#  A year in the life of GW170817: the rise and fall of a structured jet from
a binary neutron star merger

E. Troja1,2, H. van Eerten3, G. Ryan1, R. Ricci4, J. M. Burgess5,6, M. H.
Wieringa7, L. Piro8, S. B. Cenko2,1, T. Sakamoto9
1 Department of Astronomy, University of Maryland, College Park, MD
20742-4111, USA
2Astrophysics Science Division, NASA Goddard Space Flight Center, 8800
Greenbelt Rd, Greenbelt, MD 20771, USA
3Department of Physics, University of Bath, Claverton Down, Bath BA2 7AY,
United Kingdom
4INAF-Istituto di Radioastronomia, Via Gobetti 101, I-40129, Italy
5Max-Planck-Institut fur extraterrestrische Physik, Giessenbachstrasse 1,
D-85748 Garching, Germany
6 Excellence Cluster Universe, Technische Universität München, Boltzmannstraße
2, 85748 Garching,Germany
7CSIRO Astronomy and Space Science, P.O. Box 76, Epping NSW 1710, Australia
8INAF, Istituto di Astrofisica e Planetologia Spaziali, via Fosso del
Cavaliere 100, 00133 Rome, Italy
9Aoyama Gakuin University, 5-10-1 Fuchinobe, Chuoku, Sagamiharashi Kanagawa
252-5258, Japan
E-mail<EMAIL_ADDRESS>
(Accepted 12 August 2019. Received 5 August 2019; in original form 21 August
2018)

###### Abstract

We present the results of our year-long afterglow monitoring of GW170817, the
first binary neutron star (NS) merger detected by advanced LIGO and advanced
Virgo. New observations with the Australian Telescope Compact Array (ATCA) and
the Chandra X-ray Telescope were used to constrain its late-time behavior. The
broadband emission, from radio to X-rays, is well-described by a simple power-
law spectrum with index $\beta$ $\sim$0.585 at all epochs. After an initial
shallow rise $\propto$ $t^{0.9}$, the afterglow displayed a smooth turn-over,
reaching a peak X-ray luminosity of $L_{X}$$\approx$5$\times$$10^{39}$ erg s-1
at 160 d, and has now entered a phase of rapid decline, approximately
$\propto$ $t^{-2}$. The latest temporal trend challenges most models of choked
jet/cocoon systems, and is instead consistent with the emergence of a
relativistic structured jet seen at an angle of $\approx$22∘ from its axis.
Within such model, the properties of the explosion (such as its blastwave
energy $E_{K}\approx 2\times 10^{50}$ erg, jet width $\theta_{c}$ $\approx$4∘,
and ambient density $n$ $\approx$3$\times$ $10^{-3}$ cm-3) fit well within the
range of properties of cosmological short GRBs.

###### keywords:

gravitational waves – gamma-ray burst – acceleration of particles

††pubyear: 2019††pagerange:  A year in the life of GW170817: the rise and fall
of a structured jet from a binary neutron star merger –8

##  1 Introduction

On August 17th, 2017 the Advanced LIGO interferometers detected the first
gravitational wave (GW) signal from a binary neutron star (NS) merger,
GW170817, followed 1.7 s later by a short duration gamma-ray burst, GRB170817A
(Abbott et al., 2017a). Located in the elliptical galaxy NGC4993 at a distance
of $\sim$40 Mpc, GRB170817A was an atypical sub-luminous explosion. An X-ray
afterglow was detected 9 days after the merger (Troja et al., 2017). A second
set of observations, performed $\sim$15 days post-merger, revealed that the
emission was not fading, as standard GRB afterglows, but was instead rising at
a slow rate (Troja et al., 2017; Haggard et al., 2017). The radio afterglow,
detected at 16 days (Hallinan et al., 2017), continued to rise in brightness
(Mooley et al., 2018a), as later confirmed by X-ray and optical observations
(Troja et al., 2018a; D’Avanzo et al., 2018; Lyman et al., 2018; Margutti et
al., 2018).

The delayed afterglow onset and low-luminosity of the $\gamma$-ray signal
could be explained if the jet was observed at an angle (off-axis) of
$\approx$15∘-30∘. Whereas a standard uniform jet viewed off-axis could account
for the early afterglow emission, Troja et al. (2017) and Kasliwal et al.
(2017) noted that it could not account for the observed gamma-ray signal and
proposed two alternative models: a structured jet, i.e. a jet with an angular
profile of Lorentz factors and energy (see also Abbott et al., 2017a;
Kathirgamaraju et al., 2018), and a mildly-relativistic isotropic cocoon (see
also Lazzati et al., 2017; Kasliwal et al., 2017). In the latter model, the
jet may never emerge from the merger ejecta (choked jet).

The subsequent rebrightening ruled out both the uniform jet and the simple
cocoon models, which predict a sharp afterglow rise. It was instead consistent
with an off-axis structured jet (Troja et al., 2017) and a cocoon with energy
injection (Mooley et al., 2018a), characterized by a radial profile of ejecta
velocities. In Troja et al. (2018a) we developed semi-analytical models for
both the structured jet and the quasi-spherical cocoon with energy injection,
and showed that they describe the broadband afterglow evolution during the
first six months (from the afterglow onset to its peak) equally well. This is
confirmed by numerical simulations of relativistic jets (Lazzati et al., 2018;
Xie et al., 2018) and choked jets (Nakar, et al., 2018).

Several tests were discussed to distinguish between these two competing models
(e.g. Gill & Granot, 2018; Nakar, et al., 2018). Corsi et al. (2018) used the
afterglow polarization to probe the outflow geometry (collimated vs. nearly
isotropic), but the results were not constraining. Ghirlanda, et al. (2019)
and Mooley et al. (2018b) used Very Long Baseline Interferometry (VLBI) to
image the radio counterpart, and concluded that the compact source size
($\lesssim$2 mas) and its apparent superluminal motion favor the emergence of
a relativistic jet core. A third and independent way to probe the outflow
structure is to follow its late-time temporal evolution. In the case of a
cocoon-dominated emission, the afterglow had been predicted to follow a
shallow decay ($t^{-\alpha}$) with $\alpha$ $\sim$1.0-1.2 (Troja et al.,
2018a) for a quasi-spherical outflow, and $\alpha$ $\sim$1.35 for a wide-
angled cocoon (Lamb, Mandel & Resmi, 2018). A relativistic jet is instead
expected to resemble a standard on-axis explosion at late-times, thus
displaying a post-jet-break decay of $\alpha\sim 2.5$ (van Eerten & MacFadyen,
2013).

Here, we present the results of our year-long observing campaign of GW170817,
carried out with the Australian Telescope Compact Array (ATCA) in the radio,
Hubble Space Telescope (HST) in the optical, the Chandra X-ray telescope and
XMM-Newton in the X-rays. Our latest observations show no signs of spectral
evolution (Sect. 2.1) and a rapid decline of the afterglow emission (Sect.
2.2), systematically faster than cocoon-dominated/choked jet models from the
literature (Sect. 3.1). The rich broadband dataset allows us to tightly
constrain the afterglow parameters, and to compare the explosion properties of
GW170817 to canonical short GRBs (Sect. 3.2).

##  2 Observations and data analysis

Our earlier observations were presented in Troja et al. (2017, 2018a) and
Piro, et al. (2019). To these, we add a new series of observations tracking
the post-peak afterglow evolution. Table 1 lists the latest unpublished data
set, including our radio monitoring with ATCA (PI: Piro, Murphy) and X-ray
observations with Chandra, carried out under our approved General Observer
program (20500691; PI: Troja). Data were reduced and analyzed as detailed in
Troja et al. (2018a); Piro, et al. (2019). In the latest Chandra observation,
the source is detected at a count-rate of (4.9$\pm$0.9)$\times$10-4 cts s-1 in
the 0.3-8.0 keV band, corresponding to an unabsorbed flux of
(6.70$\pm$0.13)$\times$10-15 erg cm-2 s-1. We adopted a standard $\Lambda$CDM
cosmology (Planck Collaboration et al., 2018). Unless otherwise stated, the
quoted errors are at the 68% confidence level, and upper limits are at the 3
$\sigma$ confidence level.

Figure 1: Temporal evolution of the afterglow spectral energy distribution. A
single power-law segment can describe the broadband spectrum during the
different afterglow phases (rise, peak and decline). At all times, a simple
power-law model provides a good fit of the data.

###  2.1 Spectral properties

The latest epoch of X-ray observations shows a simple power-law spectrum with
index $\beta$=0.8$\pm$0.4, consistent with previous measurements, and with the
spectral index 0.4$\pm$0.3 from the late time ($t>$220 d) radio data. Figure 1
shows that, at all epochs, the broadband spectrum can be fit with a simple
power-law model with spectral index $\beta$=0.585$\pm$0.005 and no intrinsic
absorption in addition to the Galactic value $N_{H}$=7.6$\times$1020 cm-2. The
lack of any significant spectral variation on such long timescales is
remarkable. In GRB afterglows, a steepening of the X-ray spectrum due to the
gradual decrease of the cooling frequency $\nu_{c}$ is commonly detected
within a few days. Since the cooling break is a smooth spectral feature, we
used a curved afterglow spectrum (Granot & Sari, 2002) to fit the data, and
constrain its location111 The spectrum was fit with a curved spectrum leaving
the cooling frequency as a free parameter. The best fit statistics C0 was
recorded, then the location of $\nu_{c}$ was changed until the variation in
the fit statistics was equal to $C_{0}$+2.706.. We derived $\nu_{c}\gtrsim$1
keV (90% confidence level) at 260 d after the merger and $\nu_{c}\gtrsim$0.1
keV (90% confidence level) at 360 d.

For a synchrotron spectrum with $\nu_{m}<\nu_{obs}<\nu_{c}$, the measured
spectral index is related to the spectral index $p$ of the emitting electrons
(e.g. Granot & Sari, 2002) as $p$ = 2$\beta$ +1 = 2.170 $\pm$ 0.010. Figure 2
compares this value with a sample of well-constrained short and long GRB
spectra. It shows that, among short GRB spectra, GW170817/GRB170817A
represents the most precise measurement obtained so far. Such precision is
rare, but not unprecedented among long GRB afterglows.

It is tempting to interpret the accurately determined value of $p$ as an
intermediate between relativistic and non-relativistic shock acceleration,
based on theoretical considerations of plausible mechanisms (Kirk et al.,
2000; Achterberg et al., 2001; Spitkovsky, 2008), implying $\Gamma$
$\approx$2-10 for the emitting material (Margutti et al., 2018). On the other
hand, various well constrained short GRB $p$-values lie outside this
theoretical range (Fig. 2), the $p$-value distribution for the larger sample
of (long) GRBs does not appear consistent with a universal value for $p$ (Shen
et al., 2006; Curran et al., 2010), nor is there generally any evidence for
evolution of $p$ from multi-epoch spectral energy distributions (SEDs) (Fig.
1, also e.g. Varela et al. 2016) or light curve slopes. In view of these
features of the general sample of long and short GRBs (as well as other
synchrotron sources, such as blazars), direct interpretation of $p\approx
2.17$ in terms of shock-acceleration theory might be premature.

Figure 2: Spectral index $p$ of the shock-accelerated electrons for
GW170817/GRB170817A (filled circle) and a sample of of short GRBs (filled
squares) and long GRBs (open diamonds) with good afterglow constraints. The
dashed line shows the distribution inferred from XRT afterglows (Curran et
al., 2010). The hatched area shows the range of values predicted by shock
theory. Data are from: Roming et al. (2006); Soderberg et al. (2006); Mundell
et al. (2007); Resmi & Bhattacharya (2008); Xu et al. (2009); Krühler et al.
(2009); Kumar & Barniol Duran (2010); Troja et al. (2012, 2016); Fong et al.
(2015); Varela et al. (2016); Knust et al. (2017).

###  2.2 Temporal properties

We simultaneously fit the multi-color (X-ray, optical, radio) light curves by
adopting a simple power-law function in the spectral domain, and a smoothly
broken power-law in the temporal domain (Beuermann et al., 1999). The
functional form is:
$F(\nu,t)\propto\nu^{-\beta}\left[(t/t_{p})^{-\kappa\alpha_{1}}+(t/t_{p})^{\kappa\alpha_{2}}\right]^{-1/\kappa}$,
where $\beta$ is the spectral index, $\alpha_{1}$ and $\alpha_{2}$ are the
rise and decay slopes, $t_{p}$ is the peak time, and $\kappa$ is the
smoothness parameter.

We did not impose an achromatic behavior. Instead, the temporal properties
were modeled as a hierarchy where the parameters for each wavelength have an
independent value, sampled from a hyper distribution. Variations in the
hierarchy would be integrated out of the posterior if justified by the data.

Scale-family distributions were given log-normal priors and all other
parameters were given normal priors. To fit the model, we employ the NUTS
variant of Hamiltonian Monte Carlo via the Stan modeling language (Carpenter
et al., 2017). The peak time and rise slope are well constrained to
$t_{p}$=164$\pm$12 d and $\alpha_{1}$=0.90$\pm$0.06. Whereas the optical
afterglow is poorly sampled, the X-ray and radio light curves allow for better
constraints on the decay slope, $\alpha_{2}$=2.0${}^{+0.8}_{-0.5}$ (Figure 3),
and are both consistent with a rapid decline of the afterglow flux. The best
fit model and full corner plot is reported in the Supplementary Material
(Figure 7).

Figure 3: Joint marginal distribution of the decay index ($\alpha_{2}$) and
smoothness parameter ($s$) for the radio (blue) and X-ray (red) light curves.
The dashed line corresponds to $\alpha_{2}=1.35$, the steepest value predicted
by choked jet/cocoon models. Figure 4: Multi-wavelength afterglow light curves
overlaid with the Gaussian jet best fit model (solid line) and its 68%
uncertainty range (shaded areas). Radio data are from ATCA (filled symbols)
and VLA (open symbols) observations. X-ray data are from Chandra (filled
symbols) and XMM-Newton (open symbols) observations. Downward triangles are 3
$\sigma$ upper limits. The dashed line shows the expected asymptotic decline
$\propto t^{-2.5}$. Data were collected from: Troja et al. 2017, Troja et al.
2018a, Piro, et al. 2019, Hallinan et al. 2017, Lyman et al. 2018, Resmi, et
al. 2018, Margutti et al. 2018, Mooley et al. 2018a, and Alexander et al.
2018.

###  2.3 Modeling

We directly fit two semi-analytical models for structured outflows to the
data, following the description in Troja et al. (2018a). The off-axis
structured jet model assumes a Gaussian energy profile
$E\propto\exp\left(-\theta^{2}/2\theta_{c}^{2}\right)$ up to a truncating
angle $\theta_{w}$. The jet is fully determined by a set of eight parameters
$\Theta_{\mathrm{jet}}=\\{E_{0},n,\epsilon_{e},\epsilon_{B}\,p,\theta_{c},\theta_{w},\theta_{v}\\}$,
where $E_{0}$ is the on-axis isotropic-equivalent kinetic energy of the blast
wave, $n$ the circumburst density, $\epsilon_{e}$ the electron energy
fraction, $\epsilon_{B}$ the magnetic energy fraction and $\theta_{v}$ the
angle between the jet-axis and the observer’s line of sight. Following Mooley
et al. (2018a), we also fit a cocoon model with a velocity stratification of
the ejecta to allow for a slower rise and late turnover. The total amount of
energy in the slower ejecta above a particular four-velocity $u$ is modelled
as a power-law $E(>u)=E_{\mathrm{inj}}u^{-k}$. This model requires nine
parameters
$\Theta_{\mathrm{cocoon}}=\\{E_{\mathrm{inj}},n,p,\epsilon_{e},\epsilon_{B}\,M_{\mathrm{ej}},u_{\mathrm{max}},u_{\mathrm{min}},k\\}$,
where $u_{\mathrm{max}}$ is the maximum ejecta four-velocity,
$u_{\mathrm{min}}$ the minimum ejecta four-velocity, and $M_{\mathrm{ej}}$ the
initial cocoon ejecta mass with speed $u_{\mathrm{max}}$.

As described in Troja et al. (2018a), our Bayesian fit procedure utilizes the
_emcee_ Markov-chain Monte Carlo package (Foreman-Mackey et al., 2013). For
the structured jet we also include the GW constraints on the orientation
$\iota$ of the system (Abbott et al., 2017b) in our prior for $\theta_{v}$.
The results of the MCMC analysis are summarized in Table 2. The best fit jet
models is shown in Figure 4. For the corner plot see the Supplementary data
(Figure 8).

Figure 5: Late time afterglow light curves compared to different explosion
models: choked jets from numerical simulations (thicker lines, Nakar, et al.
2018), wide-angled cocoon (Lamb, Mandel & Resmi, 2018), and our best fit
models of quasi-spherical cocoon (dot-dashed line) and structured jet (solid
line). Different symbols represent different wavelengths: X-rays (circles),
optical (downward triangles; 3 $\sigma$ upper limits), and radio (diamonds)
from ATCA (filled) and VLA (empty). For plotting purposes, data were rescaled
to a common energy of 5 keV using the observed spectral slope $\beta$=0.585.

Figure 6: Afterglow parameters for GW170817 and a sample of cosmological
($z>$0.1) short GRBs with multi-wavelength afterglows. For GW170817 we report
the results based purely on the electromagnetic observations (EM), and those
incorporating the LIGO constraints on the binary inclination (EM+GW). Data are
from: Soderberg et al. (2006); Kumar & Barniol Duran (2010); Fong et al.
(2015); Troja et al. (2016, 2018b); Troja, et al. (2019) The vertical grey
band show the median values of energy $E_{k,\rm iso}$ and density $n$ for the
larger sample of Swift short GRBs (Fong et al., 2015).

##  3 Results

###  3.1 A rapid afterglow decline: constraints on the outflow structure

For jets that fail to break out (“choked jets”), the jet energy is dissipated
into a surrounding cocoon of material. This scenario is therefore included in
our group of ‘cocoon’ models (Troja et al., 2018a). The post-peak temporal
slope is a shallow decay of $\alpha\approx 1.0-1.2$ up to at least 300 days
for GRB 170817A, as can be inferred from semi-analytical modeling of the
evolution of a trans-relativistic shell (Troja et al., 2018a). Any remaining
post-turnover impact due to continued energy injection from e.g. a complex
velocity profile of the ejecta, would lead to an even shallower decay.
Afterwards, the slope will eventually become that of an expanding non-
relativistic (quasi-)spherical shell. As for the Sedov-Taylor solution of a
point explosion in a homogeneous medium, this slope translates to
$\alpha=(15p-21)/10$ for $\nu_{m}<\nu_{obs}<\nu_{c}$, when combined with a
standard synchrotron model for shock-accelerated electrons (Frail et al.,
2000). The value $p\approx 2.17$ implies $\alpha\approx 1.155$. If
$\nu_{m},\nu_{c}<\nu_{obs}$, $\alpha=(3p-4)/2\approx 1.255$ instead (e. g.
Granot & Sari, 2002). If the cocoon choking the jet is not quasi-spherical,
but merely wide-angled, some sideways spreading of the outflow may still
occur. By definition for a non-relativistic flow velocity, this will not
produce any observational features related to relativistic beaming, but the
continuous increase in working surface will give rise to additional
deceleration of the blast wave relative to the case of purely radial flow. As
a result, the temporal slope could steepen slightly by another
$\Delta\alpha\approx 0.15-0.2$ (Lamb, Mandel & Resmi, 2018), before settling
into the late quasi-spherical stage. For GW170817, this implies a maximum
$\alpha\approx 1.35$ from a cocoon-dominated / choked jet model.
By contrast, if the jet has a relativistically moving inner region (in terms
of angular distribution of Lorentz factor), the post-peak temporal slope will
be like that of an on-axis jet seen after the jet break: the entire surface of
the jet has come into view and there is no longer a contribution to the light
curve slope from a growing visible patch. Different calculations predict
various degrees of steepening (e.g. Gill & Granot, 2018): a slope
$\alpha\approx p$ according to analytical models (Sari et al., 1999), and a
somewhat steeper $\alpha\approx 2.5$ according to semi-analytical models
(Troja et al., 2018a) and hydrodynamical simulations of jets (van Eerten &
MacFadyen 2013, although the latter were done for jets starting from top-hat
initial conditions).

As derived in section 2.2, the decay slope $\alpha_{2}$ of the empirical model
exceeds the predictions of most choked jet models by a healthy margin. This is
confirmed by the comparison of the late afterglow data, from radio to X-rays,
with physical models of choked and structured jets (Figure 5). The observed
decay is consistent with the turnover of a structured jet (solid line). While
the observed $\alpha_{2}$ is not as steep as $2.5$, this is not unexpected for
a jetted flow as the transition of the light curve from rise to decay is
spread out over time (and fully captured by the direct application of a
structured jet model in section 2.3, see also Fig. 4).

The rapid decline of the afterglow therefore poses an additional challenge to
the choked jet models, in support to the results of the high-resolution radio
imaging (Mooley et al., 2018b; Ghirlanda, et al., 2019). While growing
evidence indicates that the merger remnant launched a successful relativistic
jet, the presence of a cocoon cannot be excluded. There might well be an
observable cocoon component present in the outflow even for successful jets
(Nagakura et al., 2014; Murguia-Berthier et al., 2014). Indeed, the structured
jet itself might be an indication of the presence of a cocoon (in case the
structure is not imposed by the torus upon launching, Aloy et al. 2005).
However, it would appear any such cocoon is not the dominant emission
component at late times.

###  3.2 Afterglow properties: comparison to short GRBs

The predictive power of each model can be judged by the deviance information
criteria (DIC), where lower scores correspond to greater predictive power
(Spiegelhalter et al., 2002). The Gaussian jet model fit has a DIC of 103.5
and the cocoon model fit has a DIC of 151.8, favoring the Gaussian jet as the
more predictive model. For all models and priors, the posterior value of the
electron power law slope lies around $p=2.170\pm 0.010$, fully consistent with
the value obtained by the spectral analysis (Section 2.1).

The cocoon model requires a small amount of relativistic ejecta with a
substantial Lorentz factor $\Gamma_{max}\in[6.1,200]$ (all ranges with 68%
percent confidence) followed by an energetic tail of slower ejecta with
minimum Lorentz factor $\Gamma_{min}\in[1.9,7.48]$. The high Lorentz factors
are in tension with a choked-jet scenario, where the ejecta achieve only
Newtonian velocity. The total energy, assuming a spherical blast wave, is
rather high, between $10^{51}$ and $4\times 10^{53}$ erg. The circumburst
density and shock micro-physical parameters are very poorly constrained in
this model.

The Gaussian jet (Figure 4) has a well constrained width $\theta_{c}$ =
0.06$\pm$0.02 rad (3.4∘ $\pm$ 1.1∘), and a total energy between $5\times
10^{49}$ and $1.4\times 10^{51}$ erg. The wide truncation angle is largely
unconstrained. The ambient density $n$ is constrained to be between $3\times
10^{-4}$ and $2.4\times 10^{-2}$ cm-3. The micro-physical parameters
$\epsilon_{e}\in[4\times 10^{-3},0.17]$ and $\epsilon_{B}\in[2.7\times
10^{-5},10^{-3}]$ are only somewhat constrained by the model fit. The
constraint $\nu_{c}>$ 1 keV from the lack of spectral steepening drives the
relatively small value for $\epsilon_{B}$, as
$\nu_{c}\propto\epsilon_{B}^{-3/2}$. These values were derived by assuming a
jet with a Gaussian angular profile, yet they are in good agreement with other
estimates based on different angular structures (e.g. Ghirlanda, et al.,
2019).

The viewing angle derived from the electromagnetic observations alone is
0.52$\pm$0.16 rad (30∘ $\pm$ 9∘), consistent with the constraints from prompt
emission (e.g Abbott et al., 2017a; Bégué et al., 2017), optical and radio
imaging (Mooley et al., 2018b; Ghirlanda, et al., 2019; Lamb, et al., 2019).
By adding the GW constraints on the binary inclination $\iota$ to our
modeling, we obtain $\theta_{v}$ = 0.38$\pm$0.11 rad (22∘ $\pm$ 6∘),
consistent with the LIGO estimates that informed the prior. The good agreement
of the electromagnetic and GW constraints suggest that the relativistic jet
was launched nearly perpendicularly to the orbital plane.

The year-long monitoring of GW170817 significantly reduced the allowed
parameter space of models, tightening the constraints on the afterglow
properties. This allows for a comparison with other well-studied short GRB
explosions, as presented in Figure 6. It is remarkable how the properties of
GW170817 fit within the range of short GRB afterglows. The low circumburst
densities $n\approx$0.01 cm-3 are typical of the interstellar medium, and
consistent with the location of these bursts within their galaxy light. The
electron energy fraction seems well constrained to $\epsilon_{e}\gtrsim$0.1,
whereas $\epsilon_{B}$ tends to lower values $\lesssim$0.01, and is only
loosely constrained. The narrow width of the Gaussian jet $\theta_{c}$ of
GW170817 is comparable to the half-opening angle $\theta_{j}$ inferred from
top-hat jet models of short GRBs (e.g. Troja et al., 2016), suggesting that
these GRB jets had narrow cores of similar size. The isotropic-equivalent
energy is also consistent with the measurements from other short GRBs,
although we note that most events in the sample lie above the median value of
$E_{k,iso}$ (vertical band). This is not surprising as we selected the cases
of well-sampled light curves with good afterglow constraints, thus creating a
bias toward the brightest explosions.

##  4 Conclusions

The long-term afterglow monitoring of GW170817 supports the earlier
suggestions of a relativistic jet emerging from the merger remnant, and
challenges the alternative scenarios of a choked jet. Whereas emission at
early times ($<$160 d) came from the slower and less energetic lateral wings,
the rapid post-peak decline suggests that emission from the narrow jet core
has finally entered our line of sight. The overall properties of the
explosion, as derived from the afterglow modeling, are consistent with the
range of properties observed in short GRBs at cosmological distances, and
suggest that we detected its electromagnetic emission thanks to a combination
of moderate off-axis angle ($\theta_{v}$-$\theta_{c}$ $\approx$20∘ ) and
intrinsic energy of the explosion.

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Table 1: Late-time X-ray and radio observations of GW170817 T-T0 | Facility | Exposure | $\beta$ | Flux222X-ray fluxes are corrected for Galactic absorption along the sightline.  | Frequency
---|---|---|---|---|---
(d) |  |  |  | ($\mu$Jy) | (GHz)
267 | ATCA | 11.0h | 0.8$\pm$0.8 | 30$\pm$7 | 5.5
|  |  |  | 20$\pm$6 | 9.0
|  |  |  | 30$\pm$6 | 7.25
298 | ATCA | 11.0h | -0.3$\pm$1.0 | 25$\pm$7 | 5.5
|  |  |  | 29$\pm$6 | 9.0
|  |  |  | 28$\pm$6 | 7.25
320 | ATCA | 11.5h | 0.4$\pm$0.8 | 27$\pm$8 | 5.5
|  |  |  | 22$\pm$6 | 9.0
|  |  |  | 22$\pm$5 | 7.25
359 | ATCA | 9.5h | $--$ |  $<$26 | 5.5
|  |  |  |  $<$20 | 9.0
|  |  |  |  $<$18 | 7.25
391 | ATCA | 9.5h | $--$ |  $<$33 | 5.5
|  |  |  |  $<$27 | 9.0
|  |  |  |  $<$24 | 7.25
359 | Chandra | 67.2 ks | 0.8$\pm$0.4 | 2.8${}^{+0.3}_{-0.5}$ $\times$ 10-4 | 1.2$\times$109
|  |  | 0.585 | 2.1${}^{+0.2}_{-0.3}$$\times$ 10-4 | 1.2$\times$109

Table 2: Constraints on the Gaussian jet and Cocoon model parameters. Reported are the median values of each parameter’s posterior distribution with symmetric 68% uncertainties (ie. the 16% and 84% quantiles). The first Gaussian jet column uses constraints from the afterglow alone, the second includes the LIGO constraints on the inclination angle using the Planck value of $H_{0}$. | Parameter | Jet | Jet+GW+Planck
---|---|---
$\theta_{v}$ | $0.52^{+0.16}_{-0.16}$ | $0.38^{+0.11}_{-0.11}$
$\log_{10}E_{0}$ | $52.47^{+0.81}_{-0.56}$ | $52.80^{+0.89}_{-0.65}$
$\theta_{c}$ | $0.079^{+0.026}_{-0.024}$ | $0.059^{+0.017}_{-0.017}$
$\theta_{w}$ | $0.77^{+0.47}_{-0.38}$ | $0.61^{+0.42}_{-0.31}$
$\log_{10}n$ | $-1.83^{+0.77}_{-1.0}$ | $-2.51^{+0.90}_{-0.99}$
$p$ | $2.1678^{+0.0064}_{-0.010}$ | $2.1681^{+0.0062}_{-0.0095}$
$\log_{10}\epsilon_{e}$ | $-1.13^{+0.53}_{-0.88}$ | $-1.39^{+0.62}_{-0.99}$
$\log_{10}\epsilon_{B}$ | $-4.18^{+0.85}_{-0.58}$ | $-4.00^{+1.0}_{-0.69}$
$\log_{10}E_{tot}$ | $50.24^{+0.72}_{-0.47}$ | $50.30^{+0.84}_{-0.57}$
| Parameter | Cocoon
---|---
$\log_{10}u_{\mathrm{max}}$ | $1.38^{+0.92}_{-0.6}$
$\log_{10}u_{\mathrm{min}}$ | $0.51^{+0.36}_{-0.29}$
$\log_{10}E_{\mathrm{inj}}$ | $56.1^{+3.2}_{-2.9}$
$k$ | $7.26^{+0.41}_{-0.57}$
$\log_{10}M_{\mathrm{ej}}$ | $-8.2^{2.0}_{-1.3}$
$\log_{10}n$ | $-4.5^{+2.1}_{-2.4}$
$p$ | $2.1715^{+0.0055}_{-0.0057}$
$\log_{10}\epsilon_{e}$ | $-2.0^{+1.3}_{-1.3}$
$\log_{10}\epsilon_{B}$ | $-3.4^{+1.7}_{-1.1}$
$\log_{10}E_{tot}$ | $52.4^{+1.2}_{-1.3}$
Figure 7: Corner plot for the empirical model described in Sect. 2.2. The
prior on each of these parameters is normal, the center and standard deviation
were: $\beta$ = (1,1), $\alpha_{1}$ = (0.5,1), $\alpha_{2}$ = (-2.5,1), log
$\kappa$ = (0,0.5), log $t_{\rm peak}$ = (2,2). We report the confidence
contours derived from radio (blue) and X-ray (green) data, as the optical
light curve is only poorly constrained. The best fit model is shown in the
top-left corner.  Figure 8:  Fit result for the Gaussian jet model. This
“corner plot” shows all one-dimensional (diagonal) and two-dimensional (off-
diagonal) projections of the posterior probability density function. The best-
fit value (maximum posterior probability) is shown in blue. Dotted lines mark
the 16%, 50%, and 84% quantiles of the marginalized posteriors for each
parameter.


# Model Theory of Scattered Piecewise Interpretable Hilbert Spaces

Alexis Chevalier

###### Abstract.

We study scattered piecewise interpretable Hilbert spaces from a model
theoretic point of view. We establish strong connections between the Hilbert
space structure theorems of [CH21] and the model theoretic notions of
canonical bases, one-basedness and model theoretic ranks.

Let $T$ be a complete continuous logic theory with a piecewise interpretable
Hilbert space $\mathcal{H}$ as defined in [CH21]. The main theorem of [CH21]
shows that a piecewise $\bigwedge$-interpretable subspace of $\mathcal{H}$
generated by a _scattered_ type-definable set is the orthogonal sum of
piecewise $\bigwedge$-interpretable subspaces generated by _asymptotically
free_ complete types.

Scattered piecewise interpretable Hilbert spaces were shown to capture a wide
variety of situations of interest in model theory and beyond, including
$L^{2}$-spaces associated to absolute Galois groups or definable measures,
unitary representations of $\omega$-categorical classical logic structures,
and unitary representations of algebraic groups over local fields of
characteristic $0$. In this paper, we show that these structure theorems and
the surrounding notions are very natural from the point of view of model
theory.

We establish a rough dictionary between Hilbert space notions and model
theoretic notions. We connect _weak closure_ in Hilbert spaces and _canonical
bases_ for Hilbert space types; _scattered type-definable sets_ and a local
continuous logic version of _U-rank_ ; _commuting orthogonal projections_ and
_one-basedness_. These connections are strong but they are not exact
correspondences.

In Section 1, we recall the main definitions from [CH21]. In Section 2, we
show that the partial order on the weak closure of a type-definable set $p$
has a natural model theoretic interpretation in terms of forking relations
between types. In Section 3, we study connections between scatteredness and
the rank on the weak closure of a type-definable set. We show that the
application of Von Neumann’s lemma in [CH21] is equivalent to showing one-
basedness with respect to Hilbert space forking independence. We end by
highlighting some open questions and giving some examples of interpretable
Hilbert spaces with surprising properties.

##  1\. Main definitions and notation

For convenience, we recall some key notions from [CH21]. We will use the same
notation. In this paper we always use continuous logic, as presented in
[BdlHV08] or in the appendix of [CH21].

Recall that for $M\models T$, a Hilbert space $H(M)$ is said to be piecewise
interpretable in $M$ if $H(M)$ is a direct limit of imaginary sorts of $M$.
Since all data is assumed to be definable, any piecewise interpretable $H(M)$
in $M$ gives rise to a functor $\mathcal{H}$ which takes models of $T$ to
Hilbert spaces such that for every $N\models T$, $H(N)$ is piecewise
interpretable in $N$.

We fix $\mathcal{H}$ piecewise interpretable in $T$. In this paper, we will
move freely to imaginary sorts of $T$, so we can assume that the pieces of
$\mathcal{H}$ belong to $T$ and that the direct limit maps on each piece of
$\mathcal{H}$ are isometries. Therefore, $\mathcal{H}$ is in fact _piecewise
definable._

For any $M\models T$, we will identify vectors of $H(M)$ with elements of $M$.
When $M\models T$ and $A\subseteq M$, we write $P_{A}$ for the orthogonal
projection onto the closed subspace of $H(M)$ densely spanned by $A\cap H(M)$.

Recall that a _type-definable set in $\mathcal{H}$_ is a type-definable set
_contained in a piece of $\mathcal{H}$_. If $p$ is type-definable in
$\mathcal{H}$, we write $\mathcal{H}_{p}$ for the piecewise
$\bigwedge$-interpretable subspace of $\mathcal{H}$ generated by $p$.

We recall some important notation from [CH21].

######  Definition 1.1.

For $p$ a type-definable set in $\mathcal{H}$ and $M\models T$, we write
$\mathcal{P}(p)$ for the weak closure of the set of realisations of $p$ in
$H(M)$. When $M\models T$ is $\omega_{1}$-saturated, we define the partial
order $\leq_{\mathcal{P}}$ on $\mathcal{P}(p)$ as follows: for
$u,v\in\mathcal{P}(p)$, $v\leq u$ if and only if there are
$\mathrm{bdd}$-closed subsests $A_{1},\ldots,A_{n}$ of $M$ such that
$v=P_{A_{1}}\ldots P_{A_{n}}u$.

[CH21] shows that $\mathcal{P}(p)$ is a type-definable set in a piece of
$\mathcal{H}$. Recall also from [CH21] that when $M\models T$ is
$\omega_{1}$-saturated, $\mathcal{P}(p)$ is the set of weak limit points of
indiscernible sequences in $p$, that
$\mathcal{P}(p)=\\{P_{\mathrm{bdd}(A)}a\mid a\models p,A\subseteq M\\}$ and
that $\mathcal{P}(p)$ is closed under the projections $P_{\mathrm{bdd}(A)}$
for $A\subseteq M$.

######  Definition 1.2.

We say that a type-definable set $p$ in $\mathcal{H}$ is scattered if for some
(any) $M\models T$ $\omega_{1}$-saturated, $\mathcal{P}(p)$ is locally compact
in $M$.

We say that a type-definable set $p$ in $\mathcal{H}$ is asymptotically free
if for any $M\models T$ and $a\models p$ in $M$, for any $\epsilon>0$, the
type-definable set $p(x)\cup|\langle x,a\rangle|\geq\epsilon$ is contained in
$\mathrm{bdd}(a)$. Equivalently, for $M\models T$ $\omega_{1}$-saturated and
$a,b\models p$ in $M$, we have either $\langle a,b\rangle=0$ or
$a\in\mathrm{bdd}(b)$.

In the rest of this paper, we will work with $M\models T$, a type-definable
set $p(x)$ in $\mathcal{H}$ and _$\langle x,y\rangle$ -types over $M$
consistent with $p$_. These are partial types $q$ of the form $\\{\langle
x,b\rangle=\lambda(b)\mid b\in H(M)\\}$ where $\lambda$ is a function
$H(M)\to\mathbb{R}$. Note that this is technically a partial type in an
infinite fragment of the language, as $y$ ranges over infinitely many sorts.
In the next lemma, we show that we can always restrict $y$ to range over the
same sort as $x$.

######  Lemma 1.3.

Let $\mathcal{H}$ be piecewise interpretable in $T$ and let $M\prec N\models
T$. Let $D$ be a piece of $\mathcal{H}$ and write $\mathcal{H}_{D}$ for the
piecewise interpretable Hilbert space generated by $D$, so that
$\mathcal{H}_{D}\leq\mathcal{H}$. Then $H_{D}(N)$ is orthogonal to $H(M)$ over
$H_{D}(M)$.

###### Proof.

Let $v\in H(M)$. We need to show that $P_{H_{D}(N)}v=P_{H_{D}(M)}v$. It is
enough to show that $d(v,H_{D}(N))$, the distance from $v$ to $H_{D}(N)$,
equals $d(v,H_{D}(M))$. If $d(v,H_{D}(N))=d$, then for any $\epsilon$ there is
a formula which says that there exists $w_{1},\ldots,w_{n_{\epsilon}}$ in $D$
and scalars $\lambda_{1},\ldots,\lambda_{n_{\epsilon}}$ such that
$\|v-\sum\lambda_{i}w_{i}\|<d+\epsilon$. By elementarity, we can find such
$w_{i}$ in $M$. ∎

Using the previous lemma, we lay down the convention that _a $\langle
x,y\rangle$-type over $M$ is a partial type in the unique formula $\langle
x,y\rangle$ where $y$ is in the same sort as $x$_.

##  2\. Weak closure and canonical bases

We fix $M\models T$ $\omega_{1}$-saturated and $p$ a type-definable set in
$\mathcal{H}$. Let $b\in\mathcal{P}(p)$ and let $(a_{n})$ be a sequence in $p$
converging weakly to $b$. Define $p^{b}(x)$ to be the $\langle
x,y\rangle$-type over $M$ defined by $\langle x,c\rangle=\lim_{n}\langle
a_{n},c\rangle=\langle b,c\rangle$.

######  Lemma 2.1.

For any $M\models T$ $\omega_{1}$-saturated, $\mathcal{P}(p)$ is a set of
canonical bases for $\langle x,y\rangle$-types over $M$ consistent with $p$,
where $b$ is a canonical base for $p^{b}$.

###### Proof.

$p^{b}(x)$ is a $\langle x,y\rangle$-type over $H(M)$ consistent with $p$.
Conversely, let $q$ be any $\langle x,y\rangle$-type over $H(M)$ consistent
with $p$. Let $M\prec N$ and let $a$ be a realisation of $q(x)\cup p(x)$ in
$N$. Let $c=P_{H(M)}(a)$. Then for all $v\in H(M)$, $\langle
a,v\rangle=\langle c,v\rangle$, so $q$ is definable over $c$. We only need to
check that $c\in\mathcal{P}(p)$, so that $q=p^{c}$.

By $\omega_{1}$-saturation of $M$, find $a^{\prime}\in M$ a realisation of
$p\cup(q\upharpoonright\mathrm{bdd}(c))$. Then
$P_{\mathrm{bdd}(c)}a^{\prime}=c$ but we already know that
$P_{\mathrm{bdd}(c)}a^{\prime}\in\mathcal{P}(p)$, so we are done. ∎

Remark: We are making a slightly unconventional use of the term ‘canonical
base’. Canonical bases are usually dcl-closed sets. See for example [Pil96]
for a classical discussion of canonical bases. In Lemma 2.1, nothing ensures
that distinct $b,b^{\prime}\in\mathcal{P}(p)$ are not inter-definable.
Nevertheless, we will say that $b$ is ‘the’ canonical base of $p^{b}$. With
this choice of terminology, for any $b\neq b^{\prime}\in\mathcal{P}(p)$, $b$
and $b^{\prime}$ are canonical bases for different types over $M$ even though
they may be inter-definable.

The next lemma follows directly by applying Lemma 2.1 with the type-definable
set $\mathcal{P}(p)$ instead of $p$.

######  Lemma 2.2.

$\mathcal{P}(p)$ has built-in canonical bases for $\langle x,y\rangle$-types
over models consistent with $p$: for any $M\models T$, for any $\langle
x,y\rangle$-type $q(x)$ over $M$ consistent with $\mathcal{P}(p)$, there is a
unique $b\in\mathcal{P}(p)$ in $M$ such that $q(x)$ is defined over $b$ by
$\langle x,v\rangle=\langle b,v\rangle$ for every $v\in H(M)$. Therefore
$q=p^{b}$.

We interpret Lemma 2.2 as saying that the weak closure of a type-definable set
is a natural object from a model theoretic point of view.

We now define a partial order which gives model-theoretic significance to the
partial order $\leq_{\mathcal{P}}$ on $\mathcal{P}(p)$.

######  Definition 2.3.

Define the relation $L_{1}(x,y)$ on $\mathcal{P}(p)\times\mathcal{P}(p)$ in
$M$ by saying that $L_{1}(b,c)$ if and only if $p^{b}$ extends
$p^{c}\upharpoonright\mathrm{bdd}(c)$. Define the relation $\leq_{1}$ on
$\mathcal{P}(p)\times\mathcal{P}(p)$ as the transitive closure of $L_{1}$.

For any $c\in\mathcal{P}(p)$, $p^{c}$ is the unique nonforking extension of
$p^{c}\upharpoonright\mathrm{bdd}(c)$ to $M$, since $\langle x,y\rangle$ is a
stable relation (see [BYU10] for a discussion of stability in continuous
logic). Therefore, $L_{1}(b,c)$ and $b\neq c$ if and only if $p^{b}$ is a
forking extension of $p^{c}\upharpoonright\mathrm{bdd}(c)$.

######  Lemma 2.4.

For any $b,c\in\mathcal{P}(p)$, $L_{1}(b,c)$ if and only if there exists a
small $\mathrm{bdd}$-closed $A\subseteq M$ such that $c=P_{A}b$. Hence the
partial order $\leq_{1}$ is anti-isomorphic to $\leq_{\mathcal{P}}$.

###### Proof.

For any small $\mathrm{bdd}$-closed $A$, $c=P_{A}b$ if and only if
$c=P_{\mathrm{bdd}(c)}b$ if and only if $p^{b}$ extends
$p^{c}\upharpoonright\mathrm{bdd}(c)$. ∎

In the previous lemma, we characterised $\leq_{\mathcal{P}}$ in terms of
forking extensions of $\langle x,y\rangle$-types associated to elements of
$H(M)$. Next we show that the partial order $\leq_{\mathcal{P}}$ can also be
seen as a natural partial order between the $\langle x,y\rangle$-types
themselves.

In the following definition and lemma, a $\langle x,y\rangle$-type over a
$\mathrm{bdd}$-closed subset of $M$ is a partial type in the single function
$\langle x,y\rangle$ where $y$ ranges in the same sort as $x$ over a set
$A\subseteq M$ such that $A$ is $\mathrm{bdd}$-closed _in the sort of $x$._

######  Definition 2.5.

Define the relation $L_{2}$ on $\langle x,y\rangle$-types over
$\mathrm{bdd}$-closed subsets of $M$ consistent with $\mathcal{P}(p)$ as
follows: if $r_{1}$ is over $B$ and $r_{2}$ is over $C$, then
$L_{2}(r_{1},r_{2})$ if and only if $(r_{1}\upharpoonright M)\upharpoonright
C=r_{2}$, where $r_{2}\upharpoonright M$ is the unique nonforking extension of
$r_{1}$ to $M$. Define the relation $\leq_{2}$ between types as the transitive
closure of $L_{2}$.

For the next lemma, recall that types $r_{1},r_{2}$ over $B,C$ respectively
are said to be _parallel_ if the nonforking extensions $r_{1}\upharpoonright
M$ and $r_{2}\upharpoonright M$ are equal.

######  Lemma 2.6.

Let $B,C$ be $\mathrm{bdd}$-closed subsets of $M$. Let $r_{1}$, $r_{2}$ be
$\langle x,y\rangle$-types over $B,C$ respectively consistent with $p$ with
canonical parameters $b,c$ in $\mathcal{P}(p)$. Then $L_{2}(r_{1},r_{2})$ if
and only if $L_{1}(b,c)$.

Hence $\leq_{2}$ induces a partial order on parallelism classes of $\langle
x,y\rangle$-types over small $\mathrm{bdd}$-closed subsets of $M$ consistent
with $p$. This partial order is anti-isomorphic to $\leq_{\mathcal{P}}$.

###### Proof.

If $A\subseteq M$ is a $\mathrm{bdd}$-closed subset and $r$ is a $\langle
x,y\rangle$-type over $A$ consistent with $p$, then $r$ has a canonical base
$b\in A$ and Lemma 2.2 shows that $r=p^{b}\upharpoonright A$.

Therefore $L_{2}(r_{1},r_{2})$ if and only if $p^{b}\upharpoonright C=r_{2}$
if and only if for all $v\in C$, $\langle b,v\rangle=\langle c,v\rangle$ if
and only if $P_{C}b=c$. By Lemma 2.4, this is equivalent to $L_{1}(b,c)$.

$r_{1},r_{2}$ are parallel if and only if $r_{1}\upharpoonright
M=r_{2}\upharpoonright M$ if and only if $p^{b}=p^{c}$ if and only if $b=c$,
by Lemma 2.2. Hence $\leq_{2}$ induces a partial order on parallelism classes
and these parallelism classes are in an obvious bijection with
$\mathcal{P}(p)$. ∎

##  3\. Ranks in scattered interpretable Hilbert spaces

In this section, we fix $M\models T$ a large saturated model and $p$ a type-
definable set in $\mathcal{H}$. Recall the following theorem from [CH21]:

######  Theorem 3.1 ([CH21]).

Let $p$ be a scattered type-definable set in $\mathcal{H}$. Let $A$, $B$ be
$\mathrm{bdd}$-closed subspaces of $H(M)$. Then $A\cap H_{p}(M)$ and $B\cap
H_{p}(M)$ are orthogonal over $A\cap B\cap H_{p}(M)$. Equivalently, for any
$v\in A\cap H_{p}(M)$, we have $P_{B}v=P_{A\cap B}v$. Equivalently, for any
$v\in H_{p}(M)$,

$P_{B}P_{A}v=P_{A}P_{B}v=P_{A\cap B}v$

The proof of Theorem 3.1 hinges on an application of Von Neumann’s lemma (see
[VN50]). We refer the reader to [CH21] for the proof. We show that the
conclusion of Theorem 3.1 is equivalent to a local form of one-basedness.

######  Definition 3.2.

Let $A$ be a small $\mathrm{bdd}$-closed subspace of $H(M)$ and let $q(x)$ be
a $\langle x,y\rangle$-type over $A$ consistent with $p$. We say that $q$ is
one-based in $p$ if for any realisation $a$ of $q\cup p$ in $M$, $q$ is
definable over $A\cap\mathrm{bdd}(a)\cap H_{p}(M)$.

See [Pil96] for a general treatment of one-basedness in classical logic.

######  Lemma 3.3.

$\langle x,y\rangle$-types over $\mathrm{bdd}$-closed subsets of $M$
consistent with $p$ are one-based in $p$ if and only for all
$\mathrm{bdd}$-closed $A,B\subseteq M$, $P_{A}P_{B}=P_{A\cap B}$ on
$H_{p}(M)$.

###### Proof.

Suppose first that $\langle x,y\rangle$-types consistent with $p$ are one-
based in $p$ and take $A,B\subseteq M$ $\mathrm{bdd}$-closed. Let $a$ be any
realisation of $p$ and let $b=P_{A}a\in A$. $b$ is the canonical parameter of
the $\langle x,y\rangle$-type of $a$ over $A$. Then $P_{B}P_{A}a=P_{B}b$,
which is the canonical parameter of the $\langle x,y\rangle$-type of $b$ over
$B$. By one-basedness $P_{B}b\in B\cap\mathrm{bdd}(b)\subseteq A\cap B$.
Therefore $P_{B}P_{A}a\in A\cap B$. By the same computation as in the proof of
Theorem 3.1, we deduce $P_{A}a\in A\cap B$ and hence $P_{B}P_{A}a=P_{A\cap B}$
on $H_{p}(M)$.

Conversely, let $q$ be a $\langle x,y\rangle$-type over a
$\mathrm{bdd}$-closed $A\subseteq M$ consistent with $p$. Let $a$ be a
realisation of $q\cup p$ in $M$. Let $b=P_{A}a$ be the canonical parameter of
$q$. Then $b=P_{A}P_{\mathrm{bdd}(a)}a\in A\cap\mathrm{bdd}(a)$. ∎

Remark: In [CH21], one-basedness in interpretable Hilbert spaces was found to
be the key technical fact which made it possible to derive structure theorems
for scattered interpretable Hilbert spaces. Indeed, using one-basedness, it is
easy to prove that when $p$ is scattered,
$(\mathcal{P}(p),\leq_{\mathcal{P}})$ is well-founded, and this allows a
decomposition of $\mathcal{H}_{\mathcal{P}}$ into subspaces generated by
asymptotically free sets. Recall also from [CH21] that the structure theorem
for scattered interpretable Hilbert spaces eventually leads us to the
classification of unitary representations of oligomorphic groups given in
[Tsa12].

Conversely, it is easy to see directly that if $p$ is an asymptotically free
type-definable set then $\langle x,y\rangle$-types over $\mathrm{bdd}$-closed
sets are one-based. Therefore, in the case of classical logic
$\omega$-categorical theories, one-basedness follows from the classification
theorem of [Tsa12]. This has been well-known to several authors since
Tsankov’s original result. In the $\omega$-categorical setting, the paper
[YIT17] studies conditions under which one-basedness in interpretable Hilbert
spaces gives rise to full one-basedness for stable independence in the
classical logic $\omega$-categorical setting. See Theorem 3.12 in [YIT17].

We now turn to a general discussion of properties of the partial order
$\mathcal{P}(p)$.

######  Definition 3.4.

If $(\mathcal{Q},\leq_{\mathcal{Q}})$ is an arbitrary partial order, we define
the foundation rank $F_{\mathcal{Q}}(x)$ of $x\in\mathcal{Q}$ as follows:

  1. (1)

$F_{\mathcal{Q}}(x)\geq 0$ for all $x$

  2. (2)

$F_{\mathcal{Q}}(x)\geq\lambda$ for limit ordinal $\lambda$ if
$F(x)\geq\alpha$ for all $\alpha<\lambda$

  3. (3)

$F_{\mathcal{Q}}(x)\geq\alpha+1$ if there is $y<_{\mathcal{Q}}x$ such that
$F_{\mathcal{Q}}(y)\geq\alpha$

We say that $F_{\mathcal{Q}}(x)=\infty$ if $F_{\mathcal{Q}}(x)\geq\alpha$ for
every ordinal $\alpha$ and $F_{\mathcal{Q}}(x)=\alpha$ if
$F_{\mathcal{Q}}(x)\geq\alpha$ and $F_{\mathcal{Q}}(x)\not\geq\alpha+1$.

We write $F_{\mathcal{Q}}(\mathcal{Q})=\sup\\{F_{\mathcal{Q}}(x)\mid
x\in\mathcal{Q}\\}$.

######  Proposition 3.5.

Let $F_{\mathcal{P}}$ be the foundation rank of
$(\mathcal{P}(p),\leq_{\mathcal{P}})$. If for every $x\models p$ we have
$F_{\mathcal{P}}(x)<\infty$, then $\langle x,y\rangle$-types consistent with
$p$ are one-based in $p$.

###### Proof.

This is similar to the argument behind Theorem 3.1. For any
$\mathrm{bdd}$-closed $A,B\subseteq M$, for any $a\models p$, the sequence of
alternating projections $P_{A}P_{B}\ldots P_{A}P_{B}a$ must eventually be
constant, since $F_{\mathcal{P}}$ is ordinal-valued. The argument of Theorem
3.1 shows that we must then have $P_{A}P_{B}=P_{A\cap B}$ so $\langle
x,y\rangle$-types are one-based. ∎

In light of the anti-isomorphism between $\leq_{\mathcal{P}}$ and the partial
orders $\leq_{1}$ and $\leq_{2}$, it is natural to define a local continuous
logic version of $U$-rank which captures forking inside the Hilbert space. In
the next definition, we call this the $V$-rank. The reader is referred to
[Pil96] for a general discussion of U-rank in classical logic.

######  Definition 3.6.

For any $\mathrm{bdd}$-closed subset $A$ of $M$ and $q$ a $\langle
x,y\rangle$-type over $A$ in $\mathcal{H}$, define the relation
$V(q)\geq\alpha$ for $\alpha$ an ordinal as follows:

  1. (1)

$V(q)\geq 0$ for all $q$

  2. (2)

$V(q)\geq\lambda$ for limit ordinal $\lambda$ if for every $\alpha<\lambda$,
$V(q)\geq\alpha$

  3. (3)

$V(q)\geq\alpha+1$ if there is some $B\supseteq A$ and $q^{\prime}$ over $B$
extending $q$ such that $V(q^{\prime})\geq\alpha$ and $q^{\prime}$ forks over
$A$ with respect to the function $\langle x,y\rangle$.

If $V(q)\geq\alpha$ for all $\alpha$, we say $V(q)=\infty$. We say
$V(q)=\alpha$ if $V(q)\geq\alpha$ and $V(q)\not\geq\alpha+1$.

For $a\in H(M)$, write $V(a/A)$ for the $V$-rank of the $\langle
x,y\rangle$-type of $a$ over $A$.

######  Lemma 3.7.

Let $A$ be a small $\mathrm{bdd}$-closed subspace of $H(M)$ and let $q$ be a
$\langle x,y\rangle$-type over $A$ consistent with $p$. Let $b$ be the
canonical parameter of $q$ in $\mathcal{P}(p)$. Then $V(q)$ is equal to the
foundation rank of $b$ in $(\mathcal{P}(p),\leq_{1})$.

###### Proof.

Note that $b\in A$. We have already observed that for any
$c\in\mathcal{P}(p)$, $L_{1}(b,c)$ and $b\neq c$ if and only if $p^{b}$ is a
forking extension of $p^{c}\upharpoonright\mathrm{bdd}(c)$. The lemma follows
easily. ∎

Recall from [CH21] that if $p$ is a type-definable set in an interpretable
Hilbert space $\mathcal{H}$, we say that the inner product is _strictly
definable on $p$_ if the set of scalars $\\{\langle x,y\rangle\mid x,y\models
p\\}$ is finite. We show that in this case, the $V$-rank coincides with the
Shelah $\omega$-local rank defined in [She78]. We recall the definition of
this rank here:

######  Definition 3.8 ([She78], II.1.1).

In an arbitrary theory $T$, let $\Delta(x,y)$ be a set of formulas and $p(x)$
a type-definable set, possibly with parameters. For $\alpha$ an ordinal, we
define $R_{\Delta}(p)\geq\alpha$ as follows:

  1. (1)

$R_{\Delta}(p)\geq 0$ if $p$ is consistent

  2. (2)

For $\lambda$ a limit ordinal $R_{\Delta}(p)\geq\lambda$ if
$R_{\Delta}(p)\geq\alpha$ for all $\beta<\alpha$

  3. (3)

$R_{\Delta}(p)\geq\alpha+1$ if for every finite $p^{\prime}\subseteq p$ and
every $n<\omega$ there are $\Delta$-types $(q_{m}(x))_{m<n}$ such that:

    1. (a)

for $m\neq m^{\prime}<n$, $q_{m}(x)\cup q_{m^{\prime}}(x)$ is inconsistent

    2. (b)

$R_{\Delta}(p^{\prime}\cup q_{m})\geq\alpha$ for all $m$.

We write $R_{\Delta}(p)=\alpha$ if $R_{\Delta}(p)\geq\alpha$ and
$R_{\Delta}(p)\not\geq\alpha+1$.

Since the above definition is borrowed from [She78], we chose to keep the term
‘formula’. Since we are working in continuous logic, the term ‘condition’
might be more appropriate, to use the formalism of [BYBHU08]. Alternatively,
the formalism given in the appendix of [CH21] keeps the term ‘formula’. In any
case, when the inner product is strictly definable, any expression of the form
$\langle x,y\rangle=\lambda$ behaves like a classical logic formula, since it
admits negation. Therefore in the next proposition, we will draw on results
from classical logic and we will keep our terminology in line with this.

######  Proposition 3.9.

Suppose $S$ is a piece of $\mathcal{H}$ such that the inner product is
strictly definable on $S$. Let $\Delta$ be the finite set of formulas $\langle
x,y\rangle=\lambda$ where $x,y$ range in $S$. Then for every
$a\in\mathcal{P}(S)$ and $A\subseteq M$ $\mathrm{bdd}$-closed,
$V(a/A)=R_{\Delta}(a/A)<\omega$.

###### Proof.

This is proved entirely by using results of [She78]. We sketch the proof here
for convenience. Let $q$ be a $\langle x,y\rangle$-type over $A$ consistent
with $p$. Define the rank $R^{*}_{\Delta}(p)$ in the same way as in Definition
3.8 except that we say $R^{*}_{\Delta}(q)\geq\alpha+1$ if there are
$\Delta$-types $(q_{i})_{i<\omega}$ which are pairwise inconsistent and
$R_{\Delta}^{*}(q\cup q_{i})\geq\alpha$ for all $i$.

Now all references are from [She78]. In general for $\Delta$ a finite set of
stable formulas, we can assume that $\Delta$ is a single stable formula by
II.2.1. By II.2.7, $R_{\Delta}(q)<\omega$. We see from definitions that
$R_{\Delta}^{*}(q)\leq R_{\Delta}(q)$. By II.2.9, the property
$R_{\Delta}(q)\geq n$ is type-definable. It then follows by a compactness
argument that $R^{*}_{\Delta}(q)\geq R_{\Delta}(q)$. Therefore
$R_{\Delta}(q)=R^{*}_{\Delta}(q)$.

Now it follows directly from the definition of $R^{*}_{\Delta}$ that if
$R_{\Delta}(q)=n+1$, there is a $\Delta$-type $q^{\prime}$ over some
$B\supseteq A$ such that $R_{\Delta}(p\cup q)=n$. Furthermore, II.1.2 and
III.4.1 show that $q^{\prime}$ does not fork over $A$ if and only if
$R_{\Delta}(q^{\prime})=R_{\Delta}(q)$. The proposition follows. ∎

In the remainder of this paper, we will be interested in understanding the
relations between scatteredness, one-basedness, the $F_{\mathcal{P}}$-rank,
and the $V$-rank. In [CH21], it is shown that if $p$ is scattered, then
$(\mathcal{P}(p),\leq_{\mathcal{P}})$ is well-founded so the
$F_{\mathcal{P}}$-rank is ordinal valued. We record some other connections
between these different notions.

######  Proposition 3.10.

If $p$ is scattered then for every $x\in\mathcal{P}(p)$, $V(x)<\infty$.
Equivalently, if $p$ is scattered, there is no infinite
$\leq_{\mathcal{P}}$-increasing sequence in $\mathcal{P}(p)$.

###### Proof.

Suppose that $p$ is scattered and that $(a_{n})$ is an infinite increasing
sequence in $\mathcal{P}(p)$. Then $a_{n}=P_{\mathrm{bdd}(a_{n})}a_{n+1}$ for
all $n$. By one-basedness, $\mathrm{bdd}(a_{n})\subseteq\mathrm{bdd}(a_{n+1})$
for all $n$. Write $V_{n}$ for the subspace $\mathrm{bdd}(a_{n})$ and $W_{n}$
for the orthogonal complement of $V_{n}$ in $V_{n+1}$. Then for all $n$,
$a_{n+1}=a_{n}+w_{n}$ where $w_{n}\in W_{n}$ is orthogonal to $a_{n}$. Hence
$a_{n}=a_{0}+\sum_{i=0}^{n-1}w_{i}$.

Since every $a_{n}$ is in the type-definable set $\mathcal{P}(p)$, $\|a_{n}\|$
is bounded above and the sequence $(\|a_{n}\|)$ is convergent. Now
$\|a_{n}\|^{2}=\|a_{0}\|^{2}+\sum_{k=0}^{n-1}\|w_{k}\|^{2}$ since the vectors
$(w_{k})$ and $a_{0}$ are pairwise orthogonal. For $n\geq m$,

$\|a_{n}-a_{m}\|^{2}=\|\sum_{k=m}^{n-1}w_{n}\|^{2}=\sum_{k=m}^{n-1}\|w_{n}\|^{2}=\|a_{n}\|^{2}-\|a_{m}\|^{2}$

so $(a_{n})$ is Cauchy and hence convergent to some $a\in\mathcal{P}(p)$. Note
that $a_{n}\leq_{\mathcal{P}}a$ for all $n$. Let $\epsilon>0$ and take $n$
such that $\|a-a_{n}\|<\epsilon$. There is an indiscernible sequence $(b_{k})$
in $\mathrm{tp}(a/\mathrm{bdd}(a_{n}))$ starting at $a$ and converging weakly
to $a_{n}$. For all $k$, $\|b_{k}-a\|^{2}=2\|a\|^{2}-2\langle
b_{k},a\rangle=2\|a\|^{2}-2\langle a_{n},a\rangle$. Choosing $\epsilon$ small
enough, we find that $(b_{k})$ is arbitrarily close to $a$ and hence
$\mathcal{P}(p)$ is not scattered, a contradiction. ∎

The next proposition studies a case where scatteredness fails in a strong way.

######  Proposition 3.11.

If $p$ is a complete type and is not locally compact, then
$F_{\mathcal{P}}(a)\geq\omega$ for $a\models p$.

###### Proof.

Take $a\models p$ in $M$. Note that $a\neq 0$ and $\\{b\in\mathcal{P}(p)\mid
b<a\\}$ is nonempty. Suppose that we have
$b_{0}<_{\mathcal{P}}\ldots<_{\mathcal{P}}b_{n}<_{\mathcal{P}}a$ in
$\mathcal{P}(p)$. Suppose that $\|a-b_{n}\|=\epsilon$. Since $p$ is not
locally compact around $a$, we can find an infinite indiscernible sequence
$(a_{k})$ with $a_{0}=a$ such that $\|a_{k}-a_{j}\|=\delta<\epsilon$ for all
$k\neq j$, for arbitrarily small $\delta$. Write $c$ for the weak limit of
$(a_{k})$. Then $\|a-c\|<\|a-b_{n}\|$ so $c\neq b_{n}$. Since
$b_{n}=P_{\mathrm{bdd}(b_{n})}a$ and $c=P_{\mathrm{bdd}(c)}a$, we see that
$c\notin\mathrm{bdd}(b_{n})$. Moreover,
$\|P_{\mathrm{bdd}(b_{n})}c-b_{n}\|\leq\|c-a\|=\delta$, so we can assume that
$P_{\mathrm{bdd}(b_{n})}c\notin\mathrm{bdd}(b_{n-1})$ by taking $\delta$ small
enough.

Taking into account all distances $\|b_{k+1}-b_{k}\|$, choosing $\delta$ small
enough and writing $c_{n+1}=c$, $c_{i}=P_{\mathrm{bdd}(b_{i})}c_{i+1}$, we
have $c_{0}<_{\mathcal{P}}\ldots<_{\mathcal{P}}c_{n+1}<_{\mathcal{P}}a$. This
proves that $F_{\mathcal{P}}(a)\geq\omega$. ∎

######  Proposition 3.12.

There are examples of theories $T$ with interpretable Hilbert spaces
$\mathcal{H}$ and type-definable sets $p$ satisfying any of the following:

  1. (1)

$p$ is not locally compact, $(\mathcal{P}(p),\leq_{\mathcal{P}})$ has
foundation rank $\omega$ but $(\mathcal{P}(p),\leq_{1})$ is not well-founded

  2. (2)

$p$ is not locally compact, $(\mathcal{P}(p),\leq_{1})$ has foundation rank
$\omega$, $(\mathcal{P}(p),\leq_{\mathcal{P}})$ is not well-founded, every
type in $\mathcal{P}(p)$ is locally compact and $\langle x,y\rangle$-types
consistent with $p$ are one-based in $p$.

  3. (3)

$p$ is not locally compact, $\langle x,y\rangle$-types consistent with $p$ are
one-based in $p$ but $(\mathcal{P}(p),\leq_{\mathcal{P}})$ and
$(\mathcal{P}(p),\leq_{1})$ are not well-founded.

###### Proof.

(1) Let $T$ be the classical logic theory of a collection of equivalence
relations $E_{n}$ on a set $X$ such that $E_{0}(x,y)$ is the trivial
equivalence relation $x=x$ and $E_{n+1}$ refines each $E_{n}$-class into
infinitely many infinite $E_{n+1}$-classes. Let $M\models T$ and for every
$n\geq 0$ let $(\alpha_{k}^{n})_{k<\kappa}$ be an enumeration of the
$E_{n}$-classes of $M$. Let $H$ be the free Hilbert space on the set
$\bigcup_{k,n\geq 0}\alpha_{k}^{n}$, so that $\\{\alpha_{k}^{n}\mid k,n\geq
0\\}$ is an orthonormal basis of $H$.

Define $h:X\to H$ by $h(x)=\sum_{n\geq 0}\alpha^{n}_{k(x)}/2^{n}$ where
$\alpha^{n}_{k(x)}$ is the $E_{n}$-class of $x$. Then $h$ gives rise to an
interpretation of $H$ in $M$. Let $p$ be the quotient of $X$ by the
equivalence relation $h(x)=h(y)$. We view $p$ as a subset of $h$. Then
$\mathcal{P}(p)=p\cup\\{\sum_{n\leq m}\alpha_{k(x)}^{n}/2^{n}\mid x\in X,m\geq
0\\}$.

(2) Let $T$ be the classical logic theory of a collection of equivalence
relations $(E_{n})$ on a set $X$ such that $E_{0}(x,y)$ is the equivalence
relation $x=y$ and each $E_{n+1}$-class is an infinite union of
$E_{n}$-classes. Let $M\models T$ and for every $n\geq 0$ let
$(\alpha_{k}^{n})_{k<\kappa}$ be an enumeration of the $E_{n}$-classes of $M$.
Let $H$ be the free Hilbert space on the set $\bigcup_{k,n\geq
0}\alpha_{k}^{n}$.

Define $h:X\to H$ by $h(x)=\sum_{n\geq 0}\alpha_{k(x)}^{n}/2^{n}$ where
$\alpha_{k(x)}^{n}$ is the $E_{n}$-class of $x$. Note that $h$ is injective
and that the set $h(X)$ is discrete in $H$, since $x\neq y$ implies
$\|x-y\|\geq 1$. We also have $\mathcal{P}(X)=\\{0\\}\cup\\{\sum_{n\geq
m}\alpha_{k(x)}^{n}/2^{n}\mid x\in X,m\geq 0\\}$. $\mathcal{P}(X)$ is locally
compact everywhere except around $0$.

(3) Let $T$ be the classical logic theory of an infinite set $X$. Let
$\mathcal{H}$ be the interpretable Hilbert space such that $X$ is an
orthonormal basis of $\mathcal{H}$. Let $M\models T$ be $\omega_{1}$-saturated
as a continuous logic theory. Let $(e_{n})$ be an infinite sequence in $X$ and
let $p$ be the complete type of the vector $\sum_{n\geq 0}e_{n}/2^{n}$. Then
$\mathcal{P}(p)=\\{\sum_{n\in J}e_{n}/2^{n}\mid J\subseteq\mathbb{N}\\}$ but
$\mathcal{H}$ is one-based because $X$ is scattered. ∎

The next lemma shows that the rank $\mathcal{F}_{\mathcal{P}}$ is not usually
definable.

######  Lemma 3.13.

There exists a classical logic theory $T$ with an interpretable Hilbert space
$\mathcal{H}$ and a piece $S$ such that $S$ is scattered,
$\mathcal{F}_{\mathcal{P}}(S)=\omega$, and for every complete type $p$ in $S$,
$\mathcal{F}_{\mathcal{P}}(p)<\omega$.

###### Proof.

Let $\mathcal{L}$ be the language of a family $(E_{n})_{n\geq 1}$ of binary
relations. Consider the $\mathcal{L}$-structure $M$ satisfying the following:

  1. (1)

every relation $E_{n}$ is an equivalence relation such that every
$E_{n}$-class is infinite

  2. (2)

$E_{1}$ has countably many classes, which we write $(A_{n})_{n\geq 1}$, all of
which are infinite

  3. (3)

Fix $m\geq 1$. Then for every $1\leq k<m$, $E_{k+1}$ refines $E_{k}$ on
$A_{m}$ into infinitely many equivalence classes. For every $k\geq m$,
$E_{k+1}=E_{k}$ on $A_{m}$.

Define the function $f$ on $M\times M$ by

  1. (1)

if $x,y$ are not $E_{n}$-equivalent for any $n$, then $f(x,y)=0$. If $x,y$ are
$E_{n}$-equivalent for all $n$, then $f(x,y)=1$.

  2. (2)

if $x,y$ are both in $A_{m}$ but not $E_{n}$-equivalent for all $n$, then let

$k=\min(\max\\{n\geq 1\mid x\text{ and }y\text{ are
}E_{n}\text{-equivalent}\\},m).$

We set

$f(x,y)=\frac{m-1}{m}+\frac{1}{m(m+1-k)}.$

The exact numerical values of $f$ are not essential for our argument. Note
that as $m\to\infty$, $f(x,y)\to 1$ on $A_{m}$ and that on each $A_{m}$, $f$
takes $m$ distinct values.

In a similar way to the examples given in Proposition 3.12, one can show that
$f$ is also positive semidefinite. $f$ is clearly definable in $T=Th(M)$. Let
$\mathcal{H}$ be the interpretable Hilbert space in $T$ generated by $f$.

Note that $M$ realises all type of $S$ except one, call it $p$. $f(x,y)=1$ on
$p\times p$, so $\mathcal{F}_{\mathcal{P}}(p)=0$.

Now let $q$ be a complete type realised in $M$. Then $q$ is contained in some
definable set $A_{m}$ and it is easy to see that
$\mathcal{F}_{\mathcal{P}}(q)=m$. Therefore
$\mathcal{F}_{\mathcal{P}}(S)=\omega$. ∎

Open Question: Is it possible to find $T$ and $\mathcal{H}$ with a scattered
type-definable set $p$ such that $\omega<\mathcal{F}_{\mathcal{P}}(p)<\infty$?

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