††institutetext: Department of Physics, Cornell University, Ithaca, NY 14853,
USA††institutetext: Department of Physics, Hong Kong University of Science and
Technology, Clear Water Bay,
Hong Kong, SAR, China

# The $D3$-${\bar{D3}}$-Brane Inflation Model Revisited

S.-H. Henry Tye<EMAIL_ADDRESS>
###### Abstract

Combining previous results, the $D3$-${\bar{D3}}$-brane pair potential
$U(T,\phi)$ is presented here, where the inflaton $\phi$ measures the
separation between the $D3$-brane and the anti-${D3}$-brane, and the complex
scalar mode $T$ becomes tachyonic when the annihilation of the branes happens
as they collide. Besides the distinct form of the inflationary potential, this
hybrid inflationary model differs from a typical hybrid model in 2 important
aspects: (1) $U(T,\phi)$ becomes complex when $T$ becomes tachyonic, where
${\rm{Im}}\,U(T,\phi)$ plays an important role in the dynamics towards the end
of the inflationary epoch; (2) tunnelling during the inflationary epoch can
happen; this is particularly relevant if there are multiple
$D3$-${\bar{D3}}$-brane pairs in different warped throats. Besides the
production of cosmic superstrings, the model offers the possibility of first
order phase transition that may generate large enough density perturbation for
the primordial black hole production. Stochastic gravitational wave background
from these sources remain to be fully investigated.

##  1 Introduction

It is generally believed that our universe started with a cosmic inflationary
epoch Guth:1980zm ; Linde:1981mu . In the brane world scenario in string
theory, where we live in a stack of $3$-branes (spanning the 3 large
dimensions) inside the 9 dimensional space (where the remaining 6 dimensions
are dynamically compactified), brane inflation is quite natural Dvali:1998pa .
The simplest and most natural scenario is the $D3$-brane and the
anti-$D3$(${\bar{D3}}$)-brane inflation in Type IIB string theory Dvali:2001fw
; Burgess:2001fx . However, its toy model version does not work, as the
attractive force between the $D3$-brane and the ${\bar{D3}}$-brane is too
strong in the limited compactified space Burgess:2001fx . On the other hand,
putting the $D3$-${\bar{D3}}$-brane pair in a more realistic setting works
beautifully : dynamical compactification of the 6 extra spatial dimensions
requires warped geometry Giddings:2001yu , and thus the presence of warped
throats. The $D3$-branes and the ${\bar{D}3}$-branes naturally settle in
warped throats and it is the warping that weakens the attractive force, so the
$D3$-${\bar{D3}}$-brane inflation works nicely. This is the KKLMMT model
Kachru:2003sx .

The predictions of its inflaton potential $V(\phi)$ (where the inflaton $\phi$
measures the separation of the 2 branes) made years ago agree very well with
the Cosmic Microwave Background (CMB) data obtained by the PLANCK
collaboration Planck:2016 . So it is worthwhile to examine the model more
closely. Since the full $D3$-${\bar{D3}}$-brane potential $U(T,\phi)$ of
$\phi$ and a complex $T$ mode is determined, one can check its properties in
some details and compare them with (present and future) cosmological
observations. This is especially interesting since, besides the production of
cosmic superstrings, the possibility of a first order phase transition and the
production of primordial black holes in the model after the inflationary epoch
remains to be explored. That is, the model can produce (or not produce)
phenomenons detectable in cosmological observations and so can be critically
tested. In this paper, the full potential $U(T,\phi)$ is presented, followed
by a brief discussion of its novel features. The actual phenomenological
investigation is deferred to the future.

The inflation potential $U(T,\phi)$ of the $D3$-${\bar{D3}}$-brane model
combines the results of Ref.Jones:2002sia ; Sarangi:2003sg  (presented in
Eq.(2.1, 2.2), with its simplified version in Eq.(12, 13, 28)). The model is a
function of 3 parameters, namely:
(1) the warped $D3$-brane tension $\tau_{3}$, which is determined by the
Hubble parameter $H$ and the density perturbation first observed by COBE
COBE:1992syq , and
(2) the string coupling $g_{s}\lesssim 1$. These 2 parameters are related by
the scale $M_{A}$ of the warped throat.
(3) the curvature coupling $\zeta_{\phi}H^{2}\phi^{2}$ term is an issue
encountered in all inflationary models. Here, PLANCK data Planck:2016  implies
that the parameter $|\zeta_{\phi}|<0.01$. We shall simply set
$\zeta_{\phi}=0$.
(The model also depends very weakly on the property of the warped throat in
which the $D3$-${\bar{D3}}$-brane pair sits in.)

This is a hybrid inflation model Linde:1993cn  in the sense that the slow-roll
inflation happens when $T$ sits at its minimum $T=0$ and ends when $T$ becomes
tachyonic and then rolls down the potential. It differs from the standard (and
well studied) hybrid inflation models in a number of ways:
$\bullet$ Based on a simple physical picture, the potential is well
determined. As a result of the $\phi^{-4}$ form of the attractive
(gravitational + Ramond-Ramond (RR)) potential, it predicts a scalar power
spectrum that is slightly red-tilt, in agreement with the CMB data from PLANCK
Planck:2016 .
$\bullet$ As the $D3$-brane approaches the ${\bar{D3}}$-brane (i.e., $\phi$
decreases), $T$ becomes tachyonic and the $\phi^{-4}$ potential softens to a
finite form (so the $\phi\to 0$ catastrophic divergence is absent), while
$U(T,\phi)$ develops an imaginary component, indicating the decay
(annihilation) of the $D3$-${\bar{D3}}$-brane pair. The corresponding decay
width provides a much stronger damping than that from the Hubble parameter in
the $\phi$ evolution.
$\bullet$ Even as the $D3$-${\bar{D3}}$-branes move slowly towards each other
during the inflationary epoch, $T$ can tunnel through a barrier and then roll
to infinity. Nucleation bubbles will be created during this first order phase
transition. Although this transition is unlikely to have a big cosmological
impact in the simplest $D3$-${\bar{D3}}$-brane inflation model, such a first
order phase transition phenomenon can leave behind observable signatures if
there are multiple $D3$-${\bar{D3}}$-brane pairs in different warped throats.
The presence of multiple throats with different warp factors in the dynamical
compactification of the 6 spatial dimensions is quite natural in the brane
world scenario (e.g., cf.Polchinski:1998rq ; Maartens:2003tw ; Baumann:2014nda
).

Already, the model predicts the production of cosmic superstrings that form a
network Jones:2002cv ; Sarangi:2002yt ; Jones:2003da , with properties rather
different from field theory cosmic strings Copeland:2003bj ; Jackson:2004zg .
The production of primordial black holes (PBHs) within the framework of hybrid
inflation is an interesting possibility Randall:1995dj ; Garcia-
Bellido:1996mdl ; Kawasaki:1997ju ; Abolhasani:2010kr ; Lyth:2011kj ;
Lyth:2012yp  and has been actively investigated (e.g., cf.Sasaki:2018dmp ;
Ozsoy:2023ryl ). Highly over-dense regions of inhomogeneities in the early
Universe can directly undergo gravitational collapse to form black holes
Hawking:1971ei . Physically, the collision and annihilation of a brane pair is
an explosive process, reflected in the $(T,\phi)$ evolution as well as the
presence of nucleation bubbles from a first order phase transition.
Intuitively, large density fluctuations and gravitational waves are expected
to be generated. Here we discuss briefly the general properties of the model
and its novel features.

The rest of the paper is organized as follows. Sec. 2 presents the
$D3$-${\bar{D3}}$-brane action ${\cal S}(T,\phi)$ with the potential
$U(T,\phi)$. This potential combines the result of Ref.Jones:2002sia ;
Sarangi:2003sg . Sec. 3 gives a brief review of the properties during the
inflationary epoch, where the predictions Kachru:2003sx  are in good agreement
with the PLANCK data Planck:2016 . Sec. 4 give a brief description of some of
the properties towards the end of the inflation epoch that have been
extensively studied over the years. In Sec. 5, we discuss the role of the
imaginary part ${\rm Im}\,U(T,\phi)$ in the dynamics. In Sec. 6, we discuss
qualitatively the tunnelling aspects of the model, where nucleation bubbles
from first order phase transition can come into play. Sec. 7 contains some
remarks.

##  2 The $D3$-${\bar{D3}}$-Brane Model

With the string mass scale $M_{S}$ expressed in terms of the Regge slope
$\alpha^{\prime}$, $M_{S}^{2}=1/2\pi\alpha^{\prime}$, the 6-dim
compactification volume ${\cal V}$ relates the string scale to the Newton’s
constant $G_{N}=1/8\pi M_{P}^{2}$, so $M_{P}=2.43\times 10^{18}$ GeV and

$M_{P}^{2}\sim M_{S}^{8}{\cal V}$

where $M_{S}\ll M_{P}$. The $Dp$-brane tension is given by

$T_{p}=\frac{1}{(2\pi)^{p}g_{s}\alpha^{\prime(p+1)/2}}=\frac{M_{S}^{(p+1)}}{(2\pi)^{(p-1)/2}g_{s}}$
(1)

where we expect the string coupling to lie in the range $0.1\leq g_{s}<1$.

###  2.1 The action ${\cal S}(T,\phi)$

The $D3$-${\bar{D3}}$-brane inflation takes place in a warped throat, called
the $A$-throat, while the standard model branes live in another throat, called
the $SM$-throat. In the $A$-throat, the mass scale is warped to $M_{A}$, where

$M_{A}\ll M_{S}\ll M_{P}$

so the warped $D3$-brane tension and the $D$-string tension in the A-throat
are given by, respectively,

$\tau_{3}=\frac{M_{A}^{4}}{2\pi
g_{s}},\quad\quad\tau_{1}=\frac{M_{A}^{2}}{g_{s}},$ (2)

The $D3$-${\bar{D3}}$-brane potential has 2 parameters, namely $M_{A}$ and
$g_{s}$. During the inflationary epoch, it is only $\tau_{3}$ that comes into
play.

The $D{\bar{D}}$-brane action ${\cal S}(T)$ for tachyon condensation (i.e.,
$T$ rolling down its potential) is derived in Ref.Kutasov:2000aq ;
Kraus:2000nj ; Takayanagi:2000rz  in the Boundary Superstring Field Theory
(BSFT) Witten:1992qy ; Witten:1992cr ; Shatashvili:1993kk . In
Ref.Jones:2002sia , via T-duality Myers:1999ps , the inflaton $\phi$ is
incorporated to form the action ${\cal S}(T,\phi)$ for the
$D3$-${\bar{D3}}$-brane pair system, with the Lagrangian density ${\cal
L}(T,\phi)$,111The relative position of the 2 branes in 6 spatial dimensions
involves more co-ordinates. They become (anti-)parallel after an exponential
expansion via inflation; so they can be described by a single co-ordinate
$\varphi$ as $D3$-${\bar{D3}}$-brane separation distance. We use
$\phi=\sqrt{\tau_{3}}\varphi$ as the dynamical variable with a canonical
kinetic term for slow-roll inflation.

${-\cal L}({\cal T},\phi)=e^{-{\cal T}{\bar{\cal T}}/M_{A}^{2}}{\cal
F}\left(\frac{\phi^{2}{\cal T}{\bar{\cal T}}}{\pi\tau_{3}}\right){\cal
F}\left(\frac{\partial_{\mu}{\cal T}\partial^{\mu}{\bar{\cal T}}}{\pi
M_{A}^{4}}\right)V(\phi)+\frac{\partial_{\mu}\phi\partial^{\mu}\phi}{2}$ (3)

where $V(\phi)$ is the inflaton potential, and $T$ is related to ${\cal T}$ by
a rescaling. All powers of the single derivative of $\cal T$ are contained
inside ${\cal F}$ Kraus:2000nj . The function ${\cal F}$ takes the form
Kutasov:2000aq

${\cal{F}}(x)=\frac{4^{x}x\Gamma(x)^{2}}{2\Gamma(2x)}=\frac{\sqrt{\pi}\Gamma(1+x)}{\Gamma(\frac{1}{2}+x)}$
(4)

where

$\displaystyle{\cal{F}}(x)$ $\displaystyle=1+(2\ln 2)x+\left[2(\ln
2)^{2}-\frac{\pi^{2}}{6}\right]x^{2}+{\cal O}(x^{3}),\quad\quad 0\leq x\ll 1$
(5) $\displaystyle=\sqrt{\pi x}\left[1+\frac{1}{8x}+{\cal
O}(x^{-2})\right],\quad\quad x\gg 1$ (6)

Expanding the function ${\cal F}(x)$ to leading order using Eq.(5) and setting
$V(\phi)\simeq 2\tau_{3}$, we obtain the canonical kinetic term for $T$
(around $T=0$) :

$\frac{\partial_{\mu}T\partial^{\mu}{\bar{T}}}{2}=\frac{(2\ln
2)\partial_{\mu}{\cal T}\partial^{\mu}{\bar{\cal T}}}{\pi
M_{A}^{4}}V(\phi)\simeq\frac{(4\ln 2)}{\pi^{2}g_{s}}\frac{\partial_{\mu}{\cal
T}\partial^{\mu}{\bar{\cal T}}}{2}$ (7)

Figure 1: The $D3$-${\bar{D3}}$-brane potential $U(T,\varphi)$ where
$\varphi=\phi/\sqrt{\tau_{3}}$ measures the separation distance between the 2
branes. The blue line indicates where $T$ becomes tachyonic as $\varphi$
decreases. Note that, even for a relatively large $\varphi$ where $T$ is
classically stable at $T=0$, $U(T,\varphi)\to 0$ as $T\to\infty$. This figure
is taken from Ref.Jones:2002sia .

Ignoring higher powers of $\partial_{\mu}T\partial^{\mu}{\bar{T}}$, we now
have

$\displaystyle{\cal S}(T,\phi)$ $\displaystyle=-\int
dx^{4}a^{3}(t)\left(\frac{{\dot{T}}\dot{\bar{T}}}{2}+\frac{{\dot{\phi}}^{2}}{2}+U(T,\phi)\right)$
$\displaystyle U(T,\phi)$
$\displaystyle=\exp\left[-\frac{\pi^{2}g_{s}T{\bar{T}}}{(4\ln
2)M_{A}^{2}}\right]{\cal{F}}\left(\frac{\pi g_{s}\phi^{2}T{\bar{T}}}{(4\ln
2)\tau_{3}}\right)V(\phi)$ (8)

where $a(t)$ is the cosmic scale factor and $\tau_{3}$ (2) is a function of
$M_{A}$ and $g_{s}$. The potential $U(T,\phi)$ is displayed in Figure 1 (taken
from Ref.Jones:2002sia ).

One may choose to keep all single $T$ derivative terms inside $\cal F$ in
${\cal S}(T,\phi)$(2.1) and/or express the $\phi$ kinetic term in the Dirac-
Born-Infeld form

$\tau_{3}\left(\sqrt{1-{\dot{\phi}}^{2}/\tau_{3}}-1\right)\quad\to\quad{\dot{\phi}}^{2}/2$

which reduces to the canonical form in the slow-role approximation.

We note that the mass of $T$ is given by

$m_{T}^{2}(\phi)=\pi g_{s}\left(\frac{\phi^{2}}{\tau_{3}}-\frac{\pi}{(2\ln
2)M_{A}^{2}}\right)V(\phi)\simeq 2\pi g_{s}(\phi^{2}-\phi_{c}^{2})$ (9)

so $T$ becomes a tachyon when its mass square is negative, i.e., when
$\phi^{2}<\phi_{c}^{2}$, where

$\phi_{c}^{2}=\frac{\pi\tau_{3}}{(2\ln 2)M_{A}^{2}}=\frac{M_{A}^{2}}{(4\ln
2)g_{s}}$ (10)

###  2.2 The inflaton potential

To complete, let us consider the potential $V(\phi)$ in $U(T,\phi)$ (2.1) now.
It is given by the $D3$-${\bar{D3}}$-brane tensions plus the interactive
potential ${\hat{V}}(\phi)$, which is determined (especially for
$\phi<\phi_{c}$ Sarangi:2003sg ). The term ${\hat{V}}(\phi)$ has a dual
description (e.g., cf.Polchinski:1998rq ): it can be expressed as either an
exchange of closed string modes between the 2 branes, where the lightest modes
include the graviton and the RR field, or the one-loop correction to the open
strings stretching between the 2 branes, where the lightest mode is $T$.
Choosing the latter expression, we have

$\displaystyle V(\phi)$ $\displaystyle=2\tau_{3}+{\hat{V}}(\phi)$
$\displaystyle{\hat{V}}(\phi)$
$\displaystyle=-\frac{v}{16}\left(\frac{\tau_{3}}{\phi_{c}^{2}}\right)^{2}\int_{0}^{\infty}\frac{dt}{t^{3}}\exp\left[-\frac{\pi
t\phi^{2}}{\phi_{c}^{2}}\right]Z(t)$ (11) $\displaystyle Z(t)$
$\displaystyle=e^{\pi
t}\prod_{m=1}^{\infty}\frac{(1-e^{-2\pi(m-1/2)t})^{8}}{(1-e^{-2\pi
mt})^{8}}=e^{\pi t}\ -8+36e^{-\pi t}+...$

where the first term is the $D3$-${\bar{D3}}$-brane tension that drives
inflation. At large distances (large enough $\phi$), the above integral is
dominated by small $t$, where $Z(t)\to 16t^{4}$; so the potential reduces to

$V(\phi)=2\tau_{3}+{\hat{V}}(\phi)\simeq
2\tau_{3}\left(1-v\frac{\tau_{3}}{2\pi^{2}\phi^{4}}\right),\quad\quad\phi>\phi_{c}$
(12)

and the second term comes from the gravitational plus the RR attractive
forces. The parameter $v\sim{\cal O}(1)$ measures the property of the throat.
For the Klebanov-Strassler throat Klebanov:2000hb , $v=27/16$ (e.g.,
cf.Baumann:2014nda ), a value we shall adopt in our numerical estimates.
Fortunately, the quantitative physics is insensitive to the value of $v$.

It is clear that $V(\phi)$ as given in Eq.(12) breaks down as $\phi\to 0$.
Fortunately, this disaster is actually absent in string theory. When
$\phi<\phi_{c}$, ${\hat{V}}(\phi)$ (2.2) diverges, because the lightest mode
(i.e., $e^{\pi t}$ in $Z(t)$) becomes tachyonic. This divergence is not an
infinity of the theory, but rather a signal that ${\hat{V}}(\phi)$ becomes
complex. After proper regularization by analytic continuation,
${\hat{V}}(\phi)$ develops an imaginary part as a function of $m_{T}^{2}$ (9),

${\rm{Im}}\,{\hat{V}}(\phi)={|m_{T}^{2}|^{2}}/{32\pi},$ (13)

indicating the annihilation (or decay) of the $D3$-${\bar{D3}}$-brane pair,
while the real part ${\rm{Re}}\,{\hat{V}}(\phi)$ stays finite Sarangi:2003sg .
It reaches a finite minimum ${\rm{Re}}\,{\hat{V}}(\phi_{m})$ at $\phi_{m}$
(where $\phi_{c}>\phi_{m}>0$) and $\phi$ becomes tachyonic (i.e.,
$m_{\phi}^{2}<0$) at $\phi=0$, a feature first observed in a similar situation
Garcia-Bellido:2001lbk . Note that ${\rm{Re}}\,{\hat{V}}(\phi)$ is always much
smaller than $2\tau_{3}$, so it is a reasonable approximation to treat the
Hubble value $H$ as a constant until $T$ starts to roll down. For our
qualitative discussion in this paper, we need only the overall shape of
${\rm{Re}}\,{\hat{V}}(\phi)$ for $\phi<\phi_{c}$. We shall come back to its
shape in Sec. 5.

###  2.3 Discussions

In this brane world scenario, all the energies released by the annihilation of
the $D3$-${\bar{D3}}$-brane pair can only go to $D$-strings (i.e.,
$D1$-branes) and the fundamental $F$-strings, since they are the only degrees
of freedom left, i.e., neither $D2$-branes nor $D0$-branes exist in Type IIB
string theory. They are closed string modes since there are no branes (except
$D1$-branes produced) around for the ends of open strings to end in. The
production of them can be seen more explicitly in the model :

(i) As an effective field theory, the model possesses a $U(1)$ symmetry: $T\to
e^{i\theta}T$. Since string theory has no continuous global symmetry
Banks:1988yz , such a symmetry is gauged. In fact, this gauged $U(1)$ property
leads to the presence of topological vortex-like (co-dimension 2) defects
Jones:2002sia . These defects are to be identified as $D$-strings, which will
be produced when $T$ rolls down the potential.

(ii) For large enough separation $\phi$, a $\phi^{-4}$ term in $V(\phi)$ is
generated by the open string one-loop correction. In the dual closed string
channel, this is due to the exchange of the graviton and the RR field, a
classical effect. As $\phi$ decreases, the exchange of massive closed string
modes becomes important. As $\phi$ further decreases, just as $T$ becomes
tachyonic, $V(\phi)$ develops an imaginary part Sarangi:2003sg . By optical
theorem, an ${\rm Im}\,V(\phi)$ indicates the opening of inelastic channels,
i.e., energies go to a spectrum of closed $F$-strings. That includes the
graviton, some massive string modes, string loops of cosmological sizes as
well as the light axion-like modes.

To summarize, the $D3$-${\bar{D3}}$-brane pair potential is given by
$U(T,\phi)$ (2.1), inside where the inflaton potential $V(\phi)$ (2.2) can be
simplified to Eq.(12) for $\phi\geq\phi_{c}$ (during inflation) and becomes
complex for $\phi_{c}>\phi\geq 0$ (after inflation), with imaginary part
${\rm{Im}}\,{\hat{V}}(\phi)$ (13), while the real part
${\rm{Re}}\,{\hat{V}}(\phi)$ is approximately given in Eq.(28) in Sec. 5. This
potential yields a hybrid inflation model, but with the following distinct
features that we shall briefly explore :
(1) The potential $U(T,\phi)$ is well determined in terms of only 2
parameters, namely the scale $M_{A}$ and the brane tension $\tau_{3}$, which
are related by the string coupling $g_{s}$. Other parameters (such as $v$)
play very minor roles in confronting cosmological data; this allows a critical
test of the model and check its properties.
(2) $U(T,\phi)$ develops a well determined imaginary part towards the end of
the inflationary epoch, where ${\rm{Im}}\,{\hat{V}}(\phi)$ (13) tends to damp
the $\phi$ evolution. As this damping effect is much stronger than that from
$H$, it can have a big impact on its subsequent evolution.
(3) As shown in Figure 1, $T$ can tunnel over the potential barrier and roll
to infinity; this novel feature can play an important role when there are more
than one $D3$-${\bar{D3}}$-brane pair.

##  3 The Inflationary Epoch

Let us briefly review the model’s predictions for the CMB data and fix the
value of the key parameter of the model, namely the $D3$-brane tension
$\tau_{3}$ (2). We then explain why a curvature coupling term is absent in
this $D3$-${\bar{D3}}$-brane model.

###  3.1 The slow-roll inflationary epoch - a review

When the $D3$-${\bar{D3}}$-brane separation is large enough, the $T$ mode is
stable at $|T|=0$. So the potential (may be after some damped oscillation)
reduces to $V(\phi)$ (12)

$U(T,\phi)\to
V(\phi)=2\tau_{3}\left(1-v\frac{\tau_{3}}{2\pi^{2}\phi^{4}}\right)=2\tau_{3}\left(1-\frac{C}{\phi^{4}}\right)$
(14)

where the Hubble parameter is

$H^{2}=V(\phi)/3M_{P}^{2}$ (15)

This model Kachru:2003sx  has essentially only one parameter, namely
$\tau_{3}$, which is determined by the measured density perturbation.

As the branes approach each other, $\phi$ decreases and inflation ends when
$T$ becomes tachyonic, i.e., when $\phi=\phi_{c}$ (10) (even though little
more e-folds can happen after this, as we shall see). At $\phi=\phi_{c}$, the
second term in $V(\phi_{c})$ is still only a small fraction of the leading
($2\tau_{3}$) term,

$C/\phi_{c}^{4}=\frac{27}{16(2\pi^{4})}2\pi g_{s}(2\ln 2)^{2}\simeq
0.105g_{s}$

So during slow-roll inflation, the second term in $V(\phi)$ is even smaller.
To a reasonable approximation, the slow-roll parameters are

$\displaystyle\epsilon$
$\displaystyle=\frac{M_{P}^{2}}{2}\left(\frac{V_{\phi}}{V}\right)^{2}\simeq\frac{M_{P}^{2}}{2}\left(\frac{4C}{\phi^{5}}\right)^{2}$
(16) $\displaystyle\eta$
$\displaystyle=M_{P}^{2}V_{\phi\phi}/V\simeq-\frac{20CM_{P}^{2}}{\phi^{6}}$
(17) $\displaystyle\zeta$
$\displaystyle=M_{P}^{4}\frac{V_{\phi}V_{\phi\phi\phi}}{V^{2}}$ (18)

where $V_{\phi}=\frac{\partial V}{\partial\phi}$ and so on. The density
fluctuation $\delta_{H}$ (or the power spectral density $\sqrt{{\cal{P}}_{k}}$
measured at the pivot scale $k=aH=0.05\,Mpc^{-1}$ where $a$ is the cosmic
scale factor) as measured in the CMB by PLANCK Planck:2016 ,

${\cal{P}}(k)=\delta^{2}_{H}=(\delta\rho/\rho)^{2}=4.7\times 10^{-9}$

corresponds to about $N_{e}=50-60$ e-folds before slow-roll inflation ends at
$\phi_{f}<\phi_{c}$. The number of e-folds $N$ is generated if we start at
$\phi$ and end at $\phi_{f}$,

$N=\int_{\phi}^{\phi_{f}}\frac{H}{\dot{\phi}}d\phi=\frac{1}{M_{P}^{2}}\int_{\phi_{f}}^{\phi}\frac{V}{V^{\prime}}d\phi$

Slow-roll inflation ends when $\eta\simeq-1$ at $\phi_{f}=24CM_{P}^{2}$,

$\phi^{6}-\phi_{f}^{6}=24CM_{P}^{2}(N-1)$ (19)

so the value $\phi_{i}$ is given by

$\phi_{i}^{6}\simeq 24CM_{P}^{2}N_{e}$
$\delta_{H}=\frac{1}{2\pi}\left(\frac{H^{2}}{2M_{P}^{2}\epsilon}\right)^{1/2}\simeq\frac{1}{2\pi}\left(\frac{\tau_{3}}{3M_{P}^{4}\epsilon}\right)^{1/2}\big{|}_{\phi_{i}}$
(20)

where

$\epsilon=0.0022\tau_{3}^{1/3}M_{P}^{-4/3}/N_{e}^{5/3}$

So we have

$\tau_{3}\simeq\left(4.51\times 10^{-5}M_{P}\right)^{4}$

or

$M_{A}\simeq(2\pi g_{s})^{1/4}4.5\times 10^{-5}M_{P}\simeq(2\pi
g_{s})^{1/4}\,1.08\times 10^{14}\,{\rm GeV}$ (21)

and

$H\simeq 1.18\times 10^{-9}M_{P}\simeq 2.6\times 10^{-5}(2\pi
g_{s})^{-1/4}M_{A}$ (22)

We see that $\epsilon\ll\eta$. The scalar power spectral index

$n_{s}-1=\frac{d\ln{\cal{P}}_{k}}{d\ln
k}=-6\epsilon+2\eta\simeq-\frac{5}{3N_{e}}$ (23)

so

$n_{s}=0.967-0.972,\quad{\rm for}\quad N_{e}=50-60$

while PLANCK value is $n_{s}=0.9677\pm 0.0060$ Planck:2016 . The ratio of the
tensor to scalar perturbation is negligibly small,

$r=12.4\epsilon\sim 10^{-10}$

and

$\frac{dn_{s}}{d\ln
k}=16\epsilon\eta-24\epsilon^{2}-2\zeta\simeq-\frac{5}{3N_{e}^{2}}\sim 0.0006$

These values are in good agreement with the observed values by PLANCK
Planck:2016 . Note that the predicted values reflect the particular
$-\phi^{-4}$ form of the potential.

###  3.2 Comment

In the brane world scenario, we assume all the moduli (Kähler, complex
structure and the dilaton) in the 6-dimensional flux compactification have
been stabilized Giddings:2001yu ; Kachru:2003aw . The inflaton $\phi$ covers a
small range of values (i.e., this is a small field inflationary model) so its
evolution should have little impact on the modulus values. On the other hand,
the specific compactification will have an impact on $\phi$. As we turn on
gravity (or open string one loop corrections) DeWolfe:2002nn ; Berg:2004ek ;
Baumann:2010sx , the effective action will in general contain a curvature
coupling,

$V_{\cal R}(\phi)=\frac{\zeta_{\phi}}{12}{\cal
R}\phi^{2}=\zeta_{\phi}H^{2}\phi^{2}$ (24)

where the value of the parameter $\zeta_{\phi}$ depends on the details of the
warped geometry (cf.Baumann:2014nda  for a discussion on the presence of such
a term within the string theory compactification framework). This term will
lead to a disastrously large slow-roll parameter in slow-roll inflation. This
is the $\eta$ problem present in many inflationary models.

Including this term in the $D3$-${\bar{D3}}$-brane inflation Firouzjahi:2005dh
, $n_{s}$ (23) in the model is most sensitive to this term, which becomes

$n_{s}=1+\frac{4\zeta_{\phi}}{3}-\frac{5}{3N_{e}}$

Comparing to the CMB data Planck:2016 , where $0.95<n_{s}<0.98$, we see that

$|\zeta_{\phi}|<0.01$

Phenomenologically, we can always ignore this term, i.e., set
$\zeta_{\phi}=0$. However, a term like $\zeta_{T}H^{2}T{\bar{T}}$ is entirely
possible.

##  4 Towards the end of the inflationary epoch

In this brane world scenario, all the energies released by the annihilation of
the $D3$-${\bar{D3}}$-brane pair go to closed string modes: $D$-strings and
the $F$-strings.

$\bullet$ Some energies are transferred from the closed string modes to
particles (i.e., microscopic open strings) in the stack of standard model
3-branes in the SM-throat. They reheat the observable universe to start the
hot big bang Barnaby:2004gg ; Kofman:2005yz ; Chialva:2005zy ; Frey:2005jk ;
Chen:2006ni .

$\bullet$ Some energies go to cosmological sized string loops. The $D$-string
tension is determined (2):

$G\tau_{1}=\frac{1}{M_{P}^{2}}\left(\frac{\tau_{3}}{32\pi
g_{s}}\right)^{1/2}\simeq\frac{4}{\sqrt{g_{s}}}\times 10^{-10}$

while the $F$-string tension $\mu$ is

$G\mu=g_{s}G\tau_{1}\simeq 4\sqrt{g_{s}}\times 10^{-10}$ (25)

The precise amount of large string loops produced remains to be determined.
Fortunately, if too little is produced, they evolve like $a^{-2}$; if too much
is produced, they will chop themselves up quickly. As a result, they evolve to
a scaling cosmic superstring network that scales like the radiation density
during the radiation era and like matter density during the matter dominated
era (e.g., cf.Vilenkin:2000 ). The cosmic superstrings differ from field
theory cosmic strings in 2 important aspects :

(1) A collection of $p$ $F$-strings and $q$ $D$-strings can form a $(p,q)$
bound state with tension $\mu_{(p,q)}\sim\mu\sqrt{p^{2}+q^{2}/g_{s}^{2}}$, so
there is a spectrum of string tensions, instead of a single tension
Copeland:2003bj ; the bound on $p$ allows a set of strings to end at a bead
Firouzjahi:2006vp .

(2) While the intercommutation probability $P\simeq 1$ for field theory cosmic
strings, it is $10^{-3}\leq P\leq 1$ for $F$-strings and $0.1\leq P\leq 1$ for
$D$-strings Jackson:2004zg . The amount of strings in the network must be
increased by this factor in order for an increased number of collisions per
unit length of string to offset the reduced $P$ in each collision. As a
result, the string density scales like $1/P^{0.65}\to 1/P$
Sakellariadou:2004wq ; Avgoustidis:2005nv , where the $F$-string density will
be enhanced most. Together with the fact that the number density of
$\mu_{(p,q)}$ strings goes like Tye:2005fn

$N_{(p,q)}\propto\mu_{(p,q)}^{-n},\quad\quad 6<n\leq 10$

the contribution of string bound states to the cosmic string density is
suppressed. As a result, $F$-strings, with tension $G\mu\sim 4g_{s}\times
10^{-10}$ should dominate the cosmic superstring network. The total fractional
density is about $\Omega_{superstring}\sim 8\pi{\cal G}G\mu$ where ${\cal
G}\sim 10^{3}-10^{4}$.

Their decay to gravitational waves generates the stochastic gravitational wave
background Vilenkin:1981bx ; Hogan:1984is ; Blanco-Pillado:2013qja ; Blanco-
Pillado:2017rnf , which might have already been observed in pulsar timing data
Ellis:2023tsl . Their kinks and cusps generate bursts of gravitational waves
Damour:2001bk ; Damour:2004kw ; Suresh:2023hkz  (as well as bursts from the
beads Leblond:2009fq ) to be detected in the future. Another way of detection
is via micro-lensing Chernoff:2007pd .

$\bullet$ Some energies go to the light closed string modes, like those in the
Kähler moduli, the complex structure moduli and/or the dilaton. Some can be
ultra-light axion-like particles $a_{i}$, $i=1,2,...$. Because of its very
light mass, $a_{i}$ is locked at its expectation value $\langle a_{i}\rangle$
away from its minimum during inflation. It will start rolling towards its
minimum only when the Hubble parameter becomes smaller than its mass. Its
rolling converts its vacuum energy to dark matter via the mis-alignment
mechanism. A priori, there is no reason why $a_{i}$ takes the same $\langle
a_{i}\rangle$ value everywhere. It is inflation that assures us that $\langle
a_{i}\rangle$ has the same value everywhere. Some interesting possibilities
with mass
(1) $m\sim 10^{-22}$ eV as fuzzy dark matter Hu:2000ke ; Schive:2014dra ;
Hui:2016ltb ;
(2) $m\sim 10^{-20}$ eV as a second component of the fuzz dark matter to
resolve some issues in the single component FDM Luu:2018afg ;
(3) $m\sim 10^{-30}$ eV to solve the 7Li problem in big bang nucleosynthesis
and the Hubble tension Fung:2021wbz ;
(4) $m<10^{-30}$ eV where $m<H_{today}$, so its still frozen vacuum energy
contributes to the dark energy.

##  5 The Complex Potential

The tachyon dynamics has been studied before Sen:1998sm ; Shiu:2002xp ;
Lambert:2003zr ; Chen:2003xq ; Sen:2004nf ; Leblond:2006cc ;
Brandenberger:2007ca ; Battefeld:2010rf . In string theory, both $\phi$ and
$T$ are open string modes in the $D3$-${\bar{D3}}$-brane system. Once they
annihilate, $\phi$ and $T$ no longer exist as physical degrees of freedom. In
the effective field theory approximation, we see that $|T|\to\infty$ so both
$\phi$ and $T$ drop out of any post-inflation dynamics.

As mentioned earlier, the $U(1)$ symmetry in $T{\bar{T}}$ is gauged. Below, we
shall simplify $T{\bar{T}}\to T^{2}$ where the radial $T$ is a real scalar
mode. This model contains similar structure to the hybrid inflation model
Linde:1993cn ; Randall:1995dj ; Garcia-Bellido:1996mdl , so we shall focus on
the $D3$-${\bar{D3}}$-brane model’s more novel features.

To discuss the qualitative features, let us consider the $\phi^{-4}$ term in
$V(\phi)$ (14). Naively, it diverges badly as $\phi\to 0$. Fortunately, it
does not drop to $\infty$ as $\phi\to 0$. Instead, as (real) $\phi$ decreases
so $\phi$ becomes smaller than $\phi_{c}$ (10), $V(\phi)$ (or
${\hat{V}}(\phi)$) (14) becomes complex, where ${\rm Im}\,{\hat{V}}(\phi)$ is
given in Eq.(13), while the real part smoothly approaches some finite value.
Let us now find an approximate form to capture the shape of
${\rm{Re}}\,{\hat{V}}(\phi)$.

###  5.1 ${\rm{Re}}\,{\hat{V}}(\phi)$ for $\phi<\phi_{c}$

Let ${\bar{V}}(\phi)={\rm{Re}}\,{\hat{V}}(\phi)$ be the real part of
${\hat{V}}(\phi)$ for $0\leq\phi\leq\phi_{c}$. As described in
Ref.Sarangi:2003sg , after $T$ becomes tachyonic as $\phi<\phi_{c}$, $\phi$
decreases towards $\phi_{m}>0$ where ${\bar{V}}(\phi)$ reaches its minimum
${\bar{V}}_{m}$ at $\phi=\phi_{m}$. As $\phi$ further decreases,
${\bar{V}}(\phi)$ increases and $\phi$ becomes tachyonic at $\phi=0$. Let us
parameterize these qualitative features by a simple formula

${\bar{V}}(\phi)=a-b\phi^{2}+c\phi^{4},\quad\quad 0\leq\phi\leq\phi_{c}$ (26)

where $b>0$ measures the tachyonic mass of $\phi$ at $\phi=0$. To match
${\bar{V}}(\phi)={\hat{V}}(\phi)$ at $\phi_{c}$, we have 222To match
$V_{\phi}$ and $V_{\phi\phi}$ at $\phi_{c}$, a formula more involved than
${\bar{V}}(\phi)$ (26) is required. For the sake of simplicity, this is not
needed in our approximate analysis here.

${\bar{V}}(\phi_{c})=a-b\phi_{c}^{2}+c\phi_{c}^{4}={\hat{V}}(\phi_{c})=-2\tau_{3}C/\phi_{c}^{4}=-v\tau_{3}^{2}/\pi^{2}\phi_{c}^{4}$

Ref.Sarangi:2003sg  shows that $a={\bar{V}}(\phi=0)$ is comparable to
${\hat{V}}(\phi_{c})$. In a crude approximation to simplify the analysis, we
simply set

$a={\hat{V}}(\phi_{c})\quad\to\quad c=b/\phi_{c}^{2}$

Now the minimum of ${\bar{V}}(\phi)$ is at

$\phi_{m}^{2}=b/2c=\phi_{c}^{2}/2$ (27)

and

${\bar{V}}(\phi_{m})={\hat{V}}(\phi_{c})-b\phi_{c}^{2}/4$

Next, in another approximation, we set

${\bar{V}}(\phi_{m})=2{\hat{V}}(\phi_{c})=-\frac{4\tau_{3}C}{\phi_{c}^{4}}\quad\quad|{\hat{V}}(\phi_{c})|\simeq
0.0334M_{A}^{4}$

So ${\bar{V}}(\phi)$ (26) is now given by, for $\phi_{c}\geq\phi\geq 0$,

${\bar{V}}(\phi)=c\left[\phi^{4}-\phi_{c}^{2}\phi^{2}-\frac{\phi_{c}^{4}}{4}\right],\quad\quad
c=4|{\hat{V}}(\phi_{c})|/\phi_{c}^{4}\simeq 1.03g_{s}^{2}$ (28)

Next we consider the imaginary part ${\rm{Im}}\,{\hat{V}}(\phi)$ of
${\hat{V}}(\phi)$ (13),

${\rm{Im}}\,{\hat{V}}(\phi)=\frac{|m_{T}^{2}(\phi)|^{2}}{32\pi}=\frac{\pi
g_{s}^{2}}{8}\left(\phi^{2}-\phi_{c}^{2}\right)^{2}$ (29)

where the tachyon mass $m_{T}^{2}$ (9) has been used. As pointed out in
Ref.Banks:1995ch , the appearance of such an imaginary part indicates the
opening up of inelastic channels and the violent annihilation can lead to the
production of PBHs.

##  6 The Complex Potential on the $T$ and the $\phi$ Evolution

Now we examine the $\phi$ and $T$ evolutions,

$\displaystyle\ddot{\phi}+3H\dot{\phi}+U_{\phi}$ $\displaystyle=0$ (30)
$\displaystyle\ddot{T}+3H\dot{T}+U_{T}$ $\displaystyle=0$ (31)

###  6.1 The $\phi$ Evolution

Let us first consider the $\phi$ evolution (30). Here we have

$\displaystyle{\rm Re}\,U_{\phi}$
$\displaystyle\simeq\exp\left[-\frac{\pi^{2}g_{s}T^{2}}{(4\ln
2)M_{A}^{2}}\right]2\left(\pi g_{s}T^{2}+2c\phi^{2}-c\phi_{c}^{2}\right)\phi$
$\displaystyle{\rm Im}\,U_{\phi}$
$\displaystyle=\frac{g_{s}}{4}m^{2}_{T}(\phi)\phi=\frac{\pi
g_{s}^{2}}{2}\left(\phi^{2}-\phi_{c}^{2}\right)\phi$ (32)

Note that slow-roll inflation ends when $\eta_{\phi}=-1$ at $\phi=\phi_{f}$
before it reaches $\phi_{c}$, while the potential is still stable in the $T$
direction, i.e., $\ddot{T}=\dot{T}=T=0$. So $\phi$ rolls down the potential
quickly when it passes $\phi_{c}$ until quantum diffusion $H/2\pi$ drives $T$
away from $T=0$. Let us look at the situation when $\phi$ is rolling towards
$\phi_{m}$ in the absence of quantum diffusion. For an order-of-magnitude
qualitative picture of what is going on, consider the following simplified
version for $\bar{\phi}=\phi-\phi_{m}$ around the minimum at $\phi_{m}$, where
$m$ is the mass of $\phi$ at $\phi_{m}$,

$\displaystyle\ddot{\bar{\phi}}$
$\displaystyle+3H\dot{\bar{\phi}}+(m^{2}+i\gamma){\bar{\phi}}=0$
$\displaystyle m^{2}$ $\displaystyle=2c\phi_{c}^{2}\simeq 0.74g_{s}M_{A}^{2}$
(33) $\displaystyle\gamma$ $\displaystyle={\rm
Im}\,U_{\phi}\big{|}_{\phi_{m}}\simeq-\frac{\pi g_{s}}{16\ln
2}M_{A}^{2}\simeq-0.283g_{s}M_{A}^{2}$

Let

$\lambda=m^{2}-9H^{2}/4,\quad\quad\xi=\sqrt{\lambda^{2}+\gamma^{2}}$

The solution to Eq.(6.1) is

$\displaystyle\bar{\phi}\sim$
$\displaystyle\exp\left(-\left(\frac{3H}{2}+\left[\frac{\xi-\lambda}{2}\right]^{1/2}\right)t\right)\cos\left(\left[\frac{\xi+\lambda}{2}\right]^{1/2}t\right)$
$\displaystyle\sim$
$\displaystyle\exp\left(-\left[\frac{3H}{2}+\frac{|\gamma|}{2m}\right]t\right)\cos\left(mt\right)$
(34)

where the damping term plays the role of a decay width,

$\Gamma_{\phi_{m}}=2\left[\frac{\xi-\lambda}{2}\right]^{1/2}\simeq|\gamma|/m\sim
0.329g_{s}^{1/2}M_{A}$

We may express the equations in a more familiar form, where ${\rm
Im}\,U(T,\phi)$ is replaced by the decay width $\Gamma_{\phi}$ :

$\displaystyle\ddot{\bar{\phi}}+[3H+\Gamma_{\phi}]\dot{\bar{\phi}}+{\rm
Re}\,U_{\phi}=0$ (35)

where

$\Gamma_{\phi_{m}}/3H\approx 10^{4}g_{s}^{1/4}$

That is, the damping effect coming from the imaginary part of the potential
dominates over the Hubble value. (We suspect a similar $\Gamma_{\phi}$ should
be present in any hybrid inflation model.) Note that

$0\leq\Gamma_{\phi}\leq 2\Gamma_{\phi_{m}}\quad{\rm
as}\quad\phi_{c}\geq\phi\geq 0$

As $\phi$ rolls down from $\phi_{f}$ to $\phi_{c}$, $\gamma=0$, so its motion
is damped only by $H$. Still, the slope is steep enough so this journey takes
less than one e-fold. As $\phi$ moves further towards the minimum at
$\phi_{m}$, $\Gamma_{\phi}$ grows, providing a much stronger damping than that
coming from $H$. $\Gamma_{\phi}$ grows further as $\phi$ passes $\phi_{m}$.
Here, $\phi$ oscillates with period $2\pi/m$. In one cycle, its amplitude is
suppressed by a factor of

$e^{-\Gamma_{\phi}(2\pi/m)}\simeq e^{-5}\sim 0.007$

so one expects $\phi$ to quickly settle down at $\phi_{m}$.

###  6.2 The $T$ Evolution

Next we consider the $T$ evolution (31). Since $m_{T}^{2}$ (9) is independent
of $T$, the derivative ${\rm Im}\,U_{T}=0$. We may extend $m_{T}^{2}$ to next
order to obtain a non-trivial ${\rm Im}\,U_{T}$ and examine qualitatively its
impact on the $T$ evolution. To simplify the notations, we rewrite $U(T,\phi)$
(3) with parameters $d$ and $f$,

$U(T,\phi)\simeq 2\tau_{3}e^{-dT^{2}}{\cal F}\left(f\phi^{2}T^{2}\right)$

Expanding $U(T,\phi)$ up to 4$th$ power in $T$ and re-expressing ${\rm
Re}\,U_{T}$ as

${\rm Re}\,U_{T}={\bar{m}}_{T}^{2}(T,\phi)T+\cdot\cdot\cdot$

the $T$ mass is qualitatively given by

${\bar{m}}_{T}^{2}(T,\phi)\simeq
m_{T}^{2}(\phi)+8\tau_{3}T^{2}\left(d^{2}/2-2\ln 2df\phi^{2}+\left[2(\ln
2)^{2}-\frac{\pi^{2}}{6}\right]f^{2}\phi^{4}\right)$

where ${\bar{m}}_{T}^{2}(T=0,\phi)=m_{T}^{2}(\phi)$ (9). Now, ${\rm
Im}\,U(T,\phi)$ extends to

${\rm Im}\,U(T,\phi)=|{\bar{m}}_{T}^{2}|^{2}/32\pi$

At $\phi=\phi_{m}$, we have its derivative

$\displaystyle{\rm Im}\,U(T,\phi_{m})_{T}$ $\displaystyle\simeq
m_{T}^{2}(\phi_{m})\frac{\tau_{3}T}{\pi}\left(d^{2}/2-\ln
2df\phi_{c}^{2}+\left[2(\ln
2)^{2}-\frac{\pi^{2}}{6}\right]\frac{f^{2}\phi_{c}^{4}}{4}\right)$
$\displaystyle\sim m_{T}^{2}(\phi_{m})T(-0.056g_{s})\approx
0.13M_{A}^{2}T=\sigma T$ (36)

which is qualitatively valid for $(T/M_{A})^{2}<0.4g_{s}$.

We can get a qualitative feature of the solution of Eq.(31) by considering an
approximate form ($m^{2}=-m_{T}^{2}>0$)

$\ddot{T}+3H\dot{T}+(-m^{2}+i\sigma)T=0$ (37)

In the absence of $H$ and $\sigma$, $T\propto e^{mt}\to\infty$ is the solution
we are looking for. Following the same steps as above, we have (ignoring an
oscillating factor)

$T\approx\exp\left[(-\frac{3H}{2}+m+\frac{\sigma^{2}}{8m^{3}})t\right]$

Once $T$ starts rolling, $T\to\infty$ quickly and $U(T,\phi)\to 0$
exponentially fast. So the introduction of an ${\rm Im}\,U(T,\phi_{m})_{T}$
has little impact on the qualitative behaviour on the $T$ evolution.

The density fluctuation is given by

${\cal
P}(k)\simeq\frac{1}{24\pi}\frac{U}{M_{P}^{4}\epsilon}\simeq\frac{1}{12\pi}\frac{U^{3}}{M_{P}^{6}(U_{T}^{2}+U_{\phi}^{2})}$

Before $\phi$ reaches the minimum $\phi_{m}$, the quantum diffusion $\langle
T\rangle\sim H/2\pi$ may start the $T$ rolling, which happens quickly. As a
result, ${\cal P}(k)$ may never reach a very large value during the $(T,\phi)$
evolution.

##  7 Tunnelling Properties

###  7.1 Tunneling

For fixed $\phi>\phi_{c}$, $T{\bar{T}}$ is classically stable at $T=0$.
However, as $T{\bar{T}}$ increases for fixed $\phi$, $U(T,\phi)$ rises to a
peak and then comes down so $U(T,\phi)\to 0$ as $T{\bar{T}}\to\infty$ (see
Figure 1). Although $T{\bar{T}}=0$ is classically stable, it can tunnel over
the barrier and roll to $\infty$. Although the branes are still separated by
distance $\phi>\phi_{c}$, tunnelling means a nucleation bubble forms at a
point where the branes fluctuate enough so they collide and begin to
annihilate. The growth of the bubble is slower than the inflation speed
Guth:1982pn , so the complete brane annihilation happens only after
$\phi\leq\phi_{c}$.

The probability of tunnelling is given by $P(\phi)\simeq K(\phi)e^{-S(\phi)}$,
where $S$ is the Euclidean action of the instanton (or the “bounce” solution).
$S(\varphi)$ for $Dp-{\bar{Dp}}$-brane pair system has been estimated
Jones:2002sia , for $\varphi\geq\varphi_{c}$,

$S(\varphi)\simeq 4\pi
c_{1}c_{2}^{p+1}\left[\frac{p^{p}}{(p+1)g_{s}}\frac{2\pi^{(p+1)/2}}{\Gamma(\frac{p+1}{2})}\right]\left(\frac{\varphi-\varphi_{c}}{\sqrt{\alpha^{\prime}}}\right)^{(p+1)/2}$
(38)

where $c_{1}\simeq 1.5$ and $c_{2}\simeq 0.29$. Note that the exponential
tunnelling suppression vanishes when the barrier disappears at
$\varphi=\varphi_{c}$, as expected. The thin wall approximation Coleman:1977
turns out to be good when $\varphi$ is large. Applying it to our
$D3$-${\bar{D3}}$-brane pair case ($\phi=\sqrt{\tau_{3}}\varphi$) and using
Eq.(19), where $\phi-\phi_{c}\simeq 24CM_{P}^{2}N/32\phi_{c}^{5}$ for $\phi$
close to $\phi_{c}$,

$S(\phi)\simeq
10^{3}\frac{\left(\phi-\phi_{c}\right)^{2}}{M_{A}^{2}}\sim\left(\frac{M_{P}}{M_{A}}\right)^{4}N^{2}$
(39)

where $N$ is the number of e-folds before $\phi$ reaches $\phi_{c}$, i.e.,
$\phi-\phi_{c}\propto N$. Let the probability of tunnelling per unit volume at
$\phi$ during the inflationary epoch goes like $P(\phi)\sim M_{A}^{4}\exp(-S)$
in a volume $e^{3(N_{e}-N)}/M_{A}^{3}$, we have the probability per unit time

${\cal P}(N)\sim M_{A}\exp\left[3(N_{e}-N)-(M_{P}/M_{A})^{4}N^{2}\right]$

For $N\sim N_{e}$, the second term severely suppresses the tunnelling. As
$N\to 0$, the exponential increase in volume from inflation is not able to
overcome the exponential suppression until $N$ is very small. However, the
time $\delta t\sim N/H$ available for tunnelling $T$ starts rolling is too
short to make much difference.

The above estimate does not include gravitational effect. Turning on gravity
leads us to the CDL tunnelling Coleman:1980aw . While the $H/2\pi$ fluctuation
provides some enhancement to the tunnelling rate, any radiation inside the
bubble will suppress the tunnelling process Huang:2008jr . Both the $\phi$ and
the $T$ evolutions may further complicate the dynamics of this phase
transition.

###  7.2 Double $D3$-${\bar{D3}}$-brane pairs

Since the compactified manifold presumably has a number of throats with
different warp factors, one can easily imagine $D3$-branes and
${\bar{D3}}$-branes move towards the throats and fall in. In any throat with
at least one $D3$-brane and one ${\bar{D3}}$-brane, they will approach each
other and eventually annihilate until only the remaining $D3$-branes or
${\bar{D3}}$-branes are left behind. Suppose, not only a
$D3$-${\bar{D3}}$-brane pair is present in the $A$-throat, but another
$D3$-${\bar{D3}}$-brane pair is present in another throat, say the $B$-throat.
This can be an example of double inflation Silk:1986vc , with the possibility
of PBH formation (e.g., cf.Sasaki:2018dmp ; Ozsoy:2023ryl ).

Most of the formulae for the pair in the $A$-throat applies to the $B$-throat
pair, with $M_{A}$ and $\tau_{3}$ replaced by $M_{B}$ and $\tau_{3B}$, keeping
the same string coupling $g_{s}$. Not to interfere with the properties of the
$D3$-${\bar{D3}}$-brane inflation in the $A$-throat, let the $B$-throat
warping be stronger, so

$M_{A}\gg M_{B},\quad\quad V_{A}(\phi_{A})\gg V_{B}(\phi_{B})$

In this case, the separation $\phi_{B}$ from its critical value $\phi_{Bc}$
will be frozen until $H\to\sqrt{V_{B}(\phi_{B})/3M_{P}^{2}}$. Now there is
more time for $T_{B}$ tunnelling over the $T_{B}$ barrier (see Figure 1) so
this first order phase transition is more likely to happen. Here, the
Euclidean action for $T_{B}$ tunnelling through its barrier in the $B$-throat
is given by

$S_{B}\simeq
10^{3}\frac{\left(\phi_{B}-\phi_{Bc}\right)^{2}}{M_{B}^{2}},\quad\phi_{Bc}^{2}=\frac{M_{B}^{2}}{(4\ln
2)g_{s}}$

Let us consider the general case and then the special case where the
$B$-throat is identified with the standard model $SM$-throat.

We start with the scenario when both $D3$-${\bar{D3}}$-brane pairs have
relatively large separations, i.e., $\phi_{A}>\phi_{Ac}$ and
$\phi_{B}>\phi_{Bc}$. Although the $T_{B}$ barrier height is a function of
$\phi_{B}$, it is qualitatively of order $M_{B}^{4}$. Hawking-Moss tunnelling
Hawking:1981fz  goes like, order-of-magnitude-wise,

$S_{B}\approx
24\pi^{2}M_{P}^{4}\left(\frac{1}{M_{A}^{4}}-\frac{1}{M_{A}^{4}+M_{B}^{4}}\right)\sim
24\pi^{2}M_{P}^{4}\frac{M_{B}^{4}}{M_{A}^{8}}$ (40)

With $M_{A}\simeq 5\times 10^{-5}M_{P}$ (21), we see that, for

$M_{B}\leq 10^{-10}M_{P}\quad\rightarrow\quad S_{B}\sim 0$

So the tunnelling rate $P\sim e^{-S_{B}}$ is not exponentially suppressed. In
fact, bubbles may be profusely produced. i.e., During inflation in the
$A$-throat, $H\sim 10^{-9}M_{P}$ (22) so $H>M_{B}$. In this case,
$\dot{\phi}_{B}\simeq V_{B\phi_{B}}/3H$ is negligibly small so $\phi_{B}$ is
essentially frozen, while the “B” nucleation bubbles are produced quickly.
Because of the rapid (i.e., exponential) expansion of the universe, these
bubbles do not collide. After the “A” throat inflationary epoch, the universe
enters a radiation-dominated era, when $\phi_{B}$ remains frozen. There are 2
possibilities:

(1) If $S_{B}$ remains small enough for a long enough time period, the
production of “B” nucleation bubbles remains to be efficient. Their rapid
growth can lead to bubble collisions in this radiation-dominated era and
complete the “B” throat $D3$-${\bar{D3}}$-brane annihilation without going
through another inflationary epoch.

(2) If $S_{B}$ is big enough so the bubble production is now suppressed. As
$H$ approaches $H^{2}\sim V_{B}(\phi_{B})/3M^{2}$, the universe moves from a
radiation-dominated epoch to a slow-roll $\phi_{B}$ inflationary epoch. Here,
the bubbles already produced are too far apart to collide. Eventually,
$\phi_{B}$ rolls pass $\phi_{Bc}$ so the $B$-throat brane pair collides and
annihilates. As the $T_{B}$ potential barrier disappears, the bubbles already
produced may simply fizzle out before they have any chance to collide.

Depending on the details, it remains to be studied whether this phase
transition and the formation of nucleation bubbles can yield observable
gravitational radiation and lead to big enough density perturbations for the
PBH formation, and/or other cosmological detectable signatures.

###  7.3 $D3$ \+ Stack of ${\bar{D3}}$-branes in the Standard Model throat

In general, we can consider a system of $n$ $D3$-branes and $m$
${\bar{D3}}$-branes. Such a system is quite natural in the brane world
scenario. One possibility is to identify the $B$-throat with the standard
model $SM$-throat. Suppose we live in a stack of 5 ${\bar{D3}}$-branes today,
which is big enough to accommodate the standard $SU(3)\times SU(2)\times
U_{Y}(1)$ model. In early universe, there is a $D3$-brane moving towards a
stack of 6 ${\bar{D3}}$-branes inside this throat. After colliding with the
stack of ${\bar{D3}}$-branes, the $D3$-brane annihilates with one of the
${\bar{D3}}$-branes, leaving behind the stack of 5 ${\bar{D3}}$-branes we live
in today. The energy released can go directly to the open string modes inside
the stack, i.e., SM particles that reheat our universe. In this case, the
reheating of our universe is much more efficient than transferring energy from
the $A$-throat to the $SM$ throat. Note that the electroweak phase transition
can remain second order (or weakly first order). For example, the effective
potential for the electroweak Higgs doublet $\Phi$ can take the standard form
at temperature ${\cal T}$, $V(\Phi)=\left(\kappa{\cal
T}^{2}-m^{2}\right){\bar{\Phi}}\Phi+\lambda\left(\Phi^{{\dagger}}\Phi\right)^{2}$.

One possibility is suggested by the investigation in Ref.Jones:2003ae .
Consider $(5+n)$ ${\bar{D3}}$-branes parallel to $n$ $D3$-branes. The mode $T$
transforms in the bi-fundamental $(5+n,{\bar{n}})$ representation of
$U(5+n)\times U(n)$. Since QCD plays a spectator role only, we may ignore it.
With $T$ now in the $(2+n,{\bar{n}})$ representation of

$U(2+n)\times U(n)\to U(2)\times U(n)\times U(n)$

where the $T$ in the $(1,n,{\bar{n}})$ representation rolls down the potential
(i.e., $T\to\infty$) as $n$ ${\bar{D3}}$-branes annihilate with the $n$
$D3$-branes. Any mode transforming non-trivially under either $U(n)$ will also
drop out of the spectrum. The left-over 2 ${\bar{D3}}$-branes contain the
electroweak $SU(2)$, where the $T$ in the $(2,1,1)$ representation may be
identified with the electroweak Higgs doublet $\Phi$. It will be interesting
to see how the first order phase transition impacts on the electroweak second
order phase transition.

##  8 Remarks

The recent discovery of the stochastic gravitational wave background (SGWB)
NANOGrav:2023gor ; NANOGrav:2023hde ; Xu:2023wog ; EPTA:2023sfo ; Zic:2023gta
has led to an intense investigation of its origin. Some of its possible
sources are (i) cosmic strings (e.g.,Blanco-Pillado:2017rnf ; Ellis:2023tsl ),
(ii) massive black hole binaries (e.g., cf.NANOGrav:2023gor ; Ellis:2023oxs )
and (iii) first order phase transitions (e.g.,Witten:1984rs ; Hogan:1986qda ;
Turner:1990rc ; Kamionkowski:1993fg ; Chialva:2010jt ; Hindmarsh:2015qta ;
Jiang:2015qor ; An:2020fff ). It is interesting that the
$D3$-${\bar{D3}}$-brane inflation model offers cosmic superstrings as well as
first order phase transitions as possible sources of SGWB, while the
production, the growth and/or the collision of nucleation bubbles may lead to
large density perturbations resulting in the PBH production. It is important
to explore the model in further details. As a start, the
$D3$-${\bar{D3}}$-brane potential $U(T,\phi)$ is presented in this paper. We
see that $U(T,\phi)$ has distinct novel features signifying its string theory
properties. It is also well determined, allowing detailed analyses to test the
model. With a clear physical picture behind it, one can modify or extend the
model in ways where the dynamics is still under control. Besides extending the
single $D3$-brane (and/or the single ${\bar{D3}}$-brane) to a stack, or
extending the model to multiple $D3$-${\bar{D3}}$-brane pairs in different
warped throats, one may also consider other effects :
$\bullet$ Revert to a Dirac-Born-Infeld action Alishahiha:2004eh ;
Shandera:2006ax ; the DBI action can play an important role towards the end of
inflationary epoch when $\phi$ exits the slow-roll phase.
$\bullet$ Although CMB data from PLANCK implies that the term $H^{2}\phi^{2}$
is either absent or negligibly small, one may include a
$\zeta_{T}H^{2}T{\bar{T}}$ term into $U(T,\phi)$. Among other effects, its
presence tends to shift the value of $\phi_{c}$.
$\bullet$ Other (quantum and stringy) corrections to ${\cal S}(T,\phi)$.
$\bullet$ The interplay between cosmic superstrings, PBHs and nucleation
bubbles from a first order phase transition can be important in the SGWB
production.
The success of the model’s predictions from the inflationary epoch (in
confronting the CMB data) suggests that its life after the slow-roll
inflationary epoch should be equally rich and exciting.

###### Acknowledgements.

This work is based on early collaborations with Sash Sarangi and Nick Jones,
to whom I am grateful. I thank Xingang Chen, Hassan Firouzjahi, Tao Liu and
especially Liam McAllister for valuable comments.

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