# DarkGEO: A Large-Scale Laser-Interferometric Axion Detector

Joscha Heinze<EMAIL_ADDRESS>University of Birmingham, School of Physics
and Astronomy, Birmingham B15 2TT, United Kingdom.  Alex Gill  University of
Birmingham, School of Physics and Astronomy, Birmingham B15 2TT, United
Kingdom.  Artemiy Dmitriev  University of Birmingham, School of Physics and
Astronomy, Birmingham B15 2TT, United Kingdom.  Jiří Smetana  University of
Birmingham, School of Physics and Astronomy, Birmingham B15 2TT, United
Kingdom.  Tianliang Yan  University of Birmingham, School of Physics and
Astronomy, Birmingham B15 2TT, United Kingdom.  Vincent Boyer  University of
Birmingham, School of Physics and Astronomy, Birmingham B15 2TT, United
Kingdom.  Hartmut Grote  Cardiff University, School of Physics and Astronomy,
Cardiff CF24 3AA, United Kingdom.  James Lough  Max Planck Institute for
Gravitational Physics (Albert Einstein Insitute), Hanover 30167, Germany.
Aldo Ejlli  Max Planck Institute for Gravitational Physics (Albert Einstein
Insitute), Hanover 30167, Germany.  Guido Müller  Max Planck Institute for
Gravitational Physics (Albert Einstein Insitute), Hanover 30167, Germany.
Denis Martynov  University of Birmingham, School of Physics and Astronomy,
Birmingham B15 2TT, United Kingdom.

###### Abstract

Axions and axion-like particles (ALPs) are leading candidates for dark matter.
They are well motivated in many extensions of the Standard Model and supported
by astronomical observations. We propose an iterative transformation of the
existing facilities of the gravitational-wave detector and technology testbed
GEO600, located near Ruthe in Germany, into a kilometre-scale upgrade of the
laser-interferometric axion detector LIDA. The final DarkGEO detector could
search for coincident signatures of axions and ALPs and significantly surpass
the current constraints of both direct searches and astrophysical observations
in the measurement band from ${10}^{-16}$ to
${10}^{-8}\text{\,}\mathrm{e}\mathrm{V}$. We discuss realistic parameters and
design sensitivities for the configurations of the different iteration steps
as well as technical challenges known from the first LIDA results. The
proposed DarkGEO detector will be well suited to probe the parameter space
associated with predictions from theoretical models, like grand-unified
theories, as well as from astrophysical evidence, like the cosmic infrared
background.

††preprint: APS/123-QED

##  I INTRODUCTION

A wealth of evidence from astronomical observations indicates the existence of
dark matter and, for several decades, weakly interacting massive particles
(WIMPs) were the most prominent candidate for it. A variety of observatories
have, however, not been able to detect any WIMP [1, 2, 3] and, meanwhile,
another candidate has arisen in the dark matter research focus: axions and
axion-like particles (ALPs) [4, 5, 6]. While the axion is specifically known
since 1977 as a solution to the strong charge-parity problem in quantum-
chromodynamics [7, 8, 9, 10], both types of particle emerge generically in a
plentitude of extensions of the Standard Model, e.g. from string theory and
supergravity [11, 12, 13, 14, 15]. Furthermore, there are compelling
observations from gravitational lensing which distinctly favour wave-like dark
matter and which the axion could explain [16].

This gain in significance has led to the proposal of numerous methods and
experiments to directly measure the signature of an axion or ALP, some of
which have already taken data. Among those are axion haloscopes (ADMX [17],
MADMAX [18] and DMRadio [19]), axion helioscopes (CAST [20] and IAXO [21]),
“light shining though a wall” experiments (ALPS [22] and CROWS [23]) and
magnetometers (ABRACADABRA [24]). This article considers a fairly new type of
direct axion detector based on laser interferometry. This type was proposed in
Refs. [25, 26, 27, 28, 29, 30, 31], and first engineering and observing runs
have been conducted in Refs. [32] (LIDA) and [33] (DANCE).

The most important component of these detectors is an optical travelling-wave
cavity which has a geometrical length of $5\text{\,}\mathrm{m}$ in LIDA and
$45\text{\,}\mathrm{c}\mathrm{m}$ in DANCE. Similarly to the laser-
interferometric gravitational-wave detectors [34, 35, 36], a kilometre-scale
cavity would, however, allow for a significant increase in the sensitivity due
to an increased interaction time between the laser field and the axion field
and due to additional benefits that, for instance, come with larger mirrors
and, thus, larger beam sizes.

In this article, we discuss how the gravitational-wave detector and technology
testbed GEO600, located near Ruthe in Germany, could iteratively be tranformed
into the next-generation LIDA upgrade “DarkGEO”, featuring a total of two
$600\text{\,}\mathrm{m}$ long cavities for coincidence searches with an
additional drastic increase in the operating laser power. Section II briefly
explains the signal generation in such a laser-interferometric axion detector,
in general, and Section III describes the three main configurations of DarkGEO
which can be operated during its iterative transformation, including their
design parameters and sensitivities. Finally, Section IV discusses some
technical challenges, which are known from the first LIDA results and proposes
viable solutions for them.

##  II THEORY

The laser-interferometric axion detectors can measure a signature due to the
interaction of the axions with the photons of a laser field. This interaction
is governed by the following Lagrangian term [20]

$\mathcal{L}_{a\gamma}=-\frac{g_{a\gamma}}{4}aF^{\mu\nu}\tilde{F}_{\mu\nu}\
\text{,}$ (1)

where $a$ is the axion field, $F$ is the electro-magnetic field-strength
tensor and $g_{a\gamma}$ is the axion-photon coupling coefficient. Here, an
axion with mass $m_{a}$ behaves like a coherent classical field [37]

$a(t)=a_{0}\sin\left[\Omega_{a}t+\delta(t)\right]$ (2)

with angular frequency $\Omega_{a}=2\pi f_{a}=m_{a}c^{2}/\hbar$, field
amplitude $a_{0}^{2}=2\rho_{\text{DM}}\hbar^{2}/m_{a}^{2}$, the local density
of dark matter $\rho_{\rm
DM}\approx$5.3\text{\times}{10}^{-22}\text{\,}\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{3}$$,
and the phase of the field $\delta(t)$.

The measurable effect of the interaction between a linearly polarised “main”
laser field and this axion field is a periodic rotation of the main laser
field’s polarisation axis. This rotation results from a phase $\Delta\phi$
which accumulates between the left- and right-handed circular states of
polarisation over a time period of $\tau$ (which will later be the light’s
cavity roundtrip time) [26]

$\Delta\phi(t,\tau)=g_{a\gamma}\left[a(t)-a(t-\tau)\right]\ \text{.}$ (3)

The periodic rotation of the polarisation axis can be equivalently understood
as the excitation of two coherent “sidebands” which are linearly polarised in
the direction orthogonal to the main laser field’s polarisation. These two
sidebands comprise the signal field and are shifted in frequency by
$\pm\Omega_{a}$ relative to the main laser field. If this effect occurs in an
optical cavity that is kept on resonance with the main laser field, these
sidebands build up according to [25]

$\displaystyle\begin{split}&E_{\text{sig,cav}}(\pm\Omega_{a})=-\frac{E_{\text{m,cav}}\exp\left(i\frac{\beta\mp\Omega_{a}\tau}{2}+\delta\right)}{1-\sqrt{1-2T_{\text{sig}}-l_{\text{rt}}}\exp\left[i\left(\beta\mp\Omega_{a}\tau\right)\right]}\\\
&\qquad\ \ \times
g_{a\gamma}\frac{\tau}{4}\text{sinc}\left(\frac{\Omega_{a}\tau}{4}\right)\cos\left(\frac{2\beta\mp\Omega_{a}\tau}{4}\right)\sqrt{2\tau_{a}\rho_{\text{DM}}}\
\text{.}\end{split}$ (4)

$E_{\text{m,cav}}$ is the circulating intra-cavity main field, $\beta$ takes
into account that the resonances for the polarisations of the main and signal
field may be non-degenerate, $\tau$ is the cavity roundtrip time,
$T_{\text{sig}}$ is the power transmissivity of the cavity input and output
couplers for the signal field polarisation, $l_{\text{rt}}$ is the cavity
roundtrip power loss and $\tau_{a}$ is the coherence time of the axion field.

In transmission of the cavity, the signal field can be superimposed with a
strong coherent local oscillator field by shifting a constant fraction of the
main field into the signal polarisation via a half-wave plate, yielding
$E_{\text{LO}}=~{}i\xi\sqrt{T_{\text{m}}}E_{\text{m,cav}}$ where $\xi$ is
twice the rotation angle of the half-wave plate and $T_{\text{m}}$ is the
power transmissivity of the cavity output coupler for the main field
polarisation. The amplitude spectral density $P_{\text{out}}$ of the signal is
then obtained from the beatnote between the local oscillator and the sidebands
[25]:

$P_{\text{out}}(\Omega_{a})=E_{\text{LO}}\sqrt{T_{\text{sig}}}\left[E^{*}_{\text{sig,cav}}(-\Omega_{a})-E_{\text{sig,cav}}(\Omega_{a})\right]\
\text{.}$ (5)

Here, $i\sqrt{T_{\text{sig}}}E^{(*)}_{\text{sig,cav}}(\pm\Omega_{a})$ are the
two signal sidebands after being coupled out of the cavity in transmission.
The signal-to-noise-ratio finally depends on the amplitude spectral density of
the total noise $P_{N}$ at the signal frequency $\Omega_{a}$ and on the total
measurement time $T_{\text{meas}}$:

$\text{SNR}^{2}=\left|\frac{P_{\text{out}}(\Omega_{a})}{P_{\text{N}}(\Omega_{a})}\right|^{2}\sqrt{\frac{T_{\text{meas}}}{\tau_{a}}}\
\text{.}$ (6)

For a thorough theoretical description of the operating principle, we refer to
the Refs. [26, 27, 28, 25, 29, 30].

##  III DESIGN

The GEO600 facilities are constructed to house a laser-interferometric
gravitational-wave detector in the typical Michelson topology [38]. Most
importantly, the facilities provide two $600\text{\,}\mathrm{m}$ long vacuum
tubes whose end stations are equipped with mirror suspension platforms for
vibrational isolation as well as with optical tables.

###  III.1 DarkGEO-I: linear configuration

The first iteration step of the proposed DarkGEO detector only requires minor
changes to the GEO600 design and is similar to the proposal in Ref. [31].
Instead of using a rectangular travelling-wave cavity as in LIDA and DANCE,
DarkGEO-I consists of one linear standing-wave cavity that is set up with a
length of $600\text{\,}\mathrm{m}$ in one of the vacuum tubes between the
current positions of the beamsplitter and the far end stage. This is shown in
the top schematic of Fig. 1.

Figure 1: Simplified schematic of the DarkGEO-I and -II configurations with
one linear and rectangular cavity, respectively. PBS: polarising beamsplitter,
red beam: main laser field, orange beam: signal field, dashed beam: squeezed
field. The second arm of the GEO600 facility is indicated as going upwards.

A linear cavity only offers a limited broadband sensitivity to axions and
axion-like particles because the reflection off a mirror under normal
incidence causes a phase shift of $\pi$ between the main laser field (here,
vertical polarisation) and signal field (here, horizontal polarisation).
Hence, the rotation of the polarisation axis will cancel out during one
complete roundtrip [26, 25]. However, a linear cavity can still utilise half
of a roundtrip for a modest broadband sensitivity and reach high sensitivities
around each free-spectral range.

Figure 2 assumes the parameters from Tab. 1 and shows the resulting design
sensitivity for a shot-noise limited performance. Hence,
$\left|P_{\text{N}}\right|^{2}=2\hbar\omega_{0}P_{\text{m,cav}}\sqrt{\tau_{a}/T_{\text{meas}}}$
in Eq. 6 with the optical angular frequency $\omega_{0}$ and the power
$P_{\text{m,cav}}$ of the intra-cavity main laser field. The plotted
sensitivity corresponds to the $95\text{\,}\%$ confidence level which is
obtained for $\text{SNR}=2$ in Eq. 6. DarkGEO-I could already probe a yet
unexplored region of the axion-photon parameter space around
$1\text{\,}\mathrm{n}\mathrm{e}\mathrm{V}$, or
$240\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$, which is favoured by
observations of the cosmic infrared background [39].

Please note that we assume a shot-noise limited sensitivity in the whole
measurement band which is an ambitious goal especially below frequencies of
$\sim$1\text{\,}\mathrm{H}\mathrm{z}$$. In a more advanced phase of the
DarkGEO design, a more thorough model will also include the expected couplings
of additional noise source, like technical laser noise and scattered light, to
the signal readout as well as means to mitigate them. The same holds for the
DarkGEO-II/III configuration.

The unprecedented intra-cavity power in the circulating main laser field of
$10\text{\,}\mathrm{M}\mathrm{W}$ will be achieved by operating the cavity in
the over-coupled regime and reading out the signal in reflection. The input
and output coupler transmissivities of
$20\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ and
$1\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$, respectively, yield a power
buildup factor of about 47,500 assuming a roundtrip power loss of
$20\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$. This allows for a power of
$10\text{\,}\mathrm{M}\mathrm{W}$ in combination with a laser source which
provides about $210\text{\,}\mathrm{W}$ [40]. Furthermore, an optical power of
$10\text{\,}\mathrm{M}\mathrm{W}$ on the GEO600 mirrors corresponds to a
similar optical intensity as already reached in LIDA
($4.7\text{\,}\mathrm{M}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}\mathrm{{}^{2}}$)
since beam sizes of up to a few centimetres will be possible in DarkGEO.

It is important to note that if any of these design parameters (especially the
high intra-cavity power) should not be exactly met, this would only result in
a corresponding slight reduction in the design sensitivity, and DarkGEO could
still significantly surpass the current constraints of axion searches.

The input and output mode cleaners filter out technical laser noise above
their pole frequencies as well as unwanted higher-order spatial modes. The
local oscillator field for the readout can be obtained by rotating the
polarising beamsplitter, accordingly.

Table 1:  Design parameters for the DarkGEO configurations I and II/III which the sensitivities in Fig. 2 are based on. The parameters $T_{\text{in/out}}$ refer to the labels in Fig. 1. For the configuration II/III, “m” and “sig” refer to the main laser and signal field, respectively. Parameter (DarkGEO-I) | Value | Unit
---|---|---
Wavelength | 1064 | nm
Cavity roundtrip length | 1.2 | km
Input coupler transmissivity, $T_{\text{in}}$ | 20 | ppm
Output coupler transmissivity, $T_{\text{out}}$ | 1 | ppm
Cavity roundtrip loss, $l_{\text{rt}}$ | 20 | ppm
Laser input power | 210 | W
Intra-cavity power, $P_{\text{m,cav}}$ | 10 | MW
Measurement time, $T_{\text{meas}}$ | 1 | year
Main laser field polarisation | vertical |
Signal field polarisation | horizontal |
Parameter (DarkGEO-II/III) | Value | Unit
Wavelength | 1064 | nm
Cavity roundtrip length | 1.2 | km
Input coupler transmissivity, $T_{\text{m,in}}$ | 45 | ppm
Output coupler transmissivity, $T_{\text{m,out}}$ | 1 | ppb
Input coupler transmissivity, $T_{\text{sig,in}}$ | 3000 | ppm
Output coupler transmissivity, $T_{\text{sig,out}}$ | 2.5 | ppm
Cavity roundtrip loss, $l_{\text{rt}}$ | 45 | ppm
Laser input power | 460 | W
Intra-cavity power, $P_{\text{m,cav}}$ | 10 | MW
Effective squeezing level | 10 | dB
Measurement time, $T_{\text{meas}}$ | 1 | year
Detuning, $\beta$ | 0.13 (scanned) |
Main laser field polarisation | vertical |
Signal field polarisation | horizontal |
Figure 2: Shot-noise limited design sensitivities at the $95\text{\,}\%$
confidence level to the axion-photon coupling coefficient $g_{a\gamma}$ for
DarkGEO-I and -II/-III based on the parameters in Tab. 1. The sensitivities
are compared to the CAST limit [20] as well as to the most stringent
constraints from astrophysical observations (quasar H1821+643 [41], axion star
explosions (ASE) [42], magnetic white dwarf (MWD) polarisation [43]), and to
predictions based on the cosmic infrared background (“TeV axions”)[39]. The
“detuned” case is shown as one example and corresponds to a frequency
separation of $5\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ between the
polarisation resonances in the rectangular cavity ($\beta\approx 0.13$).

###  III.2 DarkGEO-II: rectangular configuration

DarkGEO-II assumes the LIDA design with a rectangular cavity and requires
larger changes to the GEO600 facility in order to set up pairs of mirrors on
either side of the $600\text{\,}\mathrm{m}$ long vacuum tube. The laser input
and signal readout will, however, remain the same as in DarkGEO-I. This is
also shown in Fig. 1 (bottom schematic). As before, we propose a readout in
reflection of the cavity to enable a sufficient power buildup to reach a
circulating power of $10\text{\,}\mathrm{M}\mathrm{W}$ in the main laser
field, despite an assumed roundtrip loss of
$45\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$, by operating the cavity in the
over-coupled regime. Here, the design parameters yield a power buildup of
about 22,000 such that the laser source has to provide
$460\text{\,}\mathrm{W}$. While this level has not been reached yet by a
single-frequency, low-noise laser, this goal is in line with the current
research in the gravitational-wave community: the Einstein Telescope plans to
utilise a $500\text{\,}\mathrm{W}$ laser source [44], and up to
$370\text{\,}\mathrm{W}$ in the fundamental Gaussian laser mode have already
been achieved [40]. Hence, the development of a sufficiently strong laser
source by the time DarkGEO-II could be initiated is a realistic assumption.

In the rectangular configuration, the signal field experiences higher input
and output coupler transmissivities than the main laser field if the signal
field’s axis of polarisation is chosen to be horizontal. Given the parameters
from Tab. 1, the cavity’s power reflection coefficient for the signal field’s
polarisation at the resonance condition becomes about $94\text{\,}\%$, causing
only little optical loss to a field between the cavity’s input and reflection
port. Hence, we propose the injection of squeezed states of light in the
signal field’s polarisation (horizontal), as shown in Fig. 1, to mitigate
quantum shot noise. Here, shot noise is mainly caused by the vacuum
fluctuations that enter the cavity from the laser input side in the
polarisation of the signal field, are effectively reflected and reach the
signal readout. These vacuum fluctuations are replaced by squeezed vacuum as
already done in a similar way in the gravitational-wave detectors GEO600,
Advanced LIGO and Advanced Virgo [38, 45, 46]. We assume the same ambitious
squeezing level of $10\text{\,}\mathrm{d}\mathrm{B}$ as envisaged for the
next-generation gravitational-wave detectors, the Einstein Telescope [44] and
Cosmic Explorer [47], and as, so far, only achieved in tabletop experiments
[48, 49]. The squeezing level is included as a factor of $e^{-2r}$ in
$\left|P_{\text{N}}\right|^{2}$ in Eq. 6.

An additional aspect of the rectangular cavity is that the mirrors distinguish
between vertical (main laser field) and horizontal (signal field) linear
polarisation. This especially holds for the phases which the two polarisations
accumulate during one cavity roundtrip. Hence, their resonances are not
necessarily degenerate in frequency. Controlling the frequency separation
between the polarisation resonances, e.g. via an auxiliary cavity as proposed
in Ref. [25], would allow to control at which frequency, or axion mass, the
detector reaches its peak sensitivity.

Figure 2 shows the shot-noise limited sensitivities that DarkGEO-II could
reach with the parameters given in Tab. 1. The chosen example for the
“detuned” case indicates how a scan of the frequency separation will scan the
most prominent sensitivity peak through the spectrum. DarkGEO-II could
significantly surpass the most stringent constraints of the axion-photon
coupling in almost its entire measurement band from ${10}^{-16}$ to
${10}^{-8}\text{\,}\mathrm{e}\mathrm{V}$.

###  III.3 DarkGEO-III: full coincidence search

For the final iteration step, a second rectangular cavity is set up in the
second vacuum tube of the GEO600 facilities. Hence, DarkGEO-III will operate
two identical and, more importantly, independent axion detectors at the same
time, which are influenced by an axion field in the same way. The disadvantage
of the LIDA and DANCE design is that these detectors cannot be operated in a
state where they are insensitive to axions. It is, thus, non-trivial to
determine whether a potential signature is real or caused by noise. The
operation of two independent detectors, however, allows to search for
coincident signals which are significantly less likely to be caused by the
uncorrelated noise sources of the two detectors.

Please note that a true independence between the two detectors is a challenge
on its own since they will share the central vacuum chamber. In an advanced
phase of the DarkGEO design, means to mitigate any possible cross-talk have to
be specified.

##  IV CHALLENGES

###  IV.1 Non-planarity in DarkGEO-II/III

LIDA has observed a cross-talk between vertically and horizontally polarised
light inside the optical cavity [32]. When injecting a vertically polarised
laser beam into the cavity, the authors obtained an elliptically polarised
laser beam in transmission which only consisted of the injected vertical
polarisation to 82-$85\text{\,}\%$. This observation can partly be explained
by a non-planarity of the rectangular cavity, leading to a coupling of the
external orthogonal states of polarisation.

We modelled this effect using an ABCDEF matrix beam propagation method to
compute an estimate for this coupling in DarkGEO-II and -III. In this model,
the cavity has the dimensions of $L\times
l=$600\text{\,}\mathrm{m}$\times$40\text{\,}\mathrm{c}\mathrm{m}$$ to fit into
the $60\text{\,}\mathrm{c}\mathrm{m}$ wide vacuum tubes of the GEO600
facilities. Given the current GEO600 mirrors with a diameter of
$18\text{\,}\mathrm{c}\mathrm{m}$, we assume a beam waist of about
$2.2\text{\,}\mathrm{c}\mathrm{m}$ and a radius of curvature on one of the
mirrors of $R_{c}=$3000\text{\,}\mathrm{m}$$. The matrix for the free space
propagation $V_{L(l)}$ along the length of $L\,(l)$ and the matrix for the
reflection off the four mirrors $M$ are given by

$V_{L(l)}=\begin{pmatrix}1&L\,(l)&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}\ \text{,}\
M_{k}=\begin{pmatrix}1&0&0\\\
0\,\left(-\frac{2}{R_{c}}\right)&1&2\alpha_{k}\\\ 0&0&1\end{pmatrix}$ (7)

where $k=1,2,3,4$ are the four mirrors and $\alpha_{k}$ are their
misalignments. The upper limit for the mirror misalignment, causing the non-
planarity, is set to
$75\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{r}\mathrm{a}\mathrm{d}$, which
corresponds to a displacement of the beam by half of a mirror radius over a
distance of $600\text{\,}\mathrm{m}$. Finally, the eigenvalue problem of the
cavity matrix

$C=M_{1}\,V_{l}\,M_{4}\,V_{L}\,M_{3}\,V_{l}\,M_{2}\,V_{L}$ (8)

is solved for the power ratio between the light in the vertical and horizontal
polarisation in transmission of the cavity.

If a purely vertically polarised field is injected, Fig. 3 shows that the
coupling of the horizontal polarisation into the transmitted field remains
below $1\text{\,}\%$, even for a misalignment
$>$75\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{r}\mathrm{a}\mathrm{d}$$.
Hence, DarkGEO should already strongly mitigate this effect via its physical
size.

Figure 3: Cross-talk between the external vertical and horizontal states of
linear polarisation in the rectangular cavities of DarkGEO-II and -III.
Assuming that a pure vertically polarised field is injected, $p$ indicates the
contribution of the horizontal polarisation to the transmitted field due to a
mirror misalignment and due to the resulting cavity non-planarity.

###  IV.2 Drifts of the frequency separation

As mentioned in Sec. III.2, the cavity resonances of the main laser field’s
and signal field’s polarisation are likely to be separated in frequency in
DarkGEO-II and -III (detuning). While there are means to control and stabilise
this detuning [25], an estimation of how much a free-running detuning may
drift during operation is important, as well. We have conducted a test run
with LIDA where we have tracked the detuning over a time of
$141\text{\,}\mathrm{h}$.

If the polarisation of the input field is slightly rotated relative to either
the horizontal or vertical polarisation, this input field partially couples to
the respective orthogonal polarisation, as well. Most importantly, the
technical noise of this fraction of the input field will then be transmitted
through the cavity around the detuning frequency. Hence, we obtain a prominent
noise peak in the readout spectrum whose frequency coincides with the detuning
frequency and which can be tracked over time.

For this test run, we chose the linear polarisation of the input field to be
mainly horizontal, associated with a significantly lower finesse than the
vertical polarisation. This allowed for a more reliable long-term operation
and resulted in a narrower, more defined peak in the readout spectrum which
now corresponded to the higher finesse of the vertical polarisation. The
circulating intra-cavity power was about $580\text{\,}\mathrm{W}$ (intensity
of
$22\text{\,}\mathrm{k}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}\mathrm{{}^{2}}$
at the waist).

Figure 4: Evolution of the frequency separation between the cavity resonances
of the horizontal and vertical linear polarisation (detuning) during a test
run of $141\text{\,}\mathrm{h}$ (left y-axis, starting at about
$474.8\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$). The full range of the
detuning drift was about $1\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$, or
$0.2\text{\,}\%$. The evolutions of the circulating intra-cavity power and of
the lab temperature (right y-axis, ending at about 0.01 and $-0.01$,
respectively) do not show an obvious correlation over the full measurement
time.

As Fig. 4 shows, the detuning only drifted over a range of about
$1\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$, which amounts to $0.2\text{\,}\%$
of the mean detuning of $474.5\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$. Fig. 4
also compares the evolution of this detuning to the evolution of the lab
temperature and of the circulating intra-cavity power in terms of their change
relative to their respective mean value. This comparison was used to infer
whether thermo-optic changes in the mirror substrates and coatings due to the
residual absorption of the circulating laser power or changes in the ambient
temperature are the underlying cause of the drift of the detuning. While there
occasionally seems to be a correlation to the intra-cavity power over the
first $4.5\text{\,}\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}$, the detuning
especially shows a rather sharp dip over the last day even though the power is
roughly constant. There also seems to be a correlation to the lab temperature
over the last half of the time with a delay of
1-$2\text{\,}\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}$ (and no delay from day
5). However, the first peak in the detuning does not have a counterpart in the
evolution of the lab temperature. Thus, there is no obvious correlation of the
detuning to either the intra-cavity power or the lab temperature which
persists over the full measurement time.

Hence, we have not conclusively found a mechanism behind the drift of the
detuning frequency. Still, we could confirm that the detuning frequency was
quite stable and only drifted within a range that is a factor of 13.5 smaller
than LIDA’s measurement bandwidth (see the parameter $f_{\text{p,P}}$ in Ref.
[32]).

###  IV.3 Thermal effects

At the high operating optical intensity, LIDA experienced occasional drops in
the intra-cavity power in a range from $10\text{\,}\%$ up to $50\text{\,}\%$
correlated with an increase in the readout noise as well as a deterioration of
the output beam purity [32]. Even though this effect is not fully understood
yet, it is likely to be related to parametric instabilities (PIs) [50]. A PI
occurs when the main spatial laser mode which resonates in a cavity, here the
fundamental Gaussian TEM00 mode, couples to an excited vibrational eigenmode
of a mirror. This interaction leads to light being scattered into a higher-
order transverse laser mode. In addition, thermal deformations of the mirror
surfaces caused by the absorption of the high circulating laser power may
affect the cavity’s transverse mode spacing and, thus, result in a co-
resonance of the TEM00 and a higher-order mode. If this co-resonating higher-
order mode is also coupled to the TEM00 mode via the PI, this mechanism could
explain the observed thermal effect in LIDA.

If this mechanism is confirmed after additional investigation, we propose to
use acoustic mode dampers to significantly reduce the quality factors of the
mirror’s vibrational eigenmodes [51]. The risk of increasing thermal noise by
adding this channel of energy dissipation does not apply to the discussed
laser-interferometric axion detectors as the main laser field and signal field
are co-propagating and, thus, any common mode effect cancels out [25]. Still,
a more detailed future model might need to include thermally induced effects
as well, e.g. regarding birefringence that converts the vertical to the
horizontal polarisation or vice versa.

##  V Conclusion

The gravitational-wave detector and technology testbed GEO600 in Germany
provides an excellent facility to house a kilometre-scale laser-
interferometric axion detector, DarkGEO. We showed how such a transformation
could be executed in a 3-step programme. In the first step, DarkGEO-I will
operate a linear cavity to search for axions and axion-like particles around
the free-spectral ranges. DarkGEO-II will then operate a rectangular cavity
based on the LIDA topology, and DarkGEO-III will finally consist of two such
independent detectors for a coincident search. The presented design parameters
allow for sensitivities to the axion-photon coupling coefficient that are
several orders of magnitude below the current constraints within a mass range
of ${10}^{-16}$ to ${10}^{-8}\text{\,}\mathrm{e}\mathrm{V}$. DarkGEO would,
thus, be an immense and vital advance of direct axion searches. In addition,
DarkGEO would still serve as a technology demonstrator for the third
generation of gravitational-wave detectors, the Einstein Telescope and Cosmic
Explorer. This especially applies to the envisaged high-power laser source,
the unprecedented intra-cavity power and the effective squeezing level of
$10\text{\,}\mathrm{d}\mathrm{B}$ in a complex, large-scale experiment.
Moreover, we discussed the main challenges which are known from the first LIDA
results: cross-talk between the horizontal and vertical polarisation, drifts
of the detuning frequency and thermal effects inside the optical cavity. The
proposed solutions should significantly limit the impact of these effects to
enable a robust long-term operation at the simulated design sensitivities.

We acknowledge members of the UK Quantum Interferometry collaboration for
useful discussions, the support of the Institute for Gravitational Wave
Astronomy at the University of Birmingham and STFC Quantum Technology for
Fundamental Physics scheme (Grant No. ST/T006331/1, ST/T006609/1 and
ST/W006375/1). D.M. is supported by the 2021 Philip Leverhulme Prize.

## References

  * Akerib  _et al._ [2013] D. Akerib, X. Bai, S. Bedikian, E. Bernard, A. Bernstein, A. Bolozdynya, A. Bradley, D. Byram, S. Cahn, C. Camp, M. Carmona-Benitez, D. Carr, J. Chapman, A. Chiller, C. Chiller, K. Clark, T. Classen, T. Coffey, A. Curioni, E. Dahl, S. Dazeley, L. de Viveiros, A. Dobi, E. Dragowsky, E. Druszkiewicz, B. Edwards, C. Faham, S. Fiorucci, R. Gaitskell, K. Gibson, M. Gilchriese, C. Hall, M. Hanhardt, B. Holbrook, M. Ihm, R. Jacobsen, L. Kastens, K. Kazkaz, R. Knoche, S. Kyre, J. Kwong, R. Lander, N. Larsen, C. Lee, D. Leonard, K. Lesko, A. Lindote, M. Lopes, A. Lyashenko, D. Malling, R. Mannino, Z. Marquez, D. McKinsey, D.-M. Mei, J. Mock, M. Moongweluwan, M. Morii, H. Nelson, F. Neves, J. Nikkel, M. Pangilinan, P. Parker, E. Pease, K. Pech, P. Phelps, A. Rodionov, P. Roberts, A. Shei, T. Shutt, C. Silva, W. Skulski, V. Solovov, C. Sofka, P. Sorensen, J. Spaans, T. Stiegler, D. Stolp, R. Svoboda, M. Sweany, M. Szydagis, D. Taylor, J. Thomson, M. Tripathi, S. Uvarov, J. Verbus, N. Walsh, R. Webb, D. White, J. White, T. Whitis, M. Wlasenko, F. Wolfs, M. Woods, and C. Zhang, The large underground xenon (lux) experiment, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 704, 111 (2013).
  * Aprile  _et al._ [2018] E. Aprile, J. Aalbers, F. Agostini, M. Alfonsi, L. Althueser, F. D. Amaro, M. Anthony, F. Arneodo, L. Baudis, B. Bauermeister, M. L. Benabderrahmane, T. Berger, P. A. Breur, A. Brown, A. Brown, E. Brown, S. Bruenner, G. Bruno, R. Budnik, C. Capelli, J. M. R. Cardoso, D. Cichon, D. Coderre, A. P. Colijn, J. Conrad, J. P. Cussonneau, M. P. Decowski, P. de Perio, P. Di Gangi, A. Di Giovanni, S. Diglio, A. Elykov, G. Eurin, J. Fei, A. D. Ferella, A. Fieguth, W. Fulgione, A. Gallo Rosso, M. Galloway, F. Gao, M. Garbini, C. Geis, L. Grandi, Z. Greene, H. Qiu, C. Hasterok, E. Hogenbirk, J. Howlett, R. Itay, F. Joerg, B. Kaminsky, S. Kazama, A. Kish, G. Koltman, H. Landsman, R. F. Lang, L. Levinson, Q. Lin, S. Lindemann, M. Lindner, F. Lombardi, J. A. M. Lopes, J. Mahlstedt, A. Manfredini, T. Marrodán Undagoitia, J. Masbou, D. Masson, M. Messina, K. Micheneau, K. Miller, A. Molinario, K. Morå, M. Murra, J. Naganoma, K. Ni, U. Oberlack, B. Pelssers, F. Piastra, J. Pienaar, V. Pizzella, G. Plante, R. Podviianiuk, N. Priel, D. Ramírez García, L. Rauch, S. Reichard, C. Reuter, B. Riedel, A. Rizzo, A. Rocchetti, N. Rupp, J. M. F. dos Santos, G. Sartorelli, M. Scheibelhut, S. Schindler, J. Schreiner, D. Schulte, M. Schumann, L. Scotto Lavina, M. Selvi, P. Shagin, E. Shockley, M. Silva, H. Simgen, D. Thers, F. Toschi, G. Trinchero, C. Tunnell, N. Upole, M. Vargas, O. Wack, H. Wang, Z. Wang, Y. Wei, C. Weinheimer, C. Wittweg, J. Wulf, J. Ye, Y. Zhang, and T. Zhu (XENON Collaboration 7), Dark matter search results from a one ton-year exposure of xenon1t, Phys. Rev. Lett. 121, 111302 (2018).
  * Zhang  _et al._ [2018] H. Zhang, A. Abdukerim, W. Chen, X. Chen, Y. Chen, X. Cui, B. Dong, D. Fang, C. Fu, K. Giboni, F. Giuliani, L. Gu, X. Guo, Z. Guo, K. Han, C. He, S. He, D. Huang, X. Huang, Z. Huang, P. Ji, X. Ji, Y. Ju, S. Li, Y. Li, H. Lin, H. Liu, J. Liu, Y. Ma, Y. Mao, K. Ni, J. Ning, X. Ren, F. Shi, A. Tan, A. Wang, C. Wang, H. Wang, M. Wang, Q. Wang, S. Wang, X. Wang, X. Wang, Z. Wang, M. Wu, S. Wu, J. Xia, M. Xiao, P. Xie, B. Yan, J. Yang, Y. Yang, C. Yu, J. Yuan, J. Yue, D. Zhang, T. Zhang, L. Zhao, Q. Zheng, J. Zhou, N. Zhou, and X. Zhou, Dark matter direct search sensitivity of the pandax-4t experiment, Science China Physics, Mechanics & Astronomy 62, 31011 (2018).
  * Abbott and Sikivie [1983] L. Abbott and P. Sikivie, A cosmological bound on the invisible axion, Physics Letters B 120, 133 (1983).
  * Preskill  _et al._ [1983] J. Preskill, M. B. Wise, and F. Wilczek, Cosmology of the invisible axion, Physics Letters B 120, 127 (1983).
  * Dine and Fischler [1983] M. Dine and W. Fischler, The not-so-harmless axion, Physics Letters B 120, 137 (1983).
  * Peccei and Quinn [1977] R. D. Peccei and H. R. Quinn, $\mathrm{CP}$ conservation in the presence of pseudoparticles, Phys. Rev. Lett. 38, 1440 (1977).
  * Weinberg [1978] S. Weinberg, A new light boson?, Phys. Rev. Lett. 40, 223 (1978).
  * Wilczek [1978] F. Wilczek, Problem of strong $p$ and $t$ invariance in the presence of instantons, Phys. Rev. Lett. 40, 279 (1978).
  * Chadha-Day  _et al._ [2022] F. Chadha-Day, J. Ellis, and D. J. E. Marsh, Axion dark matter: What is it and why now?, Science Advances 8, eabj3618 (2022).
  * Svrcek and Witten [2006] P. Svrcek and E. Witten, Axions in string theory, Journal of High Energy Physics 2006, 051 (2006).
  * Graham and Rajendran [2013] P. W. Graham and S. Rajendran, New observables for direct detection of axion dark matter, Phys. Rev. D 88, 035023 (2013).
  * Ringwald [2012] A. Ringwald, Exploring the role of axions and other WISPs in the dark universe, Physics of the Dark Universe 1, 116 (2012), arXiv:1210.5081 [hep-ph] .
  * Ringwald [2014] A. Ringwald, Searching for axions and ALPs from string theory, Journal of Physics: Conference Series 485, 012013 (2014).
  * Farina  _et al._ [2017] M. Farina, D. Pappadopulo, F. Rompineve, and A. Tesi, The photo-philic qcd axion, Journal of High Energy Physics 2017, 95 (2017).
  * Amruth  _et al._ [2023] A. Amruth, T. Broadhurst, J. Lim, M. Oguri, G. F. Smoot, J. M. Diego, E. Leung, R. Emami, J. Li, T. Chiueh, H.-Y. Schive, M. C. H. Yeung, and S. K. Li, Einstein rings modulated by wavelike dark matter from anomalies in gravitationally lensed, Nat. Astron. , 2397 (2023).
  * Bartram  _et al._ [2021] C. Bartram, T. Braine, E. Burns, R. Cervantes, N. Crisosto, N. Du, H. Korandla, G. Leum, P. Mohapatra, T. Nitta, L. J. Rosenberg, G. Rybka, J. Yang, J. Clarke, I. Siddiqi, A. Agrawal, A. V. Dixit, M. H. Awida, A. S. Chou, M. Hollister, S. Knirck, A. Sonnenschein, W. Wester, J. R. Gleason, A. T. Hipp, S. Jois, P. Sikivie, N. S. Sullivan, D. B. Tanner, E. Lentz, R. Khatiwada, G. Carosi, N. Robertson, N. Woollett, L. D. Duffy, C. Boutan, M. Jones, B. H. LaRoque, N. S. Oblath, M. S. Taubman, E. J. Daw, M. G. Perry, J. H. Buckley, C. Gaikwad, J. Hoffman, K. W. Murch, M. Goryachev, B. T. McAllister, A. Quiskamp, C. Thomson, and M. E. Tobar (ADMX Collaboration), Search for invisible axion dark matter in the $3.3--4.2\text{ }\text{ }\mu\mathrm{eV}$ mass range, Phys. Rev. Lett. 127, 261803 (2021).
  * Caldwell  _et al._ [2017] A. Caldwell, G. Dvali, B. Majorovits, A. Millar, G. Raffelt, J. Redondo, O. Reimann, F. Simon, and F. Steffen (MADMAX Working Group), Dielectric haloscopes: A new way to detect axion dark matter, Phys. Rev. Lett. 118, 091801 (2017).
  * Brouwer  _et al._ [2022] L. Brouwer, S. Chaudhuri, H.-M. Cho, J. Corbin, W. Craddock, C. S. Dawson, A. Droster, J. W. Foster, J. T. Fry, P. W. Graham, R. Henning, K. D. Irwin, F. Kadribasic, Y. Kahn, A. Keller, R. Kolevatov, S. Kuenstner, A. F. Leder, D. Li, J. L. Ouellet, K. M. W. Pappas, A. Phipps, N. M. Rapidis, B. R. Safdi, C. P. Salemi, M. Simanovskaia, J. Singh, E. C. van Assendelft, K. van Bibber, K. Wells, L. Winslow, W. J. Wisniewski, and B. A. Young (DMRadio Collaboration), Projected sensitivity of ${\text{DMRadio-m}}^{3}$: A search for the QCD axion below $1\text{ }\text{ }\mathrm{\mu}\mathrm{eV}$, Phys. Rev. D 106, 103008 (2022).
  * Anastassopoulos  _et al._ [2017] V. Anastassopoulos _et al._ (CAST), New CAST Limit on the Axion-Photon Interaction, Nature Phys. 13, 584 (2017), arXiv:1705.02290 [hep-ex] .
  * Armengaud  _et al._ [2014] E. Armengaud, F. T. Avignone, M. Betz, P. Brax, P. Brun, G. Cantatore, J. M. Carmona, G. P. Carosi, F. Caspers, S. Caspi, S. A. Cetin, D. Chelouche, F. E. Christensen, A. Dael, T. Dafni, M. Davenport, A. V. Derbin, K. Desch, A. Diago, B. Döbrich, I. Dratchnev, A. Dudarev, C. Eleftheriadis, G. Fanourakis, E. Ferrer-Ribas, J. Galán, J. A. García, J. G. Garza, T. Geralis, B. Gimeno, I. Giomataris, S. Gninenko, H. Gómez, D. González-Díaz, E. Guendelman, C. J. Hailey, T. Hiramatsu, D. H. H. Hoffmann, D. Horns, F. J. Iguaz, I. G. Irastorza, J. Isern, K. Imai, A. C. Jakobsen, J. Jaeckel, K. Jakovčić, J. Kaminski, M. Kawasaki, M. Karuza, M. Krčmar, K. Kousouris, C. Krieger, B. Lakić, O. Limousin, A. Lindner, A. Liolios, G. Luzón, S. Matsuki, V. N. Muratova, C. Nones, I. Ortega, T. Papaevangelou, M. J. Pivovaroff, G. Raffelt, J. Redondo, A. Ringwald, S. Russenschuck, J. Ruz, K. Saikawa, I. Savvidis, T. Sekiguchi, Y. K. Semertzidis, I. Shilon, P. Sikivie, H. Silva, H. ten Kate, A. Tomas, S. Troitsky, T. Vafeiadis, K. van Bibber, P. Vedrine, J. A. Villar, J. K. Vogel, L. Walckiers, A. Weltman, W. Wester, S. C. Yildiz, and K. Zioutas, Conceptual design of the International Axion Observatory (IAXO), JINST 9, T05002 (2014).
  * Bähre  _et al._ [2013] R. Bähre, B. Döbrich, J. Dreyling-Eschweiler, S. Ghazaryan, R. Hodajerdi, D. Horns, F. Januschek, E. A. Knabbe, A. Lindner, D. Notz, A. Ringwald, J. E. von Seggern, R. Stromhagen, D. Trines, and B. Willke, Any light particle search II — Technical Design Report, Journal of Instrumentation 8 (09), T09001.
  * Betz  _et al._ [2013] M. Betz, F. Caspers, M. Gasior, M. Thumm, and S. W. Rieger, First results of the CERN Resonant Weakly Interacting sub-eV Particle Search (CROWS), Phys. Rev. D 88, 075014 (2013).
  * Salemi  _et al._ [2021] C. P. Salemi, J. W. Foster, J. L. Ouellet, A. Gavin, K. M. W. Pappas, S. Cheng, K. A. Richardson, R. Henning, Y. Kahn, R. Nguyen, N. L. Rodd, B. R. Safdi, and L. Winslow, Search for low-mass axion dark matter with abracadabra-10 cm, Phys. Rev. Lett. 127, 081801 (2021).
  * Martynov and Miao [2020] D. Martynov and H. Miao, Quantum-enhanced interferometry for axion searches, Phys. Rev. D 101, 095034 (2020).
  * DeRocco and Hook [2018] W. DeRocco and A. Hook, Axion interferometry, Phys. Rev. D 98, 035021 (2018).
  * Obata  _et al._ [2018] I. Obata, T. Fujita, and Y. Michimura, Optical ring cavity search for axion dark matter, Phys. Rev. Lett. 121, 161301 (2018).
  * Liu  _et al._ [2019] H. Liu, B. D. Elwood, M. Evans, and J. Thaler, Searching for axion dark matter with birefringent cavities, Phys. Rev. D 100, 023548 (2019).
  * Michimura  _et al._ [2020] Y. Michimura, Y. Oshima, T. Watanabe, T. Kawasaki, H. Takeda, M. Ando, K. Nagano, I. Obata, and T. Fujita, Dance: Dark matter axion search with ring cavity experiment, Journal of Physics: Conference Series 1468, 012032 (2020).
  * Nagano  _et al._ [2019a] K. Nagano, T. Fujita, Y. Michimura, and I. Obata, Axion dark matter search with interferometric gravitational wave detectors, Phys. Rev. Lett. 123, 111301 (2019a).
  * Nagano  _et al._ [2019b] K. Nagano, T. Fujita, Y. Michimura, and I. Obata, Axion dark matter search with interferometric gravitational wave detectors, Phys. Rev. Lett. 123, 111301 (2019b).
  * Heinze  _et al._ [2023] J. Heinze, A. Gill, A. Dmitriev, J. Smetana, T. Yan, V. Boyer, D. Martynov, and M. Evans, First results of the laser-interferometric detector for axions (lida) (2023), arXiv:2307.01365 [astro-ph.CO] .
  * Oshima  _et al._ [2023] Y. Oshima, H. Fujimoto, J. Kume, S. Morisaki, K. Nagano, T. Fujita, I. Obata, A. Nishizawa, Y. Michimura, and M. Ando, First results of axion dark matter search with dance (2023), arXiv:2303.03594 [hep-ex] .
  * Buikema  _et al._ [2020] A. Buikema, C. Cahillane, G. L. Mansell, C. D. Blair, R. Abbott, C. Adams, R. X. Adhikari, A. Ananyeva, S. Appert, K. Arai, J. S. Areeda, Y. Asali, S. M. Aston, C. Austin, A. M. Baer, M. Ball, S. W. Ballmer, S. Banagiri, D. Barker, L. Barsotti, J. Bartlett, B. K. Berger, J. Betzwieser, D. Bhattacharjee, G. Billingsley, S. Biscans, R. M. Blair, N. Bode, P. Booker, R. Bork, A. Bramley, A. F. Brooks, D. D. Brown, K. C. Cannon, X. Chen, A. A. Ciobanu, F. Clara, S. J. Cooper, K. R. Corley, S. T. Countryman, P. B. Covas, D. C. Coyne, L. E. H. Datrier, D. Davis, C. Di Fronzo, K. L. Dooley, J. C. Driggers, P. Dupej, S. E. Dwyer, A. Effler, T. Etzel, M. Evans, T. M. Evans, J. Feicht, A. Fernandez-Galiana, P. Fritschel, V. V. Frolov, P. Fulda, M. Fyffe, J. A. Giaime, K. D. Giardina, P. Godwin, E. Goetz, S. Gras, C. Gray, R. Gray, A. C. Green, E. K. Gustafson, R. Gustafson, J. Hanks, J. Hanson, T. Hardwick, R. K. Hasskew, M. C. Heintze, A. F. Helmling-Cornell, N. A. Holland, J. D. Jones, S. Kandhasamy, S. Karki, M. Kasprzack, K. Kawabe, N. Kijbunchoo, P. J. King, J. S. Kissel, R. Kumar, M. Landry, B. B. Lane, B. Lantz, M. Laxen, Y. K. Lecoeuche, J. Leviton, J. Liu, M. Lormand, A. P. Lundgren, R. Macas, M. MacInnis, D. M. Macleod, S. Márka, Z. Márka, D. V. Martynov, K. Mason, T. J. Massinger, F. Matichard, N. Mavalvala, R. McCarthy, D. E. McClelland, S. McCormick, L. McCuller, J. McIver, T. McRae, G. Mendell, K. Merfeld, E. L. Merilh, F. Meylahn, T. Mistry, R. Mittleman, G. Moreno, C. M. Mow-Lowry, S. Mozzon, A. Mullavey, T. J. N. Nelson, P. Nguyen, L. K. Nuttall, J. Oberling, R. J. Oram, B. O’Reilly, C. Osthelder, D. J. Ottaway, H. Overmier, J. R. Palamos, W. Parker, E. Payne, A. Pele, R. Penhorwood, C. J. Perez, M. Pirello, H. Radkins, K. E. Ramirez, J. W. Richardson, K. Riles, N. A. Robertson, J. G. Rollins, C. L. Romel, J. H. Romie, M. P. Ross, K. Ryan, T. Sadecki, E. J. Sanchez, L. E. Sanchez, T. R. Saravanan, R. L. Savage, D. Schaetzl, R. Schnabel, R. M. S. Schofield, E. Schwartz, D. Sellers, T. Shaffer, D. Sigg, B. J. J. Slagmolen, J. R. Smith, S. Soni, B. Sorazu, A. P. Spencer, K. A. Strain, L. Sun, M. J. Szczepańczyk, M. Thomas, P. Thomas, K. A. Thorne, K. Toland, C. I. Torrie, G. Traylor, M. Tse, A. L. Urban, G. Vajente, G. Valdes, D. C. Vander-Hyde, P. J. Veitch, K. Venkateswara, G. Venugopalan, A. D. Viets, T. Vo, C. Vorvick, M. Wade, R. L. Ward, J. Warner, B. Weaver, R. Weiss, C. Whittle, B. Willke, C. C. Wipf, L. Xiao, H. Yamamoto, H. Yu, H. Yu, L. Zhang, M. E. Zucker, and J. Zweizig, Sensitivity and performance of the Advanced LIGO detectors in the third observing run, Phys. Rev. D 102, 062003 (2020).
  * Bersanetti  _et al._ [2021] D. Bersanetti, B. Patricelli, O. J. Piccinni, F. Piergiovanni, F. Salemi, and V. Sequino, Advanced Virgo: Status of the Detector, Latest Results and Future Prospects, Universe 7, 322 (2021).
  * KAGRA [2022] C. KAGRA, Performance of the KAGRA detector during the first joint observation with GEO600 (O3GK), Progress of Theoretical and Experimental Physics , ptac093 (2022), https://academic.oup.com/ptep/advance-article-pdf/doi/10.1093/ptep/ptac093/45688139/ptac093.pdf .
  * Budker  _et al._ [2014] D. Budker, P. W. Graham, M. Ledbetter, S. Rajendran, and A. O. Sushkov, Proposal for a cosmic axion spin precession experiment (casper), Phys. Rev. X 4, 021030 (2014).
  * Lough  _et al._ [2021] J. Lough, E. Schreiber, F. Bergamin, H. Grote, M. Mehmet, H. Vahlbruch, C. Affeldt, M. Brinkmann, A. Bisht, V. Kringel, H. Lück, N. Mukund, S. Nadji, B. Sorazu, K. Strain, M. Weinert, and K. Danzmann, First Demonstration of 6 dB Quantum Noise Reduction in a Kilometer Scale Gravitational Wave Observatory, Phys. Rev. Lett. 126, 041102 (2021).
  * Kohri and Kodama [2017] K. Kohri and H. Kodama, Axion-like particles and recent observations of the cosmic infrared background radiation, Phys. Rev. D 96, 051701 (2017).
  * Wellmann  _et al._ [2021] F. Wellmann, N. Bode, P. Wessels, L. Overmeyer, J. Neumann, B. Willke, and D. Kracht, Low noise 400 w coherently combined single frequency laser beam for next generation gravitational wave detectors, Opt. Express 29, 10140 (2021).
  * Sisk-Reynés  _et al._ [2021] J. Sisk-Reynés, J. H. Matthews, C. S. Reynolds, H. R. Russell, R. N. Smith, and M. C. D. Marsh, New constraints on light axion-like particles using chandra transmission grating spectroscopy of the powerful cluster-hosted quasar h1821+643, Monthly Notices of the Royal Astronomical Society 510, 1264–1277 (2021).
  * Escudero  _et al._ [2023] M. Escudero, C. K. Pooni, M. Fairbairn, D. Blas, X. Du, and D. J. E. Marsh, Axion star explosions: A new source for axion indirect detection (2023), arXiv:2302.10206 [hep-ph] .
  * Dessert  _et al._ [2022] C. Dessert, D. Dunsky, and B. R. Safdi, Upper limit on the axion-photon coupling from magnetic white dwarf polarization, Phys. Rev. D 105, 103034 (2022).
  * ET Steering Committee [2020] ET Steering Committee, ET Design Report Update 2020, http://www.et-gw.eu/index.php/relevant-et-documents (2020), accessed: 2022-04-02.
  * The LIGO Scientific Collaboration [2013] The LIGO Scientific Collaboration, Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light, Nature Photonics 7, 613 (2013).
  * Acernese  _et al._ [2019] F. Acernese, M. Agathos, L. Aiello, A. Allocca, A. Amato, S. Ansoldi, S. Antier, M. Arène, N. Arnaud, S. Ascenzi, P. Astone, F. Aubin, S. Babak, P. Bacon, F. Badaracco, M. K. M. Bader, J. Baird, F. Baldaccini, G. Ballardin, G. Baltus, C. Barbieri, P. Barneo, F. Barone, M. Barsuglia, D. Barta, A. Basti, M. Bawaj, M. Bazzan, M. Bejger, I. Belahcene, S. Bernuzzi, D. Bersanetti, A. Bertolini, M. Bischi, M. Bitossi, M. A. Bizouard, F. Bobba, M. Boer, G. Bogaert, F. Bondu, R. Bonnand, B. A. Boom, V. Boschi, Y. Bouffanais, A. Bozzi, C. Bradaschia, M. Branchesi, M. Breschi, T. Briant, F. Brighenti, A. Brillet, J. Brooks, G. Bruno, T. Bulik, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, M. Canepa, G. Carapella, F. Carbognani, G. Carullo, J. Casanueva Diaz, C. Casentini, J. Castañeda, S. Caudill, F. Cavalier, R. Cavalieri, G. Cella, P. Cerdá-Durán, E. Cesarini, O. Chaibi, E. Chassande-Mottin, F. Chiadini, R. Chierici, A. Chincarini, A. Chiummo, N. Christensen, S. Chua, G. Ciani, P. Ciecielag, M. Cieślar, R. Ciolfi, F. Cipriano, A. Cirone, S. Clesse, F. Cleva, E. Coccia, P.-F. Cohadon, D. Cohen, M. Colpi, L. Conti, I. Cordero-Carrión, S. Corezzi, D. Corre, S. Cortese, J.-P. Coulon, M. Croquette, J.-R. Cudell, E. Cuoco, M. Curylo, B. D’Angelo, S. D’Antonio, V. Dattilo, M. Davier, J. Degallaix, M. De Laurentis, S. Deléglise, W. Del Pozzo, R. De Pietri, R. De Rosa, C. De Rossi, T. Dietrich, L. Di Fiore, C. Di Giorgio, F. Di Giovanni, M. Di Giovanni, T. Di Girolamo, A. Di Lieto, S. Di Pace, I. Di Palma, F. Di Renzo, M. Drago, J.-G. Ducoin, O. Durante, D. D’Urso, M. Eisenmann, L. Errico, D. Estevez, V. Fafone, S. Farinon, F. Feng, E. Fenyvesi, I. Ferrante, F. Fidecaro, I. Fiori, D. Fiorucci, R. Fittipaldi, V. Fiumara, R. Flaminio, J. A. Font, J.-D. Fournier, S. Frasca, F. Frasconi, V. Frey, G. Fronzè, F. Garufi, G. Gemme, E. Genin, A. Gennai, A. Ghosh, B. Giacomazzo, M. Gosselin, R. Gouaty, A. Grado, M. Granata, G. Greco, G. Grignani, A. Grimaldi, S. J. Grimm, P. Gruning, G. M. Guidi, G. Guixé, Y. Guo, P. Gupta, O. Halim, T. Harder, J. Harms, A. Heidmann, H. Heitmann, P. Hello, G. Hemming, E. Hennes, T. Hinderer, D. Hofman, D. Huet, V. Hui, B. Idzkowski, A. Iess, G. Intini, J.-M. Isac, T. Jacqmin, P. Jaranowski, R. J. G. Jonker, S. Katsanevas, F. Kéfélian, I. Khan, N. Khetan, G. Koekoek, S. Koley, A. Królak, A. Kutynia, D. Laghi, A. Lamberts, I. La Rosa, A. Lartaux-Vollard, C. Lazzaro, P. Leaci, N. Leroy, N. Letendre, F. Linde, M. Llorens-Monteagudo, A. Longo, M. Lorenzini, V. Loriette, G. Losurdo, D. Lumaca, A. Macquet, E. Majorana, I. Maksimovic, N. Man, V. Mangano, M. Mantovani, M. Mapelli, F. Marchesoni, F. Marion, A. Marquina, S. Marsat, F. Martelli, V. Martinez, A. Masserot, S. Mastrogiovanni, E. Mejuto Villa, L. Mereni, M. Merzougui, R. Metzdorff, A. Miani, C. Michel, L. Milano, A. Miller, E. Milotti, O. Minazzoli, Y. Minenkov, M. Montani, F. Morawski, B. Mours, F. Muciaccia, A. Nagar, I. Nardecchia, L. Naticchioni, J. Neilson, G. Nelemans, C. Nguyen, D. Nichols, S. Nissanke, F. Nocera, G. Oganesyan, C. Olivetto, G. Pagano, G. Pagliaroli, C. Palomba, P. T. H. Pang, F. Pannarale, F. Paoletti, A. Paoli, D. Pascucci, A. Pasqualetti, R. Passaquieti, D. Passuello, B. Patricelli, A. Perego, M. Pegoraro, C. Périgois, A. Perreca, S. Perriès, K. S. Phukon, O. J. Piccinni, M. Pichot, M. Piendibene, F. Piergiovanni, V. Pierro, G. Pillant, L. Pinard, I. M. Pinto, K. Piotrzkowski, W. Plastino, R. Poggiani, P. Popolizio, E. K. Porter, M. Prevedelli, M. Principe, G. A. Prodi, M. Punturo, P. Puppo, G. Raaijmakers, N. Radulesco, P. Rapagnani, M. Razzano, T. Regimbau, L. Rei, P. Rettegno, F. Ricci, G. Riemenschneider, F. Robinet, A. Rocchi, L. Rolland, M. Romanelli, R. Romano, D. Rosińska, P. Ruggi, O. S. Salafia, L. Salconi, A. Samajdar, N. Sanchis-Gual, E. Santos, B. Sassolas, O. Sauter, S. Sayah, D. Sentenac, V. Sequino, A. Sharma, M. Sieniawska, N. Singh, A. Singhal, V. Sipala, V. Sordini, F. Sorrentino, M. Spera, C. Stachie, D. A. Steer, G. Stratta, A. Sur, B. L. Swinkels, M. Tacca, A. J. Tanasijczuk, E. N. Tapia San Martin, S. Tiwari, M. Tonelli, A. Torres-Forné, I. Tosta e Melo, F. Travasso, M. C. Tringali, A. Trovato, K. W. Tsang, M. Turconi, M. Valentini, N. van Bakel, M. van Beuzekom, J. F. J. van den Brand, C. Van Den Broeck, L. van der Schaaf, M. Vardaro, M. Vasúth, G. Vedovato, D. Verkindt, F. Vetrano, A. Viceré, J.-Y. Vinet, H. Vocca, R. Walet, M. Was, A. Zadrożny, T. Zelenova, J.-P. Zendri, H. Vahlbruch, M. Mehmet, H. Lück, and K. Danzmann (Virgo Collaboration), Increasing the astrophysical reach of the advanced virgo detector via the application of squeezed vacuum states of light, Phys. Rev. Lett. 123, 231108 (2019).
  * Evans  _et al._ [2021] M. Evans, R. X. Adhikari, C. Afle, S. W. Ballmer, S. Biscoveanu, S. Borhanian, D. A. Brown, Y. Chen, R. Eisenstein, A. Gruson, A. Gupta, E. D. Hall, R. Huxford, B. Kamai, R. Kashyap, J. S. Kissel, K. Kuns, P. Landry, A. Lenon, G. Lovelace, L. McCuller, K. K. Y. Ng, A. H. Nitz, J. Read, B. S. Sathyaprakash, D. H. Shoemaker, B. J. J. Slagmolen, J. R. Smith, V. Srivastava, L. Sun, S. Vitale, and R. Weiss, A Horizon Study for Cosmic Explorer: Science, Observatories, and Community, arXiv:2109.09882 10.48550/arXiv.2109.09882 (2021).
  * Heinze  _et al._ [2022] J. Heinze, K. Danzmann, B. Willke, and H. Vahlbruch, 10 dB Quantum-Enhanced Michelson Interferometer with Balanced Homodyne Detection, Phys. Rev. Lett. 129, 031101 (2022).
  * Zander  _et al._ [2022] J. Zander, C. Rembe, and R. Schnabel, 10 db interferometer enhancement by squeezing of photon shot noise with sub-femtometer resolution and eye-safe optical power, Quantum Science and Technology 8, 01LT01 (2022).
  * Evans  _et al._ [2010] M. Evans, L. Barsotti, and P. Fritschel, A general approach to optomechanical parametric instabilities, Physics Letters A 374, 665 (2010).
  * Biscans  _et al._ [2019] S. Biscans, S. Gras, C. D. Blair, J. Driggers, M. Evans, P. Fritschel, T. Hardwick, and G. Mansell, Suppressing parametric instabilities in ligo using low-noise acoustic mode dampers, Phys. Rev. D 100, 122003 (2019).


# Reconstruction in the Horndeski theory within the scope of the Bianchi I
cosmology

Ruslan K. Muharlyamov<EMAIL_ADDRESS>Department of General Relativity and
Gravitation, Institute of Physics, Kazan Federal University, Kremlevskaya str.
18, Kazan 420008, Russia  Tatiana N. Pankratyeva<EMAIL_ADDRESS>Department
of Higher Mathematics, Kazan State Power Engineering University,
Krasnoselskaya str. 51, Kazan 420066, Russia

###### Abstract

In the previous article Eur. Phys. J. Plus 136, 590 (2021) (arXiv: 2110.15396)
we have proposed a reconstruction method for the kinetic gravity braiding
theory in the framework of the flat Friedman-Robertson-Walker spacetime. Here
we develop this method in the Bianchi I spacetime model for a subclass of the
Horndeski theory: $G_{5}\sim\phi$, $G_{2}(X)\neq 0$, $G_{3}(X)\neq 0$. The
Hubble parameter $H(t)$ and the canonical kinetic term $X(t)$ are set a
priori. The choice of the function $X(t)$ determines the anisotropic
properties of the Universe. This makes it possible to provide believable
anisotropy. The presented method allows for a realistic model of the Universe
to simply reconstruct some scalar field theory. Reconstruction example is
given for anisotropic model of a post-inflationary transition to the
radiation-dominated phase. The model is investigated for the absence ghosts
and Laplacian instabilities.

Horndeski theory, Bianchi I cosmology, dark energy

###### pacs:

04.20.Jb

##  I Introduction

To explain accelerated expansion of the Universe and other observational
facts, modifications of the gravity theory are used. The Horndeski gravity
(HG) has interesting properties Horndeski . The theory is constructed in such
a way that the motion equations are of the order of the derivative no higher
than the second. In this sense, the HG is the most general variant of the
scalar-tensor theory of gravitation. As we will see below, the action density
has a cumbersome form and contains four arbitrary functions. However, this
provides a wide field for action. Within the framework of the HG, various
cosmological and astrophysical problems are solved Sushkov0 ; Appleby0 ;
Sotiriou ; Babichev ; Maselli ; Muharlyamov .

Authors Kobayashi1  proposed such a parametrization of the action density for
the HG:

$\displaystyle
L_{H}=\sqrt{-g}\Big{(}\mathcal{L}_{2}+\mathcal{L}_{3}+\mathcal{L}_{4}+\mathcal{L}_{5}\Big{)}\,,$
(1)
$\mathcal{L}_{2}=G_{2}(\phi,X)\,,\,\mathcal{L}_{3}=-G_{3}(\phi,X)\Box\phi\,,$
$\mathcal{L}_{4}=G_{4}(\phi,X)R+G_{4X}(\phi,X)\left[(\square\phi)^{2}-(\nabla_{\mu}\nabla_{\nu}\phi)^{2}\right]\,,$
$\mathcal{L}_{5}=G_{5}(\phi,X)G_{\mu\nu}\,\nabla^{\mu}\nabla^{\nu}\phi-\frac{1}{6}G_{5X}\left[\left(\Box\phi\right)^{3}-3\Box\phi(\nabla_{\mu}\nabla_{\nu}\phi)^{2}+2\left(\nabla_{\mu}\nabla_{\nu}\phi\right)^{3}\right],$
(2)

where $g$ is the determinant of metric tensor $g_{\mu\nu}$; $R$ is the Ricci
scalar and $G_{\mu\nu}$ is the Einstein tensor; the functions $G_{i}$
($i=2,3,4,5$) depend on the scalar field $\phi$ and the canonical kinetic
term, $X=-\frac{1}{2}\nabla^{\mu}\phi\nabla_{\mu}\phi$. The following
designations are used $G_{iX}\equiv\partial G_{i}/\partial X$,
$(\nabla_{\mu}\nabla_{\nu}\phi)^{2}\equiv\nabla_{\mu}\nabla_{\nu}\phi\,\nabla^{\nu}\nabla^{\mu}\phi$,
and
$\left(\nabla_{\mu}\nabla_{\nu}\phi\right)^{3}\equiv\nabla_{\mu}\nabla_{\nu}\phi\,\nabla^{\nu}\nabla^{\rho}\phi\,\nabla_{\rho}\phi\nabla^{\mu}\phi$.

We continue the research begun in the work Muharlyamov , which considered the
kinetic gravity braiding (KGB) theory ($G_{2}(X)\neq 0,\,G_{3}(X)\neq
0,\,G_{4}=1/(16\pi)$) in the flat Friedman-Robertson-Walker spacetime (FRW).
Here we present a reconstruction method in the Bianchi I spacetime model (BI)
and in the FRW spacetime for the subclass of HG:

$G_{2}(X)\neq 0,\,G_{3}(X)\neq
0,\,G_{4}=1/(16\pi),\,G_{5}=\eta\cdot\phi/2,\,\eta=\text{const}.$ (3)

The functions $G_{2}$, $G_{3}$ are reconstructed based on the given the form
of the Hubble parameter $H(t)$ and kinetic density $X(t)$. The choice freedom
of functions $H$, $X$ allows you to regulate the issues of model stability
DeFelice1 ; Appleby1 . As shown in refs. Tahara ; Sushkov ; SushkovStar , the
presence of the function $G_{5}$ in the action density can lead to non-
standard behavior of the anisotropic Universe. During the reconstruction, the
function $X$ directly models the the anisotropy behavior of the Universe. This
moment is important to ensure believable anisotropy. For example, authors
Hawking , Hu  claim that the isotropization of the Universe occurred quite
early. At the start of primordial nucleosynthesis, the universe must already
be isotropic.

The presented method allows for a realistic model of the Universe to simply
reconstruct some scalar field theory. For example, we will reconstruct the
field theory for a post-inflationary transition to the radiation-dominated
phase. Authors R. C. Bernardo and I. Vega Bernardo  proposed their own version
of reconstruction the KGB theory for the FRW spacetime. In this theory, the
scalar charge is zero. To provide a dynamic solution $H(t)$, $X(t)$, a another
matter with a density $\rho(t)$ was introduced. In our method, the only source
is the scalar field $\phi$. The dynamics of functions $H(t)$ and $X(t)$ is
provided by the nonzero scalar charge. For the BI model, there is an
additional contribution to the dynamics of $H(t)$, $X(t)$ from the nonzero
anisotropic charges. R. C. Bernardo et al Bernardo1 ; Bernardo2  significantly
develop the ideas of reconstruction in connection with observational data.
Authors Bernardo1  reconstruct the parameters $H(t)$ from cosmic chronometers,
supernovae, and baryon acoustic oscillations compiled data sets via the
Gaussian process method and use it to draw out HG that are fully anchored on
expansion history data. In the work Bernardo2 , dark energy is studied through
the viewpoints of parametric and nonparametric analyses of late-time
cosmological data.

##  II Field equations

The metric of a Bianchi-I geometry may be written as

$ds^{2}=-dt^{2}+a^{2}_{1}(t)dx^{2}+a^{2}_{2}(t)dy^{2}+a^{2}_{3}(t)dz^{2}.$ (4)

We consider a homogeneous model: $a_{i}=a_{i}(t)$ and $\phi=\phi(t)$ . For the
action density (1) and the Bianchi-I geometry (4) the gravitation equations
have the form SushkovStar :

$G^{0}_{0}\left({\cal
G}-2G_{4X}\dot{\phi}^{2}-2G_{4XX}\dot{\phi}^{4}+2G_{5\phi}\dot{\phi}^{2}+G_{5X\phi}\dot{\phi}^{4}\right)=G_{2}-G_{2X}\dot{\phi}^{2}-$
$\displaystyle-3G_{3X}H\dot{\phi}^{3}+G_{3\phi}\dot{\phi}^{2}+6G_{4\phi}H\dot{\phi}+6G_{4X\phi}\dot{\phi}^{3}H-5G_{5X}H_{1}H_{2}H_{3}\dot{\phi}^{3}-G_{5XX}H_{1}H_{2}H_{3}\dot{\phi}^{5}\,,$
(5) ${\cal G}G^{i}_{i}-(H_{j}+H_{k})\frac{d{\cal
G}}{dt}=G_{2}-\dot{\phi}\frac{dG_{3}}{dt}+2\frac{d}{dt}(G_{4\phi}\dot{\phi})-$
$\displaystyle-\frac{d}{dt}(G_{5X}\dot{\phi}^{3}H_{j}H_{k})-G_{5X}\dot{\phi}^{3}H_{j}H_{k}(H_{j}+H_{k})\,,\,\,{\cal
G}\equiv 2G_{4}-2G_{4X}\dot{\phi}^{2}+G_{5\phi}\dot{\phi}^{2}\,.$ (6)

A dot means a derivative with respect to $t$; $H_{i}=\dot{a}_{i}/{a}_{i}$ –
the Hubble parameters;
$H=\dfrac{1}{3}\sum\limits_{i=1}^{3}H_{i}\equiv\dot{a}/{a}$ – the average
Hubble parameter with ${a}=({a}_{1}{a}_{2}{a}_{3})^{1/3}$. The triples of
indices $\\{i,j,k\\}$ take values $\\{1,2,3\\}$, $\\{2,3,1\\}$, or
$\\{3,1,2\\}$.

The equation for the scalar field $\phi(t)$ can be represented as

$\displaystyle\frac{1}{\rm a^{3}}\frac{d}{dt}({\rm a^{3}}{\cal J})={\cal
P}\,,$ (7)

with

${\cal
J}=\dot{\phi}\,\Big{[}G_{2X}-2G_{3\phi}+3H\dot{\phi}(G_{3X}-2G_{4X\phi})+$
$\displaystyle+G^{0}_{0}(-2G_{4X}-2\dot{\phi}^{2}G_{4XX}+2G_{5\phi}+G_{5X\phi}\dot{\phi}^{2})+H_{1}H_{2}H_{3}(3G_{5X}\dot{\phi}+G_{5XX}\dot{\phi}^{3})\Big{]}\,,$
(8) ${\cal
P}=G_{2\phi}-\dot{\phi}^{2}(G_{3\phi\phi}+G_{3X\phi}\ddot{\phi})+RG_{4\phi}+2G_{4X\phi}\dot{\phi}(3\ddot{\phi}H-\dot{\phi}G^{0}_{0})+$
$\displaystyle+G^{0}_{0}G_{5\phi\phi}\dot{\phi}^{2}+G_{5X\phi}\dot{\phi}^{3}H_{1}H_{2}H_{3}\,.$
(9)

For convenience, we consider the parametrization of the metric

$ds^{2}=-dt^{2}+a^{2}(t)[e^{2(\beta_{+}+\sqrt{3}\beta_{-})}dx^{2}+e^{2(\beta_{+}-\sqrt{3}\beta_{-})}dy^{2}+e^{-4\beta_{+}(t)}dz^{2}].$
(10)

The functions $e^{\beta_{+}+\sqrt{3}\beta_{-}}$ ,
$e^{\beta_{+}-\sqrt{3}\beta_{-}}$ and $e^{-2\beta_{+}}$ are the deviation from
isotropy, and $a(t)$ is the isotropic part. The Hubble parameters in the
direction of $x$, $y$ and $z$ are given by

$H_{1}=H+\dot{\beta}_{+}+\sqrt{3}\dot{\beta}_{-}\,,\,H_{2}=H+\dot{\beta}_{+}-\sqrt{3}\dot{\beta}_{-}\,,\,H_{3}=H-2\dot{\beta}_{+}\,.$
(11)

We choose the subclass of HG:

$G_{2}=G_{2}(X)\,,\,G_{3}=G_{3}(X)\,,\,G_{4}=\frac{1}{16\pi}\,,\,G_{5}=\dfrac{\eta\phi}{2}\,.$
(12)

Equations (5), (6) and (7) will change as follows

$\displaystyle
3\big{(}H^{2}-\sigma^{2}\big{)}\left(\frac{1}{8\pi}+\frac{3\eta\dot{\phi}^{2}}{2}\right)=-G_{2}+\dot{\phi}^{2}G_{2X}+3G_{3X}H\dot{\phi}^{3}\,,$
(13)
$\displaystyle\big{(}2\dot{H}+3H^{2}+3\sigma^{2}\big{)}\left(\frac{1}{8\pi}+\frac{\eta\dot{\phi}^{2}}{2}\right)+\eta
H\cdot\frac{d(\dot{\phi}^{2})}{dt}=-G_{2}+G_{3X}\dot{\phi}^{2}\ddot{\phi},$
(14)
$\displaystyle\dot{\beta}_{\pm}\left(\frac{1}{8\pi}+\frac{\eta\dot{\phi}^{2}}{2}\right)=\frac{C_{\pm}}{{a}^{3}},\,\,\sigma^{2}\equiv\dot{\beta}^{2}_{+}+\dot{\beta}^{2}_{-}\,,$
(15)
$\displaystyle\dot{\phi}\,\Big{[}G_{2X}+3HG_{3X}\dot{\phi}-3\eta(H^{2}-\sigma^{2})\Big{]}=\frac{C_{0}}{{a}^{3}},$
(16)

where $C_{0}$, $C_{+}$ and $C_{-}$ are the integration constants. Constants
$C_{\pm}$ correspond to the anisotropic charges; $C_{0}$ is the scalar charge.
The system (13), (14), (15) and (16) contains three independent equations and
five functions ($H(a)$, $X(a)$, $\sigma^{2}$, $G_{2}$, $G_{3}$). There is
freedom to choose two functions out of five. For example, we can set the law
of the the Universe development $H(a)$ and the canonical kinetic term $X(a)$.
Then we find the functions $\dot{\beta}_{\pm}$, $G_{2}(X)$ and $G_{3}(X)$. If
all charges are simultaneously $C_{0}=C_{\pm}=0$, then the system (13), (14),
(15), (16) can only have a stationary solution $H,\,X=\text{const}$,
$\dot{\beta}_{\pm}=0$. Equality $\dot{\beta}_{\pm}=0$ means that space-time
has become the FRW spacetime. In case $C_{0}\neq 0$, $C_{\pm}\neq 0$, we will
have a rich variety of cosmological models.

Equations (13), (14) and (16) have consequences

$G_{2}=-3\left(\frac{1}{8\pi}+\eta
X\right)(H^{2}[a(X)]-\sigma^{2}[a(X)])+\frac{\varepsilon
C_{0}\sqrt{2X}}{a^{3}(X)}\,,$ (17) $G_{3X}=\frac{2\eta
H}{\varepsilon\sqrt{2X}}+\frac{2}{\varepsilon\sqrt{2X}}\left(\frac{1}{8\pi}+\eta
X\right)\left(H^{\prime}_{X}+\frac{3a^{\prime}_{X}\sigma^{2}}{a(X)H}\right)+\frac{C_{0}a^{\prime}_{X}}{Ha^{4}(X)}\,,$
(18)

where $\varepsilon=\pm 1$ defines the sign of the derivative
$\dot{\phi}=\varepsilon\sqrt{2X}$. It is easy to verify that
$\dot{\phi}\ddot{\phi}=\dot{X}=aH/a^{\prime}_{X}$. As can be seen from the
corollary of equation (15)

$\dot{\beta}_{\pm}=\frac{C_{\pm}}{a^{3}\left(\frac{1}{8\pi}+\eta
X(a)\right)}\,,$ (19)

the anisotropic properties of the Universe are directly determined by the
choice of the density $X(a)$.

##  III Reduction to the isotropic case

To begin with, we consider reconstruction in isotropic space. If we set
$\dot{\beta}_{\pm}$ to zero, then we obtain the isotropic spacetime:

$ds^{2}=-dt^{2}+a^{2}(t)[dx^{2}+dy^{2}+dz^{2}]\,,$ (20)

then the field equations (13), (14) and (16) take the form

$\displaystyle
3H^{2}\left(\frac{1}{8\pi}+\frac{3\eta\dot{\phi}^{2}}{2}\right)=-G_{2}+\dot{\phi}^{2}G_{2X}+3G_{3X}H\dot{\phi}^{3}\,,$
(21)
$\displaystyle\big{(}2\dot{H}+3H^{2})\left(\frac{1}{8\pi}+\frac{\eta\dot{\phi}^{2}}{2}\right)+\eta
H\cdot\frac{d(\dot{\phi}^{2})}{dt}=-G_{2}+G_{3X}\dot{\phi}^{2}\ddot{\phi},$
(22) $\displaystyle\dot{\phi}\,\Big{[}G_{2X}+3HG_{3X}\dot{\phi}-3\eta
H^{2}\Big{]}=\frac{C_{0}}{{a}^{3}}\,.$ (23)

The restored functions (17), (18) will take the form

$G_{2}=-3\left(\frac{1}{8\pi}+\eta X\right)H^{2}[a(X)]+\frac{\varepsilon
C_{0}\sqrt{2X}}{a^{3}(X)}\,,$ (24) $G_{3X}=\frac{2\eta
H}{\varepsilon\sqrt{2X}}+\frac{2H^{\prime}_{X}}{\varepsilon\sqrt{2X}}\left(\frac{1}{8\pi}+\eta
X\right)+\frac{C_{0}a^{\prime}_{X}}{Ha^{4}(X)}\,.$ (25)

When considering the modified theories of gravity, one has to worry about the
presence of ghosts as well as Laplacian instabilities DeFelice1 ; Appleby1 .
The scalar perturbations will be stable if DeFelice1 :

$c_{S}^{2}\equiv\frac{3(2w_{1}^{2}w_{2}H-w^{2}_{2}w_{4}+4w_{1}w_{2}\dot{w}_{1}-2w_{1}^{2}\dot{w}_{2})}{w_{1}(4w_{1}w_{3}+9w_{2}^{2})}\geq
0\,,$ (26) $Q_{S}\equiv\frac{w_{1}(4w_{1}w_{3}+9w_{2}^{2})}{3w_{2}^{2}}>0\,,$
(27)

where

$\displaystyle w_{1}\equiv
2\,(G_{{4}}-2\,XG_{{4X}})-2X\,(G_{{5X}}{\dot{\phi}}H-G_{{5\phi}})\,,$ (28)
$\displaystyle
w_{2}\equiv-2\,G_{{3X}}X\dot{\phi}+4\,G_{{4}}H-16\,{X}^{2}G_{{4{XX}}}H+4(\dot{\phi}G_{{4\phi
X}}-4H\,G_{{4X}})X+2\,G_{{4\phi}}\dot{\phi}$ $\displaystyle\ \ \ \ \ \ \
\,+8\,{X}^{2}HG_{{5\phi
X}}+2H\,X\,(6G_{{5\phi}}-5\,G_{{5X}}\dot{\phi}{H})-4G_{{5{XX}}}{\dot{\phi}}X^{2}{H}^{2}\,,$
(29) $\displaystyle w_{3}\equiv
3\,X(G_{2{X}}+2\,XG_{{2XX}})+6X(3X\dot{\phi}HG_{{3{XX}}}-G_{{3\phi
X}}X-G_{{3\phi}}+6\,H\dot{\phi}G_{{3X}})$ $\displaystyle\ \ \ \ \ \ \
\,+18\,H(4\,H{X}^{3}G_{{4{XXX}}}-HG_{{4}}-5\,X\dot{\phi}G_{{4\phi
X}}-G_{{4\phi}}\dot{\phi}+7\,HG_{{4X}}X+16\,H{X}^{2}G_{{4{XX}}}-2\,{X}^{2}\dot{\phi}G_{{4\phi{XX}}})$
$\displaystyle\ \ \ \ \ \ \
\,+6{H}^{2}X(2\,H\dot{\phi}G_{{5{XXX}}}{X}^{2}-6\,{X}^{2}G_{{5\phi{XX}}}+13XH\dot{\phi}G_{{5{XX}}}-27G_{{5\phi
X}}X+15\,H\dot{\phi}G_{{5X}}-18G_{{5\phi}})\,,$ (30) $\displaystyle
w_{4}\equiv 2G_{4}-2XG_{5\phi}-2XG_{5X}\ddot{\phi}~{}.$ (31)

The speed square $c_{S}^{2}>0$ excludes the Laplacian instabilities. The
condition (27) guarantees the absence of ghosts. Similarly, for tensor
perturbations we have DeFelice1 :

$Q_{T}\equiv\frac{w_{1}}{4}>0\,,\,c^{2}_{T}\equiv\frac{w_{4}}{w_{1}}\geq 0\,.$
(32)

Taking into account (24), (25) and $G_{5}=\eta\phi/2$, we rewrite functions
$Q_{T}$, $c^{2}_{T}$, $c^{2}_{S}$ and $Q_{S}$ in the form

$Q_{T}=\frac{1}{4}\left(\frac{1}{8\pi}+\eta
X(a)\right)\,,\,c^{2}_{T}=\frac{1-8\pi\eta\cdot X(a)}{1+8\pi\eta\cdot
X(a)}\,,$ (33)

$Q_{S}=\frac{3\left(\frac{1}{8\pi}+\eta
X\right)}{w_{2}^{2}}\left[S\left(\frac{1}{\pi}+16\eta
X\right)\left(\frac{X^{5/2}H^{\prime}_{a}}{a^{4}(X^{\prime}_{a})^{2}H}-\frac{X^{3/2}}{2a^{4}X^{\prime}_{a}}\right)+\frac{4S^{2}X^{3}}{a^{8}(X^{\prime}_{a})^{2}H^{2}}\right]\,,$
(34)

$c^{2}_{S}=\frac{1}{Q_{S}}\left[-\omega_{4}+2H\left(\frac{\omega_{1}^{2}a}{w_{2}}\right)^{\prime}_{a}\right]\,,\,S=\varepsilon
C_{0}\sqrt{2}\,,$ (35)

where $w_{1}=1/(8\pi)+\eta X$, $w_{4}=1/(8\pi)-\eta X$ and

$w_{2}=2\left(\frac{1}{8\pi}+\eta
X\right)\left(H-\frac{2H^{\prime}_{a}X}{X^{\prime}_{a}}\right)-\frac{2SX^{3/2}}{a^{4}X^{\prime}_{a}H}\,.$
(36)

The Hubble parameter $H(a)$ is set from cosmological considerations. The
choice of the function $X(a)$ makes it possible to satisfy conditions (26),
(27) and (32) in the region of applicability of the model $H(a)$.

###  III.1 Post-inflationary transition to the radiation-dominated phase

Now we will show reconstruction example of the gravitating scalar field theory
for a post-inflationary transition to the radiation-dominated phase. Further,
the theory will be tested for the absence of pathologies.

We define the Hubble parameter $H(a)$ as follows

$H^{2}=\frac{8\pi\rho}{3}\,,\,\rho=\frac{2\rho_{d}}{a^{4}+1},\,\rho_{d}=\text{const}\,.$
(37)

The effective energy density $\rho$ corresponds to the equation of state

$p=\frac{\rho}{3}-\mu\rho^{2}\,,\,\mu>0$ (38)

describes the transition from the inflationary era to the radiation era in the
the early Universe Pierre . The linear term $\rho/3$ describes the radiation.
The nonlinear part $-\mu\rho^{2}$ may be due to Bose-Einstein condensates with
self-interaction. In case (38), the deceleration parameter (DP) has the form

$q(\rho)=\frac{d}{dt}\left(\frac{1}{H}\right)-1=1-\frac{3\mu\rho}{2}=1-\frac{\rho}{\rho_{d}}\,.$
(39)

The DP changes sign at point $\rho_{d}$ :

$q(\rho_{d})=0\,,\,\rho_{d}=\frac{2}{3\mu}\,.$ (40)

The Universe is accelerating when $\rho>\rho_{d}$ and decelerating when
$\rho\leq\rho_{d}$. The point $a=a_{d}=1$ corresponds to the end of inflation.
The density $\rho$ is a bounded function:

$2\rho_{d}\leftarrow\rho\rightarrow 0\,,\,\,\text{as}\,\,1\gg a\gg 1\,.$ (41)

The Planck density $\rho_{p}$ is approximately equal to $\rho_{p}\approx
2\rho_{d}$. With this in mind, we write

$\rho=\frac{\rho_{p}}{a^{4}+1}\,.$ (42)

In the early time ($a\ll 1$), the model has the quasi-de Sitter behavior:

$H^{2}\rightarrow\frac{8\pi\rho_{p}}{3}=\text{const}\,.$ (43)

At late time ($a\gg 1$) the Universe enters in the radiation era:

$H^{2}\propto\frac{1}{a^{4}}\,.$ (44)

We define the input function $X(a)$:

$X=\frac{m}{8\pi\eta a^{6}}\,,\,m=\text{const}\,.$ (45)

Looking ahead, we note that the choice of $X(a)$ will determine the behavior
of the anisotropy in the Bianchi I model.

Assumption (45) allows us to obtain from (24), (25) the functions $G_{2}$ and
$G_{3X}$:

$G_{2}=S\left(\frac{8\pi\eta}{m}\right)^{1/2}X-\frac{\rho_{p}\cdot
X^{2/3}(1+8\pi\eta X)}{\left(\frac{8\pi\eta}{m}\right)^{2/3}+X^{2/3}}\,,$ (46)
$G_{3X}=\frac{1}{4\varepsilon}\sqrt{\frac{3}{\pi\rho_{p}}}\cdot
X^{-5/6}\left[\left(\frac{8\pi\eta}{m}\right)^{2/3}+X^{2/3}\right]^{1/2}\times$
$\times\left\\{-\frac{S}{6}\left(\frac{8\pi\eta}{m}\right)^{1/2}+\frac{16\pi\rho_{p}}{3}\left[\frac{\eta
X^{2/3}}{\left(\frac{8\pi\eta}{m}\right)^{2/3}+X^{2/3}}+\left(\frac{8\pi\eta}{m}\right)^{2/3}\frac{1+8\pi\eta
X}{24\pi
X^{1/3}\left(\left(\frac{8\pi\eta}{m}\right)^{2/3}+X^{2/3}\right)^{2}}\right]\right\\}\,.$
(47)

During the radiation era ($a\gg 1$) we have a limit:

$G_{2}\simeq-\rho_{p}\left(\frac{m}{8\pi\eta}\right)^{2/3}\cdot X^{2/3}\,,$
$G_{3}\simeq-\varepsilon\sqrt{\frac{\rho_{p}}{3\pi}}\cdot\left(\frac{m}{8\pi\eta}\right)^{1/3}\cdot
X^{-1/6}+const\,.$ (48)

The quasi-de Sitter behavior ($a\ll 1$):

$G_{2}\simeq\left[-8\pi\eta\rho_{p}+S\left(\frac{8\pi\eta}{m}\right)^{1/2}\right]\cdot
X\,,$
$G_{3}\simeq\frac{\varepsilon}{2}\sqrt{\frac{3}{\pi\rho_{p}}}\cdot\left[\frac{16\pi\eta\rho_{p}}{3}-\frac{S}{6}\left(\frac{8\pi\eta}{m}\right)^{1/2}\right]\cdot
X^{1/2}+const\,.$ (49)

Figure 1: Profile of the sound speeds squared $c_{S}^{2}$ for $m=0.01$, $B=1$.
The sound speed squared is positive.  Figure 2: Profile of the function
$Q_{S}$ for $m=0.01$, $B=1$. The function $Q_{S}$ is positive.

For the model under consideration, we rewrite functions $Q_{T}$, $c^{2}_{T}$,
$Q_{S}$ and $c^{2}_{S}$ in the form:

$Q_{T}=\frac{a^{6}+m}{32\pi a^{6}}\,,\,c^{2}_{T}=\frac{a^{6}-m}{a^{6}+m}\,,$
(50)
$Q_{S}=\frac{3B(a^{6}+m)(1+a^{4})^{2}[2(a^{6}+2m)(a^{4}+3)+B(a^{4}+1)^{2}]}{8\pi
a^{6}[2(a^{6}+m)(a^{4}+3)+B(a^{4}+1)^{2}]^{2}}\,,$ (51)
$c^{2}_{S}=\frac{1}{Q_{S}}\left[-\omega_{4}+2H\left(\frac{\omega_{1}^{2}a}{w_{2}}\right)^{\prime}_{a}\right]\,,$
(52)

where

$\omega_{1}=\frac{a^{6}+m}{8\pi a^{6}}\,,\,\omega_{4}=\frac{a^{6}-m}{8\pi
a^{6}}\,,$ (53) $\omega_{2}=\frac{A[2(a^{6}+m)(a^{4}+3)+B(a^{4}+1)^{2}]}{24\pi
a^{6}(1+a^{4})^{3/2}}\,,\,A\equiv\sqrt{\frac{8\pi\rho_{p}}{3}}\,,\,B\equiv\frac{S}{A^{2}}\left(\frac{8\pi
m}{\eta}\right)^{1/2}\,.$ (54)

Conditions (32) for tensor perturbations are satisfied when $a^{6}>|m|$. In
the case $m>0$, the inequality $Q_{T}>0$ holds for any $a>0$. However,
$c^{2}_{T}$ has a negative limit, $c^{2}_{T}\rightarrow-1$, $a\rightarrow 0$.
The function $Q_{S}$, $c^{2}_{S}$ have the following approximations:

$Q_{S}\propto\frac{3Bm(12m+B)}{8\pi(6m+B)^{2}a^{6}}\,,\,a\rightarrow
0\,;\,\,Q_{S}\propto\frac{3B}{8\pi a^{2}}\,,\,\,a\rightarrow+\infty\,;$ (55)
$c^{2}_{S}\rightarrow\frac{(B-24m)(6m+B)}{3B(12m+B)}\,,\,a\rightarrow
0\,;\,\,c^{2}_{S}\propto\frac{8a^{2}}{3B}\,,\,\,a\rightarrow+\infty\,.$ (56)

The profiles of functions $c^{2}_{S}$, $Q_{S}$ are shown in Fig. 1 and Fig. 2.
Conditions (26), (27) for the scalar perturbations are satisfied.

##  IV Bianchi I model

Let us consider an anisotropic generalization of the model (37), (45). The
assumption (45) for the function $X(a)$ gives a corollary to equation (19):

$\dot{\beta}_{\pm}=\frac{8\pi C_{\pm}\cdot
a^{3}}{a^{6}+m}\,,\,\sigma^{2}=\frac{(8\pi)^{2}\sigma^{2}_{0}a^{6}}{(a^{6}+m)^{2}}\,,$
(57)

where $\sigma^{2}_{0}\equiv C_{-}^{2}+C_{+}^{2}$. In the general theory of
relativity, spatial anisotropies produce in the Einstein equations terms
proportional to $1/a^{6}$, which become dominant when one goes backwards in
time. Here, the shear scalar $\sigma^{2}$ is a non-monotonic function with the
properties

$\sigma^{2}\rightarrow 0\,\,\text{as}\,\,0\leftarrow a\rightarrow\infty\,.$
(58)

The shear scalar tend to zero for $a\rightarrow 0$ and their contribution to
the total energy balance is $\propto a^{6}$ instead of $\propto 1/a^{6}$. This
is the non-standard behavior at small $a$. The shear scalar is suppressed at
late time as well by the factor $1/a^{6}$. Taking into account anisotropy, we
generalize the assumption (37):

$H^{2}=\sigma^{2}+\frac{8\pi\rho}{3}\,,$ (59)

where

$\rho=\frac{\rho_{p}}{a^{4}+1}\,.$ (60)

The mean anisotropy parameter $\mathcal{A}$ is defined as

$\mathcal{A}=\frac{1}{3}\sum_{i=1}^{3}\left(\frac{H_{i}-H}{H}\right)^{2}=\frac{2\sigma^{2}}{H^{2}}=$
$=2\left[1+\frac{F(a^{6}+m)^{2}}{a^{6}(a^{4}+1)}\right]^{-1},\,F\equiv\frac{\rho_{p}}{24\pi\sigma^{2}_{0}}\,.$
(61)

This parameter describes the isotropization process of the Universe. The
function $\mathcal{A}$ has a limit: $\mathcal{A}\rightarrow
0\,\,\text{as}\,\,0\leftarrow a\rightarrow\infty$. Therefore, the anisotropy
effect is totally negligible at early and late times. The behavior of the
parameter $\mathcal{A}$ is shown in Fig. 3. As we can see, the selected
function $X(a)$ provides isotropization in later times. Since the anisotropy
is suppressed at early and late times, the behavior of the model (59) is
similar to that of the model (37) at these times:

$H^{2}\approx\frac{8\pi\rho}{3}\rightarrow\frac{8\pi\rho_{p}}{3}=\text{const}\,,\,a\ll
1\,.$ (62) $H^{2}\approx\frac{8\pi\rho}{3}\propto\frac{1}{a^{4}}\,,a\gg 1\,.$
(63)

There is a difference for models in intermediate times. The quasi-de Sitter
stage and the radiation era are separated by the anisotropic phase. The
profile Fig.4 compares the behavior of the Hubble parameter for models (37)
and (59). The scale factors $a$ of the two models are shown in Fig.5.

Assumption (45) allows us to obtain from (17), (18) the functions $G_{2}$ and
$G_{3X}$:

$G_{2}=S\left(\frac{8\pi\eta}{m}\right)^{1/2}X-\frac{\rho_{p}\cdot
X^{2/3}(1+8\pi\eta X)}{\left(\frac{8\pi\eta}{m}\right)^{2/3}+X^{2/3}}\,,$ (64)
$G_{3X}=\frac{1}{\varepsilon\sqrt{2X}}\left(\frac{(8\pi)^{3}\sigma^{2}_{0}\eta
X}{m(1+8\pi\eta
X)^{2}}+\frac{8\pi\rho_{p}}{3}\cdot\frac{X^{2/3}}{\left(\frac{8\pi\eta}{m}\right)^{2/3}+X^{2/3}}\right)^{-1/2}\times$
$\times\left\\{-\frac{S}{6}\left(\frac{8\pi\eta}{m}\right)^{1/2}+\frac{16\pi\rho_{p}}{3}\left[\frac{\eta
X^{2/3}}{\left(\frac{8\pi\eta}{m}\right)^{2/3}+X^{2/3}}+\left(\frac{8\pi\eta}{m}\right)^{2/3}\frac{1+8\pi\eta
X}{24\pi
X^{1/3}\left(\left(\frac{8\pi\eta}{m}\right)^{2/3}+X^{2/3}\right)^{2}}\right]\right\\}\,.$
(65)

Here, approximations (48), (49) are also valid. Stability conditions (26),
(27), (32) have been derived for the homogeneous and isotropic backgrounds.
They can be used also in the Bianchi I model case at early and late times,
since the anisotropies are then damped.

Figure 3: Profile of the parameter $\mathcal{A}$ for $m=0.01$, $F=1$.  Figure
4: Profiles of the parameter $\sqrt{\frac{3}{8\pi\rho_{p}}}\cdot H$ for
isotropic model (red) and anisotropic model (blue).  Figure 5: Scalar factor
profiles for isotropic model (red) and anisotropic model (blue) with the
initial values $a(0)=0.1$.

##  V Conclusion

We have proposed a way to reconstruct the HG theory on the Hubble evolution
and on the canonical kinetic term $X$ in the Bianchi I spacetime model. The
experience of many researchers makes it possible to set a scenario for the
evolution of the Universe a priori. The freedom of choice $X(a)$ is used to
regulate the stability of the model and the anisotropic properties of the
Universe.

We have reconstructed the scalar theory, which gives the model of post-
inflationary transition to the radiation-dominated phase in the Bianchi I
spacetime. As a basis for the reconstruction, we took a fluid with the
nonlinear equation of state. At the Universe stages with suppressed
anisotropy, this equation has the form $p=\rho/3-\mu\rho^{2}$. The term
$-\mu\rho^{2}$ may be due to Bose-Einstein condensates with repulsive
($\mu<0$) or attractive ($\mu>0$) self-interaction. The kinetic dependence
$X(a)\sim a^{-6}$ gives the following properties for the model. It is
important that in the process of expansion, the Universe is isotropized.
Interestingly, the anisotropy is also suppressed in the early times of the
Universe. At intermediate times, there is a finite burst of anisotropy with
further relaxation. The isotropic analogue of the model does not contain
ghosts and Laplacian instabilities for scalar perturbations. For the tensor
perturbations, on the interval $(0,a_{1})$, the square of the sound speed
$c^{2}_{T}$ is negative. For the anisotropic model, these judgments are valid
at stages of the Universe with suppressed anisotropy.

###### Acknowledgements.

This work is supported by the Russian Foundation for Basic Research (Grant No.
19-52-15008).

## References

  * (1) G.W. Horndeski: Int. J. Theor. Phys. 10, 363 (1974)
  * (2) S.V. Sushkov: Phys. Rev. D 80, 103505 (2009)
  * (3) S.A. Appleby, A. De Felice and E.V. Linder: JCAP 10 060 (2012)
  * (4) T. P. Sotiriou and S.-Y. Zhou: Phys. Rev. D 90 124063 (2014)
  * (5) E. Babichev, C. Charmousis and A. Lehebel: Classical and Quantum Gravity 33 154002 (2016)
  * (6) A. Maselli, H. O. Silva, M. Minamitsuji and E. Berti: Phys. Rev. D 92, 104049 (2015)
  * (7) R. K. Muharlyamov, T. N. Pankratyeva, Eur. Phys. J. Plus 136, 590 (2021)
  * (8) T. Kobayashi, M. Yamaguchi and J. Yokoyama: Prog. Theor. Phys. 126, 511 (2011)
  * (9) A. De Felice and S. Tsujikawa: JCAP 1202, 007 (2012)
  * (10) S. A. Appleby and E. V. Linder: JCAP 1203, 043 (2012)
  * (11) H. W. H. Tahara, S. Nishi, T. Kobayashi and J. Yokoyama: JCAP 07 (2018) 058, arXiv:1805.00186
  * (12) A. A. Starobinsky, S. V. Sushkov, M. S. Volkov, Phys. Rev. D 101, 064039 (2020)
  * (13) R. Galeev, R. K. Muharlyamov, A. A. Starobinsky, S. V. Sushkov, M. S. Volkov, Phys. Rev. D 103, 104015 (2021)
  * (14) Hawking S.W. and Taylor R.J., Nature. 299, 1278 (1966)
  * (15) Hu B.L. and Parker L., Physical Review. D. 17 933 (1978)
  * (16) R. C. Bernardo and I. Vega: JCAP 1910, 058 (2019)
  * (17) R. C. Bernardo and J. L. Said: JCAP09 (2021) 014, arXiv:2105.12970
  * (18) R. C. Bernardo, D. Grandon, J. L. Said and V. H. Cardenas: arXiv:2111.08289
  * (19) P.H. Chavanis: Eur. Phys. J. Plus, 129, 38 (2014)