Brane inflation in background supergravity Sayantan Choudhury1 and Supratik Pal1,2 ∗ † 1Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700 108, India 2Bethe Center for Theoretical Physics and Physikalisches Institut der Universita¨t Bonn, Nussallee 12, 53115 Bonn, Germany We propose a model of inflation in the framework of brane cosmology driven by background supergravity. Starting from bulksupergravity we construct theinflaton potential on thebraneand employ it to investigate for the consequences to inflationary paradigm. To this end, we derive the expressionsfortheimportantparametersinbraneinflation,whicharesomewhatdifferentfromtheir counterparts in standard cosmology, using the one loop radiative corrected potential. We further estimate the observable parameters and find them to fit well with recent observational data by confrontingwithWMAP7usingCAMB.Wealsoanalyzethetypicalenergyscaleofbraneinflation with our model, which resonates well with present estimates from cosmology and standard model of particle physics. 2 1 PACSnumbers: 98.80.-k;98.80.Cq;04.50.-h 0 2 n I. INTRODUCTION a J Investigations for the crucial role of Supergravity in explaining cosmological inflation date back to early eighties 5 of the last century (for two exhaustive reviews see [1] and [2] and references therein). One of the generic features ] of the inflationary paradigm based on SUGRA is the well-known η-problem, which appears in the F-term inflation h due to the fact that the energy scale of F-term inflation is induced by all the couplings via vacuum energy density. t - Precisely, in the expression of F-term inflationary potential a factor exp(K/M ) appears, leading to the second p PL slow roll parameter η 1, thereby violating an essential condition for slow roll inflation. The usual wayout is to e ≫ h impose additional symmetry to the framework. One such symmetry is Nambu-Goldstone shift symmetry [3] under [ which Ka¨hler metric becomes diagonal which serves the purpose of canonical normalization and stabilization of the volume of the compactified space. Consequently, the imaginary part of the scalar field gives a flat direction leading 4 to a successfulmodel ofinflation. An alternativeapproachis to apply noncompactHeisenberg grouptransformations v of two or more complex scalar fields where one can exploit Heisenberg symmetry [4] to solve η-problem. The role of 6 0 Ka¨hler geometry to solve η-problem in the context of N=1 SUGRA under certain constraints can be found in [5]. 2 Of late the idea of braneworlds came forward [6]. From cosmological point of view the most appealing feature of 4 branecosmologyisthatthe4dimensionalFriedmannequationsaretosomeextentdifferentfromthestandardonesdue 2. tothenon-trivialembeddingintheS1/Z2orbifold[7]. Thisopensupnewperspectivestolookatthenatureingeneral 0 and cosmology in specific. To mention a few, the role of the projected bulk Weyl tensor appearing in the modified 1 Friedmann equations has been studied extensively for metric-based perturbations [8], density perturbations on large 1 scales[9],curvatureperturbations[10]andSachs-Wolfeeffect[11],vectorperturbations[12],tensorperturbations[13] : v and CMB anisotropies [14]. Brane inflation in the above framework has also been studied to some extent [15–17]. i Apart from these phenomenological approaches, some other approaches which are more appealing in dealing with X fundamental aspects such as possible realization in string theory can be found in [18–21]. For example, an apparent r conflict between self-tuning mechanism and volume stabilization has been shown in [19], subsequently, this problem a hasbeenresolvedin[20]wherethecredentialsofthedilatonicfieldinprovidinganaturalexplanationfordarkenergy by an effective scalar field on the brane has been demonstrated using self-tuning mechanism in (4+2) dimensional bulk space time. The role of the axions as quintessential candidates has been revealed in [21]. In the Randall-Sundrum two-brane scenario [6] where the bulk is five dimensional with the fifth dimension com- pactified on the orbifold S1/Z of comovingradius R, the separationbetween the two branes give rise to a field – the 2 so-called radion – which plays a crucial role in governing dynamics on the brane. The well-known Goldberger-Wise mechanism[22] leading to severalinteresting ideas dealwith differentissues relatedto radion. Subsequently, in order toincorporateobservationallyconstraintcosmologyofthebrane,afinetuningbetweenthebranetensionofthevisible andinvisiblebranehasbeenproposed[23]. Ithasbeenpointedoutin[24,25]howtheradioncoupledwithbulkfields maygive riseto aneffective inflatonfield onthe brane. Inthe samevein, weconstructthe braneinflatonpotentialof our consideration starting from 5D SUGRA. In brane inflation the modified Friedman equations lead to a modified ∗ Electronicaddress: [email protected] † Electronicaddress: [email protected] 2 versionof the slow roll parameters [7]. So, by construction, η-problem is smoothened to some extent by modification of Friedmann equations on the brane [17, 26]. In a sense, this is a parallel approach to the usual string inflationary framework where η-problem is resolved by fine-tuning [27]. As it will appear, there is still some fine-tuning required inbraneinflation,whicharisesvia anew avataroffive-dimensionalPlanckmassbut itis softenedto someextentdue to the modified Friedman equations. As we will find in the present article the proposed model of brane inflation matches quite well with latest obser- vational data from WMAP [28] and is expected to fit well with upcoming data from Planck [29]. To this end, we explicitlyderivethe expressionsfordifferentobservableparametersfromourmodelandfurtherestimatetheirnumer- ical values finally leading to confrontation with observation using the publicly available code CAMB [30]. We have also analyzed the typical energy scale of brane inflation and found it to be in good agreementwith present estimates of cosmologicalframeworks as well as standard model of particle physics. II. MODELING BRANE INFLATION Let us consider an effective N = 1,D = 4 SUGRA inflationary potential in the brane derived from N = 2,D = 5 SUGRA in the bulk. How we have arrived at an effective N = 1,D = 4 SUGRA in the brane starting from N =2,D =5SUGRAinthebulkandthesubsequentformoftheloopcorrectedpotentialstatedineqn(2.1)hasbeen discussed in details in the Appendix. For convenience, let us express the one loop corrected renormalizable potential in terms of inflationary parameters as 4 φ φ V(φ)=∆4 1+ D +K ln , (2.1) 4 4 M M " (cid:18) (cid:18) (cid:19)(cid:19)(cid:18) (cid:19) # where we introduce new constants defined by ( C4 is negative in tree level) K4 = 29π∆24MC424, D4 =C4− 251K24. It is the Coleman Weinberg potential [31] ,[32] , provided the coupling constant satisfies the Gellmann-Low equation in the context of Renormalizationgroup [33],[34]. Here the first term is constant and physically represents the energy scale of inflation (∆). 4.´10-13 3.´10-13 4M 2.´10-13 M n iΦLI 1.´10-13 H V 0 -1.´10-13 -2.´10-13 0.0 0.2 0.4 0.6 0.8 1.0 1.2 ΦHinML FIG. 1: Variation of one loop corrected potential(V(φ)) versus inflaton field (φ) Figure(1)representstheinflatonpotentialfordifferentvaluesofC ,D andK . Fromtheobservationalconstraints 4 4 4 thebestfitmodelisgivenbythe range 0.70<D < 0.60sothatwhiledoingnumericalsweshallrestrictourselves 4 − − to this rangeofD . Inwhatfollowsourprimaryintentionwillbe toengageourselvesinmodeling braneinflationand 4 to search for its pros and cons with the above potential (2.1). We shall indeed find that brane inflation with such a potential successfully explains the CMB observations and thus leads to a promising model of inflation. Asalreadymentioned,themostappealingfeatureofbranecosmologyisthatthe4dimensionalFriedmannequations are to some extent different from the standard ones due to the non-trivial embedding in the S1/Z manifold [7]. At 2 highenergyregimeonecanneglectthecontributionfromWeyltermandconsequently,thebraneFriedmannequations are givenby [7, 35] H2 = 8πV 1+ V . The modified Freidmann equations,along with the Klein Gordonequation, 3M2 2λ PL lead to new slow roll conditions and new expressions for observable parameters as well [7, 35]. For convenience (cid:2) (cid:3) 3 throughout the analysis we define the following global functions of the inflaton field 4 L(φ)= 1+ αS(φ) , T(φ)=[1+αS(φ)], S(φ)= 1+ D +K ln φ φ , 2 { 4 4 M } M U(φ)=(cid:2)(K +4D(cid:3))+4K ln φ , E(φ)= (7K +(cid:20) 12D )+12K ln(cid:16) φ(cid:17) (cid:16), (cid:17) (cid:21) 4 4 4 M 4 4 4 M (2.2) F(φ)=h(26K +24D )+24K(cid:16) ln(cid:17)iφ , J(φh)= (50K +24D )+24K(cid:16) ln(cid:17)i φ , 4 4 4 M 4 4 4 M P˜¯(φ)=h [1+2αS(φ)L(φ)] 2αS((cid:16)φ)L(cid:17)(φi)sinh−1([2hαS(φ)L(φ)])−1/2 (cid:16) (cid:17)i − with α =∆4/λ. Incorpporating the potential of our consideration from Eq (2.1) the slow roll parameters turn out to be M2 V′ 2 1+ V U2(φ)T(φ) φ 6 ǫ = PL λ = , (2.3) V 16π V ! (1+ 2Vλ)2 2S2(φ)L2(φ)(cid:18)M(cid:19) M2 V′′ 1 E(φ) φ 2 η = PL = , (2.4) V 8π V !(1+ 2Vλ) S(φ)L(φ)(cid:18)M(cid:19) M4 V′V′′′ 1 U(φ)F(φ) φ 4 ξ = PL = , (2.5) V (8π)2 V2 !(1+ 2Vλ)2 S2(φ)L2(φ)(cid:18)M(cid:19) M6 (V′)2V′′′′ 1 U2(φ)J(φ) φ 6 σ = PL = , (2.6) V (8π)3 V3 (1+ V )3 S3(φ)L3(φ) M 2λ (cid:18) (cid:19) 1.0 1.0 0.8 0.5 0.6 -Εv ¤-Ηv 0.4 1 1 0.0 0.2 0.0 -0.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Φ , Φ FIG.2: (I)Variationofthe1-ǫV vsinflatonfieldφforC4 =−0.68,(II)Variationofthe1-|ηV|vsinflatonfieldφforC4 =−0.68 Figures(2)depicthowthefirsttwoslowrollparametersvarywiththeinflatonfieldfortheallowedrangeofD and 4 they give us a clear picture of the starting point as well as the end of the cosmic inflation. Nevertheless, Figure (2) further reveals that the η-problem is smoothened to some extent in brane cosmology. However, we are yet to figure out if there is any underlying dynamics that may lead to the solution of this generic feature of SUGRA. The number of e-foldings are defined in brane cosmology [7] for our model as a(t ) 8π φi V V f N = 1+ dφ a(t ) ≃ M2 V′ 2λ i PL Zφf (cid:18) (cid:19)(cid:18) (cid:19) M2 1 α 1 1 D αD2 1+ + 4 (1+α)(φ2 φ2)+ 4 (φ6 φ6) (2.7) ≃ U "2 2 φ2f − φ2i! 2M4 i − f 12M8 i − f # (cid:16) (cid:17) which, in the high energy regime, reduces to N αM2 1 1 . Here φ and φ are the corresponding values of ≃ 4|U| φ2i − φ2f i f the inflaton field at the start and end of inflation. h i Let us now engage ourselves in analyzing quantum fluctuation in our model and its observational imprints via primordialspectra generatedfrom cosmologicalperturbation [36]. In brane inflation the expressions for amplitude of 4 the scalar perturbation, tensor perturbation and tensor to scalar ratio [7] ,[17],[37] are given by 512π V3 V 3 M2αλS3(φ )L3(φ ) ∆2 1+ = ⋆ ⋆ , (2.8) s ≃ 75MP6L "(V′)2 (cid:20) 2λ(cid:21) #k=aH 75π2U2(φ⋆)(φ⋆)6 ∆2 32  V 1+ 2Vλ  = λα S(φ⋆)L(φ⋆), (2.9) t ≃ 75MP4L "q1+ 2λV 1+ 2Vλ − 2λV (cid:2)1+ 2Vλ(cid:3) sinh−1"q2λV(11+2Vλ)##k=aH 150π2M4 P˜¯(φ⋆)  (cid:0) (cid:1) (cid:0) (cid:1)  ∆2 8(φ )6U2(φ ) r =16 t ⋆ ⋆ . (2.10) ∆2s ≃ M6S2(φ⋆)L2(φ⋆)P˜¯(φ⋆) Here and throughout the rest of the article φ represents the value of the inflaton field at the horizon crossing and ⋆ all the global function defined in eqn(2.2) is evaluated at the horizon crossing. 70 -6.4 60 50 -6.6 40 N 30 lnDH ¤Ls-6.8 20 -7.0 10 0 -7.2 0.2 0.4 0.6 0.8 1.0 ΦHinML -10.3 -10.2 -10.1 -10.0 -9.9 -9.8 -9.7 -9.6 lnH Αs¤L FIG. 3: (I)Variation of the number of e-folding(N) vs inflation field (φ)(measured in the units of M), (II)Variation of the logarithmic scaled amplitude of the scalar fluctuation (ln(∆s)) vs logarithmic scaled amplitude of the running of the spectral index (ln(|αs|)) Figure(3(I)) represents a graphical behavior of number of e-folding versus the inflaton field in the high energy limit for different values of D and the most satisfactory point in this context is the number of e-folding lies within 4 the observational window 56 < N < 70. The end of the inflation leads to the constraint α = 2 (E )32 which is (U ) | | | | requiredfor numericalestimations. Herefigure(3(II)) representsthe logarithmicallyscaledplots ofthe physicalsetof parameter (∆ ,α )for different values of D . The plots themselves present good fit with observations. s s 4 Further, the scale dependence of the perturbations, described by the scalar and tensor spectral indices, as follows [38],[16] d(ln(∆2)) 2E(φ ) φ 2 3U(φ )T(φ ) φ 6 n 1= s (2η⋆ 6ǫ⋆)= ⋆ ⋆ ⋆ ⋆ ⋆ , s− d(ln(k)) ≃ V − V S(φ )L(φ ) M − S2(φ )L2(φ ) M ⋆ ⋆ (cid:18) (cid:19) ⋆ ⋆ (cid:18) (cid:19) d(ln(∆2)) 3U2(φ )T(φ ) φ 6 n = t 3ǫ⋆ = ⋆ ⋆ ⋆ . (2.11) t d(ln(k)) ≃− V −2S2(φ )L2(φ ) M ⋆ ⋆ (cid:18) (cid:19) where d(ln(k))=Hdt. Here one cancheck that [39] the validity of the consistency condition r =24ǫ =24ǫ⋆; n = V V t 3ǫ 3ǫ⋆ = r. − V ≃− V −8 The expressions for the running of the scalar and tensor spectral index in this specific model with respect to the logarithmic pivot scale at the horizon crossing are given by 5 α =(16ηǫ 18ǫ2 2ξ) s − − 8E(φ )U2(φ )T(φ ) φ 8 2F(φ)U(φ ) φ 4 9U4(φ )T2(φ ) φ 12 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ = , (2.12) S3(φ )L3(φ ) M − S2(φ )L2(φ ) M − 2S4(φ )L4(φ ) M ⋆ ⋆ (cid:18) (cid:19) ⋆ ⋆ (cid:18) (cid:19) ⋆ ⋆ (cid:18) (cid:19) 3E(φ )U2(φ )T(φ ) φ 8 9U4(φ )T2(φ ) φ 12 α =(6ǫη 9ǫ2)= ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ , (2.13) t − S3(φ )L3(φ ) M − 4S4(φ )L4(φ ) M ⋆ ⋆ (cid:18) (cid:19) ⋆ ⋆ (cid:18) (cid:19) One can also calculate the running of the fourth slow roll parameter as dσ = (ǫσ 2ησ), but its numerical d(ln(k)) − value turns outto be too smallto be detected evenin nearfuture for which it cantreatedas consistency conditionin brane. To estimate five dimensional Planck mass from the observationalparameters we use the relation √8πM =M = PL M53 3 . and Eq (2.8) which leads to √λ 4π q 800π4∆2U2(φ ) M = 6 s ⋆ φ . (2.14) 5 sαS3(φ⋆)L3(φ⋆) ⋆ Finally using the thermodynamic definition of density at the time of reheating ρ(t ) = π2N⋆Tb4rh in the inflaton reh 30 decay width Γ =3H(Tbreh)=3 ρ(treh) 1+ ρ(treh) ∆6 we have estimated the reheating temperature in total 3M2 2λ ≃ (2π)3M5 r the braneworld in terms of the five dimensionhal Planck miass as 3 5 M3 64M4π2Γ2 Tbreh = 5 4 1+ total 1 s4π2rN∗ M vu"s 9M56 − # u (2.15) t 2250∆2U2(φ )φ3 2M4Γ2 αS3(φ )L3(φ ) = 4 s ⋆ ⋆ 4 1+ total ⋆ ⋆ 1 , sN∗M2αS3(φ⋆)L3(φ⋆)vu"s 225π2∆2sφ6⋆U2(φ⋆) − # u t where N⋆ is the effective number of particles incorporating the relativistic degrees of freedom. III. PARAMETER ESTIMATION A. Direct numerical estimation C4 α λ φf φi N φ⋆ ∆2s ∆2t ns nt r αs αt M5 Tbreh ≃D4 ×10−14M4 M M M ×10−9 ×10−14 ×10−5 ×10−5 ×10−3 ×10−6 ×10−3M ×10−8M 0.147 70 0.158 3.126 0.951 -4.352 2.176 -0.798 -2.125 -0.70 17.389 2.553 1.017 0.158 60 0.173 1.835 6.803 0.941 -7.412 3.706 -1.142 -4.323 11.792 3.119 0.164 56 0.180 1.440 0.936 -9.447 4.723 -1.345 -5.975 0.150 70 0.161 2.902 0.951 -4.352 2.176 -0.798 -2.125 -0.65 16.757 2.632 1.036 0.161 60 0.176 1.704 6.317 0.941 -7.412 3.706 -1.142 -4.323 11.865 3.133 0.167 56 0.184 1.327 0.936 -9.447 4.723 -1.345 -5.975 0.153 70 0.165 2.679 0.951 -4.352 2.176 -0.798 -2.125 -0.60 16.099 2.758 1.057 0.165 60 0.180 1.573 5.831 0.941 -7.412 3.706 -1.142 -4.323 11.944 3.149 0.170 56 0.187 1.234 0.936 -9.447 4.723 -1.345 -5.975 TABLEI:Different observational parameters related tothecosmological perturbation for ourmodel ofinflation includingone loop radiative correction Table I represent numerical estimation for different observational parameters related to the cosmological pertur- bation as estimated from our model. Here a “ ” implies “in units of”. It is worthwhile to point out to the salient × features of those parameters in the above table as obtained from our model. 6 H0 τReion Ωbh2 Ωch2 TCMB t0 zReion Ωm ΩΛ Ωk ηRec η0 km/sec/MPc K Gyr Mpc Mpc 71.0 0.09 0.0226 0.1119 2.725 13.707 10.704 0.2670 0.7329 0.0 285.10 14345.1 TABLE II:Input parameters in CAMB TABLE III:Outputparameters from CAMB The observable parameters help us have an estimation for the brane tension to be λ (1MeV)4 provided • energyscale ofthe inflationis inthe vicinityofGUT scaleandexactlyit is ofthe ordero≫f0.2 1016GeV which × resolves Polonyi problem [40] and Gravitino problem [41]. The scalar power spectrum corresponding to different best fit values of D mentioned above is of the order of 4 • 5 105 and it perfectly matches with the observational data [28]. × The scalar spectral index for lower values of N 55 are pretty close to observational window 0.948< n < 1 s • → [28]whereasforhighervaluesofN 70thislieswellwithinthe window. Thusthissmallobservationalwindow → reveals that N 70 is more favoredin brane cosmology compared to its lower values. ≈ Thoughthe tensor to scalarratio as estimatedfromour model is wellwithin its upper bound fixedby WMAP7 • [28] (r < 0.45 at 95% C.L.), thereby facing no contradiction with observations, its value is even small to be detected in WMAP [28] or the forthcoming Planck [29]. For more discussion see [42]. For our model running of the scalar spectral index α 10 3 which is quite consistent with WMAP3 [43]. s − • Also, the running ofthe tensor spectralindex α 6 ∼10−6 may serve as an additionalobservable parameter t − ∼− × to be investigated further. Five dimensional Planck mass turns out to be M (11.792 11.944) 10 3M which is the prime input 5 − • ∼ − × for the estimation of brane reheating temperature as shown in eqn(2.15). For our model it is estimated as Tbreh (3.119 3.149) 10 8M and clearly depicts the deviation from standard cosmology. − ∼ − × B. Data analysis with CAMB In this contextwe shall make use of the cosmologicalcode CAMB[30] in order to confrontour results directly with observation. To operate CAMB, the values of the initial parameters associated with inflation are taken from the TableIforD = 0.60. AdditionallyWMAP7datasetinΛCDMbackgroundhasbeenusedinCAMBtoobtainCMB 4 angular power sp−ectrum at the pivot scale k0 = 0.002 Mpc−1. Table II and tableIII shows input from the WMAP7 dataset and the output obtained from CAMB respectively. CMB TT Angular Power Spectrum CMB TE Angular Power Spectrum CMB EE Angular Power Spectrum 6000 150 50 Best fit Best fit Best fit WMAP data set WMAP data set WMAP data set 5000 100 40 4000 50 30 2πµl(l+1)C/2[K]l 3000 2πµl(l+1)C/2[K]l 0 2πµl(l+1)C/2[K]l 20 2000 -50 10 1000 -100 0 0 -150 -10 1 10 100 1000 0 100 200 300 400 500 600 700 800 0 200 400 600 800 1000 l l l FIG. 4: Variation of CMB angular power spectrum (a)CTT, (b)CTE and (c)CEE for best fit and WMAP seven years data l l l with themultipoles l for scalar modes The curvature perturbation is generateddue to the fluctuations in the inflaton and at the end of inflation it makes horizon re-entry creating matter density fluctuations, which is the origin of the structure formation in Universe. In Fig.4(a)-Fig4(c) we confront CAMB output of CMB angular power spectrum CTT, CTE and CEE for best fit with l l l WMAP seven years data for the scalar mode. From Fig.4(a) we see that the Sachs-Wolfe plateau [44] obtained from 7 CMB TT Angular Power Spectrum CMB TE Angular Power Spectrum 700 0.5 Best fit Best fit 600 0 -0.5 500 -1 2πµl(l+1)C/2[K]l 340000 2πµl(l+1)C/2[K]l -1-.25 200 -2.5 100 -3 0 -3.5 1 10 100 1000 0 100 200 300 400 500 600 700 l l CMB EE Angular Power Spectrum CMB BB Angular Power Spectrum 0.12 0.08 Best fit Best fit 0.07 0.1 0.06 0.08 0.05 2πµl(l+1)C/2[K]l 0.06 2πµl(l+1)C/2[K]l 00..0034 0.04 0.02 0.02 0.01 0 0 0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 l l FIG. 5: Variation of CMB angular power spectrum (a)CTT, (b)CTE, (c)CEE and (d)CBB with the multipoles l for tensor l l l l mode our model is almost flat confirming a nearly scale invariant spectrum. For larger value of the multipole l, CMB anisotropy spectrum is dominated by the Baryon Acoustic Oscillations (BAO) [45] giving rise to several ups and downs in the spectrum. Also the peak positions are sensitive on the dark energy and other forms of the matter. In Fig.4(a) the first and most prominent peak arises at l = 221 at a height of 5818µK2 followed by two equal height peaks at l = 529 and l = 822. This is in good agreement with WMAP7 data for ΛCDM background apart from the two outliers at l = 21 and l = 42. The gravitational waves generated during inflation also remain constant on super Hubblescaleshavingsmallamplitudeswhichdieoffveryrapidlyduetosmallerwavelengththanhorizon. Sothesmall scale modes have no impact in the CMB anisotropyspectrum only the large scale modes have little contribution and this is obvious from Fig.5(a)-Fig.5(d) where we have plotted the CAMB output of CMB angular power spectrum CTT, CTE, CEE and CBB for best fit with WMAP7 data for the tensor mode. Thus, from the entire data analysis l l l l with CAMB, it turns out that our model confronts extremely well with WMAP7 dataset and leads to constrain the best fit value of the parameter D at 0.60. 4 − IV. DYNAMICAL SIGNATURE OF THE MODEL Letusnowengageourselvesinfindingoutthedynamicalsignatureofthemodelfromthefirstprinciple. Precisely,we areinterestedto obtaina solutionofthe modified FriedmanequationandKlein-Gordonequationin branecosmology with our proposed model. Under slow-roll approximations the inflaton field as a function of cosmic time can be expressed as φ(t)= M2 Φ˜¯(f) G¯t v 1 1+ 4D4 , (4.1) √2D4r − uu −v M4 Φ˜¯(f) G¯t 2 h iuu uu −  t t h i  8 where G¯ = 2U√2λ, Φ˜¯(f) = 1 D4φ4fe 1 + G¯t . It may be noted that in the high energy limit, the above √3M3 φ2f M4 − f (cid:16) (cid:17) 1 equation(4.1) reduces to a much tractable form φ(t)=φ 1+ 2Uφ2f 2λ(t t ) −2. f M3 3 − f (cid:20) q (cid:21) 1.0 1.´10-6 0.8 8.´10-7 ML ML tinΦHLH0.6 HtinHLH6.´10-7 4.´10-7 0.4 2.´10-7 0.2 0 7.37´109 7.38´109 7.39´109 7.4´109 7.41´109 7.42´109 7.43´109 7.37´1097.38´1097.39´1097.4´1097.41´1097.42´1097.43´109 t t FIG. 6: (I) Variation of theinflaton field (φ) with time(t), (II) Variation of theHubbleparameter (H(t))with time(t) Figure (6(I)) shows the evolution of the inflaton field under high energy approximation which shows a smooth increasing behavior of the inflaton field with respect to the inflationary time scale where the span of the scale are within the window t < t < t . In figure (6(II)) the evolution of the Hubble parameter shows deviations from the i f de-Sitter as given by the bending of the plots towards the end of inflation which leads to physically more realistic scenario so as to fit with observational data as demonstrated earlier. Substituting equation(4.1) in the modified Friedman equation in brane for our model we obtain H(t)= λ α 2+ M4 Φ˜¯(f) G¯t 2 1 1+ 4D4 (4.2) r6M  2D4 −  −v M4 Φ˜¯(f) G¯t 2  h i  uu −    t h i  which shows the time evolution as well as the susceptance of Hubble parameter in the context of brane. Consequently,the solutionofthe modifiedFriedmanequation,after rearrangingterms,givesriseto thescalefactor as follows λ α B˜ C˜ a(t)=a(t )exp 2(t t )+A˜(t t )+ (t3 t3) (t2 t2) I˜(t) (4.3) f "r6M " − f − f 3 − f − 2 − f − ## where I˜(t) = t dt (A˜+B˜t2 C˜t+1)2 1 , A˜ = M4Φ˜¯(f),B˜ = G¯2M4, C˜ = Φ˜¯(f)G¯M4. Thus the scale factor can tf − − 2D4 2D4 D4 r be obtained anRalyticahlly except for the integranid I˜(t), and it readily shows the deviation from the standardde Sitter model. However, the above form of the scale factor (4.3) is more or less sufficient to study the dynamical behavior, as represented in Figure(6(II)). As a matter of fact, the leading order contribution from Hubble parameter and the scale factor are indeed closed to de Sitter with the parameters involving brane cosmology. V. ANALYSIS OF THE ENERGY SCALE OF BRANE INFLATION Letusnowestimatethetypicalscaleofinflationinbranecosmologywiththepotentialofourconsideration. Forthis we shall make use of two initial conditions, namely, initial time t =0.737 1010M 1 and a(t )=0.369 10 1M 1. i − i − − Consequently, for N = 70 we have a(t ) = 0.929 1011M 1. Now taking×leading order contribution fr×om Eq (4.3) f − × the time corresponding to the horizon exit and re-entry can be obtained as 1 1 8D [(φ )2+2M4] t =t + 1 Φ˜¯(f) ± − 4 ⋆ , (5.1) ⋆ f G¯  − h p M4 i   9 with t =t + NM 6. Using Eq (5.1), Eq (4.1) and Eq (2.4) energy scale of brane inflation can be expressed as f i α λ q 2Eλφ2 ∆ vu4  f . (5.2) ≈ uuu|ηV|M2 1+ 2MUφ32f 23λ(t−tf)  ut (cid:20) q (cid:21) 1.4´1016 1.2´1016 1.´1016 nL vi8.´1015 e G DH6.´1015 4.´1015 2.´1015 0 0.0 0.2 0.4 0.6 0.8 1.0 Ηv¤ FIG. 7: Variation of the energy scale of inflation (∆) vs |ηV| including two roots of the horizon crossing time for the best fit model Figure (7) shows the energy scale of inflation (∆) versus the magnitude of the second slow roll parameter(η ) for v | | different values of the constant D including two feasible roots of horizoncrossing. From the figure it is obvious that 4 for two feasible roots of time corresponding to the horizon crossing an allowedregion with finite band-width appears for our proposed model. The above figure further reveals that the typical energy scale of brane inflation with our proposed model is ∆ 2 1015GeV which is supported from cosmologicalas well as particle physics frameworks. ≃ × VI. SUMMARY AND OUTLOOK Inthisarticlewehaveproposedamodelofinflationinbranecosmology. Wehavedemonstratedhowwecanconstruct an effective 4D inflationary potential starting from N =2,D =5 supergravity in the bulk which leads to an effective N = 1,D = 4 supergravity in the brane. After that we have engaged ourselves in analyzing radiative corrections of the tree level potential and the effective potential calculated from one loop correction has then been employed in estimating the observable parameters, both analytically and numerically, leading to more precise estimation of the quantities and confronting them with WMAP7 dataset using the publicly available code CAMB, which reveals consistency of our model with latest observations. The increase in precision level is worth analyzing considering the advent of more and more sophisticated techniques, both in WMAP [28] and in forthcoming Planck [29] data. WehavealsosolvedthemodifiedFriedmannequationsonthebraneleadingtoananalyticalexpressionforthescale factor during inflation. Finally we haveestimated the typical energyscale of brane inflation with the potential of our considerationandfounditto be consistentwithcosmologicalaswell asparticlephysics frameworks. This modelthus leads to an inflationary scenario in the framework of supergravity inspired brane cosmology. Adetailedsurveyofthermalhistoryoftheuniverseviareheating,baryogenesis,leptogenesiswiththeloopcorrected potential and gravitino phenomenology remains as an open issue, which may even provide interesting signatures of brane inflation. A detailed analysis on these aspects have been reported as a separate paper [46]. Acknowledgments SC thanks S. Ghosh, R. Gopakumar, A. Mukhopadhyay and B. K. Pal for illuminating discussions and Council of Scientific and Industrial Research, India for financial support through Junior Research Fellowship (Grant No. 09/093(0132)/2010). SP is supported by Alexander von Humboldt Foundation, Germany through the project “Cos- mology with Branes and Higher Dimensions” and is partially supported by the SFB-Tansregio TR33 “The Dark Universe” (Deutsche Forschungsgemeinschaft) and the European Union 7th network program “Unification in the 10 LHC era” (PITN-GA-2009-237920). Special thanks to Hans Peter Nilles for a careful reading of the manuscript and for his valuable suggestions towards the improvement of the article. VII. APPENDIX For systematic development of the formalism, let us demonstrate briefly how one can construct the effective 4D inflationarypotentialofourconsiderationstartingfromN =2,D =5 SUGRA inthe bulk whichleads toaneffective N = 1,D = 4 SUGRA in the brane. As mentioned, we consider the bulk to be five dimensional where the fifth dimension is compactified on the orbifold S1/Z of comoving radius R. The system is described by the following 2 action [47], [48] 1 +πR S = d4x dy√g M3 R 2Λ +L + δ(y y )L . (7.1) 2Z Z−πR 5" 5 (cid:0) (5)− 5(cid:1) bulk Xi − i 4i# Herethesumincludesthewallsattheorbifoldpointsy =(0,πR)and5-dimensionalcoordinatesxm =(xα,y),where i y parameterizes the extra dimension compactified on the closed interval [ πR,+πR] and Z symmetry is imposed. 2 − For N =2,D =5 supergravity in the bulk Eq (7.1) can be written as 1 +πR S = d4x dy√g M3 R 2Λ +L(5) + δ(y y )L , (7.2) 2Z Z−πR 5" 5 (cid:0) (5)− 5(cid:1) SUGRA Xi − i 4i# which is a generalization of the scenario described in [47]. Written explicitly, the contribution from bulk SUGRA in the action is given by [24] M3R(5) i 1 e(−51)L(S5U)GRA =− 52 + 2Ψ¯im˜Γm˜n˜q˜∇n˜Ψiq˜−SIJFmI˜n˜FIm˜n˜ − 2gαβ(Dm˜φµ)(Dm˜φν) +Fermionic+Chern Simons, (7.3) − Including the contribution from the radion fields χ = ψ2 and T = 1 e5˙ i 2A0 the effective brane SUGRA − 5 √2 5− 3 5 counterpart turns out to be (cid:16) q (cid:17) δ(y)L4 = e(5)∆(y) (∂αφ)†(∂αφ)+iχ¯σ¯αDαχ . (7.4) − Here∆(y)=e5δ(y)isthemodifiedDiracdeltafunction(cid:2)whichsatisfiesthenormal(cid:3)izationconditions +πRdye5∆(y)= 5˙ πR 5˙ 1, +πRdy e5 = where is the 5 dimensional volume. The Chern-Simons terms can be gauge−d away assuming πR 5˙ L L R cubic−constraints[24,25]andZ symmetry. Itisusefultodefine thefivedimensionalgeneralizedKa¨hler function(G) R 2 inthiscontextas[24,25] G= 3ln T+T† +δ(y) √2 K(φ,φ ),whichpreciselyrepresentsinteractionoftheradion − √2 T+T† † with gauge fields. Including the kin(cid:16)etic te(cid:17)rm of the five dimensional field φ the singular terms measured from the modified Dirac delta function can be rearrangedinto a perfect square thereby leading to the following expression for the action 1 +πR 1 2 S ⊃ 2 d4x dy√g5e(4)e55˙ gαβGnm(∂αφm)†(∂βφn)+ g ∂5φ− H(G)∆(y) , (7.5) Z Z−πR (cid:20) 55 (cid:16) p (cid:17) (cid:21) where H(G) = exp MG2 ∂∂φWm + ∂∂φGmMW2 †(Gnm)−1 ∂∂φWn + ∂∂φGnMW2 −3|MW2|2 . It is worthwhile to mention that from eqn(7.5) we can(cid:0)com(cid:1)p(cid:20)u(cid:16)te energy mom(cid:17)entum tens(cid:16)or for N =2,D(cid:17)=5 SUG(cid:21)RA can be expressed as T =Gn(∂ φm) (∂ φ ) g gρσ(∂ φm) (∂ φ )Gn +g55(∂ φ H(G)∆(y))2 , (7.6) αβ m α † β n − αβ ρ † σ n m 5 − h p i 1 1 T = (∂ φ H(G)∆(y))2 g gρσGn(∂ φm) (∂ φ ). (7.7) 55 2 5 − − 2 55 m ρ † σ n p