Slow-roll corrections in multi-field inflation: a separate universes approach
In view of cosmological parameters being measured to ever higher precision, theoretical predictions must also be computed to an equally high level of precision. In this work we investigate the impact on such predictions of relaxing some of the simplifying assumptions often used in these computations. In particular, we investigate the importance of slow-roll corrections in the computation of multi-field inflation observables, such as the amplitude of the scalar spectrum , its spectral tilt , the tensor-to-scalar ratio and the non-Gaussianity parameter . To this end we use the separate universes approach and formalism, which allows us to consider slow-roll corrections to the non-Gaussianity of the primordial curvature perturbation as well as corrections to its two-point statistics. In the context of the expansion, we divide slow-roll corrections into two categories: those associated with calculating the correlation functions of the field perturbations on the initial flat hypersurface and those associated with determining the derivatives of the e-folding number with respect to the field values on the initial flat hypersurface. Using the results of Nakamura & Stewart ’96, corrections of the first kind can be written in a compact form. Corrections of the second kind arise from using different levels of slow-roll approximation in solving for the super-horizon evolution, which in turn corresponds to using different levels of slow-roll approximation in the background equations of motion. We consider four different levels of approximation and apply the results to a few example models. The various approximations are also compared to exact numerical solutions.
and Jonathan White
Slow-roll corrections in multi-field inflation: a separate universes approach
University of Jyvaskyla, Department of Physics, P.O.Box 35 (YFL), FI-40014
University of Jyväskylä, Finland
Theory Center, KEK, Tsukuba 305-0801, Japan
The Graduate University for Advanced Studies (Sokendai)
Department of Particle and Nuclear Physics, Tsukuba 305-0801, Japan
Rudolf Peierls Centre for Theoretical Physics, The University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK
- 1 Introduction and outline
- 2 Background dynamics and the slow-roll approximation
3 The curvature perturbation
- 3.1 The separate universes approach and formula
- 3.2 The derivatives of
- 3.3 The derivatives of
- 3.4 The perturbations of initial conditions
- 4 Correlation functions and observables
- 5 Example Models
- 6 Summary and Conclusions
- A Background equations of motion at next-to-leading order in slow-roll
- B -independence of and its correlation functions
1 Introduction and outline
A period of inflation in the early Universe is now widely accepted as being responsible for generating the primordial density perturbations that seed all large-scale structure (LSS) in the Universe as well as the temperature fluctuations observed in the cosmic microwave background (CMB). Constraints on the amplitude and scale dependence of the primordial spectrum are already very precise, at 68% confidence level they are given as 111See Section 4 for definitions of and .
Another key property of the spectrum is its deviation from Gaussianity. In the so-called squeezed limit this can be quantified in terms of the parameter , and current constraints are still consistent with this being zero :
In addition to density perturbations, inflation is also expected to give rise to gravitational waves. At present the properties of the gravitational wave spectrum are less well constrained, with an upper limit on the amplitude essentially being given as [1, 5, 6]
at the confidence level. However, CMB polarization observations are expected to improve this constraint to the level .
The high-precision nature of current and forecast constraints necessitates that theoretical predictions used to compare candidate inflation models with the data must be equally as precise, and any approximations made in deriving these predictions must therefore be carefully scrutinised. One such approximation often made is the so-called slow-roll approximation, whereby the Hubble flow parameters defined as
are assumed to be small, namely . These two requirements are very generic: the first is necessary for (quasi-)exponential inflation, while the second ensures that inflation lasts for long enough. In the case of single-field inflation, on making an expansion in the two parameters and we find that the leading-order expression for is given as
where the subscript denotes that the quantities should be evaluated at the time at which the scale under consideration, , left the horizon during inflation. Barring a cancellation between the two terms on the right hand side of (1.5), from the observed value of given in (1.1) we conclude that . As such, in the case of single-field inflation we see that any slow-roll corrections of order or will likely represent percent-level corrections, and should thus be included if we desire percent-level precision in our theoretical predictions.
While current observations are perfectly consistent with single-field inflation, in the context of high-energy unifying theories one naively expects the presence of many fields. It is thus important that we are able to precisely determine the predictions of multi-field inflation models in order that they can be constrained using current and future data. Perhaps of particular interest in the context of multi-field inflation is the possibility of obtaining an observably large non-Gaussianity. In the case of single-field inflation it is known that the three-point function in the squeezed limit is unobservably small . This suggests that if a non-negligible three-point function were to be observed then this would be a strong indication of the presence of multiple fields during inflation. Even if not observed, however, it is still important to determine what the implications of this are for multi-field models of inflation. Indeed, while it is possible to obtain a large non-Gaussianity in such models, it is by no means a generic prediction. When considering slow-roll corrections in multi-field inflation the situation is somewhat more complicated than in the single-field case. In addition to assuming , the slow-roll approximation is often taken to imply the smallness of a larger set of expansion parameters that we label , with and being the number of fields (see Section 2 for details). One then finds that observational constraints on do not require all components of to be , such that slow-roll corrections of order may represent larger-than-percent-level corrections. If this is the case, the inclusion of next-to-leading order corrections will be crucial, and it is plausible that next-to-next-to-leading order corrections may also become important. Indeed, slow-roll corrections to the two-point statistics of scalar and tensor perturbations in multi-field inflation models have already been considered in the literature [9, 10], and in some cases corrections were shown to be on the order of .
In this paper we revisit the issue of slow-roll corrections in multi-field inflation. In particular, we are interested in the slow-roll corrections to predictions for the curvature perturbation on constant density slices, , and quantities related to its correlation functions. Our analysis makes use of the separate universes approach and formalism, which allows us to consider slow-roll corrections to the non-Gaussianity of as well as corrections to its two-point statistics. As discussed above, precise predictions for the non-Gaussianity will be key in constraining multi-field models of inflation with current and future data. The paper is structured as follows. In Section 2 we introduce the class of models under consideration and present the relevant background field equations. The slow-roll approximation and associated expansion parameters are also introduced, and the background equations of motion are expanded up to next-to-leading order in the slow-roll expansion, with details of the derivation being given in Appendix A. In Section 3 we review the separate universes approach and formalism on which our analysis is based. Rather than using a finite-difference method, as is often used, our method is based on the transport techniques introduced and developed in [11, 12, 13, 14, 15, 16, 17]. In this section we outline how these techniques can be applied under various levels of slow-roll approximation. In Section 4 we turn to the correlation functions of as given in terms of the correlation functions of field perturbations on an initial flat slice at horizon crossing. We find that slow-roll corrections arising from slow-roll corrections to the initial field perturbations can be expressed in a compact form and in terms of quantities that in principle are observable. In Section 5 we apply the results of the earlier sections to a few example models, focussing on a comparison between the various levels of slow-roll approximation. We finally summarise our results in Section 6. In this work we use the geometrized units , where is the reduced Planck mass.
2 Background dynamics and the slow-roll approximation
We consider the class of multi-field models of inflation with an action
where is the Ricci scalar associated with the spacetime metric , is the determinant of , are scalar fields and we have used the notation . in the above action is some -dependent potential. Note that we take the metric in field space to be Euclidean, so the distinction between upper and lower indices as in and is immaterial. We will use both of the notations interchangeably and summation is implied over repeated indices, unless stated otherwise.
At background level we take the metric to be of the Friedmann-Lemaitre-Robertson-Walker (FLRW) type, i.e. . The dynamics is then fully determined by the equations of motion for the scalar fields and the Friedmann equation, which are given as
where , is the total energy density of the scalar fields and we have used the notation . Taking the time derivative of (2.3) and using (2.2) we are able to find an expression for as defined in (1.4), which can in turn be used to determine . The resulting expressions are given as
For the purpose of this work it is convenient to use the e-folding number , rather than , as the time parameter, with being defined as
where a prime denotes a derivative with respect to , e.g. .
As discussed in the introduction, in the context of inflation one often makes the so-called slow-roll approximation. At the most fundamental level this approximation corresponds to assuming , which ensures that we obtain quasi-exponential inflation that lasts for long enough.222While is positive definite, may be negative, so strictly speaking we should perhaps require . However, if is negative, meaning that is decreasing with time, then the concern that inflation may not last long enough is no longer an issue. Instead, one will need to ensure that there is a suitable mechanism to end inflation. In both the single- and multi-field cases the assumption allows us to approximate the Friedmann equation (2.7) as
We thus find that the slow-roll approximation is equivalent to assuming that an attractor regime has been reached, in which the field velocity is no longer an independent degree of freedom but is given as a function of the field value . In the multi-field case, however, the situation is more complicated, as the condition only constrains the component of that lies along the background trajectory, i.e. it does not necessarily imply (or equivalently ) for all (note that there is no summation over in these expressions). Nevertheless, by the Cauchy-Schwarz inequality, the constraint will be satisfied if we assume that the magnitude of the field acceleration vector is smaller than the magnitude of the field velocity vector, namely . If we then further assume that for all – that is, we assume the field basis is not aligned in such a way that some of the field velocity components vanish or are considerably smaller than – then we can take the slow-roll approximation to mean that for all , such that (2.6) can be approximated as
In this case, we again find ourselves in an attractor regime where all field velocities are given as functions of the field positions. For later convenience we rewrite (2.11) as
Using (2.12) we are then able to derive consistency conditions for the first and second derivatives of the potential. Explicitly, the conditions become
The more stringent constraint reads
Given that , the requirement gives us the condition . This in turn means that the second term in eq. (2.15) is negligible, such that the final constraint reduces to . Introducing the notation
both of the constraints on the second derivatives of can be satisfied if the eigenvalues of are small. This in turn will be the case if we assume333For large , i.e. a large number of fields, the more stringent constraint would be appropriate.
for all and . The conditions thus constitute the slow-roll assumptions that we will make in this paper.
While (2.12) represents the lowest-order slow-roll equations of motion, in this paper we will be interested in considering next-to-leading order corrections, and we thus require the equations of motion valid to next-to-leading order in the slow-roll approximation. Details of how these can be calculated are given in Appendix A, but the final result can be written in a form almost identical to (2.12) as
The Friedmann equation at next-to-leading order is given as
Before moving on, it will turn out to be convenient later on to re-express the full equations of motion (2.6) in a more compact form that mirrors the form of eqs. (2.12) and (2.18). In order to do so, we introduce the notation
where the index runs from to . This allows us to write (2.6) as
3 The curvature perturbation
In this section we outline the method we use to determine the curvature perturbation on constant density slices, , which is based on the separate universes approach and formalism [19, 20, 21, 22, 23, 24, 25].
3.1 The separate universes approach and formula
The separate universes approach corresponds to the lowest-order approximation in a gradient expansion [22, 23]. First, one performs a smoothing of the Universe on some scale that is larger than the Hubble scale , namely . Associating the small parameter with spatial gradients, one then expands the relevant dynamical equations in powers of , and to the lowest order neglects terms of order or greater. Consequently, one finds that individual -sized patches behave as separate universes, evolving independently and according to the background field equations. On scales larger than , the differences between neighbouring patches simply result from differences in initial conditions. If we are interested in the comoving scale with wavenumber then we have . During inflation this parameter will be decreasing exponentially with time, and the separate universes approach will be applicable after the Horizon-crossing time, which is defined as the time at which .
Ultimately we are interested in computing the curvature perturbation on constant density slices, , and in the context of the separate universes approach it can be shown that is given by the number of e-foldings of expansion between a flat and constant density slice, . To see why this is, let us consider the spatial metric, which can be written as
where is the fiducial background scale factor, is the curvature perturbation and , satisfying , contains gravitational waves. Here we follow  and fix the threading such that the scalar and vector parts of are set to zero. Note that the fiducial background universe can be associated with some given location , namely . We can then consider as the effective scale factor at a given location , and the associated e-folding number is given as444The validity of this relies on the shift vector being negligible on large scales, which has been shown to be the case in e.g. [23, 22].
If we were to consider flat slices, where , then we see that the scale factor reduces to that of the fiducial background, namely , and correspondingly the e-folding number remains unperturbed, i.e. . On the other hand, if we were to consider moving between a flat slice at , with such that , and a constant density slice at , on which and such that , then we find
Note that the left-hand side of the above equation is independent of , i.e. it is independent of when we take our initial flat slice. This reflects the fact that the number of e-foldings between any two flat slices is homogeneous, and therefore does not contribute to .
If we consider the case , then given in (3.3) corresponds to the perturbation in the number of e-foldings that results from making a gauge transformation between the flat and constant-density slices at the final time . In this case we can derive a simple relation between and the density perturbation on the flat hypersurface. As the e-folding number is unperturbed on flat hypersurfaces, it is useful to use this as a time parameter. We can then decompose the density on the flat hypersurface as , where is the density of the fiducial background trajectory associated with the location , namely . Next we consider the shift in required at each location such that , that is, the time-shift required to move from the flat slice to the constant density slice [12, 11]. Assuming to be small, expanding up to second order in and solving iteratively one obtains555Note that because we are interested in computing the bispectrum of , it is enough to keep only terms up to second order in the expansion.
where and we have used
which can be determined from the fact that . This result is in agreement with that obtained in e.g. .
In the context of inflation, where we assume the dynamics to be determined by multiple scalar fields , the validity of the separate universes picture has been confirmed explicitly to all orders in perturbation theory . In this case, the perturbation can be expressed in terms of perturbations in the fields and their velocities on the flat slicing. Using the compact notation introduced in (2.21), we can expand as
which on substituting into eq. (3.4) gives an expansion of the form
In the above equation we used the same notation as in eq. (2.2), where the subscripts in coefficients and denote differentiation with respect to unperturbed fields . The explicit form of these coefficients will be given below.
While the expansion (3.7) is in terms of the field perturbations at the final time , in the context of the separate universes approach we know that the field values and velocities on the flat slicing evaluated at a given time and smoothed over a superhorizon size patch at some spatial coordinate , , are simply solutions to the background equations of motion with initial conditions given at some earlier time that are local to that patch, namely
Following e.g. [15, 27, 16], let us introduce notation in which different types of indices are used to denote quantities evaluated at different times. In particular, indices from the beginning of the Latin alphabet will be used to denote quantities evaluated at the initial time , e.g. , while indices from the end of the alphabet will be used to denote quantities at the final time , e.g. . This allows us to write
where and . In this expression are the values of fields smoothed on superhorizon patch at and (without an argument) is some fiducial trajectory in field space, which can be associated with the location .
are the perturbations of initial conditions. Thus we have an expansion of in terms of , which on substituting into (3.7) gives an expansion of the form
Note that for the same reason as discussed above, the choice of in evaluating (3.10) is arbitrary. This can be used to our advantage, as the ’s are most easily computed soon after the scales under consideration leave the horizon, corresponding to . In this case, we are able to make contact with the perhaps more familiar form of the expansion, where is expanded in terms of the field perturbations shortly after horizon crossing. Explicitly, the derivatives of with respect to the field values and velocities around horizon crossing are given as
The choice of in evaluating eq. (3.10), on the other hand, is crucial. To find the final value of , should be chosen to be some time after an adiabatic limit has been reached and freezes in. In general this can happen long after the end of inflation.
We see that there are three key elements that are required in order to determine : the derivatives of with respect to , the derivatives of the final field values with respect to the initial conditions and the perturbations in the initial conditions themselves, . The remainder of this section we will be dedicated to computing these three elements.
3.2 The derivatives of
The quantities are determined by combining eqs. (3.4) and (3.6), and for explicit expressions we need to determine explicit expressions for and . The final results will depend on the level of slow-roll approximation that we make, and we thus consider the following three cases in turn: no slow-roll approximation, leading order slow-roll approximation and next-to-leading order slow roll approximation.
3.2.1 No slow-roll approximation
In the case that no slow-roll approximation is used, the energy density is given in terms of the fields and their velocities as in eq. (2.7). Perturbing this expression up to second order in and we obtain
Similarly, perturbing the first of (3.5) we obtain
where . It is worth reiterating that while the above result is only valid up to second order in field perturbations and their temporal derivatives, no slow-roll assumption was made.
While the current form of the coefficients in eqs. (3.14)-(3.18) is perfectly acceptable, it is possible to simplify them by recognising that there are certain combinations of perturbations that decay on super-horizon scales. This observation is closely related to the relation between the constant-density and comoving surfaces in phase space. The comoving condition is defined as the requirement , where is the the energy momentum tensor associated with the multiple scalar fields. Using as the time coordinate we find that in the flat gauge and on super-horizon scales we have
This means that the comoving and constant density conditions differ by the term . As was pointed out by Sasaki & Tanaka in , and later by Sugiyama et al. in , if we take the spatial gradient of the equations of motion (2.6) and then contract this with , we find that the quantity satisfies
from which we deduce that decays as . As such, we find that even at non-linear order the comoving and constant-density conditions coincide on super-horizon scales. This result is well known at linear order in perturbation theory, and is usually shown by combining the energy and momentum constraints, see e.g. . Here, on the other hand, we simply made use of the equations of motion for the scalar fields.666Striclty speaking, we have relied on Einstein’s equations to confirm that the lapse function and the time dependence of decay on large scales.
At linear order the right-hand side of this expression can be written as a pure gradient, such that we obtain . Making use of this result we then find that to second order we obtain
which can be shown to be in agreement with the expression for in ref. . As such, we see that at second order the use of (3.24) has introduced a non-local term into the expansion of . However, taking the derivative of with respect to we find that it satisfies
where is such that at linear order in perturbations we have , from which we deduce that is decaying as . Provided and do not grow too fast after horizon exit, we thus find that also decays. As such, neglecting all decaying terms in the expansion we find that the second order expression for remains local, and we finally obtain
The fact that the non-local terms decay on super-horizon scales was shown for the two-field case in , and here we have generalised this result to the case of more than two fields.
3.2.2 Leading and next-to-leading order slow roll approximations
In the case that we make the slow-roll approximation we can use the same method as above but now we obtain our expansion of in terms of from the expressions for given in (2.9) and given (2.20) for the leading and next-to-leading order cases respectively. Note that the velocities are no-longer independent degrees of freedom, so that the expansion of is only in terms of the field fluctuations .
Perhaps a slightly quicker way to obtain the necessary results, however, is to realise that in the slow-roll case the quantities and introduced above exactly vanish. Considering first, in the slow-roll case we have
where here the superscript takes the values or to denote the leading or next-to-leading order cases respectively. Substituting these results into (3.21) we find that indeed vanishes. Similarly, turning to , at linear order we have
which on substituting into (3.25) gives . As such, the expression for given in (3.26) with the last term set to zero becomes exact, in the sense that the decaying terms involving and are exactly zero in the slow-roll case rather than just decaying. On expressing and in terms of and its derivatives, we thus obtain
Recall that our use of indices from the end of the alphabet here indicates that these are the derivatives of with respect to field values at the final time at which we wish to evaluate using eq. (3.10). For the leading-order case, i.e. taking and substituting the relevant expressions for , and , we obtain results that are in agreement with . As far as we are aware, the next-to-leading order case, with , has not been considered in the literature.
3.3 The derivatives of
Having found the coefficients and that appear in eq. (3.10) we next compute the quantities and . Once again it is possible to do this under various levels of slow-roll approximation.
3.3.1 No slow-roll approximation
Recall that the quantities and appear in the expansion of about some fiducial trajectory as in (3.9). Next, let us recall that as a consequence of the separate universes approach, the equations of motion for are of exactly the same form as those for the unperturbed fields given in eq. (2.6), which we then conveniently re-expressed in terms of as in (2.22). Perturbing eq. (2.22) up to second order we find
Then, plugging eq. (3.9) into the above result we find that and satisfy the equations
with the initial conditions and . While it is possible to give formal solutions to equations (3.38) and (3.39) – see e.g. , in practice we will solve them numerically. It is perhaps interesting to consider how much work we will have to do. Firstly, in terms of the background dynamics we will have to solve equations of motion for the fields and their velocities. Then, in solving for the perturbations, we will have to solve for the quantities and for the quantities , where we have used the fact that is symmetric in the lower two indices. Altogether we thus have equations to solve, which for large goes as . We will be able to compare this with the amount of work required when we make the slow-roll approximation.
3.3.2 Slow-roll approximation holds throughout inflation
In the case that the leading or next-to-leading order slow-roll approximation is valid throughout inflation, the field velocities are not independent degrees of freedom as they are expressed in terms of the field values via the slow-roll equations of motion. Nevertheless, the analysis goes through in exactly the same way as the non-slow-roll case considered above, but with and . Explicitly, perturbing (2.12) or (2.18) up to second order we have
Then, in analogy with (3.9), we have
where we have kept the convention that letters from the beginning and end of the alphabet are used to denote quantities evaluated at the initial and final times respectively. On substituting this expansion into (3.40) we obtain the equations
with the initial conditions and . In this case, at background level we must solve for just the scalar fields, and in solving for the perturbations we must solve for the components of and the components of , giving a total of equations to solve. For large this goes as , which is a factor of fewer than the number of equations that need to be solved in the case where no slow-roll approximation is made.
3.3.3 Slow-roll approximation holds only around horizon crossing
We finally consider the case where we only assume that the leading or next-to-leading order slow-roll approximation holds around the time that the scales of interest left the horizon, i.e. we allow for the possibility that the slow-roll approximation breaks down during the super-horizon evolution.
Note that the quantities and have mixed indices, in that runs from whereas and only run from . Substituting this expansion into (3.37) we obtain evolution equations for and as