# Model selection using time-delay lenses

Fulvio Melia,1 Jun-Jie Wei,2,3 and Xue-Feng Wu2,3
1Department of Physics, The Applied Math Program, and Department of Astronomy,
The University of Arizona, AZ 85721, USA
2Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023,
China
3School of Astronomy and Space Sciences, University of Science and Technology
of China, Hefei 230026, China John Woodruff Simpson Fellow. E-mail:
<EMAIL_ADDRESS>
###### Abstract

The sample of time-delay gravitational lenses appropriate for studying the
geometry of the Universe continues to grow as dedicated campaigns, such as the
Dark Energy Survey, the VST ATLAS survey, and the Large Synoptic Survey
Telescope, complete their census of high-redshift sources. This catalog now
includes hundreds of strong lensing systems, at least 31 of which have
reasonably accurate time delay measurements. In this paper, we use them to
compare the predictions of two competing Friedmann-Lemaître-Robertson-Walker
models: flat $\Lambda$CDM, characterized by two adjustable parameters ($H_{0}$
and $\Omega_{\rm m}$), and the $R_{\rm h}=ct$ universe (with $H_{0}$ as the
single free variable). Over the past decade, the latter has accounted for the
data better than the standard model, most recently the emergence of well-
formed galaxies discovered by JWST at cosmic dawn. Here we show that the
current sample of time-delay lenses favours $R_{\rm h}=ct$ with a likelihood
of $\sim 84\%$ versus $\sim 16\%$ for the standard model. This level of
accuracy will greatly improve as the ongoing surveys uncover many thousands
additional lens systems over the next several years.

###### keywords:

cosmological parameters – cosmology: observations – cosmology: theory –
gravitational lensing: strong – large-scale structure of the Universe

††pagerange: Model selection using time-delay lenses–References††pubyear: 2021

##  1 Introduction

The use of time-delay gravitational lenses to measure the expansion of the
Universe has been considered for almost sixty years, starting with the
original proposal by Refsdal (1964). Light rays originating from a single
quasar source travel through different gravitational potentials on either side
of a foreground lensing galaxy, so their deflection angles and travel times
probe the model-dependent angular diameter distance from the source to the
lens and the lens to the observer (Petters et al., 2001).

Of the hundreds of strong lensing systems already discovered, a small fraction
of them have exhibited measurable time delays between the various images. In
this paper, we assemble 31 of these with reasonably accurate data one can use
for model selection. About half of the lenses have been added to the catalog
only recently from the COSmological MOnitoring of GRAvItational Lenses survey
(COSMOGRAIL; Eigenbrod et al. 2005), the Dark Energy Survey (DES; Banerji et
al. 2008; Schneider 2014, the Large Synoptic Survey Telescope (LSST; Tyson et
al. 2002) project, and the VST ATLAS campaign (Koposov et al., 2014).

It is already well recognized that these strong lensing sources can yield
useful constraint on the parameters in the standard model (Paraficz & Hjorth,
2009; Suyu et al., 2013). For example, they may eventually help to resolve the
current disagreement between the Hubble constant measured at low and high
redshifts (Rathna Kumar et al., 2015; Denzel et al., 2021). But their
usefulness extends well beyond a single application to $\Lambda$CDM. They may
also help with model selection via the comparative testing of quite diverse
cosmologies, as we have already attempted to do using a much smaller sample
(of only 12 time-delay lens systems) available back then (Wei et al., 2014).

Our analysis is motivated in part by the growing tension seen between the
predictions of the standard model and the actual observations. For example,
there now exists a significant disparity between the value of the Hubble
constant, $H_{0}$, inferred locally and its measurement based on the
statistical analysis of anisotropies in the cosmic microwave background (CMB;
Riess et al. 2022).

The Hubble constant ($=67.4\pm 0.5$ km $\rm s^{-1}$ $\rm Mpc^{-1}$) measured
by Planck (Planck Collaboration et al., 2020) is in $5.0\sigma$ tension with
that measured using Type Ia supernovae, calibrated via the Cepheid distance
ladder ($H_{0}=73.04\pm 1.04$ km $\rm s^{-1}$ $\rm Mpc^{-1}$). Other early-
Universe probes, such as clustering and weak lensing, yield results similar to
the CMB (Abbott et al., 2018). Measurements based on the red giant branch
settle on an intermediate value of $H_{0}$ (Freedman et al., 2019; Freedman et
al., 2020). Given that the errors associated with these measurements are
probably realistic (Verde et al., 2019), the implied $5.0\sigma$ disparity in
the expansion of the Universe at low and high redshifts seems to refute the
standard model’s predicted evolution of the Universe.

But the problems with $\Lambda$CDM extend well beyond this well-known
inconsistency with $H_{0}$. A recently published review (Melia, 2023)
highlights at least eight independent areas where the standard model is in
significant conflict with either the observations or fundamental physical
principles. A brief survey of these inconsistencies includes the initial
entropy problem, an unknown classicalization process that must have converted
seed quantum fluctuations into the perturbations responsible for the growth of
structure, the incorrect timeline implied by the early appearance of
supermassive black holes and well-formed galaxies, and the incorrect
prediction of light-element abundances (notably the 7Li anomaly) during big
bang nucleosynthesis in this model.

It is therefore instructive to continue comparing the predictions of
$\Lambda$CDM with those of another Friedmann-Lemaître-Robertson-Walker
cosmology, known as the $R_{\rm h}=ct$ universe (Melia, 2007; Melia &
Shevchuk, 2012). Over the past 15 years, this model has been tested using over
27 different kinds of cosmological data, at both high and low redshifts, and
has been favoured over the standard model by the various model selection
criteria. A recent compilation of these results may be found in Table 2 of
Melia (2018). Its theoretical basis and a more complete examination of its
consistency with the data may be found in Melia (2020)

In this paper, we update our earlier analysis of time-delay lenses with the
improved statistics offered by a sample almost three times larger than before,
and examine whether our previous conclusion favouring $R_{\rm h}=ct$ over
$\Lambda$CDM changes as the sample size grows.

In § 2, we briefly summarize the key steps required to use time-delay lenses
in cosmological testing, and then apply them to our sample of 31 systems in §
3, where we report the outcome of the one-on-one comparison between
$\Lambda$CDM and $R_{\rm h}=ct$. We end with our conclusion in § 4.

Table 1: Time Delay Lenses

System | $\quad z_{l}$ | $\quad z_{s}$ | $\theta_{A}$ | $\theta_{B}$ | $\Delta t=t_{A}-t_{B}$ |  ${\mathcal{R}}_{\rm obs}$ (with $\eta=0.29$) | ${\mathcal{R}}_{R_{\rm h}=ct}$ | ${\mathcal{R}}_{\Lambda{\rm CDM}}$ | Refs.
---|---|---|---|---|---|---|---|---|---
|  |  | (arcsec) | (arcsec) | (days) | (Gpc) | (Gpc) | (Gpc) |
FBQ0951+2635 | 0.26 | 1.246 | $0.886\pm 0.004$ | $0.228\pm 0.008$ | $-16.0\pm 2.0$ | $1.238\pm 0.391$ | 0.947 | 0.927 | 6
SDSS J1442+4055 | 0.284 | 2.593 | $1.385\pm 0.049$ | $0.771\pm 0.028$ | $-25.0\pm 1.5$ | $1.051\pm 0.331$ | 0.893 | 0.889 | 36, 37
RX J1131-1231 | 0.295 | 0.657 | $1.898\pm 0.015$ | $1.922\pm 0.023$ | $1.6\pm 0.7$ | $0.963\pm 1.215$ | 1.509 | 1.469 | 21, 39
PG1115+0.80 | 0.311 | 1.722 | $1.12\pm 0.014$ | $0.95\pm 0.016$ | $-8.3\pm 1.5$ | $1.286\pm 0.467$ | 1.045 | 1.036 | 32, 33
QJ0158-4325 | 0.317 | 1.29 | $0.814\pm 0.038$ | $0.41\pm 0.02$ | $-22.7\pm 3.6$ | $2.491\pm 0.884$ | 1.156 | 1.141 | 21, 40
Q0957+561 | 0.36 | 1.413 | $5.220\pm 0.006$ | $1.036\pm 0.11$ | $-417.09\pm 0.07$ | $0.837\pm 0.243$ | 1.282 | 1.272 | 6, 13, 14
HS0818+1227 | 0.39 | 3.113 | $2.219\pm 0.009$ | $0.615\pm 0.003$ | $-153.8\pm 13.9$ | $1.740\pm 0.529$ | 1.140 | 1.152 | 21
SDSS J0924+0219 | 0.393 | 1.523 | $0.878\pm 0.021$ | $0.977\pm 0.072$ | $2.4\pm 3.8$ | $0.671\pm 1.203$ | 1.368 | 1.363 | 21, 41, 6
SDSS J1620+1203 | 0.398 | 1.158 | $2.277\pm 0.037$ | $0.494\pm 0.01$ | $-171.5\pm 8.7$ | $1.775\pm 0.526$ | 1.567 | 1.552 | 21, 42
DES2325-5229 | 0.400 | 2.74 | $1.23\pm 0.05$ | $1.79\pm 0.05$ | $43.8\pm 4.3$ | $1.322\pm 0.439$ | 1.191 | 1.201 | 24, 46
HE0047-1756 | 0.407 | 1.678 | $0.894\pm 0.043$ | $0.553\pm 0.027$ | $-10.4\pm 3.5$ | $1.071\pm 0.508$ | 1.371 | 1.370 | 21, 24, 38
B1600+434 | 0.414 | 1.589 | $1.14\pm 0.075$ | $0.25\pm 0.074$ | $-51.0\pm 4.0$ | $2.084\pm 0.692$ | 1.422 | 1.421 | 4, 5, 20
J1001+5027 | 0.415 | 1.838 | $1.98\pm 0.08$ | $0.93\pm 0.19$ | $-119.3\pm 3.3$ | $1.972\pm 0.651$ | 1.357 | 1.360 | 30, 31
SDSS J1335+0118 | 0.44 | 1.570 | $1.079\pm 0.031$ | $0.489\pm 0.014$ | $-56.0\pm 5.9$ | $3.004\pm 0.954$ | 1.523 | 1.524 | 21, 48
WG0214-2105 | 0.45 | 3.24 | $1.018\pm 0.225$ | $1.246\pm 0.275$ | $7.5\pm 2.8$ | $0.716\pm 1.193$ | 1.273 | 1.296 | 24, 47
HE0435-1223 | 0.454 | 1.693 | $1.298\pm 0.008$ | $1.168\pm 0.014$ | $-9.0\pm 0.8$ | $1.380\pm 0.451$ | 1.527 | 1.533 | 21, 6
Q0142-100 | 0.491 | 2.73 | $1.855\pm 0.002$ | $0.383\pm 0.005$ | $-97\pm 15.8$ | $1.411\pm 0.469$ | 1.419 | 1.445 | 6, 12, 21
SDSS J1650+4251 | 0.577 | 1.547 | $0.872\pm 0.027$ | $0.357\pm 0.042$ | $-49.5\pm 1.9$ | $3.544\pm 1.083$ | 2.079 | 2.102 | 6, 18
DES J0408-5354 | 0.597 | 2.375 | $3.626\pm 0.342$ | $3.053\pm 0.361$ | $112.1\pm 2.1$ | $1.311\pm 1.199$ | 1.758 | 1.802 | 26, 43
HE2149-2745 | 0.603 | 2.033 | $1.354\pm 0.008$ | $0.344\pm 0.012$ | $-72.6\pm 17.0$ | $1.887\pm 0.704$ | 1.890 | 1.929 | 6, 9, 21, 49
SDSS J1339+1310 | 0.609 | 2.231 | $0.580\pm 0.041$ | $1.234\pm 0.106$ | $47.0\pm 5.5$ | $1.760\pm 0.677$ | 1.835 | 1.879 | 34, 35
SDSS J0832+0404 | 0.659 | 1.116 | $1.56\pm 0.024$ | $0.435\pm 0.008$ | $-125.3\pm 18.2$ | $2.405\pm 0.784$ | 3.469 | 3.494 | 21, 45
B0218+357 | 0.685 | 0.944 | $0.057\pm 0.004$ | $0.280\pm 0.008$ | $+11.3\pm 0.2$ | $6.378\pm 1.892$ | 5.312 | 5.313 | 1, 2, 3, 19
Q1355-2257 | 0.702 | 1.370 | $0.959\pm 0.081$ | $0.267\pm 0.023$ | $-81.5\pm 11.6$ | $4.034\pm 1.500$ | 3.005 | 3.052 | 21, 6
SBS1520+530 | 0.717 | 1.855 | $1.207\pm 0.004$ | $0.386\pm 0.008$ | $-130.0\pm 3.0$ | $4.137\pm 1.204$ | 2.397 | 2.460 | 6, 16
HE1104-1805 | 0.729 | 2.319 | $1.099\pm 0.004$ | $2.095\pm 0.008$ | $152.2\pm 3.0$ | $1.978\pm 0.575$ | 2.150 | 2.223 | 2, 7, 8
SDSS J1515+1511 | 0.742 | 2.054 | $1.676\pm 0.104$ | $0.313\pm 0.019$ | $-210.2\pm 5.6$ | $3.181\pm 1.013$ | 2.338 | 2.411 | 21, 44
SDSS J1206+4332 | 0.748 | 1.789 | $1.870\pm 0.088$ | $1.278\pm 0.097$ | $-111\pm 3$ | $2.435\pm 0.891$ | 2.588 | 2.659 | 17, 23
SBS0909+532 | 0.83 | 1.377 | $0.415\pm 0.126$ | $0.756\pm 0.152$ | $+50.0\pm 3.0$ | $4.890\pm 3.415$ | 4.034 | 4.127 | 6, 15, 22
PKS1830-211 | 0.89 | 2.507 | $0.67\pm 0.08$ | $0.32\pm 0.08$ | $-26.0\pm 5.0$ | $2.837\pm 1.386$ | 2.523 | 2.640 | 10, 11
WFI J2026-4536 | 1.04 | 2.23 | $0.673\pm 0.156$ | $0.801\pm 0.184$ | $18.7\pm 4.8$ | $3.472\pm 6.795$ | 3.290 | 3.464 | 21, 6

References: (1) Carilli et al. (1993); (2) Lehár et al. (2000); (3) Wucknitz
et al. (2004); (4) Jackson et al. (1995); (5) Dai & Kochanek (2005); (6)
Kochanek et al. (2008); (7) Wisotzki et al. (1993); (8) Poindexter et al.
(2007); (9) Burud et al. (2002); (10) Lovell et al. (1998); (11) Meylan et al.
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al. (2003); (15) Dai & Kochanek (2009); (16) Auger et al. (2008); (17)
Paraficz & Hjorth (2009); (18) Vuissoz et al. (2007); (19) Biggs & Browne
(2018); (20) Burud et al. (2000); (21) Millon et al. (2020); (22) Hainline et
al. (2013); (23) Eulaers et al. (2013); (24) Millon et al. (2020); (25)
Koopmans et al. (2003); (26) Courbin et al. (2018); (27) Eulaers & Magain
(2011); (28) Fohlmeister et al. (2008); (29) Bonvin et al. (2019); (30) Rathna
Kumar et al. (2013); (31) Oguri et al. (2005); (32) Bonvin et al. (2018); (33)
Morgan et al. (2008); (34) Goicoechea & Shalyapin (2016); (35) Inada et al.
(2009); (36) Shalyapin & Goicoechea (2019); (37) Sergeyev et al. (2016); (38)
Wisotzki et al. (2004); (39) Sluse et al. (2007); (40) Morgan et al. (2008);
(41) Inada et al. (2003); (42) Kayo et al. (2010); (43) Agnello et al. (2017);
(44) Inada et al. (2014); (45) Oguri et al. (2008); (46) Ostrovski et al.
(2017); (47) Spiniello et al. (2019); (48) Oguri et al. (2004); (49) Rathna
Kumar et al. (2015).

##  2 Strong Lensing

Time-delay gravitational lenses constitute a powerful probe of the underlying
cosmology, but are limited by our imprecise knowledge of the central lens mass
distribution and other possible perturbing lenses along the line-of-sight. One
can simplify the procedure by assembling a subsample of homogeneous systems
for which one may reasonably assume the same lens model to adequately
represent the mass distribution in every case (Oguri et al., 2004). Of course,
we already know the selected sample is not perfectly homogeneous from the few
lens systems that have been modeled independently, suggesting at least some
variation in the lens structure. Taking a statistical approach, as we do here,
however, can still be useful if there is a way to quantify the non-negligible
dispersion arising from the poorly known systematics, which include the non-
uniformity of the lens itself. We discuss how this will be done in § 3. We
shall find via the use of Equation (8) that the dispersion due to this
variation of the lens mass distribution appears to be smaller than $\sim 29\%$
of the measured time-delay distance (Eq. 2). While not ideal, this relatively
modest inhomogeneity does present us with a workable approach we can use for
model selection purposes (see § 3.2).

Let the source be located at angle $\vec{\beta}$. The time delay, $\Delta
t_{i}$, for an image $i$ at angular position $\vec{\theta}_{i}$ is due to the
difference in path length along the deflected and undeflected null geodesics,
as well as the gravitational time dilation incurred by the rays traversing the
gravitational potential, $\Psi(\vec{\theta}_{i})$, of the lens:

$\Delta t_{i}={1+z_{l}\over c}{D_{A}(0,z_{s})\,D_{A}(0,z_{l})\over
D_{A}(z_{l},z_{s})}\left[{1\over
2}(\vec{\theta}_{i}-\vec{\beta})^{2}-\Psi(\vec{\theta}_{i})\right]$ (1)

(Blandford & Narayan, 1986). In this expression, $z_{l}$ and $z_{s}$ are the
lens and source redshifts, respectively, and $D_{A}(z_{1},z_{2})$ is the
angular diameter distance between $z_{1}$ and $z_{2}$. Thus, when the lens
geometry $\vec{\theta}_{i}-\vec{\beta}$ and potential $\Psi$ are known, the
time delay yields the so-called time-delay distance,

${\mathcal{R}}\equiv{D_{A}(0,z_{s})\,D_{A}(0,z_{l})\over
D_{A}(z_{l},z_{s})}\;,$ (2)

which clearly depends on the cosmological model.

One may find a replacement for $\Psi(\vec{\theta}_{i})$ by noting that the
mass distribution in lens spiral and elliptical galaxies can be represented
quite well by a power-law density profile (Rusin et al., 2003), for which

$\Psi(\vec{\theta})={b^{2}\over 3-n}\left({\theta\over b}\right)^{3-n}\;,$ (3)

in terms of the deflection scale, $b$, and index, $n$. The special case with
$n=2$ is the singular isothermal sphere (SIS), with

$b=4\pi{D_{A}(z_{l},z_{s})\sigma^{2}_{\star}\over D_{A}(0,z_{s})}\;,$ (4)

where $\sigma_{\star}$ is the velocity dispersion of the lensing galaxy.
Observations of the galaxy density distributions seem to suggest that $n$ is
often close to the isothermal value, so the SIS is both convenient and
accurate for the majority of lens galaxies (Koopmans et al., 2009). The time
delay between two images at $\vec{\theta_{A}}$ and $\vec{\theta_{B}}$ in such
a lens system is given as

$\Delta t=t_{A}-t_{B}={1+z_{l}\over
2c}\left(\theta_{B}^{2}-\theta_{A}^{2}\right){\mathcal{R}}(z_{l},z_{s})\;,$
(5)

when the velocity dispersion of an SIS is assumed in these expressions. With
this approach, the ‘observed’ time-delay distance is therefore inferred from
the expression

${\mathcal{R}}_{\rm obs}(z_{l},z_{s})={2c\over 1+z_{l}}{\Delta
t\over\left(\theta_{B}^{2}-\theta_{A}^{2}\right)}\;.$ (6)

Of course, the actual velocity dispersion $\sigma_{\star}$ may not match the
analytical value exactly, but these do appear to be very close in the majority
of lensing systems for which $\sigma_{\star}$ has been measured (Treu et al.,
2006). Nevertheless, the number of time-delay lenses for which
$\sigma_{\star}$ is known observationally is too small to make this datum
useful, so all of the analysis we carry out in this paper is based solely on
the use of Equation (5), though we allow for possible variations of the
velocity dispersion from system to system via the introduction of a systematic
error defined in Equation (8).

##  3 Time-delay lenses

###  3.1 The Sample and Methodology

Our sample of 31 time-delay gravitational lenses for which ${\mathcal{R}_{\rm
obs}}$ has been ‘measured’ with Equation (5) is shown in Table 1. All of these
are gravitationally lensed quasars, which show enough variability to permit an
accurate measurement of a time-delay between the various images. Six of them
are quadrupole image systems (RX J1131-1231, PG1115+0.80, WG0214-2105,
HE0435-1223, DES J0408-5354, and WFI J2026-4536). The rest are two-image
systems.

Columns 2–6 display the observational data, while column 7 lists the value of
${\mathcal{R}_{\rm obs}}$, together with its uncertainty
$\sigma_{\mathcal{R}}$. This dispersion is calculated from the statistical
error using the propagation equation

$\sigma_{\rm stat}=\mathcal{R}_{\rm obs}\left[\left({\sigma_{\Delta
t}\over\Delta
t}\right)^{2}+4\left({\theta_{B}\sigma_{\theta_{B}}\over\theta_{B}^{2}-\theta_{A}^{2}}\right)^{2}+4\left({\theta_{A}\sigma_{\theta_{A}}\over\theta_{B}^{2}-\theta_{A}^{2}}\right)^{2}\right]^{1/2}\;,$
(7)

and a second source of (systematic) error, $\sigma_{\rm sys}$, that takes into
account various effects giving rise to the observed scatter of individual
lenses from the assumed pure SIS profile. These include a possible rms
deviation of the lens velocity dispersion, a steepening of the mean mass
density profile compared to that of a pure SIS (Koopmans et al., 2009), and
the non-zero line-of-sight contribution, which is non-zero on average.

Figure 1: Thirty one ${\mathcal{R}}$ measurements, with error bars, compared
to the predictions of $\Lambda$CDM, with optimized parameters:
$H_{0}=85.6^{+7.0}_{-5.9}\;(1\sigma)$ km $\rm s^{-1}$ $\rm Mpc^{-1}$,
$\Omega_{\rm m}=0.50^{+0.34}_{-0.34}\;(1\sigma)$ and
$\eta=0.29^{+0.06}_{-0.05}\;(1\sigma)$ (see Fig. 2). The dashed line indicates
a perfect match between theory and observation. The reduced $\chi^{2}$ for
this fit is $0.971$ ($28$ dof).

According to Cao et al. (2012), $\sigma_{\rm sys}$ may be as large as $\sim
20\%$. Since we don’t know how large this error is a priori, we model it along
with the other variables in our maximum likelihood analysis and write it as

$\sigma_{\rm sys}\equiv\eta\,\mathcal{R}_{\rm obs}\;,$ (8)

in terms of the additional free parameter $\eta$. Its introduction is just a
convenient way to characterize this systematic uncertainty in terms of the
value of ${\mathcal{R}}_{\rm obs}$. In our maximum likelihood estimation, this
additional free parameter is optimized along with the other free parameters
individually for each model being tested. As we shall see below, however,
$\eta$ appears to be quite independent of the cosmology itself, and turns out
to have the value $\eta=0.29^{+0.06}_{-0.05}\;(1\sigma)$ for the sample of 31
lens systems used in this study, consistent with the earlier finding of Cao et
al. (2012). The total uncertainty $\sigma_{\mathcal{R}}$ in
${\mathcal{R}}_{\rm obs}$ is then found by adding $\sigma_{\rm stat}$ and
$\sigma_{\rm sys}$ in quadrature,

$\sigma_{\mathcal{R}}^{2}=\sigma_{\rm stat}^{2}+\sigma_{\rm sys}^{2}\;,$ (9)

and this is the error appearing in column 7 of Table 1.

Columns 8 and 9 in this table display the optimized model predictions for
$R_{\rm h}=ct$ and $\Lambda$CDM, respectively, which we now describe. The
angular-diameter distance in $\Lambda$CDM is a function of several parameters,
including $H_{0}$ and the mass fractions $\Omega_{\rm m}\equiv\rho_{\rm
m}/\rho_{\rm c}$, $\Omega_{\rm de}\equiv\rho_{\rm de}/\rho_{\rm c}$, and
$\Omega_{\rm r}\equiv\rho_{\rm r}/\rho_{\rm c}$, for matter, dark energy and
radiation, respectively, defined as ratios of the critical density $\rho_{\rm
c}\equiv 3c^{2}H_{0}^{2}/8\pi G$. To streamline the optimization of the fit
with this model, we adopt a minimal number of unknown variables, so we assume
that dark energy is a cosmological constant ($\Omega_{\rm
de}=\Omega_{\Lambda}$), and that the spatial curvature constant is zero. In
that case, since the contribution from radiation in the redshift range covered
in Table 1 is insignificant compared to that of the others, we simply put
$\Omega_{\rm m}+\Omega_{\Lambda}=1$. Then, the angular-diameter distance
between redshifts $z_{1}$ and $z_{2}$ ($>z_{1}$) is given by the expression

$D_{A}^{\;\Lambda{\rm CDM}}(z_{1},z_{2})={c\over
H_{0}}{1\over(1+z_{2})}\int_{z_{1}}^{z_{2}}\frac{dz}{\left[\Omega_{\rm
m}(1+z)^{3}+\Omega_{\Lambda}\right]^{1/2}}\;.$ (10)

With this application of the standard model, we thus have three free
parameters in total for the optimization: the cosmological parameters $H_{0}$
and $\Omega_{\rm m}$, and the time-delay lens dispersion parameter $\eta$.

On the other hand, the angular-diameter distance in $R_{\rm h}=ct$ (Melia,
2007; Melia & Shevchuk, 2012; Melia, 2020) depends only on $H_{0}$:

$D_{A}^{\;R_{\rm h}=ct}(z_{1},z_{2})={c\over
H_{0}}{1\over(1+z_{2})}\ln\left({1+z_{2}\over 1+z_{1}}\right)\;.$ (11)

To find the best fit to the data in Table 1, we therefore optimize two free
parameters for this model: $H_{0}$ and $\eta$.

For each model, we optimize the fit by maximizing the joint likelihood
function

$L(\sigma_{\mathcal{R}},\xi)\propto\prod_{i=1}\frac{1}{\sigma_{\mathcal{R}_{i}}}\;\exp\left[-\frac{\left(\mathcal{R}_{\rm\emph{i}}[\xi]-\mathcal{R}_{\rm
obs,\emph{i}}\right)^{2}}{2\sigma_{\mathcal{R}_{i}}^{2}}\right]\;,$ (12)

where ${\mathcal{R}}_{\rm\emph{i}}[\xi]$ is the theoretical prediction of the
time-delay distance between $z_{l,i}$ and $z_{s,i}$ for either $\Lambda$CDM or
$R_{\rm h}=ct$ and the model specific parameters $\xi$, $\mathcal{R}_{\rm
obs,\emph{i}}$ is the measured value, and $\sigma_{\mathcal{R}_{i}}$ is the
dispersion of $\mathcal{R}_{\rm obs,\emph{i}}$ given in Equation (9).

Figure 2: 1D probability distributions and 2D regions with $1\sigma$ and
$2\sigma$ contours for the best-fit $\Lambda$CDM model (see Fig. 1).

We follow this procedure using the data in Table 1 with the restriction
$0.0\leq\Omega_{\rm m}\leq 1.0$, and find that $\Lambda$CDM fits the observed
time-delay distances with a maximum likelihood corresponding to the parameter
values $H_{0}=85.6^{+7.0}_{-5.9}$ ($1\sigma$) km $\rm s^{-1}$ $\rm Mpc^{-1}$,
$\Omega_{\rm m}=0.50^{+0.34}_{-0.34}$ ($1\sigma$) and
$\eta=0.29^{+0.06}_{-0.05}$ ($1\sigma$). The best fit with the $R_{\rm h}=ct$
universe is provided by the optimized parameters $H_{0}=81.3_{-5.4}^{+6.5}$
($1\sigma$) km $\rm s^{-1}$ $\rm Mpc^{-1}$ and $\eta=0.29^{+0.06}_{-0.05}$
($1\sigma$). The $1\sigma$ and $2\sigma$ confidence intervals for these model
are shown in Figures 2 and 4, and the entries in columns 8 and 9 of Table 1
correspond to these best-fit parameters.

These results confirm a feature of the fit with $\Lambda$CDM reported on
several previous occasions (Coe & Moustakas, 2009; Linder, 2011; Treu &
Marshall, 2016). The principal dependence of the time-delay distance (Eq. 2)
is on the Hubble constant, $H_{0}$, to which it is inversely proportional. The
impact of a variation of $\Omega_{\rm m}$ on the ratio of angular-diameter
distances in this expression is much weaker. One therefore needs a much bigger
sample to constrain $\Omega_{\rm m}$ very tightly.

The observed values ${\mathcal{R}}_{\rm obs}$ are displayed in Figures 1 and
3, in comparison with those predicted by the two models. In these plots, a
perfect match would correspond to the straight dashed line. The clustering of
points towards small values of ${\mathcal{R}}$ reflects the preponderance of
lenses at relatively small redshifts. Interestingly, the optimized value of
$\eta$ is the same in both models, suggesting that the systematics are
independent of the cosmology and do indeed represent an intrinsic dispersion
of the lens potential away from a pure SIS. With its inclusion in the overall
error budget, we find that both models fit this sample of 31 time-delay lenses
quite well. The reduced $\chi^{2}$ is $0.971$ ($28$ dof) for $\Lambda$CDM in
Figure 1, and a slightly better $0.937$ ($29$ dof) for $R_{\rm h}=ct$ in
Figure 3.

###  3.2 Model Selection

Since these models formulate the angular-diameter distance in Equations (10)
and (11) differently, however, without the same number of free parameters, a
comparison of the likelihoods to determine which of them is favoured by the
data needs to be based on model selection tools. To compare the evidence for
and against competing models, the use of information criteria has become quite
common in cosmology (see, e.g., Takeuchi 2000; Liddle 2004, 2007; Tan & Biswas
2012; Schwarz 1978). Such a tool may be viewed as an enhanced ‘goodness of
fit’ criterion, extending the usual $\chi^{2}$ diagnostic by including the
number of free parameters in each model. The information criteria favour
models with fewer parameters over those with many, as long as the latter do
not provide a considerably better fit to the data.

Figure 3: Same as Fig. 1, except now for the $R_{\rm h}=ct$ universe, with
optimized parameters: $H_{0}=81.3^{+6.5}_{-5.4}\;(1\sigma)$ km $\rm s^{-1}$
$\rm Mpc^{-1}$ and $\eta=0.29^{+0.06}_{-0.05}\;(1\sigma)$ (see Fig. 4). The
reduced $\chi^{2}$ for this fit is $0.937$ ($29$ dof).

An information criterion provides the relative ranks of two or more competing
models, along with a numerical estimate of confidence that each is the best,
analogous to the likelihood (or posterior probability) in traditional
statistical inference. Information criteria are superior to the latter,
however, in that they can be applied to models that are not ‘nested,’
providing a procedure for comparing candidates that are not specializations of
each other.

An information criterion can be applied after the following kind of regression
is performed. Suppose that corresponding to the values $z_{1},\dots,z_{N}$ of
a free parameter the measured values are $h_{1},\dots,h_{N}$ of a dependent
one, with (known) error bars $\pm\sigma_{1},\dots,\pm\sigma_{N}$. These errors
are assumed to be normally distributed. Now let a model $\mathcal{M}$ predict
values $\hat{h}_{1},\dots,\hat{h}_{N}$, inferred from a formula
$\hat{h}_{i}=\hat{h}_{i}(\vec{\beta})$ involving a parameter vector
$\vec{\beta}$ comprising $f$ unknown variables, i.e.,
$\vec{\beta}=(\beta_{1},\dots,\beta_{f})$. In other words, the data model
$\mathcal{M}$ is in effect a statistical one, of the form

$h_{i}=\hat{h}_{i}({\vec{\beta}})+\sigma_{i}Z_{i}\;,$ (13)

where $Z_{1},\dots,Z_{N}$ are independent standard normal random variables.
Note that in the case of linear regression, $\hat{h}_{i}(\vec{\beta})$ would
be $\sum_{j=1}^{f}X_{ij}\beta_{j}$ for known coefficients $X_{ij}$. Typically,
$X_{ij}=\hat{h}^{(j)}(z_{i})$ for known functions
$\hat{h}^{(1)},\dots,\hat{h}^{(f)}$ of $z$.

Figure 4: Same as Fig. 2, except now for the $R_{\rm h}=ct$ universe, whose
best fit is shown in Fig. 3.

Then for model $\mathcal{M}$, the $\chi^{2}$ goodness of fit to the data is
expressible as

$\chi^{2}=\sum_{i=1}^{N}[h_{i}-\hat{h}_{i}(\vec{\beta})]^{2}/\sigma_{i}^{2}\;,$
(14)

i.e., a (weighted) sum of squared errors, and the $\chi^{2}$ per degree of
freedom,

$\chi^{2}_{\rm{dof}}=\chi^{2}/(N-f)\;.$ (15)

By necessity, we must also have $N>f$. The parameters
$(\beta_{1},\dots,\beta_{f})$ are selected to minimize the $\chi^{2}$,
producing a best fit to the data.

For the data we use in this paper, the sample is large (i.e., $>20$), so the
most appropriate one to use is the Bayesian Information Criterion (BIC;
Schwarz 1978), which tests the statistical performance of the models. It is
defined as

${\rm BIC}\equiv-2\ln L+\left(\ln N\right)f$ (16)

where, as previously noted, $N$ is the number of data points (here $31$) and
$f$ is the number of free parameters (three for $\Lambda$CDM and two for
$R_{\rm h}=ct$).

The BIC is generally considered to be a large-sample ($N\gg 1$) approximation
to the outcome of a conventional Bayesian inference procedure for selecting
the model preferred by the data. Among the available choices being tested, the
model with the lowest BIC score is the one selected by this criterion. Among
its many proponents, Liddle (2007) has argued for the use of BIC in
cosmological model selection, which has now been done to compare several
popular models against $\Lambda$CDM (see, e.g., Shi et al. 2012).

Two statistical models of the data set $(h_{1},\dots,h_{N})$, such as a ‘true’
model $\mathcal{M}_{*}$ and a second model $\mathcal{M}$, may be viewed as
probability density functions (PDF’s) on $\mathbb{R}^{N}$, say
$F_{*}(h_{1},\dots,h_{N})$ and $F(h_{1},\dots,h_{N})$, respectively. In
information theory, the discrepancy of the PDF $F$ from the PDF
$F_{*}$—effectively, a measure of distance—is given by the Kullback–Leibler
formula

$D(\mathcal{M}_{*}\|\mathcal{M})=\int_{\mathbb{R}^{n}}{\rm d}h_{1}\dots\rm
dh_{n}\,F_{*}(h)\,\ln\frac{F_{*}(h)}{F(h)}\,\geq\,0$ (17)

(using the notation that the argument $h$ stands for the entire data set
$[h_{1},\dots,h_{N}]$). The best model $\mathcal{M}$ may be selected from a
set of candidate models by choosing the one with the minimum
$D(\mathcal{M}_{*}\|\mathcal{M})$. The model $\mathcal{M}_{*}$ is not known,
but the $\mathcal{M}$ with parameters optimized by minimizing $\chi^{2}$, is
special. It turns out that the BIC of the fitted model $\mathcal{M}$
approximates $2D(\mathcal{M}_{*}\|\mathcal{M})$, up to an unimportant additive
constant. This is especially truen when $\mathcal{M}_{*}$ is a model of the
same type, with free variables $(\beta_{1}^{*},\dots,\beta_{f}^{*})$.

The quantity ${\rm BIC}/2$ is an unbiased estimator of the distance
$D(\mathcal{M}_{*}\|\mathcal{M})$. This statement is exact for linear
regression, and correct to leading order for non-linear regression. The fitted
model $\mathcal{M}$ does depend on the data set, so both
$D(\mathcal{M}_{*}\|\mathcal{M})$ and $\textrm{BIC}/2$ are random variables.
In the language of probability, a lack of bias translates to them having the
same expectation.

The extent to which the fitted information criteria represent an _accurate_
estimate of $2D(\mathcal{M}_{*}\|\mathcal{M})$ has been investigated
theoretically (see, e.g., Yanagihara & Ohmoto 2005). Their variability has
also been studied by repeatedly comparing $\Lambda$CDM to other cosmological
models (Tan & Biswas, 2012). In the case of BIC, it is known that the
magnitude of the difference $\Delta=\allowbreak{\rm BIC}_{2}-\nobreak{\rm
BIC}_{1}$ provides a numerical assessment of the evidence that model 1 is
preferred over model 2. As a general rule, the evidence is considered to be
weak if $\Delta\la 2$, mildly strong if $\Delta\approx 3$ or $4$, and quite
strong if $\Delta\ga 5$.

One may thus rank the competing models quantitatively as follows. If ${\rm
BIC}_{\alpha}$ characterizes model $\alpha$, the unnormalized confidence that
this model is correct is the ‘Bayes weight’ $\exp(-{\rm BIC_{\alpha}}/2)$. The
likelihood of model $\alpha$ being correct relative to the others is then

$P(\alpha)=\frac{\exp(-{\rm BIC_{\alpha}}/2)}{\sum_{i}\exp(-{{\rm
BIC}_{i}}/2)}\;.$ (18)

The sum in the denominator includes all of the candidates being tested
simultaneously.

The outcome of this analysis shows that $R_{\rm h}=ct$ is preferred over
$\Lambda$CDM with a $-2\ln L=444.73$ versus $444.68$ (or equivalently, with a
${\rm BIC}=451.60$ versus $454.98$), which translates into a probability of
$\sim 84\%$ versus $\sim 16\%$ of being the correct model. To complete the
discussion, we also considered how well the time-delay lens data are fit by
Planck $\Lambda$CDM, i.e., the version of the standard model with its
parameters fixed at the values optimized by Planck (Planck Collaboration et
al., 2020): $H_{0}=67.4$ km $\rm s^{-1}$ $\rm Mpc^{-1}$ and $\Omega_{\rm
m}=0.315$. Not surprisingly, a head-to-head comparison between this model and
$R_{\rm h}=ct$ reveals a much more slanted result, with the latter being
favoured by these data with a probability of $\sim 99.3\%$ versus only $\sim
0.7\%$.

##  4 Conclusion

Our use of a much larger time-delay lens sample ($31$) compared to the set of
12 systems we used in our first attempt to carry out this kind of analysis
(Wei et al., 2014), has refined the parameter search and reduced their
$1\sigma$ errors, but the optimized values themselves have changed very
little. For example, the updated value of $H_{0}$ in the context of
$\Lambda$CDM is now $85.6^{+7.0}_{-5.9}\;(1\sigma)$ km $\rm s^{-1}$ $\rm
Mpc^{-1}$ compared with $=87^{+17}_{-16}\;(1\sigma)$ km $\rm s^{-1}$ $\rm
Mpc^{-1}$. But the intrinsic dispersion (as measured by the parameter $\eta$)
persists, as indicated by its current value,
$\eta=0.29^{+0.06}_{-0.05}\;(1\sigma)$, versus $\eta=0.29^{+0.15}_{-0.09}$
($1\sigma$). As noted earlier, this suggests that the poorly known systematics
are probably due solely to deviations of the lens potential from a pure SIS,
as opposed to any cosmological influence.

Overall, our analysis of the time-delay distances using a larger sample size
has affirmed our earlier conclusion that $R_{\rm h}=ct$ is favoured by these
data over the current standard model. This catalog of lens systems is expected
to grow considerably over the coming years, improving the statistics even
further and motivating a more realistic handling of the lens mass distribution
in order to mitigate the impact of $\sigma_{\rm sys}$ well below the current
limitations.

In our previous work (Wei et al., 2014), we estimated that if the real
cosmology is $\Lambda$CDM, one would need a sample of $\sim 150$ time-delay
lenses to rule out $R_{\rm h}=ct$ at a confidence level of $\sim 99.7\%$,
while $\sim 1000$ lenses would be required to rule out $\Lambda$CDM if the
actual universe were instead $R_{\rm h}=ct$. This difference is due to the
greater flexibility afforded the standard model by its greater number of free
parameters.

Looking forward, the prospects of attaining a sample of this size look quite
promising. Several ongoing and future survey programs include a search for
strong lensing systems in their purview. According to some estimates, the
Large-aperture Synoptic Survey Telescope (LSST) will observe some 8000 lensed
quasars during its 10-year lifetime, about 3000 of which will have well-
measured time-delays (Oguri & Marshall, 2010; Collett, 2015). A subsample
useful for a detailed analysis based on the modelling of individual lenses
would require an accurate characterization of the mass distribution of the
lens galaxy, auxiliary data including high-resolution imaging, and
measurements of the stellar velocity distribution. Such requirements would
reduce the sample of strong gravitational lenses with accurate time-delay
measurements to $\sim 100$. With the statistical approach we have used in this
paper, however, in contrast to individual modelling, the latter requirements
may generally be waived, allowing us to expect a usable sample approaching
1000 or more new (LSST) lenses for future work.

But LSST is not the only survey mission promising to increase the sample size.
Euclid may provide a comparable number (Jee et al., 2016; Laureijs et al.,
2011), and the DESI Legacy Imaging Survey has already started reporting the
discovery of over 1000 new strong lensing candidates (most of which are
galaxy-galaxy pairs, rather than quasar-galaxy systems). Future work is
expected to uncover many thousands more (Huang et al., 2022). The Square
Kilometer Array (SKA) will in addition be able to compile up to $\sim 10^{5}$
lensing systems with $\sim 100\%$ reliability and known source redshifts
(Laureijs et al., 2011). Several hundred new candidates have also been
reported by the (HOLISMOKES) Hyper Suprime-Cam (HSC) survey, with thousands
more expected with continued imaging (Cañameras et al., 2021).

Our expectation is that the requisite number of strong lensing systems with
accurate time-delay measurements to rule out $R_{\rm h}=ct$ if the real
cosmology is $\Lambda$CDM should be attainable in just a few years. Compiling
a sample of $\sim 1000$ such systems to rule out $\Lambda$CDM if the real
cosmology is instead $R_{\rm h}=ct$ will take longer, but is nevertheless
quite feasible during this decade with the combined efforts of the many
ongoing and future survey missions.

## Acknowledgments

We are very grateful to the anonymous referee for their excellent review of
this paper and for making several suggestions to improve its presentation.
This work is partially supported by the National Key Research and Development
Program of China (2022SKA0130100), the National Natural Science Foundation of
China (grant Nos. 11725314 and 12041306), the Key Research Program of Frontier
Sciences (grant No. ZDBS-LY-7014) of Chinese Academy of Sciences, and the
Natural Science Foundation of Jiangsu Province (grant No. BK20221562).

## DATA AVAILABILITY STATEMENT

No new data were generated or analysed in support of this research.

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# Experimental Constraint on Stellar Electron-Capture Rates from the
${}^{88}\text{Sr}(t,{}^{3}\text{He}+\gamma){}^{88}\text{Rb}$ reaction at 115
MeV/u

J. C. Zamora  National Superconducting Cyclotron Laboratory, Michigan State
University, East Lansing, MI 48824, USA  Joint Institute for Nuclear
Astrophysics: CEE, Michigan State University, East Lansing, MI 48824, USA
R.G.T. Zegers  National Superconducting Cyclotron Laboratory, Michigan State
University, East Lansing, MI 48824, USA  Joint Institute for Nuclear
Astrophysics: CEE, Michigan State University, East Lansing, MI 48824, USA
Departament of Physics and Astronomy, Michigan State University, East Lansing,
MI 48824, USA  Sam M. Austin  National Superconducting Cyclotron Laboratory,
Michigan State University, East Lansing, MI 48824, USA  Joint Institute for
Nuclear Astrophysics: CEE, Michigan State University, East Lansing, MI 48824,
USA  D. Bazin  National Superconducting Cyclotron Laboratory, Michigan State
University, East Lansing, MI 48824, USA  B. A. Brown  National Superconducting
Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA
Joint Institute for Nuclear Astrophysics: CEE, Michigan State University, East
Lansing, MI 48824, USA  Departament of Physics and Astronomy, Michigan State
University, East Lansing, MI 48824, USA  P.C. Bender  National Superconducting
Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA
H.L. Crawford  Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
J. Engel  Department of Physics and Astronomy, The University of North
Carolina at Chapel Hill, Chapel Hill, NC 27599, USA  A. Falduto  Departament
of Physics, Central Michigan University, Mt. Pleasant, MI 48859, USA  Joint
Institute for Nuclear Astrophysics: CEE, Michigan State University, East
Lansing, MI 48824, USA  A. Gade  National Superconducting Cyclotron
Laboratory, Michigan State University, East Lansing, MI 48824, USA  Joint
Institute for Nuclear Astrophysics: CEE, Michigan State University, East
Lansing, MI 48824, USA  Departament of Physics and Astronomy, Michigan State
University, East Lansing, MI 48824, USA  P. Gastis  Departament of Physics,
Central Michigan University, Mt. Pleasant, MI 48859, USA  Joint Institute for
Nuclear Astrophysics: CEE, Michigan State University, East Lansing, MI 48824,
USA  B. Gao  National Superconducting Cyclotron Laboratory, Michigan State
University, East Lansing, MI 48824, USA  Joint Institute for Nuclear
Astrophysics: CEE, Michigan State University, East Lansing, MI 48824, USA  T.
Ginter  National Superconducting Cyclotron Laboratory, Michigan State
University, East Lansing, MI 48824, USA  C. J. Guess  Departament of Physics
and Astronomy, Swarthmore College, Swarthmore, PA 19081, USA  S. Lipschutz
National Superconducting Cyclotron Laboratory, Michigan State University, East
Lansing, MI 48824, USA  Joint Institute for Nuclear Astrophysics: CEE,
Michigan State University, East Lansing, MI 48824, USA  Departament of Physics
and Astronomy, Michigan State University, East Lansing, MI 48824, USA  B.
Longfellow  National Superconducting Cyclotron Laboratory, Michigan State
University, East Lansing, MI 48824, USA  Departament of Physics and Astronomy,
Michigan State University, East Lansing, MI 48824, USA  A.O. Macchiavelli
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA  K. Miki
Departament of Physics, Tohoku University, Sendai, Miyagi 980-8578, Japan  E.
Ney  Department of Physics and Astronomy, The University of North Carolina at
Chapel Hill, Chapel Hill, NC 27599, USA  S. Noji  National Superconducting
Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA
Joint Institute for Nuclear Astrophysics: CEE, Michigan State University, East
Lansing, MI 48824, USA  J. Pereira  National Superconducting Cyclotron
Laboratory, Michigan State University, East Lansing, MI 48824, USA  Joint
Institute for Nuclear Astrophysics: CEE, Michigan State University, East
Lansing, MI 48824, USA  J. Schmitt  National Superconducting Cyclotron
Laboratory, Michigan State University, East Lansing, MI 48824, USA  Joint
Institute for Nuclear Astrophysics: CEE, Michigan State University, East
Lansing, MI 48824, USA  Departament of Physics and Astronomy, Michigan State
University, East Lansing, MI 48824, USA  C. Sullivan  National Superconducting
Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA
Joint Institute for Nuclear Astrophysics: CEE, Michigan State University, East
Lansing, MI 48824, USA  Departament of Physics and Astronomy, Michigan State
University, East Lansing, MI 48824, USA  R. Titus  National Superconducting
Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA
Joint Institute for Nuclear Astrophysics: CEE, Michigan State University, East
Lansing, MI 48824, USA  Departament of Physics and Astronomy, Michigan State
University, East Lansing, MI 48824, USA  D. Weisshaar  National
Superconducting Cyclotron Laboratory, Michigan State University, East Lansing,
MI 48824, USA

###### Abstract

The Gamow-Teller strength distribution from 88Sr was extracted from a
$(t,{}^{3}\text{He}+\gamma)$ experiment at 115 MeV/$u$ to constrain estimates
for the electron-capture rates on nuclei around $N=50$, between and including
78Ni and 88Sr, which are important for the late evolution of core-collapse
supernovae. The observed strength below an excitation energy of 8 MeV was
consistent with zero and below 10 MeV amounted to $0.1\pm 0.05$. Except for a
very-weak transition that could come from the 2.231-MeV $1^{+}$ state, no
$\gamma$ lines that could be associated with the decay of known $1^{+}$ states
were identified. The derived electron-capture rate from the measured strength
distribution is more than an order of magnitude smaller than rates based on
the single-state approximation presently used in astrophysical simulations for
most nuclei near $N=50$. Rates based on shell-model and quasiparticle random-
phase approximation calculations that account for Pauli blocking and core-
polarization effects provide better estimates than the single-state
approximation, although a relatively strong transition to the first $1^{+}$
state in 88Rb is not observed in the data. Pauli unblocking effects due to
high stellar temperatures could partially counter the low electron-capture
rates. The new data serves as a zero-temperature benchmark for constraining
models used to estimate such effects.

###### pacs:

21.60.Cs, 23.40.-s, 25.40.Kv, 26.30.Jk

_Introduction-_ Core-collapse supernovae (CCSNe) are among the most energetic
explosions observed in the universe. They contribute to nucleosynthesis,
stimulate galactic chemical evolution, and are birth places of neutron stars
and black holes Fryer (1999); Heger  _et al._ (2003); Janka  _et al._ (2007);
Burrows (2013). A very large fraction of the energy released in CCSNe is in
the form of neutrinos, but the small fraction of energy released in the form
of visible light is important for probing the mechanism of the explosion. In
addition, CCSNe are predicted emission sites of gravitational waves
Dimmelmeier  _et al._ (2008); Richers  _et al._ (2017). Consequently, CCSNe
are attractive sites for improving our understanding of the universe through
multi-messenger studies Nakamura  _et al._ (2016). The accurate and detailed
description of relevant nuclear physics processes is key to understanding the
evolution of CCSNe and interpreting the multi-messenger signals Arcones  _et
al._ (2017).

Nuclear-weak interaction processes, in particular electron captures (EC), are
essential ingredients for simulating and understanding the dynamical evolution
of the CCSNe Bethe  _et al._ (1979); Langanke and Martínez-Pinedo (2003);
Suzuki, Toki, and Nomoto (2016). EC reactions, on nuclei in the upper $pf$ and
$pfg/sdg$-shells are particularly important during the collapse phase Hix  _et
al._ (2003). It was recently shown Sullivan  _et al._ (2016); Furusawa  _et
al._ (2017); Titus  _et al._ (2018) that ECs on a group of about 75 nuclei
around neutron number $N=50$ between and including ${}^{78}\text{Ni}$ and
${}^{88}\text{Sr}$ (hereafter referred to as the high-sensitivity region) are
responsible for about 50% of the uncertainties in characteristic parameters
such as lepton fraction, entropy, mass enclosed at core bounce, and in-fall
velocity Titus  _et al._ (2018). Also, the EC rates on nuclei in this mass
region could have a significant impact on the nucleosynthesis of trans-iron
elements produced in thermonuclear supernovae Jones  _et al._ (2019).

EC rates are derived from Gamow-Teller (GT) transition-strength
[$B(\text{GT})$] distributions in the $\beta^{+}$ direction. The EC rates
presently used for the nuclei in the high-sensitivity region rely on an
approximation that uses a single GT transition for which the strength and
excitation energy were fitted to best reproduce EC rates for nuclei in the
$pf$ shell near stability Langanke  _et al._ (2003); Raduta, Gulminelli, and
Oertel (2017). This single-state approximation does not account for strong
Pauli-blocking effects for heavier nuclei near $N=50$, even for nuclei that
are close to stability, and that could strongly reduce the EC rates for
neutron-rich nuclei in the high-sensitive region Sullivan  _et al._ (2016);
Titus  _et al._ (2018). It is important to verify such effects experimentally
and provide data to benchmark and guide theoretical calculations that are used
to estimate the EC rates for the astrophysical simulations. At high stellar
temperatures, Pauli unblocking are expected to become significant Langanke
_et al._ (2003); Langanke, Kolbe, and Dean (2001); Dzhioev  _et al._ (2010)
and it is important that models used to estimate such effects are first
validated at $T=0$. However, for nuclei in which the Gamow-Teller transitions
are not completely Pauli blocked, such as for 88Sr, such temperature-dependent
rate effects are expected to be relatively small Langanke, Kolbe, and Dean
(2001).

The only way to experimentally extract GT strength distributions in the
$\beta^{+}$ direction for neutron-rich nuclei is through the use of
($n$,$p$)-type charge-exchange (CE) reactions, as the $\beta^{+}$/EC-decay
$Q$-values for these nuclei are negative. CE experiments at intermediate beam
energies ($\gtrsim 100$ MeV/u) provide an indirect method to extract the
$B(\text{GT})$ distributions without $Q$-value constraints, based on a well-
established proportionality between the CE cross section at zero momentum
transfer and $B(\text{GT})$ Taddeucci  _et al._ (1987); Zegers  _et al._
(2007); Perdikakis  _et al._ (2011). In this letter, we present results of a
($t$,3He+$\gamma$) experiment aimed at extracting the Gamow-Teller (GT)
transition strength [$B(\text{GT})$; associated with the transfer of $\Delta
S=1$ (spin), $\Delta T=1$ (isospin), and $\Delta L=0$ (angular momentum)] from
the $N=50,Z=38$ nucleus 88Sr, which is amongst the most proton-rich nuclei in
the high-sensitive region. By combining the $(t,{}^{3}\text{He})$ CE reaction
with high-resolution $\gamma$-ray detection, even weakly excited low-lying GT
transitions that may relatively strongly impact the EC rates can be
identified, achieving a sensitivity to states with GT strengths of as small as
0.01 Noji  _et al._ (2014, 2015). This level of experimental sensitivity more
or less coincides with the limit on the applicability of the use of charge-
exchange reactions for reliably extracting GT strengths. This is due to
interference effects between the central $\sigma\tau$ and the tensor-$\tau$
components of the nucleon-nucleon force. For strengths below 0.01, such
effects complicate the clean identification of GT transitions from other
transitions, and introduce sizeable (30% for GT strengths of about 0.01)
systematic uncertainties Zegers  _et al._ (2006); Hitt  _et al._ (2009); Noji
_et al._ (2014).

The results discussed here are part of a broader effort to improve the
electron-capture rates on nuclei in the high-sensitivity region. These efforts
include additional experiments on other nuclei in this region, the
incorporation of theoretical nuclear structure models aimed at improving the
GT strength distributions used for electron-capture rate calculations, and
astrophysical simulations similar to those in Refs. Sullivan  _et al._ (2016);
Titus  _et al._ (2018).

_Experiment-_ A secondary triton beam was produced by fragmentation of a 150
MeV/u 16O primary beam from the NSCL Coupled Cyclotron Facility (CCF) on a
3525-mg/cm2 thick 9Be production target placed at the entrance on the A1900
fragment separator Morrissey  _et al._ (2003). A 99%-pure 115 MeV/u triton
beam of $4\times 10^{6}$ pps was generated with a momentum width of 0.5%
(FWHM) by using a 195-mg/cm2 thick Al degrader in the A1900 intermediate image
Hitt  _et al._ (2006). The beam was transported in dispersion-matched mode
Sherrill  _et al._ (1999); Berz  _et al._ (1993) to an isotopically enriched
88Sr (99.9%) foil with a thickness of 19.6 mg/cm2 placed at the S800
Spectrograph Bazin  _et al._ (2003) pivot point. Due to the high reactivity of
strontium, a special transfer system was used to insert the target without
coming into contact with air. 3He ejectiles produced in the reaction were
momentum analyzed and identified in the S800 focal plane Yurkon  _et al._
(1999). The particle identification was performed on an event-by-event basis
using the energy loss measured in a 5 mm-thick focal-plane scintillator and
the time-of-flight relative to the CCF radio-frequency signal. Scattering
angles and momenta of the ejectiles at the target location were reconstructed
by raytracing the angles and positions measured in two cathode-readout drift
chambers (CRDCs) in the S800 focal plane by using a fifth-order ion-optical
inverse matrix calculated in cosy infinity Makino and Berz (1999).
Subsequently, the excitation energy ($E_{x}$) of the 88Rb particles was
determined in a missing-mass calculation up to 25 MeV with a resolution of 500
keV (FWHM), which is due to the intrinsic resolution that can be achieved in
($t$,3He) experiments with a secondary triton beam and the difference in
energy loss between tritons and 3He particles in the 88Sr target.

Scattering angles in the center-of-mass (c.m.) frame were measured in the
range of $0^{\circ}<\theta_{\text{c.m.}}<5.5^{\circ}$. A total luminosity of
$2\times 10^{32}$ cm-2 was achieved over 5 days. Data were acquired for the
${}^{12}\text{C}(t,{}^{3}\text{He}){}^{12}\text{B}(1^{+},\text{g.s.})$
reaction by using a 2.6-mg/cm2 thick polystyrene (C8H8)n target. Its well-
known cross section Perdikakis  _et al._ (2011) was used to calibrate a non-
intercepting primary-beam current probe that served as an absolute measure for
the triton beam intensity during the 88Sr runs.
The Gamma-Ray Energy Tracking In-beam Nuclear Array (GRETINA) Paschalis  _et
al._ (2013); Weisshaar  _et al._ (2017), consisting of thirty-two 36-fold
segmented high-purity Ge detectors mounted on a hemisphere and providing about
$1\pi$ solid-angle coverage, was positioned around the 88Sr target. The use of
GRETINA allowed for the precise determination of $\gamma$-ray energies with a
high photopeak-detection efficiency (${\sim}4$% at 2 MeV).

_Results-_ Double-differential cross sections for the
${}^{88}\text{Sr}(t,{}^{3}\text{He})$ reaction were generated in 0.5-MeV wide
bins in $E_{x}$. The average statistical error for each bin was 5%. The
systematic error was ${\sim}7$%, dominated by the uncertainty in the triton
beam intensity. Examples for $E_{x}=2.25$ and $20.25$ MeV are shown in Figs.
1(a) and (b), respectively.

Figure 1:  (color online) Angular distributions for the
${}^{88}\text{Sr}(t,{}^{3}\text{He})$ reaction at $E_{x}=2.25$ MeV (a) and
20.25 MeV (b), fitted in the MDA. (Bottom) Double-differential cross sections
for scattering angles in the ranges of
$0^{\circ}<\theta_{\text{c.m.}}<1^{\circ}$ (c) and
$1^{\circ}<\theta_{\text{c.m.}}<2^{\circ}$ (d). The colors represent
contributions from excitations with different angular momentum transfers.

To extract the monopole contribution from the cross sections, a multipole
decomposition analysis (MDA) Bonin  _et al._ (1984); Ichimura, Sakai, and
Wakasa (2006) was performed for each bin in $E_{x}$ by fitting the
differential cross section with a linear combination of distorted wave Born
approximation (DWBA) angular distributions for angular momentum transfers of
$\Delta L=0,1,2$ and 3. The DWBA calculations were performed using the code
FOLD/DWHI Cook and Carr ; *[basedon]PETROVICH1977487;
*[modifiedasdescribedin]PhysRevC.30.1538; *fold2. The optical model potential
(OMP) parameters were taken from Ref. Kamiya  _et al._ (2003). Following Ref.
Werf  _et al._ (1989), the depths of the OMP for triton in the incoming
channel were scaled by a factor 0.85 from those for ${}^{3}\text{He}$ in the
outgoing channel. The effective nucleon-nucleon interaction of Franey and Love
Franey and Love (1985) was double-folded over the transition densities of
$t-{}^{3}\text{He}$ and ${}^{88}\text{Sr}-{}^{88}\text{Rb}$ systems. The
transition densities for $t$ and 3He were taken from variational Monte Carlo
calculations Pieper and Wiringa (2001). For the
${}^{88}\text{Sr}-{}^{88}\text{Rb}$ system, one-body transition densities
(OBTDs) were generated by using the shell-model code described below. Examples
of MDA are shown in Figs. 1(a) and (b). The MDA results for
$\theta_{\text{c.m.}}=0.67^{\circ}$ and $1.56^{\circ}$ as a function of
$E_{x}$ are shown in Figs. 1(c) and (d), respectively. For $E_{x}<8$ MeV, the
$\Delta L=0$ contribution of the cross section is consistent with zero within
the error bars ($0.07\pm 0.1$ mb/sr). For $E_{x}>10$ MeV, $\Delta L=0$
contributions are observed, but the isovector spin-monopole resonance (IVSMR)
is expected to start contributing significantly in this region space Miki  _et
al._ (2012).

The $B(\text{GT})$ strength was extracted from the $\Delta L=0$ cross section
at $\theta_{\text{c.m.}}=0^{\circ}$ by using the proportionality relation:
$\sigma_{L=0}(0^{\circ})=\hat{\sigma}_{\text{GT}}F(q,\omega)B(\text{GT})$
Taddeucci  _et al._ (1987); Zegers  _et al._ (2007); Perdikakis  _et al._
(2011). $\hat{\sigma}_{\text{GT}}$ is the GT unit cross section, which was
calculated (5.94 mb/sr) by the mass-dependent empirical relationship of Ref.
Zegers  _et al._ (2007), which has an uncertainty of 10%. $F(q,\omega)$ is a
kinematic correction factor that depends on the momentum ($q$) and energy
($\omega$) transfers, and is obtained from DWBA calculations Taddeucci  _et
al._ (1987). Its value was 1.2 (2.1) at $E_{x}=0$ (10) MeV. Fig. 2 shows the
extracted $B(\text{GT})$ distribution in the energy range from 0 to 10 MeV.
The only energy bin below the neutron separation energy with non-zero
$B(\text{GT})$ is located at $2.75\pm 0.25$ MeV, which correlates with the
locations of several $1^{+}$ states in 88Rb known from the $\beta^{-}$ decay
of 88Kr Bunting  _et al._ (1976); McCutchan and Sonzogni (2014). The summed
$B(\text{GT})$ below $E_{x}=10$ MeV is $0.10\pm 0.05$, although most of that
comes from the region above 8 MeV. This result is significantly lower than the
summed $B(\text{GT})$ of $0.7\pm 0.1$(stat.)$\pm 0.1$(sys.) measured for the
90Zr($n$,$p$) reaction up to $E_{x}=10$ MeV Yako  _et al._ (2005). Based on
transfer reaction experiments Pfeiffer  _et al._ (1986), the proton $0g_{9/2}$
occupation is 0.7 (1.0) for 88Sr (90Zr). Therefore, the decrease in GT
strength observed below $E_{x}=10$ MeV for 88Sr as compared to 90Zr is
stronger than expected based on the proton $0g_{9/2}$ occupation number only.

Figure 2:  (color online). $B(\text{GT})$ distribution extracted from MDA for
$E_{x}<10$ MeV. The error bars denote only the statistical uncertainties. The
dashed lines correspond to theoretical calculations, shell model (SM) and
QRPA, as described in the text.

Additional constraints on the GT strength can be obtained from the
($t$,3He$+\gamma$) coincident data. Fig. 3(a) shows the two-dimensional
histogram that correlates the energy of $\gamma$ rays ($E_{\gamma}$) with
$E_{x}$(88Rb). Due to the wide $E_{x}$ range covered, $\gamma$-ray transitions
from states in 88Rb, 87Rb and 86Rb were observed, as shown in Fig. 3(b). The
non observation of $\gamma$-rays from 87Kr indicates that the probability of
decay by proton emission from 88Rb was very small.

Figure 3:  (color online). (a) Two-dimensional histogram of $\gamma$-ray
energy ($E_{\gamma}$) versus excitation energy ($E_{x}$) of 88Rb. One and two-
neutron separation energies are indicated on top. (b) Projection on to the
$E_{\gamma}$ axis for $E_{x}\leq 25$ MeV. The color lines on top and bottom of
the figure represent the position of the observed transitions for 88,87,86Rb.
(c) $E_{\gamma}$ spectrum gated at $E_{x}=2.231\pm 0.422$ MeV.

By setting narrow gates on $E_{x}$ determined from the ($t$,3He) reaction, the
$\gamma$ spectrum for low $E_{x}$(88Rb) was investigated for evidence for the
decay from known $1^{+}$ states or for unknown $\gamma$ lines that could stem
from previously unknown $1^{+}$ states. No significant signals were found,
with the exception of the observation of a single event that could be due to
decay from the known 2.231-MeV $1^{+}$ state, as shown in Fig. 3(c). This
spectrum was obtained by setting a gate on $E_{x}=2.231\pm 0.422$ MeV in the
88Sr($t$,3He) spectrum, where the width of the gate corresponds to $2\sigma$
of the energy resolution. By using a Bayesian analysis Ghosh, Delampady, and
Samanta (2007) it was determined that, with an 86% probability, the credible
interval for $B(\text{GT})$ for the 2.231-MeV state ranges from 0 to 0.022,
which includes the possibility that the observed count is not due to the decay
from this state. The extracted Gamow-Teller strength from the MDA analysis in
the relevant excitation energy bin for this transition is
$0.006^{+0.02}_{-0.006}$.

The experimental results were compared to shell-model (SM) and quasi-particle
random-phase approximation (QRPA) calculations. The shell-model calculations,
performed with the code NUSHELLX Brown  _et al._ , assumed a 78Ni core and a
valence space of $(0f_{5/2},1p_{3/2},1p_{1/2},0g_{9/2})$ for protons and
$(0g_{7/2},1d_{5/2},1d_{3/2},2s_{1/2},0h_{11/2})$ for neutrons. The proton-
proton and proton-neutron two-body matrix elements (TBME) were obtained from
the jj44pna effective interaction Lisetskiy  _et al._ (2004) and a
renormalized G-matrix using the charge-dependent (CD-Bonn) nucleon-nucleon
interaction Dillmann  _et al._ (2003), respectively. The single-particle
energies were determined from the observed single-particle states in 89Sr. To
account for the model-space truncation, the result of the calculation was
scaled by a factor $\frac{1}{h}$, where $h$ is a hindrance factor that is a
product of two factors: $h_{\text{high}}$ and $h_{\text{c.p.}}$ Towner (1985).
$h_{\text{high}}$ is associated with the admixtures of two-particles two-holes
states with unperturbed energies of $2\hbar\omega$ and higher in the
oscillator basis. This factor accounts for the well-known quenching of the GT
transition strength Gaarde  _et al._ (1981); Gaarde (1985). The empirical
value for the $pf$ model space, $h_{\text{high}}=1.81$ Martínez-Pinedo  _et
al._ (1996), was used. $h_{\text{c.p.}}$ is due to the core polarization for
the $0g$ orbital. It accounts for the mixing between $0g_{9/2}$ and $0g_{7/2}$
spin-orbit partners and depends on the proton occupation number in the
$0g_{9/2}$ orbital. $h_{\text{c.p.}}$ is largest when the number of $0g_{9/2}$
protons is small Towner (1985). An occupation number of 0.58 was calculated
for the $\pi 0g_{9/2}$ shell in 88Sr by using the Ji/Wildenthal effective
interaction Ji and Wildenthal (1988), which is close to the experimental value
of 0.7 Pfeiffer  _et al._ (1986). The hindrance due to the core polarization
was taken from the results of Towner in Ref. Towner (1985) (Table 5). The
value for two protons in $0g_{9/2}$ of $h_{\text{c.p.}}=5.0$, obtained from
the $\pi+\rho$ interaction (the range for the three interactions given is 3.5
to 5.9), was used in our calculation. The $0g_{9/2}$ proton number dependence
of the hindrance factor $h_{\text{high}}\times h_{\text{c.p.}}$ leads to a
$Z$-dependent hindrance factor that is consistent with that deduced from the
$\beta^{+}$ decay of nuclei with $N=50$ ranging from 94Ru up to 100Sn Graf and
Münzel (1973); Rykaczewski  _et al._ (1985); Płochocki  _et al._ (1992); Hinke
_et al._ (2012).

The QRPA calculation was performed by using the axially-deformed Skyrme finite
amplitude method Avogadro and Nakatsukasa (2011); Mustonen  _et al._ (2014).
This method has recently been extended to odd-$A$ nuclei in the equal-filling
approximation Shafer  _et al._ (2016) and is, therefore, a candidate for
calculating GT strengths and EC rates for a large number of nuclei and
replacing the EC rates based on the single-state approximation discussed
above. The Skyrme functional and single-particle space model are the same as
those used in the global calculation of Ref. Mustonen and Engel (2016), which
fixed a single set of parameters including an effective axial-vector coupling
constant $g_{A}$ of 1.0.

The theoretical calculations shown in Fig. 2 have been folded with the
experimental resolution and the excitation energy of the first $1^{+}$ state
was matched to that of the first known $1^{+}$ state in 88Rb (at $E_{x}=2.231$
MeV). The SM and QRPA calculations both predict a strong transition to the
first $1^{+}$ state that is not observed experimentally. The summed strength
up to $E_{x}=10$ MeV is 0.12 (0.14) for the SM (QRPA) calculations. These
summed values are consistent with the present data of $0.1\pm 0.05$. The
results indicate that, besides Pauli blocking, structural effects (core
polarization) play an important role in the reduction of the GT strength.

Figure 4:  (color online). EC rates on 88Sr as function of stellar density at
a temperature of $10^{10}$ K. The shaded band with solid central curve
represents the result based on the 88Sr($t$,3He$+\gamma$) data. The dashed and
dot-dashed curves are based on the SM and QRPA calculations, respectively. The
dot-dot-dashed line represents the approximate method for estimating the EC
rate.

Stellar EC rates ($\lambda_{\text{EC}}$) were calculated based on the
formalism in Refs. Fuller, Fowler, and Newman (1980, 1982a, 1982b, 1985), in a
code previously used in Refs. Gupta  _et al._ (2007); Cole  _et al._ (2012);
Noji  _et al._ (2014, 2015). Only transitions from the ground state of 88Sr
were considered here. Fig. 4 shows the calculated EC rates (based on the
experimental and theoretical GT strength distributions) during the late stages
of CCSN, just prior to the bounce, during which the stellar density ranges
from $10^{9}$-$10^{12}$ g/cm3 and the temperature is ${\sim}10^{10}$ K.
Because no known $1^{+}$ state exists below $E_{x}=2$ MeV and the MDA analysis
also found no indication for any GT strength up to that energy, the first
transition assumed to contribute to the EC rate based on the data was the
2.231 MeV state, with an upperlimit to the strength based on the
$\gamma$-decay analysis [$B_{\text{up}}(\text{GT})=0.022$]. The $Q$-value for
EC on 88Sr is $-4.8$ MeV, which means that only at a density of $10^{11}$
g/cm3, the Fermi energy of ${\sim}15$ MeV is sufficiently high to cover the
strength distribution up to $E_{x}=10$ MeV and that details of the GT strength
distribution below that $E_{x}$ matter up to that density. The higher the
density, the less sensitive the EC rate becomes to details of the strength
distribution, as most of the strength distribution is below the Fermi energy.
Therefore, due to the presence of the relatively strong transition to the
first $1^{+}$ state, the EC rates based on the QRPA and SM calculations are
higher than the upper limit set by the data for stellar densities below
${\sim}10^{10}$ g/cm3. At higher densities, the rates based on the SM and QRPA
calculations are within the upper limit set by the data, since the summed
strengths up to 10 MeV are, within error bars, consistent. The EC rates
calculated based on the single-state approximation (with $B(\text{GT})=4.6$
and a temperature and density dependent effective excitation energy based on
Ref. Raduta, Gulminelli, and Oertel (2017)) are more than an order of
magnitude too high. Considering that 88Sr is the amongst the most proton-rich
$N=50$ nucleus in the high-sensitivity region (with the least Pauli blocking),
it is very likely that the rates based on the approximation will be also be
much too high for the other nuclei in the high sensitivity region. This has a
strong impact on the dynamical evolution during the collapse phase Sullivan
_et al._ (2016); Titus  _et al._ (2018). The drop in lepton fraction during
the collapse reduces by 10% and the enclosed mass at core bounce increases by
10% when the EC rates for nuclei in the high-sensitivity region are reduced by
a factor 10. Finally, we note that although Pauli-unblocking effects due to
the high temperature in the collapsing star should increase the EC rates
compared to rates shown in Fig. 4, the structural (core-polarization) effects
are equally important, especially for nuclei such as 88Sr in which Pauli
blocking is not complete at $T=0$, and must be considered in theoretical
models used for estimating EC rates at elevated temperatures. Therefore, the
present data provides an important zero-temperature benchmark for such
theoretical estimates.

_Summary-_ The GT transition strength in 88Sr was measured in a high
resolution $(t,{}^{3}\text{He}+\gamma)$ experiment to gain insight in EC rates
of nuclei near $N=50$ above 78Ni that are most important during the collapse
phase of massive stars prior to the supernova explosion. The extracted
$B(\text{GT})$ is consistent with zero in the energy range from 0 up to
$E_{x}=8$ MeV and sums to $0.1\pm 0.05$ up to $E_{x}=10$ MeV. SM and QRPA
calculations are consistent with this summed strength, but predict a
relatively strong transition to a low-lying state not observed in the
experiment. As the most proton-rich $N=50$ nucleus in the high-sensitivity
region, these results indicate that the EC rates based on a single-state
approximation that is used in astrophysical simulations are too high. Although
Pauli-unblocking effects due to the high stellar temperatures during the
collapse phase partially counter the lowering of the EC rates, the results
show that structural effects must be carefully considered as they
significantly lower the GT strengths and EC rates. Hence, the present data
also serves as a zero-temperature benchmark for theoretical models that can be
used to estimate temperature-dependent Pauli-unblocking effects.

_Acknowledgments-_ We thank the NSCL staff for their support. This work was
supported by the US National Science Foundation (NSF) under Cooperative
Agreement PHY-156554 (NSCL), PHY-1430152 (JINA Center for the Evolution of the
Elements), and PHY-1811855. GRETINA was funded by the US Department of Energy,
in the Office of Nuclear Physics of the Office of Science. Operation of the
array at NSCL was supported by DOE under Grants No. DE-SC0014537 (NSCL) and
No. DE-AC02-05CH11231 (LBNL). The 88Sr foil used in this research were
supplied by the US Department of Energy Office of Science by the Isotope
Program in the Office of Nuclear Physics.

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