Vol.0 (20xx) No.0, 000–000

11institutetext: Yunnan Observatories, Chinese Academy of Sciences, Kunming
650011, China<EMAIL_ADDRESS>
22institutetext: Key Laboratory for the Structure and Evolution of Celestial
Objects, Chinese Academy of Sciences, Kunming 650011, China
33institutetext: Department of Astronomy, Xiamen University, Xiamen, Fujian
361005, China<EMAIL_ADDRESS>
44institutetext: Qian Xuesen Laboratory of Space Technology, China Academy of
Space Technology, 104 Youyi Road, Haidian District, Beijing, 100094, China
55institutetext: Department of Astronomy, China West Normal University, 1
Shida Road, Shunqing District, Nanchong 637002, China

# A correction method for the telluric absorptions and application to Lijiang
Observatory

Kai-Xing Lu 1122 Zhi-Xiang Zhang 33 Ying-Ke Huang 44 An-Bing Ren 55 Liang Xu
1122 Hai-Cheng Feng 1122 Yu-Xin Xin 1122 Xu Ding 1122 Xiao-Guang Yu 1122 Jin-
Ming Bai 1122

###### Abstract

Observing a telluric standard star for correcting the telluric absorption
lines of spectrum will take a significant amount of precious telescope time,
especially in the long-term spectral monitoring project. Beyond that, it’s
difficult to select a suitable telluric standard star near in both time and
airmass to the scientific object. In this paper, we present a method of
correcting the telluric absorption lines by combining the advantages of long
slit spectroscopy. By rotating the slit, we observed the scientific object and
a nearby comparison star in one exposure, so that the spectra of both objects
should have the same telluric transmission spectrum. The telluric transmission
spectrum was constructed by dividing the observed spectrum of comparison star
by its stellar template, and was used to correct the telluric absorption lines
of the scientific object. Using the long slit spectrograph of Lijiang
2.4-meter telescope, we designed a long-term spectroscopic observation
strategy, and finished a four-year spectroscopic monitoring for a pair of
objects (an active galactic nuclei and an non-varying comparison star). We
applied this method to correct the telluric absorption lines of the long-term
monitored spectra by Lijiang 2.4-meter telescope, and investigated the
variation of the telluric absorptions at Lijiang Observatory. We found that
the telluric absorption transparency is mainly modulated by the seasonal
variability of the relative humidity, airmass and seeing. Using the scatter of
the [O III] $\lambda$5007 fluxes emitted from the narrow-line region of active
galactic nuclei as an indicator, we found that the correction accuracy of the
telluric absorption lines is 1%.

###### keywords:

techniques: telluric absorption, methods: data analysis, techniques:
spectroscopic, quasars: individual (SDSS J1536+0441)

##  1 Introduction

Ground-based optical and near-infrared spectra are contaminated by the
variable absorptions of the telluric atmosphere. In order to retrieve the
intrinsic spectral signal of the scientific object, the traditional method is
to observe a telluric standard star (early-type star) near in both time and
airmass to the scientific object, then using the telluric spectrum created
from standard star to correct the telluric absorption features by dividing the
object’s spectrum during the data reduction. Often, a rapidly rotating hot
star (spectral type A or late B) is frequently adopted as the telluric
standard because there are few and weak metal lines in the spectrum around the
telluric absorption bands.

In addition, many sophisticated methods have been developed. For example,
using G2 V star as telluric standards and the high-resolution spectrum of the
sun, Maiolino et al. (1996) developed a correction method for the telluric
absorption lines. Combining Maiolino et al. (1996)’s method with above
traditional early-type star method, Hanson et al. (1996, 2005) developed a
method to correct the telluric absorption lines, which has to observe an
A-star and a G2 V star and involves a somewhat tedious and complicated
reduction process. Vacca et al. (2003) described a new method of generating a
telluric correction spectrum by using observations of A0 V stars and a high-
resolution model spectrum of Vega, which is similar to the process described
by Maiolino et al. (1996).

Another approach is to generate a theoretical telluric absorption spectrum of
the atmospheric transmission from a line database and the observing condition.
Based on the available radiative transfer model, this approach has been
successfully developed by several groups (e.g., Seifahrt et al. 2010;
Gullikson et al. 2014; Bertaux et al. 2014; Cotton et al. 2014). Furthermore,
some groups incorporated the radiative transfer model into they own code to
fit an entire input spectrum (e.g., Husser et al. 2013; Gullikson et al. 2014;
Smette et al. 2015).

Actually, above methods have several disadvantages(e.g., Seifahrt et al. 2010;
Gullikson et al. 2014): (1) it’s difficult to find a suitable telluric
standard star near in both time and airmass to the scientific object during
the observation. (2) The strength of telluric absorption lines may
significantly changes with the varying observing conditions (such as the
weather, airmass and seeing), even if we find an ideal telluric standard star,
it is almost impossible to obtain the spectrum of scientific object and
telluric standard star under the same observing conditions. (3) if we conduct
a long-term spectral monitoring study, observing a telluric standard star near
the same time to the object, we will take a significant amount of precious
telescope time. (4) The deviation of the modelled telluric absorption spectrum
is mainly due to the limited accuracy of the meteorological data and errors in
the radiative transfer model. Therefore, in this paper, we described a new
method of correcting the telluric absorption lines from the spectrum, and used
it to investigate the variation of telluric absorptions of Lijiang
Observatory.

The paper is organised as follows. In Section 2, we described the correction
method for the telluric absorption lines of spectrum. We applied this method
to the spectra observed by Lijiang 2.4-meter telescope in Section 3, including
observation, data reduction, the telluric absorption correction, accuracy
evaluation, and investigated the telluric absorption features of Lijiang
Observatory. Section 4 and 5 are discussion and summary, respectively.

##  2 Method

###  2.1 Long slit spectroscopy

The spectrograph used in astronomy often equipped with long slit (with
different projected slit width), so that we can orient long slit to take the
spectra of the scientific object and a nearby non-varying comparison star
simultaneously. This observation method is widely used in the reverberation
mapping of active galactic nuclei to measure supermassive black hole mass and
investigate the physics of the broad-line region (e.g., Maoz et al. 1990;
Kaspi et al. 2000; Lu et al. 2019), because the spectrum of the non-varying
comparison star can be used as the fiducial spectrum to calibrate the flux of
scientific object in high precision (e.g., see Lu et al. 2019). The second
application of the comparison star spectrum is to correct the line broadening
caused by seeing and instruments (Du et al. 2016). In the application of
correcting instrument broadening, the line broadening function can be obtained
from the fitting between the comparison star spectrum and its stellar
template, then we use the broadening function to correct the broadening of the
emission-line profile through Richardson$-$Lucy deconvolution algorithm
(Richardson 1972; Lucy 1974). Here we provide a new application of the
comparison star spectrum.

###  2.2 Fitting spectrum and correcting telluric absorptions

In the scheme of long slit spectroscopy, the spectra of the scientific object
and comparison star are obtained simultaneously. Extraction processes of the
observed object and stellar spectra are similar to the works of Lu et al.
(2016a) and Lu et al. (2019) (more details refer to Section 3). The schematic
diagram of correcting the telluric absorption lines shows in Figure 1. We
first obtain the telluric transmission spectrum by fitting the spectrum of
comparison star by a stellar template. Generally, there are two approaches to
select the stellar template of comparison star: (1) selecting the most similar
stellar spectrum from the library of synthesis stellar model as the template
of comparison star (e.g., Husser et al. 2013). (2) searching the calibrated
spectra of comparison star from spectral survey database (e.g., SDSS, LAMOST)
as the template of comparison star111https://github.com/BU-
hammerTeam/PyHammer. To fit the observed spectrum of comparison star, the
selected stellar template ${\mathcal{M}}(\lambda)$ is shifted, broadened and
scaled. That is the observed spectrum ${\mathcal{S}}(\lambda)$ of comparison
star is modelled by function

${\mathcal{S}^{\prime}}(\lambda)={\mathcal{P}}(\lambda)[{\mathcal{M}}(\lambda)\otimes{\mathcal{G}}({\rm
Shift,FWHM})].$ (1)

Where, ${\mathcal{P}}$ is a Legendre polynomial that accounts for the mismatch
in the continuum shape of comparison star and stellar template,
${\mathcal{G}}$ is a Gaussian kernel with parameters of velocity and shift
that accounts for the mismatch in the line width and shift of the intrinsic
absorption lines.

It’s should be noted that the telluric absorptions from the red side of the
optical spectrum to the near infrared are mainly caused by $\rm O_{2}$ and
$\rm H_{2}O$. The main telluric absorption bands include narrow oxygen bands
(identified as the Fraunhofer $\gamma$, B, and A bands) and some comparably
weak water vapour features. The telluric absorption bands of optical region
with bandwidths (i.e., absorption windows) are presented in Table 1 (see also
Smette et al. 2015). During the fitting of the observed spectrum, the spectral
region within the telluric absorption windows (see Table 1) are masked. Then
the telluric transmission spectrum ${\mathcal{T}}(\lambda)$ in the selected
absorption windows is constructed by dividing the observed spectrum of
comparison star ${\mathcal{S}}(\lambda)$ by the modelled spectrum
${\mathcal{S}^{\prime}}(\lambda)$. Lastly, we obtained the telluric absorption
corrected spectrum by dividing the observed object spectrum
${\mathcal{O}}(\lambda)$ by the telluric transmission spectrum
${\mathcal{T}}(\lambda)$, because the spectra of comparison star and the
scientific object are obtained at the same time and with the same airmass and
seeing. In fact, this step resembles the process of IRAF telluric. Next, we
use this method to correct the telluric absorption lines of the spectra
observed by Lijiang 2.4-meter telescope and investigate the variation of the
telluric absorptions at Lijiang Observatory.

Figure 1: Schematic diagram of correction for the telluric absorption lines. Table 1: Telluric absorption bands and selected windows in the optical region. “${\rm B*}$” means the number of the telluric absorption bands (Col. 1), these absorption bands with different absorption widths (Col. 3) are caused by different absorbers (Col. 2). No | Molecular | Windows
---|---|---
(1) | (2) | (3)
B1 | $\rm O_{2}~{}(\gamma~{}band)$ | 6270 Å$\sim$6350 Å
B2 | $\rm O_{2}~{}(B~{}band)+H_{2}O$ | 6830 Å$\sim$7050 Å
B3 | $\rm H_{2}O$ | 7143 Å$\sim$7369 Å
B4 | $\rm O_{2}~{}(A~{}band)$ | 7550 Å$\sim$7750 Å
B5 | $\rm H_{2}O$ | 8080 Å$\sim$8380 Å
Figure 2: The spectrum of SDSS J153636.22+044127.0 (J1536) observed by Lijiang
2.4-meter telescope. Five telluric absorption bands in the observed spectrum
are marked by colour diagram (see Table 1).

##  3 Application

The Lijiang 2.4-meter Telescope (LJT) is located at Lijiang Observatory (E:
$100^{\circ}01^{\prime}48^{\prime\prime}$,
N:$26^{\circ}41^{\prime}42^{\prime\prime}$, altitude: 3193 meters), and is
administered by Yunnan Observatories of the Chinese Academy of Sciences. LJT
observation time is opening to the worldwide astronomical community since
2008. The structure of the 2.4-meter telescope including optical, mechanical,
motion and control systems along with its instruments are described by Fan et
al. (2015) and Wang et al. (2019). Basic observing conditions (including
seeing, wind speed, weather and so on) are presented in recent paper of Xin et
al. (2020). The Yunnan Faint Object Spectrograph and Camera (YFOSC) mounted on
the 2.4-meter telescope is a versatile instrument for spectroscopy and
photometry. As a versatile instrument, YFOSC is equipped with a series of
Grisms, long slits and filters. The wavelength coverage of these Grisms is
about 3600 Å$\sim$9000 Å, the red side of spectrum observed by YFOSC is
covered by some telluric absorption bands. Correction of the telluric
absorption lines is an essential step to extract the spectral signals.

###  3.1 Observation strategy and spectral monitoring

To estimate the correction accuracy of the telluric absorption lines using
above described method, and to investigate the variation of the telluric
absorptions at Lijiang observatory, we designed a strategy of long slit
spectroscopy and developed a long-term spectroscopic monitoring project using
Lijiang 2.4-meter telescope. In the observation of active galactic nuclei, it
is generally believed that the fluxes of forbidden line [O III] $\lambda$5007
emitted from the narrow-line region are stable in the timescale of years
(e.g., Foltz et al. 1981; Lu et al. 2016a, 2019). If the wavelength of [O III]
$\lambda$5007 emission line locate in the telluric absorption band in the
observed frame, the apparent variation of the [O III] $\lambda$5007 fluxes
will be an indicator of the telluric absorptions and the scatter of [O III]
$\lambda$5007 fluxes after correcting the telluric absorptions can also be
used to estimate the correction accuracy. Therefore, we orient the long slit
to take the spectra of the object SDSS J153636.22+044127.0 (J1536) and an non-
varying comparison star simultaneously, where J1536 is an active galactic
nuclei (AGN) with redshift 0.3889, its optical spectrum observed by Lijiang
2.4-meter telescope showed in Figure 2. There are five telluric absorption
bands in the observed spectra, which are listed in Table 1 and marked in
Figure 2. In the observed frame, the [O III] $\lambda$5007 emission line of
J1536 is in the B2 absorption band (see Figure 2 and Table 1), we can measure
the scatter of the [O III] $\lambda$5007 fluxes before and after correcting
the telluric absorption lines and use to estimate the correction accuracy of
the telluric absorption lines. On the other hand, J1536 is a candidate of
supermassive black hole binary (SMBHB, e.g., Boroson & Lauer 2009; Lu et al.
2016b; Zhang et al. 2019), except for investigating the variation of the
telluric absorptions, we will use these long-term monitored spectra to
investigate the broad-line region and optical variability physics of SMBHB.
The later will be presented in a separation paper.

We have finished a four-year spectroscopic monitoring for a pair of objects
(J1536 $vs.$ its comparison star). The observation started from 2017 and
continued to 2020, and the annual spectral sampling lasts from January to
September. All spectra of J1536 and comparison star are observed using Grism 8
mounted on the terminal of YFOSC, which provides a resolution of 78 Å mm-1
(1.5 Å pixel-1) and covers the wavelength range 5000 Å$\sim$9000 Å, which
fully guaranteed the study of long-term variation of the telluric absorptions
at Lijiang Observatory.

Figure 3: The top panel is an example of fitting observed spectrum
${\mathcal{C}}(\lambda)$ by the stellar template, the observed spectrum of
comparison star in blue, and scaled stellar template
${\mathcal{C}^{\prime}}(\lambda)$ in red. The bottom panel is the residual of
fitting regions.

###  3.2 Data reduction

Following the standard process included bias subtraction, flat-field
correction, wavelength calibration and spectrum extraction, we reduced the
two-dimensional spectroscopic data using the standard IRAF(v2.16) package. All
spectra were extracted using a uniform aperture of 20 pixels (5.7′′), and
background was determined from two adjacent regions
($+7.4^{\prime\prime}\sim+14^{\prime\prime}$ and
$-7.4^{\prime\prime}\sim-14^{\prime\prime}$) on both sides of the aperture
region. Spectral fluxes of the scientific object were calibrated by the
comparison stars in two steps. (1) We produced a fiducial spectrum of the
comparison star using the data from nights with photometric conditions. (2)
For each object/comparison star pair, we obtained a wavelength-dependent
sensitivity function by comparing the star’s spectrum to the fiducial
spectrum. Then this sensitivity function was applied to calibrate the observed
spectrum of the scientific object (also see Lu et al. 2016a, 2019).

###  3.3 Correction of the telluric absorptions

In practice, we would better correct the telluric absorption at first, and
then calibrate spectral fluxes. In this case, we do not need to mask the
telluric absorption bands again during the spectral flux calibration.
Following the method described in Section 2, we first use the Equation 1 to
model the observed spectra ${\mathcal{S}}(\lambda)$ of comparison star (in
counts). Because the wavelength coverage of all observed spectra is about 5000
Å$\sim$9000 Å, these significant absorption bands listed in Table 1 are
included, and they are masked in modelling process (see Figure 3). Then we
obtained the telluric transmission spectrum by dividing the observed spectrum
${\mathcal{S}}(\lambda)$ of comparison star by the modelled spectrum
${\mathcal{S}^{\prime}}(\lambda)$. In order to avoid introducing new errors to
the corrected spectrum of the scientific object, we set the transmission of
the wavelength regions outside the telluric absorption windows to 1 before
correcting the telluric absorption lines of the observed spectrum. This is
reasonable because $\rm O_{2}$ and $\rm H_{2}O$ molecular are fully
transparent for the photons from these wavelength regions (see Smette et al.
2015). All transmission spectra (in the telluric absorption windows)
calculated from above process were plotted annually in Figure 4, we zoom in
the B3 absorption bands and over-plotted in Figure 4. The figure clearly shows
that the telluric absorption depth presents significant variation. We first
used the telluric transmission spectra to correct the telluric absorption
lines of the observed spectra, then we obtained flux calibrated spectra using
the steps described in Section 3.2. Figure 5 is an example for comparing the
corrected spectra of telluric absorptions and uncorrected spectra, where the
top panel is the spectra of comparison star, and the bottom panel is the
spectra of J1536.

Figure 4: The transmission spectra of Lijiang Observatory obtained in Section
3.3 provide values from 0 (totally opaque) to 1 (fully transparency). The year
of observation was noted in the title of panel, a transmission spectrum
corresponds to one observation. Figure 5: Comparison of the telluric
absorption corrected spectra (see Section 3.3) and uncorrected spectra in the
observed frame. Only flux calibrated spectra show in blue, the telluric
absorption corrected and flux calibrated spectra show in red. The units of
$F_{\lambda}$ is erg s-1 cm-2 Å-1. The top panel shows the spectra of
comparison star, and the bottom panel shows the spectra of J1536.

###  3.4 Correction accuracy of the telluric absorptions

As stressed in Section 3.1, we selected a special AGN of J1536 with redshift
0.3889 to put its [O III] $\lambda$5007 emission line drop into B2 telluric
absorption bands, and developed a long-term spectroscopic monitoring, so that
we can use the scatter of the [O III] $\lambda$5007 fluxes to estimate the
correction accuracy of the telluric absorption lines. Strictly speaking, the
[O III] $\lambda$5007 emission line of J1536 drops into the red side of B2
telluric absorption bands that dominated by H2O molecular absorption.

In practice, the fluxes of forbidden line [O III] $\lambda$5007 emitted from
the narrow-line region of active galactic nuclei is a constant over
reverberation timescales ($\sim$ years), therefore it can be used as an
internal flux calibration standard (e.g, Foltz et al. 1981; Peterson et al.
1982, 2002). Based on reverberation mapping of active galactic nuclei
undertook by Lijiang 2.4-meter telescope, Lu et al. (2016a) found that the [O
III] $\lambda$5007 fluxes of NGC 5548 measured from flux calibrated spectra
show a small fluctuation at a level of 2%, which is regarded as the accuracy
of the flux calibration. While, Lu et al. (2019) recently found that the [O
III] $\lambda$5007 fluxes of Mrk 79 give rise to the apparent variation at a
level of 5% over reverberation timescales. Lu et al. (2019) further found
that, if the [O III] $\lambda$5007 emission region is an extended source, the
varying observing conditions (e.g., varying seeing) would cause extra
dispersion of the [O III] $\lambda$5007 fluxes. It should be noted that the [O
III] $\lambda$5007 emission region of NGC 5548 approximates to point source,
and the [O III] $\lambda$5007 emission region of Mrk 79 is an extended source.

To directly understand the impacts of the telluric absorptions on the
scientific spectra, we measured the [O III] $\lambda$5007 fluxes of J1536 from
the flux calibrated spectra. We used two continuum bands (4710$-$4740 Å and
5080$-$5110 Å in the rest frame) to set the continuum underneath the [O III]
$\lambda$5007 and H$\beta$ emission lines, then we integrated the continuum-
subtracted [O III] $\lambda$5007 fluxes between 4980 Å and 5025 Å, and
presented in the top panel of Figure 6. The figure shows the [O III]
$\lambda$5007 fluxes before correcting the telluric absorption lines
demonstrate apparent period variation with a scatter of 7.8%. This scatter is
larger than previous observation record (such as 5% of Mrk 79 and 2% of NGC
5548). The apparent period variation follows the seasonal variability of the
relative humidity of Lijiang Observatory (see Figure 22 of Xin et al. 2020).
Qualitatively, January is the deep winter of Lijiang Observatory, the
atmospheric relative humidity is the lowest, then begins to increase slowly,
and reaches the maximum in the rainy season (about from July to September ).
Therefore, accurate calibration of the telluric absorption lines from observed
spectrum is crucial for obtaining credible scientific data.

As a comparison, we also measured the [O III] $\lambda$5007 fluxes of J1536
from the flux calibrated and the telluric absorption corrected spectra. The
measurement results were plotted in the bottom panel of Figure 6 for comparing
with the measurements before correcting the telluric absorption lines, which
shows that the scatter of the [O III] $\lambda$5007 fluxes is 2.3%, and the
apparent period variation presented in the top panel is disappeared. As
addressed above, the fluxes of [O III] $\lambda$5007 are stable enough in the
timescale of years if the [O III] $\lambda$5007 emission region is compact. In
practice, the scatter of [O III] $\lambda$5007 fluxes (2.3%) is caused by the
accuracy of flux calibration and the accuracy of the telluric absorption
correction. If we assume that the flux calibration accuracy of J1536 is the
same with our previous experiment of NGC 5548 (i.e., 2%, Lu et al. 2016a),
this means that the correction accuracy of the telluric absorption lines using
the method described in Section 2 is 1%. In addition, in order to intuitively
inspect the differences before and after correcting the telluric absorptions,
we plotted the telluric absorption corrected and uncorrected [O III] profiles
selected randomly from observations of 2017 in Figure 7. The figure also
demonstrates the differences of the telluric absorption from the different
observations.

Figure 6: The [O III] $\lambda$5007 fluxes of J1536 in units of erg s-1 cm-2.
The top panel shows the apparent variation of the [O III] $\lambda$5007 fluxes
measured from the flux calibrated spectrum (before correcting the telluric
absorption lines), periodicity is caused by the seasonal variability of the
relative humidity. The bottom panel shows the scatter of the [O III]
$\lambda$5007 fluxes measured from the flux calibrated and the telluric
absorption corrected spectra. The year of observation was noted in the title
of panel. Figure 7: The telluric absorption corrected and uncorrected [O III]
profiles from the observation of 2017 (in the observed frame), observation
dates are noted, the units of $F_{\lambda}$ is erg s-1 cm-2 Å-1. The
differences of [O III] profiles before and after correcting the telluric
absorptions are presented clearly. Figure 8: The integral transparency
($\mathscr{T}_{\rm B1}$, $\mathscr{T}_{\rm B2}$, $\mathscr{T}_{\rm B3}$,
$\mathscr{T}_{\rm B4}$ and $\mathscr{T}_{\rm B5}$) of the telluric absorption
band (B1, B2, B3, B4 and B5, see Table 1) as a function of time, respectively.
The observation started from 2017 and continued to 2020, and the annual
spectral sampling almost lasts from January to September. The year of
observation was noted in the title of panel. Figure 9: Relationship between
the integral transparency and airmass. The results of correlation analysis are
listed in Table 2.

###  3.5 Telluric absorption features of Lijiang Observatory

In order to understand the variation of the telluric absorptions caused by the
molecular of $\rm O_{2}$, $\rm H_{2}O$ or both together at Lijiang
Observatory, we calculated the integral transparency (denoted by
$\mathscr{T}_{\rm B*}$) of each absorption band using formula of
$\int_{\lambda 1}^{\lambda 2}{\mathcal{T}}(\lambda)\,d\lambda/(\lambda
2-\lambda 1)$ (where $\lambda 1$ and $\lambda 2$ are the edges of the telluric
absorption bands, ${\mathcal{T}}(\lambda)$ is the telluric transmission
spectrum), and plotted them as the functions of time in Figure 8. Where “${\rm
B*}$” means the number of the telluric absorption band, which corresponds to
the number of Table 1. Inspecting the figure, we can see that the variation of
the integral transparency for the telluric absorption bands mainly or fully
caused by the molecular of $\rm H_{2}O$ (i.e., B2, B3 and B5 absorption bands)
is similar with the apparent period variation of the [O III] $\lambda$5007
fluxes that contaminated by the $\rm H_{2}O$ absorption (the top panel of
Figure 6). The seasonal variability of the $\rm H_{2}O$ absorption bands is
clearly visible, which is consistent with the seasonal variability of the
relative humidity (see also Figure 22 of Xin et al. 2020). While, the integral
transparency for the absorption bands caused by the molecular of $\rm O_{2}$
(i.e., B1 and B4 absorption bands) do not present obvious variation trend in
the time domain. In the varing observing conditions, the averaged integral
transparency ($\widehat{\mathscr{T}_{\rm B1}}$, $\widehat{\mathscr{T}_{\rm
B2}}$, $\widehat{\mathscr{T}_{\rm B3}}$, $\widehat{\mathscr{T}_{\rm B4}}$ and
$\widehat{\mathscr{T}_{\rm B5}}$) of the telluric absorption bands (B1, B2,
B3, B4 and B5) is (0.991, 0.961, 0.926, 0.847 and 0.964) with scatter (i.e.,
standard deviation) of (0.3%, 1.1%, 3.9%, 1.1% and 2.4%), this means that
about (1%, 4%, 7%, 15% and 4%) of photons propagated from the distant objects
are absorbed by the $\rm O_{2}$ or $\rm H_{2}O$ in these absorption windows,
respectively. We also listed the minimum and maximum of the integral
transparencies in Table 2.

We calculated the airmass in the middle of exposure time, and investigated the
relation between the integral transparency and airmass. The integral
transparency as a function of airmass was presented in Figure 9. We calculated
the Spearman correlation coefficient between the integral transparency and
airmass and listed in Table 2. These analyses show that the integral
transparency of the $\rm O_{2}$ absorption band is well correlated with
airmass, and the integral transparency of the $\rm H_{2}O$ absorption bands
also be effected by airmass. In addition, we measured the width ($\rm
FWHM_{star}$) of star’s flux distribution from the short exposure image
observed before and near the spectroscopy, and showed the relation between the
integral transparency and the $\rm FWHM_{star}$ in Figure 10. We also
calculated the Spearman correlation coefficient between the integral
transparency and the $\rm FWHM_{star}$ and listed in Table 2. In practice, the
variation of $\rm FWHM_{star}$ is mainly modulated by varying seeing, where
seeing is always changing. These analyses show that the integral transparency
of the $\rm H_{2}O$ absorption bands correlates with the variation of $\rm
FWHM_{star}$.

Table 2: Statistics of the telluric absorption features. Col (1) is the integral transparency of different absorption bands. Col (2) is the absorber. The minimum, maximum and averaged value of the integral transparencies for the different absorption bands are listed in Col (3), (4) and (5), respectively. Col (6) is the correlation analysis results ($r$, $p$) between the integral transparency and airmass. Col (7) is the correlation analysis results ($r$, $p$) between the integral transparency and the width of star’s flux distribution. $\mathscr{T}_{\rm B*}$ | Molecular | Min($\mathscr{T}_{\rm B*}$) | Max($\mathscr{T}_{\rm B*}$) | $\widehat{\mathscr{T}_{\rm B*}}$ | Airmass | $\rm FWHM_{star}$
---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7)
$\mathscr{T}_{\rm B1}$ | $\rm O_{2}~{}(\gamma~{}band)$ | 0.985 | 0.996 | 0.991$\pm$0.003 | (-0.25, 0.07) | (-0.07, 0.58)
$\mathscr{T}_{\rm B2}$ | $\rm O_{2}~{}(B~{}band)+H_{2}O$ | 0.934 | 0.976 | 0.961$\pm$0.011 | (-0.52, 3.91$\times 10^{-5}$) | (0.53, 1.85$\times 10^{-5}$)
$\mathscr{T}_{\rm B3}$ | $\rm H_{2}O$ | 0.846 | 0.979 | 0.926$\pm$0.039 | (-0.41, 1.52$\times 10^{-3}$) | (0.60, 8.36$\times 10^{-7}$)
$\mathscr{T}_{\rm B4}$ | $\rm O_{2}~{}(A~{}band)$ | 0.811 | 0.863 | 0.847$\pm$0.011 | (-0.90, 9.56$\times 10^{-21}$) | (0.28, 0.03)
$\mathscr{T}_{\rm B5}$ | $\rm H_{2}O$ | 0.917 | 0.996 | 0.964$\pm$0.024 | (-0.43, 7.71$\times 10^{-4}$) | (0.59, 1.22$\times 10^{-6}$)
Figure 10: Relationship between the integral transparency and the width of
star’s flux distribution ($\rm FWHM_{star}$, measured from the short exposure
image observed before and near the spectroscopy). The results of correlation
analysis are listed in Table 2.

##  4 Discussion

In the ground-based optical spectrum, the telluric absorption bands are caused
by two kinds of absorbers: molecular oxygen ($\rm O_{2}$) and water vapour
($\rm H_{2}O$). Based on the long slit spectroscopy, we provide a accurate
correction method for the telluric absorption lines using the observed
comparison star spectrum and stellar spectral library, the correction accuracy
is 1%. Our correction method for the telluric absorption lines has many
advantages over the traditional method (i.e., observing the telluric standard
star), especially in the study of long-term spectral monitoring. For example,
(1) the scientific object and its comparison star are observed simultaneously
in the long slit with the same observing conditions, such as the same airmass
and seeing. Therefore, It is possible that the correction accuracy of the
telluric absorption lines using this method (1%) should be better than the
traditional method; (2) in the long-term spectral monitoring project, we do
not need to take expensive telescope time to observe the telluric standard
star in each sampling. Therefore, in the research of one-epoch spectral
observation or long-term spectroscopic monitoring, this work provides a kind
of worthy observation strategy and corresponding method to correct the
telluric absorptions.

As described in Section 2, the telluric transmission spectrum is obtained from
the process of fitting the observed spectrum of comparison star by stellar
template through masking the discrete absorption windows. The discrete
absorption windows mean we can select the continuum windows in the spectral
fitting process. In this work, we only apply this method described in Section
2 to the spectrum that its wavelength covers less than 9000 Å, which is
limited by the spectrograph of YFOSC. Actually, because the wavelength ranges
from 9000 Å to around 17000 Å exist the available continuum windows (see the
synthetic telluric absorption spectrum calculated by Smette et al. (2015)
using the Line by Line Radiative Transfer Model, the telluric absorption bands
are also discrete), the application of this method can be extrapolated from
optical region to 17000 Å.

The telluric transparency of B4 absorption band caused by oxygen has the
largest value, and has the strongest correlation with airmass but it has no
correlation with the seeing. While, the telluric transparency of B1 absorption
band also caused by oxygen has the smallest value, which does not correlate
with airmass and seeing (see Section 3.5). However, not only the airmass but
also seeing correlates with the telluric transparency of water vapour
absorption (e.g., B3 and B5 bands). These results show that (1) the
absorptions of oxygen and water vapour in the optical spectrum closely
correlate with atmospheric depth, and (2) the oxygen concentration of
observation site does not change with the varying seeing while the amount of
water changes with the varying seeing. In addition, analysis results of
Section 3.5 show that the red side of optical spectrum including emission
lines and continuum would be affected by the varying telluric absorptions,
accurate correction of the telluric absorption lines should benefit from
spectral flux calibration.

##  5 Summary

In this paper, combining the advantages of long slit spectroscopy, we
described a method that can be use to accurately correct the telluric
absorption lines of spectrum. We constructed the telluric transmission
spectrum by fitting the observed spectrum of comparison star by the stellar
template, then obtained the telluric absorption corrected spectrum by dividing
the observed spectrum by the telluric transmission spectrum. We applied this
method to correct the telluric absorption lines of the long-term monitored
spectra by Lijiang 2.4-meter telescope. Using the scatter of the [O III]
$\lambda$5007 fluxes emitted from the narrow-line region of active galactic
nuclei (SDSS J153636.22+044127.0) as an indicator, we found that the
correction accuracy of this method for the telluric absorption lines is 1%. We
investigated the telluric absorption features of Lijiang Observatory, and
found that the telluric absorption transparency correlates with the relative
humidity, airmass and seeing. This work provides a kind of worthy observation
strategy and corresponding method to correct the telluric absorptions.

###### Acknowledgements.

We are grateful to the referee for useful suggestions that improved the
manuscript. This work is supported by the National Natural Science Foundation
of China (NSFC; 11991051, 12073068, 11703077, and 11803087). K.X.L.
acknowledges financial support from from the Yunnan Province Foundation
(202001AT070069), and from the Light of West China Program provided by Chinese
Academy of Sciences (Y7XB016001). L.X. acknowledges financial support from the
Light of West China Program provided by Chinese Academy of Sciences
(Y8XB018001). We acknowledge the support of the staff of the Lijiang 2.4 m
telescope. Funding for the telescope has been provided by the CAS and the
People’s Government of Yunnan Province.

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# Dipole Cosmology in $f\left(Q\right)$-gravity

Andronikos Paliathanasis<EMAIL_ADDRESS>Institute of Systems Science,
Durban University of Technology, Durban 4000, South Africa  Departamento de
Matemáticas, Universidad Católica del Norte, Avda. Angamos 0610, Casilla 1280
Antofagasta, Chile

###### Abstract

Symmetric teleparallel $f(Q)$-gravity allows for the presence of a perfect
fluid with a tilted velocity in the Kantowski-Sachs geometry. In this dipole
model, we consider an ideal gas and we investigate the evolution of the
physical parameters. The tilt parameter is constrained by the nonlinear
function $f(Q)$ through the non-diagonal equations of the field equations. We
find that the dynamics always reduce to the vacuum solutions of STEGR. This
includes the Kasner universe, when no cosmological term is introduced by the
$f(Q)$ function, and the isotropic de Sitter universe, where $f\left(Q\right)$
plays the role of the cosmological constant. In the extreme tilt limit, the
universe is consistently anisotropic and accelerated. However, the final
solution matches that of STEGR.

Symmetric teleparallel; non-metricity gravity; $f\left(Q\right)$-gravity;
dipole cosmology; dynamics

##  I Introduction

The cosmological principle states that the universe is isotropic and
homogeneous. In the framework of General Relativity, the physical space is
described by the Friedmann–Lemaître–Robertson–Walker (FLRW) geometry, the
matter source is expressed in terms of perfect fluids an the observer which is
defined to be orthogonal to the homogeneous and isotropic surface. The
analysis of cosmological observations for the early and late-time stages of
the universe gives rise to tensions in the physical parameters t1 ; t2 ; t3 .

Cosmological tensions have prompted the exploration of new gravitational
theories, including those that challenge the cosmological principle t4 . The
small anisotropies and inhomogeneities in the CMB pl2018  indicate that at the
early stages of the universe the cosmological principle was violated, that has
lead to the study of anisotropic and inhomogeneous cosmological solutions in1
; in2 ; in3 . Inhomogeneous and anisotropic cosmologies are the models where
the limit of FLRW is recovered kras .

Bianchi cosmologies contain cosmological models that have been utilized to
discuss the anisotropies of the primordial universe and its evolution toward
the observed isotropy of the present epoch Mis69 ; jacobs2 ; collins ; JB1 ;
JB2 ; jb33 . It has been shown f3  that some homogeneous models which start
from inhomogeneous models can become anisotropic in the future. However,
because of the inflation that will happen in an exponentially distant time in
the future, and in the present era the models to be still homogeneous metric
perturbations. Primordial anisotropies have been investigated before in a
various of studies pa1 ; pa2 . On the other hand, there are various studies
which investigate the small anisotropies in the recent cosmological
observations, see for instance pl1 ; pl2 ; pl3 ; pl4 . On the other hand, the
nonzero value of the spatial curvature in the early stages of the universe is
not excluded by the observations, see the discussion in val1 .

In 1972, King and Ellis ke1  investigated the existence of homogeneous
spacetimes characterized by a perfect fluid with tilted velocity. Tilted
cosmological models are characterized by non-zero expansion, rotation, and
shear of the fluid source. These characteristics led to the discovery of new
singular solutions in Bianchi cosmologies ke2 . Bianchi II tilted cosmological
models are characterized by vanishing rotation, whereas Bianchi V and Bianchi
IV universes can exist with or without rotation. Locally rotational spacetimes
with a tilted source are specifically the Bianchi V and Bianchi VII models ke1
. The presence of non-zero rotation in tilted cosmological solutions can
result in these solutions appearing inhomogeneous to another observer, even if
the original spacetime is spatially homogeneous ke1 .

The Bianchi V cosmology with a tilted ideal gas was studied in detail by
Hewitt in hew . In this work, the global dynamics of the cosmological
parameters were explored for a time-varying tilted parameter. It was
discovered that the limit of the cosmological principle exists as a future
attractor, while the Kasner spacetime can describe the asymptotic solution
near the singularity. Subsequently, a more general treatment of the dynamics
of homogeneous tilted cosmologies was presented in col0 ; col0a ; col0b . In a
related context, the asymptotic dynamics of an inhomogeneous tilted cosmology
was introduced in col1  for the analytic study of the Cosmic Microwave
Background (CMB). Tilted models have been employed in various studies in the
literature to explain observational data, as seen in works such as st1 ; st2 .
Additionally, studies on tilted cosmologies where the fluid has a time-
dependent equation of state parameter can be found in tt1 ; tt2 . These works
contribute to our understanding of tilted cosmological models.

On the other hand, alternative and modified theories of gravity have been
proposed in recent decades by cosmologists to provide explanations for various
observational phenomena. These alternative and modified gravitational theories
introduce new degrees of freedom, which in turn open up new directions in the
study of astrophysical objects and the evolution of the cosmos. These theories
aim to address unresolved questions and discrepancies in our current
understanding of gravity and cosmology, offering potential avenues for
exploring the behavior of the universe on large scales. Recently, the
Symmetric Teleparallel Equivalent of General Relativity (STEGR) Nester:1998mp
and its extensions Koivisto2 ; Koivisto3 ; gg1 ; jjd1 ; pal2  have drawn the
attention of cosmologists. In STEGR the spacetime is defined by a metric
tensor, while the autoparallels are defined with the use of a symmetric and
flat connection different from the Levi-Civita connection. Consequently, the
curvature $R_{\;\lambda\mu\nu}^{\kappa}$ and the torsion tensors
$\mathrm{T}_{\mu\nu}^{\lambda}$ for this connection are always zero, and only
the nonmetricity tensor survives
$\nabla_{\kappa}g_{\mu\nu}=Q_{\kappa\mu\nu}\neq 0~{}$Eisenhart . In STEGR the
corresponding Einstein-Hilbert Action is defined by the nonmetricity scalar
$Q$, which plays the fundamental role of STEGR and its modifications.

$f\left(Q\right)$-gravity Koivisto2 ; Koivisto3  is one of the simplest
extensions of STEGR. The gravitational Lagrangian is defined by a nonlinear
function $f$ of the nonmetricity scalar. The theory has found various
applications in cosmological studies ww0 ; ww1 ; ww2 ; ww3 ; ww5 ; ww6 ; ww8 ;
ww10 ; ww11 ; ww13 ; ww14  and in compact objects sw1 ; sw2 ; sw3 ; sw4 ; sw5
. For a recent review in $f\left(Q\right)$-gravity we refer the reader in revh
. Although $f\left(Q\right)$-gravity is charged for the appearance of ghosts
or of strong coupling in FLRW background ppr1 ; ppr2 , it provides unique
directions for the study of gravitational models. Nevertheless the
introduction of matter source or scalar fields, or the consideration of of
another geometry different from that of the FLRW background, can overpass the
limits of $f\left(Q\right)$-gravity. See for instance the discussion in stg1
for teleparallelism.

One of the main characteristics of STEGR and its generalizations is the
ambiguity in defining the connection, leading to the existence of multiple
gravitational theories for the same gravitational model. While these theories
converge to the same predictions in the limit of STEGR, they diverge in
modified theories of gravity. Specifically, for the spatially flat FLRW
geometry,there are three families of symmetric and flat connections Hohmann ;
Heis2 ; Zhao , resulting in an equal number of gravitational theories with
distinct cosmological dynamics and evolution an01 . This ambiguity in the
choice of connection has implications for the analysis of cosmological
observations, as different connections can lead to different interpretations
of observational data ww11 .

In this piece of work, we focus on the locally rotationally symmetric
Kantowski-Sachs spacetime within the framework of $f\left(Q\right)$-gravity.
Previous research has identified two families of connections that are
symmetric, flat, and invariant under the four isometries of the Kantowski-
Sachs metric ks1 . These connections result in a gravitational theory with a
modified Einstein tensor equation featuring non-diagonal components. The non-
diagonal terms can be removed through additional constraints on the
$f\left(Q\right)$ function or on the connection itself.

For the first connection, the non-diagonal terms vanish when $f\left(Q\right)$
is a linear function, leading to the recovery of STEGR. On the other hand, the
second connection eliminates the non-diagonal components by reducing the
degrees of freedom in the connection. This latter connection has been the
subject of previous studies, where exact scaling solutions were determined in
ks1 , and the asymptotic analysis of the phase-space was recently investigated
in ks2 . For further exploration of homogeneous and anisotropic spacetimes we
refer the reader in bb1 ; bb2  and references therein.

In the following, we consider the first connection for the Kantowski-Sachs
geometry in $f\left(Q\right)$-gravity; and in order the theory to survive for
a nonlinear function $f\left(Q\right)$ we introduce a perfect fluid with a
tilted velocity. Recall that while in General Relativity Kantowski-Sachs
geometry does not support a tilted velocity, that is not the case in
$f\left(Q\right)$-gravity, that is, due to the nature of the theory. We extend
the analysis of hew ; col0  for the latter gravitational model. We perform a
detailed analysis of the asymptotics for a perfect fluid with constant
equation of state parameter. We consider that the tilted parameter is time-
dependent where we investigate also the limit of extreme tilted velocity. In
General Relativity, the tilted velocity is constraint through the field
equations with the spatial curvature of the background geometry; in our
consideration the tilted velocity is constraint with the nonlinear function
$f\left(Q\right)$. Without loss of generality we make use of the scalar field
description of $f\left(Q\right)$-gravity min1 . Phase-space analysis stands as
a potent method for analytically treating nonlinear field equations and
deriving asymptotic exact solutions. Through this analysis, we aim to gain
insights into the impacts of tilted velocity in homogeneous cosmologies, while
also exploring whether the theory provides the limit of General Relativity in
the presence of a tilted observer. The structure of the paper is as follows.

In Section II, we provide the fundamental definitions of $f(Q)$-gravity.
Section III is dedicated to discussing the homogeneous and anisotropic
Kantowski-Sachs geometry. Here, we utilize previous findings to express the
dynamical degrees of freedom of the $f(Q)$ function in terms of scalar fields.
Section IV introduces the dipole cosmological model, where we consider a
pressureless perfect fluid with the tilted velocity oriented perpendicular to
the two-dimensional sphere of the background spacetime. Moving on to Section
V, we delve into the phase-space analysis for this cosmological model. Our
focus is on the power-law $f(Q)=f_{0}Q^{\alpha}$, although our main results
are generalizable to any function $f(Q)$ that asymptotically behaves as a
power-law. We particularly highlight the extreme tilt limit in this analysis.
Finally, in Section VI, we summarize our findings and draw conclusions based
on the results obtained throughout the study.

##  II $f\left(Q\right)$-gravity

We consider the four-dimensional manifold that describes physical space
characterized by the metric tensor $g_{\mu\nu}$ and a symmetric and flat
connection $\Gamma_{\mu\nu}^{\kappa}$ with the property

$\nabla_{\kappa}g_{\mu\nu}=Q_{\kappa\mu\nu}\equiv\frac{\partial
g_{\mu\nu}}{\partial
x^{\lambda}}-\Gamma_{\;\lambda\mu}^{\sigma}g_{\sigma\nu}-\Gamma_{\;\lambda\nu}^{\sigma}g_{\mu\sigma},$
(1)

in which $Q_{\kappa\mu\nu}$ is the cotorsion tensor, also known as the
nonmetricity tensor Eisenhart .

Because the connection $\Gamma_{\mu\nu}^{\kappa}$ is symmetric, there is not
any torsion component, that is,

$\mathrm{T}_{\mu\nu}^{\lambda}\equiv\Gamma_{\;\left[\mu\nu\right]}^{\lambda}=0,$
(2)

and the curvature tensor for the connection $\Gamma_{\mu\nu}^{\kappa}~{}$is
always zero

$R_{\;\lambda\mu\nu}^{\kappa}\equiv\frac{\partial\Gamma_{\;\lambda\nu}^{\kappa}}{\partial
x^{\mu}}-\frac{\partial\Gamma_{\;\lambda\mu}^{\kappa}}{\partial
x^{\nu}}+\Gamma_{\;\lambda\nu}^{\sigma}\Gamma_{\;\mu\sigma}^{\kappa}-\Gamma_{\;\lambda\mu}^{\sigma}\Gamma_{\;\mu\sigma}^{\kappa}=0.$
(3)

The gravitational Action of $f\left(Q\right)$-gravity is defined as Koivisto2
; Koivisto3

$S_{f\left(Q\right)}=\int d^{4}x\sqrt{-g}f(Q),$ (4)

where $Q$ is the nonmetricity scalar defined as Nester:1998mp

$Q=Q_{\lambda\mu\nu}P^{\lambda\mu\nu},$

tensor $P^{\lambda\mu\nu}$ is defined as

$P_{\;\mu\nu}^{\lambda}=-\frac{1}{4}Q_{\;\mu\nu}^{\lambda}+\frac{1}{2}Q_{(\mu\phantom{\lambda}\nu)}^{\phantom{(\mu}\lambda\phantom{\nu)}}+\frac{1}{4}\left(Q^{\lambda}-\bar{Q}^{\lambda}\right)g_{\mu\nu}-\frac{1}{4}\delta_{\;(\mu}^{\lambda}Q_{\nu)},$
(5)

and $Q_{\mu}=Q_{\mu\nu}^{\phantom{\mu\nu}\nu}$ and
$\bar{Q}_{\mu}=Q_{\phantom{\nu}\mu\nu}^{\nu\phantom{\mu}\phantom{\mu}}$.

If $\overset{o}{R}$ describes the Ricci scalar defined by the Levi-Civita
connection for the metric tensor $g_{\mu\nu}$, it holds that revh

$\int d^{4}x\sqrt{-g}Q\simeq\int d^{4}x\sqrt{-g}\overset{o}{R}+\text{boundary
terms.}$ (6)

This property implies that in the linear case, $f(Q)$-gravity is equivalent to
General Relativity. This theory is commonly referred to as Symmetric
Teleparallel General Relativity (STGR).

Varying of the Action Integral $S_{f\left(Q\right)}$ with respect to the
metric tensor gives the modified field equations revh

$\frac{2}{\sqrt{-g}}\nabla_{\lambda}\left(\sqrt{-g}f^{\prime}(Q)P_{\;\mu\nu}^{\lambda}\right)-\frac{1}{2}f(Q)g_{\mu\nu}+f^{\prime}(Q)\left(P_{\mu\rho\sigma}Q_{\nu}^{\;\rho\sigma}-2Q_{\rho\sigma\mu}P_{\phantom{\rho\sigma}\nu}^{\rho\sigma}\right)=0,$
(7)

where $f^{\prime}\left(Q\right)=\frac{df}{dQ}$.

On the other hand, variation with respect to the connection gives the equation
of motion revh

$\nabla_{\mu}\nabla_{\nu}\left(\sqrt{-g}f^{\prime}(Q)P_{\phantom{\mu\nu}\sigma}^{\mu\nu}\right)=0.$
(8)

The equation of motion for the connection is possible be identically
satisfied, in this case we shall say that the connection is defined in the
“coincidence gauge”.

It is important to note that the connection and the metric tensor have
different transformation rules, and the connection is coordinate-dependent.
While the flatness of the connection implies that all its components can be
zero in a particular coordinate system, the choice of a specific metric tensor
already defines the coordinate system. In this specific coordinate system, if
the equation of motion for the connection is not trivially zero, we say that
the connection is defined in the “non-coincidence gauge”. On the other hand,
it has been shown that in the presence of spatial curvature the limit of
General Relativity is recovered for connections defined in the non-coincidence
gauge Zhao .

##  III Kantowski-Sachs geometry

We introduce the Kantowski-Sachs spacetime expressed in the Misner variables
where the line element is given by

$ds^{2}=-dt^{2}+a^{2}\left(t\right)\left(e^{2b\left(t\right)}dr^{2}+e^{-b\left(t\right)}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right)\right).$ (9)

Function $a\left(t\right)$ is the scale factor which defines the radius of the
space and function $b\left(t\right)$ defines the anisotropy. Isotropization is
recovered when $b\left(t\right)=0$. In this limit the Kantowski-Sachs
spacetime describes the closed FLRW geometry. For the comoving observe, the
Hubble function is defined as $H=\frac{\dot{a}}{a}$.

Kantowski-Sachs universe ks1p  is described by an homogeneous and anisotropic
line element with a topology $R\times S^{2}$ and admits four Killing vector
fields which act on spacelike hypersurfaces WE . In the isotropization limit
the Kantowski-Sachs universe is reduced to that of the closed FLRW universe.
There are a plethora of applications in the literature where the Kantowski-
Sachs space has been used to explain the physical phenomena wa1 ; wa2 ; wa3 ;
wa5 ; wa4 ; wa7 ; wa8 . Another important characteristic of Kantowski-Sachs
universe is that it can be the limit of inhomogeneous and anisotropic
geometries kras . Kantowski-Sachs geometry has some important characteristics,
indeed there can be barrel, pancake, cigar singularities or isotropic
structure based on the initial conditions col . That properties makes
Kantowski-Sachs an important mathematical structure for the connection of
different eras in the cosmic evolution. For a fruitful discussion on the
matter we refer the reader to col10 .

The line element (9) admits four isometries, they are

$\xi_{1}=\partial_{\varphi}~{},~{}\xi_{2}=\cos\varphi\,\partial_{\theta}-\cot\theta\sin\varphi\partial_{\varphi}~{},$
(10)
$\xi_{3}=\sin\varphi\partial_{\theta}+\cot\theta\cos\varphi\partial_{\varphi}~{},~{}\xi_{4}=\partial_{x}~{}.$
(11)

The requirement the connection $\Gamma_{\mu\nu}^{\kappa}$ to be flat, and to
inherit the isometries of the Kantowski-Sachs geometry leads to two families
of connections ks1 . These two connections are defined in the non-coincidence
gauge, where dynamical degrees of freedom are introduced by the connection in
the field equations.

In the following we employ the connection with non-zero coefficients

$\begin{split}&\Gamma_{\;tt}^{t}=\gamma_{2},\quad\Gamma_{\;tt}^{r}=\frac{1}{c_{1}}\left(\dot{\gamma}_{1}-\gamma_{1}\gamma_{2}+\gamma_{1}^{2}\right),\quad\Gamma_{\;tr}^{r}=\Gamma_{\;t\theta}^{\theta}=\Gamma_{\;tz}^{z}=\gamma_{1},\quad\Gamma_{\;\theta\theta}^{r}=-\frac{1}{c_{1}},\\\
&\Gamma_{\;rr}^{r}=\Gamma_{\;r\theta}^{\theta}=\Gamma_{r\varphi\;}^{\varphi}=c_{1},\quad\Gamma_{\;\varphi\varphi}^{r}=-\frac{\sin^{2}\theta}{c_{1}},\quad\Gamma_{\;\varphi\varphi}^{\theta}=-\cos\theta\sin\theta,\quad\Gamma_{\;\theta\varphi}^{\varphi}=\cot\theta.\end{split}$
(12)

where $c_{1}$ is a nonzero constant and
$\gamma_{1}\left(t\right),~{}\gamma_{2}\left(t\right)$ are functions.

The non-metricity scalar for the latter connection and the metric (9) is
expressed as follows

$Q=-6H^{2}+\frac{3}{2}\dot{b}^{2}+2\frac{e^{b}}{a^{2}}+3\left(3H\gamma_{1}+\dot{\gamma}_{1}\right),$
(13)

###  III.1 Vacuum field equations

The nonzero components of the gravitational field equations (7) in the vacuum
are

$tt:$

$f^{\prime}(Q)\left(3H^{2}+\frac{k}{a^{2}}e^{b}-\frac{3}{4}\dot{b}^{2}\right)+\frac{1}{2}\left(f(Q)-Qf^{\prime}(Q)\right)+\frac{3}{2}\gamma_{1}\dot{Q}f^{\prime\prime}(Q)=0.$
(14)

$tr:$

$\frac{3}{2}c_{1}\dot{Q}f^{\prime\prime}=0.$ (15)

$rr:$

$f^{\prime}\left(Q\right)\left(\ddot{b}+3H\dot{b}-2\dot{H}-3H^{2}-\frac{e^{b}}{a^{2}}-\frac{3}{4}\dot{b}^{2}\right)-\frac{1}{2}\left(f\left(Q\right)-Qf^{\prime}\left(Q\right)\right)+\dot{Q}f^{\prime\prime}\left(Q\right)\left(\dot{b}-2H+\frac{3}{2}\gamma_{1}\right)=0.$
(16)

$\theta\theta,\varphi\varphi:$

$f^{\prime}\left(Q\right)\left(\ddot{b}+3H\acute{b}+4\dot{H}+6H^{2}+\frac{3}{2}\dot{b}^{2}\right)+\left(f\left(Q\right)-Qf^{\prime}\left(Q\right)\right)+\dot{Q}f^{\prime\prime}\left(Q\right)\left(4H+\dot{b}-3\gamma_{1}\right)=0.$
(17)

Furthermore, the equation of motion (8) for the connection takes the following
expression

$\left(a^{3}f^{\prime\prime}\left(Q\right)\dot{Q}\right)^{\cdot}=0,$ (18)

which is a conservation law.

We observe the nondiagonal equation (15) gives the constraint
$\dot{Q}f^{\prime\prime}=0$, that is, $f\left(Q\right)=f_{1}Q+f_{2}$ or
$Q=Q_{0}$. In these two cases the cosmological solution is that of General
Relativity with (or without) the cosmological constant term.

For this specific connection, the only allowed solution in vacuum corresponds
to that of General Relativity. However, due to the nature of the nonmetricity
theory, it is possible to have fluids with a tilted observer in the context of
the Kantowski-Sachs geometry. This is the scenario we investigate in the
following discussion.

At this point we remark that an equivalent way to write the field equations is
by using a scalar field description min1 .

We introduce $\phi=f^{\prime}\left(Q\right)$ and
$V\left(\phi\right)=\left(f(Q)-Qf^{\prime}(Q)\right)~{}$, then the field
equations read

$tt:$

$\phi\left(3H^{2}+\frac{k}{a^{2}}e^{b}-\frac{3}{4}\dot{b}^{2}\right)+\frac{1}{2}V\left(\phi\right)+\frac{3}{2}\gamma_{1}\dot{\phi}=0,$
(19)

$tr:$

$\frac{3}{2}c_{1}\dot{\phi}=0.$ (20)

$rr:$

$\phi\left(\ddot{b}+3H\dot{b}-2\dot{H}-3H^{2}-\frac{e^{b}}{a^{2}}-\frac{3}{4}\dot{b}^{2}\right)-\frac{1}{2}V\left(\phi\right)+\dot{\phi}\left(\dot{b}-2H+\frac{3}{2}\gamma_{1}\right)=0,$
(21)

$\theta\theta,\varphi\varphi:$

$\phi\left(\ddot{b}+3H\dot{b}+4\dot{H}+6H^{2}+\frac{3}{2}\dot{b}^{2}\right)+\frac{1}{2}V\left(\phi\right)+\dot{\phi}\left(4H+\dot{b}-3\gamma_{1}\right)=0.$
(22)

and the equation of motion for the connection

$\left(a^{3}\dot{\phi}\right)^{\cdot}=0.$ (23)

##  IV Dipole Cosmology

We introduce the perfect fluid energy-momentum tensor ke1 ; ke2

$T_{\mu\nu}=\left(\rho+p\right)\tilde{u}_{\mu}\tilde{u}_{\nu}+pg_{\mu\nu},$
(24)

in which $\tilde{u}^{\mu}$ is the tilted observer, that is,
$\tilde{u}^{\mu}\tilde{u}_{\mu}=-1$, and
$\tilde{u}^{\mu}=\cosh\beta\left(t\right)\partial_{t}+\frac{e^{-b}}{a}\sinh\beta\left(t\right)\partial_{r}$.
Function $\beta\left(t\right)$ is the tilted parameter, when
$\beta\left(t\right)=0$, $\tilde{u}^{\mu}$ is reduced to the comoving observer
$u^{\mu}=\partial_{t}$.

For this observer, the energy momentum tensor (24) reads

$T_{\mu\nu}=\begin{pmatrix}\cosh^{2}\beta\left(\rho+p\right)-p&-ae^{b}\sinh\beta\cosh\beta\left(\rho+p\right)&0&0\\\
-ae^{b}\sinh\beta\cosh\beta\left(\rho+p\right)&a^{2}e^{2b}\left(\cosh^{2}\beta\left(\rho+p\right)-\rho\right)&0&0\\\
0&0&a^{2}e^{-b}p&0\\\ 0&0&0&a^{2}e^{-b}p\sin^{2}\theta\end{pmatrix}.$ (25)

We assume that the matter source is minimally coupled to gravity, thus, we
derive the conservation equations

$\displaystyle\dot{\rho}+3\left(1+\frac{p}{\rho}\right)H\rho+\left(1+\frac{p}{\rho}-2\frac{p}{\rho}\coth^{2}\beta\right)\tanh\beta~{}\dot{\beta}\rho$
$\displaystyle=0,$ (26)
$\displaystyle\dot{p}+\left(1+\frac{p}{\rho}\right)\rho\left(H+\dot{b}+\coth\beta~{}\dot{\beta}\right)$
$\displaystyle=0.$ (27)

From equation (27) it is clear that the evolution of the tilted parameters
$\beta\left(t\right)$ is related to the anisotropic scale factor
$b\left(t\right)$.

For the tilted observer we define the expansion rate ke1

$\tilde{\theta}=3\cosh\beta~{}H+\sinh\beta~{}\dot{\beta},$ (28)

the shear

$\tilde{\sigma}^{2}=\left(\sqrt{\frac{3}{2}}\cosh\beta~{}\dot{b}+\sqrt{\frac{2}{3}}\sinh\beta~{}\dot{\beta}\right)^{2},$
(29)

and the deceleration parameter ke1
$\tilde{q}=-1-\frac{3}{\tilde{\theta}^{2}}\tilde{u}^{\mu}\theta_{;\mu}$, as

$\tilde{q}=-1-\frac{\left(\cosh\beta~{}H+\frac{1}{3}\sinh\beta~{}\dot{\beta}\right)^{\cdot}}{\left(\cosh\beta~{}H+\frac{1}{3}\sinh\beta~{}\dot{\beta}\right)^{2}}.$
(30)

Finally, the field equations for the tilted observer reads

$\displaystyle\phi\left(3H^{2}+\frac{k}{a^{2}}e^{b}-\frac{3}{4}\dot{b}^{2}\right)+\frac{1}{2}V\left(\phi\right)+\frac{3}{2}\gamma_{1}\dot{\phi}$
$\displaystyle=\cosh^{2}\beta\left(1+w_{m}\right)\rho-w_{m}\rho~{},$ (31)
$\displaystyle\frac{3}{2}c_{1}\dot{\phi}$
$\displaystyle=-ae^{b}\sinh\beta\cosh\beta\left(1+w_{m}\right)\rho~{},$ (32)
$\displaystyle\phi\left(-2\dot{H}-3H^{2}-\frac{3}{4}\dot{b}^{2}-\frac{e^{b}}{3a^{3}}\right)-2\dot{\phi}H+\frac{3}{2}\dot{\phi}\gamma_{1}-V\left(\phi\right)$
$\displaystyle=\left(w_{m}+\frac{1}{3}\left(1+w_{m}\right)\sinh^{2}\beta\right)\rho~{},$
(33)
$\displaystyle\phi\left(\ddot{b}+3H\dot{b}-\frac{2}{3}\frac{e^{b}}{a^{2}}\right)-\dot{b}\dot{\phi}$
$\displaystyle=\frac{2}{3}\left(1+w_{m}\right)\sinh^{2}\beta\rho~{},$ (34)
$\displaystyle\left(a^{3}\phi\right)^{\cdot}$ $\displaystyle=0~{},$ (35)

where $w_{m}$ is the equation of state parameter for the perfect fluid, that
is, $p=w_{m}\rho$.

Equation (32) states that $\dot{\phi}\neq 0,~{}$i.e.
$\dot{Q}f^{\prime\prime}\neq 0,$ when there is a nonzero contribution of the
tilted fluid source in the universe. Only in this case the
$f\left(Q\right)$-theory introduce an “exotic” matter source in the field
equations.

##  V Asymptotic solutions

In this section, we conduct a comprehensive analysis of the phase-space for
the cosmological field equations (31)-(35) considering a tilted observer with
the energy-momentum tensor (24).

For the perfect fluid, we assume it to be pressureless, i.e.,
$p=0,~{}w_{m}=0$, representing the dark matter component of the universe.
Consequently, equations (26) and (27) simplify as follows

$\displaystyle\dot{\rho}+\left(3H+\tanh\beta~{}\dot{\beta}\right)\rho$
$\displaystyle=0,$ (36)
$\displaystyle\rho\left(H+\dot{b}+\coth\beta~{}\dot{\beta}\right)$
$\displaystyle=0.$ (37)

In the limit where $\beta=0$, then we end with the cosmological model of STEGR
with a matter source, where the $\Lambda$CDM model is recovered in the
isotropic limit.

We introduce the new dependent dimensionless variables

$\Sigma=\frac{\dot{b}}{2D}~{},~{}x=\frac{3\dot{\phi}}{2\phi
D}~{},~{}y=\frac{V\left(\phi\right)}{\phi
D^{2}}~{},~{}z=\frac{\gamma}{D}~{},~{}\mu=\sinh\beta~{},~{}\eta=\frac{H}{D}$
(38)
$\omega_{m}=\frac{\left(1+\cosh\left(2\beta\right)\right)}{6\phi}\frac{\rho}{D^{2}}~{},~{}\lambda=\phi\frac{V_{,\phi}}{V}~{},~{}u=\left(\frac{a}{D}\right)^{3}~{},~{}D=\sqrt{H^{2}+\frac{e^{b}}{3a^{2}}},$
(39)

and the independent variable $dt=Dd\tau$.

In terms of the new variables the field equations takes the form of the
following system

$\displaystyle\frac{d\Sigma}{d\tau}$
$\displaystyle=\left(\left(\eta^{2}-1\right)\left(\frac{2\sqrt{3}}{3}x\Sigma+\Sigma^{2}-1\right)+\frac{3}{2}\eta\Sigma\left(y-xz+\Sigma^{2}-1\right)+\frac{\mu^{2}\omega_{m}\left(2+\eta\Sigma\right)}{2\left(1+\mu^{2}\right)}\right)~{},$
(40) $\displaystyle\frac{dx}{d\tau}$
$\displaystyle=x\left(\left(\eta^{2}-1\right)\left(\frac{2\sqrt{3}}{3}x+\Sigma\right)+\frac{1}{2}\eta\left(3\left(y-xz+\Sigma^{2}-1\right)+\frac{\mu^{2}\omega_{m}}{1+\mu^{2}}\right)\right)~{},$
(41) $\displaystyle\frac{dy}{d\tau}$
$\displaystyle=\frac{1}{3}y\left(x\left(2\sqrt{3}\left(\lambda-1+2\eta^{2}\right)-9\eta
z\right)+6\left(\eta^{2}-1\right)\Sigma+3\eta\left(3\left(1+y+\Sigma^{2}\right)+\frac{\mu^{2}\omega_{m}}{1+\mu^{2}}\right)\right)~{},$
(42) $\displaystyle\frac{dz}{d\tau}$ $\displaystyle=\frac{3}{2}\eta
z\left(y-xz\right)+\frac{\sqrt{3}}{3}\left(4\eta^{2}+\left(\lambda
y-2\left(1+\Sigma^{2}\right)\right)\right)$
$\displaystyle+z\left(\frac{\eta}{3}\left(\eta\left(2\sqrt{3}x+3\Sigma\right)+\frac{9}{2}\left(\Sigma^{2}-1\right)+\frac{\mu^{2}\omega_{m}}{1+\mu^{2}}\right)-\Sigma\right)~{},$
(43) $\displaystyle\frac{d\mu}{d\tau}$
$\displaystyle=-\mu\left(\eta+2\Sigma\right)~{},~{}$ (44)
$\displaystyle\frac{d\eta}{d\tau}$
$\displaystyle=\frac{1}{2}\left(\eta^{2}-1\right)\left(1+3y+\frac{x}{3}\left(4\sqrt{3}\eta-z\right)+2\eta\Sigma+3\Sigma^{2}+\frac{\mu^{2}\omega_{m}}{1+\mu^{2}}\right)~{}.$
(45)

Furthermore, from equations (31) and (32) we determine the algebraic
constraints

$\displaystyle\omega_{m}-1+y+xz-\Sigma^{2}~{}$ $\displaystyle=0,$ (46)
$\displaystyle\frac{\sqrt{3}}{9}c_{1}\left(1+\mu^{2}\right)x-u\left(\eta^{2}-1\right)\mu\omega_{m}$
$\displaystyle=0.$ (47)

As far as the parameter $\lambda$ is concerned, we consider to be always a
constant, which corresponds to the power-law potential $V\left(\phi\right)$.
We remark that a power-law potential corresponds to a power-law
$f\left(Q\right)$ function. Although this is a special function
$f\left(Q\right)$ our analysis stands in the limit when the power-law term of
a given function $f\left(Q\right)$ dominates the cosmological dynamics.

The canonical anisotropic parameter
$\tilde{\Sigma}=\left(\frac{\tilde{\sigma}}{\tilde{\theta}}\right)$ and the
deceleration parameter $\tilde{q}$ are defined for the tilted observer by
expressions (29) and (30) in terms of the dimensionless variables become

$\tilde{\Sigma}^{2}=\left(\frac{3\Sigma+\mu^{2}\left(\Sigma-\eta\right)}{\eta\left(3+2\mu^{2}\right)-2\mu^{2}\Sigma}\right)^{2},$
(48) $\displaystyle\frac{\tilde{q}+1}{\Delta}$
$\displaystyle=9y\left(3+2\mu^{2}\left(3+2\mu^{2}\right)\right)+3\left(3\left(1+2\mu^{2}\right)\right)^{2}+\eta^{2}\left(6+4\mu^{2}\left(2+\mu^{2}\right)\right)$
$\displaystyle-8\eta\mu^{2}\left(2+\mu^{2}\right)\Sigma+\left(9-4\mu^{2}\left(1+2\mu^{2}\right)\Sigma^{2}\right)$
$\displaystyle+x\left(4\sqrt{3}\left(1+\mu^{2}\right)\left(\eta\left(3+2\mu^{2}\right)-2\mu^{2}\Sigma\right)-9z\left(3+4\mu^{2}\right)\right).$
(49)

where
$\Delta=\left(6\sqrt{1+\mu^{2}}\left(\eta\left(3+2\mu^{2}\right)-2\mu^{2}\Sigma^{2}\right)^{2}\right)$.

###  V.1 Stationary points

We compute the stationary points of the dynamical system (40)-(45). Each
stationary point corresponds to an asymptotic solution for the cosmological
model under study. To reconstruct the cosmological history predicted by the
model, we derive the cosmological parameters at these stationary points.
Additionally, we determine the stability properties of these points.

The stationary points
$\left(\Sigma\left(P\right),x\left(P\right),y\left(P\right),z\left(P\right),\mu\left(P\right),\eta\left(P\right)\right)~{}$are

  * •

$T_{1}=\left(-1,0,0,\frac{4}{\sqrt{3}},\mu_{1},2\right)$

describes an asymptotic solution with physical parameters
$\omega_{m}\left(T_{1}\right)=0$,
$\tilde{\Sigma}^{2}\left(T_{1}\right)=\frac{1}{4}$ and
$\tilde{q}\left(T_{1}\right)=-1+\frac{1}{2\sqrt{1+\mu_{1}^{2}}}$. Hence, the
asymptotic solution describes an accelerated Kantowski-Sachs universe. The
eigenvalues of the linearized system around the stationary point are
$\left\\{-6,3,3,3,0,0\right\\}$, from where we infer that $T_{1}$ is a saddle
point.

  * •

$T_{2}=\left(1,0,0,-\frac{4}{\sqrt{3}},\mu_{2},-2\right),$

describes an asymptotic solution with the same physical properties of point
$T_{1}$; that is, $\omega_{m}\left(T_{2}\right)=0$,
$\tilde{\Sigma}^{2}\left(T_{2}\right)=\frac{1}{4}$ and
$\tilde{q}\left(T_{2}\right)=-1+\frac{1}{2\sqrt{1+\mu_{2}^{2}}}$. The
eigenvalues of the linearized system are $\left\\{6,-3,-3,-3,0,0\right\\}$,
from where we infer that $T_{2}$ is always a saddle point.

These two points are the unique stationary points with a nonzero tilted
parameters. The following points have $\mu\left(\Sigma\right)=0$.

  * •

$B_{1}=\left(-1,0,0,z_{1},0,-1\right),$

with physical parameters $\omega_{m}\left(B_{1}\right)=0$,
$\tilde{\Sigma}^{2}\left(B_{1}\right)=1$ and $\tilde{q}\left(B_{1}\right)=0$,
describes the Bianchi I spacetime, in particular the vacuum Kasner solution
given by General Relativity. We calculate the eigenvalues
$\left\\{-6,-6,-3,3,0,0\right\\}$. Hence, $B_{1}$ is a saddle point.

  * •

$B_{2}=\left(-1,0,0,z_{2},0,1\right),$

with $\omega_{m}\left(B_{2}\right)=0$,
$\tilde{\Sigma}^{2}\left(B_{2}\right)=1$, $\tilde{q}\left(B_{2}\right)=0$ and
eigenvalues $\left\\{6,3,2,1,0,0\right\\}$. Point $B_{2}$ describes an
unstable Kasner solution. Specifically $B_{2}$ is a source point.

  * •

$B_{3}=\left(1,0,0,z_{3},0,-1\right),$

with $\omega_{m}\left(B_{3}\right)=0$,
$\tilde{\Sigma}^{2}\left(B_{3}\right)=1$, $\tilde{q}\left(B_{3}\right)=0$ and
eigenvalues $\left\\{-6,-3,-2,-1,0,0\right\\}$. The asymptotic solution
describes the Kasner spacetime of General Relativity. Because of the zero
eigenvalues we study the stability properties of point $B_{3}~{}$numerically.
In Figs. 1 and 2 we present the two-dimensional phase-space portraits where
point $B_{3}$ lies. We observe that for initial conditions where
$z\left(B_{3}\right)\leq 0$ and$~{}x\left(B_{3}\right)>0$, point $B_{3}$ is an
attractor.

  * •

$B_{4}=\left(1,0,0,z_{3},0,1\right),$

describes an asymptotic solution with physical parameters
$\omega_{m}\left(B_{4}\right)=0$, $\tilde{\Sigma}^{2}\left(B_{4}\right)=1$,
$\tilde{q}\left(B_{4}\right)=0$ while the eigenvalues of the linearized system
around the point are $\left\\{6,6,3,-3,0,0\right\\}$. Hence, $B_{4}$ is a
saddle point which provides the Kasner solution of STEGR.

  * •

$K_{1}=\left(\frac{1}{2},0,-\frac{3}{4},-\frac{2+\lambda}{2\sqrt{3}},0,\frac{1}{2}\right)\text{,}$

corresponds to the asymptotic solution of Kantowski-Sachs spacetime with a
cosmological constant of General Relativity. Indeed, the physical parameters
are $\omega_{m}\left(K_{1}\right)=0$,
$\tilde{\Sigma}^{2}\left(K_{1}\right)=1$, $\tilde{q}\left(K_{1}\right)=-1$.
Furthermore, the eigenvalues of the linearized system around the point are
$\left\\{-3,-\frac{3}{2},-\frac{3}{2},-\frac{3}{2},-\frac{3}{2},\frac{3}{2}\right\\}$,
which means that $K_{1}$ is a saddle point.

  * •

$K_{2}=\left(-\frac{1}{2},0,-\frac{3}{4},\frac{2+\lambda}{2\sqrt{3}},0,-\frac{1}{2}\right),$

describes an accelerated Kantowski-Sachs spacetime with the same physical
properties of point $K_{1}$, that is, $\omega_{m}\left(K_{2}\right)=0$,
$\tilde{\Sigma}^{2}\left(K_{2}\right)=1$, $\tilde{q}\left(K_{2}\right)=-1$ and
eigenvalues
$\left\\{3,\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{3}{2},-\frac{3}{2}\right\\}$.
Thus, $K_{2}$ is also a saddle point.

  * •

$F_{1}=\left(0,0,0,\frac{4}{3\sqrt{3}},0,1\right),$

with $\omega_{m}\left(F_{1}\right)=1$,
$\tilde{\Sigma}^{2}\left(F_{1}\right)=0$, $\tilde{q}\left(F_{1}\right)=0$, and
eigenvalues $\left\\{3,1,-1,-\frac{3}{2},-\frac{3}{2},-\frac{3}{2}\right\\}$.
The asymptotic solution is that of spatially flat FLRW universe dominated by
the matter source. Moreover, from the eigenvalues we conclude that $F_{1}$ is
always a saddle point.

  * •

$F_{2}=\left(0,0,0,-\frac{4}{3\sqrt{3}},0,-1\right),$

has the same physical quantities with point $F_{1}$, that is,
$\omega_{m}\left(F_{2}\right)=1$, $\tilde{\Sigma}^{2}\left(F_{2}\right)=0$,
$\tilde{q}\left(F_{2}\right)=0$. The eigenvalues are calculated
$\left\\{-3,-1,1,\frac{3}{2},\frac{3}{2},\frac{3}{2}\right\\}$, that is,
$F_{2}$ is always a saddle point.

  * •

$F_{3}=\left(0,0,-1,\frac{\lambda-2}{2\sqrt{3}},0,-1\right),$

and physical parameters $\omega_{m}\left(F_{3}\right)=0$,
$\tilde{\Sigma}^{2}\left(F_{3}\right)=0$, $\tilde{q}\left(F_{3}\right)=-1$,
describe a FLRW universe with a cosmological constant. The eigenvalues are
$\left\\{3,3,3,3,2,1\right\\}$, which means that that $F_{3}$ is always a
source.

  * •

$F_{4}=\left(0,0,-1,-\frac{\lambda-2}{2\sqrt{3}},0,1\right),$

describes the de Sitter universe, with $\omega_{m}\left(F_{4}\right)=0$,
$\tilde{\Sigma}^{2}\left(F_{4}\right)=0$, $\tilde{q}\left(F_{4}\right)=-1$ and
eigenvalues $\left\\{-3,-3,-3,-3,-2,-1\right\\}$. Point $F_{4}$ is the unique
attractor global of the cosmological model.

The above results are summarized in Table 1. In Figs. 3 and 4 we present the
qualitative evolution for the dynamical variables and for the physical
variables $\tilde{\Sigma}^{2}$ and $\tilde{q}$ for two different sets of
initial conditions where points $B_{3}$ and $F_{4}$ are attractors.

Figure 1: 2D Phase-space portraits for the dynamical system (40)-(45) on the surfaces where point $B_{3}$ exists. The plots are for $\lambda=1$. It follows that $B_{3}$ is an attractor. Figure 2: 2D Phase-space portraits for the dynamical system (40)-(45) on the surfaces where point $B_{3}$ exist. The plots are for $\lambda=1$. It follows that $B_{3}$ is an attractor. Figure 3: Qualitative evolution of the dynamical variables for the system (40)-(45) for $\lambda=1$ and initial conditions $\left(\Sigma_{0},y_{0},z_{0},\mu_{0},\eta_{0}\right)=\left(\frac{1}{2},-1,-1,0.2,0.1\right)$. Solid line is for $x_{0}=2$ where the Kasner solution described by point $B_{3}$ is an attractor and dashed line is for $x_{0}=-2$ where the de Sitter universe is the attractor. Figure 4: Qualitative evolution of the physical variables physical variables $\tilde{\Sigma}^{2}$ and $\tilde{q}$ for the numerical solutions of Fig. 3. Table 1: Stationary points and physical parameters. Point | Spacetime | Tilted | $\mathbf{\omega}_{m}$ | $\mathbf{\tilde{\Sigma}}^{2}$ | $\mathbf{\tilde{q}}$ | Stability
---|---|---|---|---|---|---
$T_{1}$ | Kantowski-Sachs | Yes | $0$ | $\frac{1}{4}$ | $-1+\frac{1}{2\sqrt{1+\mu^{2}}}$ | Saddle
$T_{2}$ | Kantowski-Sachs | Yes | $0$ | $\frac{1}{4}$ | $-1+\frac{1}{2\sqrt{1+\mu^{2}}}$ | Saddle
$B_{1}$ | Kasner | No | $0$ | $1$ | $0$ | Saddle
$B_{2}$ | Kasner | No | $0$ | $1$ | $0$ | Saddle
$B_{3}$ | Kasner | No | $0$ | $1$ | $0$ | Attractor
$B_{4}$ | Kasner | No | $0$ | $1$ | $0$ | Saddle
$K_{1}$ | Kantowski-Sachs | No | $0$ | $1$ | $-1$ | Saddle
$K_{2}$ | Kantowski-Sachs | No | $0$ | $1$ | $-1$ | Saddle
$F_{1}$ | FLRW | No | $1$ | $0$ | $0$ | Saddle
$F_{2}$ | FLRW | No | $1$ | $0$ | $0$ | Saddle
$F_{3}$ | FLRW | No | $0$ | $0$ | $-1$ | Source
$F_{4}$ | FLRW | No | $0$ | $0$ | $-1$ | Attractor

###  V.2 Extreme limit

We investigate the case of the extreme tilted observer, that is,
$\mu\rightarrow+\infty$. At the extreme tilted, the deceleration parameter
$\tilde{q}$ and the anisotropic parameter $\tilde{\Sigma}^{2}$ are simplified
as $\tilde{q}=-1$ and $\tilde{\Sigma}^{2}=\frac{1}{4}$. Thus at the extreme
tilt scenario the observable universe is always anisotropic and accelerated.

In order to perform this analysis we employ the change of variable
$\mu=\frac{1}{\hat{\mu}}$, and we determine the stationary points for the
dynamical system (40)-(45) with $\hat{\mu}=0$.

The new stationary points
$\left(\Sigma\left(P\right),x\left(P\right),y\left(P\right),z\left(P\right),\hat{\mu}\left(P\right)\rightarrow
0,\eta\left(P\right)\right)$ are

  * •

$\displaystyle\hat{B}_{1}$
$\displaystyle=\left(-1,0,0,z_{1},0,-1\right)~{},~{}\hat{B}_{2}=\left(-1,0,0,z_{2},0,1\right)~{},~{}$
$\displaystyle\hat{B}_{3}$
$\displaystyle=\left(1,0,0,z_{3},0,-1\right)~{},~{}\hat{B}_{4}=\left(1,0,0,z_{3},0,1\right)~{},~{}$

which correspond to Kasner solutions. The corresponding eigenvalues of the
linearized system for each point are
$\hat{B}_{1}:\left\\{-6,-6,-3,0,0,0\right\\}$,
$\hat{B}_{2}:\left\\{6,4,2,-1,0,0\right\\}$,
$\hat{B}_{3}:\left\\{-6,-4,-2,1,0,0\right\\}$ and
$\hat{B}_{4}:\left\\{6,6,3,0,0,0\right\\}$. Hence, $\hat{B}_{2}$,
$\hat{B}_{3}$ are always saddle points and $\hat{B}_{4}$ is a source point. n
Figs. 1 and 2 we present the two-dimensional phase-space portraits where point
$\hat{B}_{1}\,\ $is defined. We observe that point $\hat{B}_{1}$ is always a
saddle point.

  * •

$\hat{K}_{1}=\left(\frac{1}{2},0,-\frac{3}{4},-\frac{2+\lambda}{2\sqrt{3}},0,\frac{1}{2}\right)~{},~{}\hat{K}_{2}=\left(-\frac{1}{2},0,-\frac{3}{4},\frac{2+\lambda}{2\sqrt{3}},0,-\frac{1}{2}\right)~{},$

correspond to Kantowski-Sachs solutions with a cosmological constant term. The
eigenvalues are
$\hat{K}_{1}:\left\\{-3,-3,-\frac{3}{2},-\frac{3}{2},\frac{3}{2},\frac{3}{2}\right\\}$
and
$\hat{K}_{2}:\left\\{3,3,\frac{3}{2},\frac{3}{2},-\frac{3}{2},-\frac{3}{2}\right\\}$,
from where we infer that that the asymptotic solutions are always unstable and
the $\hat{K}_{1},~{}\hat{K}_{2}$ are saddle points.

  * •

$\hat{F}_{3}=\left(0,0,-1,\frac{\lambda-2}{2\sqrt{3}},0,-1\right)~{},~{}\hat{F}_{4}=\left(0,0,-1,-\frac{\lambda-2}{2\sqrt{3}},0,1\right),$

describe spatially flat FLRW spacetimes dominated by the cosmological
constant. We calculate the eigenvalues
$\hat{F}_{3}:\left\\{4,3,3,3,2,-1\right\\}$ and
$\hat{F}_{4}:\left\\{-4,-3,-3,-3,-2,1\right\\}$. Consequently the stationary
points are always saddle points.

  * •

$\hat{D}_{1}=\left(2,0,-3,\frac{2+\lambda}{\sqrt{3}},0,-1\right)~{},~{}\hat{D}_{2}=\left(-2,0,-3,-\frac{2+\lambda}{\sqrt{3}},0,-1\right)$

describe Bianchi I spacetimes with a cosmological constant, nonzero matter
component $\omega_{m}\left(\hat{D}_{1,2}\right)=-6$ and eigenvalues
$\hat{D}_{1}:\left\\{\frac{3}{2}\left(1+\sqrt{17}\right),\frac{3}{2}\left(1-\sqrt{17}\right),6,3,3,3\right\\}$,
$\hat{D}_{2}:\left\\{-\frac{3}{2}\left(1+\sqrt{17}\right),-\frac{3}{2}\left(1-\sqrt{17}\right),-6,-3,-3,-3\right\\}$.
Hence, the points are always saddle points.

The above results for the extreme tilted scenario are summarized in Table 2.

Figure 5: 2D Phase-space portraits for the dynamical system (40)-(45) on the surfaces where point $\hat{B}_{1}$ exists. The plots are for $\lambda=1$. We observe that $\hat{B}_{1}$ is a saddle point. Figure 6: 2D Phase-space portraits for the dynamical system (40)-(45) on the surfaces where point $\hat{B}_{1}$ exists. The plots are for $\lambda=1$. We observe that $\hat{B}_{1}$ is a saddle point. Table 2: Stationary points and physical parameters at the extreme tilted scenario. Point | $\mathbf{\omega}_{m}$ | $\mathbf{\tilde{\Sigma}}^{2}$ | $\mathbf{\tilde{q}}$ | Stability
---|---|---|---|---
$\hat{B}_{1}$ | $0$ | $\frac{1}{4}$ | $-1$ | Saddle
$\hat{B}_{2}$ | $0$ | $\frac{1}{4}$ | $-1$ | Saddle
$\hat{B}_{3}$ | $0$ | $\frac{1}{4}$ | $-1$ | Saddle
$\hat{B}_{4}$ | $0$ | $\frac{1}{4}$ | $-1$ | Source
$\hat{K}_{1}$ | $0$ | $\frac{1}{4}$ | $-1$ | Saddle
$\hat{K}_{2}$ | $0$ | $\frac{1}{4}$ | $-1$ | Saddle
$\hat{F}_{3}$ | $0$ | $\frac{1}{4}$ | $-1$ | Saddle
$\hat{F}_{4}$ | $0$ | $\frac{1}{4}$ | $-1$ | Saddle
$\hat{D}_{1}$ | $\neq 0$ | $\frac{1}{4}$ | $-1$ | Saddle
$\hat{D}_{2}$ | $\neq 0$ | $\frac{1}{4}$ | $-1$ | Saddle

##  VI Conclusions

In General Relativity, and consequently in STEGR, the Einstein tensor for the
Kantowski-Sachs universe is diagonal. This implies that the matter source
allowed in this universe should be described by a diagonal energy-momentum
tensor. In the case of a perfect fluid, the velocity should align with that of
the comoving observer, perpendicular to the three-dimensional spacelike
hypersurface.

However, in modified theories like STEGR within a Kantowski-Sachs geometry, it
is possible to consider a tilted velocity for the matter source, resulting in
a dipole universe with a Kantowski-Sachs geometry. To explore this concept,
within the framework of $f(Q)$-gravity, we examined a perfect fluid with a
velocity tilted perpendicular to the two-dimensional spacelike sphere. Our
investigation focused on the phase-space analysis, determining the global
evolution of the physical parameters.

In this model, initial conditions can allow for the fluid to possess a tilted
velocity, implying a violation of the cosmological principle in the early
universe. However, the unique attractors of the model describe the limit of
STEGR with or without a cosmological constant. Specifically, these attractors
correspond to the anisotropic Kasner universe and the isotropic de Sitter
universe. Hence, this dipole gravitational model in $f\left(Q\right)$-gravity
leads to universe with zero spatially curvature. For initial conditions where
in the asymptotic limit $Q\rightarrow 0$, the attractor is the anisotropic
spacetime, while the isotropic de Sitter universe is recovered when
$Q\rightarrow Q_{0},~{}Q_{0}$ is a nonzero constant. Consequently, we can
infer that the cosmological principle is an attractor of
$f\left(Q\right)$-gravity.

In the previous analysis we considered the power-law $f\left(Q\right)$-theory,
which leads to the power-law potential $V\left(\phi\right)$ and the constant
parameter $\lambda$. Nevertheless, for arbitrary function $f\left(Q\right)$,
that is, scalar field potential $V\left(\phi\right)$, parameter $\lambda$ is
not a constant. It is evolution it is given the differential equation

$\frac{d\lambda}{d\tau}=\frac{2}{3}\lambda
x\left(\lambda\Gamma\left(\lambda\right)+\left(1-\lambda\right)\right),~{}$
(50)

with
$\Gamma\left(\lambda\right)=\frac{V_{,\phi\phi}V}{\left(V_{,\phi}\right)^{2}}$.
Hence, the stationary points for the new dynamical system should solve
equation $\lambda
x\left(\lambda\Gamma\left(\lambda\right)+\left(1-\lambda\right)\right)=0$. We
end with two families of points, those with $\lambda=\lambda_{0}$ such that
$\lambda_{0}\left(\lambda_{0}\Gamma\left(\lambda_{0}\right)+\left(1-\lambda_{0}\right)\right)=0$,
and the points with $x=0$. While the general dynamics and evolution are
affected by a general potential, it is clear that there are not new families
of solutions for arbitrary potential, and the analysis for the power-law
function covers all main families of solutions, as discussed for the FLRW case
in an01 .

While this work specifically considers $f(Q)$-geometry, these results are
generalizable to other modified symmetric teleparallel theories, as well as to
other Bianchi models. It will be of interest to investigate the existence of
spherical symmetric solutions with this kind of matter source.

###### Acknowledgements.

The author thanks Prof. Sibu Moyo, Prof. Nikolaos Dimakis, the Stellenbosch
University and the Universidad de La Frontera for the hospitality provided
while part of this work was carried out. AP thanks the support of VRIDT
through Resolución VRIDT No. 096/2022, Resolución VRIDT No. 098/2022.

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