Conformal Cyclic Evolution of Phantom Energy Dominated Universe

Home , Cycles of Time, Initial singularity, Phantom energy

RESEARCH Revista Mexicana de F´ısica 66 (2) 209–223 MARCH-APRIL 2020

Conformal cyclic evolution of phantom energy dominated universe

S. Natarajan PG and Research Department of Physics Sree Sevugan Annamalai College Devakottai India R. Chandramohan PG and Research Department of Physics Sree Sevugan Annamalai College, Devakottai, India, PG Department of Physics, Vidhyagiri College of Arts and Science, Puduvayal, India R. Swaminathan PG Department of Mathematics, Vidhyagiri College of Arts and Science, Puduvayal, India Received 10 July 2019; accepted 3 September 2019

From the Wheeler-Dewitt solutions, the scale factor of the initial universe is discussed. In this study, scale factors from Wheeler-Dewitt solutions, loop quantum gravity, and phantom energy dominated stages are compared. Certain modifications have been attempted in scale factor and quantum potentials driven by canonical quantum gravity approaches. Their results are discussed in this work. Despite an increment of phantom energy density, avoidance of Big Rip is reported. Scale factors predicted from various models are discussed in this work. The relationship between scale factors and the smooth continuation of Aeon is discussed by the application of conformal cyclic cosmology. Quantum potentials for various models are correlated and a correction parameter is included in the cosmological constant. Phantom energy dominated, final stage non-singular evolution of the universe is found. Eternal increment of phantom energy density without interacting with dark matter is reported for the consequence of the evolution of the future universe. Also, the non-interacting solutions of phantom energy and dark matter are explained. As the evolution continues even after the final singularity is approached, the validity of conformal cyclic cosmology is predicted. Non zero values for the scale factor for the set of eigenvalues are presented. Results are compared with supersymmetric classical cosmology. The non-interacting solutions are compared with SiBI solutions.

Keywords: Phantom energy; Wheeler-Dewitt theory; scale factor quantization; loop quantum cosmology; cosmological constant; non interacting phantom cosmology.

PACS: 98.80.Qc; 02.40.Xx; 98.80.k; 98.80.Bp DOI: https://doi.org/10.31349/RevMexFis.66.209

1. Introduction background-independent resolution, the loop quantum gravity provides a tool to understand the quantum nature of the Thr scale factor determines the evolution of the universe. At Big Bang. Time and critical density dependence of the scale the initial stages, it is closest to its minimum value. At later factor are determined by loop quantum cosmology. Simi- stages, the scale factor approaches infinity, classically. Here, larly, pre-Big Bang scenario is characterized by the confor- Table I shows the divergence of various singularities. mal cyclic cosmology. This confirms the cyclic evolution of The classical cosmological model does not provide de- the universe. In such resolutions, the conservation of infor- tailed information about the dynamics of the universe at the mation is required to be understood. In recent days, Penrose singularity, due to the vanishing scale factor. Particularly, the expressed his ideas about the confirmation of the existence classical analysis of the singularity does not make sense. The of Hawking points from previous Aeon, through the CMB Wheeler-Dewitt analysis and the LQC provides detailed in- radiation [1]. formation about Hubble parameter at the initial stages of the The classical singularity is a mathematical point, in which universe. the cosmological parameters such as energy conditions, pres- The initial singularity is the mysterious problem from sure, and curvature face divergences [2]. Every classical cos- the theoretical standpoint predictions. The classical analysis mological scenario avoids the initial singularity. The singu- did not resolve the initial singularity; quantum treatment is larity can be analysed through a quantum mechanical per- required. To analyse the initial singularity. With the help of spective. The quantum mechanical solutions provide the un- derstanding of initial conditions of the universe. Dynamics of TABLE I. Divergence of various cosmological parameters. the universe is approached with many theoretical models such Type Divergence as Big Bang [3], Ekpyrotic [4], cyclic [5], Lambda CDM [7] and many more. The initial conditions of the universe resem- I a,ρ,p ble like quantum mechanical vacuum fluctuations [8]. The II p Wheeler-Dewitt model was proposed for the quantization of III ρ,p the gravity [9]. It is applied for the quantum analysis of initial IV p˙ stages of the universe. 210 S. NATARAJAN, R. CHANDRAMOHAN AND R. SWAMINATHAN

From such solutions, the universe is said to be created leads to the continued evolution of the universe without ap- spontaneously from quantum vacuum [10]. The quantum proaching the final singularity. universe that pops out of a vacuum does not require any Big Bang-like singularities [11]. Similarly, such a scenario re- 2. Conformal Cyclic Cosmology quires no initial boundary conditions. In contrast to other models, the LQG makes consistent results over renormal- The Conformal Cyclic Cosmology (CCC) provides alterna- ization of quantum parameters for quantizing the gravity, to tive explanations for existing cosmological models [20-22]. avoid infrared ultraviolet divergences [12]. The Loop Quan- The CCC predicts the evolution of the universe as cycles or tum Cosmology resolves the classical singularity and exam- Aeons. The repeated cycles are referred to as Aeons. Each ines the universe quantum mechanically [13]. In LQC, the Aeon starts from a Big Bang and ends up with a Big Crunch. quantum nature of the Big Bang is analysed [14]. The loop The CCC follows conformal structure over the metric struc- quantum model suggests that at singularity, the quantum uni- ture. The CCC approaches the initial stages of the universe in verse bounces back as the density approaches the critical a lower entropic state which was higher for the remote part of level. Hence, the Big Bang is replaced with a Big Bounce iin the previous Aeon [22]. The universe without inflation is pre- the loop quantum cosmological scenario [15]. The formation dicted by conformal cyclic cosmology. The CCC explains the of the quantum mechanical universe from the quantum vac- imbalance between the thermal nature of radiation and mat- uum is discussed here with the help of the Wheeler-Dewitt ter in the earlier phases of the universe. The suppression of theorem. With scale factor quantization, the behaviour of the gravitational degrees of freedom is a consequence of remote quantum cosmological constant is also discussed. A time- past Aeon which has low gravitational entropy. Thus the uni- varying cosmological parameter is proposed to have a consis- verse is proposed as the conformal evolution over the Aeon. tent value for the cosmological constant, which is currently a It is possible to cross over the Aeons, due to the conformal in- need of the hour. The scale factor solutions are discussed for variance of physics of massive particles which survived from the k = 0 model. past Aeon. The materials of galactic clusters remain as the form of Hawking radiation. Due to the evaporation, massive To construct a local supersymmetric quantum cosmo- particles can be available as photons for the future Aeons, logical model an alternative procedure is presented in [16]. after the survival from Big Bang singularity. The cosmo- A superfield formulation is introduced and applied to the logical constant Λ is suggested as invariant in all Aeons. In Friedmann-Robertson-Walker FRW model. Scalar field cos- CCC the inflation is said to be absent. The Λ overtakes the mologies with perfect fluid in a Robertson-Walker metric are inflation, that provides exponential expansion. The CCC is discussed in [17]. Asymptotic solutions for the final Fried- consistent with LCDM [23], without implementing inflation. mann stage with simple potentials are found in the same The CCC does not violate the second law of thermodynam- work. It also predicts that the perfect fluid and curvature ics [24]. The conformal cyclic cosmological model topolog- may affect the evolution of the universe. Scalar phantom en- ically analyses the past and future singularities as surfaces ergy as a cosmological dynamical system is reported in [18]. with smooth boundaries. Hence, future and past boundaries Three characteristic solutions can be identified for the canon- are equated with the same kind of topology. Relationship be- ical formalism. Effect of local supersymmetry on cosmology tween past and future singularities mainly exist in the scale is discussed in [19]. The authors discuss the case of a su- factor. Though the singularities come under various types, persymmetric FRW model in a flat space in the superfield the initial Big Bang and future Big Rip are compared in Ta- formulation. ble III. This comparison is of interest for of the Big Rip is The interactions of phantom energy and matter fluid are considered as the end of the evolution of the universe. So the reported in [20]. Avoidance of cosmic doomsday in front of scale factors from Big Bang and Big Rip can be mapped con- phantom energy is reported in the same. What will happen for formally with each other. Comparisons of scale factor from the universe, soon after such singularity is approached? Will loop quantum cosmology and classical super symmetrical so- Type I singularity stop the evolution of the universe? Will lutions are discussed in this work. it be the endpoint of all universes? Answers for these ques- Information on the existence of Hawking radiation from tions are discussed in this work. Evolution of the universe is previous Aeon can be obtained from Hawking points. Such explained by the various cosmological models. Hawking radiation obtained from Hawking points might be Initially, this work starts with a basic introduction on con- the remnant of the super-massive blackhole of the previous formal cyclic cosmology. In the next section, an introduc- Aeon. Such existence can be proven by mathematical and ex- tion to phantom energy and dark energy is provided. In the perimental solutions such as cosmic microwave background next section, an overview for loop quantum cosmology is dis- [1]. The conformal cyclic cosmology is theorized with the cussed. We discuss quantization procedures. In the next sec- Aeons. The universe evolves as Aeons. Evolution begins tion, we analyze the solutions for Wheeler-Dewitt equations. from a singularity grows with exponential expansion, reaches A comparison between Wheeler-Dewitt cosmology and clas- the critical density and then they end up with singularity sical supersymmetric cosmology is drawn. In later sections, again. Before proceeding into such mappings, some intro- we provide solutions for non-interacting dark energy which duction about fundamental topics are required.

Rev. Mex. F´ıs. 66 (2) 209–223 CONFORMAL CYCLIC EVOLUTION OF PHANTOM ENERGY DOMINATED UNIVERSE 211

3. Dark energy and Phantom energy Various cosmological singularities are reported in [31]. The scale factor has the following characterization.

The dark energy is a mysterious form of the energy content η0 η1 a(t) = c0|t − t0| + c1|t − t0| + ..., (1) of the universe. The equation of state ω determines the acceleration of the universe. From Einstein’s field equations, where η0 < η1 < ..., c0 > 0 cosmological constant is proposed for the consequences for The geodesics are parameterized as the accelerated expansion of the universe [25]. In general, s the cosmological constant value is smaller than the expected P 2 t0 = δ + , (2) value from quantum gravity. Though the technical conflicts a2(t) exist between the theory and experiment, cosmological constant is included in Einstein’s field equations. and P The phantom energy has the values of equation of state r0 = ± , (3) parameter ω < −1. The phantom energy violates the null a2(t)f(r) energy conditions. The phantom energy is proposed as a can- with didate for the sustainability of traversable wormholes [26]. 1 Wormholes metric with the dominance of phantom energy is f 2(r) = . (4) studied in [27]. Symmetric distribution of phantom energy in 1 − kr2 a static wormhole is also reported. The phantom energy has It has been used for simplicity a constant geodesic motion super-luminal properties. Such properties are similar to those P and δ = 0 and 1 for null and time like geodesics. For null predicted by supergravity or higher derivative gravitational geodesics δ = 0 theories [28-30]. Zt In the evolutionary phase of the universe, the singularities 0 0 can be classified by the diverging parameters. Big Rip singu- a(t)t = P = a(t)t , (5) larity appears in a finite time. The finite-time singularity will t0 occur with weak conditions of ρ > 0 and ρ + 3P > 0 in an P = P (τ − τ0). (6) expanding universe. A Type I singularity has a scale factor divergence along If η0 > 0, the scale factor vanishes at t0. Hence, there with the associated to energy density and pressure. Type II will be either a Big Bang or Big Crunch. If η0 = 0, then the singularities are referred to as sudden feature singularities. scale factor will be finite at t0. A sudden future singularity Divergence of pressure with finite scale factor and energy will appear in the evolution of the universe. If η0 < 0, then density occurs. Type III is referred to as big freeze singu- the universe will face Big Rip at t0. If the cosmological mod- larity. In Type III energy density and pressure diverge with els exist with η0 ≤ −1, then the null geodesics will avoid a finite scale factor. Type IV singularity is referred to as the Big Rip singularity. In our solutions, η0 → (−∞, 0), big separation singularity, where energy density pressure and η1 → (η0, ∞), k = 0, ±1, and c0 → (0, ∞). Various cos- scale factor remain finite, but the time derivative of pressure mological singularities [32-34] are reported in Table III. or energy density diverge. The universe can be extended af- ω = const is not the only option to obtain an accelerated ter it approaches singularities of either Type II or IV, b ecause cosmic expansion. There are dark energy parameterizations they are relatively weak. The strong Big Rip singularities is that can cross the phantom divided-line with success and also resolved here using loop quantum cosmology. Among these reproduce ω = −1. singularities Type zero is experienced by the universe at the earlier stages and Type I is encountered at later times. TABLE III. Strength of singularities is determeined by the various parameters.

η0 η1 k c0 Tipler Krolak´ TABLE II. Types of singularities. (−∞, 0) (η0, ∞) 0, ±1 (0, ∞) Strong Strong 0 (0, 1) 0, ±1 (0, ∞) Weak Strong Type Singularity 0 [1, ∞) 0, ±1 (0, ∞) Weak Weak Type 0 Big Crunch or Big Bang singularity (0, 1) (η0, ∞) 0, ±1 (0, ∞) Strong Strong Type I Big Rip singularity 1 (1, ∞) 0, 1 (0, ∞) Strong Strong Type II Sudden future singularity 1 (1, ∞) −1 (0, 1) ∪ (1, ∞) Strong Strong Type III Future scale factor singularity 1 (1, 3) −1 1 Weak Strong Type IV Big separation singularity 1 [3, ∞) −1 1 Weak Weak Type V ω singularity, little rip pseudo singularity (1, ∞)(η0, ∞) 0, ±1 (0, ∞) Strong Strong

Rev. Mex. F´ıs. 66 (2) 209–223 212 S. NATARAJAN, R. CHANDRAMOHAN AND R. SWAMINATHAN

TABLE IV. Dark energy parameterizations with best ﬁts and σ−distances values using SNe Ia JLA data. Referred from [35]. Model Parameterization 2 2 3 LCDM H (z) = H0 [Ωm(1 + z) + (1 − Ωm)] 2 2 3 3(1+w0+w1) −3w1z Linear H (z) = H0 [Ωm(1 + z) + (1 − Ωm)(1 + z) e ] −3w1z 2 2 3 3(1+w0+w1) 1+z CPL H (z) = H0 [Ωm(1 + z) + (1 − Ωm)(1 + z) ×e ] 2 2 3 3(1+w0) 2 3w1/2 BA H (z) = H0 [Ωm(1 + z) + (1 − Ωm)(1 + z) ×(1 + z ) h i 9(w0−w0.5)z 2 2 3 3(1−2w0+3w0.5) 1+z LC H (z) = H0 [Ωm(1 + z) + (1 − Ωm)(1 + z) ×e ] 2 3w1z 2 2 3 3(1+w0) 2(1+z)2 JBP H (z) = H0 [Ωm(1 + z) + (1 − Ωm)(1 + z) ×e ] ½ h i¾ w0 2 2 3 3 1+ 1+w ln (1+z) WP H (z) = H0 Ωm(1 + z) + (1 − Ωm)(1 + z) 1

Six bidimensional dark energy parameterizations are For LDGP model wq = −1, the crossing occurs at a = ∞. studied and tested with available SNe Ia and BAO data [35]. At a redshift z∗ > zc, we have ρeff(z∗) = 0. Hence, the Obtained results are in favour of the LCDM model. Various phantom GR picture of QDGP diverges. parameterizations are reported in the same reference. The Ã ! Friedmann-Raychaudri equation is 1/3(21+wq) 4Ωrc Ωm µ ¶2 z = − 1. (13) H(z) ∗ 2 2 Ωq E(z) = = 8πG(ρm + ρDE) H0 3 × [Ω0m(1 + z) + Ω0DEf(z)], (7) The future Big Rip can be avoided due to the parameters H, H˙ → 0 as a → ∞. This asymptotic behavior reﬂects the 2 a¨ H fact that the phantom effects might have been dumped. The = [Ωm + ΩDE(1 + 3Ω)], (8) a 2 total equation of state parameter is deﬁned by where H(z) is the Hubble parameter, G the gravitational constant, and the subindex 0 indicates the present-day values for 1 + ωtot(z) the Hubble parameter and matter densities. For dark energy Ω (1 + z)3 + (1 + ω )Ω (1 + z)3(1+ωq ) = m p q q . (14) ρDE(z) = ρ0(DE)f(z), (9) E(z)[ Ωrc + E(z) with   This shows that ω (z) ≥ −1. Zz eff 1 + w(˜z) For the quantization of the singularity, the basic under- f(z) = exp 3 dz˜ . (10) 1 +z ˜ standing of quantum geometry is required. The quantum geo- 0 metric analysis, done via loop quantum geometry is discussed For quiessence models, w = const. in the next session. Solution of f(z) is therefore, f(z) = (1 + z)3(1+w). (11) 4. Loop quantum cosmology - a brief analysis For cosmological constant, w = −1 and f = 1. In addition to the simplest models in which the universe The canonical quantum gravitational formalism is based on contains only cold dark matter and a cosmological constant, the quantization of the metric. It attempts to quantize the class of braneworld models can lead to a phantom-like accel- phase space as a Hilbert space. In canonical formalism, the eration of the late universe [36]. This model does not require phase space variables are replaced by operators. But the any phantom matter. The quintessence leads to a crossing of formalism faces some constraints. There are Hamiltonian, the phantom divide-line w = 1. This model avoids the future diffeomorphism and Gauss constraints. To solve these con- Big Rip by decreasing the Hubble parameter. straints, Loop Quantum Gravity is proposed. The model has For wq > −1, a smooth crossing of the phantom divide numerous success in resolving the initial singularity. The occurs at a redshift zc that depends on the values of the free model provides background free solutions for the canonical parameters Ωm, Ωq and wq. zc is obtained as gravity approaches. The Loop Quantum Gravity explains the p discreet nature of the spacetime. On cosmological scales, Ω Ω 3wq m rc (1 + zc) E(zc) = . (12) Loop Quantum Cosmology is proposed for the unperturbed (1 + wq)Ωq evolution of the universe.

Rev. Mex. F´ıs. 66 (2) 209–223 CONFORMAL CYCLIC EVOLUTION OF PHANTOM ENERGY DOMINATED UNIVERSE 213

Loop Quantum Cosmology is based on the canonical The parameter λ determines the minimum eigenvalue pa- quantum gravitational formalism. The Loop Quantum Grav- rameter of LQG and the discreteness of quantum geometry ity does not require renormalization; this makes LQG stand [38]. The parameter is denoted as special over other quantization approaches. From canonical √ 1 quantization, one can expect non zero discrete values for ge- λ = 2( 3πγ) 2 lpl. (22) ometric quantities as quantum observables. Hence, the Loop quantization approach provides non zero values for area and If the Hamiltonian constraint is vanished Heff = 0, it volume. Area and volume in LQG are formulated as opera- leads to tors. sin2(λβ) 8πγ2G = ρ, (23) A → A,ˆ (15) λ2 3 V → V.ˆ (16) with H The constraints reduce the possibilities of quantization. ρ = eff , (24) Setting constraints as c = 0, then the solution of the quantum V evolution will be the energy density. From the Hamiltonian equation,

cˆ1ψ = 0. (17) V˙ = V, In general, every cosmological scenario faces singulari- ∂ 3 sin(λβ) H = −4πGγ H = cos(λβ)V. (25) ties during the initial stages of evolution. The initial condi- eff ∂β eff γ λ tions of the universe possess strong singularity. The initial Big Bang singularity bears high curvatures and energy den- This equation can call the modified Friedmann equation sity divergences. To resolve this singularity conditions, LQC as is equipped with the application of LQG theory. LQG at- µ ¶ ˙ 2 tempts to derive non-perturbative and background indepen- 2 V 8πG ρ H = 2 = ρ 1 − , (26) dent quantization of general relativity. In LQC there is a 9V 3 ρcrit straightforward link between full theory and the cosmolog- where the critical density is inferred as ical models, which is in contrast to other cosmological approaches. The full theory of quantum gravity is required to 3 ρ = ∼ 0.41ρ . (27) be constructed. The LQC is based on symmetrical reduc- crit 8πGγ2λ2 pl tion. But this methodology faces mathematical problems in full theory. So, current research is happening on symmetrical, Similarly, the Raychaudhuri equation is also modified ˙ non-symmetrical models and their relationships. with the help of β = β, Heff In Robertson-Walker metric for a flat (k = 0) homoge- µ ¶ µ ¶ a¨ −4πG ρ ρ neous isotropic universe in Loop Quantum Gravity (LQC) is = ρ 1 − − 4πρG 1 − 2 , (28) a 3 ρcrit ρcrit ds2 = −dt2 − a2(t)(dr2 + r2(dθ2 + sin2 θdφ2), (18) which holds the conservation law with a(t) scale factor and t is the proper time. The effective Hamiltonian in LQC ρ˙ = 3H(ρ + p), (29) −3 sin2(λβ) H = V. (19) where the pressure is eff 8πγ2G λ2 −∂H The details of quantum dynamics are provided by the ef- p = matt . (30) ∂V fective Hamiltonian, and Hmatt provides the matter Hamilto- nian. Some new quantum variables, β and V , can be intro- Hubble parameter in gravitational Hamiltonian is reduced in the quantum regime. The conjugate variables β and placed by the holonomies. They are non linear functions of V satisfy the commutation relation, a and a˙. The spacetime behaves with discreteness, near the 3 Planckian scale. Quantum replacement for classical size a v0 {β, V } = 4πGγ. (20) is written as nv [39]. Where V = a3 and γ = 0.2375 is the Barbaro-Immirizi n(t)v(t) = a(t)3V , (31) parameter [37]. The phase space variable from classical dy- 0 namics is where v is the mean size units in 3D space and n the num- a˙ ber of sections in the region. Implementation of discreteness β = γ . (21) a of quantum geometry results in the dynamics. It depends on

Rev. Mex. F´ıs. 66 (2) 209–223 214 S. NATARAJAN, R. CHANDRAMOHAN AND R. SWAMINATHAN the patch size and independent of the number of patches n. [41,42]. Holonomies are represented as In LQC, the scalar field is considered as an internal clock. The Hubble distance is defined as ˜ C = γa.˙ (32) a H−1 = . (43) These modified equations express the Ricci flow on FRW a˙ background. We may discuss about the curvatures. Invariant In LQG formalism, the canonical part satisfies form of Ricci curvature will be 3 µ ¶ {H, v0a } = 4πG. (44) a¨ R = 6 H2 + a The scale factor is quantized with Ashtekar variables [14] µ ¶ as follows, ρ = 8πGρ 1 − 3ω + 2 (1 + 3ω) . (33) − 1 ρcrit i 3 i Aa = cV0 ˚ωa. (45) From the Eq. (33), it can be observed that the curvature Its conjugate momenta is represented as scalar approaches negative values for the chosen parameters − 2 p such as ρ = ρcrit and ω < −1. Hence Anti-Desitter kind of Ea = pV 3 ˚q˚ea, (46) future universe may appear. Also, the conformal Aeon will i 0 i face regulated future singularity as AdS like singularity. with

Similarly, the Ricci components can be written in terms 2 3 2 of lapse function N and scale factor as defined in [40], |p| = V0 a , (47) Ã ! 1 a˙ a¨ a˙ k a˙ N˙ c = γV 3 , (48) R = 6 + + − . (34) 0 N N 2 a N 2 a2 a2 a N 3 which satisfies The Friedmann equation can be obtained in terms of 8πGγ Ashtekar variables as {c, p} = . (49) 3V0 3 p H = − (γ2(c − Γ)2 + Γ2) |p| Here, ˚ωi and ˚ea are fiducial triads and q is fiducial metric. 8πG a i ab These equations reveal the relationships between triads and + Hmatter(p) = 0, (35) scale factors. where c b = 1 , (50) 2 1 |p| 3 ˜ Γ = V0 Γ, (36) 3 |p| 2 v = sgn(p) . (51) Γ˜ =c ˜ − γa,˜˙ (37) 2πG

(ı(l c/V (1/3)) and The holonomy e 0 0 becomes shift operator in

1 p. The quantization is conﬁrmed by promoting the Poisson 3 c = V0 c.˜ (38) bracket into a commutator. The Poisson bracket of the volume with connection components can be represented as In general, the AdS universe expands eternally. But there p B c exists a maximum cut-off by loop quantum cosmological so- i 3 ijk Ej Ek {A , | det E|d x} = 2πG² ²abc p . (52) lutions. Those solutions predict that when the energy density a | det E| reaches ρcrit then the universe will nucleate future Aeon. With The metric of an isotropic slice is

p 2 i i ω = , (39) qij = a δIJ = e e . (53) ρ I J P Here ei is a co-triad. The geometry of spatial slice is as λ → 0 which results in G~ → 0, then the equations will I encoded in the structural metric q . The canonical momenta behave classicaly as ab Kab is driven by extrinsic curvature. Diffeomorphisms con- 8πG straints produce deformations of a spatial slice. This may be H2 = ρ, (40) 3 connected to the decoherence of quantized space in the sin- a¨ −4πG gularity. In LQG the geometrical formalism is deﬁned by the = (ρ + 3p), (41) ea a 3 triad formalism i , not by the spatial metric. Summations over the indices connect the triads vector ﬁelds. The advan- R = 8πG(ρ + 3p), (42) tage of choosing the triad over metric is that the triad vectors

Rev. Mex. F´ıs. 66 (2) 209–223 CONFORMAL CYCLIC EVOLUTION OF PHANTOM ENERGY DOMINATED UNIVERSE 215 can be rotated without changing the metric. This entails ad- 5. Wheeler-Dewitt solution for initial scale fac- ditional gauge freedom with group SO(3) acting on i. tor To understand the space of metrics or structure tensors, Ashtekar variables are introduced. Wheeler-DeWitt equations attempted to quantize the initial Triads can be written as denstized form. That the densi- singularity. Like Einstein’s field equation Wheeler-DeWitt tized triads conjugate extrinsic curvature coefficient. equation is also a field equation. The Wheeler-Dewitt approach attempts to quantize gravity by connecting General a b ki = Kabei , (54) Relativity and Quantum Mechanics. The Wheeler-Dewitt equation resolves the Hamiltonian constraint using metric {ki (x),Eb(y)} = 8πGδb δi δ(x, y). (55) a j a j variables. The curvature is replaced with Ashteaker connections. Mini super-space models can be implemented to explain the emergence of the universe [45-47]. Action of mini super- i i i Aa = Γa + γka. (56) space is defined as Z 1 √ Ashteaker connections conjugate to triads will be S = R −gd4x. (65) 16πG i b b i {Aa(x),Ej (y)} = 8πGδaδjδ(x, y). (57) Metric of the mini super-space is defined by Hence, the spin connection will be 2 2 2 2 2 2 µ ¶ ds = σ [N (t)dt − a (t)dΩ3]. (66) i ijk b k 1 c i i Γ = ² e ∂[ae ] + e e ∂[ce ] . (58) a j e 2 k a b The mini super-space is considered to be homogeneous and isotropic. dΩ2 is metric on 3 sphere, N 2(t) is the lapse func- The spatial geometry is obtained from densitized triads. 3 tion. σ2 is normalization parameter. From Eqs. (65) and (66), a b ab the Lagrangian is derived as Ei Ei = q det q. (59) µ ¶ N a˙ 2 An expression for the inverse scale factor is L = a k − , (67) 2 N 2 q M = √ IJ (60) IJ det q and the momentum is ei ei −aa˙ = I J (61) p = . (68) (det e) a N 1 = δ . (62) The canonical form of the Lagrangian can be written as a IJ L = paa˙ − NH, (69) The inverse scale factor MIJ can be quantized to volume operator [43]. The bounded operator will be where µ ¶ p q q 2 µ ¶ 16 1 1 −1 p2 MIJ,j = 4 Vj − V 1 − V 1 a 2 2 j+ 2 j− 2 H = + ka . (70) γ lp 2 2 2 a ³q q ´2 + δIJ V 1 − V 1 . (63) The Wheeler-Dewitt theory determines the evolution of the j+ 2 j− 2 universe. The Hamiltonian is in the form Quantization of the inverse scale factor does not diverge at the singularity. Even though the volume operator diverges Hψ = 0, (71) approaches to zero, the corresponding inverse scale factor doesn’t approach zero. and Vˆ µ ¶ There are eigenstates of volume operator with eigen- ∂ ∂ values [44], p2 = −a−p ap . (72) a ∂a ∂a s µ ¶ 3 1 1 2 2 Then the WDWE is changed as [48,49] Vj = (γlp) j j + (j + 1). (64) 27 2 µ ¶ 1 ∂ ∂ ap − ka2 ψ(a) = 0. (73) Those results for eigenspectrum and scale factor values ap ∂a ∂a are plotted in Fig. 2. The full Hamiltonian depends upon the patch volume but Here, K = 0, +1, and -1 for flat, closed, and open bub- not the number of patches or the total volume. bles, respectively. The quantum trajectories can be obtained,

Rev. Mex. F´ıs. 66 (2) 209–223 216 S. NATARAJAN, R. CHANDRAMOHAN AND R. SWAMINATHAN from quantum ﬁeld theory and non-relativistic perspectives Here, κ2 = 8πG

[45,50]. It is represented as From Eq. (77), and keeping p = −2, and (b1/b2) > 0 ∂L ∂S Hence, = aa˙ = , (74) ∂a˙ ∂a (t+t0) b1 −1 ∂S a(t) = e b2 . (83) a˙ = . (75) a ∂a Scale factors from Wheeler-Dewitt and supersymmetric The inﬂation occurs for the selected values of p = −2 or cosmology can be compared. Comparing Eqs. (82) and (83), 4. The quantum vacuum experiences exponential expansion which is triggered by quantum potential [9]. The expansion (t+t0) b1 is analyzed for a ﬂat, i.e. k = 0 bubble. The analytic solution a(t)=e b2 =a0 for Eq. (73) is,  s   1 3 1−p 2 2˙ 2 4 a   κ ϕ0a0 κ   ψ(a) = ib − b , (76) + 3 + (t − t0) . (84) 1 1 − p 2 6 32 where b1 and b2 are arbitrary constants. The scale factor can be determined as Scale factor of the bubble universe is compared with the clas-  ¡ ¢ sical super symmetric solutions. At t = t0, b1 (3 − |1 − p|  b2  1 (3−|1−p|) (2t0) ×(t + t0)) , |1 − p| 6= 0, 3 b1 a(t) = (77) a(t0) = e b2 , (85)   (t+t0) b1 e b2 , |1 − p| = 3. if (b1/b2) = 1 then As discussed earlier for p = −2 or 4, then (b1/b2) > 0. 2t0 Quantum potential corresponding to the small scale factor is a0 = e . (86)

b1 1 Q(a → 0) = − 3 − (78) Hence, the Eq. (82) becomes b2 a  s   1 Hence, the classical potential V (a) cancels out. The effect of 3 quantum potential on vacuum bubbles resembles to the scalar κ2ϕ2e˙4t0 κ4 a(t)=e2t0 + 3  0 +  (t−t ) . (87) filed potential [51] or cosmological constant [52]. The effec- 6 32 0 tive cosmological constant for a k = 0 bubble is in the order of µ ¶ This equation provides a modified scale factor for the dis- b Λ ∼ 3 1 . (79) cussed solutions above. b2 The universe will expand rapidly for a scale factor a ¿ 1 and it will stop its expansion for a À 1. Quantum potential 6. Non interacting solutions plays the role of cosmological constant, which consequences for the exponential expansion. The conformal cyclic cosmology (CCC) has different solutions for the evolution of the universe. The model predicts 5.1. Comparison of results to supersymmetric classical that the universe evolves as conformal cycles. Hence, the cosmology initial singularity can be modified. As from the conformal model, the initial singularity is smooth and has finite surface. Results obtained for the function ψ are compared to the Similarly, the final singularity is also smooth with finite sur- Eq. (76) and solution from Ref. [53]. is expected to have face. To obtain such finiteness, the initial singularity must the form of a WKB solution, be subject to expansion and the final singularity must cope ψ = e(Sϕ+Sa). (80) with contraction. Instead of future Big Crunch, one may be curious about the conformal mapping between phantom dom- Comparing Eqs. (76) and (80) leads to the solution of inated at final stages of the universe and initial stages of the the scale factor. For supersymmetric cosmology the solutios universe. The phantom dominance final stages of the universe obtained is are characterized below. ³ ´3 a0 The scale factor by phantom dominated final stages with ϕ˙(t) =ϕ ˙ , (81) 0 a ω < −1 can be obtained from the relation Ã ! 1 2 2˙ 2 4 2 µ µ ¶¶ 2 κ ϕ a κ 3(1+ω) a3(t) = a3 + 3 0 0 + (t − t ). (82) t 0 6 32 0 a(t) = a(tm) −ω(1 + ω) . (88) tm

Rev. Mex. F´ıs. 66 (2) 209–223 CONFORMAL CYCLIC EVOLUTION OF PHANTOM ENERGY DOMINATED UNIVERSE 217

The scale factor will blown up with time While crossing the phantom divide line the interaction parameter to be discussed for phantom and dark matter. ωtm t = . (89) The universe will avoid phantom dominated Big Rip due (1 + ω) to the interaction between dark matter and the phantom en- Friedmann equation in terms of matter fluid ρm and phan- ergy. The energy conversion makes the conversion between tom field φ can be written as phantom energy to the dark matter. Then the universe is suggested to face an accelerated expansion phase. 8πG H2 = (ρ + ρ ). (90) For the universe to continue its evolution by adding up 3 x m the phantom energy, the interaction between dark matter and Energy density and pressure of the phantom field is ob- phantom energy must be nullified; the interaction between the tained by the following equations. dark matter and the phantom energy depends on the coupling constant. 1 ˙2 ρx = − φ + V (φ), (91) 2 1 − ² C = r x , (102) 1 (1 + r)2 P = − φ˙2 − V (φ), (92) x 2 which determines the type of interaction. If the coupling con- here V (φ) indicates phantom field potential. stant is positive transformation of energy between phantom The interaction between dark matter and dark energy can energy to dark matter happens. The positivity of the coupling be explained with interaction term Γ, constant is confirmed by ²x < 1. Hence, the future increment of phantom energy density will be deduced. But the Γ =ρ ˙ + 3H² ρ , (93) x x x x CCC and LQC requires the final state of the universe must be Γm =ρ ˙m + 3H²mρm, (94) equivalent to the initial density by the topological structure, so the eternal increment of the phantom energy is necessarily with required. ρx + Px Hence, the coupling constant term can be modified with ²x = 1 + ωx = − , (95) ρx 1 − ²x ρ + P NC = Nr , (103) ² = 1 + ω = m m , (96) (1 + r)2 m m ρ m Nr − Nr² NC = x . (104) where ρx in the energy density of dark energy and ρm the (1 + r)2 energy density of dark matter. The term ² has values as ²x ≤ 0 and 1 ≤ ²m ≤ 2. The Setting LHS to zero energy density ratio between dark matter and the phantom energy field gives additional freedom for the interaction γ, 0 = Nr − Nr²x. (105) ρ r = m . (97) Rearranging, ρx

The interaction term is redeﬁned here based on the phan- 0 = Nr − Nr(1 + ωx), (106) tom energy considerations. The interaction can be explained 0 = Nr − Nr + Nrω . (107) as x

2 Then Γ = 3HC (ρx + ρm), (98) where C is the coupling constant [54]. When the coupling 0 = Nrωx. (108) constant is positive, the energy will be transferred from dark energy to dark matter. When the coupling constant is nega- Setting ω 6= 0 and N 6= 0, Hence tive, the dark energy will be transferred from dark matter to dark energy. r = 0, (109) A similar kind of interaction can be found in [19]. or rρx γ = 3H(²m − ²x) , (99) 1 + r C = 0. (110) with For the case r = 0, ² = 1 + ω (100) m m ρ m = 0. (111) ²x = 1 + ωx. (101) ρx

Rev. Mex. F´ıs. 66 (2) 209–223 218 S. NATARAJAN, R. CHANDRAMOHAN AND R. SWAMINATHAN

Hence, ρx has to attain maximum values as compared with ρm. So the interaction between phantom and cold dark matter stops. Yielding the existence of eternally increasing phantom energy theoretically predicted from the Eq. (111). The phantom dominated scale factor can be written as

µ ¶ 2 t 3σ a(t) = a(tm) 1 − σ + σ , (112) tm with ² r σ = x . (113) 1 + r For ω < −1, the scale factor approaches its maximum values with non interacting solutions (r = 0, C = 0). Hence,

a(t) → amax. (114) FIGURE 1. Increment of phantom energy over time is shown in the The dark energy interaction can be discussed within a ﬁgure. As time increases the phantom energy keep on increases. LQG model. Here, ωx > −1 is quintessence mode and Referred from [58]. ωx < −1 is phantom mode. Density perturbations in the universe can also be domi- the phantom energy density will increase over time. This can nated by the Chaplygin gas, which has negative pressure [55]. be understood with the help of Fig. 1. Therefore, the energy In addition to non-interacting solutions of quintessence and density will approach to maximum values. phantom, Chaplying gases can fulﬁll such requirements. The When the phantom energy density reaches values greater Chaplygin gas can be a possible candidate for dark energy than the matter-energy density ρp À ρm, the future universe [56]. The Chaplygin gas has an equation of state will approach the Big Rip. For the spatially FRW universe, A the Friedmann equation can be written as P = − , (115) ρ 8πG with A is a positive constant. H2 = (ρ + ρ ), (118) 3 p m

7. Final stages of the universe where ρm is the density of matter field and ρp is the density of the phantom field. If the increasing phantom energy density In the classical case, values for the scale factor can be ob- approaches the values ρp ∼ ρcrit, (where ρcrit ∼ 0.41ρpl) tained from [57] then the universe should bounce back as it does in Type 0 or µ ¶ 1 (2A + Bρ1−α)ρ1−α Big Bang singularity. As suggested from [19], the Big Rip a = a0 exp . (116) singularity can be avoided. But the implementation of loop 6 AB(1 − α) quantum modification of classical analysis is required to ex- Depending upon the choice of parameters, future singu- plain such a phenomenon. larities will appear. The maximum value for the scale factor The LQC has modified Friedmann equations [59] such as can be obtained from Eq. (116). In this case, no strong sin- µ ¶ gularities will appear for the values of (1/2) < α < (3/4). 8πG ρ For the values α = 0.8, A = 1 and B = 1, the scale factor H2 = ρ 1 − . (119) 3 ρ will face extreme values. crit Usually the Type I singularity faces the dominance from From the modified Friedmann equations, one can understand phantom energy. The density of phantom energy increases that the universe will bounce back, once if the critical den- over time. sity is attained. The increasing phantom energy density may Equation (116) can be rewritten as µ come close to Planck density values. This scenario is ob- 1 tained by the non-interacting phantom energy model. Then, a = a exp 0 6 instead of blow off, the universe will bounce back at the later ¶ times of the evolution. (2A + B(ρ + ρ )1−α)(ρ + ρ )1−α × m p m p . (117) Smooth mapping between the Eqs. (77) , (117), (45), AB(1 − α) (112) and (133) is required for the conformal evolution of the At the very final stages, the standard model predicts that universe. At the initial stages, the universe has the scale fac- the future universe will have a very low matter density. But tor as obtained from Eqs. (77) and (117) and at later stages, it has been modified with Eqs. (45) and (112).

Rev. Mex. F´ıs. 66 (2) 209–223 CONFORMAL CYCLIC EVOLUTION OF PHANTOM ENERGY DOMINATED UNIVERSE 219

8. Phantom dominated possible state of uni- As per the classical evolution at the Big Rip, the universe verse will get a continuous increment of the scale factor, energy density, and pressure. The LQC has the maximum value for At vanishing or diverging scale factors, the universe under- the energy density and Hubble rate. When the energy density goes a Big Bang or Big Rip singularity, accordingly. The approaches value close to the critical density, the dynamical loop quantum universe behaves as the de Sitter universe in effects will be lead by the quantum effects. The acceleration such regimes [12]. The universe itself appears to be a de parameter a¨ approaches to a negative value and the Hubble Sitter space instead of the universe tunnelling into de Sitter rate approaches zero. Instead of universe ripping apart by fi- space [60]. The dynamic and geodesic equations do not have nite time, it will re-collapse and evolution will continue for cut offs in LQC as the energy density and Hubble rate are future cycles of the universe. Curvature Independence Ricci bonded together. In general, LQC invokes all singularities scalar will be and that holds only weak singularities and curvature singu- µ ¶ a¨ k larities. Previous attempts to resolve the Big Bang singular- 2 R = 6 H + + 2 . (125) ity, such as the Wheeler-Dewitt theory which didn’t resolve a a the nature of singularity. The classical trajectory of Wheeler- Hence, Dewitt solution leads to a Big Bang singularity, that requires µ 4πG k a different theory rather than Wheele-Dewitt’s work to solve R = 6 (ρ + 3p) + the Big Bang singularity. The LQC satisfies such require- 3 a2 µ ¶ ments. It resolves all the classical singularities and elaborates 8πG ρ the possibility of extension of space beyond the classical sin- + + kχ (ρ + 3p) 3 ρcrit gularity. µ ¶ ¶ From Eq. (27), it has been shown that the critical den- kχ 2ξk ρ 1 + 2 − 2 + kχ − . (126) sity will be in the order of ∼ 0.41ρpl . The time dependent γ ∆ γ ∆ ρcrit ρ curvature scalar will be in the order of As ∆ → 0, Ricci scalar approaches zero. Here, K is the cur- R = 8πG(ρ + 3p). (120) vature index. χ is different values for K = −1 and K = +1 [61], ( In the classical Big Bang singularity, the scale factor, en- 2 2 −2 ergy density, and the curvature invariants vanish. In a Type 0 sin µ(1 + γ )µ , for K = −1 χ = 2 2 (127) singularity the null energy condition (ρ + p) > 0 is satisfied. −γ µ , for K = +1 But in a Type I singularity, the null energy conditions are [62,63]. violated. Despite the divergence in the existence of energy Here, density and pressure, the Type III singularity has finiteness √ in the scale factor. Hence, Type I singularity is resolved as ∆ → µ2p = 4 3πγl2, (128) Type III singularity and the universe will have the upper limit p for the scale factor. As the universe approaches the upper this is the minimum eigenvalue of the area operator. Then, limit of the scale factor, the energy density also approaches 2 the maximum limit derived from LQC. Instead of completely fop = sin µ − µ¯ sin(¯µ) cos(¯µ), (129) ripping off, the universe will bounce back from the Big Rip, revealing that , it is possible that the Big Rip singularity can while the energy density approaches ρpl → 0.41ρpl. Hence, the Big Rip singularity is resolved. Here, in a similar way, be resolved. the Hubble rate divergence may also be resolved. In classical Big Rip solutions, the Hubble rate diverges. Meanwhile, in 9. Relating quantum potential Λ with N LQC, the Hubble rate has its maximum numerical value as The quantization is confirmed by promoting Poisson bracket 1 |Hmax| = , (121) into commutator relations. In LQC, the inverse volume quan- 2γλ tization provides discrete values. The differentiation equation with converts and difference equation. 3r 3r 3r 3 d|p| 2 (|p + ∆p| 2 − |p − ∆p| 2 ) ρ = , (122) → . (130) max 8πGγ2λ2 dp 2∆p and The LQC model with FRW solutions has many salient mathematical advantages. The LQC replaces the Big Bang 2 2 λ = ∆lpl, (123) with a Big Bounce. There are many similarities and dif- √ ferences between Wheeler-Dewitt theory and LQC. In gen- ∆ = 4 3πγ. (124) eral, the classical relativity works very well, either until the

Rev. Mex. F´ıs. 66 (2) 209–223 220 S. NATARAJAN, R. CHANDRAMOHAN AND R. SWAMINATHAN

2 scalar curvature reaches ∼ (0.15π/lpl) or the matter den- equal values in energy density. Hence, the Big Rip induced sity reaches 0.01ρpl. The classical evolution breaks down initial stages is also possible. at the singularity. But the loop quantum evolution analyses The quantum potential from the WDW Eq. (78) can be such scenario, as an extension of previous cycles of the uni- treated with detailed mathematical analysis. The Eq. (78) verse. The LQC introduces symmetry reduction formalism. can be modified with the help of Eq. (108). The Wheele-Dewitt theory agrees with LQC with finite ac- curacy. The LQC, provides non-zero eigenvalues for the area b1 N(t) Λ ∼ − 3 , (137) gap ∆, and it introduces the elementary cell V. The dynamics b2 a of LQC is analyzed with the implementation of fiducial triads here N(t) is a time-varying parameter that keeps the phantom a i ab ˚ei and cotriads ˚ωa that defines a flat metric ˚q . energy to be invariant throughout the evolution. The effective In LQC, p and a are related to the scale factor as Hamiltonian is c = γa,˙ (131) sin2 λβ H = −3V + V ρ. (138) lqc γ2λ2 p = σa2, (132) The Wheeler-Dewitt equation has the Hamiltonian as ex- where σ = ±1 is the orientation factor. The LQC is an explained from Eq. (70). The loop quantum version of the actly solvable model. Hence, the scalar field is deployed as Hamiltonian is Eq. (138). The LQC renormalizes the cosmo- internal time [64]. As per the LQC formalism, it has been logical constant quantum mechanically. proposed that the quantum bounce is generic. There is an µ ¶ upper bound for the matter density. There is a fundamen- Λ Λ0 = Λ 1 − . (139) tal discreteness of space-time, which is derived from its loop 8πGρcrit quantum nature. Loop quantum cosmological analysis can be introduced by Wheeler-Dewitt solutions, for the universe to This solution modifies the FRW equations as emerge from vacuum also. The scale factor has the values for Λ0 k = 0 model as from the Eq. (77). For the universe to be cre- H2 = . (140) 3 ated out of vacuum, the singularity can be treated with LQC. The matter bouncing scale factor from LQC is given by The resolution is free from the classical potential V (a). The phantom energy, which is the function of critical density, µ ¶ 1 3 3 2 is scrutinized with LQC formulations. Earlier works suggest a = ρ t + 1 . (133) 4 mb 4 c that the quantum potential should be proportional to a . This introduces more errors in obtaining meaningful values of the Matter bouncing scale factor is proportional to the critical cosmological constant. Hence, we have modified the Λ pa- density, which is not included the Wheeler-Dewitt solutions. rameter with Eq. (137). The Hubble rate for the matter bouncing scenario is

1 9.1. Eddington-inspired Born-Infeld theory of gravity ρct H (t) = 2 . (134) solutions mb 3 2 4 ρct + 1 The late tome universe will face the Big Rip at a finite time. The matter density at the bouncing scenario is also de- Such a scenario is referred to as cosmic doomsday. It can rived from LQC formalisms. be analysed via the modified theory of classical and quantum ρ gravity [65]. Eddington-inspired Born-Infeld singularity so- ρ (t) = c . (135) mb 3 t2 + 1 lutions also can play a vital role in analyzing cosmological 4 singularities. EiBI model confirms the availability of auxil- Universe bounces back with the values of energy density, iary finite scale factor in Big Rip like singular stages. obtained from Eq. (135). The cosmological constant is dis- The EiBI action is defined as [66] cussed as a quantum potential in WDW at Eq. (78). The Z ·q ¸ 2 4 √ quantum potential is the cause for the accelerated expansion SEiBI = d x |gµν +Rµν (Γ)|λ g +Sm(g), (141) of the quantum vacuum bubbles. Though the cosmological k constant is quantized with LQC calculations, it requires mod- where k is a constant which is assumed to be positive. The ifications to make it constant throughout the evolution. Big Bang singularity is removed by the EiBI model. Simi- The Eq. (135) can be modified via the following way. larly, the late time Big Rip (little Rip, Little Sibling Big Rip) 1 can also be avoided in this formalism. ρ (t) = + ρ . (136) mb 3 t2 + 1 c The future Big Rip is avoided for a scale factor of the 4 auxiliary metric as suggested from Eq. (42) of [65]. This makes the energy density of the universe in initial If there is a minimum length (and maximum density) at and later times as equal. Both t → 0 and t → ∞ provide early times on homogeneous and isotropic space-times, then

Rev. Mex. F´ıs. 66 (2) 209–223 CONFORMAL CYCLIC EVOLUTION OF PHANTOM ENERGY DOMINATED UNIVERSE 221 such predictions will lead to an alternative theory of the Big not included. From Eqs. (77) and (133) the scale factors for Bang [66]. k = 0 model have been equated. The solution for scale factor A modified Friedman equation is obtained for the EiBi depends upon the selection of cosmological variables. model as · ¸ From the WDW model stanspoint, there is the possibil- 1 1 p ities, for the scale factor to vanish as per the chosen values 3H2 = kρ − 1 + √ (kρ + 1)(3 − kρ)3 k 3 3 of the parameters. But LQC avoid such a scenario. Even · ¸ at the singularity, LQC processes the non zero scale factor. (kρ + 1)(3 − kρ)3 × , (142) Such results are the consequences of discreteness of quan- (3 − k2ρ2)2 tized spacetime. The minimum value for the scale factor is obtained as Hence, the Hubble parameter is modified with loop quantum cosmology from Eq. (134). Compared to the Eq. −32 1 ab ∼ 10 (k) 4 a0. (143) (75), the universe bounces back at singularity with matter bouncing energy densities, that which is calculated from Eq. and the minimum length is predicted to be (135). The numerical predictions represents the value for µ ¶ −4 the matter density at the bounce back, which is ρcric ∼ ρ0 ab = . (144) 0.41ρ ρ pl. The Wheeler-Dewitt quantum potential resolves b time-varying scale factors. After the time-varying parame- Replacing ρb with ρcrit obtained from LQG, then at en- ter N(t) is included in the quantum potential Eq. (78) and it ergy densities ρb = 0.41ρpl, the universe will bounce back. becomes Eq. (137), obtained results of scale factors are com- Similarly, the minimum scale factor obtained from the EiBI pared with EiBI theory and classical supersymmetric cosmol- calculations ab provides the same values as the minimal scale ogy. By comparing, it has been understood that the scale fac- factor values predicted from the LQC. Both theories confirm tor acts as a function of the trigonometric tangent function. the non-singular initial stages and singularity-free gravita- EiBI-inspired modified scale factor is reported in Eq. (147). tional collapse. The minimum scale factor values are predicted from LQG A tensor instability in the Eddington inspired Born-Infeld and are consistent with the minimum scale factor, which is Theory of Gravity is reported in [67]. The modified scale fac- predicted from EiBI theory. tor is obtained as The time-varying parameter N(t) confirms the consis- 2 tent value for the cosmological constant over the cosmo- a = ab[1 + tan Υη], (145) logical evolution. The cosmological constant which is pro- where η is conformal time and posed for the accelerated expansion of the universe behaves s like a quantum potential. Hence, future bounce is available 2 with increasing phantom field. The FRW equations modified Υ = a . (146) b 3|k| with the cosmological constant and their quantized results are obtained from the LQC Eqs. (139) and (140) respectively. Relating the Eqs. (133) and (145) leads the scale factor The regular cosmological constant is renormalized within the as LQC framework. This behaves as a function of the critical

µ ¶ 1 " Ãµ ¶ 1 ! # density. 3 3 3 3 a= ρ t2+1 1+ tan2 ρ t2+1 η . (147) 4 c 4 c

Equation (147) is the modified scale factor obtained from 11. Conclusion loop quantum and EiBi solutions. The modified scale factor with the effect of EiBI solutions is obtained. Although the WDW analysis attempted to quantize the singularity solutions, it also encountered a divergence problem. 10. Discussion Loop quantum cosmological analysis confirms the existence of a non zero scale factor at the initial stages of the universe From the solutions of Eq.(136) is has been understood that by transforming the scale factor as an operator. the final stages of the universe will have the possibility to attain the critical energy density with values near to Planck The author in [19] discussed that the interaction between density. Further increment of energy density is forbidden. the phantom energy and dark matter lead to reduced den- Hence, the universe will bounce back to the formation of a sity to a certain critical level. Consequently, the Big Rip is new Aeon. This solution predicts the avoidance of a Big Rip. avoided in the future universe. But we express an alternative In this work LQC-based work, modified solutions are to such conclusion; the universe will evolve even after the Big implemented for the Wheeler-Dewitt solutions. Earlier, in Rip, which means that the universe will continue its evolution Wheeler-Dewitt solution, the critical density parameter was in some other way. Interaction between the phantom

Rev. Mex. F´ıs. 66 (2) 209–223 222 S. NATARAJAN, R. CHANDRAMOHAN AND R. SWAMINATHAN

confirmed. The mapping between the scale factors of various models and various stages of the universe can be understood by the procedure of quantization. From the Eq. (33), if the values of cosmological parameters have specified values, such as ρ = ρcrit and ω < −1, the universe will have negative curvature instead of zero curvature. Hence, AdS kind of future universe might have ap- peared. Non zero values for the scale factor for the set of eigenvalues can be understood from the Fig. 2. The quantized scale factor never approaches zero at the initial stages of the universe, in spite of classical scale factor facing zero at an initial singularity. This could be the initial adjustment for the conformal mapping of initial singularity in conformal cyclic cosmology. Similarly, the Hamiltonian matter bounce is compared FIGURE 2. Nonzero scale facor is predicted by LQG, as a funtion with the critical density parameter. The universe bounces of Planck length. back with the densities ρcri ∼ 0.41ρpl. The cosmological constant is quantized and renormalized within LQC formal- energy and dark matter will be nonexisent while the coupling ism. Hence, inspired by the CCC model, the late time avoid- constant approaches zero. Then, the energy density of the ance of Big Rip and continuing evolution can be held. phantom energy will continue to increase eternally. At later Loop Quantum Cosmology provides some modifications stages, the phantom energy density will be equal to the crit- on scale factor regularization, quantum potential, and Hamil- ical density. Subsequently there is a possibility to bounce tonian formulation. Additionally, included cosmological pa- back for the future evolution. The role of N(t) in Eq. (137) rameter on phantom energy, gives a consistent value for phan- on cosmological evolution is to conserve the phantom energy. tom energy throughout the evolution of the universe. Also, Hence the phantom energy remains unperturbed throughout one can understand that the modified scale factor from EiBI the evolution. Also, the non-interacting solutions of phantom theory can act as a function of the critical density. energy and dark matter hold the eternal nature of phantom An explanatory theory for the quantum emergence of the energy (Eq. (110)). As the evolution continues even after universe via conformal cyclic evolution is attempted and en- the final singularity is approached, the viability of CCC, is visaged.

1. D. An, K.A. Meissner, P. Nurowski and R. Penrose, arXiv 13. A. Ashtekar, T. Pawlowski, P. Singh, Phys. Rev. D 73 (2006) preprint arXiv:1808.01740 (2018). 124038. 2. C.W. Misner, K.S. Thorne, J.A. Wheeler, D.I. Kaiser, Gravita- 14. I. Agullo, A. Corichi, in Springer Handbook of Spacetime, tion (Princeton University Press, New Jersey, 2017). edited by A. Ashtekar and V. Petkov (Springer, Berlin, 2014), pp. 809-839. 3. D. Tytler, J.M. O’Meara, N. Suzuki, D. Lubin, in Particle 15. O. Obregon,´ J. J. Rosales, V. I. Tkach, Phys. Rev. D 53 (1996) Physics and the Universe, edited by L. Bergstrom,¨ C. Fransson, R1750(R). and P. Carlson (World Scientiﬁc, Sweden, 2001), pp. 12-31. 16. L. P. Chimento, A. S. Jakubi, Int. J. Mod. Phys. D 5 (1996) 71. 4. J. Khoury, B.A. Ovrut, P.J. Steinhardt, and N. Turok, Phys. Rev. 17. L. A. Urena-L˜ opez,´ J. Cosmol. Astropart. Phys. 2005 (2005) D 64 (2001) 123522. 013. 5. L. Baum, P. H. Frampton, Phys. Rev. Lett. 98 (2007) 071301. 18. C. Escamilla-Rivera, O. Obregon,´ L.A. Urena-L˜ opez,´ in AIP 6. J. Wambsganss, P. Bode, J. P. Ostriker, Astrophys. J. Lett. 606 Conference Proceedings 1256 (2010) 262-266. (2004) L93. 19. Z. K. Guo, Y. -Z. Zhang, Phys. Rev. D 71 (2005) 023501. 20. R. Penrose, Proceedings of EPAC Edinburgh, 2006, edited by 7. E .P. Tryon, Nature 246 (1973) 396. C. Prior (EPAC, Edinburgh, 2006), p. 2759. 8. B.S. DeWitt, Phys. Rev. 160 (1967) 1113. 21. R. Penrose, Cycles of time (Vintage, London, 2011). 9. D. He, D. Gao, and Q. Cai, Phys. Rev. D 89 (2014) 083510. 22. R. Penrose, Found. Phys. 48 (2018) 1177. 10. A. Vilenkin, Phys. Lett. B 117 (1982) 25. 23. E.F. Bunn, N. Sugiyama, ePrint: arXiv: astro-ph/9407069 (1994). 11. V. Taveras, Phys. Rev. D 78 (2008) 064072. 24. R. Penrose, in AIP Conference Proceedings 1446 (2012) 233- 12. P. Singh, Class. Quantum Gravity 26 (2009) 125005. 243.

Rev. Mex. F´ıs. 66 (2) 209–223 CONFORMAL CYCLIC EVOLUTION OF PHANTOM ENERGY DOMINATED UNIVERSE 223

25. D. Kramer, H. Stephani, M. MacCallum, E. Herlt, Exact solu- 46. N. Pinto-Neto, F. Falciano, R. Pereira, E. S. Santini, Phys. Rev. tions of Einstein’s field equations (Cambridge University Press, D 86 (2012) 063504. Cambridge, 1980). 47. S.P. Kim, ePrint: arXiv preprint gr-qc/9703065 (1997). 26. F. S. N. Lobo, Phys. Rev. D 71 (2005) 084011. 48. A. Vilenkin, Phys. Rev. D 50 (1994) 2581. 27. S. Sushkov, Phys. Rev. D 71 (2005) 043520. 49. S.W. Hawking, Nucl. Phys. B 239 (1984) 257. 28. H. P. Nilles, Phys. Rep. 110 (1984) 1. 50. L. P. Grishchuk, Class. Quantum Grav. 10 (1993) 2449. 29. M. Pollock, Phys. Lett. B 215 (1988) 635. 51. J. B. Hartle, S. W. Hawking, and T. Hertog, J. Cosmol. As- 30. P. H. Frampton, Phys. Lett. B 555 (2003) 139. tropart. Phys. 2014 (2014) 015. 31. L. Fernandez-Jambrina,´ R. Lazkoz, in Proceedings of the 52. D. Coule, Class. Quantum Grav. 22 (2005) R125. MG12 Meeting on General Relativity, Paris, 2009, edited by 53. C. Escamilla-Rivera, O. Obregon,´ L.A. Urena-L˜ opez,´ J. Cos- T. Damour, R. Jantzen, and R. Ruffini (World Scientific, Paris, mol. Astropart. Phys. 2010 (2010) 011. 2009), p. 1887. 54. W. Zimdahl, D. Pavon,´ and L. P. Chimento, Nucl. Phys. B 521 32. L. Fernandez-Jambrina,´ and R. Lazkoz, Phys. Rev. D 70 (2004) (2001) 133. 121503(R). 55. J.C. Fabris, S. V. B. Gonçalves, and P. E. de Souza, Gen. Rela- 33. L. Fernandez-Jambrina,´ and R. Lazkoz, Phys. Lett. B 670 tiv. Gravit. 34 (2002) 53. (2009) 254. 56. V. Gorini, A. Kamenshchik, and U. Moschella, Phys. Rev. D 67 34. L. Fernandez-Jambrina,´ and R. Lazkoz, Phys. Rev. D 74 (2006) (2003) 063509. 064030. 57. P. Singh, and F. Vidotto, Phys. Rev. D 83 (2011) 064027. 35. C. Escamilla-Rivera, Galaxies 4 (2016) 8. 58. A. K. Yadav, Astrophys. Space Sci. 361 (2016) 276 . 36. L.P. Chimento, R. Lazkoz, R. Maartens, and I. Quiros, J. Cos- 59. P. Singh, Class. Quantum Grav. 29 (2012) 244002. mol. Astropart. Phys. 2006 (2006) 004. 60. D. Perlov, and A. Vilenkin, Cosmology for the Curious 37. J. Ganc, and E. Komatsu, Phys. Rev. D 86 (2012) 023518. (Springer, 2017), pp. 333-341. 38. A. Ashtekar, and E. Wilson-Ewing, Phys. Rev. D 78 (2008) 61. B. Ratra, and P. J. Peebles, Phys. Rev. D 37 (1988) 3406. 064047. 62. H.P. Nilles, Phys. Rep. 110 (1984) 1. 39. M. Bojowald, Class. Quantum Grav. 26 (2009) 075020. 63. M. Pollock, Phys. Lett. B 215 (1988) 635. 40. M. Bojowald, Living Rev. Relativ. 11 (2008) 4. 64. A. Ashtekar, A. Corichi, P. Singh, Phys. Rev. D 77 (2008) 41. C. Cattoen¨ and M. Visser, Class. Quantum Grav. 22 (2005) 024046. 4913. 65. I. Albarran, M. Bouhmadi-Lopez,´ C.-Y. Chen, and P. Chen, 42. L. Fernandez-Jambrina, R. Lazkoz, Phys. Rev. D 74 (2006) Phys. Lett. B 772 (2017) 814. 064030. 66. M. Banados, and P. G. Ferreira, Phys. Rev. Lett. 105 (2010) 43 . T. Thiemann, Class. Quantum Grav. 15 (1998) 1281. 011101. 44. M. Bojowald, Class. Quantum Grav. 17 (2000) 1509. 67. C. Escamilla-Rivera, M. Banados, and P. G. Ferreira, Phys. Rev. 45. N. Pinto-Neto, Class. Quantum Grav. 30 (2013) 143001. D 85 (2012) 087302.

Rev. Mex. F´ıs. 66 (2) 209–223