Semi-classical treatment of k -essence effect on cosmic temperature
aa r X i v : . [ g r- q c ] J un Semi-classical treatment of k -essence effect on cosmic temperature Abhijit Bandyopadhyay , Debashis Gangopadhyay and Arka Moulik Department of PhysicsRamakrishna Mission Vivekananda UniversityBelur Math, Howrah 711202, India
Abstract
A phenomenological model is described for Cosmic Microwave Background Radiation (CMBR)evolution with dark energy an essential ingredient in the form of a k − essence scalar field. Thefollowing features of this evolution can be successfully obtained from this model: (a) the observed variation of the rate of change of scale factor a ( t ), i.e. ˙ a , with time and (b) the observed value ofthe epoch when the universe went from a decelerating phase to an accelerated phase. These twofeatures have been matched with graphical transcriptions of SNe Ia data. The model also indicatesthat the evolution is sensitive to the presence of inhomogeneity and this sensitivity increases as onegoes further into the past. Further, the value of the inhomogeneity parameter determines the epochof switch over to an accelerated phase. A positive value of inhomogeneity parameter leads to switchover at earlier epochs, while a negative value leads to switch over at later epochs. If the value of theinhomogeneity parameter is a bit negative then the crossover point from deceleration to accelerationgives better agreement with the observed value. Observations of luminosity distances of the type Ia Supernovae (SNe Ia) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,13] indicate that the universe is presently undergoing a phase of accelerated expansion. Overwhelmingsupport exists from other independent observations like Cosmic Microwave Background anisotropiesmeasured with WMAP satellite [14] and Planck satellite [15], Baryon Acoustic Oscillations(BAO) [16,17, 18] and measurement of oscillations present in the matter power spectrum through large scale surveys[19].One of the theoretical approaches in explaining the observed late time acceleration of the universe isthe presence of dark energy which correspond to a negative pressure in the ideal perfect fluid model ofa Friedmann-Lemaitre-Robertson-Walker (FLRW) universe. Recent measurements in Planck Satelliteexperiment [15] suggest dark energy contributes 68.3% of the total content of present universe. Theissue of origin of dark energy can be addressed in the framework of k − essence scalar field model of darkenergy which involve actions with non-canonical kinetic terms. In a k − essence scalar field model, thekinetic energy dominates over the potential energy associated with the scalar field. Literature on darkenergy and k − essence models can be found in [20, 21, 22, 23, 24].To begin with (Sec. 2), we shall consider a model [25] where the scalar field is homogeneous, i.e. φ ( t, x ) ≡ φ ( t ) and the FLRW metric has zero curvature constant, i.e. the universe is flat. A Lagrangian Email: [email protected] Email: [email protected] Email: [email protected] k − essence field that we shall use has [25] two generalised coordinates q ( t ) = ln a ( t ) ( a ( t ) isthe scale factor) and a scalar field φ ( t ) with a complicated polynomial interaction between them. Inthis Lagrangian, q has a standard kinetic term while φ does not have a kinetic part and occurs purelythrough the interaction term. The general form of k − essence Lagrangian is assumed to be a function L = − V ( φ ) F ( X ) with X = g µν ∇ µ φ ∇ ν φ where ∇ µ is the covariant derivative, X does not dependexplicitly on φ to start with and V ( φ ) is taken to be a constant. In [26], X was shown to satisfy a generalscaling relation, viz. X ( dFdX ) = Ca ( t ) − with C a constant. [25] incorporates the scaling relation of[26].In [27] it was shown that the Lagrangian in the above model, under certain assumptions reduces tothat of a harmonic oscillator on the half-plane with time dependent frequency. The quantum mechanicalamplitude for q to evolve from a value q a ( t a ) to q b ( t b ) was computed and using the fact that the scalefactor a ( t ) is inversely proportional to the cosmic temperature T a at a given epoch t a , the quantumamplitude is transformed into an amplitude for evolving from ln T a to ln T b (Sec. 2).In this work we shall show that the above quantum mechanical amplitude is a plausible phenomeno-logical model of CMBR evolution.Again, the latest results obtained from the Planck probe [15] have firmly established that inhomo-geneity effects in CMBR do not have their origins in non-gaussianities. Hence alternative theoreticalapproaches to inhomogeneities are desirable. In this context we shall show that a phenomenologicalinput can be introduced in the above model to take into account inhomogeneities. This is done bymaking the scalar field inhomogeneous (Sec. 2).This phenomenological model has been developed along the following lines keeping the observationalscenario in mind. First a combined analysis of SNe Ia data and Observational Hubble data is done toobtain graphical transcriptions of (a) the behaviour of the scale factor a ( t ) with time (b) the behaviourof the rate of change of scale factor ˙ a ( t ) with time and (c) the second derivative of a ( t ) with respect totime viz. ¨ a ( t ).The observational values thus obtained are then used as inputs in the model described as follows.Values of a ( t ) obtained at specific epochs are plugged into the expression for the quantum amplitudeto obtain the probability profile of the evolution with time. The obtained profile is then like the profileof the expectation value of microscopic quantum fluctuations, remembering that the expectation valueis proportional to the transition probability.The following features of this evolution can be successfully obtained from this model:(a) The observed variation of the rate of change of scale factor a ( t ), i.e. ˙ a , depicted in middle panelof Fig. 2, matches with the probability profile obtained theoretically from the model after pluggingin observed values of ˙ a at corresponding epochs, Fig. 4.(b) The observed value of the epoch when the universe went from a decelerating phase to an acceleratedphase, Fig. 2, is nearly the same as obtained from the theoretically obtained profile , Fig. 4.(c) There is a qualitative indication that the probability is sensitive to the presence of inhomogeneity.This sensitivity increases as one goes further into the past. This is seen in Fig. 4 and 6 where thesolid line represents homogeneity while the dotted lines denote the presence of inhomogeneity.(d) The value of the inhomogeneity parameter determines the epoch of switch over to an acceleratedphase. A positive value leads to switch over at earlier epochs, while a negative value leads toswitch over at later epochs. 2e) If the value of the inhomogeneity parameter is a bit negative then the crossover point from de-celeration to acceleration gives better agreement with the observed value. The homogeneous case(solid line in the figure) seems to be roughly the mean curve with respect to positive and negativevalues for the inhomogeneity parameter. This is seen in Fig. 6. k -essence Lagrangian for scalar field We recall briefly the content of references [25, 27]. The Lagrangian L (or the pressure p ) is taken as L = − V ( φ ) F ( X ) (1)The energy density is ρ = V ( φ )[ F ( X ) − XF X ] (2)with F X ≡ dFdX and in the present work V ( φ ) = V is a constant ( > k − essence field is( F X + 2 XF XX ) ¨ φ + 3 HF X ˙ φ + (2 XF X − F ) V φ V = 0 (3) H = ˙ a ( t ) /a ( t ) is the Hubble parameter. Isotropy and homogeneity imply φ ( x, t ) ≡ φ ( t ), and so X = ˙ φ .For V ( φ ) = constant, one has the scaling law solution [26] XF X = Ca − (4)Using Eq. (4), the zero-zero component of Einstein’s field equations and homogeneity and isotropyan expression for the Lagrangian is obtained as [25] L = − c ˙ q − c V ˙ φe − q (5)where a ( t ) = e q ( t ) , c = 3(8 πG ) − , c = 2 √ C , (we shall always take the positive square root of C ) andthe scalar potential V is a constant.Smaller values of q mean that we are going back to smaller values of a i.e. to earlier epochs.Expanding the exponential and keeping terms up to O ( q ) one has [27] L = − M h ˙ q + 12 πGg ( t ) q i − g ( t ) (6)where M = πG = m l π , g ( t ) = 2 √ CV ˙ φ , m P l is the Planck energy and we use ~ = c = 1 ( c is speed oflight, ~ is Planck’s constant). The last term, g ( t ), is a total derivative. Dropping this term and theminus sign in front we finally write the Lagrangian as L = (cid:18) πG (cid:19) h ˙ q + n πG √ CV ˙ φ o q i (7)A possible solution for q is obtained when − πG √ CV ˙ φ is a positive number. Writing πG = m π ≡ M and − πG √ CV ˙ φ ≡ Ω ( t ) (Ω is real), we rewrite the Lagrangian as L = M (cid:2) ˙ q − Ω ( t ) q (cid:3) (8)3he harmonic oscillator with time-dependent frequency (Eq. 8) can now be used as our cosmologicalmodel for estimating quantum fluctuations of the temperature using path integral technique. Write thedynamical variable as q ( t ) = q cl ( t ) + y ( t ) where y ( t ) (0 < y ( t ) < ∞ ) is the fluctuation over the classicalvalue q cl ( t ). This corresponds to a time dependent oscillator in the half plane [28, 29, 30, 31]. Then thequantum mechanical amplitude for q to evolve from a value q a ( t a ) to q b ( t b ) is given by [32] h q a , t a | q b , t b i = F ( t b , t a ) exp (cid:18) i ~ S cl (cid:19) (9)where S cl = R t b t a L cl dt = R t b t a dt M (cid:2) ˙ q − Ω ( t ) q (cid:3) and F ( t a , t b ) is calculated following [28]. The fluctua-tions y ( t ) satisfy the differential equation ¨ y + Ω ( t ) y = 0 , (10)which will have quasi-periodic solutions for real Ω . Consider two independent quasi-periodic solutionsof Eq. (10) y ( t ) = ψ ( t ) sin ξ ( t, t a ) ; y ( t ) = ψ ( t ) sin ξ ( t b , t ) (11)with the boundary conditions y ( t a ) = 0 ; y ( t b ) = 0 (12)and where ψ ( t ) satisfies the Ermakov-Pinney equation [33]¨ ψ + Ω ( t ) ψ − ψ − = 0 (13)with ξ ( t, s ) defined as ξ ( t, s ) ≡ ν ( t ) − ν ( s ) = Z ts ψ − ( t ′ ) dt ′ (14) ψ ( t ) and ν ( t ) respectively represent the amplitude and phase of the time dependent oscillator. Thefluctuation factor F ( t b , t a ) is then given by F ( t b , t a ) = " M p ( ˙ ν a ˙ ν b )2 πi ~ sin ξ ( t b , t a ) / (15)So the amplitude is (for relevant boundary conditions q ( t a ) = q a , q ( t b ) = q b ), h q b , t b | q a , t a i = " M p ( ˙ ν a ˙ ν b )2 πi ~ sin ξ ( t b , t a ) / (cid:20) exp (cid:18) iS +cl ~ (cid:19) − exp (cid:18) iS − cl ~ (cid:19)(cid:21) (16)where S ± cl = ˙ ψ b q b ψ b − ˙ ψ a q a ψ a ! + 1sin ξ ( t b , t a ) h ( ˙ ν b q b + ˙ ν a q a ) cos ξ ( t b , t a ) ∓ p ˙ ν b ˙ ν a q b q a i (17)We assume ˙ ν ≪
1, i.e. time rate of change of phase is small. Also, in a homogeneous universe, thetemperature of the background radiation is inversely proportional to the scale factor i.e. T ( t ) ∼ a ( t ) .4hen to lowest order in ˙ ν , one has the probability for the logarithm of scale factor or logarithm ofinverse temperature evolution as P ( t b , t a ) ≡ P ( b, a ) = |h q a , t a | q b , t b i| ≡ (cid:12)(cid:12)(cid:12)(cid:12)(cid:28) ln 1 T b , t b (cid:12)(cid:12)(cid:12) ln 1 T a , t a (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) = m π ~ c ! q a q b ( ˙ ν a ˙ ν b ) / sin ξ ( t b , t a ) ! = 3 m π (ln T a ) (ln T b ) ( ˙ ν a ˙ ν b ) / sin ξ ( t b , t a ) (18)where ~ and c have been put equal to unity. Expanding the function (sin ξ ) − in a Taylor series about ξ = 0 we write P ( b, a ) = 3 m π (ln T a ) (ln T b ) ( ˙ ν a ˙ ν b ) / ξ ( t b , t a ) (cid:20) ξ ( t b , t a ) + 17120 ξ ( t b , t a ) + · · · (cid:21) = 3 m π (ln T a ) (ln T b ) " p ( b, a ) + p ( b, a ) + p ( b, a ) + · · · (19)where p ( b, a ) = ( ˙ ν a ˙ ν b ) / ξ ( t b , t a ) , p ( b, a ) = 12 ( ˙ ν a ˙ ν b ) / ξ ( t b , t a ) , p ( b, a ) = 17120 ( ˙ ν a ˙ ν b ) / ξ ( t b , t a ) , · · · (20)Choose ψ ( t ) = e γt where γ is a constant and 0 < γ < ψ , ξ ( t, s ) = ν ( t ) − ν ( s ) = R ts e − γt dt = − γ (cid:2) e − γt − e − γs (cid:3) and therefore, ν ( t ) = − (1 / γ ) e − γt ; ˙ ν ( t ) = e − γt .Using the above expressions for the choice ψ ( t ) = e γt , and expanding different functions as a poly-nomial of γ we obtain( ˙ ν a ˙ ν b ) / = e − γ ( t b + t a ) = (cid:20) − γ ( t b + t a ) + 92 γ ( t b + t a ) + · · · (cid:21) (22) ξ ( t b , t a ) = ν ( t b ) − ν ( t a ) = Z t b t a e − γt dt = ( t b − t a ) (cid:20) − γ ( t b + t a ) + 23 γ ( t b + t b t a + t a ) − · · · (cid:21) (23)1 ξ = 1( t b − t a ) h t b + t a ) γ + 2 (cid:2) t a + 2 t b + 5 t b t a (cid:3) γ + · · · i (24)1 ξ = 1( t b − t a ) (cid:20) t b + t a ) γ + 13 (cid:2) t a + t b + 4 t a t b (cid:3) γ + · · · (cid:21) (25)5sing Eqs. (22),(23), (24) and (25) we calculate the terms p ( b, a ), p ( b, a ), p ( b, a ) · · · appearing in Eq.(19) as p ( b, a ) = ( ˙ ν a ˙ ν b ) / ξ ( t b , t a ) = 1( t b − t a ) " −
12 ( t b − t a ) γ + · · · (26) p ( b, a ) = 12 ( ˙ ν a ˙ ν b ) / ξ ( t b , t a )= 12( t b − t a ) " − t b + t a ) γ + 16 (cid:0) t b + 11 t a + 26 t a t b (cid:1) γ + · · · (27) p ( b, a ) = 17120 ( ˙ ν a ˙ ν b ) / ξ ( t b , t a )= 17120 ( t b − t a ) " − t b + t a ) γ + 16 (cid:0) t b + 49 t a + 94 t b t a (cid:1) γ + · · · (28)The inhomogeneous situation is obtained when relevant quantities have spatial dependence, i.e.dependence on x ≡ ( r, θ, ϕ ), where ( r, θ, ϕ ) being the comoving coordinates appearing in the FLRWmetric. Then X = 12 g µν ∂ µ φ∂ ν φ = 12 (cid:20) ( ∂ t φ ) − a ( ∂ r φ ) − a r ( ∂ θ φ ) − a r sin θ ( ∂ ϕ φ ) (cid:21) (29)Now, for k -essence fields the kinetic energy term dominates over the potential energy i.e. | ∂ t φ | dom-inates over the square of the r , θ and ϕ derivatives of the field φ , so that, X ≈ ˙ φ ( t, x ). We write φ ( t, x ) = φ ( t ) · φ ( x ) = φ ( t ) · (1 + g ( x )) ∼ φ ( t )(1+ f ), where we have assumed an expansion g ( x ) = Σ f n x n , n = 0 , · · · , ∞ with f ≡ f . The thing to remember is that f is always nonzero and small i.e. 0 < | f | < f being zero means homogeneous universe. Hencein this work the inhomogeneity is introduced through a phenomenological parameter whose non zerovalue signifies the presence of inhomogeneity. Here the inhomogeneity is introduced through a functionwhich need not be well behaved everywhere so that it need not be constant. However, the first term inthe series expansion of this (analytic) function is non-zero, more specifically is small i.e. lies betweenzero and unity. This particular approach will ensure a phenomenological computation of effects of in-homogeneity. In this work, we are working out the zeroth order (in spatial dependence) correction. Butthe formalism is sufficiently general to calculate up to higher orders. We then have X = ˙ φ (1 + f ) andfor constant V ( φ ) the form of the Eq. (3) still remains the same and the validity of the scaling relation XF X = Ca − is again ensured. Moreover, in the derivation of Eq. (4), nowhere was it assumed that φ ishomogeneous. The crucial assumption was that V ( φ ) is a constant. So, the presence of inhomogeneitydoes not change the scaling relation.Imposing these constraints the Lagrangian Eq. (7) reduces to L = −√ Ca − V ( ∂ t φ ( t, x )) − (cid:18) πG (cid:19) H (30)where we have dropped a total derivative term as before. Writing φ ( t, x ) ∼ φ ( t )(1 + f ) and proceeding6s in the previous section taking V ( φ ) = constant = V we have L = M h ˙ q + n πG √ CV ˙ φ o (cid:16) f (cid:17) q i = M h ˙ q − Ω ( t ) (cid:16) f (cid:17) q i = M (cid:2) ˙ q − Ω f ( t ) q (cid:3) (31)where Ω f ( t ) ≡ (1 + f ) / Ω( t ).It is but natural that the scale factor cannot take the same value when inhomogeneity is present.Technically this means that now it has to be some different function of the time t . To distinguishthese two cases, i.e. homogeneous and inhomogeneous, we write the scale factor in the presence ofinhomogeneity as a f ( t ) instead of a ( t ) which denotes the homogeneous scenario. In the same spirit, thefunctions ν f , ψ f , ξ f are also different functions of time from their homogeneous counterparts , viz. , ν, ψ and ξ .To estimate quantum fluctuations of the temperature in the presence of inhomogeneity ( f ), wefirst assert that the dynamical variable q ( t ) = ln a ( t ) will be modified in presence of inhomogeneitywhich we denote by the notation q f ( t ) = ln a f ( t ). Then we write this as q f ( t ) = q f, cl ( t ) + y f ( t ) where y f ( t ) (0 < y f ( t ) < ∞ ) is the fluctuation over the classical value q f, cl ( t ). We take q f, cl = q cl , so that q f ( t ) = q cl ( t ) + y f ( t ) and y f ( t ) gives the fluctuation (in presence of inhomogeneity) over the sameclassical value q cl . Then the quantum mechanical amplitude for q f to evolve from a value q fa ( t a ) to q fb ( t b ) is given by h q fa , t a | q fb , t b i = F f ( t b , t a ) exp (cid:18) i ~ S f, cl (cid:19) (32)where S f, cl = R L f, cl dt = R t b t a M h ˙ q − Ω f ( t ) q i which is structurally same as S cl but with Ω replaced byΩ f ; and the fluctuation factor F f ( t a , t b ) can again be calculated following [28]. The fluctuations y f ( t )now satisfies the differential equation ¨ y f + Ω f ( t ) y f = 0 . (33)As before, two independent quasi-periodic solutions of Eq. (33) (for real Ω f ) can be considered as y f ( t ) = Ψ f ( t ) sin ξ f ( t, t a ) , y f ( t ) = Ψ f ( t ) sin ξ f ( t b , t ) (34)with boundary conditions y f ( t a ) = 0 ; y f ( t b ) = 0 (35)where, Ψ f ( t ) satisfies the Ermakov-Pinney equation¨Ψ f + Ω f ( t )Ψ f − Ψ − f = 0 (36)with ξ f ( t, s ) defined as ξ f ( t, s ) ≡ ν f ( t ) − ν f ( s ) = Z ts Ψ − f ( t ′ ) dt ′ (37)7ere Ψ f ( t ) and ν f ( t ) respectively represent the amplitude and phase of the time dependent oscillator.The fluctuation factor F f ( t b , t a ) in presence of inhomogeneity is then given by F f ( t b , t a ) = M q ˙( ν f ) a ˙( ν f ) b πi ~ sin ξ f ( t b , t a ) / (38)So the amplitude is (for relevant boundary conditions q f ( t a ) = q fa , q f ( t b ) = q fb ), h q fb , t fb | q a , t a i = M q ˙( ν f ) a ˙( ν f ) b πi ~ sin ξ f ( t b , t a ) / h exp iS + f, cl ~ ! − exp iSf − f, cl ~ ! i (39)where S ± f, cl = ˙Ψ fb q fb Ψ fb − ˙Ψ fa q fa Ψ fa ! + 1sin ξ f ( t b , t a ) h ( ˙ ν fb q fb + ˙ ν fa q fa ) cos ξ f ( t b , t a ) ∓ p ˙ ν fb ˙ ν fa q fb q fa i (40)For small ˙ ν f ≪
1, to lowest order in ˙ ν , the probability of transition for logarithm of scale factor (inversetemperature) in presence of inhomogeneity would be P f ( b, a ) = |h q fa , t a | q fb , t b i| = 3 m π (ln T fa ) (ln T fb ) ( ˙ ν fa ˙ ν fb ) / sin ξ f ( t b , t a ) (41)We now choose to write Ψ f ( t ) = (1 + f ) ψ ( t ) (42)so that for f = 0, Ψ f ( t ) → ψ ( t ) and Ω f ( t ) → Ω( t ) and we get back homogeneous scenario. Also forsuch a choice the quantities ν f and ξ f appearing in Eq. (41) can be expressed explicitly in terms ofrespective quantities ν and ξ corresponding to the homogeneous scenario in manner described below.If we choose e γ f ( t ) to represent a solution of the Ermakov-Pinney equation [Eq. (36)], then in orderthe Eq. (42) is satisfied, γ f ( t ) will be related to the parameter γ (recall that: e γt represents a solutionof the Ermakov-Pinney equation (Eq. (13)) corresponding to the homogeneous case) by the followingrelation γ f ( t ) = γt + ln(1 + f ) (43)Note here that the way we choose to incorporate the inhomogeneity into our scheme retains theErmakov-Pinney structure of the differential equations which is the central feature of the formalismdescribed in Sec. 2. This will ensure that new solutions thereby obtained will obey all the mathematicalproperties of Ermakov theory regarding the form of the nonlinear equations.It is quite evident that Ψ reduces to ψ ( t ) if f = 0 and then γ f ( t ) reduces to γt and we get back theoriginal situation. Therefore we can unambiguously use the previous formalism with P ( b, a ) replacedby P f ( b, a ), ν ( t )’s replaced by ν f ( t )’s and ξ ( t b , t a ) replaced by ξ f ( t b , t a ) respectively. Now with Ψ f ( t ) =8 γ f ( t ) the new ν f ( t ) and ξ f ( t b , t a ) are related to corresponding functions for homogeneous case by thefollowing relations ν f ( t ) = ν ( t )[1 + f ] (44) ξ f ( t b , t a ) = ν f ( t b ) − ν f ( t a ) = ν ( t b ) − ν ( t a )[1 + f ] (45)Using Eqs. (44) and (45) in Eq. (41) we get P f ( b, a ) = 3 m π (ln T a ) (ln T b ) " p f ( b, a ) + p f ( b, a ) + p f ( b, a ) + · · · (46)where the quantities p f , p f , p f , · · · appearing in the above equation is expressible in terms of theirhomogeneous counterparts as p f ( b, a ) = [ ˙ ν f ( t a ) ˙ ν f ( t b )] / ξ f ( t b , t a ) = [ ˙ ν ( t a ) ˙ ν ( t b )] / / (1 + f ) ξ ( t b , t a ) / (1 + f ) = p ( b, a ) p f ( b, a ) = 12 [ ˙ ν f ( t a ) ˙ ν f ( t b )] / ξ f ( t b , t a ) = 12 [ ˙ ν ( t a ) ˙ ν ( t b )] / / (1 + f ) ξ ( t b , t a ) / (1 + f ) = p ( b, a )(1 + f ) = p ( b, a )(1 − f + O ( f )) p f ( b, a ) = 17120 [ ˙ ν f ( t a ) ˙ ν f ( t b )] / ξ f ( t b , t a )= p ( b, a )(1 + f ) = p ( b, a )(1 − f + O ( f ))... p nf ( b, a ) = p n ( b, a )(1 + f ) n = p n ( b, a )(1 − nf + O ( f ))In terms of γ , t a and t b the above expression becomes P f ( b, a ) = 3 m π (ln T a ) (ln T b ) × " t b − t a ) h −
12 ( t b − t a ) γ + · · · i + 12( t b − t a )(1 + f ) h − t b + t a ) γ + 16 (cid:0) t b + 11 t a + 26 t a t b (cid:1) γ + · · · i + 17120 ( t b − t a )(1 + f ) h − t b + t a ) γ + 16 (cid:0) t b + 49 t a + 94 t b t a (cid:1) γ + · · · i (47) Measurement of luminosity distance of the type Ia Supernovae (SNe Ia) during nearly last two decadesestablishes that the universe is presently undergoing a phase of accelerated expansion. Observation of9 .6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 α β σ σ best-fit Figure 1: Allowed regions in the parameter space α − β at 1 σ and 2 σ confidence level from the combinedanalysis of SNe Ia data and OHDBaryon Acoustic Oscillations (BAO), Cosmic Microwave Background (CMB) radiations, power spectrumof matter distributions in the universe provide other independent evidence in favour of this late-timecosmic acceleration. However, the SNe Ia data remain the key observational ingredient in determiningtime evolution of the scale factor a ( t ) in the late-time phase of evolution of the universe. Besides theSNe IA data, observational data based on measurement of differential ages of the galaxies by GeminiDeep Deep Survey GDDS [8], SPICES and VDSS surveys also provide dependence of Hubble parameterwith redshift. We have extracted the this time evolution from the combined analysis of SNe Ia dataand observational Hubble data. Here we briefly describe the methodology we use for the combined analysis of SNe IA and ObservationalHubble Data (OHD). We use a closed form parametrisation of the luminosity distance of supernova, d L , as a function of the redshift as [13] d L ( α, β, z ) = cH (cid:18) z (1 + αz )1 + βz (cid:19) (48)where c is the speed of light and H the value of the Hubble parameter at the present epoch definedthrough the dimensionless quantity h by H = 100 h km s − Mpc − . The luminosity distance is relatedto the distance modulus µ as µ th ( α, β, z ) = 5 log h D L ( α, β, z ) i + µ = 5 log h (cid:18) z (1 + αz )1 + βz (cid:19) i + µ (49)10here, D L ( α, β, z ) ≡ H c d L ( α, β, z ) = (cid:18) z (1 + αz )1 + βz (cid:19) (50)is a dimensionless quantity called the Hubble free luminosity distance and µ = 42 . − h . Fromdifferent compilations of SNe Ia observations by different groups - HST + SNLS + ESSENCE [1, 2, 3],SALT2 and MLCS data [4], UNION [5] and UNION2 data [6]. provide the values of the distancemodulus for different values of the redshift from the SNe Ia observations. The observed values of thedistance modulus µ obs ( z i ) corresponding to measured redshifts z i are given in terms of the absolutemagnitude M and the apparent magnitudes m obs ( z i ) by µ obs ( z i ) = m obs ( z i ) − M . (51)To obtain the best-fit values of the parameters α and β from SNe Ia observations we perform a likelihoodanalysis whose methodology has been discussed in detail in [7]. This involves minimization of a suitablychosen χ function with respect to the parameters α and β . We give below a brief outline of themethodology of χ -analysis adopted here for the analysis of SNe Ia data. The χ function is defined asthe function of the parameters α , β and M ′ ≡ µ + M (called nuisance parameter) as χ ( α, β, M ′ ) = N X i =1 ( µ obs ( z i ) − µ th ( α, β, z i )) σ i = N X i =1 h D L ( α, β, z i ) − m obs ( z i ) + M ′ i σ i (52)where σ i ’s are the uncertainties in observations of distance modulus µ obs ( z i )’s, and N is the total numberof data points. The values of the parameters α and β (appearing in parametrisation of luminositydistance) which fits the SNe IA data best, are those which minimizes the χ function after the parameter M ′ is marginalised over. Expanding the χ function as χ = P ( α, β ) + 2 Q ( α, β ) M ′ + RM ′ (53)where P ( α, β ) = N X i =1 (5 log ( D L ( α, β, z i )) − m obs ( z i )) σ i (54) Q ( α, β ) = N X i =1 (5 log ( D L ( α, β, z i )) − m obs ( z i )) σ i (55) R = N X i =1 σ i (56)we observe that, the χ have a minimum at M ′ = − Q/R and its value at the minimum is ¯ χ ( α, β ) = P − Q /R . To obtain the best-fit value of the parameters α and β its then enough to minimize the11unction ¯ χ ( α, β ) with respect to α and β only since the effect of marginalisation over M ′ gets takencare of in the above consideration. So the χ -function for analysis of SNe IA data used here is χ ( α, β ) = P ( α, β ) − Q ( α, β ) R (57)where P ( α, β ), Q ( α, β ) and R are given by Eqs. (54), (55) and (56) respectively.Besides the SNe IA data, compilation of the observational data based on measurement of differentialages of the galaxies by Gemini Deep Deep Survey GDDS [8], SPICES and VDSS surveys provide thevalues of the Hubble parameter at 15 different redshift values [9, 10, 11, 12]. The χ function for theanalysis of this observational Hubble data can be defined as χ ( α, β ) = X i =1 (cid:20) H ( α, β ; z i ) − H obs ( z i )Σ i (cid:21) , (58)where H obs is the observed Hubble parameter value at z i with uncertainty Σ i .Varying the parameters α and β freely we minimize the global χ function which is defined as χ ( α, β ) = χ ( α, β ) + χ ( α, β ) . (59)The values of the parameters α and β at which minimum of χ is obtained are the best-fit values ofthese parameters for the combined analysis of the observational data from SNe Ia and ObservationalHubble Data (OHD). We also find the 1 σ and 2 σ ranges of the parameters α and β from the analysisof the observational data discussed above. In this case of two parameter fit, the 1 σ (68.3% confidencelevel) and 2 σ (95.4% confidence level) allowed ranges of the parameters correspond to χ ≤ χ + ∆ χ ,where ∆ χ = 2 . .
17) denotes the 1 σ (2 σ ) spread in χ corresponding to two parameter fit.In this work we have considered the SNe Ia data from HST+SNLS+ESSENCE (192 data points)[1, 2, 3] and Observational Hubble Data from [9, 10, 11, 12] (15 data points). The best fit for thecombined analysis of the SNe Ia data and OHD is obtained for the parameters values α = 1 . , β = 0 .
55 (60)with a minimum χ of 204.94. In Fig. 1 we have shown the regions of the α − β parameter space allowedat 1 σ and 2 σ confidence levels from the analysis. Using the values of α and β (Eq. (60)) as obtained from the analysis we can determine the timedependence of the scale factor and the Hubble parameter during the late time evolution of the universe.For a flat universe, which is consistent with the current bounds from PLANCK and WMAP dataon the ratio of the energy density in curvature to the critical density, | Ω K | < .
01 (95% confidencelevel) (PLANCK) the Hubble parameter H ( z ) corresponding to a redshift z is directly related to theluminosity distance through the relation E ( z ) ≡ H ( z ) H = (cid:20) ddz (cid:18) D L ( z )1 + z (cid:19)(cid:21) − (61)12 t / t t / t t / t -0.6-0.4-0.200.20.40.6 a .. a a . inflectionpoint Figure 2: Plot of a (left panel), ˙ a (middle panel) and ¨ a (right panel) against t corresponding to thebest-fit values of parameters α and β obtained from analysis of SN data.From the equations H = ˙ aa and a a = 1 + z we get dt = − dz (1 + z ) H = − dz (1 + z ) H E ( z ) (62)which on integration gives t ( z ) t = 1 − H t Z z dz ′ (1 + z ′ ) E ( z ′ ) (63)where t is the time corresponding to present epoch. Taking the best-fit values of α and β , we useEq. (50) to numerically evaluate D L ( z ) at different z values. Using this in Eq. (61) we then evaluate E ( z ) as a function of z which can further be used in Eq. (63) to perform the integration numerically toobtain time t as a function of redshift z . From the z − t ( z ) relationship thus obtained and the equation a /a = 1 + z we eliminate z to obtain the scale factor a as a function of t .In Fig. (2) we have shown the time dependence of the scale factor as obtained from the analysisof the observational data following technique described above. The left panel shows plot of a ( t ) vs t/t where the value of scale factor at present epoch has been normalised to unity a ( t ) = 1. Theobserved supernova Ia events have redshifts ranging between 0 < z ∼ < . . ∼ < t/t <
1. In the obtained t -dependence of a ( t ) there exists a point of inflection at t/t ≈ . z ≈ . a against t . In the right panel (¨ a vs t ) again there is the signature of crossover to an accelerated phase ofexpansion. We now investigate various aspects of the transition probability (Eq. (47)) and implications of itsdependences on the two parameters of the theory - γ (appearing in the solution of Ermakov-Pinney13 -4 -3 -2 -1 γ -0.04-0.0200.020.040.060.080.10.12 P ( t a , t b , γ ) - P ( t a , t b , γ = ε ) P ( t a , t b , γ = ε ) t b = 1 (present epoch),for all curves t a = 0.3t a = 0.4t a = 0.5t a = 0.6t a = 0.7 Figure 3: Plot of [ P γ ( t a , t b ) − P γ → ǫ ( t a , t b )] /P γ → ǫ ( t a , t b ) vs γ for t b = 1 (present epoch) and for differentchosen values of t a . We take ǫ = 10 − .equation) and f (phenomenological parameter as a measure inhomogeneity), as well as the epochs( t a , t b ) between which the transition is considered. To calculate the transition probability, we use thefact microwave background temperature at time t , T ( t ) ∝ a ( t ) and we make use of time dependence of a ( t ) as obtained from the combined analysis of SNe Ia and observational Hubble data described in Sec.3. To find the γ sensitivity of the theory we investigate the γ -dependence of the the quantity P ( γ )defined as P ( γ ) ≡ P γ ( t b , t a ) − P γ = ǫ ( t b , t a ) P γ = ǫ ( t b , t a ) (64)where we take ǫ = 10 − . For a given t a , t b the quantity P ( γ ) is a measure of fractional change in thetransition probability due to variation of the parameter γ from the arbitrarily chosen small value 10 − .In figure (3) we plot P ( γ ) vs γ , for fixed values of t a , t b and taking f = 0. For all the plots we havekept fixed t b = 1 (present epoch) and have shown the plots for different chosen values of t a . We seefrom the figure that for small γ , viz. γ < . P remains effectively same for all epochs implying thatbasic results of the model are insensitive to the values of γ .The dependence of the transition probability on the epochs t a and t b , between which the transitionhas been considered is shown in Fig. 4 where we plot P γ ( t a , t b , f ) against t a for three chosen values ofthe phenomenological parameter f viz.
0, 0.2 and 0.6. We have taken γ = 10 − for the plots, but theplots will remain same for all values of γ < .
01 as already seen from results presented in Fig. 3.The plots of Fig. 4 show that the probability of transition from an epoch t to the present epoch t = 1 falls slowly with t in the decelerated phase of expansion of the universe. It attains a minimumat a value of t near t ≈ . t in the acceleratedphase of expansion and rises sharply with t as it approaches more towards the present epoch. So thenature of t -dependence of the probability P ( t, t = 1) has a profile similar to that of ˙ a , as evident fromthe middle panel of Fig. 2. 14 .2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t a P ( t a , t b ; f ) f = 0f = 0.2f = 0.6 t a t b = ( p r e s en t epo c h ) Figure 4: Plot of P ( t a , t b , f ) vs t a for t b = 1 (present epoch) for the homogeneous case and two non-zerobenchmark values of inhomogeneity parameter f The nature of the Fig. 4 is further corroborated by an interesting plot. In the left panel of Fig.5 the solid line is the plot of P ( t a , t b = 1) vs t a with the t − dependence of a ( t ) taken as a ( t ) ∼ t / for all epochs. This corresponds to a decelerated expansion of the universe dominated by matter. Thedotted line is the plot of P ( t a , t b = 1) vs t a with the t − dependence of a ( t ) as a ( t ) ∼ e Ht for all epochs.This corresponds to a dark energy driven accelerated expansion of universe. The t a -dependence of P ( t a , t b = 1) obtained in the theory is given by Eq. (47). The t a dependence enters in the probabilitythrough a multiplicative term (ln T a ) ∼ (ln a ( t a )) ≡ η (say), and through the term in the squarebracket η , say. For low values of γ used in our computation, the term η increases with t a as t a approaches the present epoch ( t b = 1), while the term η goes as (ln t a ) for a ∼ t / (matter dominateduniverse) and as t a for a ∼ e Ht (dark energy dominated universe). Since the time parameter t we use isnormalised to 1 at present epoch, t a is fractional, and (ln t a ) decreases as t a approaches t b = 1, while t a always increases with t a in the domain under consideration. The fact that the probability is a productof η and η , there is a resultant behaviour which determines the turning point (minimum) at t a ∼ . η dominates and we are in the accelerated phase.In the right panel of Fig. 5 we plot P ( t a , t b = 1) vs t a with a ( t ) ∼ t / up till t a ∼ .
5. Thatis, the behaviour of a ( t ) is taken to be that of a matter dominated universe. After t a ∼ . a ( t ) ∼ e Ht and compute P ( t a , t b = 1). Here we are taking a dark energy dominatedscenario. Note that the graph mimics Fig. 4 to a great extent except for a discontinuous portion around t a ∼ .
5. This is expected as the change over from a decelerating to an accelerated phase is bound to beassociated with a discontinuity which cannot be analytically obtained from a phenomenological model.Mathematically also this is expected because there cannot be a smooth transition from t / behaviourto that of e Ht . However the overall behaviour ( i.e. a transition from decelerating to an acceleratingphase) is reflected in the graphical transcription.From Fig. 4 we also get a qualitative indication how the presence of inhomogeneity affects theevolution of transition probability between different temperatures. This is shown more comprehensivelyin Fig. 6 where we have again plotted P ( t a , t b = 1) vs t a (left panel) for different values of f , bothpositive and negative. For comparison we have also shown the plot of ˙ a ( t ) vs t obtained from analysis15 .2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t a P ( t a , t b ; f ) a ~ t a ~ e Ht t a Figure 5: Left panel: Plot of P ( t a , t b ) vs t a for t b = 1 (present epoch) for the homogeneous case with a ( t ) ∼ t / for all all epochs (solid line) and with a ( t ) ∼ e Ht for all epochs (dotted line). Right panel:Plot of P ( t a , t b ) vs t a for t b = 1 (present epoch) for the homogeneous case with a ( t ) ∼ t / for t < . a ( t ) ∼ e Ht for t > . t a P ( t a , t b ; f ) t a t b = ( p r e s en t epo c h ) f=0f=0.2f=0.3 f = - 0.3f = - 0.2 a . Figure 6: Left panel: Plot of P ( t a , t b ) vs t a for t b = 1 (present epoch) for the different values of f , Rightpanel : Plot of ˙ a ( t ) vs t obtained from analysis of observational data.of observational data in the right panel. Fig. 6 shows that the probability is sensitive to the presenceof inhomogeneity. This sensitivity increases as one goes further into the past and larger absolute valueof the inhomogeneity parameter, | f | , leads to larger departure from probability values correspondingto the homogeneous scenario. Also the value of the inhomogeneity parameter determines the epoch ofswitch over from a decelerated to an accelerated phase of expansion of the universe. A positive valueleads to switch over at earlier epochs, while a negative value leads to switch over at later epochs. Weobtain better agreement with the observed value of crossover point t ≈ .
53 for a negative value of16nhomogeneity parameter f ≈ − . In this work a phenomenological model has been developed to study the evolution of the universe in thecontext of the CMBR. A key ingredient of the model is the presence of dark energy through a scalar fieldwhose kinetic energy dominates i.e. a k − essence scalar field. We first develop the observational evidencethrough a rigorous graphical transcription of SNe Ia data. This is depicted in Fig. 2. Subsequently aLagrangian model of dark energy (obtained from very general considerations, Sec. 2) is used to explainthe evidence depicted in Fig. 2.The approach taken is as follows. The Lagrangian (Eq. (8)) is that of a time dependent oscillatorand the dynamical variables is q = ln a ( t ). We compute the the quantum fluctuations h q a , t a | q b , t b i whichis tantamount to computing the correlations between the logarithm of the temperatures at two epochs t a and t b where we have used the association between the scale factor a ( t ) and cosmic temperature ata particular epoch, v iz., T a ∼ a ( t a ) .What is remarkable is that the probability of transition between the logarithm of the temperaturesln T a at t = t a and ln T b at t = t b (present epoch) follows a similar profile as that of ˙ a ( t ). This is shownin Fig. 4. Another point of note is that the crossover from a decelerating phase to an accelerated phaseoccurs at precisely at the same epoch, v iz., i.e. t a = 0 . a ( t ) and as a ( t + dt ) ∼ a ( t ) + ˙ a ( t ) dt , therefore the probability of transition should beproportional to ˙ a . Figs. 2 and 4 seem to confirm this fact.Our phenomenological model successfully explains the observed variation of ˙ a . This conclusionfollows from the fact that the variation of ˙ a matches with the probability profile obtained theoreticallyfrom the model after plugging in observed values of ˙ a at corresponding epochs. Moreover, the observedvalue of the epoch when the universe went from a decelerating phase to an accelerated phase, it is nearlythe same as that obtained from the theoretically obtained profile.This model also throws light on how inhomogeneity may affect the CMBR evolution. There isa qualitative indication that the probability is sensitive to the presence of inhomogeneity and thissensitivity increases as one goes further into the past. Also the value of the inhomogeneity parameterdetermines the epoch of switch over to an accelerated phase. A positive value leads to switch over atearlier epochs, while a negative value leads to switch over at later epochs. Better agreement with theobserved value of crossover point is obtained for a small negative value of the inhomogeneity parameter.This is seen from Fig. 6. References [1] A. G. Riess et al., Astrophys. J. 659, 98 (2007)[2] W. M. Wood-Vasey et al., Astrophys. J. 666, 694 (2007)[3] T. M. Davis et al., Astrophys. J. 666, 716 (2007)[4] R. Kessler et al., Astrophys. J. Suppl. 185 , 32 (2009)[5] M. Kowalski et al. Astrophys. J. 686, 749 (2008)176] R. Amanullah et al., Astrophys. J. 716, 712 (2010)[7] L. Xu, Y. Wang, JCAP 1006, 002 (2010), S. Nesseris and L. Perivolaropoulos, Phys. Rev. D
010 (2009).[8] R. G. Abraham et al., Astron. J. 127, 2455 (2004)[9] J. Simon, L. Verde and R. Jimenez, Phys. Rev. D 71, 123001 (2005)[10] E. Gaztanaga, A. Cabre, L. Hui, Mon. Not. Roy. Astron. Soc. 399, 166 (2009)[11] A. G. Riess et al., Astrophys. J. 699, 539 (2009)[12] D. Stern, R. Jimenez, L. Verde, M. Kamionkauski, S. A. .Stanford, JCAP 1002, 008 (2010)[13] T. Padmanabhan and T. R. Choudhury, Mon. Not. Roy. Astron. Soc. 344, 823 (2003)[14] E. Komatsu et al , Astrophys. J. Suppl. , 18 (2011)[15] P. A. R. Ade et al. [Planck Collaboration], arXiv:1303.5062 [astro-ph.CO]. P. A. R. Ade et al. [Planck Collaboration], arXiv:1303.5072 [astro-ph.CO]. P. A. R. Ade et al. [Planck Collaboration],arXiv:1303.5075 [astro-ph.CO].[16] S. Cole et al.
Mon. Not. Roy. Astron. Soc. , 505 (2005)[17] G. Huetsi, Astron. Astrophys. , 891 (2006)[18] W. J. Percival et al. , Astrophys. J. , 51 (2007)[19] D. Eisenstein, Astrophys. J, , 560 (2005)[20] V.Sahni,Dark matter and dark energy, Lect.Notes Phys.653 141 (2004) [astro-ph/0403324]; 843 111(2006)[astro-ph/0602117];T.Padmanabhan,Dark energy: mystery of the millenium, AIP Conf.Proc.861 179 (2006) [astro-ph/0603114];T.Padmanabhan,Dark energy and gravity, Gen.Rel.Grav. 40529 (2007) [arXiv:0705.2533];E.J.Copeland,M.Sami and S.Tsujikawa, Dynamics of dark energy,Int.Jour.Mod.Phys. D15 1753 (2006) [hep-th/0603057].[21] P.J.E.Peebles and B.Ratra, The cosmological constant and dark energy, Rev.Mod.Phys. 75 559(2003) ; T.Padmanabhan, Cosmological constant-the weight of the vacuum, Physics Reports 380 235(2003) [hep-th/0212290].[22] M.Malquarti,E.J.Copeland,A.R.Liddle and M.Trodden,A new view of k-essence, Phys.Rev. D67123503 (2003) [astro-ph/0302279] M.Malquarti,E.J.Copeland and A.R.Liddle, K-essence and the co-incidence problem, Phys.Rev. D68 023512 (2003) [astro-ph/0304277]; L.Mingzhe and X.Zhang, K-essence leptogenesis, Phys.Lett. B573 20 (2003) [hep-ph/0209093]; J.M.Aguirregabiria,L.P.Chimentoand R.Lazkoz, Phan- tom k-essence cosmologies, Phys.Rev. D70 023509 (2004) [astro-ph/0403157].1823] L.P.Chimento and R.Lazkoz,Phys.Rev. D71 023505 (2005); L.P.Chimento,M.Forte and R.Lazkoz,Mod.Phys.Lett. A20 2075 (2005); R.Lazkoz,Int.Jour.Mod.Phys. D14 635 (2005) [gr-qc/0410019];H.Kim, Phys.Lett. B606 223 (2005); J.M.Aguirregabiria,L.P.Chimento and R.Lazkoz, Phys.Lett.B631 93 (2005); H.Wei and R.G.Cai,Phys.Rev. D71 043504 (2005) [hep-th/0412045]; C.Armendariz-Picon and E.A.Lim, JCAP 0508 7 (2005).[24] L.R.Abramo and N.Pinto-Neto,Phys.Rev. D73 063522 (2006); A.D.Rendall,Class.Quant.Grav. , 121 (2008)[26] R.J.Scherrer, Phys.Rev.Lett , 231 (2010)[28] D.C.Khandekar and S.V.Lawande,Phys.Rep.
115 (1986).[29] H.Ezawa,J.R.Klauder and L.A.Shepp, J.Math.Phys.
783 (1975).[30] B.Simon, J.Functional Analysis and Applications
295 (1973).[31] T.E.Clark, R.Menikoff and D.H.Sharp, Phys.Rev.
D22
Quantum Mechanics and Path Integrals (McGraw Hill, New York,1965).[33] V.P.Ermakov, Univ.Izv.Kiev1