Abstract
The power spectrum of Mtheory cascade inflation is derived. It possesses three distinctive signatures: a decisive power suppression at small scales, oscillations around the scales that cross the horizon when the inflaton potential jumps and stepwise decrease in the scalar spectral index. All three properties result from features in the inflaton potential. Cascade inflation realizes assisted inflation in heterotic Mtheory and is driven by nonperturbative interactions of M5branes. The features in the inflaton potential are generated whenever two M5branes collide with the boundaries. The derived smallscale power suppression serves as a possible explanation for the dearth of observed dwarf galaxies in the Milky Way halo. The oscillations, furthermore, allow to directly probe Mtheory by measurements of the spectral index and to distinguish cascade inflation observationally from other string inflation models.
HUTP06/A0025
Power Spectrum and Signatures for Cascade Inflation
Amjad Ashoorioon^{1}^{1}1 and Axel Krause^{2}^{2}2
Department of Physics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA
George P. & Cynthia W. Mitchell Institute for Fundamental Physics,
Texas A&M University, College Station, TX 77843, USA
1 Introduction
For a long time it seemed difficult to connect inflation to stringtheory. In its lowenergy approximation stringtheory is described by supergravity. Inflation based on the Fterm potentials of 4dimensional supergravities resulting from string/Mcompactifications suffers from a large slowroll parameter . The origin of this problem traces back to the appearance of the Kählerfactor in the Fterm potential. New possibilities to address this problem arose with the advent of Dbranes [1]. They allowed to identify the inflaton with open string modes such as the geometrical distance between two Dbranes [2].
An inflaton requires a very shallow potential. Hence, a priori, moduli serve as natural candidates. To provide them with a nontrivial potential supersymmetry needs to be broken which can be done in various ways. One might add anti Dbranes to the open string sector [3] or supersymmetry breaking fluxes to the closed string sector [4]. Also the inclusion of nonperturbative instanton effects leads to spontaneous supersymmetry breaking in the lowenergy supergravity [5]. Assuming just a single inflaton, the task for deriving inflation from stringtheory becomes then finding a way of breaking supersymmetry which leaves the inflaton with a sufficiently flat potential while endowing all other moduli with steep stabilizing potentials. All standard methods of breaking supersymmetry generate, however, steep potentials, not flat ones. One way to generate a flat inflaton potential nevertheless is to study braneantibrane inflation in warped backgrounds with the inflaton being identified with the braneantibrane distance [3]. Warped geometries arise in the presence of branes and fluxes. The eventual stabilization of the volume modulus, however, modifies the inflaton potential and renders it too steep for inflation unless finetuning is applied [6].
Here, we focus on an alternative mechanism to generate inflation in M/stringtheory, the multi brane inflation proposal [7], [8] (see also [9]). One starts with a multi inflaton scenario associating one inflaton with each brane position. The presence of several branes is indeed generically enforced by tadpole cancellation conditions. The interesting advantage of this mechanism lies in the fact that the potentials for the individual inflatons need no longer be flat. The reason is that the Hubble friction experienced by every inflaton becomes large – simply by increasing the number of inflatons – regardless of the steepness of the potentials. This had first been pointed out in [10] in the context of 4dimensional FriedmannRobertsonWalker (FRW) cosmologies based on exponential potentials which generate powerlaw inflation. The premise under which this mechanism operates is the suppression of strong crosscouplings among the inflatons. This suppression is given in multi brane inflation models since interactions between nonneighboring branes which could generate crosscouplings are suppressed by longer distances. In Mtheory cascade inflation [8] there is an exponential suppression of such crosscouplings since interactions between the relevant M5branes arise from nonperturbative open M2instantons.
In this work, after highlighting the needed ingredients of Mtheory cascade inflation, we focus on the determination of its power spectrum and the resulting observable signatures. Beyond demonstrating the compatibility of the power spectrum with present cosmological constraints, we find that it exhibits three distinctive signatures – power suppression at small distances, stepwise decrease in the spectral index and oscillations in the spectrum. The power suppression which follows in cascade inflation from Mtheory dynamics might serve as an explanation for the scarceness of observed dwarf galaxies in the Milky Way halo, as suggested in [11], [12]. This is not explained by standard cosmology which overpredicts their abundance by an order of magnitude. The oscillations and stepwise decreases, on the other hand, provide a unique signature which allow to probe Mtheory observationally by measuring the spectral index. It furthermore clearly distinguishes Mtheory cascade inflation observationally from other string inflation models.
2 Cascade Inflation
Let us first provide a brief review of cascade inflation in Mtheory. The starting point is Mtheory on [13], [14], also known as heterotic Mtheory, compactified on a CalabiYau threefold down to an supergravity in four dimensions. It is a warped compactification due to the presence of flux sourced by the 10dimensional boundaries and additional spacetimefilling M5branes [15], [16]. The warped background which reduces the size of along the interval plays an important role in cascade inflation and its exit [7], [8] as well as in the possible formation of heterotic cosmic strings towards its end [17]. Potentials which are generated for the moduli by flux and several nonperturbative effects [5], [18], [19], [20] are too steep for inflation. It has therefore been suggested in [7] to use the joint effect of the interactions between several M5branes to increase the system’s Hubble friction and thus obtain inflation via the assisted inflation idea [10]. Indeed, having M5branes is generically required to satisfy the tadpole cancellation condition. The M5branes are spacetime filling and wrap a genus zero holomorphic twocycle on to preserve the same supersymmetry as the background. For simplicity, a single rigid twocycle is considered. Being BPS objects, the M5branes interact only at quantum level via open M2instantons which stretch between them along and wrap . For a single rigid twocycle the sum over instantons trivializes and no integral over the moduli space of the twocycle needs to be carried out. There are two ways to show that the distribution of M5branes approaches dynamically an equidistant distribution along . The first is based on energy minimization [7]. Setting the Kähler covariant derivatives of the superpotential with respect to the complex M5brane position fields to zero is tantamount to a partial energy minimization and implies that supersymmetry is not broken in the moduli sector. Geometrically, , enforces the M5branes to be distributed equidistantly in the large volume limit. A second way is to show that the equidistant distribution corresponds to an attractor solution [8].
The moduli fields of interest to us are the complex M5brane position moduli next to the complex volume and orbifoldsize moduli and . Twice their real parts are denoted , and . Moreover, it’s convenient to define which enters the supergravity’s Kähler potential capturing the backreaction of the M5branes on the background geometry. For an equidistant distribution
(1) 
is independent of and is identified with the inflaton. is the interval size, the position of the th M5brane. Note that for an equidistant distribution the multi inflaton system effectively reduces to a single inflaton one. In the large volume limit – specified by the inequality – where cascade inflation occurs, the potential for the canonically normalized inflaton becomes
(2) 
where
(3) 
are the energy scale and power parameter and the CalabiYau intersection number. For more details on the derivation we refer the reader to [7], [8].
The combined nonperturbative interaction of M5branes thus leads to an exponential potential for whose cosmological FRW evolution leads to power law inflation [21] for which the growth of the FRW scale factor^{1}^{1}1Note the typographical difference between cosmic time and the modulus .
(4) 
is entirely determined by the parameter . A period of inflation occurs if . It is therefore important to note that with M5branes one finds a scaling [7], [8] which easily allows to be larger than one and shows the generation of inflation for sufficiently many M5branes. This can also be seen from the slowroll parameters which decrease parameterically like and hence become small when becomes large. In fact is bounded below and above. The bound from below follows from , the prerequisite for inflation. The bound from above descends from the requirement to work in the large volume regime where and noticing that grows with . For typical parameter values one finds [7].
Let us now describe the cascade inflation phase [8]. The repulsive M2interactions between the M5branes cause them to spread over the interval until the two outermost M5branes hit the boundaries. The ensuing nonperturbative small instanton transition transforms the outermost M5branes into small instantons on the boundaries [22]. More precisely, the small instantons are described by a torsion free sheaf, a singular bundle. The singular torsion free sheaf can then be smoothed out to a nonsingular holomorphic vector bundle by moving in moduli space [23]. This process changes the topological data on the boundaries while the number of M5branes participating in the inflationary bulk dynamics drops to . The small instanton transitions can be either chirality or gauge group changing [23]. We are considering the first case in which a change in the third Chern class of the visible boundary’s vector bundle changes the number of fermion generations during the transition. This opens up the attractive possibility of reducing dynamically the number of generations during the cascade inflation phase, given that most compactifications exhibit a large number of generations far greater than three. Notice that for the chirality changing transition the gauge group will not change during the transition and unwanted relics are not produced.
The cascading process starts when the first two outermost M5branes hit the boundaries and no longer participate in the bulk dynamics. The remaining M5branes will continue to spread until the second most outermost M5branes hit the boundaries in a second transition and so on. The successive stepwise drop of the number of M5branes by two marks the cascade inflation phase [8]. Between each of these transitions we have a potential of the form (2) giving powerlaw inflation but with stepwise decreasing values for and thus different parameters and after each transition. The cascade inflation process comes to an end when the number of M5branes, given by
(5) 
in the th phase, drops below a critical value in the th phase determined by the exit condition
(6) 
Throughout the cascading process the inflaton will always be identified with the M5brane separation and grows continuously. The evolution during cascade inflation is thus a series of consecutive powerlaw inflation phases
(7) 
Matching the scale factor at the transition times determines the prefactors to be
(8) 
The scale factor, but not the Hubble parameter, is therefore continuous at the transition times . The onset time of inflation, , is determined by inverting the exact powerlaw inflation solution for in the initial phase and noting that . The result is
(9) 
Similarly, by inverting the solution for one obtains for the transition times [8]
(10) 
from which the number of efoldings generated during cascade inflation follows
(11) 
WMAP threeyear results indicate that the scalar spectral index, , is [25]. For powerlaw inflation one has in the initial phase. Adopting typical values of and for the case of an unbroken hidden [7], has to lie within the interval to satisfy the spectral index constraint. Of course, the initial number of M5branes can be larger than this upper bound, with the proviso that the resulting , at the scales of our Hubble radius, lies within the interval given by the WMAP data set. Taking the central value, , one finds M5branes.
The scale of inflation, , can be at most of order the grand unified (GUT) scale to have gravitational waves under control. Assuming instant reheating one needs about 60 efoldings to solve the problems of standard BigBang (SBB) cosmology. Mapping the cascade inflation model, with the above values for and , to GUTscale inflation, requires . One might lower the required minimal number of efoldings by either lowering the reheating temperature, , or . However, for the above choices of and , lowering requires larger values for which seem to be nongeneric. The details of reheating have yet to be worked out for cascade inflation, nonetheless, we assume instant reheating and therefore . We should note here that the above values of , , and are not the only values that lead to GUT scale inflation which satisfy the theoretical and observational constraints. Surfing the landscape of parameters allows us to choose different sets of parameters. For example, one can also achieve a GUTscale inflation by choosing or (3000, 20, 1000, 123). As we will see, different values for these parameters determine the location of the resulting oscillations in the power spectrum. Henceforth, we will proceed with the initial values although the qualitative features do not change with other choices of parameters.
Starting initially with M5branes in the bulk, we find
(12) 
in Planckian units. The total number of efoldings is
(13) 
which is much larger than the number of efoldings required to solve the horizon and flatness problems of SBB. Most of the inflationary expansion takes place within the first powerlaw phase in which none of the M5branes has yet collided with the boundaries. However, we are interested in the last 60 efoldings of expansion which are within our observable horizon.
3 Power Spectrum of Cascade Inflation
Inflation, besides solving the flatness and horizon problems of standard cosmology, provides a causal mechanism to generate the seed for large scale structures of the universe. Temperature fluctuations of the cosmic microwave background radiation (CMBR) – the afterglow of the BigBang – are believed to be generated by quantum fluctuations of the field(s) responsible for inflation. WMAP alone indicates a flat dominated universe with nearly scaleinvariant power spectrum with [25]. Any viable inflationary model should be able to produce a power spectrum compatible with these observations.
During inflation two types of perturbations are produced: scalar (density) perturbations and tensor perturbations (gravitational waves). These two types of perturbations are both responsible for the temperature anisotropy of the CMBR. Let us focus on scalar perturbations. The evolution of Fourier components of scalar perturbations, , is known to be governed by the equation [26]
(14) 
where
(15) 
is the Fourier component of the gauge invariant Mukhanov variable , which is a combination of scalar perturbations of the metric and inflaton [26]. is proportional to the curvature perturbation of the comoving hypersurface [27]
(16) 
In the equations above, prime (dot) denotes differentiation with respect to the conformal (cosmic) time, (). The solutions to the mode equation (14) are normalized so that they satisfy the Wronskian condition
(17) 
Ultimately the scalar power spectrum is defined as
(18) 
The power spectrum should be evaluated in the limit where the mode goes well outside the horizon. To recover the ordinary quantum field theory result at very short distances much smaller than the curvature scale, we require that the mode approaches the BunchDavies vacuum when
(19) 
During each powerlaw phase equation (14) simplifies to a Bessel equation
(20) 
where
(21) 
and is the conformal time corresponding to . The Bessel equation has the following general solution
(22) 
where and are the first and second Hankel functions of order . Starting from the first powerlaw phase and demanding that the mode satisfies equation (19) at the beginning of cascade inflation, one can determine and
(23) 
Through the transition from one powerlaw phase to the next, the scale factor is continuous but the Hubble parameter is not. This happens because the potential has steps at the transitions. Potentials with step occur in supergravity motivated models of inflation where the inflaton lies within the hidden sector and is gravitationally coupled to a visible sector which contains the standard model [24]. Spontaneous supersymmetry breaking that occurs in the visible sector changes the mass of the inflaton and leads to a sudden downward steps in the inflaton potential. During inflation, the resulting change in the potential is compensated by an increase in the inflaton kinetic energy. Therefore the slowroll approximations become unreliable around the transition points [28]. Despite some similarity that exists between the cascade inflation model and these supergravity motivated inflation models, there is an eminent difference. In cascade inflation, the differences in potential energy after each step are transferred to the boundaries as the two outermost M5branes dissolve into them via small instanton transitions. Thus the kinetic energy of the inflaton fields, whose role is played by the separations of the M5branes in the bulk, will not get modified by the existence of such jumps in the potential. That is why we can still approximate the evolution by a powerlaw, even instantaneously after the transitions. Since the Hubble parameter decreases whereas the scale factor remains continuous through the transitions, the size of the Hubble radius increases slightly. Therefore some modes that have just gone outside the horizon, are recaptured and start oscillating again. As we will see these oscillations will be translated to oscillations in the power spectrum later.
Focusing on adiabatic perturbations, the three curvature perturbations of the comoving hypersurface, , is continuous and differentiable through the transitions [29]. This allows us to determine the th Bogoliubov coefficients in terms of th ones and calculate the power spectrum in the limit when all modes are far outside the horizon. The left graph in fig.1 displays the power spectrum for the modes that have crossed the horizon at least once during the last 60 efolds of inflation and are still outside the horizon at the end of inflation. Since the Hubble parameter and , drops in each step, the total amplitude of perturbations decreases as well. The modes that cross the horizon twice during the transitions display oscillationary behavior. Actually, the oscillations last for an interval of much larger than the interval crossed by the horizon twice. As the right graph in fig.1 shows, for the first step with approximately drop in amplitude of the potential, the oscillations last for as much as three decades of . The left graph in fig.2 presents vs. for the first five inflationary bouts. Aside from the superimposed oscillations, the modes pick up the value of spectral index of the bout during which they cross the horizon. Of course, this is only true for the first few bouts that last long enough to let the oscillations fade away. For the last inflationary bouts which last much less than an efolding, this inference breaks down. The amplitudes of perturbations suppresses significantly and we have a very red spectrum at such scales. As the right graph in fig.2 demonstrates, the period of oscillations in the power spectrum decreases as increases.
body simulations of structure formation use the assumption of a scaleinvariant power spectrum and predict the number of dwarf galaxies in the Milkyway halo an order of magnitude larger than observed. As suggested in [11], a sudden downswing of the power spectrum at small scales can explain this discrepancy and ameliorates the disagreement between cuspy simulated halos and smooth observed halos. Since in cascade inflation, the power spectrum drops in each step, its value at small scales is smaller than what a simple extrapolation of the spectrum at large scales predicts. With the above choice of parameters, the first downturn occurs at about Mpc. A suppression at such a scale ameliorates the problem of the dearth of dwarf galaxies without violating constraints from the Lymanalpha forest. Of course, one should mention that there are also other explanations for the dearth of power at small scales, see for e.g. [12].
Recently, Covi et. al. have tried to explain measured deviations of WMAP threeyear data from featureless power spectrum [30], using potentials with step. They found interesting constraints on the location, magnitude and gradient of possible step in the inflaton potential. Especially, they noticed that current data allude to the existence of one step at the location of WMAP low glitches and one at smaller scales, beyond the second peak. In particular, the feature at small scales can mimic the effect of baryonic oscillations and reduce the estimate of , baryon density. As mentioned earlier, the freedom of the choice of number of M5branes and other Mtheory parameters in our model, can change the location and magnitude of the resulted oscillations. It will be interesting if one uses the results of that paper, to derive Mtheory parameters or at least put some constraints on them.
One should also note that in generation of such oscillations from supergravity models with hidden sector inflation, one had to assume that symmetry breaking phase transitions happen during inflation [24]. In cascade inflation this assumption will necessarily be true since the transitions are generated by the very multi M5brane dynamics which also drives inflation. Hence, features in the potential will inevitably be produced.
Acknowledgments
We are grateful to R. Allahverdi, R. Brandenberger, R. Easther, J. Khoury, W. Kinney, R. Mann and R. Woodard for helpful discussions. A.A. is supported by the Natural Sciences & Engineering Research Council of Canada. A.K. is supported by the National Science Foundation under grant PHY0354401 and the University of Texas A&M.
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