“Black Universe” epoch in String Cosmology
“Black Universe” epoch in String Cosmology
|Alex Buchel and Lev Kofman|
|Department of Applied Mathematics|
|University of Western Ontario|
|London, Ontario N6A 5B7, Canada|
|Perimeter Institute for Theoretical Physics|
|Waterloo, Ontario N2J 2W9, Canada|
|CITA, University of Toronto|
|60 St. George st.|
|Totonto, Ontario M5S 3H8, Canada|
1 Introduction: Warped geometry, Duality and Cosmology
There are various high dimensional formulations of particle physics theories, such as fundamental string theory or phenomenological high dimensional constructions. Yet, most often string theory cosmology is reduced to the conventional dimensional universe, perhaps with some unconventional ingredients. Low energy effective description is obtained from the high dimensional theory by integration over compact inner dimensions
An example recipe for string theory cosmology is to start with KKLT  (or large volume) compactification in the type IIB theory, include multiple throats with conifolds (attached to the bulk Calabi-Yau) to provide the hierarchy of masses and couplings, and to engineer inflation in this background geometry with branes or moduli fields. In such a cosmological scenario at the end point of inflation, reheating, energy is supposed to stream through the labyrinth of the compact manifold to find, in one of its corners, the light standard model particles which eventually heat and fill up the universe [3, 5, 6, 7]. For instance, one of the scenario is a warped brane inflation which occurs in the short (inflationary) throat at the GUT energy scale which contains an anti-brane at the tip attracting the mobile brane . Brane-antibrane annihilation terminates inflation and energy cascades into KK modes associated with the short throat. Suppose the SM particles are localized at the probe brane(s) in another, long SM throat of TeV scale. It is often assumed that KK modes of the short throat tunnel into the long throat and subsequently transfer their energy into SM sector. All the calculations can be done in ten dimensional warped throat geometry (with the radial coordinate along of the throat)
in the supergravity approximation and then reduced to the effective four dimensional picture, according to the prescription (1.1). For the warped throats, one can use well-studied Klebanov-Strassler  solution with the deformed conifold at the tip, to estimate KK masses, tunneling and decay rates, etc.
The ten dimensional warped throat supergravity solution (like KS geometry) has a dual four dimensional gravity-free gauge theory description. Two or more throats attached to the bulk CY will be dual to two or more gauge theories
where is an interaction term due to the high dimensional operators, , are the fields in two gauge theories. Excitations of KK modes in supergravity correspond to the glueball excitations in the dual picture (1.3), while the tunneling of KK modes between the throats has a dual description in terms of the high dimensional operator .
A specific feature of the cosmological onset is that the two throats containing interacting excitations is a highly dynamic system with the huge hierarchy of the energy scales. Indeed, each of the gauge theories , has its own critical energy (or critical temperature ) which separates the lower energy confinement phase from the higher energy deconfinement plasma phase. On the other hand, the phase transitions at correspond to the emergence of the strong gravity regime in the supergravity dual, manifested in the appearance of a black hole (black brane) horizon across the throat, heuristically similar to the AdS/BH solution
where is related to the radial direction in (1.2). Therefore, to understand string cosmology scenario with two throats of the inflationary and SM scales one has to go well beyond the simple picture of KK modes tunneling between throats and to recall the higher dimensional description of the throat geometry which can contain the BH horizon. Moreover, the time-dependent cosmological scenario addresses to the string theory the questions how the BH horizon in the throat appears, evolves and eventually disappears, and what may be the interesting consequences of this higher dimensional epoch of the evolution of the universe. This is very different from the simple picture of KK modes tunneling between throats. Moreover, as the BH horizon screens the tip of the SM throat (perhaps also the branes that live in the throat), the integration in Eq. (1.1) is not valid so that we have to deal with the higher dimensional theory. These are the problems we will try to formulate and discuss in this paper.
Let us recall the story of the high temperature phase transition /BH horizon duality. Although the motivation of each original study of such a duality was different, we will look at those from the perspective of our string theory cosmology scenario. Phase transition/BH duality is deeply connected with the correspondence between supergravity in the and CFT at high temperature . In the context of supergravity throat solutions and gauge theories, gravity dual to the supersymmetric gauge theory111Assuming unbroken chiral symmetry. is the Klebanov-Tseytlin throat solution in the type IIB theory with fluxes  which has a singularity at the conifold. In  it was proposed that gauge theory with the restored chiral symmetry above the critical temperature corresponds to the Schwarzschild horizon which cloaks the naked singularity of the KT solution or tip of the throat in the KS solution. While the original purpose in  was to resolve the singularity of the KT solution, its implications may go well beyond this. Klebanov-Strassler solution with deformed conifold resolves the singularity problem of KT solution at zero temperature. However, it is the BH horizon that provides a resolution of the KT singularity at high temperature. Technically, it is a challenge to find analytic BH throat solutions in ten dimensional type IIB theory with fluxes. Supergravity solution with the regular BH horizon in the KT throat was constructed in [14, 15, 16]. Gauge theory with the broken chiral symmetry at low temperature is dual to the regular throat solution, while gauge theory with restored chiral symmetry at is dual to the generalized throat solution with the regular BH horizon across the throat. Finite temperature gauge fields dynamics corresponds to the first order phase transition at .
Originally, the KS throat solution was constructed as a supergravity dual to the cascading gauge theory. However, because this self-consistent geometry describes the throat with the regular tip at the finite distance, and because this throat can be smoothly embedded in the CY manifold [17, 18], KS solution is often evoked by cosmologists as the model of the throat attached to the compact space.
Correspondence between BH horizon and the phase transition was also discussed in the very different context, namely, in the phenomenological five-dimensional Randall-Sundrum model. While usually the four-dimensional effective theory at the brane is derived from the bulk+branes system in the spirit of Eq. (1.1), there was a proposal in  to extrapolate the ideas of holography for the RS braneworld and particle phenomenology on the brane. Original high-temperature plasma phase was proposed to be dual to the bulk AdS/BH horizon in five dimensions, while EW phase transition at the brane should be dual to the disappearance of the horizon .
In the following we recall the cascading gauge theory side of the duality. After that we explicitly evaluate the critical energy density for a given throat. This will allow us to lay down the new cosmological scenario which contains higher dimensional stage of formation and later disappearance of the higher dimensional BH horizon. Similarly, this dynamics is relevant for the theoretical “thought experiment”, if one is not interested in cosmology. Finally we discuss potential cosmological problems and consequences of this cosmological model and address interesting theoretical issues.
2 Gauge Theory Perspective
The gauge theory dual to a local warped geometry is a so called “cascading gauge theory” introduced in [21, 12, 10]. This cascading gauge theory can be thought of as a specific supersymmetric gauge theory, with a number of colors which runs logarithmically with the energy scale [10, 13, 22, 23]
where is the strong coupling scale of the cascading gauge theory. Despite having an infinite number of degrees of freedom in the ultraviolet, cascading gauge theory is holographically222In the UV the cascading gauge theory ’t Hooft coupling becomes strong and thus renormalization of this gauge theory must be addressed in the framework of the dual gravitational description. renormalizable as a four dimensional quantum field theory . At a given scale the gauge theory has two chiral superfields , in the representation, and two fields , in the representation. The superpotential of the model is
The two gauge group factors have gauge couplings and . Under the renormalization group flow the sum of the coupling does not run
while the difference is
where is the anomalous dimension of operators . It is clear from (2.3), (2.4) that starting at some energy scale and flowing either to the UV or the IR one inevitably encounters a Landau pole: one of the two gauge couplings will become infinitely large. In  it was argued that extension of the RG flow past the infinite couplings is achieved by a cascade of self-similar Seiberg duality  transformations on the strongly coupled gauge group factor. At each duality step for the RG flow to the UV, and for the RG flow to the IR, leading to the effective logarithmic running of the number of colors (2.1).
Cascading gauge theory confines in the IR; it has a classical symmetry which is explicitly broken to by the anomaly, and then it is broken spontaneously by the gluino condensate to . In  it was pointed out that at sufficiently high temperature, the R-symmetry of the gauge theory is restored; moreover, this restoration is accompanied by a first order deconfinement phase transition333The high temperature deconfinement state of the cascading gauge theory was studied in [13, 14, 15].. Such a phase transition was recently identified in . A summary of the proposal  and the detailed analysis  is that a cascading gauge theory at equilibrium has a critical energy density, which we refer to as , such that for density it is in a deconfined phase with entropy.
Several comments are in order before we finish the gauge theory discussion.
First, the deconfinement phase transition at assumes that the boundary gauge theory is in flat space-time, or at least . Here, is the Ricci scalar of the background metric.
One expects a chiral symmetry restoration ( but without the deconfinement ) phase transition in curved space-time. For a cascading gauge theory on this was demonstrated in  while for the cascading gauge theory in this was discussed in . In both cases the background curvature serves as a regulator that cuts off the IR physics associated with the gluino condensate. By studying a -brane probe in the deformed geometry444In the case this was done in . it is easy to see that the spectrum of fluctuations in the deformed throat is gapped.
In principle, for a cascading gauge theory in flat space-time one could imagine two separate phase transitions: a confinement/deconfinement one, and the chiral symmetry restoration transition. For the scenario discussed in this paper it is important whether or not there is a deconfined phase with a broken chiral symmetry. Such a phase, were it to exist, is expected to have a critical energy density higher than . A detailed analysis  indicates that such a phase is not realized. In other words: the hot cascading gauge theory plasma cools though a first order phase transition where the chiral symmetry is broken only when the theory confines.
3 Critical Energy Density in the Throat
In this Section we turn to the throat geometry. We will compute the critical energy density for weakly curved 3+1 dimensional space-time (modeling our universe) that is necessary to hide it behind the horizon in the ambient warped throat geometry. Each throat has its own critical density, which corresponds to horizon formation, but for our scenario we will focus on for the SM throat.
We assume that our universe (in the low energy limit where the energy density in all throats is much lower than ) is part of the KS geometry which away from the tip is locally described by the KT geometry
and and are the integer numbers associated with the quantization conditions of the form fields fluxes of the type IIB theory which generate the throat geometry (3.1). The local geometry (3.1) is holographically dual to the four-dimensional cascading gauge theory reviewed in the previous section. For large enough energy density this cascading gauge theory undergoes a deconfining phase transition [13, 16], which is reflected in the formation of a Schwarzschild horizon555As we will explain in the following Sections, in our scenario this energy density is accumulated from the “gravitational collapse” of the excited KK modes initially produced from the annihilation in the inflationary throat. in (3.1). The type IIB supergravity background corresponding to a Schwarzschild horizon in asymptotic geometry (3.1) takes form 666The frames are defined as in , such that the metric on a unit size is given by .:
where , and are some functions of the radial coordinate . There is also a dilaton, and form fields ( see  for details). Notice that the radial coordinate in (3.4) is gauged-fixed777Because we gauge-fixed the radial coordinate, there is a constraint equation coming from the equation of motion of this variable; this equation can be solved to determine . so that
The new radial coordinate is related asymptotically to in (3.1) as follows
Type IIB supergravity equations of motion in the background metric (3.4) are solved, subject
to the following boundary conditions:
i) near the boundary ( or ) the black hole metric (3.4) approaches the Klebanov-Tseytlin geometry (3.1);
ii) the hypersurface is a regular Schwarzschild horizon of the metric (3.4); the latter is equivalent to requiring that all the warp factors , and are positive at .
For temperature of order the black hole thermodynamics can be studied only numerically. It was found in  that there is a critical temperature, corresponding to a critical energy density , such that the free energy density of the black hole (3.4) vanishes precisely at , and becomes positive for . Thus we expect a first order deconfinement/confinement phase transition in the cascading gauge theory plasma once its energy density becomes less than . This first order phase transition (which occurs via bubble nucleation and percolation) shall be dual to the inhomogeneous melting of the BH horizon (which before this is translationally invariant in the directions).
From the point of view of the usual dimensional GR, the disappearance of the horizon is rather unusual. However, the disappearance of a higher dimensional horizon is a familiar phenomenon, and we shall comment on it. A crucial observation is that at the supergravity side, there is another geometry, besides the black hole geometry (3.4), which also asymptotes to (a Euclidean version) of (3.1): the KS geometry with a Euclidean time direction identified with a period of . The latter solution has no horizon, it has a zero free energy, and is dual to a thermal gas of the confined cascading gauge theory. The coexistence of these two solutions provides the possibility for transitions between horizon and no-horizon geometrical phases (through instantons, see.e.g. ). The supergravity dual of the phase transition bears similarity to the Hawking-Page transition, which was identified by Witten as a supergravity dual to a (kinematic) confinement/deconfinement transition of the SYM plasma on a three-sphere . There is a notable difference however: while the Hawking-Page-Witten transition occurs in a finite volume999The transition disappears in the infinite volume., the phase transition in the cascading gauge theory occurs in the infinite volume. Thus, we expect it to proceed via the nucleation of bubbles of the stable phase, which further expand and ’remove’ the horizon.
Next, we evaluate quantitatively. In the notation of , a Schwarzschild horizon is formed in (3.1) at the critical energy density (at the gauge theory side) which is given by equation (see also Eq. (4.10) of )
where the 5-dimensional Newton’s constant is
(see Eq. (5.12) of ).
where is a brane tension and is the hierarchy warp factor of the throat . The four-dimensional energy-density of KK modes (located at the tip of the throat) is identified with the energy-density of the glueballs in the dual picture. Also notice that the bigger energy corresponds to the bigger portion of the throat cloacked by the horizon.
4 Theory of Multiple-throat Tunneling Revisited
Suppose compact manifold contains throats of significantly different warpings which are attached to the bulk, as sketched in the Figure 1. Suppose the short throat is associated with the higher energy scale and is populated with Kaluza-Klein excitations. It is well known that the typical KK masses are of the order of
where in the warping factor at the tip of the throat, is the radius scale of the angular coordinates at the tip, for the metric (3.1) it is . Wave functions of the KK modes in the warped geometry are exponentially peaked around the tip of the throat, as illustrated in the Figure 1.
Let us set up the thought experiment with the static geometry of the Figure 1, with the flat outer space. Suppose the short throat is filled up with the KK excitations of the mass (4.1) and four-dimensional energy density . For definiteness we can take , so that GeV (index 1 attributes parameters to the short throat, while index 2 to the long throat). Suppose . KK modes from the short throat begin to tunnel into the long throat. Putting aside interesting subtleties (see e.g. [3, 6, 7, 8]) let us use the estimation of  for the inter-throat tunneling rate
which gives us sec. During this time the long throat is filling up with KK excitations, each with mass much lighter than . If we choose , , then GeV. Those modes again are accumulating101010We can neglect the tunneling of KK modes from the long (SM) throat back to the short (inflationary) throat as the latter is suppressed compare to (4.2) by a factor . around the tip of the long throat, with ever increasing four-dimensional energy density . However, gravitational backreaction of KK modes on the geometry of the long-throat becomes significant as soon as is approaching from below the critical density of the long throat. Let us estimate from Eq. (3.13). For example, choosing , from we get , i.e. the scale of the EW phase transition. This is much lower than the original energy density of the excitations in the system. It means that long before the time KK modes accumulating in the long throat will completely change the geometry of that throat. Actually, it happens instantly 111111This follows from the energy balance equation ..
While it is immensely difficult to follow the detailed complicated metamorphose of the compact space in the self-consistent supergravity formalism, the dual picture in the gauge theory side suggests that increasing energy of the plasma above corresponds to the formation of the BH horizon in the throat geometry. Since the chiral phase transition in the gauge theory plasma is a first order one, the formation of the BH horizon will be dual to the formation of the bubbles of the new phases which eventually percolate.
Therefore we conjecture the following picture of the throat BH formation. The horizon begins to cloak the tip of the throat and propagates, further screening a bigger and bigger part of the throat. This may occur in the manner of the Choptuik critical collapse . Formation of the horizon may occur not uniformly in space, but in patches which later percolate into the uniform horizon (translationary-invariant in three-dimensional outer space). As KK modes from the short throat continue to tunnel into the long throat, BH there absorbs all of them and it happens much faster that the tunneling time (this timing is defined by the cross-section of the high dimensional black hole).
All together, in our thought experiment we start with the excitations of the KK modes in the short throat and end up with the high dimensional Black Hole cloaking the tip of the long throat. From the four dimensional perspective, there are no particles, but uniformly distributed energy density from the high dimensional BH. Again, the dual picture suggest this form of the energy density has the radiation dominated equation of state but without radiation! Recall similar situation in the RS braneworlds with the AdS/Schwarzschild bulk geometry where four-dimensional “dark radiation” is associated with the projection of the five-dimensional Weyl tensor .
5 “Black Universe” Cosmological Scenario
The setting of the thought experiment of the previous Section takes place naturally in the popular string theory cosmological model. Indeed, in this model, the warped brane inflation is based on the brane-antibrane interaction in the short (inflationary) throat, which provides very shallow effective four-dimensional inflationary potential . Four dimensional energy density of the branes is . At the end of inflation the brane-antibrane pair annihilates and releases energy into KK modes excitations, e.g. [2, 3, 4]. KK modes from the short throat begin to tunnel into the long throat. The results of the previous section (where four dimensional outer space is not expanding) indicate that the cosmological scenario with the multiple throat geometry of the inner manifold drastically differs from what was considered earlier in the literature on warped brane string theory cosmological model.
In fact, expansion of the universe makes the multiple-throat cosmological scenario even more involved, and in some aspects different from the scenario without cosmological expansion considered of the previous Section.
During branes inflation in the short throat the Hubble parameter is GeV. After inflation this value decreases with time as . Thus, initially expansion of the universe is significant and would dilute the energy density of KK modes in the long throat, which are much lighter than that of the short throat (). KK modes of the long throat behave as radiation, while KK modes in the short throat have matter equation of state. As a result the tunneling will happen only after the Hubble rate drops below the value equal to the tunneling rate , GeV. Energy density in the long throat at this moment is
in our example .
However, large value of the Hubble parameter generates mass gap of the KK modes . For instance, it is known that massive gravitons in the four-dimensional de Sitter geometry have the mass gap . As long as is large, KK modes of the long throat are not light, but become lighter and lighter as decreases. As drops below , those KK modes can be treated as radiation and we return to the same estimate (5.1) 121212Above estimate is correct provided that the problem of the angular KK modes  is resolved and the decay of KK modes into gravitons is suppressed compare to the tunneling time, see  for details. Our choice of parameters respect these conditions.
which is in our example. Thus tunneling KK modes from the short throat collapse into a black hole in the long throat, see Figure 1. Our universe enters the Black Universe (BU) phase of its cosmological evolution. Because of the horizon, dimensional reduction (1.1) is not relevant. Meanwhile, high dimensional strong gravity theory is very complicated.
Yet, the further evolution of the Black Universe phase can be understood with the magic of duality. Had the long throat cloaked with the horizon been infinitely long, it would have been holographically dual to the deconfined Klebanov-Tseytlin plasma (discussed in Section 2) in Minkowski space-time without gravity. In this case, however, the four dimensional Planck mass would be infinitely large. Gluing the long throat to a compact manifold produces a finite , and in the dual picture this corresponds to the coupling of the hot KT plasma to 4d gravity [35, 36, 19]. In other words, the compact manifold with strong gravity in two throats is dual to the four dimensional theory (1.3) where we will add the four-dimensional gravity . Energy density of the deconfined KT plasma will drive adiabatic expansion of the background space-time, while redshifting itself as radiation. Such an expansion continues until the plasma energy density redshifts to the critical energy density of the first order confinement transition (3.13). The Hubble scale at the beginning of radiation-dominated expansion is
while the Hubble scale at the confinement transition is
to an excellent approximation expanding KT plasma can be considered to be in thermal equilibrium in (almost) flat space-time during the whole period to expansion, up to the confinement phase transition.
For the Black Universe evolution dual to the KT plasma expansion (cooling) with the subsequent first order confinement phase transition, we expect that the energy density of the black hole in the long throat will dilute, according to the four-dimensional radiation dominated cosmology. In higher dimensional picture this corresponds to the recession of the horizon in the direction towards imaginary tip of the throat. As the black hole horizon energy density redshifts below the critical energy density, a first order phase transition must take place that would remove the horizon from the long throat and expose the tip of the throat131313A ’horizon removal’ transition was discussed previously in . It is not clear that our transition is similar. We comment in the conclusion how one can study such transition in the context of gauge/string duality.. The disappearance of the horizon will occur in patches in accordance with the dual picture of the first order phase transitions.
Instead of the geometry with the horizon, the long throat will be filled up with the KK excitations. If there were light SM fields, say, attached to probe brane(s) around the tip of the long throat, they would be produced due to the decay these KK modes. It is at this stage that our universe is ’born’, and the SM hot FRW cosmology follows. In this case one could estimate the reheating temperature of the universe in the scenario. The energy density at the phase transition is Assuming that all this energy is available for the reheating of the SM, (where is the number of the Standard Model degrees of freedom), we find a relatively low reheating temperature GeV.
However, this may be irrelevant, becase there is a new problem in the cosmological scenario with hierarchical throats. Indeed, it was assumed for sucessful phenomenology that the probe brane(s) containing SM field are located in the long throat from the very beginning. Meanwhile, emergence of the Black Brane horizon cloaks the long throat together with any probe branes located there. In a sense, the SM sector becomes screened from the theory. Later on, the geometrical phase transition with melting horizon and re-appearance of the compact throat solution is not accompanied by the re-appearance of the SM brane. The SM sector may be missing after the “Black Universe” epoch.
We presented a cosmological scenario when excited KK modes produced from the brane/antibrane annihilation in the inflationary throat tunnel to a Standard Model throat and cloak it with a Schwarzschild horizon. This suggests a model when string theory inflation is followed the ‘Black Universe’ epoch. This is significantly different from the models of reheating after string theory inflation previously considered in the literature. However, dual picture suggest that the ‘Black Universe’ epoch can be simply described by the expanding universe filled with the hot plasma composed of light particles of a (hidden sector) gauge theory 2 in deconfinement phase with degrees of freedom. These originate from the rapid decay of the massive particles of another, confined (hidden sector) gauge theory 1. After the first order phase transition, the gauge theory 2 is described by the confined phase. If light SM particles are present in the theory, the corresponding particles (glueballs) decay into SM particles with relatively low reheat temperature .
However, for the hierarchical multiple throat scenarios we identify the problem of the SM sector disappearance: the probe brane with SM fields, which was initially placed in the long throat, will be absorbed by the horizon together with a segment of the long throat. After the end of the end “Black Universe” epoch and re-appearnce of the long throat geometry, it is not clear how the probe could re-appear. Similarly, it is not clear to us what is the dual gauge theory interpretation of the screening and re-appearance of the SM sector in the theory like (1.3).
There are various directions for further study. First, one has to resolve the problem of the SM sector. One potential resolution would be to keep SM brane far enough from the tip of the long throat so that is will be not swallowed by the horizon. On the cosmology side, we expect a very drastic phenomena to occur at the epochs of the black brane ’horizon emergence’ as well as the ’horizon removal’ transitions.
’Horizon removal’ phase transition is dual to the confinement/deconfinement phase transition of the long-throat dual gauge theory plasma, coupled to the 4d gravity. How one would study such a phase transition on the gravity side of the gauge/string correspondence? In  if was proposed how to study the gravity dual to a gauge theory in de-Sitter space-time. In our cosmological scenario the dual gauge theory plasma couples to an FRW cosmology, driven by the plasma energy density. Thus, extending ideas of , we should try to set-up the boundary metric to that of the appropriate FRW cosmology. The full ten-dimensional geometry should then be reconstructed requiring the nonsingularity, along the line of the gravity dual to the boost-invariant expansion of the supersymmetric Yang-Mills plasma [37, 38, 39, 40].
We would like to thank Neil Barnaby, Renata Kallosh, Igor Klebanov, Volodya Miransky, Rob Myers, Marco Peloso, Joe Polchinski, Sergei Prokushkin and Dam Son for valuable discussions. We would like to thank KITP Santa Barbara for hospitality, where this work was initiated during the Program ’Nonequilibrium Dynamics in Particle Physics and Cosmology”. AB’s research at Perimeter Institute is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI. AB gratefully acknowledges further support by an NSERC Discovery grant and support through the Early Researcher Award program by the Province of Ontario. LK was supported by NSERC and CIFAR.
-  S. Kachru, R. Kallosh, A. Linde and S. P. Trivedi, Phys. Rev. D 68, 046005 (2003) [arXiv:hep-th/0301240].
-  N. Barnaby, C. P. Burgess and J. M. Cline, JCAP 0504, 007 (2005) [arXiv:hep-th/0412040].
-  L. Kofman and P. Yi, Phys. Rev. D 72, 106001 (2005) [arXiv:hep-th/0507257].
-  A. R. Frey, A. Mazumdar and R. C. Myers, Phys. Rev. D 73, 026003 (2006) [arXiv:hep-th/0508139].
-  J. F. Dufaux, L. Kofman and M. Peloso, arXiv:0802.2958 [hep-th].
-  X. Chen and S. H. Tye, JCAP 0606, 011 (2006) [arXiv:hep-th/0602136].
-  D. Chialva, G. Shiu and B. Underwood, JHEP 0601, 014 (2006) [arXiv:hep-th/0508229].
-  B. v. Harling, A. Hebecker and T. Noguchi, JHEP 0711, 042 (2007) [arXiv:0705.3648 [hep-th]].
-  S. Kachru, R. Kallosh, A. Linde, J. M. Maldacena, L. McAllister and S. P. Trivedi, JCAP 0310, 013 (2003) [arXiv:hep-th/0308055].
-  I. R. Klebanov and M. J. Strassler, JHEP 0008, 052 (2000) [arXiv:hep-th/0007191].
-  E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998) [arXiv:hep-th/9803131].
-  I. R. Klebanov and A. A. Tseytlin, Nucl. Phys. B 578, 123 (2000) [arXiv:hep-th/0002159].
-  A. Buchel, Nucl. Phys. B 600, 219 (2001) [arXiv:hep-th/0011146].
-  A. Buchel, C. P. Herzog, I. R. Klebanov, L. A. Pando Zayas and A. A. Tseytlin, JHEP 0104, 033 (2001) [arXiv:hep-th/0102105].
-  S. S. Gubser, C. P. Herzog, I. R. Klebanov and A. A. Tseytlin, JHEP 0105, 028 (2001) [arXiv:hep-th/0102172].
-  O. Aharony, A. Buchel and P. Kerner, Phys. Rev. D 76, 086005 (2007) [arXiv:0706.1768 [hep-th]].
-  O. Aharony, Y. E. Antebi and M. Berkooz, Phys. Rev. D 72, 106009 (2005) [arXiv:hep-th/0508080].
-  M. K. Benna, A. Dymarsky, I. R. Klebanov and A. Solovyov, arXiv:0712.4404 [hep-th].
-  N. Arkani-Hamed, M. Porrati and L. Randall, JHEP 0108, 017 (2001) [arXiv:hep-th/0012148].
-  P. Creminelli, A. Nicolis and R. Rattazzi, JHEP 0203, 051 (2002) [arXiv:hep-th/0107141].
-  I. R. Klebanov and N. A. Nekrasov, Nucl. Phys. B 574, 263 (2000) [arXiv:hep-th/9911096].
-  M. Krasnitz, arXiv:hep-th/0011179.
-  O. Aharony, A. Buchel and A. Yarom, JHEP 0611, 069 (2006) [arXiv:hep-th/0608209].
-  O. Aharony, A. Buchel and A. Yarom, Phys. Rev. D 72, 066003 (2005) [arXiv:hep-th/0506002].
-  N. Seiberg, Nucl. Phys. B 435, 129 (1995) [arXiv:hep-th/9411149].
-  A. Buchel and A. A. Tseytlin, Phys. Rev. D 65, 085019 (2002) [arXiv:hep-th/0111017].
-  A. Buchel, Phys. Rev. D 65, 125015 (2002) [arXiv:hep-th/0203041].
-  A. Buchel and R. Roiban, Phys. Lett. B 590, 284 (2004) [arXiv:hep-th/0311154].
-  O. Aharony and A. Buchel, to appear.
-  S. B. Giddings, S. Kachru and J. Polchinski, Phys. Rev. D 66, 106006 (2002) [arXiv:hep-th/0105097].
-  S. Dimopoulos, S. Kachru, N. Kaloper, A. E. Lawrence and E. Silverstein, Phys. Rev. D 64, 121702 (2001) [arXiv:hep-th/0104239].
-  M. W. Choptuik, Phys. Rev. Lett. 70 (1993) 9.
-  T. Shiromizu, K. i. Maeda and M. Sasaki, Phys. Rev. D 62, 024012 (2000) [arXiv:gr-qc/9910076]; S. Mukohyama, Phys. Lett. B 473, 241 (2000) [arXiv:hep-th/9911165].
-  A. Higuchi, Class. Quant. Grav. 8, 1961 (1991); 8, 1983 (1991).
-  E. Witten, unpublished, H. L. Verlinde, Nucl. Phys. B 580, 264 (2000) [arXiv:hep-th/9906182].
-  S. S. Gubser, Phys. Rev. D 63, 084017 (2001) [arXiv:hep-th/9912001].
-  R. A. Janik and R. Peschanski, Phys. Rev. D 73, 045013 (2006) [arXiv:hep-th/0512162].
-  R. A. Janik, Phys. Rev. Lett. 98, 022302 (2007) [arXiv:hep-th/0610144].
-  M. P. Heller and R. A. Janik, Phys. Rev. D 76 (2007) 025027 [arXiv:hep-th/0703243].
-  P. Benincasa, A. Buchel, M. P. Heller and R. A. Janik, Phys. Rev. D 77, 046006 (2008) [arXiv:0712.2025 [hep-th]].