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A bulk inflaton from large-volume extra dimensions

Brian Greenea,b, Daniel Kabatc, Janna Levina,d, and Dylan Thurstone aInstitute for Strings, Cosmology, and Astroparticle Physics (ISCAP) bDepartments of Physics and Mathematics, Columbia University, NY, NY 10027 cDepartment of Physics and Astronomy, Lehman College, CUNY, Bronx, NY 10468 dDepartment of Physics and Astronomy, Barnard College, Columbia University, NY, NY 10027 and eDepartment of Mathematics, Barnard College, Columbia University, NY, NY 10027

The universe may have extra spatial dimensions with large volume that we cannot perceive because the energy required to excite modes in the extra directions is too high. Many examples are known of such manifolds with a large volume and a large mass gap. These compactifications can help explain the weakness of four-dimensional gravity and, as we show here, they also have the capacity to produce reasonable potentials for an inflaton field. Modeling the inflaton as a bulk scalar field, it becomes very weakly coupled in four dimensions and this enables us to build phenomenologically acceptable inflationary models with tunings at the few per mil level. We speculate on dark matter candidates and the possibility of braneless models in this setting.

Modern theories suggest that although the universe appears to have three spatial dimensions, there may in fact be more. As is well-known, if the extra dimensions are sufficiently small, they would escape observation. If the extra dimensional volume were large, however, a number of attractive features emerge, including an appealing explanation for the small value of Newton’s constant. But familiar intuition suggests that as the internal volume grows, it becomes energetically easier to excite modes in the extra directions–the mass gap to the Kaluza-Klein states decreases. This raises the question: why does the universe appear to be three dimensional? Or, put another way, why haven’t we seen the Kaluza-Klein states?

A standard response to this question is to focus on fields that are localized on a 3-brane so they do not probe the Kaluza-Klein states. However, as an alternative response, we point to an infinite number of examples that circumvent the familiar intuition. We will discuss known examples of spaces that have a large mass gap and a large volume. Consequently, even fields that did live in the bulk would find the lowest Kaluza-Klein state energetically difficult to excite.

The essential reason why some surfaces have large

FIG. 1: The lowest mode on the surface made by linking doughnuts together wobbles between the two halves divided by the curve . As links are added, the tone gets lower as the two symmetric areas grow.

minimum eigenvalue is related to the question famously posed by Mark Kac in the 1966 paper, “Can you hear the shape of a drum” [1]. While two drums can sound the same, as was shown nearly 30 years later [2], some features of the drum can be heard–you can ring out the eigenmodes of the Laplacian by banging the manifold [3, 4]. A reasonable guess is that the bigger the drum the lower the tone. For instance, imagine the lowest frequencies on a surface made from stringing together doughnuts as drawn in Fig. 1. The lowest tone will result when roughly half of the surface wobbles out of phase with the other half. This conforms with Cheeger’s bound on the minimum eigenvalue [5], which for a two-dimensional surface has the form, inf ` where is the length of a path that divides the surface into two areas, A1 and A2, and the infimum is taken over all area dividing paths. In the case of the string of doughnuts, the minimum (non-zero) eigenvalue does indeed go down with area. The larger the area, the lower the tone.

However, there are counter-examples. For instance there are hyperbolic spaces, as we’l elaborate, that correspond to large mass gap and large volume. Compactification on these spaces, and the associated cosmology, has been studied in [6–9]. While in two dimensions these spaces are topologically equivalent to a string of doughnuts, they are not metrically equivalent. There are no thin bottlenecks that divide the space into roughly equal parts, so there is no mode that wobbles a large area of the surface at once. The lowest tone amounts to wobbling a small area. In another analogy, like waves in a pond full of barriers, the eigenmodes can only excite small areas at a time due to the intricate arrangement of holes. No matter how big you make the drum by adding more handles and holes, the lowest tone does not get any lower.

Large-volume extra dimensions can be put to good use in diluting the strength of gravity, thereby accounting for arXiv:1001.1423v2 [hep-th] 30 Jan 2010

the small value of Newton’s constant. Besides this phenomenological advantage, they are a curious intellectual possibility: at every point in space there might be some large transverse volume that we simply cannot perceive, not because we’re confined to a brane, and not because the internal dimensions are small, but because it is simply too costly to do so at the low energies of our everyday experience. We discuss the mathematical constructions in §I.

As additional motivation for considering these spaces, they provide an attractive inflaton in the form of a bulk scalar field. We discuss this in general in §II and study a concrete model in §III. Inflation in a large volume, large gap compactification has the following attractive features: (1) a suppression of the 4d coupling constant so the inflaton potential is flattened, (2) a 4d description which remains valid, even during inflation, thanks to the large gap, (3) a 4d vacuum expectation value (vev) for the inflaton driven up to the 4d Planck scale M4, (4) an inflaton mass at the fundamental scale of the bulk M,

(5) inflation which takes place at an intermediate energy density ∼ M2M24, and (6) a standard cosmological evolution protected from copious and disruptive K mode production by energetics.

These models have some more speculative advantages.

The inflaton is very weakly coupled, which means it can double as a dark matter candidate. It is also tempting to revive the Kaluza-Klein idea in this context and construct a braneless model in which we are prohibited from detecting the extra dimensions by the large mass gap. We return to these possibilities in section §IV.

First we review the familiar arguments about the energetic expense of exciting modes in the internal space.

Consider the action for a scalar field in higher dimen- sions:∫

Here M is the fundamental scale of the higherdimensional description, and we have included an overall factor of Mn so that all fields, masses, and coupling constants will have the same units as in (3 + 1)-dimensions. Take the case of a product geometry for the N = 3 + n spatial dimensions M(3) × M(n), where M(n) is some compact internal manifold and M(3) represents our large dimensions so M(3), even if it is finite, is so large it is effectively R3. The metric is then separable, ds2 = GIJdxIdxJ = ηµνdxµdxν + b2hijdyidyj (3)

Here we have pulled out of the internal metric a dimensionful scale factor, b, that sets the size and curvature radius of the internal manifold and µ,ν = 0...3 while i,j = 4...N. Varying the action (2) gives the (N + 1)- dimensional wave equation for each φ mode, where = ηµν∂µ∂ν is the usual (3+1)-dimensional wave operator and the extra-dimensional Laplace-Beltrami op- erator 4 = hij∇i∇j is replaced by its eigenvalues with use of the Helmholtz equation

The last term in Eq. (4) is the usual Kaluza-Klein tower of heavy mass states, m2k = m2 + k2b−2. For instance, in the case of a circle S1 of size b, the masses, as dic- tated by eigenvalues of the Laplacian, are m ∼ n/b for n ∈ Z, which illustrates the well-known fact that for larger b the modes are easier to excite. In the absence of a brane, the circle would have to be smaller than b ∼ TeV−1 ∼ 10−16mm to hide excitations of standard model fields from experiments.

There is, however, an alternative mechanism for hiding the Kaluza-Klein modes. The intuition that the minimum energy mode will necessarily decrease into an observable domain as the volume of the internal space increases cannot be applied to all manifolds. Indeed there are an infinite number of compact manifolds whose minimum eigenvalue is large, implying a large mass gap, despite a having large volume. We consider these now.

A convenient starting point is the set of twodimensional surfaces with constant negative curvature, as summarized in the Ricci scalar R = −2/b2. The Gauss- Bonnet theorem connects the area of these spaces with their topology:

where g is the genus and b, again, is a dimensionful scale factor. The larger the genus, the larger the area of the surface for the same value of b. In most familiar examples, such as the string of doughnuts, the minimum eigenvalue goes down with the area for fixed b. But there is an extensive literature on the construction of Riemann surfaces (specifically, oriented compact surfaces with a metric of constant negative curvature) of arbitrary genus that possess a large first eigenvalue: large in the sense that the minimum non-zero eigenvalue is bounded below by the curvature scale b−2, and is independent of the area even as the area goes to infinity for fixed b [10–15].

First consider hyperbolic space Hn (with curvature −1). In n dimensions the square-integrable eigenvalue spectrum of Hn is k ∈ [(n − 1)/2,∞]. The corresponding eigenmodes define a complete set of states in which to expand the function φ. Although these squareintegrable eigenmodes do vary over lengths greater than the curvature radius, correlations beyond the curvature radius are exponentially damped. For this reason, these square-integrable modes are often referred to as subcurvature modes.

There are also super-curvature modes, modes with eigenvalues k < (n − 1)/2. These correspond to eigenmodes that are not square-integrable on Hn and are generally not considered in the expansion of fields. So it might seem as though there is an intrinsic mass gap even for the simply connected infinite hyperbolic plane: could we live with a transverse Hn and not know it? As mathematicians and physicists have both emphasized (see [16] and references therein), physical processes that generate random Gaussian fields in the early universe require contributions from both sub-curvature and super-curvature modes. We might therefore expect cosmological processes to probe the light part of the spectrum down to k = 0 for the infinite hyperbolic spaces, in which case we could not hide from the existence of the extra dimensions.

To hide the extra dimensions, we consider compact hyperbolic surfaces (with n = 2). It was originally conjec- tured by Buser in 1978 that the minimum eigenvalue k1 would go to zero for large genus [12]. However, he later disproved his own conjecture by exhibiting surfaces of arbitrarily large genus with minimum eigenvalue squared k21 ≥ 3/16 [13]. The surfaces in Buser’s proof come from number theoretic constructions. This therefore gives us

Riemann surfaces with arbitrarily large genus g, and therefore arbitrarily large area, that maintain a large mass gap, to use the physics lexicon. Since the work of Buser, the number-theoretic lower bound has been improved slightly to k21 ≥ 171/784 (for the same surfaces) [17], while the construction of surfaces was improved by Brooks and Makover to allow surfaces of arbitrary genus with first eigenvalue obeying nearly the same bound [10]. If Selberg’s conjecture that the square of the minimum can be replaced by 1/4 [14] is ever proven, then the theorem of Ref. [10, 1] would deliver the bound for these same surfaces.

In a separate construction, Brooks and Makover show that in fact a random surface has large first eigenvalue. More precisely, take a large number N of equilateral triangles and glue them together in a random way by pairing up the edges to obtain a triangulated surface. The resulting surface has a canonical conformal structure, and by the Uniformization Theorem there is a unique hyperbolic metric in the conformal class. Then there is a constant

C so that this hyperbolic metric will satisfy k21 ≥ C with a probability that goes to 1 as N goes to infinity. (How- ever, they do not give an explicit value for C, and their proof would probably give a very bad bound.) This then shows that for surfaces that are “random” in a certain sense the first eigenvalue behaves moderately well. For our purposes, it is more important that we have a good bound on k21 than that the surfaces be generic. We therefore continue with the number-theoretic surfaces, and will use Selberg’s conjectured bound k21 ≥ 1/4, al- though the difference between 171/784 and 1/4 has negligible impact on our application.

In practice then, there are surfaces of arbitrarily large genus, with area A ∼ 4pigb2 and a minimum eigenvalue bounded from below as in Eq. (7). For b = TeV−1, the mass gap, kb−1 ≥ TeV/2, is too large to overcome except in the highest energy settings and yet the area is large if g is large. For a g ∼ 1030, A ∼ (m)2. Despite such a large area, we would be unable to excite modes in the higher dimensions and would experience a 4d universe. Only at the energy scales of the Large Hadron Collider (LHC) could we expect to witness excitation of modes in the bulk.

These 2-surfaces are illustrative but there are presumably similar constructions in higher dimensions. Three dimensional hyperbolic internal spaces of arbitrarily large volume are known [18] and have the particularly nice feature of being rigid – all metrical quantities are fixed by the topology and the requirement of constant curvature [19]. In other words, if the volume is stabilized, all moduli would be stabilized as a result of the rigidity.

So far, we have consider only the Laplacian (scalar) spectrum. Spinors also need to see a large mass gap in a realistic theory. The Dirac eigenspectrum is less well studied and it is not yet known if the large genus hyperbolic surfaces discussed above have a suitable spectrum. Amman, Humbert, and Jammes have constructed surfaces (of any genus, with bounded volume) with a zero mode followed by an arbitrarily large gap in the Dirac spectrum [20, 21], although these surfaces (dubbed “Pinocchio surfaces”, formed by stretching out a long nose from the surface) do not have a suitable Laplacian spectrum.

Although we have focused on hyperbolic spaces, there are other constructions. For instance one can obtain a large gap on a flat 2-torus, simply by allowing the complex structure to degenerate [2–25]. Another example, which gives the desired Kaluza-Klein tower for both scalars and fermions, is a rectangular n-torus of volume ∼ bn with n 1. The mass gap stays fixed even as the volume can be sent to infinity by sending the number of dimensions to infinity. This is less remarkable than the hyperbolic construction: each individual direction is small and the large volume is simply a result of a large number of dimensions. On the down side, there is a huge spinor degeneracy since the number of spinors grows exponentially with the number of dimensions. Still, the ndimensional torus demonstrates the existence of a space that has the required large mass gap for both scalars and fermions.

One phenomenological advantage to having a large volume is that it weakens the observed force of gravity in four dimensions. But any other bulk interactions will be suppressed as well. In this section we use this to help construct inflationary potentials [26, 27].

We begin with a φ4 theory in the bulk, with action∫

(8) where bulk quantities carry a B. Integrating over the internal dimensions the action becomes∫

where we canonically normalize the kinetic term by redefining φ = V1/2φB. Here

is a dimensionless measure of the volume of the internal space, and the 4d couplings and vev’s are related to bulk values through

It follows that masses are the same in the bulk and the 4d description.

These simple equations highlight the main features of large-volume compactification: we naturally get models with tiny couplings and huge vev’s. To give a sense of scale we need to compare to the gravitational action under dimensional reduction:∫

M24R(4) + | ] |

Here M is the underlying higher-dimensional scale and the effective reduced four-dimensional Planck mass is

Leaving M the unknown, this requires the volume adjust

Taking M ∼ TeV, for example, we would have a scalar field with

Thus the coupling in the 4d theory is minute. The vev is at the 4d Planck scale, while the mass is much below Planck scale. Intriguingly, this implies that if there exist fundamental scalar fields in the bulk their interactions should be brutally suppressed. We would not easily observe such scalar fields, as indeed we do not. Furthermore, any scalar field potential would be exceedingly flat as slow-roll inflation requires: a very small coupling and a very large vacuum expectation value. And, neatly enough, any remnant scalar particles from the early universe would be dark matter candidates, with a mass set by the underlying higher-dimensional Planck scale M.

We note that although φ has mass set by the bulk scale

M, inflation occurs at a much higher energy scale. Near the maximum of the potential, where φ v, the effective 4d energy density is where we’re assuming the bulk energy density λBv4B ∼ M4 ∼ TeV4. So an intriguing observation about the in- flaton potential is that the energy scale of inflation would be 1010 GeV despite being driven by a field with an electroweak scale mass.

Although the choice M ∼ TeV is natural from the point of view of electroweak physics, the resulting inflationary scale ∼ 1010 GeV does not generate density perturbations of the required magnitude. Instead, as we’l see in the next section, the observed density perturbations favor the existence of an intermediate fundamental scale, with M ∼ 1011 GeV and V ∼ 1014.

In this section we study a concrete model of bulk inflation and show that we can get a reasonable power spectrum, density perturbations of the right magnitude, and the requisite number of e-folds, all with tunings of the inflaton potential at the few per mil level.

We emphasize that any reasonable potential could be chosen for the inflaton. For simplicity we take a potential of the form where λ,v are set as in Eqs. (16). Setting α = 0 recovers the usual φ4 potential. We could equally well have used a potential of the Coleman-Weinberg type [28] or any other variant of inflaton potential. There are various phenomenological constraints that must be satisfied.

Power spectrum First we quantify the naturalness of V as a slow-roll inflaton potential using the parameters described in Refs. [29–3]. Slow-roll inflation is a consistent assumption if the slope and the curvature of the potential are small as

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