Modified Theory of Gravity and the History of Cosmic Evolution

Home , Deceleration parameter

arXiv:1408.1524v1 [astro-ph.CO] 7 Aug 2014 eeeae atrdmntdearqie ogv way give to required long era a dominated of matter cosmic phase decelerated time intermediate late of the universe the including to the form acceleration, era of radiation particular early picture in from and a evolution starting gravity Here, continuous of up a theory take present modified literature. to we the corresponding work, action in present appeared red-shift the never continuous high date from a universe till However, the of picture presently evolutionary the data. mod- explaining attractive cosmological for most suitable available the years, and of recent one inflation in unify be els early to to with out appears turned acceleration 5] hence [4, cosmic gravity time of as late theory known not Modified is energy nature yet. dark exact invoke whose to universe is dominated acceleration cosmic of explanation time viable expansion late most The of present accelerated. the con- is [3] universe that data the equivocally WMAP by almost CMBR of firm anisotropy supernovae and of 2] observations, surveys [1, redshift cosmological high at available those particularly presently the All Introduction 1 hydrogen redshift. the low to at explanation reionization an helium and provide radiation- of history might of theory era phase the modified like second Thus by The a explained singlehandedly. fluid. is gravity term, evolution ideal linear cosmic an a of and com- term, term a squared containing three-half curvature gravity re- of of been theory has bination era modified radiation in of alized phase second a era by dominated pass- followed matter acceleration Friedmann-like long cosmic a time through late ing to through radi- era Friedmann-like ation early from transition continuous A Abstract 3 2 1 sanyal [email protected] [email protected] 0 lcrncaddress: Electronic oie hoyo rvt n h itr fcsi evolutio cosmic of history the and gravity of theory Modified [email protected] 3 1 et fPyis agprClee usiaa,Ida-742213 - India Murshidabad, College, Jangipur Physics, of Dept. , 2 et fPyis nvriyo ayn,Nda ni 741235 - India Nadia, Kalyani, of University Physics, of Dept. .Modak B. 1 asi Sarkar Kaushik , uut9 2021 9, August 1 2 osrcuefrain tre ic matter-radiation since at started equality formation, structure to htn n h ayn h lcrn nparticular. around at in and electrons cools it the expands universe - thermal the baryons As between hot the scattering remains and standard Thomson’s photons Universe to the due era. enters opaque dominated universe radiation followed cosmo- the era Big-Bang of inflationary reheating, history early the by After of account evolution. brief logical a present us Let r eope ihteosto eobnto r at era recombination of onset the radiation z and with Matter decoupled radiation. over are takes matter 7] [6, naclrto fteUies 9-2]adrotation and [9]-[24] Universe to the successful fits, of very SNe-Ia as acceleration were such dark an data of theories cosmological source standard the such explain as [4] since role fourth crucial (see energy, such a plays acceleration particular, gravity In cosmic order reviews). time comprehensive early late for unifies role successfully with the and phenomenological inflation energy plays higher dark which a dynamical including terms, of invariant gravity is curvature Einstein’s gravity order of of generalization theory After Modified this reionization. which of accelerating. starts at universe epoch the reionization epoch the of phase The redshift the called and low hydrogen is ionized. neutral at happened been the occurred have change when have helium phase with mechanism, second must some filled a universe by is means, suggests the This (IGM) of also medium observation plasma. intergalactic ionized Present the that structure twinkling. around to at start started rise universe the give z of instabilities dawn evolution The the the formation. era, dominated with the matter During this developed of universe. the At the course with in occurs long atoms light. hydrogen neutral of of in of change source formation universe so phase any first the The a without to epoch way age giving today. dark transparent called observe becomes we process the as CMBR, the n bi ua Sanyal Kumar Abhik and r ∼ ∼ 0 hntefis eeainsasadquasars and stars generation first the when 20, 00[]adtepoosfe temforming stream free photons the and [8] 1080 z 3145 = − +140 139 6,3196 [6], 3 − +134 133 [7]. n z ∼ 3200 curves for galaxies [25, 26]. It was also suggested In the following section, we construct the model of F (R) that the standard general relativity together with theory of gravity, write down the field equations and ex- Dark-Matter and Dark-Energy may be distinguished press them in the form suitable for numerical solution. from Rn approaches with gravitational microlensing In section 3, we briefly review the presently available [27]. cosmological data. In section 4, we proceed to present numerical solutions, which are depicted in the graphs. In section 5, we demonstrate weak energy limit by trans- In the very early universe, a renormalizable theory of forming the action in canonical form, firstly taking into gravity [28] also requires higher order curvature invari- account an additional tensor degree of freedom and then 2 µν ant terms like, R and Rµν R , in addition to a linear a scalar degree of freedom. In section 6, we present per- term, generated by one-loop quantum gravitational turbation equation. Section 7 is dedicated to the under- corrections. Likewise at the end, a particular form standing of the observed late time radiation era. Section of f(R) is therefore also necessary in particular, to 8 concludes our work. establish the claims of modified theory of gravity in the late universe. Such an attempt has been made invoking Noether symmetry. In vacuum or with pressureless dust 3 2 The model: f(R) R 2 has been found invoking Noether symmetry ∝ F (R) R1+δ theory of gravity suffered initial setback in the Robertson-Walker line element [29, 30]. In ∝ fact, despite many possible attempts, e.g. taking into under synthesis of light elements, shift of the horizon size account a scalar-tensor theory of gravity in addition at matter-radiation equality and perihelion-precession and also considering different anisotropic models [31], observation of Mercury [36]. All these data together 19 attempting Noether gauge symmetry [32, 33] and puts up severe constraint on δ, viz. 0 <δ< 7.2 10− . × treating Born-Infeld action being coupled to f(R) Further, solar system also puts up a severe constraints [34], no other symmetry has been found to exists for on alternative theories of gravity [37, 38]. Particularly, 3 f(R) theory of gravity. Therefore R 2 is in particular for an action a very special form of f(R) and so it is required to A = √ gd4xRn (1) explore the cosmological consequence of such term. − Nevertheless, despite claims in favour of such a form Z of f(R) [35], it shows an un-physical evolution like the gravitational potential [27] in the weak field limit is 3 a t 4 in the radiation era, a being the scale factor. expressed as [36] The∝ situation has improved when a linear term is added Gm r β and it was found to evolve like Friedmann solution Φ(r)= 1+ (2) − 2r r (a √t) in the radiation era [31]. However, general " c # analytical∝ solution in the matter dominated era for such where, r is an arbitrary parameter varying within the an action does not exist. Here, we therefore present c range (1 104)AU, taking into account the velocity of numerical solution of the field equations corresponding the earth− to be 30 Km s 1 [37] while β is related to n to an action containing a combination of curvature − as squared term (R2), a linear term (R), a three-half term 3 (R 2 ) and taking both radiation and matter (baryonic 12n2 7n 1 √36n4 + 12n3 83n2 + 50n +1 β = − − − − . and non-baryonic) into account. Note that it is not 6n2 4n +2 µν − necessary to incorporate Rµν R term due to the fact (3) µν 1 2 that Rµν R 3 R is a total derivative in 4-dimension. For the action− under consideration, deceleration pa- Clearly, for n = 1, β = 0, and Newtonian gravitational rameter (q) versus redshift (z) plot clearly shows yet field is recovered. Any other value of n, which apprecia- another radiation era (q = 1) in the late universe, in bly differs from 1 is ruled out from light bending data in addition to the early radiation era followed by a long the sun limb and planetary periods [37]. The problem Friedmann-like matter dominated era (q 0.5). This was alleviated [39] by considering an action in the form ≈ m n 4 late time radiation-like evolution might at least be [βR + αR + γR− ]√ gd x, m > 0,n > 0, which partially responsible for reionization of neutral atoms passes solar test and therefore− is suitable to explain the present in the IGM. Acceleration of the universe follows cosmologicalR evolution right from the inflationary era thereafter. In the process, the complete history of through to late time accelerated epoch. At the initial cosmological evolution from radiation dominated era stage, Rm term dominates and a de-Sitter solution is till date, has been successfully demonstrated. realizable for m = 2 in particular, explaining inflationary epoch without invoking phase transition [40, 41].

2 In the middle, the linear term dominates giving way to where f,R is the derivative of f(R) with respect to R, the standard BBN and structure formation and finally now reads for the model (4) under consideration, n R− term dominates and late stage of accelerated cosmological expansion is realized, without invoking dark 1 n 2α Rµν Rgµν energy. However, R− term is not distinguished at all, − 2 since neither it is generated by one-loop quantum grav- 1 3/2 itational corrections nor from any other physical conse- +3βMP √RRµν +√R gµν √R;µ;ν R gµν − − 3 quence. Rather, it was considered just to invoke late 3 1 time accelerated expansion. On the contrary, R 2 term +4γ RR + Rg R R2g = T . µν µν − ;µ;ν − 4 µν µν appeared as a consequence of Noether symmetry in R- W metric both in vacuum and in matter dominated era (7) [29]-[34]. Further, in contrast to other powers of R, no 3 Note that in the flat Robertson-Walker metric decay of earth radius has been observed for R 2 term [42]. It therefore appears that the gravitational action ds2 = dt2 + a(t)2 dr2 + r2(dθ2 + sin2 θdφ2) , (8) corresponding to the following form of f(R) theory − and in the absence of Einstein-Hilbert term, R2 term and 4 A = √ g d x [f(R)+ matter] cosmological constant term in the action, the field equa- − L Z (4) tion can be expressed in terms of deceleration parameter 3 4 2 2 = √ g d x [αR + βMP R + γR + matter], (q) and the Hubble parameter (H) as 3q ˙ + 2(1 q)(1 + − L 2q)H = 0. Thus, an analytical solution in the early− vac- Z 3 is more suitable to explain cosmic evolution right from uum dominated universe, when R 2 term dominates over the very early stage, till date, since it satisfies all the others is given by strong conditions necessary for a viable f(R) theory 1 C2 a(t)2 of gravity. In the above, matter is the matter La- a(t)2 = (At + B)4 C2 ; q = − , (9) grangian which contains barotropicL perfect fluid in the 2 − C2 +2a(t)2 form of radiation and pressureless dust together with 2 where A, B and C are constants. Above solution (9) MP 1 CDM. α(= 2 = 16πG ), β, γ in the above action stand indicates power law inflation and is similar to those for dimensionless coupling constants and Λ stands for presented by Capozziello [35] and Sarkar et al [44]. cosmological constant. Here, we would like to mention Nevertheless, the same solution is admissible even at that Starobinsky model naturally explains inflationary the late stage of cosmic evolution taking baryonic and stage and reheating following the mechanism of particle dark matter into account [44]. This clearly indicates production via scalaron decay, exploiting gravity only 3 that R 2 term is compatible to generate either an [40, 43]. In the Starobinsky’s action being expressed in inflaton field in the early universe or dark energy in Jordan frame 3 the late. However, R 2 term is not an outcome of a M 2 R2 renormalizable theory of gravity, rather, as already S = p √ gd4x R + S , (5) − 2 − − 6µ2 m mentioned, it appears invoking Noether symmetry of Z f(R) theory of gravity. Therefore it should not be 5 where, µ = 1.3 10− M is a parameter being fixed × p treated to explain inflation. Rather it should be treated by the normalization of scalar perturbation amplitude, as dark energy. Nevertheless, when treated as dark an additional degree of freedom, viz. scalaron plays the energy, the early radiation and matter dominated era role of inflaton field. The scalaron slow rolls and is re- do not track Friedmann like solution, giving rise to the sponsible for inflationary stage producing a flat power problems in explaining Nucleosynthesis and structure spectrum of perturbation. However, its oscillation re- formation. The problem was alleviated by coupling 3 heats the universe. Thus, the action under considera- R 2 with a linear term (Einstein-Hilbert) [31]. The tion explains very early stage of cosmological evolution. solution obtained in the process [31] tracks Friedmann Here we take up the above action to enunciate the fact like evolution in the radiation dominated era. Although that after the reheating is over, the universe being at exact analytical solution in the matter dominated era radiation dominated era, evolves smoothly to a matter was not found, a particular solution indicated late time dominated era due to the presence of the linear term and 3 cosmic acceleration, which is promising. Therefore, a late time acceleration is realized via R 2 term. Field in the absence of exact analytical solution of the equations corresponding to f(R) theory of gravity, viz. field equation (7), here we simulate numerical solutions 1 taking both radiation and pressureless dust into account. (R + g )f g f(R)= T (6) µν µν − ∇µ∇ν ,R − 2 µν µν

3 Now for the purpose of obtaining numerical solution, we 3 Presently available data: express the field equations in the flat Robertson-Walker metric (8) taking Hubble function H(z) as a function of the red-shift parameter z. In the process, the trace and the time-time component of the field equations (7) (un- 2 MPl 1 der the choice, 8πG =1, ie., α = 2 = 2 ) are expressed The present value of the Hubble parameter is 1 1 as H0 = 73.8 2.5 Km.s− Mpc− , as reported by Riess et al in± 2011 [45]. In view of its standard 3 1 1 2 H′ 3√3 − 2 form, viz. H0 = 100h Km.s− Mpc− , it implies 6H 2 (1 + z) + β√H 2H H′(1 + z) − H 2√2 − 0.713 h 0.763. Now under the choice of unit h i h i ≤ ≤ h 1 4 2 2 4 2 8πG = c = 1, H0 = 9.78 Gyr− and therefore 64H +(1+ z) 3(1 + z) H′ 2(1 + z)HH′ 22H′ 0.073 H 0.078. This means that taking the age of − ≤ 0 ≤ 2 2 2 the universe t = 13.7Gyr, H t lies within the range, h 9(1 + z)H′′ +3 H 47H′ +(1+ z) 2H′H′′′ 0 0 0 − 2 3 1 1 H0t0 1.07, which is fairly good. H′′ 18(1 + z)H′H′′ 12H 14(1 + z)− H′ ≤ ≤ − − − 2 3 +(1 +z)H′′′ 4H′′) 72γH(1 + z) (1 + z) H′ − − i h2 + H(1 + z)H′ 4(1 + z)H′′ 7H′ + H 6H′ Considering 7-year WMAP data, BAO data and the − present value of Hubble parameter H0 altogether, the 3 +(1+ z) H′′′(1 + z) 4H′′ =(1+ z) ρm0 present value of the effective state parameter has been − constrained to w = 1.10 0.14 by Komatsu et al i (10) e0 in 2010 [6]. This implies− that± the present value of the deceleration parameter q0 (obtained in view of equation

2 (13)) lies within the range 1.36 q0 1.24. Note H 3 4 − ≤ ≤ − =Ωm0(1 + z) +Ωr0(1 + z) +Ωc . (11) that high redshift type Ia Supernovae has not been H0 taken into account which makes q0 more negative. Further, the range of effective state parameter and In the above equations dash (′) stands for derivative with correspondingly the deceleration parameter have been respect to the redshift parameter z, ρm0 is the present fixed in view of ΛCDM model only, and so these should value of matter density, Ωm0 and Ωr0 are the present values of matter and radiation density parameters re- not be treated as experimental data. spectively, while Ωc is the contribution of the higher or- 3 der curvature invariant terms R2 and R 2 to the density parameter which acts as the source of dynamical dark energy and is given by Before the end of its operation in 2013, the 9-year WMAP [46, 47] had measured the third acoustic peak in the temperature power spectrum (TT) with fair preci- 3 3 H 2 H′′ sion. As a result, much tighter constraints on the density Ωc = β 3(1 + z) − 2  H′  H parameters have been presented by Larson et al [7]. In r H2 2 (1 + z) 0 − H the context of the flat ΛCDM model, the total matter H 2 qH H4 density (which is the sum of the physical baryon density +(1 + z)2 ′ 7 ′ +4 12γ (z + 1) H − H − H2 and the cold dark matter density) has been constrained 0 2 +.0056 2 to Ωmh = 0.1334 .0055. Thus the density parameter 2H (z) H (z) 4H (z) − (z + 1) ′′ + ′ ′ in view of the Hubble parameter data lies within the H(z) H(z) ! − H(z) ! range 0.2197 Ωm 0.2734. Allowing tensor modes ≤ ≤ (12) in the context of ΛCDM model, the primordial power spectrum constraints the dark energy density parame- aa¨ ter to 0.726 ΩΛ 0.788, which is the same as ob- Additionally, the deceleration parameter q = a˙ 2 can ≤ ≤ be expressed in terms of the Hubble parameter− or the tained adding axion type iso-curvature perturbation [6]. Finally, curvaton type iso-curvature perturbation con- effective state parameter (we) as straints it in the limit 0.738 Ω 0.794. All these ≤ Λ ≤ H 3w +1 data together, restricts the matter density parameter q =(1+ z) ′ 1= e (13) H − 2 to 0.206 Ωm 0.274. Keeping all these parameters within the≤ specified≤ range, we are now in a position to which are useful to find numerical solution. present numerical solutions of the field equations.

4 4 Numerical solution: 1.0

To obtain H(z) as a solution of the ﬁeld equation (10) 0.5 containing up-to third derivative, it is required to set three boundary conditions viz. H0,H0′ ,H0′′. For this 0.0 1.0 1.0 ---> purpose, we undertake the following scheme. Setting q 0.5 0.8 the present values of the Hubble and deceleration pa- 0.0 0.6 -0.5 0.4 -0.5 rameters H0 and q0 by hand, H0′ is obtained in view of 0.2 -1.0 equation (13). H0′′ may then be found in view of equa- 0.0 0 2 4 6 8 10 1500 2000 2500 3000 tion (12), provided the coupling parameters β, γ and -1.0 the present value of the density parameter Ωc0 are set 0 200 400 600 800 1000 a-priori. z--->

Figure 1: The plot q versus z for β = 2.903 and γ =0.0001 (Case-I) depicts that the universe was in pure radiation era at z > 3200. Deceleration parameter falls off from the matter-radiation equality epoch to q 0.9 4.1 Case - I, [β = 2.903 γ = 0.0001, H0 = at z 1100 - the decoupling epoch. It falls even sharply≈ ≈ 0.074, Ωc0 =0.74, q0 = 0.5] thereafter and a Friedmann type (q 0.5) matter dom- − inated era is reached at around z ≈ 200. The deceleration parameter starts increasing≈ slowly from around In the present case, we set β = 2.903, H0 = 0.074 z = 20 and it is peaked (q = 1) at around z =2.5. Late and the limiting present value of the effective state time acceleration starts at around z =1.39. Thereafter parameter, w = 2 , which fixes q = 0.5, in it crosses the phantom divide line at around z = 1 and e0 − 3 0 − view of equation (13). Finally the density parameters makes a second transition out of it at z =0.5. 5 Ω =0.26 and Ω =8 10− are taken into account, m0 r0 × which set Ωc0 = 0.74. Thus we find H0′ = 0.037 and H0′′ = 0.103458. Using these parametric values, the trace equation− (10) is solved numerically and three in 4.2 Case - II, [β = 9.3 γ = 0.0001, H0 = one different plots of q versus z (Figure-1) have been 0.074, Ωc0 =0.74, q0 = 0.6] presented in both high, medium and low redshift regions. − Here, we increase the value of β substantially, so that lower value of the effective state parameter is admissi- Figure-1 depicts that at z > 3200 the universe was ble. To enunciate, we take β =9.3, H0 =0.074 and the 1 purely in radiation dominated era (q = 1, w = ) due present value of effective state parameter, we0 = 0.733, e 3 − to the presence of real photons which form CMBR today. which fixes q0 = 0.6, in view of equation (13). Finally − 5 From around z = 3200, the matter-radiation equality, the density parameters Ωm0 =0.26 and Ωr0 =8 10− × the deceleration parameter falls off to q 0.9 at the de- are taken into account, which set Ωc0 = 0.74. Thus we ≈ coupling era around z 1100 as shown in high redshift find H0′ = 0.0296 and H0′′ = 0.0609391. Using these − plot in figure-1 (right≈ inset). Thereafter, the decelera- parametric values, the trace equation (10) is again tion parameter q falls sharply with the redshift param- solved numerically and three in one different plots of q eter z to enter exact Friedmann type matter dominated versus z (Figure-2) have been presented in both high, era (q 0.5, we = 0) at around z = 200. The deceler- medium and low redshift regions, as before. ation parameter≈ falls a little below 0.4 and then starts increasing very slowly again at around z 20 reaching Figure-2 depicts the same behaviour as figure-1, viz. at 1 ≈ the peak with q = 1, we = 3 at around z 2.5, as z > 3200 the universe was purely in radiation domi- ≈ 1 is evident from the low redshift plot (left inset). Tran- nated era (q = 1, we = 3 ) due to the presence of sition to an accelerated phase starts around z 1.39. real photons which form CMBR today. From around Thereafter it crosses the phantom divide line at≈ around z = 3200, the matter-radiation equality, the decelera- z = 1 and makes a second transition to come out of it tion parameter falls off to q 0.83 at the decoupling at z =0.5. era around z 1100 as shown≈ in high redshift plot in figure-2 (right≈ inset). Thereafter, the deceleration pa-

5 1.0

0.5

0.0 1.0 1.0 1.0 0.5 0.8 0.5 0.8 0.0 --->

q - 0.6 0.5 0.6 0.0 -1.0 --->

---> 0.4

q - -0.5 0.4 1.5 q -0.5 -2.0 0.2 0.2 -2.5 -1.0 0.0 0 5 10 15 20 25 30 1500 2000 2500 3000 0.0 -1.0 0 5 10 15 20 25 30 1500 2000 2500 3000 z---> z--->

0 200 400 600 800 1000 z---> Figure 3: The plot q versus z for β = 0.22 and γ = 0.000001 (Case-III) depicts that the universe− was Figure 2: The plot q versus z for β =9.3 and γ =0.0001 in pure radiation era at z > 3200. Deceleration param- (Case-II) depicts that the universe was in pure radia- eter falls of from the matter-radiation equality epoch to q 0.77 at z 1100 - the decoupling epoch. It tion era at z > 3200. Deceleration parameter falls oﬀ ≈ ≈ from the matter-radiation equality epoch to q 0.83 at falls oﬀ even sharply thereafter and a Friedmann type z 1100 - the decoupling epoch. It falls even≈ sharply (q 0.5) matter dominated era is reached at around ≈ z ≈200. The deceleration parameter starts increasing thereafter and a Friedmann type (q 0.5) matter dom- ≈ inated era is reached at around z ≈ 250. The decel- slowly from around z = 25 and it is peaked (q = 1) eration parameter starts increasing≈ slowly from around at around z = 0.85. Late time acceleration starts at z = 20 and it is peaked (q = 1) at around z =3.2. Late around z =0.25. time acceleration starts at around z = 2. Thereafter it crosses the phantom divide line at around z = 1.5 and makes a second transition out of it at z =0.5.

5 and Ω = 8 10− , which set Ω = 0.777, one rameter q falls sharply with the redshift parameter z r0 × c0 to enter exact Friedmann type matter dominated era finds H0′ = 0.1216 and H0′′ = 1.37486. Using these parametric values,− the trace equation (10) is again (q 0.5, we = 0) at around z = 250. The deceleration≈ parameter falls a little below 0.4 and then starts solved numerically and the plots of q versus z are increasing very slowly again at around z 20 reaching presented (Figure-3) in both high, medium and low 1 ≈ redshift regions. the peak with q =1, we = 3 at around z 3.2, as is evident from the low redshift plot (left inset).≈ Transition to an accelerated phase starts around z 2. Thereafter ≈ it crosses the phantom divide line at around z =1.5 and Figure-3 as before, depicts that at z > 3200 the universe makes a second transition to come out of it at around 1 was purely in radiation dominated era (q =1, we = 3 ) z = 0.5. It is to be mentioned that larger value of β is due to the presence of real photons which form CMBR required to obtain lower present value of effective state today. From around z = 3200, the matter-radiation parameter. equality, the deceleration parameter falls off to q 0.77 at the decoupling era around z 1100 as shown in≈ high redshift plot (right inset). Thereafter,≈ the deceleration 4.3 Case - III, [β = 0.22 γ = 0.000001, parameter q falls sharply with the redshift parameter − z to enter exact Friedmann type matter dominated era H0 =0.076, Ωc0 =0.777, q0 = 2.6] − (q 0.5, we = 0) at around z = 200. The deceleration Interestingly enough, the same features as above are parameter≈ falls a little below 0.5 and then starts increas- observed taking even negative values of the coupling ing very slowly at around z 25 and reaches the peak parameter β. For example, choosing β = 0.22, with q = 1, w = 1 again at≈ around z 0.85, as is ev- − e 3 ≈ H0 = 0.076, the present value of the effective state ident from the low redshift plot (left inset). Transition parameter, w = 2.07 (q = 2.6). Finally, taking to an accelerated phase starts around z 0.25. e0 − 0 − ≈ into account the density parameters Ωm0 = 0.223

6 4.4 Case-IV: [β = 0.22, γ = 0, H0 = − 0.076, Ωc0 =0.794, q0 = 2.8] −

To show that the feature remains unaltered, we have 1.0 made little change in the data corresponding to case-III, in respect of Ωc0 and q0. With the above data, the 1.0 0.5 boundary conditions H0′ ,H0′′ have been found as before and the z versus q plot has been presented in ﬁgure-4. 0.5 The ﬁgure again depicts that after a long Friedmann- 0.0

---> 0.0 like matter dominated era with q 0.5, the deceleration q ≈ parameter starts increasing and the late time radiation -0.5 -0.5 like era (q = 1) is realized at z 0.8. Late time ≈ acceleration starts at around z = 0.25 (inset). Further, -1.0 matter-radiation equality is clearly visible in the high -1.0 0 5 10 15 20 redshift plot, since the deceleration parameter falls off 0 500 1000 1500 2000 2500 3000 from its value q =1at z = 3200, to q 0.79 at z = 1100. z---> ≈ The feature remains unaltered in the range 0.15 < − β < 0.24 which constraints 0.70 < Ωc0 < 0.81. Al- though− the chosen present value of deceleration parameter appears to be low, it does not make any problem Figure 4: The low redshift plot of q versus z for β = since as already mentioned, it is model dependent. The 0.22 (Case-IV) depicts identical feature as in case-III. feature remains unaltered even for γ < 0. For exam- The− only difference is that at low redshift, the decel- ple, setting γ = 0.000002, β = 0.22, H0 = 0.076 eration parameter is now peaked (q = 1) at z 0.8. − − and the present value of the effective state parameter, Universe then smoothly transits towards late time≈ ac- we0 = 2.2 (q0 = 2.8), together with the density pa- celeration starting at around z =0.25 (inset). − − 5 rameters Ω = 0.21 and Ω = 8 10− , which set m0 r0 × Ωc0 = 0.79, we find H0′ = 0.1368 and H0′′ = 1.5390. Except for the fact that the peak− q = 1, ie. the late time radiation era is realized at around z 0.94, the feature derivative of the extrinsic curvature tensor [48] or the remains unaltered. ≈ said tensor itself [49]. It might also be a scalar mode (the scalaron) obtained under scalar-tensor equivalence Thus, it has been possible to explain the history of via conformal transformation [50]. Here we shall discuss, cosmic evolution right from the radiation dominated era the fate of the present model under weak field limit fol- at z > 3200 till date, in the modified theory of gravity lowing both the methodology one-by-one. containing a linear term, a curvature squared term 3 together with (R 2 ) term in the presence of an ideal 5.1 Canonical formulation with a tensor fluid and CDM. A recent Friedmann-type radiation era mode 1 (q = 1, we = 3 ) is clearly the outcome of the curvature 3 In a series of articles Sanyal and Modak, Sanyal and term R 2 , since CMBR photons do not play any role at his co-workers had developed the formalisms of Boul- this epoch. ware [48] and Horowitz [49] to produce a canonical theory of Einstein-Hilbert action being modified by curvature squared term in Robertson-Walker minisuperspace model [51]-[55]. In particular, canonical formulation of 5 Weak energy Limit Einstein-Hilbert action being modified by scalar curvature squared term in Robertson-Walker metric appears At this stage it is important to discuss the behaviour in the literature [56] as. of f(R) theory of gravity in the weak field limit. f(R) theory of gravity gives rise to fourth derivatives in the A = h˙ πij + K˙ Πij N dtd3x, (14) field equations. To get rid of such complexity, canon- ij ij − H ical formulation is necessary under the introduction of Z an additional degree of freedom. This additional de- where, the basic variables hij and Kij are the metric gree of freedom might be a tensor mode obtained under on 3-space and extrinsic curvature while πij and Πij variation of the action with respect to the highest Lie are canonical momenta respectively. In the above, N is

7 the lapse function, while is the Hamiltonian. Here The canonical momenta are, we show that such canonicalH formulation is also possible 3 ˙ for an action containing R 2 term. For simplicity, we N 3z ˙ z˙ pz = Q˙ Q , pQ = z,˙ pN = Q, (21) drop out matter and curvature squared term and take − − N − 8N√z − −N up action (4), as and the Hamilton constraint equation is,

3 4 2 A1 = √ g d x [αR + β1R ]+ σ1 + σ2, (15) N˙ 3z ˙2 − H = Q˙ z˙ zQ˙ 2kN 2Q Z c − − N − 16N√z − 3 (22) where, β1 = βMP and σ1 = 2α K√hd x, σ2 = 2 4 8α N 3 3 3 + Q Nk√z. 2β1 Kf ′(R)√hd x are the Gibbons-Hawking-York 2 R 27β1 − 4 term and the boundary term required to supplement higherR order curvature invariant term respectively. Now, In view of the deﬁnitions of canonical momenta (21) 2 under the choice hij = a = z the Ricci scalar takes the ˙ 2 6 z¨ 2 k z˙N˙ ˙ N 3z ˙ form, R = N 2 ( 2z + N z 2zN ) and the above action pQpz = Qz˙ + zQ˙ + (23) now reads − N 8N√z

z¨√z √zz˙N˙ one obtains the following relation, A1 = + Nk√z 2N − 2N 2 ˙ 2 2 Z " ˙ N 3z ˙ 3z ˙ 3 (16) Qz˙ zQ˙ = pQpz + ˙ 2 − − N − 16N√z − 16N√z √3β1 z˙N 2 (24) + z¨ +2kN dt + σ1 + σ2. 3 2 2αN 2 − N  = pQpz + p ! − 16N√z Q  Under integration by parts the first terms in the above which allows to express the Hamiltonian constraint action yields a counter term that gets canceled with σ1 equation in terms of the phase space variables as and we are left with 3 8α2N 4 2 2 2 3 z˙ Hc = pQpz + pQ 2kN Q + 2 Q A = + Nk√z − 16N√z − 27β1 1 −4N√z (25) 3 Z 3 √ 2 (17) Nk z =0. ˙ − 4 √3β1 z˙N 2 + 2 z¨ +2kN dt + σ2. 2αN − N !  Now in order to express the Hamiltonian in terms of the basic variables, let us choose  Now introducing the auxiliary variable as z˙ ∂A ∂A ∂x˙ px 1 x = ,Q = = = and p = z˙ = Nx ˙ 2 N ∂z¨ ∂x˙ ∂z¨ N Q − − ∂L 3√3β1 z˙N 2 Q = = 2 z¨ +2kN , (18) (26) ∂z¨ 4αN − N ! to express equation (25) as one can express above action in the canonical form as 2 ˙ 2 4 3 2 8α 3 3 4 N 2 8α N 3 Hc = N xpz + x 2kpx + p k√z A1 = Qz¨ zQ˙ +2kN Q Q 2 x 2 16√z − 27β1 − 4 3 " − N − 27β1 Z (19) =0= N , 3z ˙2 3 H + Nk√z dt + σ2. (27) −16N√z 4 Now the first term in (19) is integrated by parts and It is now straightforward to express the action (19) as [sincez ˙ = Nx; therefore, we substitutez ¨ = Nx˙ + Nx˙ the total derivative term gets canceled with σ2. We are px then finally left with (the overall constant term has been in the first term of (19),z ˙ = Nx,Q = N in the second 3 3 3 z˙ absorbed in the action) and third terms, px = N Q in the fourth and x = N in the fifth] ˙ 2 4 ˙ N 2 8α N 3 A = Qz˙ zQ˙ +2kN Q 2 Q 3 − − N − 27β A = [zp ˙ z +xp ˙ x N ] dt d x Z " 1 (20) − H Z (28) 3z ˙2 3 + Nk√z dt. = h˙ πij + K˙ Πij N dtd3x, −16N√z 4 ij ij − H Z 8 which is the required canonical form, where in addition formally equivalent theory is dealt with, to get infor- to the three-space metric hij , the extrinsic curvature mation regarding the weak field limit of f(R) theory of tensor Kij play the vital role towards canonical formu- gravity. For example, the action lation. Apart from the two familiar mass-less spin-2 gravitons arising out of the linearized field energies of A = αf(R)√ gd4x (33) these particle excitations, the additional degree of free- − Z dom leads to a pair of massless spin-2 particles. There- fore, the model under consideration does not contain may be cast in the following Brans-Dicke form of action ghost degree of freedom. It is now required to check if without the help of conformal transformation action (4) admits Newtonian gravity so that it might satisfy solar test under weak field approximation which A = √ gd4x[φR V (φ)], (34) − − is valid at low energy limit. For this purpose, one can Z 2 always set γ = 0, since in no way R term influences the where, V (φ) = φχ f(χ) and χ = R. Clearly, one − solar test. In weak field approximation gµν = ηµν + hµν , observes that this analogy has been established at the where h 1. Retaining only linear terms in h we | µν | ≪ µν cost of vanishing Brans-Dicke parameter ω. Since it have is well-known that Brans-Dicke parameter should be 1 1 large enough and particulary ω , to satisfy solar R h and R h, where h = hµ . (29) → ∞ µν ≃ 2 µν ≃ 2 µ constraint, so under conformal transformation f(R) theory fails to satisfy solar test. For this reason f(R) The time-time component of field equation is theory of gravity had initially been ruled out. However, rigorous calculation of Newtonian limit of f(R) theory 1 1/2 3β1 3 of gravity taking into account correct analogy between R g R + β R (3R Rg )+ R− 2 00 − 2 00 1 00 − 00 2 f(R) and scalar-tensor theory, has proved that it is too 1 1 early to make final conclusion [60] as there are other RR R R;λ g RR + R R = T . − 2 ;λ 00 − ;0;0 2 ;0 ;0 00 techniques to establish scalar-tensor equivalence. One h i (30) such technique is Palatini formalism, in which canonical formulation reduces the field equations to second order In static background spacetime, equation (30) with only by considering metric and connection as independent linear term in hµν yields (terms containing derivatives variables. Although Palatini formalism is identical to of R have been discarded as they will contain third and the metric formalism for general theory of relativity, it fourth derivatives of Φ, which will have no counterparts differs by and large for higher order theory of gravity. in Poisson equation.) Particularly, scalar-tensor equivalence has been established with a non-zero Brans-Dicke parameter [61, 62]. ▽2h ρ. (31) 00 ≃ Thus Solar test might not fall short in this formalism. This raised interest to understand the situation deeply or considering next higher order term in hµν , equation under metric variation formalism also, which is our (30) gives present concern.

2 1 2 2 1 2 ▽ h00 +3β1 ▽ h ▽ h00 ▽ h ρ, (32) Another way to establish scalar-tensor equivalence is 2 − 6 ≃ possible under conformal transformation [50], which r again replaces higher (fourth) order theory to second (see appendix for detailed calculation). Since at low order, by the introduction of a scalar degree of free- energy limit Poisson equation is obtained, as in the case dom, dubbed as scalaron. In this technique, the action of general theory of relativity, so Newtonian gravity (33) under a conformal transformation gµν f,Rgµν = is valid at weak energy limit. This is one important −2ηφ → M technique to test the viability of f(R) theory of gravity e Pl gµν reads under weak energy approximation. 1 A = αR ∂ φ∂,µφ V (φ) √ gd4x (35) − 2 ,µ − − Z

5.2 Canonical formulation with a scalar 1 (Rf,R f) 2 − with η = √ and V (φ) = α f . In the process, mode - the Chameleon Mechanism − 6 ,R a technique dubbed as chameleon mechanism had been Canonical formulation of f(R) theory of gravity is also invoked. Here our aim is to check if under chameleon possible via scalar-tensor equivalence. Usually, such a mechanism our present model passes solar test. For this

9 purpose, following [57] we express action (4) as of magnitude smaller than that on earth. Correspond- ing Compton wavelength is λ(bulk) 0.42mm. Taking ≈ 4 case-II on the contrary, for which β = 9.3, the mass A = √ gd x αR + βMP F (R) − of the scalaron on earth is found to be mF (earth) Z 2 (36) 2 ≈ 3 γR 8.34 10 eV . Corresponding Compton wavelength is 2 F (R)= R + . × 7 βMP λ(earth) 2.36 10− mm, which is negligibly small to produce any≈ correction× to the Newtonian gravity. The and compute the trace of the corresponding ﬁeld equa- mass of the scalaron on the bulk, on the other hand is 4 tion as, mF (bulk) 2.638 10− eV , producing compton wavelength λ(bulk≈ ) 0×.75mm. It is important to mention 1 αR T ≈ F = 2F (R) RF + + . (37) that quantum stability bound gives 5 10−13eV as the ,R 3 − ,R βM 6βM × P P lower limit to the mass of the scalaron on bulk [59]. Al-

∂Ve though the Compton wavelength corresponding to the Expressing the above equation as F,R = , where, ∂F,R bulk is not appreciably large in either case, but we have Ve is the effective scalaron potential, the mass of the observed from the graphs (I through IV) that Fried- scalaron field may be calculated as mann solutions have been modified appreciably indicat- 2 ing possibility for long range interaction. It is important 2 ∂ Ve βMP F,R + α R mF = 2 = (38) to mention that the scalaron mass obtained considering ∂F 3MP βF,RR − 3 1 34 ,R R− theory of gravity is of the order of 10− eV . This value is too small and the corresponding Compton wave- In view of the definition of F (R) given in (36), the length is larger than the size of the universe and is ruled scalaron mass corresponding to the present model in Jor- out by quantum stability criterion [59]. On the contrary, dan frame reads, the mass of scalaron at bulk in the present model is at 1 par with quantum stability bound [59]. Although β < 0 R 4α 2 m = + √R (39) shows the same cosmological behaviour and can not be F 3 9βM P ruled out following weak energy limit studied in section The same above expression (39) may also be obtained (5.1), however, the scalaron mass is negative and there- considering the wave equation in the Einstein frame un- fore is plagued with tachyon or ghost degree of freedom. der conformal transformation, following [59] and then Thus, the present model passes the solar test with con- translating it back to Jordan frame, by multiplying fidence for β > 0. 2 mEinstein by f,R. However, for this purpose, we need 3 2 to take F (R) = R + 16πG(βMP R 2 + γR ), instead. 6 Perturbation about back- Now to study the viability of the chameleon mechanism we need to compare the masses of the scalaron both on ground curvature: earth and at the bulk (cosmological scale). For this purpose, we need to know the value of the Ricci scalar R It is believed that higher order theory of gravity modi- on earth and on the bulk. For the sake of simplicity, fies deeply the spectrum of perturbation. Therefore let we take help of the Friedmann equation to estimate R, us study the issue in brief. Taking R = Rb + Rp, where, which for pressureless dust (p = 0) reads Rb and Rp are the background and perturbed curvature scalars respectively, the dynamics of perturbed curva- ρ R = . (40) ture scalar has been evaluated by Nojiri and Odintsov 2 2 MP [4]. In the present case it reads (in the absence of R term i.e. γ = 0), 3 Now in air ρ = 10− g/cc, while in the bulk, it is 29 10− g/cc. In the unit c = 1, the value of the Ricci scalar R¨p +3HR˙ p + RpV (Rb)=0, (41) on earth and on the bulk may be calculated in view of 40 2 66 2 equation (40) as Re 10− eV and Rb 10− eV where, respectively. Therefore,≈ taking the value β≈= 2.903, as in case-I, we find the mass of the scalaron on earth to 1 R˙ 2 4 2 V (R )= b + R √R . (42) be m (earth) 1.49 103eV . Corresponding Compton b 2 b b F 2 Rb 9 − 9β1 ≈ × 7 wavelength is λ(earth) 1.32 10− mm, which is negligibly small to produce≈ any correction× to the Newtonian Equation (41) implies that perturbed space-time is os- gravity. In contrast, the mass of the scalaron on the cillatory with decaying amplitude, suggesting that the 4 bulk is m (bulk) 4.72 10− eV , which is seven order background space-time remains unaffected. F ≈ ×

10 7 A possible interpretation of tensor of an electromagnetic ﬁeld respectively, under ap- late time radiation era propriate choice of unit. Thus it can be shown that

µ µ 1 δ 3 Since everything is well behaved, it is therefore impor- A˙ = JµA + (AδA ) R (46) ;µ 2 − 4 tant to make a thorough study to understand the con- 3 2 µ µ σ sequence of gravity including R term on the late time where, A˙ = A ;σA . Tµν therefore looks very much cosmic evolution, particularly the nature of the graphs like an energy-momentum tensor equivalent to that of µ at low red-shift where the peak (q = 1), corresponding a source-free (Jµ = 0) or interaction free (JµA = 0) σδ to the late time radiation era is found. electro-magnetic field, satisfying the relation F Aσ = 1 √ ;δ √ 2 ( R) . Note that since R is constant on the space- σδ 7.1 The field equation like surface, so F Aσ is a time-like vector. With this understanding, one can now clearly observe that if the It is important to note that all the important aspects effective cosmological constant term Λe(R) dominates of higher order gravity have been explored only through at the early epoch, inflation would be realized. In the scalar-tensor equivalence under conformal transforma- middle, if the perfect fluid energy momentum tensor Tµν tion or using an auxiliary variable χ = R, as already dominates, then due to the presence of the interaction demonstrated. This reduces the theory to a minimally term (2α +3β1√R +4γR), a continuous transition from or non-minimally coupled scalar-tensor theory of gravity Friedmannn-like radiation dominated era (a √t with and the scalar is treated as a real scalar field. Likewise, 1 ∝ the effective state parameter we = ) to matter dom- here we reduce higher order theory under consideration 3 inated era (we 0) would be realized. At the later to linear gravity being non-linearly coupled to a tensor stage of cosmological→ evolution, if the effective electro- field, viz, the elctromagnetic field exhibiting and estab- magnetic energy-momentum tensor Tµν dominates, then lishing the equivalence. It is also important to note that one should expect yet another phase of radiation era no transformation is necessary for this purpose, rather 1 (we = 3 ). Finally, at the very late stage of cosmolog- one can simply cast the field equation (7) in the follow- ical evolution, if the effective electro-magnetic energy- ing interesting form momentum tensor Tµν falls off sharply at a much faster 1 rate than the effective cosmological constant Λe, so that Gµν = [Tµν +Λe(R)gµν + Tµν ] Λ again overtakes T , then an accelerated expansion √ e µν 2α +3β1 R +4γR might be realized. These facts have been demonstrated (43) in the spatially flat Robertson-Walker line element in the figures 1 through 4. where, Gµν is the Einstein’s tensor, Tµν is the energy- momentum tensor corresponding to matter Lagrangian. 3 Λ (R)= β1 R 2 + 9 √R γ (R +3R) acts as an 7.2 Gravitational wave equation: e − 2 2 − effective dynamical cosmological constant, being the In this section, we explore to understand the role of D’Alembertian operator. Finally, the last and the most the typical late time radiation era on cosmic evolu- T interesting term µν , given by tion. For this purpose, let us construct gravitational 1 wave equation assuming linear term in hµν in the left Tµν =3β1 √R;µ;ν (√R)gµν hand side of (43). Thus under the gauge condition − 4 µ 1 µ (44) (h h δ ),µ = 0, (43) effectively yields, 1 ν − 2 ν +4γ R;µ;ν gµν R . − 4 1 1 1 h hη = 3β g (√R);λ 2 µν − 2 µν − 1 4 µν ;λ is clearly traceless. Further, one can show trivially that, σδ 3 under the choice, FσδF = √R, β1 3 9 (47) 2 (√R) R 2 + (√R);λ g − ;µ;ν − 2 2 ;λ µν T σ 1 σδ i µν = Eµν = Fµ Fνσ FσδF (45) γ (Rg 4 (R g R)) + T = J − 4 − µν − ;µ;ν − µν µν µν The term (√R) is a symmetric tensor, so F σF where g = η + h , h < 1. In the wave equa- ;µ;ν µ νσ µν µν µν | µν | is also symmetric in view of equation (45) which holds tion (47), Jµν acts as the source term which produces for both the symmetric and antisymmetric nature of the the gravitational wave and its behavior depends on its tensor Fµν . Assuming it to be antisymmetric with the strength. More precisely, the amplitude of the gravita- choice Fµν = Aµ;ν Aν;µ, Fµν and Eµν may be in- tional wave should be calculated from the transverse- terpreted as the field− tensor and the energy-momentum traceless part of the space-space component of the

11 1 J1 early reionization on the baryonic components of the universe. If reionization is described as an instanta- 6 10 neous increment of the intergalactic medium (IGM) temperature, a key role is supposed to have been 104 played by Compton cooling at redshift z > 10, which counteracts any heating of the gas. A late reionization 100 is therefore required at zreion < 10 to suﬃciently reduce the number of luminous dwarf satellites around our

z Galaxy. The temperature at this epoch increases up 3.0 3.2 3.4 3.6 to 31.6 eV , which is sufficient to ionize hydrogen. The 1 4 absorpion spectra of SDSS (Sloan Digital Sky Survey) Figure 5: A plot of J 1 in eV , versus z (Case-II) shows a 1 quasars at z 6 indicate that the neutral fraction of peak at z 3.2 where, J 1 =1.3 107 eV 4, i.e. (J 1 ) 4 1 1 hydrogen (reionization∼ energy being 13.6 eV) increases T 60 eV≈, which is sufficient to× reionize intergalactic≃ significantly at z > 6 [63]-[68] and the UV spectrum of medium.≈ quasars implies that helium (reionization energy being 54.4 eV) is fully ionized only recently, at z 2.7. Since

1 ≈ J1 low energy CMBR photons (T 0.4 eV at z 1080) falls far below reionization energy,∼ it is usually assumed∼ that the ultraviolet radiation and mechanical energy 5 10 that preheated and reionized most of the hydrogen and helium in the IGM, ending the “dark ages”, are due to 1000 early generation (10

1 4 Figure 6: A plot of J 1 in eV , versus z (Case-IV) shows a couple of peaks at z 0.7712 and z 0.7721 where, 1 In subsection 7.1, we have shown that the modified 1 6 4 ≈ 1 4 ≈ J 1 =5.36 10 eV , i.e. (J 1) T 48 eV , which is theory of gravity under consideration, may be looked × ≃ ≈ sufficient to reionize intergalactic medium. upon as to induce an effective electro-magnetic field tensor Tµν in the field equation. This means other than scalar-tensor equivalence, higher order theory may also be looked upon as tensor-tensor equivalence and the energy-momentum tensor, which is the source of the photons corresponding to such a theory might well in- gravitational waves. Here, in figure-5, we therefore teract with the atoms to ionize them. The source of present a plot of space-space component J 1 versus the 1 the gravitational wave calculated from the transverse- redshift parameter z corresponding to the Case-II (ne- traceless part of the space-space component of the glecting the contribution of energy momentum tensor of energy-momentum tensor has been found to produce ideal fluid). The peak at z 3.2 of the source term is of 1 ≈ strong enough gravitational waves ( 60 eV, for β > 0, the order of (J 1 ) 4 T 60 eV and are located around 1 figure-5) to ionize both hydrogen and∼ helium (case-II) the peak of the deceleration≃ ≈ parameter q = 1. In figure- and ( 48 eV, for β < 0, figure-6) to ionize hydrogen 6, we present a plot of space-space component J 1 versus 1 (case-IV).∼ Therefore there is an indication that the ef- the redshift parameter z corresponding to case-IV. The fective electro-magnetic field might be responsible for peaks at z 0.7712 and z 0.7722 of the source term ≈ 1 ≈ reionizing IGM. So, at least from classical point of view, are of the order of (J 1 ) 4 T 48 eV and are located 1 modified theory of gravity shades some light in the issue around the peak of the deceleration≃ ≈ parameter q = 1. of reionization of the IGM. Indeed, for a definite claim in this respect, it is required to specify a mechanism for 7.3 Does late time radiation era reion- producing the first sources or energy injection into the izes IGM? standard model plasma in the form of the actual production of UV photons from the gravitational sector. Present discrepancy between the abundance of galac- For this purpose, it is required to solve the Schrödinger tic subhaloes predicted by N-Body simulations with equation in the presence of gravitational Waves, which those observed in the ‘Local Group’, suggests an is beyond the scope of the present work.

12 8 Conclusion: the space-space component of the energy-momentum tensor shows that its strength is sufficient to reionize f(R) theory of gravity has been taken seriously in recent both hydrogen and helium in the IGM. years, to explain late time cosmological evolution. Here an action containing a linear term a curvature squared Weak energy limit of the model under consideration has term and a three-half term together with baryonic also been established following canonical formulation of matter has been taken into account to describe the the model with tensor-tensor mode and scalar-tensor cosmological evolution. Starobinsky inflation is one of mode. The mass of the scalaron in the second case the minimal models which naturally explains inflation shows Newtonian correction is insignificant in the solar and reheating from geometry itself, without invoking a system, since the Compton wavelength is very small (of scalar field. In fact, Starbinsky inflation is so powerful the order of nanometre) while it falls within the limit that adding three Majorana fermions to the standard of quantum stability bound in the bulk. model, it is possible to explain neutrino oscillation, inflation, reheating, dark matter generation and baryon 3 In view of all these, we conclude that Modified theory asymmetry of the universe [76]. The importance of R 2 of gravity should be taken up even more seriously to term has already been established, since no other form understand if the primary investigations done here are of f(R) is admissible in view of Noether symmetry. relevant. 3 When such a term (R 2 ) is added to the Starobinsky term, rest of the history after initial stage of the cosmological evolution is explained naturally. A smooth and continuous transition from early radiation era 9 Appendix(Calculation for weak via matter-radiation equality (z 3200), decoupling energy limit): (z 1100) through to late time≈ cosmic acceleration ≈ after a long matter dominated era is clearly visible Field equation for action (4) reads, from the graphs. Additionally, a late time radiation era 3 1 has also been observed, which is of particular interest. R 2 R Rg + β R2 (3R Rg ) 3 µν − 2 µν 1 µν − µν Since field equation (43) shows that a part of R 2 term clearly acts like an effective energy-momentum 3 ;λ 1 + β g RR R R;λ (48) tensor of an electromagnetic field, therefore this late 2 1 µν ;λ − 2 ;λ time radiation era might be responsible for reionizing 3 1 3 β RR R R = R 2 T IGM. However, we have not presented a solid proof in − 2 1 ;µ;ν − 2 ;µ ;ν µν this connection by solving Schrödinger equation in the presence of gravitational wave equation. Nevertheless, The trace of the above equation being multiplied by 1 if gravitational wave corresponding to modified theory 2 gµν , leads to of gravity is responsible for reionization then, in the 1 5 1 9 2 3 present model it ends at around z 3 for β > 0 and gµν R gµν β1R + β1gµν R R ≈ − 2 − 2 4 (49) z 0.8 for β < 0, keeping all other cosmological data 9 1 3 ≈ β g R R;σ = R 2 Tg at par with observations. − 8 1 µν ;σ 2 µν Now, the (0,0) component of the difference of equations The seven year CMBR data has presented reionization 3 2 result whose profile is a smooth ramp in the redshift (48) and (49) when divided by R , finally gives space and the parameter ∆ changes the slope of the z 1 R 3 3 ramp about its midpoint in such a way as to preserve R + β R 2 3R g β g R− 2 (RR 00 1 00 − 2 00 − 4 1 00 total optical depth. Adding ∆z as a parameter to ;σ 3 3 1 the basic ΛCDM model and varying it in the range +R R ) β R− 2 RR R R ;σ − 2 1 ;0;0 − 2 ;0 ;0 0.5 < ∆z < 15, the redshift of reionization has been found to be zreion = 10.5 1.2 [7], using CAMB (Code 3 1 ± g R R;σ = T Tg for Anisotropies in the Microwave Background) [77]. −4 00 ;σ 00 − 2 00 Our result fits perfectly with such data as all the graphs (50) show reionization epoch starting at around zreion 10, (q > 0.5). The peak depicts the end of reionization≈ Assuming only linear term, one can use the following at z 3 for β > 0 and z 0.8 for β < 0, which fit relations in equation (28) earlier≈ data [71]-[75]. The≈ source of the gravitational 1 2 1 2 ;0 wave calculated from the transverse-traceless part of R00 = h00, R = h, R;0R =0,g00 = (1+h00) 2∇ 2∇

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