arXiv:1003.4991v1 [hep-ph] 25 Mar 2010 fe e er fisivnin oeiec ftension phe- gravitational of of evidence tests no or invention, observations that, its current is of models with years ADD and ten RS1 after the of feature remarkable h re fTV even TeV, of order the stewre atr lal,b rprycosn the choosing properly by Clearly, factor. distance warped the is h lnkbaei eae otemass the to related is brane Planck the hn[] npriua,Arkani-Hamed since particular, studied In extensively been [2]. have then and proposed were h iil TV rn by brane (TeV) visible the M h apdfco,frwihtemass the used which RS for dimensions, extra factor, large differ- warped using is the of problem Instead hierarchy [5]. the warped. ent solve but to model, homogeneous, repro- mechanism RS1 not is the The are as dimensions to referred Newtonian extra often the 4D model, this the In brane and (invisible) duced. energy Planck gravitational the low included, near at is localized that brane be showed can the [5] effects of (RS) self-gravity Sundrum the and if Randall model, ent l esaladmyee eo h cl fmillimeters of size scale typical the a on For be even may [4]. necessar- and not small need be dimensions ily extra the that out pointed lnkms,and mass, Planck ie xr iesos( dimensions extra given where ∗ -iesoa udmna lnkmass Planck fundamental D-dimensional to uae stelredffrnei antdsbetween scales, magnitudes electroweak in for- and be difference Planck can large problem the (SM) The the Model [1]. as Standard decades mulated few the last beyond the during of forces ing problem hierarchy lcrncades anzhong˙[email protected] address: Electronic D orsleti rbe,i 98baewrdscenarios world brane 1998 in problem, this resolve To Introduction M a ea o steeetoeksae nadiffer- a In scale. electroweak the as low as be can pl M by y pl riodbae nsrn/-hoyadtercosmological their and string/M-Theory in branes Orbifold c ( M ∼ ewe h w rns n a lower can one branes, two the between ASnmes 98.80.Cq,11.25Mj,11.25.Y6 numbers: th PACS of branes between acceleration interaction realized. transient the easily to late rather due also a theory, the that correctio exponentia of The find are feature TeV. modes we of KK , gap gravitational mass to o order a spectrum higher have the the can and from and brane, discrete visible is the modes on fa warped localized dimension, is extra Gravity large the combining by addressed n -hoyon M-Theory and eycnevtv siain a eo h re f0 of order the of be can estimation, conservative very D ogsadn rbe npyisis physics in problem long-standing A : nti re eot esmaieorrcn tde nbran in studies recent our summarize we report, brief this In 10 = 16 hc a enoeo h andriv- main the of one been has which ,  M M CPCSE,PyisDprmn,Byo nvriy Waco University, Baylor Department, Physics GCAP-CASPER, eV T EW pl 2 /R m ( R eoe h four-dimensional the denotes ) ∼ D 0 D fteetadmnin,the dimensions, extra the of ssilo h re of order the of still is eV T − m ≥ 4 S  ) if 6), = 1 1 h lcrwa scale. electroweak the ) / /Z ( e D 2 − − nsc eus efidta h aini tbeadismass, its and stable is radion the that find we setups, such In . ky M 2) R c lal,frany for Clearly, . pl m m slreenough, large is tal et /M m 0 0 M where , esrdon measured esrdon measured EW D Dtd oebr2,2018) 21, November (Dated: AD [3] (ADD) srelated is ≃ M nhn Wang Anzhong e pl 10 m − A . the ky 16 to c , n npriua ttenwybitLreHdo Col- Hadron Large newly-built [6]. the are (LHC) at lider they colliders particular important, particle in More high-energy and at [2]. testable found experimentally been has nomena oiiecsooia osat(C,adteohris the brane DGP and other the [12] as the such models gravity, and of theories tiny (CC), modified a from constant and gen- [11] cosmological within quintessence as constructed positive such is (GR), one relativity eral classes: two into divided [10]. cosmology pro- the modern have of of one acceleration challenges is the biggest them of understanding and mystery origin implications, complete and found nature a established cos- The well remains late now physics [9]. the is underlying While universe the the [8], of structure. acceleration scale mic large radiation and background microwave (CMB) cosmic ev- the independent from and supernovae idence of by studies subsequently detailed confirmed more and [7], measurements novae universe the of ation hoyo rvt,s ti aua oebdsc models such embed to natural is quantum it consistent con- so a present, gravity, to for the of bet important At best theory very picture.” our phenomenologi- is “bigger is theory a is string/M it in far models Therefore, so these sider out nature. However, carried in [2]. work cal shall articles the we review of so the references, most to diffi- un-biased very readers of is refer list it simply the a that fact, provide studied of the to extensively matter cult of so a one been As been has physics. continuously field of and frontiers so, active most or decade last the the as considered been cosmology. of has model” dom- model “standard matter current this dark cold periods, inflationary early and inated an radiation with subsequently because together exactly and that, the is It triumph of this 8]. one [7, of far all be with so consistent might out has is carried CC and observations model crisis, tiny convincing the no of A resolutions far simplest so yet. that constructed, say to been fair is it ever, . 1 nte motn rbe is problem important Another aiu oeshv enpooe 9,wihcnbe can which [9], proposed been have models Various rn ol cnro aebe nesvl tde in studied intensively been have scenarios world Brane ∼ ∗ 0 . 1GV h irrh rbe a be can problem hierarchy The GeV. 01 tr n eso opigmechanisms. coupling tension and ctor, n uk onigeryuies is universe early bouncing A bulk. and h rvttoa auaKen(KK) Kaluza-Klein gravitational the f l upesd pligsc setups such Applying suppressed. lly nvresest etegeneric the be to seems universe e omlg nbt tigtheory string both in cosmology e st h DNwoinpotential Newtonian 4D the to ns X76798-7316 TX , rtosre rmTp asuper- Ia Type from observed first , f ( R olna rvt 1] How- [13]. gravity nonlinear ) applications h aecsi acceler- cosmic late the iha with 2

VFHzcL The radion masse is given by [22] 12 2 1/2 α1 α2 11 1 ∂ VΦ M11 R mϕ = 2 ≈ Mpl, (1) 2 ∂ϕ Mpl lpl 10      

where α1 = 3 and α2 = 2. For M ≃ 1 TeV and R ≃ 9 −22 10 m, we find that mϕ ≃ 0.1 GeV . In the framework 8 of string, the radion mass is also given by Eq. (1) but now with α =8/3 and α =5/3 [24]. 7 1 2 Localization of Gravity and 4D Effective Newtonian 6 Potential: In contrast to the RS1 model in which the gravity is localized on the invisible brane [5], we find z 12 14 16 18 20 c that the gravity in the framework of both string [24] and M theory [22] is localized on the visible brane, because FIG. 1: The radion potential in the framework of the HW in the present setup the warped factor increases as one heterotic M Theory [22]. approaches the visible brane from the invisible one. The spectrum of the gravitational KK modes is discrete and the corresponding masses are given by [22, 24, 26] into string/M theory. In fact, the invention of branes lpl mn ≃ nπ Mpl, (2) [3, 5] was originally motivated by string/M theory [14]. yc However, relatively much less efforts have been devoted   to such studies [15]. One of the main reasons is that where yc is the distance between the two branes. Thus, −19 such an embedding is non-trivial and frequently ham- for yc ≃ 10 m we have m1 ≃ 1 TeV. pered by the complexity of string/M theory. In such a The 4D Newtonian potential, on the other hand, takes setup, two fundamental issues are: (a) the stability of the the form, large number of moduli resulting from the compactifica- 2 ∞ tion; and (b) the localization of gravity on the branes. In M M M − U(r)= G 1 2 1+ pl e mnr |ψ (z )|2 , (3) the Horava-Witten (HW) heterotic M theory, the moduli 4 r M 3 n c 5 n=1 ! are naturally split into two families, depending on the X type of harmonic form, (1, 1) and (2, 1). Various mecha- −19 where ψn(yc) ≃ 2/yc. Clearly, for yc ≃ 10 m and nisms to stabilize these moduli have been proposed. In r ≃ 10 µm, the high order corrections to the 4D Newto- particular, one may use the internal fluxes, introduced nian potential arep exponentially suppressed, and can be et al by Kachru (KKLT) originally for the moduli sta- safely neglected. bilization of type IIB string [16], to stabilize the (2, 1) The hierarchy problem: This problem can also be ad- moduli. Brane stretching between the two boundaries, on dressed in our current setups, but the mechanism is a the other hand, can fix the (1, 1) moduli [17], while gaug- combination of the ADD large extra dimension [3] and ino condensation may fix the volume of the 3 Calabi-Yau the RS warped factor mechanisms [5], together with the manifold [18]. In addition, to fix the distance (radion) brane tension coupling scenario [29], and the 4D Newto- between the two branes, Goldberger-Wise (GW) mecha- nian constant is given by [22, 24], nism [19] and Casimir energy contributions [20] have also been widely used. gk G = , (4) N β1 β2 In the past couple of years, we have studied orbifold 48πM R branes in the framework of the 11D HW heterotic M where g denotes the tension of the brane, (β ,β ) = 1 k 1 2 Theory on S /Z2 [14], developed by Lukas et al [15] by (18, 12) for the HW heterotic M-Theory [22], and compactifying the 11D HW theory on a 6D Calabi-Yau (β1,β2) = (16, 10) for the [24]. space [21, 22], and orbifold branes in the framework of It is interesting to note that the 4D effective cosmolog- (type II) string theory [23–26], as well as orbifold branes ical constant can be cast in the form [21, 24], in the RS setup [27, 28]. In particular, we investigated in detail the three important issues: (i) the radion stabil- β1 β2 Λ4 M R 4 ity and radion mass; (ii) the localization of gravity and ρΛ = =3 M . (5) 8πG M l pl high-order Yukawa corrections; and (iii) the hierarchy 4  pl   pl  problem. −22 For R ≃ 10 m and M10 ≃ 1 T eV , in both cases we −47 4 Radion Stability and masses: Using the Goldberger have ρΛ ∼ ρΛ,ob ≃ 10 GeV . and Wise (GW) mechanism [19], we found that the ra- Cosmological Applications: Applying such setups to dion is stable. Fig. 1 shows the radion potential in the cosmology, we found the generalized Friedmann-like case of HW heterotic M Theory [22]. A similar result was equations on each of the two orbifold branes, and showed also obtained in the framework of string theory [24]. that the late acceleration of the universe is transient, due 3

V4(φ) denoting the potential on the brane. Clearly, at the 1.5 early time the last term in the right-hand side of Eq. (6) dominates, and there always exists a non-zero minimum 1 ai > 0 at which H(ai) = 0. For a

0.5 Similar results can be also obtained in the setup of string theory [23–25]. a**/a 0 Concluding Remarks: With all these remarkable fea- tures, it is very desirable to investigate other aspects of these models. In particular, in our previous studies, we −0.5 have not addressed the issue of supersymmetry. Working with TeV scale, a distinctive feature is the possibilities

−1 of finding observational signals to LHC [6]. Meanwhile, 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 a/a in the framework of brane cosmology, significant devia- 0 tions come from the early universe [2]. To explain the ∗∗ late cosmic acceleration of the universe, various models FIG. 2: The acceleration a /a in the framework of string 1 have been proposed [9]. While different models can give theory on S /Z2 [25]. the same late time accelerated expansion, the growth of matter perturbation they produce usually differ [30]. Re- to the interaction of the bulk and the branes. Fig. 2 cently, the use of the growth rate of matter perturbation shows the future evolution of the acceleration of the uni- in addition to the expansion history of the Universe to verse in the framework of string [25]. Similar result was differentiate dark energy models and modified gravity at- also obtained in M-Theory [21]. tracted much attention. Therefore, it would be very im- Bouncing universe can be also constructed. In par- portant to study perturbations of the cosmological mod- ticular, in the setup of M-Theory [21, 22], the general- els in our string/M theory setups. ized Friedmann equations for moving branes in a five- dimensional bulk with a 4-dimensional Poincare symme- try take the forms, Acknowledgments 2 2πG 2 1 H = (ρ + τφ +2ρΛ) − 2 12 , (6) 3ρΛ 25L a ρ˙ +τ ˙φ +3H(ρ + p)= −6H(2ρΛ + ρ + τφ), (7) The author would like to thank T. Ali, R.-G. Cai, G. Cleaver, M. Devin, Y.-G. Gong, N.O. Santos, P. Vo and −2 −φ where L is a constant, and τφ ≡ 6ǫακ5 e +V4(φ), with Q. Wu for collaborations.

[1] I. Antoniadis, arXiv:0710.4267. [13] S. Capozziello, S. Carloni, and A. Troisi, [2] R. Maartens, Living Rev. Relativ. 7 arXiv:astro-ph/0303041; T.P. Sotiriou and V. Faraoni, (2004); arXiv:astro-ph/0602415; D. Wands, arXiv:0805.1726. arXiv:gr-qc/0601078; K. Koyama, Class. Quant. [14] H. Horava and E. Witten, Nucl. Phys. B460, 506 (1996); Grav. 24, R231 (2007). 475, 94 (1996). [3] N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. [15] A. Lukas, et al., Phys. Rev. D59, 086001. Lett. B429, 263 (1998); Phys. Rev. D59, 086004 (1999); [16] S. Kachru, M.B. Schulz, and S. Trivedi, JHEP, 10, 007 and I. Antoniadis, et al., Phys. Lett., B436, 257 (1998). (2003); S. Kachru, et al., Phys. Rev. D68, 046005 (2003). [4] D.J. Kapner, et al., Phys. Rev. Lett. 98, 021101 (2007). [17] E. Buchbinder and B.A. Ovrut, Phys. Rev. D69, 086010 [5] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (2004); V. Braun and B.A. Ovrut, JHEP, 07, 035 (2006). (1999). [18] M. Dine, et al., Phys. Lett. B156, 55 (1985); P. Horava, [6] L. Randall and M.B. Wise, arXiv:0807.1746. Phys. Rev. D54,7561 (1996); A. Lukas, B. A. Ovrut, and [7] A. G. Riess et al.,Astron. J. 116, 1009 (1998); S. Perl- D. Waldram, Phys. Rev. D57, 7529 (1998); J. Gray, A. mutter et al.,Astrophys. J. 517, 565 (1999). Lukas, and B. Ovrut, Phys. Rev. D76, 126012 (2007). [8] E. Komatsu et al, arXiv:1001.4538. [19] W.D. Goldberger and M.B. Wise, Phys. Rev. Lett. 83, [9] J.A. Frieman, AIP Conf. Proc. 1057, 87 (2008). 4922 (1999). [10] A. Albrecht, et al., arXiv:astro-ph/0609591. [20] M. Fabinger and P. Horava, Nucl. Phys. B580, 243 [11] R. R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. (2000); A. Flachi and T. Tanaka, arXiv:0906.288. Lett. 80, 1582 (1998). [21] G.-Y. Gong, A. Wang, and Q. Wu, Phys. Lett. B663, [12] G. R. Dvali, G. Gabadadze and M. Porrati, Phys. Lett. 147 (2008). B484, 112 (2000); C. Deffayet, ibid., 502, 199 (2001). [22] Q. Wu, G.-Y. Gong, and A. Wang, JCAP, 03, 029 (2009). 4

[23] A. Wang and N.O. Santos, Phys. Lett. B669, 127 (2008). [28] A. Wang, R.-G. Cai, and N.O. Santos, Nucl. Phys. B797, [24] A. Wang and N.O. Santos, Inter. J. Mod. Phys. A in 395 (2008). press (2010) [arXiv:0808.2055]. [29] J. Cline, C. Grojean, and G. Servant, Phys. Rev. Lett. [25] Q. Wu, N.O. Santos, P. Vo, and A. Wang, JCAP, 09, 004 83, 4245 (1999); C. Csaki et al, Phys. Lett. B462, 34 (2008). (1999). [26] M. Devin, T. Ali, G. Cleaver, A. Wang, and Q. Wu, [30] Y.-G. Gong, M. Ishak, and A. Wang, Phys. Rev. D80, JHEP, 10, 095 (2009). 023002 (2009). [27] A. Wang, Phys. Rev. D66, 024024 (2002).