Abstract
We study the simple gauge invariant model \({f^2}FF\) as a way to generate primordial magnetic fields (PMF) in natural inflation (NI). We compute both magnetic and electric spectra generated by the \({f^2}FF\) model in NI for different values of model parameters and find that both de Sitter and power law expansion lead to the same results at sufficiently large number of e-foldings. We also find that the necessary scale invariance property of the PMF cannot be obtained in NI in first order of slow roll limits under the constraint of inflationary potential, \(V\left( 0 \right) \simeq 0\). Furthermore, if this constraint is relaxed to achieve scale invariance, then the model suffers from the backreaction problem for the co-moving wave number, \(k \lesssim 8.0\times 10^{-7} \mathrm {Mpc^{-1}}\) and Hubble parameter, \(H_i \gtrsim 1.25\times 10^{-3} \mathrm {M_\mathrm{{Pl}}}\). The former can be considered as a lower bound of k and the later as an upper bound of \(H_i\) for a model which is free from the backreaction problem. Further, we show that there is a narrow range of the height of the potential \(\Lambda \) around \({\Lambda _{\min }} \approx 0.00874{M_{\mathrm{{Pl}}}}\) and of k around \({k_{\min }} \sim 0.0173\mathrm{{Mp}}{\mathrm{{c}}^{ - 1}}\), at which the energy of the electric field can fall below the energy of the magnetic field. The range of k lies within some observable scales. However, the relatively short range of k presents a challenge to the viability of this model.
Similar content being viewed by others
References
Starobinsky, A.A.: Phys. Lett. B 91, 99 (1980)
Guth, A.H.: Phys. Rev. D 23, 347 (1981)
Sato, K.: MNRAS 195, 467 (1981)
Linde, A.D.: Phys. Lett. B 108, 389 (1982)
Neronov, A., Vovk, I.: Science 328, 73–75 (2010)
Fujita, T., Mukohyama, S.: arXiv:1205.5031v3 [astro-ph.CO]
Tavecchio, F., et al.: arXiv:1004.1329v2 [astro-ph.CO]
Ichiki, K., Takahashi, K., Sugiyama, N.: arXiv:1112.4705v1 [astro-ph.CO]
Ade, P.A.R., et al.: (Planck intermediate results. XXXIII), arXiv:1411.2271v1 [astro-ph.GA]
Esseya, W., Andob, S., Kusenko, A.: Astropart. Phys. 35, 135 (2011).arXiv:1012.5313
Kahniashvili, T.: Phys. Rev. D 82, 083005 (2010)
Ade, P.A.R., et al.: Planck 2015 Results. XIX. Constraints on Primordial Magnetic Fields. arXiv:1502.01594v1 [astro-ph.CO]
Subramanian, K., Barrow, J.: Phys. Rev. D 58, 083502 (1998). arXiv:astro-ph/9712083v1
Jedamzik, K., Katalinić, V., Olinto, A.: Phys. Rev. D 57, 3264 (1998). arXiv:astro-ph/9606080v2
Moss, D., Sokoloff, D.: arXiv:1307.0142v1 [astro-ph.CO]
Grasso, D., Rubinstein, H.: Phys. Rep. 348, 163 (2001). arXiv:astro-ph/0009061v2
Widrow, L.: Rev. Mod. Phys. 74, 775 (2002). arXiv:astro-ph/0207240v1
Barrow, J., Maartens, R., Tsagas, C.: Phys. Rep. 449, 131 (2007). arXiv:astro-ph/0611537v4
Kandus, A., Kunze, K., Tsagas, C.: arXiv:1007.3891v2 [astro-ph.CO]
Ryu, D., et al.: Space Sci. 166, 1 (2012)
Widrow, L., et al.: Space. Sci. Rev. 166, 37 (2012)
Yamazaki, D., et al.: Phys. Rep. 517, 141 (2012)
Durrer, R., Neronov, A.: arXiv:1303.7121v2 [astro-ph.CO]
Turner, M.S., Widrow, L.M.: Phys. Rev. D 37, 2743 (1988); [SPIRES]
Bamba, K., Yokoyama, J.: Phys. Rev. D 69, 043507 (2004)
Bamba, K., Yokoyama, J.: Phys. Rev. D 70, 083508 (2004)
Ratra, B.: GRP-287/CALT-68-1751 (1991)
Ratra, B.: Astrophys. J. 391, L1 (1992)
Martin, J., Yokoyama, J.: JCAP 0801, 025 (2008).arXiv:0711.4307v1 [astro-ph].CO
Subramanian, K.: Astron. Nachr. 331, 110 (2010).arXiv:0911.4771v2 [astro-ph.CO]
Himmetoglu, B.: Vector Fields During Cosmic Inflation: Stability Analysis and Phenomenological Signatures. PhD Thesis (2010)
Kanno, S., Soda, J., Watanabe, M.A.: JCAP 0912, 009 (2009). arXiv:0908.3509 [astro-ph.CO]
Demozzi, V., Mukhanov, V., Rubinstein, H.: JCAP 0908, 025 (2009).arXiv:0907.1030v1 [astro-ph.CO]
Ferreira, R.J.Z., Jain, R.K., Sloth, M.S.: JCAP 10, 004 (2013).arXiv:1305.7151v3 [astro-ph.CO]
Linde, A.: Phys. Lett. B129, 177 (1983)
Vilenkin, A.: arXiv:gr-qc/0409055v1
Martin, J., Ringeval, C., Vennin, V.: arXiv:1303.3787v3 [astro-ph.CO]
Lucchin, F., Matarrese, S.: Phys. Rev. D 32, 1316 (1985)
Komatsu, E., et al.: (WMAP7) Astrophys. J. Suppl. 192, 18,(2011). arXiv:1001.4538v3 [astro-ph.CO]
Hinshaw, G., et al.: Astrophys. J. Suppl. 208, 19 (2013); (WMAP9). arXiv:1212.5226 [astro-ph.CO]
Ade, P.A.R., et al.: Planck 2013 results. XXII. Constraints on inflation. Astron. Astrophys. 571, A22 (2014). arXiv:1303.5082v2 [astro-ph.CO]
Linde, A.: arXiv:1402.0526v2 [hep-th]
Ade, P.A.R., et al. BICEP2 collaboration. Phys. Rev. Lett. 112, 241101 (2014). arXiv:1403.3985
Martin, J., Trotta, C.R.R., Vennin, V.: JCAP 03, 039 (2014)
Martin, J., et al.: Phys. Rev. D 90, 063501 (2014).arXiv:1405.7272v2 [astro-ph.CO]
Kallosh, R., Linde, A., Westphal, A.: Phys. Rev. D 90, 023534 (2014).arXiv:1405.0270v1 [hep-th]
Kallosh, R., Linde, A., Roest, D.: arXiv:1405.3646v1 [hep-th]
Kobayashi, T., Seto, O.: Phys. Rev. D 89, 103524 (2014).arXiv:1403.5055v2 [astro-ph.CO]
Chiba, T., Kohri, K.: arXiv:1406.6117v1 [astro-ph.CO]
Freese, K., Frieman, J.A., Olinto, A.V.: Phys. Rev. Lett. 65, 3233 (1990)
Freese, K., Kinney, W.H.: arXiv:1403.5277v3 [astro-ph.CO]
Ade, P.A.R., et al.: Planck Intermediate Results. XXX. The Angular Power Spectrum of Polarized Dust. arXiv:1409.5738v1 [astro-ph.CO]
Ade, P.A.R., et al.: A Joint Analysis of BICEP2/Keck Array and Planck Data 2015. arXiv:1502.00612v1 [astro-ph.CO]
Ade, P.A.R., et al.: Planck 2015 Results. XX. Constraints on Inflation. arXiv:1502.02114v1 [astro-ph.CO]
AlMuhammad, A., Lopez-Mobilia, R.: The Early Universe \({f^2}FF\) Model of Primordial Magnetic Field at Large Field Inflation (submitted to publication)
AlMuhammad, A.: Inflationary Magnetogenesis in R2-Inflation After Planck (2015) (submitted to publication)
Liddle, A.R. Lyth. D.H.: Cosmological Inflation and Large-Scale Structure. Cambridge University Press (2000)
Liddle, A.R., Parsons, P., Barrow, J.D.: Phys. Rev. D 50, 7222 (1994). arXiv:astro-ph/9408015v1
Martin, J., Schwarz, D.: Phys. Rev. D 62, 103520 (2000). arXiv:astro-ph/9911225v2
Mukhanov, V.: Physical Foundation of Cosmology. Cambridge University Press (2005)
Martin, J., Ringeval, C.: Phys. Rev. D 82, 023511 (2010).arXiv:1004.5525v2 [astro-ph.CO]
Acknowledgments
We would like to thank Bharat Ratra for useful comments. This work is supported in part by the Department of Physics and Astronomy in The University of Texas at San Antonio.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
AlMuhammad, A.S., Lopez-Mobilia, R. Primordial magnetic field generated in natural inflation. Gen Relativ Gravit 47, 134 (2015). https://doi.org/10.1007/s10714-015-1978-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10714-015-1978-1