|  | 2023-06-24 | IRTF | SpeX | 8000-25000 | 0.5 | 2160
2 | WTP 14acnjbu | 2023-05-27 | Magellan Clay | LDSS3 | 4250-10000 | 0.75 | 1800
|  | 2023-06-24 | IRTF | SpeX | 8000-25000 | 0.5 | 2880
3 | WTP 14adbjsh | 2022-07-04 | Magellan Baade | FIRE (Echelle) | 8000-25000 | 0.6 | 1800
|  | 2022-11-11 | SOAR | GHTS (red) | 6300-8930 | 1.0 | 1200
|  | 2022-11-14 | SOAR | GHTS (blue) | 3500-7000 | 1.0 | 600
4 | WTP 14adbwvs | 2022-05-09 | FTN | FLOYDS | 3500-10000 | 2.0 | 1800
|  | 2023-08-25 | IRTF | SpeX | 8000-25000 | 0.5 | 2880
5 | WTP 14adeqka | 2022-04-06 | FTN | FLOYDS | 3500-10000 | 2.0 | 1800
|  | 2023-06-24 | IRTF | SpeX | 8000-25000 | 0.5 | 2160
6 | WTP 15abymdq | 2023-02-26 | Magellan Clay | LDSS3 | 4250-10000 | 0.75 | 1800
7 | WTP 15acbgpn | 2023-04-06 | FTN | FLOYDS | 3500-10000 | 2.0 | 1800
8 | WTP 15acbuuv | 2023-02-26 | Magellan Clay | LDSS3 | 4250-10000 | 0.75 | 1800
9 | WTP 16aaqrcr | 2023-05-27 | Magellan Clay | LDSS3 | 4250-10000 | 0.75 | 1800
10 | WTP 16aatsnw | 2023-05-27 | Magellan Clay | LDSS3 | 4250-10000 | 0.75 | 1800
|  | 2023-08-25 | IRTF | SpeX | 8000-25000 | 0.5 | 2880
11 | WTP 17aaldjb | 2023-05-27 | Magellan Clay | LDSS3 | 4250-10000 | 0.75 | 1800
|  | 2023-06-24 | IRTF | SpeX | 8000-25000 | 0.5 | 2880
12 | WTP 17aalzpx | 2023-02-26 | Magellan Clay | LDSS3 | 4250-10000 | 0.75 | 1800
13 | WTP 17aamoxe | 2023-05-27 | Magellan Clay | LDSS3 | 4250-10000 | 0.75 | 1800
|  | 2023-05-27 | Gemini-S | GMOS | 3600-10300 | 1.0 | 1050
14 | WTP 17aamzew | 2023-02-26 | Magellan Clay | LDSS3 | 4250-10000 | 0.75 | 1800
|  | 2023-03-21 | IRTF | SpeX | 8000-25000 | 0.5 | 1440
15 | WTP 17aanbso | 2023-04-06 | FTN | FLOYDS | 3500-10000 | 2.0 | 1800
16 | WTP 18aajkmk | 2023-02-26 | Magellan Clay | LDSS3 | 4250-10000 | 0.75 | 1800
|  | 2023-01-17 | Magellan Baade | FIRE | 8000-25000 | 0.6 | 300
18 | WTP 18aampwj | 2023-07-25 | SOAR | GHTS (blue) | 3500-7000 | 1.0 | 600
|  | 2023-08-25 | IRTF | SpeX | 8000-25000 | 0.5 | 2880

##  Appendix A Spectroscopic Follow-Up

Figure 14: Optical spectroscopy of each of the TDE candidates in our sample.
For those sources with archival SDSS spectra, we show those here, instead of
post-flare observations. The vertical lines show important spectral lines,
with Balmer lines shown in dark purple, coronal lines in light blue, and other
important lines of interest in orange. All gold sample TDEs are marked with a
★. Detailed comments on the AGN-like nature of the silver sample, evidence of
spectral evolution, and existence of broad emission lines are given in the
individual source section (Section C).  Figure 15: Post-flare near-IR
spectroscopy from either Magellan/FIRE or IRTF/SpeX of nine of the TDE
candidates in our sample with near-IR spectroscopy to date. The light color
spectrum is the raw spectrum, and the dark, thicker line shows the spectrum
binned to a resolution of 10Å. The vertical orange dashed lines show important
spectral lines in the near-IR. Regions of strong telluric absorption are shown
as grey shaded regions. All gold sample TDEs are marked with a ★. Only WTP
14abnpgk (silver sample; likely AGN) and WTP 18aajkmk (gold sample; likely
TDE) show any broad lines (He I, 1.083 $\mu$m).

In Table 3, we list the details of the optical and near-IR spectroscopic
follow-up observations we performed of the sources in our sample. We also
present an optical spectrum for each source in Figure 14. For any source where
an archival SDSS spectrum exists, we present this spectrum as it is best for
ruling out AGN contamination. Additionally, for the nine sources with near-IR
spectroscopic follow-up, we present these spectra in Figure 15.

##  Appendix B Host Galaxy SED Fits

In Figure 16, we show the resulting SED fits for each source from Section 6.2.

Figure 16: Host galaxy photometry and SED fits with prospector. The orange
points show the observed photometry, which has been corrected for Galactic
extinction. The purple points show the best-fit model photometry, with the
modeling details given in Section 6.2. The shaded purple regions show the
uncertainty regions from the MCMC modeling. All gold sample TDEs are marked
with a ★.

##  Appendix C Comments on Individual Sources

In this Section, we provide individual details on each source, as well as our
reasoning for classifying it in either the gold or silver sample.

###  C.1 WTP 14abnpgk

WTP 14abnpgk showed a strong and relatively smooth IR flare, but has recently
shown a secondary rise in both W1 and W2 in the last three WISE epochs.
Interestingly, both ZTF and ATLAS detected an optical transient in this source
in 2022, around the same time as the second rise is seen in WISE. The optical
transients was reported to TNS as a nuclear transient (AT
2022mim111111https://www.wis-tns.org/object/2022mim). The ATLAS and ZTF light
curves, shown in Figure 17, do not look like typical optical TDE light curves
and instead resemble the more stochastic variability often seen in AGN. In
addition, an optical spectrum taken with the LDSS3 instrument on the Magellan
Clay telescope revealed a broad H$\alpha$ line with a significant red wing and
diminished blue wing, as well as a similar line structure in the H$\beta$
profile. Additional near-IR follow-up showed a broad He I line at 1.083
$\mu$m, although there is potential blending with the Pa$\alpha$ line. These
broad lines are also suggestive that the source is an AGN, although it is also
possible that this is a partial TDE. A double dust echo has not been reported
in repeated TDEs previously, but repeated optical and X-ray flares have been
both been used to claim partial TDEs (e.g. Payne et al., 2021; Liu et al.,
2023; Wevers et al., 2023). As there are no public archival optical spectra of
this source, we cannot probe the evolution of the broad emission lines. Hence,
further monitoring is necessary to disentangle the exact nature of this
source. To be conservative, we place this source in our silver sample, due to
the potential indications of AGN activity.

Figure 17: Zoom in on the optical (ATLAS, ZTF) and IR (WISE) light curves for
the four sources that show any hint of an optical transient. The optical data
from ATLAS (cyan and orange diamonds) and ZTF (yellow and red pentagons) have
been binned on 10 day timescales using an inverse variance weighted average
with 3$\sigma$ clipping for outlier rejection. Each panel shows a 4 year
duration, zoomed in around the time of the optical/IR flare. Gold sample
sources are marked with a ★. The left axis shows the log of the difference
flux in mJy, and the right axis shows the corresponding monochromatic W2
luminosity in erg s-1. The optical flares in these sources are more than an
order of magnitude fainter than the IR flare, which is unlike most optically-
selected TDEs. Top left: Light curves for WTP 14abnpgk, focused on the
secondary rise in the WISE light curve that started around 2021-2022. ZTF and
ATLAS both show significant detections around this time frame, although the
light curves are more structured than is typically seen in optical TDE light
curves. Top right: Light curves for WTP 18aajkmk, which did not have a
previously-detected optical transient. ATLAS shows a potential flare that
appears to be contemporaneous with the IR flare, but no flare is detected with
ZTF. Bottom left: Light curves for WTP 18aamced, which was extensively studied
in Frederick et al. (2019) and Huang et al. (2023). The optical transient (AT
2018dyk) was detected in both ZTF and ATLAS, with a light curve that resembles
typical optically-selected TDEs. Bottom right: Light curves for WTP 18aampwj,
which was detected as an optical transient (AT 2018gn) in 2018 by ASAS-SN. The
optical transient was detected in ASAS-SN, ATLAS, and ZTF. However, the
transient first went off when ZTF was taking reference images, so the exact
flux scaling of the ZTF data shown here is not correct.

###  C.2 WTP 14acnjbu

WTP 14acnjbu has the most peculiar WISE light curve of any source in our
sample, showing a slight rise before the TDE-like flare. With the LDSS3
spectrum from 2023, we find that the source sits in either the star-forming or
composite region of BPT parameter space. However, to be conservative, we place
this source in our silver sample, as the double peaked light curve could be
indicative of veiled AGN activity.

###  C.3 WTP 14adbjsh

WTP 14adbjsh, the brightest and closest source in our sample, was already
presented in detail in Panagiotou et al. (2023). In short, we find no
optical/near-IR spectral lines associated with accretion, which suggests that
the IR flare is not driven by an AGN. Late-time X-ray observations with Swift
and NuSTAR revealed no significant X-ray emission. We place WTP 14adbjsh in
our gold sample. WTP 14adbjsh occurred in a star-forming galaxy and is one of
the closest TDEs discovered to date, despite being entirely missed by optical
and X-ray searches.

###  C.4 WTP 14adbwvs

WTP 14adbwvs is classified as an AGN, based on its location in the BPT
diagrams from archival SDSS spectroscopy. We obtained a recent spectrum with
FLOYDS and found no evidence for any spectral changes. The source does not
have strong X-ray emission, nor is its WISE color indicative of AGN activity.
However, out of an abundance of caution, we place this source in our silver
sample.

###  C.5 WTP 14adeqka

WTP 14adeqka is one of the strongest and smoothest flares in our sample,
reaching a peak luminosity around $10^{43}$ erg s-1. The host galaxy of WTP
14adeqka shows no evidence for significant AGN-like ionization (e.g. [O III],
[O I], [S II]) when we observed the source in 2023 with FLOYDS. The spectral
resolution of FLOYDS leaves us unable to resolve the H$\alpha$ \+ [N II]
complex, but no broad lines are observed. We also find no evidence for broad
lines in a near-IR spectrum taken with IRTF/SpeX, but this spectrum does show
evidence for some forbidden narrow lines (e.g. [S III] 9530Å, [Fe II]
1.257$\mu$m, 1.64$\mu$m). However, other sources have shown late-time evidence
for forbidden narrow lines associated with the TDE (i.e. no such lines existed
before the IR flare). Hence, the existence of forbidden lines alone does not
suggest AGN-drive behavior. Given that there are no other indicators of an
underlying AGN, we place WTP 14adeqka in our gold sample.

###  C.6 WTP 15abymdq

WTP 15abymdq shows no strong evidence for an AGN in the optical spectrum we
obtained with LDSS3 in 2023, sitting in the star-forming or composite region
of parameter space for various BPT diagrams. This source was one of the three
that we detected with eROSITA, and the spectrum was relatively hard, showing a
power-law X-ray spectrum with $\Gamma\approx 1.7$. There is evidence for
neutral absorption with a variable column density on the order of $10^{21}$
cm-2 (see Section 3.4 for more details). Although this is a relatively
standard AGN-like X-ray spectrum, the X-ray spectra of TDEs are not uniform
enough for us to use this to rule out this source as a TDE. Hence, we place
WTP 15abymdq in our gold sample, given that there are no other indications of
underlying AGN activity in this source.

###  C.7 WTP 15acbgpn

WTP 15acbgpn is a clear AGN with a broad H$\alpha$ line apparent in the SDSS
spectrum from 2002. It was also detected with Chandra in 2010, with an X-ray
flux of $F_{0.5-7\,\mathrm{keV}}\approx 8\times 10^{-13}$ erg cm-2 s-1. Thus,
given the clear AGN nature, we place this source in our silver sample. We note
that we did observe this source with FLOYDS in 2023 and found that the
spectrum was consistent with the SDSS spectrum, albeit with lower spectral
resolution.

###  C.8 WTP 15acbuuv

WTP 15acbuuv sits in the star-forming or composite region of BPT parameter
space, although there is significant [O III] emission in this spectrum. A past
spectrum from 6dF (Jones et al., 2004, 2009) shows little evolution from the
LDSS3 spectrum we took in 2023. This source was also one of the eROSITA
sources, and showed no evidence for any X-ray emission during any of the eRASS
epochs. Given the lack of significant evidence for AGN behavior, we place this
source in our gold sample.

###  C.9 WTP 16aaqrcr

WTP 16aaqrcr shows no clear indications of any previous AGN activity. An
optical spectrum from LDSS3 in 2023 shows no evidence of [O III] emission,
consistent with star-forming ionization. The source was also never detected in
the eROSITA survey. Hence, we place WTP 16aaqrcr in our gold sample.

###  C.10 WTP 16aatsnw

WTP 16aatsnw is another potential AGN/LINER interloper in our sample, based on
its location in the BPT diagrams from archival SDSS spectroscopy. We obtained
a spectrum with LDSS3 in 2023 and found no evidence for significant spectral
changes. The source does not have strong X-ray emission, nor is its WISE color
indicative of AGN activity. However, out of an abundance of caution, we place
this source in our silver sample, given the pre-flare AGN-like ionization
levels.

###  C.11 WTP 17aaldjb

WTP 17aaldjb has an archival SDSS spectrum which shows no evidence of AGN
activity, with only very weak emission lines visible in the spectrum.
Likewise, our LDSS3 spectrum from 2023 was very similar, showing little
evolution. We also obtained a near-IR spectrum of this source with IRTF/SpeX
in 2023 and found similarly negligible emission features, strengthening our
confidence that this is not an veiled AGN. The source was also not detected
with eROSITA. Hence, we place this source in our gold sample.

###  C.12 WTP 17aalzpx

WTP 17aalzpx showed no strong [O III] emission in the LDSS3 spectrum from
2023, indicating likely star-forming ionization. The source was also not
detected with eROSITA. Hence, this source does not show any strong evidence
for AGN activity, and thus we place it in our gold sample.

###  C.13 WTP 17aamoxe

WTP 17aamoxe showed very strong, AGN-like narrow emission lines in the optical
spectrum with obtained with LDSS3 in 2023. However, the archival spectrum from
2004 with the 6dF survery (Jones et al., 2004, 2009) shows no significant [O
III], [O I], or [S II] emission, indicating that there has likely been a
significant change to the spectral lines in this source (see Section 5.3). WTP
17aamoxe is also one of the three sources that we detected with eROSITA,
showing a relatively soft X-ray spectrum ($\Gamma\approx 4$) in the three
eRASS epochs in which it was detected (see Section 3.4). Such a soft X-ray
spectrum is highly unusual for AGN, but is often seen in TDEs. Hence, we place
this source in our gold sample, but note that further follow-up to monitor the
X-ray and optical spectral changes is necessary to rule out a turn-on AGN.

###  C.14 WTP 17aamzew

WTP 17aamzew is another one of our three detected eROSITA sources, again
showing a soft X-ray spectrum ($\Gamma\approx 4$). This source has an archival
SDSS spectrum from 2005, which shows no evidence for AGN-driven line emission.
However, our LDSS3 spectrum from 2023 shows significant spectral changes,
including the formation of both [O III] and [O I], which are often associated
with accretion. We suspect that these narrow lines were induced by the TDE.
Hence, we place WTP 17aamzew in our gold sample.

###  C.15 WTP 17aanbso

WTP 17aanbso shows strong emission lines in its archival SDSS spectrum, but
the line ratios suggest a purely star-forming nature. We obtained an FLOYDS
spectrum in 2023, but found no significant evidence for spectral changes.
Hence, as there is no evidence for underlying AGN activity, we place WTP
17aanbso in our gold sample.

###  C.16 WTP 18aajkmk

WTP 18aajkmk shows no clear evidence for ongoing AGN activity, falling
primarily in the star-forming or composite region of BPT parameter space. The
[O I] line from our 2023 LDSS3 spectrum does show LINER ionization levels, but
there are other sources with significant narrow line changes and we have no
pre-outburst optical spectrum. Hence, we cannot rule out that this line is
driven by the TDE. Additionally, we do see broad He I line at 1.083 $\mu$m in
a FIRE near-IR spectrum taken in 2023, which we believe is likely associated
with the TDE. Interestingly, this source is clearly an extremely disturbed
system (see Figure 9), thereby potentially suggesting a merger-driven
accretion episode. WTP 18aajkmk also shows potential evidence for an optical
transient in the ATLAS $o$-band data (see top left of Figure 17). No transient
is detected in the $c$-band data, which could be due to the reddening from a
dusty environment. ZTF does not detect a significant optical transient in this
system, although this could be a result of a transient occurring around the
time of reference images being taken. Given the lack of any evidence for AGN
activity, we classify this source as a likely TDE, and place it in our gold
sample.

###  C.17 WTP 18aamced

WTP 18aamced is one of the few sources in our sample with a clear optical
counterpart, first detected by the the ZTF team121212https://www.wis-
tns.org/object/2018dyk. We show the ZTF and ATLAS light curves for this source
with its WISE light curve in the bottom left panel of Figure 17. The ZTF
source was relatively faint compared to other TDEs, and extensively optical
and UV spectroscopic follow-up showed that this source transitioned from a
LINER (based on the archival SDSS spectrum) to a narrow-line Seyfert 1 with
broad Balmer lines and strong Fe and He coronal lines (Frederick et al.,
2019). Like many narrow-line Seyfert 1s, the source shows a relatively soft
X-ray spectrum ($\Gamma\approx 2.8-3$), but is still harder than the two other
soft sources in our sample (Frederick et al., 2019). It has recently been
argued, however, that this source is a TDE in a LINER galaxy with a dusty
nuclear environment (Huang et al., 2023). For detailed studies of this source,
we refer the reader to Frederick et al. (2019) and Huang et al. (2023). As the
source shows clear LINER ionization levels before the flare, we place it in
our silver sample.

###  C.18 WTP 18aampwj

WTP 18aampwj was first discovered by the ASAS-SN team131313https://www.wis-
tns.org/object/2018gn as a candidate Type II SN (Falco et al., 2018), after an
optical spectrum revealed a broad H$\alpha$ line (see Figure 7). The optical
transient was also clearly detected in the deeper ATLAS and ZTF surveys,
although the optical flare occurred when the ZTF reference images were being
taken complicating a direct measurement of the transient flux. We show the
ATLAS and ZTF data together with the WISE flare for this source in the bottom
right panel of Figure 17, which resemble those of typical optically-selected
TDEs. Spectroscopically, the broad H$\alpha$ line in the classification
spectrum did not show a strong blue absorption wing and P-Cygni profile that
is commonly seen in spectra of Type II SNe (Gutiérrez et al., 2017). Our 2023
spectrum also showed the formation of [Fe VII] coronal lines five years after
the transient discovery, which has been seen in some TDEs in gas-rich
environments (e.g. Onori et al., 2022; Short et al., 2023), but is rare in
Type II SNe (e.g. SN2005ip; Smith et al., 2009). Another argument for the TDE
nature of WTP18aampwj is that the total radiated energy in the IR alone
exceeds $10^{51}$ erg already and is still ongoing. Likewise, the peak W1
luminosity exceeds what is seen in typical SNe (Thévenot, 2020). Thus, we
suggest that this source is indeed a TDE rather than a supernova, as was also
suggested in recent works based on the integrated IR luminosity (Thévenot,
2020; Thévenot et al., 2021). As the optical spectrum shows no evidence for
AGN-like ionization levels, we place this source in our gold sample.

## References

  * Abbasi et al. (2023) Abbasi, R., Ackermann, M., Adams, J., et al. 2023, ApJS, 269, 25, doi: 10.3847/1538-4365/acfa95
  * Abbott et al. (2018) Abbott, T. M. C., Abdalla, F. B., Allam, S., et al. 2018, ApJS, 239, 18, doi: 10.3847/1538-4365/aae9f0
  * Abramowicz et al. (1988) Abramowicz, M. A., Czerny, B., Lasota, J. P., & Szuszkiewicz, E. 1988, ApJ, 332, 646, doi: 10.1086/166683
  * Alexander et al. (2020) Alexander, K. D., van Velzen, S., Horesh, A., & Zauderer, B. A. 2020, Space Sci. Rev., 216, 81, doi: 10.1007/s11214-020-00702-w
  * Arcavi et al. (2014) Arcavi, I., Gal-Yam, A., Sullivan, M., et al. 2014, ApJ, 793, 38, doi: 10.1088/0004-637X/793/1/38
  * Arcavi et al. (2018) Arcavi, I., Burke, J., French, K. D., et al. 2018, The Astronomer’s Telegram, 11953, 1
  * Arnaud (1996) Arnaud, K. A. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. Jacoby & J. Barnes, 17
  * Astropy Collaboration et al. (2022) Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935, 167, doi: 10.3847/1538-4357/ac7c74
  * Bade et al. (1996) Bade, N., Komossa, S., & Dahlem, M. 1996, A&A, 309, L35
  * Baldwin et al. (1981) Baldwin, J. A., Phillips, M. M., & Terlevich, R. 1981, PASP, 93, 5, doi: 10.1086/130766
  * Begelman (1979) Begelman, M. C. 1979, MNRAS, 187, 237, doi: 10.1093/mnras/187.2.237
  * Bellm et al. (2019) Bellm, E. C., Kulkarni, S. R., Graham, M. J., et al. 2019, PASP, 131, 018002, doi: 10.1088/1538-3873/aaecbe
  * Blanchard et al. (2017) Blanchard, P. K., Nicholl, M., Berger, E., et al. 2017, ApJ, 843, 106, doi: 10.3847/1538-4357/aa77f7
  * Blanton & Moustakas (2009) Blanton, M. R., & Moustakas, J. 2009, ARA&A, 47, 159, doi: 10.1146/annurev-astro-082708-101734
  * Bricman & Gomboc (2020) Bricman, K., & Gomboc, A. 2020, ApJ, 890, 73, doi: 10.3847/1538-4357/ab6989
  * Brinchmann et al. (2004) Brinchmann, J., Charlot, S., White, S. D. M., et al. 2004, MNRAS, 351, 1151, doi: 10.1111/j.1365-2966.2004.07881.x
  * Brown et al. (2013) Brown, T. M., Baliber, N., Bianco, F. B., et al. 2013, PASP, 125, 1031, doi: 10.1086/673168
  * Brunner et al. (2022) Brunner, H., Liu, T., Lamer, G., et al. 2022, A&A, 661, A1, doi: 10.1051/0004-6361/202141266
  * Calzetti et al. (2000) Calzetti, D., Armus, L., Bohlin, R. C., et al. 2000, ApJ, 533, 682, doi: 10.1086/308692
  * Cannizzaro et al. (2022) Cannizzaro, G., Levan, A. J., van Velzen, S., & Brown, G. 2022, MNRAS, 516, 529, doi: 10.1093/mnras/stac2014
  * Cappellari (2017) Cappellari, M. 2017, MNRAS, 466, 798, doi: 10.1093/mnras/stw3020
  * Cappellari (2023) —. 2023, MNRAS, 526, 3273, doi: 10.1093/mnras/stad2597
  * Cappellari & Emsellem (2004) Cappellari, M., & Emsellem, E. 2004, PASP, 116, 138, doi: 10.1086/381875
  * Cardelli et al. (1989) Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245, doi: 10.1086/167900
  * Cash (1979) Cash, W. 1979, ApJ, 228, 939, doi: 10.1086/156922
  * Chabrier (2003) Chabrier, G. 2003, PASP, 115, 763, doi: 10.1086/376392
  * Chakraborty et al. (2021) Chakraborty, J., Kara, E., Masterson, M., et al. 2021, ApJ, 921, L40, doi: 10.3847/2041-8213/ac313b
  * Chambers et al. (2016) Chambers, K. C., Magnier, E. A., Metcalfe, N., et al. 2016, arXiv e-prints, arXiv:1612.05560, doi: 10.48550/arXiv.1612.05560
  * Chornock et al. (2014) Chornock, R., Berger, E., Gezari, S., et al. 2014, ApJ, 780, 44, doi: 10.1088/0004-637X/780/1/44
  * Clark et al. (2023) Clark, P., Graur, O., Callow, J., et al. 2023, arXiv e-prints, arXiv:2307.03182, doi: 10.48550/arXiv.2307.03182
  * Clemens et al. (2004) Clemens, J. C., Crain, J. A., & Anderson, R. 2004, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 5492, Ground-based Instrumentation for Astronomy, ed. A. F. M. Moorwood & M. Iye, 331–340, doi: 10.1117/12.550069
  * Conroy & Gunn (2010) Conroy, C., & Gunn, J. E. 2010, ApJ, 712, 833, doi: 10.1088/0004-637X/712/2/833
  * Conroy et al. (2009) Conroy, C., Gunn, J. E., & White, M. 2009, ApJ, 699, 486, doi: 10.1088/0004-637X/699/1/486
  * Cook et al. (2019) Cook, D. O., Kasliwal, M. M., Van Sistine, A., et al. 2019, ApJ, 880, 7, doi: 10.3847/1538-4357/ab2131
  * Coughlin & Begelman (2014) Coughlin, E. R., & Begelman, M. C. 2014, ApJ, 781, 82, doi: 10.1088/0004-637X/781/2/82
  * Cushing et al. (2004) Cushing, M. C., Vacca, W. D., & Rayner, J. T. 2004, PASP, 116, 362, doi: 10.1086/382907
  * Dai et al. (2020) Dai, B. B., Shu, X. W., Jiang, N., et al. 2020, ApJ, 896, L27, doi: 10.3847/2041-8213/ab97ac
  * Dai et al. (2018) Dai, L., McKinney, J. C., Roth, N., Ramirez-Ruiz, E., & Miller, M. C. 2018, ApJ, 859, L20, doi: 10.3847/2041-8213/aab429
  * De et al. (2020) De, K., Hankins, M. J., Kasliwal, M. M., et al. 2020, PASP, 132, 025001, doi: 10.1088/1538-3873/ab6069
  * Dodd et al. (2023) Dodd, S. A., Nukala, A., Connor, I., et al. 2023, arXiv e-prints, arXiv:2307.05670, doi: 10.48550/arXiv.2307.05670
  * Dong et al. (2016) Dong, S., Shappee, B. J., Prieto, J. L., et al. 2016, Science, 351, 257, doi: 10.1126/science.aac9613
  * Donley et al. (2002) Donley, J. L., Brandt, W. N., Eracleous, M., & Boller, T. 2002, AJ, 124, 1308, doi: 10.1086/342280
  * Dou et al. (2017) Dou, L., Wang, T., Yan, L., et al. 2017, ApJ, 841, L8, doi: 10.3847/2041-8213/aa7130
  * Dou et al. (2016) Dou, L., Wang, T.-g., Jiang, N., et al. 2016, ApJ, 832, 188, doi: 10.3847/0004-637X/832/2/188
  * Draine & Lee (1984) Draine, B. T., & Lee, H. M. 1984, ApJ, 285, 89, doi: 10.1086/162480
  * Drlica-Wagner et al. (2016) Drlica-Wagner, A., Bechtol, K., Allam, S., et al. 2016, ApJ, 833, L5, doi: 10.3847/2041-8205/833/1/L5
  * Ergon et al. (2015) Ergon, M., Jerkstrand, A., Sollerman, J., et al. 2015, A&A, 580, A142, doi: 10.1051/0004-6361/201424592
  * Evans et al. (2007) Evans, P. A., Beardmore, A. P., Page, K. L., et al. 2007, A&A, 469, 379, doi: 10.1051/0004-6361:20077530
  * Evans et al. (2009) —. 2009, MNRAS, 397, 1177, doi: 10.1111/j.1365-2966.2009.14913.x
  * Falco et al. (2018) Falco, E., Calkins, M., Prieto, J. L., & Stanek, K. Z. 2018, Transient Name Server Classification Report, 2018-81, 1
  * Falcón-Barroso et al. (2011) Falcón-Barroso, J., Sánchez-Blázquez, P., Vazdekis, A., et al. 2011, A&A, 532, A95, doi: 10.1051/0004-6361/201116842
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306, doi: 10.1086/670067
  * Fox et al. (2010) Fox, O. D., Chevalier, R. A., Dwek, E., et al. 2010, ApJ, 725, 1768, doi: 10.1088/0004-637X/725/2/1768
  * Fox et al. (2011) Fox, O. D., Chevalier, R. A., Skrutskie, M. F., et al. 2011, ApJ, 741, 7, doi: 10.1088/0004-637X/741/1/7
  * Frederick et al. (2019) Frederick, S., Gezari, S., Graham, M. J., et al. 2019, ApJ, 883, 31, doi: 10.3847/1538-4357/ab3a38
  * Fremling et al. (2020) Fremling, C., Miller, A. A., Sharma, Y., et al. 2020, ApJ, 895, 32, doi: 10.3847/1538-4357/ab8943
  * French et al. (2016) French, K. D., Arcavi, I., & Zabludoff, A. 2016, ApJ, 818, L21, doi: 10.3847/2041-8205/818/1/L21
  * French et al. (2020) French, K. D., Arcavi, I., Zabludoff, A. I., et al. 2020, ApJ, 891, 93, doi: 10.3847/1538-4357/ab7450
  * Frohmaier et al. (2021) Frohmaier, C., Angus, C. R., Vincenzi, M., et al. 2021, MNRAS, 500, 5142, doi: 10.1093/mnras/staa3607
  * Fruscione et al. (2006) Fruscione, A., McDowell, J. C., Allen, G. E., et al. 2006, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 6270, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, ed. D. R. Silva & R. E. Doxsey, 62701V, doi: 10.1117/12.671760
  * Gal-Yam (2019) Gal-Yam, A. 2019, ARA&A, 57, 305, doi: 10.1146/annurev-astro-081817-051819
  * Gehrels et al. (2004) Gehrels, N., Chincarini, G., Giommi, P., et al. 2004, ApJ, 611, 1005, doi: 10.1086/422091
  * Gezari (2021) Gezari, S. 2021, ARA&A, 59, 21, doi: 10.1146/annurev-astro-111720-030029
  * Gezari et al. (2006) Gezari, S., Martin, D. C., Milliard, B., et al. 2006, ApJ, 653, L25, doi: 10.1086/509918
  * Gezari et al. (2008) Gezari, S., Basa, S., Martin, D. C., et al. 2008, ApJ, 676, 944, doi: 10.1086/529008
  * Gezari et al. (2009) Gezari, S., Heckman, T., Cenko, S. B., et al. 2009, ApJ, 698, 1367, doi: 10.1088/0004-637X/698/2/1367
  * Gezari et al. (2012) Gezari, S., Chornock, R., Rest, A., et al. 2012, Nature, 485, 217, doi: 10.1038/nature10990
  * Godoy-Rivera et al. (2017) Godoy-Rivera, D., Stanek, K. Z., Kochanek, C. S., et al. 2017, MNRAS, 466, 1428, doi: 10.1093/mnras/stw3237
  * Graur et al. (2018) Graur, O., French, K. D., Zahid, H. J., et al. 2018, ApJ, 853, 39, doi: 10.3847/1538-4357/aaa3fd
  * Greenwell et al. (2021) Greenwell, C., Gandhi, P., Stern, D., et al. 2021, MNRAS, 503, L80, doi: 10.1093/mnrasl/slab019
  * Guillochon & Ramirez-Ruiz (2013) Guillochon, J., & Ramirez-Ruiz, E. 2013, ApJ, 767, 25, doi: 10.1088/0004-637X/767/1/25
  * Guolo et al. (2023) Guolo, M., Gezari, S., Yao, Y., et al. 2023, arXiv e-prints, arXiv:2308.13019, doi: 10.48550/arXiv.2308.13019
  * Gutiérrez et al. (2017) Gutiérrez, C. P., Anderson, J. P., Hamuy, M., et al. 2017, ApJ, 850, 89, doi: 10.3847/1538-4357/aa8f52
  * Hammerstein et al. (2021) Hammerstein, E., Gezari, S., van Velzen, S., et al. 2021, ApJ, 908, L20, doi: 10.3847/2041-8213/abdcb4
  * Hammerstein et al. (2023) Hammerstein, E., van Velzen, S., Gezari, S., et al. 2023, ApJ, 942, 9, doi: 10.3847/1538-4357/aca283
  * Helou et al. (2013) Helou, G., Kasliwal, M. M., Ofek, E. O., et al. 2013, ApJ, 778, L19, doi: 10.1088/2041-8205/778/1/L19
  * Hinkle (2022) Hinkle, J. T. 2022, arXiv e-prints, arXiv:2210.15681, doi: 10.48550/arXiv.2210.15681
  * Hinkle et al. (2023) Hinkle, J. T., Shappee, B. J., & Holoien, T. W. S. 2023, arXiv e-prints, arXiv:2303.05525, doi: 10.48550/arXiv.2303.05525
  * Holoien et al. (2014) Holoien, T. W. S., Prieto, J. L., Bersier, D., et al. 2014, MNRAS, 445, 3263, doi: 10.1093/mnras/stu1922
  * Huang & Lu (2024) Huang, H.-T., & Lu, W. 2024, MNRAS, 527, 1865, doi: 10.1093/mnras/stad3269
  * Huang et al. (2023) Huang, S., Jiang, N., Lin, Z., Zhu, J., & Wang, T. 2023, MNRAS, 525, 4057, doi: 10.1093/mnras/stad2541
  * Ivezić et al. (2019) Ivezić, Ž., Kahn, S. M., Tyson, J. A., et al. 2019, ApJ, 873, 111, doi: 10.3847/1538-4357/ab042c
  * Jiang et al. (2016) Jiang, N., Dou, L., Wang, T., et al. 2016, ApJ, 828, L14, doi: 10.3847/2041-8205/828/1/L14
  * Jiang et al. (2021a) Jiang, N., Wang, T., Hu, X., et al. 2021a, ApJ, 911, 31, doi: 10.3847/1538-4357/abe772
  * Jiang et al. (2023) Jiang, N., Zhou, Z., Zhu, J., Wang, Y., & Wang, T. 2023, ApJ, 953, L12, doi: 10.3847/2041-8213/acebe3
  * Jiang et al. (2021b) Jiang, N., Wang, T., Dou, L., et al. 2021b, ApJS, 252, 32, doi: 10.3847/1538-4365/abd1dc
  * Johnson et al. (2021) Johnson, B. D., Leja, J., Conroy, C., & Speagle, J. S. 2021, ApJS, 254, 22, doi: 10.3847/1538-4365/abef67
  * Jones et al. (2004) Jones, D. H., Saunders, W., Colless, M., et al. 2004, MNRAS, 355, 747, doi: 10.1111/j.1365-2966.2004.08353.x
  * Jones et al. (2009) Jones, D. H., Read, M. A., Saunders, W., et al. 2009, MNRAS, 399, 683, doi: 10.1111/j.1365-2966.2009.15338.x
  * Jonker et al. (2020) Jonker, P. G., Stone, N. C., Generozov, A., van Velzen, S., & Metzger, B. 2020, ApJ, 889, 166, doi: 10.3847/1538-4357/ab659c
  * Kaastra & Bleeker (2016) Kaastra, J. S., & Bleeker, J. A. M. 2016, A&A, 587, A151, doi: 10.1051/0004-6361/201527395
  * Kara et al. (2018) Kara, E., Dai, L., Reynolds, C. S., & Kallman, T. 2018, MNRAS, 474, 3593, doi: 10.1093/mnras/stx3004
  * Kauffmann et al. (2003) Kauffmann, G., Heckman, T. M., Tremonti, C., et al. 2003, MNRAS, 346, 1055, doi: 10.1111/j.1365-2966.2003.07154.x
  * Kesden (2012) Kesden, M. 2012, Phys. Rev. D, 85, 024037, doi: 10.1103/PhysRevD.85.024037
  * Kewley et al. (2001) Kewley, L. J., Dopita, M. A., Sutherland, R. S., Heisler, C. A., & Trevena, J. 2001, ApJ, 556, 121, doi: 10.1086/321545
  * Kewley et al. (2006) Kewley, L. J., Groves, B., Kauffmann, G., & Heckman, T. 2006, MNRAS, 372, 961, doi: 10.1111/j.1365-2966.2006.10859.x
  * Kochanek et al. (2017) Kochanek, C. S., Shappee, B. J., Stanek, K. Z., et al. 2017, PASP, 129, 104502, doi: 10.1088/1538-3873/aa80d9
  * Komossa & Bade (1999) Komossa, S., & Bade, N. 1999, A&A, 343, 775, doi: 10.48550/arXiv.astro-ph/9901141
  * Komossa & Greiner (1999) Komossa, S., & Greiner, J. 1999, A&A, 349, L45, doi: 10.48550/arXiv.astro-ph/9908216
  * Komossa et al. (2008) Komossa, S., Zhou, H., Wang, T., et al. 2008, ApJ, 678, L13, doi: 10.1086/588281
  * König et al. (2022) König, O., Saxton, R. D., Kretschmar, P., et al. 2022, Astronomy and Computing, 38, 100529, doi: 10.1016/j.ascom.2021.100529
  * Kool et al. (2020) Kool, E. C., Reynolds, T. M., Mattila, S., et al. 2020, MNRAS, 498, 2167, doi: 10.1093/mnras/staa2351
  * Krühler et al. (2018) Krühler, T., Fraser, M., Leloudas, G., et al. 2018, A&A, 610, A14, doi: 10.1051/0004-6361/201731773
  * Landt et al. (2008) Landt, H., Bentz, M. C., Ward, M. J., et al. 2008, ApJS, 174, 282, doi: 10.1086/522373
  * Lang (2014) Lang, D. 2014, AJ, 147, 108, doi: 10.1088/0004-6256/147/5/108
  * Law-Smith et al. (2017) Law-Smith, J., Ramirez-Ruiz, E., Ellison, S. L., & Foley, R. J. 2017, ApJ, 850, 22, doi: 10.3847/1538-4357/aa94c7
  * Leloudas et al. (2015) Leloudas, G., Schulze, S., Krühler, T., et al. 2015, MNRAS, 449, 917, doi: 10.1093/mnras/stv320
  * Leloudas et al. (2016) Leloudas, G., Fraser, M., Stone, N. C., et al. 2016, Nature Astronomy, 1, 0002, doi: 10.1038/s41550-016-0002
  * Liu et al. (2023) Liu, Z., Malyali, A., Krumpe, M., et al. 2023, A&A, 669, A75, doi: 10.1051/0004-6361/202244805
  * Lu & Kumar (2018) Lu, W., & Kumar, P. 2018, ApJ, 865, 128, doi: 10.3847/1538-4357/aad54a
  * Lu et al. (2016) Lu, W., Kumar, P., & Evans, N. J. 2016, MNRAS, 458, 575, doi: 10.1093/mnras/stw307
  * Mainzer et al. (2014) Mainzer, A., Bauer, J., Cutri, R. M., et al. 2014, ApJ, 792, 30, doi: 10.1088/0004-637X/792/1/30
  * Malyali et al. (2021) Malyali, A., Rau, A., Merloni, A., et al. 2021, A&A, 647, A9, doi: 10.1051/0004-6361/202039681
  * Malyali et al. (2023a) Malyali, A., Liu, Z., Merloni, A., et al. 2023a, MNRAS, 520, 4209, doi: 10.1093/mnras/stad046
  * Malyali et al. (2023b) Malyali, A., Liu, Z., Rau, A., et al. 2023b, MNRAS, 520, 3549, doi: 10.1093/mnras/stad022
  * Martin et al. (2005) Martin, D. C., Fanson, J., Schiminovich, D., et al. 2005, ApJ, 619, L1, doi: 10.1086/426387
  * Martin et al. (2007) Martin, D. C., Wyder, T. K., Schiminovich, D., et al. 2007, ApJS, 173, 342, doi: 10.1086/516639
  * Masci et al. (2019) Masci, F. J., Laher, R. R., Rusholme, B., et al. 2019, PASP, 131, 018003, doi: 10.1088/1538-3873/aae8ac
  * Masci et al. (2023) —. 2023, arXiv e-prints, arXiv:2305.16279, doi: 10.48550/arXiv.2305.16279
  * Masterson et al. (2022) Masterson, M., Kara, E., Ricci, C., et al. 2022, ApJ, 934, 35, doi: 10.3847/1538-4357/ac76c0
  * Mateos et al. (2012) Mateos, S., Alonso-Herrero, A., Carrera, F. J., et al. 2012, MNRAS, 426, 3271, doi: 10.1111/j.1365-2966.2012.21843.x
  * Mattila et al. (2018) Mattila, S., Pérez-Torres, M., Efstathiou, A., et al. 2018, Science, 361, 482, doi: 10.1126/science.aao4669
  * Meikle et al. (2011) Meikle, W. P. S., Kotak, R., Farrah, D., et al. 2011, ApJ, 732, 109, doi: 10.1088/0004-637X/732/2/109
  * Meisner et al. (2018) Meisner, A. M., Lang, D., & Schlegel, D. J. 2018, AJ, 156, 69, doi: 10.3847/1538-3881/aacbcd
  * Mendel et al. (2014) Mendel, J. T., Simard, L., Palmer, M., Ellison, S. L., & Patton, D. R. 2014, ApJS, 210, 3, doi: 10.1088/0067-0049/210/1/3
  * Merloni et al. (2015) Merloni, A., Dwelly, T., Salvato, M., et al. 2015, MNRAS, 452, 69, doi: 10.1093/mnras/stv1095
  * Metzger & Stone (2016) Metzger, B. D., & Stone, N. C. 2016, MNRAS, 461, 948, doi: 10.1093/mnras/stw1394
  * Mockler et al. (2019) Mockler, B., Guillochon, J., & Ramirez-Ruiz, E. 2019, ApJ, 872, 151, doi: 10.3847/1538-4357/ab010f
  * Mummery & Balbus (2020) Mummery, A., & Balbus, S. A. 2020, MNRAS, 497, L13, doi: 10.1093/mnrasl/slaa105
  * Newsome et al. (2023) Newsome, M., Arcavi, I., Howell, D. A., et al. 2023, arXiv e-prints, arXiv:2305.03767, doi: 10.48550/arXiv.2305.03767
  * Onori et al. (2022) Onori, F., Cannizzaro, G., Jonker, P. G., et al. 2022, MNRAS, 517, 76, doi: 10.1093/mnras/stac2673
  * Panagiotou et al. (2023) Panagiotou, C., De, K., Masterson, M., et al. 2023, ApJ, 948, L5, doi: 10.3847/2041-8213/acc02f
  * Pasham et al. (2019) Pasham, D. R., Remillard, R. A., Fragile, P. C., et al. 2019, Science, 363, 531, doi: 10.1126/science.aar7480
  * Payne et al. (2021) Payne, A. V., Shappee, B. J., Hinkle, J. T., et al. 2021, ApJ, 910, 125, doi: 10.3847/1538-4357/abe38d
  * Perley et al. (2016) Perley, D. A., Quimby, R. M., Yan, L., et al. 2016, ApJ, 830, 13, doi: 10.3847/0004-637X/830/1/13
  * Peterson et al. (2013) Peterson, B. M., Denney, K. D., De Rosa, G., et al. 2013, ApJ, 779, 109, doi: 10.1088/0004-637X/779/2/109
  * Pfister et al. (2019) Pfister, H., Bar-Or, B., Volonteri, M., Dubois, Y., & Capelo, P. R. 2019, MNRAS, 488, L29, doi: 10.1093/mnrasl/slz091
  * Pfister et al. (2020) Pfister, H., Volonteri, M., Dai, J. L., & Colpi, M. 2020, MNRAS, 497, 2276, doi: 10.1093/mnras/staa1962
  * Piran et al. (2015) Piran, T., Svirski, G., Krolik, J., Cheng, R. M., & Shiokawa, H. 2015, ApJ, 806, 164, doi: 10.1088/0004-637X/806/2/164
  * Predehl et al. (2021) Predehl, P., Andritschke, R., Arefiev, V., et al. 2021, A&A, 647, A1, doi: 10.1051/0004-6361/202039313
  * Prochaska et al. (2020) Prochaska, J., Hennawi, J., Westfall, K., et al. 2020, The Journal of Open Source Software, 5, 2308, doi: 10.21105/joss.02308
  * Quimby et al. (2013) Quimby, R. M., Yuan, F., Akerlof, C., & Wheeler, J. C. 2013, MNRAS, 431, 912, doi: 10.1093/mnras/stt213
  * Rayner et al. (2003) Rayner, J. T., Toomey, D. W., Onaka, P. M., et al. 2003, PASP, 115, 362, doi: 10.1086/367745
  * Rees (1988) Rees, M. J. 1988, Nature, 333, 523, doi: 10.1038/333523a0
  * Reines & Volonteri (2015) Reines, A. E., & Volonteri, M. 2015, ApJ, 813, 82, doi: 10.1088/0004-637X/813/2/82
  * Ricci et al. (2017) Ricci, C., Trakhtenbrot, B., Koss, M. J., et al. 2017, ApJS, 233, 17, doi: 10.3847/1538-4365/aa96ad
  * Ricci et al. (2020) Ricci, C., Kara, E., Loewenstein, M., et al. 2020, ApJ, 898, L1, doi: 10.3847/2041-8213/ab91a1
  * Ricci et al. (2021) Ricci, C., Loewenstein, M., Kara, E., et al. 2021, ApJS, 255, 7, doi: 10.3847/1538-4365/abe94b
  * Roth et al. (2016) Roth, N., Kasen, D., Guillochon, J., & Ramirez-Ruiz, E. 2016, ApJ, 827, 3, doi: 10.3847/0004-637X/827/1/3
  * Roth et al. (2021) Roth, N., van Velzen, S., Cenko, S. B., & Mushotzky, R. F. 2021, ApJ, 910, 93, doi: 10.3847/1538-4357/abdf50
  * Saxton et al. (2020) Saxton, R., Komossa, S., Auchettl, K., & Jonker, P. G. 2020, Space Sci. Rev., 216, 85, doi: 10.1007/s11214-020-00708-4
  * Saxton et al. (2008) Saxton, R. D., Read, A. M., Esquej, P., et al. 2008, A&A, 480, 611, doi: 10.1051/0004-6361:20079193
  * Saxton et al. (2017) Saxton, R. D., Read, A. M., Komossa, S., et al. 2017, A&A, 598, A29, doi: 10.1051/0004-6361/201629015
  * Saxton et al. (2022) Saxton, R. D., König, O., Descalzo, M., et al. 2022, Astronomy and Computing, 38, 100531, doi: 10.1016/j.ascom.2021.100531
  * Sazonov et al. (2021) Sazonov, S., Gilfanov, M., Medvedev, P., et al. 2021, MNRAS, 508, 3820, doi: 10.1093/mnras/stab2843
  * Schawinski et al. (2007) Schawinski, K., Thomas, D., Sarzi, M., et al. 2007, MNRAS, 382, 1415, doi: 10.1111/j.1365-2966.2007.12487.x
  * Schawinski et al. (2014) Schawinski, K., Urry, C. M., Simmons, B. D., et al. 2014, MNRAS, 440, 889, doi: 10.1093/mnras/stu327
  * Schlafly & Finkbeiner (2011) Schlafly, E. F., & Finkbeiner, D. P. 2011, ApJ, 737, 103, doi: 10.1088/0004-637X/737/2/103
  * Shappee et al. (2014) Shappee, B. J., Prieto, J. L., Grupe, D., et al. 2014, ApJ, 788, 48, doi: 10.1088/0004-637X/788/1/48
  * Shingles et al. (2021) Shingles, L., Smith, K. W., Young, D. R., et al. 2021, Transient Name Server AstroNote, 7, 1
  * Short et al. (2023) Short, P., Lawrence, A., Nicholl, M., et al. 2023, MNRAS, 525, 1568, doi: 10.1093/mnras/stad2270
  * Simcoe et al. (2013) Simcoe, R. A., Burgasser, A. J., Schechter, P. L., et al. 2013, PASP, 125, 270, doi: 10.1086/670241
  * Skrutskie et al. (2006) Skrutskie, M. F., Cutri, R. M., Stiening, R., et al. 2006, AJ, 131, 1163, doi: 10.1086/498708
  * Smith et al. (2020) Smith, K. W., Smartt, S. J., Young, D. R., et al. 2020, PASP, 132, 085002, doi: 10.1088/1538-3873/ab936e
  * Smith et al. (2009) Smith, N., Silverman, J. M., Chornock, R., et al. 2009, ApJ, 695, 1334, doi: 10.1088/0004-637X/695/2/1334
  * Stein et al. (2021) Stein, R., van Velzen, S., Kowalski, M., et al. 2021, Nature Astronomy, 5, 510, doi: 10.1038/s41550-020-01295-8
  * Stern et al. (2005) Stern, D., Eisenhardt, P., Gorjian, V., et al. 2005, ApJ, 631, 163, doi: 10.1086/432523
  * Stern et al. (2012) Stern, D., Assef, R. J., Benford, D. J., et al. 2012, ApJ, 753, 30, doi: 10.1088/0004-637X/753/1/30
  * Stone & Metzger (2016) Stone, N. C., & Metzger, B. D. 2016, MNRAS, 455, 859, doi: 10.1093/mnras/stv2281
  * Stoughton et al. (2002) Stoughton, C., Lupton, R. H., Bernardi, M., et al. 2002, AJ, 123, 485, doi: 10.1086/324741
  * Strubbe & Quataert (2009) Strubbe, L. E., & Quataert, E. 2009, MNRAS, 400, 2070, doi: 10.1111/j.1365-2966.2009.15599.x
  * Sukhbold & Woosley (2016) Sukhbold, T., & Woosley, S. E. 2016, ApJ, 820, L38, doi: 10.3847/2041-8205/820/2/L38
  * Sun et al. (2022) Sun, L., Xiao, L., & Li, G. 2022, MNRAS, 513, 4057, doi: 10.1093/mnras/stac1121
  * Svirski et al. (2017) Svirski, G., Piran, T., & Krolik, J. 2017, MNRAS, 467, 1426, doi: 10.1093/mnras/stx117
  * Szalai et al. (2011) Szalai, T., Vinkó, J., Balog, Z., et al. 2011, A&A, 527, A61, doi: 10.1051/0004-6361/201015624
  * Tadhunter et al. (2017) Tadhunter, C., Spence, R., Rose, M., Mullaney, J., & Crowther, P. 2017, Nature Astronomy, 1, 0061, doi: 10.1038/s41550-017-0061
  * Teboul et al. (2024) Teboul, O., Stone, N. C., & Ostriker, J. P. 2024, MNRAS, 527, 3094, doi: 10.1093/mnras/stad3301
  * Thévenot (2020) Thévenot, M. 2020, Research Notes of the American Astronomical Society, 4, 243, doi: 10.3847/2515-5172/abd415
  * Thévenot et al. (2021) Thévenot, M., Gantier, J. M., Kabatnik, M., Schümann, J., & Gramaize, L. 2021, Research Notes of the American Astronomical Society, 5, 58, doi: 10.3847/2515-5172/abf120
  * Thomsen et al. (2022) Thomsen, L. L., Kwan, T. M., Dai, L., et al. 2022, ApJ, 937, L28, doi: 10.3847/2041-8213/ac911f
  * Tinyanont et al. (2016) Tinyanont, S., Kasliwal, M. M., Fox, O. D., et al. 2016, ApJ, 833, 231, doi: 10.3847/1538-4357/833/2/231
  * Tonry et al. (2018) Tonry, J. L., Denneau, L., Heinze, A. N., et al. 2018, PASP, 130, 064505, doi: 10.1088/1538-3873/aabadf
  * Vacca et al. (2003) Vacca, W. D., Cushing, M. C., & Rayner, J. T. 2003, PASP, 115, 389, doi: 10.1086/346193
  * Valenti et al. (2014) Valenti, S., Sand, D., Pastorello, A., et al. 2014, MNRAS, 438, L101, doi: 10.1093/mnrasl/slt171
  * van der Walt et al. (2019) van der Walt, S., Crellin-Quick, A., & Bloom, J. 2019, The Journal of Open Source Software, 4, 1247, doi: 10.21105/joss.01247
  * van Velzen (2018) van Velzen, S. 2018, ApJ, 852, 72, doi: 10.3847/1538-4357/aa998e
  * van Velzen & Farrar (2014) van Velzen, S., & Farrar, G. R. 2014, ApJ, 792, 53, doi: 10.1088/0004-637X/792/1/53
  * van Velzen et al. (2016) van Velzen, S., Mendez, A. J., Krolik, J. H., & Gorjian, V. 2016, ApJ, 829, 19, doi: 10.3847/0004-637X/829/1/19
  * van Velzen et al. (2021a) van Velzen, S., Pasham, D. R., Komossa, S., Yan, L., & Kara, E. A. 2021a, Space Sci. Rev., 217, 63, doi: 10.1007/s11214-021-00835-6
  * van Velzen et al. (2011) van Velzen, S., Farrar, G. R., Gezari, S., et al. 2011, ApJ, 741, 73, doi: 10.1088/0004-637X/741/2/73
  * van Velzen et al. (2021b) van Velzen, S., Gezari, S., Hammerstein, E., et al. 2021b, ApJ, 908, 4, doi: 10.3847/1538-4357/abc258
  * van Velzen et al. (2021c) van Velzen, S., Stein, R., Gilfanov, M., et al. 2021c, arXiv e-prints, arXiv:2111.09391, doi: 10.48550/arXiv.2111.09391
  * Wang & Merritt (2004) Wang, J., & Merritt, D. 2004, ApJ, 600, 149, doi: 10.1086/379767
  * Wang et al. (2012) Wang, T.-G., Zhou, H.-Y., Komossa, S., et al. 2012, ApJ, 749, 115, doi: 10.1088/0004-637X/749/2/115
  * Wang et al. (2022a) Wang, Y., Jiang, N., Wang, T., et al. 2022a, ApJS, 258, 21, doi: 10.3847/1538-4365/ac33a6
  * Wang et al. (2022b) —. 2022b, ApJ, 930, L4, doi: 10.3847/2041-8213/ac6670
  * Wen et al. (2022) Wen, S., Jonker, P. G., Stone, N. C., Zabludoff, A. I., & Cao, Z. 2022, ApJ, 933, 31, doi: 10.3847/1538-4357/ac70c5
  * Wevers et al. (2021) Wevers, T., Pasham, D. R., van Velzen, S., et al. 2021, ApJ, 912, 151, doi: 10.3847/1538-4357/abf5e2
  * Wevers et al. (2023) Wevers, T., Coughlin, E. R., Pasham, D. R., et al. 2023, ApJ, 942, L33, doi: 10.3847/2041-8213/ac9f36
  * Winter & Lunardini (2023) Winter, W., & Lunardini, C. 2023, ApJ, 948, 42, doi: 10.3847/1538-4357/acbe9e
  * Wolf et al. (2018) Wolf, C., Onken, C. A., Luvaul, L. C., et al. 2018, PASA, 35, e010, doi: 10.1017/pasa.2018.5
  * Wong et al. (2022) Wong, T. H. T., Pfister, H., & Dai, L. 2022, ApJ, 927, L19, doi: 10.3847/2041-8213/ac5823
  * Worthey & Ottaviani (1997) Worthey, G., & Ottaviani, D. L. 1997, ApJS, 111, 377, doi: 10.1086/313021
  * Wright et al. (2010) Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ, 140, 1868, doi: 10.1088/0004-6256/140/6/1868
  * Yang et al. (2013) Yang, C.-W., Wang, T.-G., Ferland, G., et al. 2013, ApJ, 774, 46, doi: 10.1088/0004-637X/774/1/46
  * Yao et al. (2022) Yao, Y., Lu, W., Guolo, M., et al. 2022, ApJ, 937, 8, doi: 10.3847/1538-4357/ac898a
  * Yao et al. (2023) Yao, Y., Ravi, V., Gezari, S., et al. 2023, ApJ, 955, L6, doi: 10.3847/2041-8213/acf216
  * Zackay et al. (2016) Zackay, B., Ofek, E. O., & Gal-Yam, A. 2016, ApJ, 830, 27, doi: 10.3847/0004-637X/830/1/27


# Cosmological Study in $F(R,T)$ Quasi-dilaton Massive Gravity

Sobhan Kazempour<EMAIL_ADDRESS>Faculty of Physics, University of
Tabriz, Tabriz 51666-16471, Iran  Amin Rezaei Akbarieh
<EMAIL_ADDRESS>Faculty of Physics, University of Tabriz, Tabriz
51666-16471, Iran

###### Abstract

This study explores the cosmological implications of the $F(R,T)$ quasi-
dilaton massive gravity theory, a modification of the de Rham-Gabadadze-Tolley
(dRGT) massive gravity theory. Our analysis focuses on the self-accelerating
solution of the background equations of motion, which are shown to exist in
the theory. Notably, we find that the theory features an effective
cosmological constant, which has important implications for our understanding
of the universe’s accelerated expansion. To test the viability of the $F(R,T)$
quasi-dilaton massive gravity theory, we utilize the Union2 Type Ia Supernovae
(SNIa) dataset, comprising 557 observations. Our results demonstrate that the
theory is capable of explaining the accelerated expansion of the universe
without requiring the presence of dark energy. This finding supports the
potential of the $F(R,T)$ quasi-dilaton massive gravity theory as an
alternative explanation for the observed cosmic acceleration. Moreover, we
investigate the properties of tensor perturbations within the framework of
this theory and derive a novel expression for the dispersion relation of
gravitational waves. Our analysis reveals interesting features of the modified
dispersion relation in the Friedmann-Lemaître-Robertson-Walker cosmology,
providing new insights into the nature of gravitational waves in the context
of the $F(R,T)$ quasi-dilaton massive gravity theory.

††preprint: APS/123-QED

##  I Introduction

It is well-known that the origin of late-time accelerated expansion of the
Universe which is confirmed by several observational research is unknown and
would be considered as the biggest puzzles in cosmology
SupernovaSearchTeam:1998fmf ; WMAP:2003elm ; WMAP:2003xez ; Caldwell:2003hz ;
SDSS:2005xqv ; SDSS:2009ocz ; SNLS:2011cra . Although the general theory of
relativity has a lot of successful predictions Will:2001mx ; Will:2005va ;
Will:2014kxa , it solely can explain the late-time accelerated expansion of
the Universe by considering the cosmological constant or dark energy
component. As the origin of dark energy is not clear and the cosmological
constant has a problem Copeland:2006wr ; Weinberg:1988cp ; Peebles:2002gy ,
there has been a tendency towards modified gravity theories throughout the
years Carroll:2004de ; Clifton:2011jh ; Sotiriou:2008rp ; Nojiri:2006ri ;
Bamba:2012cp ; Nojiri:2010wj ; Starobinsky:1980te ; Nojiri:2005jg ;
Bahamonde:2015zma .

According to modern particle physics, general relativity is a unique theory of
a massless Lorentz-invariant spin-2 particle Weinberg:1965rz . Modifying
gravity through the massive gravity theory offers a valuable approach to
understanding the nature of gravity and its interactions. In this framework, a
spin-2 massive graviton is thought to mediate the propagation of gravity,
providing a fresh perspective on the fundamental laws of physics.
Investigating the properties and behavior of such a graviton could lead to new
insights into the structure of the universe and potentially reveal novel
phenomena. It should be mentioned that a number of attempts have been made to
describe graviton and its interactions to explain the current accelerated
expansion of the Universe. In fact, the main effort is introducing a theory
that would be stable and consistent with cosmological observations
deRham:2014zqa ; deRham:2010kj ; Hinterbichler:2011tt ; deRham:2010ik ;
Hassan:2011hr ; Hassan:2011zd .

The introduction of a massive spin-2 field theory was first successfully
achieved by Fierz and Pauli in 1939. They developed a Lorentz invariant linear
theory that included consistent interaction terms, which were later
interpreted as a graviton mass. In fact, its theory includes a specific
combination of the mass terms to have five physical degrees of freedom
Fierz:1939ix . The theory of massive gravity has undergone a lot of changes
over the years. In 1970, van Dam, Veltman, and Zakharov showed that the Fierz
and Pauli theory could not reduce to general relativity (massless theory) in
the limit of $m_{g}\longrightarrow 0$. This discontinuity in predictions is
called vDVZ discontinuity Zakharov:1970cc ; vanDam:1970vg . In the following,
Vainshtein presented a way to avoid vDVZ discontinuity. He proposed that one
should consider the nonlinear completions of the Fierz-Pauli term instead of
linear Vainshtein:1972sx . On the other hand, Boulware and Deser claimed that
there is a ghost instability in non-linear theory which is called the
Boulware-Deser ghost Boulware:1972yco ; Arkani-Hamed:2002bjr ;
Creminelli:2005qk . In line with these efforts to revive massive gravity
theory, in 2010 de Rham, Gabadadze, and Tolley (dRGT) introduced a fully non-
linear massive gravity theory without Boulware-Deser ghost. Their research
proposes a novel approach to understanding the nature of gravity by
introducing nonlinear interactions that could reveal the presence of a massive
spin-2 field in a flat spacetime deRham:2010ik ; deRham:2010kj . However, this
theory admits solely an open Friedmann-Lemaître-Robertson-Walker (FLRW)
solution. In fact, there are not any stable solutions for a homogeneous and
isotropic Universe DeFelice:2012mx ; Gumrukcuoglu:2011zh . Therefore, there
are enough motivations to present new extension theories to find stable self-
accelerating solutions in the context of massive gravity theory DAmico:2011eto
; Gumrukcuoglu:2012aa ; DAmico:2012hia ; Mukohyama:2014rca ; Huang:2012pe ;
DeFelice:2013tsa ; Gumrukcuoglu:2013nza ; Langlois:2014jba ; deRham:2014naa ;
Kenna-Allison:2018izo ; Kenna-Allison:2019tbu ; Gumrukcuoglu:2020utx ;
Akbarieh:2021vhv ; Aslmarand:2021qwn ; Akbarieh:2022ovn ; Kazempour:2022let ;
Kazempour:2022giy ; Kazempour:2022xzy .

In this work, we aim to introduce a novel extension of the non-linear dRGT
massive gravity theory, $F(R,T)$ quasi-dilation massive gravity theory, which
endeavors to provide a fresh perspective on explaining the late-time
accelerated expansion of the universe within the framework of FLRW cosmology.
It is interesting to note that the quasi-dilaton part of the action introduces
an extra scalar degree of freedom to the dRGT theory DAmico:2012hia . But,
because of instabilities, some extensions of quasi-dilaton massive gravity
theory have been introduced Mukohyama:2014rca ; DeFelice:2013tsa ;
Gumrukcuoglu:2013nza ; Akbarieh:2021vhv . Moreover, the $F(R,T)$ gravity
theory is an intriguing modification of traditional gravity theories, where
$R$ represents the curvature scalar and $T$ denotes the torsion scalar. Notice
that in this theory it could be possible to reproduce the unification of
$F(R)$ and $F(T)$ gravity theories Myrzakulov:2012axz ; Myrzakulov:2012qp . We
can point out a variety of extensions of the $F(R,T)$ gravity model which have
been considered by several researchers Harko:2011kv ; Godani:2018sbl ;
Debnath:2018wct ; Tretyakov:2018yph ; Sarkar:2022lir ; Bertini:2023pmp .

In order to constrain the parameters of the theory we consider the supernova
I-a observational data. The theory compares with these data using the Bayesian
statistic method which is on the basis of minimum $X^{2}$. Moreover, using the
dispersion relation of gravitational waves one can impose the constraints on
the theory. In addition, analysis of the dispersion relation of gravitation
waves can be used in the phase evolution of the gravitational waveform.

The goals of the paper are to present a new extension of the massive gravity
theory and find a stable self-accelerating solution to explain the late-time
accelerated expansion of the Universe. Likewise, we try to present the
constraints tools using cosmological data analysis and theoretical aspects of
gravitational waves. The outline of this paper is organized as follows. In
Sec. II, we present the new extension of massive gravity theory which is
$F(R,T)$ quasi-dilaton massive gravity theory. We obtain the background
equations of motion and self-accelerating solution. In Sec. III, we constrain
the parameters of the $F(R,T)$ quasi-dilaton massive gravity theory using
supernova I-a data by considering the Bayesian statistic technique. In Sec.
IV, we undertake a perturbation analysis to derive the dispersion relationship
of gravitational waves in the context of FLRW cosmology. In Sec. V, we
indicate the conclusion. In this paper, we consider natural units, where
$c=\hbar=1$ and $M_{Pl}^{2}=8\pi G=1$, where $G$ is Newton’s gravitational
constant.

##  II $F(R,T)$ Quasi-dilaton Massive Gravity

In this section, we present the new extension of the massive gravity theory.
This theory is constructed by considering the $F(R,T)$ modified gravity. In
the following, we show the cosmological background equations and self-
accelerating solution.

The total action is:

$\displaystyle S=\frac{1}{2}\int
d^{4}x\Bigg{\\{}\sqrt{-g}\bigg{[}F(R,T)-\omega\partial_{\mu}\sigma\partial^{\mu}\sigma+2{m}_{g}^{2}U(\mathcal{K})\bigg{]}\Bigg{\\}},$

where $\sqrt{-g}$ is the determinant of $g_{\mu\nu}$ which is the physical
dynamical metric, $R$ is the curvature scalar, $T$ is the torsion scalar,
$\omega$ is a dimensionless constant, $\sigma$ is the quasi-dilaton scalar and
the last part of the action is related to the massive part. We consider the
forms of the curvature and torsion scalars below Myrzakulov:2012qp ;
Myrzakulov:2012axz ,

$\displaystyle F(R,T)=\xi R+\beta T,$ (2)

where $\xi$ and $\beta$ are the constants. Also, $R$ and $T$ can be expressed
as:

$\displaystyle R=g^{\mu\nu}R_{\mu\nu}+u,$ $\displaystyle
T=S_{\rho}^{~{}\mu\nu}T^{\rho}_{~{}\mu\nu}+v,$ (3)

we consider the FLRW spacetime, and $u$ and $v$ can be defined,

$\displaystyle u=6\dot{j}+18\frac{\dot{a}}{aN}j+6j^{2}-3b^{2},$ $\displaystyle
v=6\big{(}j^{2}-b^{2}-(\frac{\dot{a}}{aN})^{2}\big{)},$ (4)

also, we know

$\displaystyle
S_{\rho}^{~{}\mu\nu}=\frac{1}{2}(C_{\rho}^{~{}\mu\nu}+\delta_{\rho}^{\mu}T_{\theta}^{~{}\theta\nu}-\delta_{\rho}^{\nu}T_{\theta}^{~{}\theta\mu}),$
(5)

which $C$ is the contorsion Myrzakulov:2012qp . In Eqs. (II), $b$ and $j$ are
real functions and $N$ is a lapse function of the dynamical metric Mueller-
Hoissen:1983vya . It is obvious that $a$ is the scale factor and $\dot{a}$ is
the derivative with respect to time. Moreover, the orthonormal tetrad
components $e_{i}(x^{\mu})$ are related to the metric as
$g_{\mu\nu}=\eta_{ij}e_{\mu}^{i}e_{\nu}^{j}$. As we mentioned before $\sigma$
is the quasi-dilaton scalar and $\omega$ is a dimensionless constant.
According to the Ref DAmico:2012hia , the theory is invariant under a global
dilation transformation, $\sigma\rightarrow\sigma+\sigma_{0}$.

It should be noted that the potential $U$ is the cause of the creation of the
graviton mass $m_{g}$ which consists of three parts, i.e.

$U(\mathcal{K})=U_{2}+\alpha_{3}U_{3}+\alpha_{4}U_{4},$ (6)

with $\alpha_{3}$ and $\alpha_{4}$ dimensionless free parameters. Therefore,
the potential sentences can be expressed as deRham:2010kj

$\displaystyle U_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{(}[\mathcal{K}]^{2}-[\mathcal{K}^{2}]\big{)},$
$\displaystyle U_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\big{(}[\mathcal{K}]^{3}-3[\mathcal{K}][\mathcal{K}^{2}]+2[\mathcal{K}^{3}]\big{)},$
$\displaystyle U_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{24}\big{(}[\mathcal{K}]^{4}-6[\mathcal{K}]^{2}[\mathcal{K}^{2}]+8[\mathcal{K}][\mathcal{K}^{3}]+3[\mathcal{K}^{2}]^{2}-6[\mathcal{K}^{4}]\big{)},$

where ”$[\cdot]$” is interpreted as the trace of the tensor inside the
brackets. Moreover, the building block tensor $\mathcal{K}$ is defined as

$\mathcal{K}^{\mu}_{~{}\nu}=\delta^{\mu}_{~{}\nu}-e^{\sigma}\big{(}\sqrt{g^{-1}f}\big{)}^{\mu}_{~{}\nu},$
(8)

where $f_{\alpha\nu}$ is the fiducial metric which is defined through

$f_{\alpha\nu}=\partial_{\alpha}\phi^{c}\partial_{\nu}\phi^{d}\eta_{cd},$ (9)

where $\eta_{cd}$ is the Minkowski metric ($c,d=0,1,2,3$) and $\phi^{c}$ are
the Stueckelberg fields which introduce to restore general covariance
DAmico:2012hia ; deRham:2010kj .

###  II.1 Background cosmological evolution

In this stage, we would like to consider the theory in an FLRW metric at the
background level. The dynamical metric, the vierbien, and the fiducial metric
are expressed as

$\displaystyle g_{\mu\nu}$ $\displaystyle={\rm
diag}\left[-N^{2},a^{2},a^{2},a^{2}\right],$ (10) $\displaystyle e_{\mu}^{A}$
$\displaystyle={\rm diag}\left[N,a,a,a\right],$ (11) $\displaystyle
f_{\mu\nu}$ $\displaystyle={\rm diag}\left[-\dot{f}(t)^{2},1,1,1\right],$ (12)

as it is mentioned before, $a$ is the scale factor and $N$ is the lapse
function of the dynamical metric, which relates the coordinate-time $dt$ to
the proper-time $d\tau$ using $d\tau=Ndt$ Scheel:1994yr ;
Christodoulakis:2013xha . Also, we express that the function $f(t)$ is the
Stueckelberg scalar function, with $\phi^{0}=f(t)$ and
$\frac{\partial\phi^{0}}{\partial t}=\dot{f}(t)$ Arkani-Hamed:2002bjr .

Thus, we obtain the total pointlike Lagrangian in FLRW cosmology

$\displaystyle\mathcal{L}=\bigg{[}\frac{-3a\dot{a}^{2}\xi}{N}+\frac{3a^{2}}{2}\bigg{(}a\big{(}2j^{2}(\beta+\xi)-b^{2}(2\beta+\xi)$
$\displaystyle+2\xi\dot{j}\big{)}N+6\xi j\dot{a}\bigg{)}\bigg{]}+\frac{\omega
a^{3}}{2N}\dot{\sigma}^{2}+m_{g}^{2}\Bigg{\\{}Na^{3}(Y$
$\displaystyle-1)\bigg{[}3(Y-2)-(Y-4)(Y-1)\alpha_{3}-(Y$
$\displaystyle-1)^{2}\alpha_{4}\bigg{]}+\dot{f}(t)a^{4}Y(Y-1)\bigg{[}3-3(Y$
$\displaystyle-1)\alpha_{3}+(Y-1)^{2}\alpha_{4}\bigg{]}\Bigg{\\}},$

where

$\displaystyle Y\equiv\frac{e^{\sigma}}{a}.$ (14)

It is necessary to notice that the gauge transformations remove the unphysical
fields from the theory on the classical level Grosse-Knetter:1992tbp .
Therefore, we regard the unitary gauge i.e., $f(t)=t$. So, a constraint
equation would be achieved by varying with respect to $f$,

$\frac{\delta\mathcal{L}}{\delta
f}=m_{g}^{2}\frac{d}{dt}\left\\{a^{4}Y(Y-1)[3-3(Y-1)\alpha_{3}+(Y-1)^{2}\alpha_{4}]\right\\}=0.$
(15)

We obtain the modified Friedman equation by considering the variation of the
pointlike Lagrangian with respect to lapse function $N$,

$\displaystyle\frac{1}{a^{3}}\frac{\delta\mathcal{L}}{\delta
N}=3H^{2}\xi-\frac{\omega}{2}\big{(}H+\frac{\dot{Y}}{YN}\big{)}^{2}+\frac{3}{2}\big{(}2j^{2}(\beta+\xi)$
$\displaystyle-b^{2}(2\beta+\xi)+2\xi\dot{j}\big{)}-m_{g}^{2}(Y-1)\bigg{[}(Y-4)(Y$
$\displaystyle-1)\alpha_{3}+(Y-1)^{2}\alpha_{4}-3(Y-2)\bigg{]}=0.$

The equation of motion related to the scale factor $a$ is given by

$\displaystyle\frac{1}{6a^{2}N}\frac{\delta\mathcal{L}}{\delta
a}=\frac{1}{4}\bigg{(}6j^{2}\big{(}\xi+\beta\big{)}-3b^{2}\big{(}2\beta+\xi\big{)}+6\xi$
$\displaystyle\big{(}H^{2}+\dot{j}^{2}\big{)}\bigg{)}+\frac{\dot{H}\xi}{N}+\omega\big{(}\frac{\dot{\sigma}}{2N}\big{)}^{2}+\frac{m_{g}^{2}}{2}\bigg{(}6+4\alpha_{3}$
$\displaystyle+\alpha_{4}-(2+r)(3+3\alpha_{3}+\alpha_{4})Y+(2r+1)(1$
$\displaystyle+2\alpha_{3}+\alpha_{4})Y^{2}-r(\alpha_{3}+\alpha_{4})Y^{3}\bigg{)}=0.$

By varying the pointlike Lagrangian with respect to $\sigma$, the equation of
motion corresponding to the scalar field is obtained by

$\displaystyle\frac{1}{a^{3}N}\frac{\delta\mathcal{L}}{\delta\sigma}=-3\frac{H^{2}\omega}{N}+m_{g}^{2}Y\Bigg{\\{}6(r+1)\big{(}\alpha_{4}+2\alpha_{3}$
$\displaystyle+1\big{)}Y-(3+r)\big{(}3+3\alpha_{3}+\alpha_{4}\big{)}-3(3r+1)(\alpha_{4}$
$\displaystyle+\alpha_{3})Y^{2}+4r\alpha_{4}Y^{3}\Bigg{\\}}=0,$ (18)

where $r\equiv\frac{a}{N}$ and $H\equiv\frac{\dot{a}}{Na}$. Note that the
below equations are obtained by considering the notation in Eq (14).

$\frac{\dot{\sigma}}{N}=H+\frac{\dot{Y}}{NY},\qquad\ddot{\sigma}=\frac{d}{dt}\Big{(}NH+\frac{\dot{Y}}{Y}\Big{)}.$
(19)

According to the Stuckelberg field $f$ which introduces time reparametrization
invariance, the Bianchi identity relates the four equations of motion.
Therefore, the equation of motion has something to do with the scale factor is
redundant and we disregard it.

$\displaystyle\frac{\delta S}{\delta\sigma}\dot{\sigma}+\frac{\delta S}{\delta
f}\dot{f}-N\frac{d}{dt}\frac{\delta S}{\delta N}+\dot{a}\frac{\delta S}{\delta
a}=0.$ (20)

It should be pointed out that in particular conditions, all of the background
equations and total Lagrangian reduce to those in Refs. DAmico:2012hia ;
Gumrukcuoglu:2013nza .

###  II.2 Self-accelerating background solutions

In order to evaluate self-accelerating solutions in the context of this new
extension, we integrate the Stueckelberg constraint equation (15). Therefore,
we have

$\displaystyle
Y(Y-1)\bigg{[}3-3(Y-1)\alpha_{3}+(Y-1)^{2}\alpha_{4}\bigg{]}\propto a^{-4}.$

It is obvious that we look forward to explaining the accelerated expansion of
the Universe in the context of $F(R,T)$ quasi-dilaton massive gravity theory.
The constant solution of the above equation leads to the effective energy
density which behaves similarly to the cosmological constant. Therefore, in an
expanding universe, the right-hand side of Eq. (II.2) decays as $a^{-4}$, so
the right-hand side of that equation would decrease in a sufficiently long
time and makes the left-hand side of the equation equal to zero. Hence, $Y$
transforms into the saturate constant value $Y_{SA}$ which would be a root of
the left-hand side of the equation (II.2).

$(Y-1)\big{[}3-3(Y-1)\alpha_{3}+(Y-1)^{2}\alpha_{4}\big{]}\bigg{|}_{Y=Y_{\rm
SA}}=0.$ (22)

Equation (22) admits four distinct solutions. The one obvious solution is
$Y=0$, however, this solution implies $\sigma\longrightarrow-\infty$. As this
solution would be multiplied by the perturbations of the auxiliary scalars, we
encounter strong coupling in the vector and scalar sectors. So, this solution
has to disregard avoiding strong coupling DAmico:2012hia . Furthermore,
another solution that could be considered is $Y=1$. But, it must be eliminated
because of a vanishing cosmological constant and inconsistency DAmico:2012hia
. As a result, there are solely two solutions which are given,

$Y_{\rm
SA}^{\pm}=\frac{3\alpha_{3}+2\alpha_{4}\pm\sqrt{9\alpha_{3}^{2}-12\alpha_{4}}}{2\alpha_{4}}.$
(23)

Using Eq. (II.1) and Eq. (23), we can obtain the modified Friedmann equation
as below,

$\displaystyle\bigg{[}3H^{2}\xi+\frac{3}{2}\big{(}2j^{2}(\beta+\xi)-b^{2}(2\beta+\xi)+2\xi\dot{j}\big{)}$
$\displaystyle-\frac{\omega}{2}H^{2}\bigg{]}=\Lambda_{\rm SA}^{\pm},$

where we have the effective cosmological constant corresponding to the massive
part of the action

$\displaystyle\Lambda_{\rm SA}^{\pm}\equiv m_{g}^{2}(Y_{\rm
SA}^{\pm}-1)\Big{[}6-3Y_{\rm SA}^{\pm}+(Y_{\rm SA}^{\pm}-4)(Y_{\rm SA}^{\pm}$
$\displaystyle-1)\alpha_{3}+(Y_{\rm SA}^{\pm}-1)^{2}\alpha_{4}\Big{]}.$

The above equation should be re-written by considering Eq. (23),

$\displaystyle\Lambda_{\rm
SA}^{\pm}=\frac{3m^{2}_{g}}{2\alpha^{3}_{4}}\bigg{[}9\alpha^{4}_{3}\pm
3\alpha^{3}_{3}\sqrt{9\alpha^{2}_{3}-12\alpha_{4}}-18\alpha^{2}_{3}\alpha_{4}$
$\displaystyle\mp
4\alpha_{3}\alpha_{4}\sqrt{9\alpha^{2}_{3}-12\alpha_{4}}+6\alpha^{2}_{4}\bigg{]}.$
(25)

Hence, $H^{2}$ in given by equation (II.2),

$\displaystyle
H^{2}=\frac{-6j^{2}(\xi+\beta)+3b^{2}(\xi+2\beta)-6\xi\dot{j}+2\Lambda_{\rm
SA}^{\pm}}{6\xi-\omega}.$ (26)

At the end of this subsection, we obtain $r_{\rm SA}$ using Eq. (II.1),

$\displaystyle r_{\rm SA}=1+\frac{H^{2}\omega}{m_{g}^{2}Y_{\rm
SA}^{2\pm}\big{(}\alpha_{3}Y_{\rm SA}^{\pm}-\alpha_{3}-2\big{)}}.$ (27)

Here, it should be expressed that we substituted $\alpha_{4}$ using the
Stueckelberg equation (II.2). About the main result of this subsection, we
presented that this theory consists of self-accelerating solutions with an
effective cosmological constant ie., $\Lambda_{\rm SA}^{\pm}$. Likewise, it is
shown that we do have not any strong coupling, also the theory possesses a
well-behaved self-accelerating solution that can explain the accelerated
expansion of the Universe.

##  III Cosmological Data

It is obvious that the exploration of type I-a supernovae showed the
accelerated expansion of the Universe Copeland:2006wr ; Frieman:2008sn ;
Perlmutter:2003kf ; Yang:2019fjt . In the following, using the Union2
supernovae I-a dataset consisting of 557 SNIa Amanullah:2010vv , we would like
to assess the $F(R,T)$ quasi-dilaton massive gravity theory. Notice that the
main goal is to constrain the theory using the cosmological data. The results
of the SNIa dataset should be represented in terms of $\mu_{\rm obs}$, and
could be compared with the predictions of the model.

$\displaystyle\mu_{\rm th}(z_{i})=5\log_{10}D_{L}(z_{i})+\mu_{0},$ (28)

note that $\mu_{0}=42.38-5\log_{10}\mathcal{H}$ ($\mathcal{H}$ would be the
Hubble constant $H_{0}$ in units of $100\,{\rm km/s/Mpc}$), also we have the
luminosity distance as below,

$\displaystyle D_{L}(z)=(1+z)\int_{0}^{z}\frac{dx}{E(x;p)},$ (29)

moreover, it is clear that $E=H/H_{0}$ and $p$ represents the model
parameters. Notice that $X^{2}$ is given by

$X_{\mu}^{2}(p)=\sum_{i}\frac{[\mu_{\rm obs}(z_{i})-\mu_{\rm
th}(z_{i})]^{2}}{\sigma^{2}(z_{i})},$ (30)

here $\sigma$ corresponds to $1\sigma$ error and the parameter $\mu_{0}$ is a
nuisance parameter and is independent of the data points. We know that
$X_{\mu}^{2}$ in equation (30) make minimize it with respect to $\mu_{0}$
Nesseris:2005ur ; DiPietro:2002cz .

$X_{\mu}^{2}(p)=\tilde{A}-2\mu_{0}\tilde{B}+\mu_{0}^{2}\tilde{C},$ (31)

where

$\displaystyle\tilde{A}(p)=\sum_{i}\frac{[\mu_{\rm obs}(z_{i})-\mu_{\rm
th}(z_{i};\mu_{0}=0,p)]^{2}}{\sigma_{\mu_{\rm obs}}^{2}(z_{i})},$
$\displaystyle\tilde{B}(p)=\sum_{i}\frac{\mu_{\rm obs}(z_{i})-\mu_{\rm
th}(z_{i};\mu_{0}=0,p)}{\sigma_{\mu_{\rm obs}}^{2}(z_{i})},$
$\displaystyle\tilde{C}=\sum_{i}\frac{1}{\sigma_{\mu_{\rm obs}}^{2}(z_{i})}.$
(32)

It should be explained that for $\mu_{0}=\frac{\tilde{B}}{\tilde{C}}$,
equation (31) has a minimum at

$\tilde{X}_{\mu}^{2}(p)=\tilde{A}(p)-\frac{\tilde{B}^{2}(p)}{\tilde{C}}.$ (33)

As it is clear that $X_{\mu,{\rm min}}^{2}=\tilde{X}_{\mu,{\rm min}}^{2}$, it
could be considered minimizing $\tilde{X}_{\mu}^{2}$ which is independent of
$\mu_{0}$. We should pay attention that the best-fit model parameters are
determined by minimizing $X^{2}=\tilde{X}_{\mu}^{2}$. Meanwhile, we know that
the corresponding $\mathcal{H}$ should be determined by
$\mu_{0}=\frac{\tilde{B}}{\tilde{C}}$ for the best-fit parameters.

By considering equation (26), the change of variables $a=\frac{1}{1+z}$ and
$\frac{d}{dt}=-H(z+1)\frac{d}{dz}$, the dimensionless Hubble parameter for
this case should be written. Thus, the solution to the asymptotic state in
small red-shifts should be given as,

$\displaystyle
E=\frac{H(z)}{H_{0}}=1-\frac{\mathcal{M}z}{2(\mathcal{M}+\mathcal{D})}+\frac{\big{(}3\mathcal{M}^{2}+4\mathcal{M}\mathcal{D}\big{)}z^{2}}{8(\mathcal{M}+\mathcal{D})^{2}}+.....,$

where

$\displaystyle\mathcal{M}=\frac{-6j^{2}(\xi+\beta)-6\xi\dot{j}}{6\xi-\omega},$
$\displaystyle\mathcal{D}=\frac{3b^{2}(\xi+2\beta)+2\Lambda_{\rm
SA}^{\pm}}{6\xi-\omega}.$ (35)

###  III.1 Union2 I-a

In this step, we plotted the $X^{2}$ and likelihoods as functions of
parameters $\mathcal{M}$ and $\mathcal{D}$. According to calculations, the
best fit has $X_{\rm min}^{2}=546.947$, and the best-fit parameters are,

$\displaystyle\mathcal{M}=$ $\displaystyle-0.037,$ (36)
$\displaystyle\mathcal{D}=$ $\displaystyle 0.070.$ (37)

Consequently, it should be mentioned the best fit for the $F(R,T)$ quasi-
dilaton massive gravity is $\mathcal{H}=0.693$.

Figure 1: The distance modulus diagram for the best fit (red solid line) of
the parameters of the $F(R,T)$ quasi-dilaton massive gravity theory in
comparison with the 557 Union2 SNIa data points (blue dots). Figure 2: The
$X^{2}$ functions of parameter $\mathcal{M}$ for the $F(R,T)$ quasi-dilaton
massive gravity. Figure 3: The likelihood as functions of parameter
$\mathcal{M}$ for the $F(R,T)$ quasi-dilaton massive gravity. Figure 4: The
$X^{2}$ functions of parameter $\mathcal{D}$ for the $F(R,T)$ quasi-dilaton
massive gravity. Figure 5: The likelihood as functions of parameter
$\mathcal{D}$ for the $F(R,T)$ quasi-dilaton massive gravity.

According to Figs (1-5), the result of fitting the $F(R,T)$ quasi-dilaton
massive gravity theory with the cosmological data gives us the best values of
model parameters.

Figure 6: The best-fit diagram for the luminosity distance $D_{L}$ with
respect to velocity $V$ for the $F(R,T)$ quasi-dilaton massive gravity.

In Fig (6), we plotted the velocity $V$ versus luminosity distance $D_{L}$ for
the $F(R,T)$ quasi-dilaton massive gravity theory. For calculating the
velocity from redshift, one should use $V=c-\frac{2c}{(z+1)^{2}+1}$.

To test the viability of the $F(R,T)$ quasi-dilaton massive gravity theory, we
utilize the Union2 supernovae I-a dataset consisting of 557 SNIa. We employ
the Bayesian statistics method to constrain the model parameters using the
SNIa data. Specifically, we plot the $X^{2}$ and likelihood functions as
functions of the parameters $\mathcal{M}$ and $\mathcal{D}$, and determine the
best-fit values of these parameters by minimizing $X^{2}$.

Our analysis reveals that the best-fit values of the model parameters are
$\mathcal{M}=-0.037$ and $\mathcal{D}=0.070$, with a minimum $X^{2}$ value of
$546.947$. Using these parameter values, we calculate that the best fit of
$\mathcal{H}=0.693$ for the theory. In Fig. (1) we show the distance modulus
diagram $\mu$ versus redshift $z$ and the best fit for the $F(R,T)$ quasi-
dilaton massive gravity theory using supernova I-a data. Furthermore, we plot
the velocity $V$ versus luminosity distance $D_{L}$ for the $F(R,T)$ quasi-
dilaton massive gravity theory, and obtain a best-fit diagram in comparison
with supernova I-a data.

Our results provide strong evidence in favor of the $F(R,T)$ quasi-dilaton
massive gravity theory as a viable explanation for the accelerated expansion
of the Universe. In fact, these results exhibit good agreement between the
theoretical predictions and observational data. These findings lend support to
the idea that the $F(R,T)$ quasi-dilaton massive gravity theory can
successfully describe late-time cosmic acceleration. The derived values of the
model parameters and the Hubble constant provide further credence to this
theory. Future investigations involving larger and more diverse datasets will
help to reinforce or challenge these conclusions, ultimately contributing to a
deeper understanding of the fundamental laws governing the behavior of the
Universe.

##  IV Perturbations Analysis

In this section, we demonstrate the perturbation analysis of $F(R,T)$ quasi-
dilaton massive gravity. It is worth noting that the significance of such
analysis is that the stability conditions of the solutions can be determined.
As we would be interested in quadratic perturbations, we should expand the
physical metric $g_{\mu\nu}$, the vierbein and the scalar field in terms of
small fluctuations $\delta g_{\mu\nu}$, $\delta e^{A}_{\mu}$ and
$\delta\sigma$ around the background solution $g_{\mu\nu}^{(0)}$ and $\delta
e^{A(0)}_{\mu}$:

$g_{\mu\nu}=g_{\mu\nu}^{(0)}+\delta g_{\mu\nu},$ (38)
$e^{A}_{\mu}=e^{A(0)}_{\mu}+\delta e^{A}_{\mu},$ (39)
$\sigma=\sigma^{(0)}+\delta\sigma.$ (40)

Note that one can keep all terms up to quadratic order. We consider tensor
perturbations around background $\delta g_{ij}=a^{2}h_{ij}^{TT}$. Also, the
tensor perturbations are transverse $\partial^{i}h_{ij}=0$, and traceless
$h_{i}^{~{}i}=0$. We should pay attention that all perturbations are functions
of time and space, and they are consistent with the transformations under
spatial rotations Kahniashvili:2014wua ; DeFelice:2013tsa .

Besides, one can write the actions expanded in Fourier plane waves, namely
$\vec{\nabla}^{2}\rightarrow-k^{2}$, $d^{3}x\rightarrow d^{3}k$. Also, the
spatial indices are raised and lowered by $\delta^{ij}$ and $\delta_{ij}$.
Hence, as all calculations are done in the unitary gauge, one does not need to
specify gauge-invariant combinations Gumrukcuoglu:2013nza .

###  IV.1 Dispersion Relation of GWs

Notice that we obtain the perturbed action at the second order separately for
the different parts. The $F(R,T)$ part is given as

$\displaystyle S^{(2)}_{\rm F(R,T)}=\frac{1}{8}\int
d^{3}kdta^{3}N\Bigg{\\{}\xi\bigg{[}\frac{\dot{h}_{ij}\dot{h}^{ij}}{N^{2}}-\Big{(}\frac{k^{2}}{a^{2}}+\frac{4\dot{H}}{N}$
$\displaystyle+6H^{2}+u\Big{)}h^{ij}h_{ij}\bigg{]}+\beta\bigg{[}\frac{\dot{h}_{ij}\dot{h}^{ij}}{N^{2}}-\Big{(}\frac{k^{2}}{a^{2}}$
$\displaystyle+3H^{2}+v\Big{)}h^{ij}h_{ij}\bigg{]}\Bigg{\\}}.$

Meanwhile, the quasi-dilaton part of the perturbed action reads as

$S^{(2)}_{\rm Quasi-dilaton}=-\frac{1}{8}\int
d^{3}kdta^{3}N\Bigg{[}\left(\frac{\omega}{N^{2}}\dot{\sigma}^{2}\right)h^{ij}h_{ij}\Bigg{]},$
(42)

also, the massive gravity part becomes

$\displaystyle S^{(2)}_{\rm massive}=\frac{1}{8}\int
d^{3}kdta^{3}Nm_{g}^{2}\bigg{[}(\alpha_{3}+\alpha_{4})rY^{3}$
$\displaystyle-(1+2\alpha_{3}+\alpha_{4})(1+3r)Y^{2}$
$\displaystyle+(3+3\alpha_{3}+\alpha_{4})(3+2r)Y$
$\displaystyle-2(6+4\alpha_{3}+\alpha_{4})\bigg{]}h^{ij}h_{ij}.$ (43)

In the end, by considering the above terms, the second-order perturbed action
for tensor perturbations $S^{(2)}_{\rm total}=S^{(2)}_{\rm
F(R,T)}+S^{(2)}_{\rm Quasi-dilaton}+S^{(2)}_{\rm massive}$, becomes

$\displaystyle S^{(2)}_{\rm total}=\frac{1}{8}\int
d^{3}k\,dt\,a^{3}N\Bigg{\\{}\frac{\dot{h}^{ij}\dot{h}_{ij}}{N^{2}}(\xi+\beta)$
$\displaystyle-\bigg{[}\frac{k^{2}}{a^{2}}(\xi+\beta)+M_{\rm
GW}^{2}\bigg{]}h^{ij}h_{ij}\Bigg{\\}},$ (44)

where

$\displaystyle M^{2}_{\rm
GW}=\bigg{(}\frac{4\dot{H}}{N}+6H^{2}+u\bigg{)}\xi+\bigg{(}3H^{2}+v\bigg{)}\beta$
$\displaystyle+\frac{\omega}{N^{2}}\dot{\sigma}^{2}+\varpi,$

and with

$\displaystyle\varpi=\frac{m_{g}^{2}}{(Y_{\rm SA}^{\pm}-1)}\Bigg{\\{}Y_{\rm
SA}^{\pm}\bigg{[}18+8\alpha_{3}+Y_{\rm SA}^{\pm}\bigg{(}2\alpha_{3}Y_{\rm
SA}^{\pm}+Y_{\rm SA}^{\pm}-r_{\rm SA}\big{(}3(\alpha_{3}+2)$
$\displaystyle+Y_{\rm SA}^{\pm}(\alpha_{3}Y_{\rm
SA}^{\pm}-4\alpha_{3}-3)\big{)}-8\alpha_{3}-10\bigg{)}\bigg{]}-2(\alpha_{3}+3)\Bigg{\\}}.$

The last relation is obtained using (23) to substitute $\alpha_{3}$ and
$\alpha_{4}$. It is obvious the equation (IV.1) determines the modified
dispersion relation of gravitational waves in $F(R,T)$ quasi-dilaton massive
gravity. In order to guarantee the stability of long-wavelength gravitational
waves, the mass square of gravitational waves should be positive i.e., $M_{\rm
GW}^{2}>0$. But, if the mass square would be negative, it is unstable and can
be considered as tachyonic. So, the instability could take the age of the
Universe to develop by considering the mass of the tachyon in the order of the
Hubble scale.

Notice that the main result of this section is the modified dispersion
relation of gravitational waves. They showed the propagation of gravitational
perturbations in the FLRW cosmology in the $F(R,T)$ quasi-dilaton massive
gravity. It is worth mentioning that cosmological events such as gravitational
wave observations can be used for testing modified propagation. The
modification introduced in this theory will lead to additional effects on the
phase evolution of the gravitational waveform. Moreover, these extra
contributions could be detected with accurate matched-filtering techniques in
the data analysis. It is interesting to note that Nishizawa and Arai
investigated the modified propagation of gravitational waves in the
cosmological context Nishizawa:2017nef ; Arai:2017hxj ; Nishizawa:2019rra .
The result of this paper also demonstrates theory-specific analysis in
$F(R,T)$ quasi-dilaton massive gravity which is complementary to their work.
At the end of this part, I should like to point out that future space-based
gravitational-wave detectors which are sensitive, provide the testing of
crucial properties of gravitation at different wavelengths.

##  V Conclusions

In this study, we have successfully extended the dRGT massive gravity theory
to develop the innovative $F(R,T)$ quasi-dilaton massive gravity theory. Our
exhaustive investigation has led to the derivation of the complete set of
field equations for a Friedmann-Lemaître-Robertson-Walker (FLRW) background,
paving the way for a detailed examination of the late-time acceleration of the
Universe.

To validate the predictions of the $F(R,T)$ quasi-dilaton massive gravity
theory, we embarked on a rigorous evaluation by comparing its forecasts with
the Union2 Supernovae Type Ia (SNIa) dataset, comprising 557 observations. Our
analysis revealed an exceptional agreement between the theoretical framework
and the empirical data, strongly supporting the potential of the model in
accurately describing the late-time cosmic acceleration. Furthermore, we were
able to determine the best-fit parameter, yielding $h=0.693$, via a thorough
fitting process.

Motivated by the quest to deepen our understanding of gravitational phenomena,
we conducted a comprehensive exploration of tensor perturbations. This
endeavor allowed us to unveil the underlying principles governing the mass of
the graviton within the context of the $F(R,T)$ quasi-dilaton massive gravity
theory. Our findings facilitated the derivation of the dispersion relation for
gravitational waves and an examination of the propagation properties of
gravitational perturbations in FLRW cosmology. These critical investigations
hold significant implications for the current era of gravitational wave
astronomy, where novel approaches to understanding the behavior of gravity are
being actively pursued.

As we mentioned before, it is important to highlight that our work complements
previous studies, such as those by Nishizawa and Arai, who investigated the
modified propagation of gravitational waves in the cosmological context. Our
research provides a theory-specific analysis in $F(R,T)$ quasi-dilaton massive
gravity, offering valuable insights into the nature of gravity in diverse
environments.

As we conclude this paper, it is essential to acknowledge the profound impact
that future space-based gravitational-wave detectors will have on our
understanding of the universe. With their heightened sensitivity, they will
enable the testing of crucial properties of gravitation at various
wavelengths, further expanding our knowledge of the cosmos.

## Acknowledgements

This work is based upon research funded by the University of Tabriz, Iran
National Science Foundation (INSF), and Iran National Elites Foundation
(INEF), under project No. 4014244.

The authors are grateful to A. Emir Gumrukcuolu for useful comments. Also, the
authors are grateful to Nishant Agarwal for notes and codes related to tensor
perturbations.

## References

  * (1) A. G. Riess et al. [Supernova Search Team], Astron. J. 116, 1009-1038 (1998) doi:10.1086/300499 [arXiv:astro-ph/9805201 [astro-ph]].
  * (2) D. N. Spergel et al. [WMAP], Astrophys. J. Suppl. 148, 175-194 (2003) doi:10.1086/377226 [arXiv:astro-ph/0302209 [astro-ph]].
  * (3) E. Komatsu et al. [WMAP], Astrophys. J. Suppl. 148, 119-134 (2003) doi:10.1086/377220 [arXiv:astro-ph/0302223 [astro-ph]].
  * (4) R. R. Caldwell and M. Doran, Phys. Rev. D 69, 103517 (2004) doi:10.1103/PhysRevD.69.103517 [arXiv:astro-ph/0305334 [astro-ph]].
  * (5) D. J. Eisenstein et al. [SDSS], Astrophys. J. 633, 560-574 (2005) doi:10.1086/466512 [arXiv:astro-ph/0501171 [astro-ph]].
  * (6) W. J. Percival et al. [SDSS], Mon. Not. Roy. Astron. Soc. 401, 2148-2168 (2010) doi:10.1111/j.1365-2966.2009.15812.x [arXiv:0907.1660 [astro-ph.CO]].
  * (7) M. Sullivan et al. [SNLS], Astrophys. J. 737, 102 (2011) doi:10.1088/0004-637X/737/2/102 [arXiv:1104.1444 [astro-ph.CO]].
  * (8) C. M. Will, Living Rev. Rel. 4, 4 (2001) doi:10.12942/lrr-2001-4 [arXiv:gr-qc/0103036 [gr-qc]].
  * (9) C. M. Will, Living Rev. Rel. 9, 3 (2006) doi:10.12942/lrr-2006-3 [arXiv:gr-qc/0510072 [gr-qc]].
  * (10) C. M. Will, Living Rev. Rel. 17, 4 (2014) doi:10.12942/lrr-2014-4 [arXiv:1403.7377 [gr-qc]].
  * (11) E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753-1936 (2006) doi:10.1142/S021827180600942X [arXiv:hep-th/0603057 [hep-th]].
  * (12) S. Weinberg, Rev. Mod. Phys. 61, 1-23 (1989) doi:10.1103/RevModPhys.61.1
  * (13) P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559-606 (2003) doi:10.1103/RevModPhys.75.559 [arXiv:astro-ph/0207347 [astro-ph]].
  * (14) S. M. Carroll, A. De Felice, V. Duvvuri, D. A. Easson, M. Trodden and M. S. Turner, Phys. Rev. D 71, 063513 (2005) doi:10.1103/PhysRevD.71.063513 [arXiv:astro-ph/0410031 [astro-ph]].
  * (15) T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis, Phys. Rept. 513, 1-189 (2012) doi:10.1016/j.physrep.2012.01.001 [arXiv:1106.2476 [astro-ph.CO]].
  * (16) T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys. 82, 451-497 (2010) doi:10.1103/RevModPhys.82.451 [arXiv:0805.1726 [gr-qc]].
  * (17) S. Nojiri and S. D. Odintsov, eConf C0602061, 06 (2006) doi:10.1142/S0219887807001928 [arXiv:hep-th/0601213 [hep-th]].
  * (18) K. Bamba, S. Capozziello, S. Nojiri and S. D. Odintsov, Astrophys. Space Sci. 342, 155-228 (2012) doi:10.1007/s10509-012-1181-8 [arXiv:1205.3421 [gr-qc]].
  * (19) S. Nojiri and S. D. Odintsov, Phys. Rept. 505, 59-144 (2011) doi:10.1016/j.physrep.2011.04.001 [arXiv:1011.0544 [gr-qc]].
  * (20) A. A. Starobinsky, Phys. Lett. B 91, 99-102 (1980) doi:10.1016/0370-2693(80)90670-X
  * (21) S. Nojiri and S. D. Odintsov, Phys. Lett. B 631, 1-6 (2005) doi:10.1016/j.physletb.2005.10.010 [arXiv:hep-th/0508049 [hep-th]].
  * (22) S. Bahamonde, C. G. Böhmer and M. Wright, Phys. Rev. D 92, no.10, 104042 (2015) doi:10.1103/PhysRevD.92.104042 [arXiv:1508.05120 [gr-qc]].
  * (23) S. Weinberg, Phys. Rev. 138, B988-B1002 (1965) doi:10.1103/PhysRev.138.B988
  * (24) C. de Rham, Living Rev. Rel. 17, 7 (2014) doi:10.12942/lrr-2014-7 [arXiv:1401.4173 [hep-th]].
  * (25) C. de Rham, G. Gabadadze and A. J. Tolley, Phys. Rev. Lett. 106, 231101 (2011) doi:10.1103/PhysRevLett.106.231101 [arXiv:1011.1232 [hep-th]].
  * (26) K. Hinterbichler, Rev. Mod. Phys. 84, 671-710 (2012) doi:10.1103/RevModPhys.84.671 [arXiv:1105.3735 [hep-th]].
  * (27) C. de Rham and G. Gabadadze, Phys. Rev. D 82, 044020 (2010) doi:10.1103/PhysRevD.82.044020 [arXiv:1007.0443 [hep-th]].
  * (28) S. F. Hassan and R. A. Rosen, Phys. Rev. Lett. 108, 041101 (2012) doi:10.1103/PhysRevLett.108.041101 [arXiv:1106.3344 [hep-th]].
  * (29) S. F. Hassan and R. A. Rosen, JHEP 02, 126 (2012) doi:10.1007/JHEP02(2012)126 [arXiv:1109.3515 [hep-th]].
  * (30) M. Fierz and W. Pauli, Proc. Roy. Soc. Lond. A 173, 211-232 (1939) doi:10.1098/rspa.1939.0140
  * (31) V. I. Zakharov, JETP Lett. 12, 312 (1970)
  * (32) H. van Dam and M. J. G. Veltman, Nucl. Phys. B 22, 397-411 (1970) doi:10.1016/0550-3213(70)90416-5
  * (33) A. I. Vainshtein, Phys. Lett. B 39, 393-394 (1972) doi:10.1016/0370-2693(72)90147-5
  * (34) D. G. Boulware and S. Deser, Phys. Rev. D 6, 3368-3382 (1972) doi:10.1103/PhysRevD.6.3368
  * (35) N. Arkani-Hamed, H. Georgi and M. D. Schwartz, Annals Phys. 305, 96-118 (2003) doi:10.1016/S0003-4916(03)00068-X [arXiv:hep-th/0210184 [hep-th]].
  * (36) P. Creminelli, A. Nicolis, M. Papucci and E. Trincherini, JHEP 09, 003 (2005) doi:10.1088/1126-6708/2005/09/003 [arXiv:hep-th/0505147 [hep-th]].
  * (37) A. De Felice, A. E. Gumrukcuoglu and S. Mukohyama, Phys. Rev. Lett. 109, 171101 (2012) doi:10.1103/PhysRevLett.109.171101 [arXiv:1206.2080 [hep-th]].
  * (38) A. E. Gumrukcuoglu, C. Lin and S. Mukohyama, JCAP 03, 006 (2012) doi:10.1088/1475-7516/2012/03/006 [arXiv:1111.4107 [hep-th]].
  * (39) G. D’Amico, C. de Rham, S. Dubovsky, G. Gabadadze, D. Pirtskhalava and A. J. Tolley, Phys. Rev. D 84, 124046 (2011) doi:10.1103/PhysRevD.84.124046 [arXiv:1108.5231 [hep-th]].
  * (40) A. E. Gumrukcuoglu, C. Lin and S. Mukohyama, Phys. Lett. B 717, 295-298 (2012) doi:10.1016/j.physletb.2012.09.049 [arXiv:1206.2723 [hep-th]].
  * (41) G. D’Amico, G. Gabadadze, L. Hui and D. Pirtskhalava, Phys. Rev. D 87, 064037 (2013) doi:10.1103/PhysRevD.87.064037 [arXiv:1206.4253 [hep-th]].
  * (42) S. Mukohyama, JCAP 12, 011 (2014) doi:10.1088/1475-7516/2014/12/011 [arXiv:1410.1996 [hep-th]].
  * (43) Q. G. Huang, Y. S. Piao and S. Y. Zhou, Phys. Rev. D 86, 124014 (2012) doi:10.1103/PhysRevD.86.124014 [arXiv:1206.5678 [hep-th]].
  * (44) A. De Felice and S. Mukohyama, Phys. Lett. B 728, 622-625 (2014) doi:10.1016/j.physletb.2013.12.041 [arXiv:1306.5502 [hep-th]].
  * (45) A. E. Gümrükçüoğlu, K. Hinterbichler, C. Lin, S. Mukohyama and M. Trodden, Phys. Rev. D 88, no.2, 024023 (2013) doi:10.1103/PhysRevD.88.024023 [arXiv:1304.0449 [hep-th]].
  * (46) D. Langlois, S. Mukohyama, R. Namba and A. Naruko, Class. Quant. Grav. 31, 175003 (2014) doi:10.1088/0264-9381/31/17/175003 [arXiv:1405.0358 [hep-th]].
  * (47) C. de Rham, L. Heisenberg and R. H. Ribeiro, Class. Quant. Grav. 32, 035022 (2015) doi:10.1088/0264-9381/32/3/035022 [arXiv:1408.1678 [hep-th]].
  * (48) M. Kenna-Allison, A. E. Gümrükçüoǧlu and K. Koyama, Phys. Rev. D 99, no.10, 104032 (2019) doi:10.1103/PhysRevD.99.104032 [arXiv:1812.05496 [hep-th]].
  * (49) M. Kenna-Allison, A. E. Gümrükçüoglu and K. Koyama, Phys. Rev. D 101, no.8, 084014 (2020) doi:10.1103/PhysRevD.101.084014 [arXiv:1912.08560 [hep-th]].
  * (50) A. E. Gumrukcuoglu, R. Kimura and K. Koyama, Phys. Rev. D 101, no.12, 124021 (2020) doi:10.1103/PhysRevD.101.124021 [arXiv:2003.11831 [gr-qc]].
  * (51) A. R. Akbarieh, S. Kazempour and L. Shao, Phys. Rev. D 103, 123518 (2021) doi:10.1103/PhysRevD.103.123518 [arXiv:2105.03744 [gr-qc]].
  * (52) S. M. Aslmarand, A. R. Akbarieh, Y. Izadi, S. Kazempour and L. Shao, Phys. Rev. D 104, no.8, 083543 (2021) doi:10.1103/PhysRevD.104.083543 [arXiv:2107.09919 [gr-qc]].
  * (53) A. R. Akbarieh, S. Kazempour and L. Shao, Phys. Rev. D 105, no.2, 023501 (2022) doi:10.1103/PhysRevD.105.023501 [arXiv:2203.00901 [gr-qc]].
  * (54) S. Kazempour and A. R. Akbarieh, Phys. Rev. D 105, no.12, 123515 (2022) doi:10.1103/PhysRevD.105.123515 [arXiv:2204.05595 [gr-qc]].
  * (55) S. Kazempour, A. R. Akbarieh, H. Motavalli and L. Shao, Phys. Rev. D 106, no.2, 023508 (2022) doi:10.1103/PhysRevD.106.023508 [arXiv:2205.10863 [gr-qc]].
  * (56) S. Kazempour, A. R. Akbarieh and E. N. Saridakis, Phys. Rev. D 106, no.10, 103502 (2022) doi:10.1103/PhysRevD.106.103502 [arXiv:2207.13936 [gr-qc]].
  * (57) R. Myrzakulov, Gen. Rel. Grav. 44, 3059-3080 (2012) doi:10.1007/s10714-012-1439-z [arXiv:1008.4486 [physics.gen-ph]].
  * (58) R. Myrzakulov, Eur. Phys. J. C 72, 2203 (2012) doi:10.1140/epjc/s10052-012-2203-y [arXiv:1207.1039 [gr-qc]].
  * (59) T. Harko, F. S. N. Lobo, S. Nojiri and S. D. Odintsov, Phys. Rev. D 84, 024020 (2011) doi:10.1103/PhysRevD.84.024020 [arXiv:1104.2669 [gr-qc]].
  * (60) N. Godani, Int. J. Geom. Meth. Mod. Phys. 16, no.02, 1950024 (2018) doi:10.1142/S0219887819500245
  * (61) P. S. Debnath, Int. J. Geom. Meth. Mod. Phys. 16, no.01, 1950005 (2018) doi:10.1142/S0219887819500051 [arXiv:1907.02238 [gr-qc]].
  * (62) P. V. Tretyakov, Eur. Phys. J. C 78, no.11, 896 (2018) doi:10.1140/epjc/s10052-018-6367-y [arXiv:1810.11313 [gr-qc]].
  * (63) P. Sarkar, Ashmita and P. K. Das, Phys. Dark Univ. 40, 101190 (2023) doi:10.1016/j.dark.2023.101190 [arXiv:2212.01782 [gr-qc]].
  * (64) N. R. Bertini and H. Velten, Phys. Rev. D 107, no.12, 124005 (2023) doi:10.1103/PhysRevD.107.124005 [arXiv:2303.09699 [gr-qc]].
  * (65) F. Mueller-Hoissen and J. Nitsch, Phys. Rev. D 28, 718-728 (1983) doi:10.1103/PhysRevD.28.718
  * (66) M. A. Scheel, S. L. Shapiro and S. A. Teukolsky, Phys. Rev. D 51, 4208-4235 (1995) doi:10.1103/PhysRevD.51.4208 [arXiv:gr-qc/9411025 [gr-qc]].
  * (67) T. Christodoulakis, N. Dimakis and P. A. Terzis, J. Phys. A 47, 095202 (2014) doi:10.1088/1751-8113/47/9/095202 [arXiv:1304.4359 [gr-qc]].
  * (68) C. Grosse-Knetter and R. Kogerler, Phys. Rev. D 48, 2865-2876 (1993) doi:10.1103/PhysRevD.48.2865 [arXiv:hep-ph/9212268 [hep-ph]].
  * (69) J. Frieman, M. Turner and D. Huterer, Ann. Rev. Astron. Astrophys. 46, 385-432 (2008) doi:10.1146/annurev.astro.46.060407.145243 [arXiv:0803.0982 [astro-ph]].
  * (70) S. Perlmutter and B. P. Schmidt, Lect. Notes Phys. 598, 195-217 (2003) doi:10.1007/3-540-45863-8_11 [arXiv:astro-ph/0303428 [astro-ph]].
  * (71) Y. Yang and Y. Gong, JCAP 06, 059 (2020) doi:10.1088/1475-7516/2020/06/059 [arXiv:1912.07375 [astro-ph.CO]].
  * (72) R. Amanullah, C. Lidman, D. Rubin, G. Aldering, P. Astier, K. Barbary, M. S. Burns, A. Conley, K. S. Dawson and S. E. Deustua, et al. Astrophys. J. 716, 712-738 (2010) doi:10.1088/0004-637X/716/1/712 [arXiv:1004.1711 [astro-ph.CO]].
  * (73) S. Nesseris and L. Perivolaropoulos, Phys. Rev. D 72, 123519 (2005) doi:10.1103/PhysRevD.72.123519 [arXiv:astro-ph/0511040 [astro-ph]].
  * (74) E. Di Pietro and J. F. Claeskens, Mon. Not. Roy. Astron. Soc. 341, 1299 (2003) doi:10.1046/j.1365-8711.2003.06508.x [arXiv:astro-ph/0207332 [astro-ph]].
  * (75) T. Kahniashvili, A. Kar, G. Lavrelashvili, N. Agarwal, L. Heisenberg and A. Kosowsky, Phys. Rev. D 91, no.4, 041301 (2015) [erratum: Phys. Rev. D 100, no.8, 089902 (2019)] doi:10.1103/PhysRevD.91.041301 [arXiv:1412.4300 [astro-ph.CO]].
  * (76) A. Nishizawa, Phys. Rev. D 97, no.10, 104037 (2018) doi:10.1103/PhysRevD.97.104037 [arXiv:1710.04825 [gr-qc]].
  * (77) S. Arai and A. Nishizawa, Phys. Rev. D 97, no.10, 104038 (2018) doi:10.1103/PhysRevD.97.104038 [arXiv:1711.03776 [gr-qc]].
  * (78) A. Nishizawa and S. Arai, Phys. Rev. D 99, no.10, 104038 (2019) doi:10.1103/PhysRevD.99.104038 [arXiv:1901.08249 [gr-qc]].