Holographic cosmology from a system of M2–M5 branes

Abstract

In this paper, we analyze the holographic cosmology using a M2–M5 brane configuration. In this configuration, a M2-brane will be placed in between a M5-brane and an anti-M5-brane. The M2-brane will act as a channel for energy to flow from an anti-M5-brane to a M5-brane, and this will increase the degrees of freedom on the M5-brane causing inflation. The inflation will end when the M5-brane and anti-M5-brane get separated. However, at a later stage the distance between the M5-brane and the anti-M5-bran can reduce and this will cause the formation of tachyonic states. These tachyonic states will again open a bridge between the M5-branes and the anti-M5-branes, which will cause further acceleration of the universe.

Introduction

It is widely known that the entropy of the black holes is proportional to the area of the horizon, and its temperature is proportional to the surface gravity. Thus, a link between gravity and thermodynamics has been established. This link has become the basis of the Jacobson formalism  [1]. In the Jacobson’s formalism, the Einsteins equations are obtained from the first law of thermodynamics. It has been possible to derive the Friedmann equations from the Clausius relation using this Jacobson formalism  [2]. In deriving the Friedmann equations, the entropy is assumed to be proportional to the area of the cosmological horizon. In fact, motivated by the Jacobson formalism, it has been proposed that the gravity is an entropic force  [3]. In fact, Einstein’s equations have been obtained from this entropic force formalism. Furthermore, as the entropy of the bulk of the black hole is proportional to the area of the boundary, it has led to the development of the holographic principle. The holographic principle states that the number of degrees of freedom of a region in space is the same as the number of degrees of freedom on the boundary of that region.

The holographic principle has motivated the development of the holographic cosmology  [4], [5], [6], [7], [8]. The holographic cosmology is based on the idea that the difference between the degrees of freedom in a region and the degrees of freedom on the boundary surrounding that region drives the expansion of the universe. The holographic cosmology has been studied in the context of Lovelock gravity, with special emphasis on the Gauss–Bonnet gravity  [9]. The holographic cosmology for the brane world models  [10], scalar–tensor gravity  [11], and $f\left(R\right)$   [12], has also been discussed. It may be noted that this analysis was performed using a thermodynamic description of the brane world models, holographic scalar–tensor cosmology, and holographic $F\left(R\right)$ cosmology  [13]. The holographic cosmology has also been generalized to the Friedmann–Robertson–Waker universe with an arbitrary spatial curvature [14]. This generalization has been performed for non-flat universes by using the aerial volume instead of the proper volume  [15]. In this context, the Friedmann equation for the Lovelock gravity with an arbitrary spatial curvature have also been studied  [16].

The holographic cosmology has been studied using the BIonic solution  [17]. This BIon solution is a configuration of a D3-brane and an anti-D3-brane with a wormhole in between them. This action for the D-branes is a non-linear action called the Dirac–Born–Infeld (DBI) action, and the non-linearity of this action is important in constructing this BIonic solution  [18], [19], [20], [21]. The F-string end on a point of a D-brane in this BIonic solution, and the F-string charge gets associated with the world-volume electric flux carried by the D-brane. The D3-brane of the BIon has been identified with our universe and the BIon solution has been used for analyzing the holographic cosmology  [22], [23], [24], [25]. In this solution, first a BIon forms from black F-strings. Then the degrees of freedom of the D3-brane increase as energy flows from an anti-D3-brane into the D3-brane. However, as the D3-brane moves away from the anti-D3-brane, then the spike of the D3-brane gets separated from the spike of the anti-D3-brane spike, and the inflation ends at this stage. Finally, when the D3-brane comes close to the anti-D3-brane, a new wormhole forms due to the tachyonic states. This wormhole also increases the degrees of freedom on the D3-brane, and this causes late time acceleration of the universe.

It may be noted that the F1–D3 intersection is U-dual to a system of M2–M5 branes  [26], [27]. The M2-branes intersecting with M5-branes have been analyzed in the supergravity regime  [28], [29]. This analysis has been done using the blackfold approach. Thus, it has been possible to recover the $1/4$-BPS self-dual string solution  [30], [31], as a three-funnel solution of an effective five-brane world volume theory  [32], [33], [34], [35]. The finite temperature effects for non-extremal self-dual string solution solutions and wormhole solutions interpolating between stacks of M5-branes and anti-M5-branes have also been studied. These solutions define a BIon solution in M-theory  [36], [37]. It would be interesting to perform a similar analysis for this M-theory system. So, it is possible to study the holographic cosmology using a M2–M5 brane system. Thus, we will first study a M5-brane connected to an anti-M5-brane by a M2-brane. This will cause the degrees of freedom to flow into the M5-brane causing inflation. The inflation will end when the M5-brane and the anti-M5-brane get separated. However, at a later stage when the M5-brane approaches the anti-M5-brane, tachyonic states will be formed. These tachyonic states will form another bridge between the M5-brane and the anti-M5-brane. We will investigate the inflation in this model of M2–M5 branes, along with the consequences of the formation of these tachyonic states.

It is possible for the four dimensional universe to emerge from compactification of M5-branes. However, in this paper, we will not discuss such a compactification, and we will discuss the inflation using M5-brane geometry. In fact, inflation has already been studied using the M5-brane geometry  [38], [39], [40], [41]. However, in this paper, we study the inflation in the M-theory using the recently proposed proposal of holographic cosmology  [4], [5], [6], [7], [8]. It may be noted that it is possible for the branes moving in the extra dimensional bulk to collide with each other. Such collisions have been studied in the context of Ekpyrotic universe  [42], [43], [44], [45], [46], [47]. In this paper, we will also use the formalism of holographic cosmology  [38], [39], [40], [41], to discuss the state of the universe before such a collision occurs. It may be noted that such a model has also been studied using BIonic solution  [22], [23], [24], [25], and such a system is U-dual to a system of M2–M5 branes  [26], [27]. So, this system is actually U-dual to the BIonic cosmology. In other words, this is the M-theory version of the holographic cosmology, which has so far only been studied in the string theory  [22], [23], [24], [25].

We would like to point out the fact that F1–D3 intersection is U-dual to a system of M2–M5 branes is only mentioned as an observation at this stage. The analysis in this paper seems to be similar to the analysis performed for using BIonic solution, so there might be a deeper relation between such a duality. However, at this stage we only mention this as a motivation to study inflation using a M2–M5-brane system. We would also like to point out that the bulk is populated by M5-branes and anti-M5-branes. So, a random collision between a M5-brane and an anti-M5-brane can occur. However, this will be a collision between a random M5-brane with a random anti-M5-brane in the bulk. Such collision between random branes occurs in Ekpyrotic universe  [42], [43], [44], [45], [46], [47]. However, in this model, we can calculate the state of the universe just before such a collision occurs.

The paper is organized as follows. In Section  2, we discuss the holographic inflationary cosmology using the system of M2–M5 brane. In Section  3, we analyze the tachyonic states of the M2–M5 brane system. Finally, in Section  4, we will summarize our main results. We will also discuss possible extension of the results obtained in this paper.