Recently, the Planck collaboration has reported a hemispherical power asymmetry in the CMB [1], which conformed the result of WMAP, but it has better precision. Such asymmetry has also been found by estimating the power spectrum in the two hemispheres by using the quadratic maximum likelihood [2]. In addition, the Planck collaboration has also reported a power deficit in the low-l\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l$$\end{document} CMB power spectrum at l≲40\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l\lesssim 40$$\end{document} [1] with the statistical significance 2.5∼3σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2.5\sim 3\sigma $$\end{document}, which is not concordant with the Planck best-fit model, although the data points are still consistent well with the cosmic variance.

The Planck data have larger statistical significance than the WMAP data, which makes it difficult to attribute the anomalies to the foregrounds, e.g. [3, 4]. Thus it seems that these anomalies should have an underlying and common physical origin, which deserves to be considered seriously.

The CMB power asymmetry might be modeled as a dipole modulation of the power [5, 6], see also [7], which results from a superhorizon perturbation crossing the observable Universe [8, 9]. This modulation can be explained in light of the spatial change of the spectrum of primordial curvature perturbation R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R}$$\end{document},1PR1/2(k,x)=1+A(k)p^·xxlsPR1/2(k),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal{P}^{1/2}_\mathcal{R}(k,\mathbf {x})=\left( 1+ A(k){\hat{\mathbf {p}}\cdot \mathbf {x}\over x_\mathrm{ls}}\right) \mathcal{P}^{1/2}_{\mathcal{R}}(k), \end{aligned}$$\end{document}where p^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\mathbf {p}}$$\end{document} is the unit vector of the dipole modulation direction, xls\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_\mathrm{ls}$$\end{document} is the distance to the last scattering surface, PR(k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{P}_{\mathcal{R}}(k)$$\end{document} is the power spectrum with index nR(k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\mathcal{R}(k)$$\end{document}, and A(k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A(k)$$\end{document} is the amplitude of modulation, which is [9, 10]2A(k)=|∇PR1/2(k,x)|PR1/2xls=1-ϵnR(k)-12kLxlsPR,L1/2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A(k)&= {|\nabla \mathcal{P}^{1/2}_{\mathcal{R}}(k,\mathbf {x})|\over \mathcal{P}^{1/2}_{\mathcal{R}}}\,x_\mathrm{ls}\nonumber \\&= \left( 1-\epsilon \right) \left[ {n_\mathcal{R}(k)-1\over 2}\right] k_\mathrm{L} x_\mathrm{ls}\mathcal{P}^{1/2}_{\mathcal{R},\mathrm{L}}, \end{aligned}$$\end{document}where PR,L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{P}_{\mathcal{R},\mathrm{L}}$$\end{document} is the amplitude of the power spectrum of the modulating mode kL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_\mathrm{L}$$\end{document}, and ϵ=-H˙/H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon =-{\dot{H}}/H^2$$\end{document}. We have (kLxls)PR,L1/2≲0.1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k_L x_\mathrm{ls})\mathcal{P}^{1/2}_{\mathcal{R},\mathrm{L}}\lesssim 0.1$$\end{document} [8, 9, 11, 12].

In single field inflationary scenario, the spectrum ninf-1∼0.04\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{\mathrm{inf}}-1\sim 0.04$$\end{document} is almost scale invariant. Thus on large angular scales the amplitude of the modulation is too small to fit the observation [8, 9]. In addition, the almost scale invariance of the inflationary spectrum also fails to explain the power deficit on large angular scales.

However, it could be observed that a large amplitude of the modulation consistent with the observations actually requires the breaking of the scale invariance of power spectrum on large angular scales, while simultaneously such a breaking also helps to explain the power suppression on corresponding scales, e.g. [10]. In this angle of view, the anomalies on large angular scales may be a hint of the pre-inflationary physics, which might be relevant with the initial singularity, e.g. [13–16].

Here, we will show that an inflationary model, in which before the slow-roll inflation the Universe is in a superinflationary phase, can generate a large-scale cutoff in the primordial power spectrum, which may account for not only the power suppression on large angular scales, but also a large dipole power asymmetry in the CMB.

It is generally thought that the pre-inflationary physics ought to be controlled by a fundamental theory, e.g. string theory. How to embed the inflationary scenario into string theory has been a significant issue, which has been studied intensively; see Reference [17]. Thus it is intriguing and might be naturally expected that a stringy mechanism of inflation could give rise to the CMB anomalies on large angular scales, e.g. [4, 18] with a string landscape, and also [19, 20] with a fast-roll phase in fiber inflation [21]. For how to involve the degrees of freedom of the standard model; see e.g. Reference [22, 23]. We will discuss an implementation of our model in string theory, based on References [24, 25].

We first will calculate the primordial perturbation generated in such an inflationary model, and identify the corresponding modulating mode from a superinflationary phase.

The equation of the curvature perturbation R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal R$$\end{document} in momentum space is3uk″+cs2k2-z″zuk=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_k^{\prime \prime } +\left( c^2_s k^2-{z^{\prime \prime }\over z}\right) u_k = 0, \end{aligned}$$\end{document}after uk≡zRk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_k \equiv z\mathcal{R}_k$$\end{document} is defined, where ′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$'$$\end{document} is for the derivative with respect to the conformal time η=∫dt/a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta =\int \mathrm{d}t/a$$\end{document}, z≡a2MP2ϵ/cs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z\equiv {a\sqrt{2M_P^2\epsilon }/ c_s}$$\end{document}. We have cs2=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_s^2=1$$\end{document} for a canonical scalar field.

The Universe initially is in a superinflationary phase with ϵPre-inf∼-O(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon _{\mathrm{Pre-inf}}\sim -\mathcal{O}(1)$$\end{document}; hereafter, it will get into an inflationary phase with ϵinf≪1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon _\mathrm{inf}\ll 1$$\end{document}. We will neglect the matching details for simplicity. Thus in conformal time, after adopting an instantaneous matching, we have4a≃a01-2H0η,forthesuperinflationa01-H0η,fortheinflation,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} a&\simeq {a_0\over \sqrt{1-2\mathcal{H}_0\eta }}, \,\,\,\mathrm{for}\,\,\mathrm{the }\,\,\mathrm{superinflation} \nonumber \\&{a_0\over 1-\mathcal{H}_0\eta }, \,\,\, \mathrm{for} \,\,\mathrm{the }\,\,\mathrm{inflation}, \end{aligned}$$\end{document}where η<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta <0$$\end{document} in the superinflationary phase and η>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta >0$$\end{document} in the inflationary phase, respectively, and a=a0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=a_0$$\end{document} for η=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta =0$$\end{document} is set, H0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{H}_0$$\end{document} is the comoving Hubble length at matching time η=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta =0$$\end{document}, which sets the inflationary energy scale by Hinf=H0/a0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_\mathrm{inf}=\mathcal{H}_0/a_0$$\end{document}. Here, ϵPre-inf=-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon _{\mathrm{Pre-inf}}=-1$$\end{document} is applied. In principle, another value with |ϵPre-inf|≳1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\epsilon _{\mathrm{Pre-inf}}| \gtrsim 1$$\end{document} may also be used, which, however, hardly would alter the result qualitatively. The evolution of the superinflationary phase with arbitrary ϵ<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon <0$$\end{document} and the primordial perturbation generated have been studied earlier in Reference [26]. The case with ϵ≪-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon \ll -1$$\end{document} corresponds to the slow expansion scenario of the primordial universe, which has been proposed earlier in Reference [27] and investigated in detail in Reference [28–30].

When k2≫z″z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k^2\gg {z^{\prime \prime }\over z}$$\end{document}, the perturbation is deep inside its horizon, we have uk∼12ke-ikη\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_k\sim {1\over \sqrt{2k}} e^{-ik\eta }$$\end{document}. In the superinflationary phase,5z″z≃3H02(1-2H0η)2.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {z^{\prime \prime }\over z}\simeq {3\mathcal{H}_0^2\over (1-2\mathcal{H}_0 \eta )^2}. \end{aligned}$$\end{document}When k2≪z″z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k^2\ll {z^{\prime \prime }\over z}$$\end{document}, the solution of Eq. (3) is6uk=π(1-2H0η)8H0H1(1)-kη+k2H0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {u_k}=\sqrt{\pi (1-2\mathcal{H}_0\eta )\over 8\mathcal{H}_0}H_1^{(1)}\left( -k\eta +{k\over 2\mathcal{H}_0}\right) , \end{aligned}$$\end{document}where H1(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_1^{(1)}$$\end{document} is the first-order Hankel function of the first kind.

In the inflationary phase,7z″z≃2H02(1-H0η)2.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {z^{\prime \prime }\over z} \simeq {2\mathcal{H}_0^2 \over (1-\mathcal{H}_0\eta )^2}. \end{aligned}$$\end{document}When k2≪z″z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k^2\ll {z^{\prime \prime }\over z}$$\end{document}, i.e. -kη+k/H0≪1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-k\eta +{k/\mathcal{H}_0}\ll 1$$\end{document}, the solution of Eq. (3) is8uk=-kη1-1H0η×C1H3/2(1)-kη+kH0+C2H3/2(2)-kη+kH0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&{u_k} = \sqrt{-k\eta }\,\sqrt{1 -{1\over \mathcal{H}_0\eta }}\nonumber \\&\quad \times \left( C_1 H_{3/2}^{(1)}\left( -k\eta +{k\over \mathcal{H}_0}\right) +C_2 H_{3/2}^{(2)}\left( -k\eta +{k\over \mathcal{H}_0}\right) \right) ,\nonumber \\ \end{aligned}$$\end{document}where H3/2(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{3/2}^{(1)}$$\end{document} is the 3/2th-order Hankel function of the first kind, H3/2(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{3/2}^{(2)}$$\end{document} is the 3/2th-order Hankel function of the second kind, C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_1$$\end{document}, C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_2$$\end{document}∼1/H0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim 1/\sqrt{\mathcal{H}_0}$$\end{document} are only dependent on k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}.

We require that all physical quantities continuously pass through the matching surface. The continuity of the curvature perturbation gives9C1=iπe-ik/H016H01-H0kiH0(1)k2H0-H2(1)k2H0-πe-ik/H08H01-2H02k2-2H0kiH1(1)k2H0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C_1&= { \frac{i\pi e^{-ik/\mathcal{H}_0}}{16 \sqrt{\mathcal{H}_0}}\left( 1-\frac{\mathcal{H}_0}{k}i\right) } \left[ H^{(1)}_0\left( \frac{k}{2\mathcal{H}_0 }\right) -H^{(1)}_2\left( \frac{k}{2\mathcal{H}_0}\right) \right] \nonumber \\&- \frac{\pi e^{-ik/\mathcal{H}_0}}{8\sqrt{\mathcal{H}_0}}\left( 1-\frac{2\mathcal{H}_0^2}{k^2}-\frac{2\mathcal{H}_0}{k}i\right) H^{(1)}_1\left( \frac{k}{2\mathcal{H}_0}\right) ,\end{aligned}$$\end{document}10C2=-iπeik/H016H01+H0kiH0(1)k2H0-H2(1)k2H0-πeik/H08H01-2H02k2+2H0kiH1(1)k2H0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C_2&= { -\frac{i\pi e^{ik/\mathcal{H}_0}}{16\sqrt{\mathcal{H}_0}}\left( 1+\frac{\mathcal{H}_0}{k}i\right) } \left[ H^{(1)}_0\left( \frac{k}{2\mathcal{H}_0 }\right) -H^{(1)}_2\left( \frac{k}{2\mathcal{H}_0}\right) \right] \nonumber \\&- \frac{\pi e^{ik/\mathcal{H}_0}}{8\sqrt{\mathcal{H}_0}}\left( 1-\frac{2\mathcal{H}_0^2}{k^2}+\frac{2\mathcal{H}_0}{k}i\right) H^{(1)}_1\left( \frac{k}{2\mathcal{H}_0}\right) , \end{aligned}$$\end{document}where H0(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_0^{(1)}$$\end{document} is the zeroth-order Hankel function of the first kind and H2(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_2^{(1)}$$\end{document} is the second-order Hankel function of the first kind.

Thus the power spectrum of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal R$$\end{document} is11PR=k32π2ukz2=PRinf2πkC1-C22,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal{P}_\mathcal{R} = {k^3\over 2\pi ^2}\left| {u_k\over z}\right| ^2=\mathcal{P}_\mathcal{R}^\mathrm{inf} {2\over \pi }k\left| C_1 -C_2\right| ^2, \end{aligned}$$\end{document}where PRinf=Hinf24π2MP2ϵinf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{P}_\mathcal{R}^\mathrm{inf} = {H_\mathrm{inf}^2\over 4 \pi ^2 M_P^2 \epsilon _\mathrm{inf}}$$\end{document} is that of the standard slow-roll inflation, which may has a slight red spectrum consistent with the observation, and C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_1$$\end{document} and C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_2$$\end{document} are determined by Eqs. (9) and (10), respectively. The spectrum index of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal R$$\end{document} is nR=ninf+dln(k|C1-C2|2)dlnk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\mathcal{R}= n_\mathrm{inf}+{d\ln {(k|C_1 -C_2|^2)}\over d\ln k}$$\end{document}.

In Reference [31], the perturbation from a superinflationary phase was also calculated. However, it is assumed that before the superinflationary phase a nonsingular bounce appears, which is not required here.

Here, H0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{H}_0$$\end{document} is the comoving Hubble length at matching surface η=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta =0$$\end{document}. The modulating mode corresponds to that on large scales k≪H0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ll \mathcal{H}_0$$\end{document}, which is generated during the superinflationary evolution.

We may expand the Hankel functions in terms of k≪H0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ll \mathcal{H}_0$$\end{document} and have12PRk<H0≃PRinf2kπH01+k212H02lnkH02∼kH0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal{P}_\mathcal{R}^{k < \mathcal{H}_0} \simeq \mathcal{P}_\mathcal{R}^\mathrm{inf}\frac{2k}{\pi \mathcal{H}_0}\left( 1+\frac{k^2}{12\mathcal{H}^2_0}\ln \frac{k}{\mathcal{H}_0}\right) ^2 \sim {k\over \mathcal{H}_0}. \end{aligned}$$\end{document}The details of the calculations are given in the appendix. Thus the spectrum is strongly blue tilt.

Meanwhile, at intermediate and small angular scales, i.e. k≫H0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \gg \mathcal{H}_0$$\end{document}, we have13PRk>H0≃PRinf1-3H08ksin2kH0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal{P}_\mathcal{R}^{k> \mathcal{H}_0}\simeq \mathcal{P}_\mathcal{R}^\mathrm{inf}\left( 1-\frac{{3\mathcal{H}_0}}{{8k}}\sin \left( \frac{2k}{\mathcal{H}_0}\right) \right) . \end{aligned}$$\end{document}Thus the spectrum is almost scale invariant but modulated with a small oscillation, which is the standard result of slow-roll inflationary evolution. We plot PR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{P}_\mathcal{R}$$\end{document} in Eq. (11) in Fig. 1, which is consistent with our analytical result. Here, it is just the superinflationary evolution that brings the modulating mode with 1-ϵPre-inf≳1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-\epsilon _{\mathrm{Pre-inf}}\gtrsim 1$$\end{document} and nR-1≳1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\mathcal{R}-1\gtrsim 1$$\end{document} on large angular scales. Fig. 1

Best-fit primordial power spectrum of curvature perturbations for the pure power law (dashed) and our model (solid) using Planck+WP data

We will show the fit of our model to the CMB TT spectrum, and also the corresponding signals in the TE and EE power spectra.

The slow-roll inflationary spectrum PRinf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{P}_\mathcal{R}^\mathrm{inf}$$\end{document} in Eq. (11) may be parameterized as a power law with PRinf=Ainf(k/k0)ninf-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{P}_\mathcal{R}^\mathrm{inf}\!=\!A_\mathrm{inf}(k/k_0)^{n_\mathrm{inf}-1}$$\end{document}, where Ainf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_\mathrm{inf}$$\end{document} is the amplitude of perturbation; see [32] for possible features in the primordial power spectrum and [33–35] for the general shape reconstructed from the CMB data. We follow Reference [1] and choose the pivot scale to be k0=0.05Mpc-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_0=0.05\, \mathrm{Mpc}^{-1}$$\end{document}, roughly in the middle of the logarithmic range of scales probed by Planck.

We assume that the late-time cosmology is the standard flat Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document}CDM model described by four free cosmological parameters: Ωbh2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _bh^2$$\end{document}, Ωch2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _ch^2$$\end{document}, Θs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta _s$$\end{document} and τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document}. Here h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h$$\end{document} is the dimensionless Hubble parameter such that H0=100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_0 = 100$$\end{document} h km s-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} Mpc-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} (noting that here H0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_0$$\end{document} is not related with the cutoff scale H0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{H}_0$$\end{document}), Ωbh2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _b h^2$$\end{document} and Ωch2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _ch^2$$\end{document} are the physical baryon and dark matter densities relative to the critical density at the present day, respectively, Θs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta _s$$\end{document} is the ratio of the sound horizon to the angular diameter distance at the photon decoupling, and τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} is the Thomson scattering optical depth due to reionization.

We modify the numerical Boltzmann code CAMB [36] to calculate the lensed TT, TE, EE power spectra and two-point correlation function, and show the results in Fig. 2. The blue dashed curves show the pure power law while the black solid curves show our model (11) with the best-fit value of ln(H0/Mpc-1)=-7.47\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ln (\mathcal{H}_0/\mathrm{Mpc}^{-1})=-7.47$$\end{document}. We see that the TT, TE, and EE spectra for our model are suppressed in the range l<6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l<6$$\end{document}, compared to the pure power law. Since the corresponding signals are induced in the TE and EE spectra, the ongoing Planck polarization data are expected to improve the constraints on the model parameter H0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{H}_0$$\end{document}. As shown in [37], the polarization data can be used to test the parity asymmetry of the CMB pattern. Note that there is a small bump around l∼10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l\sim 10$$\end{document} in the TT spectrum due to oscillations of the primordial power spectrum at large scales. The predicted two-point correlation function at θ>50∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta > 50^\circ $$\end{document} fits the Planck data much better than the pure power-law spectrum [38, 39]. Fig. 2

Best-fit TT (upper left), TE (lower left), EE (lower right) power spectra, and two-point correlation function (upper right) for the pure power law (dashed) and our model (solid) using Planck+WP data. The red points show the Planck data with 1σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} errors

How to embed such an inflationary model into string theory is interesting. We will discuss an implementation of our model in string theory. In warped compactifications with the brane/flux annihilation [58], the effective potential controlling the relevant evolution may potentially support a cosmological inflation [24, 25]. However, we find that there may be a superinflationary phase before the slow-roll inflation.

In a ten dimensional CY manifold with a warped KS throat, the metric of the warped throat is14ds2=1f(r)ds(4)2+f(r)(dr2+r2ds(5)2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathrm{d}s^2= {1\over \sqrt{f(r)}}\mathrm{d}s^2_{(4)}+\sqrt{f(r)}(\mathrm{d}r^2+r^2 \mathrm{d}s_{(5)}^2) \end{aligned}$$\end{document}for r<r∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r<r_*$$\end{document}, where r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r$$\end{document} is the proper distance to the tip of the throat, ds(5)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{d}s^2_{(5)}$$\end{document} is the angular part of the internal metric, and f(r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(r)$$\end{document} is the warp factor, which has a minimal value at r0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_0$$\end{document} and is determined by β≡r0R∼e-2πK3gsM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \equiv {r_0\over R}\sim e^{-{2\pi \mathcal{K}\over 3g_s \mathcal{M}}}$$\end{document}, in which R4=27π4gsNα′2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^4 ={27\pi \over 4} g_s N\alpha ^{\prime 2}$$\end{document}, N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N$$\end{document} equals the product of the fluxes M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{M}$$\end{document} and K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{K}$$\end{document} for the RR and NSNS three-forms, respectively, gs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_s$$\end{document} is the string coupling and α′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha ^\prime $$\end{document} is set by the string scale. When r>r∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r>r_*$$\end{document}, this metric can be glued to the metric of the bulk of the compact space, which is usually taken to be a CY manifold. When r0<r<r∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_0< r <r_{*}$$\end{document}, f(r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(r)$$\end{document} is approximately f(r)=(Rr)4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(r)= ({R\over r})^4$$\end{document}.

We follow Reference [58]. When p(≪M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\ll \mathcal{M})$$\end{document}D3¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{ D3 }$$\end{document}-branes sit at the tip of KS throat, the system is a nonsupersymmetric NS5-brane “giant graviton” configuration, in which the NS5-brane warps a S2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^2$$\end{document} in S3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^3$$\end{document}, and carries p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document} unites flux, which induces the D3¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{ D3 }$$\end{document}-charge. S2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^2$$\end{document} is inclined to expand as a spherical shell in S3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^3$$\end{document}, which may be parameterized by an angle 0⩽ψ⩽π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 \leqslant \psi \leqslant \pi $$\end{document}, in which ψ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi =0$$\end{document} corresponds to the north pole of S3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^3$$\end{document} and ψ=π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi =\pi $$\end{document} is the south pole. The angular position may be regarded as a scalar in the world volume action, which describes the motion of the NS5-brane across the S3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^3$$\end{document}. The effective potential controlling the relevant evolution is15Veff(ψ)=Mβ4T3b02sin4ψπ2+V~2(ψ)+V~(ψ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} V_\mathrm{eff}(\psi )= \mathcal{M}\beta ^4 T_3\left( \sqrt{{b_0^2 \sin ^4{\psi }\over \pi ^2}+{\tilde{V}}^2(\psi )}+{\tilde{V}}(\psi )\right) \end{aligned}$$\end{document}with b0≃0.9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_0\simeq 0.9$$\end{document}, where V~(ψ)=pM-ψ-sin(2ψ)2π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{V}}(\psi ) ={p\over \mathcal{M}}-{\psi -{\sin {(2\psi )}\over 2}\over \pi }$$\end{document} and T3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_3$$\end{document} is the D3¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{ D3 }$$\end{document}-brane tension. This potential is plotted in Fig. 4 with respect to ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document}.

In the regime with p/M<0.08\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p/\mathcal{M} < 0.08$$\end{document}, the metastable bound state forms, which corresponds to a static NS5-brane wrapping a S2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^2$$\end{document} in S3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^3$$\end{document}.

This metastable bound state corresponds to ψ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi = 0$$\end{document} and Veff(0)=2pβ4T3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ V_\mathrm{eff}(0)=2p\beta ^4 T_3$$\end{document}; see Fig. 4. The true minimum is at ψ=π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi = \pi $$\end{document}, in which the potential energy is 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0$$\end{document}.

In the regime p/M≳0.08\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p/\mathcal{M} \gtrsim 0.08$$\end{document}, this metastable state disappears, which implies that the nonsupersymmetric configuration of p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}D3¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{ D3 }$$\end{document}-branes becomes classically unstable and will relax to a supersymmetric minimum by a classical rolling of ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document} along its potential. This classical rolling may lead to a slow-roll inflation, which has been studied in detail in Reference [25]. When ψ=π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi =\pi $$\end{document}, in which the potential energy is 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0$$\end{document}, the inflation will end. The result of this evolution is M-p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{M}-p$$\end{document} D3-branes instead of the original p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}D3¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{ D3 }$$\end{document}-branes appearing at the tip of the throat, while the three-form flux K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{K}$$\end{document} is changed to K-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{K}-1$$\end{document}, i.e. we have brane/flux annihilation [58].

During the period before the slow-roll inflation, in which p/M<0.08\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p/\mathcal{M} < 0.08$$\end{document}, the Hubble expansion of the Universe is given by16H2=2pβ4T33,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} H^2= {2p\beta ^4 T_3 \over 3}, \end{aligned}$$\end{document}where 8π/MP2=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$8\pi /M_P^2= 1$$\end{document}. When D3¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{ D3 }$$\end{document}-branes are pulled into the throat continuously, the metastable minimum will increase [59], see Fig. 4, which implies that H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H$$\end{document} will increase rapidly during this period.

Thus the parameter ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document} is17ϵPre-inf=-H˙H2∼-p˙2Hp.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \epsilon _{\mathrm{Pre-inf}} =-{{\dot{H}}\over H^2}\sim -\left( {{\dot{p}}\over 2Hp}\right) . \end{aligned}$$\end{document}Thus in units of Δt=1/H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t= 1/H$$\end{document}, we approximately have |ϵPre-inf|∼Δp2p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\epsilon _{\mathrm{Pre-inf}}| \sim {\Delta p\over 2p}$$\end{document}, where Δp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta p$$\end{document} is the change of p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document} in unit of 1/H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/H$$\end{document}.

We assume Δp2p≳1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta p\over 2p}\gtrsim 1$$\end{document}, which may be consistent with M∼104\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{M}\sim 10^4$$\end{document} and pI∼O(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{I}\sim \mathcal{O}(1)$$\end{document}, where pI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{I}$$\end{document} is the initial number of D3¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{ D3 }$$\end{document}-branes at the tip of the KS throat. Here, all the moduli is assumed to be fixed, and the interaction between D3¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{ D3 }$$\end{document}-branes has been also neglected for simplicity.Fig. 4

The figure of the potential Eq. (15). When D3¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{ D3 }$$\end{document}-branes are pulled into the throat continuously, the metastable minimum will rise inch by inch