Global monopoles can change Universe's topology

If the Universe undergoes a phase transition, at which global monopoles are created or destroyed, topology of its spatial sections can change. More specifically, by making use of Myers' theorem, we show that, after a transition in which global monopoles form, spatial sections of a spatially flat Universe become finite and closed. This implies that global monopoles can change topology of Universe's spatial sections (from infinite and open to finite and closed). We emphasize that global monopoles cannot alter topology of the space-time manifold.


INTRODUCTION
The question of global properties (topology) of our Universe is a fascinating one, and it has been attracting attention for a long time. Yet only as-of-recently the data have been good enough to put observational constraints on the Universe's topology. It is often stated that the general FLRW metric ds 2 = −c 2 dt 2 + a 2 (t)dr 2 1 − κr 2 + a 2 (t)r 2 [dθ 2 + sin 2 (θ)dφ 2 ], (1) where c is the speed of light and 0 ≤ r < ∞, 0 ≤ θ ≤ π, 0 ≤ φ < 2π are spherical coordinates, implies that, if the curvature of spatial sections κ is 1. negative (κ < 0), then the spatial sections of the Universe are hyperbolic and infinite (not compact), 2. zero (κ = 0), then the spatial sections are flat and infinite and 3. positive (κ > 0), then the spatial sections are finite (compact) and homeomorphic to a three dimensional sphere.
The first observational evidence that supports that we live in a flat and infinite universe was presented in 2000 by the balloon experiments Boomerang [1] and Maxima [2]. The most recent bound on κ [3] is obtained when recent BAO observations are combined with the Planck data [4] and the polarization data from the WMAP satellite (WP) Ω κ = −0.003 ± 0.003, , where H 0 ≃ 68 km/s/Mpc, showing a slight (one σ) preference for κ > 0 (the same conclusion is reached when the BAO data are dropped [4], 0.006 > Ω κ > −0.086).
Eq. (2) implies a lower bound on the curvature radius of spatial sections, R c = 1/ |κ| ≥ 60 Gpc. In spite of this slight preference for a positive spatial curvature, a robust * a.marunovic@uu.nl, t.prokopec@uu.nl conclusion remains "that our Universe is spatially flat to an accuracy of better than a percent" (cited from page 42 of Ref. [4]).
Even if the Universe is spatially flat, one can impose periodic boundary conditions, making the Universe effectively finite and thus determining Universe's global topology. Although different scenarios have been considered in literature (for a recent review see [5], and for somewhat older reviews see [6,7]), no evidence has been found that would favor any of the proposed models.
The above considerations make an implicit assumption that the spatial curvature of the Universe is given and that it cannot be changed throughout the history of our Universe. In this letter we argue that this assumption ought to be relaxed, and we propose a dynamical mechanism: formation of global monopoles at an early universe phase transition, by which the (average, measured) spatial curvature of the Universe can change in the sense that it will become positive if it starts slightly negative or zero. This claim will leave many readers with a queasy feeling since, when κ changes from κ ≤ 0 to κ > 0, spatial sections change from infinite (hyperbolic or parabolic) to finite (elliptic), thus changing topology of spatial sections. One should keep in mind that all this happens at space-like hyper-surfaces of constant time, and hence it is not in contradiction with any laws of causality. And yet it does leave us with an uncomfortable feeling that 'somewhere there' distant spatial sections of the Universe are reconnecting, thereby changing them from infinite to finite and periodic. Even though not directly observable today, this process can have direct consequences for our future. Indeed, when an observer in that reconnected Universe sends a (light) signal, it will eventually arrive from the opposite direction. Furthermore, the future of a spatially finite (compact) universe can change from infinite and uneventful to finite and singular (namely, if cosmological constant is zero such a universe will end up in a Big Crunch singularity). Because of all of these reasons, a tacit consensus has emerged that no topology change is possible in our Universe. We argue in this work that this consensus needs to be reassessed.
In fact, the idea that the curvature of spatial sections could change can be traced back to the work of Krasinski [8] based on Stephani's exact solution [9] to Einstein's equations. Even though Krasinski has argued that the curvature of spatial sections could dynamically change, he has not offered any mechanism by which such a change could occur [10]. In this letter we provide such a dynamical mechanism.
A particularly instructive case to consider is the maximally symmetric de Sitter space, whose geometry can be clearly visualized from its five dimensional (flat, Minkowskian) embedding (see figure 1), Thus de Sitter space is geometrically a four dimensional hyperboloid H 4 , and its symmetry is the five dimensional Lorentz group, SO(1, 4), which has -just like the Poincaré group of the symmetries of Minkowski space -10 symmetry generators. This means that de Sitter space also has 10 global symmetries (Killing vectors). Common coordinates on de Sitter space (3)  Krasinski has, however, pointed out that there are also de Sitter coordinates in which κ changes in time. Both cases, when κ changes from negative to positive, and v.v. are possible. An example of the metric when κ(t) changes from negative to positive can be easily inferred from [8], where r 0 , c, H are constants. That this is a de Sitter space can be checked, for example, by evaluating the Riemann tensor. One finds where R = 12H 2 /c 2 ≡ 12/R 2 H is the Ricci curvature scalar, H = const. is the Hubble parameter and R H = c/H is the Hubble radius. Relation (5) holds uniquely for maximally symmetric spaces such as de Sitter space. The curvature of spatial sections of de Sitter in (4) can be inferred from the Riemann curvature of spatial sections, from which we infer, which means that κ < 0 for t < 0, κ = 0 for t = 0, and κ > 0 for t > 0. Note that topology of spatial sections changes at t = 0. For t < 0 the sections are three dimensional hyperboloids, with a time dependent (physical) throat radius r c (t) = r 3/2 0 / √ −ct, for t = 0 they are paraboloids and for t > 0 they are three-spheres with a (time-dependent) radius, r c = r figure 1). Consequently, topology of spatial sections changes at t = 0, as can be seen in figure 1 [11]. A similar (albeit inhomogeneous) construction is possible on FLRW space-times. While this shows that there are observers for which topology of spatial sections of an expanding space-time changes, it does not tell us how to realize such a change, and whether such a change is possible in a realistic setting. This is what we address next.

THE MODEL
The action for gravity we take to be the Einstein-Hilbert (EH) action, where G N is the Newton constant, R is the Ricci curvature scalar, and g is the determinant of the metric tensor g µν . In our model the EH action is supplemented by the action that governs the dynamics of global monopoles, with the Higgs type of O(3) symmetric potential where repeated indices a indicate a summation over a = 1, 2, 3. µ is a mass parameter and λ φ is a self-coupling. The scalar field φ = (φ a ) (a = 1, 2, 3) consists of 3 real components, such that the action (9) is O(3)-symmetric. When µ 2 < 0 the vacuum exhibits a field condensate, φ a φ a ≡ φ 2 0 = −µ 2 /λ φ , which spontaneously breaks the O(3) symmetry of the action to an O(2), such that the resulting vacuum manifold M has a symmetry of the two dimensional sphere, M = O(3)/O(2) ∼ S 2 . The two excitations along the two orthogonal directions of S 2 are the two massless Goldstone bosons, while the excitation orthogonal to M is massive, m 2 (φ 0 ) = 2λ φ φ 2 0 = −2µ 2 . The potential (10) is chosen such that the energy density of the (classical) vacuum is V (φ 0 ) = 0.
It is well known that the action (9-10) permits non-vacuum classical solutions known as global monopoles [13]. Global monopoles have a non-vanishing vacuum energy, but they do not decay as they are stabilised by topology. The simplest such solution of the equations of motion is a hedgehog-like spherically symmetric solution of the form, φ(t, r ) = φ(r) (sin θ cos ϕ, sin θ sin ϕ, cos θ) T , where θ, ϕ and r are spherical coordinates. One can show [14] that the topological charge (also known as the winding number) of that solution is unity, and that it is stable under small field perturbations.
Global monopoles are generically created at a phase transition by the Kibble mechanism [13] (at least of the order one per Hubble volume) if the effective field mass matrix, [m 2 eff (φ a = 0)] ab = ∂ 2 V eff /∂φ a ∂φ b changes from having all positive eigenvalues to at least one negative eigenvalue [15,16], which can be realised in e.g. a hot Big Bang.
One important and defining property of global monopoles is their solid deficit angle [17] (see also the Appendix), which extends to the particle horizon associated with the monopole creation event.

GLOBAL AND LOCAL PROPERTIES OF SPACE-TIME
As shown in the Appendix, at sufficiently large distances, the metric of a monopole can be approximated by a FLRW metric with a solid deficit angle. This deficit angle generates a spatial Ricci tensor that breaks spatial homogeneity of the FLRW space-time and decays as , where ∆ is the deficit angle and m φ the monopole mass. In the limit of a large number of monopoles per Hubble volume all with ∆ ≪ 1 and if in the absence of monopoles the spatial curvature is zero, a local observer that is sufficiently far from any individual monopole will observe a metric that can be approximated by a FLRW metric with a positive spatial curvature. Hence, an observer in such a universe will tend to conclude that the Universe is spatially compact and finite.
In this section we show that a (spatially flat) cosmological space filled with randomly distributed global monopoles must have spatial sections that are closed, and therefore its spatial geometry is that of a three dimensional sphere. This conclusion is reached based on the well known Myers' theorem [18], which states that for any Riemannian manifold whose Ricci curvature R j i is positive and limited from below as, the distance function d(x; x ′ ) is limited from above by d(x; x ′ ) < π/ √ R min in any number of (spatial) dimensions. This then implies that the manifold is globally closed and its radius of curvature is limited from above as r c ≤ π/ √ R min . This powerful theorem relates local properties of Riemannian manifolds to their global properties. In particular, when applied to global monopoles, that, at asymptotically large distances, the minimum component of the Ricci is (see Eq. (22) in the Appendix), Assume that the monopole-monopole correlation function is that of randomly (Poisson) distributed monopoles with an average distance squared, Let us consider a sphere of radius ar ∼ aσN 1/3 , in which the average number of monopoles is, N ∼ (ar/σ) 3 ∼ 1/∆, which is the number needed to close the space (here we neglect factors of order unity, such as the volume of the 3-dimensional unit sphere, Vol(S 3 ) = 2π 2 ). The Poisson distribution then implies that the maximum distance between two monopoles is ar max ∼ aσ[ln(∆ −1 )] 1/3 . Now, according to Myers' theorem, a spatial section of the Universe filled with monopoles is a closed inhomogeneous manifold with a (comoving) radius of curvature, r c m φ aσ 2 ∆ −1/2 [ln(∆ −1 )] 2/3 . While this represents an upper bound, the actual Universe's curvature radius will be smaller, probably of the order r c ∼ σ/ √ ∆, where a dilute monopole gas is assumed, maσ ≫ 1.
To conclude, due to their deficit angle, formation of global monopoles can have an impact on global (topological) properties of the (spatial sections of the) Universe. In particular, a (spatially flat) universe filled with global monopoles will have a closed geometry and, in the limit of many monopoles where each has a small deficit angle, it will closely resemble a closed universe with a constant positive spatial curvature κ. We have thus shown that, if before a phase transition at which global monopoles form, spatial sections of the Universe are flat or slightly negatively curved and therefore infinite, after the phase transition the Universe will have on average a positive curvature and its spatial sections will be homeomorphic to a three-sphere and hence compact.
Even though the Universe filled with global monopoles resembles a FLRW universe with κ > 0, it is an inhomogeneous universe with an uncertain future (at the moment it is not clear to us whether the Universe will end up in a Big Crunch or it will expand forever). Furthermore, an observer residing sufficiently close to a monopole, on top of the usual Hubble flow will feel a repulsive gravitational force that points away from the monopole core (see e.g. [19]). By studying physical effects of this force a local observer will be able to distinguish between a homogeneous universe and a universe filled with global monopoles. Next, while on a clump of matter photons and particles get deflected by an angle that depends on their velocity, sufficiently far from a global monopole the deflection angle will be equal for (relativistic and nonrelativistic) massive particles and photons, i.e. it will be independent on particle's speed.
Furthermore, measuring spatial curvature on the largest observable scales (such as it is done by modern CMB observations) can provide an upper limit on the number of monopoles in our horizon. Since CMB observatories such as the Planck and WMAP satellites measure κ on the Hubble scale, they can be reinterpreted as the upper limit on the total solid deficit angle in our Hubble volume generated by global monopoles, where −(Ω κ ) max represents the upper limit on −Ω κ = κc 2 /H 2 0 allowed by the observations and H 0 ≃ 68 km/s/Mpc is the Hubble parameter today.
(17) In presence of a global monopole (11) the asymptotic behaviour of the A and B functions in (17) is [14], (f 0 and g 0 are constants) and that of φ(r) in (11) is, with φ 2 0 = −µ 2 /λ φ , see Eqs. (9)(10)(11). With these in mind, one obtains, for non-vanishing components of the stressenergy tensor the following asymptotic form, where m 2 φ = 2λ φ φ 2 0 ≫ H 2 . On the other hand, the Einstein equations for the manifold's spatial sections are, Upon comparing these with (20) one obtains that the monopole contributes to the relevant components of the spatial Ricci curvature tensor as, where ∆ denotes the solid deficit angle.