Abstract
We propose an extension of natural inflation, where the inflaton potential is a general periodic function. Specifically, we study elliptic inflation where the inflaton potential is given by Jacobi elliptic functions, Jacobi theta functions or the Dedekind eta function, which appear in gauge and Yukawa couplings in the string theories compactified on toroidal backgrounds. We show that in the first two cases the predicted values of the spectral index and the tensor-to-scalar ratio interpolate from natural inflation to exponential inflation such as R 2- and Higgs inflation and brane inflation, where the spectral index asymptotes to n s = 1 − 2/N ≃ 0.967 for the e-folding number N = 60. We also show that a model with the Dedekind eta function gives a sizable running of the spectral index due to modulations in the inflaton potential. Such elliptic inflation can be thought of as a specific realization of multi-natural inflation, where the inflaton potential consists of multiple sinusoidal functions. We also discuss examples in string theory where Jacobi theta functions and the Dedekind eta function appear in the inflaton potential.
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References
A.H. Guth, The inflationary universe: a possible solution to the horizon and flatness problems, Phys. Rev. D 23 (1981) 347 [INSPIRE].
K. Sato, First order phase transition of a vacuum and expansion of the universe, Mon. Not. Roy. Astron. Soc. 195 (1981) 467 [INSPIRE].
A.A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett. B 91 (1980) 99 [INSPIRE].
R. Brout, F. Englert and E. Gunzig, The creation of the universe as a quantum phenomenon, Annals Phys. 115 (1978) 78 [INSPIRE].
D. Kazanas, Dynamics of the universe and spontaneous symmetry breaking, Astrophys. J. 241 (1980) L59 [INSPIRE].
A.D. Linde, A new inflationary universe scenario: a possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems, Phys. Lett. B 108 (1982) 389 [INSPIRE].
A. Albrecht and P.J. Steinhardt, Cosmology for grand unified theories with radiatively induced symmetry breaking, Phys. Rev. Lett. 48 (1982) 1220 [INSPIRE].
A.A. Starobinsky, Spectrum of relict gravitational radiation and the early state of the universe, JETP Lett. 30 (1979) 682 [INSPIRE].
A.D. Linde, Chaotic inflation, Phys. Lett. B 129 (1983) 177 [INSPIRE].
K. Freese, J.A. Frieman and A.V. Olinto, Natural inflation with pseudo-Nambu-Goldstone bosons, Phys. Rev. Lett. 65 (1990) 3233 [INSPIRE].
Planck collaboration, P.A.R. Ade et al., Planck 2013 results. XXII. Constraints on inflation, Astron. Astrophys. 571 (2014) A22 [arXiv:1303.5082] [INSPIRE].
BICEP2 collaboration, P.A.R. Ade et al., Detection of B-mode polarization at degree angular scales by BICEP2, Phys. Rev. Lett. 112 (2014) 241101 [arXiv:1403.3985] [INSPIRE].
Planck collaboration, R. Adam et al., Planck intermediate results. XXX. The angular power spectrum of polarized dust emission at intermediate and high galactic latitudes, arXiv:1409.5738 [INSPIRE].
F.L. Bezrukov and M. Shaposhnikov, The standard model Higgs boson as the inflaton, Phys. Lett. B 659 (2008) 703 [arXiv:0710.3755] [INSPIRE].
B.L. Spokoiny, Inflation and generation of perturbations in broken symmetric theory of gravity, Phys. Lett. B 147 (1984) 39 [INSPIRE].
F. Lucchin and S. Matarrese, Power law inflation, Phys. Rev. D 32 (1985) 1316 [INSPIRE].
A.B. Goncharov and A.D. Linde, Chaotic inflation in supergravity, Phys. Lett. B 139 (1984) 27 [INSPIRE].
D.S. Salopek, J.R. Bond and J.M. Bardeen, Designing density fluctuation spectra in inflation, Phys. Rev. D 40 (1989) 1753 [INSPIRE].
R. Fakir and W.G. Unruh, Improvement on cosmological chaotic inflation through nonminimal coupling, Phys. Rev. D 41 (1990) 1783 [INSPIRE].
E.D. Stewart, Inflation, supergravity and superstrings, Phys. Rev. D 51 (1995) 6847 [hep-ph/9405389] [INSPIRE].
G.R. Dvali and S.H.H. Tye, Brane inflation, Phys. Lett. B 450 (1999) 72 [hep-ph/9812483] [INSPIRE].
M. Cicoli, C.P. Burgess and F. Quevedo, Fibre inflation: observable gravity waves from IIB string compactifications, JCAP 03 (2009) 013 [arXiv:0808.0691] [INSPIRE].
K. Nakayama, F. Takahashi and T.T. Yanagida, Polynomial chaotic inflation in the Planck era, Phys. Lett. B 725 (2013) 111 [arXiv:1303.7315] [INSPIRE].
K. Nakayama, F. Takahashi and T.T. Yanagida, Polynomial chaotic inflation in supergravity, JCAP 08 (2013) 038 [arXiv:1305.5099] [INSPIRE].
R. Kallosh, A. Linde and A. Westphal, Chaotic inflation in supergravity after Planck and BICEP2, Phys. Rev. D 90 (2014) 023534 [arXiv:1405.0270] [INSPIRE].
K. Nakayama, F. Takahashi and T.T. Yanagida, Polynomial chaotic inflation in supergravity revisited, Phys. Lett. B 737 (2014) 151 [arXiv:1407.7082] [INSPIRE].
M. Czerny and F. Takahashi, Multi-natural inflation, Phys. Lett. B 733 (2014) 241 [arXiv:1401.5212] [INSPIRE].
M. Czerny, T. Higaki and F. Takahashi, Multi-natural inflation in supergravity, JHEP 05 (2014) 144 [arXiv:1403.0410] [INSPIRE].
M. Czerny, T. Higaki and F. Takahashi, Multi-natural inflation in supergravity and BICEP2, Phys. Lett. B 734 (2014) 167 [arXiv:1403.5883] [INSPIRE].
N. Arkani-Hamed, H.-C. Cheng, P. Creminelli and L. Randall, Extra natural inflation, Phys. Rev. Lett. 90 (2003) 221302 [hep-th/0301218] [INSPIRE].
D. Croon and V. Sanz, Saving natural inflation, JCAP 02 (2015) 008 [arXiv:1411.7809] [INSPIRE].
N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485] [hep-th/9407087] [INSPIRE].
N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [INSPIRE].
M. Berg, M. Haack and B. Körs, Loop corrections to volume moduli and inflation in string theory, Phys. Rev. D 71 (2005) 026005 [hep-th/0404087] [INSPIRE].
D. Baumann et al., On D3-brane potentials in compactifications with fluxes and wrapped D-branes, JHEP 11 (2006) 031 [hep-th/0607050] [INSPIRE].
D. Cremades, L.E. Ibáñez and F. Marchesano, Yukawa couplings in intersecting D-brane models, JHEP 07 (2003) 038 [hep-th/0302105] [INSPIRE].
D. Cremades, L.E. Ibáñez and F. Marchesano, Computing Yukawa couplings from magnetized extra dimensions, JHEP 05 (2004) 079 [hep-th/0404229] [INSPIRE].
M. Mariño, R. Minasian, G.W. Moore and A. Strominger, Nonlinear instantons from supersymmetric p-branes, JHEP 01 (2000) 005 [hep-th/9911206] [INSPIRE].
S. Hamidi and C. Vafa, Interactions on orbifolds, Nucl. Phys. B 279 (1987) 465 [INSPIRE].
B. Dundee, S. Raby and A. Westphal, Moduli stabilization and SUSY breaking in heterotic orbifold string models, Phys. Rev. D 82 (2010) 126002 [arXiv:1002.1081] [INSPIRE].
H. Abe, T. Kobayashi and H. Otsuka, Natural inflation with and without modulations in type IIB string theory, arXiv:1411.4768 [INSPIRE].
R. Schimmrigk, Automorphic inflation, arXiv:1412.8537 [INSPIRE].
T. Kobayashi and F. Takahashi, Running spectral index from inflation with modulations, JCAP 01 (2011) 026 [arXiv:1011.3988] [INSPIRE].
B. Feng, M.-z. Li, R.-J. Zhang and X.-m. Zhang, An inflation model with large variations in spectral index, Phys. Rev. D 68 (2003) 103511 [astro-ph/0302479] [INSPIRE].
F. Takahashi, The spectral index and its running in axionic curvaton, JCAP 06 (2013) 013 [arXiv:1301.2834] [INSPIRE].
M. Czerny, T. Kobayashi and F. Takahashi, Running spectral index from large-field inflation with modulations revisited, Phys. Lett. B 735 (2014) 176 [arXiv:1403.4589] [INSPIRE].
K.N. Abazajian, G. Aslanyan, R. Easther and L.C. Price, The knotted sky II: does BICEP2 require a nontrivial primordial power spectrum?, JCAP 08 (2014) 053 [arXiv:1403.5922] [INSPIRE].
Y. Wan et al., Single field inflation with modulated potential in light of the Planck and BICEP2, Phys. Rev. D 90 (2014) 023537 [arXiv:1405.2784] [INSPIRE].
Q.E. Minor and M. Kaplinghat, Inflation that runs naturally: gravitational waves and suppression of power at large and small scales, Phys. Rev. D 91 (2015) 063504 [arXiv:1411.0689] [INSPIRE].
A. de la Fuente, P. Saraswat and R. Sundrum, Natural inflation and quantum gravity, arXiv:1412.3457 [INSPIRE].
Planck collaboration, P.A.R. Ade et al., Planck 2013 results. XVI. Cosmological parameters, Astron. Astrophys. 571 (2014) A16 [arXiv:1303.5076] [INSPIRE].
R. Easther and H. Peiris, Implications of a running spectral index for slow roll inflation, JCAP 09 (2006) 010 [astro-ph/0604214] [INSPIRE].
H. Abe, T. Kobayashi, K. Sumita and Y. Tatsuta, Gaussian Froggatt-Nielsen mechanism on magnetized orbifolds, Phys. Rev. D 90 (2014) 105006 [arXiv:1405.5012] [INSPIRE].
J. Polchinski, String theory. Vol. 1: An introduction to the bosonic string, Cambridge University Press, Cambridge U.K. (1998) [INSPIRE].
I. Antoniadis, K. Benakli and M. Quirós, Finite Higgs mass without supersymmetry, New J. Phys. 3 (2001) 20 [hep-th/0108005] [INSPIRE].
R. Blumenhagen, B. Körs, D. Lüst and S. Stieberger, Four-dimensional string compactifications with D-branes, orientifolds and fluxes, Phys. Rept. 445 (2007) 1 [hep-th/0610327] [INSPIRE].
H. Jockers and J. Louis, The effective action of D7-branes in N = 1 Calabi-Yau orientifolds, Nucl. Phys. B 705 (2005) 167 [hep-th/0409098] [INSPIRE].
K.-I. Izawa and T. Yanagida, Dynamical supersymmetry breaking in vector-like gauge theories, Prog. Theor. Phys. 95 (1996) 829 [hep-th/9602180] [INSPIRE].
K.A. Intriligator and S.D. Thomas, Dynamical supersymmetry breaking on quantum moduli spaces, Nucl. Phys. B 473 (1996) 121 [hep-th/9603158] [INSPIRE].
S. Kachru, R. Kallosh, A.D. Linde and S.P. Trivedi, De Sitter vacua in string theory, Phys. Rev. D 68 (2003) 046005 [hep-th/0301240] [INSPIRE].
T.W. Grimm and J. Louis, The effective action of N = 1 Calabi-Yau orientifolds, Nucl. Phys. B 699 (2004) 387 [hep-th/0403067] [INSPIRE].
V. Balasubramanian, P. Berglund, J.P. Conlon and F. Quevedo, Systematics of moduli stabilisation in Calabi-Yau flux compactifications, JHEP 03 (2005) 007 [hep-th/0502058] [INSPIRE].
J.P. Conlon, F. Quevedo and K. Suruliz, Large-volume flux compactifications: moduli spectrum and D3/D7 soft supersymmetry breaking, JHEP 08 (2005) 007 [hep-th/0505076] [INSPIRE].
K. Choi and K.S. Jeong, Supersymmetry breaking and moduli stabilization with anomalous U(1) gauge symmetry, JHEP 08 (2006) 007 [hep-th/0605108] [INSPIRE].
M. Haack, D. Krefl, D. Lüst, A. Van Proeyen and M. Zagermann, Gaugino condensates and D-terms from D7-branes, JHEP 01 (2007) 078 [hep-th/0609211] [INSPIRE].
P.G. Camara, L.E. Ibáñez and A.M. Uranga, Flux induced SUSY breaking soft terms, Nucl. Phys. B 689 (2004) 195 [hep-th/0311241] [INSPIRE].
P.G. Camara, L.E. Ibáñez and A.M. Uranga, Flux-induced SUSY-breaking soft terms on D7-D3 brane systems, Nucl. Phys. B 708 (2005) 268 [hep-th/0408036] [INSPIRE].
M. Berg, M. Haack and B. Körs, String loop corrections to Kähler potentials in orientifolds, JHEP 11 (2005) 030 [hep-th/0508043] [INSPIRE].
A. Collinucci, F. Denef and M. Esole, D-brane deconstructions in IIB orientifolds, JHEP 02 (2009) 005 [arXiv:0805.1573] [INSPIRE].
H. Abe, T. Higaki and T. Kobayashi, Remark on integrating out heavy moduli in flux compactification, Phys. Rev. D 74 (2006) 045012 [hep-th/0606095] [INSPIRE].
O. DeWolfe, A. Giryavets, S. Kachru and W. Taylor, Type IIA moduli stabilization, JHEP 07 (2005) 066 [hep-th/0505160] [INSPIRE].
T.W. Grimm, Non-perturbative corrections and modularity in N = 1 type IIB compactifications, JHEP 10 (2007) 004 [arXiv:0705.3253] [INSPIRE].
T.W. Grimm, Axion inflation in type-II string theory, Phys. Rev. D 77 (2008) 126007 [arXiv:0710.3883] [INSPIRE].
R. Kallosh and A.D. Linde, Landscape, the scale of SUSY breaking and inflation, JHEP 12 (2004) 004 [hep-th/0411011] [INSPIRE].
H. Hayashi, R. Matsuda and T. Watari, Issues in complex structure moduli inflation, arXiv:1410.7522 [INSPIRE].
S. Kachru et al., Towards inflation in string theory, JCAP 10 (2003) 013 [hep-th/0308055] [INSPIRE].
D. Baumann, A. Dymarsky, I.R. Klebanov, L. McAllister and P.J. Steinhardt, A delicate universe, Phys. Rev. Lett. 99 (2007) 141601 [arXiv:0705.3837] [INSPIRE].
D. Baumann, A. Dymarsky, I.R. Klebanov and L. McAllister, Towards an explicit model of D-brane inflation, JCAP 01 (2008) 024 [arXiv:0706.0360] [INSPIRE].
H.-Y. Chen, L.-Y. Hung and G. Shiu, Inflation on an open racetrack, JHEP 03 (2009) 083 [arXiv:0901.0267] [INSPIRE].
O. Lebedev, H.P. Nilles and M. Ratz, De Sitter vacua from matter superpotentials, Phys. Lett. B 636 (2006) 126 [hep-th/0603047] [INSPIRE].
E. Dudas, C. Papineau and S. Pokorski, Moduli stabilization and uplifting with dynamically generated F-terms, JHEP 02 (2007) 028 [hep-th/0610297] [INSPIRE].
H. Abe, T. Higaki, T. Kobayashi and Y. Omura, Moduli stabilization, F-term uplifting and soft supersymmetry breaking terms, Phys. Rev. D 75 (2007) 025019 [hep-th/0611024] [INSPIRE].
R. Kallosh and A.D. Linde, O’KKLT, JHEP 02 (2007) 002 [hep-th/0611183] [INSPIRE].
E. Dudas, Y. Mambrini, S. Pokorski and A. Romagnoni, Moduli stabilization with Fayet-Iliopoulos uplift, JHEP 04 (2008) 015 [arXiv:0711.4934] [INSPIRE].
E. Dudas, Y. Mambrini, S. Pokorski, A. Romagnoni and M. Trapletti, Gauge versus gravity mediation in models with anomalous U(1)’s, JHEP 03 (2009) 011 [arXiv:0809.5064] [INSPIRE].
T. Li, Z. Li and D.V. Nanopoulos, Chaotic inflation in no-scale supergravity with string inspired moduli stabilization, Eur. Phys. J. C 75 (2015) 55 [arXiv:1405.0197] [INSPIRE].
T. Li, Z. Li and D.V. Nanopoulos, Natural inflation with natural trans-Planckian axion decay constant from anomalous U(1) X , JHEP 07 (2014) 052 [arXiv:1405.1804] [INSPIRE].
T. Li, Z. Li and D.V. Nanopoulos, Aligned natural inflation and moduli stabilization from anomalous U(1) gauge symmetries, JHEP 11 (2014) 012 [arXiv:1407.1819] [INSPIRE].
K.A. Intriligator and N. Seiberg, Lectures on supersymmetric gauge theories and electric-magnetic duality, Nucl. Phys. Proc. Suppl. B 45 (1996) 1 [hep-th/9509066] [INSPIRE].
A. Font, L.E. Ibáñez, D. Lüst and F. Quevedo, Supersymmetry breaking from duality invariant gaugino condensation, Phys. Lett. B 245 (1990) 401 [INSPIRE].
A. Font, L.E. Ibáñez, D. Lüst and F. Quevedo, Strong-weak coupling duality and nonperturbative effects in string theory, Phys. Lett. B 249 (1990) 35 [INSPIRE].
H. Abe, T. Kobayashi and H. Otsuka, Towards natural inflation from weakly coupled heterotic string theory, arXiv:1409.8436 [INSPIRE].
J.E. Kim, H.P. Nilles and M. Peloso, Completing natural inflation, JCAP 01 (2005) 005 [hep-ph/0409138] [INSPIRE].
M. Fukugita and T. Yanagida, Baryogenesis without grand unification, Phys. Lett. B 174 (1986) 45 [INSPIRE].
T. Asaka, K. Hamaguchi, M. Kawasaki and T. Yanagida, Leptogenesis in inflaton decay, Phys. Lett. B 464 (1999) 12 [hep-ph/9906366] [INSPIRE].
T. Asaka, K. Hamaguchi, M. Kawasaki and T. Yanagida, Leptogenesis in inflationary universe, Phys. Rev. D 61 (2000) 083512 [hep-ph/9907559] [INSPIRE].
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Higaki, T., Takahashi, F. Elliptic inflation: interpolating from natural inflation to R 2-inflation. J. High Energ. Phys. 2015, 129 (2015). https://doi.org/10.1007/JHEP03(2015)129
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DOI: https://doi.org/10.1007/JHEP03(2015)129