TOK-HEP051220 Angular Power Spectrum and Dilatonic Inflation in Modular-Invariant Supergravity Mitsuo J. Hayashi∗ Department of Physics, Tokai University, 1117 Kitakaname, Hiratsuka, Kanagawa, 259-1292, Japan Shiro Hirai† Department of Digital Games, Osaka Electro-Communication University, 1130-70 Kiyotaki, Shijonawate, Osaka, 575-0063, Japan Yusuke Okame‡ Graduate School of Science and Technology, Tokai University, 1117 Kitakaname, Hiratsuka, Kanagawa, 259-1292, Japan 6 0 0 Tomoki Watanabe§ 2 Advanced Research Institute for Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo, 169-8555, Japan n (Dated: February 2, 2008) a J Theangularpowerspectrumisinvestigatedinthemodelofsupergravity,incorporatingthetarget- 3 space duality and the non-perturbative gaugino condensation in the hidden sector. The inflation 2 and supersymmetry breaking occur at once by the interplay between the dilaton field as inflaton and the condensate gauge-singlet field. The model satisfies the slow-roll condition which solves 1 the η-problem. When the particle rolls down along the minimized trajectory of the potential at a v dualityinvariantfixedpointT =1,wecanobtainthee-foldvalue∼57. Andthenthecosmological 0 parametersobtained from ourmodelwellmatch withtherecentWMAPdatacombined withother 9 experiments. The TT and TE angular power spectra also show that our model is compatible with 1 thedataforl>20. However,thebestfitvalueofτ inourmodelissmallerthanthatoftheΛCDM 1 model. These results suggest that, among supergravity models of inflation, the modular-invariant 0 supergravity seems to open a hope toconstruct therealistic theory of particles and cosmology. 6 0 PACSnumbers: 04.65.+e,11.25.Mj,11.30.Pb,12.60.Jv,98.80.Cq / h p - I. INFLATIONARY COSMOLOGY explain the contents of the universe?: Baryonic matter p 4%,Darkmatter23%,Darkenergy73%andsoon. These e problemsseemtorequirefarricherstructuresofcontents h Since WMAP combined with the other experiments : demonstrated on February, 2003 that the big bang and thanthoseofthestandardtheoryofparticles. Andmore v phenomenologically,iv) Is the model consistent with the i inflation theories continue to be true[1], the constraints X observedCMBangularpowerspectra? Inparticular,the on the cosmological parameters, such as the spectral in- inflaton should satisfy the slow-roll condition in order r dex and its running as well as the ratio of the tensor to a that the model predict the nearly scale-invariant spec- thescalar,havebeenimprovedbycombiningWMAPand tralindexaswellasthesufficientnumberofe-folds. (See Lyα forest[2, 3]. Recently the polarization-temperature ref.[5] for the recent review on the theories of inflation.) angular cross power spectrum of the cosmic microwave background (CMB) from the 2003 Flight of Boomerang Althoughrecentlythe inflationtheoriesinstring theo- was also published[4]. riesareextensivelyandpromissinglyinvestigatedbyvar- From the theoretical viewpoint, it is customary to iousauthors[5],here weconcentrateonthe frameworkof introduce scalar field(s) called inflaton into inflation a supergravity inspired by superstrings, following pre- models[5,6];thereare,however,severalproblemsincon- vious papers[7, 8, 9]. The well-known difficulty of su- structing successful theories: i) What is it, the inflaton? pergravity is that the potential form gives arise the η- ii) What kind of theoretical frameworks is the most ap- problem[8], which breaks the slow-roll condition. The propriate as the theory of particle physics, inflation and string-inspired supergravity is derived from the d = 10 the recently observed accelerating universe? iii) How to heterotic string by dimensional reduction to N =1, d= 4 supergravity[7], whose typical features are: i) No-scale structure at the tree level. ii) E E gauge group (one 8 8 × ofthe E iscalledthe hiddensectorofthe gaugegroup). ∗Electronicaddress: [email protected] 8 iii)Non-perturbativegauginocondensationinthehidden †Electronicaddress: [email protected] ‡Electronicaddress: [email protected] sectorcanbreakthesupersymmetry. iv)Modularinvari- §Electronicaddress: [email protected] ance, acting on a single modulus T, valid at any string- 2 loop order (Target-space duality)[8, 9, 10, 11, 12, 13]. and the gauge function f is ab Inthispaper,theangularpowerspectruminmodular- invariantsupergravityisinvestigatedinthemodelwhere f =δ S. (3) ab ab we had shown that the inflation and the supersymme- try breaking occur at once by the interplay between the Inorderto constructthe effective theoryofgauginocon- dilaton field as the inflaton and the condensated gauge- densation, we introduce the composite superfield Y of singlet field rolling down the inflationary trajectory, free the gaugino condensation[11, 13]: from the η-problem[9]. Y3 =δ WaǫαβWb/S3 =(λλ+ )/S3, (4) ab α β 0 ··· 0 II. A STRING-INSPIRED SUPERGRAVITY where λ is the gaugino fields in the Hidden sector. The effective K¨ahler potential and superpotential in- First of all, for the self-containedness, we will review corporating modular invariant one-loop corrections are the idea of the construction of the effective theory of given as[11]: gauginocondensation,incorporatingthetarget-spacedu- ality, following ref.[11], where the gaugino condensation hasbeendescribedbyaduality-invarianteffectiveaction K = ln(S+S∗) 3ln T +T∗ Y 2 Φi 2 , (5) − − −| | −| | forthegauge-singletgauginoboundstatescoupledtothe (cid:0) (cid:1) fundamental fields as the dilaton S and moduli T. On and the other hand, in ref.[10], the gaugino-condensate has beenreplacedby its vacuumexpectationvalue to yielda W =3bY3ln ceS/3bYη2(T) +W (6) matter duality-invariant“truncated”actionthatdependsonthe h i fundamental fields only. The equivalence between these whereη isDedekind’sη function,cisafreeparameterin two approaches had been proved in refs.[12]. thetheoryandb= β0 (β istheone-loopbeta-function Assuming that the compactificationof the superstring 96π2 0 coeffiients). theorypreservesN =1supersymmetry,theeffectivethe- Since S+S∗ =α′m2 , the choice: ory should be of the general type of N =1 supergravity h i pl coupled to gauge and matter fields. The most general form of Lagrangian in N = 1 supergravity at the tree- [e−K/3S0S¯0]θ=θ¯=0 =[S+S¯]θ=θ¯=0, (7) level is[7](See also [8, 14, 15, 16]): is correspondedto the conventionalnormalizationof the 1 = e−K/3S S¯ + S3W + f WaǫαβWb , gravitationalaction: L −2 0 0 D 0 F ab α β F where thehK¨ahler potiential(cid:2)K is(cid:3)given(cid:2)by (cid:3)(1) Lgrav ∼[e−K/3S0S¯0]θ=θ¯=0R. (8) K = ln(S+S∗) 3ln T +T∗ Φ 2 , (2) Then, the scalar potential is obtained as follows: i − − −| | (cid:0) (cid:1) 3(S+S∗)Y 4 2 V(S,T,Y) = | | 3b2 1+3ln ceS/3bYη2(T) (T +T∗ Y 2)2 −| | (cid:12) h i(cid:12) Y 2 (cid:12)(cid:12) 2 (cid:12)(cid:12) η′(T) 2 η′(T) η′(T) ∗ + | | S+S∗ 3bln ceS/3bYη2(T) +6b2 Y 2 2(T +T∗) + + , (9) T +T∗−|Y|2(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − h i(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | | " (cid:12)(cid:12)(cid:12)(cid:12)η(T)(cid:12)(cid:12)(cid:12)(cid:12) η(T) (cid:18)η(T)(cid:19) #! where matter fields are neglected. stable minimum at (Y ,S ) (0.00646,0.435) (See min min ∼ Fig.1). Therefore, we may conclude that inflation arises by III. INFLATIONARY TRAJECTORY AND the evolution of dilaton field S and supersymmetry is STABILITY IN MODULAR VARIABLE T brokenby the condensatedfield Y, providedit begins at the unstable saddle point and slowly rolls down to the The potential is modular invariant and shown to be minimum. stationary at the self-dual points T = 1 and T = eiπ/6. We found that the potential V(S,Y) at T = 1 has a The inflationary trajectory will be well approximated 3 successful theory of inflation should explain the mecha- nism of the reheating process, we remain this reheating problem for later work in the framework of the present model. The values of ǫ and η are obtained numerically in 1.5×10−7 S SS V 1×10−7 8 tFhige.sl3owfix-rionlglctohnedpitairoanmisetweresllcsa=ti1sfi8e3da,nadndbt=he5.η5-;pwroebfilenmd 5×10−8 6 can just be avoided. 0 S 4 00..000022 00..000044 2 1 YY 00..000066 0.8 0 00..000088 | S S 0.6 η | , 0.4 S FIG. 1: The plot of V(S,Y) at fixed T =1 (self-dual point) ǫ 0.2 with c = 183, b = 5.5.The stable minimum of VY(S) = 0 and a saddle point exist. We can see a valley of the poten- 0 tial and a stable minimum of VY(S) = 0 at (Ymin,Smin) ∼ 2 4 6 8 10 12 (0.00646,0.435). S FIG. 3: The evolution of the slow-roll parameters. The solid by the equation: curve represents ǫS whereas the dashed curve denotes |ηSS|, whichdemonstratethatthepotentialV(S)issufficientlyflat. Y (S) 0.00663e−S/16.2. (10) min ∼ Inflation endsat S ∼1.98 in ourmodel. In Fig. 2, we have shown a plot of V(S) minimized with respect to Y. As shown by Ferrara et al.[11], su- persymmmetry is broken by the hidden sector gaugino The potential V is stable at the self-dual point T = 1 condensation[13] because F λλ =0. in arbitrary points in the inflationary trajectory for our h| |i∝h| |i6 choice of the parameters c and b. By choosing the three points, i.e., horizon exit, end of inflation and the stable 3 minimumandinsertingthosevaluesS, Y atthesepoints 2.5 into the original V(S,Y,T), we will here demonstrate 2 V that the potential V(T) has minima exactly at T = 1 × 1.5 and hence is stable at these typical stages in the infla- 0 10 1 tionary trajectory. The variations of V(T) are obtained 1 0.5 numerically in Figs. 4 and 5 at the fixed parameters 0 c=183 and b=5.5. 0 2 4 6 8 10 12 14 S FIG. 2: The plot of V(S) minimized with respect to Y. The minimum value of the potential is V(Smin)∼−9.3×10−13. 22..55××1100−−77 Oneofthemainpurposesofthispaperistoprovethat VV((TT)) 11..55××1100−−77 the dilaton field plays the role of the inflaton field. 55××1100−−88 The slow-rollparameters (in Planck units mpl/√8π = 00..55 1 1) are defined by: 11 1 ∂ V 2 ∂ ∂ V RRee11TT..55 0 Im T α α β ǫα = 2 V , ηαβ = V . (11) 22 -1 (cid:18) (cid:19) 22..55 The slow-rollconditiondemands both values to be lower than 1. It is the end of inflation, when the slow-roll pa- rameterǫαreachesthevalue1. Afterpassingthroughthe FIG.4: The3DplotaroundtheminimumofV(T)asafunc- end of inflation, “matter” may be produced during the tion of complex variable T for Smin and Ymin. Im T = 0 is oscillationsaroundtheminimumofthepotential(reheat- obviously stable. ing) with the critical density, i.e. Ω = 1. Although any 4 10−7×V(ReT) spectrum. The value of ns∗ is consistent with the recent observations; the best fitting of them (WMAPext, 2dF- GRSandLymanα)forpowerlawΛCDMmodelsuggests 3.0 [1, 2, 3, 4] n (k )=0.96 0.02. s ∗ ± Finally,estimatingthespectrumofthedensitypertur- bation caused by slow-rolling dilaton[19]: 1.0 1 V3 , (18) Re T PR ∼ 12π2∂V2 0.5 1 1.5 2 2.5 3 we find 2.1 10−9. R∗ P ∼ × This result matches the measurements as well. Inci- FIG.5: TheplotsofV(T)atImT =0atthreerepresentative dentally speaking, the energy scale V 10−10 is also inflationary stages. It is obvious that T = 1 is stable. The within the constrained range obtained∼by Liddle and solid, dashed and dotted curves represent the stages at the Leach[17]. horizon exit, the end of inflation and the stable minimum respectively. Gravitational waves are inevitable consequence of all inflationalmodels. Now the tensor purturbationandthe gravitationalwave spectrum is given as: Number of e-folds at which a comovingscale k crosses the Hubble scale aH during inflation is given by: H 2 2 =8 = V. (19) Pgrav 2π 3π2 k 1 (1016GeV)4 1 Vk (cid:18) (cid:19) N(k) 62 ln ln + ln , ∼ − a0H0 − 4 Vk 4 Vend In SRA, the spectral index of Pgrav is givenby the slow- (12) roll parameters ǫ and η as: where we assume V = ρ . We focus on the scale end reh n = 2ǫ. (20) k∗ = 0.05 Mpc−1 and the inflationary energy scale is T − V 10−10 (1016GeV)4 as shown in Fig. 2, therefore The ratio r between and is given as R grav ∼ ∼ P P thenumberofe-foldswhichcorrespondstoourscalemust be around 57. r = Pgrav =16ǫ= 8n . (21) T − On the other hand, using the slow-roll approximation R P (SRA), N is also calculated by: Thegravitationalwavespectrumdoes notevolveandre- mainsfrozen-inasamasslessfieldevenafterthehorizon- S2 V exit, independent of the scalar perturbations[18]. Con- N dS. (13) ∼− ∂V trarytothisfact,theprimordialcurvaturefluctuation ZS1 R evolution is given by the product between the transfer Wecouldhaveobtainedthenumber 57,byintegrating function T (k) and : from S 1.98 to S 10.46, fix∼ing the parameters r R end ∗ ∼ ∼ (m) c=183 and b=5.5, i.e. our potential has the ability to k =Tr(k) k. (22) R R produce the cosmologically plausible number of e-folds. Therefore, the ratio r evolves as Here S is the value corresponding to k . ∗ ∗ Next, the scalar spectral index standing for a scale (m) dependence of the spectrum of density perturbation and Pgrav = 8Tr2nT (23) − its running are defined by: (cid:18) PR (cid:19) up to the present time. This result will be used in the dln R calculation of the angular power spectra. n 1 = P , (14) s − dlnk dn s α = . (15) s IV. THE ANGULAR POWER SPECTRUM OF dlnk THE MODEL These are approximated in the slow-rollparadigm as: Inourmodelwecancalculatethe angularpowerspec- n (S) 1 6ǫ +2η , (16) s ∼ − S SS trumtocomparewiththeWMAPanalysisandtheother α (S) 16ǫ η 24ǫ2 2ξ2 , (17) s ∼ S SS − S − (3) experimental data[1, 2, 3, 4, 20]. The multipoles alm of the CMB anisotropy are defined by where ξ is an extra slow-roll parameter that includes (3) trivial third derivative of the potential. Substituting S m=l ∗ δT 1in0t−o4t.heseequations,wehavens∗ ∼0.95andαs∗ ∼−4× ∆T ≡ T = l>0m=−lalmYlm(e), (24) X X Because n is not equal to 1 and α is almost negligi- ble, our modsel supports the model witsh tilted power law alm = dΩn∆T(n)Yl∗m(e), (25) Z 5 where Y (n) are the sperical harmonic functions evalu- which is allowable within the experimental error, while lm atedinthe directionn. The multipoles with l 2 repre- the total χ2 value takes minimum at τ =0.07. ≥ sent the intrinsic anisotropy of the CMB. If the CMB The angularpowerspectra of our model are presented temperature fluctuation ∆T is Gaussian distributed, inFig. 6forTT mode,Fig. 7forTE modeatthe values then each a is an independent Gaussian deviate with τ = 0.17, 0.07 and Fig.8 for TE mode at the values lm τ = 0.17, 0.13, 0.07 for l 20 with more detailed data. alm =0, (26) BecausetheTE spectraal≤mostcompletelycoincidewith h i those of the ΛCDM model for 20 < l < 500, we have and shownthespectrumforl <50intheFigs.7and8,where a distinction between our model and the ΛCDM model halma∗l′m′i=δll′δmm′Cl, (27) can be seen. Moreover, the best fit value of τ takes 0.13 for TE mode in our model whereas 0.17 in the ΛCDM where C is the ensemble average power spectrum, or, l model. We would like to emphasize this result as one of the angularpowerspectrum of the CMB. Ingeneral,the the characteristic features in our model. cosmological information is encoded in the standard de- For TT mode, although both our model and the viations and correlations of the coefficients: ΛCDM model can explain almost satisfactorily the hXYi= aXlmaYl′m′∗ =δll′δmm′ClXY. (28) WsupMpArePssidoantao,f tthheersepreecmtrauimnsastolmaregeinacnognusilsatrensccayleisn(lth=e If we use the sp(cid:10)herical exp(cid:11)ansion of the form for an 2,3)[20]. These data points appear to be the reasonwhy arbitrary function g(x): the best fit of τ even becomes 0 for TT mode and 0.07 for the total χ2 value, lower than 0.13 which is the best ∞ 2 χ2 value for TE mode. g(x)= dk g (k) kj(kx)Y (θ,φ), (29) lm l lm In summary, the model we have here investigated is π Z0 Xlm r compatible with the present observational data for l > 20, whereas there remains some problems unexplained where j is the spherical Bessel function,(θ,φ) is the di- l rection of x, then the angular power spectrum CTT and for small l. l the temperature-polarizationcrosspowerspectrum CTE l will be given by TABLE I: The χ2 values for the TT and TE spectrum and theirtotalsum. Thebestfitisatτ =0.13.forTE modeand ∞ dk CTT = 4π T2(k,l) (k) , (30) at τ =0.07 for thetotal sum. l Z0 Θ PR k τ TT TE Total CTE = 4π ∞T (k,l)T (k,l) (k)dk, (31) 0.17 986.92 456.73 1443.65 l Θ E PR k 0.14 982.10 456.50 1438.60 Z0 0.13 980.87 456.48 1437.35 where T and T are the transfer functions andΘ is the 0.12 980.00 456.72 1436.71 Θ E brightness function. 0.08 977.87 458.28 1436.15 Now we will show the behavior of those power spectra 0.07 977.40 458.63 1436.03 in our model. 0.06 977.31 459.51 1436.81 0.01 976.47 464.53 1441.00 The scalar spectral index is n (k ) = 0.95 and the s ∗ 0.00 976.26 465.35 1441.61 running index is α (k ) = 0.0004 at k = 0.05 Mpc−1 s ∗ ∗ − as already shown. The tensor-to-scalar ratio is assumed as r = 16ǫ = 8n = 0.00923 (ǫ = 0.00058 at k = T ∗ (cid:12)(cid:1)(cid:0)(cid:0)(cid:2)(cid:5)(cid:0)(cid:6) 0.05 Mpc−1). (cid:13)(cid:14)(cid:15)(cid:16) We will use the CMBFAST[21], where we have as- (cid:11)(cid:1)(cid:0)(cid:0)(cid:2)(cid:5)(cid:0)(cid:6) (cid:17)(cid:16)(cid:18)(cid:19) sumed the cosmological parameters to be: Ω = 1 for (cid:20)(cid:21)(cid:0)(cid:1)(cid:4)(cid:12) the total enegy density, ωΛ = 1 and ΩΛ = 0t.o7t3 for the 2K )(cid:10)(cid:1)(cid:0)(cid:0)(cid:2)(cid:5)(cid:0)(cid:6) (cid:20)(cid:21)(cid:0)(cid:1)(cid:0)(cid:12) − dark energy, Ωb =0.046 and Ωcdm =0.224 for the bary- π ((cid:9)(cid:1)(cid:0)(cid:0)(cid:2)(cid:5)(cid:0)(cid:6) onic and dark matter density, h = 0.71 for the Hubble 2 constant. The angular TT power spectra were normal- TT/(cid:8)(cid:1)(cid:0)(cid:0)(cid:2)(cid:5)(cid:0)(cid:6) C l ized with respect to 11 data points in the WMAP data 1)(cid:7)(cid:1)(cid:0)(cid:0)(cid:2)(cid:5)(cid:0)(cid:6) from l = 65 to l = 210 and the same values are used in + the analysis of the angular TE spectrum. l(l(cid:4)(cid:1)(cid:0)(cid:0)(cid:2)(cid:5)(cid:0)(cid:6) By using the likelihood method[22], we calculated the (cid:0)(cid:1)(cid:0)(cid:0)(cid:2)(cid:3)(cid:0)(cid:0) χ2 values for the TT and TE spectrum and their total (cid:4) (cid:4)(cid:0) (cid:4)(cid:0)(cid:0) (cid:4)(cid:0)(cid:0)(cid:0) (cid:4)(cid:0)(cid:0)(cid:0)(cid:0) sum, which are shown at Table I. l The ΛCDM modelshowsthat the best fitvalue of τ is 0.17 by the same method. On the other hand, the best FIG. 6: Temperature angular power spectrum (TT). fitofourmodel seemsrealizedatτ =0.13forTE mode, 6 tonfieldastheinflatonandthecondensategauge-singlet (cid:4) field. Our model is compatible with the angular power (cid:7)(cid:8)(cid:9)(cid:10) spectra of the WMAP data for l > 20, whereas there (cid:11)(cid:10)(cid:12)(cid:13) remains some problems unexplained for small l, as the ) (cid:3) (cid:14)(cid:15)(cid:2)(cid:16)(cid:1)(cid:17) ΛCDM model is. The best fit value of τ in our model is 2K (cid:14)(cid:15)(cid:2)(cid:16)(cid:2)(cid:17) smaller than 0.17 in the ΛCDM model. µ ( 2 π(cid:1) It appears that supergravity is one of the most plau- TE/ sible frameworks to explain the new physics, including C l theundetectedobjects,suchastheinflaton,darkmatter ) (cid:2) +1 and dark energy. Particularly, since the inflaton field is (l concernedwiththePlanckscalephysics,thedilatonfield (cid:0)(cid:1) seems to be the most presumable candidate of the infla- (cid:2) (cid:1)(cid:2) (cid:3)(cid:2) (cid:4)(cid:2) (cid:5)(cid:2) (cid:6)(cid:2) ton. Amongthepossibesupergravitymodelsofinflation, l themodularinvariantmodelhererevisitedseemstoopen a hope to construct the realistic theory of particles and FIG.7: Temperature-polarizationcrosspowerspectrum(TE) cosmology. for l<50. Forfurtherinvestigations,i)Weshouldconsideronthe effects ofthe hidden sectormassive matter overinflation and the supersymmetry breaking[12, 13]. ii) It will be interesting to understand what kind of phenomena are (cid:23) (cid:25)(cid:26)(cid:27)(cid:28) obtained from the S-duality invariant theory[15]. Fur- (cid:29)(cid:28)(cid:30)(cid:31) (cid:19) thermore, iii) What kind of theories can shed light on !(cid:22)"(cid:21)# ) !(cid:22)"(cid:21)(cid:19) the greatproblemofthe darkenergyanddarkmatterto 2K (cid:20) !(cid:22)"(cid:22)# understandtheir originandtheir relationto the recently (µ(cid:21) observedaccelratinguniverse[25,26]and[1,2,3,4]? iv) π 2 Gravitino, inflatino and axion production and their ef- TE/ (cid:22) fectsshouldbetraced[24]. v)Braneworldcosmologyand C l M-theoreticalapproachmightbepromising. (SeeLinde’s )(cid:18)(cid:21) 1 lecture in ref.[5], ref.[15, 23] and references therein.) + (l(cid:18)(cid:20) These problems and the reheating after inflation will be our further tasks[5]. (cid:18)(cid:19) (cid:22) (cid:24) (cid:21)(cid:22) (cid:21)(cid:24) (cid:20)(cid:22) l FIG.8: Temperature-polarizationcrosspowerspectrum(TE) for l<20 with more detailed data. For larger values of l our model almost completely coincide with theΛCDM model. Acknowledgments V. 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