Eur. Phys. J. C (2018) 78:849 https://doi.org/10.1140/epjc/s10052-018-6322-y

Regular Article - Theoretical Physics

Does fractal describe a complete cosmic scenario ?

Dipanjana Das1,a, Sourav Dutta2,b, Abdulla Al Mamon1,c, Subenoy Chakraborty1,d 1 Department of Mathematics, Jadavpur University, Kolkata, West Bengal 700032, India 2 Department of Pure Mathematics, Ballygunge Science College, 35, Ballygunge Circular Rd, Ballygunge, Kolkata, West Bengal 700019, India

Received: 5 February 2018 / Accepted: 11 October 2018 / Published online: 22 October 2018 © The Author(s) 2018

Abstract The present work deals with evolution of the frac- tions (for FLRW model) has been done either by introducing tal model of the universe in the background of homogeneous (in the right hand side) some exotic [14,15] (known as and isotropic FLRW - geometry. The cosmic sub- DE) having large negative pressure or by modifying the geo- strum is taken as perfect fluid with barotropic equation of metric part of the field equations i.e., considering modified state. A general prescription for the deceleration parameter gravity theories (considering general form of the Lagrangian is determined and it is examined whether the deceleration density than EinsteinÐHilbert or by higher dimensional theo- parameter may have more than one transition during the evo- ries) [39]. To have an idea of what the “fractal effects” could lution of the fractal universe for monomial form of the fractal possible be, an attempt is made in the present work to explain function as a function of the scale factor. Finally, the model the present (observed) cosmic acceleration in a gravity theory has been examined by making comparison with the observed with imprints of fractal effects. data. One of the challenging issues in the present day theo- retical physics is to formulate a consistent theory of quan- tum gravity. Besides the major attempts namely string theory 1 Introduction and (loop) , other independent investigations such as causal dynamical triangulations [30], asymptotically The homogeneous and isotropic FLRW (Friedmann Lemaitre safe gravity [31], spin-foam models [32,33] and HoˇravaÐ Robertson Walker) model without the Lambda term in stan- Lifshitz gravity [34,35] have received some attention. All dard relativistic has been put in a big question these theories exhibit a running spectral dimension ds [26Ð mark for the last two decades due to a series of observa- 29] of such that ds is smaller than four in the tional evidences [1Ð14]. However, there are various attempts UV scale [16,17,35]. Systems whose effective dimensional- to go beyond the simple hypothesis of homogeneity and ity changes with the scale may have fractal behaviour, even isotropy and to include the effect of spatial inhomogeneities if they are defined on a smooth manifold. A lower spectral in the metric. These observational data are interpreted in dimension can be associated with scenarios with improved the framework of the Friedmann solutions of the Einstein renormalization but it is not true in general. field equations and it is found that there should be an accel- In fractal universe time and space-coordinates scale μ eration. This implies the presence of a Lambda-like term isotropically i.e., [x ]=−1,μ= 0, 1,...,D − 1 and the in the Friedmann equations or alternatively, this could be standard measure in the action is replaced by a non-trivial due to the effect of large scale spatial inhomogeneities [12]. measure (usually appears in LebesgueÐStieltjes integrals): Thus this challenge of cosmology has been addressed in the D → ρ( ), [ρ]=− α =− , frame work of Einstein gravity by introducing inhomoge- d x d x D D neous space time model [36Ð38] or by modifying the Ein- where D is the topological (positive integer) dimension of stein gravity itself in the usual homogeneous and isotropic the embedding space-time and the parameter α>0 roughly FLRW model. The modifications in the Einstein field equa- corresponds to the fraction of states preserved at a given time during the evolution of the system. One can obtain such struc- a e-mail: [email protected] ture through variable effective dimensionality of the universe b e-mail: [email protected] at different scales. This feature can be obtained simply by c e-mail: [email protected] introducing fractal action as on net fractals and they can be d e-mail: [email protected] approximated by fractal integrals [16,17]. Usually, the mea- 123 849 Page 2 of 9 Eur. Phys. J. C (2018) 78 :849 sure is considered on a fractal set as general Borel probability and measure ρ.SoinD dimensions, (M,ρ)denotes the metric space-time M equipped with measure ρ.Hereρ is absolutely v˙ v 6(H˙ + H 2) =−8πG(ρ +3p)+6H −2wv˙2 +3 +2, continuous with v v   (5) dρ = d D x v(x), where ‘’ is the usual D’Alembertian operator and matter some multidimensional Lebesgue measure and v is the is chosen as perfect fluid with barotropic equation of state: weight function (fractal function). p = γρ. For simplicity we choose  = 0 and units are In a fractal space-time the total action of Einstein gravity chosen such that 8πG = 1. The above field equations can can be written as be written as [19] = + , S Sg Sm (1) 2 3H = ρ + ρ f = ρt , (6) where and  M2 √   p D μ ˙ 2 Sg = d x v(x) −g R − 2 − w∂μv∂ v , (2) 2H + 3H =−p − p =−p , (7) 2 f t (ρ , ) is the gravitational part of the action [16Ð18] and where f p f are termed as energy density and thermo- dynamic pressure of the effective (hypothetical) fractal fluid  √ and have the expressions D Sm = d x v(x) −gLm, (3) w v˙ 2 ρ f = v˙ − 3H , (8) is the action of the matter part minimally coupled to grav- 2 v − 1 ity. Here M = (8πG) 2 is the reduced Planck mass and w p and stands for fractal parameter in the action in (i.e., Eq. (2)). The w v˙ v¨ paper is organized as follows: Sect. 2 presents fractal universe 2 p f = v˙ + 2H + . (9) as standard cosmology with interacting two fluids. In Sect. 3, 2 v v fractal cosmology has been shown to be equivalent as a par- ticle creation mechanism in the context of non equilibrium The energy conservation equation in a fractal universe thermodynamical prescription and temperature of the indi- takes the form [16,17,19,20] vidual fluids has been determined in terms of creation rate   v˙ parameter. Also entropy variation of the individual fluids as ρ˙ + 3H + (ρ + p) = 0. (10) well as the total entropy variation has been evaluated for this v open thermodynamical system. A detailed study of the evo- v lution of the deceleration parameter and its possible ranges Note that if the fractal function is chosen as constant then ρ = = in different stage of evolution has been studied in Sect. 4.In from Eqs. (8) and (9) f 0 p f and Eqs. (6), (7) and (10) Sect. 5, the model has been compared with observational data represent the usual Einstein field equations and the energy using contour plots and variation of the deceleration param- conservation law for Einstein gravity in FLRW model. eter has been shown graphically. The paper ends with a brief From the field equations (6) and (7), due to Bianchi iden- summary in Sect. 6. tity we have

ρ˙t + 3H(ρt + pt ) = 0. (11) 2 Fractal universe as interacting two fluids in Einstein gravity Now from Eqs. (10) and (11) one can write the individual matter conservation equations as In homogeneous, isotropic and flat FLRW space-time model v˙ the variation of the action (in Eq. (1)) with respect to the met- ρ˙ + H(ρ + p) =− (ρ + p) = Q, 3 v (12) ric tensor gives the Friedmann equations in a fractal universe as and v˙ w 2 2 3H = 8πGρ − 3H + v˙ + , (4) ρ˙ + H(ρ + p ) =−Q. v 2 f 3 f f (13) 123 Eur. Phys. J. C (2018) 78 :849 Page 3 of 9 849

Thus, the modified Friedmann equations in fractal uni- equations (12) and (13) can be written as verse can be interpreted as Friedmann equations in Einstein ρ˙ + (ρ + + ) = , gravity for an interacting two fluid system of which one is 3H p pc 0 the usual normal fluid under consideration and other is the ρ˙f + 3H(ρ f + p f + pcf ) = 0, (19) effective fractal fluid and the interaction term is given by v˙ with Q =−v (ρ + p). However, one can write the above conservation equations v˙ pc = (ρ + p) =−pcf . (20) (12) and (13) in non-interacting form with effective equation 3Hv of state parameters as In the context of non-equilibrium thermodynamics this ( ) v˙ γ ef f = γ + (1 + γ), (14) dissipative pressure may be assumed to be due to particle 3Hv creation process. So the particle number conservation rela- (ef f) v(˙ 1 + γ) γ = γ f − r, (15) tions take the form f 3Hv ρ where r = is the dimensionless coincidence parameter ˙ + = , ρ f n 3Hn n (21) p and γ = f is the equation of state parameter for effective f ρ f fractal fluid. and Usually, for interacting two fluid system the interaction term Q should be positive as energy is transferred to the dark n˙f + 3Hn f =− f n f . (22) matter (the usual fluid here). Here positivity of Q implies that the fractal function v should decrease with the evolution Here ‘n’ represent the normal fluid particle density with (i.e., v<˙ 0). Thus, if the present normal fluid is chosen as ‘n f ’ corresponds to number density of the effective particles cold dark matter (i.e., γ = 0) then γ (ef f) < 0 i.e., we have (termed as fractal particle) for the effective fluid. Note that effectively some exotic nature of the matter component due these effective particles are introduced to map the present to fractal cosmology. However, if the fluid is chosen as hot model to one for a standard GR universe with some particle dark matter (i.e., γ>0) then it may or may not remain hot species. It is assumed that normal fluid particles are created dark matter in fractal cosmology, depending on the explicit (i.e., the particle creation rate >0) and the effective ‘frac- choice of the fractal function. tal’ particles are annihilated (i.e., f < 0) in course of the Moreover, using the conservation equations (12) and (13) evolution. For simplicity if we assume the non-equilibrium the time evolution of the coincidence parameter may be writ- thermodynamical process to be adiabatic (i.e., isentropic) ten as then the dissipative pressures are related to the particle cre- ation rates linearly as [21] dr 1 + γ dv = r (γ f − γ)− (1 + r) , (16) dτ v dτ f pc =− (ρ + p) and pcf = (ρ f + p f ). (23) 3H 3H with τ = 3lna. As a result the time variation of the total energy density can be written as Now comparing Eqs. (20) and (23) the particle creation rates are given by dρ rγ + γ T =−ρ + f . v˙ v(ρ˙ + ) τ T 1 + (17) p d 1 r = and f =− . (24) v v(ρ f + p f )

3 Fractal universe as particle creation mechanism and Thus, in fractal universe the particle creation rate is related temperature of the fluid components to the fractal function by the above relations. For the combined single fluid as pct = pc + pcf = 0so The Friedmann equations (6) and (7) of the last section for the particle creation rate for the resulting single fluid should the fractal universe can be written as vanish identically. Hence effective single fluid forms a closed system. 2 3H = ρ + ρ f =−ρt , Further, due to particle creation mechanism there is an ˙ 2 energy transfer between the two fluid components and as a 2H + 3H =−(p + pc) − (p f + pcf ), (18) result, the two subsystems may have different temperatures with pcf =−pc, the dissipative pressure (bulk viscous pres- and thermodynamics of irreversible process comes into the sure) for the fluid components. So the energy conservation picture. 123 849 Page 4 of 9 Eur. Phys. J. C (2018) 78 :849

Using Euler’s thermodynamical relation: nTs = ρ + p, entropies of the normal fluid and effective fractal fluid then the evolution of the temperature of the individual fluid is using Clausius relation to the individual fluids one obtains given by [22,23] dSn = dQ = dE + dV ,   T p ˙ γ˙ dt dt dt dt T (ef f) =−3H γ + + , (25) and T 3H (1 + γ) dS dQ dE dV T f = f = f + p , and f dt dt dt f dt   ( , = ρ ) ( , = ρ ) ˙ where Q E V and Q f E f f V are the corre- T ( ) γ˙ f =− γ ef f − f + f , sponding amount of heat and energy of the two fluid com- 3H f (26) T f 3H (1 + γ f ) = 4 π 3 ponents with V 3 Rh being the volume bounded by the horizon. Assuming the whole universe bounded by the hori- γ (ef f) γ (ef f) where and f defined in Eqs. (14) and (15) can zon to be isolated in nature, the heat flow (Qh) across the now be written in terms of particle creation rate as horizon is balanced by the heat flow through the two fluid components i.e.,

(ef f) (ef f) f γ = γ − (1+γ),γ = γ f + (1+γ f ). (27) =−( + ). 3H f 3H Qh Q Q f

Now integrating Eqs. (25) and (26) the temperature of the Moreover, from the point of view of non-equilibrium thermo- individual fluid component can be written as [22] dynamics, it is desirable to consider the contributions from    irreversible fluxes of energy transfer to the total entropy vari- a da ation as [44] T = T0(1 + γ)exp −3 γ 1 − , a 3H a 0   dST dSn dSf dSh ˙ ¨ ˙ ¨ a = + + − A Q Q − A Q Q , f da dt dt dt dt f f f h h h T f = T0(1 + γ f )exp −3 γ f 1 + , (28) a 3H a 0 where Sh stands for the entropy of the horizon, A f and Ah are where T0 is the common temperature of the two fluids in the constants in the energy transfer between the fluid compo- equilibrium configuration and a0 is the value of the scale nents within the horizon and across the horizon respectively. factor in the equilibrium state. In particular, using Eq. (24) Normally, for FLRW model, trapping (i.e., apparent) horizon for (in terms of the fractal function) the temperature of the is chosen as the boundary of the universe. However, in the normal fluid for constant γ can have the following explicit perspective of the present accelerating phase of the universe expression one may consider event horizon as the bounding horizon.

 − γ  −γ a 3 v T = T0(1 + γ) . (29) a0 v0 4 Evolution of fractal universe and deceleration parameter Usually, at very early stages of the evolution of the uni- verse one should have T f < T and then with the evolution of At first, the variation of the deceleration parameter will be = ρ the universe, both the cosmic fluids attain an equilibrium era examined in fractal cosmology. By introducing 2 ρ 3H at a = a0 with T = T f = T0. In the subsequent evolution of and = f , the density parameters for the normal fluid f 3H 2 the universe one has a > a0 and T f > T due to continuous and the effective fractal fluid one gets flow of energy from the effective fractal fluid (chosen as DE) to the normal fluid (considered as DM) and consequently, the + f = 1, (30) thermodynamical equilibrium is violated. Now, from thermo- dynamical point of view, one may consider the equilibrium from the Friedmann equation (6). Then the deceleration temperature T0 as the (modified) Hawking temperature [43] parameter q takes the form [24] 2 = H Rh | , i.e., T0 a=a0 1 3 2π q = + ( γ + f γ f ). (31) 2 2 where Rh is the geometric radius of the horizon, bounding the universe. Now solving Eqs. (30) and (31)for and f one obtains Now, one can investigate the entropy variation of both the flu- ids under consideration as well as the total entropy variation 2q − (1 + 3γ f ) = , (32) of the isolated thermodynamical system. If Sn and S f be the 3(γ − γ f ) 123 Eur. Phys. J. C (2018) 78 :849 Page 5 of 9 849 and with   u γ (ef f)(x) (1 + γ)− 2q μ( ) = − = . u exp 3 dx and f (33) 1 x 3(γ − γ f ) ⎡ ⎤  ( ) u γ ef f ( ) ⎣ f x ⎦ Due to non-negativity of the density parameters, the decel- γ(u) = exp −3 dx . (40) x eration parameter has a lower bound and an upper bound i.e., 1 As a result the density parameters take the forms + γ ( + γ) 1 3 f ≤ ≤ 1 . q (34) μ( ) γ(μ) 2 2 = u = , μ + γ and f μ + γ (41) During the dominance of the effective fractal fluid (i.e., 1 ≤ ≤ 1 and 0 ≤ ≤ 1 ) the above inequality 2 f 2 and hence the expression for the deceleration parameter (34) modifies to becomes

1 + 3γ 1 3 f ≤ ≤ + (γ + γ ). 1 3 μγ + γγf q f (35) q(u) = + . (42) 2 2 4 2 2 μ + γ γ =− In particular, if f 1 (i.e., on the phantom barrier) Now differentiation of the above equation with respect to then one has u with the help of the Eq. (40)forμ and γ one obtains (after 1 a bit simplification) −1 ≤ q ≤− .   4 γ dγ dq = 3 d + f + 9 − (γ − γ )2 Further, if one consider the above two fluid system as f f f du 2 du du 2u effective single fluid system then effective equation of state v(˙ 1 + γ)(1 + r) − (γ − γ f ) . (43) parameter for the single fluid is given by 3vH

γe = γ + f γ f = γ f + (γ − γ f ), Normally, the equation of state parameter γ is positive and decreases with the evolution (or remains constant) i.e., γ and as expected it is independent of the interaction term Q. dγ ≤ d f dq du 0 while du may have any sign. Thus, du may change Also the expression for the deceleration parameter takes the sign more than once during the evolution i.e., q(u) = 0may form have more than one real solution. Hence universe may have more than one transition from deceleration to acceleration 1 q = (1 + 3γe), (36) and vice versa during the process of evaluation of the uni- 2 verse. However, it should be noted that if v˙ = 0 i.e., v is a constant then fractal effects will be eliminated and one has the usual definition of the deceleration parameter for a single dq ≤ 0 i.e., q is a decreasing function of u and q(u) = 0 fluid. du has only one real solution. Therefore, one may conclude that Suppose tc be the time instant at which both the fluids fractal effect has a significant impact in the transition from have identical energy densities i.e., decelerating phase to accelerating phase or in the reverse way. ρ(tc) = ρ f (tc) = ρc (say), (37)

= ( ) and ac a tc be the value of the scale factor at this instant. 5 Present fractal model and the observed data Suppose one introduces a dimensionless variable [24,25] In fractal cosmology the cosmic fluid in the form of perfect a u = . (38) fluid with constant equation of state: p = γρ,(γ a constant) a c is chosen. Regarding the choice of the fractal function v (i.e., measure weight) the monomial [16,17]oftheform and then integrating the continuity equations (12) and (13) using the Eqs. (14) and (15) one has the expressions for −β v(t) = v t , (44) energy densities as 0

ρ ρ is chosen with β = 4(1 − α).Hereα>0, is related to the ρ = c μ(u) and ρ = c γ(u), (39) u3 f u3 Hausdorff dimension of the physical space-time. At short 123 849 Page 6 of 9 Eur. Phys. J. C (2018) 78 :849 scale UV regime if the inhomogeneities are small then the it is given by modified Friedmann equations (4) and (5) can be interpreted  ( + ) z v( )  ( ) as background for perturbations rather than a self consistent ( ) = 1 z z dz , ( ) = H z . dL z  h z (50) dynamics. Now eliminating ρ between (4) and (5)(using H0 0 h(z ) H0 the equation of state p = γρ) the gravitational constraint μ equation so obtained is a Riccati equation in H, which gives By definition, the distance modulus for any SNIa at a redshift z, is written as the background expansion without knowledge of the matter   content. Hence this over determination of the dynamics rules H d (z) μ(z) = 5log 0 L + μ out the vacuum solution (i.e., ρ = 0 = p). So the above 10 1Mpc 0 measure may be justified at early . On the otherhand, if where, μ is a nuisance parameter that should be marginal- the choice of the measure weight is chosen as a monomial of 0 ized. Thus the expression for χ 2 has the form: the scale factor i.e.,

580 μth(z ,μ ,θ)− μobs(z ) 2 v = v m, < , χ 2 (μ ,θ)= i 0 i , 0a m 0 (45) SNIa 0 2 σμ(zi ) i=1 v then this choice of is similar to the above choice (i.e., in where, μth and μobs denote the theoretical distance modulus Eq. (44)) in the matter dominated era as scale factor grows as and observed distance modulus respectively. Also, θ denotes power-law form. So the parameter ‘m’ is related to the Haus- any model parameter and σμ denotes the uncertainty in the dorff dimension of the physical space-time. Then the cosmic distance modulus measurements for each data point. In this evolution of the fractal universe for the above functional form work, we have used the Union2.1 compilation data [50] con- v of the fractal function ‘ ’ has been studied as follows:. taining 580 data points for μ at different redshifts. The above monomial form of the fractal function can be Following Ref. [51] and marginalizing μ0, one can write used in the energy conservation equation (10) to obtain: (θ)2 χ 2 (θ) = (θ) − Y , −μ SNIa X ρ = ρ0a ,μ= (1 + γ)(m + 3), (46) Z(θ) where, and consequently the Hubble parameter can be expressed in 580 th obs 2 terms of the scale factor as μ (zi ,μ0 = 0,θ)− μ (zi ) X(θ) = , −μ 2 2ρ a σμ(zi ) H 2 = 0 , i=1 ( + ) − w 2v2 2m 6 1 m m 0a 580 th obs   μ (zi ,μ0 = 0,θ)− μ (zi ) 2 −μ ( + ) − w 2v2 Y (θ) = , H a 6 1 m m 2 = 0 0 (47) σμ(zi ) ( + ) − w 2v2 2m i=1 6 1 m m 0a and where H0 is the present value of H(z). In this case, the deceleration parameter takes the form 580 1 Z(θ) = .  2 σμ(zi ) 1 wm2v2(1 − γ)a2m i=1 q =−1 + 0 2 + m 2 • χ 2  for Hubble data Measurements of Hubble parameter are also useful to con- +3(1 + γ)(1 + m) − m + m2 . (48) straint the parameters in the cosmological models. The rele- Finally, the present fractal model has also been compared vant χ 2 for the normalized Hubble parameter data set is given with the observed data, namely Type Ia Supernova (SNIa) by constraints and data of the observational Hubble parameter. 30 hth(z ,θ)− hobs(z ) 2 The total χ 2 function is defined as, χ 2 (θ) = i i . H σ 2(z ) i=1 h i χ 2 = χ 2 + χ 2 , (49) t SNIa H The error in h(z) is given by [52]  χ 2     where the individual for each data set is calculated as σ 2 σ 2 follows: σ = h H + H0 . h H H • χ 2 for SNIa data 0 The present phase of accelerated expansion was first pointed Here, one has used the 30 data points of H(z) measure- out by the data from SNIa observations. In the SNIa obser- ments [40Ð42,45Ð49]. The present value of H(z) is taken vations, the luminosity distance plays an important role and from ref [53]. 123 Eur. Phys. J. C (2018) 78 :849 Page 7 of 9 849

0.30

0.74 0.35 0.76

0.40 0.78

0.80 0.45 m m

0.82 0.50

0.84

0.55 0.86

0.35 0.30 0.25 0.20 0.15 0.38 0.37 0.36 0.35 0.34 0.33 γγ

Fig. 1 The figure shows the 1σ and 2σ confidence contours for the v0 = 3.5andw =−0.5, w = 0 respectively. In each panel, the large m choice of v = v0a using SNIa+Hubble data sets. The left and right dot represents the best-fit values of the model parameters panels represent the γ − m model parameter for the choices

For the (SNIa+Hubble) data set, one can obtain the best- fit values of parameters by minimizing the χ 2 function, as given in Eq. (49). In this work, the 1σ and 2σ confidence contours on γ − m model parameter spaces are plotted using the combined SNIa+Hubble data set, as shown in Fig. 1.It deserves to mention here that the χ 2 analysis has been done by fixing v0 = 3.5 and w =−0.5, w = 0 respectively. The best-fit values are obtained as γ =−0.257 and m =−0.441 (χ 2 = 582.55) for w =−0.5, while it is obtained as γ = −0.352 and m =−0.796 (χ 2 = 598.93) for w = 0. Finally, using these best-fit values for γ and m the evolution of the deceleration parameter has been shown graphically in Fig. 2 with w =−0.5 only.

6 Results and discussions

The present work is an attempt of examining the fractal cos- mological model from the point of view of recent observa- Fig. 2 The figure shows the evolution of q using the best-fit values tions. Rather than introducing any dark fluid the fractal FRW of the parameters γ and m arising from the combined analysis of SNIa+Hubble data set for w =−0.5 from Fig. 1 model is chosen for the description of the universe. The mod- ified Friedmann equations are shown to be equivalent to the usual Friedmann equations for interacting two fluid system- of which one is the usual fluid and the other is termed as is determined both for the individual fluids as well as for the effective fractal fluid. Also it has been shown that the present total entropy of the system bounded by the horizon. model can be considered as a particle creation mechanism in The fractal function is chosen as a power law form (i.e., which the normal matter particles are created while the effec- monomial) of the scale factor and the deceleration parameter tive fractal particles are annihilated in course of evolution . is evaluated. It is found from Fig. 2 that the deceleration From thermodynamical consideration the temperature of parameter has one transition for the above monomial choice the individual fluids are evaluated and the entropy variation of the fractal function. From the graph one may conclude 123 849 Page 8 of 9 Eur. Phys. J. C (2018) 78 :849 that the present cosmological model with monomial form of 7. U. Seljak, Cosmological parameter analysis including SDSS Ly- the fractal function describes the evolution of the universe alpha forest and galaxy bias: constraints on the primordial spectrum from early inflationary era to the matter dominated phase for of fluctuations, neutrino mass, and dark energy. Phys. Rev. D 71, w =−. 103515 (2005). arXiv:astro-ph/0407372 the choice 0 5. It is to be noted that for this graph 8. D.N. Spergel, Wilkinson Microwave Anisotropy Probe (WMAP) of deceleration (q) parameter, one uses the best fit value of three year results: implications for cosmology. Astrophys. J. Suppl. the parameters. For w = 0,q is a negative constant, so this 170, 377 (2007). arXiv:astro-ph/0603449 choice of w describes only a constant accelerating era. Thus, 9. A.C.S. Readhead, Extended mosaic observations with the Cosmic Background Imager. Astrophys. J. 609, 498 (2004). the fractal cosmological model can describe the evolution arXiv:astro-ph/0402359 from the early inflationary era to the matter dominated epoch. 10. J.H. Goldstein, Estimates of cosmological parameters using the From the comparison with the observation (in Fig. 1)itis CMB angular power spectrum of ACBAR. Astrophys. J. 599, 773 found that the parameter m is always negative (−0.52 < (2003). arXiv:astro-ph/0212517 < − . σ 11. E. Komatsu, Five-year Wilkinson Microwave Anisotropy Probe m 0 36) with in 2 confidence level and it is consistent (WMAP) observations: cosmological interpretation. Astrophys. J. with the second law of thermodynamics. Therefore from the Suppl. 180, 330 (2009). arXiv:0803.0547 [astro-ph] above analysis one may conclude that the monomial form 12. D.J. Eisenstein, Detection of the baryon acoustic peak in the large- of the fractal function is consistent with recent observations scale correlation function of SDSS luminous red galaxies. Astro- phys. J. 633, 560 (2005). arXiv:astro-ph/0501171 particularly for describing the early and intermediate era of 13. B. Jain, A. Taylor, Cross-correlation tomography: measuring dark evolution. energy evolution with weak lensing. Phys. Rev. Lett. 91, 141302 (2003). arXiv:astro-ph/0306046 Acknowledgements Author DD thanks Department of Science and 14. V. Sahni, The Cosmological constant problem and quintessence. Technology (DST), Govt. of India for awarding Inspire research fel- Class. Quantum Gravity 19, 3435 (2002). arXiv:astro-ph/0202076 lowship (File No: IF160067). SD thanks to the financial support from 15. T. Padmanabhan, Cosmological constant: the weight of the vac- Science and Engineering Research Board (SERB), Govt. of India, uum. Phys. Rep. 380, 235 (2003). arXiv:hep-th/0212290 through National Post-Doctoral Fellowship Scheme with File No: 16. G. Calcagni, Quantum field theory, gravity and cosmology in a PDF/2016/001435. Also, SD thankful to the Department of Mathemat- fractal universe. JHEP 1003, 120 (2010). arXiv:1001.0571 [hep- ics, Jadavpur University where a part of the work was completed. SC th] thanks Science and Engineering Research Board (SERB) for award- 17. G. Calcagni, Fractal universe and quantum gravity. Phys. Rev. Lett. ing MATRICS Research Grant support (File No: MTR/2017/000407) 104, 251301 (2010). arXiv:0912.3142 [hep-th] and Inter University Center for Astronomy and Astrophysics (IUCAA), 18. G. Calcagni, Multi-scale gravity and cosmology. JCAP 12, 041 Pune, India for their warm hospitality as a part of the work was done (2013). arXiv:1307.6382 [hep-th] during a visit. 19. S. Haldar, J. Dutta, S. Chakraborty, A comparative study of different entropies in fractal universe. arXiv:1601.01055 [gr-qc] Open Access This article is distributed under the terms of the Creative 20. A. Sheykhi, Z. Teimoori, B. Wang, Thermodynamics of fractal Commons Attribution 4.0 International License (http://creativecomm universe. Phys. Lett. B 718, 1203 (2013) ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, 21. S. Saha, A. Biswas, S. Chakraborty, Particle creation and non- and reproduction in any medium, provided you give appropriate credit equilibrium thermodynamical prescription of dark fluids for uni- to the original author(s) and the source, provide a link to the Creative verse bounded by an event horizon. Astrophys. Space Sci. 356, 141 Commons license, and indicate if changes were made. (2015). arXiv:1507.08224 [physics.gen-ph] Funded by SCOAP3. 22. S. Chakraborty, A. Biswas, Interacting dark fluid in the universe bounded by event horizon: a non-equilibrium prescription. Gen. Relativ. Gravity 46, 1712 (2014). arXiv:1210.0080 [physics.gen- ph] 23. J. Zhou, B. Wang, D. Pavon, E. Abdalla, A Preliminary analysis of References the energy transfer between the dark sectors of the Universe. Mod. Phys. Lett. A 24, 1689 (2009). arXiv:0807.3128 [gr-qc] 1. A.G. Riess, Observational evidence from supernovae for an accel- 24. S. Pan, S. Chakraborty, Will there be again a transition from accel- erating universe and a cosmological constant. Astron. J. 116, 1009 eration to deceleration in course of the dark energy evolution of (1998). arXiv:astro-ph/9805201 the universe?. Eur. Phys. J. C 73, 2575 (2013). arXiv:1303.5602 2. S. Perlmutter, Measurements of Omega and Lambda from [gr-qc] 42 high redshift supernovae. Astrophys. J. 517, 565 (1999). 25. J. Ponce de Leon, Cosmological model with variable equations arXiv:astro-ph/9812133 of state for matter and dark energy. Class. Quantum Gravity 29, 3. W.J. Percival, The 2dF Galaxy Redshift Survey: the power spec- 135009 (2012). arXiv:1204.0589 [gr-qc] trum and the matter content of the Universe. Mon. Not. R. Astron. 26. S. Alexander, R. Orbach, Density of states on fractals: fractons. J. Soc. 327, 1297 (2001). arXiv:astro-ph/0105252 Phys. Lett. (Paris) 43, 625 (1982) 4. P. Astier, The Supernova legacy survey: measurement of omega 27. R. Rammal, G. Toulouse, Random walks on fractal structures and (m), omega (lambda) and W from the first year data set. Astron. percolation clusters. J. Phys. Lett. (Paris) 44, 13 (1983) Astrophys. 447, 31 (2006). arXiv:astro-ph/0510447 28. J. Ambjrn, J. Jurkiewicz, Y. Watabiki, On the fractal structure of 5. A.G. Riess, Type Ia supernova discoveries at z>1 from the Hub- two-dimensional quantum gravity. Nucl. Phys. B 454, 313 (1995). ble Space Telescope: evidence for past deceleration and con- arXiv:hep-lat/9507014 straints on dark energy evolution. Astrophys. J 607, 665 (2004). 29. D. ben-Avraham, S. Havlin, Diffusion and Reactions in Fractals arXiv:astro-ph/0402512 and Disordered Systems (Cambridge University Press, Cambridge, 6. M. Tegmark, Cosmological parameters from SDSS and WMAP. 2000) Phys. Rev. D 69, 103501 (2004). arXiv:astro-ph/0310723 123 Eur. Phys. J. C (2018) 78 :849 Page 9 of 9 849

30. J. Ambjrn, J. Jurkiewicz, R. Loll, Spectral dimension of the uni- 44. S. Saha, A. Biswas, S. Chakraborty, Particle creation and non- verse. Phys. Rev. Lett. 95, 171301 (2005). arXiv:hep-th/0505113 equilibrium thermodynamical prescription of dark fluids for uni- 31. O. Lauscher, M. Reuter, Fractal spacetime structure in asymptoti- verse bounded by an event horizon. Astrophys. Space Sci. 356(1), cally safe gravity. JHEP 10, 050 (2005). arXiv:hep-th/0508202 141 (2015). arXiv:1507.08224 [physics.gen-ph] 32. L. Modesto, Fractal structure of loop quantum gravity. Class. Quan- 45. C. Zhang, H. Zhang, S. Yuan, T.J. Zhang, Y.C. Sun, Four new tum Gravity 26, 242002 (2009). arXiv:0812.2214 observational H(z) data from luminous red galaxies in the Sloan 33. D. Benedetti, Fractal properties of quantum spacetime. Phys. Rev. Digital Sky Survey data release seven. Res. Astron. Astrophys. Lett. 102, 111303 (2009). arXiv:0811.1396 14(10), 1221 (2014). arXiv:1207.4541 [astro-ph.CO] 34. P. Horava, Quantum gravity at a Lifshitz point. Phys. Rev. D 79, 46. D. Stern, R. Jimenez, L. Verde, M. Kamionkowski, S.A. Stan- 084008 (2009). arXiv:0901.3775 ford, Cosmic chronometers: constraining the equation of state 35. P. Horava, Spectral dimension of the universe in quantum grav- of dark energy. I: H(z) measurements. JCAP 1002, 008 (2010). ity at a Lifshitz point. Phys. Rev. Lett. 102, 161301 (2009). arXiv:0907.3149 [astro-ph.CO] arXiv:0902.3657 47. L. Samushia, The clustering of galaxies in the SDSS-III DR9 36. T. Buchert, Is there proof that backreaction of inhomogeneities baryon oscillation spectroscopic survey: testing deviations from  is irrelevant in cosmology?. Class. Quantum Gravity 32, 215021 and general relativity using anisotropic clustering of galaxies. Mon. (2015). arXiv:1505.07800 [gr-qc] Not. R. Astron. Soc. 429, 1514 (2013). arXiv:1206.5309 [astro- 37. T. Buchert, Toward : focus on inhomogeneous ph.CO] geometry and its non-perturbative effects. Class. Quantum Gravity 48. T. Delubac et al., [BOSS Collaboration], Baryon acoustic oscilla- 28, 164007 (2011). arXiv:1103.2016 [gr-qc] tions in the Ly forest of BOSS DR11 quasars. Astron. Astrophys. 38. T. Buchert, A Cosmic equation of state for the inhomogeneous uni- 574, A59 (2015). arXiv:1404.1801 [astro-ph.CO] verse: can a global far-from-equilibrium state explain dark energy ?. 49. X. Ding, M. Biesiada, S. Cao, Z. Li, Z.H. Zhu, Is there evi- Class. Quantum Gravity 22, 113Ð119 (2005). arXiv:gr-qc/0507028 dence for dark energy evolution? Astrophys. J. 803(2), L22 (2015). 39. T. Clifton, Modified gravity and cosmology. Phys. Rep. 513, 189 arXiv:1503.04923 [astro-ph.CO] (2012) 50. N. Suzuki, Astrophys. J. 746, 85 (2012). arXiv:1105.3470 [astro- 40. M. Moresco, Improved constraints on the expansion rate of the ph.CO] Universe up to z 1.1 from the spectroscopic evolution of cosmic 51. R. Lazkoz, S. Nesseris, L. Perivolaropoulos, Exploring cosmolog- chronometers. JCAP 1208, 006 (2012). arXiv:1201.3609 [astro- ical expansion parametrizations with the gold SnIa dataset. JCAP ph.CO] 0511, 010 (2005). arXiv:astro-ph/0503230 41. M. Moresco, Raising the bar: new constraints on the Hubble param- 52. A. Al Mamon, S. Das, A divergence free parametrization of decel- eter with cosmic chronometers at z 2. Mon. Not. R. Astron. Soc. eration parameter for scalar field dark energy. Int. J. Mod. Phys. D 450(1), L16 (2015). arXiv:1503.01116 [astro-ph.CO] 25(03), 1650032 (2016). arXiv:1507.00531 [gr-qc] 42. J. Simon, L. Verde, R. Jimenez, Constraints on the redshift depen- 53. A.G. Riess, Astrophys. J. 826(1), 56 (2016). arXiv:1604.01424 dence of the dark energy potential. Phys. Rev. D 71, 123001 (2005). [astro-ph.CO] arXiv:astro-ph/0412269 43. S. Chakraborty, Is thermodynamics of the universe bounded by the event horizon a Bekenstein system? Phys. Lett. B 718, 276 (2012). arXiv:1206.1420 [gr-qc]

123