The Universe as a Cosmic String

The Universe as a Cosmic String

Florian Niedermann florian.niedermann@physik.uni-muenchen.de Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität, Theresienstraße 37, 80333 Munich, Germany    Robert Schneider robert.bob.schneider@physik.uni-muenchen.de Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität, Theresienstraße 37, 80333 Munich, Germany    Stefan Hofmann stefan.hofmann@physik.uni-muenchen.de Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität, Theresienstraße 37, 80333 Munich, Germany    Justin Khoury jkhoury@sas.upenn.edu Center for Particle Cosmology, Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA
July 13, 2019
Abstract

The cosmology of brane induced gravity in six infinite dimensions is investigated. It is shown that a brane with Friedmann-Robertson-Walker symmetries necessarily acts as a source of cylindrically symmetric gravitational waves, so called Einstein-Rosen waves. Their existence essentially distinguishes this model from its codimension-one counterpart and necessitates solving the nonlinear system of bulk and brane-matching equations. A numerical analysis is performed and two qualitatively different and dynamically separated classes of cosmologies are derived: degravitating solutions for which the Hubble parameter settles to zero despite the presence of a non-vanishing energy density on the brane and super-accelerating solutions for which Hubble grows unbounded. The parameter space of both the stable and unstable regime is derived and observational consequences are discussed: It is argued that the degravitating regime does not allow for a phenomenologically viable cosmology. On the other hand, the super-accelerating solutions are potentially viable, however, their unstable behavior questions their physical relevance.

pacs:
04.50.-h, 98.80.-k, 95.36.+x, 04.25.D-

I Introduction

We are in the golden age of observational cosmology, in which General Relativity (GR) is being put to the test at the largest observable distances Jain and Khoury (2010); Joyce et al. (2014). Consequently, it has become an important task to develop consistent competitor theories which modify CDM predictions on cosmological scales. Moreover, there is still no fundamental understanding of the dark sector, which constitutes the main part of the energy budget in the CDM model. The most pressing issue from a theory standpoint is the cosmological constant problem (see Weinberg (1989) for a seminal work and Burgess and van Nierop (2013) for a more recent discussion). This provides a strong motivation to look for consistent infrared modifications of gravity.

A prominent candidate is the model of brane induced gravity (BIG) Dvali et al. (2000); Dvali and Gabadadze (2001) according to which our four dimensional universe (the brane) and all its matter content is localized in a -dimensional infinite space-time (the bulk). Despite the fact that the extra dimensions are infinite in extent, 4D gravity is nevertheless recovered at short enough distances on the brane, thanks to an intrinsic Einstein-Hilbert term (or brane induced gravity term) on the brane. This results in a modification of gravity characterized by a single length scale which discriminates between two gravitational regimes: a conventional 4D regime on scales , for which the Newtonian potential is proportional to up to small corrections; and a -dimensional regime on scales , for which gravity on the brane is effectively weakened and the scaling becomes . In order to be in accordance with gravitational measurements on solar system scales, the cross-over scale has to be large enough, e.g., for lunar laser ranging experiments demand  Afshordi et al. (2009). Thus, cosmology represents the ideal playground for testing these theories.

Brane induced gravity models are interesting also for other reasons. At the linear level, the effective 4D graviton is a resonance, i.e., an infinite superposition of massive graviton states. Historically it turned out to be notoriously difficult to give a mass to the 4D graviton on a nonlinear level without introducing Boulware-Deser ghost instabilities (for recent reviews, see Hinterbichler (2012); de Rham (2014)). This has been achieved recently with dRGT gravity de Rham et al. (2011). Extra dimensional constructions, such as BIG, offer promising arenas to devise ghost-free examples. Another motivation comes from the degravitation approach to the cosmological constant problem Dvali et al. (2003, 2002); Arkani-Hamed et al. (2002); Dvali et al. (2007); de Rham et al. (2008). The massive/resonant graviton leads to a weakening of the gravitational force law at large distances, which makes gravity effectively insensitive to a large cosmological constant. There are linear de Rham et al. (2008) and nonlinear Charmousis et al. (2001) indications for that claim.

The best-known and most extensively studied example is the Dvali-Gabadadze-Porrati (DGP) model Dvali et al. (2000), corresponding to . The cross-over scale in this case is given by , where is the bulk Planck scale. For cosmology, the DGP setup gives rise to a modified Friedmann equation Deffayet (2001), , featuring an additional term controlled by . Accordingly, the modification can be neglected for early times and large curvature (), whereas it becomes significant at late times and small curvature (). The plus and minus sign correspond to two different branches of solutions, the “normal” and the “self-accelerating” branch, respectively. The former is characterized by a weakening of gravity since the energy density gets effectively reduced, while the latter describes a gravitational enhancement. The self-accelerated branch is widely believed to suffer from perturbative ghost instabilities Luty et al. (2003); Nicolis and Rattazzi (2004); Koyama (2005); Charmousis et al. (2006); Gregory et al. (2007); Gorbunov et al. (2006). The normal branch is perturbatively stable. Confronting DGP with cosmological observations yields a rather stringent bound on the cross-over scale:  Lombriser et al. (2009).

A natural generalization of the DGP model are higher-codimension scenarios () Dvali and Gabadadze (2001). Several difficulties have impeded their development:

  • According to claims in the literature, the model propagates a linear ghost on a Minkowski background Dubovsky and Rubakov (2003); Hassan et al. (2011), which questions the quantum consistency of the whole theory.

  • Bulk fields are generically divergent at the position of a higher-codimension brane and require a regularization prescription.

  • A non-trivial cosmology on the brane implies the existence of gravitational waves which are emitted into the bulk. (In , the symmetries of the geometry imply a static bulk, because there is a generalization of Birkhoff’s theorem to planar symmetry Taub (1951). However, no such theorem exists for cylindrical symmetry, and Einstein-Rosen waves Einstein and Rosen (1937) are in fact a counter-example.) Including these waves in the dynamical description makes it much more difficult to solve the full system.

The first point, which clearly would be the most severe, was recently proven to be wrong Berkhahn et al. (2012). Through a detailed constraint analysis, it was shown rigorously in Berkhahn et al. (2012) that the would-be ghost mode is not dynamical and is instead subject to a constraint. This is analogous to the conformal mode of standard 4D GR. For the positive definiteness of the Hamiltonian was explicitly shown in Berkhahn et al. (2012). Consequently, in a weakly coupling regime on a Minkowski background the model is healthy. This result offered a new window of opportunity for investigating consistently modified cosmologies at the largest observable scales.

In the present paper we explore cosmological solutions in the simplest case: brane induced gravity in dimensions. Those solutions are obviously interesting for observational purposes, but they also test the non-perturbative stability of the model.

To overcome the second issue listed above, we introduce in Sec. II a regularization which replaces the infinitely thin brane by a hollow cylinder of finite size . We stabilize this size by introducing an appropriate azimuthal pressure. The microscopical origin of this pressure component is not specified, but we check a posteriori whether the required source is physically reasonable (i.e., whether it satisfies the standard energy conditions).

We first check the consistency of our framework by deriving known solutions for a static cosmic string in 6D in Sec. III. Based on these solutions the geometry of the setup is illustrated and a distinction between sub- and super-critical branes is motivated.

According to the third issue listed above, which is discussed in more detail in Sec. IV, a key feature of the higher codimensional models is the existence of bulk gravitational waves which are emitted by the brane and affect its dynamics. For they correspond to a higher-dimensional generalization of Einstein-Rosen waves. Consequently, we must resort to numerics, introduced in Sec. V, to find the most general solutions.

We then solve Einstein’s field equations in the bulk in the presence of FRW matter (and brane induced gravity terms) on the brane and present the results in Sec. VI. We stress that these solutions have been derived from the full system of nonlinear Einstein equations without making any approximations or additional assumptions other than having FRW symmetries on the brane and a source-free bulk. This result makes it possible for the first time to discuss the phenomenological viability of the six dimensional BIG model with respect to cosmological observations.

Depending on the model parameters, we find two qualitatively different classes of solutions:

  • Degravitating solutions for which the system approaches the static cosmic string solution, i.e., the 4D Hubble parameter becomes zero despite the presence of a non-vanishing on-brane source.

  • Super-accelerating solutions for which Hubble grows unbounded for late times.

The solution of the first type constitutes the first example of a dynamically realized degravitation mechanism. Accordingly, the brane tension is shielded from a 4D observer by exclusively contributing to extrinsic curvature. We dismiss the second type due to its pathological run-away behavior. In addition, the effective energy density that sources 6D gravity turns negative for these solutions. This bears strong resemblance with the self-accelerating branch in the DGP model and thus questions their perturbative quantum stability.

It is shown that the degravitating and super-accelerating solutions are separated by a physical singularity. Thus, it is not possible to dynamically evolve from one regime to the other. We derive an analytic expression for the separating surface in parameter space. This in turn allows us to derive a necessary condition to be in the degravitating regime:

(1)

with the circumference of the cylinder and the crossover scale111Here and henceforth, refers to the 6D crossover scale, defined below in (22).. However, a phenomenologically viable solution has to fulfill two requirements: First, for early times which ensures that the deviation from standard Friedmann cosmology is small. Second, in order to be insensitive to unknown UV physics that led to the formation of the brane. Obviously, these two conditions are incompatible with the bound (1). As a consequence of these considerations, the degravitating solutions are ruled out phenomenologically.

We conclude in Sec. VIII with some remarks on super-critical energy densities. A number of technical results have been relegated to a series of appendices. In particular, we repeat the analysis with a different regularization scheme in Appendix A to check the insensitivity of our results to the regularization details.

We adopt the following notational conventions: capital Latin indices denote six-dimensional, small Latin indices five-dimensional, and Greek indices four-dimensional space-time indices. Small Latin indices run over the three large spatial on-brane dimensions and corresponding vectors are written in boldface. The space-time dimensionality of some quantity is sometimes made explicit by writing . Our sign conventions are “” as defined (and adopted) in Misner et al. (1973). We work in units in which .

Ii The model

The action of the BIG model in dimensions is the sum of three terms:

(2)

The first term,

(3)

describes Einstein-Hilbert gravity in infinite space-time dimensions. The bulk Planck scale is denoted by . The bulk is assumed to be source-free; in particular, the bulk cosmological constant is set to zero for simplicity. The second term is the induced gravity term on a codimension- brane:

(4)

This describes intrinsic gravity on the brane, with denoting the induced metric. To match standard GR in the 4D regime, is identified as the usual 4D Planck scale. From the effective field theory point of view, the BIG term can be thought to arise from integrating out heavy matter fields on the brane. The last term in (2), , is the action for matter fields localized on the brane, which by definition couple to .

Henceforth we will focus on , corresponding to the codimension case.

ii.1 Regularization schemes

In general, a localized codimension-two source leads to a singular geometry, i.e., the bulk metric diverges logarithmically at the position of the brane. This is well known for static solutions, reviewed in Sec. III. For the pure tension case, the space-time develops a conical singularity—the bulk geometry stays flat arbitrarily close to the brane but diverges exactly at the brane. For more general static and non-static solutions we have to deal with curvature singularities other than the purely conical one. These singularities can be properly dealt with by introducing a certain brane width.

In this work, we adopt a regularization which consists of blowing up the brane to a circle of circumference  Kaloper and Kiley (2007); Burgess et al. (2009). In other words, the brane is now a codimension-one object, with topology . The matter fields are smeared out on the . This amounts to the substitution

(5)

where , and is the five dimensional induced metric.

Furthermore, in the main body of the paper, we follow a static regularization scheme, which makes the evolution completely insensitive to the geometry inside the regularized brane. This scheme can be viewed from two, equivalent perspectives:

  • The brane is a boundary of space-time, and there is no interior geometry to speak of. This is the hollow cylinder perspective. In this case, the equations of motion consist of Einstein’s field equations in the exterior, supplemented by Israel’s junction conditions Israel (1966, 1967) at the brane,

    (6)

    where is the extrinsic curvature tensor. In the second line, we have extracted from a cosmological constant along . This is necessary to ensure that the deficit angle vanishes when .

  • The brane has an interior geometry, such that the junction condition now becomes

    (7)

    where . However, to ensure that the interior region does not introduce any dynamics on the brane, we demand that is equal to a constant value corresponding to a static cylinder:

    (8)

    With this choice, the junction condition (7) agrees with (6), and the two descriptions give identical brane geometry and exterior space-time. We will not be concerned with the brane interior.

A priori one naturally expects that the solutions thus obtained should not depend sensitively on the details of the regularization, as long as the characteristic time scale (, in the case of interest) is much longer than the radius of the circle, i.e.,

(9)

We explicitly check this expectation in Appendix A, by studying a different regularization scheme called dynamical regularization. In this scheme, the gravitational dynamics are fully resolved inside the cylinder. We find that the time-averaged Hubble evolution on the brane agrees with the static regularization result in the limit (9).

Let us stress that only by performing this fully self-consistent GR analysis, which in particular implements regularity at the symmetry axis, was it possible to quantify the effect of having some interior dynamics and thus to show that our results are regularization independent. Moreover, this analysis revealed that the static regularization corresponds to the favorable case where the effects of the interior dynamics are minimized and perfectly smoothed out. The presentation in the main part of the paper therefore uses the simpler static regularization. The interested reader is referred to the Appendix A for more details.

ii.2 Bulk geometry

The assumed symmetries are homogeneity, isotropy and (for simplicity) spatial flatness along the three spatial brane dimensions, as well as axial symmetry about the brane. As shown in Appendix C, given these symmetries and the fact that the space-time is empty away from the brane, the bulk metric can be brought to the form:

(10)

Note that by formally replacing in the first and last term and , we recover the ansatz that was used by Einstein and Rosen to derive the existence of cylindrically symmetric waves in GR Einstein and Rosen (1937) (see also, e.g.Marder (1958)). The additional factor 3 in the generalized case simply counts the dimensionality of the symmetry axis. In the remainder of the paper we will refer to (10) as the Einstein-Rosen coordinates.

The Einstein field equations in the exterior (vacuum) region become

(11a)
(11b)
(11c)

The fact that obeys the linear222Despite the linearity of this equation, the complete brane-bulk system is still highly nonlinear due to the junction conditions, discussed below. 2D wave equation (11a) makes the coordinate choice (10) unique and especially convenient for numerical implementation.

ii.3 Brane geometry

The induced cosmological metric on the brane is

(12)

where the subscript “0” denotes evaluation at the brane position. The scale factor is recognized as , with Hubble parameter . The proper time is related to the “bulk” time via

(13)

where

(14)

with describing the position of the brane in the extra-dimensional space, and . Here and henceforth, dots refer to .

To recover 4D gravity in the appropriate regime, we assume that the proper circumference (divided by ) is stabilized:

(15)

The justification is clear: A realistic defect would have some underlying bulk forces to keep its core stable. Technically, this is imposed by introducing a suitable azimuthal pressure component . We must of course check a posteriori whether the pressure thus inferred satisfies physically reasonable energy conditions, such as the Null Energy Condition.

As an immediate consequence of the stabilization condition, the 4D Planck mass,

(16)

is constant. Moreover, (15) implies , which allows us to rewrite (14) as

(17)

The symmetries of our system allow for a fluid ansatz of the localized 5D energy-momentum tensor

(18)

where the overall factor is such that defines a 4D energy-momentum tensor. Fixing also implies that the energy density and pressure satisfy the standard 4D conservation equation

(19)

ii.4 Junction conditions

In the next step, we explicitly evaluate the junction conditions (6). The outward-pointing unit normal vector is given by . It is straightforward to show that has components

(20a)
(20b)
(20c)

Using (15), (16) and (18), the component of the junction conditions gives a modified Friedmann equation

(21)

where is given by (17), and denotes the cross-over scale

(22)

The modification to the standard Friedmann equation is controlled by this cross-over scale. Assuming , one can already tell that in the regime where the modification is negligible and the model reproduces the standard 4D evolution. When becomes of order , however, the modification becomes important and we expect a transition to a 6D regime. This is of course the way the model was engineered to work in the first place. It is also very similar to the 5D (DGP) case, where the modification term is simply , with the appropriate 5D crossover scale . But the crucial difference is that in the 6D case, the modification term cannot be directly expressed in terms of on-brane quantities like . It knows something about the bulk geometry through its dependence on , and in order to make quantitative predictions one has to solve the bulk Einstein equations (11) as well.

The component of the junction conditions, combined with the vacuum Einstein equations (11b) and (11c) in the limit , can be expressed as

(23)

where

(24)

and

(25a)
(25b)

In our analysis, we will see that the sign of allows to discriminate between a stable and an unstable class of solutions.

The closed set of equations describing the bulk-brane system comprises the bulk equations of motion (11), the energy conservation equation (19) and the Friedmann equation (21). The equation (23) follows from these, as usual. For the purpose of numerical implementation, however, we will integrate the equation. The Friedmann equation will only be implemented at the initial time and later on will serve as a numerical consistency check.

Finally, the component of the junction conditions can be used to determine the azimuthal pressure:

(26)

In our analysis, we will compute explicitly to check, for instance, whether the equation of state along the azimuthal direction satisfies the Null Energy Condition.

Before investigating the dynamical solutions, let us pause to recover the well-known static solutions from our setup.

Iii Static Solutions

The static case constitutes an important check of the above equations and will provide a first physical insight into the geometry of the system333Note that in this case the static regularization (used in the main text) and the dynamical one (discussed in Appendix A) coincide by construction. Indeed, the only non-singular static geometry inside the cylinder is Minkowski space, hence the extrinsic curvature at the inner boundary is exactly the one given by (8)..

For a purely static solution and all metric functions solely depend on . The exterior field equations (11) yield the solution

(27)

By rescaling coordinates tangential to the brane, we can set without loss of generality. The remaining constants and are determined by the junction conditions (21) and (23):

(28a)
(28b)

where is the equation of state. Here we have introduced the critical density . The third junction condition (26) then becomes

(29)

Note that (28a) is ill-defined for ; we will come back to this point shortly. The line element for the exterior reads

(30)

Since the brane induced terms vanish identically for static configurations, this solution is the direct generalization of the exterior metric of a static cylinder in 4D, first derived by Levi-Civita Levi-Civita (1919) and later reviewed for example in Thorne (1965).

Consider the case of pure 4D tension on the brane:

(31a)
(31b)

The coordinate rescaling yields the famous wedge geometry in Gaussian normal coordinates, characterized by the deficit angle :

(32)

where

(33)

Note that this solution corresponds to the generalization of the cosmic string geometry Vilenkin (1981); Hiscock (1985) to 6D. The coordinates cover again the whole space-time including the interior. A well-known fact about this solution is that the intrinsic brane geometry is flat and the energy on the brane only affects the extrinsic curvature, thereby creating a deficit angle. This property makes the higher codimensional models in particular interesting with respect to the cosmological constant problem because is effectively “filtered out” from the perspective of a brane observer; see Burgess and van Nierop (2013) and Dvali et al. (2003) in the case of large or infinite extra dimensions, respectively.

For sub-critical tensions we find for the ratio of physical radius and circumference: for and for . In an embedding picture this corresponds to a capped cone, as shown in Fig. 1. In the critical limit , the embedding geometry becomes “cylindrical”.

In the super-critical case, , the circumference decreases for and vanishes for a certain radius , implying the existence of a second axis. However, in general the geometry is not elementary flat at that position, i.e., , which indicates the existence of a naked singularity. It has been argued that this (conical) singularity is an artifact of the static approximation and is resolved once the full dynamics are taken into account Cho (1998).

(a)
(b)
(c)
Figure 1: Embedding diagrams of the regularized static geometry in the case of a pure tension brane. The circle at describes the brane. As the tension approaches the critical value, the deficit angle approaches , and the bulk geometry becomes cylindrical (b). For super-critical tensions, a naked singularity develops in the bulk a finite distance away from the brane.

The derivation of the junction conditions in the Einstein-Rosen language is not compatible with the super-critical scenario. This is clear in the static case, as already mentioned, since (28a) does not allow a real solution for in the super-critical regime. See Appendix C for a more detailed discussion of this point in the context of dynamical solutions, and Niedermann and Schneider (2014) for a detailed investigation of super-critical cosmic strings. We henceforth exclude the super-critical regime from our analysis.

Iv Interlude: Bulk-brane dynamics

The analysis of cosmological solutions on the brane is greatly complicated by the fact that the assumed symmetries allow for axially symmetric gravitational waves propagating in the bulk. This is unlike the much-studied codimension-one case, where the assumption of planar symmetry enforces a version of Birkhoff’s theorem Taub (1951): The only vacuum 5D solutions are Minkowski or Schwarzschild. The Schwarzschild mass parameter enters the brane Friedmann equation as the coefficient of a “dark radiation” term. In particular, the brane Friedmann equation is completely local.

The codimension-two case of interest is qualitatively very different. The bulk field equations (11) explicitly show that in this case gravitational waves are in fact compatible with all the symmetries. As a consequence, it would be possible to prepare a wave packet in the bulk that reaches the brane at some arbitrary time. Since the amplitude of the wave is given by the metric function , while the 4D scale factor is determined by , the 4D cosmological evolution will inevitably be influenced by such a wave packet. As a result, it cannot be possible to derive a closed local on-brane evolution equation for , without imposing additional restrictions on the bulk geometry.

What could these restrictions be? As a first guess, one could try to assume a flat bulk geometry, just as could be done in the DGP case. After all, this is also what happens in the static pure tension solution. However, it turns out that this is no longer possible after one demands to have non-trivial dynamics. To show this, let us try to set the - and -components of the Riemann tensor to zero, which is a necessary condition for flatness. This in turn demands

(34)

The only solution to these equations is indeed the trivial configuration .

So a dynamical codimension-two brane inevitably curves the extra-dimensional space-time, and since the on brane geometry will be time-dependent, so will be the bulk geometry. In other words, gravitational waves are not only possible for a non-trivial cosmology in this setup, but in fact necessary.

One could still try to arrive at a closed on-brane system by implementing an “outgoing wave condition” at the outer boundary of the brane to exclude incoming bulk waves. Physically, this is clearly a necessary condition because we assume a source-free, infinite bulk. However, it is well known that such a condition is necessarily non-local (in time) in the case of cylindrically symmetric waves (see Givoli (1991) for a review, and Hofmann et al. (2013) for a discussion in the context of GR). Moreover, because the coordinate position of the brane will in general be time dependent, the resulting on-brane system would be non-local both in space and time. It is clear that solving such a non-local system would not be any easier than solving the full bulk system from the start. In other words, if one tried to accommodate for all allowed bulk configurations in the on-brane system, one would end up with not only one, but infinitely many “constants of integration”. This is what makes the codimension-two problem much harder to solve.

Therefore, there seems to be no way around solving the full bulk geometry in order to see what 4D cosmology emerges in the codimension-two BIG model. This can in general only be done numerically, and we will do so in the next sections.

V Numerical implementation and Initial Data

We now turn to the numerical implementation of the full brane-bulk system (11), (19) and (21). Solutions were obtained by specifying initial data, as explained below, and numerically integrating this initial value problem forward in time. Since the dynamical bulk equation (11a) is nothing but the standard (flat space) cylindrically symmetric scalar wave equation, it is straightforward to find a stable integration scheme for the PDE part of the problem. There is only a slight complication stemming from the matching procedure. Even though the physical brane size is fixed, its coordinate position is generally time-dependent. Therefore, if one chooses a fixed spatial grid size in the bulk (as we do), one has to allow to lie in between those grid points. We deal with this problem by using some suitable interpolation scheme. The details of the numerical implementation can be found in Appendix B.

The numerical integration starts at some initial time , . Let us denote all functions evaluated at this time with a subscript . Through a global rescaling of coordinates, we can always set on the brane initially, i.e.,

(35)

Consequently, the initial brane position is

(36)

In the bulk we must specify the initial radial profile and its time derivative . To be definite, as initial profile we choose the static profile given by (27), namely

(37)

where the constant is given by (28b) with . In particular, for a cosmological constant (), we get , and hence . Note that by choosing the static profile we are not putting any potential energy into the bulk gravitational field initially.

At the brane position, the velocity profile is related to the initial Hubble parameter via

(38)

Extending this to the bulk, we write

(39)

where is some profile function satisfying the boundary condition . To minimize the amount of kinetic energy put into the gravitational field initially, which could impact the brane cosmology for long times, we will choose profile functions which are sharply localized around the brane. For definiteness, we will focus on a Gaussian profile of width ,

(40)

With these choices, we expect the on-brane evolution to rapidly become insensitive to the initial conditions.

This completes the specification of initial data. Indeed, the remaining variable, , is fixed by the constraint (21), together with the relation (17)444The full radial profile can be calculated from (11b), but is actually not needed for the evolution of . Only enters through the junction conditions, and it can be calculated at later times from its initial value using (11b) and (11c) only locally at the brane position.. Specifically,

(41)

Note that this equation does not always have a (real) solution for . The existence of a real solution places an upper bound on the energy density:

(42)

Since the constraint has to hold for all times, we were able to drop the subscript . We will refer to this as the criticality bound, separating the sub- and super-critical regimes. As soon as (42) is violated, the initial constraint cannot be fulfilled. The reason is that in this parameter regime the Einstein-Rosen coordinates as used in our derivation are no longer valid. The interested reader is referred to Appendix C for more details. Since this super-critical regime is not compatible with our coordinate choice, it will not be considered in this paper.

As a check on  (42), note that it correctly reproduces the static criticality bound in the static limit . In the dynamical case, however, the bound is more general. In particular, for , i.e., without the induced gravity terms, the bound becomes stronger—the critical point is reached for a smaller value of than in the static case. Physically, the reason is that for , there is additional kinetic energy in the system. For , on the other hand, the induced gravity terms can absorb (or “shield”) part of the energy density from the bulk, thereby allowing much larger values for than in the static case.

The final ingredient is the choice of grid spacing for the numerical calculation. We use a scheme in which the temporal and radial grid spacing is the same and constant:

(43)

The system can then be evolved forward in time using (11) and (23) for any given , , , , and equation of state parameter . (In fact, the quantities , and enter the equations only in the combinations and , so only two of them need to be specified while the third one is degenerate.) The constraint equation (21) can be used as an important consistency check for the numerical solver. Further details of the numerical implementation are given in Appendix B. In what follows we will present the results.

Vi Numerical Solutions

(a) The Hubble parameter on the brane exhibits degravitation. It starts out positive and asymptotically tends to zero.
(b) The radial profile for at different values of . The dots indicate the brane position as a function of time.
(c) Equation of state of that is needed to keep the brane circumference fixed for . It never falls below the value corresponding to unphysical matter.
(d) The effective energy density, , as “seen” by 6D GR. This approaches a positive value consistent with the static solution.
Figure 2: Example of a degravitating solution.
(a) The Hubble parameter on the brane grows in time, indicating super-acceleration.
(b) The radial profile of the function at different values of . At fixed , grows in time.
(c) Equation of state of that is needed to keep the brane circumference fixed. It is negative and falls rapidly below .
(d) The effective energy density, as “seen” by 6D GR, becomes negative. This is interpreted as the source of the physical instability.
Figure 3: Example of a super-accelerating solution.
(a) Behavior of solutions for different choices of and . The green region (Region (1)) shows stable solutions; the red region (Region (2)) shows unstable solutions. The solid line in between corresponds to . The gray region corresponds to super-critical solutions, which are not covered in our analysis.
(b) Zoom into the small blue rectangle depicted in Fig. (a)a. The yellow/orange regions show solutions which hit the singularity at in a finite time. The dashed lines have been inferred from the numerical results.
(c) The evolution of for and different values of . The color/numerical labels of the curves match those of Fig. (b)b. The yellow (4) and the orange (5) lines hit the singularity at in finite time, while the green (1) and red (2) curves avoid the singularity.
Figure 4: (color online) Results of the numerical stability analysis of the model.

We have found two, qualitatively different classes of solutions, depending on the initial conditions. The first class, called degravitating solutions, features a geometry which at late times approaches the static profile. In particular, on the brane. The second class, called super-accelerating solutions, features a run-away behavior for the Hubble parameter on the brane. The source for this apparent instability is an effective energy density on the brane which violates the Null Energy Condition.

After describing a fiducial degravitating (Sec. VI.1) and super-accelerating (Sec. VI.2) solution, we will discuss the regions of parameter space spanned by each class in Sec. VI.3.

vi.1 A degravitating solution

As a first example, let us consider a 4D cosmological constant source () with parameters555For completeness, the width of the initial Gaussian profile (40) is set to , and the step size for integration (43) is .

(44)

For this choice, the energy density lies in the sub-critical regime. Meanwhile, the cross-over scale is smaller than the initial Hubble radius, hence we expect a large modification to standard 4D gravity. This can be seen directly from the Friedmann equation (21): The modification term is controlled by .

The results of the numerics are depicted in Fig. 2. Fig. (a)a shows the Hubble parameter on the brane as a function of time. (The numerical error estimates for , discussed in Appendix B.2, are smaller than the line thickness.) We see that initially decreases to negative values, turns around and approaches zero at late times. This confirms that the static solutions of Sec. III have a finite basin of attraction. This is one of the central results of this work: it is the first example of dynamical degravitation, and demonstrates how the brane tension can be absorbed into extrinsic curvature while the intrinsic brane geometry tends to flat, Minkowski space. The evolution of the bulk geometry, characterized by , is shown in Fig. (b)b. The initial configuration, as discussed in the last section, leads after a few time steps to a rather narrow Gaussian profile. As time evolves, we see that describes a two dimensional gravitational wave that moves outwards, gets more and more diluted and asymptotically settles to a constant.

It remains to check the physicality of the azimuthal pressure component required for stabilization. The equation of state corresponding to this pressure component is shown in Fig. (c)c. The equation of state satisfies the Null Energy Condition (), and is therefore physically reasonable. At late times, , which is consistent with the static solution for a 4D cosmological constant—see (31b). Figure (d)d shows the effective energy density (including the brane induced terms) that sources the 6D bulk gravity theory, . This quantity remains positive at all times, which indicates a healthy source from the bulk perspective. At late times, , and approaches , which is consistent with a static solution with brane density given by (44).

We have repeated the analysis with a dust or radiation component on the brane and found similar behavior. The system approaches the corresponding static, deficit-angle solutions at late times. The azimuthal pressure and effective density are healthy at all times.

vi.2 A super-accelerating solution

Consider once again a 4D cosmological constant source (), with the same parameters as before except for a somewhat larger value of :

(45)

In this case we find completely different behavior. The Hubble parameter on the brane, shown in Fig. (a)a, grows monotonically in time, which indicates an effective violation of the Null Energy Condition. This growth propagates into the bulk, as can be seen from Fig. (b)b: the wave function grows in time at any .

This pathological behavior is reflected in the azimuthal pressure , whose equation of state (Fig. (c)c) becomes less than and tends to . Such an equation of state violates the Null Energy Condition and is rather unphysical. This suggests that no consistent stabilization mechanism exists for a super-accelerating solution. One might wonder whether this apparent instability is solely due to this strange azimuthal component required to fix the brane circumference. We found that this is not the case. In Appendix A.3.4, we show that fixing by hand, and therefore allowing the circumference to evolve in time, still results in super-acceleration.

The instability can be clearly seen by looking at the effective energy density that sources 6D gravity. As shown in Fig. (d)d, starts out positive but eventually turns around and reaches negative values. This behavior bears resemblance to the DGP model, where the self-accelerating branch leads to a negative effective energy density Gregory et al. (2007). The self-accelerating branch is widely believed to contain a ghost in the spectrum Luty et al. (2003); Nicolis and Rattazzi (2004); Koyama (2005); Charmousis et al. (2006); Gregory et al. (2007); Gorbunov et al. (2006). Although the study of perturbations is beyond the scope of this paper, we also expect that the super-accelerating solutions in 6D are likely to have ghosts. (The instability is even more severe in our case, since decreases monotonically at late times whereas it is bounded below in DGP.) Note that this instability uncovered here is a nonlinear result which can only be inferred from the full Einstein equations. On a Minkowski background the linear 6D model is stable Berkhahn et al. (2012).

vi.3 Contour plot

As the above examples show emphatically, our 6D model yields qualitatively very different solutions, depending on the choice of parameters. To study this more systematically, we now perform a scan over and , keeping fixed. This will allow us, in particular, to understand the border delineating degravitating and super-accelerating solutions.

The results are shown in Fig. (a)a, where each dot corresponds to one set of parameters for which we ran the numerics. The green region (also labeled (1)) corresponds to degravitating solutions. As in the example of Sec. VI.1, the brane Hubble parameter tends to zero at late times, and the effective energy density is always positive. The red region (also labeled (2)) indicates super-accelerating solutions. As in Sec. VI.2, grows unbounded, while eventually becomes negative, indicating a classical instability. Finally, the gray region (labeled (3)) corresponds to parameter choices for which the criticality bound (42) is violated. As explained earlier, our coordinate system is ill-defined in this case, and hence we cannot make any statements about solutions in this region.

It turns out that the border between the stable and unstable regions matches perfectly the location in parameter space where

(46)

first introduced in (24), vanishes. This is drawn as a solid line in Fig. (a)a. In the degravitating regime, is negative, and in the super-accelerating regime it is positive. Since appears in the denominator on the right-hand side of the equation (23), the evolution of becomes ill-defined when vanishes. The system hits a (physical) singularity, where the numerics of course break down.

To better understand the boundary between the stable and unstable regions, Fig. (b)b zooms in on the boxed region of Fig. (a)a. For parameters sufficiently close to the line, dynamically approaches zero after a short time, and the system hits a singularity. The basin of attraction for the singularity corresponds to the yellow region (labeled (4)), in which case one starts in the “healthy” region, and the orange region (labeled (5)), in which case one starts in the “unstable” region. This is shown in more detail in Fig. (c)c. This yellow-orange attractor region of the singularity, which is hardly visible in Fig. (a)a, can be broadened by injecting more energy into the bulk initially. This can be achieved by widening the initial Gaussian velocity profile.

We checked that these results are largely unchanged if one uses dust () or radiation () on the brane. Furthermore, we repeated the entire analysis for a different value of the circumference, namely  , and found similar agreement. In particular, the border between the stable and unstable regimes again coincides with the line in parameter space.

vi.4 Interpretation

The main lesson from the above analysis can be summarized as follows: For sub-critical energy densities, the model is stable if and only if the function . Using the constraint (21) to eliminate , this stability condition can be cast into the form

(47)

If this bound is violated, the model is unstable. The stable and unstable regions are separated by a physical singularity, so it is not possible to evolve dynamically from one region to the other.

It is instructive to compare this result with the analogous situation in the DGP model. In that case, the modified Friedmann equation reads Deffayet (2001)

(48)

where . The sign corresponds to the “normal” branch and the sign to the “self-accelerated” branch. At initial time, this can be rewritten as

(49)

The ratio , which is the 5D analogue of , is fixed (up to the choice of branch) for a given crossover scale . Therefore, the DGP parameter space is only one-dimensional. This difference is due to the fact that in 6D there additional freedom in choosing the initial deficit angle. The resulting DGP “contour” plot, shown in Fig. 5, is remarkably similar to the 6D setup. The green line corresponds to the normal branch of DGP; this branch is stable, and the effective density is positive. The red line is the self-accelerated branch. On this branch, is always larger than , and is always negative.

Figure 5: The “contour” plot for the DGP model consists of two disjoint lines. The green line is the normal branch, which is stable. The red line is the self-accelerated branch, which is unstable.

Our results generalize this peculiarity of the DGP model to codimension-two. The main differences are: (i) the stable/unstable solutions lie on disconnected branches in the DGP model, whereas they are separated by a physical singularity in 6D; (ii) there is no criticality bound on in DGP, hence no gray region.

Vii Phenomenology

The stable/degravitating (green) region of Fig. (a)a is bounded from above by the critical bound (42), and from below by the stability bound (47). Since we have analytic expressions for both borders, we can discuss how this stable region depends on model parameters. Of particular interest is whether phenomenologically viable points can lie inside this region.

(a)
(b)
(c)
Figure 6: Contour plots for different values of . The dotted lines correspond to the dynamical regularization discussed in Appendix A. The color scheme is the same as in Fig. 4.

Fig. 6 shows three contour plots for different values of . In the limit , the degravitating region gets squeezed towards the axis, while approaching from below. The dotted lines are the corresponding boundaries for the dynamical regularization discussed in Appendix A. As decreases, the dotted and solid lines approach each other, implying that the two regularization schemes agree in this limit, as expected.

The bounds (42) and (47) imply that sub-critical, stable solutions exists if and only if

(50)

This bound can also be derived in the dynamical regularization, in which case it is only a necessary condition.

For phenomenological reasons, we need to reproduce standard 4D cosmological evolution on the brane, at least at early times. Indeed, if instead , then the system will exhibit a 6D behavior. On the other hand, we must have , as mentioned in (9), in order for brane physics to admit an effective 4D description. Clearly, these two requirements— and —are mutually incompatible, given (50). In other words, the model admits no (sub-critical) solutions that are both stable and phenomenologically viable.

In the super-accelerating (red) region of Fig. (a)a, on the other hand, there is no problem with achieving arbitrarily large values of . Fig. 7 shows the Hubble evolution for different values of (black curves), compared to the standard 4D evolution (blue curve). The matter consists of dust and cosmological constant, with

(51)

As expected, the larger the value, the longer the standard evolution is traced. Once the modification kicks in, however, the evolution becomes unstable and super-accelerating. This instability, accompanied by a negative effective energy density, should be regarded as strong indications against the physical relevance of those solutions. We expect fluctuations around such backgrounds to exhibit ghost instabilities, analogous to the DGP model. It would of course be worthwhile to verify this expectation through explicit calculation. While it would be desirable to further verify this last claim, we think that our current results already suggests that the super-accelerating solutions should not be regarded as consistent alternative cosmologies.

Figure 7: The Hubble evolution for different values of the cross-over scale (black curves), compared to the standard 4D evolution (blue curve). Since , these curves all lie deep inside the super-accelerating/red region. As the value of is increased, the solution traces the 4D evolution for longer.

Viii Conclusion

In this work, the cosmology of the brane induced gravity model in dimensions has been investigated. The existence of bulk gravitational waves, and the fact that a (nontrivial) FRW codimension-two brane cannot be embedded in a Minkowski bulk, makes it impossible to derive a local on-brane Friedmann equation as in the DGP case. Therefore, we solved the full (nonlinear) system of bulk-brane equations numerically.

We found that the model can show two qualitatively different behaviors: either the solutions degravitate, i.e., they dynamically approach the static deficit angle solution, or they super-accelerate, i.e., the Hubble parameter grows unbounded. This instability originates from the effective energy density , which sources six-dimensional GR, becoming negative in those cases. It is very likely—though we have not shown this in the present work—that perturbations around those solutions would allow for ghosts, on top of the classical instability of the background itself. It would certainly be desirable to verify this claim; one strong indication for it is that this is exactly what happens in the DGP case: ghosts are present in fluctuations around the self-accelerated branch, which also has . But in 6D the instability already shows up in the background solution, which is why we already consider them physically irrelevant.

Whether a solution degravitates or super-accelerates depends on the three independent (dimensionless) parameters , and . We were able to derive an analytic expression that determines the border between the two regimes and showed that it corresponds to a physical singularity. Thus, a solution can never dynamically evolve from one regime to the other.

Unfortunately, it turned out that the stable, degravitating solutions are not phenomenologically viable because they never lead to an almost 4D behavior, and thus could never match the past history of our universe which is very well described by the standard FRW evolution. On the other hand, phenomenologically interesting parameters , which are indeed able to mimic a 4D evolution, always lead to an instable behavior once the modification sets in. Unless there is some way to make sense of those instable solutions—which seems very unlikely—we conclude that the BIG model in is ruled out (for sub-critical energy densities).

It should be noted that we have not investigated super-critical energy densities. An effective field theory (EFT) analysis in Appendix D shows that for large enough values of the regularization scale () this constitutes the remaining window in parameter space which could allow for a phenomenologically interesting solution. Finally, we have not considered a cosmological constant in the bulk. It might be interesting to check how relaxing this assumption would change the size of the healthy region in parameter space.

Acknowledgements.
We thank Felix Berkhahn, Gia Dvali and Michael Kopp for helpful discussions. FN and RS would like to thank the Department of Physics and Astronomy at the University of Pennsylvania for its hospitality in the course of this work. The work of SH was supported by the DFG cluster of excellence ‘Origin and Structure of the Universe’ and by TRR 33 ‘The Dark Universe’. The work of FN and RS was supported by the DFG cluster of excellence ‘Origin and Structure of the Universe’. JK is supported in part by NSF CAREER Award PHY-1145525 and NASA ATP grant NNX11AI95G.

Appendix A Dynamical regularization

In the dynamical regularization, the space-time geometry in the interior of the cylinder is resolved and its dynamical impact on the brane evolution is properly taken into account. This has the advantage that the regularity condition at the axis can be implemented and thus one obtains a fully self-consistent and non-singular solution of the (modified) Einstein equations in the whole space-time. Note that in the static regularization, the brane dynamics was not influenced by an interior geometry because the system (brane + exterior bulk) was closed by defining the brane as the boundary of space-time (7) or, equivalently, setting the extrinsic curvature in the interior to its static value (8). The geometrically more consistent boundary condition is the one that ensures regularity at the axis. However, this has the drawback that one has to specify more initial data, and that the solutions will become more sensitive to those initial conditions, because gravitational waves that are reflected at the axis can influence the on-brane evolution. However, it turns out that the solutions obtained in the two regularizations agree very well, up to small oscillations in the dynamical case which are caused by the initial conditions. This result shows that the static regularization is indeed an efficient way to get rid of the dependency on the interior geometry, but without affecting the evolution on the time-scales we are actually interested in.

In this section, we give the details of the dynamical regularization. We will consider the case in which the brane circumference is fixed (as in the main text), but also the case in which the brane circumference becomes time-dependent. The latter case serves as a proof that super-acceleration in the stabilized scenario is not caused by the (unphysical) equation of state of . We present the numerical results in Sec. A.3 and compare them to the ones obtained in the static regularization, which were shown in the main body of the paper.

a.1 Brane bulk dynamics

As discussed in Appendix C, the Einstein-Rosen coordinates (10) can only be introduced in vacuum regions of space-time. Since the interior of the cylinder is also source-free, we can use the same metric ansatz there. However, the energy-momentum tensor that is localized on the brane then implies that the interior and exterior coordinate patches will not be continuously connected. To distinguish them, we will put tildes on all coordinates and functions that live in the interior, so the line element inside is

(52)

with and being functions of . (The coordinates and are continuous, so there is no need for tildes on them.) Einstein’s field equations inside the cylinder take of course the same form as outside, equation (11) with the replacement .

Regularity at the axis implies the condition

(53)

and elementary flatness, i.e., the absence of a conical singularity, requires

(54)

Denoting with the brane position in the interior coordinate patch, and defining analogously to (14), continuity of the metric at the position of the brane yields

(55a)
(55b)
(55c)

The extrinsic curvature at the exterior and interior boundary of the cylinder are calculated using the outward pointing normal vectors

(56)
(57)

respectively. Using this, Israel’s junction conditions (7) become:

(58a)
(58b)
(58c)

Here “tilde” is shorthand for repeating all the terms in the square brackets, but with tildes on all functions and variables. Note that we have not assumed , and so the brane induced gravity terms on the left-hand side of (58) receive contributions not only from , but also from . Furthermore, the energy conservation equation now reads

(59)

where the five-dimensional source terms are related to the four dimensional ones by . As a consistency check, one can verify that this conservation equation follows from the junction conditions (58), together with the vacuum Einstein equations (11).

Finally, it will be convenient to work with instead of and in (58). To this end, note that the definition of in (15) implies and so (14) can be written as

(60)

Another straightforward calculation gives

(61)

Equations (60) and (61) similarly hold for the tilde quantities, i.e., for .

As before, the equations of motion are only closed after specifying an equation of state for both of the two pressure components and . For the former we will again assume a fixed (but arbitrary) linear equation of state . For the latter, we will consider two different possibilities: is chosen to stabilize the brane circumference, exactly as it was done in the main part of this work; . Let us now further discuss the two cases separately.

a.1.1 Fixed brane width

As in the main text, we set and use the junction condition (58c) only to infer the value of that is needed to stabilize the brane. The remaining junction conditions take the form:

(62a)
(62b)

with

(63)

the function is the one defined in (25a) and

(64)

The modified Friedmann equations (62) are very similar to the ones of the static regularization, (21) and (23), with the crucial difference that now the quantities and enter, which are determined by the interior bulk evolution. In this way, the brane evolution is now influenced by the space-time dynamics inside the cylinder. Furthermore, note that the function is now slightly modified to .

a.1.2 Vanishing azimuthal pressure

For the brane circumference will in general be time-dependent, and so the energy conservation equation (59) now implies

(65)

As a consequence, the dimensionally reduced quantity scales exactly as before.

We can still formally introduce the four dimensional Planck-scale and the crossover-scale as in (16) and (22) respectively, but one has to keep in mind that they will now be functions of time as well. Specifically, they scale with as .

The junction conditions then become

(66a)
(66b)
(66c)

with the following definitions:

(67a)
(67b)