling 555The value of the dimensionless non-minimal coupling though in itself quite large, is much smaller than the ratio , where is the Electroweak scale. [24, 25]. The model does not require an additional degree of freedom beyond the and fits the observational data quite well . Reheating after inflation in this model has been studied in detail [25, 26] and quantum corrections to the potential at very high energies have been shown to be small . In this section we assess the generality of Higgs inflation (in the Einstein frame) and determine the range of initial conditions which gives rise to adequate inflation (with ) for a given value of the initial energy scale.
Action for Higgs Inflation
where is the Ricci scalar and is the metric in the Jordan frame. The potential for the Higgs field is given by
where is the vacuum expectation value of the Higgs field
and the Higgs coupling constant has the value . Furthermore
where is a mass parameter given by 
being the non-minimal coupling constant whose value
agrees with observations  (see Appendix A). For the above values666Note that the observed vacuum expectation value of the Higgs field is much smaller than the energy scale of inflation and hence we neglect it in our subsequent calculations. of and , one finds , so that
We now transfer to the Einstein frame by means of the following conformal transformation of the metric 
where the conformal factor is given by
After the field redefinition the action in the Einstein frame is given by 
Limiting cases of the potential in the Einstein Frame
For one finds
where is given by (Appendix A)
Generality Analysis of Higgs inflation in the Einstein Frame
As we have seen, Higgs inflation in the Einstein frame can be described by a minimally coupled canonical scalar field with a suitable potential . We have analysed two different limits of the potential which is asymptotically flat and has plateau like arms for . One notes that when , has a tiny kink with amplitude . This kink is much smaller than the maximum height of the potential and can be neglected for all practical purposes. (This is simply a reflection of the fact that the inflation energy scale is much larger than the electro-weak scale.) We have numerically evaluated the potential defined in (4.10) & (4.11) and compared it with the approximate form given in equation (4.15); see figure 14. The difference between the two potentials is shown in figure 15. One finds that the maximum fractional difference between the two potentials is only which justifies the use of (4.15) for further analysis.
During Higgs inflation, the slow-roll parameter is given by
since slow-roll ends when , one finds
We study the generality of Higgs inflation in the Einstein frame by plotting the phase-space diagram for the potential (4.15) and determining the region of initial conditions which lead to adequate inflation (i.e. ). Our results are shown in figure 16 and a zoomed-in view is presented in figure 17.
We see that the phase-space diagram for Higgs inflation has very interesting properties. The asymptotically flat arms result in robust inflation as expected. However it is also possible to obtain adequate inflation if the inflaton commences from . This is because the scalar field is able to climb up the flat wings of . This property is illustrated in figure 16 by lines originating in the central region, which are slanted and hence can converge to the slow-roll inflationary separatrics resulting in adequate inflation. This feature is not shared by chaotic inflation where one cannot obtain adequate inflation by starting from the origin (provided the initial energy scale is not too large, i.e. .)
This does not however imply that all possible initial conditions lead to adequate inflation in the Higgs scenario. As shown in figure 18 there is a small region of initial field values denoted by which does not lead to adequate inflation if and have opposite signs (dashed red lines). By contrast, the solid blue lines in the same figure show the region of that results in adequate inflation independently of the direction of the initial velocity . The dependence of and on the initial energy scale is shown in table 4 (also see figure 18). Note the surprising fact that the value of remains virtually unchanged as increases.
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The results of figures 16, 17 and 18 lead us to conclude that there is a region lying close to the origin of , namely , where one gets adequate inflation regardless of the direction of . One might note that this feature is absent in the power law family of potentials described in the previous section (compare figure 18 with figures 3, 8, 12). values that lead to partially adequate inflation, We therefore conclude that a wide range of initial conditions can generate adequate inflation in the Higgs case, which does not support some of the conclusions drawn in .
Finally we would like to draw attention to the fact that the phase-space analysis performed here for Higgs inflation is likely to carry over to the T-model -attractor potential , since the two potentials are qualitatively very similar.
4.2 Initial conditions for Higgs Inflation in the non-canonical framework
The class of initial conditions leading to sufficient inflation widens considerably if we choose to work with scalar fields possessing a non-canonical kinetic term.
The Lagrangian for this class of models is 
which reduce to the canonical form when . The two Friedmann equations now acquire the form
and the equation of motion of the scalar field becomes