The String Soundscape at Gravitational Wave Detectors

We argue that gravitational wave (GW) signals due to collisions of ultra-relativistic bubble walls may be common in string theory. This occurs due to a process of post-inflationary vacuum decay via quantum tunnelling within (Randall-Sundrum-like) warped throats. Though a specific example is studied in the context of type IIB string theory, we argue that our conclusions are likely more general. Many such transitions could have occurred in the post-inflationary Universe, as a large number of throats with exponentially different IR scales can be present in the string landscape, potentially leading to several signals of widely different frequencies -- a soundscape connected to the landscape of vacua. Detectors such as eLISA and AEGIS, and observations with BBO, SKA and EPTA (pulsar timing) have the sensitivity to detect such signals, while at higher frequency aLIGO is not yet at the required sensitivity. A distribution of primordial black holes is also a likely consequence, though reliable estimates of masses and $\Omega_{\rm pBH} h^2$ require dedicated numerical simulations, as do the fine details of the GW spectrum due to the unusual nature of both the bubble walls and transition.


I. INTRODUCTION
The recent direct detection of gravitational waves (GW) by LIGO [1] opens a new mode of physical exploration. Although the potential of GW detectors to study astrophysical objects has been deeply investigated [2], their potential for exploring Beyond-the-Standard-Model (BSM) physics is still in a relative infancy. Prime examples studied include the physics of inflation [3][4][5][6][7], the presence of strongly first order thermal phase transitions in the early Universe, e.g. non-SM electroweak (see [8][9][10][11][12][13] for early work and [14] for a recent overview), and the possibility of probing the existence of axions [15].
We argue that GW detectors may provide a powerful tool to interrogate the nature of short-distance physics, particularly string theory, in a way unrelated to the process of inflation: specifically, GW signals from postinflationary vacuum decay are a natural feature of the type IIB (and likely more general) string landscape. Our conclusions rely on the observation that flux compactifications in type IIB string theory can contain a large number of highly warped regions [16][17][18][19][20], often referred to as throats, with physics related to that of Randall-Sundrum models [21,22] (see Fig. 1). Under rather general assumptions, later made more precise, a throat can present a metastable vacuum in which supersymmetry (SUSY) is locally broken, along with a true locally-SUSY-preserving vacuum [23], to which it eventually decays.
Specifically, we explore the process of vacuum decay, taking place within a given throat, via the process of zerotemperature quantum tunnelling. We study the effect of the nucleation of bubbles of the true vacuum in the early Universe, and argue that resulting ultra-relativistic bubble wall collisions may lead to observable GW signals. The frequency of the GW produced will be different for different throats, since it depends sensitively on its details, most of all on the gravitational warp factor (i.e. red-shift), w IR 1, setting the relation between the IR energy scale of the tip of the throat and the string scale M s . Since a large number of throats with exponentially different warp factors can be present in the string landscape (see e.g. [24]), GW signals with very different frequencies can be produced.
Space-based experiments, such as BBO [25,26], eLISA [27], or AEGIS [28], and pulsar timing arrays like SKA [29] or EPTA [30], are well suited for detection of these soundscape signals, both in terms of frequency range and sensitivity. Larger compactification volumes and smaller w IR both shift the frequency peak of the signal towards smaller values, making pulsar timing array detectors optimal for probing very large volume and/or very strongly warped scenarios, while in the higher frequency range where aLIGO [31] operates, even the strongest GW signal is unlikely to be detectable by current technology.
Another likely consequence of the ultra-relativistic bubble wall collisions is the production of primordial black holes (pBHs) [32][33][34][35][36]. This pBH production process, and the fine details of the high-frequency portion of the GW spectrum itself, is sensitive to the peculiarities of the bubble wall and vacuum decay dynamics that apply in our case (the dynamics are different than those of both thermal phase transitions, and previous studies of inflation-terminating quantum tunnelling vacuum decay). A reliable calculation of the pBH mass distribution and of Ω pBH h 2 requires dedicated numerical simulations, as does the GW spectrum in the frequency domain beyond the peak position. If the production of pBHs is both highly efficient, and has a mass distribution that extends above 10 9 g, then the maximum amplitude of GWs observable today can be constrained. On the other hand, the possible production of pBHs in the mass regions where pBHs may still comprise a significant fraction of the dark matter (DM) density provides another motivation for detailed studies. We emphasise that the study we present here is just a first step towards understanding the rich physics of the string soundscape. (a) Cartoon of a type IIB flux compactification featuring a large number of warped regions (throats) some of which will be of Klebanov-Strassler (KS) type [37]. (b) Closeup of a KS throat (topologically ∼ = S 3 × S 2 × R) with 3-form RR and NSNS flux quanta on the A-cycle and on the B-cycle respectively. The fluxes lead to a tip warp factor wIR. In the locally SUSY-breaking false vacuum anti-D3 branes are localized at the tip [23].

II. FALSE VACUUM DECAY
Outline of early Universe history -We assume that after inflation ends, the visible sector (i.e. the Standard Model (SM) plus any other states in significant thermal contact) gets reheated to some temperature T rh . We conservatively take T rh 4 MeV so that Big Bang nucleosynthesis (BBN) can proceed undisrupted. (This assumption may be relaxed within more general early Universe histories.) But other sectors, such as those living at the end of highly warped throats, may not be reheated to the same degree and, in general, one would expect many, though possibly not all, of those hidden sectors being left at temperatures T T rh . This is a natural assumption given that we do not observe a much larger DM-to-baryon ratio, nor a significant number of excess relativistic degrees of freedom, ∆N ef f . Thus in the following we take, for simplicity, the throat sector under consideration to be at temperature T th = 0. (Strictly, all we require is that T th is much smaller than all mass scales present in the problem, in which case a T th = 0 treatment suffices. The case T th = 0 may lead to a thermally assisted quantum tunnelling decay, or a purely thermal transition if T th is high enough, similar to the Randall-Sundrum case [38][39][40]. These possibilities are studied in upcoming work [41].) This is a self-consistent assumption since the IR dynamics of a throat are known to be sufficiently (though not absolutely) sequestered from the dynamics of the rest of the compactification [42].
In this set-up, the throat sector remains in a metastable vacuum so long as the probability of nucleating a bubble of true vacuum in a Hubble volume in a Hubble time is much smaller than one, i.e. Γ/H(T ) 4 1, where Γ is the decay rate per unit volume (independent of temperature since T th = 0) and H(T ) is the Hubble rate, dependent on the visible sector temperature T . Only when Γ/H(T ) 4 ≈ 1 does the decay occur. (We remark that as the decay takes place when Γ/H(T ) 4 ≈ 1, one does not need to worry about counting of negative fluctuation modes of the bounce solution [43], since in this regime it is proven that one and only one negative mode is present and therefore the Euclidean bounce solution is guaranteed to correctly compute the false vacuum decay [44]. This feature is not assured in the much different parameter regime considered in [23], so it is not clear if in [23] a physical decay rate has been calculated or not.) Throughout, we will assume that the radiation energy density of the visible sector, ρ rad (T ), dominates the false vacuum energy density of the throat [78]. Defining we thus take α(T ) 1 for all temperatures of interest. This ensures that the Universe is always radiation dominated (RD), so that a second inflationary phase never takes place and phenomenologically disastrous, large late time density perturbations do not occur [45,46]. Notice that since ρ rad (T ) ∼ T 4 , α(T ) decreases as the Universe expands, so even if at the time of bubble wall collision α c ≡ α(T c ) ≈ 1 (α c will set the strength of the GW signal), at the epoch of bubble nucleation α n ≡ α(T n ) < 1.
Both the nucleation temperature T n (set by Γ), and the bubble properties, depend on the microphysics of our specific type IIB model to which we now briefly turn. Readers primarily interested in the resulting GW phenomenology may jump ahead to the next sub-section.
Metastable throats -The authors (KPV) of [23] considered the dynamics of p anti-D3 branes (D3) in a Klebanov-Strassler (KS) throat [37] (see Fig. 1). In this conifold geometry (topologically ∼ = S 3 ×S 2 ×R), M units of RR 3-form flux pierce the A-cycle ( ∼ = S 3 of the conifold), whereas K units of NSNS 3-form flux pierce the dual B-cycle which extends in the non-compact directions in the local model (i.e. into the bulk of the geometry when embedded into a compact manifold). These fluxes result in a tip warp factor w IR ∼ exp(−2πK/3M g s ) where g s is the string coupling [16]. Ignoring both backreaction local to the throat, and back-reaction arising from other distant parts of the compactification (we later comment on these issues), in [23] it was argued that if the ratio p/M was smaller than a certain critical value r c = (π − 3 + b 4 0 )/(4π) ≈ 0.08 then the system features a metastable vacuum in which SUSY is locally broken by the D3-branes. Decay to the true SUSY-preserving vacuum, with no D3-branes, (M − p) D3-branes, and (K − 1) NSNS flux quanta could only take place quantum mechanically, or through a thermal transition.
The fluxes are constrained by a 'tadpole condition', a generalisation of Gauss' law that relates, in a 6dimensional Calabi-Yau (CY) manifold, the tadpole L (depending on the topology of the CY, and other types of D-branes and O-planes), the number of D3 branes and the RR and NSNS background fluxes: . In addition, the string coupling, g s , is stabilised supersymmetrically by a function of these flux choices. To study a type IIB effective action (as we do here) requires a string coupling g s 1. The correct SM gauge couplings can easily be accommodated with g s ∼ O(10 −2 ) [48].
In the metastable vacuum, the system is not well described in terms of individual D3 branes, but rather as an NS5 brane. This NS5 brane (a 5-dimensional object) has 3 non-compact spatial dimensions, the remaining 2 being wrapped around an S 2 contained in the S 3 of the conifold geometry. The position of this S 2 within the S 3 is described by an angular variable ψ. The state of the system can then be encapsulated by the dynamics of a scalar field ψ, initially in a false vacuum ψ f v ∈ [0, π/4), and whose value in the true vacuum is ψ tv = π (where the radius of the NS5 brane is zero). The Lagrangian describing this system (setting M s ≡ 1/ √ α = 1 and in red-shifted units, so hiding the warp factor w IR ) is [23] with µ 3 = (2π) −3 and b 2 0 ≈ 0.93266. When p/M < r c the potential of this system has a local minimum below ψ = π/4, while for p/M ≥ r c there exists only the minimum at ψ tv = π. (We refer the reader to [23] for further details.) Note that the local non-compact KPV set-up, used in this letter, suffers from back-reaction of D3-branes on the geometry. This issue has led to a long-standing debate (see [49][50][51][52][53][54][55][56][57][58][59][60] for a selection of issues), but as of now there is no definitive full string theory calculation on this issue. We work under the assumption that the EFT point of view in [60] is appropriate, and hence that the local back-reaction will not change the qualitative features of this system but only small quantitative changes will occur (likely resulting in a somewhat smaller value of r c ).
Bubble nucleation -KPV [23], considered the case p/M r c where the false vacuum decay rate (if a decay) leads to a lifetime τ ≫ 10 10 yrs. We instead focus on the case p/M r c , close to the regime of classical instability. We thus introduce δ defined as We stress that our consideration of one or more throats close to classical instability is not unreasonable: given the large number of throats typically present in the type IIB string landscape, some of them can find themselves in this near-to-critical situation. (We return to this below.) Following Coleman [61], the decay rate per unit volume can be written as Γ ∼ m 4 e −B , where B is defined where ψ B is the field configuration that extremises the Euclidean action (the bounce solution), and ψ f v refers to the static false vacuum configuration. The pre-factor results from the (square-root) determinant of the quadratic fluctuation operator around the bounce, the associated mass scale m being sufficiently well approximated by the curvature around the barrier. We find (restoring powers of M s and w IR ) Moreover, since the metastable vacuum is close to classical instability, the vacuum energy density, ρ vac , is much larger than the barrier height, and thus one should expect the bubbles at nucleation to show a thick-wall profile (rather than a thin-wall profile as considered in [23] for the case p/M ≈ 0). In this regime, ψ B (R) (R ≡ R · M s w IR , with R being the SO(4)-invariant Euclidean radius) will change slowly withR, an observation that allows the bounce equation to be simplified as which can be solved numerically using the undershoot/overshoot method (we utilise [62]). (In Fig. 2 we show the critical bubble profile for two different values of δ. As expected, the bubbles show a thick-wall profile at the time of nucleation, and the value of ψ at the centre of the bubble is initially well below the true vacuum value ψ tv = π.) Once the bounce solution ψ B (R) has been obtained, numerical evaluation of the bounce action B becomes straightforward. We obtain where f (δ) ≈ 0.38 δ 1/2 + 6.0 δ (for δ ≈ 10 −4 to 10 −2 ). Some comments are in order. The reader might be worried that gravitational effects may be important when studying the decay process [63]. This is not the case, as in the region of parameter space we consider the radius of the bubbles at the time of nucleation (the critical radius R c ) always satisfies R c H(T n ) −1 . Secondly, one might question our entitlement to vary δ essentially continuously: since p refers to the number of D3 branes at the tip of the throat, and M to the units of RR flux, one would think that only discrete values of their ratio, and thus discrete values of δ, can be considered. However, small, but important corrections appear when embedding the local KPV setup we utilise into a complete, global compactification manifold depending on the suppression of couplings between local and global modes (sequestering) [64,65]. For us, this dependence of local parameters (e.g. w IR , and especially δ) on bulk properties, including the enormously large number of distant flux values {K i , M i }, effectively produces a very finely grained discretum [66], justifying our choice of varying an apriori discrete parameter δ continuously.

III. GRAVITATIONAL WAVES
Expansion of bubbles -At nucleation, the bubble walls are spherical and at rest [79]. After a time ∼ R c they are expanding with relativistic velocities and, because of the absence of a thermal plasma with which they meaningfully interact, they continue to accelerate to ultra-relativistic velocities. Since we consider a zerotemperature tunnelling process, production of gravitational radiation from the decay arises solely from collision of bubbles: effects like sound-waves or turbulence in the thermal plasma, which modify the GW spectrum in the case of thermal transitions [14], are, to a good approximation, not present. Moreover, unlike in most thermal phase transitions where the temperatures at bubble nucleation and bubble collision are very close to each other, in our situation this is not the case. Bubbles of critical size are nucleated at temperature T n (or time t n ), when Γ/H(t n ) 4 ≈ 1. Since the radius of this critical bubbles is R c H(t n ) −1 , we can treat them as point- like, and therefore the average separation between nearest bubbles is d(t n ) ∼ H(t n ) −1 . so after a time ∆t R c their radius is R(t) ∼ ∆t. On the other hand, the centre of two bubbles has expanded further apart in the way that corresponds to a RD Universe d(t n + ∆t) ∼ H(t n ) −1 a(t n + ∆t)/a(t n ) = 2 t n (t n + ∆t) and so the bubble walls finally collide at a time ∆t ≈ 1.6t n after nucleation. This translates into a temperature at collision, T c = T n t n /t c ≈ T n / √ 2.6 ≈ 0.62 T n . This distinction between T n and T c becomes important for the GW spectrum, since it is T c that will set the frequency peak of the corresponding GW signal. Notice that T c ≈ 0.62T n leads to α n ≈ α c (T c /T n ) 4 ≈ 0.15α c < α c .
Collision of bubbles -Emission of GWs occurs (predominantly) when the bubbles collide. Assuming a RD Universe during and after the decay, we can estimate the present frequency peak of the GW signal as [13,67] f 0 ∼ 10 −5 Hz g * (T c ) 100 where t * refers to the duration of the transition. Given that ∆t ∼ H(T n ) −1 , we expect t * H(T c ) = O(1), and so we take t * H(T c ) ≈ 1 for illustration in Fig. 3 and Fig. 4.
(Accurate determination of t * requires numerical simulations. Note that, although at T c the dominant collision is between ultra-relativistic bubbles of size ∼ H(t n ), there are also O(10 − 100) small, semi-relativistic, thickwalled, bubbles with which each large bubble collides.) Fig. 3 shows the frequency of the GW signal arising from bubble wall collisions, as a function of the tip warp factor w IR . The frequency of the GW signal can span virtually the whole range of parameter space that will be Given the unusual nature of the kinetic term of the scalar field ψ (Eq. (2)), a precise expression for the energy density in gravitational radiation observed today, as a function of frequency, would need a dedicated numerical simulation. However, we believe the peak amplitude is well approximated by the usual expression for bubble wall collisions in the case of ultra-relativistic bubbles [67,68]: In Fig. 4, we show an approximation to the expected signals for three different values of the frequency peak. Although the peak position should be well approximated by Eq. (7), the profile of the signal as a function of frequency beyond the peak requires dedicated numerical simulation. In the present work, we have used the usual high-f dependence purely for illustration (see [14] for details) .

IV. FATE OF THE VACUUM ENERGY
Although enough to produce a potentially observable GW signal, the fraction of the vacuum energy den-sity that converts into gravitational radiation is small, Eq. (8). The remaining O(1) fraction of the false vacuum energy density must therefore have a different fate. In the following, we outline some of the possibilities. We stress that this is a very model dependent issue, and unambiguous statements can only be made on a case-by-case basis.
Dark radiation -It is possible that some degrees of freedom in the throat will remain massless or very light, even after SUSY breaking. If some fraction of the vacuum energy density was transferred into these massless states (either directly or indirectly after the decay of some other massive throat states), they will behave as dark radiation (DR), and constraints from bounds on the number of effective neutrino species, ∆N ef f , will apply. Since initially T th ≈ 0, we find that the contribution to ∆N ef f from massless throat states is [80] where η DR is the fraction of vacuum energy density that gets transferred into massless states in the throat.
The contribution to ∆N ef f from the SM sector is ∆N (SM ) ef f ≈ 0.046, and Planck has measured ∆N ef f ≈ 0.15 ± 0.23 [69]. Thus, even for η DR ≈ 1 and α c ≈ 0.1, the prediction for ∆N (th) ef f is small enough to be compatible with data (and an observable GW signal may still be produced, as shown in Fig. 4). On the other hand, α c ≈ 1 would yield too large a value of ∆N (th) ef f . Black hole production -The large concentration of energy that takes place when bubbles collide likely leads to the formation of pBHs [32][33][34][35][36]. Thus, the stringent constraints from energy injection into the thermal plasma due to evaporating pBHs can apply [70].
Depending on the fraction of energy transferred into pBHs of a certain mass m BH , certain range of collision temperatures (and therefore of GW frequencies) could be ruled out, specially for masses 10 9 g m BH 10 18 g, where the constraints are strongest. Determining the extent to which pBH production affects our conclusions would require a detailed numerical simulation, taking into account the details of the bubble wall profiles at collision. We note that the production of pBHs with a mass 10 17 − 10 18 g accounting for the DM is a possibility, although its feasibility depends of the fraction of energy density transferred into pBHs. Formation of pBHs with masses ∼ 30M (the other possibly open DM window [71]) does not seem possible if the Universe remains RD during and after the transition, but less standard cosmological scenarios may allow for this possibility. We return to this in future work [41].
Non-pBH dark matter -If sufficiently stable states exist in the throat sector, and suitable fraction of ρ vac is transferred to these, they could account for the DM in the Universe, or a component of it. Although DM candidates arising from warped throats have been studied [72][73][74], and they are indeed a very natural possibility, whether a given throat can accommodate a DM candidate depends sensitively on the details of the sector living on the throat, as well as on other aspects such as how SUSY is broken in the rest of the manifold. Due to how model dependent this issue is, we cannot be more concrete regarding the possibility of successful generation of DM in the scenarios we consider, or regarding potential constraints from DM overproduction. However, we note that this is an interesting possibility for DM model building that would lead to a DM candidate with features linked to those of the GW signal discussed in Section III [41].
Final word -GW detection experiments will help shape the future of physics in the coming century. We believe that they provide an exciting opportunity for the investigation of fundamental physics linked to the highest energy scales.