# The backreaction effect of sound speed resonance in DBI inflation.

Bichu Li<EMAIL_ADDRESS>Deep Space Exploration Laboratory/School of
Physical Science, University of Science and Technology of China, 96 Jinzhai
Road, Hefei, Anhui 230026, P.R. China  CAS Key Laboratory for Researches in
Galaxies and Cosmology/Department of Astronomy, School of Astronomy and Space
Science, University of Science and Technology of China, 96 Jinzhai Road,
Hefei, Anhui 230026, P.R. China  Chao Chen<EMAIL_ADDRESS>Jockey Club
Institute for Advanced Study, The Hong Kong University of Science and
Technology, Clear Water Bay, Kowloon, Hong Kong, P.R. China  Bo Wang
<EMAIL_ADDRESS>Deep Space Exploration Laboratory/School of Physical
Science, University of Science and Technology of China, 96 Jinzhai Road,
Hefei, Anhui 230026, P.R. China  CAS Key Laboratory for Researches in Galaxies
and Cosmology/Department of Astronomy, School of Astronomy and Space Science,
University of Science and Technology of China, 96 Jinzhai Road, Hefei, Anhui
230026, P.R. China

###### Abstract

We examine the backreaction effect of the enhanced small-scale scalar
perturbations from the sound speed resonance (SSR) mechanism for primordial
black hole formation in Dirac-Born-Infeld (DBI) inflation Within the
perturbative regime, the backreaction effect of perturbations on the
background dynamics can be described by an effective action after integrating
out the perturbation sector. Starting with the effective field theory of a
specific DBI inflation model that incorporates SSR, we obtain the one-loop
effective action by integrating out the scalar perturbations at the quadratic
level. Using the effective Friedmann equations derived from this one-loop
effective action, we solve the Hubble parameter with backreaction and the
effective perturbation dynamics on this background as well. Our numerical
findings reveal that, for a viable parameter space, the backreaction effect
results in a relative correction to the Hubble parameter of approximately
$10^{-7}$, whereas the relative correction to the slow-roll parameter can vary
between $-0.3$ and $0.1$, before gradually converging to $10^{-7}$.
Furthermore, our results show that the backreaction effect on SSR sound speed
causes a slight reduction in the resonant peak of the curvature power
spectrum, and the subsequent PBH formation predicted by the SSR mechanism
remains almost unchanged.

##  I Introduction

The conventional perturbative approach to the classical Einstein’s field
equation relies on the balance of the perturbative orders between geometry and
matter sectors, and implicitly assumes the negligible interference among
different orders. However, the intrinsic nonlinearity of Einstein’s gravity
inevitably allows linear cosmological perturbations around a homogeneous and
isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) Universe to affect on the
background dynamics, and this backreaction effect of linear perturbations
arise as second and higher orders in the perturbation expansion. It is well
known that gravitational waves (GWs) could affect the dynamics of the
background on which they are propagating, and thus one can define the
effective energy-momentum tensor of GWs by expanding Einstein’s field equation
up to the second order and taking a spatial average Brill and Hartle (1964);
Isaacson (1968a, b); Abramo  _et al._ (1997); Giovannini (2020). In addition,
considerable attention has been given to the backreaction effect of scalar
perturbations in relation to whether small-scale inhomogeneities have an
impact on the global FLRW evolution, which may give a new solution to
cosmological coincidence problem, i.e., the acceleration epoch begins roughly
at the same moment that non-linear structures form Adamek  _et al._ (2019);
Schander and Thiemann (2021). Besides the above-mentioned classical approach
to the cosmological backreaction problem, semi-classical and quantum
approaches are also investigated Schander and Thiemann (2021); Hu and
Verdaguer (2020).

Intuitively, the backreaction effect is expected to be proportional to the
amplitude of perturbations. In the context of primordial black hole (PBH)
formation within the inflationary Universe, the small-scale curvature
perturbations are required to be amplified around seven orders of magnitude
larger than the large-scale perturbations constrained by cosmic microwave
background (CMB) radiation experiments. The amplification mechanism requires a
deviation from the standard slow-roll inflation, at least for a period with a
few e-folds that accounts for PBH formation Sasaki  _et al._ (2018); Carr and
Kuhnel (2020); Escrivà  _et al._ (2022). So far, there are plenty of
inflationary models that successfully enhance the small-scale primordial
curvature perturbations during inflation and the resulting overdense regions
can collapse into PBHs, such as a non-attractor phase during inflation Garcia-
Bellido and Ruiz Morales (2017); Germani and Prokopec (2017); Motohashi and Hu
(2017); Pattison  _et al._ (2017); Di and Gong (2018); Byrnes  _et al._
(2019); Passaglia  _et al._ (2019); Xu  _et al._ (2020); Tasinato (2021); Fu
_et al._ (2020); Inomata  _et al._ (2022a); Kawai and Kim (2021); Zheng  _et
al._ (2022); Balaji  _et al._ (2022); Qiu  _et al._ (2023); Fu and Chen
(2023); Kasai  _et al._ (2022); Fu and Wang (2023); Pi and Wang (2023); Cai
_et al._ (2023), parametric resonance Cai  _et al._ (2018); Chen and Cai
(2019); Cai  _et al._ (2020); Ashoorioon  _et al._ (2021); Zhou  _et al._
(2020); Chen  _et al._ (2020); Cai  _et al._ (2021a), additional contributions
from isocurvature perturbations Linde  _et al._ (2013); Braglia  _et al._
(2020); Liu (2023); Meng  _et al._ (2023); Chen  _et al._ (2023a); Ferrante
_et al._ (2023); Ge  _et al._ (2023) and potentially large non-Gaussianity
Ezquiaga  _et al._ (2020); Atal  _et al._ (2020); Figueroa  _et al._ (2021);
Cai  _et al._ (2021b, 2022); Matsubara and Sasaki (2022); Gow  _et al._
(2022); Pi and Sasaki (2023), A natural question arises: if such enhanced
small-scale scalar perturbations have a certain impact on the background
dynamics, and further alter the efficiency of amplification of perturbations
expected in the original models?

For the ultra-slow-roll inflation, the backreaction effect was investigated in
Ref. Cheng  _et al._ (2022) under two concrete models using Hartree
factorization, and authors found that the backreaction can boost the curvature
perturbation on superhorizon scales. The lattice simulation of the axion-U(1)
inflation model in Ref. Caravano  _et al._ (2022) shows that the non-
Gaussianity of curvature perturbation is sensitive to the backreaction.
Reference Yu  _et al._ (2023) considers the backreaction of a spectator field
on an inflationary background using Hartree approximation. Moreover, several
recent works on the one-loop corrections from curvature perturbations Inomata
_et al._ (2022b); Kristiano and Yokoyama (2022); Riotto (2023); Choudhury  _et
al._ (2023a, b); Firouzjahi (2023); Franciolini  _et al._ (2023); Cheng  _et
al._ (2023); Fumagalli (2023) and tensor perturbations Chen  _et al._ (2023b);
Ota  _et al._ (2022) show the potential importance of the one-loop corrections
in the context of PBH formation. Reference Gong and Mylova (2022) studied the
backreaction in the hybrid inflation by perturbatively expanding the action up
to the one-loop order and integrating out the perturbation to obtain an
effective one-loop action from the path integral formalism. In this work, we
investigate the backreaction effect of the sound speed resonance (SSR)
mechanism which enhances small-scale curvature perturbations through the
parametric resonance triggered by a time-oscillating sound speed of inflaton
Cai  _et al._ (2018). Reference Chen  _et al._ (2020) has shown that the SSR
mechanism can be realized in the framework of Dirac-Born-Infeld (DBI)
inflation. Following the method presented in Ref. Gong and Mylova (2022), we
integrate out scalar perturbations at the quadratic order using the path
integral formalism, and derive the modified Friedmann equations that measure
the backreaction effect of perturbations on the background dynamics. We will
show the backreaction corrections on the Hubble parameter, slow-roll parameter
and sound speed, and further investigate backreaction on the enhancement of
power spectrum.

This paper is organized as follows. In Sec. II, we briefly review the SSR
realized in DBI inflation. In Sec. III.1, we calculate the effective Friedmann
equations from the one-loop effective action and provide numerical results for
background parameters including the Hubble parameter, slow-roll parameter, and
also the sound speed of curvature perturbations. In Sec. III.3, we use the
results from Sec. III.1 to calculate the curvature power spectrum with
backreaction. Finally, we summarize the results in Sec. IV.

##  II Realization of SSR in DBI inflation

In this section, we will review the realization of SSR within DBI inflation
Chen  _et al._ (2020). In the original SSR mechanism Cai  _et al._ (2018), the
sound speed of inflaton is assumed to be time-oscillating during a few
e-folds,

$c_{s}^{2}(\tau)=1-2\xi\left[1-\cos(2k_{*}\tau)\right]~{},~{}\tau>\tau_{s}~{},$
(1)

where $\tau$ is the conformal time, $\xi$ and $k_{*}$ are the small amplitude
and frequency of the sound speed oscillation, respectively. $\xi<1/4$ is
required such that $c_{s}^{2}$ is positively defined, and the oscillation
begins at $\tau_{s}$, where $k_{*}$ is chosen to be deep inside the Hubble
horizon with $|k_{*}\tau_{s}|\gg 1$. Consequently, the resonance of the
curvature perturbations happens inside the Hubble horizon and leads to PBH
formation during the later radiation-dominated epoch.

The DBI model contains a non-canonical kinetic term, which can be used to
realize a non-trivial sound speed. The DBI action is written as

$S_{\text{DBI}}=\int\mathrm{d}^{4}x\sqrt{-g}\left[\frac{1}{f(\phi)}\left(1-\sqrt{1-2f(\phi)X}\right)-V(\phi)\right]~{},$
(2)

where $X\equiv-\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$, and
$f(\phi)$ is the so-called warp factor. The DBI action (2) is inspired by
string theory with the inflaton field regarded as the radial position of
branes moving inside a warped throat Born and Infeld (1934); Silverstein and
Tong (2004); Alishahiha  _et al._ (2004), and it has rich cosmological
phenomenology Cai and Wang (2010); Cai  _et al._ (2011); Cai and Xue (2009);
Cai and Wang (2010). The sound speed in the DBI model can be expressed as

$c_{s}^{2}=1-f(\phi)\dot{\phi}^{2}~{},$ (3)

where the dot denotes the derivative with respect to the cosmic time.

To realize a time-oscillating sound speed in the form of Eq. (1), Ref. Chen
_et al._ (2020) introduces an oscillating term into the well-studied anti-de
Sitter warp factor $f(\phi)=\lambda/\phi^{4}$ with $\lambda$ a constant,

$f(\bar{\phi})={\lambda\over\bar{\phi}^{4}}\big{[}1-\Theta\left(\bar{\phi}-\bar{\phi}_{s}\right)C(\bar{\phi})\big{]}~{},$
(4)

where $\bar{\phi}(\tau)$ is the homogeneous background of inflaton field
$\phi$, and $\bar{\phi}_{s}\equiv\bar{\phi}(\tau_{s})$ is evaluated at the
beginning of sound speed oscillation. The oscillation function $C(\phi)$ is
defined as

$C(\bar{\phi})\equiv\cos\left\\{2k_{*}\tau_{s}\exp\left[{H(1-\epsilon)\sqrt{\lambda}\over\sqrt{2\xi}}\left({1\over\bar{\phi}}-{1\over\bar{\phi}_{s}}\right)\right]\right\\}~{}.$
(5)

where $H$ is the Hubble parameter.
$\Theta\left(\bar{\phi}-\bar{\phi}_{s}\right)$ is the Heaviside step function,
which divides the inflationary epoch into the non-oscillating
($c_{s}^{2}=1-2\xi$ for $\tau\leq\tau_{s}$) and oscillating (i.e., Eq. (1))
phases. The functional form (5) is derived from the background evolution of
inflaton $\bar{\phi}(\tau)$,

$\bar{\phi}(\tau)\simeq\left({1\over\bar{\phi}_{i}}+{\sqrt{2\xi}\over
H(1-\epsilon)\sqrt{\lambda}}\ln\frac{\tau}{\tau_{i}}\right)^{-1}~{},$ (6)

in a quasi de-Sitter spacetime $a(\tau)\simeq 1/(\epsilon-1)H\tau$ where
$\epsilon\equiv-\dot{H}/H^{2}$ is the slow-roll parameter. $\bar{\phi}_{i}$ is
the value taken at the beginning moment of inflation $\tau_{i}$. Although this
solution is solved by non-oscillating sound speed, it is considered applicable
to both oscillating and non-oscillating phases Chen  _et al._ (2020). The
theoretical viability of this DBI model to realize SSR is also investigated in
Ref. Chen  _et al._ (2020).

##  III Backreaction in SSR-DBI inflation

###  III.1 Backreaction effect on the background

In order to obtain an effective background Lagrangian with backreaction, a
partition functional consisting of a DBI inflaton and an external source
$J(x)$ is introduced as

$Z[J]=\int\mathcal{D}\phi\exp\left[iS_{\text{DBI}}+i\int\mathrm{d}^{4}x\sqrt{-g(x)}J(x)\phi(x)\right]~{},$
(7)

where $\int\mathcal{D}[\phi]$ is the functional integral over all possible
inflaton field configurations. Using the semiclassical approach within the
path integral formalism Burgess (2020), one can perturbatively expand $Z[J]$
in Eq. (7) with Eq. (2) around the classical background $\bar{\phi}$, up to
the quadratic order in terms of the field perturbation $\phi_{\rm per}$ (i.e.,
$\phi=\bar{\phi}+\phi_{\rm per}$. The first-order term of $Z[J]$ vanishes due
to the classical equation of motion for $\phi_{\rm per}$). Within the
framework of the effective field theory (EFT) of inflation Cheung  _et al._
(2008), the inflation field perturbation $\phi_{\rm per}$ is “eaten” by
graviton (namely the unitary gauge $\phi_{\rm per}=0$), one can write down the
most general inflation action merely based on metric which respects the
unbroken spatial diffeomorphism. After using the Stückelberg trick, the time
diffeomorphism can be restored and the quadratic action for the scalar degree
of freedom of gravity is given by Cheung  _et al._ (2008)

$S^{(2)}=\int\mathrm{d}t\mathrm{d}^{3}xa^{3}(t)\left[\alpha(t)\dot{\pi}^{2}-\beta(t){(\nabla\pi)^{2}\over
a^{2}(t)}\right]~{},$ (8)

where the dot refers the derivative with respect to the cosmic time. $\pi$
corresponds to the broken time translation symmetry. The explicit forms of the
coefficients $\alpha(t),\beta(t)$ in the above action can be obtained by
matching the SSR-DBI model with EFT action (see Appendix A for details). After
integrating out the quadratic action (8) in the partition functional (7), one
can obtain the effective Lagrangian for background evolution as,

$\mathcal{L}_{\text{tot}}\equiv\mathcal{L}_{\text{bg}}+\mathcal{L}_{\text{br}}~{},$
(9)

which describes the backreaction effect and is calculated as follows (see
Appendix B for details),

$\displaystyle\mathcal{L}_{\text{bg}}$
$\displaystyle=\frac{1}{2}M_{\mathrm{Pl}}^{2}R_{\text{tot}}+f(\bar{\phi}_{\text{tot}})^{-1}\left(1-\sqrt{1-f(\bar{\phi}_{\text{tot}})\dot{\bar{\phi}}_{\text{tot}}^{2}}\right)-V(\bar{\phi}_{\text{tot}})~{},$
(10) $\displaystyle\mathcal{L}_{\text{br}}$
$\displaystyle={c_{s,\mathrm{bg}}^{-3}\over
128\pi^{2}}\Bigg{[}\Lambda_{k}^{2}\left(\Lambda_{k}^{2}-2\mathcal{M}^{2}\right)+2\mathcal{M}^{4}\ln\left(1+{\Lambda_{k}^{2}\over\mathcal{M}^{2}}\right)-2\Lambda_{k}^{4}\ln\left({\Lambda_{k}^{2}+\mathcal{M}^{2}\over\Lambda_{M}^{2}}\right)\Bigg{]}~{},$
(11)

where we have used the subscript “tot/bg” to represent the background
with/without backreaction, and the subscript “br” refers to the backreaction
term arising from the integrated perturbation sector. For example, the total
background inflaton field is written as $\bar{\phi}_{\rm tot}=\bar{\phi}_{\rm
bg}+\bar{\phi}_{\rm br}$, where $\bar{\phi}_{\rm bg}$ follows the evolution
(6). In Eq. (11), $\Lambda_{k}\equiv c_{s,\mathrm{bg}}^{-1}H_{\text{tot}}$ is
a physical cutoff at the sound horizon which is introduced to prevent the
divergence of $Z[J]$ as Appendix B shows, and $H_{\text{tot}}\equiv
H_{\text{bg}}+H_{\text{br}}$.
$\mathcal{M}^{2}\equiv\frac{1}{4}\left(3-2s\right)^{2}H_{\text{tot}}^{2}-H_{\text{tot}}\dot{s}$
acts as an effective mass implicated by Eq. (44), and
$s\equiv\dot{c}_{s,\mathrm{bg}}/(H_{\text{tot}}c_{s,\mathrm{bg}})$ is a
dimensionless parameter representing the rate of change of the sound speed
during inflation in presence of the backreaction. $\Lambda_{M}^{2}$ is a
parameter with the unit of the mass square Gong and Mylova (2022), whose value
is taken such that $\mathcal{L}_{\text{br}}$ vanishes at the initial time of
inflation $\tau_{i}$ when SSR and its backreaction have not occurred. Here we
assume the backreaction correction on background is homogeneous Mukhanov  _et
al._ (1997), thus above parameters do not depend on the spatial coordinate.

From the total Lagrangian $\mathcal{L}_{\text{tot}}$ in Eq. (9), we obtain the
effective Friedman equations as follows,

$\displaystyle\dot{H}_{\text{bg}}=$ $\displaystyle-{1\over
2M_{\text{Pl}}^{2}}c_{s,\mathrm{bg}}^{-1}\dot{\bar{\phi}}_{\text{bg}}^{2}~{},$
(12) $\displaystyle\dot{H}_{\text{br}}=$ $\displaystyle{1\over
2M_{\text{Pl}}^{2}}\frac{\delta\mathcal{L}_{\text{br}}}{\delta\dot{\bar{\phi}}_{\text{bg}}}\dot{\bar{\phi}}_{\text{bg}}~{},$
(13)

where $\delta/\delta\dot{\bar{\phi}}_{\text{bg}}$ refers to the variation with
respective to $\dot{\bar{\phi}}_{\text{bg}}$.

In principle, one can integrate the above equations to give the Hubble
parameter with/without backreaction. However, since it involves the
complicated integration of Eqs. (5) and (6), it is hard to investigate the
effect of backreaction analytically. In the remaining part of this paper, we
shall adopt the numerical method and analyze the modification of the SSR-DBI
inflation background by the backreaction effect. As an illustration, we shall
show the evolution of the Hubble parameter, slow-roll parameter, sound speed
and power spectrum of curvature perturbations with and without backreaction
for comparison.

For simplicity, we use $\bar{\phi}_{\rm bg}$ instead of $\bar{\phi}_{\rm tot}$
as the temporal parameter, using the relation $\dot{\bar{\phi}}_{\rm
bg}=\sqrt{\frac{2\xi}{\lambda}}\bar{\phi}_{\rm bg}^{2}$ indicated by Eq. (6).
According to Eqs. (12) and (13), we obtain the approximation,

$\frac{\mathrm{d}H_{\text{tot}}}{\mathrm{d}\bar{\phi}_{\text{bg}}}\simeq\frac{\mathrm{d}H_{\text{bg}}}{\mathrm{d}\bar{\phi}_{\text{bg}}}+\frac{1}{2M_{\mathrm{Pl}}^{2}}{\delta\mathcal{L}_{\text{br}}\over\delta\dot{\bar{\phi}}_{\text{bg}}}~{}.$
(14)

On the right-hand side, the background term is given by Chen  _et al._ (2020)

${\mathrm{d}H_{\text{bg}}\over\mathrm{d}\bar{\phi}_{\rm bg}}=-\sqrt{{\xi\over
2\lambda\left[1-2\xi\left(1-\Theta(\bar{\phi}_{\rm
bg}-\bar{\phi}_{s})C(\bar{\phi}_{\rm bg})\right)\right]}}{\bar{\phi}_{\rm
bg}^{2}\over M_{\text{Pl}}^{2}}~{},$ (15)

while the backreaction term is calculated as

$\displaystyle\frac{1}{2M_{\mathrm{Pl}}^{2}}\frac{\partial\mathcal{L}_{\text{br}}}{\partial\dot{\bar{\phi}}_{\mathrm{bg}}}$
$\displaystyle=\frac{c_{s,\text{bg}}^{-3}}{256\pi^{2}M_{\mathrm{Pl}}^{2}\dot{\bar{\phi}}_{\mathrm{bg}}}\left[-4\Lambda_{k}^{2}+4\mathcal{M}^{2}\ln\left(1+\frac{\Lambda_{k}^{2}}{\mathcal{M}^{2}}\right)\right]\left(8s^{2}H_{\text{tot}}^{2}-6sH_{\text{tot}}^{2}-2H_{\text{tot}}\dot{s}\right)c_{s,\text{bg}}^{-2}$
(16)
$\displaystyle+\frac{3\left(c_{s,\text{bg}}^{-2}-1\right)c_{s,\text{bg}}^{-3}}{256\pi^{2}M_{\mathrm{Pl}}^{2}\dot{\bar{\phi}}_{\rm
bg}}\left[\Lambda_{k}^{2}\left(\Lambda_{k}^{2}-2\mathcal{M}^{2}\right)+2\mathcal{M}^{4}\ln\left(1+\frac{\Lambda_{k}^{2}}{\mathcal{M}^{2}}\right)-2\Lambda_{k}^{4}\ln\left(\frac{\Lambda_{k}^{2}+\mathcal{M}^{2}}{\Lambda_{M}^{2}}\right)\right]~{},$

where $c_{s,\text{bg}}^{2}=1-f(\bar{\phi}_{\rm
bg})\dot{\bar{\phi}}_{\text{bg}}^{2}$ based on Eq. (3).

Figure 1:  The evolutions of Hubble parameters $H_{\text{tot}}$ (red solid)
and $H_{\text{bg}}$ (gray dashed) in terms of $\bar{\phi}_{\rm bg}$. The black
dashed vertical line refers to $\bar{\phi}_{s}$. The subplot shows the ratio
of backreaction correction $\Delta_{H}\equiv H_{\text{br}}/H_{\text{bg}}$
around $\bar{\phi}_{s}$. The parameters are taken as Chen  _et al._ (2020):
$\lambda=2\times 10^{9}$, $\epsilon=10^{-3}$, $\xi=0.1$, $k_{*}=10$,
$\tau_{s}=-14$, $N_{s}=19.752$ and $H_{i}/M_{\mathrm{Pl}}=10^{-5}$.

Numerically integrating Eq. (14) gives the evolution of $H_{\text{tot}}$,
which is shown by the red curve in Fig. 1. The ratio $\Delta_{H}\equiv
H_{\text{br}}/H_{\text{bg}}$ is shown in the subplot by the blue curve. The
parameters taken in this paper follow Ref. Chen  _et al._ (2020):
$\lambda=2\times 10^{9}$, $\epsilon=10^{-3}$, $\xi=0.1$, $k_{*}=10$,
$\tau_{s}=-14$, $N_{s}=19.752$111 Here we choose a different $\tau_{s}$ with
Chen  _et al._ (2020) to obtain a curvature spectrum with a resonant peak
$\mathcal{P}_{\zeta}(k_{peak})\sim 0.01$ Green and Kavanagh (2021) and
$H_{i}/M_{\mathrm{Pl}}=10^{-5}$ at the initial time $\tau_{i}$, then we
calculate $\Lambda_{M}^{2}/M_{\mathrm{Pl}}\simeq 1.7\times 10^{-5}$. It is
clearly seen from Fig. 1 that $H_{\text{tot}}$ is fairly close to $H_{\rm bg}$
during the whole inflationary era, which demonstrates a negligible correction
on the background expansion from the backreaction. At the non-oscillating
stage before SSR occurs ($\bar{\phi}_{\rm bg}<\bar{\phi}_{\rm s}$), the
backreaction effect almost vanishes as expected, due to the tiny curvature
perturbations. While during the oscillating stage, the backreaction grows to
around $\Delta_{H}\sim 10^{-7}$ as a result of the enhanced curvature
perturbations. Hence, we conclude that even if the peak of curvature
perturbations are enhanced by around seven orders of magnitude through the SSR
mechanism, their backreaction on the background expansion can be neglected.
This is physically reasonable since only a narrow range of scalar mode
functions around $k_{*}$ is enhanced in SSR, and their contribution to the
background energy density comes from only a small fraction of the whole
Fourier modes.

Figure 2: The evolutions of slow-roll parameters $\epsilon_{\text{tot}}$ (red
solid) and $\epsilon_{\text{bg}}$ (gray dashed) in terms of $\bar{\phi}_{\rm
bg}$. The ratio
$\Delta_{\epsilon}\equiv\epsilon_{\text{br}}/\epsilon_{\text{bg}}$ is
displayed in the subplot by the blue curve.

Fig. 2 shows the numerical result for the slow-roll parameter with/without
backreaction defined as
$\epsilon_{\text{tot}}=-\dot{H}_{\text{tot}}/H_{\text{tot}}^{2}$ and
$\epsilon_{\text{bg}}=-\dot{H}_{\text{bg}}/H_{\text{bg}}^{2}$, and also shows
the ratio of them defined as

$\displaystyle\Delta_{\epsilon}\equiv{\epsilon_{\text{br}}\over\epsilon_{\text{bg}}}$
$\displaystyle={1+\dot{H}_{\text{br}}/\dot{H}_{\text{bg}}\over(1+\Delta_{H})^{2}}-1~{},$
(17)

where $\epsilon_{\text{br}}\equiv\epsilon_{\text{tot}}-\epsilon_{\text{bg}}$.
The evolution of $\dot{H}_{\text{br}}/\dot{H}_{\text{bg}}$ is shown in Fig. 3,
which oscillates when $\bar{\phi}_{\rm bg}\gtrsim\bar{\phi}_{s}$, and then
decays to $\dot{H}_{\text{br}}/\dot{H}_{\text{bg}}\simeq-10^{-7}\ll 1$ for
larger $\bar{\phi}$, since
$\dot{H}_{\text{br}}/\dot{H}_{\text{bg}}=\frac{\mathrm{d}}{\mathrm{d}t}\left(\Delta_{H}H_{\text{bg}}\right)/\dot{H}_{\text{bg}}$
and $\Delta_{H}\rightarrow-10^{-7}$ as shown in Fig. 1. Thus, we obtain

$\displaystyle\Delta_{\epsilon}$ $\displaystyle\simeq-2\Delta_{H}=2\times
10^{-7}~{}~{}~{}\text{ for large $\bar{\phi}_{\mathrm{bg}}$}~{},$ (18)

which is consistent with the numerical result in Fig. 2, where the
backreaction effect on the slow-roll parameter experiences an oscillation at
the beginning of the oscillating phase and then converges to zero.

Figure 3:  The evolution of $\dot{H}_{\text{br}}/\dot{H}_{\text{bg}}$ during
SSR-DBI inflation.

Using the above results, we also investigate the backreaction effect on the
background energy density
$\rho=-T^{0}{}_{0}=-\left(\frac{\partial\mathcal{L}}{\partial\left(\partial_{0}\phi_{\alpha}\right)}\partial_{0}\phi_{\alpha}-g^{0}{}_{0}\mathcal{L}\right)$
and pressure
$P=\frac{1}{3}T^{i}{}_{i}=\frac{1}{3}\left(\frac{\partial\mathcal{L}}{\partial\left(\partial_{i}\phi_{\alpha}\right)}\partial_{i}\phi_{\alpha}-g^{i}{}_{i}\mathcal{L}\right)$.
By the decomposition of $\rho=\rho_{\text{bg}}+\rho_{\text{br}}$ and
$P=P_{\text{bg}}+P_{\text{br}}$, one writes the backreaction terms as follows,

$\displaystyle\rho_{\text{br}}$
$\displaystyle=\rho_{\text{bg}}\left(2\Delta_{H}+\Delta_{H}^{2}\right)~{},$
(19) $\displaystyle P_{\text{br}}$
$\displaystyle=\rho_{\text{bg}}\Delta_{H}\left({2\over
3}\frac{\dot{H}_{\text{br}}}{\dot{H}_{\text{bg}}}-\Delta_{H}-2\right)~{}.$
(20)

Therefore, the equation of state of the backreaction term is given by

$\displaystyle\omega_{\text{br}}\equiv{P_{\text{br}}\over\rho_{\text{br}}}$
$\displaystyle={\Delta_{H}\left({2\over
3}\dot{H}_{\text{br}}/\dot{H}_{\text{bg}}-\Delta_{H}-2\right)\over
2\Delta_{H}+\Delta_{H}^{2}}~{}.$ (21)

Since $\dot{H}_{\text{br}}/\dot{H}_{\text{bg}}\ll 1$ and $\Delta_{H}\ll 1$ as
discussed above, only considering leading terms, Eq. (21) becomes

$\displaystyle\omega_{\text{br}}\simeq-1~{}.$ (22)

And since at leading order $\rho_{\text{br}}\simeq
2\Delta_{H}\rho_{\text{bg}}$, where $\Delta_{H}\leq 0$ according to the
numerical result from Fig. 1, and
$\rho_{\text{bg}}=\frac{3H_{\mathrm{bg}}^{2}}{8\pi G}>0$, thus one has
$\rho_{\text{br}}<0$. Therefore the backreaction effect of perturbation
behaves like a negative cosmological constant, which is similar to the case of
chaotic inflation Brandenberger (2002).

###  III.2 Backreaction effect on the sound speed

Next, we investigate the backreaction effect on the sound speed $c_{s}^{2}$ as
the enhancement of the curvature perturbation $\zeta$ is sensitive to
$c_{s}^{2}$ in SSR Cai  _et al._ (2018); Chen  _et al._ (2020). In the next
subsection, we shall further show how the power spectrum of $\zeta$ in the
SSR-DBI inflation is influenced by the backreaction modification of
$c_{s}^{2}$. In this paper, we assume the functional form of the warp factor
does not receive any correction from the backreaction, and the sound speed
with backreaction in SSR-DBI inflation is thus given by

$c_{s,\text{tot}}^{2}=1-f(\bar{\phi}_{\text{tot}})\dot{\bar{\phi}}_{\text{tot}}^{2}~{},$
(23)

where $\bar{\phi}_{\text{tot}}(\tau)$ is written as

$\displaystyle\bar{\phi}_{\text{tot}}(\tau)\equiv$
$\displaystyle\left(\frac{1}{\bar{\phi}_{i}}+\frac{\sqrt{2\xi}}{H_{\text{tot}}(1-\epsilon_{\text{tot}})\sqrt{\lambda}}\ln\frac{\tau}{\tau_{i}}\right)^{-1}~{},$
(24)

The derivative of $\bar{\phi}_{\text{tot}}$ under the slow-roll approximation
$(\epsilon_{\text{tot}},~{}\epsilon_{\text{bg}})\ll 1$ and up to the first
order of $\Delta_{H}$, is calculated as

$\displaystyle\dot{\bar{\phi}}_{\text{tot}}(\tau)\simeq$
$\displaystyle\left[1+\left(1-2\frac{\bar{\phi}_{\text{bg}}}{\bar{\phi}_{i}}\right)\left(\Delta_{H}-\epsilon_{\text{br}}\right)\right]\dot{\bar{\phi}}_{\text{bg}}$
(25)
$\displaystyle+\left(1-\frac{\bar{\phi}_{\text{bg}}}{\bar{\phi}_{i}}\right)\bar{\phi}_{\text{bg}}\left(\dot{\Delta}_{H}-\dot{\epsilon}_{\text{br}}\right)~{}.$

Notice that in Eq. (14), we do not consider the backreaction in
$\bar{\phi}_{\rm tot}$ when calculating $H_{\rm br}$. In principle, one can
use $\bar{\phi}_{\rm tot}$ to replace $\bar{\phi}_{\rm bg}$ to obtain a more
precise prediction of $H_{\rm br}$. More detailed research shall be carried
out in our future work.

Based on the form of $\phi_{\text{tot}}$ in Eq. (24), the oscillation term
$C\left(\phi_{\text{tot}}(\tau)\right)$ in Eq. (5) is not affected by the
backreaction. Thus the correction on the warp factor $f(\phi_{\text{tot}})$ in
Eq. (4) can be expanded as
$f(\bar{\phi}_{\text{tot}}(\tau))=\left(1-4\frac{\bar{\phi}_{\text{br}}(\tau)}{\bar{\phi}_{\text{bg}}(\tau)}\right)f(\bar{\phi}_{\text{bg}}(\tau))$.
Using the above results, the sound speed with backreaction becomes

$c_{s,\text{tot}}^{2}=c_{s,\text{bg}}^{2}+\left(1-c_{s,\text{bg}}^{2}\right)\left[2\Delta_{H}-2\left(\bar{\phi}_{\mathrm{bg}}-\frac{\bar{\phi}_{\mathrm{bg}}^{2}}{\bar{\phi}_{i}}\right)\frac{\dot{\Delta}_{H}-\dot{\epsilon}_{\text{br}}}{\dot{\bar{\phi}}_{\text{bg}}}+\left(2-4\frac{\bar{\phi}_{\text{bg}}}{\bar{\phi}_{i}}\right)\epsilon_{\text{br}}\right]~{}.$
(26)

The evolutions of the sound speed $c_{s,\text{tot}/\text{bg}}^{2}$ are shown
in Fig. 4. The subplot shows the comparison with the correction manually
amplified to show the difference more explicitly. It can be seen that the
correction is large at the beginning of the oscillation stage and decays
afterward, which is consistent with the behavior of $\epsilon_{\text{br}}$
shown in Fig. 2. It is also worth noting that the backreaction does not cause
the $c_{s}$ to become larger than untiy.

Figure 4:  The numerical results of sound speeds $c_{s,\text{tot}}^{2}$ (red
solid) and $c_{s,\text{bg}}^{2}$ (gray dashed), and their ratio is shown by
the blue curve in the subplot.

###  III.3 Backreaction effect on the curvature perturbations

With the backreaction effects on the background quantities and the sound speed
given in previous sections, we shall examine the backreaction on the power
spectrum of curvature perturbations $\zeta$, which is given by

$\mathcal{P}_{\zeta}(k)=\frac{k^{3}}{2\pi^{2}}|\zeta_{k}|^{2}~{},$ (27)

and can be decomposed as $\mathcal{P}^{\rm
tot}_{\zeta}=\mathcal{P}_{\zeta}^{\text{bg}}+\mathcal{P}_{\zeta}^{\text{br}}$.
The dynamics of $\zeta$ satisfies the Mukhanov-Sasaki equation Sasaki (1986);
Mukhanov (1988)

$\frac{\mathrm{d}^{2}v_{k}}{\mathrm{d}\tau^{2}}+\left(c_{s,\text{tot}}^{2}k^{2}-\frac{1}{z}\frac{\mathrm{d}^{2}z}{\mathrm{d}\tau^{2}}\right)v_{k}=0~{},$
(28)

with $v\equiv z\zeta$ and $z=\sqrt{2\epsilon_{\text{tot}}}a/c_{s,\text{tot}}$.
By taking the initial condition as the Bunch-Davies vacuum
$v_{k}(\tau)=e^{-i\left(1-2\xi\right)^{2}k\tau}/[\sqrt{2k}\left(1-2\xi\right)^{1/4}]$,
we numerically solved $\zeta$ at the horizon-crossing time $\tau=1/k$. The
result for the power spectrum is shown in Fig. 5. We find that the peak with
backreaction $\mathcal{P}^{\rm tot}_{\zeta}(k_{\rm peak})$ is smaller than
that without backreaction $\mathcal{P}^{\rm bg}_{\zeta}(k_{\rm peak})$, where
$k_{\mathrm{peak}}$ denotes the position of resonant peak. Subsequently, the
relative correction $\left(\mathcal{P}^{\rm tot}_{\zeta}(k_{\rm
peak})-\mathcal{P}^{\rm bg}_{\zeta}(k_{\rm peak})\right)/\mathcal{P}^{\rm
bg}_{\zeta}(k_{\rm peak})\approx-4.3\times 10^{-4}$ is negative. Therefore,
the backreaction effect does not break the perturbativity of SSR mechanism.

Figure 5:  The numerical results of the curvature power spectra
$\mathcal{P}^{\rm tot}_{\zeta}$ (red solid) and $\mathcal{P}^{\rm bg}_{\zeta}$
(gray dashed). The small subplot shows a zoom-in to the power spectra around
the peak.

##  IV Conclusions

In this paper, we investigate the backreaction effect of the enhanced small-
scale curvature perturbations in the SSR-DBI inflation. We find that the
backreaction effects on both the background and perturbations can be neglected
and do not spoil the previous results in the SSR-DBI inflation model in the
absence of backreaction.

As a starting point, we review the SSR-DBI inflation model Chen  _et al._
(2020) and adopt the methods presented in Ref. Gong and Mylova (2022) to
calculate the backreaction effects of perturbations on the background. After
integrating out the quadratic curvature perturbation action, we obtain the
effective Lagrangian (11) which describes the backreaction effect on the
inflationary background dynamics.

Based on this effective Lagrangian, we numerically solve the evolution of the
Hubble parameter $H$, the slow-roll parameter $\epsilon$, and the sound speed
$c_{s}^{2}$ with backreaction. We find that the backreaction effect causes a
tiny correction to the Hubble parameter in an order of $10^{-7}$ compared to
the original results in Ref. Chen  _et al._ (2020), as shown in Fig. 1. For
the slow-roll parameter, as displayed in Fig. 2, the correction resulting from
the backreaction effect exhibits an oscillation that begins with an amplitude
between $-0.3$ and $0.1$, but gradually decreases to a value as small as
$-2\Delta_{H}\sim 10^{-7}$. Additionally, the results presented in Fig. 4 tell
us that the sound speed is slightly altered by curvature perturbations.
Applying the sound speed with/without backreaction to the Mukhanov-Sasaki
equation (28), we numerically solve the curvature perturbation $\zeta$ and
plot the corresponding power spectra $\mathcal{P}_{\zeta}$ in Fig. 5. We
conclude that the backreaction effect on the amplification efficiency of
curvature perturbations and the PBH formation predicted by the SSR mechanism
is fairly small.

###### Acknowledgements.

We would like to thank Yi-Fu Cai, Maria Mylova, Xin Ren, Chon Man Sou, Yi
Wang, Xuan Ye and Yuhang Zhu for valuable suggestions and discussions. This
work is supported in part by National Key R&D Program of China
(2021YFC2203100), by NSFC (11961131007, 12261131497, 12003029), by CAS young
interdisciplinary innovation team (JCTD-2022-20), by 111 Project for
“Observational and Theoretical Research on Dark Matter and Dark Energy”
(B23042), by Fundamental Research Funds for Central Universities, by CSC
Innovation Talent Funds, by USTC Fellowship for International Cooperation, by
USTC Research Funds of the Double First-Class Initiative, by CAS project for
young scientists in basic research (YSBR-006). C.Chen is supported by the
Jockey Club Institute for Advanced Study at HKUST. We acknowledge the use of
computing clusters LINDA and JUDY of the particle cosmology group at USTC.

##  Appendix A EFT description of SSR-DBI inflation

In this appendix, we shall give an EFT description for SSR-DBI inflation. EFT
provides a generic description of metric perturbations around a quasi-de
Sitter background Cheung  _et al._ (2008). Even without the knowledge of the
full UV theory, EFT can remove the unnecessary high-energy degrees of freedom,
while keeping track of their influence on the low-energy physics Weinberg
(1979); Polchinski (1984, 1992); Georgi (1993); Weinberg (2021); Burgess
(2020); Green (2022). As the inflation vacuum is time-evolving, the general
EFT action for inflation can be built by operators with broken-time
translation symmetry Cheung  _et al._ (2008); Weinberg (2008). For the single
field inflation model $\phi(t,\vec{x})=\phi_{0}(t)+\delta\phi(t,\vec{x})$, one
can work in the unitary gauge $\delta\phi(t,\vec{x})=0$ Cheung  _et al._
(2008). Besides the Riemann tensor $R_{\mu\nu\rho\sigma}$ that is already
invariant under diffeomorphism transformation, operators satisfying this
reduced symmetry in unitary gauge include tensors with upper index 0: $g^{00}$
and $R^{00}$, extrinsic curvature tensor $K_{\mu\nu}\equiv
h_{\mu}{}^{\sigma}\nabla_{\sigma}n_{\nu}$ (where
$n_{\mu}=\frac{\partial_{\mu}t}{\sqrt{-g^{\mu\nu}\partial_{\mu}t\partial_{\nu}t}}$
and $h_{\mu\nu}\equiv g_{\mu\nu}+n_{\mu}n_{\nu}$ are the normal vector and the
induced metric of the hypersurface, respectively), arbitrary time-dependent
function $f(t)$ and the covariant derivatives of the aforementioned operators.
As $R^{00}$ can be written in terms of $g^{00}$ and $h^{\mu\nu}$, while
$n^{\mu}$ can be written in terms of $g^{00}$ and $K_{\mu\nu}$, the most
generic action is given by the following form Cheung  _et al._ (2008),

$S=\int
d^{4}x\sqrt{-g}F\left(R_{\mu\nu\rho\sigma},g^{00},K_{\mu\nu},\nabla_{\mu},t\right)~{},$
(29)

which can be expanded perturbatively as

$\displaystyle S_{\text{EFT}}=$
$\displaystyle\int\mathrm{d}t\mathrm{d}^{3}x\sqrt{-g}\left[\frac{M_{\mathrm{Pl}}^{2}}{2}R-c(t)g^{00}-\Lambda(t)+{M_{2}(t)^{4}\over
2!}(g^{00}+1)^{2}+{M_{3}(t)^{4}\over 3!}(g^{00}+1)^{3}+\cdots\right.$ (30)
$\displaystyle-{\bar{M}_{1}(t)^{3}\over 2}(g^{00}+1)\delta
K_{\mu}{}^{\mu}-{\bar{M}_{2}(t)^{2}\over 2}(\delta
K_{\mu}{}^{\mu})^{2}-{\bar{M}_{3}(t)^{2}\over 2}\delta K_{\nu}{}^{\mu}\delta
K_{\mu}{}^{\nu}+\cdots$
$\displaystyle\left.+m_{1}^{2}(t)h^{\mu\nu}\partial_{\mu}g^{00}\partial_{\nu}g^{00}+\lambda_{1}(t)\delta
R^{2}+\lambda_{2}(t)\delta R_{\mu\nu}\delta
R^{\mu\nu}+\mu_{1}^{2}(t)\left(1+g^{00}\right)\delta R+\cdots\right]~{},$

where $M_{\text{Pl}}$ is the reduced Planck mass, $R$ is the Ricci scalar,
$c(t)$, $\Lambda(t)$, $M_{2,3}(t)$, $\bar{M}_{1,2,3}(t)$, $\lambda_{1,2}(t)$,
$\mu_{1}(t)$, $m_{1}(t)$ are arbitrary time-dependent functions. The broken
time diffeomorphism is restored by performing an infinitesimal transformation
$t\rightarrow t+\pi(x)$, which will induce transformations for time-dependent
functions in the action as

$f(t)\rightarrow
f\left(t+\pi\left(t,x^{i}\right)\right)=f(t)+\dot{f}(t)\pi\left(t,x^{i}\right)+\frac{1}{2}\ddot{f}(t)\pi\left(t,x^{i}\right)^{2}+\ldots~{},$
(31)

and requiring $\pi$ transform as
$\pi(x)\rightarrow\widetilde{\pi}(\widetilde{x}(x))=\pi(x)-\xi^{0}(x)$ under
the time translation $t\rightarrow\tilde{t}=t+\xi^{0}(x)$. This procedure is
called the Stükelberg trick. The $\pi$ field that arises from the broken time
diffeomorphism is the Goldstone boson, and is related to the curvature
perturbation via relation $\zeta=-H\pi$ Cheung  _et al._ (2008). By invoking
Arnowitt-Deser-Misner decomposition
$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=-N^{2}dt^{2}+h_{ij}\left(dx^{i}+N^{i}dt\right)\left(dx^{j}+N^{j}dt\right)$
and solving the constraints for Lagrangian multipliers: the lapse function $N$
and the shift functions $N^{i}$, one obtains a quadratic action describing the
dynamics of $\pi$ Cheung  _et al._ (2008); Gleyzes  _et al._ (2013),

$S_{\pi}^{(2)}=\int\mathrm{d}t\mathrm{d}^{3}xa^{3}(t)\left[\alpha(t)\dot{\pi}^{2}-\beta(t){(\nabla\pi)^{2}\over
a^{2}(t)}\right]~{},$ (32)

where the coefficients are defined as

$\displaystyle\alpha(t)$
$\displaystyle=c(t)+2M_{2}(t)^{4}+\frac{3}{4}{\bar{M}_{1}(t)^{6}\over
M_{\mathrm{Pl}}^{2}}~{},$ (33) $\displaystyle\beta(t)$
$\displaystyle=c(t)-\frac{1}{4}{\bar{M}_{1}(t)^{6}\over
M_{\mathrm{Pl}}^{2}}+\frac{1}{2}\left(\dot{M}_{1}(t)^{3}+H\bar{M}_{1}(t)^{3}\right)~{}.$
(34)

We refer readers to Cabass  _et al._ (2023) for a recent review on EFT in
cosmology. The sound speed $c_{s}$ can be read directly from the action (32),

$c_{s}^{2}(t)={\beta(t)\over\alpha(t)}~{}.$ (35)

By matching with the DBI action, one can directly obtain the EFT coefficients
as

$\displaystyle c(t)$
$\displaystyle=\Lambda(t)=\frac{1}{2}\dot{\bar{\phi}}^{2}\left(1-f(\bar{\phi})\dot{\bar{\phi}}^{2}\right)^{-1/2}~{},$
(36) $\displaystyle M_{n}(t)^{4}$
$\displaystyle=\frac{1}{4}f(\bar{\phi})\dot{\bar{\phi}}^{4}\left(1-f(\bar{\phi})\dot{\bar{\phi}}^{2}\right)^{-3/2}~{},$
$\displaystyle\bar{M}_{1}(t)^{3}$
$\displaystyle=\lambda_{1,2}(t)=\mu_{1}^{2}(t)=m_{1}^{2}(t)=0~{},$
$\displaystyle\bar{M}_{2}(t)^{2}$
$\displaystyle=-\bar{M}_{3}(t)^{2}=-M_{\mathrm{Pl}}^{2}~{},$

and coefficients $\alpha(t)$ and $\beta(t)$ in Eq. (32) become

$\displaystyle\beta_{\text{DBI}}(t)$
$\displaystyle=\frac{1}{2}\dot{\bar{\phi}}^{2}\left(1-f(\bar{\phi})\dot{\bar{\phi}}^{2}\right)^{-1/2}~{},$
(37) $\displaystyle\alpha_{\text{DBI}}(t)$
$\displaystyle=\frac{1}{2}\dot{\bar{\phi}}^{2}\left(1-f(\bar{\phi})\dot{\bar{\phi}}^{2}\right)^{-3/2}~{}.$

with the above coefficients, Eq. (35) also gives the sound speed in DBI
inflation $c_{s}^{2}=1-f(\bar{\phi})\dot{\bar{\phi}}^{2}$. Notice that the
above EFT description works when $H$ is below the energy scale

$E^{4}\sim
16\pi^{2}M_{\mathrm{Pl}}^{2}|\dot{H}|\frac{c_{s}^{5}}{1-c_{s}^{2}}\sim
8\pi^{2}H^{2}M_{\mathrm{Pl}}^{2}\frac{\epsilon}{\xi}~{},$ (38)

namely, the non-renormalizable interactions become insignificant Cheung  _et
al._ (2008). In Eq. (38), an approximation of time-averaged sound speed
$c_{s}^{2}=1-2\xi$ is adopted. For typical parameter choices in the SSR-DBI
inflation, $H/M_{\rm Pl}\sim 10^{-5}$ and $\epsilon=0.01$, the condition (38)
is satisfied.

##  Appendix B One-Loop effective action

Here we derive the Lagrangian with backreaction effect from the one-loop
effective action in the SSR-DBI model following the method presented by Ref.
Gong and Mylova (2022), in which the backreaction is obtained by integrating
out the high-order perturbations. To get this effective action, we start from
the partition function $Z[J]$ in Eq. (7) with the perturbed inflation field as
$\phi=\bar{\phi}+\phi_{\rm per}$, which can be perturbatively expanded around
the classical background up to the second order as

$Z[J]=\int\mathcal{D}\phi\exp\left\\{i\int\mathrm{d}^{4}x\sqrt{-g}\left(\mathcal{L}[\bar{\phi}]+J\bar{\phi}\right)+{1\over
2}\int\sqrt{-g}\mathrm{d}^{4}x\sqrt{-g}\mathrm{d}^{4}y{\delta^{2}\mathcal{L}[g,\phi]\over\delta\phi(x)\delta\phi(y)}\Big{|}_{\bar{\phi}}\phi_{\rm
per}(x)\phi_{\rm per}(y)\right\\}~{}.$ (39)

The last term in the above formula represents the contribution from the
perturbation $\phi_{\text{per}}$, where the integral over $\phi_{\text{per}}$
can be carried out by converting the quadratic term to a Gaussian integral
Peskin and Schroeder (1995),

$\displaystyle\int\mathcal{D}\eta\exp\left({i\over
2}\int\mathrm{d}^{4}x\mathrm{d}^{4}y\eta(x)\mathcal{A}(x,y)\eta(y)\right)=\text{const}\times\left(\det\left[-i\mathcal{A}(x,y)\right]\right)^{-1/2}~{},$
(40)

where $\mathcal{A}(x,y)$ is an operator, and the constant can be dropped by
the normalization of vacuum state Peskin and Schroeder (1995). The effective
action for $\bar{\phi}$ is Gong and Mylova (2022)

$\displaystyle\Gamma[\bar{\phi}]=$ $\displaystyle-i\ln
Z[J]-\int\mathrm{d}^{4}x\sqrt{-g}J\bar{\phi}~{}.$ (41)

Substituting Eqs. (39) and (40) into the above, and using the identity
$\ln\left[\det(-i\mathcal{A})\right]=\text{Tr}\left[\ln(-i\mathcal{A})\right]$
Peskin and Schroeder (1995), Eq. (41) becomes

$\displaystyle\Gamma[\bar{\phi}]=\int\mathrm{d}^{4}x\sqrt{-g}\mathcal{L}[\bar{\phi}]-\frac{1}{2}\mathrm{Tr}\left[\ln\left(-i\mathcal{A}\right)\right]\equiv\int\mathrm{d}^{4}x\sqrt{-g}\left(\mathcal{L}_{\text{bg}}[\bar{\phi}]+\mathcal{L}_{\text{br}}[\bar{\phi}]\right)~{},$
(42)

for the SSR-DBI model, the background and the one-loop correction are

$\mathcal{L}_{\text{bg}}=\mathcal{L}_{\text{DBI}}[\bar{\phi}]~{},\quad\mathcal{L}_{\text{br}}=-\frac{i}{2}\int{\mathrm{d}^{4}k\over(2\pi)^{4}}\ln\left(-i\mathcal{A}\right)~{},$
(43)

where $k$ denotes the physical momentum and we have taken the continuous limit
of summation. The explicit form of $\mathcal{A}$ can be obtained from Eq. (32)
using integration by parts,

$S^{(2)}=\int\mathrm{d}\tau\mathrm{d}^{3}x\eta(\tau,\mathbf{x})\left[\frac{\mathrm{d}^{2}}{\mathrm{d}\tau^{2}}-c_{s}(\tau)^{2}{\nabla^{2}\over
a^{2}(\tau)}+\mathcal{M}^{2}\right]\eta(\tau,\mathbf{x})~{},$ (44)

where $\eta\equiv\pi\left(a^{2}\alpha\right)^{-1/2}$ is a normalized variable,
the effective mass term is
$\mathcal{M}^{2}=\frac{1}{4}\left(3-2s\right)^{2}H_{\text{tot}}^{2}-H_{\text{tot}}\dot{s}$.
The operator $\mathcal{A}$ can be read by comparing Eq. (44) with Eq. (40),

$\mathcal{A}=\frac{\mathrm{d}^{2}}{\mathrm{d}\tau^{2}}-c_{s}(\tau)^{2}{\nabla^{2}\over
a^{2}(\tau)}+\mathcal{M}^{2}~{},$ (45)

Substituting Eq. (45) to Eq. (43), one gets

$\mathcal{L}_{\text{br}}=-\frac{1}{2}c_{s}^{-3}\left.\frac{\partial}{\partial\gamma}\left[\int{\mathrm{d}^{4}\tilde{k}^{\prime}\over(2\pi)^{4}}{1\over\left(\tilde{k}^{\prime}{}^{2}+\mathcal{M}^{2}\right)^{\gamma}}\right]\right|_{\gamma=0}~{},$
(46)

where $k^{\prime}{}^{0}\equiv ik^{0}$ and $k^{\prime}{}^{i}\equiv c_{s}k^{i}$,
and the identity
$\int\frac{\mathrm{d}^{4}\tilde{k}}{(2\pi)^{4}}\ln\left(\tilde{k}^{2}+\mathcal{M}^{2}\right)=\left.-\frac{\partial}{\partial\gamma}\left[\int\frac{\mathrm{d}^{4}\tilde{k}}{(2\pi)^{4}}\frac{1}{\left(\tilde{k}^{2}+\mathcal{M}^{2}\right)^{\gamma}}\right]\right|_{\gamma=0}$
is used Gong and Mylova (2022). To prevent the divergence of the above
integration, we introduce a cutoff at the sound horizon $\Lambda_{k}\equiv
c_{s}^{-1}aH$ and only the modes of $|k^{\prime}|<\Lambda_{k}$ contribute to
the integration, since the resonant mode of curvature perturbation in SSR-DBI
inflation reaches its maximum at the sound horizon and the backreaction effect
shall be relatively small for $|k^{\prime}|>\Lambda_{k}$. So we carry out the
integration for $|k^{\prime}|<\Lambda_{k}$ and obtain Eq. (11).

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