Thick self-gravitating plane-symmetric domain walls

We investigate a self-gravitating thick domain wall for a λΦ potential. The system of scalar and Einstein equations admits two types of non-trivial solutions: domain wall solutions and false vacuum-de Sitter solutions. The existence and stability of these solutions depends on the strength of the gravitational interaction of the scalar field, which is characterized by the number ǫ. For ǫ ≪ 1, we find a domain wall solution by expanding the equations of motion around the flat spacetime kink. For “large” ǫ, we show analytically and numerically that only the de Sitter solution exists, and that there is a phase transition at some ǫmax which separates the two kinds of solution. Finally, we comment on the existence of this phase transition and its relation to the topology of the domain wall spacetime. INTRODUCTION The spacetime of cosmological domain walls has now been a subject of interest for more than a decade since the work of Vilenkin and Ipser and Sikivie [1,2], who used Israel’s thin wall formalism [3] to compute the gravitational field of an infinitesimally thin planar domain wall. This revealed the gravitating domain wall as a rather interesting object: although the scalar field adopts a static solitonic form, the spacetime cannot be static if one imposes a reflection symmetry around the defect [1,2], but displays a de Sitter expansion in any plane parallel to the wall. Moreover, there is a cosmological event horizon at finite proper distance from the wall; this horizon provides a length scale to the coupled system of the Einstein and scalar field equations for a thin wall. After the original work by Vilenkin, Ipser and Sikivie [1,2] for thin walls, attempts focussed on trying to find a perturbative expansion in the wall thickness [4,5]. With the proposition by Hill, Schramm and Fry [6] of a late phase transition with thick domain walls, there was some effort in finding exact thick solutions [7,8]; however, these walls were ∗E-mail address: [email protected] †E-mail address: [email protected] ‡E-mail address: [email protected] 1 supposed to be thick by virtue of the low temperature of the phase transition, which means that the scalar couples very weakly to gravitation. The suggestion that the cores of defects created near the Planck time could undergo an inflationary expansion [9,10] then reopened the question of thick domain walls (where now thick means relative to the wall’s natural de Sitter horizon). This time, the high temperature of the phase transition ensures that the scalar field in this case interacts very strongly with gravity. Here, we consider gravitating thick domain wall solutions with planar and reflection symmetry in the Goldstone model; a more detailed discussion can be found in our paper [11], where we consider general potentials and de Sitter/anti-de Sitter background spacetimes. DOMAIN WALLS IN FLAT SPACETIME In order to fix the notation let us first briefly review a domain wall in flat space-time [12]. Consider a flat metric ηab with signature (+,−,−,−) and spacetime coordinates x a = {t, x, y, z}. The matter Lagrangian will be given by the λΦ Lagrangian L = η ∇aΦ∇bΦ− V (Φ) (1) V (Φ) = λ ( Φ − η )2 , (2) where Φ = Φ(z) is a real scalar field. For a potential with a non-trivial degenerate set of minima, such as (2), one gets domain wall solutions because the vacuum manifoldM = {±η} is not connected. We scale out the dimensionful symmetry breaking scale η by letting X = Φ/η and we set the wall’s width w = 1 √ λη to unity (thus measuring distances in wall units rather than Planck units). Then the matter langrangian takes the simplified form L = −(X ′)2 − (X − 1) (3) with equation of motion X ′′ − 2X(X − 1) = 0. (4) The well known kink solution is given by X(z) = tanhz, an odd function approaching exponentially the true vacua ±1 as z → ±∞ (see figure 1). GRAVITATION AND GENERAL SETTING OF THE PROBLEM Now let us couple our flat space-time domain wall solution to gravity in a minimal way. Consider a domain wall with local planar symmetry, reflection symmetry around the wall’s core at z = 0. The matter langrangian is given by, L = g∇aX∇bX − (X 2 − 1) (5) and coupled to gravity via a metric admitting these symmetries, of components gab. We suppose again that X = X(z), a static field and the metric gab is given by, ds = A(z) dt −B(z, t) (