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Non-singular Bounce from a Phase transition

The single field Ekpyrosis of Khoury and Steinhardt \cite{khoury}-\cite{khoury1} admits a Jordan frame scalar field with a double well potential. After the end of the ekpyrotic phase, as energy density increases we find a phase transition scenario to a symmetric phase at the conformal coupling fixed point leads to a non-singular bounce in Einstein frame.


INTRODUCTION
The density fluctuations in the observed Cosmic Microwave Background [3]- [4] requires a nearly scale invariant curvature power spectrum within an early Hubble radius, see [5]. The current paradigm to provide these fluctuations, as well as a flat, homogeneous and isotropic universe, is inflation [4]. Inflation can be realised using effective models of canonical quantum scalar fields with a plateau potential in Einstein frame [5]- [8]. From a similar scalar field plateau potential, one can also obtain nearly invariant curvature perturbations in slowly evolving (contracting or expanding) solutions, [9]- [10]. For the inflation solution, conceptual challenges such as super-Planckian problem, multiverse problem aside [5]- [7], the isotropic power loss of 5 ∼ 10% in the small multipole l < 40 range, suppression of quadruple and dipolar asymmetry [4], are notable concerns though they are arguably within statistical variance. Attempts to address these issues usually require pre-"slow-roll inflation" physics such as a constant roll phase [11]- [12], a fast roll phase [13], a "slowly expanding" phase or a "slowly contracting" phase [9]. The slowly contracting phase we focus on in this work is the "Ekpyrotic" phase in [1]- [2], which requires a bounce to connect the contracting phase to the subsequent expanding phase, whether it be a radiation dominant phase or an inflation phase. A bounce can be non-singular if it occurs at energy scales sufficiently lower than the Planck scale, where the evolution of the classical background is more trustable.
To obtain a bounce mechanism presents a major challenge. After the Ekpyrotic phase, the Einstein frame Hubble constant is H E < 0 and the scale factor of the universe is in a slowly contracting phase. For an effective bounce to occur, one requires a phase whereḢ E > 0 and after the bounce a subsequent field roll in a non negative potential V ≥ 0, see [14]. In the scalar field kinetic energy dominant epoch, with a shrinking scale factor a, the field kinetic energy has the largest power ǫ in a −2ǫ amongst cosmological fluids, spatial curvature and static field potential. This slowly contracting phase is shown to be an effective smoothing mechanism [15] and addresses the gravitational entropy problem [16]. A contracting universe will keep contracting unless some "additional" conditions exist that leads toḢ E > 0. One early approach is by allowing, for a short time before the bounce, the kinetic or gradient term to take the wrong sign with its cocomitant violation of Null Energy Conditon (NEC) [17]. More careful analysis [18] shows that the cosmological perturbations along the transition between dark energy field and phantom field is unstable. Some successes are found in a class of models with Kinetic Gravity Braiding [19] and Galileons [20]- [21], though challenge remains [22]. Other notable solution involves higher derivatives operator [23]. Attempts to obtain a bounce from canonical scalar fields include using a generic positive curvature [24] and adding string inspired negative energy density term [25]. In Jordan frame, one can implement a bounce without Null Energy Condition (NEC) violation, [26]- [27]. Yet these solutions are shown to be dual to inflation in Einstein frame. A Jordan frame model with a conformal coupling involving the Higgs field is also considered in [29]- [30], in which the Einstein frame scalar field plateau potential is described as a double well potential in Jordan frame. A phase transition is effected when the quartic coupling of the scalar field passes between a positive value region and a negative value region. In this work, we consider a Jordan frame double well potential which goes through a phase transition at conformal coupling.
As we consider a bounce after the end of Ekpyrotic phase in [1]- [2], we recall their results for completeness. Here one considers a canoncial scalar field with a lifted plateau potential in Einstein frame where M pl is the Planck mass, V = −V 0 e −cϕ/Mp is the negative potential and V 1 , usually taken as V 1 = V 0 , is the lift to take the overall potential to positive density at large ϕ. This potential provides an effective model for Inflation [8].
However, the same model also admits two other solution branches, namely adiabatic Ekpyrosis and a slow expansion followed by a slow contraction [10].
Ekpyrosis scenario [32]- [34], where a period of ultra slow scale factor contraction, with equation of state parameter ǫ ≫ 1, can be shown to produce scale invariant power spectrum fluctuations. For potential in Eq.(1), we take M pl = 1, with scale factor a(t) and Einstein frame Hubble constant H E , where the cosmic time t is negative and increasing toward 0 − . At large c, the field has a tachyonic roll given by the equation Given that −t ek−beg and −t ek−end are the points where Ekpyrotic phase begins and ends, Eq.(2) has the following scaling solution for larger |t| ≫ |t ek−beg | where t ek−beg is the time at which the pre-Ekpyrotic phase ends and the Ekpyrotic phase begins. The equation of state parameter ǫ changes from a small value at t 2 ≫ 1 towards a large constant c 2 2 as time grows towards t ek−beg .
In this "Pre-Ekpyrotic" phase, the Hubble radius is large and the small momentum perturbations grow outside of a curvature scale (a reminiscence of the correlation length) that is smaller than the Hubble radius. These scale invariant perturbations which is red-tilted stay inside the Hubble radius and can only exit the Hubble radius during a subsequent Ekpyrotic (contracting Hubble radius) phase. During the ekpyrotic contraction, the red-tilt spectrum will exit the Hubble radius first to be followed by the blue-tilt "ekpyrotic" spectrum.
Cosmological observations of non-gaussianity and a small red-tilt requires that c(ϕ) in Eq.(1) to be very large (∼ 10 28 ) [1] in both the "pre-Ekpyrotic" phase and at the beginning of the Ekpyrotic phase. Also c ′ (ϕ) > 0 (that is c decreases as ϕ decreases) is required to reduce c(ϕ) to O(1) where the Ekpyrotic phase ends. After c(ϕ) takes O(1) value, we haveḢ E = − 1 2φ 2 . To match observations that the Hubble radius is expanding at late time.Ḣ E needs to change from a negative value to positive value (Ḣ E > 0). Typically the connection between the contracting phase to expanding phase (the bounce) is described schematically as a → a,ȧ → −ȧ, ϕ → ϕ,φ →φ.
In this work, by noting that the Einstein frame canonical scalar field with a plateau potential in Eq.(1) admits a corresponding Jordan frame scalar field with a double well potential, we consider that the system is in a broken symmetric phase. As energy density increases we study the scenario that this Jordan frame scalar field goes through a phase transition where the non-minimal coupling bounces off its conformal value while its mass term goes through zero. We find thatḢ E > 0 appears naturally in the broken symmetric phase leading to a bounce into the symmetric phase. In the next section we recapitulate the correspondence of Eq.(1) in Jordan frame. In section 3 and 4, we consider the scaling region in the broken symmetric and the symmetric phase respectively. In section 5 and 6, we consider the bounce and the subsequent field roll in the effective potential in the symmetric phase. We end with a conclusion and discussion.
The Jordan frame double well potential Starting with a Jordan frame action the Einstein frame action is given by (see for example [36]) where for K ≤ 6 (where K = 6 is the conformal coupling value), where the positive kinetic energy corresponds to K < 0. The Einstein frame canonical field ϕ is related to Jordan frame field φ by Here "c" is no longer a free parameter but is related to the non-conformal coupling "K" by the equation such that for near conformal coupling value K < ∼ 6, the parameter c takes on a very large value.
In Scalar-Tensor theory modelling of cosmological evolution from high energy density, as energy density ρ reduces, K(ρ) < 6 is usually taken as a fine-tunable parameter, while the parameters in potential V J is either taken to be fixed or to vary with K, see for example [37].
We consider a a double well potential in Jordan frame, its Einstein frame action is given by where the potential V E matches the plateau potential of Eq.(1) if we set λ = V 1 and 1 2 m 2 = V 0 . The positive potential with V 0 = − 1 2 m 2 is the simplest choice in many scalar-tensor theories. [37] In the Ekpyrotic phase where a(τ ) ∼ constant, the Einstein frame equation of state parameter is given by Although our primary attention is in the Einstein frame dynamics, the Jordan frame dynamics, especially the scaling behaviour of the parameters play a critical role. The equation between H J and H E is given as We note that from Eq.(19) that the "Pre-Ekpyrotic" phase of [1]- [2] corresponds to a stable 3H 2 E = λ where the sum of the negative potential and the kinetic term is small comparing to λ In Jordan frame, this corresponds to a nearly conformal coupling K ≃ 6 and a large φ 2 for some fixed m 2 . As ϕ (and φ 2 ) decreases, when the negative potential dominates over λ, one enters the "Ekpyrotic phase" .
At the beginning of the Ekpyrotic phase, starting with a large positive field (ϕ > 0), a negative field speed (φ < 0) and large negative time t, the evolution of system in Einstein frame follows the work of [1]- [2] with solution in Eq.(3). From Eq.(16), the unnaturally large value c ∼ 10 28 can be easily understood in Eq.(13) as K having a near conformal coupling value. As K moves away from the conformal coupling value to 6 > K > 16/3, we have ǫ > 3 towards the end of Ekpyrotic phase. From Eq.(11) we take the corresponding starting φ at φ > 0, withφ = (cφ)φ < 0. In this scenario the Jordan frame field φ rolls down from a large positive field to its bottom (corresponding to the Einstein frame pre-Ekpyrotic phase) and rolls up the potential towards the φ = 0 (corresponding to the Ekpyrotic and Kinetic energy phase). At the end of Ekpyrotic phase, K is still not far away from the conformal coupling value. We note that scale invariant curvature spectrum is also obtained in [26]- [27] using Jordan frame model with nearly scale invariant potentials, though this spectrum are inflation related.
After the Ekpyrotic phase, the scale factor continues to shrink in the Einstein frame (which is also true in Jordan frame for slow varying φ > 0). The Hubble constant increases towards the Planck scale. As the canonical kinetic term dominates over the potential with decreasing scale factor, the energy density which scales as ∼ a −6 can only continue to increase since none of the known cosmological components can slow it down. To avoid reaching the Planck energy density, one way is to consider the existence of a high energy density fixed point below the Planck scale. Jordan frame theories in the form of Eq. (9) are used to describe the universe's evolution from very high energy densities to low energy densities over a large energy density scales. Conformally invariant fixed point K = 6 and m 2 = 0 provides an important asymptotic solution. In [38], for a positive potential it is found that as m 2 φ 2 → 0, the system under functional renormalisation group flow will reach the conformally invariant infra red fixed point described above. In principle, the equilibrium renormalised field φ remains finite, however, at finite −ϕ, for c in cϕ = ln φ 2 to reach the conformal value (c → ∞) it is also required that φ 2 → 0 as m 2 → 0. This conformally invariant infra red fixed point can be transformed into the Einstein-Hilbert form. There are indications [38]- [40] that this is is also an UV fixed point for gravitational constant and cosmological constant flows .
With an unimpeded energy density rise, the Hubble radius can shrink to the Planck scale. Alternatively, due to the double well potential, before the system reaches the Planck energy density, it could go through a phase transition at the conformal point into a symmetric phase with a single well potential described by Eq. (21). A second order phase transition in quantum gravity around some critical field value φ 2 c has also been proposed in [41]. In a flat space R J = 0 scenario, this is the most common mechanism used for phase transition studies [42]. In cosmological evolution, as the energy scale varies across a large range that a scalar field with a given symmetry is expected to go through phase transitions, which could lead to inflation [43] as well as in the production of cosmological defects [44].
More specifically, we consider the scenario that as the energy density increases the system enters into the scaling (Ginsberg) region of a phase transition, in which system parameters follow scaling laws. Its broken symmetric phase (renormalised) mass term −m 2 φ 2 < 0 in Eq.(14) will go through zero into the symmetric phase scaling region with +m 2 φ 2 > 0. If the phase transition fixed point is effectively at the conformal invariant fixed point, as the renormalised mass goes through zero, the coupling K approaches the conformal coupling limit and then moves away in its subsequent evolution. The end result is that in Einstein frame, the mass term m 2 (ρ) has an energy density dependence, with a derivative w.r.t to t and ρ given by When ρ increases over time, we haveρ > 0 and the renormalised mass flow is m 2 ,ρ < 0. As c 2 ∝ (6 − K) −1 , the corresponding rate of change for c is c ,ρ > 0 towards the critical point.

Scaling region in the Broken symmetric phase
From the above discussion, in the scaling region of the broken symmetric phase, we write that the mass term is dependent on the energy density of the Einstein frame ρ, as so that Keeping in mind that it is the parameters in Jordan frame that scale as energy density increases, we differentiate Eq. (24) w.r.t the time in Einstein frame t, We note that where respectively m 2 ,ρ and λ ,ρ are the renormalised mass flow and renormalised coupling flow due to energy density increases, while c ,ρ is a result of the non-conformal coupling flow towards the conformal value. We can rewrite Eq.(26) asḢ We assume that scaling laws here is qualitatively similar to the flat space canonical λφ 4 theory [42], we have where a 0 > 0 and λ 0 > 0 are constants, and the exponent l > 0 and γ > 0 describes the effect of mass renormalisation and quartic coupling renormalisation respectively. (We note that in 4-dimensions, the mean field value for l is l MF = 1 for the flat space λφ 4 theory, and it is also possible that λ goes to zero logarthimically. . The possibility that λ is an non-trivial does not alter the result of the analysis below.) In Einstein frame, for the potential to rise as energy density increases towards the phase transition, the mass term m 2 /φ 2 will need to reduce towards zero. Also at finite ϕ < 0 with c → ∞ towards the phase transition. A consistent choice for φ and c will be (We note that φ 2 can also diminish logarithmically as φ 2 ∝ −(ln(ρ c − ρ)) −1 , but this also does not change the result of the following analysis.) For the exponent l > q > 0, as ρ → ρ c from below, the mass term Also we have As c diverges logarthimically, we have near the critical point We note that the energy density of the patch with scale factor a(t) is ρ(a) = a −2ǫ . Its time derivative is given bẏ where a p is some reference scale that a > a p . The equation of state parameter ǫ from Eq.(5) in Einstein Frame is given by in the broken symmetric phase Starting with large field speeds and a decreasing mass term m 2 /φ 2 , in the scaling region ǫ will remain at ǫ > 3 and we can assume thatǫ ∼ 0. For a constant ǫ, from Eq.( 28) we obtaiṅ (If φ 2 → 0 logarithmically, we have q = φ 2 < 1.) In flat space λφ 4 theory, the exponent γ of the renormalised coupling λ is found to be very small and the field renormalisation is also weak that q ≪ 1. It is therefore reasonable to assume that γ is smaller than the exponent of the mass term in Eq. (32), so that γ ≪ (l − q). In this case, in the scaling region far away from the critical point, the m 2 /φ 2 term dominates over the λ term,Ḣ E remains negative. As the m 2 /φ 2 term diminishes rapidly near the critical point, at some point the λ term in Eq.(37) dominates, (taking initiallyφ 2 ∼ ρ,) whereḢ E becomes positive. A small λ and γ would requiresḢ E > 0 to occur very close to the critical point.

Scaling region in the Symmetric phase
Kibble [44]- [45] considers that in a large physcial system going through a phase transition, the correlation length does not have enough time to reach infinity before the phase transition is completed. In this scenario, phase transition occurs when the mass term − 1 2 m 2 ϕ 2 is frozen and simply flips to + 1 2 m 2 ϕ 2 after a finite time. For a small Hubble radius this may be problematic. If the cut-off scale factor a p occurs at Planck scale (∼ (10 19 GeV ) 4 ), a phase transition at energy scale ∼ (10 14 GeV ) 4 may involve finite size effects or modification of scaling laws. For our purpose, as in flat space the quartic coupling exponent γ is much smaller than the exponent of the mass term l − q, we assume that the modification of scaling exponents does not affect our result. We also note that when the Hubble radius becomes nearly infinite around the phase transition point, the Kibble mechanism is perfectly valid around this region.
In the symmetric phase, the Einstein frame Hubble constant turns positive, H E > 0. We consider a scenario that for a short time the energy density can continue to grow such thatρ > 0. In the symmetric phase the potential is given by with Hubble parameter We assume for convenience that the scaling laws on both sides of the phase transition are similar (which is true if the model is in flat space), m 2 and λ scales as Assume also that We haveḢ We see that for H E > 0 andρ > 0,Ḣ E > 0 is maintained for a very smallφ 2 .

The bounce
We recall the Eq.(36)ρ The Friedmann equation 3H 2 E = 1 2φ 2 + V (ϕ) suggests that if H E goes to zero continuously, H E = 0 will be reached in the broken symmetric phase where V (ϕ) < 0.
If this occurs in a region where ǫ > 3 is still nearly constant, as H E reaches H E = 0 − , we havė TheḢ E > 0 term in Eq.(37) is switched off completely. At this point H E goes back to the background energy density That is the energy density can continue to increase asρ > 0 ifǫ < 0 and a > a p . This is possible due to the fact that the value of ǫ comes down from a very large value to ǫ < 3 as H E passes through H E = 0. At some point where the system reaches H E > 0 andḢ E = 0 in the symmetric phase. The bounce from this point leads naturally to an expanding phase. When the exit potential is V E < 1 2φ 2 , we have ǫ > 2 the system enters the radiation epoch directly. If V E >φ 2 , we have ǫ < 1 and the system will go through a short inflationary epoch.
We also notice that near the bounce where H E ∼ 0 andḢ E ∼ 0, the Einstein frame Scalar curvature R E = 12H 2 E + 6Ḣ E is small. One can also show that as the field rollφ nearly stops that R J ∼ 0. At this point, the background metric is nearly flat and for a λφ 4 theory the phase transition structure around a critical point is well understood [42].

After the bounce
After the bounce, the field speed relation between Einstein frame and Jordan frame is given bẏ We consider the point where −ϕ ∼ O(1). We take that c remains positive, and that the Jordan frame field lands at some large field φ > 0 in the single-well potential. For φ to roll down the symmetric potential, we need the kinetic energy term to have the correct sign which is K < 0 (and c 2 < 1/3), one hasċ c ≪ 1, the Einstein frame (negative) field speedφ < 0 corresponds toφ < 0 (and dφ dtJ < 0), so that ϕ will continue its roll towards the ϕ → −∞ while φ rolls towards φ ∼ 0.
It is important to note that as ϕ → −∞, the Einstein frame potential reduces to λ asymptotically, only when the m 2 term has an energy density dependence, such that as ρ → 0, m 2 (ρ) → 0 faster than e −cϕ = φ −2 → ∞. This is possible only in an expanding phase where energy density reduction is fast and the Jordan frame system parameters flow into a conformal infra-red fixed point in the symmetric phase. We recall that a conformal infra-red fixed point for the positive potential in Eq.(51) is found in [38]. In [46], it is observed recently that a non-minimal coupling model at late time universe has value K ∼ 6. However, one still needs a stable (or slow varying) φ > 0 for the Newton's constant to be well defined.
One way is to note that the field acceleration (52) shows that both terms on the R.H.S. provide friction on the field speedφ < 0. A sufficently slowφ allows for a choice of the dynamical Newtonian constant G = φ −2 .
Alternatively, it is shown in [47] that in flat space, starting at low energies with a scale invariant (massless) λφ 4 theory with a scale invariant quantum measure proposed by de-Alfaro, Fubini and Furlan [48] - [49]. The scale invariant potential is spontaneously broken by the quantum measure and the field at potential minimum takes up a non-zero value. This spontaneous broken potential minimum plays the role of the infra-red fixed point of the theory. As the energy density increases the potential becomes (via renormalisation) a massive λφ 4 theory in its symmetric phase. If the Jordan frame scalar has a scale invariant λφ 4 potential with a scale invariant quantum measure, the infra-red fixed point in Jordan frame will have a minimum at φ > 0. That a Jordan frame scalar with a scale invariant λφ 4 potential can be broken spontaneously by the Coleman-Weinberg mechanism is discussed in [50]- [52], although it is not clear whether it supports a massive λφ 4 at high energy densities.

Summary and Discussion
A lifted plateau potential has observational support in the inflationary paradigm. Khoury and Steinhardt [1] show that from the same plateau potential, one can obtain similar red-tilted curvature spectrum from a contracting universe scenario. Once inside the kinetic dominant phase, some new physics is required to effect a connection into the familiar expanding universe phase. Although there are now some potentially viable mechanisms, they are highly non-trivial in comparision to that of a slow roll inflation. Here we start from the post-Ekpyrotic phase of [1]. In Jordan frame, the plateau potential resembles a double well potential. A key observation is that phenomenologically the parameter c takes an unnaturally large value at pre-ekpyrotic phase and crosses many orders of magnitude to reach O(1) at the end of Ekpyrotic phase, which in Jordan frame simply means the non-minimal coupling is initially very close to the conformal coupling value but moves away by a fraction (upto 12%) towards the end of Ekpyrotic phase. This indicates that conformal coupling plays a crucial role in this Ekpyrotic model. The kinetic energy dominance epoch provides an unimpeded source of energy density increase, we consider the possibility that the scalar field can go through a second order phase transition to a single well potential at below Planckian energy density. From the work of [38]- [39] that if we start with the Jordan frame model with a positive mass term in a quartic scalar potential, as energy density decreases the model admits a conformal invariant fixed point at zero mass. This provides a partial support for a phase transition at a conformal fixed point. A cosmological phase transition for a scalar field is a much studied system, in which there are regions at both sides of phase transition where system parameters scale adiabatically with similar scaling laws. As energy density passes the critical energy density, the parameters will freeze and the mass parameter squared changes sign. Within this paradigm, we show that we can obtainḢ E > 0 naturally near the phase transition point. As the equation of state parameter goes to infinity at H E = 0, its rapid decline once H E enters the H E > 0 region leads to a continuation of energy density increase into the symmetric phase. As the equation of state parameter decline finishes, the energy density increase stops and the bounce occurs which takes H E > 0 to its expanding background at the same energy density. The subsequent field roll to ϕ → −∞ in Einstein frame can resemble that in the standard Ekpyrotic paradigm provided that its low energy state also corresponds to a conformal invariant infra-red fixed point in Jordan frame. In this case, the Einstein frame asymptotic potential value is given by the renormalised quartic coupling of the Jordan frame scalar field, which is naturally very small near a conformal invariant fixed point.