Staggered Multi-Field Inflation
We investigate multi-field inflationary scenarios with fields that drop out of the model in a staggered fashion. This feature is natural in certain multi-field inflationary setups within string theory; for instance, it can manifest itself when fields are related to tachyons that condense, or inter-brane distances that become meaningless when branes annihilate. Considering a separable potential, and promoting the number of fields to a smooth time-dependent function, we derive the formalism to deal with these models at the background and perturbed level, providing general expressions for the scalar spectral index and the running. We recover known results of e.g. a dynamically relaxing cosmological constant in the appropriate limits. We further show that isocurvature perturbations are suppressed during inflation, so that perturbations are adiabatic and nearly Gaussian. The resulting setup might be interpreted as a novel type of warm inflation, readily implemented within string theory and without many of the shortcomings associated with warm inflation.
To exemplify the applicability of the formalism we consider three concrete models: assisted inflation with exponential potentials as a simple toy model (a graceful exit becomes possible), inflation from multiple tachyons (a constant decay rate of the number of fields and negligible slow roll contributions turns out to be in good agreement with observations) and inflation from multiple M5-branes within M-theory (a narrow stacking of branes yields a consistent scenario).
- I Introduction
- II Staggered Inflation: Fields Becoming Obsolete during Inflation
- III Applications
- IV Conclusions
An inflationary epoch of the early universe is widely accepted as the most efficient mechanism to solve the flatness and the horizon problem, while providing a nearly scale invariant spectrum of scalar perturbations, in agreement with observations. However, the embedding of inflation driven by a single scalar field within string theory (presently, the only known self-consistent theory of quantum gravity) has proven to be challenging, since it requires an extremely flat potential. In addition, the presence of many dynamic fields in string theory renders single field models less appealing because all but one degree of freedom need to be already stabilized at the onset of inflation. Though far from being simple or generic, a partially successful implementation is the KKLMMT construction Kachru:2003sx ().
To ameliorate the problem of fine tuning the potential, Liddle et. al. proposed assisted inflation in Liddle:1998jc () (see also Malik:1998gy (); Kanti:1999vt () and follow ups), a certain type of multi-field inflation whereby the presence of many fields increases the Hubble friction, allowing for steeper potentials to still drive slow roll inflation. One shortcoming, however, is the need for fine tuned initial conditions for particular potentials, especially in the absence of an attractor solution (for instance in N-flation Dimopoulos:2005ac (); see Calcagni:2007sb () for a more general stability analysis of assisted inflation). The presence of many degrees of freedom in string theory has, over the years, galvanized the emergence of a variety of models implementing assisted inflation; these include inflation from multiple tachyons Majumdar:2003kd (), from multiple M5-branes within M-theory Becker:2005sg () or from axions Dimopoulos:2005ac () among others, see e.g. Cline:2005ty (); Ward:2007gs (); Grimm:2007hs (); Panigrahi:2007sq (); Piao:2002vf (); Mazumdar:2001mm () for a small selection.
To make predictions in these models, it is common to introduce a single effective degree of freedom, visualized as the length of the trajectory in field space, see e.g. the review Wands:2007bd (). Utilizing this approach correctly recovers adiabatic perturbations, but not isocurvature/entropy modes. The latter ones can be pictured as perturbations perpendicular to the trajectory and develop whenever several degrees of freedom are present Gordon:2000hv (). Although non-Gaussianities may be produced (see the review Wands:2007bd () for details), it is possible to show that they are generically slow roll suppressed Battefeld:2006sz (); Battefeld:2007en () (see however Byrnes:2008wi ()). With this in mind, is it then feasible to discriminate a given multi-field model from its single field analog?
In this article, we investigate a, so far largely ignored (see however Ashoorioon:2006wc ()), discriminating property of multi-field models, namely the possibility that fields decay during inflation. By ”decay” we mean that individual fields suddenly become obsolete, while their energy is converted into other forms, such as radiation. This is a widespread feature of models within string theory: consider, for instance, an association of inflatons with distances between branes that are located in some internal space. If a brane annihilates during inflation, for example as a result of its dissolution into a boundary brane or a collision with an anti-brane, the distance to the just mentioned brane becomes meaningless. Of course, energy does not vanish, but is converted into a different type during the annihilation. A further example is inflation driven by tachyons, which can condense during inflation in a staggered fashion.
The disappearance of fields causes an additional decrease of the potential energy during inflation, which can be even more important than the reduction of potential energy induced by slow roll. Hence, leading order corrections to observable parameters that are sensitive to the slow roll parameters, such as the scalar spectral index, are possible. Indeed, one can construct models of inflation entirely without a slow roll phase, similar to inflation without inflatons as proposed in Watson:2006px ().
In order to compute these effects we assume a simple separable potential and promote the number of fields to a time dependent function. We further smooth out so that we can introduce a continuous decay rate . This approach is only justified if the number of fields and are large enough so that several fields disappear in any given Hubble time. Further, any signal due to a rapid drop in the potential energy, such as a ringing in the power spectrum or additional non-Gaussianities Covi:2006ci (); Ashoorioon:2006wc (); Chen:2006xjb (), cannot be recovered by this approach. To guarantee energy-momentum conservation (), we are forced to introduce an additional component of the total energy momentum tensor that takes over the energy of the disappearing fields. The ratio of its energy density to the one in the remaining inflatons yields an additional small parameter appearing alongside the usual slow roll parameters. As a consequence, the equation of motion of the effective field is modified. The resulting set-up is reminiscent of warm inflation Berera:1995ie (), and might indeed be seen as a new, less problematic implementation of warm inflation within string theory.
At the perturbed level, we consider adiabatic and entropy perturbations and show that the latter ones are suppressed in the models of interest. Focusing on the Mukhanov variable, we derive the scalar spectral index and its running for general decay rates and number of fields. We recover the usual slow roll result as well as Watson:2006px () (a dynamically relaxing cosmological constant) in the appropriate limits, but we also find leading order corrections to the slow roll result in general. The running remains second order in small parameters and is, therefore, too small to be observed.
To exemplify the applicability of the formalism we make a detailed study of three concrete models; firstly, as an instructive toy model, we consider assisted inflation with exponential potentials and a nonzero, constant decay rate, which we insert by hand. provides a graceful exit to inflation at the cost of requiring somewhat flatter potentials or more fields so that the model remains consistent with observations. If the decay rate is too big () the scalar spectral index becomes unacceptably red, but all smaller rates work well. Secondly, we investigate inflation from multiple tachyons, as proposed in Majumdar:2003kd (). Here, tachyons get displaced from the top of their potential by thermal or quantum mechanical fluctuations, causing them to condense during inflation. We find that the best motivated and least fine tuned setup consists of a constant decay rate and tachyons close to the top of their potential so that their slow roll evolution yields negligible contributions to observable parameters. The resulting spectral index is , where is the number efolds at which we evaluate . This is in good agreement with the observations of the CMBR, Komatsu:2008hk (). Lastly, we look at inflation from multiple M-5 branes within M-theroy Becker:2005sg (). Here, inflatons are associated with the inter-brane distance between M5-branes located within an orbifold . Whenever a brane comes close to a boundary brane it dissolves into the boundary via a small instanton transition. Given the model’s parameter ranges, which encompass, for instance, the concrete potential and the maximal number of branes, we show that inflation comes to an end within a few efolds after the first outermost brane disintegrates. Hence, only an initial narrow stacking of branes (involves fine tuning) permits a consistent scenario, and the usual slow roll expressions apply. Based upon these findings, cascade inflation as investigated in Ashoorioon:2006wc () is irrelevant for the cosmological scales that are observed in the CMBR, given the parameters put forward in Becker:2005sg ().
It is evident that whether or not corrections due to are important is model-dependent, ranging from being the primary ingredient, as in the tachyon case, to being not pertinent, as in the M5-brane case. Thus, the many implementations of multi-field inflation within string theory should be thoroughly re-investigated.
The outline of this paper is as follows: In section II, we derive the formalism to deal with dacaying fields, at the background level (sec. II.1) and the perturbed one (sec. II.2). We compute the spectral index and the running in section II.2 and II.3. Along the way, we draw comparisons to warm inflation in section II.1.1 and comment on isocurvature modes as well as non-Gaussianities in section II.4. We then shift gears and focus on applications, that is we investigate concrete models: assisted inflation (sec. III.1), inflation from tachyons (sec. III.2) and inflation from multiple M5-branes (sec. III.3); summaries of the conclusions for each model can be found at the end of their respective subsection. We conclude in section IV.
Ii Staggered Inflation: Fields Becoming Obsolete during Inflation
Consider scalar fields with canonical kinetic terms and a separable potential 111Note that is not the SUSY superpotential., so that the action reads
For ease of notation, we set the reduced Planck mass equal to one throughout . Assume that the fields all evolve according to the same potential and that they start out from an identical initial value 222These assumptions simplify our treatment considerably, but are not crucial and could be relaxed while retaining the effect of staggered inflation., so that
In this case, an effective single field model with and describes correctly the inflationary phase (if is sufficiently flat) as well as the production of adiabatic perturbations (we comment on entropy perturbations in section II.4). To guarantee slow roll inflation, we demand
where a prime on or denotes a derivative with respect to or respectively. So far, this is merely a simple model of assisted inflation Liddle:1998jc (), see also Malik:1998gy (); Kanti:1999vt () and follow ups.
Given this setup, we would like to investigate the consequences of individual fields dropping out of the model, more or less instantaneously. By dropping out we mean that a field decays while its energy is converted into a different form, for example radiation. This may seem artificial at first glance, but it is actually quite generic in multi-field models of inflation within string theory 333See Chialva:2008zw () for a related application within chain inflation.. For instance, in the model of Becker:2005sg () (see also Krause:2007jr (); Ashoorioon:2006wc ()) the inflatons are related to the distances between adjacent M5-branes. These branes are located along an orbifold and slowly separate from each other in the orbifold direction, corresponding to the inflationary phase in the effective four dimensional description. Since the orbifold is quasi-static during this regime, the outermost branes will collide with the orbifold-fixed planes at some point in time, dissolving into the boundaries through small instanton transitions. Thus, the degrees of freedom associated with the distances to these just dissolved branes become obsolete during inflation. Of course, the energy associated with the branes does not vanish, but gets converted to other degrees of freedom, such as radiation. Another example is inflation driven by multiple tachyons as proposed in Majumdar:2003kd (); here, tachyons roll slowly away from the top of their potential, but fluctuations may displace a given field far enough to condense suddenly, again making obsolete this degree of freedom during inflation. We come back to these two concrete models later on, after developing the formalism to deal with decaying fields. For additional multi-field models see e.g. Cline:2005ty (); Ward:2007gs (); Grimm:2007hs (); Dimopoulos:2005ac (); Panigrahi:2007sq (); Piao:2002vf (); Mazumdar:2001mm () and Wands:2007bd () for a review.
It is worth stressing at this point that the disappearance of a given field in the models considered in this paper is a sudden, but continuous process. This means potential problems associated with first order phase transitions, as discussed in some detail in Watson:2006px (), are absent. To model the disappearance of individual fields we promote to a time dependent function. We further smooth out 444It should be noted that by smoothing out , we will not be able to recover additional features in the power-spectrum that are directly related to sudden drops in the potential. For instance, features could consist of a ringing or additional non-Gaussianities Covi:2006ci (); Ashoorioon:2006wc (); Chen:2006xjb (). However, such signals depend crucially on the detailed physics of the fields’ dissapearance, a rather badly understood and model dependent issue for the current generation of setups. As a consequence, these features will be quite hard to estimate within an effective single field description and we will not address them further in this article. so that we can introduce a continuous decay rate , which is to be determined from the underlying model. This smoothing is the key simplifying assumption in our approach, and it is only viable if the number of fields is large and the decay rate is such that within any given Hubble time a few fields become obsolete. For the cases we are interested in, this rate is small compared to the Hubble parameter during inflation (inflation ends quickly otherwise). This means we can introduce the small parameter
The time dependence of induces an additional decrease in the energy of the effective inflaton. To be concrete, the continuity equation of needs to be modified to account for the energy loss in the inflaton sector due to decaying fields , so that
Here and throughout the subscript denotes the entire inflaton sector. It is worthwhile mentioning that the individual evolve according to the standard slow roll equation of motion , as long as they are present. To retain , we must allow for an additional component within the energy budget 555Note that , so that energy always flows from the inflaton field sector into . satisfying
We use the subscript “”, since we have radiation in mind ( with ); this choice seems most natural to us, considering the models we are interested in, but to remain as general as possible we keep arbitrary throughout.
The condition implies that makes up only a small fraction of the total energy density during inflation. Thus, we may introduce the small parameter
Naturally, is not independent of , and one can show under mild assumptions that during inflation (see below). Further, using slow roll of the individual fields , one can show that the total energy and pressure of the effective inflaton obey . This, along with and the definitions above, leads to
during slow roll (the “” always denotes equality to first order in small parameters such as or ). Taking the derivative with respect to time of the Friedmann equation and using (11) as well as (12), yields the Hubble slow roll parameter
We have carefully avoided the use of the Klein Gordon equation for , since it gets modified by the presence of . To derive this modification, one can use in the continuity equation (11) and make the usual slow roll approximations, or take the time derivative of directly with so that
where we introduced the short hand notation
As expected, the usual slow roll equation of motion is recovered from (15) in the limit . The limit , that is , corresponds to a dynamically relaxing cosmological constant, which is discussed in great detail in Watson:2006px ().
During inflation, , as well as change very slowly, so that (12) can be integrated to with . This means that the additional component of the energy budget approaches a scaling solution for which and . Thus, we can use
ii.1.1 Comparison to Warm Inflation
At this point, we would like to comment on similarities to warm inflation Berera:1995ie (), where the scalar field’s interaction with other particles (through which the scalar field transfers some of its energy into a thermal bath) prevents the temperature from rapidly reaching zero. Within warm inflation, the motion of the scalar field is described by the modified Klein-Gordon equation (assuming slow roll)
The extra friction term represents an additional energy loss of the scalar field stemming from particle creation. This equation needs to be compared to our equation (15): in our case, an increase/decrease of the right hand side is present, depending on the sign of , while in (18), the right hand side is always decreased by .
A more concrete realization of warm inflation based on thermal viscosity has been (critically) examined in Yokoyama:1998ju (). In this study, part of the inflaton’s energy is converted into radiation through a viscosity term. As a consequence, one can show that the energy density satisfies
with . Comparing (19) to our case (8) we see a similar modification: an additional positive term, potentially counterbalancing the dilution of due to redshifting; however, the origin of this term differs: in our case it is due to the transfer of potential energy, whereas that of (19) arises from infusing kinetic energy.
Though the friction term in (19) could at first glance be large enough to allow for successful warm inflation, a more careful examination by Yokoyama and Linde revealed Yokoyama:1998ju () that warm inflation is not feasible in this framework, since changes significantly over the relaxation time of the relevant particles, violating an adiabaticity condition required for the validity of (19). Nevertheless, albeit the presence of many problems like the one just mentioned, warm inflation remains an active field of research.
Staggered multi-field inflation, as introduced in the present paper, might be seen as an independent (less problematic) realization of warm inflation, which can be embedded into string theory.
ii.2 The Scalar Spectral Index
In order to compute the scalar spectral index we focus on adiabatic perturbations, that is, for the time being we neglect isocurvature perturbations. The latter ones arise due to the presence of the many scalar fields and , but one can show that they are suppressed during inflation. The applicability of this approximation is discussed in section II.4.
The canonical degree of freedom that diagonalizes the action of adiabatic scalar perturbations is the Mukhanov variable Mukhanov:1990me (). It is related to the curvature perturbation on uniform density hyper-surfaces via
Here, with and is the equation of state parameter, while is the adiabatic sound speed, which becomes on large scales. In terms of , the power spectrum is given by
and the scalar spectral index is
The Mukhanov variable satisfies the simple equation of motion Mukhanov:1990me ()
where a prime denotes a derivative with respect to conformal time , . If the Hubble slow roll parameter in (13) is evolving slowly, which is the case during inflation, we can approximate
treating as a constant and imposing the Bunch-Davies vacuum at (), we get
where is an irrelevant phase factor, . Hence, the curvature perturbation reads
which can be expanded on large scales to
Plugging this into (23) and taking the logarithmic derivative, one can read off the scalar spectral index to
Thus, we only have to compute and identify in order to get . Since is second order in small parameters, we need to focus on from (22) only. To leading order in and , the equation of state parameter reads
In order to take the derivatives with respect to conformal time, we need
where we introduced
to leading order in small parameters (we assume that is of the same order as ). Using from (26) so that we can read off . The resulting scalar spectral index is
This is our first major result. In the limit , so that and , we recover the slow roll result . On the other hand, if (that is ) and , we recover the case of a dynamically relaxing cosmological constant of Watson:2006px () .
The running can be computed by applying666At horizon crossing , resulting in ; we use that is evolving slowly during inflation, that is . to (39). Using again and
In the limit , we recover the standard slow roll result777Note a recurring sign mistake in the literature, e.g. in the popular review Lyth:1998xn () or textbook Liddle:2000cg (); see Kosowsky:1995aa () for the correct expression. . On the other hand, if , the running reduces to . In any event, the running remains second order in small parameters, rendering it un-observably small.
ii.4 Isocurvature (Entropy) Perturbations
So far we focused on adiabatic perturbations only, neglecting entropy perturbations entirely. To check if entropy perturbations are indeed negligible, we follow Malik:2002jb () (see also Kodama:1985bj (); Watson:2006px ()). The perturbation of the total pressure is in general
where is the entropy density, (we use cosmic time in this section in order to avoid confusion of this with conformal time) and is the adiabatic sound speed, which reduces to on large scales. The non-adiabatic pressure perturbation may thus be defined as
In a multi-component fluid, there are two contributions to , a relative one between the fluid components and an intrinsic one within each fluid Malik:2002jb (). In our case, we have two main components, the effective inflaton (composed of fields) with , and the additional component , for instance given by radiation, with a constant equation of state parameter .
Let us first discuss , which can be written as Malik:2002jb ()
where and the relative entropy perturbation is defined as
We have already seen in the discussion before (17) that approaches a scaling solution during inflation for which is small (). Since , the relative non-adiabatic pressure perturbation is heavily suppressed and can safely be ignored during inflation 888Note that remains small and finite in the limit of small , because the curvature perturbations and remain small; see Watson:2006px ()..
Second, consider the intrinsic contributions
Since we have and . Therefore , that is the intrinsic non-adiabatic pressure vanishes identically. However, for the effective inflaton the case is less clear, since its equation of state (and thus the intrinsic sound speed) changes and is actually composed of components. We first note that the individual fields approach an equation of state during inflation, so that the equation of state parameter becomes nearly constant for each of the fields. Thus we have where . Therefore, the intrinsic non-adiabatic pressure within each inflaton field is . Second, the components’ relative non-adiabatic pressure perturbations contribute to . But if we look at the relative non-adiabatic pressure contributions between the fields we conclude that since . Thus, the total non-adiabatic pressure is negligible during inflation and we are justified to focus on adiabatic perturbations 999There is one caveat to the above arguments: during the short intervals when fields decay the evolution of the decaying field is rapid and its equation of state changes – it would indeed be interesting to investigate the productions of isocurvature perturbations (and non-Gaussianities) during these instances, which can not be recovered with our approach since we employ a smooth Ashoorioon ().. These are correctly recovered by focusing on the Mukhanov variable , as we did in section II.2.
One further comment might be in order: due to the suppression of entropy perturbations during inflation, we do not expect any large additional non-Gaussianities (NG) caused by . Further, since multi-field inflationary models generically yield comparable NG to their single field analogs Battefeld:2006sz (); Battefeld:2007en (), and NG are suppressed during slow roll inflation (see however Byrnes:2008wi ()), we do not expect any measurable NG within the setups discussed in this paper. One caveat to this argument consists of the short intervals whenever one of the inflaton fields decays (see e.g. Chen:2006xjb () for NG from steps in a potential in single field inflation). We cannot exclude the production of NG at these instances, but we do not anticipate them either, because the process is very similar to preheating and generically, preheating (e.g. instant preheating) does not cause large NG Enqvist:2005qu () (see however the possibility of larger NG in tachyonic preheating Enqvist:2005qu (); Barnaby:2006cq (); Barnaby:2006km ()). Nevertheless, given a better microphysical understanding of how inflatons become obsolete, e.g. by investigating the brane annihilation in Majumdar:2003kd (); Becker:2005sg (), one can and should check the validity of this expectation.
We would like to compute the scalar spectral index in (39) and the running in (II.3) within a couple of models. First, we extend assisted inflation with exponential potentials Liddle:1998jc () by incorporating a non-zero , which provides a graceful exit of inflation; this phenomenological model has the advantage of being instructive and simple. Next, we consider two concrete models Majumdar:2003kd (); Becker:2005sg () which have the feature of decaying fields during inflation already build in, at the price of being more complicated to treat. Our approach consists of extracting the potential slow roll parameters , and , as well as with in order to apply (39) and (II.3).
iii.1 Staggered Assisted Inflation
Consider the original proposal of assisted inflation Liddle:1998jc (), where the scalar fields have identical exponential potentials
so that the potential for the single effective field reads
with (we assume identical initial values for all ). Power law inflation () results for large enough , which can be achieved even with steep potentials if . Note that the single field solution is an attractor during inflation Malik:1998gy () (see Calcagni:2007sb () for a general discussion of stability in multi-field inflation). The slow roll parameters in the above model are
However, the above model has a graceful exit problem: inflation never ends because and are constant. This problem can be alleviated by introducing a non-zero , so that the number of fields decreases during inflation. Let’s for simplicity take a constant rate so that . Consequently, inflation comes to an end when becomes of order one, at which point decreases rapidly during a Hubble time so that the assistance effect diminishes.
Since a shift in the individual fields can be absorbed into a redefinition of , we can set at efolds before the end of inflation, without loss of generality. Hence , and the scalar spectral index in (39) becomes 101010A shift in the fields causes both, a shift in since changes, and a shift in since changes; however, an observable such as the scalar spectral index remains unaffected.
where we used and . Similarly, the running in (II.3) reads
where we also used and .
These predictions differ from the corresponding slow roll ones: the additional energy loss in the inflaton sector due to causes a redder spectrum with a running that is second order in the epsilons (such a running is well below current observational limits 111111Since the running is generically second order in the slow roll parameters and , we will not comment on it further in section III.2 and III.3.). The physical reason for the difference in the spectrum is the smooth graceful exit caused by a non-zero decay rate, resulting in . Note that for a sharp exit from inflation to the reheating era, which could be modeled by a step function with , we anticipate no corrections.
In figure 1, we plot lines in the - plane, showing the interval of WMAP5 Komatsu:2008hk () . Given , it is evident that in order to fit from observations in assisted inflation we need a smaller . To put it another way, the smooth graceful exit of inflation introduced by comes at the price of either requiring flatter potentials or more fields. Further, the spectral index lies outside the region if is too large (), even if were identical to zero. This means the half-time needs to be considerably larger than the Hubble time during inflation, in order to be consistent with observations ().
In the phenomenological model above, we inserted a non-zero decay rate by hand. As we shall see in the next sections, more concrete models within string theory actually force us to have , rendering the above model more natural.
iii.2 Inflation from Multiple Tachyons
In Majumdar:2003kd (), brane anti-brane pairs () where considered, giving rise to tachyons due to a gauge symmetry. However, these tachyons are generically all coupled to each other, making them not well suited for assisted inflation. To alleviate this problem, Davis and Majumdar proposed to focus on the Abelian part of , resulting in uncoupled tachyons 121212Our notation differs from Majumdar:2003kd (), where tachyons are denoted by and time by ., ; admittedly, this choice is to some extent unphysical Majumdar:2003kd (), but offers an instructive model. Within this setup, the potential has the form Ohmori:2001am ()
which is valid in close proximity to . Here, and Ohmori:2001am (), but the brane tension is model dependent. For large sufficient inflation can result if the tachyons are initially close to zero. For simplicity, we assume that they all start out from the same initial value, in line with our assumptions in section II.
Whenever a tachyon gets displaced far enough from the origin (be it due to its slow roll evolution or a dislocation caused by either thermal or quantum mechanical fluctuations Majumdar:2003kd ()) the perturbative potential in (54) becomes unreliable, and the tachyon condenses quickly. This condensation corresponds to the annihilation of a brane anti-brane pair. Inflation ends when all tachyons condense. This condensation is also expected to be responsible for reheating, but a concrete study is lacking in the literature. Giving the model the benefit of the doubt, we assume that whenever a brane anti-brane pair annihilates, its energy is indeed converted into some type of relativistic matter with (from the four dimensional point of view). Thus, the model of Majumdar:2003kd () is exactly of the type we examined in section II.
Next, we extract the slow roll parameters as well as specify the rate at which the tachyons condense in order to compute . If all tachyons condense at the same time, we have during inflation and the standard slow roll expressions apply. However, given the sensitivity of condensation to thermal and quantum mechanical dislocations of the tachyons, a staggered fashion of condensation is expected. In Majumdar:2003kd () three types of such a staggered condensation were proposed:
Tachyons condense serially so that and ( in the notation of Majumdar:2003kd ()). This is the case if tachyons condense at wildly different times.
Tachyons condense in a staggered fashion, but a handful survive and drive an extended phase of slow roll inflation, rendering again during the cosmological relevant phase of inflation. (We do not examine this case further.)
Because the applicability of the potential (and thus slow roll) is questionable for large values of , we first examine cases one and two in a simplified setup with all tachyons sitting very close to the top of their potentials (, ). Whenever a tachyon gets displaced by a fluctuation, it is assumed to condense immediately. In later sections, we incorporate slow roll.
iii.2.1 , Negligible Slow Roll Contributions
Similar to section III.1, we consider a constant condensation rate so that the number of tachyons decreases exponentially in time
where we take at efolds before the end of inflation. Once the number of fields is depleted, that is once , inflation ends. To be concrete, we take . Then, using and , the number of efolds becomes
we get from (39)
which is within the error bars of WMAP5 Komatsu:2008hk ().
If we fine-tune the brane tension such that vanishes at the minimum, , we obtain
Thus, for , we need tachyons, which is somewhat large; however, for we achieve the desired amount of inflation and a spectral index within observational bounds with a few hundred tachyons.
It should be noted that neglecting slow roll might actually be the best motivated case: we only trust the tachyon potential close to ; then the potential is indeed very flat so that can be neglected 131313Note that in the present case the amplitude of perturbations (set by the COBE normalization) is determined by the decay rate, that is (just as in Watson:2006px ()), and not by .. As soon as an individual tachyon gets dislodged a bit, it should quickly condense and drop out of the model, leading to the above estimate for .
Here, the number of fields is decreasing linearly
so that inflation ends around . Analogous to (58), the number of efolds becomes
so that . Since we get from (39)
smaller than in the case and close to the boundary of WMAP5 Komatsu:2008hk (). Tuning the brane tension again to , we obtain
As a result, in order to achieve the desired efolds of inflation with a few hundred fields we need .
iii.2.2 , with Slow Roll
In case all tachyons condense at once we have during inflation and the usual slow roll expressions apply. In order to provide a concrete example we assume , tune the brane tension to and take
even for , which is stretching the applicability of the potential 141414A value of corresponds to individual field values of order if . Since in order for (67) to apply, we are reaching the limit of its applicability..
Inflation ends when either or become of order one. For our potential, first at
The value of at efolds before the end of inflation can then be computed numerically from
Given , the slow roll parameters and follow straightforwardly. Since the only free parameter is the number of fields, we can fine-tune to yield the desired scalar spectral index. To be concrete, for we get and so that , resulting in , matching WMAP5 Komatsu:2008hk ().
If the full tachyon potential should become steeper before or the tachyons condense collectively at some smaller value , the corresponding shifts to lower values, causing and to decrease further. As a consequence, even less fields are needed to match .
iii.2.3 , with Slow Roll
Based on the last two sections, we expect the contributions to in (39) by the slow roll parameters and to be of comparable magnitude if the tachyons do not start out too close to the origin and we have as well as a decay rate of order (or ). For even smaller rates, becomes negligible and the usual slow roll results apply, whereas larger values cause a premature end of slow roll inflation. On the other hand, if the tachyons start out very close to the origin, we can neglect the slow roll contribution altogether, just as in section III.2.1. To quantify these statements, we take the slow roll setup with as well as fields and slowly turn on in order to show its effect on .
First, we determine the end of slow roll inflation, which occurs whenever , or becomes of order one (or itself becomes of order one). Assuming is still valid at the end of inflation, one can show that before and if , where
Since for fields and we are primarily interested in small decay rates, we determine from . Given , we can determine the time , and thus the field value from the requirement
using and a numerical solution to the equation of motion for in (15). Once is known, we can straightforwardly compute , and at efolds before the end of inflation. Using these in (39), we plot the resulting spectral index over in figure 2. As expected, the decay rate becomes important for , quickly driving the scalar spectral index outside the observationally favored region Komatsu:2008hk () as increases further.