# Gauge invariant quantum backreaction in $U(1)$ axion inflation

Davide Campanella Galantia,b<EMAIL_ADDRESS>Pietro
Conzinuc,b<EMAIL_ADDRESS>Giovanni Marozzia,b
<EMAIL_ADDRESS>Simony Santos da Costad,e,b
<EMAIL_ADDRESS>aDipartimento di Fisica, Università di Pisa,
Largo B. Pontecorvo 3, 56127 Pisa, Italy
bIstituto Nazionale di Fisica Nucleare, Sezione di Pisa, Italy
cTheory Division, CERN, CH-1211 Geneva 23, Switzerland
dDipartimento di Fisica, Università di Trento, Via Sommarive 14, 38123 Trento,
Italy
eTrento Institute for Fundamental Physics and Applications (TIFPA), Via
Sommarive 14, 38123 Trento, Italy

###### Abstract

We evaluate the quantum backreaction due to a gauge field coupled to a pseudo
scalar field driving a slow-roll inflationary stage, the so-called axion
inflation. The backreaction is evaluated for the first time using a gauge
invariant approach, going to second order in perturbation theory and taking in
consideration inflaton fluctuations as well as scalar perturbations of the
metric. Within our gauge invariant, but observer-dependent approach, we
naturally consider as physical observer the one comoving with the inflaton
field. Considering the effective expansion rate consequent to the gauge fields
bacreaction, we observe that the backreaction effect becomes significant quite
rapidly, moving the system out of the perturbative regime and into what is
often referred to the strong backreaction regime. This behavior also applies
to the parameter which dictates the production of the gauge fields. The space-
time dynamic is initially mainly influenced by the helicity contribution, with
a significant response to the energy density which occurs much later with
respect to the number of e-folds. As a final result, we see that the evaluated
backreaction prolongs the inflationary period much more compared to the
scenarios previously studied neglecting scalar metric perturbations.

##  I Introduction

The inflationary paradigm was originally proposed to solve the problems
associated with the Hot Big-Bang standard theory Guth (1981); Linde (1982) via
an early stage of accelerated cosmological expansion. Immediately afterwords
it was also realized that already a single field inflationary model could
furnish a causal mechanism to generate nearly scale invariant spectra of
scalar and tensor fluctuations Mukhanov and Chibisov (1981); Starobinsky
(1979), elevating the model to one of the most recognized and accepted
candidates to describe the early universe.

This first success leads to a continuous and intense investigation of
different inflationary models, together with the investigation of the
fundamental nature of the inflaton field, its possible interaction with other
fields, and the predictions and phenomenology associated to the possible
different models (see for example Martin  _et al._ (2014) for a large
collection of models). Recent literature Anber and Sorbo (2010); Sorbo (2011);
Anber and Sorbo (2012); Barnaby  _et al._ (2011) has generated significant
interest in the behavior of a pseudo-scalar inflaton when coupled to U(1)
abelian gauge fields. This possibility extends models of inflation driven by a
axion-like particle, such as natural inflation, first proposed in Freese  _et
al._ (1990); Adams  _et al._ (1993). Axions copiously appear in string theory
Svrcek and Witten (2006); Kim (1987); Marsh (2016) providing a remarkable
inflaton candidate in UV complete theory.

Within this framework, the inflaton/gauge fields coupling brings several
interesting effects with a rich phenomenology (see, for example, Ballardini
_et al._ (2019); Anber and Sorbo (2010); Pajer and Peloso (2013); Maleknejad
_et al._ (2013); Barnaby  _et al._ (2012); Turner and Widrow (1988); Garretson
_et al._ (1992); Adshead  _et al._ (2016); Sobol  _et al._ (2019); Adshead
_et al._ (2015); McDonough  _et al._ (2016); Domcke and Mukaida (2018);
Barnaby  _et al._ (2011); Domcke  _et al._ (2020); Caravano  _et al._ (2022);
Hashiba  _et al._ (2022); Ishiwata  _et al._ (2022); Lozanov  _et al._
(2019)). Furthermore, let us mention how inflation driven by a pseudo-scalar
is the perfect touchstone to incorporate parity violation during a quasi-
exponential expansion (see e.g. Bartolo  _et al._ (2015)).

The core of the physical process concerning the coupling between the pseudo-
scalar inflaton $\phi$ and the gauge fields $A_{\mu}$ relies on the so-called
Chern-Simons-like operator. As a consequence of the coupling through such
operator, quantum vacuum fluctuations of the gauge field $A_{\mu}$ are
amplified into physical excitations, since $A_{\mu}$ acquires a time-dependent
background as a consequence of the slow-rolling of the inflaton. In this way,
the field $\phi$ will slow down because its energy is utilized for the
production of the quantum gauge fields: this dissipative effect should make
inflation lasts longer depending on how much the gauge coupling is strong
Anber and Sorbo (2010); Sorbo (2011); Anber and Sorbo (2012); Barnaby  _et
al._ (2011). To resume, as a consequence of the pseudo-scalar/gauge fields
coupling, the inflaton field provokes an amplification of the quantum
fluctuations of $A_{\mu}$ which backreact on the space-time dynamic.

Let us mention that, since we are in the framework of QFT in curved space-time
the observables present in the model show (as usual) ultraviolet (UV)
divergences and have to be regularized through a renormalization scheme
Birrell and Davies (1984); Parker and Toms (2009). Here we will use the
adiabatic regularization approach Zeldovich and Starobinsky (1971); Parker and
Fulling (1974); Fulling and Parker (1974); Bunch (1980); Anderson and Parker
(1987) in order to compute the energy density and helicity integral of the
gauge fields. In particular, we will consider in our computation the finite
results obtained through a new adiabatic renormalization scheme which use a
physically motivated comoving infrared (IR) cut-off, as described in Animali
_et al._ (2022), in order to remove some unphysical IR divergences that one
would obtain through the standard adiabatic subtraction for the considered
model (see Ballardini  _et al._ (2019)).

In this manuscript we evaluate the gauge fields backreaction using a gauge
invariant observer dependent averaging prescription Gasperini  _et al._ (2009)
and looking to a set of gauge invariant effective field equations as defined
in Gasperini  _et al._ (2010) (see also the seminal works Buchert (2000,
2001)). Furthermore, following Finelli  _et al._ (2011) we will work within a
perturbative framework and go up to the second perturbative order in the
evaluation of the cosmological backreaction to obtain the first non-zero
contribution.

The set of effective equations of Gasperini  _et al._ (2010) describes, in
particular, the evolution of the effective expansion rate of the universe as
measured by a given observer. Within the framework of single field inflation
is then natural to consider as such observer the one comoving with the
inflaton field, being the inflaton field the only available clock in this
framework. In this approach, where one looks to cosmological observables,
Einstein equations are satisfied at any perturbative order (as one would
expect) and the gauge fields backreaction on the expansion rate is a
consequence of their coupling with the matter and metric scalar fluctuations
to second perturbative order. The above approach differs from the standard one
proposed in Anber and Sorbo (2010, 2012); Barnaby  _et al._ (2012), and
further developed in Gorbar  _et al._ (2021); Ballardini  _et al._ (2019);
Adshead  _et al._ (2016); Sobol  _et al._ (2019); Adshead  _et al._ (2015);
McDonough  _et al._ (2016); Domcke and Mukaida (2018); Domcke  _et al._
(2020); Hashiba  _et al._ (2022); Ishiwata  _et al._ (2022); Lozanov  _et al._
(2019). In the case of Anber and Sorbo (2010, 2012); Barnaby  _et al._ (2012)
the backreaction of the gauge fields is studied adding their contribution to
the background Friedmann equations (with Einstein equation effectively
considered only at the background level) and to the background equation of
motion of the pseudo scalar inflaton field. Namely, in Anber and Sorbo (2010,
2012); Barnaby  _et al._ (2012) is taken in consideration only the gauge
fields contribution, but are neglected the induced scalar perturbations of
matter and metric sectors, although the inflaton fluctuations were then added
in Domcke  _et al._ (2024) and simulations in lattice were provided in
Caravano  _et al._ (2022); Caravano (2022); Figueroa  _et al._ (2023a, b).

The manuscript is organized as follows. In Section II we describe the
theoretical framework of the U(1)-axion inflation model, where a pseudo-scalar
inflaton is coupled to abelian gauge fields. In Section III we introduce the
gauge invariant, observed dependent, approach to the cosmological backreaction
problem we shall use, and explicitly obtain the related effective Hubble
expansion rate. In Section IV we detail the second-order perturbative
equations for scalar perturbations taking in consideration their coupling with
the gauge fields fluctuations. In Section V we introduce the class of the
comoving observers in single-field inflationary models and describe the form
of the backreaction terms coming from the gauge fields contribution. In
Section VI we solve the second-order equation of motion for the pseudo-scalar
inflaton, considering the gauge fields contribution, and present the
analytical expression for the gauge fields backreaction on the effective
expansion rate. In Section VII we introduce the numerical methods used to
evaluate the magnitude of the gauge invariant backreaction effect, and plot
the consequent results. Finally, in Section VIII we present our final remarks
and conclusions. In Appendix A we give the expressions for the energy density
and the helicity integral of the gauge fields used in the manuscript while in
Appendix B we write the second order perturbative expression for Einstein
tensor and energy-momentum tensor.

##  II $U(1)$-axion inflation model

Let us consider a pseudo-scalar inflaton coupled to U(1) abelian gauge fields
described by the following action

$\displaystyle
S=\int\differential^{4}x\sqrt{-g}\left[-\dfrac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)-\dfrac{1}{4}F_{\mu\nu}F^{\mu\nu}-\dfrac{g\phi}{4}F^{\mu\nu}\tilde{F}_{\mu\nu}\right]\,,$
(2)

where $\phi$ is the axion-like field,
$\tilde{F}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$,
$F_{\alpha\beta}=\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}$,
$A_{\mu}$ is the gauge fields. Furthermore, the coupling constant $g$ can be
expressed in terms of the axion decay constant $f$ by the relation
$g=\alpha/f$, with $\alpha$ a dimensionless parameter. The last term in the
action (2) is a Chern-Simon like term, thanks to this coupling term the axion
field drives the gauge fields amplification.

To make contact with past literature, let us first consider the case in which
we set to zero metric perturbations. Therefore, we consider a spatially flat
Friedmann- Lemaître-Robertson-Walker (FLRW) background metric

$ds^{2}=-dt^{2}+a^{2}(t)d\textbf{x}^{2}\,,$ (3)

with $a(t)$ the scale factor, and we will evaluate the impact of the gauge
fields production on the background Euler-Lagrange and Einstein equations.
Starting from the Euler-Lagrange equations, using the electro-magnetic-
notation (bold letters stand for 3 dimensional vectorial quantities)

$\displaystyle\textbf{E}=-\dfrac{\textbf{A}^{\prime}}{a^{2}}\,,\qquad\qquad\textbf{B}=\frac{\boldsymbol{\nabla}\times\textbf{A}}{a^{2}}\,,$
(4)

one finds Anber and Sorbo (2010)

$\displaystyle\phi^{\prime\prime}+2aH\phi^{\prime}-\laplacian{\phi}+a^{2}V_{\phi}(\phi)=ga^{2}\textbf{E}\cdot\textbf{B}\,,$
(5)
$\displaystyle\textbf{E}^{\prime}+2aH\textbf{E}-\boldsymbol{\nabla}\times\textbf{B}=-g\phi^{\prime}\textbf{B}-g\boldsymbol{\nabla}\phi\times\textbf{E}\,,$
(6)
$\displaystyle\boldsymbol{\nabla}\cdot\textbf{E}=-g\quantity(\boldsymbol{\nabla}\phi)\cdot\textbf{B}\,,$
(7)

while from the Bianchi identities one obtains

$\displaystyle\textbf{B}+2aH\textbf{B}+\boldsymbol{\nabla}\times\textbf{E}=0\,,$
(8) $\displaystyle\boldsymbol{\nabla}\cdot\textbf{B}=0\,,$ (9)

where the prime indicates the derivative with respect to the conformal time
and $V_{\phi}=\derivative{V}{\phi}$.

On studying only the impact of the gauge fields on the inflationary
background, we take the inflaton as homogeneous, $\boldsymbol{\nabla}\phi=0$.
As a consequence, in the Coulomb gauge
$(A_{0}=0\,,\;\boldsymbol{\nabla}\cdot{\bf{A}}=0)$, we obtain the following
equation of motion for the gauge field

$\partialderivative[2]{\textbf{A}}{\tau}-\laplacian{\textbf{A}}-g\phi^{\prime}\boldsymbol{\nabla}\times{\textbf{A}}=0\,.$
(10)

We then promote $\textbf{A}(\tau,\textbf{x})$ to a quantum operator

$\hat{\textbf{A}}(\tau,\textbf{x})=\sum_{\lambda=\pm}\mathop{\text{\Large$\int$}}\nolimits\dfrac{d^{3}k}{(2\pi)^{3}}\quantity[\boldsymbol{\epsilon}_{\lambda}(\textbf{k})A_{\lambda}(\tau,\textbf{k})\hat{a}_{\lambda}(\textbf{k})e^{i\textbf{k}\textbf{x}}+\text{h.c.}]\,,$
(11)

where we denote with $\boldsymbol{\epsilon}_{\pm}$ the polarization vectors of
the gauge field, and choose as a basis the circular polarization transverse to
the direction of propagation fixed by the (comoving) momentum k. The helicity
vectors satisfy then the following relations

$\begin{matrix}&\textbf{k}\cdot\boldsymbol{\epsilon}_{\pm}(\textbf{k})=0\,,&&\quad\textbf{k}\times\boldsymbol{\epsilon}_{\pm}(\textbf{k})=\mp
i\absolutevalue{\textbf{k}}\boldsymbol{\epsilon}_{\pm}(\textbf{k})\,,\\\ \\\
&\boldsymbol{\epsilon}^{\ast}_{\pm}(\textbf{k})=\boldsymbol{\epsilon}_{\pm}(-\textbf{k})\,,&&\boldsymbol{\epsilon}^{\ast}_{\lambda}(\textbf{k})\cdot\boldsymbol{\epsilon}_{\sigma}(\textbf{k})=\delta_{\lambda\sigma}\,.\end{matrix}$
(12)

Finally, the equation of motion for the Fourier modes $A_{\pm}$ reads

$\derivative[2]{\tau}A_{\pm}(\tau,k)+\quantity(k^{2}\mp
kg\phi^{\prime})A_{\pm}(\tau,k)=0\,.$ (13)

To understand the impact of the gauge fields on the Einstein equations one has
to evaluate their energy-momentum tensor. Starting from the initial action we
then obtain the following result

$T_{\mu\nu}^{(F)}=F_{\rho\mu}F^{\rho}_{\hskip
2.84544pt\nu}+g_{\mu\nu}\dfrac{\textbf{E}^{2}-\textbf{B}^{2}}{2}\,,$ (14)

where the energy density and pressure terms, descend from the $(0,0)$ and
$(i,j)$ components of $T_{\mu\nu}^{(F)}$. These are given by

$\displaystyle T_{00}^{(F)}$
$\displaystyle=\dfrac{\textbf{E}^{2}+\textbf{B}^{2}}{2}\,,$ (15)
$\displaystyle T_{ij}^{(F)}$
$\displaystyle=-E_{i}E_{j}-B_{i}B_{j}+\delta_{ij}\dfrac{\textbf{E}^{2}+\textbf{B}^{2}}{2}\,.$
(16)

Let us also note that the off-diagonal term $T^{0\hskip 1.42271pt(F)}_{\hskip
2.84544pti}=F_{j}^{\hskip 2.84544pt0}F^{j}_{\hskip 2.84544pti}$ can be
rewritten as

$T^{0\hskip 1.42271pt(F)}_{\hskip 2.84544pti}=F_{j}^{\hskip
2.84544pt0}F^{j}_{\hskip
2.84544pti}=\epsilon_{ijk}E^{j}B^{k}=\quantity(\textbf{E}\times\textbf{B})_{i}\,,$
(17)

where $\epsilon_{ijk}$ is the Levi-Civita tensor.

Taking into account only the backreaction of the gauge fields on the
background, we can then rewrite the Friedmann equations in the Hartree
approximation as (see e.g. Anber and Sorbo (2010))

$\displaystyle
H^{2}=\dfrac{1}{3m^{2}_{Pl}}\quantity[\dfrac{\dot{\phi}^{2}}{2}+V(\phi)+\dfrac{\langle\textbf{E}^{2}+\textbf{B}^{2}\rangle}{2}]\,,$
(18)
$\displaystyle\dot{H}=-\dfrac{1}{2m^{2}_{Pl}}\quantity[\dot{\phi}^{2}+\dfrac{2}{3}\langle\textbf{E}^{2}+\textbf{B}^{2}\rangle]\,,$
(19)

and the background inflaton equation of motion correspondent to Eq.(5) becomes
(in proper time):

$\ddot{\phi}+3H\dot{\phi}+V_{\phi}=g\langle\textbf{E}\cdot\textbf{B}\rangle\,.$
(20)

with ${\langle\cdots\rangle}$ the (quantum) expectation value. The exact
expression for the energy density ${{\langle{\bf E}^{2}+{\bf
B}^{2}\rangle}\over 2}$ and the so-called helicity integral ${\langle\bf
E\cdot B\rangle}$ that we will use in the following to evaluate analitically
and numerically the backreaction are the ones obtained in Animali  _et al._
(2022). For the sake of completeness and clarity we report them in Appendix A.

##  III Gauge invariant quantum backreaction

Cosmological backreaction is a consequence of the non-linearity of the
Einstein equations. Given a background cosmological solution of metric and
matter, the addition of small amplitude fluctuations leads to perturbative
equations which depend on the perturbative order considered. As an example, to
the quadratic order we obtain a quadratic first order source for our second
order perturbations, and so on at higher perturbative order. An observer in a
homogeneous universe would measure different physical features from those
perceived by dynamical observers living in a perturbed space-time. This effect
is indeed what we call the cosmological backreaction. The cosmological
backreaction induced by quantum fluctuations along the inflationary epoch have
been, in particular, the subject of several investigations relying on many
different approaches Mukhanov  _et al._ (1997); Abramo  _et al._ (1997); Unruh
(1998); Abramo and Woodard (1999, 2002a, 2002b); Geshnizjani and Brandenberger
(2002); Losic and Unruh (2005); Finelli  _et al._ (2004). In particular,
starting from the observation that a backreaction effect is meaningful only
when “measured” by a given observer, it was constructed a covariant and gauge
invariant (GI) approach, following Gasperini  _et al._ (2009, 2010), within
the framework of quantum backreaction in the early universe in Finelli  _et
al._ (2011). This approach was then further applied to different cases within
the framework of slow-roll inflation (see Brandenberger  _et al._ (2018);
Marozzi and Vacca (2014, 2013); Marozzi  _et al._ (2013); Marozzi and Vacca
(2012)). In practice, by using gauge invariant (but observer dependent)
averaged quantities, as defined in Gasperini  _et al._ (2009), one can
describe the space-time evolution by a set of gauge invariant effective
observer-dependent equations Gasperini  _et al._ (2010). Going into more
details, one can construct a general non-local observable by taking quantum
averages of a scalar field $S(x)$ over a space-time hypersurface
$\Sigma_{A_{0}}$ where $A(x)$, another time-like scalar field, assumes a
constant value $A_{0}$. By setting a barred coordinate system
$\bar{x}^{\mu}=(\bar{t},\textbf{x})$, in which the scalar field $A$ is
homogeneous, the above physical gauge invariant observable is then defined by
Gasperini  _et al._ (2009, 2010)

$\langle
S\rangle_{A_{0}}=\dfrac{\langle\sqrt{\absolutevalue{\overline{\gamma}(t_{0},\textbf{x})}}\hskip
2.84544pt\overline{S}(t_{0},\textbf{x})\rangle}{\langle\sqrt{\absolutevalue{\overline{\gamma}(t_{0},\textbf{x})}}\rangle}\,,$
(21)

where $t_{0}$ is the time $\bar{t}$ for which
$\bar{A}(\bar{x})=A^{(0)}(\bar{t})=A_{0}$, and
$\absolutevalue{\overline{\gamma}(t_{0},\textbf{x})}$ is the determinant of
the induced three dimensional metric on $\Sigma_{A_{0}}$.

The four vector

$n^{\mu}=-\dfrac{\partial^{\mu}A}{\sqrt{-\partial^{\nu}A\partial_{\nu}A}}\,,$
(22)

settles then the class of observers and the foliation of the space-time.
Let us now look to the effective scale factor
$a_{eff}=\langle\sqrt{\absolutevalue{\overline{\gamma}}}\rangle^{\frac{1}{3}}$,
which describes the expansion of the space-time as seen by the observers
comoving with the hypersurface $\Sigma_{A_{0}}$. This effective scale factor
satisfies the following gauge independent effective equation Gasperini  _et
al._ (2010)

$\quantity(\dfrac{1}{a_{eff}}\partialderivative{a_{eff}}{A_{0}})^{2}=\dfrac{1}{9}\biggl{\langle}\dfrac{\Theta}{\sqrt{-\partial^{\mu}A\partial_{\mu}A}}\biggl{\rangle}^{2}_{A_{0}}\,,$
(23)

where $\Theta=\nabla_{\mu}n^{\mu}$ is the expansion scalar of the time-like
congruence $n^{\mu}$. As a consequence of the above gauge invariant
construction, the space-time dynamics can be studied in any gauge, simply by
solving the matter and Einstein equations in the chosen gauge.

Following Finelli  _et al._ (2011), to evaluate the backreaction within a
perturbative framework we consider cosmological perturbation theory up to the
second order, so we write the general perturbed metric around a FLRW space-
time as

$\displaystyle g_{00}=-1-2\alpha-2\alpha^{(2)}\,,$ $\displaystyle
g_{i0}=-\dfrac{a}{2}\quantity(\beta_{,\hskip
1.42271pti}+B_{i})-\dfrac{a}{2}\quantity(\beta^{(2)}_{,\hskip
1.42271pti}+B^{(2)}_{i})\,,$ $\displaystyle
g_{ij}=a^{2}\biggl{[}\delta_{ij}\quantity(1-2\psi-2\psi^{(2)})+D_{ij}\quantity(E+E^{(2)})$
$\displaystyle+\dfrac{1}{2}\quantity(\chi_{i,j}+\chi_{j,i}+h_{ij})+\dfrac{1}{2}\quantity(\chi^{(2)}_{i,j}+\chi^{(2)}_{j,i}+h^{(2)}_{ij})\biggr{]}\,,$
(24)

where $D_{ij}=\partial_{i}\partial_{j}-\delta_{ij}\nabla^{2}/3$ and, for
clearness, we write first order quantities without an upper script. We then
have that $\alpha$, $\beta$, $\psi$ and $E$ are scalar perturbations, $B_{i}$
and $\chi_{i}$ are transverse vectors ($\partial^{i}B_{i}=0$ and
$\partial^{i}\chi_{i}=0$), and $h_{ij}$ is a traceless and transverse tensor
($\partial^{i}h_{ij}=0$ and $h^{i}_{i}=0$). In the follow we also neglect
vector perturbations which decay dynamically.

At the same time, the inflaton can be written to second order as

$\phi(t,\textbf{x})=\phi(t)+\varphi(t,\textbf{x})+\varphi^{(2)}(t,\textbf{x})\,.$
(25)

Let us note how the ten degrees of freedom in the metric (24) are redundant.
To fix a gauge corresponds to reduce such degrees of freedom, typically
removing two scalars and one vector perturbations. The cases of interests for
us are the uniform field gauge (UFG), defined by setting
$\phi(t,\textbf{x})=\phi(t)$ and by another conditions (we consider
$g_{i0}=0$), and the uniform curvature gauge (UCG), defined by
$g_{ij}=a^{2}\left[\delta_{ij}+\frac{1}{2}\left(h_{ij}+h^{(2)}_{ij}\right)\right]$.

Applying the long wavelength (LW) limit for scalar fluctuations 111In the long
wavelength limit one neglects terms like
$\sim|\vec{\nabla}\varphi|^{2}/(aH)^{2}$ in evaluating cosmological
observables. This is a consequence of the fact that in the Fourier space such
terms goes like $k^{2}/(aH)^{2}$ and are negligible in the infrared, while
their ultraviolet tail is almost erased by the renormalization process. one
then gets Finelli  _et al._ (2011)

$\displaystyle\bar{\Theta}=3H-3H\bar{\alpha}-3\dot{\bar{\psi}}+\dfrac{9}{2}H\bar{\alpha}^{2}+3\bar{\alpha}\dot{\bar{\psi}}-6\bar{\psi}\dot{\bar{\psi}}$
$\displaystyle\qquad\qquad\qquad-3H\bar{\alpha}^{(2)}-3\dot{\bar{\psi}}^{(2)}-\dfrac{1}{8}h_{ij}\dot{h}^{ij}\,,$
(26)
$\displaystyle-\partial_{\mu}\bar{A}\partial^{\mu}\bar{A}=1-2\bar{\alpha}+4\bar{\alpha}^{2}-2\bar{\alpha}^{(2)}\,,$
(27)
$\displaystyle\sqrt{\absolutevalue{\overline{\gamma}}}=a^{3}\quantity(1-3\bar{\psi}+\dfrac{3}{2}\bar{\psi}^{2}-\dfrac{1}{16}h^{ij}h_{ij}-3\bar{\psi}^{(2)})\,.$
(28)

Then by substituting Eqs. (26), (27) and (28) inside Eq. (23)), and using Eq.
(21), one obtains the general form of the effective expansion for the class of
observers defined by the scalar field $A(t,\textbf{x})$. Thus, neglecting
tensor perturbations and in the LW limit, we obtain Finelli  _et al._ (2011)

$H^{2}_{eff}=\quantity(\dfrac{1}{a_{eff}}\partialderivative{a_{eff}}{A_{0}})^{2}=H^{2}\quantity[1+\dfrac{2}{H}\langle\bar{\psi}\dot{\bar{\psi}}\rangle-\dfrac{2}{H}\langle\dot{\bar{\psi}}^{(2)}\rangle]\,.$
(29)

Let us stress that, since the above result is by construction gauge invariant,
we can now use the dynamical solutions for inflaton and metric fluctuations in
any frame of our choice.

##  IV Second order perturbative equations

Hereafter, we want to evaluate the gauge fields backreaction taking in
consideration their coupling with both matter and metric scalar perturbations,
and using the gauge invariant approach described above. Therefore, with the
aim of considering the weak backreaction regime we promote the gauge fields
$A_{\mu}$ to first order perturbations and consistently consider the scalar
sector up to second order in perturbation theory. In fact, within the
assumption that gauge fields have no background, such first order fields
evolve independently from the matter-gravity scalar perturbation sector. The
equations of motion that govern the dynamics of $A_{\mu}$ are unchanged, and
one has a coupling among the two different sectors only to second order.

To evaluate how the gauge fields backreaction impact on the effective Hubble
factor $H_{eff}$ of Eq.(29) we have so to solve Einstein and Euler-Lagrange
equations up to second order taking in consideration the above coupling. As
described in the previous section, the approach used to obtain $H_{eff}$ is
gauge invariant and we can fix the gauge without lose of generality.

Let us so choose the uniform curvature gauge (UCG), where, as mentioned, we
fix $\psi=E=0$ at first and second order in Eq. (24). The Einstein tensor
terms remain the usual ones (first obtained in Finelli  _et al._ (2004) and
reported in Appendix B), while the energy-momentum tensor is modified by the
presence of the gauge fields. As said, we consider the weak backreaction
regime and neglect the background contribution of the gauge fields. As a
consequence, we will treat expressions like $\textbf{E}^{2}$,
$\textbf{B}^{2}$, $\textbf{E}\cdot\textbf{B}$ as second order perturbations.
The gauge fields contributions to the enegy-momentum tensor $T_{\mu\nu}^{(F)}$
can then be obtained using the background metric and coincide with the one
evaluated in Eqs. (14)$-$(16). This will act as a second order source for
inflaton and metric fluctuations. Furthermore, the axionic coupling between
the pseudo-scalar inflaton and the gauge fields introduces a new term in the
Euler-Lagrange equation (see Eq.(5)).

As said, first order scalar perturbations are decoupled from the gauge fields
contribution, while from the momentum constraint to second order one finds
(the scalar contributions below and in the follow were first evaluated in
Finelli  _et al._ (2004))

$\alpha^{(2)}=4\pi G\dfrac{\dot{\phi}}{H}\varphi^{(2)}+s+s_{F}\,,$ (30)

with

$\displaystyle s$ $\displaystyle=-4\pi
G\epsilon\varphi^{2}+2\alpha^{2}-\dfrac{\absolutevalue{\boldsymbol{\nabla}\beta}^{2}}{8}+\dfrac{1}{\laplacian}\biggl{[}\dfrac{4\pi
G}{H}\boldsymbol{\nabla}\cdot\quantity(\dot{\varphi}\boldsymbol{\nabla}\varphi)$
$\displaystyle\qquad\qquad+\dfrac{1}{4aH}\quantity(\alpha^{,\hskip
1.42271ptkj}\beta_{,\hskip 1.42271ptkj}-\alpha^{,\hskip 1.42271ptk}_{,\hskip
1.42271ptk}\beta^{,\hskip 1.42271ptj}_{,\hskip 1.42271ptj})\biggr{]}\,,$ (31)
$\displaystyle s_{F}$ $\displaystyle=-\dfrac{4\pi
G}{H}\dfrac{1}{\laplacian}\quantity[\boldsymbol{\nabla}\cdot\quantity(\textbf{E}\times\textbf{B})]\,.$
(32)

From the equivalence between the curvature part and the matter sector of the
Einstein equations to second order, we obtain

$\displaystyle\dfrac{H}{a}\laplacian{\beta^{(2)}}=$ $\displaystyle 8\pi
G\dfrac{\dot{\phi}^{2}}{H}\derivative{t}\quantity(\dfrac{H}{\dot{\phi}}\varphi^{(2)})-\quantity(Q+Q_{F})+$
$\displaystyle\qquad\qquad\qquad+16\pi GV(s+s_{F})\,,$ (33)

where

$\displaystyle Q$
$\displaystyle=12H^{2}\alpha^{2}-\dfrac{3}{4}H^{2}\absolutevalue{\boldsymbol{\nabla}\beta}^{2}-\dfrac{H}{a}\quantity(\alpha_{,\hskip
1.42271pti}\beta^{,\hskip 1.42271pti}+2\alpha\laplacian{\beta})+$
$\displaystyle-\dfrac{1}{8a^{2}}\quantity(\beta_{,\hskip
1.42271ptij}\beta^{,\hskip 1.42271ptij}-\quantity(\laplacian{\beta})^{2})-8\pi
G\Biggl{[}\dfrac{\dot{\varphi}^{2}}{2}+\dfrac{\absolutevalue{\boldsymbol{\nabla}\varphi}^{2}}{2a^{2}}+$
$\displaystyle+\dfrac{1}{2}V_{\phi\phi}\varphi^{2}+2\dot{\phi}^{2}\alpha^{2}-2\dot{\phi}\alpha\dot{\varphi}-\dfrac{1}{8}\dot{\phi}^{2}\absolutevalue{\boldsymbol{\nabla}\beta}^{2}\Biggr{]}\,,$
(34) $\displaystyle Q_{F}$ $\displaystyle=-8\pi
G\quantity(\dfrac{\textbf{E}^{2}+\textbf{B}^{2}}{2})\,.$ (35)

On the other hand, the equation of motion for the second order inflaton
perturbation, due to the presence of the gauge fields, reads

$\displaystyle\ddot{\varphi}^{(2)}+3H\dot{\varphi}^{(2)}-\dfrac{\laplacian{\varphi^{(2)}}}{a^{2}}+$
$\displaystyle\qquad+\quantity[V_{\phi\phi}+2\dfrac{\dot{H}}{H}\quantity(3H-\dfrac{\dot{H}}{H}+2\dfrac{\ddot{\phi}}{\dot{\phi}})]\varphi^{(2)}=D\,,$
(36)

where

$\displaystyle D$
$\displaystyle=g(\textbf{E}\cdot\textbf{B})+R+\dot{\phi}\quantity(\dot{s}+\dot{s}_{F})-2V_{\phi}\quantity(s+s_{F})$
$\displaystyle+\dfrac{\dot{\phi}}{2H}\quantity[\quantity(Q+Q_{F})-16\pi
GV(s+s_{F})]\,,$ (37)

and

$\displaystyle R$
$\displaystyle=-\dfrac{1}{2}V_{\phi\phi\phi}\varphi^{2}-2\dot{\phi}\alpha\dot{\alpha}+\dot{\alpha}\dot{\varphi}+\dfrac{2}{a^{2}}\alpha\laplacian{\varphi}-2V_{\phi\phi}\alpha\varphi$
$\displaystyle+\dfrac{\dot{\phi}}{2a}\alpha_{,\hskip 1.42271pti}\beta^{,\hskip
1.42271pti}-\dfrac{1}{4}V_{\phi}\absolutevalue{\boldsymbol{\nabla}\beta}^{2}+\dfrac{1}{a^{2}}\alpha_{,\hskip
1.42271pti}\varphi^{,\hskip 1.42271pti}-\dfrac{H}{a}\beta_{,\hskip
1.42271pti}\varphi^{,\hskip 1.42271pti}$
$\displaystyle-\dfrac{1}{2a}\dot{\varphi}\laplacian{\beta}+\dfrac{\dot{\phi}}{4}\beta_{,\hskip
1.42271pti}\dot{\beta}^{,\hskip 1.42271pti}-\dfrac{1}{2a}\varphi_{,\hskip
1.42271pti}\dot{\beta}^{,\hskip 1.42271pti}-\dfrac{1}{a}\beta_{,\hskip
1.42271pti}\dot{\varphi}^{,\hskip 1.42271pti}\,.$ (38)

We have therefore obtained perturbative equations, for the pseudo-scalar
inflaton fluctuations and metric fluctuations, to second order including the
contribution given by the gauge fields.

##  V Comoving observers

To obtain the gauge fields quantum backreaction, we have now to choose the
observer w.r.t. such backreaction has to be evaluated. While for late time
cosmological observables the choice of the right observer depends on how an
observation is done, for the case of a single field inflationary model we have
a well defined, and physically motivated, choice. During such early universe
phase the only dynamical matter field that we can employ as a clock is the
inflation field itself, hence the quantum backreaction should be evaluated
with respect to the observer that uses such clock. Namely, we have to consider
the so-called comoving observer, i.e. the observer who sees an homogeneous
inflaton field for which $\varphi={\varphi}^{(2)}=0$. To evaluate the
backreaction as experienced by the above observer in our framework, let us
first note that in Finelli  _et al._ (2011); Marozzi and Vacca (2013) was
shown how in a single field model of inflation, taking in consideration the
scalar sector and with no coupling between the inflaton and other fields, the
class of comoving observers do not see on the effective Hubble rate any
quantum backreaction effect, in the LW limit, for any potential $V(\phi)$ and
to all orders in the slow-roll parameters. Therefore, for studying the total
backreaction in our particular model, it is enough to take in consideration
only the backreaction coming from the gauge fields.

In our framework the gauge fields contribute only to the second order
equations of motion, while do not change the dynamic of first order
perturbations. As a consequence, hereafter we can neglect the first-order
square scalar contributions to the backreaction and focus only on the second
order gauge fields contribution to Eq. (29).

Moving from a general coordinate system to the barred one, we can determine
the expressions associated with equation (21) using the following coordinate
transformation up to second order Bruni  _et al._ (1997)

$x^{\mu}\rightarrow\tilde{x}^{\mu}=x^{\mu}+\epsilon^{\mu}_{(1)}+\dfrac{1}{2}\quantity(\epsilon^{\nu}_{(1)}\partial_{\nu}\epsilon^{\mu}_{(1)}+\epsilon^{(\mu)}_{(2)})\,,$
(39)

where $\epsilon^{\mu}_{(1)}$ and $\epsilon^{\mu}_{(2)}$ are the first order
and second order generators, given by

$\epsilon^{\mu}_{(1)}=\quantity(\epsilon^{0}_{(1)},\partial^{i}\epsilon_{(1)}+\epsilon^{i}_{(1)})\,,\quad\epsilon^{\mu}_{(2)}=\quantity(\epsilon^{0}_{(2)},\partial^{i}\epsilon_{(2)}+\epsilon^{i}_{(2)})\,,$
(40)

with $\partial_{i}\epsilon^{i}_{(1)}=\partial_{i}\epsilon^{i}_{(2)}=0$.
They enter in the associated first order and second order gauge
transformations of a general tensor $T$ as (see, for example Marozzi (2011)
for the detailed trasformation of inflaton and metric fluctuations)

$\displaystyle
T^{(1)}\rightarrow\tilde{T}^{(1)}=T^{(1)}-L_{\epsilon_{(1)}}T^{(0)}\,,$ (41a)
$\displaystyle
T^{(2)}\rightarrow\tilde{T}^{(2)}=T^{(2)}-L_{\epsilon_{(1)}}T^{(1)}+$
$\displaystyle\qquad\qquad\quad\quad\quad+\tfrac{1}{2}\left(L^{2}_{\epsilon_{(1)}}T^{(0)}-L_{\epsilon_{(2)}}T^{(0)}\right)\,,$
(41b)

where $L_{\epsilon_{(1,2)}}$ is the Lie derivative respect the vector
$\epsilon^{\mu}_{(1,2)}$.

As said, in our framework we can neglect first order scalar contribution in
our calculations (they have no impact on the backreaction). By neglecting
them, we then obtain the following simplified gauge transformation to second
order for a general scalar $A$

$A^{(2)}\rightarrow\bar{A}^{(2)}=A^{(2)}-\frac{1}{2}\epsilon^{0}_{(2)}\dot{A}^{(0)}\,,$
(42)

while for the second order metric scalar perturbations we have

$\displaystyle\bar{\alpha}^{(2)}$ $\displaystyle=$
$\displaystyle\alpha^{(2)}-\frac{1}{2}\dot{\epsilon}^{0}_{(2)}\,,$ (43a)
$\displaystyle\bar{\beta}^{(2)}$ $\displaystyle=$
$\displaystyle\beta^{(2)}-\frac{1}{a}\epsilon^{0}_{(2)}+a\dot{\epsilon}_{(2)}\,,$
(43b) $\displaystyle\bar{\psi}^{(2)}$ $\displaystyle=$
$\displaystyle\psi^{(2)}+\frac{H}{2}\epsilon_{(2)}^{0}+\frac{1}{6}\nabla^{2}\epsilon_{(2)}\,,$
(43c) $\displaystyle\bar{E}^{(2)}$ $\displaystyle=$ $\displaystyle
E^{(2)}-\epsilon_{(2)}\,.$ (43d)

The observer which sees an homogeneous inflaton field must have
$\bar{\varphi}^{(2)}=0$, hence the $\bar{x}^{\mu}$ reference system will
correspond to the UFG. The UFG conditions starting from the UCG are then
obtained by choosing

$\epsilon^{0}_{(2)}=2\frac{\varphi^{(2)}}{\dot{\phi}}\quad,\quad\epsilon_{(2)}=\int{\rm
dt}\left(\frac{1}{a^{2}}\epsilon^{0}_{(2)}-\frac{1}{a}\beta^{(2)}\right)\,.$
(44)

By taking into account only the purely second order terms and working in the
long wavelength limit, we then have (by substituting Eq. (44) in Eq. (43c))

$\bar{\psi}^{(2)}=\dfrac{H}{\dot{\phi}}\varphi^{(2)}\,.$ (45)

The contribution that goes in the backreaction of Eq. (29) is then given by

$-\dfrac{2}{H}\langle\dot{\bar{\psi}}^{(2)}\rangle=-\dfrac{2}{H}\biggl{<}\quantity[\quantity(\dfrac{\dot{H}}{\dot{\phi}}-H\dfrac{\ddot{\phi}}{\dot{\phi}^{2}})\varphi^{(2)}+\dfrac{H}{\dot{\phi}}\dot{\varphi}^{(2)}]\biggr{>}\,.$
(46)

##  VI Second order solution in $U(1)$-axion inflation

To evaluate the gauge field backreaction we have now to solve the equation of
motion for the pseudo-scalar inflaton to second order within the UCG.

For our purpose it is enough to take in consideration only the gauge fields
contribution to $\varphi^{(2)}$. Therefore, starting from Eq.(IV), using
$\ddot{\varphi}^{(2)}\sim\dot{H}/H\dot{\varphi}^{(2)}$, working at leading
order in the slow-roll parameters and in the long-wavelength approximation, we
obtain

$\displaystyle
3H\dot{\varphi}^{(2)}+\left[3\dot{H}-\frac{3}{2}H\frac{\ddot{H}}{\dot{H}}\right]\varphi^{(2)}$
$\displaystyle\simeq
g(\textbf{E}\cdot\textbf{B})+\dot{\phi}\dot{s}_{F}-2V_{\phi}s_{F}+$
$\displaystyle+\dfrac{\dot{\phi}}{2H}\quantity(Q_{F}-\frac{2}{M_{Pl}^{2}}Vs_{F})\,,$
(47)

where we have used $M^{2}_{Pl}=\frac{1}{8\pi G}$. The above equation can also
be written as

$\displaystyle
3\derivative{t}\quantity(H\varphi^{(2)})-\frac{3}{2}H\frac{\ddot{H}}{\dot{H}}\varphi^{(2)}$
$\displaystyle\simeq
g(\textbf{E}\cdot\textbf{B})+\dot{\phi}\dot{s}_{F}-2V_{\phi}\,s_{F}$
$\displaystyle+\dfrac{\dot{\phi}}{2H}\quantity(Q_{F}-\frac{2}{M_{Pl}^{2}}Vs_{F})\,.$
(48)

At this point, to obtain the gauge fields contribution to
$\langle\varphi^{(2)}\rangle$, we need to do first the v.e.v. of Eq.(48) and
then integrate the obtained equation with respect to time. The first step
produces the following dynamical equation

$\displaystyle\derivative{t}\quantity(H\langle\varphi^{(2)}\rangle)-$
$\displaystyle\frac{H}{2}\frac{\ddot{H}}{\dot{H}}\langle\varphi^{(2)}\rangle\simeq$
$\displaystyle\frac{g}{3}\langle\textbf{E}\cdot\textbf{B}\rangle-\frac{\dot{\phi}}{6HM_{Pl}^{2}}\dfrac{\langle\textbf{E}^{2}+\textbf{B}^{2}\rangle}{2}\,.$
(49)

where we have used that

$\langle Q_{F}\rangle=-8\pi
G\dfrac{\langle\textbf{E}^{2}+\textbf{B}^{2}\rangle}{2}\,,$ (50)

and the fact that the vacuum expectation value of $s_{F}$ vanishes because
$\langle\boldsymbol{\nabla}\cdot(\textbf{E}\times\textbf{B})\rangle=0$ for the
isotropy of the background. Namely, we have

$\langle
s_{F}\rangle=-\Big{\langle}\dfrac{1}{\laplacian}[\boldsymbol{\nabla}\cdot(\textbf{E}\times\textbf{B})]\Big{\rangle}=0\,.$
(51)

In order to solve Eq.(49), let us now do the following ansatz

$\langle\varphi^{(2)}\rangle=\frac{\gamma(H)}{\dot{H}}\frac{\dot{\phi}}{6HM_{Pl}^{2}}{{\langle{\bf
E}^{2}+{\bf B}^{2}\rangle}\over 2}+\frac{\sigma(H)}{\dot{H}}{g\over
3}{\langle\bf E\cdot B\rangle}\,,$ (52)

where $\gamma(H)$ and $\sigma(H)$ are unknown functions to be determined. At
leading order in slow-roll, and in the chaotic $m^{2}\phi^{2}$ inflation limit
222For this case we have $\frac{\ddot{H}}{\dot{H}H}={\cal
O}\left(\frac{\dot{H}}{H^{2}}\right)^{2}$. We use this limit because, as
described in Sec. VII, we will evaluate the backreaction considering natural
inflation with $f\gg M_{Pl}$. In this case, natural inflation is really well
approximated by this chaotic potential., we obtain then the following
differential equations for such functions:

$\displaystyle H\partial_{H}\gamma+\gamma
H\partial_{H}\ln\rho_{\text{EM}}=-1\,,$ (53a) $\displaystyle
H\partial_{H}\sigma+\sigma(1+H\partial_{H}\ln\rho_{\text{Hel}})=1\,,$ (53b)

with solution

$\displaystyle\gamma(H)$
$\displaystyle=-{1\over\rho_{\text{EM}}}\int{\rho_{\text{EM}}\over
H}\differential H\,,$ (54a) $\displaystyle\sigma(H)$ $\displaystyle={1\over
H\rho_{\text{Hel}}}\int\rho_{\text{Hel}}\differential H\,.$ (54b)

where we have defined &ρ_EM ≡⟨E^2+B^2 ⟩2 , ρ_Hel ≡⟨E⋅B ⟩ . Finally, we obtain

${\langle\varphi^{(2)}\rangle}=-\frac{1}{6}\frac{\dot{\phi}}{H\dot{H}M_{Pl}^{2}}\int{\rho_{\text{EM}}\over
H}\differential H+{g\over
3}\frac{1}{H\dot{H}}\int\rho_{\text{Hel}}\differential H\,.$ (55)

At this point we are able to evaluate the effective Hubble parameter

$\displaystyle H^{2}_{\text{eff}}=$ $\displaystyle\left({1\over
a_{\text{eff}}}\partialderivative{a_{\text{eff}}}{A_{0}}\right)^{2}=$
$\displaystyle=$ $\displaystyle\left[1-{2\over\dot{\phi}}\left({\dot{H}\over
H}-{\ddot{H}\over
2\dot{H}}\right){\langle\varphi^{(2)}\rangle}-{2\over\dot{\phi}}{\langle\dot{\varphi}^{(2)}\rangle}\right]$
$\displaystyle\simeq$
$\displaystyle\,H^{2}\left[1-{2\over\dot{\phi}}\left({\dot{H}\over
H}{\langle\varphi^{(2)}\rangle}+{\langle\dot{\varphi}^{(2)}\rangle}\right)\right]\,.$
(56)

The contribution $\left<\dot{\varphi}^{(2)}\right>$ can be explicitly
calculated. Starting from Eq. (55), and at leading order in slow-roll, we
obtain

$\left<\dot{\varphi}^{(2)}\right>\simeq-\frac{\dot{H}}{H}\left<{\varphi}^{(2)}\right>-\frac{1}{6}\frac{\dot{\phi}}{H^{2}M_{Pl}^{2}}\rho_{\text{EM}}+{g\over
3}\frac{1}{H}\rho_{\text{Hel}}\,.$ (57)

Finally, we have

$H^{2}_{\text{eff}}=H^{2}\left[1+{1\over
3H^{2}}\left(\frac{1}{M_{Pl}^{2}}{{\langle{\bf E}^{2}+{\bf B}^{2}\rangle}\over
2}-{g^{2}\over\xi}{\langle\bf E\cdot B\rangle}\right)\right]\,,$ (58)

or, considering that we are in the weak-backreaction regime, we can also write

$H_{\text{eff}}=H\left[1+{1\over
6H^{2}}\left(\frac{1}{M_{Pl}^{2}}{{\langle{\bf E}^{2}+{\bf B}^{2}\rangle}\over
2}-{g^{2}\over\xi}{\langle\bf E\cdot B\rangle}\right)\right]\,,$ (59)

where we define $\xi\equiv\frac{g\dot{\phi}}{2H}$.

##  VII Numerical analysis

Let us now investigate the magnitude of the obtained gauge invariant
backreaction effect, which includes both metric and matter scalar
perturbations. To this aim we perform the integration of the homogeneous field
equation

$\ddot{\phi}+3H\dot{\phi}+V_{\phi}=0\,,$ (60)

written in terms of the effective number of e-fold $N_{\text{eff}}$ defined by

$\differential N_{\text{eff}}\equiv H_{\text{eff}}\leavevmode\nobreak\
\differential t\,,$ (61)

where $H_{\text{eff}}$ is given by Eq. (59). The complete system of equations,
in terms of $\xi$, reads then

$\displaystyle\xi^{\prime}+\left(\tfrac{H^{\prime}}{H}+3\tfrac{H}{H_{\text{eff}}}\right)\xi+\tfrac{g}{2}\tfrac{V_{,\phi}}{HH_{\text{eff}}}=0\,,$
(62a) $\displaystyle\phi^{\prime}=\tfrac{2}{g}\tfrac{H}{H_{\text{eff}}}\xi\,,$
(62b)

where now a prime states a derivative w.r.t. $N_{\text{eff}}$. The above
integration can then be compared with what is usually done in literature Anber
and Sorbo (2010), i.e. the integration of the equation

$\ddot{\phi}+3H\dot{\phi}+V_{\phi}=g\langle\textbf{E}\cdot\textbf{B}\rangle\,.$
(63)

The main difference between the two approaches is that in our case the
backreaction effect is implicitly encoded in the e-fold number of Eq.(61),
where we are taking in consideration also the induced scalar fluctuations.

The missing one ingredient to close the system is the inflationary model,
which we choose to be natural inflation, first proposed in Freese  _et al._
(1990); Adams  _et al._ (1993). For this model we have

$V(\phi)=\Lambda^{4}\left[1-\cos\left(\frac{\phi}{f}\right)\right]\,.$ (64)

The correspondent slow-roll parameters are given by

$\displaystyle\epsilon$ $\displaystyle=\dfrac{1}{16\pi
G}\quantity(\dfrac{V^{\prime}}{V})^{2}=\frac{1}{2}\left(\frac{M_{Pl}}{f}\right)^{2}\left[\frac{\sin(\phi/f)}{1-\cos(\phi/f)}\right]^{2}\,,$
(65) $\displaystyle\eta$ $\displaystyle=\dfrac{1}{8\pi
G}\quantity(\dfrac{V^{\prime\prime}}{V})=\left(\frac{M_{Pl}}{f}\right)^{2}\frac{\cos(\phi/f)}{1-\cos(\phi/f)}\,,$
(66)

and we then have the following relation

$\eta=\epsilon-\frac{1}{2}\left(\frac{M_{Pl}}{f}\right)^{2}\,.$ (67)

Assuming to be in the slow-roll regime $\epsilon\ll 1$, we also have $\eta\ll
1$ only if $f^{2}\gg M_{Pl}^{2}$. In the following, we indeed assume $f^{2}\gg
M_{Pl}^{2}$ to have a model consistent with observation Akrami  _et al._
(2020). As mentioned, in this limit natural inflation can be well approximated
by chaotic inflation. The detailed study of the backreaction, when the
condition $f^{2}\gg M_{Pl}^{2}$ does not hold, is postponed to future
investigation.

The initial conditions necessary to solve the system can be calculated as done
in absence of backreaction, being the backreaction negligible at the initial
instants of the inflationary phase. To better compare our results with
previous findings (e.g. Domcke  _et al._ (2016)), we choose these initial
conditions to obtain a 60 e-folds long inflationary phase including the
backreaction as evaluated by Eq.(63). Let us then use the number of e-folds to
find the value of the scalar field at the beginning of inflation. In the
absence of backreaction we have

$N=\int_{t_{i}}^{t_{f}}H\differential
t=\int_{\phi_{f}}^{\phi_{i}}\frac{V}{V_{\phi}}\differential\phi\,.$ (68)

To obtain $\phi_{i}$ we need the field value at the end of inflation, which is
obtained with the condition $\epsilon=1$. Following Domcke  _et al._ (2016),
we then obtain $60$ e-folds including backreaction, if without backreaction we
require a $N=42$ e-folds long inflationary phase. Thus, we computed $\phi_{i}$
using $N=42$ and then applied it to find
$\xi_{i}=-\frac{g}{2}\frac{V_{\phi}}{V}|_{\phi_{i}}\leavevmode\nobreak\
$333The equation of motion in the slow-roll regime without backreaction reads
as  $\phi^{\prime}+\frac{V_{\phi}}{V}=0\,.$ (69) . In addition, we considered
the values of $g=60/M_{Pl}$ and $f=5M_{Pl}$ throughout the numerical analysis.

Finally, one can integrate the system of Eqs. (62a) and (62b) to obtain the
solutions for $\phi$ and $\xi$, and then evaluate the consequent effective
expansion rate as given by Eq. (59). The results are shown in black lines in
Fig. LABEL:Fig1a and Fig. LABEL:Fig1b. For completeness, we also included in
the figures the results of solving the Friedmann and Klein-Gordon equations,
both in the absence of backreaction (blue line) and in the case when only the
gauge field production is taken into account for the backreaction (red line),
which means using Eq. (63) (see Domcke  _et al._ (2016); Ballardini  _et al._
(2019)). Let us note that for our case, the impact of the backreaction on the
effective expansion rate (Fig. LABEL:Fig1a) becomes large before with respect
to the case in which only the gauge fields production is taken into account.
It requires then a higher number of e-folds for the magnitude of the effect to
change significantly and lead inflation to its end. This is also noticed on
the parameter $\xi$ (Fig. LABEL:Fig1b), which controls the influence of the
gauge fields on the inflaton dynamics. It starts to change at $N\approx 15$
but evolves much slower compared with the standard dynamics and the other
backreaction considered (this latter starts to differ from the standard
dynamics at $N\approx 35$).

Other interesting quantities to examine are the relative densities, since we
can identify when each becomes important during cosmic evolution. Rewriting
Eq. (59) as

$3H^{2}_{\text{eff}}=\frac{\dot{\phi}^{2}}{2}+V(\phi)+{{\langle{\bf
E}^{2}+{\bf B}^{2}\rangle}\over 2}-{g^{2}\over\xi}{\langle\bf E\cdot
B\rangle}\,,$ (70)

we can identify

$\displaystyle\rho_{\text{V}}$ $\displaystyle\equiv
V(\phi),\qquad\quad\rho_{\text{k}}\equiv\frac{\dot{\phi}^{2}}{2},\quad\rho_{\text{EM}}\equiv{{\langle{\bf
E}^{2}+{\bf B}^{2}\rangle}\over 2},$ $\displaystyle\rho_{\text{EB}}$
$\displaystyle\equiv-{g^{2}\over\xi}{\langle\bf E\cdot
B\rangle},\qquad\mathrm{and}\quad\,\,\,\leavevmode\nobreak\
\quad\rho_{T}=3H^{2}_{\text{eff}}\,.$ (71)

The evolution of the relative densities with the number of e-folds can be seen
in Fig. 5. We show three cases: in Fig. LABEL:Fig:densitiesa one can see their
evolution in the absence of backreaction. Indeed, we have the classic slow-
roll approximation for the inflaton dynamics, with the potential energy
dominating the kinetic term until the very end of inflation. In the second
case, shown in Fig. LABEL:Fig:densitiesb, the gauge fields production is the
only source for the backreaction. Here, one can observe the inflaton energy’s
decay into the gauge field production by increasing the $\rho_{\text{EM}}$
component. Lastly, in Fig. LABEL:Fig:densitiesc, we show the new results
associated to our approach as described in Eqs.(70) and (71). We always start
with the potential energy dominating the other densities. However, it is
crucial to note that the contribution of the helicity term,
$\rho_{\text{EB}}$, is consistently larger than $\rho_{\text{EM}}$ and
$\rho_{\text{k}}$ from the beginning. Notably, $\rho_{\text{EB}}$ exceeds the
potential energy at an early stage ($N\sim 15$), while $\rho_{\text{EM}}$
surpasses $\rho_{\text{V}}$ when $N\sim 200$. Furthermore, from $N\sim 15$ we
also have that $\rho_{\text{EM}}$ becomes comparable to the kinetic term.

Figure 2: Time evolution for the expansion rate (a) and $\xi$ parameter (b),
considering $g=60/M_{Pl}$ and $f=5M_{Pl}$. We show the results of solving the
Friedmann and Klein-Gordon equations in the absence of backreaction (blue
dotted lines), the case when only the gauge fields production is taken into
account (red dot-dashed lines), and the current work in solid black lines,
where we consider also metric and field scalar perturbations to calculate an
effective expansion rate. Figure 5: Relative density evolution for the cases:
(a) no backreaction, (b) backreaction due to only gauge fields production, and
(c) new effective backreaction which includes both field and metric scalar
perturbations. In all cases, we considered $\rho_{\text{V}}=V(\phi)$,
$\displaystyle\rho_{\text{k}}=\tfrac{\dot{\phi}^{2}}{2}$,
$\displaystyle\rho_{\text{EM}}=\tfrac{{\langle{\bf E}^{2}+{\bf
B}^{2}\rangle}}{2}$, and
$\displaystyle\rho_{\text{EB}}=-\tfrac{g^{2}}{\xi}{\langle\bf E\cdot
B\rangle}$, where $\phi$ is the one obtained solving the corresponding system.
When required, we considered $g=60/M_{Pl}$ and $f=5M_{Pl}$.

##  VIII Discussion and Conclusions

In this manuscript, we have reconsidered the cosmological backreaction problem
for a pseudo-scalar field $\phi$ driving inflation coupled to gauge fields via
a Chern-Simons-like operator, the so-called $U(1)$-axion inflation model (see
Anber and Sorbo (2010); Sorbo (2011); Anber and Sorbo (2012); Barnaby  _et
al._ (2011)). We have built our investigation starting from two pillars.
First, from the idea that to obtain a meaningful description of a physical
phenomenon, like backreaction, in a cosmological context one should always
consider gauge invariant observables. Otherwise, the effect of perturbations
might be attributed to a gauge artifact, making any physical interpretation
extremely challenging. Second, since cosmological backreaction arises from the
non-linearity of Einstein’s equations, for a consistent description within
perturbation theory, one has to include both matter and metric fluctuations in
the framework, being all these fluctuations coupled among them. Therefore,
starting from the covariant and gauge invariant approach introduced in
Gasperini  _et al._ (2009, 2010), we have studied the effective evolution of
the scalar factor as measured by an observer comoving with the inflaton field.
The analytical expression obtained, valid at the leading order in the slow-
roll approximation, is shown in Eq.(59), while the backreaction impact on the
model and space-time evolution is quantified in Fig. 2 and Fig. 5.

The backreaction obtained within our gauge-invariant approach, considering
inflaton and scalar metric fluctuations, is quite different with respect to
the one obtained with standard approaches, which consider only the gauge
fields contribution including, at maximum, the inflaton fluctuations. Looking
at Fig. 2 we can note how the backreaction begins to be significant well
before the standard case. Such backreaction became non-perturbative already
after $N\sim 10-15$ e-folds, while in the standard case, one has a non-
perturbative impact of the backreaction only at $N\sim 35$ e-folds. At the
same time, one can see how, after the backreaction became non-perturbative,
both the effective Hubble factor and the parameter $\xi$, which controls the
influence of the gauge fields on the inflaton dynamics, take a quite larger
number of e-folds to change considerably their values. Namely, they evolve
much slower compared with the standard dynamics and the case where only the
gauge fields backreaction is included. As a consequence, the inflationary
period lasts a much larger number of e-folds, up to $10^{3}$ and more. In Fig.
5 we show the evolution of the relative densities with the number of e-folds.
Among others, one interesting thing that can be noted from these figures is
that the backreaction effect is dominated up to almost the end of inflation
from the contribution associated with the helicity integral. Indeed, such
contribution becomes larger than the one associated with the potential energy
for $N\sim 15$ and, as a consequence, the backreaction becomes dominant and
non-perturbative, this confirms the above considerations based on Fig. 2.

Correspondingly, both Fig. 2 and Fig. 5 demonstrate that the cosmological
backreaction became non-perturbative already after $N\sim 10-15$ e-folds. When
this happens our assumption of considering the gauge fields contribution as a
perturbative contribution is not valid anymore. Having said that, to go beyond
this weak backreaction regime, considering all the possible contributions, one
possibility could be to define an effective background for the gauge fields’
contribution and write a perturbative theory on top of it, to take into
consideration the coupling between the scalar fluctuations of the
matter/gravity sector with the gauge fields effective fluctuations
consistently. The above approach is necessary to consider, in such a case, not
only the backreaction of the gauge fields on the inflationary dynamics but
also the backreaction that the scalar fluctuations of the matter/gravity
sector have on the gauge fields’ production. This non-trivial issue will be
addressed in a forthcoming paper. Moreover, this above extension would be
crucial to accurately evaluate the impact of backreaction also on the
perturbation spectra. In this regard, it is noteworthy that, as discussed in
Durrer  _et al._ (2024), scalar metric perturbations significantly affect the
spectrum of the comoving curvature perturbation.

To conclude, let us underline how only the approach proposed in this
manuscript takes into consideration consistently both the scalar fluctuations
of the matter/metric sector and their coupling with the gauge fields. This
approach gives us the possibility to study the weak backreaction regime and
understand when this is not valid anymore, without missing part of the effect
coming from scalar induced peturbations. This also shows how the contribution
coming from metric perturbations cannot be neglected as done in previous
literature, and we believe such contribution will likely be important also
when one moves beyond the standard perturbative approach and goes into the
strong backreaction regime.

## Acknowledgements

We wish to thank Chiara Animali and Matteo Braglia for useful discussions.
DCG, PC, GM and SSC are supported in part by the Istituto Nazionale di Fisica
Nucleare (INFN) through the Commissione Scientifica Nazionale 4 (CSN4), GM and
SSC under the Iniziativa Specifica (IS) Theoretical Astroparticle Physics
(TAsP) and the IS Quantum Fields in Gravity, Cosmology and Black Holes (FLAG),
while DCG and PC under the IS TAsP. The work of GM was partially supported by
the research grant number 2022E2J4RK “PANTHEON: Perspectives in Astroparticle
and Neutrino THEory with Old and New messengers” under the program PRIN 2022
funded by the Italian Ministero dell’Università e della Ricerca (MUR). SSC
acknowledges support from the Fondazione Cassa di Risparmio di Trento e
Rovereto (CARITRO Foundation) through a Caritro Fellowship (project “Inflation
and dark sector physics in light of next-generation cosmological surveys”).
The work of PC is supported by a Della Riccia foundation grant. PC is thankful
to the CERN Theoretical Physics Department for the hospitality and financial
support provided during the development of this project.

##  Appendix A \- Gauge fields contribution

The Fourier mode functions $A_{\pm}$ of the gauge fields satisfy the following
equation of motion

$\frac{{\rm d}^{2}}{{\rm d}\tau^{2}}A_{\pm}(\tau,k)+\left(k^{2}\mp
kg\phi^{\prime}\right)A_{\pm}(\tau,k)=0\,.$ (72)

Under the assumption of de Sitter expansion for the background (i.e.
$a(\tau)=-1/(H\tau)$ with $\tau<0$, $H=\text{const.}$ and
$\dot{\phi}=\text{const.}$), we can rewrite the above equation in terms of the
constant parameter $\xi\equiv g\phi^{\prime}/(2a(\tau)H)=g\dot{\phi}/(2H)$. In
this approximation the analytical solution of Eq.(72) is given in terms of
Whittaker $W-$ functions

$A_{\pm}(\tau,k)=\frac{1}{\sqrt{2k}}e^{\pm\pi\xi/2}W_{\pm
i\xi,\frac{1}{2}}(-2ik\tau)\,.$ (73)

The gauge fields contribution can be given in terms of two physical
quantities. The vacuum expectation value of the energy density of the gauge
fields

$\displaystyle\dfrac{\langle\textbf{E}^{2}+\textbf{B}^{2}\rangle}{2}=\mathop{\text{\Large$\int$}}\nolimits$
$\displaystyle\dfrac{dk}{(2\pi)^{2}a^{4}}k^{2}\Bigl{[}\absolutevalue{A^{\prime}_{+}}^{2}+\absolutevalue{A^{\prime}_{-}}^{2}+k^{2}\bigl{(}\absolutevalue{A_{+}}^{2}+\absolutevalue{A_{-}}^{2}\bigr{)}\Bigr{]}\,,$
(74)

and the so-called helicity integral

$\langle\textbf{E}\cdot\textbf{B}\rangle=-\mathop{\text{\Large$\int$}}\nolimits\dfrac{dk}{(2\pi)^{2}a^{4}}k^{3}\partialderivative{\tau}\quantity(\absolutevalue{A_{+}}^{2}-\absolutevalue{A_{-}}^{2})\,.$
(75)

These integrals diverge for large $k$, as usual in quantum field theories in
curved backgrounds Birrell and Davies (1984); Parker and Toms (2009).

One can perform a proper renormalization procedure by subtracting to the bare
results of the energy density and of the helicity integral their adiabatic
counterparts as described in Animali  _et al._ (2022).

The final renormalized result for the energy density is Animali  _et al._
(2022)

$\begin{split}{{\langle{\bf E}^{2}+{\bf B}^{2}\rangle}\over
2}=&\,\frac{2H^{4}}{960\pi^{2}}+\frac{H^{4}\xi^{2}\left(-1185\xi^{4}+(330+4\sqrt{15})\xi^{2}+435+4\sqrt{15}\right)}{960\pi^{2}\left(1+\xi^{2}\right)}\\\
&-\frac{3H^{4}\xi^{2}\left(5\xi^{2}-1\right)\log\left(15/4\right)}{64\pi^{2}}+\frac{H^{4}\xi\left(30\xi^{2}-11\right)\sinh(2\pi\xi)}{64\pi^{3}}\\\
&-\frac{3H^{4}\xi^{2}\left(5\xi^{2}-1\right)(\psi(-1-i\xi)+\psi(-1+i\xi))}{32\pi^{2}}\\\
&+\frac{3iH^{4}\xi^{2}\left(5\xi^{2}-1\right)(\psi^{(1)}(1-i\xi)-\psi^{(1)}(1+i\xi))\sinh(2\pi\xi)}{64\pi^{3}}\,,\end{split}$
(76)

while for the helicity integral we obtain

$\begin{split}\langle\mathbf{E}\cdot\mathbf{B}\rangle=&\,\frac{H^{4}\xi\left(705\xi^{2}-330-\sqrt{15}\right)}{240\pi^{2}}+\frac{3H^{4}\xi\left(5\xi^{2}-1\right)\log\left(15/4\right)}{32\pi^{2}}\\\
&+\frac{3H^{4}\xi\left(5\xi^{2}-1\right)(\psi(1-i\xi)+\psi(1+i\xi))}{16\pi^{2}}\\\
&+\frac{3iH^{4}\xi\left(5\xi^{2}-1\right)(-\psi^{(1)}(1-i\xi)+\psi^{(1)}(1+i\xi))\sinh(2\pi\xi)}{32\pi^{3}}\\\
&+\frac{H^{4}\left(11-30\xi^{2}\right)\sinh(2\pi\xi)}{32\pi^{3}}\,.\end{split}$
(77)

where $\psi(x)$ is the Digamma function and
$\psi^{(1)}(x)\equiv\text{d}\psi(x)/\text{d}x$ .

##  Appendix B \- Einstein and energy-momentum tensor to second order

Within the perturbative framework introduced in the main text, the Einstein
tensor expanded to second order in the UCG is given by (see Finelli  _et al._
(2004)):

$\displaystyle G^{0}_{0}$ $\displaystyle=$ $\displaystyle G^{0(0)}_{0}+\delta
G^{0(1)}_{0}+\delta G^{0(2)}_{0}$ $\displaystyle=$
$\displaystyle-3H^{2}-\frac{H}{a}\nabla^{2}\beta+6H^{2}\alpha$
$\displaystyle-\frac{H}{a}\nabla^{2}\beta^{(2)}+6H^{2}\alpha^{(2)}-12H^{2}\alpha^{2}+\frac{3}{4}H^{2}|\vec{\nabla}\beta|^{2}$
$\displaystyle+\frac{H}{a}\left(\vec{\nabla}\alpha\cdot\vec{\nabla}\beta+2\alpha\nabla^{2}\beta\right)$
$\displaystyle+\frac{1}{8a^{2}}\left(\beta_{,ij}\beta^{,ij}-\left(\nabla^{2}\beta\right)^{2}\right)\,,$
$\displaystyle G^{0}_{i}$ $\displaystyle=$ $\displaystyle G^{0(0)}_{i}+\delta
G^{0(1)}_{i}+\delta G^{0(2)}_{i}$ (78) $\displaystyle=$
$\displaystyle-2H\alpha_{,i}-2H\alpha_{,i}^{(2)}+8H\alpha\alpha_{,i}-\frac{1}{2a}H\alpha_{,i}\nabla^{2}\beta$
$\displaystyle+\frac{1}{2a}\vec{\nabla}\alpha\cdot\vec{\nabla}\beta_{,i}-\frac{H}{2}\vec{\nabla}\beta_{,i}\cdot\vec{\nabla}\beta-\frac{1}{4}\nabla^{2}\dot{\chi}_{i}^{(2)}\,,$

$\displaystyle G^{i}_{j}$ $\displaystyle=$ $\displaystyle G^{i(0)}_{j}+\delta
G^{i(1)}_{j}+\delta G^{i(2)}_{j}$ (79) $\displaystyle=$
$\displaystyle\delta^{i}_{j}\left\\{-(3H^{2}+2\dot{H})+2\alpha(3H^{2}+2\dot{H})+2H\dot{\alpha}\right.$
$\displaystyle\left.+\frac{1}{a^{2}}\nabla^{2}\alpha-\frac{H}{a}\nabla^{2}\beta-\frac{1}{2a}\nabla^{2}\dot{\beta}\right.$
$\displaystyle\left.+2\alpha^{(2)}(3H^{2}+2\dot{H})+2H\dot{\alpha}^{(2)}+\frac{1}{a^{2}}\nabla^{2}\alpha^{(2)}-\frac{H}{a}\nabla^{2}\beta^{(2)}-\frac{1}{2a}\nabla^{2}\dot{\beta}^{(2)}\right.$
$\displaystyle\left.+\frac{H}{a}\vec{\nabla}\alpha\cdot\vec{\nabla}\beta+\frac{H}{2}\vec{\nabla}\beta\cdot\vec{\nabla}\dot{\beta}+\left(\frac{1}{4}|\vec{\nabla}\beta|^{2}-4\alpha^{2}\right)\left(3H^{2}+2\dot{H}\right)-\right.$
$\displaystyle\left.-8H\alpha\dot{\alpha}+\left(\frac{\dot{\alpha}}{2a}+2\frac{H}{a}\alpha\right)\nabla^{2}\beta-\frac{2}{a^{2}}\alpha\nabla^{2}\alpha+\frac{\alpha}{a}\nabla^{2}\dot{\beta}-\frac{1}{a^{2}}|\vec{\nabla}\alpha|^{2}\right.$
$\displaystyle\left.+\frac{1}{8a^{2}}\left(\beta^{,\ell m}\beta_{,\ell
m}-(\nabla^{2}\beta)^{2}\right)\right\\}+\left\\{\frac{1}{2a}\dot{\beta}^{,i}_{,j}+\frac{H}{a}\beta^{,i}_{,j}-\frac{1}{a^{2}}\alpha^{,i}_{,j}\right.$
$\displaystyle\left.+\frac{1}{2a}\dot{\beta}^{(2)\,,i}_{\,\,\,\,\,\,\,\,\,,j}+\frac{H}{a}\beta^{(2)\,,i}_{\,\,\,\,\,\,\,\,\,,j}-\frac{1}{a^{2}}\alpha^{(2)\,,i}_{\,\,\,\,\,\,\,\,\,,j}+\frac{1}{a^{2}}\alpha^{,i}\alpha_{,j}-\frac{H}{a}\beta^{,i}\alpha_{,j}+\frac{2}{a^{2}}\alpha\alpha^{,i}_{,j}\right.$
$\displaystyle\left.-\frac{2}{a}H\alpha\beta^{,i}_{,j}-\frac{1}{2a}\dot{\alpha}\beta^{,i}_{,j}-\frac{1}{a}\alpha\dot{\beta}^{,i}_{,j}+\frac{1}{4a^{2}}\left(\nabla^{2}\beta\beta^{,i}_{,j}-\beta^{,i}_{,k}\beta^{,k}_{,j}\right)\right.$
$\displaystyle\left.+\frac{3}{4}H\left(\dot{\chi}^{(2),i}_{j}+\dot{\chi}^{(2)i}_{\,\,\,\,\,\,\,\,,j}+\dot{h}^{(2)i}_{\,\,\,\,\,\,\,j}\right)+\frac{1}{4}\left(\ddot{\chi}_{j}^{(2),i}+\ddot{\chi}^{(2)i}_{\,\,\,\,\,\,\,\,,j}+\ddot{h}^{(2)i}_{\,\,\,\,\,\,\,j}\right)\right.$
$\displaystyle\left.-\frac{1}{4a^{2}}\nabla^{2}h^{(2)i}_{\,\,\,\,\,\,\,j}\right\\}\,.$

On the other hand, the axionic coupling between the pseudo-scalar inflaton and
the gauge fields changes the energy-momentum tensor, introducing new terms. Up
to the second order in perturbation theory, one obtains

$\displaystyle T^{0}_{\hskip 2.84544pt0}=$ $\displaystyle\leavevmode\nobreak\
T^{0\hskip 1.42271pt(0)}_{\hskip 2.84544pt0}+\delta T^{0\hskip
1.42271pt(1)}_{\hskip 2.84544pt0}+\delta T^{0\hskip 1.42271pt(2)}_{\hskip
2.84544pt0}$ $\displaystyle=$
$\displaystyle-\quantity[\dfrac{\dot{\phi}^{2}}{2}+V]+\dot{\phi}^{2}\alpha-\dot{\phi}\dot{\varphi}-V_{\phi}\varphi+\dot{\phi}^{2}\alpha^{(2)}-\dot{\phi}\dot{\varphi}^{(2)}$
$\displaystyle-
V_{\phi}\varphi^{(2)}-\dfrac{\dot{\varphi}^{2}}{2}-\dfrac{\absolutevalue{\boldsymbol{\nabla}\varphi}^{2}}{2a^{2}}-\dfrac{1}{2}V_{\phi\phi}\varphi^{2}-2\dot{\phi}^{2}\alpha^{2}+2\dot{\phi}\alpha\dot{\varphi}$
$\displaystyle+\dfrac{1}{8}\dot{\phi}^{2}\absolutevalue{\boldsymbol{\nabla}\beta}^{2}-\dfrac{\textbf{E}^{2}+\textbf{B}^{2}}{2}\,,$
$\displaystyle T^{0}_{\hskip 2.84544pti}=$ $\displaystyle\leavevmode\nobreak\
T^{0\hskip 1.42271pt(0)}_{\hskip 2.84544pti}+\delta T^{0\hskip
1.42271pt(1)}_{\hskip 2.84544pti}+\delta T^{0\hskip 1.42271pt(2)}_{\hskip
2.84544pti}$ $\displaystyle=$ $\displaystyle-\dot{\phi}\varphi_{,\hskip
1.42271pti}-\dot{\phi}\varphi^{(2)}_{,\hskip
1.42271pti}-\dot{\varphi}\varphi_{,\hskip
1.42271pti}+2\dot{\phi}\alpha\varphi_{,\hskip 1.42271pti}+F_{j}^{\hskip
2.84544pt0}F^{j}_{\hskip 2.84544pti}\,,$ $\displaystyle T^{i}_{\hskip
2.84544ptj}=$ $\displaystyle\leavevmode\nobreak\ T^{i\hskip
1.42271pt(0)}_{\hskip 2.84544ptj}+\delta T^{i\hskip 1.42271pt(1)}_{\hskip
2.84544ptj}+\delta T^{i\hskip 1.42271pt(2)}_{\hskip 2.84544ptj}$
$\displaystyle=$
$\displaystyle\quantity(\dfrac{\dot{\phi}^{2}}{2}-V)\delta^{i}_{\hskip
2.84544ptj}+\quantity(\dot{\phi}\dot{\varphi}-\dot{\phi}^{2}\alpha-
V_{\phi}\varphi)\delta^{i}_{\hskip 2.84544ptj}+$
$\displaystyle+\Biggl{[}\dot{\phi}\dot{\varphi}^{(2)}-\dot{\phi}^{2}\alpha^{(2)}-V_{\phi}\varphi^{(2)}+\dfrac{1}{2}\Biggl{(}\dot{\varphi}^{2}-4\dot{\phi}\alpha\dot{\varphi}-\dfrac{1}{4}\dot{\phi}^{2}\absolutevalue{\boldsymbol{\nabla}\beta}^{2}$
$\displaystyle+4\dot{\phi}^{2}\alpha^{2}+\dfrac{1}{a}\dot{\phi}\boldsymbol{\nabla}\varphi\cdot\boldsymbol{\nabla}\beta-\dfrac{\absolutevalue{\boldsymbol{\nabla}\varphi}^{2}}{a^{2}}-V_{\phi\phi}\varphi^{2}\Biggr{)}\Biggr{]}\delta^{i}_{\hskip
2.84544ptj}$ $\displaystyle+\dfrac{1}{a^{2}}\varphi^{,\hskip
1.42271pti}\varphi_{,\hskip 1.42271ptj}-\dfrac{\dot{\phi}}{2a}\beta^{,\hskip
1.42271pti}\varphi_{,\hskip
1.42271ptj}-E^{i}E_{j}-B^{i}B_{j}+\delta^{i}_{\hskip
2.84544ptj}\dfrac{\textbf{E}^{2}+\textbf{B}^{2}}{2}\,.$ (80)

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