Emergent de Sitter Cosmology from Decaying AdS

Recent developments in string compactifications demonstrate obstructions to the simplest constructions of low energy cosmologies with positive vacuum energy. The existence of obstacles to creating scale-separated de Sitter solutions indicates a UV/IR puzzle for embedding cosmological vacua in a unitary theory of quantum gravity. Motivated by this puzzle, we propose an embedding of positive energy Friedmann-Lemaitre-Robertson-Walker cosmology within string theory. Our proposal involves confining 4D gravity on a brane which mediates the decay from a non-supersymmetric false AdS5 vacuum to a true vacuum. In this way, it is natural for a 4D observer to experience an effective positive cosmological constant coupled to matter and radiation, avoiding the need for scale separation or a fundamental de Sitter vacuum.


INTRODUCTION
Since the discovery of dark energy two decades ago, string theory has been faced with the challenge of reproducing a small positive vacuum energy. The dominant approach has been the reliance on a landscape of different vacua [1] equipped with a transition mechanism such that the anthropic principle selects our vacuum [2]. This approach became calculable in string theory with the construction of KKLT [3], in which one can achieve a landscape of scale-separated vacua, with both positive and negative cosmological constant (CC), by tuning flux numbers. However, issues have recently been raised which indicate that the naïve application of supersymmetry-breaking and non-perturbative effects necessary in the construction of the landscape is insufficient [4][5][6][7][8]. These concerns suggest that not even a single rigorous string vacuum has actually been constructed and further hints that string theory abhors de Sitter space and any solution with a positive vacuum energy will suffer from instabilities.
Given these difficulties, it is reasonable to consider the possibility that neither metastability nor scale separation can be achieved in string theory in the way envisioned. It seems, therefore, that something completely different is needed. In order to construct an alternative, we will take motivation from work that received considerable attention around the turn of the millennium just before the idea of the string landscape started to flourish: braneworlds. In this context, the cosmology we see as 4D observers is not due to vacuum energy, but rather arises as an effective description on a dynamical object embedded in a higher dimensional space.
In the scenario developed by Randall and Sundrum (RS) [9,10], two identical AdS 5 vacua are glued together across a three-brane. The 5D graviton has a zero mode confined on the brane that gives rise to an effective 4D gravity despite the existence of large extra dimensions; this solves the issue of finding scale-separated vacua. We will consider a variation of this scenario that starts with a metastable false AdS 5 vacuum that non-perturbatively decays to a supersymmetric true AdS 5 vacuum through bubble nucleation. Here, a spherical brane separates the two phases with an inside and an outside, 1 and 4D observers confined to the brane experience an effective dS 4 . This scenario is further motivated by recent arguments that all non-supersymmetric AdS vacua must possess such an instability in a consistent theory of quantum gravity [11,12].
In section 2, we discuss the physics experienced by an observer riding on an expanding bubble that nucleated in a first-order phase transition. In section 3 we discuss the restriction of gravity to the brane, performing a consistency check that the 4D Newton's constant matches the expected value. In section 4 we sketch a concrete construction of a de Sitter braneworld using explicit type IIB string vacua; a metastable vacuum breaking all supersymmetries first obtained by Romans [13] is shown to have a decay channel to an orbifold of the maximally supersymmetric vacuum. We conclude by commenting on some future developments of this scenario.

A WORLD ON A SHELL
Let us consider the cosmology on top of an expanding bubble of true AdS 5 vacuum (with CC: Λ − = −6k 2 − = −6/L 2 − ) that has nucleated in the background of a false AdS 5 vacuum (Λ + = −6k 2 + = −6/L 2 + ), where k − > k + . While such constructions have been studied (see e.g. [14][15][16]), the majority of the literature focuses on RS-like scenarios which connect two insides, or the spacetime has an exact Z 2 symmetry. The bubble nucleation process demands an inside/outside construction where there can be no Z 2 symmetry across the bubble. In global coordinates, the AdS 5 metric inside (−) and outside (+) of the bubble The 4D CC as a function of tension. The vertical line corresponds to a change of branch; tensions below(above) the threshold are compatible with junctions between an inside and an outside(inside). The inside/inside branch is given by taking the sum of the terms in (3). The extremal tensions are marked with points, the extremal tension for inside/inside branch,σext, is given by flipping the sign in (4).
is given by where f ± (r) := 1+k 2 ± r 2 . In terms of proper time τ on the shell located at r = a(τ ), the induced metric takes the Friedmann-Lemaître-Robertson-Walker (FLRW) form: In order for this composite spacetime to be a solution of the Einstein equations, the stress-energy tensor on the shell needs to source a jump in extrinsic curvature. This results in a tension of the shellà la Israel-Lanczos-Sen [17][18][19]: where the dot denotes derivative with respect to τ . In Fig. 1 we plot the induced CC as a function of the tension. In general, the resulting Friedman equation will be nonlinear in the shell tension. However, when the 5D CC's, k ± , are large compared to the 4D Hubble parameter, the tension of the shell approaches (from below) the extremal tension which results in a flat shell Expanding (3) in = 1 − σ/σ ext gives the usual Friedmann equation plus small corrections: with the identifications This shows that in order to have an expanding (dS) bubble, the tension needs to be sub-extremal i.e. σ < σ ext . The bubble nucleates withȧ = 0 with its radius set by the 4D CC. As a consequence, the universe starts out with a size comparable to the horizon scale, with the subsequent expansion further reducing the curvature. The presence of mass in the bulk modifies the AdS 5 metric via f (r) = 1 + k 2 ± r 2 − 4G 5 m ± /(πr 2 ), where m − is the mass inside the bubble and m + is the effective mass measured outside the shell. Through the junction condition, one sees that matter in the bulk contributes a radiation term to the Friedmann equatioṅ where we drop terms higher order in and m ± /k ± . Adding matter is a bit more subtle. Usually, the rest mass of matter remains constant under expansion so that its energy density only decreases with volume as 1/a 3 . How can we reproduce such a behavior in the shellworld? Matter confined to the shell will yield a 1/a 4 contribution to the Friedmann equation (the same as radiation) because the gravitational redshift as the shellworld climbs up through the AdS-throat reduces the effective rest mass. The way to get the desired 1/a 3 is to construct massive particles as strings ending on the shell hanging down from a UV brane. As the shell climbs up the throat it eats the strings and the massive particles represented by the end points are supplied with the required potential energy to keep a constant rest mass.
To see this in detail, let us consider a case where there is a homogeneous distribution of massive particles across the shell universe. These are then end points of strings stretching out from the shell towards a brane further up in the throat. For simplicity, we do not consider any contributions from radiation, and find the metric to be given by 2 where η is the effective tension of all the strings stretching out from the brane. Evaluating the expressions on top of the shell, we find effective 4D-matter with density ρ = η/a 3 k + .
The picture of the massive particle is sustainable only as long as the shell has strings to eat. Once it reaches the top, colliding with the UV brane, physics completely changes. For this to happen in the far future, the distance between the two approaching branes needs to be astronomical as measured in r = a(τ ). However, due to the AdS-metric in the surrounding space, the distance between two branes positioned at a 1 and a 2 translates into a logarithmically small proper distance, d = L ln(a 2 /a 1 ), where L is the microscopic AdS-scale. Even with a large number of e-foldings, the distance remains microscopic. In fact, the radial direction could be part of a compact throat whose size is of the order of the other compact directions.
Even though the proper length of the strings is small, their mass is enormous, E = µ a, where µ is the tension. This is, however, not the mass that is relevant for the 4D observer as the above analysis using the junction conditions shows. What would happen if all of these massive particles were to annihilate into massless radiation sitting directly on the shellworld? 5D energy conservation requires m + just outside of the shell to be equal to the total mass the strings had before they vanished. In addition m − , evaluated on top of the shell, will increase dramatically to represent the mass that went inside of the shell or been captured by it. What is relevant to the 4D observer is the difference m + − m − , which will be determined through 4D energy conservation and is continuous in the process. In this way, all processes on the shellworld will be like shadows of processes taking place in 5D involving much larger energies.
One might wonder about the stability of the universe on top of the shell. An obvious decay channel would be another bubble of true vacuum nucleating on top of the shell as was considered in [22]. However, this decay channel seems to be absent giving a strong indication in favor of the stability of these bubbles. We intend to analyze the issue of stability in detail in an upcoming work [23].
At this point it might be interesting to estimate the number of degrees of freedom on our shellworld. To do so, take a bubble of true vacuum centered in AdS-Schwarzchild and consider how thermal equilibrium is established. If the metric outside the bubble is AdS-Schwarzschild with a mass parameter m = m + , the effective temperature just outside the shell as measured by a distant observer sitting at r 1/k + scales as T 4 + ∼ m + k 2 + /r 4 . In the interior of the bubble, all mass is in the form of a black hole with mass m − (not necessarily equal to m + ) with temperature For the shell to be in equilibrium we need T − = T + = T , leading to Since k − > k + , we find that m − < m + , indicating that the black hole has lost some mass due to the presence of the shell. Finally, using these conditions, the energy density for radiation on the shell at a = r can be estimated as implying that the number of degrees of freedom is pro-

SHELLWORLD GRAVITY
We now turn our attention to local gravitational physics on the 4D shell at late times when the curvature of the bubble is negligible in a local patch, i.e. a(τ ) is large. Furthermore, for simplicity, we consider a Poincaré patch of the asymptotically AdS spacetime written in domainwall coordinates: withã(z) := exp(zk ± ). In RS-like constructions,ã(z) is chosen to be exp(zk − ) for z < z RS and exp(−zk + ) for z > z RS so that the warp factor falls off exponentially on both sides of the brane. When expressed in terms of Poincaré coordinates, this yields a volcano-shaped effective potential for the graviton modes (see the central region of Fig. 2). In contrast, our setup consists of an inside and an outside and therefore the warp factor increases towards the boundary. This results in a potential as shown in the unshaded region of Fig. 2. For such configurations, the zero mode of the graviton is not normalizable due to divergence at the boundary of AdS [28]. If we place a cutoff brane near the boundary, the non-normalizable mode causes both branes to bend [29]. This effect can be interpreted as localized sources induced on the branes by the bulk modes. We identify these sources as the end points of the strings stretched between the branes. The strings result in a relation between these two sources which ensures continuity of the zero mode across the branes. One example of this setup is depicted in Fig. 2 where a RS brane atã(z) =ã RS plays the role of a cut-off brane. 4 The RS brane has Z 2 symmetry across it, which gives a mirror shell. Our shell nucleates atã b ã RS and expands with time to approach the RS brane.
Consider fluctuations to the metric (11) of the form g µν → g µν +ã 2 (z)h µν , where h µν is transverse and traceless. The radial component of h µν obeys a Schrödinger like equation of the form where we have changed coordinates from domain wall coordinate z to the Poincaré coordinate w. The potential V (w) is plotted in Fig. 2. Momentum space graviton mode propagators with zero energy (but with non-zero 3 momentum p) satisfy the following equation [30] − p 2 a 2 + which has the following solutions where I 2 and K 2 are Bessel functions. Variation of the extrinsic curvature across the shell gives an additional boundary condition where prime denotes derivative with respect toã and Σ b is a source term on the shell. The boundary conditions, (15) require χ (p, z) to be proportional to K 2 in all regions 5 as shown in Fig. 2. This gives where G 4 = (2k + k − ) / (k − − k + ) is the 4D Newton's constant induced on the shell by the bulk geometry. This is in agreement with the gravitational constant obtained from the Friedmann equation (6), providing a consistency check for our scenario. The tensor Σ b appearing in (16) consists of two parts where the first term Σ brane b := T µν − T γ µν /3 contains the contribution from the local stress energy tensor T µν on the shell with induced metric γ µν . The contribution Σ brane b comes from matter confined to the shell, which, according to (16), would contribute with a negative to the energy density just as it does in the Friedmann equation. 5 In [30], the addition of I 2 was required to ensure that χ decays away from the brane. However, presence of stretched strings and a cut-off brane in our scenario allows us to have growing modes i.e. pure K 2 away from the brane.ã The other piece, Σ str b , arises from the bending effect of the strings on the shell. Viewing the effect of mass as a localized deformation imparted by the endpoint of the stretched string, the contribution from strings is of the form Σ str b ∼ −(α + /k + − α − /k − ), where α ± is the energy carried by the strings. This follows from the consistency with Friedman equation when one identifies α ± with m ± . Therefore, Σ str b yields a positive contribution to χ b in (16). The above arguments imply that not only are the strings important for the existence of a non-vanishing zero mode, but also to ensure a well defined propagator realizing localized gravitational effects on the shellworld.
Lastly, applying a boundary condition analogous to (15) across the RS brane shows that the matter sources on the shell and the RS brane also need to be related by Σ RS k + = 2 Σ b / (1/k + − 1/k − ) as one expects due to the strings stretching between them.
It is worth mentioning that these string sources in the bulk give an effective 5D geometry outside the bubble that mimics the CHR construction [31]. Gravitational collapse in 4D of the string endpoints result in an unstable black string solution in 5D. This is the source of a longstanding problem in satisfactorily realizing black holes in a braneworld scenario [32]. However, the 5D uplift of black shells proposed in [33,34] avoid the instability by construction and may provide a way to realize black hole solutions on the shellworld. Work in this direction is in progress and will appear in future work [23].

CONSTRUCTION IN STRING THEORY
To realize this scenario in a string theory construction, we propose the decay of the non-supersymmetric Romans vacuum [13] to the supersymmetric AdS 5 × S 5 vacuum via the nucleation of a spherical (p, q)5 brane. The Romans vacuum is given by the reduction of type IIB over a 5D Sasaki-Einstein manifold seen as a U(1) fibration over a Kähler-Einstein base, in close analogy to the Pope-Warner reductions from eleven to four dimensions [35]. The non-supersymmetric vacuum results from taking the base manifold to be CP 2 and a non-trivial relative stretching of the fiber to the base is supported by threeform flux. This solution has been identified with the SU(3) × U(1) invariant critical point of 5D supergravity [36,37].
While the consistent truncation to 5D supergravity is perturbatively stable at the SU(3) × U(1) critical point [38,39], it is still possible that there are tachyonic modes that have been truncated out [40]. Due to the lack of supersymmetry the instability is likely and has been taken as a foregone conclusion [41]. However, it is also expected that one can remove the offending tachyonic modes by taking an orbifold. The orbifold of the Romans vacuum will decay into AdS 5 × S 5 /Z k , which is in general not supersymmetric and has a bubble of nothing nonperturbative instability [42]. However, bubbles of nothing nucleating within the true vacuum must remain inside the lightcone describing the braneworld trajectory and will therefore not affect our scenario.
In addition to the self-dual five-form flux present in the supersymmetric vacuum, the Romans vacuum has nonzero three-form flux G 3 = F 3 + τ H 3 , which breaks all supersymmetry and leads to the squashing of the S 5 . The ten-dimensional metric is given by the product AdS 5 × M 5 , where M 5 is a specific fibration over CP 2 where L AdS (L 5 ) parametrizes the AdS (KK) scale, ν is the modulus controlling the relative fiber/base stretching, and e 5 is a one-form satisfying de 5 = J, where J is the self-dual Kähler form for the unit CP 2 .
The axion C (0) is set to zero and the three-and fiveform fluxes are given by where K is a holomorphic two-form on CP 2 and φ is the fiber coordinate. Solving the ten-dimensional equations of motion results in the following SUSY : L AdS = L 5 , ν = 1 , α 5 = 4 g s L 5 . SUSY : L AdS = 2 11/10 3 3/5 L 5 , ν = 2 3

1/5
, In order to make contact with the supergravity literature we will measure the curvature in the two vacua in 5D Planck units: M 3 Pl = L 5 5 π 3 /(g 2 s (2π) 7 α 4 ). Furthermore, we are interested in decay via nucleation of five-branes which remove G (3) but leave F (5) untouched. The flux quantization condition for F (5) Thus, holding N 5 fixed and using Planck units we have: The important result of these trivial manipulations is the hierarchy of potential energy (V ∝ L −2 AdS ) between the two vacua: indicating that the supersymmetric vacuum lies below the non-supersymmetric vacuum.
In order to determine if there is a non-perturbative decay channel, we need to check if there is a fundamental brane in the theory which is able to realize the desired flux shift and has a tension σ < σ ext . Using (22) we can compute σ ext from (4): While the precise embedding of the five-branes which mediate this decay depends on the orbifold needed to ensure perturbative stability and will be left to future work, we are able to demonstrate the existence of such a decay channel by using supergravity to obtain the tension of the requisite five-brane. We use the superpotential and potential derived in [37] (see also [38,43]): The maximally supersymmetric critical point is at ρ = 1, χ = 0, and the Romans vacuum is located at ρ = 1, χ * = arccosh(2)/2. However one immediately notices that the hierarchy of the vacua is reversed with respect to (22) 5d SUGRA: This is due to the fact that moving in the χ-direction corresponds to a deformation of the internal manifold that is not captured by the 5D theory. Specifically, this deformation does not preserve N 5 . However, since the superpotential is linear in the fluxes, rescaling the superpotential such that the hierarchy (22) is recovered will also amount to holding fixed the five-form flux. Thus, we should use the superpotential of (24) at the supersymmetric critical point, and W = 2 −2/3 W at the critical point corresponding to the Romans vacuum.
In order to deduce the tension of the fundamental (p, q)5-brane that is available to meditate our decay, we use the fact that there should be a BPS brane that sources the F (3) and H (3) flux. The tension of this BPS brane will be given by the junction condition for an extremal brane (4) where k + and k − are associated to V + = −g 2 W 2 (1, χ * )/3 and V − = −g 2 W 2 (1, 0)/3. Again, because the superpotential is linear in flux, and we have rescaled W such that the difference relative to W is entirely due to the change in three-form flux, using the supersymmetric values for the potential will give the effective 5D tension for the BPS brane that sources this change in flux numbers.
Lastly, in order to compare the tension of the BPS brane to the extremal tension (23), we note that the gauge coupling is fixed to g = 2/L SUSY AdS so that the potential reproduces L AdS = (4πg s N 5 ) 1/4 √ α at the supersymmetric critical point where V − = −3/L 2 AdS . Thus, we find Comparing with (23) one finds σ BPS < σ ext so that the fundamental brane that sources the correct charge also can facilitate the decay via a spherical bubble with finite Euclidean action. 6 The rescaling of W that moves the brane tension away from the BPS value can also be understood as accounting for the backreation of the moduli corresponding to the deformation of the S 5 in the nonsupersymmetric vacuum, which is not a priori captured by the 5D supergravity. Note that this is a non-trivial 6 The presence of the decay channel in this toy model does not guarantee that one can tune σ to near-extremal values resulting in small 4D CC and linearized Friedmann equation.
check which posed an obstacle to embedding the canonical RS scenario in string theory [44]. A stringy realization of the refined setup presented in the previous section where we expect 4D gravity confined on the brane is sketched in Fig. 3. The further study of the precise brane embedding in the orbifolded geometry remains an interesting topic for future research.

CONCLUSION AND OUTLOOK
In this letter we have considered the perspective of a 4D observer sitting on the interface separating a true vacuum bubble from a false AdS 5 vacuum in the exterior. This analysis turns out to yield an interesting way of understanding FLRW cosmology within a time independent framework. In particular this offers us the opportunity to embed this construction into string theory and resolve the unitarity puzzle posed by top-down constructions of de Sitter space in a UV complete theory of quantum gravity. This leaves open many channels for future work. First, we will explore the possible decay channels for the effective 4D de Sitter. Second, we would like to further understand the localization of gravity in the presence of strings, including the expected modifications at both small and large momenta. Third, we hope to use the black shell construction to realize black holes on the shellworld. Finally, we would like to further specify the embedding in string theory, specifically finding the effective action for probe (p, q)5 branes.