A Terminal Velocity on the Landscape:
Particle Production near Extra Species Loci in Higher Dimensions
We investigate particle production near extra species loci (ESL) in a higher dimensional field space and derive a speed limit in moduli space at weak coupling. This terminal velocity is set by the characteristic ESL-separation and the coupling of the extra degrees of freedom to the moduli, but it is independent of the moduli’s potential if the dimensionality of the field space is considerably larger than the dimensionality of the loci, . Once the terminal velocity is approached, particles are produced at a plethora of nearby ESLs, preventing a further increase in speed via their backreaction. It is possible to drive inflation at the terminal velocity, providing a generalization of trapped inflation with attractive features: we find that more than sixty e-folds of inflation for sub-Planckian excursions in field space are possible if ESLs are ubiquitous, without fine tuning of initial conditions and less tuned potentials. We construct a simple, observationally viable model with a slightly red scalar power-spectrum and suppressed gravitational waves; we comment on the presence of additional observational signatures originating from IR-cascading and individual massive particles. We also show that moduli-trapping at an ESL is suppressed for , hindering dynamical selection of high-symmetry vacua on the landscape based on this mechanism.
- I Introduction
- II Quantum Trapping in Moduli Space
III Moduli as Inflatons
- III.1 , Suppressed Trapping
- III.2 , a Terminal Velocity on the Landscape
- III.3 Phenomenology of Trapped Inflation in Higher Dimensions,
- III.4 Additional Observational Signatures
- III.5 A Bending Field Trajectory
- IV Conclusion
- A Trapping Probability, Improved Estimate
- B Examples
- C Drifting through Moduli Space during Inflation
The presence of many light fields, or moduli, is a common feature of string theory. At late times, the expectation values of these fields determine low energy observables; hence, their evolution is heavily constraint from the time of nucleosynthesis, although they are expected to be dynamical in the very early universe. For these reasons, moduli trapping is an important aspect of inflationary model building in string theory, see HenryTye:2006uv (); Cline:2006hu (); Burgess:2007pz (); McAllister:2007bg (); Baumann:2009ni () for reviews; often all but one degree of freedom are stabilized by construction during inflation, but how they stabilize is seldom addressed.
A possible dynamical stabilization process is the String Higgs effect Bagger:1997dv (); Watson:2004aq (), which singles out certain locations in moduli space. At these loci additional degrees of freedom become light, often due to the presence of an enhanced symmetry, and are produced Kofman:2004yc (); Watson:2004aq () if the moduli approach the location. The process of particle production is identical to the one examined in preheating Traschen:1990sw (); Kofman:1997yn (), see Bassett:2005xm (); Kofman:2008zz () for reviews. Backreaction of these new states on the moduli can be dramatic: moduli can be trapped Kofman:2004yc (); Watson:2004aq (); Patil:2004zp (); Patil:2005fi (); Cremonini:2006sx (), or, if moduli drive inflation, the inflaton velocity can decrease temporarily Chung:1999ve (), as in trapped inflation Kofman:2004yc (); Green:2009ds (); Silverstein:2008sg (). A concrete realization of the string Higgs effect can be found in a system of parallel -branes Witten:1995im (), whose separation is a modulus field from the four-dimensional point of view. If they come close to each other, strings stretching between the branes become massless, gauge symmetries are enhanced and the branes stick to each other due to backreaction – the modulus is trapped. For a small selection of other examples, see Seiberg:1994rs (); Seiberg:1994aj (); Intriligator:1995au (); Strominger:1995cz (); Witten:1995ex (); Katz:1996ht (); Bershadsky:1996nh (); Witten:1995gx ().
In this paper, we examine the consequences of particle production near such extra species loci (ESL) if the dimensionality of moduli space is large compared to the dimensionality of the locus, . is natural in string theory, leading to the notion of a landscape Susskind:2003kw (), yet the effect of ESLs has primarily been investigated for Kofman:2004yc (); Watson:2004aq (); Greene:2007sa (). Based on geometric arguments, we show in Sec. II that trapping is suppressed if the dimensionality of moduli space is larger than the dimensionality of a given locus (). This result is expected, since there is no classical attraction towards ESLs and it is improbable to run head on into an ESL if . This means that the mere presence of ESLs does not guarantee a dynamical preference of high symmetry states for moduli Kofman:2004yc (); Dine:2000ds (); Dine:1998qr (). However, in Sec. III.2.1 we show that the presence of many loci with a characteristic inter-ESL distance leads to a general speed limit, or terminal velocity, on moduli-space. At strong coupling, speed limits are known Silverstein:2003hf (), leading to DBI-inflation Alishahiha:2004eh (). Here, we derive a speed limit at weak coupling caused by the combined backreaction of particles produced at many ESLs in the vicinity of the trajectory.
We take a bottom up approach, assuming the viability of low-energy effective field theory and treating the characteristic distance between ESLs, as well as their dimensionality, as free parameters. Furthermore, we model the additional light degrees of freedom by a massless scalar field that couples to the moduli via interactions of the type , as in Kofman:2004yc (). In this notation, the speed limit takes the simple form , given that the classical trajectory is reasonably straight (we allow for a classical potential for the moduli)and is large. The presence of a terminal velocity offers the opportunity to drive inflation by moduli with hitherto unsuitable potentials.
This type of inflation is a generalization of trapped inflation Kofman:2004yc (); Green:2009ds (), which recently re-surfaced as monodromy inflation Silverstein:2008sg (), but with several crucial differences: firstly, particle production at any given ESL is minor because ESLs are not approached too closely, in contrast to the one-dimensional case in Green:2009ds (); there, extra species points (ESPs) are encountered head on and all particle production occurs at a single ESP at any given time. Secondly, in the large limit the velocity at which the trajectory is traversed is independent of the slope of the potential as long as the slope is steep enough. Again, this differs from Green:2009ds (), where the slope still determines the speed and observational parameters, such as the scalar spectral index.
We investigate this type of trapped inflation in higher dimensions in Sec. III.3, where we encounter several attractive features: focusing on a model with quadratic potentials for the moduli, more than sixty e-folds of inflation can be achieved with sub-Planckian field excursions if is small enough. The inflationary scale is set by the COBE normalization, but the dynamics is not sensitive to the actual shape of the potential as long as a single, reasonably straight classical trajectory with large slope is present. This is, in a sense, opposite to setups needed for slow-roll inflation, where the potential needs to be shallow over long ranges in field space. As a consequence, the -problem is alleviated. Furthermore, no fine-tuning of the initial speed is needed: if the initial speed is large, particle production is strong, causing strong backreaction and temporary trapping; after the produced particles are diluted away due to the expansion of the universe, the speed picks up again and the terminal velocity is approached from below. In this sense, trapped inflation is an attractor solution.
The ongoing particle production during inflation has several additional observational consequences, Sec. III.4.1 and III.4.2. Since -particles scatter off the inflaton condensate, they cause IR-cascading and additional contributions to the power-spectrum Barnaby:2009mc (); Barnaby:2009dd (); furthermore, -particles dilute and become massive once the trajectory moves away from their ESL of production. Thus, one needs to consider the effect of individual massive particles that are coupled to the moduli, which could lead to additional circular cold-spots in the CMBR Itzhaki:2008ih (); Fialkov:2009xm (); Kovetz:2010kv (). We plan to investigate both effects, as well as the generation of non-Gaussianities, in a future publication, accompanied by a more rigorous string theoretical implementation of trapped inflation in higher dimensions.
Readers familiar with the notion of particle production, as discussed in Kofman:2004yc (), and a primary interest in the terminal velocity and consequences for inflation may skip Sec. II and go directly to the main part of this work, Sec. III. On the other hand, readers with a primary interest on a dynamical selection principle on the landscape will find Sec. II and Appendix C useful (these sections review Kofman:2004yc () and contain minor extensions and consequence which are implied to in Kofman:2004yc () but not fully spelled out).
Ii Quantum Trapping in Moduli Space
When are moduli affected by the presence of extra species loci? After estimating the distance that moduli can move classically in Sec. II.1, we review the notion of quantum moduli trapping in Sec. II.2 as developed in Kofman:2004yc () and estimate the trapping probability in Sec. II.3. We comment on drifting moduli in Sec. II.4 and Appendix C. Readers familiar with Kofman:2004yc () may want to skip Sec. II.1 and Sec. II.2.
ii.1 Length of the Field-Trajectory
Let’s start by reviewing the effect of Hubble damping on the evolution of moduli fields, as discussed in Kofman:2004yc (). Consider a -dimensional moduli space with freely moving fields (, ), that is, without a potential that could trap the fields classically. Additionally, assume a flat moduli space metric, leading to a straight trajectory. Defining as the effective field along this trajectory, its equation of motion is
where a dot denotes a derivative with respect to cosmic time and is the Hubble parameter. For a constant equation of state parameter the Hubble parameter becomes with , and (1) can be integrated to
If the universe is dominated by freely moving scalar fields, that is if , the length of the field trajectory
On the other hand, if that is , equation (2) yields
which is bounded from above by . Since , we have so that .
To get intuition for the expected values of in realistic scenarios, let’s consider concrete numbers; in an inflating universe with the upper bound becomes , less than regardless of the initial velocity. However, if is closer to , the length of the trajectory can increase above . For instance, consider a phase dominated by kinetic energy,
Thus, an extended kinetic phase is needed to achieve super-Planckian values for . For example, if moduli have an initial Planckian kinetic energy () and if they dominate until high-scale slow-roll inflation sets in around (), we get and . Conceivably, moduli could also dominate after inflation ended i.e. down to the SUSY breaking scale , so that and .
In summation, if the kinetic energy of the moduli does not dominate the energy density of the universe, the moduli travel a limited stretch in field space until they come to rest due to Hubble friction, ; for instance, they do not travel far during inflation or in a radiation/matter dominated universe 111They could also be displaced by quantum mechanical fluctuations during inflation, see Appendix C.1.. If the moduli’s kinetic energy dominates, i.e. before a standard high scale inflationary phase or in the interval after inflation but before SUSY breaking, they can travel farther ( and ). In the following sections we keep in mind the interval
ii.2 Quantum Trapping
Even in the absence of a classical potential, moduli can be trapped near locations where additional particle species become light. These additional degrees of freedom are produced quantum mechanically if the field trajectory gets close enough to an extra species’ locus (ESL), and their backreaction can force the trajectory in an orbit around it. Since Hubble damping in an expanding universe causes an inward spiraling trajectory, the moduli get trapped. The origin of the additional light degrees of freedom are often enhanced symmetries at the locus Kofman:2004yc (); Watson:2004aq (), so that ESL or ESP can also denote enhanced symmetry locus or point. Quantum moduli trapping was proposed in Kofman:2004yc (); Watson:2004aq () and was recently investigated in Greene:2007sa (). A field theoretical description of this phenomenon is identical to the resonant particle production after inflation, usually referred to as preheating Traschen:1990sw (); Kofman:1997yn (), see i.e. Bassett:2005xm (); Kofman:2008zz () for reviews. Our first goal is to estimate a characteristic distance to the ESL below which moduli trapping is common.
Consider a single ESP at the origin where additional light states appear. We model this situation by introducing an new scalar field coupled to the freely moving moduli via a quadratic interaction
without any bare mass, as in Kofman:2004yc (). Classically, along with a straight trajectory in moduli space is a solution. Starting with this solution, the trajectory and the ESP span a plane in moduli space. Hence, we can redefine the fields so that the straight trajectory takes the simple form
where we ignored Hubble friction so that . Hence, to discuss the effect of a single ESL, the problem reduces to the two dimensional case investigated in Kofman:2004yc (). Efficient particle production in a Fourier mode occurs if the non-adiabaticity parameter
satisfies Kofman:1997yn ()
Since the effective mass of the -field is given by we see that this requirement is satisfied if the -field is light. Once -particles are produced, they induce a classical confining potential for the moduli, which gets stronger with successive bursts of particle production. Evaluating (14) at the point where the ESP is closest, we conclude that trajectories with impact parameters smaller than
have a strong chance to get trapped near the ESP (see Fig. 1 for a schematic). The value of , and correspondingly the value of the characteristic impact parameter below which trapping is common, is model dependent. In the case of a higher dimensional ESL the discussion is analogous, with denoting the minimal distance between the straight classical trajectory and the ESL. Since , the value of the characteristic impact parameter decreases drastically once fields slow down.
The actual trapping event is complicated by Hubble damping as well as the fragmentation of fields once backreaction becomes important. Except in the simplest one-dimensional cases, a proper discussion of trapping requires numerical simulations, as in preheating Prokopec:1996rr (); Felder:2000hq (); Frolov:2008hy (); Khlebnikov:1996zt (); Khlebnikov:1996mc ().
In this section, we take a binary point of view and assume that the moduli get trapped once they approach an ESL to within a distance smaller than (we go beyond this binary treatment in Sec. III). We further treat as a constant222We will find that ; hence, trapping becomes less likely as decreases due to Hubble friction, so that we overestimate by treating as a constant over the whole trajectory. A better estimate could be achieved by integrating a probability density over the trajectory using . Since the latter is model dependent, we will be satisfied with the overestimate caused by treating (the correction usually introduces factors of order one). .
ii.3 Trapping Probability
As we saw in section II.1, Hubble damping limits the overall path length to in an expanding universe. If the field trajectory approaches an ESL to within the characteristic impact parameter , trapping of the moduli is common, see Sec. II.2 and Fig. 1. We denote the dimensionality of the ESL by (), so that corresponds to an ESP, to a line, etc.
Further, we assume that ESLs are spread over moduli space with a characteristic distance (we make this more precise below). For , the probability of trapping is less than one, provided that the start out at a random initial position and velocity. Our goal is to estimate the dependence of this probability on the ratios and , as well as the scaling with the moduli space’s and ESLs’ dimensions, and . Throughout this section, we take (see Sec. III.2 for ).
ii.3.1 Extra Species Points (ESP, )
We assume that ESPs are common and distributed over field space with a number density of . Consider a D-dimensional sphere of radius centered at , enclosing the region that the moduli can access, see Fig. 2. In this sphere we find
ESPs that are potentially reachable by . Here
is the volume of a D-dimensional unit-sphere, and is the Gamma-function ( for large ). Since the majority of the volume in a sphere is close to its surface for large , the distance from to the majority of the accessible ESPs is given by a distance close to , the radius of the sphere. Thus, we approximate the distance of all ESPs to the center of the sphere by . A lower bound for the probability of trapping can then be estimated by the ratio of the sphere’s surface blocked by the ESPs to the total surface area
is the surface area of a D-dimensional unit-sphere, see Fig. 3 for a schematic. Here, we assume that ESPs are sparse enough to prevent overlapping of blocked regions on the sphere’s surface so that our results are only valid for . Further, we take , since the one dimensional case is trivial: the probability of getting trapped is if and for . (18) is a lower bound, since even a few ESPs close to the center of the sphere, which we treated as if they where sitting on the surface, can block out a large fraction of the surface. We come back to this issue in appendix A.
With (16) the lower bound on the probability reads
In line with expectations based on Kofman:2004yc (), the more ESPs are within reach, the higher the probability of trapping. However, since and we see that trapping becomes heavily suppressed for ; thus, for large , the starting point needs to be within close range of an ESP, that is within a distance of order or less, to have a reasonable chance to get trapped. This should be compared to the one dimensional case, where the relevant distance is : the modulus field is likely to be trapped as long as a single ESP is within reach.
The suppression has important consequences for vacuum selection on the landscape. When the notion of quantum moduli trapping near ESPs was investigated in Kofman:2004yc (); Watson:2004aq (), a dynamical selection principle to single out certain vacua associated with additional symmetries on the landscape, as used in Dine:2000ds (); Dine:1998qr (), was a strong motivation. However, for large we see that needs to start out or be accurately aimed at an ESP in order to get trapped near it, hindering this dynamical selection. This argument remains true for loci of higher dimensions as long as and .
ii.3.2 Extra Species Loci (ESL, general )
There are many cases where symmetries are enhanced (and additional light states appear) not at points, but at loci of higher dimensionality. We can easily generalize the derivation of the last section to arbitrary . We first assume a distribution of the ESLs on moduli-space with a characteristic distance , similar to the case of ESPs; for example, if , one can visualize the loci as a string network with characteristic inter-string distance . Thus, we can define an analog to the number density in the ESP case via , so that the blocked (by ESLs) surface area of the D-dimensional sphere with radius is bigger than . Here we treated all loci as if they were located at the surface of the sphere (hence, we find again a lower bound for the probability) and that they are sparse enough to ignore overlapping. As a consequence, the trapping probability satisfies
For we observe a suppression of the trapping probability . Furthermore, as , providing yet another suppression. See Appendix B to see how this result applies to two simple examples.
The factor of is absent if we estimate the probability more accurately by means of a volume argument, see Appendix A, which does not require any artificial shifting of ESLs, yielding
ii.4 Comments on Drifting Moduli
During inflation any classical movement of moduli is quickly damped by Hubble friction, yet quantum mechanical fluctuations can still cause fields to drift through moduli space given that Hubble induced corrections to the moduli’s masses are negligible and no classical potential is present. If we follow the field value in a given Hubble patch, we can observe a random walk. Hence, one might expect an increased trapping probability, since a random walk is more volume filling that a straight trajectory. Since the Hausdorff dimension of a random walk is , that is a random walk is area filling, we expect the trapping probability to be slightly less suppressed, instead of . This expectation is correct333Note that in appendix C.3 has a different meaning: it is not related to the speed of the fields, but it is set by the distance at which the extra degrees of freedom can be described by a massless scalar in field theory. However, the scaling is also valid for if a classical trajectory is intertwined on small scales so that it can be approximated by a random walk in dimensions., see appendix C.3. We note that if the fields perform a random walk as described in appendix C, and trapping does occur, then the trapping event has to take place before the last sixty e-folds of inflation; in that case, our entire Hubble patch today lies in a region that was inside the Hubble patch at the time of trapping and as a result, the moduli fields have the same value everywhere in our observable universe.
However, we are primarily interested in the limit , leading to a strongly suppressed trapping probability if . Then the mere presence of ESLs does not guarantee late time values of moduli fields near these locations; see appendix C.4 for more details.
In section III.2 we investigate the case , in which case we expect particle production to be important; we further allow for a classical potential for the , so that they may be responsible for inflation.
Iii Moduli as Inflatons
So far, we focused on fields whose potential energy is negligible compared to both, their kinetic energy and (from here on we set ). If the potential energy dominates, a phase of moduli-driven inflation results (see McAllister:2007bg () for a recent review on inflationary models in string theory); successful moduli trapping goes hand in hand with (p)reheating, while failed trapping events can still have interesting consequences.
If moduli serve as inflatons, a successful trapping event is unwanted if it occurs during the last sixty e-folds of inflation. However, failed trapping events during inflation, that is some particle production at an ESL, slow down the fields and render steeper potential viable for slow-roll inflation. This is the idea behind trapped inflation Kofman:2004yc (); Green:2009ds (), recently resurfaced as a special case of single field monodromy inflation Silverstein:2008sg () with sub-Planckian field excursions. Here, ESPs are repeatedly encountered by the inflaton as the angular modulus in a string compactification revolves around a periodic direction Silverstein:2008sg (), see also Brandenberger:2008kn (). Due to the classical (smooth) potential, the inflaton does not get trapped for long (even though and ) but it slows down until inflation dilutes the produced particles sufficiently and the field picks up speed again. After leaving the ESP, the produced -particles quickly become heavy, re-scatter off the homogeneous inflaton condensate and cause a cascade of inflaton fluctuations in the IR Barnaby:2009mc (); Barnaby:2009dd (). These fluctuations can dominate over the primordial ones and lead to observable signatures Barnaby:2009mc (); Barnaby:2009dd (), such as a bump in the power-spectrum for each failed trapping event and potentially large non-Gaussianities.
One may wonder if it is conceivable to increase the number of fields in trapped inflation, for as long as each of the inflatons comes close to ESLs during inflation, the above mentioned effects will be present.
iii.1 , Suppressed Trapping
Since the probability of getting close enough to ESLs depends on the type of movement through moduli space, we need to discern between three classes of inflation
Fields with a smooth potential. A smooth classical trajectory in field space results, without any tunneling events.
An overall smooth potential with small random bumps and dips. The classical trajectory can be described by a biased random walk; the overall drift is determined by the averaged potential, while the superimposed Brownian motion is caused by scattering events on the bumps, see i.e. Tye:2008ef (); Huang:2008jr (); Tye:2009ff ().
If the trajectory is intertwined, as in case two, we need to discern between the drift velocity , which is again small, and the velocity along the trajectory . These two velocities are comparable if small scattering events dominate; thus, we expect again an absence of moduli trapping if . However, if , for example due to frequent large angle scattering events, we note that is not suppressed. The trapping probability of the resulting biased random walk lies between the classical one in (22) and from (102). To be concrete, if , where is the mean free path between two scattering events (not ), is the number of e-folds and is the number of scattering events per e-fold, we may approximate . On the other hand, if the classical drift is negligible, , we may use .
In case three, the trajectory is well described by an unbiased random walk so that (102) is applicable. However, the step width , number of steps per e-fold , and need to be reconsidered, because the trajectory is the result of repeated tunneling/scattering events.
Regarding the viability of trapped inflation in the presence of multiple fields, we note that if is applicable, there needs to be a series of closed dimensional ESL-hypersurfaces encompassing the origin to which is heading; since , the trapping probability (22) is not suppressed, particle production can occur and trapped inflation may commence. If the fields perform a random walk, the dimensionality of ESLs can be smaller (). Thus, trapped inflation, as envisioned in Kofman:2004yc (); Green:2009ds () requires ESLs of dimensionality close to .
However, as we shall see in Sec.III.2, a compelling type of trapped inflation occurs naturally if is large and due to the presence of a terminal velocity, even if .
iii.2 , a Terminal Velocity on the Landscape
Consider that the field-trajectory encounters a steep region on the landscape, unsuitable for slow-roll inflation. The classical potential leads to an increasing speed so that becomes large. Such an increase in causes the effective trapping radius to grow. For simplicity, we consider ESPs only, that is so that it is extremely unlikely to run head on into an ESP for large (a generalization to ESLs is straightforward as long as ).
Once , the trajectory is bound to come within range of neighboring ESPs. For example, if , the inflaton becomes susceptible to ESPs, one ahead and one behind; if is large, the trajectory becomes within reach of many ESPs, of order .
Around each of these ESPs, particles are produced (see Sec. III.2.1 for the computation). As the inflatons continue to roll, there is an increasing number of particles in the wake of the trajectory 444The particles are subsequently diluted due to the expansion of the universe and also back-scatter and decay, see Sec. III.2.1. . As we shall derive below, the effect of each individual ESP is small and no actual trapping occurs if is large, but the combined backreaction of all produced particles leads to a resistive force opposite to . This situation is comparable to that of friction/viscosity in a fluid.
For large , this friction increases sharply once , that is around
preventing a further increase in and leading to a terminal velocity on the landscape, primarily determined by the density of ESPs and the coupling strength between the inflaton condensate and the new light degrees of freedom. Thus, depending on and , inflatons could evolve slowly even if the potential is steep. Since no single ESP has a strong influence on , we expect the trajectory to be largely unchanged, that is, still follows the potential gradient, but with a maximal speed determined by balancing the drag force, not Hubble friction, with the driving force of the potential. A similar argument can be made for ESLs as long as . Note that this terminal velocity is present at weak coupling, opposite to the speed limit at strong coupling in Silverstein:2003hf () which led to DBI-inflation Alishahiha:2004eh ().
We conclude that driving inflation on the landscape may not require a flat potential after all. If ESLs are ubiquitous, fields evolve slowly due to quantum backreaction. This type of inflation is a generalization of one-dimensional trapped inflation, with the important difference that no actual trapping occurs. Furthermore, whereas ESPs need to be aligned neatly like pearls on a string if , which may or may not be natural Silverstein:2008sg (), trapped inflation can result for randomly distributed ESPs in higher dimensions. Further, the terminal velocity becomes independent of the slope of the potential in the large limit.
In the following we analyze the physical processes behind higher dimensional trapped inflation in more detail, before providing a phenomenological model and computing some observational consequences.
iii.2.1 Particle Production near ESPs and Computation of the Terminal Velocity
Let us consider particle production at a single ESP which the trajectory passes by at a distance at with constant velocity (see Sec.II.2 for the notation) 555Since particle production near an ESP is exponentially suppressed the larger the distance to the ESP, the majority of particles are produced during a short time-frame, when the distance is minimal; this justifies treating as a constant while particles are produced.. Ignoring backreaction onto the trajectory while is close to the ESP, we can compute the number density of particles with wave-number after the encounter, , to Kofman:1997yn (); Kofman:2004yc ()
where we incorporated the subsequent dilution for due to the expansion of the universe. The corresponding energy density is
where is the location of the ESP, , and
We assume that -particles are sufficiently stable to ignore their decay, but the unavoidable scattering off the inflaton condensate lowers the number density over time by an additional factor of at most Kofman:2004yc (). For simplicity, we ignore this factor in the following 666The presence of a terminal velocity and even its value in the large limit does not change if this factor is kept..
Before we compute the backreaction onto , let us investigate up to which distance we need to consider ESPs. If we find ourself at a random position in field space, one might guess that only nearest neighbors roughly a distance away need to be considered. However, there are two effects which render the effective impact parameter bigger than . Consider a D-dimensional sphere of radius in field space with an ESP density ; solving gives the radius above which we expect at least one ESP inside the sphere. Since and using the asymptotic form of the Gamma function for large arguments, , we arrive at
considerably larger than for .
The effect of ESPs that are further away is exponentially suppressed according to (27), but their increased number can counterbalance this suppression to some degree. As a consequence, we need to consider not only nearest neighbor ESPs. Let us compare the energy density of -particles at ESPs that were produced at up to a distance of perpendicular to the trajectory; we are interested in their energy density when , see Fig. 4. Noting that for large almost all ESPs are separated from perpendicular to the trajectory and that the majority of ESPs can be found near the rim of the cylinder with radius in Fig. 4, we can approximate the ratio of the -particles’ energy density near the rim to the one of particles at nearest neighbor ESPs by
This ratio is maximal for
where we ignored terms of order and used . Thus, the energy density is dominated by -particles a distance
away, which sets the characteristic impact parameter for the majority of relevant ESPs.
Next, we compute the backreaction onto the inflationary trajectory in the Hartree approximation Traschen:1990sw (); Kofman:1997yn (); Kofman:2004yc (). We first note that the combined effect of ESPs in the directions perpendicular to the trajectory becomes exceedingly suppressed with increasing : since , the distribution of ESPs near the rim of the cylinder in Fig. 4 can be well approximated by a continuous ESP density; this approximation is similar to treating electric charges distributed on a macroscopic metal cylinder as a continuous surface charge density. Then, due to the symmetry of the configuration, the net force perpendicular to the trajectory vanishes, but a resistive force opposite to remains. We assume that the potential is such that the trajectory is straight over times-scales of interest 777If during the last sixty e-folds of inflation, the COBE normalization imposes , see forward to (59). We deduce that the trajectory needs to be straight over , which should be sub-Planckian. Hence, the terminal velocity needs to be (32) for internal consistency., that is over so that . Denoting with the field along the trajectory, the equation of motion for becomes
with from (31) and we took as well as , so that . For there is a strong exponential suppression , so that backreaction is negligible and trapped inflation is impossible, in agreement with our prior heuristic arguments. We recognize
as the aforementioned critical velocity of order above which backreaction from particles near ESPs becomes important. In the regime the drag increases sharply: assuming888Using the COBE normalization, see forward to (59), yields and the condition imposes an upper bound on the terminal velocity of (37) indicating that the potential needs to be steep enough for to provide a speed limit. In essence, this is the opposite of the requirement for a potential to support slow-roll inflation. in (33), we can compute the terminal velocity by equating the drag with the driving force caused by the potential gradient to
where we defined
Note that in the limit of ; as a consequence, the terminal velocity becomes independent of the slope of the potential. Further, keeping only the leading order terms in , we recognize that the terminal velocity is simply
in the large limit, in agreement with our heuristic arguments of section III.2. This is a remarkably simple result, providing a general speed limit on the landscape entirely determined by the distribution of ESPs and the coupling strength of the additional light degrees of freedom to the moduli, but independent of the slope of the potential.
If the classical trajectory is intertwined on small scales () the above derivation is inapplicable; we comment on this regime in Sec. III.5.
iii.2.2 Different Regimes of Trapped Inflation
Given a reasonably straight classical trajectory, there are two distinct regimes of inflation that could occur once becomes of order :
The classical trajectory is followed with a constant velocity close to ; this requires a large to prevent overshooting and extended oscillations around the terminal velocity.
The classical trajectory is followed with a velocity that oscillates around , never settling into the constant velocity regime before inflation ends.
To decide which type of inflation takes place, let us estimate the relevant time scale at which backreaction of particles near ESPs becomes stronger than the classical force due to the potential. In this section we neglect the expansion of the universe; as the velocity approaches from below we expand
before backreaction becomes important; here, we defined and set (particles start to be produced at ). Hence, the characteristic time-scale for the classical force is
On the other hand, the force due to backreaction increases in time as
Here, we used before solving for , and in the last step we kept leading order terms in only and used as well as the asymptotic form of the LambertW function for large arguments, . For
backreaction dominates before the classical potential has a chance to significantly alter the velocity. As a consequence, the classical trajectory is followed with a constant velocity close to once reaches , without prolonged oscillations around . (48) along with (46) imposes a lower bound on the dimensionality of field space, . If we take a steep potential , large999Constraints from IR cascading require , see Sec.III.4.1. couplings and ubiquitous ESPs so that and (32) is satisfied, we get . If the potential happens to be shallower, as in Sec.III.3, lower values of suffice, whereas smaller values of require slightly bigger values of . Note that regardless of the slope, we need at least , since we ignored terms of order throughout.
If is less then the minimal value imposed by (48), one might still get inflation of the second type: the velocity overshoots, even more light particles are produced since increases, backreaction sets in and the velocity is forced101010The velocity may actually turn around so that the trajectory becomes temporarily trapped near up until particles are diluted sufficiently by the expansion of the universe and increases again. again well below , mimicking the speed of cars during rush hour. Due to the changing velocity, we expect bumps in the power-spectrum as well as strong additional non-Gaussianities, rendering this regime less appealing. We leave this class of models to future studies.
iii.3 Phenomenology of Trapped Inflation in Higher Dimensions,
Consider a simple multi-field inflationary model, with and for all . If we denote the effective field along the trajectory with , we get . Let us consider a regime that does not yield slow-roll inflation, so that field values remain sub-Planckian (we use if not stated otherwise in this section). If the trajectory is in a -dimensional space () with ubiquitous ESPs (or more generally ESLs), we concluded in section III.2.1 that the speed of is bounded from above by the terminal velocity , see equation (38). Furthermore, in the large limit , independent of the slope of the potential; as a working example that satisfies all constraints in this paper, we choose and ; then the coupling constant is below the bound originating from IR-cascading (see forward to Sec. III.4.1), is well below the bound in (32) originating from the requirement that the distance over which the trajectory has to be straight be sub-Planckian, inflation lasts long enough, see forward to (55), and (37) is satisfied so that Hubble friction is subdominant. The dimensionality of moduli space needs to be above the bound imposed by (48) to guarantee that rolls at the terminal velocity 111111Since the COBE normalization sets the inflationary scale in (59), the mass becomes for ; then the bound on imposed by (48) becomes since . Note that the slope satisfies so that the terminal velocity is indeed a speed limit. One can further check that (49) as required for inflation to take place (-particles dominate the equation of motion for , but their energy remains subdominant.)., but the value of does not enter directly into observables as long as it is large enough ( is sufficient in this section). The Hubble slow evolution parameter becomes
where we used the COBE normalization in (59) to set the inflationary scale. Since should be much smaller than one, we find an upper bound for the terminal velocity
which coincides with the one in (32). Inflation takes place up until , that is for . The energy density at this instant is , corresponding to an upper bound on the reheating temperature121212The actual reheating temperature depends on several factors: if preheating via parametric resonances (unlikely, see Sec:B.2) and/or tachyonic instabilities takes place, the reheating temperature is close to this upper bound, which puts it at odds with bounds originating from thermal relics, such as gravitinos (, see Ellis:1984eq (); Khlopov:1984pf (); Kawasaki:2008qe () or Battefeld:2009sb () for a brief review). If reheating takes place via the old theory of reheating Dolgov:1982th (); Abbott:1982hn (); Kofman:1997yn (), the inflatons continue to oscillate around until becomes comparable to their decay rate(s) . An estimate of the reheating temperature is then given by . of where we used as a estimate for the number of relativistic degrees of freedom Kofman:1997yn () and reinstated . To put this another way, to guarantee a high enough temperature for nucleosynthesis, say , we need a terminal velocity of at least
which is a mild lower bound on .
The number of e-folds becomes
where we neglected . As a consequence, we can write
from demanding that in order to solve the usual problems (flatness, horizon, etc.) of the big bang. We chose the inter-ESP distance in hindsight to satisfy this observational constraint. For different potentials, the numerical pre-factor in (54) changes, but the scaling remains. Note that is smaller than in the slow-roll case where .
iii.3.1 Observational Consequences
Let us focus on adiabatic perturbations, which are properly described by (the Fourier modes of) the Mukhanov variable with (see Bassett:2005xm () for a review). Here is the curvature perturbation on uniform density surfaces. We are interested in evaluating its power-spectrum, which gets imprinted onto fluctuations of the cosmic microwave background radiation after inflation. If evolves slowly, which is the case during inflation, we can approximate
where is conformal time, and , solve the equation of motion for
In the end, the amplitude of the power-spectrum at horizon crossing () becomes
to leading order in , which is set by the COBE normalization Komatsu:2008hk (). This means we need an inflationary scale of
Further, the scalar spectral index reads
where is defined by
and we used . The power-spectrum is closer to scale invariance than in the slow-roll case, since .
In the simplest realization of trapped inflation in higher dimensions with , the Hubble slow evolution parameter is suppressed by an additional power of (the number of e-folds) if compared to a slow-roll setup. As a consequence, the scalar power-spectrum is closer to a scale invariant one, in tension with the WMAP5 results Komatsu:2008hk () at the level, and the tensor to scalar ratio is unobservably small. Thus, one could firmly rule out this model of trapped inflation in higher dimensions if error bars around tighten or primordial gravitational waves of order be observed.
However, we worked with a simple potential resulting in a straight trajectory. In general, one might not expect such a smooth potential on the landscape, but one leading to a curved trajectory. In such a case we can not neglect isocurvature perturbations, since they feed into the adiabatic mode with every twist and turn, leading to additional deviations from scale invariance and potentially observable non-Gaussianities. Further, if the trajectory twists and turns on small scales, our derivation of the terminal velocity is not applicable any more. We comment on this regime in Sec.III.5.
We would like to emphasize that the potential does not influence the speed of the effective field along the trajectory in trapped inflation if is large. Hence, less tuning of the potential is needed, but we require a high dimensionality of field space (so that backreaction increases sharply at and (48) is satisfied) and ubiquitous ESPs/ESLs (, so that is sufficiently small, and inflation lasts long enough, see (55)). Further, we impose to prevent perturbations from IR-cascading to dominate, see Sec. III.4.1. Other conditions imposed throughout are
the trajectory should not be strongly curved over a distance covered in a few Hubble times, (32),
the slope of the potential needs to be large enough to guarantee that Hubble friction can be ignored, (37),
the potential energy needs to dominate over the kinetic one, which in turn needs to dominate over , to guarantee that the universe inflates (49),
the reheating temperature needs to be large enough for nucleosynthesis to take place, (52).
These requirements are all satisfied in the simple model presented in Sec. III.3.
Another advantage of trapped inflation over ordinary slow-roll inflation is its independence on the initial velocity, a quantity that requires fine tuning in many slow-roll models. If the initial velocity happens to be bigger than , a plethora of light states is produced rapidly whose backreaction causes a sudden decrease of well below , potentially leading to temporary trapping near the initial location. Once -particles are diluted by the expansion of the universe, the speed picks up again due to the classical potential and settles into the constant velocity regime of trapped inflation if is large enough.
iii.4 Additional Observational Signatures
We would like to comment on two additional observational signatures brought forth by particle production during inflation.
A crucial aspect of trapped inflation is the ongoing production of particles during inflation near ESPs. As -particles become heavy, they re-scatter off the homogeneous inflaton condensate and cause a cascade of inflaton fluctuations in the IR Barnaby:2009mc (); Barnaby:2009dd (), leading to observable signatures in the power-spectrum. For particle production at a single ESP, an additional bump like feature at results, which can be approximated by Barnaby:2009mc (); Barnaby:2009dd ()
where the amplitude scales with the coupling ; for example, comparison with lattice field theory in a simple, one dimensional inflationary model yields for . Note that the shape in (66) is only reliable for . Bumps with an amplitude of up to of the primordial power-spectrum on scales probed by current CMBR experiments are still consistent with observations Barnaby:2009dd (), leading to an upper limit on the coupling of .
However, in the case of trapped inflation as investigated in Sec.III.3, particles are produced at a multitude of ESPs. The resulting superposition of bumps leads to a constant contribution to the power-spectrum. It is expected that this contribution, albeit scale invariant, is strongly non-Gaussian and thus ruled out as the dominant one Barnaby:2009dd () 131313A computation of Non-Gaussianities from IR cascading is in preparation by the authors of Barnaby:2009dd (); observational bounds on non-Gaussianities might yield stronger constraints on particle production during inflation than considerations based on the power-spectrum..
In Barnaby:2009dd () observational bounds on the amplitude of features were given under the assumption that all ESPs lie on the trajectory, as in one dimensional monodromy inflation Silverstein:2008sg (), resulting in a superposition of bumps with identical amplitudes. For a large ESP or bump density ( with ), it was found that the constant contribution scales as ; consequently, the maximally allowed amplitude is proportional to , roughly , which provides and upper bound on .
This may sound alarming, since trapped inflation in higher dimensions relies on a multitude of ESPs, roughly , at any given time. There is however a crucial difference, rendering the conclusions of Barnaby:2009dd () only indirectly applicable. The -particles that dominate the energy density in Sec III.2.1 are produced at ESPs a distance away from the trajectory. As a consequence, particle production at any single ESP is weak; to be concrete, the particle number at each ESP is exponentially suppressed