Emergent Cosmology Revisited
aa r X i v : . [ a s t r o - ph . C O ] J u l Prepared for submission to JCAP
Emergent Cosmology Revisited
Satadru Bag, a Varun Sahni, a Yuri Shtanov, b,c
Sanil Unnikrishnan d a Inter-University Centre for Astronomy and Astrophysics, Pune 411007, India b Bogolyubov Institute for Theoretical Physics, Kiev 03680, Ukraine c Department of Physics, Taras Shevchenko Kiev National University, Kiev, Ukraine d Department of Physics, The LNM Institute of Information Technology, Jaipur 302031, IndiaE-mail: [email protected], [email protected], [email protected],[email protected]
Abstract.
We explore the possibility of emergent cosmology using the effective potential for-malism. We discover new models of emergent cosmology which satisfy the constraints posedby the cosmic microwave background (CMB). We demonstrate that, within the framework ofmodified gravity, the emergent scenario can arise in a universe which is spatially open/closed.By contrast, in general relativity (GR) emergent cosmology arises from a spatially closed past-eternal Einstein Static Universe (ESU). In GR the ESU is unstable, which creates fine tuningproblems for emergent cosmology. However, modified gravity models including Braneworldmodels, Loop Quantum Cosmology (LQC) and Asymptotically Free Gravity result in a sta-ble ESU. Consequently, in these models emergent cosmology arises from a larger class ofinitial conditions including those in which the universe eternally oscillates about the ESUfixed point. We demonstrate that such an oscillating universe is necessarily accompanied by graviton production . For a large region in parameter space graviton production is enhancedthrough a parametric resonance, casting serious doubts as to whether this emergent scenariocan be past-eternal.
Keywords:
Effective potential, Einstein Static Universe, Inflation, Emergent Cosmology,CMB constraints on inflation, Particle production
ArXiv ePrint: ontents
The inflationary scenario has proved to be successful in describing a universe which is re-markably similar to the one which we inhabit. Indeed, one of the central aims of the ongoingeffort in the study of cosmic microwave background (CMB) observations is to converge onthe correct model describing inflation [1].However, despite its very impressive achievements, the inflationary paradigm leavessome questions unanswered. These pertain both to the nature of the inflaton field and to thestate of the universe prior to the commencement of inflation. Indeed, as originally pointedout in [2], inflation (within a general relativistic setting) could not have been past eternal.This might be seen to imply one of several alternative possibilities including the following:1. The universe quantum mechanically tunnelled into an inflationary phase.– 1 –. The universe was dominated by radiation (or some other form of matter) prior toinflation and might therefore have encountered a singularity in its past.3. The universe underwent a non-singular bounce prior to inflation. Before the bouncethe universe was contracting.4. The universe existed ‘eternally’ in a quasi-static state, out of which inflationary expan-sion emerged.One should point out that, at the time of writing, none of the above possibilities isentirely problem free. Nevertheless, our focus in this paper will be on the last option, namelythat of an
Emergent Cosmology .The idea of an emergent universe is not new and an early semblance of this concept canbe traced back to the seminal work of Eddington [3] and Lemaˆıtre [4], which was based onthe Einstein Static Universe [5]. Indeed, in 1917, Einstein introduced the idea of a closed andstatic universe sourced by a cosmological constant and matter. Subsequently it was foundthat: (a) the observed universe was expanding [6], (b) the Einstein Static Universe (ESU)was unstable. It therefore became unlikely that ESU could describe the present universe butallowed for our universe to have emerged from a static ESU-phase in the past.With the discovery of cosmic expansion Einstein distanced himself from his own earlyideas referring to them, years later, as his biggest blunder [7].Interest in the ESU subsequently waned, although models in which the ESU featuredas an intermediate stage — called loitering — received a short-lived burst of attention inthe late 1960’s, when it was felt that a universe which loitered at z ≃ - - - - - - a U H a L E m e r g e n t U n i v e r s e C on t r ac ti ng U n i v e r s e a E ESU t a H t L E m e r g e n t U n i v e r s e C on t r ac ti ng U n i v e r s e Einstein Static Universe
Figure 1 . The effective potential (left panel) is schematically shown for a universe consisting of twocomponents, one of which satisfies the strong energy condition ρ + 3 P ≥
0, while the other violates it.The maxima of the effective potential corresponds to the Einstein Static Universe (ESU) for whichthe expansion factor is a constant. However ESU is unstable to small perturbations and can thereforebe perturbed either into an accelerating emergent cosmology, or into a contracting singular universe(right panel).
An Einstein Static Universe (ESU) is possible to construct provided the universe is closedand is filled with at least two components of matter: one of which satisfies the strong energycondition (SEC) ρ + 3 P ≥
0, whereas the other violates it. The cosmological constant with P = − ρ presents us with an example of the latter, as does a massive scalar field which couplesminimally to gravity.In order to appreciate the existence of ESU, and therefore of emergent cosmology (EC),consider the following set of equations which describe the dynamics of a FRW universe: (cid:18) ˙ aa (cid:19) = κ X i =1 ρ i − k a , k = 0 , ± (cid:18) ¨ aa (cid:19) = − κ X i =1 ( ρ i + 3 P i ) , (2.2)where κ = 8 πG = M − p . We shall assume that the pressure of the first component satisfiesthe SEC ( w ≥ − / w < − / w = P/ρ is the parameter of equation of state.Equation (2.1) can be recast in terms of the effective potential U ( a ) as follows [15]:12 ˙ a + U ( a ) = E ≡ − k , (2.3)where U ( a ) = − κ a X i =1 ρ i . (2.4)– 3 –he ESU arises when the following two conditions are simultaneously satisfied:˙ a = 0 , ¨ a = 0 . (2.5)Substituting ˙ a = 0 into (2.3) gives U ( a E ) = − k , (2.6)where a E is the scale factor for the ESU. The second condition implies that the ESU corre-sponds to an extremum of U ( a ) (see FIG. 1):¨ a = 0 ⇒ U ′ ( a E ) = 0 . (2.7)In this case, one finds the following relationship between the curvature of the ESU, a − E , andthe densities ρ i for a closed universe ( k = 1): ρ ρ = − w w , κρ = (cid:18) w w − w (cid:19) a E . (2.8)For a closed ΛCDM ESU, we have ρ ≡ ρ m , κρ = Λ, w = 0, w = −
1, and equation (2.8)yields the familiar results Λ = κ ρ m = 1 a E . (2.9)On the other hand, for a closed radiation + Λ ESU, we have ρ ≡ ρ r , κρ = Λ, w = 1 / w = −
1, and one finds Λ = κρ r = 32 a E . (2.10)The form of U ( a ) in Fig. 1 immediately suggests that the ESU with U ′ ( a E ) = 0, U ′′ ( a E ) < Emergent Cosmology . In this latter case, an exact solutiondescribing the emergence of a Λ-dominated accelerating universe from a radiation + Λ basedESU is given by [16] a ( t ) = a i " √ ta i ! / , (2.11)where a i = 3 / (2Λ) = 3 / (2 κρ r ).The above example provided us with a toy model for the emergent scenario in which‘inflation’ takes place eternally. A more realistic scenario can be constructed if one replacesΛ by the inflaton [9] thereby allowing inflation to end and the universe to reheat. A spatially homogeneous scalar field minimally coupled to gravity is described by the La-grangian density L = 12 ˙ φ − V ( φ ) , (2.12)and satisfies the evolution equation¨ φ + 3 H ˙ φ + V ′ ( φ ) = 0 . (2.13)– 4 –he energy density and pressure of such a field are given, respectively, by ρ φ = 12 ˙ φ + V ( φ ) , (2.14a) P φ = 12 ˙ φ − V ( φ ) , (2.14b)and, by virtue of (2.13), satisfy the usual energy conservation equation˙ ρ φ = − H ( ρ φ + P φ ) . (2.15) Figure 2 . The potential (2.16) can give rise to an emergent cosmology commencing in a
EinsteinStatic Universe (ESU). ESU is followed by inflation after which φ oscillates and the universe reheats.Since the potential is symmetric, emergent cosmology can be realized by the inflaton rolling eithertowards the right, or the left. An emergent universe can be constructed if the inflaton potential V ( φ ) has one (ormore) flat wings. • Consider first the potential V ( φ ) = V tanh p ( λφ/M p ) ; p = 1 , , .... (2.16)Large absolute values of | λφ | ≫ M p lead to a flat potential with V ( φ ) ≃ V ; see Fig. 2.In this case, the equation of motion (2.13) becomes ¨ φ + 3 H ˙ φ ≃ . Since H = 0 in anESU, one immediately finds ˙ φ = const. Consequently, a scalar field sustaining an ESUbehaves exactly like a two-component fluid, with one component being stiff matter withequation of state P = ρ = ˙ φ /
2, while the other is the cosmological constant Λ ≡ κV .Substituting ρ = ˙ φ / ρ = V , w = 1, w = − φ = V = 2 κa E , (2.17)which demonstrates that the kinetic term must be precisely matched to the asymptoticvalue of the potential term in an ESU scenario; see also [9]. Note that the post-ESUinflationary phase can last for a sufficiently long duration, as illustrated in Fig. 3.– 5 – i t f t . . . . . ln[ a ( t ) /a i ]˙ φ /V ( φ ) Figure 3 . The emergent scenario is illustrated by the scale factor (red curve) and slow roll parameter(green curve). Inflation commences when t ∼ t i and lasts for 120 e-folds. Note that the (quasi-flat)left branch of the potential in (2.16) needs to have a small positive slope in order for the ESU to endin inflation. If this branch has a negative slope, then the ESU will end in a contracting universe – seeFig. 1. Next, we shall demonstrate that emergent cosmology based on (2.16) is consistent withrecent measurements of the Cosmic Microwave Background (CMB) made by the Plancksatellite [1]; see also [17].As shown in Fig. 2, ESU ends and inflation commences once the inflaton field φ beginsto roll down its potential. In the slow-roll approximation, the scalar ( n S ) and tensor( n T ) spectral indices and the tensor-to-scalar ratio ( r ) are given by n S − − p λ (cid:16) p + 8 N pλ + p p λ (cid:17)(cid:16) N pλ + p p λ (cid:17) − , (2.18) n T = − p λ (cid:16) N pλ + p p λ (cid:17) − , (2.19) r = 128 p λ (cid:16) N pλ + p p λ (cid:17) − , (2.20)respectively, where N is the number of e -folds counted from the end of inflation. Thevalues of n S and r are plotted as a function of λ in Fig. 4. We find that CMB constraintsare easily satisfied if λ > .
1. – 6 – .001 0.01 0.1 1 100.920.940.960.981.00 Λ n s p = = = Λ r p = p = = Figure 4 . In the left panel, the scalar spectral index n S is plotted as a function of λ for three differentvalues of the parameter p in the potential (2.16) while, in the right panel, the value of tensor-to-scalaris plotted as a function of λ for the same set of values for p . The number of e -folds is N = 60. Notethat when λ > .
1, the scalar spectral index approaches a constant value of 0 . r decreasesas λ − . The shaded region refers to 95% confidence limits on n S and r determined by Planck [1]. - - - - Λ V M p p = p = p = Figure 5 . The value of V is plotted as a function of λ for three different values of the parameter p in the potential (2.16). The number of e -folds is taken to be N = 60. In this case, equations (2.18)–(2.20) are simplified, respectively, to n S − ≃ − N , (2.21) n T ≃ − N λ , (2.22) r ≃ N λ , (2.23)which are independent of the value of p in (2.16).Interestingly, the above expression for n S is exactly the same as in Starobinsky’s R + R model [18]. In fact, for λ = 1 / √
6, the expressions for n T and r match those in theStarobinsky inflation ! – 7 –he value of the parameter V in the potential (2.16) can be fixed using CMB normal-ization, which gives P S ( k ∗ ) = 2 . × − at the pivot scale k ∗ = 0 .
05 Mpc − [1]. Onefinds V M p = 192 π p λ (cid:16) N pλ + p p λ + 1 (cid:17) p P S ( k ∗ ) (cid:20)(cid:16) N pλ + p p λ (cid:17) − (cid:21) p +1 , (2.24)which is plotted as a function of λ in Fig. 5. Since λ > . n S and r , it follows that V < − M p . • Consider next the potential V ( φ ) = V (cid:20) exp (cid:18) β φM p (cid:19) − (cid:21) . (2.25)shown in Fig. 6 and earlier discussed in [10].For suitable values of V and β , inflation can occur both from the (flat) left branch andfrom the (steep) right branch of this potential.In this section, we shall focus on the left branch since this is the branch which leads toan ESU, and hence to emergent cosmology in the GR context. For β > .
1, one finds n S − ≃ − N − β N , (2.26) n T ≃ − β N , (2.27) r ≃ β N . (2.28)If N ≥
60, then β > .
14 is required in order to satisfy the CMB bound r < .
11. Thevalue of V is determined from the relation V M p = 12 π (cid:18) P S ( k ∗ ) N β (cid:19) . (2.29)Substituting here β = 0 . V = 2 . × − M p , n S ≃ .
96 and r ≃ . Planck results [1]. Note that Starobinsky’s R + R modelof inflation [18] can equivalently be described by a scalar field with potential (2.25) butwith β = p /
3; see [19].To summarize, we have demonstrated that viable models of emergent cosmology arepossible to construct in a general relativistic setting. Unfortunately, although our modelseasily satisfy CMB constraints, they run into severe fine-tuning issues — since the initial valueof ˙ φ i must precisely match the asymptotic form of the inflaton potential — a consequenceof the ESU being unstable in GR; see (2.17). (Note that emergent cosmology can also besustained by a spinor field, as demonstrated in [20].)As we shall show in the next section, emergent cosmology based on modified gravitycan circumvent this difficulty since an ESU can be associated with a stable fixed point in thiscase. – 8 – igure 6 . Potentials (2.25) and (3.33) can give rise to an emergent cosmology from a Einstein StaticUniverse (ESU). If the ESU is GR-based, as in section 2, then inflation will occur from the left (A)branch. For modified gravity models explored in section 3, inflation can occur from both A and Bbranches.
In this section we focus on emergent cosmology based on modified gravity. We shall confineour discussion to three distinct models of modified gravity, namely: (i) Braneworld cosmologyand its generalizations, (ii) Loop Quantum Cosmology (LQC), and (iii) a phenomenologicalmodel of asymptotically free gravity.
Consider first the following generalization of the Einstein equations which is known to giverise to a non-singular bouncing cosmology [21]: H = κ ρ (cid:26) − (cid:18) ρρ c (cid:19) m (cid:27) − k a , κ = 8 πG = M − p , (3.1)¨ aa = − κ (cid:20) ( ρ + 3 P ) − (cid:26) (3 m + 1) ρ + 3( m + 1) P (cid:27) (cid:18) ρρ c (cid:19) m (cid:21) . (3.2)The bounce occurs near the critical density ρ c . When m = 1, these equations reduce to thefollowing set of equations describing the dynamics of a Friedmann–Robertson–Walker (FRW)metric on a brane [11] H = κ ρ (cid:26) − ρρ c (cid:27) − k a , (3.3)¨ aa = − κ (cid:26) ( ρ + 3 P ) − ρρ c (2 ρ + 3 P ) (cid:27) , (3.4)where ρ c is related to the five-dimensional Planck mass [11]. Both (3.1), (3.2) & (3.3), (3.4)reduce to the FRW limit when ρ c → ∞ , while (3.3), (3.4) are valid in LQC when k = 0 [12].– 9 –n the presence of several components one replaces ρ = P i ρ i and P = P i P i in all theseequations.The emergent scenario which we discuss in this paper is based on ‘normal’ matter thatsatisfies the strong energy condition, along with a scalar field that violates it. As pointedout in the previous section, emergent cosmology arises when the scalar field potential has aflat wing along which V ( φ ) is effectively a constant, which we denote by Λ. Equation (3.3)in such a two-component universe becomes H = (cid:18) κ ρ + Λ3 (cid:19) (cid:18) − ρρ c − Λ κρ c (cid:19) − k a . (3.5)For a spatially closed universe, this equation can be recast as (2.3) with the effective potential U ( a ) = − a (cid:18) κ ρ + Λ3 (cid:19) (cid:18) − ρρ c − Λ κρ c (cid:19) , (3.6)where κ ρ = Aa l , l = 3 (1 + w ) , w > − / . (3.7)The effective potential U ( a ) in (3.6) exhibits a pair of extremes (a maximum and a mini-mum). The scale factor and the energy density corresponding to these extremes are given,respectively, by a l ± = 3 A ( l −
2) ( κρ c − κρ c − Λ) " ± s −
16 ( l −
1) Λ ( κρ c − Λ)( l − ( κρ c − , (3.8) ρ ± = ( l − κ ( l −
1) ( κρ c − " ± s −
16 ( l −
1) Λ ( κρ c − Λ)( l − ( κρ c − . (3.9)Note that a − < a + and ρ − < ρ + . The extremes can exist as long as the term under thesquare root in (3.8) and (3.9) remains positive, which imposes the following conditions on Λ :Λ < Λ crit = κρ c (cid:20) − r qq + 4 (cid:21) , where q = 16 ( l − l − , (3.10)or Λ > κρ c . (3.11)For the first condition in (3.10), ( a − , ρ + ) is associated with the minimum of U ( a ) while( a + , ρ − ) is associated with the maximum of U ( a ). On the other hand, for the second conditionin (3.11) ( i.e. Λ > κρ c ), only a + in (3.8) and ρ − in (3.9) survive, and account for a minimain U ( a ). However, according to (3.5), a spatially closed universe ( k = 1) does not supporta solution with Λ ≥ κρ c . Hence this minima is shown, in Appendix A, to correspond to anESU in a spatially open universe ( k = − k = 1) for whichwe can obtain static (or oscillating) solutions in the regime Λ < Λ crit < κρ c ; i.e. accordingto (3.10). As noted in the previous section, a scalar field rolling along a flat potential with V ′ ( φ ) ≡
0, behaves just like stiff matter plus a cosmological constant. Our analysis in thefollowing subsections will therefore focus on this important case.– 10 – .1.1 The effective potential in the presence of stiff matter only
From (3.8) and (3.9), one can see that, as Λ tends to zero, a + approaches infinity and ρ − declines to zero, but both a − and ρ + remain finite. In other words, in the absence of thecosmological constant, the (unstable) maximum of the effective potential U ( a ) disappears,while the stable minimum remains in place. This new feature of brane cosmology distinguishesit from GR, in which the effective potential has no extreme value when Λ = 0. As we shallsee later, the persistence of a stable minimum of U ( a ) in the absence of Λ carries over toother modified gravity models as well.In the absence of the cosmological constant, and with a universe consisting only of stiffmatter ( P = ρ ), the minimum of U ( a ) is determined from (3.8) to be at a − = 152 Aκρ c . (3.12)As mentioned earlier in (2.5) – (2.7), the Einstein Static Universe (ESU) arises when ˙ a = 0and ¨ a = 0. As a result, the ESU is associated with an extremum of U ( a ), so that U ′ ( a E ) = 0and U ( a E ) = − k / a E (the scale factor at the ESU) isthe critical value of a − for which the minimum of U ( a ) lies on the E = − k / U ( a ) is determined from (3.9) to be ρ + = ρ E = 25 ρ c , (3.13)where ρ c is the braneworld constant defined in (3.3), (3.4). The fact that ρ + is independentof A implies that, for a universe which is oscillating about a − , the density of matter at U ( a − ) is the same as in ESU. (This interesting result is independent of the equation of state ofmatter and holds for l ≥ ρ + by ρ E , the matter density atESU. Note that for a closed universe ( k = 1) if U ( a − ) < −
12 , then ESU will no longer be asolution of the field equations. Instead, the universe will oscillate around U ( a − ). With thisresult taken into account, the minimum in the effective potential U ( a ) becomes U ( a − ) = − κ a − ρ E (cid:18) − ρ E ρ c (cid:19) , (3.14)while the ESU condition (2.6) for a closed universe ( k = 1) takes the form U ( a E ) = −
12 = − κ a E ρ E (cid:18) − ρ E ρ c (cid:19) . (3.15)Comparing (3.14) and (3.15) we easily find U ( a − ) = − a − a E . (3.16)By using (3.13) and (3.15), a E is determined to be a E = 252 κρ c . (3.17)By using (2.3), with k = 1, the condition to have a physical solution becomes U ( a − ) ≤ − ⇒ a − ≥ a E ⇒ A ≥ A E , (3.18)– 11 – - - - - a U H a L Bouncing Universe a E a - E =- (cid:144) a B1 a B2 Figure 7 . The effective potential is shown for a universe consisting of stiff matter with P = ρ . Twovalues of the parameter A are chosen. For A = A E (black curve), the minimum of U ( a ) lies preciselyon the E = − / a E , ρ E ) which corresponds to the Einstein StaticUniverse (ESU). For A > A E , the minimum of U ( a ) shifts towards higher a and lies below the E = − / a − , ρ E ), i.e. aroundthe minimum of the effective potential (red curve). Note that the value of the energy density at theminimum of U ( a ) is a fixed quantity and is given by (3.13). From the figure it is clear that the motionof a closed universe is always bounded, and the turning points of its scale factor are a B and a B . Bycontrast, a spatially flat/open universe bounces at small a but need not turn around and collapse. Itis important to note that the above form of the potential U ( a ) is robust and remains qualitativelyunchanged if stiff matter is replaced by any other form of matter satisfying the SEC, i.e., for ρ ∝ A/a l with l ≥ κ = 1, ρ c = 1. where A E is the value of the parameter A in (3.7) for which the ESU conditions are met. Inthe last inequality of (3.18), we have used (3.12) and the fact that a E is simply the value of a − at ESU.The effective potential is shown in Fig. 7 for a universe containing stiff matter. As a → U ( a ) diverges causing the universe to bounce and avoid the big bang singularity.For large a , the braneworld equations (3.3), (3.4) reduce to the GR limit ( ρ/ρ c ≪ U ( a )asymptotically approaches zero. The motion of a spatially closed universe is therefore alwaysbounded. The minima in U ( a ) represent stable universes. Two values of the parameter A in (3.7) are considered. For A = A E , the universe is static (ESU) at a = a E (black curve),while, for A > A E , the universe oscillates around a = a − (red curve). For stiff matter and Λ, the scale factor and stiff matter density at the extremes of U ( a ) aredetermined by setting l = 6 in (3.8) and (3.9). Consequently, one obtains a ± = 3 A ( κρ c − κρ c − Λ) " ± s −
5Λ ( κρ c − Λ)( κρ c − (3.19)– 12 –nd ρ ± = 15 κ ( κρ c − " ± s −
5Λ ( κρ c − Λ)( κρ c − . (3.20)As before, ( a − , ρ + ) is associated with the minimum of U ( a ) while ( a + , ρ − ) is associated withthe maximum of U ( a ), and a − < a + , ρ − < ρ + . The unstable ESU associated with themaximum is GR-like and was previously encountered in section 2. Hereafter we shall focuson the stable ESU that is associated with the minimum of U ( a ).It follows from (3.19) that a ± increases as the parameter A increases, while ρ ± is constantfor a given Λ. This empowers (3.16) and (3.18) to hold true even in the completely generalcase when Λ = 0; see (3.8) and (3.9). For a certain value of the parameter A = A E , ESUconditions are satisfied and the corresponding scale factor can be determined using (2.6) and(3.20) to be a E = 3( κρ + + Λ) (cid:18) − ρ + ρ c − Λ κρ c (cid:19) . (3.21)Note that the condition for the existence of two extreme values in U ( a ), namely (3.10),is simplified to Λ < Λ crit = κρ c " − √ . (3.22)The form of the effective potential is very sensitive to the value of Λ, as shown inFig. 8(a). From this figure, we see that the potential U ( a ) diverges as a →
0, which allowsthe universe to bounce at small a and avoid the big bang singularity. For Λ = 0, there existsonly a single minimum in U ( a ), as discussed in section 3.1.1. The maximum, representing anunstable fixed point, appears when Λ >
0. As Λ increases, the maxima and the minima in U ( a ) approach each other. They merge when Λ = Λ crit , which results in an inflection pointin the effective potential, see Fig. 8(b). For Λ > Λ crit , the potential has no extreme valueat finite a , which is indicative of the absence of fixed points in the corresponding dynamicalsystem. While figures 8(a) and 8(b) have been constructed for matter with P = ρ (in view ofpossible links to the kinetic regime during pre-inflation), the main results of our analysis,based on (3.8), (3.9) and (3.10), should remain qualitatively true for any matter componentwith w > − /
3, with Λ crit depending upon w . The energy conservation equation (2.15) for stiff matter reads˙ ρ = − Hρ . (3.23)Differentiating (3.5) with respect to time leads to the Raychaudhuri equation:˙ H = − κ (cid:20)(cid:18) ρ − Λ κ (cid:19) − ( κρ + Λ) κρ c (cid:18) ρ − Λ κ (cid:19)(cid:21) − H . (3.24) Note that U ( a ) exhibits a minimum also for Λ > κρ c . However in this case U ( a ) > – 13 – = L= L= - - - - a U H a L E =- (cid:144) I II (a) L (cid:144)H ΚΡ c L a , Ρ a + a - Ρ + Ρ - L = L c r it (b) Figure 8 . (a) The effective potential is plotted for different values of Λ for a two-component universeconsisting of stiff matter and a cosmological constant. (i) For Λ = 0, there is only a single minimumrepresenting a stable fixed point denoted by I (black); see section 3.1.1. (ii) For Λ >
0, there appearsa maximum associated with the unstable fixed point, denoted by II (red). (iii) As Λ further increases,these two fixed points approach each other as illustrated in Fig. 8(b). At Λ = Λ crit , they merge toproduce an inflection point in U ( a ), and, for Λ > Λ crit , no fixed points are present in the system(green line in left panel). (b) The scale factor corresponding to the minimum (maximum) of U ( a ),denoted by I (II) in Fig. 8(a) and by a − ( a + ) in (3.8), is plotted as a function of Λ in the right panel.The right panel also shows the value of the stiff matter density at the minimum (maximum) of U ( a ),and denoted by ρ + ( ρ − ) in (3.9). As Λ increases, the two fixed points, stable and unstable, movetowards each other and merge at Λ = Λ crit , beyond which no fixed point exists. Units of κ = 1 and ρ c = 1 are assumed together with a suitable choice of A , for purposes of illustration. The unit alongthe y-axis is arbitrary. The fixed points of the dynamical system described above are characterized by˙ ρ = 0 ⇒ H = 0 and ˙ H = 0 ⇒ ¨ a = 0 , (3.25)precisely the same conditions as for ESU. For Λ >
0, the two fixed points in the ( ρ, H ) planeare ( ρ + ,
0) and ( ρ − , ρ ± are given by (3.20). The condition for the existence of thesefixed points was described earlier in (3.22). For Λ = 0, only one fixed point ( ρ + ,
0) exists. Toanalyse stability, the nonlinear dynamical system should be linearized using the linearizationtheorem (for instance, as described in [22]) near the two fixed points. The eigenvalues ofthe Jacobian matrix of the linearized system (as described in Appendix B) at the two fixedpoints are λ I,II = 4 κρ ± (cid:20) − ρ c (cid:18) ρ ± + 2Λ κ (cid:19)(cid:21) , (3.26)where the indices I and II stand for the two fixed points ( ρ + ,
0) and ( ρ − , ρ ± from (3.20), the properties of the eigenvalues at the two fixed pointsare listed in Table 1 for Λ < Λ crit in (3.22).For the fixed point II ( ρ − , ρ + ,
0) areimaginary and complex conjugates of each other, suggesting that the fixed point in this caseis of centre type (stable), as expected. But the linearization theorem does not guarantee this– 14 –ixed point ( ρ, H ) Eigenvalues StabilityI ( ρ + , λ I < ρ − , λ II > Table 1 . Eigenvalues of the Jacobian matrix and stability of the fixed points for stiff matter and acosmological constant. for linearized systems of centre type [22], hence this result needs to be confirmed numerically.The phase portrait for general matter with w > − / ρ = − Hρ (1 + w ) , (3.27)while the Raychaudhuri equation is given by˙ H = − κ (cid:20) ρ w ) − Λ κ − ( κρ + Λ) κρ c (cid:26) (2 + 3 w ) ρ − Λ κ (cid:27)(cid:21) − H . (3.28)Again, for 0 < Λ < Λ crit the system possesses two fixed points I, II in the ( ρ, H ) plane,given by ( ρ + ,
0) and ( ρ − , ρ ± and Λ crit were determined in (3.9) and(3.10). The eigenvalues of the linearized system at the two fixed points are found to be λ I,II = (1 + 3 w ) (1 + w ) κ ρ ± (cid:20) − ρ c (cid:26) w )(1 + 3 w ) ρ ± + 2Λ κ (cid:27)(cid:21) . (3.29)Using the values of ρ ± from (3.9), it can be shown that, for ( ρ − , λ II >
0, and realeigenvalues of opposite sign confirm the fixed point II to be a saddle. For I, the eigenvaluesare again imaginary and complex conjugates of each other, given by λ I <
0. Although thissuggests that the fixed point I is a centre, the linearization theorem does not assure this,hence this result needs to be confirmed numerically.We have plotted the phase space for stiff matter and Λ using (3.23) and (3.24). Sincewe focus on the spatially closed case, we need to add the additional constraint H < (cid:18) κ ρ + Λ3 (cid:19) (cid:18) − ρρ c − Λ κρ c (cid:19) , (3.30)which follows from (3.5). The resulting phase portrait is shown in Fig. 9(a) for Λ = 0. Notethat only the centre type stable fixed point I exists in this case, as expected.For Λ >
0, the unstable saddle point II appears along with the centre I. As Λ increases,these two fixed points move towards each other. The phase portrait for a typical value of Λis illustrated in Fig. 9(b). When Λ reaches its critical value in (3.22), i.e., at Λ = Λ crit , thestable and unstable points merge giving rise to an inflection point in the effective potential U ( a ), as discussed earlier. For Λ > Λ crit , fixed points are absent, and flows in the phaseportrait are depicted in Fig. 9(c), suggesting that, after the bounce, the matter densitydeclines monotonically as the universe expands. Note that for Λ > κρ c , the dynamical system admits a centre type stable fixed point given by ( ρ − , – 15 – .0 0.2 0.4 0.6 0.8 1.0 - - - Ρ H I (a) For Λ = 0 - - Ρ H III (b) For Λ = 0 . κρ c - - Ρ H (c) For Λ = 0 . κρ c > Λ crit Figure 9 . (a) Phase portrait of a universe consisting only of stiff matter. In this case, only thecentre type fixed point I is present. The motion of the spatially closed braneworld is oscillatory andbounded. (b)
Phase portrait for a closed braneworld consisting of stiff matter and a cosmologicalconstant: 0 < Λ < Λ crit . For Λ > crit , giving rise to aninflection point in U ( a ). (c) Phase portrait for stiff matter with Λ > Λ crit . There is no fixed point inthe dynamical system. In the numerical calculation we assume, for simplicity, that κ = 1 with ρ c = 1. The above discussion showed that: (i) the instability associated with ESU in the GR contextcan be avoided by studying such a model in the braneworld context; (ii) in this case, inaddition to the unstable critical point II reminiscent of GR-based ESU, there appears astable critical point I around which the universe can oscillate.The construction of a realistic emergent cosmology requires that the universe be ableto exit its oscillatory phase. In order to do this, the inflationary potential must be chosenjudiciously, so that:[A] V ( φ ) have an asymptotically flat branch where V ( φ ) ≃ V . This permits the appearanceof the stable minimum I in the effective potential U ( a ). At this minimum, whichcorresponds to the ESU, the field’s kinetic energy and the scale factor of the universe– 16 –re given, respectively, by˙ φ = 25 ( ρ c − V ) " s − V ( ρ c − V )( ρ c − V ) (3.31)and a E = 3 M p ρ (1 − ρ/ρ c ) . (3.32)The corresponding value of ρ can be determined from (2.14a) using the value of ˙ φ in(3.31). Note that (3.31) is identical to (3.20) for ρ + , while (3.32) is the same as (3.21).[B] V ( φ ) should increase monotonically beyond some value of φ so that the effective cos-mological constant, mimicked by V , increases with time. This allows the stable andunstable fixed points to merge and the ESU phase to end. Thereafter the universe in-flates in the usual fashion. The potential described by (2.25), which is shown in Fig. 6and was earlier discussed in [10] and [13], clearly satisfies this purpose.It was earlier shown, in section 2, that the (flat) left wing of V ( φ ) can give rise toemergent cosmology in GR, provided inflation occurs from the (A) branch in Fig 6. Forbraneworld-based emergent cosmology one requires inflation to take place from the muchsteeper (B) branch of Fig. 6. This leads to a problem since along this branch the potential in(2.25) becomes an exponential, which is ruled out by recent CMB constraints [1]. Therefore,to examine this scenario further, we replace (2.25) by the following potential which adequatelyserves our purpose: V ( φ ) = V (cid:20) − θ ( φ ) (cid:18) φM (cid:19) γ (cid:21) , γ > , (3.33)where θ ( φ ) is a step function: θ ( φ ) = 0 for φ <
0, and θ ( φ ) = 1 for φ ≥
0, which ensures that V ( φ ) = V for φ < , φ = M ,V (cid:18) φM (cid:19) γ for φ ≫ M . (3.34)Note that this potential qualitatively resembles the one shown in Fig. 6.Once the scalar field begins to roll up its potential, V ( φ ) (mimicked by Λ in the previoussection) increases, and the stable and unstable fixed points in Fig. 9(b) move towards eachother (see Fig. 8(b)), merging when V = V crit , where V crit = ρ c " − √ . (3.35)For V > V crit , there is no fixed point, and the universe escapes from attractor I.For positive H in (2.13), the scalar field experiences a very large damping which causesit to stop climbing the potential V ( φ ). The inflationary regime commences once the scalarfield begins to roll slowly down its potential.The system of equations (2.13), (2.14) and (3.3), (3.4) has been integrated numericallyin the context of spatially closed universe ( k = 1) using the potential given in (3.33). The– 17 – i t f t l n [ a ( t ) / a i ] Emergent AEmergent B t i t − . . . . . l n [ a ( t ) / a i ] Emergent AEmergent B
Figure 10 . Top panel:
Time evolution of the scale factor a ( t ): t i and t f stand for the beginningand end of inflation, respectively. Emergent A: the universe was at ESU prior to inflation.
EmergentB: the universe was perturbed slightly away from ESU before inflation.
Bottom panel shows amagnified view of the top panel prior to inflation and demonstrates that universe B oscillates aboutthe minimum of the effective potential. κ = 1 and ρ c = 1 are assumed. θ ( φ ) function is realized numerically by (1 + tanh( τ φ )) / τ . Our resultsare shown in Fig. 10 for the following two cases: Emergent A: the universe was at ESU priorto the inflation;
Emergent B: the universe was perturbed away from ESU before inflation.The beginning and end of inflation are denoted by t i and t f , respectively, on the time axis.The lower panel to Fig. 10 shows a zoomed-in view of the scale factor prior to inflation. Thispanel demonstrates that universe A stays at ESU ( a E ) eternally in the past, whereas universeB exhibits oscillations around the minimum ( a − ) of the effective potential U ( a ). (Note that a − is slightly displaced from a E , as illustrated in Fig. 7 for Λ = 0.) The discerning readermay realize at this point that in Emergent A, the fine-tuning requirement on the field’skinetic energy, given by (3.31), is similar in spirit to that imposed in GR-based Emergent– 18 – i t f t . . . . . V ( φ ) Emergent AEmergent B
Figure 11 . Time evolution of V ( φ ) in the emergent scenario described by (3.33). κ = 1, ρ c = 1 areassumed. cosmology, namely (2.17). In other words, given an effective potential with a minimum, itsmuch more ‘likely’ for the universe to oscillate about the ESU (Emergent B) than to sitexactly at the ESU fixed point (Emergent A). The time evolution of the inflaton potential V ( φ ) is illustrated in Fig. 11. The CMB constraints for this scenario can easily be satisfied,as shown in Appendix C. In this section we discuss the emergent scenario within the context of a theory of gravitywhich becomes asymptotically free at high energies. In this phenomenological model thegravitational constant depends upon the matter density as follows: G ( ρ ) = G exp ( − ρ/ρ c ).As a consequence, the FRW equations become H = κ ρe − ρ/ρ c − k a , (3.36)˙ H = κ w ) ρe − ρ/ρ c (cid:20) ρρ c − (cid:21) + k a , (3.37)where κ = 8 πG and G is the asymptotic value of the gravitational constant: G ( ρ →
0) = G . It is easy to see that the low-energy limit of this theory is GR, while at intermediateenergies 0 ≪ ρ < ρ c the field equations (3.36), (3.37) resemble those for the braneworld (3.3),(3.4). From (3.36), (3.37) we find that at large densities gravity becomes asymptotically free: G ( ρ ) → ρ ≫ ρ c . In this case: • The universe will bounce if k = 1. • The universe will ‘emerge’ from Minkowski space ( ˙ a = 0 , ¨ a = 0 as a →
0) if k = 0. • The universe will ‘emerge’ from the Milne metric ( a ∝ t as a →
0) if k = − = L= L= - - - - a U H a L E =- (cid:144) Figure 12 . The effective potential, U ( a ), is shown for Asymptotically free gravity for various valuesof Λ (in units κ = 1 and ρ c = 1). For Λ = 0 there is only a single minima in U ( a ), shown by the blackline. A maxima appears when Λ > crit which results in an inflection point in U ( a ). For Λ > Λ crit there isno extrema in U ( a ) (green line). In all three cases the big bang singularity is absent.The effective potential for this theory is U ( a ) = − κ a ρe − ρ/ρ c , (3.38)where the evolution of ρ is given is (3.7). In order to link the emergent scenario with inflationwe shall consider ρ to represent the density of the inflaton field. Recall that the inflatonmoving along a flat direction ( V ′ = 0) behaves like a fluid consisting of two non-interactingcomponents, namely stiff matter and the cosmological constant. Hence ρ in (3.36), (3.37) &(3.38) is effectively replaced by ρ φ ≡ ρ stiff + ρ Λ where ρ stiff ∝ a − and ρ Λ = Λ /κ .If Λ = 0 then the effective potential in (3.38) exhibits a single minimum shown by theblack line in Fig. 12. As noted in section 2, the existence of an ESU implies the simultaneousimplementation of the following conditions:1. ¨ a = 0 ⇒ U ′ ( a E ) = 0,2. ˙ a = 0 ⇒ U ( a E ) = − k / k = 1) dominated by stiff matter (Λ = 0) one finds thatthe scale factor and density associated with ESU are given by ρ E = 23 ρ c , a E = 92 κρ c e / . (3.39)– 20 –ncluding Λ >
0, one finds that the form of U ( a ) changes to accommodate both a maximaand a minima. The scale factor and density associated with these extrema are given by a ± = 3 A κρ c (2 κρ c − " ± s − κρ c (2 κρ c − , (3.40) ρ ± = (2 κρ c − κ " ± s − κρ c (2 κρ c − . (3.41)Again ( a − , ρ + ) corresponds to the minimum, while ( a + , ρ − ) is associated with the maximumin U ( a ). Note that ρ ± are independent of the value of the parameter A defined in (3.7).This implies that for a universe which oscillates about a − , the matter density at U ( a − ) is thesame as in the ESU. Recall that this situation was earlier encountered for the Braneworld insection 3.1 and implies that equations (3.16) and (3.18) hold in the present case too. Oncemore we shall focus on the stable ESU corresponding to the minima. The ESU is given bythe scale factor a E = 3( κρ + + Λ) exp (cid:20) κρ + + Λ κρ c (cid:21) . (3.42)As Λ increases the minima and maxima in (3.40) approach each other, merging whenΛ = Λ crit = 43 κρ c − √ ! , (3.43)which results in an inflection point in U ( a ). The effective potential in Asymptotically freegravity shown in Fig. 12 resembles that in the braneworld model in Fig. 8. We therefore findthat, as in the braneworld, an emergent scenario can be supported by the inflaton potentialin (3.33). In Loop Quantum Cosmology (LQC), the FRW equation for a spatially closed universe( k = 1) consisting of matter which satisfies the SEC and Λ is [12, 14] H = (cid:18) κ ρ + Λ3 − a (cid:19) (cid:18) − ρρ c − Λ κρ c + 3 κρ c a (cid:19) , (3.44)where ρ c ∼ M p . In this case the effective potential depends on the curvature, and for k = 1one determines it from (2.3) to be U ( a ) = − (cid:18) κ ρa + Λ6 a − (cid:19) (cid:18) − ρρ c − Λ κρ c + 3 κρ c a (cid:19) − , (3.45)where ρ is given by (3.7). The ESU conditions (2.5) provide two possibilities for an EinsteinStatic Universe in this scenario, which have been listed in Table 2. An unstable ESU appearsfor Λ > > κρ c [14].As discussed earlier, a scalar field driven scenario, in which the scalar rolls along a flatpotential ( V ′ = 0), is equivalent to stiff matter together with a cosmological constant Λ. Theeffective potential in this case is shown for various values Λ in Fig. 13. It is worth noting thatevery extremum (maxima or minima) in U ( a ) does not necessarily result in an ESU sinceevery solution of ¨ a = 0 may not support ˙ a = 0. Note that unlike the braneworld case in section 3.1, this asymptotically free gravity model admits nominima in U ( a ) for large Λ, i.e. for Λ > κρ c . – 21 –ixed point (ESU) Λ ρ a E Unstable Λ > ρ GR = 2Λ κ (1 + 3 w ) a GR = 1 + 3 w Λ(1 + w )Stable Λ > κρ c ρ LQC = 2(Λ − κρ c ) κ (1 + 3 w ) a LQC = 1 + 3 w (Λ − κρ c )(1 + w ) Table 2 . Density and scale factor for the
Einstein Static Universe (ESU) in LQC. The stable ESUis denoted by ‘LQC’ while the unstable fixed point resembles the ‘GR’ case. L= L= L= - - - - a U H a L E =- (cid:144) (a) L= L= L= - - - - - a U H a L E =- (cid:144) (b) Figure 13 . The effective potential for LQC is plotted for small values of Λ in the left panel (a) ,and for large values of Λ in the right panel (b) . In both cases one assumes that the universe consistsof stiff matter in addition to the cosmological constant. This combination mimics the behavior of ascalar field rolling along a flat direction in the potential ( V ′ = 0) as described by (3.33) or (2.16). Wechoose a typical value of the parameter A in (3.7) while assuming M p = 1 with ρ c ∼ M p . The leftpanel indicates that inflation could proceed via the potential (3.33) illustrated in Fig. 6. Whereas theright panel supports inflation described by (2.16) and illustrated in Fig. 2. Figures 13(a) and 13(b) demonstrate that the emergent scenario in LQC can arise intwo distinct ways:1. As shown in Fig. 13(a) a minimum in U ( a ) can exist for small values of Λ. As Λincreases this minimum gets destabilized. This indicates that an inflaton potentialsuch as (3.33) discussed earlier in the braneworld context, could also give rise to anemergent scenario in LQC. Note that unlike the braneworld case, a stable ESU doesnot exist in LQC for small Λ – see Table 2. However this does not prevent the universefrom oscillating about the minimum of U ( a ), thereby giving rise to a stable emergentscenario. It is easy to see that since ρ ≪ ρ c during inflation, the CMB constraints onthe parameters of the inflaton potential in (3.33) will be similar to those discussed inAppendix C in the braneworld context.2. The presence of a minimum in U ( a ) is illustrated in Fig. 13(b) for large values of Λ:Λ > κρ c . This minimum is associated with a stable ESU as demonstrated in Table 2.– 22 –s Λ decreases this minimum gets destabilized. This suggests that a potential such as(2.16), earlier discussed in the GR context, could give rise to an emergent scenario inLQC.Possibility 2 might however be problematic in two respects:As noted in Table 2, the value of V ( ≡ Λ /κ ) in the flat wing of the emergent cosmologypotential must be larger than ρ c ∼ M p in order for LQC effects to successfully driveemergent cosmology.(i) V > M p might question the semi-classical treatment pursued by us in this section.(ii) While the potential (2.16) can successfully drive an emergent scenario in LQC,CMB bounds derived in section 2.2 suggest V < − M p , which conflicts with theLQC requirement V & M p .While (i) lies outside the scope of the present paper, we demonstrate in Appendix Dthat CMB constraints can be satisfied even with V > M p provided the scalar fieldLagrangian possesses non-canonical kinetic terms [23].All of the emergent scenarios discussed in this section passed through a prolonged (for-mally infinite) duration quasi-static stage during which the universe was located either atthe Einstein Static fixed point (ESU) or oscillated around it. In the next section we examinethe semi-classical properties of the oscillatory universe, focusing especially on its impact ongraviton production. In a flat FRW universe, each of the two polarization states of the graviton behaves as amassless minimally coupled scalar field [24]. This is also true for the massless gravitonmodes in the higher-dimensional theories (see, e.g., [25] for the case of braneworld model).While the conformal flatness of the FRW space-time ensures that the creation of conformallycoupled fields (including photons) does not happen, no such suppression mechanism exists forfields that couple non-conformally to gravity [26, 27]. Indeed, it is well known that gravitonsare generically created in a FRW universe, and this effect has been very well studied in thecontext of inflation [28–30]. One, therefore, naturally expects gravity waves to be created inan oscillating universe such as the one examined in the previous section, in the context ofemergent cosmology. That this is indeed the case will be demonstrated below.In the emergent scenario, the universe oscillates around a fixed value of the scale factorfor an indefinite amount of time before (gradually changing value of) the potential ends theoscillatory regime and leads to inflation. While oscillating, the universe produces gravitons bya quantum-mechanical process. A large graviton density (compared with that of the existingmatter), can disrupt the oscillatory regime making it difficult for emergent cosmology tobe past-eternal. Here we investigate this issue in the context of the braneworld scenariodescribed in section 3.1. While the CMB bounds in section 2.2 were derived using GR, this would also be a good approximationto the LQC case in which ρ ≪ ρ c once inflation commences, and LQC effects can be ignored. – 23 – .1 Resonant particle production Tensor metric perturbations (or gravity waves) in general relativity are described by thequantities h ij defined as δg ij = a h ij , i, j = 1 , , . (4.1)The tensor h ij is transverse and traceless and obeys the following equations [31]: h ′′ ij + 2 H h ′ ij − ( ▽ − k ) h ij = 0 , (4.2)where H = a ′ /a , and k (= 0 , ±
1) is the spatial curvature of the FRW universe. Here, theprime denotes differentiation with respect to the conformal time η . For a spatially closeduniverse ( k = 1), we make the rescaling h ij = χ ij /a and pass to the generalized Fouriertransform on the three-sphere χ ij ( η, x ) = X nℓ χ nℓ ( η ) Y nℓij ( x ) , (4.3)where Y nℓij ( x ) are the normalized transverse and traceless tensor eigenfunctions of the Lapla-cian operator on a unit three-sphere. They are labeled by the main quantum number n andby the collective quantum numbers ℓ = { p, l, m } , which have the following meaning [32] : n = 3 , , , . . . (main quantum number) ,p = 1 , ,l = 2 , , . . . , n − ,m = − l, − l + 1 , . . . , l (angular momentum projection) . (4.4)The eigenvalues − k n of the Laplacian operator ∇ for transverse traceless tensor modes ona unit three-sphere depend only on n and are given by [32] k n = n − , n = 3 , , , . . . . (4.5)Equation (4.2) then leads to the following equations for the Fourier coefficients χ nℓ ( η ) : χ ′′ nℓ + (cid:18) k n + 2 − a ′′ a (cid:19) χ nℓ = 0 . (4.6)Sufficiently close to the minimum of the effective potential, the universe can exhibitoscillatory motion. Subject to a small perturbation, the oscillatory motion with frequency ω satisfies a ( t ) = a − + δa cos ωt , (4.7)where the frequency is given by ω = d U ( a − ) da , (4.8)and the amplitude is δa = − ω (cid:20) U ( a − ) + 12 (cid:21) . (4.9)To be able to use (4.2) and (4.6), we express this motion in terms of the conformal time η : η = Z dta ( t ) = Z dta − + δa cos ωt ≈ ta − for δaa − ≪ . (4.10)– 24 –ow, the functions a and a ′′ /a can be calculated as a ( η ) = a − + δa cos ζη , (4.11a) a ′′ a = − δaa − ζ cos ζη , (4.11b)where ζ = a − ω (4.12)is the frequency in the conformal time η .We would like to apply the theory of parametric resonance, as described in [33], toequation (4.6). This equation has the general form that was under consideration in [33] : χ ′′ nℓ + (cid:2) ζ n + ǫg ( ζη ) (cid:3) χ nℓ = 0 , (4.13)where g ( x ) is a 2 π -periodic function, and ǫ is a convenient small parameter. In our case, wehave ζ n = k n + 2 = n − , n = 3 , , , . . . , (4.14) g ( x ) = ζ cos x , (4.15)and ǫ = δaa − . (4.16)Equation (4.13) with the function g ( x ) given by (4.15) is just the Mathieu equation. Itis known that the first resonance band for this equation, which is dominant for small valuesof ǫ , lies in the neighbourhood of the frequency ζ res = ζ , (4.17)and the resonant amplification takes place for eigenfrequencies satisfying the condition ∆ n < g . (4.18)Here, ∆ n = 1 ǫ (cid:0) ζ n − ζ (cid:1) , (4.19)and g = 12 ζ (4.20)is the Fourier amplitude of the harmonic function g ( x ).Within the resonance band (4.18), particle production proceeds exponentially with time,so that the number of quanta in the mode grows as N n = 11 − ∆ n /g sinh µ n η , (4.21)where µ n = ǫζ q g − ∆ n . (4.22)The condition (4.18) for instability in the first resonance band can be expressed as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ n − (cid:18) ζ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ǫζ . (4.23)– 25 – .1.1 Radiation-dominated universe with Λ = 0From the viewpoint of the effective potential U ( a ), the created gravitons behave as radiation.Therefore, for simplicity of the analysis, we consider a radiation-dominated universe, in whichthe produced gravitons just increase the existing radiation energy density (i.e., their effectwill consist in the increase of A in (3.7)). For further simplification, we first consider thecase with no cosmological constant. The frequency of oscillations (with respect to the cosmictime t ) turns out to be constant in this case : ω = d U ( a − ) da = 49 κρ c = 49 M p ρ c . (4.24)Therefore, adding more to the existing radiation energy density (increasing the quantity A in (3.7)) does not affect ω . The energy densities at the extremes of the effective potential arealso constant (independent of A in (3.7)) : ρ + = 13 ρ c , ρ − = 0 . (4.25)The second equation implies that there are no maxima in the effective potential (or themaximum is reached as a → ∞ ).From (2.6) and (4.25), the scale factor and the quantity ζ can be evaluated for the ESU : a E = 272 κρ c = 272 M p ρ c , (4.26) ζ E = ω a E = 6 . (4.27)Eventually, the terms appearing in (4.13) can be calculated as ζ = ω a − = 6 (cid:18) a − a E (cid:19) ≥ ǫ = (cid:18) δaa − (cid:19) = 16 − M p ρ c a − = 16 (cid:18) − a E a − (cid:19) ≤ . (4.29)Although ω is independent of graviton production, we can see from (3.8) that a − increases asmore and more gravitons are produced. This leads to an increase both in ζ and in ǫ (although ǫ has an asymptotic saturation value 1 / √ x = (cid:18) a − a E (cid:19) ≥ . (4.30)In this case, the resonance condition (4.23) has the form (cid:18) ζ n − x (cid:19) < x ( x − . (4.31)In Table 3, the resonance intervals of the quantity a − /a E along with the correspondingintervals of ǫ are shown for several lower modes ( ζ n is given by (4.14) and (4.5)) by solvingthe inequality (4.31) in terms of x . As a graviton mode gets excited, the quantity a − /a E – 26 – n a − /a E ǫ − − − − − − Table 3 . Resonance intervals of the quantities a − /a E and ǫ for several lower modes in a radiation-dominated universe. increases due to the growth of A in (3.8). The resonant intervals of a − /a E for neighboringvalues of n overlap, as demonstrated in Table 3 for several lowest values of n . It is clearthen that, if the initial value of a − /a E is sufficiently large, so that it falls in any of theresonance regions, then this will lead to excitation of all modes, one by one, resulting ina monotonous increase in a − /a E due to graviton production. In this case, the oscillatoryregime is unstable with respect to graviton production. However, if the initial amplitude ofoscillations is sufficiently small, so that ǫ < . a − /a E < . ǫ in Table 3 turn out to be not much smaller than unity, they arestill considerably small, and we believe that our analysis in this case is qualitatively correct. For a constant scalar field potential V ( φ ), the kinetic energy density ˙ φ in (2.14) evolves as a − , similarly to the energy density of stiff matter. Therefore, when dealing with inflationbased on a scalar field, it is a good idea to consider the case of a stiff-matter dominateduniverse. It was observed previously that the value a − of the scale factor corresponding tothe minimum of the effective potential actually increases as more and more gravitons areadded to the existing stiff-matter energy density. This was also verified numerically. Below,we analyze both the simple case with Λ = 0 and the more realistic case with non-zero Λ. [A] No cosmological constant, Λ = 0The analysis for stiff-matter dominated universe is carried out in a manner similar to thatof the universe filled with radiation that was under investigation in section 4.1.1. Here, thefrequency in cosmic time ω again turns out to be constant (independent of A in (3.7)) andis given by ω = d U ( a − ) da = 85 κρ c = 85 M p ρ c . (4.32)The value ρ E of the stiff matter energy density at the minimum of the effective potential wasalready calculated in (3.13). Using the value of a E from (3.17), we figure out the value of ζ at the ESU : ζ E = ω a E = 20 . (4.33)Again, the terms appearing in (4.23) can be calculated as ζ = ω a − = 20 (cid:18) a − a E (cid:19) ≥
20 (4.34)– 27 –nd ǫ = 1 ω (cid:18) κρ c − a − (cid:19) = 120 (cid:18) − a E a − (cid:19) ≤ . (4.35)Introducing the quantity x as in (4.30), and using (4.34) and (4.35), we present theresonance condition (4.23) in the form (cid:0) ζ n − x (cid:1) < x ( x − . (4.36) ζ n a − /a E ǫ − − − − − − Table 4 . Resonance intervals of the quantities a − /a E and ǫ for several lower modes in a stiff-matterdominated universe. The resonance intervals of a − /a E and ǫ , given by inequality (4.36), are listed for severallowest modes in Table 4. Again, the overlapping of the resonance bands indicates that, oncea graviton mode is excited, the monotonous growth in a − /a E will lead to resonant excitationof the next modes. However, for a − /a E < . ǫ < . [B] Stiff matter with cosmological constant ΛIf the value of the flat wing of the scalar-field potential is non-negligible, then, in our modelof stiff matter, we must also introduce the cosmological constant Λ. As before, the frequency ω is independent of the quantity A in (3.7), ω = d U ( a − ) da = − κρ + (cid:18) − κρ c (cid:19) + 553 κρ ρ c − Λ3 (cid:18) − Λ κρ c (cid:19) , (4.37)while ρ + and a E are already given in (3.20) and (3.21), respectively. The terms appearing in(4.23) can be expressed more generally as ζ = ω a − = ζ E (cid:18) a − a E (cid:19) ≥ ζ E (4.38)and ǫ = 1 ω (cid:18) a E − a − (cid:19) = 1 ζ E (cid:18) − a E a − (cid:19) ≤ ζ E . (4.39)In deriving (4.39), we used equation (3.16).For a general value of Λ, the value of ζ E cannot be evaluated analytically. For Λ = 0, thecalculation is presented earlier in this subsection 4.1.2. In the opposite limit Λ → κρ c (1 / −√ / a E = 15 κρ c , ω = 0 , ζ E = 0 . (4.40)It is impossible to satisfy the resonance condition (4.23) in this limit of Λ, hence no resonantgraviton production takes place. Now, as Λ increases a − also increases while ω decreases.– 28 – .00 0.02 0.04 0.06 0.08 0.10 0.1205101520 L (cid:144)H ΚΡ c L Ζ E L = L c r it Figure 14 . ζ E vs Λ in the natural units κ = 1 and ρ c = 1. Fig. 14 shows that ζ E actually decreases with increasing Λ, and ζ E goes to zero as Λ ap-proaches the limiting value specified in (3.22).For convenience, along with the variable x defined in (4.30), we also introduce thevariable y = ζ E /
4. Then ζ ζ E (cid:18) a − a E (cid:19) = xy , (4.41)and, for a general ζ E , the resonance condition (4.23) becomes (cid:0) ζ n − yx (cid:1) < yx ( x − . (4.42)Solving the inequality with respect to x for a specified value ζ n , we get the allowed rangeof x to excite the respective mode. The point y = 1 is critical, so one needs to distinguishbetween two cases.(i) For y >
1, the range of allowed x lies in the interval ( x − , x + ), where the end pointsare given by x ± = (cid:18) a − a E (cid:19) = 2 ζ n −
12 ( y − " ± s − ζ n ( y − ζ n − y . (4.43)(ii) For 0 < y <
1, also taking into account that x >
1, we find that the resonantproduction takes place in the domain x > x − = 2 ζ n − − y ) "s ζ n (1 − y )(2 ζ n − y − . (4.44)It is worth noting that, for any mode, x + in (4.43) diverges as y → x − in (4.44) blows up as y → → Λ crit , as expected). Note that thelower boundary in (4.43) and (4.44) is, by expression, the same function of y in differentdomains, so is denoted by the same symbol x − .The lowest mode ( ζ n = 8) is of particular interest because it determines the conditionthat no gravitons are resonantly produced. This mode is excited for the range of x determined– 29 – st resonance band2nd resonance band3rd resonance band L (cid:144)H ΚΡ c L a - a E quiescent particle production region L = L c r it Figure 15 . The resonance bands of the quantity a − /a E for three lowest graviton modes are shownas a function of the cosmological constant Λ. Again, the natural unit of κ = 1 is assumed along with ρ c = 1. The “quiescent particle production region” below the first resonance domain corresponds tothe region of parameters in which no graviton mode is in resonance. by x ± = 152( y − " ± s − y − y , y > , (4.45) x − = 152(1 − y ) "s − y )225 y − , < y < . (4.46)The resonance bands of the quantity a − /a E for three lowest modes are plotted as afunction of Λ in Fig. 15. The values of y in (4.43) and (4.44) for different values of Λ arecalculated from (3.20), (3.21) and (4.37). The overlapping bands (spanning all values ofΛ < Λ crit ) again imply that, once a particular graviton mode is in the resonance, all thesubsequent modes eventually will be resonantly excited. However, the range of a − /a E whichdoes not excite any resonance mode actually expands with increasing Λ. This region, belowthe first resonance band, is indicated as the “quiescent particle production region”. In this section, we investigate more thoroughly the case where no resonant production ofgravitons takes place, i.e., where no frequency ζ n lies in the resonance band determined bycondition (4.23). Although the graviton modes are not excited resonantly , their excitation– 30 –till takes place. Our aim is to show that this effect is rather small and does not destroy thestability of the oscillating universe.Since all graviton modes are assumed to be out of the resonance band, we can applyperturbation theory for the calculation of their average occupation numbers. From the generaltheory of excitation of a harmonic oscillator with time-dependent frequency Ω( η ), it is knownthat these occupation numbers are expressed through the complex Bogolyubov coefficients α and β . These coefficients, for boson fields, obey the normalization condition | α | − | β | = 1 , (4.47)and satisfy the following system of equations (see, e.g., [33]): α ′ = Ω ′ e +2 i R Ω dη β , β ′ = Ω ′ e − i R Ω dη α . (4.48)If initially (for convenience, we set η = 0) the oscillator is not excited, then one can set α (0) = 1 , β (0) = 0 . (4.49)The average excitation number (average number of particles) at the time η is then given by N ( η ) = | β ( η ) | . (4.50)In the case of graviton production, from (4.6), (4.11b) and (4.13), the time-dependentfrequency of the corresponding harmonic oscillator can be identified asΩ n = ζ n + ǫζ cos ζη . (4.51)For sufficiently small value of ǫ and outside the resonance band, we haveΩ ′ n n ≈ − ǫζ sin ζηζ n . (4.52)To the first order in ǫ , from (4.48) and (4.49) we then have the following equation for β : β ′ n ( η ) ≈ − ǫζ sin ζηζ n e − iζ n η = iǫζ ζ n h e − i (2 ζ n − ζ ) η − e − i (2 ζ n + ζ ) η i . (4.53)Now integrating (4.53) from 0 to η , we have β n ( η ) ≈ ǫζ ζ n " e − i (2 ζ n + ζ ) η − ζ n + ζ − e − i (2 ζ n − ζ ) η − ζ n − ζ . (4.54)Hence, | β n ( η ) | ≈ (cid:18) ǫζ ζ n (cid:19) " e − i (2 ζ n + ζ ) η − ζ n + ζ − e − i (2 ζ n − ζ ) η − ζ n − ζ × " e i (2 ζ n + ζ ) η − ζ n + ζ − e i (2 ζ n − ζ ) η − ζ n − ζ (4.55)– 31 –r, | β n ( η ) | ≈ (cid:18) ǫζ ζ n (cid:19) (cid:18) sin ( ζ n + ζ/ η ( ζ n + ζ/ + sin ( ζ n − ζ/ η ( ζ n − ζ/ + 12 [ ζ n − ( ζ/ ] [cos (2 ζ n + ζ ) η + cos (2 ζ n − ζ ) η − cos 2 ζη − (cid:19) . (4.56)In the resonance, as ζ n → ζ/
2, the second term in (4.54) dominates. When the resonancecondition is not met, we can safely assume that ζ n ≫ ζ/ | β n ( η ) | ≈ (cid:18) ǫζ ζ n (cid:19) ζ n (cid:20) ζ n η + cos 2 ζ n η −
12 (1 + cos 2 ζη ) (cid:21) = (cid:18) ǫζ (cid:19) sin ζηζ n . (4.57)Now, to calculate the graviton energy density ρ grav , the quantity | β n ( η ) | should besummed over the graviton modes with the energy of graviton taken into account, and thendivided by the volume of the space, which is a − . We recall that the graviton modes on athree-sphere are labeled by the quantum numbers (4.4). The frequency depends only on n .Summing over l and m , we get n − X l =2 l X m = − l = n − X l =2 (2 l + 1) = n − . (4.58)Then, taking into account two polarizations and the fact that the energy of a graviton is ζ n /a − , we have ρ grav = 2 a − (cid:18) ǫζ (cid:19) sin ζη ∞ X n =3 n − ζ n , = 2 a − (cid:18) ǫζ (cid:19) sin ζη ∞ X n =3 n − n − / , = 2 a − (cid:18) ǫζ (cid:19) sin ωt ∞ X n =3 n − n − / . (4.59)The sum in this expression is convergent, and can be estimated as ∞ X n =3 n − n − / ≈ . . (4.60)The graviton energy density turns out to be periodic with time as indicated in (4.59),with period equal to half the period of oscillation of the universe. The energy density ofproduced gravitons is shown as a function of conformal time η in Fig. 16 for a typical value of ζ = √
20, i.e., for the case of stiff-matter dominated universe, with no cosmological constant,oscillating very close to the ESU (see (4.33) and (4.38)).From Fig. 14 and in the view of (4.38), it is apparent that, for small values of Λ, thequantity ζ/ ζ = √ Η Ρ g r a v Ρ Figure 16 . The energy density ρ grav as a function of conformal time η in the case of gravitonproduction far away from the resonance, when ζ n ≫ ζ res = ζ/
2. Here, we choose ζ = 20 along with ρ = 2 a − (cid:18) ǫζ (cid:19) . assumption is not strictly valid, a more general expression for ρ grav can be calculated directlyfrom (4.56) : ρ grav = 2 a − (cid:18) ǫζ (cid:19) ∞ X n =3 n − ζ n (cid:18) sin ( ζ n + ζ/ η ( ζ n + ζ/ + sin ( ζ n − ζ/ η ( ζ n − ζ/ + 12 (cid:16) ζ n − ( ζ/ (cid:17) [cos (2 ζ n + ζ ) η + cos (2 ζ n − ζ ) η − cos 2 ζη − . (4.61)In Figs. 17(a) and 17(b), the quantity ρ grav given by (4.61) is shown as a function of theconformal time η for ζ = 6 and ζ = 20, which can represent a radiation and stiff-matterdominated universe, respectively, with Λ = 0, while oscillating very close to the ESU. It isnormalized by the quantity ρ = 132 ǫ ζ ω = 132 (cid:18) ǫ ζ E − ǫ ζ E (cid:19) ω = 132 a E (cid:18) ǫ ζ E − ǫ ζ E (cid:19) , (4.62)where we have used the relation (see (4.38) and (4.39)), ǫ ζ = ǫ ζ E − ǫ ζ E . It is worth noting that, in the limit ǫ → /ζ E , the quantity ρ (hence, also ρ grav ) becomesinfinite. This is consistent with (4.59) and (4.61) since, in this limit, a − (hence, also ζ ) isalso infinite (see (4.39)).Now we have to compare the amount of graviton produced to the existing matter density.Although Figs. 17(a) and 17(b) show some departure from Fig. 16, we see that, in all cases,the energy density of the gravitons is small. Indeed, let us denote the summation in (4.61) by s (in (4.59), it is the summation multiplied by the time-dependent part). From (4.25), (3.13)– 33 – Η Ρ g r a v Ρ (a) ζ = 6 Η Ρ g r a v Ρ (b) ζ = 20 Figure 17 . The energy density ρ grav as a function of conformal time η with (a) ζ = 6 and (b) ζ = 20 in the case of non-resonant graviton production. These two cases represent a universe filledwith radiation and stiff matter, respectively, with Λ = 0 and oscillating very closely to the ESU.Again, ρ = 2 a − (cid:18) ǫζ (cid:19) . and (3.20), it is clear that ρ c is a good estimation for the existing matter density. Hence, theratio of the energy density of the produced gravitons to that of existing matter is ρ grav ρ c ≈ s (cid:18) ǫζ (cid:19) a − ρ c = 2 s (cid:18) ǫζ (cid:19) ζ E a E ρ c = sω ρ c (cid:18) ǫ ζ E − ǫ ζ E (cid:19) = s a E ρ c (cid:18) ǫ ζ E − ǫ ζ E (cid:19) . (4.63)In the last step, we used the general expression for ζ from (4.38). Now we can see that ρ grav ρ c ≪ ǫ on the right-hand side (all otherquantities have finite values). For a typical case of stiff-matter dominated universe withΛ = 0, we can estimate the ratio using (3.17) and (4.33), ρ grav ρ c ≈ s ( ǫζ ) a E M p = 85 (cid:18) sǫ − ǫ (cid:19) ρ c M p ≪ . (4.64)Thus, for small deviations from the Einstein Static Universe, the maximum possibleenergy density of gravitons produced during the eternal oscillations in the emergent scenarioappears to be tiny compared to the existing matter density, which ensures the stability ofthe model. In this paper we have shown how the effective potential formalism can be used to study thedynamical properties of the emergent universe scenario. Within the GR setting, the effectivepotential has a single extreme point, a maximum, which corresponds to the unstable
EinsteinStatic Universe (ESU). Extending our analysis to modified gravity theories we find that anew minimum in the effective potential appears corresponding to a stable
ESU. These resultsare in broad agreement with earlier studies which also pointed out the appearance of an ESUin the context of extensions to GR [13, 14, 34]; also see [35].– 34 –hile in GR, the emergent scenario can only occur if the universe is closed, we showthat this restriction does not apply to certain modified gravity models in which the emergentscenario can occur in spatially closed as well as open cosmologies.The appearance of a stable minimum in the effective potential considerably enlargesthe initial data set from which the universe could have ‘emerged’. In this case, in additionto being precisely located at the minimum (ESU) – which requires considerable fine tuningof initial conditions, the universe can oscillate about it. Furthermore, we show that theexistence of an ESU, while being conducive for emergent cosmology, is not essential for it. Insection 3.3 this is demonstrated for LQC for which a stable ESU exists only for Λ > κρ c [14].We demonstrate that even for Λ ≪ κρ c , when a stable ESU no longer exists, the universecan still oscillate about the minimum of its effective potential allowing an emergent scenarioto be constructed.However an oscillating universe is always accompanied by graviton production. Whilethe magnitude of this semi-classical effect depends upon parameters in the effective poten-tial, for a large region in parameter space this effect can be very large, casting doubts as towhether such an emergent scenario could have been past-eternal. (The instability of emer-gent cosmology to quantum effects has also been recently investigated in [36].) Althoughgraviton production has been discussed in detail for an effective potential derived from thebraneworld scenario, the effect itself is semi-classical and generic, and would be expected toaccompany any emergent scenario in which the universe emerges from an oscillatory state.One might also note that in the emergent scenarios discussed in this paper, the post-emergent universe inflates by well over 60 e-folds. Consequently any feature associated withthe transition from an ESU to inflation is pushed to scales much larger than the presenthorizon. However it could well be that in some emergent scenarios this is not the case, andthe transition from the ESU to inflation takes place fewer than ∼
60 e-folds from the endof inflation. In this case the spectrum of inflationary perturbations would differ from thoseconsidered in this paper on large scales, and may contain a feature in the CMB anisotropyspectrum, C ℓ , at low values of ℓ . Acknowledgments
V.S. and Yu.S. acknowledge support from the India-Ukraine Bilateral Scientific Cooperationprogramme. The work of Yu.S. was also partially supported by the SFFR of Ukraine GrantNo. F53.2/028. Graviton production is small, and does not stand in the way of emergent cosmology being past eternal,only if the universe oscillates very near the minimum of its effective potential. (Naturally, there is no particleproduction for a universe located precisely at the minimum of U ( a ), i.e. for the ESU.) But this situationmight require considerable fine tuning of parameters. – 35 – Emergent scenario in a spatially open Braneworld
As mentioned in section 3.1, the braneworld admits a minimum in the effective potential U ( a ) if Λ > κρ c ; see (3.8) and (3.9). Considering stiff matter along with Λ > κρ c , from theequations (3.19) and (3.20), one finds that only ( a + , ρ − ) survive which now account for the minimum given by a = 3 A (2Λ − κρ c )Λ (Λ − κρ c ) " s − κρ c )(2Λ − κρ c ) , (A.1) ρ − = ρ E = 15 κ (2Λ − κρ c ) "s − κρ c )(2Λ − κρ c ) − . (A.2) L= L= L= - a U H a L E = (cid:144) Figure 18 . The effective potential for the braneworld in (3.6) is plotted for large values of Λ (in units κ = 1, ρ c = 1). For Λ > κρ c , a minima in U ( a ) appears which can give rise to an ESU provided theuniverse is open ( k = − ≤ κρ c the minimum disappears which allows the universe to exitthe ESU and also to inflate (for a suitable choice of the inflaton potential). According to (3.5), when Λ > κρ c , a static solution can exist only for a spatially openuniverse ( k = − ρ − is the energy density at the ESU, hence denoted by ρ E in (A.2).The scale factor at ESU is calculated for the spatially open universe using (3.5) and (A.2) tobe a E − = ( κρ E + Λ)3 (cid:18) ρ E ρ c + Λ κρ c − (cid:19) . (A.3)The effective potential in this case is plotted for various Λ in Fig. 18 which illustratesthe emergence scenario in a spatially open universe. For Λ > κρ c the universe can eitherbe an ESU or can oscillate around the minimum of U ( a ) at a + . When Λ = κρ c , the mini-mum disappears and, for large values of a , U ( a ) asymptotically approaches zero from above.Therefore Λ < κρ c destabilizes the ESU and can result in inflation for the potential V ( φ ) in(2.16). After inflation commences the curvature term, k /a , rapidly declines to zero resultingin a spatially flat universe. (See [37] for a discussion of an emergent scenario in a spatiallyflat braneworld.) – 36 – Linearization near a fixed point
A two dimensional non-linear system given by˙ x i = X i ( x , x ) where i = 1 , , (B.1)can be linearized in the neighbourhood of its simple fixed point ( ζ, η ) as [22],˙ x i ≈ A ij x j where A ij = ∂X i ∂x j (cid:12)(cid:12)(cid:12) ( ζ,η ) . (B.2)It should be noted that the linearization theorem guarantees that the linearized system in theneighbourhood of a fixed point is qualitatively equivalent to the non-linear system as long asthe former does not suggest a centre type linearization.For braneworld consisting of two component fluid: stiff matter with Λ, the linearizedsystem is given by the co-efficient matrix in (B.2), A = − ρ ± − κ (cid:26) − ρ c (cid:18) ρ ± + 2Λ κ (cid:19)(cid:27) The stability of fixed points of a linear system depends upon the eigenvalues of theJacobian matrix A (see chapter 2 of [22]). For a two dimensional system, if both eigenvaluesare real and of opposite sign then the fixed point is a saddle in the phase portrait. On theother hand, if the eigenvalues are imaginary and complex conjugates of each other then thefixed point is likely to be a centre, but this must be confirmed numerically. C CMB constraints on the Emergent Scenario in Braneworld Cosmology
The parameter γ in the potential (3.33) can be constrained using the CMB bounds on thescalar spectral index n S and the tensor-to-scalar ratio r from the recent Planck mission [1]. Asthe scalar field rolls down the potential from φ >> M , the potential (3.33) is approximatelythe same as in the case of chaotic inflation models with V ( φ ) ∝ φ γ . Therefore, in the slowroll limit one finds that n S − − γ + 1) γ + 2 N ,n T = − γγ + 2 N ,r = 16 γγ + 2 N . (C.1)At 95% CL
Planck data allows n S within the range [0 . − . n S to lie in thisallowed range with N = 60, the parameter γ must lie in the following range: [0 . − . Planck data also indicate that r < .
12 at 95% CL when BAO data isincluded [1]. Using Eq. (C.1) and with N = 60, r < .
12 can be realised only if γ ≤ . φ >> M can lead to both n S within the range [0 . − . r < .
12 if the parameter γ is within the following range:0 . ≤ γ ≤ . .02 0.05 0.10 0.20 0.50 1.00 2.0010 - - - - - - Γ V o M p M = - M p M = - M p M = M p Figure 19 . The CMB normalized value of V in the potential (3.33) is plotted as a function of γ forthree different values of M . In this figure we have taken the number of e-folds N = 60. For example, if γ = 0 .
8, one gets n S ≃ .
97 and r ≃ . Planck results [1]. Note that in the potential (3.33), we have assumed that φ >> M during inflation. Inflation ends at φ = √ γ M p . Therefore, the approximation φ >> M isreasonable if M << M p .The constant V can be fixed using the CMB normalization which indicate that P S ( k ∗ ) =2 . × − at the pivot scale k ∗ = 0 .
05 Mpc − [1]. The expression for V in terms of P S ( k ∗ )is given by V M p = (cid:18) π γ P S ( k ∗ )[2 γ ( γ + 2 N )] γ +1 (cid:19) (cid:18) MM p (cid:19) γ . (C.3)Using the above equation, it turns out that V = 8 × − M p for γ = 0 . M = 10 − M p .In Fig. 19, the CMB normalized value of V is plotted as a function of γ . One might notethat in deriving (C.1) we assumed ρ << ρ c during inflation, with ρ c defined in Eq. (3.3).At 60 e-folds before the end of inflation, one finds ρ ≃ V ( φ ) = 3 . × − M p for γ = 0 . M . Therefore, the approximation ρ << ρ c during inflation isvalid provided ρ c >> − M p . D CMB constraints on the Emergent Scenario in LQC
As noted in section 3.3, the emergent scenario in LQC can proceed in two distinct ways:(i) If a minimum in the effective potential exists for small values of the inflaton potential.In this case the minimum gets destabilized as the inflaton potential increases , as shownin Fig. 13(a). A canonical scalar field potential such as (3.33) can accomplish this whilesatisfying CMB constraints.(ii) If a minimum in the effective potential exists for large values of the inflaton potential.In this case the minimum gets destabilized as the inflaton potential decreases , as shown inFig. 13(b). While an inflaton potential such as (2.16) does accomplish this, it fails to satisfyCMB bounds if the scalar field has canonical kinetic terms . One therefore needs to turn to non-canonical scalars in order to construct a working example of the emergent scenario inthis case, which forms the focus of this appendix.– 38 –onsider the following non-canonical scalar field Lagrangian [23] L ( X, φ ) = X (cid:18) XM (cid:19) α − − V ( φ ) , X = 12 ˙ φ , (D.1)where α is a dimensionless parameter ( α ≥
1) while M has dimensions of mass. The canonicalLagrangian (2.12) corresponds to α = 1 in (D.1). The energy density and pressure aremodified for the non-canonical Lagrangian as follows: ρ φ = (2 α − X (cid:18) XM (cid:19) α − + V ( φ ) , (D.2a) P φ = X (cid:18) XM (cid:19) α − − V ( φ ) . (D.2b)The modified scalar field equation of motion is given by¨ φ + 3 H ˙ φ α − (cid:18) V ′ ( φ ) α (2 α − (cid:19) (cid:18) M ˙ φ (cid:19) α − = 0 . (D.3)As long as V ( φ ) is constant, the non-canonical scalar is equivalent to two non-interactingfluids: Λ (which mimics the constant potential V ) plus matter with equation of state w = 12 α − . (D.4)Thus the non-canonical formalism allows for a wider range of possibilities for the equationof state: 0 ≤ w <
1. For instance α = 2 ⇒ w = 1 /
3, and the non-canonical scalar plays therole of a radiation+Λ filled universe. The corresponding effective potential U ( a ) is similar tothat shown for the canonical scalar in Fig. 13(b), while the scale factor at ESU is determinedfrom Table 2 to be a LQ = 3(Λ − κρ c ) / λ φ << M p , so that (2.16) can be approximated as V ( φ ) ≃ V λ p (cid:18) φM p (cid:19) p , (D.5)which allows the problem to be tackled analytically.Following [23] we find that in this case n S − − (cid:18) σ + p N σ + p (cid:19) ,n T = − p N σ + p ,r = (cid:18) √ α − (cid:19) (cid:18) p N σ + p (cid:19) , (D.6)where σ = α + p ( α − α − . (D.7)– 39 – Α n s p = = p = Α r p = p = p = Figure 20 . The scalar spectral index n S (left panel) and the tensor-to-scalar ratio r (right panel)are shown as functions of α , for p = 1 and p = 2 in (2.16). The number of e-folds is fixed to N = 60.The shaded region refers to 95% confidence limits on n S and r determined by Planck [1]. - - - - Α V M p p = = - M p Λ =
Λ =
Λ = - - - Α V M p p = = - M p Λ = . Λ = . Λ = . Figure 21 . The CMB normalized value of V in the potential (2.16) is plotted as a function of α . Inthe left panel, the value of the parameter M in (D.1) is set to be 10 − M p , whereas M = 10 − M p inthe right panel. In Fig. 20, the scalar spectral index n S and the tensor-to-scalar ratio r are plotted asfunctions of α . Note that r decreases as α increases. Substituting α = 2 in (D.6), we find n S = 0 .
97 and r = 0 .
076 for p = 1 ,n S = 0 .
96 and r = 0 .
11 for p = 2 , (D.8)which satisfy the Planck requirements n S ∈ [0 . − .
98] and r < .
12 at 95% CL [1].The value of V can be fixed using CMB normalization viz. P S ( k ∗ ) = 2 . × − atthe pivot scale k ∗ = 0 .
05 Mpc − [1]. For (D.1) and (2.16), the expression for V in terms of– 40 – S ( k ∗ ) is given by V M p = (cid:18) λ p (cid:19) (cid:18) π p P S ( k ∗ ) √ α − (cid:19) "(cid:18) α p (cid:19) (cid:18) µ (cid:19) α − pσ (2 α − (cid:18) N σ + p (cid:19) σ + pσ σ (2 α − α , (D.9)where µ = M/M p . In Fig. 21 the CMB normalized value of V is plotted as a function of α for different values of λ and M . One finds that when M = 10 − M p the value of V is nearlyindependent of α , whereas V increases dramatically with increasing α , when M < − M p .This result also holds when p >
1. Furthermore, for fixed values of α and M , V increasesas λ decreases, as shown in Fig. 22. From these figures its clear that one can have V & M p by appropriately choosing the model parameters α , λ and M . For instance, α = 2 and M = 10 − M p result in V & M p when λ . . × − for p = 1 ,V & M p when λ . . × − for p = 2 , (D.10)together with the Planck -consistent values for n S and r quoted in (D.8). - - Λ V M p Α = = - M p p = p = Figure 22 . The CMB normalized value of V in (2.16) is plotted as a function of λ . The twoparameters α and M in the Lagrangian (D.1) have been fixed to α = 2 and M = 10 − M p . As mentioned earlier, these results are valid only when λφ << M p during inflation. Inthis case, N e-folds prior to the end of inflation, one finds φ ( N ) = C / (2 σ ) (2 N σ + p ) / (2 σ ) M p , (D.11)where σ was defined in (D.7) and C is given by C = ( p µ α − α ! M p V λ p !) α − . (D.12)For α = p = 2, the above equations give φ = 0 . M p , 60 e-folds before the end of inflation,making the approximation λφ << M p perfectly reasonable provided λ < n S and r even when V & M p , provided that the semi-classical treatment followed by us is allowed in this regime.– 41 – eferences [1] P. A. R. Ade et al. [Planck Collaboration], Planck 2013 results. XXII. Constraints on inflation ,arXiv:1303.5082 [astro-ph.CO].[2] A. Borde, A. H. Guth and A. Vilenkin,
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