R. K. Tiwari & Rameshwar Singh

Role of conharmonic flatness in Friedmann cosmology R. K. Tiwari & Rameshwar Singh Astrophysics and Space Science An International Journal of Astronomy, Astrophysics and Space Science ISSN 0004-640X Volume 357 Number Astrophys Space Sci (015) 357:1-5 DOI 10.1007/s10509-015-360-1 1 3

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Astrophys Space Sci (015) 357:130 DOI 10.1007/s10509-015-360-1 ORIGINAL ARTICLE Role of conharmonic flatness in Friedmann cosmology R.K. Tiwari 1 Rameshwar Singh 1 Received: 3 March 015 / Accepted: 19 April 015 Springer Science+Business Media Dordrecht 015 Abstract In this paper, investigation of conharmonically flat FRW space-time is considered in presence of perfect fluid as a matter source. To get the deterministic model, we have employed the condition between vacuum density Λ and the Hubble parameter H as Λ = 3βH, where β is a constant (Astrophys. Space Sci. 314:83, 008). Physical features of the model are discussed in detail. An estimate of the statefinder and snap parameters is also carried out for the model. The observed values of these parameters agree ΛCDM model. We see that the model is in agreement with recent observations. Keywords Conharmonically flat Snap parameter De-Sitter universe ΛCDM model 1 Introduction Conharmonic transformations are a special type of conformal transformations preserving the harmonicity of functions. These type of transformations were introduced by Ishii (1957) and is now studied from various points of view. It is well known that such transformations possess a tensor invariant, the so called conharmonic curvature tensor, that is it shows the classical symmetry properties of the Riemannian curvature tensor. The Riemannian spaces are said to be conformal spaces if their metrics g ij and gij satisfy the re- B R. Singh singhrameshwar86@yahoo.in R.K. Tiwari rishitiwari59@rediffmail.com 1 Department of Mathematics, Govt. Model Science College Rewa, 486001 M.P., India lation g ij = e φ g ij (1) (where φ is a real valued differentiable function of coordinates) and the correspondence between the spaces is called a conformal transformation. A function with vanishing Laplacian is referred to as harmonic function. In general, a harmonic function is not tensor invariant. The condition under which a harmonic function transforms into harmonic function has been studied by Ishii (see Ishii 1957), who introduced the conharmonic transformation as a subgroup of the conformal transformation satisfying the relation φ i ;i + φ ;iφ i ; = 0 () where (;) denotes the covariant differentiation with respect to metric g ij. A rank four tensor L l ij k that retains its invariant form under conharmonic transformation for an n-dimensional Riemannian differentiable manifold is given by L l ij k = Rl ij k 1 ( gij Rk l n g ikrj l + δl k R ij δj l R ik) (3) where Rij l k is the Riemannian curvature tensor, R ij is the Ricci tensor. The curvature tensor defined by (3) isknown as conharmonic curvature tensor. A space-time in which L l ij k vanishes at each point is known as conharmonically flat space-time. Thus this tensor represents the deviation of the space-time from conharmonic flatness. A conharmonically flat space-time is an Einstein space and is of constant curvature. The significance of the spaces of constant curvature is very well known in cosmology. If we assume that the universe is isotropic and homogeneous, which is known as cosmological principle, the simplest cosmological model

130 Page of 5 Astrophys Space Sci (015) 357:130 of the universe is obtained. Isotropy defines the equivalence of all the spatial directions, while homogeneity means the universe is same everywhere that is it is impossible to distinguish one place in the universe from the other. That is in rest system of matter there is no preferred point and no preferred direction, the three dimensional space being constituted in the same way everywhere. The cosmological solutions of Einstein field equations which contain a three dimensional space-like surface of a constant curvature are the Robertson-Walker metrics, while a four dimensional space of constant curvature is the de-sitter model of the universe. de-sitter model possesses a three dimensional space of constant curvature and thus belongs to Robertson-Walker metrics. These discussions clearly point towards the relativistic significance of the conharmonic curvature tensor and conharmonic flatness. Conharmonic curvature and its significance to four dimensional space-time in general relativity has been discussed by Siddiqui and Ahsan (010). Ahsan and Siddiqui (009) also studied the nature of concircular curvature tensor with perfect fluid in the space-time of general relativity. The results of the data released from the observations of Type Ia supernovae (SNe Ia) show that the present universe is passing through an accelerating phase of expansion (Reiss et al. 1998, 004; Perlmutter et al. 1998). This inflationary phase of the universe is attributed to a mysterious component with negative pressure, dubbed as dark energy, which constitutes with approximately 3/4 ofthe critical density. A simple candidate for dark energy can be a cosmological constant in the classical FRW model. However, the cosmological constant is facing some well known puzzles like fine-tuning problem, coincidence problem. In order to explain the nature of dark energy, a variety of theoretical models have been proposed in the literature such as quintessence, k-essence, phantom energy, chaplygin gas etc. (Ratra and Peebles 1998; Chiba et al. 000; Caldwell 00; Piazza and Tsujikawa 004). But, the exact nature of dark energy still remains a mystery (Copeland et al. 006). The models of the universe with evolving Λ- term are becoming more important as they can solve the cosmological constant problem in natural manner. The cosmological models with phenomenological Λ-decay scenarios have been considered by many authors (Chen and Wu 1990; Gabriel and Le Denmat 1999; Pradhan and Aotemshi 00; Pradhan and Yadav 00; Singh 006; Vishwakarma 1996, 000; Tiwari 009, 010, 011; Tiwari and Jha 009). A number of researchers (Abdel Rahaman 1990, 199;Carvalho et al. 199; Silveira and Waga 1994; Waga1993; Overduin 1999; Berman and Som 1990) have taken keen interest to study the models with decaying Λ-term in different physical contexts. In the present work, we have explored the significance of the flat FRW model in the context of conharmonic curvature tensor in presence of perfect fluid as a source of matter distribution, in the framework of general relativity. Exact solution of Einstein s field equations is obtained by a special law for the decay of cosmological term as Λ = 3βH (where β is a constant). Physical properties of the model have been discussed in detail. Metric and field equations For isotropic and homogeneous world models (Robertson 1935; Walker1936), the metric of the space-time have the form [ dr ds = dt R ] (t) 1 kr + r dω (4) where dω = dθ + sin θdφ, R(t) is the scale factor which describes how the distance between any two world lines change with cosmic time t, r is the comoving radial coordinate. The θ and φ parameters are the usual azimuthal and polar angles of spherical coordinates with 0 θ π and 0 φ π. The coordinates (t,r,θ,φ) are called comoving coordinates. Also k is the curvature of the space which can take the values ( 1, 0, 1) representing open, flat, closed universe respectively. The expression for Einstein s relativistic field equations (c = 1 = 8πG) is given by R ij 1 Rg ij = T ij Λg ij (5) We assume that the energy-momentum tensor of a perfect fluid has the form T ij = (ρ + p)u i u j pg ij (6) where ρ is the energy density, p is the isotropic pressure and u i is the four velocity vector with u i u i = 1. For the perfect fluid, we take the barotropic equation of state as p = ωρ (7) where ω is the equation of state parameter such that 0 ω 1. For a relativistic four dimensional space-time, the conharmonic curvature tensor can be written as L l ij k = Rl ij k ( 1 gij Rk l g ikrj l + δl k R ij δj l R ) ik (8) The conharmonically flat space-time gives R l ij k = g ij R l k g ikr l j + δl k R ij δ l j R ik (9)

Astrophys Space Sci (015) 357:130 Page 3 of 5 130 Contracting Eq. (9) with j = l and taking summation over j, we find R ik = 1 4 Rg ik (10) In view of Eq. (10), the Einstein s field equations (5) reduce to the form R ij = 1 3 T ij Λ 3 g ij (11) For the flat (k = 0) FRW metric (4) and energy-momentum tensor (6), the Einstein s field equations (11) read as ρ Λ = 9 ( H + H ) (1) p + Λ = 3 ( H + 3H ) (13) where H = Ṙ/R is the directional Hubble parameter. Here and elsewhere an over-head dot denotes ordinary differentiation with respect to cosmic time t. 3 Solution of field equations Equations (7), (1) and (13) are a system of three equations with four unknown parameters H, p, ρ and Λ. Hence one extra constraint is required to pack the system completely. There is significant theoretical evidence for the phenomenological Λ-decay scenarios considered by a number of authors (Wang and Meng 005; Ray et al. 009) during last two decades. Al-Rawaf and Taha (1996), Al-Rawaf (1998), Overduin and Cooperstock (1996) proposed cosmological term of the form Λ = β R R, where β is a constant. Following the same decay law Arbab (003) have investigated cosmic acceleration with positive cosmological constant and also analyze the implication of model built-in cosmological constant. Pradhan et al. (005) have reported the Kaluza- Klein type Robertson Walker (RW) cosmological models by considering three different forms of variable Λ: Λ ( ṘR ), Λ R R and Λ ρ. Recently, Kumar and Srivastava (013) have studied FRW cosmological model for conharmonically flat space-time. We consider the following ansatz for the decay of Λ-term (Singh et al. 008; Jamil and Debnath 011) Λ = 3βH (14) where β is a constant. Here β represents the ratio between vacuum and critical densities. From Eqs. (7), (1) (14), one can find that H + (1 + ω)(3 β) H = 0 (15) (1 + 3ω) which integrates to give H(t)= 1 + 3ω [(1 + ω)(3 β)t + c 1 (1 + 3ω)] (16) From Eq. (16), one can easily obtain the value of scale factor R(t) as R(t) = c [ (1 + ω)(3 β)t + c1 (1 + 3ω) ] 1+3ω (3 β)(1+ω) (17) where c 1 and c are constants of integration. The isotropic pressure p and the cosmic volume V for the model comes out to be 6ω(1 + 3ω)(3 β) p(t) = [(1 + ω)(3 β)t + c 1 (1 + 3ω)] (18) V(t)= c 3 [ (1 + ω)(3 β)t + c1 (1 + 3ω) ] 3(1+3ω) (3 β)(1+ω) (19) We can obtain the energy density ρ and expansion scalar θ as ρ(t)= θ(t)= 6(1 + 3ω)(3 β) [(1 + ω)(3 β)t + c 1 (1 + 3ω)] (0) 3(1 + 3ω) [(1 + ω)(3 β)t + c 1 (1 + 3ω)] Cosmological constant Λ can be evaluated as Λ(t) = (1) 3β(1 + 3ω) [(1 + ω)(3 β)t + c 1 (1 + 3ω)] () The deceleration parameter, q = R R/Ṙ, can be obtained as (1 + ω)(3 β) q = 1 + (3) (1 + 3ω) From Eqs. (16) (3), we observe that the spatial volume V and the radius scale factor R is zero at t = (1+3ω)c 1 (1+ω)(3 β) = t c, which shows that the universe in the present model has singularity at t = t c.nowast increases, spatial volume V and radius scale factor R also increase whereas the parameters (p,ρ,h,θ,λ) decrease and as t approaches to infinite, V and R become infinitely large while the parameters (p,ρ,h,θ,λ) drop to zero. This shows that the universe in the model starts with a finite time big-bang singular state and expands with cosmic time t. The condition for such a behavior of the universe in the model is β<3, c 1 > 0, c > 0. We also observe that for c 1 = c = 0, R = 0, V = 0. This indicates that the universe has initial singularity for c 1 = c = 0. Furthermore, it can be seen that for β>0, Λ>0whereas p>0, ρ>0forβ<3. The density parameter Ω = ρ ρ c = (3 β) (1+3ω), which gives that ρ c >ρ for β> 5 3ω and ρ c <ρ for β< 5 3ω, however, for β = 5 3ω, ρ c = ρ. Theratio between vacuum and matter densities is given by ρ v ρ = β(1+3ω) (3 β). It is worth to mention here that a positive value of Λ- term in the derived model of the universe leads to repulsion with negative effective mass density. Hence we can see

130 Page 4 of 5 Astrophys Space Sci (015) 357:130 that in presence of positive Λ, the expansion of the universe in the model will tend to accelerate whereas the expansion of the universe with negative Λ will slow down, stop and reverse. Recent observations in cosmology (Schmidt et al. 1998; Garnavich et al. 1998) suggest the cosmological constant to require a small positive value with magnitude Λ(Għ/c 3 ) 10 13. Redshift of Type Ia supernovae indicate that our universe may be accelerating with induced cosmological density through the cosmological term Λ, which shows that our model is in reasonable agreement with recent observations. Also from Eq. (3), we see that q>0forβ< 1+ω which shows the decelerating phase of expansion of the universe. For β = 1+ω, the deceleration parameter q is zero indicating that deceleration in the expansion halts i.e. every galaxy moves with constant speed while for β> 1+ω, q<0which indicates the accelerating phase of expansion of the universe in the model. Thus our model exposes deceleration and acceleration phases of the universe both. It can be seen that for β = 3, we have H 1 c 1, ρ 0, Λ 9, q = 1. This exposes that for β = 3, the universe in c1 the derived model corresponds exactly to de-sitter universe. Also for this case (β = 3) the universe has the largest value of Hubble s parameter hence the fastest rate of expansion (Tiwari and Rameshwar 014). This is assumed to be the future scenario of the observable universe. 3.1 Statefinder parameters The cosmological constant problem and cosmic coincidence problem are considered as the most challenging and fascinating problems in the study of dynamics of the universe in the context of general theory of gravity. In order to overcome the puzzles arising due to complex and astonishing behavior of Λ-term, different dark energy models have been examined by researchers from time to time. Sahni and Alam (Sahni et al. 003; Alametal.003) in an attempt to differentiate between different dark energy models have introduced a pair of dimensionless geometrical parameters which they called statefinder parameters play a crucial role in cosmology. The statefinder parameters may be defined as... R r = RH 3 (4) s = r 1 3(q 1 ) (5) where R is the expansion scale factor, H is the Hubble parameter, q is the deceleration parameter and over-head dot here and everywhere indicates differentiation with respect to time t. The variable r is also named as jerk parameter. For a ΛCDM model, the statefinder pair {r, s} have the value {1, 0}. r = s = For the derived model, we have [ + β(1 + ω)][ 5 3ω + β(1 + ω)] (1 + 3ω) (6) [ + β(1 + ω)][ 5 3ω + β(1 + ω)] (1 + 3ω) 3(1 + 3ω)[ 3(1 + 3ω) + (1 + ω)(3 β)] (7) For β = 3, Eqs. (6) and (7) give r = 1, s = 0. Thus we obtain {r, s}={1, 0}, which shows that the model is in agreement with ΛCDM model. Now let us define and discuss cosmic snap parameter 3. Cosmic snap parameter The cosmic snap parameter is another important parameter in cosmology to examine the models close to ΛCDM. This parameter is defined as (Visser 004, 005) s = 1 d 4 R RH 4 dt 4 (8) The cosmic snap parameter can be located in the fourthorder term of the Taylor series expansion of the scale factor around R 0 : R(t) R 0 = 1 + H 0 (t t 0 ) 1 q 0H 0 (t t 0) + 1 6 r 0H 3 0 (t t 0) 3 + 1 4 s 0 H 4 0 (t t 0) 4 + O [ (t t 0 ) 5] (9) where subscript 0 denotes the present day value of the concerned quantity. Equation (8) can be rewritten as s = ṙ r( + 3q) (30) H where q is the deceleration parameter and r is the jerk parameter. For a flat ΛCDM model s = ( + 3q) as r = 1 and the departure of the quantity ds dq from 3 gives a measure of the evolution of the universe deviating from ΛCDM dynamics (Rapetti et al. 007; Poplawski 006). For the present model, β = 3 implies q = 1soEq.(30) gives s = 1. Thus the model is in impressive agreement with recent observations. 4 Conclusion In this paper, we have investigated a spatially homogeneous and isotropic flat FRW model with vanishing conharmonic curvature tensor in presence of perfect fluid as a source of

Astrophys Space Sci (015) 357:130 Page 5 of 5 130 (1+ω)(3 β) matter distribution of the universe. A special variation law Λ = 3βH (where β is a constant) is employed to obtain the unknown parameters of the Einstein s field equations which describe a cosmological scenario in which the universe in the present model has singular state at some finite time t = t c = (1+3ω)c 1 with big-bang. Hubble Parameter H, isotropic pressure p, energy density ρ, expansion scalar θ, and cosmological constant Λ all diverge at t = t c with scale factor R and spatial volume V zero and the parameters (p,ρ,h,θ,λ) infinite whereas all these parameters (p,ρ,h,θ,λ) converge to zero as t tends to infinity with V and R infinite. It is to note that cosmological constant Λ is found to be positive and decreasing function of cosmic time t i.e. Λ 1 t. Such a behavior of Λ is observed by a number of authors (Berman 1990; Berman and Gomide 1990; Amirhashchi 011; Yadav 01).Thesameisalso predicted by Type Ia supernovae (S Ne Ia) observations. The deceleration parameter q is positive for β< 1+ω, which indicates the decelerating phase of expansion of the universe. q is zero for β = 1+ω, which shows that the universe expands with constant expansion rate and q is negative for β< 1+ω, which exposes the accelerating phase of the universe. Therefore the present model shows deceleration and acceleration phases of the universe. The positive value of Λ and negative value of q is required to solve the age problem. It is interesting to note here that for β = 3, we have H c 1 1, ρ 0, Λ 9, q = 1. This shows that for β = 3, c1 the universe in the derived model corresponds exactly to de- Sitter universe. 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