# Testing Asteroseismic Radii of Dwarfs and Subgiants with Kepler and Gaia

C. L. Sahlholdt1,2, V. Silva Aguirre1, L. Casagrande3, J. R. Mosumgaard1,M.
Bojsen-Hansen1
1Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus
University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark
2Lund Observatory, Department of Astronomy and Theoretical Physics, Box 43,
SE-221 00 Lund, Sweden
3Research School of Astronomy & Astrophysics, Mount Stromlo Observatory, The
Australian National University, ACT 2611, Australia E-mail:
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We test asteroseismic radii of Kepler main-sequence and subgiant stars by
deriving their parallaxes which are compared with those of the first Gaia data
release. We compute radii based on the asteroseismic scaling relations as well
as by fitting observed oscillation frequencies to stellar models for a subset
of the sample, and test the impact of using effective temperatures from either
spectroscopy or the infrared flux method. An offset of 3%, showing no
dependency on any stellar parameters, is found between seismic parallaxes
derived from frequency modelling and those from Gaia. For parallaxes based on
radii from the scaling relations, a smaller offset is found on average;
however, the offset becomes temperature dependent which we interpret as
problems with the scaling relations at high stellar temperatures. Using the
hotter infrared flux method temperature scale, there is no indication that
radii from the scaling relations are inaccurate by more than about 5%. Taking
the radii and masses from the modelling of individual frequencies as reference
values, we seek to correct the scaling relations for the observed temperature
trend. This analysis indicates that the scaling relations systematically
overestimate radii and masses at high temperatures, and that they are accurate
to within 5% in radius and 13% in mass for main-sequence stars with
temperatures below 6400 K. However, further analysis is required to test the
validity of the corrections on a star-by-star basis and for more evolved
stars.

###### keywords:

Asteroseismology – Stars: oscillations – Stars: fundamental parameters –
Parallaxes

††pubyear: 2017††pagerange: Testing Asteroseismic Radii of Dwarfs and
Subgiants with Kepler and Gaia–Testing Asteroseismic Radii of Dwarfs and
Subgiants with Kepler and Gaia

##  1 Introduction

With the advent of remarkably precise space-based photometry in recent years,
the field of asteroseismology has seen great progress as a means of
determining precise stellar parameters. For a star showing solar-like
oscillations, two global parameters can be determined from the power spectrum,
namely the mean large frequency separation, $\langle\Delta\nu\rangle$111Since
we use the mean value throughout this paper we drop the brackets and simply
write $\Delta\nu$., and the frequency of maximum oscillation power,
$\nu_{\mathrm{max}}$. These parameters follow a set of approximate scaling
relations, tying them to the mass and radius of the star, which makes it
straightforward to estimate properties of any star with detected solar-like
oscillations. Therefore, asteroseismology holds great potential for
determining precise stellar parameters, and it is important to verify that the
results are also accurate.

Direct measurements of radii and masses are challenging to obtain which makes
it difficult to perform large-scale tests of the scaling relations. Still,
empirical tests have been carried out for smaller samples, and the scaling
relation for the stellar radius has been shown to be accurate to within 4% for
main-sequence and subgiant stars based on comparisons with results from
interferometry (Huber et al., 2012; White et al., 2013). The scaling relation
for the mass can be tested with eclipsing binaries in which one of the
components shows solar-like oscillations. Gaulme et al. (2016) compared both
masses and radii based on eclipse analyses with those from the scaling
relations for a sample of 10 red giants. They found that the scaling relations
overestimate radii and masses by about 5% and 15% respectively, emphasising
the need for further investigation of the accuracy of the scaling relations.
In order to test the results of asteroseismology for larger stellar samples,
less direct comparisons must be employed. Silva Aguirre et al. (2012) used the
scaling relations in combination with broadband photometry to derive stellar
distances which they compared to Hipparcos parallaxes. Based on their sample
of 22 main-sequence stars, they found the asteroseismic distances, and thereby
also the radii, to be accurate to within 5%.

With the recent first data release from the Gaia mission (GDR1; Gaia
Collaboration et al. 2016), a new opportunity to test asteroseismic radii has
arisen. The Gaia data have significantly increased the number of solar-like
oscillators with precise parallax measurements and allow for new observational
constraints to be put on the accuracy of asteroseismic radii. A first
comparison between seismic and Gaia parallaxes was carried out by De Ridder et
al. (2016) who found good agreement for 22 dwarfs and subgiants; however, for
938 red giants they could reject the 1:1 relation at the 95% confidence limit.
Huber et al. (2017) used the Gaia parallaxes to test the scaling relations for
a sample of 2200 Kepler stars spanning evolutionary stages from the main
sequence to the red-giant branch and found the radii to be accurate to within
5%. Gaia parallaxes have also been used to test the radii of the 66 main-
sequence stars of the Kepler LEGACY sample (Lund et al., 2017; Silva Aguirre
et al., 2017) which were derived by fitting the individual frequencies to
stellar models. This test showed a systematic offset between seismic and Gaia
parallaxes, with the seismic parallaxes being about 0.25 mas larger on
average. This is in good agreement with the findings of Stassun & Torres
(2016, hereafter ST16) who derived parallaxes based on radii of eclipsing
binaries. Davies et al. (2017) compared Gaia parallaxes to those of red clump
stars by adopting a common absolute magnitude based on litterature values.
They found an offset which increases with parallax and reaches the ST16 value
for the largest parallaxes of their sample (${{\sim}1.6}$ mas).

It is possible that some of these offsets are caused partly by the temperature
scale which affects both the radii themselves and the transformation between
radii and parallaxes. Indeed, Huber et al. (2017) showed that the use of
different temperature scales for main-sequence and subgiant stars can change
the mean parallax difference by about 2% for results based on the scaling
relations. They also found that the offsets identified by ST16 and Davies et
al. (2017) were overestimated for stars with parallaxes ${\lesssim 5}$ mas.

In this paper we take a closer look at the comparison between seismic and Gaia
parallaxes for the Kepler main-sequence and subgiant stars. This includes an
investigation of the impact of the temperature scale on the results of both
the scaling relations and the model fits to individual frequencies. We
rederive stellar parameters for the LEGACY sample using different effective
temperatures and extend the sample by including the Kages stars (Davies et
al., 2016; Silva Aguirre et al., 2015). Additionally, the stellar parameters
given by the scaling relations are compared to the ones obtained from the
analysis of individual frequencies. Based on the assumption that the
individual frequencies give the most accurate stellar parameters obtainable by
asteroseismology, we seek possible corrections to the scaling relations,
mainly as a function of effective temperature.

##  2 Samples and Data

We consider two different (but partially overlapping) samples of Kepler stars
which will be referred to as the main sample and the frequency sample. The
main sample is defined as all stars which have $\Delta\nu$ and
$\nu_{\mathrm{max}}$ from Chaplin et al. (2014), effective temperatures and
metallicities obtained from spectroscopy by Buchhave & Latham (2015), and
parallaxes from the GDR1 (Lindegren et al., 2016). The frequency sample
consists of all of the stars in the Kages and LEGACY samples with parallaxes
in the GDR1, except for the two Kages stars that showed signs of mixed dipole
modes. For most of this sample, $T_{\mathrm{eff}}$ and
$[\mathrm{Fe}/\mathrm{H}]$ are also taken from Buchhave & Latham (2015);
however, not all of the LEGACY stars were included in that study. For the ones
missing (11 stars in total), $T_{\mathrm{eff}}$ and $[\mathrm{Fe}/\mathrm{H}]$
are taken from the same spectroscopic sources as used in the original LEGACY
analysis by Silva Aguirre et al. (2017). This results in a main sample of 449
dwarf and subgiant stars with a completely homogeneous set of observables, and
a frequency sample of 86 dwarf stars. Additionally, for all stars analysed,
photometry in the infrared $JHK_{s}$ filters was collected from the Two Micron
All-Sky Survey (2MASS) catalogue (Skrutskie et al., 2006), and photometry in
the $griz$ filters, as well as $E(B-V)$, were collected from the Kepler Input
Catalogue (KIC; Brown et al. 2011). The $griz$ photometry was transformed to
the SDSS scale using the corrections by Pinsonneault et al. (2012).

The two samples have 52 stars in common for which the same temperatures and
metallicities are used, but the asteroseismic observables $\Delta\nu$ and
$\nu_{\mathrm{max}}$ have been determined separately. For the main sample, the
asteroseismic observables have been determined based on one month of short
cadence Kepler data for each star (see Chaplin et al. (2014) for further
details). The stars of the frequency sample are a sub-sample of the stars
which were chosen to be observed for longer, and for most of them the
asteroseismic observables are based on at least 12 months of data. As a
result, the values from the frequency sample are an order of magnitude more
precise. The median relative uncertainties for the stars in common are 0.2% in
$\Delta\nu$ and 0.5% in $\nu_{\mathrm{max}}$ for the frequency sample, and for
the main sample the corresponding values are 2.0% and 4.3%. With 449 stars in
the main sample, 86 in the frequency sample, and 52 in common between them,
there are 483 unique stars which will be referred to collectively as the full
sample.

Figure 1: Observed $\Delta\nu$ and spectroscopic temperature of the main and
frequency samples. The lines show evolutionary tracks computed with GARSTEC
for models with solar metallicity and masses as labelled.

Both samples are shown in Figure 1 in the space of observed $\Delta\nu$ and
spectroscopic $T_{\mathrm{eff}}$ along with evolutionary tracks computed using
the Garching Stellar Evoluion Code (GARSTEC; Weiss & Schlattl 2008). With the
visual aid of the evolutionary tracks, it is clear that the stars in the
frequency sample are generally less evolved than the stars in the main sample.
This is mainly due to the fact that the stars of the frequency sample have
been selected to be on the main sequence in order to avoid the complications
of modelling mixed dipole modes which show up in the oscillation spectra of
more evolved stars.

Figure 2: Difference between the two sets of temperatures (IRFM and
spectroscopic) which have been collected for the full sample. On average, the
IRFM temperatures are higher and the dot-dashed line shows the mean difference
which is 60 K.

Finally, a second effective temperature has been derived for the full sample
using the Infrared Flux Method (IRFM) as implemented by Casagrande et al.
(2006); Casagrande et al. (2010) (the procedure is described in subsection
3.2). In Figure 2 the differences between the IRFM and spectroscopic
temperatures are shown. The scatter is quite large (about 150 K) and, on
average, the IRFM temperatures are higher than the spectroscopic ones by about
60 K; however, this mean offset does not apply to the entire range of
temperatures of the sample. If only the stars with temperatures below 5500 K
are considered, which are mostly subgiants (see Figure 1), the mean offset is
instead around just 10 K (but still with a large scatter). For stars with
temperatures above 5500 K, the mean offset agrees with the overall mean of 60
K which is also the case for both the main and frequency samples when
considered individually.

##  3 Methods

###  3.1 Radii

The scaling relations for the average asteroseismic parameters $\Delta\nu$ and
$\nu_{\mathrm{max}}$ are given by (Ulrich, 1986; Brown et al., 1991)

$\displaystyle\frac{\Delta\nu}{\Delta\nu_{\sun}}$
$\displaystyle\simeq\left(\frac{M}{M_{\sun}}\right)^{1/2}\left(\frac{R}{R_{\sun}}\right)^{-3/2}\;,$
(1) $\displaystyle\frac{\nu_{\mathrm{max}}}{\nu_{\mathrm{max},\sun}}$
$\displaystyle\simeq\left(\frac{M}{M_{\sun}}\right)\left(\frac{R}{R_{\sun}}\right)^{-2}\left(\frac{T_{\mathrm{eff}}}{T_{\mathrm{eff},\sun}}\right)^{-1/2}\;,$
(2)

which can be rearranged to give the scaling relations for the radius and mass

$\displaystyle\frac{R}{R_{\sun}}$
$\displaystyle\simeq\left(\frac{\nu_{\mathrm{max}}}{\nu_{\mathrm{max},\sun}}\right)\left(\frac{\Delta\nu}{\Delta\nu_{\sun}}\right)^{-2}\left(\frac{T_{\mathrm{eff}}}{T_{\mathrm{eff},\sun}}\right)^{1/2}\;,$
(3) $\displaystyle\frac{M}{M_{\sun}}$
$\displaystyle\simeq\left(\frac{\nu_{\mathrm{max}}}{\nu_{\mathrm{max},\sun}}\right)^{3}\left(\frac{\Delta\nu}{\Delta\nu_{\sun}}\right)^{-4}\left(\frac{T_{\mathrm{eff}}}{T_{\mathrm{eff},\sun}}\right)^{3/2}\;.$
(4)

The scaling relations have been applied to both stellar samples using the two
different effective temperature scales resulting in two sets of stellar
parameters. The solar values are taken to be
$\nu_{\mathrm{max,\sun}}=3090~{}\mathrm{\upmu Hz}$,
$\Delta\nu_{\sun}=135.1~{}\mathrm{\upmu Hz}$ (Huber et al., 2011), and
$T_{\mathrm{eff,\sun}}=5777~{}\mathrm{K}$.

In order to take advantage of the individual frequencies, the stars of the
frequency sample have been fitted to a grid of GARSTEC stellar models with
theoretical oscillation frequencies computed using the Aarhus adiabatic
oscillation package (ADIPLS, Christensen-Dalsgaard 2008). The stellar models
have been computed with the OPAL equation of state (Rogers & Nayfonov, 2002),
OPAL opacitites (Iglesias & Rogers, 1996) with low-temperature opacities by
Ferguson et al. (2005), the NACRE compilation of nuclear reaction rates
(Angulo et al., 1999), and the solar mixture of Grevesse & Sauval (1998).
Convection is implemented in the mixing length formalism and we use a solar-
calibrated mixing length parameter of $\alpha_{\mathrm{mlt}}=1.7917$. The
solar-calibrated initial composition of $Y_{i}=0.2705$ and $Z_{i}=0.0189$ has
been combined with the Big Bang nucleosynthesis values of $Y_{0}=0.2482$ and
$Z_{0}=0$ (Steigman, 2010) to get a helium enrichment law of $\Delta Y/\Delta
Z=1.179$. This has been used in all models to define the initial helium
abundance for a given initial metallicity $[\mathrm{Fe}/\mathrm{H}]$.

Two grids of stellar models have been computed: one including microscopic
diffusion and the other including convective overshoot using the diffusion
formalism implemented in GARSTEC with an efficiency parameter of $f=0.016$
(see Weiss & Schlattl 2008, section 3.1.5). The diffusion grid spans masses of
0.70–1.30 $M_{\sun}$ and the overshoot grid spans masses of 1.00–1.80
$M_{\sun}$, both in steps of 0.01 $M_{\sun}$. Initial metallicities from -0.65
to +0.55 dex in steps of 0.05 dex are included in the overshoot grid and for
the diffusion grid the initial values have been increased slightly, thus
covering the range -0.60 to +0.65 dex, in order to account for the decrease in
surface metallicity with evolution. Both grids cover $\Delta\nu$, as
calculated from Equation 1, in the range 13–180 $\upmu$Hz, which spans the
entire observed range including some room for inaccuracies in the scaling
relations. All stars were fitted to both model grids and for the ones that
returned masses in the range of overlap (1.00–1.30 $M_{\sun}$), we choose, by
visual inspection, the fit for which the probability distribution of the mass
is not cut off by the edge of the grid. If none of the two fits hit the edge,
we compare the observed and modelled oscillation frequencies, effectively
choosing the grid for which the best fitting model has the highest likelihood.
Small individual diffusion grids were computed for the three stars with
metallicities lower than -0.60 dex.

For all fits to the grids, metallicities and temperatures are included in
addition to the asteroseismic observables. Like for the scaling relations,
this gives two sets of stellar parameters corresponding to the use of the two
different temperature scales. Theoretical stellar oscillations are affected by
systematic errors caused by improper modelling of the outer stellar layers in
current stellar models. Therefore, when fitting the observed oscillation
frequencies to the model grids, we have to correct the model frequencies or
use certain ratios of frequency differences which have been shown to be
insensitive to the surface layers (Roxburgh & Vorontsov, 2003). Using the
Bayesian Stellar Algorithm (BASTA, see Silva Aguirre et al. 2015) we have
fitted the stars to both frequency ratios and to individual frequencies after
applying the correction introduced by Ball & Gizon (2014) for the surface
effect. These two methods give very similar results with a mean radius
difference of less than 0.1% and a scatter of 0.7%. Thus, the choice between
fitting to frequency ratios or corrected individual frequencies has no
significant impact on the final results, and the stellar parameters based on
fits to frequency ratios will be used in the following.

The grid-based method has also been applied to the stars of the main sample by
fitting the observed $\Delta\nu$ to the mean frequency separation of the
radial model frequencies, $\Delta\nu_{\mathrm{fit}}$, calculated as described
by White et al. (2011). This quantity is also sensitive to the surface effect;
therefore, we have scaled the grid values by the ratio of the observed solar
frequency separation, $\Delta\nu_{\sun}=135.1~{}\mathrm{\upmu Hz}$, and that
of the solar model calibration,
$\Delta\nu_{\mathrm{fit,sunmod}}=136.1~{}\mathrm{\upmu Hz}$. By applying this
scaling, the surface effect is assumed to increase the value of $\Delta\nu$ by
a constant fraction of 1.007 for all stars. Note that $\nu_{\mathrm{max}}$ has
not been included in these model fits in order to get a set of stellar
parameters independent of the scaling relations. It turns out, however, that
it makes no difference whether it is included or not because the typical
relative uncertainty on $\nu_{\mathrm{max}}$ of 4.3% for the stars of the main
sample is too large to impose any significant constraint on the models.

###  3.2 Distances

In order to calculate asteroseismic distances, and thereby parallaxes, we have
used the distance modulus based on the magnitudes in the $griz$ and $JHK_{s}$
photometric filters, and the implementation is divided into two steps. The
first step of the calculation is to determine the reddening in a way to make
it consistent with the final distance. This is done by an iterative procedure
starting with the value of $E(B-V)$ from the KIC. This value of the color
excess is used as input for the bolometric correction software by Casagrande &
VandenBerg (2014) (as implemented in BASTA) together with the observed surface
properties $T_{\mathrm{eff}}$ (spectroscopic or IRFM depending on which was
used to determine the stellar properties), $[\mathrm{Fe}/\mathrm{H}]$, and the
surface gravity $\log g$ obtained from the asteroseismic analysis. The output
is the extinction-corrected bolometric corrections for each of the photometric
filters. With each observed magnitude and theoretical bolometric correction,
the distance is calculated using the luminosity from asteroseismology and the
absolute bolometric magnitude of the Sun $M_{\mathrm{bol},\sun}=4.75$. The
final distance is determined by taking the median of the values from the
different filters. With this distance, the color excess is updated using the
3D dust map by Green et al. (2015) and the process is iterated twice, at which
point the color excess has converged to within 0.01 mag.

The second step of the distance calculation is to determine the distances and
uncertainties using the final color excess obtained in step one. For this
purpose, we apply Monte Carlo sampling using the uncertainties on all input
parameters assuming Gaussian error distributions. The values and uncertainties
for $T_{\mathrm{eff}}$, $[\mathrm{Fe}/\mathrm{H}]$, and the $JHK_{s}$
magnitudes are taken from the input catalogues. No individual uncertainties
are given for the $griz$ magnitudes in the KIC so they are simply all set to
0.02 mag which is the level of precision found by Brown et al. (2011) based on
repeatability of the photometry. Uncertainties on the luminosity and surface
gravity are taken from the results of the asteroseismic analysis.

We have also combined the asteroseismic scaling relations with the IRFM in the
way described by Silva Aguirre et al. (2012) to derive a set of self-
consistent temperatures (the ones introduced in section 2), radii, angular
diameters, and hence distances. For this procedure the 3D dust map by Green et
al. (2015) has also been used iteratively. We applied the IRFM twice using
different input photometry in the visible. One set was obtained using the
$B_{T}$ and $V_{T}$ bands of the Tycho-2 catalogue (Høg et al., 2000) and the
other set made use of the $griz$ bands from the KIC. In both cases, the 2MASS
$JHK_{s}$ bands were used in the infrared. This was done since either set of
visible photometry may be problematic in some cases. For example, for the two
brightest stars of this study, 16 Cygni A & B, we find the parallaxes to be
underestimated when using the KIC photometry (compared to the Gaia
observations), but not when using Tycho-2 photometry, which indicates problems
with the photometry in the $griz$ filters for these stars. On the other hand,
the quality of the Tycho-2 photometry is worse for the faintest stars. An
initial comparison between the two sets of temperatures showed large
discrepancies with differences above 200 K for about 15% of the stars.
Therefore, we decided to use a third set of temperatures for comparison in
order to choose, on a star-by-star basis, which of the two temperatures from
the IRFM to use. The third set was calculated from 2MASS photometry only,
which is usually good over a wide range of magnitudes, using a color-
temperature calibration linking $J-K_{s}$ to $T_{\mathrm{eff}}$. For each
star, the final temperature was chosen to be the IRFM temperature which was
closest to the one from the 2MASS color calibration. With this criterion, the
KIC ($griz$) set of temperatures and angular diameters was used for 53% of the
stars and the Tycho-2 ($B_{T}V_{T}$) set was used for the remaining 47%. We
could of course have used the 2MASS temperatures directly; however, this would
defeat the purpose of using the IRFM which is to obtain a set of self-
consistent temperatures, radii, and distances.

In principle, the IRFM parameters should be derived anew when using the
individual frequencies, instead of the scaling relations, to derive stellar
parameters. However, in practice, the radii which are used to get the
distances do not change enough for the changes in reddening to be significant.
A test showed that for the majority of the frequency sample the difference in
$E(B-V)$ is at the level of 0.001 mag. This level of agreement translates to a
difference in $T_{\mathrm{eff}}$ of just a few kelvin which is well below the
statistical uncertainties.

##  4 Results

###  4.1 Parallax comparisons for the frequency sample

Figure 3: Absolute differences between seismic and Gaia parallaxes as a
function of Gaia parallax for the frequency sample. The dot-dashed lines are
linear fits to the data which are weighted according to the uncertainties and
the thick dashed line is the linear relation found by ST16 based on radii of
eclipsing binaries (the lines curve due to the logarithmic x-axis). Grey
shaded areas indicate the 1$\sigma$-region of the linear fits. The left- and
right-hand columns show results based on frequency ratio fits and the scaling
relations, respectively. The rows show different methods for calculating
seismic distances (and hence parallaxes) as labelled.

Based on the seismic distances, $d_{\mathrm{seis}}$, we simply calculate the
seismic parallaxes as $\varpi_{\mathrm{seis}}=1/d_{\mathrm{seis}}$. In Figure
3 the absolute differences between the seismic and Gaia parallaxes are plotted
as a function of the Gaia parallaxes for the frequency sample. The columns
show the two different methods used to derive stellar properties, and the rows
show the different methods for calculating distances. A number of outliers
were identified in the original study of the LEGACY sample, and all of the
stars they found to be problematic have been excluded from this comparison.
They quoted several sources to argue that the offsets are mainly due to
contaminated photometry. For example, some of the stars are binaries and their
flux measurements include the contributions from both components (see the
discussion in Silva Aguirre et al. 2017, section 5.3).

If we first consider the results based on fits to frequency ratios (left-hand
column), there is good agreement overall and individual differences are
generally within the statistical uncertainties. However, there is also a trend
towards seismic parallaxes being too large on average. For the results based
on spectroscopic temperatures, the weighted mean difference is
$\Delta\varpi=(0.21\pm 0.04)~{}\mathrm{mas}$, and the offset also seems to be
distance dependent as indicated by the linear fit to the data which is
weighted according to the plotted uncertainties. Based on this fit, the
difference as a function of parallax is approximately given by

$\Delta\varpi=(0.03\varpi+0.03)~{}\mathrm{mas}\,.$ (5)

This is similar to what was found by ST16 based on radii of eclipsing binaries
(dashed lines in the figure), but their linear relation predicts a slightly
larger offset at low parallaxes (with an intersection at 0.16 mas) which is
just outside the $1\sigma$-region of what is found here. For parallaxes
${\gtrsim}10~{}\mathrm{mas}$ the two linear relations agree. A distance
dependent absolute offset means that the relative offset is constant across
all distances. The weighted mean parallax ratio is $1.029\pm 0.006$, meaning
that the seismic parallaxes are overestimated by about 3% assuming that the
Gaia parallaxes are unbiased.

An overestimation of the seismic parallaxes is exactly what we would expect to
get if the temperatures are underestimated. Higher temperatures will lead to
higher luminosities which increase the derived distances and lowers the
parallaxes. Comparing the stellar parameters obtained with the two sets of
temperatures, the increase of about 60 K on average leads to best-fitting
models which are ${\sim}2\%$ more luminous. The mean change in radius,
however, is below 0.1% which means that the entire increase in luminosity is
due to the best-fitting models being hotter. Comparing the upper two panels,
which differ only in the adopted temperatures, we see that the parallaxes are
still too high on average. Surprisingly, with the IRFM temperatures, the
weighted mean ratio has not decreased. In fact, it has increased slightly (but
not significantly) to $1.031\pm 0.006$ for the results using the distance
modulus. As argued above, higher luminosities should lead to lower parallaxes
and better overall agreement with the Gaia data. Based on the proportionality
from the distance modulus $\varpi\propto(L/L_{\odot})^{-1/2}$, the parallaxes
are expected to decrease by about $1\%$ due to the luminosity increase of
$2\%$. If we instead compare the median ratios, the results are $1.041$ and
$1.033$ with the spectroscopic and IRFM temperatures, respectively. This is
more in line with our expectations and indicates that the parallaxes have
decreased overall, but the stars with high weights (i.e. low statistical
uncertainties) have retained their seismic parallax value. In any case, there
is a systematic offset at the level of $3\%$ regardless of the adopted
temperature scale. A comparison between the two lower panels shows that the
two different methods used to calculate distances agree very well. When
comparing the two sets of seismic parallaxes directly, they are found to agree
within 0.2% on average even though different photometry was used in the
visible for two-thirds of the stars (which is the fraction of the frequency
sample for which the Tycho-2 photometry was used to derive the IRFM
temperatures).

Figure 4: Ratios of seismic and Gaia parallaxes as a function of effective
temperature (left) and metallicity (right) for the frequency sample. The data
are shown as weighted mean bins with equal separation. Errorbars on the binned
data are shown for the results based on frequency ratios to give a sense of
the uncertainty at a given value on the x-axis. The most metal-poor star of
the sample ($[\mathrm{Fe}/\mathrm{H}]=-0.92$) has been excluded in the right-
hand panel.

Turning to the results based on the scaling relations (right-hand column of
Figure 3), the mean parallax offset has decreased compared to the results
based on ratio fits. For the spectroscopic temperatures, the weighted mean
difference is $\Delta\varpi=(0.14\pm 0.05)~{}\mathrm{mas}$ which still implies
that the seismic parallaxes are too high. In this case, however, the
temperature scale does make a difference since it enters explicitly into the
scaling relations, and with IRFM temperatures and angular diameters the offset
decreases to just $\Delta\varpi=(0.05\pm 0.05)~{}\mathrm{mas}$, consistent
with no offset. Note also that for both temperature scales the slope of the
linear fit is not significantly different from zero, and for the IRFM
temperatures the offset is clearly not compatible with the results of ST16 at
low parallaxes. This stands in contrast to the results based on ratio fits
where the slope is significant regardless of the adopted temperatures.

These results corroborate the findings of Huber et al. (2017), namely that the
hotter IRFM temperatures lead to better agreement between seismic (from
scaling relations) and Gaia parallaxes, and that the offset identified by ST16
may be overestimated for low parallaxes. However, the offset found by Silva
Aguirre et al. (2017) for the LEGACY sample has not been reduced by the
inclusion of the Kages stars or by the change in temperature. This leads to an
interesting tension between the scaling relations giving good agreement with
the Gaia parallaxes and the fits to frequency ratios showing a constant offset
of a few percent.

The differences between the results from the scaling relations and the ratio
fits, and the impact of the temperature scale, become clearer when the results
are plotted together as a function of temperature as shown in the left-hand
panel of Figure 4. To avoid clutter, only the weighted mean bins are shown,
and the error bars, which are of similar magnitude for all data points at a
given temperature, are only shown for the ratio fits. At the lowest
temperatures, the scaling relations generally agree well with the ratio fits
for both temperature scales. However, at higher temperatures the scaling
relations deviate from the ratio fits and result in lower parallaxes. So the
main reason that the parallaxes from the scaling relations show overall better
agreement with Gaia seems to be a deviation from the ratio fits at high
temperatures. There are also differences between the two applications of the
scaling relation (upper and lower panel) due to the different temperatures. At
high temperatures ($T_{\mathrm{eff}}>5800~{}\mathrm{K}$) the IRFM results give
lower seismic parallaxes than the spectroscopic results and vice versa at the
lower temperatures. This is what makes the IRFM results compatible with the
Gaia data (on average), and it is simply a reflection of differences in the
temperature scales.

The right-hand panel of Figure 4 shows the parallax comparison as a function
of metallicity where no significant trends are found. Here the difference
between the scaling relations and the ratio fits show up as a constant offset
which does not depend on the metallicity.

Since the scaling relations are anchored to the solar values, they are
naturally thought to be most accurate for stars like the Sun. Therefore, it is
interesting that the parallaxes based on the scaling relations agree so well
with the ones from frequency ratios at near-solar temperatures and only
deviate at the highest temperatures considered in this study. This indicates
that either the asteroseismic radii are underestimated at temperatures around
that of the Sun, and the scaling relations become more accurate at higher
temperatures, or, which seems more likely, that there is a bias in either the
Gaia parallaxes or the calculation of seismic parallaxes, and that the scaling
relations overestimate the radii at high temperatures. If the deviation
between the two methods at high temperatures is indeed due to inaccuracies in
the scaling relations, we are still left with the problem of explaining a 3%
offset in seismic parallaxes based on fits to frequency ratios.

Figure 5: Same as Figure 3 but for the main sample with the grid-based results
based on $\Delta\nu_{\mathrm{fit}}$ instead of frequency ratios.

The Gaia parallaxes are known to suffer from position and colour dependent
systematic errors at a level of $\pm$0.3 mas (Lindegren et al., 2016). This
may explain part of the observed offset; however, it seems unlikely to explain
a constant fractional offset such as the one found here. A fractional offset
may instead be introduced in the conversion between seismic radii and
parallaxes. Huber et al. (2017) tested the use of different ways to determine
bolometric corrections and found that it can lead to systematic errors of
around 1% in parallax due to the use of different model atmospheres and
methods for extracting the synthetic fluxes. It is also possible that the
extinction values of the 3D map are biased. In order to decrease the seismic
parallaxes and bring them into agreement with the Gaia values, the degree of
extinction has to be decreased. However, the stars considered here are not
very distant ($d<1~{}\mathrm{kpc}$), and the median extinction in the visible
is only about 0.03 mag based on the extinction map. For comparison, a
difference of about 0.06 magnitudes is necessary in order to explain a
relative difference of 3% in parallax.

It is also worth discussing what could lead to underestimated seismic radii
from the ratio fits, if that is in fact the cause of the offset. The most
obvious sources of bias in the stellar parameters are the model physics that
have been fixed based on e.g. a solar calibration. For example, the mixing
length parameter $\alpha_{\text{mlt}}$ is fixed at the solar-calibrated value
of $1.791$ in all models. However, 3D simulations of convection have shown
that the convective efficiency varies with temperature and surface gravity
(e.g. Trampedach et al. 2014). Additionally, there is a known degeneracy
between the initial helium abundance and the stellar mass (Silva Aguirre et
al., 2015) and, for the grids used in this study, the initial helium abundance
is fixed for any given metallicity by the enrichment law $\Delta Y/\Delta
Z=1.179$. Finally, the overshoot efficiency directly influences the obtained
masses for stars with convective cores.

Now the question is whether the effects of fixing the model physics are large
enough to explain the observed parallax offset. In the original study of the
LEGACY sample (Silva Aguirre et al., 2017), a number of different stellar
model grids and fitting algorithms were applied. Most notably, a number of the
algorithms left the values of $\alpha_{\text{mlt}}$ and the initial helium
abundance as free parameters to be optimized during the fit. Even with this
variety of methods, the overall agreement between the radii was good (with
typical mean offsets between the methods below 1%) and the parallax offset
compared to the Gaia values was seen with every method. Therefore, model
physics are only able to explain part of the offset unless more fundamental
features of the models like the opacities or the equation of state need
adjustments.

All things considered, it is not possible to point to any one part of the
analysis as the sole source of the observed offset. It may be caused by any
combination of the different factors that have been discussed, including of
course the seismic radii. What can be said, however, is that there are no
indications of deviations from the scaling relation for the radius by more
than 5% based on these parallax comparisons.

###  4.2 Parallax comparisons for the main sample

In Figure 5 the absolute differences between the seismic and Gaia parallaxes
are shown for the main sample. One feature which must be mentioned is the
tendency for the offset to be systematically positive at the lowest
parallaxes. This happens because the Gaia parallaxes are less precise than the
seismic ones, and therefore scatter to lower values causing a positive offset.
Similarly, when they scatter to higher values the offset is negative, and the
combined effect is a diagonal edge at low parallaxes.

Like we saw for the ratio fits of the frequency sample, the grid-based method
results in absolute offsets which increase with parallax. Due to the diagonal
edge at low parallaxes, only stars with
$\varpi_{\mathrm{Gaia}}>2.5~{}\mathrm{mas}$ have been included in the linear
fits. With spectroscopic temperatures the weighted mean difference is
$\Delta\varpi=(0.28\pm 0.02)~{}\mathrm{mas}$, and the linear fit is given
approximately by

$\Delta\varpi=0.06\varpi+0.07\,.$ (6)

This slope is double the value found for the ratio fits and by ST16. The
weighted mean relative offset has also doubled to $(6.0\pm 0.4)\%$.

Figure 6: Same as Figure 4 but for the main sample with the grid-based results
based on $\Delta\nu_{\mathrm{fit}}$ instead of frequency ratios. The two most
metal-poor stars of the sample ($[\mathrm{Fe}/\mathrm{H}]=-0.92,-1.75$) have
been excluded in the right-hand panel.

The temperature carries more weight in the fits to $\Delta\nu_{\mathrm{fit}}$
of the main sample than it did for the fits to frequency ratios, and the
change to IRFM temperatures has a larger impact. Changing from spectroscopic
temperatures and the distance modulus to IRFM temperatures and angular
diameters, the weighted mean parallax offset is reduced to $(3.3\pm 0.4)\%$,
bringing it much closer to the results of the frequency sample and ST16. The
combination of IRFM temperatures and the distance modulus gives an offset of
$(4.9\pm 0.4)\%$ which falls right in between the two extremes. Thus, the mean
offset gradually decreases when going from the upper to the lower panels of
Figure 5.

Like for the frequency sample, the scaling relations lower both the overall
offset between the parallaxes and the slopes of the linear fits. For the
spectroscopic and IRFM temperatures, the weighted mean differences are
$\Delta\varpi=(0.16\pm 0.02)~{}\mathrm{mas}$ and $(0.07\pm
0.02)~{}\mathrm{mas}$, respectively, in agreement with the offsets found with
the scaling relations for the frequency sample, but with lower uncertainties
owing to the larger sample size. The slope is only just significant within one
standard deviation for the spectroscopic temperatures, but for the IRFM
temperatures it is insignificant. It is also seen once again that the relation
from ST16 overestimates the offset at low parallaxes compared to the results
found here with both temperature scales.

In Figure 6 the weighted mean bins from the scaling relations and the fits to
$\Delta\nu_{\mathrm{fit}}$ are compared directly for the two sets of
temperatures. The scaling relations follow the fits to
$\Delta\nu_{\mathrm{fit}}$ at low temperatures where the seismic parallaxes
show the best agreement with Gaia. For higher temperatures, i.e.
$T_{\mathrm{eff}}\gtrsim 6000~{}\mathrm{K}$ the two methods begin to
increasingly deviate. Note how similar the trend is to the one for the
frequency sample (Figure 4) if the temperatures below 5500 K are ignored. In
this region, both samples agree that the seismic parallaxes based on the
scaling relations change with temperature while the methods independent of the
scaling relations show a nearly constant offset. Also like for the frequency
sample, there are no significant trends with metallicity.

###  4.3 Corrections to the scaling relations

If we assume that the parallax differences seen in Figure 4 and Figure 6 are
solely due to errors in the seismic radii, the scaling relation for the radius
needs a correction, depending on the temperature, with a maximum value of
about 5%. For example, using the results from the IRFM, the radii are, on
average, underestimated at the solar temperature and overestimated at the
highest temperatures in the sample. The best agreement between the parallaxes
is found for the subgiants in the main sample which suggests that little or no
correction is needed to the scaling relation for these stars.

The fact that both the parallaxes from the scaling relations and from
frequency modelling show an offset at the solar temperature (where the scaling
relations are thought to be accurate for dwarf stars) leads us to consider the
alternate assumption that there is a bias in either the Gaia parallaxes or in
the quantities involved in transforming the seismic radii into distances (i.e.
bolometric corrections and extinctions). The parallax comparisons show that
the radii of the scaling relations deviate from those of the grid-based
analyses, which are independent of the scaling relations, at high effective
temperatures. Even though the scaling relations give parallaxes which agree
better with the Gaia data on average, it seems most likely that the trend with
temperature reflects a problem with the scaling relations. In the following,
we attempt to quantify this temperature trend and correct the scaling
relations for it. We will use the stellar parameters obtained from the ratio
fits (with IRFM temperatures) as reference values. Despite the constant
parallax offset of 3%, these radii and masses are very precise and they are
internally consistent within the adopted stellar models. Thus we consider them
to be the best available substitute for the true values. However, it should be
kept in mind that the validity of the corrections we obtain for the scaling
relations depend on the assumption that the parallax offset of 3%, between
parallaxes from Gaia and from fitting to frequency ratios, is not due to
systematic errors in the seismic radii. We also assume that the temperature
trend is not due to a potential temperature dependent bias in the IRFM
temperatures.

We take the stellar radii, masses, and temperatures from the best-fitting
models and calculate the model values $\Delta\nu_{\mathrm{scaling}}$ and
$\nu_{\mathrm{max,scaling}}$ based on the scaling relations (Equation 1 and
Equation 2). It is interesting whether or not the deviations between
$\Delta\nu_{\mathrm{scaling}}$, $\nu_{\mathrm{max,scaling}}$ and the observed
values depend on the physical parameters of the stars. If they do, it is
possible to define corrections to the scaling relations which will bring the
radii and masses calculated from them into line with the ones found from the
ratio fits. Since this analysis does not depend on the Gaia parallaxes we have
included the LEGACY and Kages stars which were not a part of the GDR1,
increasing the sample size from 86 to 95 stars.

As we saw in the parallax comparisons, the difference between the radii
obtained from the ratio fits and those given by the scaling relations varies
as a function of temperature. This suggests that corrections to the scaling
relations for $\Delta\nu$ and $\nu_{\mathrm{max}}$ should at least depend on
temperature. For $\Delta\nu$ this is no surprise since previous studies (e.g.
White et al. 2011) already showed that the difference between
$\Delta\nu_{\mathrm{fit}}$ and $\Delta\nu$ from the scaling relation is mainly
a function of temperature. However, using the masses and radii of the ratio
fits it will be possible to investigate potential corrections to the scaling
relation for $\nu_{\mathrm{max}}$ as well. Defining the two functions
$f_{\Delta\nu}(T_{\mathrm{eff}})$ and
$f_{\nu_{\mathrm{max}}}(T_{\mathrm{eff}})$ as the temperature dependent
corrections to the scaling relations for $\Delta\nu$ and $\nu_{\mathrm{max}}$,
respectively, the corrected versions of Equations 1 and 2 are given by

$\displaystyle\frac{\Delta\nu}{\Delta\nu_{\sun}}$
$\displaystyle\simeq\left(\frac{M}{M_{\sun}}\right)^{1/2}\left(\frac{R}{R_{\sun}}\right)^{-3/2}f_{\Delta\nu}(T_{\mathrm{eff}})\;,$
(7) $\displaystyle\frac{\nu_{\mathrm{max}}}{\nu_{\mathrm{max},\sun}}$
$\displaystyle\simeq\left(\frac{M}{M_{\sun}}\right)\left(\frac{R}{R_{\sun}}\right)^{-2}\left(\frac{T_{\mathrm{eff}}}{T_{\mathrm{eff},\sun}}\right)^{-1/2}f_{\nu_{\mathrm{max}}}(T_{\mathrm{eff}})\;,$
(8)

from which the corrected scaling relations for the mass and radius follow

$\displaystyle\frac{M}{M_{\sun}}$
$\displaystyle\simeq\left(\frac{\nu_{\mathrm{max}}}{\nu_{\mathrm{max},\sun}}\right)^{3}\left(\frac{\Delta\nu}{\Delta\nu_{\sun}}\right)^{-4}\left(\frac{T_{\mathrm{eff}}}{T_{\mathrm{eff},\sun}}\right)^{3/2}f_{\nu_{\mathrm{max}}}^{-3}(T_{\mathrm{eff}})f_{\Delta\nu}^{4}(T_{\mathrm{eff}})\;,$
(9) $\displaystyle\frac{R}{R_{\sun}}$
$\displaystyle\simeq\left(\frac{\nu_{\mathrm{max}}}{\nu_{\mathrm{max},\sun}}\right)\left(\frac{\Delta\nu}{\Delta\nu_{\sun}}\right)^{-2}\left(\frac{T_{\mathrm{eff}}}{T_{\mathrm{eff},\sun}}\right)^{1/2}f_{\nu_{\mathrm{max}}}^{-1}(T_{\mathrm{eff}})f_{\Delta\nu}^{2}(T_{\mathrm{eff}})\;.$
(10)

The combinations of correction factors in these equations can be defined as
correction factors for the mass
$f_{M}=f_{\nu_{\mathrm{max}}}^{-3}f_{\Delta\nu}^{4}$, and the radius
$f_{R}=f_{\nu_{\mathrm{max}}}^{-1}f_{\Delta\nu}^{2}$. Corrections that apply
to the frequency sample can be obtained by comparing the observed and modelled
asteroseismic parameters as a function of temperature. Only the stars with
effective temperatures higher than 5500 K will be used in this analysis due to
the poor sample size at lower temperatures.

Figure 7: Ratio of observed and scaling relation $\Delta\nu$ (upper panel) and
$\nu_{\mathrm{max}}$ (lower panel) as a function of IRFM effective temperature
with error bars indicating the uncertainties on the observed values. The
scaling relation values are taken from the best-fitting models of the ratio
fits and depend on the models’ masses, radii, and temperatures. The data are
color coded according to surface gravity and the dashed lines are the best-
fitting relations discussed in the text. The shaded areas indicate the
bootstrapped uncertainties of the fits.

In Figure 7, the ratios of observed and scaling relation $\Delta\nu$ and
$\nu_{\mathrm{max}}$ are shown as a function of observed effective temperature
and color coded by surface gravity. The error bars only include the
statistical uncertainty on the observed values. Inspired by the functional
form of the correction found by White et al. (2011) a second order polynomial
has been fitted to the data for $\Delta\nu$. The fit follows the data nicely,
and the residual scatter is at a level of ${\sim}1\%$. For this limited range
of surface gravities, there is no strong indication that the offset depends on
the value of $\log g$. Based on the fit, the correction factor for $\Delta\nu$
is given by

$f_{\Delta\nu}(T_{\mathrm{eff}})=-2.52\left(\frac{T_{\mathrm{eff}}}{10^{4}\mathrm{K}}\right)^{2}+2.90\left(\frac{T_{\mathrm{eff}}}{10^{4}\mathrm{K}}\right)+0.17\;,$
(11)

which applies in the range of temperatures and surface gravities covered in
the figure. This expression gives a small correction of
$f_{\Delta\nu}(T_{\mathrm{eff},\sun})=1.002$ at the solar temperature which is
not ideal considering that the scaling relations are anchored to the solar
values. One could add a constraint on the fit to make it hit unity at the
solar temperature, but since the deviation is within the uncertainty (shaded
area in the figure), we have chosen not to do so.

For $\nu_{\mathrm{max}}$ the uncertainties are larger and the data points are
much more scattered at any given temperature. Although the correlation with
temperature is less clear, the offset does seem to increase slightly with
increasing temperature. The linear fit shown in the figure implies a
correction factor of

$f_{\nu_{\mathrm{max}}}(T_{\mathrm{eff}})=0.397\left(\frac{T_{\mathrm{eff}}}{10^{4}\mathrm{K}}\right)+0.771\;.$
(12)

Like for the $\Delta\nu$ correction, the fit hits unity at the solar
temperature within the uncertainty and has not been constrained. The large
scatter around this relation is partly due to the uncertainties on the
observed values, but it may also suggest that the offset depends on more than
just temperature. Most of the stars with low surface gravities seem to fall
below the linear fit which indicates that the correlation can be improved by
including a $\log g$ term. A dependence on surface gravity is reasonable
considering it is the other factor, besides the temperature, which enters into
the $\nu_{\mathrm{max}}$ scaling relation. Additionally, a dependence on
metallicity is not unthinkable considering that Viani et al. (2017) recently
pointed out that the approximation usually adopted for the acoustic cut-off
frequency $\nu_{\mathrm{ac}}$ (and therefore also $\nu_{\mathrm{max}}$) lacks
a factor related to the mean molecular weight. However, no significant
correlations with these parameters have been found. A direct comparison
between $\nu_{\mathrm{max,obs}}/\nu_{\mathrm{max,scaling}}$ and the
metallicity, showing no significant correlation, is given in Figure 8.
Similarly, no significant correlation is found with $\log g$. Thus, the
dependence of the correction on surface gravity and metallicity is not very
strong, and the simple relation in Equation 12 has been adopted as the
correction to $\nu_{\mathrm{max}}$ in the following.

Figure 8: Ratio of observed and scaling relation $\nu_{\mathrm{max}}$ as a
function of spectroscopic metallicity with errorbars indicating the
uncertainty on the observed values. The scaling relation values are taken from
the best-fitting models of the ratio-fits and depend on the models’ masses,
radii, and temperatures (using the IRFM set). The data are color coded
according to surface gravity and the dashed line is the best linear fit. The
shaded area indicates the bootstrapped uncertainty of the fit. Figure 9:
Corrections to $\Delta\nu$ (black dashed), $\nu_{\mathrm{max}}$ (black dot-
dashed), and the resulting corrections to the scaling relations for the radius
(blue) and mass (green).

Figure 9 shows the corrections obtained for $\Delta\nu$ and
$\nu_{\mathrm{max}}$ as well as the corrections they imply for the scaling
relations giving the mass and radius. The correction to $\nu_{\mathrm{max}}$
is greater than the correction to $\Delta\nu$ at all temperatures; however,
due to the greater exponent on $\Delta\nu$ in the scaling relations, the two
corrections have an almost equal impact on the stellar radius and mass at the
highest temperatures. These corrections imply that the scaling relations
overestimate the radii and masses of main-sequence stars by $(8\pm 1.5)\%$ and
$(22\pm 4)\%$, respectively, at a temperature of 6600 K. However, they also
show that for main-sequence stars with temperatures below 6400 K (which is 85%
of the stars in this study) radii and masses from the scaling relations are
accurate to within 5% and 13%, respectively. Due mostly to the large scatter
around $f_{\nu_{\mathrm{max}}}$, these numbers represent averages for the
frequency sample and do not apply to each of the stars individually.

Coelho et al. (2015) performed a test of the $\nu_{\mathrm{max}}$ scaling
relation using the same method applied here, but to about 400 of the Kepler
dwarfs with fits to $\Delta\nu_{\mathrm{fit}}$ instead of frequency ratios.
They only found a constant offset from the scaling relation of a few percent
and no significant trend with temperature. Even though the results presented
here are based on frequency ratios which give more precise (and likely
accurate) stellar parameters than $\Delta\nu_{\mathrm{fit}}$, the sample size
is also considerably smaller, and it is not unthinkable that the trend
disappears with more observations. In fact, if all data points below a
temperature of 5800 K are removed from Figure 7, the linear fit to
$\nu_{\mathrm{max}}$ is instead completely flat suggesting a constant offset
of about 3% independent of the temperature. An updated analysis in the future
using individual frequencies for a larger sample of main-sequence stars would
help shed more light on this issue. There is also the potential of including
subgiant stars with individual frequencies, in order to extend this analysis
to later evolutionary stages, but detailed frequency fitting of subgiants
still takes a lot of effort due to their complex oscillation spectra including
mixed modes (see e.g. Grundahl et al. (2017); Brandão et al. (2011)).

It is interesting that none of the results has shown any indications that the
scaling relations need a metallicity-dependent correction. As mentioned
previously, Viani et al. (2017) pointed out that the approximation of the
acoustic cutoff frequency, which is used to derive the scaling relation for
$\nu_{\mathrm{max}}$, introduces a systematic offset in radius that depends on
metallicity due to the omission of a factor related to the mean molecular
weight. In fact, the offset should be ${\sim}6\%$ in radius between the lowest
and highest metallicities of the samples considered in this study. This is at
the same level as the observed temperature dependence and it should be
detectable with the current precision of the parallaxes. However, no such
trend is seen in the parallax comparisons for either of the samples or in the
direct comparison between $\nu_{\mathrm{max}}$ from observations and the
scaling relations. It is possible that inaccuracies in the relation between
$\nu_{\mathrm{max}}$ and $\nu_{\mathrm{ac}}$ somehow makes up for the
inaccuracies in $\nu_{\mathrm{ac}}$, but until reliable theoretical
predictions of $\nu_{\mathrm{max}}$ can be made, it is not possible to know
for sure. Still, it would be an interesting continuation of this work to
include the proposed metallicity correction to $\nu_{\mathrm{max}}$ in the
grids of stellar models and test its impact on the obtained radii.

Finally, it is important to keep in mind the assumptions that have led to the
proposed corrections to the scaling relations. They depend directly on the
actual accuracy of the radii and masses given by the ratio fits which have
been used as reference values. While the corrections remove the observed trend
with temperature in the parallax comparison, there is still a constant offset
of 3% which remains unexplained, and the corrections should be applied with
this caveat in mind.

##  5 Conclusions

The main results of comparing asteroseismic and Gaia parallaxes can be summed
up as follows.

Parallaxes based on the fits to frequency ratios are, on average, about 3%
higher than the Gaia parallaxes which leads to an absolute offset which
increases with parallax. The offset does not depend on the adopted
temperatures or the method used to calculate distances.

For parallaxes based on radii from the scaling relations, the temperature
scale does affect the comparison. Using IRFM temperatures and angular
diameters instead of spectroscopic temperatures and the distance modulus
decreases the offset by about 2%, bringing the parallaxes into agreement with
Gaia on average. However, the offset becomes temperature dependent. Within
about 200 K of the solar temperature, the scaling relations agree on the
offset found from the ratio fits, and at both lower and higher temperatures
the offset decreases.

None of the results indicated any dependence of the offset on metallicity.

Due to the indirect nature of the parallax comparison, it is difficult to
convert it directly into an overall accuracy of the seismic radii. In the end,
the results indicate that the scaling relation for the radius is accurate to
within 5% for main-sequence and subgiant stars. No tighter constraints can be
placed on the analysis until more precise and accurate parallaxes become
available from future Gaia data releases. What is more interesting then is the
fact that the fractional offset between seismic parallaxes based on the
scaling relations and Gaia parallaxes change as a function of temperature.
Based on the good agreement between the scaling relations and the ratio fits
at the solar temperature, it seems most likely that the deviation between the
offsets at other temperatures reflect inaccuracies in the scaling relations,
rather than improved accuracy.

By adopting the stellar parameters obtained from ratio fits as reference
values, we have defined corrections to the scaling relations for $\Delta\nu$
and $\nu_{\mathrm{max}}$. It is important to note that the validity of these
corrections depend on the actual accuracy of the reference values. In adopting
the radii from the ratio fits as reference values we assume that the 3%
parallax offset is due to a systematic error in the Gaia parallaxes or in the
bolometric corrections and extinctions used to convert the seismic radii into
distances (or perhaps a combination of these). The corrections are found to
mainly be functions of effective temperature, and no significant dependence on
metallicity is found despite the fact that the scaling relation for the
acoustic cut-off frequency needs to be corrected for the star’s mean molecular
weight. This may be due to inaccuracies in the scaling between
$\nu_{\mathrm{max}}$ and $\nu_{\mathrm{ac}}$, and a test of the effect of
adding a mean molecular weight term to the scaling relation in the grid-based
method would help reveal its impact on the stellar parameters.

The obtained corrections imply that the scaling relations systematically
overestimate both radii and masses for main-sequence stars at super-solar
temperatures. For stars with temperatures below 6400 K, the deviations from
the scaling relations stay below a level of 5% in radius and 13% in mass. The
correction to $\nu_{\mathrm{max}}$ is questionable due to a large scatter
around the relation, and the fact that the correlation with temperature is
very weak if the low-temperature stars are excluded. An extension of the
analysis to later evolutionary stages by the inclusion of subgiants would help
shed light on the validity of the proposed linear correlation with
temperature.

## Acknowledgements

Funding for the Stellar Astrophysics Centre is provided by The Danish National
Research Foundation (Grant agreement No. DNRF106). V.S.A. acknowledges support
from VILLUM FONDEN (research grant 10118). L.C gratefully acknowledges support
from the Australian Research Council through Discovery Program DP150100250 and
Future Fellowship FT160100402. This work has made use of data from the
European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia),
processed by the Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This publication makes use
of data products from the Two Micron All Sky Survey, which is a joint project
of the University of Massachusetts and the Infrared Processing and Analysis
Center/California Institute of Technology, funded by the National Aeronautics
and Space Administration and the National Science Foundation.

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# Pressure evolution of the normal- and superconducting-state properties of
the line-nodal material CaSb2 revealed by 123Sb nuclear quadrupole resonance

H. Takahashi<EMAIL_ADDRESS>Department of Physics,
Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan  S.
Kitagawa  Department of Physics, Graduate School of Science, Kyoto University,
Kyoto 606-8502, Japan  K. Ishida  Department of Physics, Graduate School of
Science, Kyoto University, Kyoto 606-8502, Japan  A. Ikeda  Maryland Quantum
Materials Center and Department of Physics, University of Maryland, College
Park, Maryland 20742, USA  Toyota Riken-Kyoto University Research Center
(TRiKUC), Kyoto 606-8501, Japan  S. R. Saha  Maryland Quantum Materials Center
and Department of Physics, University of Maryland, College Park, Maryland
20742, USA  S. Yonezawa  Department of Physics, Graduate School of Science,
Kyoto University, Kyoto 606-8502, Japan  J. Paglione  Maryland Quantum
Materials Center and Department of Physics, University of Maryland, College
Park, Maryland 20742, USA  Canadian Institute for Advanced Research, Toronto,
Ontario, Canada M5G 1Z8  Y. Maeno  Department of Physics, Graduate School of
Science, Kyoto University, Kyoto 606-8502, Japan  Toyota Riken-Kyoto
University Research Center (TRiKUC), Kyoto 606-8501, Japan

###### Abstract

CaSb2 is the Dirac line-nodal material that exhibits a superconducting (SC)
transition at 1.7 K. In spite of its conventional SC state at ambient
pressure, the transition temperature $T_{\mathrm{c}}$ shows a peak structure
against hydrostatic pressure. We performed ac magnetic susceptibility and
123Sb nuclear quadrupole resonance (NQR) measurements on single-crystalline
CaSb2 under pressures up to 2.08 GPa. $T_{\mathrm{c}}$ monotonically increased
in this pressure region, which is consistent with a previous study. We
observed continuous broadening of the NQR spectrum against pressure, which is
a sign of unique compression behavior of the lattice. In the normal state, the
nuclear spin-lattice relaxation rate $1/T_{1}$ is proportional to temperature
in all pressure values; typical of a metal. However, $1/T_{1}T$ in the normal
state is independent of pressure, indicating that the density of states at the
Fermi energy $N(E_{\mathrm{F}})$, which is one of the parameters governing
$T_{\mathrm{c}}$, is insensitive to pressure. From these results, we conclude
that $N(E_{\mathrm{F}})$ does not govern the origin of the enhancement in
$T_{\mathrm{c}}$. This is unusual for a weak electron-phonon coupling
superconductor. In the SC state, we revealed that the SC gap becomes larger
and more isotropic under pressure.

Introduction. Materials with nontrivial topological features have been the
central part of the research target in condensed matter physics. In
particular, topological superconductivity has been attracting much attention
since it could be a host of exotic bound states. One of the promising
candidates for bulk topological superconductivity is superconductivity in
semimetals with nontrivial topological numbers. In such materials, chemical
doping is not necessary for the emergence of superconductivity. This ensures
“clean” topological superconductivity. So far, many studies have been
performed to discover superconductivity in topological semimetals such as
Dirac [1, 2, 3, 4, 5], Weyl [6, 7, 8, 9, 10, 11, 12, 13, 14], and line-nodal
semimetals [15, 16, 17, 18, 19, 20, 21, 22, 23]. Dirac and Weyl semimetals
have topologically protected point nodes in their bulk electronic bands, while
nodes form lines in the $\bm{k}$ space in line-nodal semimetals. In terms of
superconductivity, line-nodal semimetals seem to be especially advantageous
due to their moderate density of states (DOS). As a matter of fact, a greater
number of superconducting (SC) line-nodal semimetals exhibit superconductivity
without pressure or other techniques [15, 16, 17, 18, 19, 21, 22, 23].
Therefore, the line-nodal semimetal is a good platform for exploring
topological superconductivity and for understanding the relationship between
nodal lines and superconductivity in detail.

CaSb2 is a superconductor [23] which was predicted as a line-nodal material
[24]. It crystalizes in a monoclinic structure with a nonsymmorphic space
group ($P2_{1}/m$, No. 11, $C^{2}_{2h}$) with two nonequivalent
crystallographic Sb sites, Sb(1) and Sb(2) [Fig. 1(a)]. In most line-nodal
materials, line nodes are gapped by spin-orbit coupling (SOC). However, the
line nodes in CaSb2 are protected against SOC by nonsymmorphic crystalline
symmetry [24], making CaSb2 suitable for investigating the superconductivity
in line-nodal materials.

First-principles calculation predicts that CaSb2 has two kinds of Fermi
surfaces: a topologically trivial three-dimensional hole pocket that stems
from Sb(2) and a pair of quasi-two-dimensional electron pockets related to the
nodal lines that stems from Sb(1) [25]. Some of the authors of this Letter
observed both of them by quantum oscillation measurements [26]. Regarding
superconductivity, we succeeded in observing the nuclear quadrupole resonance
(NQR) signals of 121/123Sb(1) and revealed a full-gap $s$-wave SC behavior at
ambient pressure [25]. A full-gap behavior was also reported from magnetic
penetration depth measurements [27]. Microfocused angle-resolved photoemission
spectroscopy ($\mu$-ARPES) reported that the existence of electron pockets is
sample dependent and pointed out that topologically trivial hole pockets play
a dominant role in superconductivity [28]. These results indicate that, at
least at ambient pressure, the SC state of CaSb2 is conventional.

On the contrary, the deviation from the Bardeen-Cooper-Schrieffer (BCS)
behavior in specific heat was reported [29]. Furthermore, we observed
nonmonotonic behavior of the SC transition temperature $T_{\mathrm{c}}$
against hydrostatic pressure, indicating the unconventional character of the
SC state in CaSb2 [30]. The origin of the pressure dependence of
$T_{\mathrm{c}}$ remains unexplained. We note that this kind of behavior in
$T_{\mathrm{c}}$ is also observed in a SC line-nodal metal NaAlSi [31], whose
origin is not revealed either. We consider that it is important to clarify the
origin of nonmonotonic change in $T_{\mathrm{c}}$ to understand the SC
properties of CaSb2 and line-nodal materials, particularly those relating to
band topology.

In this Letter, we report ac magnetic susceptibility measurements and NQR
measurements on a 123Sb(1) site, at which the nodal behavior is expected,
under hydrostatic pressures up to 2.08 GPa to reveal the origin of the
increase in $T_{\mathrm{c}}$. The increase in $T_{\mathrm{c}}$ by pressure
reported for polycrystalline samples was reproduced in single crystals. In the
normal state, the nuclear spin-lattice relaxation rate $1/T_{1}$ is
proportional to temperature, namely the value of $1/T_{1}T$ is constant in all
pressure values. The constant value of $1/T_{1}T$, associated with DOS at the
Fermi energy $E_{\mathrm{F}}$, is almost pressure independent. This indicates
that $N(E_{\mathrm{F}})$ (DOS at $E_{\mathrm{F}}$) is pressure independent.
From this result, we conclude that the change in $T_{\mathrm{c}}$ is not
ascribed to any changes in $N(E_{\mathrm{F}})$. In the SC state, a higher
coherence peak was observed under pressure, suggesting that anisotropy in the
SC full gap is reduced under pressure.

Methods.

Figure 1: (Color online) (a) Crystal structure of CaSb2 with two nonequivalent
Sb sites drawn by VESTA [32]. (b) Temperature dependence of the ac
susceptibility $\chi_{\mathrm{AC}}$ measured by a self-inductance method using
a frequency range of 83.5-89.7 MHz under various pressure values. (c) Pressure
dependence of $T_{\mathrm{c}}$. The dashed lines are visual guidance identical
to those in Ref. [30].

We used single crystals of CaSb2 with a slablike shape grown by the self-flux
method as reported in Ref. [26]. The ac magnetic susceptibility
$\chi_{\mathrm{AC}}$ was measured by the NQR tank circuit to determine
$T_{\mathrm{c}}$. A standard spin-echo technique was used for the 123Sb NQR
measurements (nuclear spin $I=7/2$). The value of $1/T_{1}$ was obtained by
measuring the time dependence of the spin-echo intensity after saturation of
the nuclear magnetization $M(t)$. The obtained data were fitted by the
theoretical relaxation function [33]

$\displaystyle m(t)=$
$\displaystyle\frac{2}{21}\exp\left(-\frac{3t}{T_{1}}\right)+\frac{25}{154}\exp\left(-\frac{10t}{T_{1}}\right)$
$\displaystyle+\frac{49}{66}\exp\left(-\frac{21t}{T_{1}}\right)$ (1)

in the same way as Ref. [25] where $m(t)=[M(\infty)-M(t)]/M(\infty)$. Pressure
was applied using a NiCrAl-BeCu piston-cylinder-type cell with Daphne 7373 oil
as the pressure medium. The diameter of the initial sample space is 3.7 mm.
Pressure was determined by a Pb manometer using the relationship [34]

$P\ \mathrm{(GPa)}=\frac{7.18\ \mathrm{(K)}-T_{\mathrm{c}}\
\mathrm{(K)}}{0.364\ \mathrm{(K/GPa)}}.$ (2)

Results and Discussion. Figure 1(b) shows the temperature dependence of
$\chi_{\mathrm{AC}}$ measured at various pressures. A clear diamagnetic signal
due to the SC transition was observed. The width of the transition, which is
shown with an errorbar, becomes broader up to 1.10 GPa. The broadening of the
SC transition can be attributed to the inhomogeneity of pressure, while the
origin of the sharpening beyond 1.10 GPa is not clear. $T_{\mathrm{c}}$ is
defined as the onset temperature of the SC transition. The pressure dependence
of $T_{\mathrm{c}}$ is plotted in Fig. 1(c). In this pressure range,
$T_{\mathrm{c}}$ monotonically increases. This is consistent with the previous
result shown with the dashed line [30].

We observed a 123Sb(1) NQR spectrum corresponding to the $\pm
3/2\leftrightarrow\pm 5/2$ transition. The site assignment of the Sb(1) NQR
signal was described in a previous study [25]. The NQR spectra measured at 4.2
K under various pressure values are presented in Fig. 2(a). The full width at
half maximum (FWHM) at ambient pressure is 0.079 MHz. This is about $1/3$ of
FWHM in the powder samples [25], indicating a much higher quality of the
single-crystalline samples. The spectrum at 0 GPa is compared with that of the
powder samples in the inset of Fig. 2(a). As pressure increases, the peak
frequency decreases. This indicates that the electric field gradient around
the observed nuclei varied owing to compression. In addition, the linewidth of
the spectrum monotonically increases with applying pressure as shown in Fig.
2(b). If the environment around the observed nuclei is homogeneous throughout
the compressed sample, the linewidth would not increase. In general,
measurements under pressure are influenced by the inhomogeneity of pressure,
which may cause the broadening of the NQR spectra. When using a piston-
cylinder-type cell and Daphne 7373 oil, it was reported that pressure
inhomogeneity is small up to 2.2 GPa above which Daphne 7373 oil becomes solid
at room temperature, and that the inhomogeneity is almost constant between 1
and 2.2 GPa [35, 36]. Therefore, the observed continuous increase in the FWHM
above 1.10 GPa cannot be attributed to the inhomogeneity of pressure. Another
possibility is plastic deformation of the crystal. We can exclude this for the
following reason. After the compression process, we relieved pressure and
measured the NQR spectrum at ambient pressure again. The shift of the
resonance frequency was much smaller than the original FWHM and the value of
FWHM was almost unchanged compared to the change caused by pressure,
indicating the shrinkage is reversible in the pressure region below 2.08 GPa.
We believe that the nonsaturating broadening of the NQR spectrum reflects the
unique features of the compression of CaSb2. Specifically, a lattice anomaly
occurring at higher pressure as seen in Cd2Re2O7 is anticipated [37]. In order
to understand the evolution of the structural parameters and electric field
gradient against pressure, the detection of all NQR signals is desirable.

Figure 2: (Color online) (a) 123Sb(1) NQR spectrum of CaSb2 measured at 4.2 K
under various pressure values. Inset: 123Sb(1) NQR spectrum at 0 GPa with that
of the powder samples. (b) Pressure dependence of peak frequency and FWHM of
the NQR spectrum.

Figure 3(a) shows the normalized temperature $T/T_{\mathrm{c}}$ dependence of
$1/T_{1}T$ at each pressure. Above $T_{\mathrm{c}}$, $1/T_{1}T$ is almost
constant in all pressure values. This so-called Korringa behavior is a
characteristic of Fermi liquid-systems. Considering $1/T_{1}T\propto
N(E_{\mathrm{F}})^{2}$, pressure independent $1/T_{1}T$ indicates that
$N(E_{\mathrm{F}})$ is independent of pressure. This suggests that the
increase in $T_{\mathrm{c}}$ cannot be explained by that in
$N(E_{\mathrm{F}})$. We discuss this point later.

We measured $1/T_{1}T$ below $T_{\mathrm{c}}$ at 0 and 1.68 GPa to compare the
SC properties under pressure with those at ambient pressure. In the SC state,
a clear coherence peak was observed just below $T_{\mathrm{c}}$ at 1.68 GPa as
shown in Fig. 3(a). This suggests that BCS superconductivity continues in this
pressure region. We also normalize $1/T_{1}T$ with the mean value of that
above $T_{\mathrm{c}}$ and plot it for 1.68 GPa along with that at ambient
pressure [25] as shown in Fig. 4. The normalized coherence peak becomes higher
under pressure. We fitted the data using the BCS model [38],

$\frac{T_{1\mathrm{n}}}{T_{1\mathrm{s}}}=\frac{2}{k_{\mathrm{B}}T}\int_{0}^{\infty}dE\
N_{\mathrm{s}}^{2}(E)\left[1+\frac{|\Delta(T)|^{2}}{E^{2}}\right]f(E)\left[1-f(E)\right],$
(3)

where $T_{1\mathrm{n}}$ ($T_{1\mathrm{s}}$) is $T_{1}$ in the normal (SC)
state, $N_{\mathrm{s}}(E)$ is the quasiparticle DOS in the SC state,
$\Delta(T)$ is the $T$-dependent energy gap, and $f(E)$ is the Fermi
distribution function. The effect of the energy broadening in
$N_{\mathrm{s}}(E)$ was introduced by the convolution with a rectangular
broadening function whose width and height are $2\delta$ and $1/2\delta$,
respectively. $\delta$ corresponds to the distribution of the SC gap. Although
the existence of two SC gaps was discussed from the temperature dependence of
the SC upper critical field $H_{\mathrm{c2}}$ [26] and the magnetic
penetration depth [27], a single gap was sufficient for the fitting of
$1/T_{1}T$ in the SC state. (We also performed the fitting with a two-gap
model, but the two obtained gaps had nearly the same values.) The results are
shown as dashed curves in Fig. 4. At 1.68 GPa, we obtained
$\Delta(0)/k_{\mathrm{B}}T_{\mathrm{c}}=1.72$ and $\delta/\Delta(0)=0.17$.
Compared with the values at ambient pressure
[$\Delta(0)/k_{\mathrm{B}}T_{\mathrm{c}}=1.52$ and $\delta/\Delta(0)=0.35$]
[25], the size of the SC gap became larger and the distribution of the gap
became smaller. The distribution of the gap mainly comes from the intrinsic
gap anisotropy and the inhomogeneity of the sample. We note that the data just
below $T_{\mathrm{c}}$ measured in the single crystalline samples at ambient
pressure were almost the same as those in the powder sample [25]. To be more
specific, there is little change in the height of the coherence peak, which is
a decisive factor for $\delta/\Delta(0)$. Thus, the reduction of the
distribution is not due to the sample quality, but to the suppression of the
gap anisotropy under pressure. The origin of a more isotropic gap by applying
pressure is unclear, but it might be related to the pressure dependence of
phonons. We consider that the electron scattering by low-energy phonons might
be enhanced because of the pressure-induced structural fluctuations, averaging
the electronic states over the Fermi surfaces. As a result, the effective
electron-electron interaction is averaged out and becomes momentum
independent, making the SC gap more isotropic. The enhancement of the electron
scattering by phonons can also explain the broadening of the NQR spectrum by
the disturbance of the electric field gradient.

Figure 3: (Color online) (a) $T/T_{\mathrm{c}}$ dependence of $1/T_{1}T$ under
various pressure values. (b) Pressure dependence of observed Korringa value
(black squares) and calculated ones (red circles). Figure 4: (Color online)
$T/T_{\mathrm{c}}$ dependence of normalized $1/T_{1}T$ in the low-temperature
region at ambient pressure and 1.68 GPa. Dashed curves show the fitting
results using the BCS model.

Here, we discuss the origin of the enhancement in $T_{\mathrm{c}}$ by applying
pressure. In weak-coupling BCS superconductors, $T_{\mathrm{c}}$ is expressed
as

$T_{\mathrm{c}}=1.13\frac{\hbar\omega_{\mathrm{D}}}{k_{\mathrm{B}}}\exp\left[-\frac{1}{N(E_{\mathrm{F}})V}\right],$
(4)

where $\omega_{\mathrm{D}}$ is the Debye frequency, $V$ is the effective
interaction between electrons mediated by phonons, and $k_{\mathrm{B}}$ is
Boltzmann’s constant. Accordingly, $N(E_{\mathrm{F}})$ is one of the
parameters governing $T_{\mathrm{c}}$. In conventional metallic systems,
$1/T_{1}$ follows the Korringa law expressed as

$\frac{1}{T_{1}T}=\frac{\pi}{\hbar}A^{2}N(E_{\mathrm{F}})^{2}k_{\mathrm{B}},$
(5)

where $\hbar$ and $A$ are Dirac’s constant and the hyperfine coupling
constant, respectively. By measuring the value of $1/T_{1}T$, we can detect
the change in $N(E_{\mathrm{F}})$ and confirm whether or not the increase in
$T_{\mathrm{c}}$ is related to $N(E_{\mathrm{F}})$ [39, 40]. In CaSb2, we
observed a conventional metallic behavior originating from Fermi-surface
portions away from the topological line nodes above $T_{\mathrm{c}}$, and BCS
SC behavior below $T_{\mathrm{c}}$ [25]. Therefore, we can apply the above
method to CaSb2. In our experimental results, $1/T_{1}T$ was unchanged by
pressure, indicating that $N(E_{\mathrm{F}})$ is independent of applied
pressure. Note that the increase in $N(E_{\mathrm{F}})$ accompanied by the
decrease in the hyperfine coupling, which results in constant $1/T_{1}T$, is
not expected in weak-coupling electron systems. In most cases, the hyperfine
coupling is kept constant or increased by compression in nonmagnetic metals
[41, 42, 43]. Since CaSb2 has multiple bands crossing the Fermi energy, if
some of the bands have weak hyperfine coupling, those bands may not be seen by
NQR. However, the possibility that the pressure-dependent band is essentially
invisible is also excluded because $1/T_{1}T$ at 1.68 GPa shows a clear
coherence peak just below the enhanced $T_{\mathrm{c}}$. Our measurements
reliably captured the electronic states changed by pressure. For a more
detailed discussion, we estimated the expected pressure variation of
$1/T_{1}T$ by assuming that the increase in $T_{\mathrm{c}}$ is caused only by
that in $N(E_{\mathrm{F}})$. Here, we used
$\hbar\omega_{\mathrm{D}}/k_{\mathrm{B}}=231$ K [29] and assumed that
$\omega_{\mathrm{D}}$ and $V$ are pressure independent. The result is shown in
Fig. 3(b) with experimental data. Clearly, there is a discrepancy between
calculation and experimental data. From this result, we conclude that pressure
dependence of $T_{\mathrm{c}}$ does not arise from $N(E_{\mathrm{F}})$. As in
Eq. (4), $T_{\mathrm{c}}$ is tuned by three parameters $N(E_{\mathrm{F}})$,
$\omega_{\mathrm{D}}$, and $V$. Therefore, the origin is considered to be the
Debye frequency $\omega_{\mathrm{D}}$ or the effective interaction $V$. This
is surprising because in a weak electron-phonon coupling superconductor, the
pressure dependence of $V$ and $\omega_{\mathrm{D}}$ is usually weaker than
that of $N(E_{\mathrm{F}})$ [44]. Regardless of the origin, there should be a
change in the lattice properties. The peak in $T_{\mathrm{c}}$ suggests that
the pressure variation of the lattce properties would have an anomaly. As the
superconductivity of CaSb2 is likely to be mediated by electron-phonon
coupling, this anomaly might cause interesting phenomena such as a drastic
change of the SC state. In particular, it is predicted that the SC symmetry
other than the $s$ wave in this material is topologically nontrivial [45].
Another fascinating possibility for the realization of topological
superconductivity is the formation of interband pairing. In CaSb2, the
interband coupling could be strong because of the band degeneracy at the nodal
lines. If such coupling becomes stronger by pressure, the interband pairing
would be promoted. Notably, $T_{\mathrm{c}}$ is always enhanced when interband
pairing is additionally formed [46]. An experimental investigation that can
probe the pressure variation of $\omega_{\mathrm{D}}$ or $V$ such as a
specific-heat measurement is desired.

Conclusions. In conclusion, we performed ac magnetic susceptibility and
123Sb(1) NQR measurements on single-crystal CaSb2 to investigate the origin of
the enhancement in $T_{\mathrm{c}}$ by pressure. We confirmed the enhancement
in $T_{\mathrm{c}}$ of the current samples up to 2.08 GPa. The NQR spectral
peak of the single-crystal samples is much sharper than that of the powder
samples, ensuring the higher quality of the single crystals. We observed
nonsaturating spectral broadening against pressure that probably reflects the
unique compression behavior of CaSb2, rather than the pressure inhomogeneity.
At all pressure values, $1/T_{1}$ exhibits Korringa behavior above
$T_{\mathrm{c}}$. The constant value of $1/T_{1}T$ against pressure indicates
that $N(E_{\mathrm{F}})$ is independent of pressure. This result excludes an
enhancement of $N(E_{\mathrm{F}})$ from the possible causes of the enhancement
in $T_{\mathrm{c}}$. According to the BCS theory, the remaining candidates are
$\omega_{\mathrm{D}}$ and $V$, both of which are related to the crystal
lattice. As for the SC properties, the gap becomes larger and more isotropic
by applying pressure. For further investigations of the pressure evolution of
the electronic state, NQR measurements at higher pressures with a different
type of cell would be useful. It is an interesting future issue to clarify the
origin of the peak in $T_{\mathrm{c}}$ against pressure in CaSb2, as well as
its SC symmetry in a higher-pressure region.

Acknowledgments. We would like to thank the Research Center for Low
Temperature and Materials Sciences, Kyoto University for the stable supply of
liquid helium. This work was supported by Grant-in-Aid for Scientific Research
on Innovative Areas from MEXT, JSPS Core-to-core program (JPJSCCA20170002),
JSPS KAKENHI (No. JP17H06136, No. JP20H00130, No. JP15K21732, No. JP15H05745,
No. JP20KK0061, No. JP19H04696, No. JP19K14657, No. JP20H05158, No.
JP21K18600, No. JP22H04933, No. JP22H01168, No. JP23H01124, No. JP22H04473,
and No. JP23H04861), ISHIZUE 2023 of Kyoto University Research Development
Program, the Murata Science Foundation, the Kyoto University Foundation, and
Toyota Physical and Chemical Research Institute. Research at the University of
Maryland was supported by the U.S. Department of Energy Award No. DE-
SC-0019154 (low temperature characterization), the Gordon and Betty Moore
Foundation’s EPiQS Initiative through Grant No. GBMF9071 (materials
synthesis), and the Maryland Quantum Materials Center. This work was also
supported by JST SPRING, Grant No. JPMJSP2110.

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