# Impact of the absorber coupling design for Transition Edge Sensor X-ray
Calorimeters

M. de Wit<EMAIL_ADDRESS>L. Gottardi  E. Taralli  K. Nagayoshi  M.L. Ridder
H. Akamatsu  M.P. Bruijn  R.W.M. Hoogeveen  J. van der Kuur  K. Ravensberg  D.
Vaccaro  J-R. Gao  Faculty of Applied Science, Delft University of Technology,
Lorentzweg 1, 2628 CJ Delft, The Netherlands  J-W.A. den Herder  NWO-I/SRON
Netherlands Institute for Space Research, Niels Bohrweg 4, 2333 CA Leiden, The
Netherlands

###### Abstract

Transition Edge Sensors (TESs) are the selected technology for future
spaceborne X-ray observatories, such as Athena, Lynx, and HUBS. These missions
demand thousands of pixels to be operated simultaneously with high energy-
resolving power. To reach these demanding requirements, every aspect of the
TES design has to be optimized. Here we present the experimental results of
tests on different devices where the coupling between the x-ray absorber and
the TES sensor is varied. In particular, we look at the effects of the
diameter of the coupling stems and the distance between the stems and the TES
bilayer. Based on measurements of the AC complex impedance and noise, we
observe a reduction in the excess noise as the spacing between the absorber
stem and the bilayer is decreased. We identify the origin of this excess noise
to be internal thermal fluctuation noise between the absorber stem and the
bilayer. Additionally, we see an impact of the coupling on the superconducting
transition in the appearance of kinks. Our observations show that these
unwanted structures in the transition shape can be avoided with careful design
of the coupling geometry. Also the stem diameter appears to have a significant
impact on the smoothness of the TES transition. This observation is still
poorly understood, but is of great importance for both AC and DC biased TESs.

##  I Introduction

Transition Edge Sensors (TESs) are very sensitive detectors of energy that
utilize the sharpness of a superconductor normal-to-superconducting
transitionIrwin and Hilton (2005); Ullom and Bennett (2015); Gottardi and
Nagayashi (2021). TESs are the selected technology in a number of either
planned or proposed space-borne observatories, such as Athena Barret  _et al._
(2020), Lynx Gaskin and Swartz (2019), and HUBS Cui  _et al._ (2020). These
missions aim for thousands of pixels to be operated simultaneously with high
energy resolving power. The limited resources available for a satellite-based
instrument, such as electrical power, cooling power, and allowable weight,
require the use of sophisticated multiplexing schemes, such as Time Division
Multiplexing (TDM) Durkin  _et al._ (2019), Frequency Division Multiplexing
(FDM) Akamatsu  _et al._ (2020), and others Morgan  _et al._ (2016); Bennett
_et al._ (2019); Nakashima  _et al._ (2020). In this paper, we focus on TES
based X-ray micro-calorimeters developed for the Athena X-ray Integral Field
Unit (X-IFU) instrument Pajot  _et al._ (2018). X-IFU will consist of over
3000 multiplexed pixels with a planned spectral resolution of 2.5 eV for 7 keV
X-rays.

In order to achieve the demanding requirements set by the X-IFU instrument,
every aspect of the TES pixel design has to be optimized. Over the years much
work has been done towards this goal. Research topics included the impact of
the thermal conductance on the TES transition width Hays-Wehle  _et al._
(2016); Morgan  _et al._ (2017); Zhang  _et al._ (2019), the impact of the
normal resistance by changing the TES size and aspect ratio Gottardi  _et al._
(2017); Sakai  _et al._ (2018); Morgan  _et al._ (2019); De Wit  _et al._
(2020), the noise-mitigating properties of normal-metal features Ullom  _et
al._ (2004); Sadleir  _et al._ (2011); Miniussi  _et al._ (2018); Wakeham  _et
al._ (2018), and how the energy deposited in the absorbed is transferred to
the bilayer Kilbourne  _et al._ (2007); Smith  _et al._ (2012, 2014). In this
paper, we expand on this last point by studying how the detailed design of the
coupling between the X-ray absorber and the TES bilayer affects the properties
of the TES micro-calorimeters. The basic layout of the TESs are explained in
Sec. II of this paper.

Previous work has shown that the design of the coupling geometry could have a
big impact on the detector properties. It was demonstrated that the stems
introduce effects of non-equilibrium superconductivity via the proximity
effect, affecting the critical temperature and critical current, and influence
the small-signal transition parameters $\alpha=\frac{T}{R}\frac{\partial
R}{\partial T}$ and $\beta=\frac{I}{R}\frac{\partial R}{\partial I}$, and the
detector energy resolution and time constants Bandler  _et al._ (2008);
Sadleir  _et al._ (2011); Smith  _et al._ (2013, 2014). Note that these
effects were reported for stems with a very large footprint on the bilayer,
while in this work, we study the impact of much smaller stems, which will
allow us to pick up on more subtle differences than those reported before. We
mainly investigate the effect of the diameter of the two stems that connects
the absorber to the bilayer, and the spacing between these two. We measured
the complex impedance and noise, which are used to estimate the internal
thermal conductance in the TES and how this is influenced by the geometry. We
compare the small-signal limit energy resolution, estimated from the measured
noise and transition shape, with the measured Noise Equivalent Power (NEP),
and find a good qualitative agreement between the two.

##  II Detectors and Measurement Setup

Figure 1: (a) Schematic of the layout of the TESs. We vary only two aspects of
the design: the diameter of the absorber stem $D$, and the spacing between the
bilayer and the stem $X$. (b) Overhead view of the 4 different designs
studied: A single design with $D=10~{}\upmu$m and $X=0~{}\upmu$m, and three
designs with $D=5~{}\upmu$m and varying spacing, $X=0$, $5$, and
$10~{}\upmu$m. The green structures are the niobium leads.

We have fabricated an $8\times 8$ TES array with four different geometries for
the absorber-bilayer coupling. The basic layout of the detectors is shown
schematically in Fig. 1(a). The TESs are based on a bilayer made of 35 nm Ti
and 200 nm Au, with a sheet resistance of 26 m$\Omega$/$\square$. The bilayer
is patterned to be an $80\times 10~{}\upmu$m2 rectangle with a normal
resistance of approximately 200 m$\Omega$. Niobium leads are used to apply a
voltage bias to the devices. The bilayers are grown on a 0.5$~{}\upmu$m thick
silicon nitride membrane. The TES is connected to an Au X-ray absorber via two
central absorber stems with a height of 3.5 $\upmu$m, one at each edge of the
bilayer. Four additional stems (not shown in Fig. 1(a)) with diameter 5
$\upmu$m are used to support the absorber on the membrane. The absorber has a
thickness of 2.35 $\upmu$m and a heat capacity of 0.85 pJ/K at 90 mK. Further
details about the TESs and the fabrication procedure are given by Nagayoshi
_et al._ Nagayoshi  _et al._ (2019).

To study how the detailed coupling between the absorber and the bilayer
affects the detector properties, we vary the geometry of the two absorber
stems, as schematically illustrated in Fig. 1(b). We look at the possible
impact of the diameter of the two central stems, by comparing two designs with
diameters $D=10~{}\upmu$m and $D=5~{}\upmu$m, both fabricated such that the
edge of the bilayer is aligned with the edge of the outer perimeter of the
stem. Additionally, we look at whether the precise position of the stems is of
importance for the final detector performance, by comparing three designs with
the same diameter $D=5~{}\upmu$m but different spacing between the edge of the
bilayer and the edge of the stems, $X=0$, $5$, and $10~{}\upmu$m. The arm
connecting the stem and the bilayer is an integral part of the bilayer,
deposited in the same step of the fabrication. All other properties of the
devices are identical to allow for direct comparison.

We characterize the devices using an FDM setup, with which we bias a single
pixel at a time within the superconducting transition using an alternating
current (AC). The resistance of the TES at a particular bias point is
generally specified with respect to the normal resistance $R_{n}$. Each of the
investigated pixels is connected to a high-Q LC resonator Bruijn  _et al._
(2012) with a bias frequency between 1-5 MHz. Further details about the FDM
setup are given elsewhere Akamatsu  _et al._ (2016); De Wit  _et al._ (2020).
Most of the results presented in the remainder of this paper do not depend on
the bias frequency, and are valid for both AC and DC biased pixels. Therefore,
we focus on presenting results from pixels measured at low bias frequencies
($<2~{}$MHz), as these results are typically more straightforward to interpret
and compare to the literature.

The setup is mounted at the mixing-chamber of a cryogen-free dilution
refrigerator, and is stabilized at a temperature of 50 mK. External magnetic
fields are shielded using a room-temperature $\mu$-metal shield and a lead and
cryoperm shield at the cold stage. Both the setup and the cryoperm shield are
suspended from a kevlar vibration isolation stage for mechanical decoupling
Gottardi  _et al._ (2019). A more extensive overview of the setup is given
elsewhere De Wit  _et al._ (2020).

For all devices measurements of the current-voltage characteristic (IV-curve)
at different bath temperatures are used to determine the critical temperature
and thermal conductance of the detectors. The averaged thermal conductance for
all devices is 46.5 pW/K with a standard deviation of 4.6 pW/K at a critical
temperature of $81.8\pm 0.2~{}$mK. Given the fact that the thermal conductance
of this type of devices is generally proportional to the phonon-emitting
perimeter of the TES Hoevers  _et al._ (2005); Hays-Wehle  _et al._ (2016); De
Wit  _et al._ (2020), the thermal conductance is expected to increase for
larger $X$, with a difference of approximately 7 pW/K between the $X=0$ and
$10~{}\upmu$m devices. This predicted difference has not been resolved in the
measured data. We have observed no effect of the stem geometry on the current
distribution, nor in the temperature dependence of the critical current.

##  III Impact of Stem-Bilayer Distance

We fully characterize the properties of the transition by measuring the
complex impedance of the TES Lindeman  _et al._ (2004). For various points in
the transition, a small modulation is added to the bias current with a
frequency ranging from $10~{}$Hz$~{}-~{}50~{}$kHz, where we record the TES
response for each frequency. This data is combined with knowledge of the heat
capacity, which is dominated by the X-ray absorber, and thermal conductance to
extract the dimensionless derivatives of the resistance with respect to the
temperature $\alpha$ and to the current $\beta$. We focus on these parameters,
as they describe the shape of the superconducting transition and can be used
to obtain, among other things, the bandwidth and sensitivity of the detectors.
Details of this measurement and analysis under AC bias are given elsewhere
Taralli  _et al._ (2019); Gottardi  _et al._ (2021). Representative data for
$\alpha$ and $\beta$ for low bias frequency devices with varying length of the
arm between the absorber and the bilayer is shown in Fig. 2.

Figure 2: Results of complex impedance and noise measurements for low bias
frequency pixels with different stem-bilayer separations. $\alpha$ (a) and
$\beta$ (b) extracted from complex impedance measurements for the relevant
part of the normal to superconducting transition. (c) Ratio of
$\alpha$/$\beta$ computed for the same bias points as those shown in (a) and
(b).

All the devices show a similar ratio between $\alpha/\beta$, visible in Fig.
2(c). However, there are clear differences between the devices low in the
resistive transition, as visible in Figs. 2(a) and (b). The $X=5~{}\upmu$m and
$X=10~{}\upmu$m devices show high peaks in both $\alpha$ and $\beta$, signs of
sharp features or kinks in the superconducting transition. No kinks are
observed for the $X=0~{}\upmu$m devices for all measured bias frequencies.
Kinks are regularly observed in devices with normal metal stripes on top of
the TES Smith  _et al._ (2015); Wakeham  _et al._ (2018). Here we show that
also the normal metal of the absorber stems might play a role in the precise
location of these kinks in the transition. For the shortest arms with
$X=0~{}\upmu$m, the kinks are absent or might have shifted to lower resistance
values outside of the region of interest. The removal of kinks from the
relevant region of the parameter space is of utmost importance for achieving
uniform performance in large arrays of TESs.

Figure 3: Example noise spectra measured for the different $D=0~{}\upmu$m
pixels, measured at $R\sim 0.15~{}R_{n}$. Dashed lines represent the
calculated noise models: squid noise (purple), phonon and additional low
frequency noise (orange), voltage noise from RSJ model (green), and ITFN
(red). The total of all noise sources is given by the solid black line.

Apart from the complex impedance, for each bias point we have measured the
current noise spectrum. Example noise spectra for the three $D=5~{}\upmu$m
pixels measured at $R=0.15~{}R_{n}$ are shown by the blue points in Fig. 3.
The roll-off visible at high frequencies results from the bandwidth of the
electrical circuit.

We can compare the measured noise spectra with the calculated contributions
from various noise sources. The principal sources of noise include the white
readout noise of the SQUID, low frequency phonon noise coming from thermal
fluctuations between the TES and the bath (and other low frequency noise
sources such as vibrations and stray photons, all summed under the name
additional low frequency noise), and Johnson noise from the TES resistance. It
has recently been shown that for this type of devices the Johnson noise is
best described by the Resistively-Shunted-Junction (RSJ) model Kozorezov  _et
al._ (2012); Gottardi  _et al._ (2021); Wessels  _et al._ (2021), given by:

$S_{V,RSJ}=4k_{B}TR\left(1+\frac{5}{2}\beta+\frac{3}{2}\beta^{2}\right),$ (1)

with $k_{B}$ being the Boltzmann constant. These main sources of noise are
shown as the various dashed lines in Fig. 3. However, the sum of these noise
sources is clearly insufficient to fully explained the measured noise.
Therefore, we express the factor of difference between the measured noise and
the predicted noise using the parameter $M^{2}$, defined by:

$M^{2}=\left(S_{V,data}-S_{V,RSJ}\right)/S_{V,data}$ (2)

Fig. 4(a) shows $M^{2}$ for various bias points in the transition for the
different designs, showing a clear rise in $M^{2}$ as $X$ increases.

In order to explain the additional noise in the kHz range, we interpret it as
resulting from internal thermal fluctuation noise (ITFN) within the detector
itself Wakeham  _et al._ (2019); Gottardi  _et al._ (2021). ITFN occurs when
there are distributed heat capacities inside the detector connected by
internal thermal impedances. Such a complex thermal structure results in
thermal fluctuations between the different heat capacities. To confirm this
hypothesis we add another noise contribution coming from the ITFN, given by
$S_{P,\rm{itfn}}=4k_{B}T^{2}G_{\rm{itfn}}$, where $G_{\rm{itfn}}$ is a thermal
conductance located within the TES Palosaari  _et al._ (2012); Maasilta
(2012). This contribution is shown as the red dashed lines in Fig. 3. Note
that the shape of the ITFN is similar to the RSJ Johnson noise due to the
conversion of power to current noise via the power-to-current responsivity and
the TES circuit inductance. $G_{\rm{itfn}}$ is treated as a free parameter to
be varied such that $M^{2}$ is minimized. The resulting $G_{\rm{itfn}}$ is
shown in Fig. 4(b). As expected, the internal thermal conductance decreases as
$X$ increases. At $R/R_{n}\sim~{}20$%, the extracted $G_{iftn}$ increases from
$\sim 14~{}$nW/K for $X=10~{}\upmu$m to $\sim 18~{}$nW/K for $X=5~{}\upmu$m
and finally to $\sim 27~{}$nW/K for $X=0~{}\upmu$m.

Figure 4: (a) Excess noise factor defined with respect to expected noise in
the RSJ model $S_{V,RSJ}$ for the same bias points show in Fig. 2. (b)
$G_{\rm{itfn}}$ required to minimize the excess noise factor for each bias
point.

The internal thermal conductance is believed to be a combination of the
internal thermal conductance of the bilayer $G_{bl}$, and the limited thermal
conductance of the arm between the absorber stem and the bilayer, $G_{arm}$.
Using the simplest possible thermal model in which the different thermal
components are connected in parallel, $G_{arm}$ can be estimated from the
measured $G_{\rm{itfn}}$ using:

$G_{arm}=\left(\frac{1}{G_{\rm{itfn}}}-\frac{1}{G_{bl}}\right)^{-1}$ (3)

The internal thermal conductance of the bilayer is estimated by using the
Wiedemann-Franz law, $G_{bl}=L_{0}T/R_{\square}$, with $L_{0}=2.45\cdot
10^{-8}~{}\Omega WK^{-2}$ and $R_{\square}$ the sheet resistance of the
bilayer Gottardi  _et al._ (2021). For the $80\times 10\upmu$m2 devices,
$G_{bl}=77$ nW/K. The estimated $G_{arm}$ is shown as the magenta points in
Fig. 5.

To understand the origin of the limited thermal conductance of the arm, we
estimate it in two limiting regimes. The upper limit is given by the normal
state thermal conductance of the arm $G_{NS}$, which again is estimated using
the Wiedemann-Franz law. In this case the resistance of the arm is calculated
using an effective length $X+D/2$, the full length of the arm including the
end part covered by the absorber stem. This upper limit is shown in Fig. 5 as
the red dashed lines. The lower limit is obtained by assuming that the arm
remains superconducting. As cooper pairs cannot transport heat the thermal
conductance is purely governed by the fraction of depaired electrons, which is
dictated by the temperature dependent superconducting gap $\Delta(T)$ and the
thermal excitation energy $k_{B}T$. Thus, the thermal conductance can be
estimated using:

$G_{SC}=G_{NS}~{}\exp{\left(-\frac{\Delta(T)}{k_{B}T}\right)}$ (4)

In order to evaluate this equation knowledge about the temperature dependence
of the superconducting gap is required. For a conventional s-wave
superconductor at $T\approx T_{C}$, this dependence is given by

$\Delta(T)\approx 1.74\Delta(0)\sqrt{1-\frac{T}{T_{C}}},$ (5)

with $\Delta(0)=1.764~{}k_{B}T_{C}$ the gap at T = 0 Tinkham (1996). However,
this is not valid in the presence of the proximity effect, which significantly
alters the size of the gap close to the $T_{C}$ of the bilayer. Therefore, we
simplify the problem and derive a lower limit for the thermal conductance by
neglecting the temperature dependence of the gap and using
$\Delta(T)=\Delta(0)$. The resulting $G_{SC}$ is shown as the blue dashed
lines in Fig. 5. Indeed the values for $G_{arm}$ extracted from the
experimental data fall between these two boundaries, supporting the
interpretation that the arm partially remains superconducting with a
$\Delta(T)$ that is suppressed compared to the ideal $T=0$ value, but remains
finite.

Figure 5: Estimated value of $G_{arm}$, assuming a constant bilayer thermal
conductance given by the Wiedemann-Franz law (green dashed line). Blue dashed
line shows the arm thermal conductance when it remains superconducting
(ignoring the temperature dependence of the gap). The red dashed line shows
the arm thermal conductance when it is in the normal state (and thus follows
the Wiedemann-Franz law). We take the effective length of the arm equal to
$X+D/2$.

The previous analysis clearly indicates that a small but significant reduction
of the excess noise can be achieved by keeping the stem as close to the
bilayer as possible, or perhaps by even placing the stems on top of the
bilayerWakeham  _et al._ (2020). Note that after reducing the ITFN, the
performance of this type of devices under these conditions will remain limited
by the noise resulting from the weak-link effect as described by the RSJ
model.

##  IV Impact of Stem-Bilayer Diameter

The main effects of the stem diameter are visible in the measured $\alpha$,
shown in Fig. 6, and excess noise, shown in Fig. 7. In these figures, we focus
on the designs with $X=0~{}\upmu$m. Three different sets of pixels are
selected, measured at (a) low, (b) mid, and (c) high bias frequencies, the
precise values of which are indicated in the legend of Fig. 6.

Figure 6: Measured $\alpha$ versus $R/R_{n}$ for pixels with different stem
diameters. We show three sets of pixels at different bias frequencies: (a) 1.1
- 1.5 MHz, (b) 2.4 - 2.6 MHz, and (c) 4.1 - 4.3 MHz.

Changing the stem diameter from $D=10$ to $D=5~{}\upmu$m does not change the
overall trend of $\alpha$. However, there is a striking difference between how
$\alpha$, and consequently $\beta$ and $M^{2}$, changes throughout the
transition. The $D=10~{}\upmu$m devices show large oscillations of $\alpha$
for different bias points. These oscillations are typically attributed to the
weak-link effect in these detectors Gottardi  _et al._ (2018); De Wit  _et
al._ (2020) resulting from the interaction between the high $T_{C}$ Nb leads
and the low $T_{C}$ bilayer. However, the amplitude of the oscillations is
much smaller for all $D=5~{}\upmu$m devices, except for those measured at the
highest bias frequencies as visible in Fig. 6(c). In order to confirm this
observation and to understand the underlying mechanism for why the smaller
stem diameter improves the smoothness of the superconducting transition, more
devices with different stem diameters should be studied.

Another interesting feature of the smaller stem diameter is visible in the
excess noise in Fig. 7(a). For all the devices studied in this work with
$D=10~{}\upmu$m, a peak in the excess noise is visible between
$R=0.2~{}-~{}0.5R_{n}$. This peak is never observed for any of the devices
with $D=5~{}\upmu$m. These devices do show a bump in the transition around
$R\approx 0.1R_{n}$, visible in Fig. 2, but in those cases the increase in
alpha is correlated with an increase in $M^{2}$. This is not the case for the
$D=10~{}\upmu$m devices, a fact that becomes clear when looking at Fig. 7(b),
in which we plot $M^{2}/\alpha$. The stark difference in the ratio of $M^{2}$
and $\alpha$ between the $D=10~{}\upmu$m data (red points) and the other data
signifies the difference in nature of the bump in the excess noise.

Figure 7: (a) Excess noise $M^{2}$ and (b) $M^{2}$ / $\alpha$ for low freq
pixels with different stem diameters. In (a), data are shown for the
$X=0,~{}D=10~{}\upmu$m (red) and $X=0,~{}D=5~{}\upmu$m (green) devices. In
(b), also the data for the $X=5~{}\upmu$m (blue) and $X=10~{}\upmu$m (yellow)
devices are shown as a reference.

The exact origin for the excess noise bumps is unclear, but we believe that
it’s related to a scattering or interference effect of the quasi-particles
interacting with the normal metal stems. A new set of devices is currently
under fabrication to study these effects in more detail.

##  V NEP and energy resolution

Arguably the most important figure of merit for a micro-calorimeter is the
energy resolution, i.e. the capacity to resolve different photon energies. In
this section, we demonstrate the effect of the absorber coupling geometry in
two ways. First, we evaluate the integrated noise-equivalent power (NEP) at
5.9 keV for bias points between $R=0.05~{}-~{}0.30R_{n}$. The NEP is generally
a good predictor for the achievable energy resolution in the large-signal
limit. Secondly, we can use $\alpha$, $\beta$, and $M^{2}$, as shown in Secs.
III and IV, to calculate the expected energy resolution in the small-signal
limit for each bias point, using:

$\Delta
E=2\sqrt{2\ln{2}}\sqrt{4k_{B}T^{2}\frac{C}{\alpha}\sqrt{\left(1+\frac{5}{2}\beta+\frac{3}{2}\beta^{2}\right)\left(1+M^{2}\right)}}$
(6)

The results for the NEP and expected energy resolution are shown in Fig. 8 for
four different pixels, one of each design, measured at low bias frequencies.
In absolute terms, a relatively large difference is observed between the NEP
and estimated $\Delta E$. This is most likely the result from the fact that
our detectors are working outside of the small-signal limit for 5.9 keV
X-rays. Note that the increase in NEP and $\Delta E$ seen in the
$X=0~{}\upmu$m device at higher $R/R_{n}$ is caused by the bump in the excess
noise shown in Fig. 7.

Figure 8: (a) Measured NEP and (b) small-signal limit energy resolution
calculated using Eq. 6. The NEP is measured at the optimal bias frequency for
each value of $R/R_{n}$.

The NEP and $\Delta E$ qualitatively give the same results, and confirm the
trends previously identified. The best energy resolutions are expected for the
devices with $X=0~{}\upmu$m. As the length of the arm between the absorber and
the bilayer increases, the energy resolution becomes worse. This observation
holds over the full bias frequency range. The increase in the energy
resolution is about $0.2~{}-~{}0.4$ eV, and clearly indicates that maximizing
the thermal conductance between the absorber and the bilayer is of great
importance to optimize the resolution. The effect of the stem diameter is much
less pronounced, with only a very small reduction of less than 0.1 eV for both
the NEP and $\Delta E$. This is not completely unexpected. If we assume that
the gold in the absorber stem has a electrical resistivity
$\rho_{Au}=1$n$\Omega$m, similar to previously reported values Wakeham  _et
al._ (2019), the Wiedemann-Franz law can be used to estimate a thermal
conductance of the stem (with height 3.5 $\upmu$m). This leads to
$G_{stem}=11~{}\upmu$W/K and $45~{}\upmu$W/K for the $5~{}\upmu$m and
$10~{}\upmu$m diameter stems, respectively. For both diameters, $G_{stem}$ is
much higher that the estimated thermal conductance of the arm, and therefore
does not play a significant role in the noise.

##  VI Conclusions

We have studied the impact of the detailed geometry for the thermal coupling
between the X-ray absorber and the bilayer, with a particular focus on the
diameter of the two stems that connect the absorber to the TES bilayer, and
the spacing between the stems and the bilayer.

Based on measurements of the noise, we found that the noise is reduced for
shorter arms, an observation that we explain in terms of a higher internal
thermal conductance between the absorber and the TES. Additionally, we observe
that the position of kinks in the superconducting transition is affected by
the spacing between the stems. While a more thorough investigation is required
to fully understand the responsible mechanism, the effect appears to be
related to previously reported observations in devices with added normal-metal
structures on top of the bilayer. Comparing the devices with different stem
diameters resulted in the striking observation that the smoothness of the
transition seems to be significantly improved for the smaller stem diameter.
Testing of additional devices of different geometry should help to confirm
this observation. These devices are currently under fabrication.

The conclusion of this study is that the stems should be placed as close to
the bilayer as possible in order to reduce the internal thermal fluctuation
noise, leading to an improved energy resolution of up to 0.4 eV. This
observation is consistent with the results from similar studies Wakeham  _et
al._ (2020). The optimization of the stem diameter is more complicated. For
the energy resolution, increasing the diameter could further reduce the ITFN
and give better energy resolutions in certain bias points (even though the
effect will be quite modest). However, our results suggest that reducing the
stem diameter, or perhaps only using a single supporting stem, might greatly
improve the smoothness and uniformity of the normal-to-superconducting
transition, a feature that is of great importance for the successful operation
of large number of TESs in any multiplexing scheme.

##  VII Acknowledgement

SRON is supported financially by NWO, the Netherlands Organization for
Scientific Research. This work is partly funded by European Space Agency (ESA)
and coordinated with other European efforts under ESA CTP contract ITT
AO/1-7947/14/NL/BW.

##  VIII Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## References

  * Irwin and Hilton (2005) K. Irwin and G. Hilton, Transition-Edge Sensors, in  _Topics Appl. Phys._, Vol. 99 (2005) pp. 63–150.
  * Ullom and Bennett (2015) J. N. Ullom and D. A. Bennett, Review of superconducting transition-edge sensors for x-ray and gamma-ray spectroscopy, Superconductor Science and Technology 28, 084003 (2015).
  * Gottardi and Nagayashi (2021) L. Gottardi and K. Nagayashi, A review of X-ray microcalorimeters based on superconducting transition edge sensors for astrophysics and particle physics, Applied Sciences (Switzerland) 11, 10.3390/app11093793 (2021).
  * Barret  _et al._ (2020) D. Barret, A. Decourchelle, A. Fabian, M. Guainazzi, K. Nandra, R. Smith, and J.-W. den Herder, The Athena space X-ray observatory and the astrophysics of hot plasma†, Astronomische Nachrichten 341, 224 (2020), arXiv:1912.04615 .
  * Gaskin and Swartz (2019) J. A. Gaskin and D. A. Swartz, Lynx X-Ray Observatory: an overview, Journal of Astronomical Telescopes, Instruments, and Systems 5, 1 (2019).
  * Cui  _et al._ (2020) W. Cui, L. B. Chen, B. Gao, F. L. Guo, H. Jin, G. L. Wang, L. Wang, J. J. Wang, W. Wang, Z. S. Wang, Z. Wang, F. Yuan, and W. Zhang, HUBS: Hot Universe Baryon Surveyor, Journal of Low Temperature Physics 199, 502 (2020).
  * Durkin  _et al._ (2019) M. Durkin, J. S. Adams, S. R. Bandler, J. A. Chervenak, S. Chaudhuri, C. S. Dawson, E. V. Denison, W. B. Doriese, S. M. Duff, F. M. Finkbeiner, C. T. Fitzgerald, J. W. Fowler, J. D. Gard, G. C. Hilton, K. D. Irwin, Y. I. Joe, R. L. Kelley, C. A. Kilbourne, A. R. Miniussi, K. M. Morgan, G. C. O’Neil, C. G. Pappas, F. S. Porter, C. D. Reintsema, D. A. Rudman, K. Sakai, S. J. Smith, R. W. Stevens, D. S. Swetz, P. Szypryt, J. N. Ullom, L. R. Vale, N. A. Wakeham, J. C. Weber, and B. A. Young, Demonstration of athena X-IFU compatible 40-row time-division-multiplexed readout, IEEE Transactions on Applied Superconductivity 29, 1 (2019).
  * Akamatsu  _et al._ (2020) H. Akamatsu, L. Gottardi, J. van der Kuur, C. P. de Vries, M. P. Bruijn, J. A. Chervenak, M. Kiviranta, A. J. van den Linden, B. D. Jackson, A. R. Miniussi, K. Ravensberg, K. Sakai, S. J. Smith, and N. Wakeham, Progress in the Development of Frequency-Domain Multiplexing for the X-ray Integral Field Unit on Board the Athena Mission, Journal of Low Temperature Physics 199, 737 (2020).
  * Morgan  _et al._ (2016) K. M. Morgan, B. K. Alpert, D. A. Bennett, E. V. Denison, W. B. Doriese, J. W. Fowler, J. D. Gard, G. C. Hilton, K. D. Irwin, Y. I. Joe, G. C. O’Neil, C. D. Reintsema, D. R. Schmidt, J. N. Ullom, and D. S. Swetz, Code-division-multiplexed readout of large arrays of TES microcalorimeters, Applied Physics Letters 109, 112604 (2016).
  * Bennett  _et al._ (2019) D. A. Bennett, J. A. B. Mates, S. R. Bandler, D. T. Becker, J. W. Fowler, J. D. Gard, G. C. Hilton, K. D. Irwin, K. M. Morgan, C. D. Reintsema, K. Sakai, D. R. Schmidt, S. J. Smith, D. S. Swetz, J. N. Ullom, L. R. Vale, and A. L. Wessels, Microwave SQUID multiplexing for the Lynx x-ray microcalorimeter, Journal of Astronomical Telescopes, Instruments, and Systems 5, 1 (2019).
  * Nakashima  _et al._ (2020) Y. Nakashima, F. Hirayama, S. Kohjiro, H. Yamamori, S. Nagasawa, A. Sato, S. Yamada, R. Hayakawa, N. Y. Yamasaki, K. Mitsuda, K. Nagayoshi, H. Akamatsu, L. Gottardi, E. Taralli, M. P. Bruijn, M. L. Ridder, J. R. Gao, and J. W. Den Herder, Low-noise microwave SQUID multiplexed readout of 38 x-ray transition-edge sensor microcalorimeters, Applied Physics Letters 117, 3 (2020).
  * Pajot  _et al._ (2018) F. Pajot, D. Barret, T. Lam-Trong, J.-W. den Herder, L. Piro, M. Cappi, J. Huovelin, R. Kelley, J. M. Mas-Hesse, K. Mitsuda, S. Paltani, G. Rauw, A. Rozanska, J. Wilms, M. Barbera, F. Douchin, H. Geoffray, R. H. den Hartog, C. Kilbourne, M. Le Du, C. Macculi, J. M. Mesnager, and P. Peille, The Athena X-ray Integral Field Unit (X-IFU), Journal of Low Temperature Physics 193, 901 (2018).
  * Hays-Wehle  _et al._ (2016) J. P. Hays-Wehle, D. R. Schmidt, J. N. Ullom, and D. S. Swetz, Thermal Conductance Engineering for High-Speed TES Microcalorimeters, Journal of Low Temperature Physics 184, 492 (2016).
  * Morgan  _et al._ (2017) K. M. Morgan, C. G. Pappas, D. A. Bennett, J. D. Gard, J. P. Hays-Wehle, G. C. Hilton, C. D. Reintsema, D. R. Schmidt, J. N. Ullom, and D. S. Swetz, Dependence of transition width on current and critical current in transition-edge sensors, Applied Physics Letters 110, 2 (2017).
  * Zhang  _et al._ (2019) X. Zhang, S. M. Duff, G. C. Hilton, P. J. Lowell, K. M. Morgan, D. R. Schmidt, and J. N. Ullom, Controlling the thermal conductance of silicon nitride membranes at 100 mK temperatures with patterned metal features, Applied Physics Letters 115, 052601 (2019).
  * Gottardi  _et al._ (2017) L. Gottardi, H. Akamatsu, J. van der Kuur, S. J. Smith, A. Kozorezov, and J. Chervenak, Study of TES-Based Microcalorimeters of Different Size and Geometry Under AC Bias, IEEE Transactions on Applied Superconductivity 27, 1 (2017).
  * Sakai  _et al._ (2018) K. Sakai, J. S. Adams, S. R. Bandler, J. A. Chervenak, A. M. Datesman, M. E. Eckart, F. M. Finkbeiner, R. L. Kelley, C. A. Kilbourne, A. R. Miniussi, F. S. Porter, J. S. Sadleir, S. J. Smith, N. A. Wakeham, E. J. Wassell, W. Yoon, H. Akamatsu, M. P. Bruijn, L. Gottardi, B. D. Jackson, J. van der Kuur, B. J. van Leeuwen, A. J. van der Linden, H. J. van Weers, and M. Kiviranta, Study of Dissipative Losses in AC-Biased Mo/Au Bilayer Transition-Edge Sensors, Journal of Low Temperature Physics 193, 356 (2018).
  * Morgan  _et al._ (2019) K. M. Morgan, D. T. Becker, D. A. Bennett, W. B. Doriese, J. D. Gard, K. D. Irwin, S. J. Lee, D. Li, J. A. Mates, C. G. Pappas, D. R. Schmidt, C. J. Titus, D. D. Van Winkle, J. N. Ullom, A. Wessels, and D. S. Swetz, Use of Transition Models to Design High Performance TESs for the LCLS-II Soft X-Ray Spectrometer, IEEE Transactions on Applied Superconductivity 29, 1 (2019).
  * De Wit  _et al._ (2020) M. De Wit, L. Gottardi, E. Taralli, K. Nagayoshi, M. L. Ridder, H. Akamatsu, M. P. Bruijn, M. D’Andrea, J. Van Der Kuur, K. Ravensberg, D. Vaccaro, S. Visser, J. R. Gao, and J. W. Den Herder, High aspect ratio transition edge sensors for x-ray spectrometry, Journal of Applied Physics 128, 10.1063/5.0029669 (2020).
  * Ullom  _et al._ (2004) J. N. Ullom, W. B. Doriese, G. C. Hilton, J. A. Beall, S. Deiker, W. D. Duncan, L. Ferreira, K. D. Irwin, C. D. Reintsema, and L. R. Vale, Characterization and reduction of unexplained noise in superconducting transition-edge sensors, Applied Physics Letters 84, 4206 (2004).
  * Sadleir  _et al._ (2011) J. E. Sadleir, S. J. Smith, I. K. Robinson, F. M. Finkbeiner, J. A. Chervenak, S. R. Bandler, M. E. Eckart, and C. A. Kilbourne, Proximity effects and nonequilibrium superconductivity in transition-edge sensors, Physical Review B 84, 184502 (2011).
  * Miniussi  _et al._ (2018) A. R. Miniussi, J. S. Adams, S. R. Bandler, J. A. Chervenak, A. M. Datesman, M. E. Eckart, A. J. Ewin, F. M. Finkbeiner, R. L. Kelley, C. A. Kilbourne, F. S. Porter, J. E. Sadleir, K. Sakai, S. J. Smith, N. A. Wakeham, E. J. Wassell, and W. Yoon, Performance of an X-ray Microcalorimeter with a 240 $\mu$m Absorber and a 50 $\mu$m TES Bilayer, Journal of Low Temperature Physics 193, 337 (2018).
  * Wakeham  _et al._ (2018) N. A. Wakeham, J. S. Adams, S. R. Bandler, J. A. Chervenak, A. M. Datesman, M. E. Eckart, F. M. Finkbeiner, R. L. Kelley, C. A. Kilbourne, A. R. Miniussi, F. S. Porter, J. E. Sadleir, K. Sakai, S. J. Smith, E. J. Wassell, and W. Yoon, Effects of Normal Metal Features on Superconducting Transition-Edge Sensors, Journal of Low Temperature Physics 193, 231 (2018).
  * Kilbourne  _et al._ (2007) C. A. Kilbourne, S. R. Bandler, A. D. Brown, J. A. Chervenak, E. Figueroa-Feliciano, F. M. Finkbeiner, N. Iyomoto, R. L. Kelley, F. S. Porter, and S. J. Smith, Uniform high spectral resolution demonstrated in arrays of TES x-ray microcalorimeters, UV, X-Ray, and Gamma-Ray Space Instrumentation for Astronomy XV 6686, 668606 (2007).
  * Smith  _et al._ (2012) S. J. Smith, J. S. Adams, M. E. Eckart, C. N. Bailey, S. R. Bandler, J. A. Chervenak, F. M. Finkbeiner, R. L. Kelley, C. A. Kilbourne, F. S. Porter, and J. E. Sadleir, Small pitch transition-edge sensors with broadband high spectral resolution for solar physics, Journal of Low Temperature Physics 167, 168 (2012).
  * Smith  _et al._ (2014) S. J. Smith, J. S. Adams, S. R. Bandler, S. E. Busch, J. A. Chervenak, M. E. Eckart, F. M. Finkbeiner, R. L. Kelley, C. A. Kilbourne, S. J. Lee, J. P. Porst, F. S. Porter, and J. E. Sadleir, Characterization of Mo/Au transition-edge sensors with different geometric configurations, Journal of Low Temperature Physics 176, 356 (2014).
  * Bandler  _et al._ (2008) S. R. Bandler, R. P. Brekosky, A. D. Brown, J. A. Chervenak, E. Figueroa-Feliciano, F. M. Finkbeiner, N. Iyomoto, R. L. Kelley, C. A. Kilbourne, F. S. Porter, J. Sadleir, and S. J. Smith, Performance of TES X-ray microcalorimeters with a novel absorber design, Journal of Low Temperature Physics 151, 400 (2008).
  * Smith  _et al._ (2013) S. J. Smith, J. S. Adams, C. N. Bailey, S. R. Bandler, S. E. Busch, J. A. Chervenak, M. E. Eckart, F. M. Finkbeiner, C. A. Kilbourne, R. L. Kelley, S.-J. Lee, J.-P. Porst, F. S. Porter, and J. E. Sadleir, Implications of weak-link behavior on the performance of Mo/Au bilayer transition-edge sensors, Journal of Applied Physics 114, 074513 (2013).
  * Nagayoshi  _et al._ (2019) K. Nagayoshi, M. L. Ridder, M. P. Bruijn, L. Gottardi, E. Taralli, P. Khosropanah, H. Akamatsu, S. Visser, and J.-R. Gao, Development of a Ti/Au TES Microcalorimeter Array as a Backup Sensor for the Athena/X-IFU Instrument, Journal of Low Temperature Physics 199, 943 (2019).
  * Bruijn  _et al._ (2012) M. P. Bruijn, L. Gottardi, R. H. den Hartog, H. F. C. Hoevers, M. Kiviranta, P. A. J. de Korte, and J. van der Kuur, High-Q LC Filters for FDM Read out of Cryogenic Sensor Arrays, Journal of Low Temperature Physics 167, 695 (2012).
  * Akamatsu  _et al._ (2016) H. Akamatsu, L. Gottardi, J. van der Kuur, C. P. de Vries, K. Ravensberg, J. S. Adams, S. R. Bandler, M. P. Bruijn, J. A. Chervenak, C. A. Kilbourne, M. Kiviranta, A. J. van der Linden, B. D. Jackson, and S. J. Smith, Development of frequency domain multiplexing for the X-ray Integral Field unit (X-IFU) on the Athena, Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray 9905, 99055S (2016).
  * Gottardi  _et al._ (2019) L. Gottardi, H. J. van Weers, J. Dercksen, H. Akamatsu, M. P. Bruijn, J. R. Gao, B. D. Jackson, P. Khosropanah, J. van der Kuur, K. Ravensberg, and M. Ridder, A six-degree-of-freedom micro-vibration acoustic isolator for low-temperature radiation detectors based on superconducting transition-edge sensors, Review of Scientific Instruments 90, 10.1063/1.5088364 (2019).
  * Hoevers  _et al._ (2005) H. F. Hoevers, M. L. Ridder, A. Germeau, M. P. Bruijn, P. A. De Korte, and R. J. Wiegerink, Radiative ballistic phonon transport in silicon-nitride membranes at low temperatures, Applied Physics Letters 86, 1 (2005).
  * Lindeman  _et al._ (2004) M. A. Lindeman, S. Bandler, R. P. Brekosky, J. A. Chervenak, E. Figueroa-Feliciano, F. M. Finkbeiner, M. J. Li, and C. A. Kilbourne, Impedance measurements and modeling of a transition-edge-sensor calorimeter, Review of Scientific Instruments 75, 1283 (2004).
  * Taralli  _et al._ (2019) E. Taralli, P. Khosropanah, L. Gottardi, K. Nagayoshi, M. L. Ridder, M. P. Bruijn, and J. R. Gao, Complex impedance of TESs under AC bias using FDM readout system, AIP Advances 9, 045324 (2019).
  * Gottardi  _et al._ (2021) L. Gottardi, M. de Wit, E. Taralli, K. Nagayashi, and A. Kozorezov, Voltage Fluctuations in ac Biased Superconducting Transition-Edge Sensors, Physical Review Letters 126, 217001 (2021).
  * Smith  _et al._ (2015) S. J. Smith, J. S. Adams, S. R. Bandler, G. Betancourt-Martinez, J. A. Chervenak, M. E. Eckart, F. M. Finkbeiner, R. L. Kelley, C. A. Kilbourne, S. J. Lee, F. S. Porter, J. E. Sadleir, and E. J. Wassell, Uniformity of kilo-pixel arrays of transition-edge sensors for X-ray astronomy, IEEE Transactions on Applied Superconductivity 25, 10.1109/TASC.2014.2369352 (2015).
  * Kozorezov  _et al._ (2012) A. Kozorezov, A. A. Golubov, D. D. E. Martin, P. A. J. de Korte, M. A. Lindeman, R. A. Hijmering, J. van der Kuur, H. F. C. Hoevers, L. Gottardi, M. Y. Kupriyanov, and J. K. Wigmore, Electrical Noise in a TES as a Resistively Shunted Conducting Junction, Journal of Low Temperature Physics 167, 108 (2012).
  * Wessels  _et al._ (2021) A. Wessels, K. Morgan, J. D. Gard, G. C. Hilton, J. A. B. Mates, C. D. Reintsema, D. R. Schmidt, D. S. Swetz, J. N. Ullom, L. R. Vale, and D. A. Bennett, A model for excess Johnson noise in superconducting transition-edge sensors, Applied Physics Letters 118, 202601 (2021).
  * Wakeham  _et al._ (2019) N. A. Wakeham, J. S. Adams, S. R. Bandler, S. Beaumont, J. A. Chervenak, A. M. Datesman, M. E. Eckart, F. M. Finkbeiner, R. Hummatov, R. L. Kelley, C. A. Kilbourne, A. R. Miniussi, F. S. Porter, J. E. Sadleir, K. Sakai, S. J. Smith, and E. J. Wassell, Thermal fluctuation noise in Mo/Au superconducting transition-edge sensor microcalorimeters, Journal of Applied Physics 125, 164503 (2019).
  * Palosaari  _et al._ (2012) M. R. Palosaari, K. M. Kinnunen, I. J. Maasilta, M. Ridder, J. van der Kuur, and H. F. Hoevers, Analysis of impedance and noise data of an x-ray transition-edge sensor using complex thermal models, Journal of Low Temperature Physics 167, 129 (2012).
  * Maasilta (2012) I. J. Maasilta, Complex impedance, responsivity and noise of transition-edge sensors: Analytical solutions for two- and three-block thermal models, AIP Advances 2, 10.1063/1.4759111 (2012).
  * Tinkham (1996) M. Tinkham,  _Introduction to superconductivity_ , 2nd ed. (McGraw-Hill, Inc., 1996).
  * Wakeham  _et al._ (2020) N. A. Wakeham, J. Adams, S. Bandler, S. Beaumont, M. Chang, J. Chervenak, M. Eckart, F. Finkbeiner, J. Ha, R. Hummatov, R. L. Kelley, C. A. Kilbourne, A. R. Miniussi, H. Muramatsu, F. S. Porter, J. E. Sadleir, K. Sakai, S. J. Smith, and E. J. Wassell, The impact of transition-edge sensor design on internal thermal fluctuation noise and thermal conductance, Poster Presentation ASC (2020).
  * Gottardi  _et al._ (2018) L. Gottardi, S. J. Smith, A. Kozorezov, H. Akamatsu, J. van der Kuur, S. R. Bandler, M. P. Bruijn, J. A. Chervenak, J. R. Gao, R. H. den Hartog, B. D. Jackson, P. Khosropanah, A. R. Miniussi, K. Nagayoshi, M. Ridder, J. E. Sadleir, K. Sakai, and N. Wakeham, Josephson Effects in Frequency-Domain Multiplexed TES Microcalorimeters and Bolometers, Journal of Low Temperature Physics 193, 209 (2018).


#  $f(T)$ Cosmology with Nonzero Curvature

Andronikos Paliathanasis<EMAIL_ADDRESS>Institute of Systems Science,
Durban University of Technology, Durban 4000, South Africa  Instituto de
Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia
5090000, Chile

###### Abstract

We investigate exact and analytic solutions in $f\left(T\right)$ gravity
within the context of a Friedmann–Lemaître–Robertson–Walker background space
with nonzero spatial curvature. For the power law theory
$f\left(T\right)=T^{n}$ we find that the field equations admit an exact
solution with a linear scalar factor for negative and positive spatial
curvature. That Milne-like solution is asymptotic behaviour for the scale
factor near the initial singularity for the model
$f\left(T\right)=T+f_{0}T^{n}-2\Lambda$. The analytic solution for that
specific theory is presented in terms of Painlevé Series for $n>1$. Moreover,
from the value of the resonances of the Painlevé Series we conclude that the
Milne-like solution is always unstable while for large values of the
indepedent parameter, the field equations provide an expanding universe with a
de Sitter expansion of a positive cosmological constant. Finally, the presence
of the cosmological term $\Lambda$ in the studied $f\left(T\right)$ model
plays no role in the general behavior of the cosmological solution and the
universe immerge in a de Sitter expansion either when the cosmological
constant term $\Lambda$ in the $f\left(T\right)$ model vanishes.

Teleparallel cosmology; exact solutions; open universe

###### pacs:

98.80.-k, 95.35.+d, 95.36.+x

##  I Introduction

Teleparallel theory of gravity ein28 ; cc  and its modifications fer1 ; fer2 ;
fer3 ; fer4 ; fer5 ; fer6 ; fer7 ; myr11  have attracted the attention of
cosmologists over the last year because it can provide a geometric explanation
for the explanation of the recent observations sp1 ; fg5 , for a recent review
we refer the reader to rew1 . The modified teleparallel theories of gravity
belong to the family of theories in the Lorentz symmetry is violated li1 ; lv1
. In teleparallelism, the fundamental connection is the curvature-less
Weitzenböck connection Weitzenb23  while the torsion scalar $T$ is used for
the definition of the gravitational Action Integral Hayashi79 . In contrary to
General Relativity in which the Levi-Civita connection and the Ricciscalar $R$
are the fundamental geometric objects of the theory.

When the gravitational Action Integral is linear on the torsion scalar $T$,
then the theory is equivalent to General Relativity and it is know as TEGR.
However, there are various originally proposed gravitational Lagrangians such
is the Teleparallel dark energy theory, the $f\left(T\right)$ theory and its
extensions, see for instance fer5 ; fer6 ; fer7 ; myr11 ; ss1 ; ss2 ; mm1 ;
mm2 ; mm3 ; mm4 .

In this study we consider the $f\left(T\right)$ gravity in a
Friedmann–Lemaître–Robertson–Walker background space. The theory has been
proposed originally as a geometric dark energy candidate in a spatially flat
background space fer1 . However, during the last years it has been found that
it can explain various eras of the cosmological history. In $f\left(T\right)$
cosmology, the field equations are of second-order and they have only one
dependent variable, the scale factor of the underlying geometry. Recently, in
ftcur  the case of nonzero spatial curvature for the background space has been
considered. The authors investigated the existence of bounces and static
solutions, while the conditions for the existence of these solutions were
investigated.

We focus on the existence of exact and analytic solutions for the cosmological
field equations in $f\left(T\right)$ cosmology where the spatial curvature is
nonzero. A background space with nonzero spatial curvature is not excluded by
the inflationary scenario in1 ; in1a . Indeed, inflation is immune to negative
curvature while the energy density which corresponds to the curvature can be
nonzero in the pre-inflationary era in2 ; in3 . Moreover, we shall focus on
the existence of the de Sitter expansion as described by a positive
cosmological constant. Such an expansion is necessary because it provides a
rapid expansion for the size of the universe such that the latter effectively
loses its memory on the initial conditions, which means that the de Sitter
expansion solves the “flatness”, “horizon” and monopole problem f1 ; f2 . The
cosmic “no-hair” conjecture states that all expanding universes with a
positive cosmological constant admit as an asymptotic solution the de Sitter
universe nh1 ; nh2 . The plan of the paper is as follows.

In Section II, we present the field equations for the cosmological model of
our consideration. In Section III we consider the power-law
$f\left(T\right)=T^{n}$ theory where we find a generalized Milne solution that
exists for the field equations for negative and positive spatial curvature. A
more general function $f\left(T\right)$ is considered in Section IV.
Specifically we select the $f\left(T\right)=T+f_{0}T^{n}-2\Lambda$ model,
which has been proposed as a dark energy candidate. We write the field
equations where we find that they admit a movable singularity. Near the
singularity and for $n>1$ the field equations are dominated by the $T^{n}$
term which means that the generalized Milne solution describes the scale
factor at the singularity. Furthermore, we study if the field equations admit
the Painlevé property. We find that for $n>1$ the field equations pass the
Painlevé test and the resonances give that the analytic solution can be
expressed by a Right Painlevé Series. The latter indicates an expanding
universe which leads to a de Sitter expansion. Finally, in Section V we
summarize our results and we draw our conclusions.

##  II $f\left(T\right)$ Cosmology with nonzero curvature

The modified teleparallel modified theory of gravity known as
$f\left(T\right)$ theory is a second-order theory which violates the Lorentz
symmetry for any nonlinear function $f$ li1 . The fundamental geometric
objects of the theory are the vierbein fields $e^{i}(x^{k})$ which define the
unholonomic frame of the theory. For the vierbein fields it follows
$g(e_{i},e_{j})=e_{i}.e_{j}=\eta_{ij}$, while in a coordinate system
$e^{i}(x^{k})=h_{\mu}^{i}(x^{k})dx^{i}$ such that
$g_{\mu\nu}(x^{k})=\eta_{ij}h_{\mu}^{i}(x^{k})h_{\nu}^{j}(x^{k})$ where
$h_{\mu}^{i}$ is the dual basis of the theory. The invariant which is used for
the definition of the Lagrangian in teleparallelism is the scalar $T$ for the
curvatureless Weitzenböck connection Weitzenb23 . Because of the existence of
the unholonomic tensor the non-null torsion is defined as

$T_{\mu\nu}^{\beta}=\hat{\Gamma}_{\nu\mu}^{\beta}-\hat{\Gamma}_{\mu\nu}^{\beta}=h_{i}^{\beta}(\partial_{\mu}h_{\nu}^{i}-\partial_{\nu}h_{\mu}^{i})\;.$
(1)

where the scalar $T$ is given by the expression

$T=\frac{1}{2}(K_{\beta}^{\mu\nu}+\delta_{\beta}^{\mu}T_{\theta}^{\theta\nu}-\delta_{\beta}^{\nu}T_{\theta}^{\theta\mu})T_{\mu\nu}^{\beta}$
(2)

where $K_{\beta}^{\mu\nu}$ equals the difference of the Levi Civita connection
in the holonomic and the unholonomic frame and it is defined as$\
K_{\beta}^{\mu\nu}=-\frac{1}{2}(T_{\beta}^{\mu\nu}-T_{\beta}^{\nu\mu}-T_{\beta}^{\mu\nu}).$

The Action Integral in $f\left(T\right)$ theory is defined as

$S_{f\left(T\right)}=\frac{1}{16\pi G}\int{d^{4}xef(T)+S}_{m},$ (3)

in which ${S}_{m}$ is the Action Integral of the matter source and $G$ is
Newton’s constant. In the case where $f\left(T\right)$ is a linear function,
the teleparallel equivalence of General relativity, with or without the
cosmological constant, is recovered.

Variation with respect to the vierbein fields of the gravitational Action
Integral (3) provides the field equations fer1

$\displaystyle
e^{-1}\partial_{\mu}(e{S}_{i}^{\mu\nu})f^{\prime}(T)-h_{i}^{\lambda}T_{\mu\lambda}^{\beta}S_{\beta}^{\nu\mu}f^{\prime}(T)$
$\displaystyle+S_{i}^{\mu\nu}\partial_{\mu}(T)f^{\prime\prime}(T)+\frac{1}{4}h_{i}^{\nu}f(T)=4\pi
Gh_{i}^{\beta}T_{\beta}^{\nu}$ (4)

in which $f^{\prime}\left(T\right)=\frac{df\left(T\right)}{dT}$,
$f^{\prime\prime}\left(T\right)=\frac{d^{2}f}{dT^{2}}$, $T_{\mu\nu}$ includes
the contribution of the matter source in the field equation and
${S_{i}}^{\mu\nu}={h_{i}}^{\beta}S_{\beta}^{\mu\nu}$ where
$S_{\beta}^{\mu\nu}=\frac{1}{2}(K_{\beta}^{\mu\nu}+\delta_{\beta}^{\mu}T_{\theta}^{\theta\nu}-\delta_{\beta}^{\nu}T_{\theta}^{\theta\mu})$.

In the case of a spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW)
universe

$ds^{2}=dt^{2}-a^{2}\left(t\right)\left(dx^{2}+dy^{2}+dz^{2}\right),$ (5)

then when we select the diagonal unholonomic frame
$e^{A}=\left(dt,a\left(t\right)dx,a\left(t\right)dy,a\left(t\right)dz\right)$,
the field equations (4) are in agreement with the Action integral (3).

However, when we consider a nonzero spatial curvature,$~{}K\neq 0,$ in the
FLRW universe,

$ds^{2}=dt^{2}-a^{2}\left(t\right)\left(dr^{2}+\sin^{2}\left(\phi\right)\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)\right),~{}K=1~{},$ (6)
$ds^{2}=dt^{2}-a^{2}\left(t\right)\left(dr^{2}+\sinh^{2}\left(\phi\right)\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)\right),~{}K=-1~{},$ (7)

a nondiagonal frame should be considered. Indeed, the selected unholonomic
frame is
$e_{i}^{A}=\left(dt,a\left(t\right)E^{r}\left(K\right),a\left(t\right)E^{\theta}\left(K\right),a\left(t\right)E^{\phi}\left(K\right)\right)$,
in which for $K=1$, ff1

$E^{r}\left(K=1\right)=-\cos\theta dr+\sin r\sin\theta\left(\cos rd\theta-\sin
r\sin\theta d\phi\right)~{},$ (8) $\displaystyle E^{\theta}\left(K=1\right)$
$\displaystyle=\sin\theta\cos\phi dr-\sin r\left(\sin r\sin\phi-\cos
r\cos\theta\cos\phi\right)d\theta+$ $\displaystyle-\sin r\sin\theta\left(\cos
r\sin\phi+\sin r\cos\theta\cos\phi\right)d\phi~{},$ (9) $\displaystyle
E^{\phi}\left(K=1\right)$ $\displaystyle=-\sin\theta\sin\phi dr-\sin
r\left(\sin r\cos\phi+\cos r\cos\theta\sin\phi\right)d\theta+$
$\displaystyle-\sin r\sin\theta\left(\cos r\cos\phi-\sin
r\cos\theta\sin\phi\right)d\phi.$ (10)

On the other hand, for
$K=-1,~{}E^{r}\left(K\right),~{}E^{\theta}\left(K\right)$ and
$E^{\phi}\left(K\right)$ are defined as

$E^{r}\left(K=-1\right)=\cos\theta dr+\sinh r\sin\theta\left(-\cosh
rd\theta+i\sinh r\sin\theta d\phi\right)~{},$ (11) $\displaystyle
E^{\theta}\left(K=-1\right)$ $\displaystyle=-\sin\theta\cos\phi dr+\sinh
r\left(i\sinh r\sin\phi-\cos r\cos\theta\cos\phi\right)d\theta+$
$\displaystyle+\sinh r\sin\phi\left(\cosh r\sin\phi+i\sinh
r\cos\theta\cos\phi\right)d\phi~{},$ (12) $\displaystyle
E^{\phi}\left(K=-1\right)$ $\displaystyle=\sin\theta\sin\phi dr+\sinh
r\left(i\sinh r\cos\phi+\cosh r\cos\theta\sin\phi\right)d\theta+$
$\displaystyle+\sinh r\sin\theta\left(\cosh r\cos\phi-\sinh
r\cos\theta\sin\phi\right)d\phi~{}.$ (13)

For this frame, the scalar $T$ is derived

$T=6\left(\frac{K}{a^{2}}-H^{2}\right),~{}H=\frac{\dot{a}}{a}~{},~{}\dot{a}=\frac{da}{dt}$
(14)

while the field equations (4) for a isotropic fluid source with energy density
$\rho$ and pressure $p$ are ff1

$6\left(H^{2}+\frac{K}{a^{2}}\right)f^{\prime}+\left(f\left(T\right)-Tf^{\prime}\right)=2\rho~{},$
(15)
$-4f^{\prime}\left(2\dot{H}+3H^{2}\right)+4\left(\dot{H}+Ka^{-2}\right)\left(12H^{2}f^{\prime\prime}+f^{\prime}\right)-f\left(T\right)=2p~{},$
(16)

where we observe that of linear function $f\left(T\right)$, the usual
Friedmann’s equations are recovered.

We continue with the investigation of analytic and exact solutions for the
field equations (15), (16) for functional forms of $f\left(T\right)$ of
specific interests.

##  III Generalized Milne universe

In the following we assume that the spacetime is vacuum, that is, $\rho=0$ and
$p=0$. Therefore, gravitational field equations become

$6\left(H^{2}+\frac{K}{a^{2}}\right)f^{\prime}+\left(f\left(T\right)-Tf^{\prime}\right)=0~{},$
(17)
$-4f^{\prime}\left(2\dot{H}+3H^{2}\right)+4\left(\dot{H}+Ka^{-2}\right)\left(12H^{2}f^{\prime\prime}+f^{\prime}\right)-f\left(T\right)=0.$
(18)

From the latter, we are able to define the effective energy density for the
geometric fluid to be
$\rho_{f\left(T\right)}=-\frac{\left(f\left(T\right)-Tf^{\prime}\right)}{2f^{\prime}}$.

In the following we consider the power-law function
$f\left(T\right)=T^{n}~{}$ls1 , and we investigate the existence of Milne-like
solutions.

Indeed for the power-law $f\left(T\right)$ function, the field equation (17),
(18) are written as follows

$a^{-2n}\left(K-\dot{a}^{2}\right)^{n-1}\left(K+\left(2n-1\right)\dot{a}^{2}\right)=0~{},$
(19)
$a^{-2n}\left(K-\dot{a}^{2}\right)^{n-2}\left(\left(2n-3\right)\left(K-\dot{a}^{2}\right)\left(K+\left(2n-1\right)\dot{a}^{2}\right)-2na\left(K-\left(2n-1\right)\dot{a}^{2}\right)\ddot{a}\right)=0~{}.$
(20)

Hence, for $n>1$, we have the (positive scale factor) solutions
$a_{A}\left(t\right)=\sqrt{K}t$ and
$a_{B}\left(t\right)=\sqrt{\frac{K}{1-2n}}t$. The first solution, namely
$a_{A}\left(t\right),$ is a real function when $K>0,$ while
$a_{B}\left(t\right)$ is a real solution and physically accepted when $K<0$.
The second solution $a_{B}\left(t\right)$ describes the generalized Milne
universe, since $K<0$, where the geometric terms which follow by
$f\left(T\right)$ in the field equations mimic the curvature filed such that
to modify the effective curvature from $K\rightarrow\frac{K}{1-2n}$. Moreover,
$a_{A}\left(t\right)$ is a new solution for positive curvature where the
geometric fluid mimics the curvature term with a negative effective energy
density.

Conversely, $n<1$, there exists only the generalized Milne solution
$a_{B}\left(t\right)~{}$but in this case the solution exists for $K>0$. These
solutions are the general solutions for the field equations.

It is important to mention that in contrary to the spatially flat FLRW
universe in which the vacuum solution in $f\left(T\right)$ gravity is the
Minkowski universe ftb1 , that is, the vacuum solution of General Relativity;
in the presence of the curvature term the vacuum solutions are different from
that of General Relativity.

##  IV Analytic solutions

We proceed by considering the more general functions $f\left(T\right)$ with
special interests in cosmology and astrophysics. Specifically, for the
function $f\left(T\right)$ we consider the dark energy model
$f\left(T\right)=T+f_{0}T^{n}$ known as dark torsion ls2 , and the case of the
dark torsion model with the cosmological constant
$f\left(T\right)=T+f_{0}T^{n}-2\Lambda~{}$which has been constraint by local
gravitational systems ls3 .

For these two models, we observe that Milne or Milne-like exact solutions do
not exist as for the power-law theory. Thus, other mathematical techniques
should be applied for the derivation of solutions. We follow the approach
applied in ftb1  on the application of the singularity analysis for the
construction of analytic cosmological solutions.

The necessary property in order for the singularity analysis to work is the
existence of a movable pole for the differential equation. Hence, we consider
$a\left(t\right)=\phi_{0}\left(t\right)\phi\left(t\right)^{p}$, where
$\phi\left(t\right)_{|t\rightarrow t_{0}}=0$, and $p$ should be a negative
rational number. Parameter $p$ and function $\phi_{0}\left(t\right)$ are
derived by choosing the master equation of our study, i.e. equation (18) to be
dominated by the singular terms. This is known as the first step in the
Ablowitz-Ramani-Segur (ARS) algorithm Abl1 ; Abl2 ; Abl3 . In singularity
analysis the solution is expressed in terms of Painlevé Series, thus it is
necessary to find the integration constants and the step of the Painlevé
Series as we move far from the singularity .

The second step of the algorithm is based on the determination of the
resonances which provide information about the position of the integration
constants of the solution buntis . Hence, for the leading order behaviour
$a_{l}\left(t\right)~{}$that we have found, we replace
$a\left(t\right)=a_{l}\left(t\right)\left(1+\varepsilon\phi\left(t\right)^{S}\right)$
and we linearize around $\varepsilon^{2}\rightarrow 0$. From the leading-order
terms of the latter equation we extract a polynomial where by assuming that it
is zero, we find the values of the resonances $S$. The number of independent
values of $S$ should be equal to the order of the differential equation, since
as we mentioned before the resonances are related with the integration of
constants of the solution. Furthermore, the resonance $S=-1$ should exist in
order for the singularity to be a simple pole. The third and final step of the
ARS algorithm, known as consistency test, describes how to write the Painlevé
Series and replace it in the differential equation in order to test that it is
a solution. For more details on the ARS algorithm and for an extended
discussion we refer the reader to leachan .

###  IV.1 Dark torsion$~{}f\left(T\right)=T+f_{0}T^{n}$

For the dark torsion model with $f\left(T\right)=T+f_{0}T^{n}$, the modified
Friedmann equations become

$0=12H^{2}\left(1+nf_{0}T^{n-1}\right)+T+f_{0}T^{n}~{},$ (21) $\displaystyle
0$
$\displaystyle=-4\left(1+nf_{0}T^{n-1}\right)\left(2\dot{H}+3H^{2}\right)-T-f_{0}T^{n}$
$\displaystyle+4\left(1+nf_{0}T^{n-1}\left(12\left(n-1\right)H^{2}+T\right)\right)\left(\dot{H}+Ka^{-2}\right)~{}.$
(22)

We replace scalar $T$ from (14), and we assume that the leading-order
behaviour is described by the
$a\left(t\right)=\phi_{0}\left(t\right)\phi\left(t\right)^{p}$. Usually,
$\phi\left(t\right)$ is assumed to be the linear function
$\phi\left(t\right)=\left(t-t_{0}\right)$, however that it is not necessary b5
.

For $n>1,$ from equation (22) we find that the leading-order terms is for
$p=1$. However, $p=1,$ it is not acceptable because such value for $p$ does
not provide a singular behaviour. In order to overpass that we follow the
approach described in leachan  and we consider the new variable
$b\left(t\right)=\left(a\left(t\right)\right)^{-1}$.

Hence, in the new variable we find that the leading-order behaviour of
equation (22) is
$b\left(t\right)=\Phi_{0}\left(t\right)\Phi\left(t\right)^{p}$, where $p=-1$
and
$K=\left(\frac{\dot{\Phi}\left(t\right)}{\Phi_{0}\left(t\right)}\right)^{2}$
or
$K=-\left(1-2n\right)\left(\frac{\dot{\Phi}\left(t\right)}{\Phi_{0}\left(t\right)}\right)^{2}$.
In the case where $\Phi\left(t\right)$ is a linear function, then the leading-
order behaviour is that of the generalized Milne solution whish is described
by the power-law term $T^{n}$ of the model. From the second step of the ARS
algorithm we find the resonances $S=-1$ and $S=1$, that is, the analytic
solution is expressed in terms of the Painlevé Series

$b\left(t\right)=\Phi_{0}\left(t\right)\left(\Phi\left(t\right)\right)^{-1}+\Phi_{1}\left(t\right)+\Phi_{2}\left(t\right)\Phi\left(t\right)+\Phi_{3}\left(t\right)\left(\Phi\left(t\right)\right)^{2}+...~{}.$
(23)

In order to perform the consistency test we select a specific value for the
exponent $n$. Consider that $n=2$, then by replacing (23) in (22) we end up
with the following constraint equations.

For positive value $K$, i.e. for the leading-order behaviour with
$K=\left(\frac{\dot{\Phi}\left(t\right)}{\Phi_{0}\left(t\right)}\right)^{2}$
we find that $\Phi_{1}\left(t\right)$ is an arbitrary function, and

$\Phi_{2}\left(t\right)=\frac{\left(216f_{0}K\left(\Phi_{1}\right)^{2}-72f_{0}\sqrt{K}\dot{\Phi}_{1}-1\right)\dot{\Phi}^{2}+72f_{0}\ddot{\Phi}+24f_{0}\dot{\Phi}\left(9\sqrt{K}\Phi_{1}\ddot{\Phi}-\Phi^{\left(3\right)}\right)}{72f_{0}\sqrt{K}\dot{\Phi}^{3}}~{},~{}\text{etc.~{}.}$
(24)

Moreover, from the constraint equation (21) it follows that

$\Phi_{1}\left(t\right)=-\frac{1}{2\sqrt{K}}\frac{\ddot{\Phi}}{\dot{\Phi}}.$
(25)

For negative curvature and for
$K=-3\left(\frac{\dot{\Phi}\left(t\right)}{\Phi_{0}\left(t\right)}\right)^{2}$
we apply the same procedure and we find the constraint equation for the
integration function

$\Phi_{1}\left(t\right)=-\frac{1}{2}\sqrt{-\frac{3}{K}}\frac{\ddot{\Phi}}{\dot{\Phi}}$
(26)

and

$\Phi_{2}\left(t\right)=\frac{\left(1-24f_{0}\left(5K\Phi_{1}^{2}+3\sqrt{-3K}\dot{\Phi}_{1}\right)\right)\dot{\Phi}^{2}+144f_{0}\ddot{\Phi}+24f_{0}\dot{\Phi}\left(5\sqrt{-3K}\Phi_{1}\ddot{\Phi}-3\Phi^{\left(3\right)}\right)}{72f_{0}\sqrt{-3K}\dot{\Phi}^{3}}~{},~{}\text{etc.~{}.}$
(27)

On the other hand, for $n<1$ we replace
$b\left(t\right)=\left(a\left(t\right)\right)^{-1}$ and we find the leading-
order behaviour
$b\left(t\right)=\sqrt{\frac{1-2n}{K}}\Phi\left(t\right)^{-1}$, which is valid
for $K>0$, when $\Phi\left(t\right)$ is a real function. We apply the same
procedure as before and we find the resonances to be $S=-1$ and $S=2n-3$, that
is, the second resonance is negative which indicates that the analytic
solution is expressed in terms of the Left Painlevé Series

$b\left(t\right)=\Phi_{0}\left(t\right)\left(\Phi\left(t\right)\right)^{-1}+\Phi_{1}\left(t\right)\left(\Phi\left(t\right)\right)^{-2}+\Phi_{2}\left(t\right)\left(\Phi\left(t\right)\right)^{-3}+\Phi_{3}\left(t\right)\left(\Phi\left(t\right)\right)^{-4}+...~{}.$
(28)

However, by replacing in the field equations the latter solution we find that
the latter Painlevé Series does not solve the differential equations, thus,
the singularity analysis fails.

Let us assume now the case with linear function
$\Phi\left(t\right)=\left(t-t_{0}\right)$ where the leading-order behaviour is
that of the generalized Milne universe, and let us write the analytic
solutions. For $n>1$, the analytic solution is expressed in terms of the
Puiseux Series

$b\left(t\right)=b_{0}\left(t-t_{0}\right)^{-1}+b_{1}+b_{2}\left(t-t_{0}\right)+b_{3}\left(t-t_{0}\right)^{2}+...~{}.$
(29)

Hence, for $n=2\,,~{}$and $K=1$ we find the coefficients

$b_{0}=1~{},~{}b_{1}=0~{},~{}b_{2}=-\frac{1}{72f_{0}}~{},~{}b_{3}=0~{},~{}b_{4}=\frac{1}{3240f_{0}^{2}}~{},~{}b_{5}=0~{},~{}\text{etc.~{}.}$
(30)

On the other hand for $K=-1~{}$it follows

$b_{0}=\sqrt{3}~{},~{}b_{1}=0~{},~{}b_{2}=\frac{1}{27\sqrt{3}f_{0}}~{},~{}b_{3}=0~{},~{}b_{4}=\frac{17}{77760\sqrt{3}f_{0}^{2}}~{},~{}b_{5}=0~{},~{}\text{etc.~{}.}$
(31)

Someone can use another function $\Phi\left(t\right)$ to write the analytic
solution. However, for every smooth function $\Phi\left(t\right)$ near the
singularity $\Phi\left(t\right)\simeq t-t_{0}$ such that
$\Phi\left(t_{0}\right)\simeq 0$.

For $n>1$, from (29) we observe that as far as we move from the singularity,
then the right parts of the Series dominate, which indicate that when
$t_{0}>0$, the behaviour $\left(t-t_{0}\right)^{-1}$ is unstable. As far as
the scale factor $a\left(t\right)$ is concerned, it can be easily constructed
by using function $b\left(t\right)$.

Indeed for $n=2$ and $K=1$ it follows

$a\left(t\right)=\left(t-t_{0}\right)\left(1+\frac{1}{72f_{0}}\left(t-t_{0}\right)^{2}-\frac{1}{8640f_{0}^{2}}\left(t-t_{0}\right)^{4}+...\right),$
(32)

while for $K=-1$ we calculate the scale factor

$a\left(t\right)=\frac{\left(t-t_{0}\right)}{\sqrt{3}}\left(1-\frac{1}{216f_{0}}\left(t-t_{0}\right)^{2}-\frac{1}{19440f_{0}^{2}}\left(t-t_{0}\right)^{4}+...\right).$
(33)

###  IV.2 Dark torsion with cosmological constant
$f\left(T\right)=T+f_{0}T^{n}-2\Lambda$

In the presence of the cosmological constant in the dark torsion model, the
singularity analysis provides the same results as before, with $\Lambda=0$.
The only difference is in the values of the coefficients of the analytic
solution where the cosmological constant is introduced. In order to make it
clear we present the solution for the scale factor $a\left(t\right)$ for $n=2$
and $K=1,~{}K=-1$ as before.

For $K=1$, the scale factor is

$a\left(t\right)=\left(t-t_{0}\right)\left(1+\frac{1}{72f_{0}}\left(t-t_{0}\right)^{2}-\frac{1+12f_{0}\Lambda}{8640f_{0}^{2}}\left(t-t_{0}\right)^{4}+...\right),$
(34)
$a\left(t\right)=\frac{\left(t-t_{0}\right)}{\sqrt{3}}\left(1-\frac{1}{216f_{0}}\left(t-t_{0}\right)^{2}-\frac{1+9f_{0}\Lambda}{19440f_{0}^{2}}\left(t-t_{0}\right)^{4}+...\right).$
(35)

from which it is obvious the the contribution of the cosmological constant is
significant far from the movable singularity.

##  V Conclusions

In this study we investigated the existence of exact and analytic solutions
for the modified teleparallel $f\left(T\right)$ gravitational theory in a FLRW
background space with nonzero spatial curvature. For the $f\left(T\right)$
function we assumed the proposed power-law models. We found that there exists
a movable singularity for the field equations which describes a singular
scaling solution for the field equations. The asymptotic behaviour for the
scale factor is that of a Milne-like scale factor, for negative and positive
spatial curvature for the background space. These generalized Milne solutions
are the analytic solutions for the power-law $f\left(T\right)=T^{n}$ theory.

However, for the $f\left(T\right)=T+f_{0}T^{n}-2\Lambda$ model, for the
derivation of the analytic solution, because of the existence of the movable
singularity, we applied the singularity analysis and specifically the ARS
algorithm. We proved that the modified Friedmann equations possess the
Painlevé property and the analytic solution for the scalar factor are
expressed in Right Puiseux Series, where the first term is the generalized
Milne behaviour. Because the Puiseux solutions are Right, that is, as far as
we move from the initial singularity the scale factor increases, consequently,
we can easily derive the attractor for the field equations and the solution
for large values of the independent parameter. In addition, the presence of
the cosmological constant term plays no role in the general evolution for the
dynamical system.

For large values of a nonconstant scale factor, from (14) it follows
$T\simeq-6H^{2}$ , that is, equation (21) becomes

$0\simeq
H^{2}\left(-2\left(1+nf_{0}\left(-6H^{2}\right)^{n-1}\right)-1+f_{0}\left(-6H^{2}\right)^{n-1}\right).$
(36)

Hence, for it follows
$H\left(a\right)=-6\left(\frac{1}{f_{0}\left(1-2n\right)}\right)^{\frac{1}{n-1}}$,
for $n\neq\frac{1}{2}$. Thus, this model leads to an expanding universe, as we
found from the singularity analysis without necessary to consider a
cosmological constant term.

It is important to mention here that the limit of General Relativity is not
recovered for a nonlinear function $f\left(T\right)$. Indeed, General
Relativity is recovered in $f\left(T\right)$ gravity, when it holds
$f\left(T\right)_{|T\rightarrow 0}=0$, and
$Tf^{\prime}\left(T\right)_{|T\rightarrow 0}=0$. Thus, from (14) it is clear
that the General Relativity vacuum solution does not hold when $T=0.$

We found that in the presence of the spatial curvature in $f\left(T\right)$
gravity the universe becomes inflationary, while a generalized Milne-like
exact solution was found in the power-law model. In a future study we plan to
consider the asymptotic scale factor near to the movable singularity as a toy
model for the cosmological observations with a background space with spatial
curvature.

###### Acknowledgements.

This work is based on the research supported in part by the National Research
Foundation of South Africa (Grant Numbers 131604).

## References

  * (1) A. Einstein 1928, Sitz. Preuss. Akad. Wiss. p. 217; ibid p. 224 [Translated by A. Unzicker and T. Case, (preprint: arXiv: physics/0503046)]
  * (2) L. Combi and G.E. Romero, Annalen der Physik 530, 1700175 (2018)
  * (3) R. Ferraro and F. Fiorini, Phys. Rev. D 75, 084031 (2007)
  * (4) R. Ferraro and F. Fiorini, Phys. Rev. D 78, 124019 (2008)
  * (5) G.R. Bengochea and R. Ferraro, Phys. Rev. D 79, 124019 (2009)
  * (6) C. Xu, E.N. Saridakis and G. Leon, JCAP 07, 005 (2012)
  * (7) R. Myrzakulov, Gen. Rel. Gravit. 44, 3059 (2012)
  * (8) A. Paliathanasis, JCAP 08, 027 (2017)
  * (9) M. Krssak, R. J. van den Hoogen, J. G. Pereira, C. G. Boehmer and A. A. Coley, Class. Quantum Grav. 36, 183001 (2019)
  * (10) R. Myrzakulov, FRW Cosmology in F(R,T) gravity, EPJC 72, 1 (2012)
  * (11) E. Di Valentino, O. Mean, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D.F. Mota, A.G. Riess and J. Silk, In the Realm of the Hubble tension - a Review of Solutions, (2021) [arXiv:2103.01183]
  * (12) R.C. Nunes, JCAP 05, 052 (2018)
  * (13) S. Bahamonte, K.F. Dialektopoulos, C. Escamilla-Rivera, G. Farrugia, V. Gakis, M. Hohmann, J.L. Said, J. Mifsud and E. Di Valentino, Teleparallel Gravity: From Theory to Cosmology, arXiv:2106.13793 (2021)
  * (14) B. Li, T.P. Sotiriou and J.D. Barrow, Phys. Rev. D 83, 064035 (2011)
  * (15) T.P. Sotiriou, B. Li and J.D. Barrow, Phys. Rev. D 83, 104030 (2011)
  * (16) R. Weitzenböck, Invarianten Theorie, Nordhoff, Groningen (1923)
  * (17) K. Hayashi and T. Shirafuji, Phys. Rev. D 19, 3524 (1979)
  * (18) S. Chakrabarti, J.L. Said and G. Farrugia, EPJC 77, 815 (2017)
  * (19) M. Caruana, G. Farrugia and J.L. Said, EPJC 80, 640 (2020)
  * (20) J. Yang, Y.-L. Li, Y. Zhong, and Y. Li, Phys. Rev. D 85, 084033 (2012)
  * (21) W.-D. Guo, Q.-M. Fu, Y.-P. Zhang, and Y.-X. Liu, Phys. Rev. D 93, 044002 (2016)
  * (22) J. Wang, W.-D. Guo, Z.-C. Lin, and Y.-X. Liu, Phys. Rev. D 98, 084046 (2018)
  * (23) Q. Tan, W.-D. Guo, Y.-P. Zhang, and Y.-X. Liu, Eur. Phys. J. C 81, 373 (2021)
  * (24) M.A. Skugoreva and A.V. Toporensky, EPJC 80, 1054 (2020)
  * (25) G. Steigman and M.S. Turner, Phys. Lett. B 12, 295 (1983)
  * (26) N. Turok and S.W. Hawking, Phys. Lett. B 432, 271 (1998)
  * (27) G. Aslanyan and R. Easther, Phys. Rev. D 91, 123523 (2015)
  * (28) G.J. Mathews, I.-S. Suh, N.Q. Lan and T. Kajino, Phys. Rev. D 92, 123514 (2015)
  * (29) K. Sato, MNRAS 195, 467 (1981)
  * (30) J.D Barrow and A. Ottewill, J. Phys. A 16, 2757 (1983)
  * (31) G.W. Gibbons and S.W Hawking, Phys. Rev. D 15, 2738 (1977)
  * (32) S.W. Hawking and J.G. Moss. Phys. Lett. B 110, 35 (1982)
  * (33) F. Fiorini and R. Ferraro, Int. J. Mod. Phys.: Conf. Series 3, 227 (2011)
  * (34) A. Paliathanasis, J.D. Barrow and P.G.L. Leach, Phys. Rev. D 94, 023525 (2016)
  * (35) S. Basilakos, S. Capozziello, M. De Laurentis, A. Paliathanasis and M. Tsamparlis, Phys. Rev. D 88, 103526 (2013)
  * (36) G.R. Bengochea, Phys. Lett. B 695, 405 (2011).
  * (37) L. Iorio, N. Radicella and M.L. Ruggiero, JCAP 08, 021, (2015)
  * (38) M.J. Ablowitz, A. Ramani and H. Segur, Lettere al Nuovo Cimento 23, 333 (1978)
  * (39) M.J. Ablowitz, A. Ramani and H. Segur, J. Math. Phys. 21, 715 (1980)
  * (40) M.J. Ablowitz, A. Ramani and H. Segur, J. Math. Phys. 21, 1006 (1980)
  * (41) A. Ramani, B. Grammaticos and T. Bountis, Physics Reports, 180, 159 (1989)
  * (42) A. Paliathanasis and P.G.L. Leach, Int. J. Geom. Meth. Mod. Phys. 13, 1630009 (2016)
  * (43) R. Conte, The Painlevé Property: One Century Later, Conte Robert ed, CRM Series in Mathematical Physics, Springer-Verlag, New York, (1999)