Tensor-to-scalar ratio in punctuated inflation

Rajeev Kumar Jain, Pravabati Chingangbam, L. Sriramkumar, and Tarun Souradeep

Phys. Rev. D 82, 023509 – Published 14 July 2010

Abstract

Recently, we have shown that scalar spectra with lower power on large scales and certain other features naturally occur in punctuated inflation, i.e. the scenario wherein a brief period of rapid roll is sandwiched between two stages of slow roll inflation. Such spectra gain importance due to the fact that they can lead to a better fit of the observed CMB anisotropies, when compared to the conventional, featureless, power law spectrum. In this paper, with examples from the canonical scalar field as well as the tachyonic models, we illustrate that, in punctuated inflation, a drop in the scalar power on large scales is always accompanied by a rise in the tensor power and, hence, an even more pronounced increase in the tensor-to-scalar ratio $r$ on these scales. Interestingly, we find that $r$ actually exceeds well beyond unity over a small range of scales. To our knowledge, this work presents for the first time, examples of single scalar field inflationary models wherein $r ≫ 1$ . This feature opens up interesting possibilities. For instance, we show that the rise in $r$ on large scales translates to a rapid increase in the angular power spectrum, $C_{ℓ}^{BB}$ , of the $B$ -mode polarization of the CMB at the low multipoles. We discuss the observational implications of these results.

Received 19 February 2010

DOI:https://doi.org/10.1103/PhysRevD.82.023509

Authors & Affiliations

Rajeev Kumar Jain^1,*, Pravabati Chingangbam^2,†, L. Sriramkumar^1,‡, and Tarun Souradeep^3,§

¹Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India
²Korea Institute for Advanced Study, 207–43 Cheongnyangni 2-dong, Dongdaemun-gu, Seoul 130-722, Korea
³IUCAA, Post Bag 4, Ganeshkhind, Pune 411 007, India

^*Present address: Department of Theoretical Physics, University of Geneva, 24 Quai Ernest-Ansermet, CH-1211, Geneva 4, Switzerland; rajeev.jain@unige.ch
^†Present address: Astrophysical Research Center for the Structure and Evolution of the Cosmos, Sejong University, 98 Gunja Dong, Gwangjin gu, Seoul 143-747, Korea; prava@kias.re.kr
^‡sriram@hri.res.in
^§tarun@iucaa.ernet.in

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 82, Iss. 2 — 15 July 2010

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
The scalar power spectrum $P_{S} (k)$ (the solid black line) and the tensor power spectrum $P_{T} (k)$ (the dashed black line) have been plotted as a function of the wave number $k$ for the cases of $n = 3$ (on the left) and $n = 4$ (on the right). These spectra correspond to the following values of the potential parameters: $m = 1.5368 \times 10^{- 7}$ and $λ = 6.1517 \times 10^{- 15}$ (corresponding to $ϕ_{0} = 1.9594$ ) for $n = 3$ case and $m = 1.1406 \times 10^{- 7}$ and $λ = 1.448 \times 10^{- 16}$ (corresponding to $ϕ_{0} = 2.7818$ ) for $n = 4$ case. The red curve in these plots is the spectrum (9) with the exponential cutoff, whose parameters have been arrived at by a simple visual comparison with the numerically evaluated scalar spectrum. It corresponds to $A_{S} = 2 \times 10^{- 9}$ , $n_{S} = 0.945$ , $α = 3.35$ , and $k_{*} = 2.4 \times 10^{- 4} {Mpc}^{- 1}$ in the $n = 3$ case, while $A_{S} = 2 \times 10^{- 9}$ , $n_{S} = 0.95$ , $α = 3.6$ , and $k_{*} = 9.0 \times 10^{- 4} {Mpc}^{- 1}$ in the case of $n = 4$ . Note that the vertical blue lines denote $k_{*}$ . We should mention that, in the two slow roll regimes, the spectral amplitudes evaluated in the slow roll approximation [cf. Eqs. (5, 6)] match the above exact numerical spectra quite well. The horizontal dotted lines indicate the maximum value of the tensor amplitude that can arise in these MSSM-motivated, punctuated inflationary models.Reuse & Permissions
Figure 2
The tensor-to-scalar ratio $r (k)$ for the cases of $n = 3$ (the solid black line) and $n = 4$ (the dashed black line) has been plotted as a function of the wave number $k$ . These plots have been drawn for the same choice of parameters as in the previous figure. The vertical solid and dashed blue lines denote the $k_{*}$ corresponding to the $n = 3$ and $n = 4$ cases, respectively. Note that, despite the rise at the larger wavelengths, the tensor-to-scalar ratio remains smaller than $10^{- 4}$ for modes of cosmological interest (i.e. for $k ≳ k_{*}$ ). For this reason, in our earlier work [13], we had ignored the tensor contribution when comparing with the WMAP 5-year data. Interestingly, we find that there arises a domain wherein the tensor-to-scalar ratio $r (k)$ is actually much greater than unity. To highlight this feature, we have included the horizontal green line which denotes $r = 1$ . In the Appendix, we have plotted the evolution of the scalar and tensor amplitudes for a mode from this domain.Reuse & Permissions
Figure 3
The scalar power spectrum $P_{S} (k)$ (the solid black line) and the tensor power spectrum $P_{T} (k)$ (the dashed black line) have been plotted (on the left) as a function of the wave number $k$ for the hybrid inflation model described by the potential (10). The corresponding tensor-to-scalar ratio $r (k)$ has also been plotted (on the right). As in the previous figure, the horizontal green line in the right graph denotes $r = 1$ . These spectra correspond to following values of the potential parameters: $M = 2.6 \times 10^{- 5}$ and $B = 0.552$ . The solid red curve in the left graph is the exponential cutoff spectrum (9) corresponding to $A_{S} = 2 \times 10^{- 9}$ , $n_{S} = 1.0$ , $k_{*} = 2.2 \times 10^{- 3} {Mpc}^{- 1}$ , and $α = 3.5$ . The vertical blue line in both the graphs denotes $k_{*}$ . We should mention that the blue tilt in the scalar spectrum is very small and, hence, is not evident from the figure. Moreover, we find that, as in the previous MSSM-motivated examples, in the two slow roll regimes, the amplitudes of the spectra calculated in the slow roll approximation agree very well with the exact numerical spectra. Further, it should be noted that the amplitude of the tensors in the first slow roll phase determines the maximum tensor amplitude that can arise in such a punctuated inflationary scenario.Reuse & Permissions
Figure 4
The scalar and the tensor power spectra, the corresponding tensor-to-scalar ratio as well as the exponential cutoff spectrum have been plotted for the tachyonic punctuated inflation model exactly in the same fashion as in the previous figure. These figures correspond to the following values of the parameters: $λ = 10^{- 13}$ , $T_{0} = 3.55 \times 10^{7}$ , $x_{1} = 10$ , $A_{S} = 2 \times 10^{- 9}$ , $n_{S} = 0.96$ , $k_{*} = 5.5 \times 10^{- 4} {Mpc}^{- 1}$ , and $α = 3$ . We should also point out that, as in the earlier cases, in the two slow roll regimes, the spectral amplitudes, when evaluated in the slow roll approximation, are in good agreement with the exact numerical spectra. Again, as in the previous examples, the maximum value of the tensor amplitude that can arise in such punctuated inflationary scenarios is determined by its value in the first slow roll phase.Reuse & Permissions
Figure 5
The $B$ -mode CMB angular power spectrum $C_{ℓ}^{BB}$ has been plotted as a function of the multipole $ℓ$ for the best fit values of the $n = 3$ (the solid black line) and the $n = 4$ (the dashed black line) cases of the MSSM-motivated model. For comparison, we have also plotted the $C_{ℓ}^{BB}$ for the concordant cosmological model with a strictly scale invariant tensor spectrum and a tensor-to-scalar ratio of $r = 0.01$ (the solid blue line), $r = 2 \times 10^{- 8}$ (the solid red line), and $r = 10^{- 7}$ (the dashed red line). The latter two curves match the $n = 3$ and $n = 4$ cases of the MSSM-motivated model at the small angular scales, and they help in highlighting the effects of punctuated inflation at the lower multipoles.Reuse & Permissions
Figure 6
The evolution of the amplitudes of the curvature perturbation $R_{k}$ (in blue) and the tensor perturbation $U_{k}$ (in red) has been plotted as a function of the number of $e$ -folds $N$ for the best fit values of the $n = 3$ , MSSM-motivated model. These perturbations correspond to the mode $k = 10^{- 5} {Mpc}^{- 1}$ , and the arrow denotes the time when the mode leaves the Hubble radius. Notice that, as expected, the tensor amplitude freezes at its value near Hubble exit. In contrast, the amplitude of the curvature perturbation is suppressed on super-Hubble scales. Evidently, it is this behavior of the curvature perturbation that leads to the large tensor-to-scalar ratio associated with the mode.Reuse & Permissions

Physical Review D

covering particles, fields, gravitation, and cosmology