Phenomenology of D-Brane Inflation with General Speed of Sound
A characteristic of D-brane inflation is that fluctuations in the inflaton field can propagate at a speed significantly less than the speed of light. This yields observable effects that are distinct from those of single-field slow roll inflation, such as a modification of the inflationary consistency relation and a potentially large level of non-Gaussianities. We present a numerical algorithm that extends the inflationary flow formalism to models with general speed of sound. For an ensemble of D-brane inflation models parameterized by the Hubble parameter and the speed of sound as polynomial functions of the inflaton field, we give qualitative predictions for the key inflationary observables. We discuss various consistency relations for D-brane inflation, and compare the qualitative shapes of the warp factors we derive from the numerical models with analytical warp factors considered in the literature. Finally, we derive and apply a generalized microphysical bound on the inflaton field variation during brane inflation. While a large number of models are consistent with current cosmological constraints, almost all of these models violate the compactification constraint on the field range in four-dimensional Planck units. If the field range bound is to hold, then models with a detectable level of non-Gaussianity predict a blue scalar spectral index, and a tensor component that is far below the detection limit of any future experiment.
- I Introduction
- II DBI Inflation
- III Microscopic Constraints
- IV The Flow Formalism for D-brane Inflation
- V Results
- VI Conclusion
- A Monte-Carlo Algorithm for DBI Inflation
Understanding the physics of inflation (1) is one of the main challenges of fundamental physics and modern cosmology. Since string theory remains the most promising candidate for a UV completion of the Standard Model that unifies gauge and gravitational interactions in a consistent quantum theory, it seems natural to search within string theory for an explicit realization of the inflationary scenario.
This search has so far revealed two distinct classes of inflationary models which identify the inflaton field with either open string modes (e.g. brane inflation (2); (3); (4), DBI inflation (5)), or closed string modes (e.g. Kähler moduli inflation (6), racetrack inflation (7), N-flation (8)).
The brane inflation scenario (2); (3) in particular has received considerable theoretical and phenomenological interest. At the same time, precise cosmological observations (9) have made the detailed predictions of inflation testable.
In this paper we study the general phenomenology of D-brane inflation models with arbitrary speed of sound. The theoretical ansatz is motivated by brane–anti-brane dynamics in warped spaces as described by the Dirac-Born-Infeld (DBI) action. Relativistic dynamics of the brane motion
deviations of the propagation speed of inflaton fluctuations from the speed of light. These effects generically result in a large level of non-Gaussianity of primordial fluctuations (10) and modify the inflationary single-field consistency relation (11). We study these exciting observational signatures using a generalization of the inflationary flow formalism. This allows us to explore an ensemble of D-brane inflation models parameterized by the evolution of the Hubble parameter and the speed of sound as polynomial functions of the inflaton field.
The outline of the paper is as follows: In §II we review the DBI mechanism of D-brane inflation and discuss its basic cosmological predictions. In §III we discuss microscopic consistency constraints that these models have to satisfy. In particular, we review and generalize the field range bound of (12). In §IV we apply the inflationary flow formalism to D-brane inflation. We derive the generalized flow equations and use them to study the phenomenology of an ensemble of brane inflation models with arbitrary speed of sound. §V contains a summary of our main results. We conclude in §VI. An Appendix gives further details of the Monte-Carlo algorithm. We use natural units throughout the paper, where and . Interesting related work has appeared in (13) and (14).
|Inverse sound speed||equations (6) and (65) / Monte-Carlo|
|, ,||Slow variation parameters||equations (68) to (72)|
|Scalar power spectrum||We use|
|Tensor power spectrum|
|Tensor-to-scalar ratio||equation (21); derived at|
|Scalar spectral index||equation (19); derived at|
|Tensor spectral index||equation (20); derived at|
|NG parameter||equation (27); DBI prediction.|
|Number of -folds|
|Radial coordinate||equation (2)|
|Monte Carlo inflaton||at the UV end of the throat.|
|D3-brane tension||equation (35)|
|Warp factor||equation (1) and §III|
|Warped brane tension||. Same as in Ref. (13)|
|String scale||; equation (36)|
|4d Planck mass|
|,||Flux on - and -cycles||integer quantum numbers|
NOTES.—A summary of variables and functions related to our description of DBI inflation and observables.
Ii DBI Inflation
ii.1 Review of Warped D-brane Inflation
Since a quantum theory of (super)strings is naturally defined in ten spacetime dimensions, any realistic description of cosmology and particle phenomenology requires compactification of six of the space dimensions. Different compactification geometries lead to different low energy effective theories in four dimensions. In addition, the extra dimensions have to be stabilized, since even a small time-dependence in the size and shape of the extra dimensions can manifest itself in a variation of fundamental coupling constants and induce observable fifth forces.
Recently, Kachru, Kallosh, Linde and Trivedi (KKLT) (15) provided a framework for stabilizing the extra dimensions of type IIB string theory in the presence of background fluxes and non-perturbative effects (see (16) for a review of flux compactification). The flux background fixes the shape (complex structure moduli) of the extra dimensions, but leaves the overall size (Kähler moduli) unfixed (17). As (15) showed, the size of the compact space may be stabilized by the inclusion of non-perturbative effects, e.g. gaugino condensation on D7-branes. In addition to stabilizing the shape of the extra dimensions, the background fluxes lead to strong warping of the spacetime. This warping provides an elegant mechanism to produce exponential hierarchies (a realization of the Randall-Sundrum scenario in string theory (18)) and has important consequences for the dynamics of brane motion in the warped space.
The line element of a warped flux compactification of type IIB string theory to four dimensions takes the following form
where is the warp factor and and . Typically, the internal space will have one or more conical throats sourced by the background fluxes, i.e. regions in which the metric is locally of the form
for some five-manifold which forms the base of the cone.
If the background contains suitable fluxes, the metric in the throat
region can be highly warped, .
The KKLMMT scenario (3) refers to brane inflation (2) in a warped throat of a KKLT flux compactification (15).
In particular, most models of brane inflation are concerned with the dynamics of a mobile D3-brane that fills four-dimensional spacetime and is point-like in the compact space.
Here, the inflaton field is identified with the geometrical separation between a D3-brane and an anti-D3-brane. The anti-brane is fixed at the tip of the throat (), while the brane moves from large radius () to small radius ().
Here is the inflaton field, is the tension of the D3-brane and is the rescaled warp factor of the background spacetime. For a homogeneous background is the canonical kinetic term of the inflaton. The potential for the brane motion arises from moduli stabilization effects (3); (20) and the Coulombic brane–anti-brane interaction. In the slow-roll limit , the DBI action (4) reduces to the following familiar form
This slow-roll limit can be understood as the non-relativistic motion of the brane in the presence of a weak force from a flat potential. The relativistic limit of brane motion in a warped background is characterized by the parameter (defined in analogy to the Lorentz factor of relativistic particle dynamics)
Positivity of the argument of the square-root in (6) imposes a local speed limit on the brane motion, . The presence of strong warping, , in the throat can make this maximal speed of the brane much smaller than the speed of light. When is close to this speed limit, then is large.
From the inflaton action (3) we find the homogeneous energy density in the field
while the pressure is
and source the dynamics of the homogeneous background spacetime, , as described by the Friedmann equations
Accelerated expansion () requires smallness of the variation of the Hubble parameter , as quantified by the parameter
From the expression for the equation of state parameter (12) we see that although the brane moves relativistically in the DBI limit, , inflation still requires that the potential energy dominates over the kinetic energy of the brane . This is possible because the kinetic energy of the brane is suppressed by the large warping of the internal space, .
As discussed in Ref. (21), the slow roll (non-relativistic brane motion) and the DBI (ultra-relativistic brane motion) limits are connected continuously by an intermediate regime where the relativistic effects are small, , but non-negligible.
ii.2 Cosmological Observables
This is the speed at which fluctuations of the inflaton propagate relative to the homogeneous background. In addition, we define slow variation parameters in analogy with the standard slow roll parameters for inflation with canonical kinetic term
To first order in the slow variation parameters, the basic cosmological observables as a function of wavenumber are (23)
Scalar perturbations freeze when they exit the sound horizon , while tensor perturbations freeze when . Over a limited range of scales it is appropriate to parameterize deviations from perfect scale-invariance by the following spectral indices
Notice the dependence of on the evolution of the speed of sound as captured by the parameter . The tensor-to-scalar ratio in these models is
The dependence of (21) on the speed of sound implies a modified consistency relation
The standard slow roll predictions for the cosmological perturbation spectra are recovered in the limit , .
The non-trivial structure of the kinetic term in the action (4) leads to striking observational signatures of relativistic DBI inflation. In particular, it was observed (5) that this generically leads to a very large non-Gaussianity of the primordial fluctuations.
Let us give a brief qualitative description of the physical origin of this non-Gaussianity
and its first order variation under
This indicates that self-couplings of are enhanced by factors of arising from expansion of the square root of the DBI action. Non-Gaussianities come from the third-order interactions in due to the perturbation . The leading effect can be estimated by considering the ratio of the cubic perturbation to the matter Lagrangian, , to the quadratic perturbation, , neglecting mixing with gravitational perturbations. Since matter self-interactions must dominate in order to obtain significant non-Gaussian features this provides a rough estimate of the effect. In the large limit the leading contribution to the non-Gaussianity parameter scales as (5)
We observe that the magnitude of non-Gaussianities scales with the Lorentz factor . Observational constraints on primordial non-Gaussianities therefore lead to interesting constraints on the magnitude of relativistic DBI effects.
The non-Gaussianity of fluctuations in DBI inflation was estimated in the large limit in (5) and computed more precisely for the general case in (10). Observational tests of the non-Gaussianity of the primordial density perturbations are most sensitive to the three-point function of the comoving curvature perturbations . It is usually assumed that the three-point function has a form that would follow from the field redefinition
where is Gaussian. The scalar parameter then quantifies the amount of non-Gaussianity. It is a function of three momenta which form a triangle in Fourier space. Here we cite results for the limit of an equilateral triangle. The general shape of non-Gaussianities in DBI inflation may be found in (10). Slow roll models predict (25) , which is far below the detection limit of present and future observations. For generalized inflation models represented by the action (3) with general pressure function , one finds (10)
Measurements of primordial non-Gaussianity therefore constrain the speed of sound during the time when CMB scales exit the horizon during inflation. This implies an upper bound on from the observed upper limit on the non-Gaussianity of the primordial perturbations. Using the WMAP limit (26), (95% confidence level), one finds . Ref. (27) forecasts constraints on primordial non-Gaussianity from Planck of at the level, which translates to .
Although not inconceivable, it will be challenging to experimentally test this unique prediction of brane inflation. For the standard slow-roll case with , the combination of next-generation CMB polarization measurements of and estimates of based on the direct detection of gravitational waves fails to test this consistency relation to better than 50% for even the most optimistic experimental scenarios (28).
Iii Microscopic Constraints
The philosophy of this paper is to take maximal theoretical guidance from the string theoretic origin of D-brane inflation models, but use a phenomenological Ansatz to capture the largest set of possible scenarios in a single framework. In particular, we allow considerable freedom on the functional form of the inflaton potential and the background warp factor . Ultimately, both and should of course be derived from an explicit string compactification (see (22)). Here, we take the approach of studying a general set of functional forms for these quantities, restricted only by cosmological constraints (§II.2) and a minimal set of microscopic consistency requirements. In this section, we describe the microscopic constraints that we impose on our models.
iii.1 Warped Background
Schematically, the warp factor in equation (1) is determined by the solution of the Laplace equation
where is the six-dimensional Laplacian and parameterizes 3-form flux which sources the warping. Intuitively, equation (31) is just like the equation for an electrostatic potential on the compact Calabi-Yau space in the presence of a ‘charge density’. The functional form of the warp factor hence depends on how the ‘charge density’ is distributed. If it is localized in one region, then will tend to decrease monotonically away from it. We call this the one-throat situation. should reach some finite value (i.e. it should not diverge) at the bottom of the throat, since otherwise the warped string and brane tensions which scale as would vanish. In addition, to avoid a naked singularity should not vanish.
The Klebanov-Strassler (KS) geometry (29) is an explicit non-compact ten-dimensional solution to type IIB supergravity in the presence of background fluxes. The KS spacetime decomposes into the form (1) with the internal space (2) given by a cone over . Far from the tip of the throat the KS solution is well approximated by with the warp factor determined by the Green’s function of (31)
Here, parameterizes the dimensionless volume of with unit radius. Typically, , e.g. . and are the string coupling and the string length, respectively and is the minimal radial coordinate at the tip of the throat,
The integers and denote flux quanta on the and cycles of the throat. The exact KS warp factor is non-singular at the tip (29). The AdS warp factor (32) forms the basis for many theoretical studies of brane motion in warped throat regions. However, other warped solutions are possible (and some are known), so in the spirit of our phenomenological approach and to retain maximal generality we allow to be a free function subject only to minimal theoretical constraints. Of course, we appreciate that it is therefore not guaranteed that all (or even most) warp factors we consider in this study have explicit microscopic realizations. More detailed theoretical constraints on the functional form of than the ones presented in this section are beyond the scope of this paper. For a more complete theoretical understanding of the phenomenology of DBI inflation, such a theoretical study is essential.
If one considers a scenario with two (or more) throats, corresponding to two localized ‘charge’ distributions in (31), then will not be monotonic overall (similarly to the electric potential of two positive charges). It will reach a minimum between the throats (‘charges’) where . However, within each throat it should behave monotonically, as in the explicitly known examples of gauge/gravity duality like the KS solution. Let us remark that although we will focus our attention to single throat scenarios with monotonic warp factors, the methodology of the present paper is easily generalized to studies of multi-throat scenarios.
Warped Brane Tension
The dynamics of a D3-brane in a warped throat region is determined by the brane potential and the warped brane tension . In the following we relate the scale of the warped tension to microscopic parameters. The D3-brane tension in four-dimensional Planck units is
where defines the string scale. The string scale is related to the four-dimensional Planck mass (or the Newton constant ) via the (warped) compactification volume
Notice that the four-dimensional Planck mass increases as the (warped) volume of the internal manifold is increased. The warp factor in the throat region lies in the following range , where
is the warp factor at the tip of the throat and defines the region where the throat is glued into a bulk space. Small string coupling and large allow exponentially large warping, i.e. small . Using , we find
To facilitate comparison with the analysis of Bean et al. (13), let us consider the same fiducial values for the string coupling and the string scale, , . This implies fixing the D3-brane tension to have the following value, . Allowing the warp factor to be in the range then implies for the fiducial range of . However, we note that by decreasing the string scale (increasing the volume of the internal space) and/or increasing the warping, one can make much larger. In general, we treat the magnitude of as a free parameter of the models.
iii.2 Bound on the Field Range
Since the inflaton field for D-brane inflation acquires a geometrical meaning, there exist geometrical restrictions on the allowed range of . In particular, the finite size of the compact extra dimensions restricts the field variation of the canonical inflaton in four-dimensional Planck units. Naively, since , one might imagine that the inflaton field range can be increased by simply scaling up the radial dimension of the throat. However, this increases the volume of the compact space, and by equation (36), changes the four-dimensional Planck mass. For the warped cones that form the basis of most explicit theoretical models, the inflaton field range in four-dimensional Planck units in fact decreases as the radial dimension of the cone is increased.
Using the conservative bound that the total warped volume (which receives contributions from the throat and the bulk) is at least as big as the warped volume of the throat only, i.e.
For the cut-off AdS warp factor
this implies that
Since , these microscopic considerations imply the following bound on the total field variation during warped D-brane inflation (12)
where for theoretical consistency the flux integer has to be much greater than unity. In all explicitly known examples, super-Planckian field variations are therefore microscopically disallowed and brane-inflation models that predict should be viewed with suspicion.
For ultra-relativistic DBI inflation () with a quadratic potential one can show that the normalization of the primordial scalar spectrum requires (12)
Since typically , observations therefore imply very large , allowing only very small field variations. However, Ref. (12) also derived an upper limit on in terms of the observational limits on non-Gaussianity and tensors
The numerical analysis of Bean et al. (13) shows that the intuition gained from the explicit example of (12) extends to the non-analytic, intermediate DBI regime. In particular, Bean et al. find that imposing the microscopic bound (46) dramatically reduces the parameter space of viable DBI models in the intermediate and ultra-relativistic regime. In this paper we study if this conclusion continues to hold when is allowed to be a free function.
For general , the total field variation during inflation is bounded by , or
Typically, . In this paper we compute by direct numerical integration of our output warp factors . Models that violate (49) cannot be embedded in a consistent string compactification. We find that most models discussed in this paper violate this condition, which highlights the theoretical challenge of constructing microscopically viable models of UV DBI inflation that are consistent with observations.
iii.3 Microscopic Bound on Tensors
The field range bound has important implications for the expected level of gravitational waves from brane inflation models (12). By the Lyth bound (30), the constraint on the evolution of the inflaton (46), or the generalized bound (49), is related to the maximal observable gravitational wave signal from inflation. Let us recall the argument and generalize it to general speed of sound theories. Restricting to the homogeneous mode we find from (14) that
where . Notice the non-trivial generalization of the standard slow roll result through the factor . For DBI inflation this factor happens to be unity, since , so that the Lyth bound remains the same as for slow roll inflation
quantifies the support of the integral (52) and denotes the tensor-to-scalar ratio evaluated on CMB scales. The value of depends on the evolution of after CMB scales have exited the horizon during inflation. In slow roll models this evolution is typically small (second order in slow roll), with observational data on CMB scales requiring , leading to the prediction of an unobservably small level of gravitational waves, (12).
iii.4 Implications for Relativistic DBI
Lidsey and Huston (14) derived an interesting generalization of the field range bound of (12) that is useful in the DBI limit (). Here we briefly review their argument. Let be the field variation when observable scales are generated during inflation, corresponding to -foldings of inflationary expansion (this corresponds to the CMB multipole range ). Then the integral in the bound on the Planck mass (42) can be approximated as follows
Here, we have bounded the integral by a small part of the Riemann sum, defined and used . Equation (42) then becomes
Next, we note that the warped tension can be expressed in terms of the scalar power spectrum , the tensor-to-scalar ratio , and the non-Gaussianity parameter . can therefore be related to observables
For slow roll models with this is not a very useful constraint. However, for relativistic DBI models with the bound (58) is independent of . Since, , (from observations) and (from theory) we conclude that super-Planckian field variation is inconsistent with observations. The Lyth bound (53) can now be written as
The observed level of scalar fluctuations therefore implies that the tensor amplitude is unobservably small for relativistic DBI models if the field range bound of (12) is applied (unless is made unnaturally small). We emphasize that the bound (60) does not apply to slow roll models since has been assumed in its derivation.
Interestingly, Lidsey and Huston also derived a lower limit on for models of UV DBI inflation with and (14)
With the work of (12), (13) and (14) there is now a growing body of evidence that the best motivated theoretical models of relativistic (UV) DBI inflation are in tension with the data if microscopic constraints are applied consistently. In this paper we reach conclusions that are consistent with this and show that these problems persist even if considerable freedom is allowed for the functional form of the brane potential and the background warp factor .
Iv The Flow Formalism for D-brane Inflation
To study the ensemble of inflationary models specified by the theoretical ansatz of the previous section, we adapt the inflationary flow formalism (31); (32); (33). In this section we derive the generalized flow equations for brane inflation. In the next section we present our numerical results.
iv.1 Inflation in the Hamilton-Jacobi Approach
Let us recall the Hamilton-Jacobi (HJ) approach to inflationary dynamics and apply it to warped brane inflation. For a spatially flat FRW spacetime, the scale factor is determined by the Friedmann equation
where and follow from the DBI action (4).
where primes denotes derivatives with respect to . In the HJ-formalism the Hubble expansion rate is considered the fundamental quantity. The master equation relating and follows from equations (62) and (63)
From equation (64) one finds
Given and , the inflaton potential is
Notice the following important consequence of the Hamilton-Jacobi equations (66) and (67): For any specified function and , it produces a potential and a warp factor which admits the given and as an exact inflationary solution.
iv.2 Inflationary Flow Equations
Analogous to the Hubble slow roll (HSR) parameters, we can write down a set of slow variation parameters for brane inflation that are defined in terms of derivatives of and with respect to :
where . For notational convenience we define , , and . Note that is related to in (15) by . The trajectories of these parameters are governed by a set of coupled first order differential equations. Using the relation
This set of differential equations defines the flow equations for brane inflation. Notice that while the flow parameters depend on , the flow equations do not depend on explicitly.
As pointed out by Liddle (34), the flow equations have an analytic solution. Truncating the hierarchy of flow equations so that the last non-zero terms are and ensures that = and = at all times (along with all the higher order terms). From Eqs. (71) and (72), it then follows that higher derivatives vanish at all times:
We thus arrive at polynomials of order and in respectively for the functions and ,
So far, we haven’t specified an initial value for . If we truncate the series as above, any set of flow parameters spans an dimensional space. However, the evolution equations define the flow parameters as functions of . Consequently, the set of distinct trajectories spans an dimensional space, effectively fibering the space of initial conditions for the flow hierarchy. However, if the flow parameters are specified at the ambiguity is removed.
From the definitions of the flow parameters, the coefficients and can be written in terms of the initial values of the flow parameters
The sign convention we choose is as follows. To define the direction of time with respect to the number of -folds before the end of inflation , we choose to increase as one goes backward in time; i.e., , so that as .
Furthermore, we choose to have the same sign as . This is equivalent to choosing in our notation; the sign of specifies in which direction the field is rolling.
Note that in our present work is not fixed to be a specific function. Instead, is a derived quantity that is determined through equation (66) via Monte-Carlo descriptions of and . Similarly, is also a derived function determined through and using equation (67). In this sense, our work differs significantly from the recent work of Bean et al. (13) in that they fix both and to particular (well-motivated) forms. We allow a large range of warp-factors through a specification of a general sound speed captured by , and a large range of dynamics captured by . Thus, the observable distributions shown in Bean et al. (13) for quantities such as the scalar spectral index and the tensor-to-scalar ratio are under a fixed theoretical model whose internal parameters (such as the normalization of the AdS warp factor) are allowed to vary. This can be considered a ‘top-down’ approach. Our work adopts the complementary ‘bottom-up’ viewpoint that, while some (or even many) of our warp factors may not have microscopic realizations, it is still interesting to investigate the phenomenology of the set of warp factors, potentials, and observables that are generally allowed by existing cosmological data.
iv.3 Monte-Carlo Algorithm for DBI Inflation
Now we shall outline the algorithm used to produce the numerical results of §V; technical details of the specific implementation are described in Appendix A.
The algorithm is designed to capture the dynamics of DBI inflation from the time the brane enters the mouth of the throat and inflation begins, until inflation ends (either through a tachyonic transition near the tip of the throat or due to slow roll ending). Because we explicitly define the direction of motion as the brane moving into the throat ( decreases), the algorithm as described below only applies to monotonic warp factors (single throat scenarios). However, it is straightforward to generalize the method to include the brane moving out of the throat ( increases) and thereby model multi-throat scenarios with non-monotonic warp factors; we plan to explore this possibility in future work.
A key feature of the algorithm is a well-defined physical “location” for , corresponding to the mouth of the throat, . Here and in the following, the subscript denotes evaluation at .
There are three phenomenological classes of DBI inflation that we would like to consider in this work. We classify models by their value of at CMB scales:
slow roll DBI:
ultra-relativistic DBI: .
We are particularly interested in the limit of the DBI classes above which are consistent with the microphysical bound (49) on the field range. In order to sample these different regimes efficiently, we consider different combinations of priors on and in our Monte-Carlo simulations, as follows:
Small initial conitions: We assume that the initial speed of the brane is very small, so
for and draw randomly from the narrow flat prior . This prior is effective at sampling the slow roll and intermediate DBI regimes well.
General initial conditions: We relax the above assumption on the initial speed of the brane, drawing randomly from the broad flat prior . This prior is effective at sampling the intermediate and ultra-relativistic DBI regimes well.
General initial conditions: We allow a key observable quantity, the tensor-to-scalar ratio, to take a wide range of values by drawing from the broad flat prior . Unless specified otherwise, this is the standard prior we will use in §V.
Small initial conditions: As we will find below, the general prior generates DBI models that are consistent with the current cosmological constraints, but which strongly violate the microphysical bound (49). In order to quantify the limit on the tensor-to-scalar ratio at which the field range bound is satisfied for the different classes of models, we draw from the narrow flat prior .
The rest of the algorithm is identical for all the cases that we consider. We Monte-Carlo over , the initial amplitude of the power spectrum of scalar density perturbations at and its value at the CMB scale