# A SYSTEMATIC STUDY OF EFFECTS OF STELLAR ROTATION, AGE SPREAD AND BINARIES
ON COLOR-MAGNITUDE DIAGRAMS WITH EXTENDED MAIN-SEQUENCE TURN-OFFS

Zhongmu Li1,2, Caiyan Mao1, Liyun Zhang3, Xi Zhang1, Li Chen1 1Institute for
Astronomy and History of Science and Technology, Dali University, Dali 671003,
China<EMAIL_ADDRESS>2Key Laboratory for the Structure and Evolution of
Celestial Objects, Chinese Academy of Sciences, Kunming 650011, China
3College of Science/Department of Physics & NAOC-GZU-Sponsored Center for
Astronomy Research, Guizhou University, Guiyang 550025, China

###### Abstract

Stellar rotation, age spread and binary stars are thought to be three most
possible causes of the peculiar color-magnitude diagrams (CMDs) of some star
clusters, which exhibit extended main-sequence turn-offs (eMSTOs). It is far
from getting a clear answer. This paper studies the effects of three above
causes on the CMDs of star clusters systematically. A rapid stellar
evolutionary code and a recently published database of rotational effects of
single stars have been used, via an advanced stellar population synthesis
technique. As a result, we find a consistent result for rotation to recent
works, which suggests that rotation is able to explain, at least partially,
the eMSTOs of clusters, if clusters are not too old ($<$ 2.0 Gyr). In
addition, an age spread of 200 to 500 Myr reproduces extended turn-offs for
all clusters younger than 2.5 Gyr, in particular, for those younger than 2.2
Gyr. Age spread also results in extended red clumps (eRCs) for clusters
younger than 0.5 Gyr. The younger the clusters, the clearer the eRC
structures. Moreover, it is shown that binaries (including interactive
binaries) affect the spread of MSTO slightly for old clusters, but they can
contribute to the eMSTOs of clusters younger than 0.5 Gyr. Our result suggests
a possible way to disentangle the roles of stellar rotation and age spread,
i.e., checking the existence of CMDs with both eMSTO and eRC in clusters
younger than 0.5 Gyr.

Stars: evolution — Hertzsprung-Russell (HR) and C-M diagrams — globular
clusters: general

##  1 INTRODUCTION

Color-magnitude diagrams (CMDs) with extended main-sequence turn-offs
(eMSTOs), which were observed by Hubble Space Telescope (HST) in the star
clusters of the Large Magellanic Cloud (LMC) and Small Magellanic Cloud (SMC),
have received much attention in recent years. Such CMDs challenge the widely
accepted thoughts of star clusters, i.e., simple stellar population (SSP) with
a single age and metallicity (Mackey & Broby Nielsen, 2007; Mackey et al.,
2008; Milone et al., 2009; Piatti, 2013; Girardi et al., 2009). Great efforts
have been made to explain the observation. Many factors are thought to be the
reasons for the observed eMSTOs, including a spread of chemical abundance
(Mackey et al., 2008; Goudfrooij et al., 2009; Piotto et al., 2005, 2007),
capture of field stars (Mackey et al., 2008; Goudfrooij et al., 2009), merger
of existing star clusters (Mackey & Broby Nielsen, 2007), formation of a
second generation of stars from the ejecta of first generation asymptotic
giant branch stars (D’Ercole et al., 2008; Goudfrooij et al., 2009), binary
stars (e.g., Milone et al. 2009; Yang et al. 2011), observational selection
and uncertainty effects (Keller et al., 2011), mixture of stars with and
without overshooting (Girardi et al., 2011), differential reddening (Platais
et al., 2012), age spread (e.g., Girardi et al. 2011; Rubele et al. 2011;
Richer 2013), stellar rotation (e.g., Bastian & de Mink 2009), and combination
of binaries and rotation (Li et al., 2012b). A spread of age (Milone et al.,
2010; Girardi et al., 2011), rotation(Bastian & de Mink, 2009), and a
combination of rotation and binary (Li et al., 2012b) are finally thought of
as three most possible causes of the special CMDs, although some against
points are insisted by other works such as Mucciarelli et al. (2008),
Goudfrooij et al. (2011a), Goudfrooij et al. (2011b), Glatt et al. (2008),
Rubele et al. (2010), and Girardi et al. (2013). The work of Li et al. (2015a)
indicates a degeneracy of the effects of stellar binarity and rotation.

Recently, many works have investigated the effects of some possible factors
separately. In particular, stellar rotation has been widely concerned (e.g.,
Yang et al. 2013; Li et al. 2014b; Jiang et al. 2014; Goudfrooij et al. 2014;
Brandt & Huang 2015; Niederhofer et al. 2015; D’Antona et al. 2015; Bastian et
al. 2016). Yang et al. (2013) studied the effect of rotation on massive stars
via their own stellar model. They concluded that rotation does not affect the
CMD of star clusters younger than about 0.7 Gyr. A limitation of that work is
that binary evolution was not taken into account, and a comparison to the
effects of age spread and binaries was not made. However, an opposite
conclusion was gained later by Brandt & Huang (2015), which took the stellar
model of Georgy et al. (2013) and has considered gravity darkening. The works
of Niederhofer et al. (2015) and D’Antona et al. (2015) also give support to
the conclusion of Brandt & Huang (2015), but binary evolution was not taken
into account either. Li et al. (2014b) declare that NGC1651 is a genuine
simple stellar population from the observational thickness of the sub-giant
branch, but Li et al. (2015a) and in particular, Li et al. (2015b) brought
forward some doubts to it according to the best-fit results based on detailed
CMD comparisons. Correnti et al. (2014) also suggest that age spread can
better explain some CMDs than rotation. In this case, whether the peculiar
CMDs of star clusters result from stellar rotation remains unclear (Bastian et
al., 2016; Niederhofer et al., 2016). More works on the causes of eMSTOs and
stellar population types of star clusters are needed. In addition, most star
clusters possibly contain a lot of binaries, and the effects of binary
evolution, rotation and age spread are somewhat degenerated (Li et al.,
2015a). Thereby, in order to explain the observed CMDs of star clusters in a
right way, it is necessary to study the effects of stellar rotation, binary
evolution, and age spread simultaneously. This work, therefore, revisits the
effects of the above three factors on the CMDs of clusters with various ages.
Besides the application of a new population synthesis technique (Li et al.,
2015a), another improvement of this work is that a few thousand stars are
assumed for each cluster, which leads to comparable star numbers to the most
observed CMDs with eMSTO. Finally, the roles of stellar rotation, age spread,
and resolved and unresolved binaries are shown clearly and compared to each
other.

The layout of the paper is as follows: in section 2, we outline the model
assumptions and computation techniques. Then we show the effects of rotation,
binaries and age spread respectively in sections 3, 4 and 5. We finally
conclude this work with section 6.

##  2 MODEL ASSUMPTIONS AND COMPUTATION TECHNIQUES

In order to model the CMDs of star clusters in detail, we use an advanced
stellar population synthesis (hereafter ASPS) technique, which was brought
forward by Prof. Zhongmu Li and has taken into account stellar binarity,
rotation, star formation history and observational uncertainties
simultaneously (Li et al., 2015b). The model assumptions and computation
techniques are introduced as follows.

###  2.1 Initial Mass Function

Following some previous works (e.g., Li & Han 2008a, b; Li et al. 2012a,
2015a), we take the initial mass function (IMF) of Salpeter (1955)
($\phi(m)\propto m^{-2.35}$) for stellar population models. This IMF is widely
used in all kinds of stellar population synthesis studies. The lower and upper
mass limits of stars are set to 0.1 and 100 $\mathrm{M}_{\odot}$ respectively,
as stars with masses out of this range are too faint or evolve too fast to be
observed. Although the Salpeter IMF is not so accurate for low mass ($<$ 1
$\mathrm{M}_{\odot}$) stars, it will not affect the result because only bright
CMD parts are used in this work.

###  2.2 Star Sample

We build up stellar populations based on a series of basic simple stellar
populations (SSPs) with half single stars and half binaries. Each basic SSP
contains 100 000 components. The star sample of basic populations is generated
as follows. The mass of the primary component of a binary is generated
following the selected IMF, and the mass of the secondary component is then
calculated by taking a random secondary-to-primary mass ratio ($q$), which
obeys a uniform distribution within 0–1. Because eccentricity affects stellar
evolution slightly (Hurley et al., 2002), a random eccentricity ($e$) within
0–1 is assigned to each binary. The separations ($a$) of two binary components
are given by a simple shape:

$an(a)=\left\\{\begin{array}[]{lc}\alpha_{\rm sep}(a/a_{\rm 0})^{\rm m}&a\leq
a_{\rm 0};\\\ \alpha_{\rm sep},&a_{\rm 0}<a<a_{\rm 1},\\\ \end{array}\right.$
(1)

where $\alpha_{\rm sep}\approx 0.070$, $a_{\rm 0}=10R_{\odot}$, $a_{\rm
1}=5.75\times 10^{\rm 6}R_{\odot}=0.13{\rm pc}$ and $m\approx 1.2$ (Han et
al., 1995). This process leads to about 50 per cent binaries with orbital
periods less than 100 yr in a simple population. We call such simple
populations basic models and build up other stellar populations based on these
models. This method is used by many previous works, e.g., Zhang et al. (2004)
and Li & Han (2008a).

Because binary fraction is different for various star clusters, we build
stellar populations with different binary fractions. In a simple way, we
change the binary fraction of a population by removing some random single
stars or binaries from basic models. This allows one to build stellar
populations with arbitrary binary fraction between 0 and 100 per cent easily.
Note that all stars with orbital periods larger than 100 yr are considered as
single stars, as their components hardly exchange mass in their evolution
(Hurley et al., 2002).

###  2.3 Stellar Evolution

After the generation of star sample, we evolve all stars using the rapid
stellar evolution code of Hurley & Tout (1998) and Hurley et al. (2002)
(Hurley code). This code calculates the evolution of stars using some fitting
formulae, which are based on the stellar models computed by Pols et al.
(1998). Many stellar evolutionary parameters, e.g., effective temperature,
surface gravity, and luminosity, can be computed by Hurley code. There are two
advantages to take Hurley code for this work. First, both the evolution of
single and binary stars can be calculated via this code. Most binary
interactions such as mass transfer, mass accretion, common-envelope evolution,
collisions, supernova kicks, angular momentum loss mechanism, and tidal
interactions are taken into account in its binary evolution mode. Second, it
allows one to calculate the evolution of stars much faster than traditional
stellar evolutionary codes, but with enough accuracy (less than 5 per cent in
stellar luminosity, radius and core mass) for population synthesis works. Some
default parameters of Hurley code, which have been tested by Hurley & Tout
(1998) and Hurley et al. (2002), are used for this work. Because binary
evolution changes both the evolutionary tracks and main-sequence lifetimes of
stars and will directly change the CMDs of star clusters, it is necessary to
include binary evolution in CMD studies. We will be shown that binary
evolution is helpful for understanding the eMSTOs of clusters younger than 0.5
Gyr because some scatter stars form in this evolution mode. Therefore, it is
appropriate to choose Hurley code for this work. For convenience, a
metallicity of $Z$ = 0.008 is taken for simulated clusters.

###  2.4 Treatment of Stellar Rotation

Hurley code does not take stellar rotation into account, so we add the effect
of rotation on effective temperature and luminosity to some stars in the mass
range of 1.6 $\sim$ 15 $\mathrm{M}_{\odot}$ when necessary. It makes us able
to study star clusters younger than about 2 Gyr, in which CMDs with eMSTO are
usually observed. Because the database of Georgy et al. (2013) does not
include stars less massive than 1.7 $\mathrm{M}_{\odot}$, the effect of
rotation on stars between 1.6 and 1.7 $\mathrm{M}_{\odot}$ is simply
interpolated by assuming zero effect for 1.6 $\mathrm{M}_{\odot}$. However, in
order to give a reliable conclusion, only the results of clusters younger than
1.7 Gyr are used for this paper.

####  2.4.1 Inclusion of Rotation Effect

A recent result of Georgy et al. (2013) is chosen for calculating the effect
of rotation, because this database is particularly designed for constructing
synthetic stellar populations. The database is derived from the Geneva code,
which includes a full parameterization of angular momentum transport and wind
loss. This database has been used by many previous works such as Li et al.
(2015a) and Brandt & Huang (2015). The procedure to include rotation effect is
as follows. First, the changes of surface temperature and luminosity, which
are caused by stellar rotation, are calculated by comparing the evolutionary
tracks of rotating stars to those of their non-rotating counterparts. These
changes depend on metallicity, mass, rotation rate and age. Then such changes
are added to the evolutionary parameters of non-rotating stars, which are
computed via Hurley code. A limitation in the database of Georgy et al.
(2013), i.e., the stellar mass range of 1.7 $\sim$ 15 $\mathrm{M}_{\odot}$,
should be noted, as it makes us unable to study the rotational effect on
clusters older than about 1.7 Gyr reliably. Because the database of Georgy et
al. (2013) does not contain the cases for some masses, metallicities, rotation
rates, and ages, the original values are interpolated to match our needs. Note
that the effect of rotation on the main-sequence lifetimes of stars is
naturally included in our treatment. The reason is that the database of Georgy
et al. (2013) has given the change of evolutionary parameters caused by
rotation at various ages. The changes of main-sequence lifetimes can be
described by the evolutionary parameters. For example, on the Hertzsprung-
Russell (HR) diagram, a rotating star will be shown to get a turn-off point at
an older age compared to its non-rotating counterpart. Thereby, when we
correct for the evolutionary parameters of rotating stars, the effect of
rotation on main-sequence lifetime has been included. However, unlike the work
of Brandt & Huang (2015), gravity darkening has not been accounted in this
work. According to a test of Dr. Timothy D. Brandt (later Brandt), this
affects our result slightly, although gravity darkening leads to a small extra
extension of turn-offs.

####  2.4.2 Ration Rate Distribution

In order to calculate the effects of stellar rotation on the synthetic CMDs of
simulated star clusters properly, we need to assume a distribution of stellar
rotation rate ($\omega=\Omega/\Omega_{\rm crit}$) (RRD), which affects the CMD
parts near MSTO obviously. Fortunately, the works of Royer et al. (2007) and
Zorec & Royer (2012) have determined the statistical distributions of B9–F2
type stars. This enables us to assign random rotation rates to the member
stars of a population directly following some observational distributions
rather than a theoretical one. For the sake of convenient use of the observed
distributions, we fitted the results of Royer et al. (2007) (hereafter Royer
distribution) via some polynomial functions (see Fig. 1 and Table 1) and then
use these functions to represent the RRDs of B9–F2 stars. The comparison of
fitting distributions to the original results of Royer et al. (2007) indicates
that these fitting functions can reproduce the observed distributions
accurately. Stars of earlier types than B9 ($>$
$\thicksim$2.5$\mathrm{M}_{\odot}$) are assumed to have the same RRD as the
Royer result for B9 stars. An obvious trend is that the fraction of stars
decreases quickly with increasing rotation rate for $\omega>0.7$. This means
that there are only a small fraction of stars with $\omega>0.7$. Note the
results will be similar if the more recent result of Zorec & Royer (2012) is
used instead of Royer et al. (2007), because these two works, in fact, give
similar distributions for the stellar rotation rate and rotational velocity of
stars.

Figure 1: Fitting distributions of rotation rates ($\omega\geq$ 0.05) of B9-F2 type stars. The fitting formulae are based on the result of Royer et al. (2007), and are described by Table 1. $p$ and $\omega$ are star fraction and rotation rate respectively. The paper uses the B9 distribution for more massive stars. Table 1: Fitting formulae for rotation rate distributions (RRD) of B9 to F2 type stars. All distributions are described by $\frac{dp}{d\omega}=f_{0}+\sum{\frac{a_{i}}{w_{i}}.exp(-2*(\frac{\omega-c_{i}}{w_{i}})^{2})}$, where $p$ is star fraction and $\omega$ is rotation rate. Type | $f_{0}$ | $a_{1}$ | $w_{1}$ | $c_{1}$ | $a_{2}$ | $w_{2}$ | $c_{2}$
---|---|---|---|---|---|---|---
B9 | 0 | 0.91994 | 0.85301 | 0.62004 | 0.11451 | 0.07828 | 0.13371
A0-A1 | 0 | 0.29196 | 0.25220 | 0.20447 | 0.61923 | 0.61374 | 0.67926
A2-A3 | -0.99083 | 0.12616 | 0.22893 | 0.06704 | 1.99339 | 0.84887 | 0.59142
A4-A6 | 0 | 0.21978 | 0.12983 | 0.49718 | 0.60092 | 0.33494 | 0.70526
F0-F2 | 0 | 0.06730 | 0.06988 | 0.10557 | 0.75391 | 0.45133 | 0.55747
| $f_{0}$ | $a_{1}$ | $w_{1}$ | $c_{1}$ | $a_{2}$ | $w_{2}$ | $c_{2}$
A7-A9 | 0 | 0.02195 | 0.02689 | 0.07478 | 0.01771 | 0.03942 | 0.20553
|  | $a_{3}$ | $w_{3}$ | $c_{3}$ | $a_{4}$ | $w_{4}$ | $c_{4}$
|  | 0.10796 | 0.08934 | 0.41528 | 0.65859 | 0.29589 | 0.66827

###  2.5 Atmosphere Library

The stellar evolutionary parameters ([Fe/H], Teff, $\log g$, $\log L$) are
transformed into colors and magnitudes via the atmosphere library of Lejeune
et al. (1998) (BaSeL). Because this wide-wavelength band coverage (including
the well used $B$, $V$, $I$ and $K$) library was well calibrated and has been
widely used in a good deal of research works, it is an excellent choice for
this work.

##  3 EFFECT OF STELLAR ROTATION

It is necessary to check the reliability of the treatment of stellar rotation
first, as the combination of results of Hurley et al. (2002) and Georgy et al.
(2013) is used. Our test finally shows that the treatment of rotation in this
work is in well agreement with other works, e.g., Brandt & Huang (2015). Fig.
2 shows a CMD of a simulated cluster with $Z$ = 0.08 and 1.0 Gyr. The RRD of
Royer et al. (2007) is taken for this cluster and all stars are assumed to be
single stars. We see that this CMD is close to the result of Brandt & Huang
(2015).

Figure 2: Synthetic CMD of a single star stellar population with metallicity
($Z$) of 0.008 and age of 1 Gyr. The rotation effect of stars more massive
than 1.6 $\mathrm{M}_{\odot}$ is considered. All stars are assumed to follow
the RRD of Royer et al. (2007).

Figs. 3-10 then show the effect of rotation on the CMDs of simulated star
clusters with various ages from 0.2 to 1.7 Gyrs. Three binary fractions
($f_{\rm b}$ = 0.3, 0.5 and 0.7) are taken for clusters on account of the
existence of binaries in most young and intermediate-age clusters (e.g. Elson
et al. 1998). In addition, we vary rotator fraction ($f_{\rm r}$) from 0 to
Royer distribution because some star clusters without eMSTO have been
observed, and such clusters may contain less rotating stars than the result of
Royer et al. (2007). In fact, the RRD of stars in the LMC clusters is unknown,
and it is possible that some star clusters include more rotating stars than
the observation of Royer et al. (2007). We therefore enlarge the fraction of
stars with large rotation rate by taking a Gaussian distribution with mean and
standard deviation of 0.7 and 0.1 for rotation rate (hereafter Gaussian
distribution). As a whole, four rotator fractions ($f_{\rm r}$), i.e., 0, 0.5
Royer, Royer, and Gaussian are finally adopted for simulated star clusters.
The models with $f_{\rm r}$ = 0 contain no rotating stars. Those with $f_{\rm
r}$ = 0.5 Royer contain half non-rotating stars and half stars with rotation
rates following Royer distribution. For the case of $f_{\rm r}$ = Royer, all
stars obey Royer distribution. Taking various rotator fractions will produce
various kinds of CMDs, and this is helpful for understanding the role of
rotation clearly. In order to enable readers to compare the figures, Figs.
3-10 are plotted with the same size. Observational errors are not considered
to avoid their confusion.

Figs. 3 and 4 show the effect of rotating stars on the CMDs of young (0.2 and
0.4 Gyrs) star clusters. When binary fraction is small (0.3), stellar rotation
leads to only a narrow eMSTO for clusters of 0.2 Gyr, while it extends the
MSTO of clusters of 0.4 Gyr more. At the same time, binaries (including both
resolved and unresolved ones) supply significant extension on the turn-off of
such clusters. Obvious eMSTO structures are formed by including a large
fraction (e.g., 0.7) of binaries. In particular, eMSTOs become clearer when
both binaries and rotating stars are included.

Figs. 5 and 6 give the result for some clusters with ages of 0.5 and 0.8 Gyrs.
We see that rotation results in eMSTOs. The extension of MSTO is obviously
larger than the cases of 0.2–0.4 Gyrs and it increases with stellar age.
Moreover, the extension of MSTO seems very sensitive to RRD. The second case
of rotation, $f_{\rm r}$ = 0.5 Royer, leads to the largest extension. Binaries
are shown to be much less important to form eMSTO, although a few scatter
stars are generated from binaries.

Figs. 7 and 8 present the results for 1.0 and 1.2 Gyr-old simulated clusters.
The CMDs of populations with rotators become significantly different from
those of populations of non-rotating stars. If only the effect of binary stars
(including resolved and unresolved binaries) is taken into account, MSTOs are
very thin (top panels), although some scatter stars (e.g., blue stragglers and
red stragglers) are generated. The turn-offs become significantly thick when
some rotators are included in star clusters. In particular, maximum eMSTOs are
observed in clusters containing half rotating stars following the RRD of Royer
et al. (2007). Large eMSTOs can also be observed when taking Royer
distribution for all stars. One can look at the two middle lines of Figs. 7
and 8 for the details. For these clusters, their CMD parts near turn-off look
like “golf-club” (Girardi et al., 2011), in which the part above turn-off
point is wider than the lower part in color direction. The synthetic CMDs are
similar to those observed in many star clusters, e.g., NGC1399.

Figs. 9 and 10 show the cases of simulated clusters with ages of 1.5 and 1.7
Gyrs. Similar to previous figures, we are shown that stellar rotation is able
to generate eMSTOs. However, the extension of MSTOs becomes smaller compared
to clusters with ages around 1.1 Gyr (Figs. 7 and 8).

It is event from the above results that overall stellar rotation have some
effects on the CMDs of clusters between 0.2 and 1.7 Gyr, but the effect
depends on stellar age. The effect of rotation on MSTO increases with age from
0.2 to about 1.2 Gyr, then it decreases with increasing age. The result is
similar to the work of Brandt & Huang (2015). Rotation makes both the turn-off
and main sequence near turn-off wider than the non-rotating case. For
simulated clusters with ages between 1.0 and 1.5 Gyr, the CMD part including
turn-off and main sequence looks like a “golf-club” (see, e.g., 0.5 Royer),
which is similar to the observed results of many clusters, e.g., NGC 1651, NGC
1868, and NGC 1399. Therefore, stellar rotation is able to reproduce, at least
partially, the eMSTOs of star clusters. Our conclusion agrees well with the
works of Brandt & Huang (2015), D’Antona et al. (2015) and Niederhofer et al.
(2015), but is partially different from Yang et al. (2013). The different
conclusion of Yang et al. (2013) for young clusters possibly results from
their stellar evolution model. Note that if gravity darkening effect is
considered, the results will be more similar to the observed CMDs, because
gravity darkening effect can slightly extend MSTOs, according to the test of
Brandt.

From this work, we find that the role of rotation depends on many factors,
e.g., stellar evolution model, RRD and the mass range of rotating stars. In
more detail, some opposite results will be obtained from the stellar evolution
models of Yang et al. (2013) and Georgy et al. (2013). RRD affects the CMD
shapes of simulated clusters significantly (see Figs. 6–8), and considering
stars less massive than 1.7 $\mathrm{M}_{\odot}$ or not leads to various roles
of rotation for clusters older than about 1.7 Gyr. This therefore calls for
more works on these factors, in order to unfold the stellar populations of
star clusters with eMSTOs. However, gravity darkening does not significantly
affect our main conclusions.

Figure 3: Effects of rotating stars and binaries on CMDs of simulated star
clusters. Color and magnitude are in mag. All clusters have the same
metallicity ($Z$ = 0.008) and age (0.2 Gyr). Stars of each cluster (about 6000
stars) formed in a single star burst. Binary fractions ($f_{\rm b}$) in left,
middle, and right columns are 0.3, 0.5 and 0.7 respectively. From top to
bottom, the fraction of rotating stars ($f_{\rm r}$) increases. “rot-bsSSP”
means SSP with a fraction of rotating stars (including binary and single
stars). “eMSTO” denotes extended main-sequence turn-offs. “BS” and “RS” mean
blue stragglers and red stragglers. Binaries closer than 2500 AU are assumed
to be unresolved. This separation corresponds to 0.05′′ (approximately $HST$
resolution) at a distance of 50 kpc. Figure 4: Similar to Fig. 3, but for
simulated star clusters with the age of 0.4 Gyr. Figure 5: Similar to Fig. 3,
but for simulated star clusters with the age of 0.5 Gyr. Figure 6: Similar to
Fig. 3, but for simulated star clusters with the age of 0.8 Gyr. Figure 7:
Similar to Fig. 3, but for simulated star clusters with the age of 1.0 Gyr.
Figure 8: Similar to Fig. 3, but for simulated star clusters with the age of
1.2 Gyr. Figure 9: Similar to Fig. 3, but for simulated star clusters with the
age of 1.5 Gyr. Region A is an extension of MS that is caused by unresolved
binaries. Region B consists of scatter stars that form from unresolved
binaries. Region C is a part that is different from observed eMSTO CMDs. Red
points represent the CMD of a stellar population without any binaries. Figure
10: Similar to Fig. 9, but for simulated star clusters with the age of 1.7
Gyr.

##  4 EFFECT OF BINARY STARS

Some previous works (e.g., Li et al. 2014b, 2015a) have mentioned that binary
star is not the main cause of eMSTOs, but no work supplies a clear figure
about the effects of different types of binaries. This section gives an answer
via Figs. 3–10. Figs. 3 and 4 suggest that resolved and unresolved binaries
may contribute to the eMSTOs of young ($<$ 0.5 Gyr) clusters. Binary stars
lead to some blue stragglers (scatters on the upper left of turn-off, e.g., BS
in Fig. 3), red stragglers (scatters on the lower right of turn-off, e.g., RS
in Fig. 3), and scatter stars above the turn-off (e.g., Figs. 9 and 10).
However, the role of binaries depends on stellar age. For clusters older than
0.5 Gyr, scatter stars from binaries affect the turn-off shape very slightly
(Figs. 5–10). This implies that binaries is certainly not the main cause of
eMSTO of most intermediate-age clusters. As a part of binaries, resolved or
interactive binaries are correspondingly not the main cause of eMSTO. The top
panels of Figs. 9 and 10 show the effect of unresolved binaries in a clearer
way. It is obvious that unresolved binaries (black stars in region “A”) make
the MS wider compared to a population of single stars (red points), as some
unresolved binaries locate on the right of MS. In addition, a few unresolved
binaries locate above the turn-off (region “B”). However, the turn-off
structure of a population of non-rotating binaries is obviously different from
the observed CMDs, because such binary populations contain few stars in region
“C” but many stars are observed in star clusters with peculiar CMDs. Such
stars are actually an important part of eMSTOs. Therefore, binaries, including
interactive binaries, resolved binaries, and unresolved binaries are not the
main cause of eMSTO, although they can contribute to the eMSTOs of clusters
younger than 0.5 Gyr.

##  5 EFFECT OF AGE SPREAD

In this section, we study the effect of age spread on the CMDs of star
clusters with various ages. We assume that clusters form their stars within
200–500 Myrs, and the binary fractions ($f_{\rm b}$) of them change from 0.3
to 0.7. The ages of youngest stars in clusters are used as stellar population
ages, and they are given between 0.2 and 2.5 Gyr. As some examples, the main
results are presented in Figs. 11 to 17. Note that only a few separated star
bursts with age interval of 100 Myr are taken for building composite stellar
populations, but the star formation of real clusters could be continuous
bursts. A homogeneous star formation history is assumed for all simulated
clusters, although it is possibly different for real clusters. This results in
separated isochrones, and the figures look somewhat differently from the CMDs
caused by rotation (Figs. 3–10). In addition, the age spread of most star
clusters younger than 0.8 Gyr is possibly less than about 40 per cent, but in
order to supply some comparisons with older clusters, the CMDs with spreads up
to 500 Myr are shown here. This is helpful for better understanding the
dependence of effect of age spread on stellar age, and also the difference
between the effects of age spread and rotation.

We find that age spread has a significant effect on the CMDs of all clusters
younger than 2.0 Gyr (Figs. 11–16). The main effect of age spread is to
thicken the main-sequence turn-off part. In detail, the turn-off of a cluster
with multiple star bursts has obviously spread in both color and magnitude,
which is obviously different from that of an SSP of stars with the same
metallicity and age (see top panels of Figs. 3 to 10 for comparisons). As we
see, the shape of CMD part consisting of main sequence and turn-off looks like
a “golf-club”, if a cluster is younger than 2.0 Gyr and has age spreads larger
than 200 Myr. However, we see that an age spread less than 500 Myr affects the
CMDs of clusters older than about 2.5 Gyr slightly (Fig. 17). Even though an
age spread of 500 Myr is assumed, the turn-offs of clusters with age of 2.5
Gyr are similar to that of simple populations if the typical observational
errors ($\sim$ 0.014 mag) in color and magnitude are taken into account.

We can conclude from Figs. 11 to 17 that age spread does not obviously affect
the thickness of main-sequence part below turn-off, for all ages. This is a
key difference between the effects of age spread and stellar rotation, because
besides turn-off part, rotation widens the main sequence fainter than turn-
off, as shown in e.g., Figs. 4–6. This is possibly useful for checking whether
age spread exists in clusters.

Meanwhile, we observe that age spread leads to obvious extended red clumps
(eRC) (Figs. 11 and 12) for clusters younger than 0.5 Gyr. This is clearly
different from the effect of stellar rotation (see Figs. 3 and 4 for
comparison) and implies that we can possibly check the existence of age spread
from the shapes of red clumps of star clusters.

Figure 11: Effects of age spread and binary fraction on CMDs of simulated star
clusters. All clusters have the same metallicity ($Z$ = 0.008). No rotating
star is included in these clusters, and binaries closer than 2500 AU are
assumed to be unresolved. The age of latest star burst (i.e., youngest
population component), is assumed to be 0.2 Gyr. “$t$” gives the age ranges of
stars. From top to bottom panels, age spread is different from 200 to 500 Myr.
eMSTO and eRC mean extended main-sequence turn-off and extended red clump,
respectively. Binary fractions ($f_{\rm b}$) in left, middle, and right
columns are 0.3, 0.5 and 0.7 respectively. “norot-bsCSP” means CSP consisting
of non-rotating binaries and single stars. Figure 12: Similar to Fig. 11, but
for simulated star clusters with youngest population component of 0.4 Gyr.
Figure 13: Similar to Fig. 11, but for simulated star clusters with youngest
population component of 0.5 Gyr. Figure 14: Similar to Fig. 11, but for
simulated star clusters with youngest population component of 1.0 Gyr. Figure
15: Similar to Fig. 11, but for simulated star clusters with youngest
population component of 1.5 Gyr. Figure 16: Similar to Fig. 11, but for
simulated star clusters with youngest population component of 2.0 Gyr. Figure
17: Similar to Fig. 11, but for simulated star clusters with youngest
population component of 2.5 Gyr.

##  6 CONCLUSION

This paper investigates the effects of stellar rotation, age spread and binary
stars on the CMDs of simulated star clusters with various ages, via the ASPS
technique. It appears that binary star is not the main cause of eMSTO of most
clusters older than 0.5 Gyr, although a few blue stragglers and red stragglers
are produced by binary evolution, and unresolved binaries widen the MS toward
redder color. However, binaries seem important for the eMSTOs of younger
(e.g., $<$ 0.5 Gyr) clusters. Meanwhile, stellar rotation is able to explain,
at least partially, the eMSTOs of clusters. Rotation can somewhat widen the
turn-offs of clusters younger than 0.5 Gyr, and extends the turn-offs of
clusters between 0.5 and 1.7 Gyrs significantly. Because the rotation effect
of stars outside the mass range of 1.6–15 $\mathrm{M}_{\odot}$ is not taken
into account, this work does not study the effect of rotation on the CMDs of
clusters older than 1.9 Gyr. Rotation seems difficult to form CMD shapes like
“golf-club” in clusters significantly older than about 1.9 Gyr, because the
turn-off stars of such clusters, which are less massive than 1.6
$\mathrm{M}_{\odot}$, usually rotates much more slowly, and the effect of
rotation decreases with increasing age from about 1.2 Gyr (see also Brandt &
Huang 2015). This agrees well with the observational result, i.e., there is
little evidence for eMSTOs in clusters of 2 Gyr in age or older. If less
massive stars (e.g., those between 1.0 and 1.6 $\mathrm{M}_{\odot}$) are
considered, rotation widens the main sequence part below turn-off for all
simulated clusters. In addition, age spread can reproduce eMSTO for all
clusters younger than 2.5 Gyr, but it does not obviously affect the thickness
of main-sequence part fainter than turn-off. In particular, age spread results
in “golf-club” shapes in the regions near turn-off for all clusters from about
1.0 Gyr to 2.2 Gyr (tested but not shown in this paper). It also results in a
significant extension of the red clumps (eRC) of clusters younger than 0.5
Gyr. As a whole, the effects of binary stars and stellar rotation depend on
stellar age, obviously.

It can be concluded that the eMSTOs of clusters younger than 0.5 Gyr possibly
result from age spread or a combination of binaries and stellar rotation. The
eMSTOs of older clusters may be caused mainly by age spread or stellar
rotation. It is also possible that the observed features are related to all
the three factors.

Although we have used a recent result of Georgy et al. (2013) to calculate the
effects of rotation on luminosity and surface temperature of stars, it still
remains some uncertainties. This may somewhat affect the synthesized CMDs. Our
conclusion of effect of stellar rotation agrees well with recent works such as
Brandt & Huang (2015), Niederhofer et al. (2015) and Milone et al. (2016).
Even if another database of rotating stars, Ekström et al. (2012), is used
instead of Georgy et al. (2013), the results will be similar, because the two
databases are from the same code. Note that although the works of Bastian & de
Mink (2009), Ekström et al. (2012), and Georgy et al. (2013) suggest the
possibility of rotation to explain eMSTOs of star clusters, the rotation
effect of two recent works, i.e., Ekström et al. (2012) and Georgy et al.
(2013) is different from Bastian & de Mink (2009) (see Brandt & Huang 2015).
Besides the stellar evolutionary models, RRD affects the final CMDs evidently.
This calls for accurate distributions for clusters in LMC and SMC. Moreover,
this work considers the effects of binaries and rotating stars separately, but
they are actually related. Although such treatment is reasonable by now due to
the lack of a stellar evolutionary code, in which both binaries and rotation
have been taken into account, it is certainly necessary to check the results
using some new codes in the future.

This work has been supported by the Chinese National Science Foundation (Grant
No. 11203005, 11563002, and U1431114), and the open projects of Key Laboratory
for the Structure and Evolution of Celestial Objects, Chinese Academy of
Sciences, and the Key Laboratory of semi analytical systems of galaxies and
Cosmology, Shanghai. The authors are grateful to Dr. Timothy D. Brandt for
useful comments and some test results, and Drs. Zhanwen Han and Xiaoying Pang
for suggestion.

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# A Scattering Result of the Radial Defocusing Cubic Schrodinger Equation in
the Hyperbolic Space with Rough Data

Chutian Ma<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the defocusing cubic Schrödinger equation on three
dimensional hyperbolic space $\mathbb{H}^{3}$ with radial initial data in the
Sobolev Space $H^{s}(0<s<1)$. Our main result is that the initial value
problem is globally wellposed and scatters for $\frac{15}{16}<s<1$. This is an
extension of the high-low method of Bourgain and the work of Staffilani and Yu
to the three dimensional hyperbolic space.

##  1\. Introduction

In this paper we study the cubic dofocusing Schrödinger equation on the
hyperbolic space $\mathbb{H}^{3}$ with initial data that is radial and lies in
the Sobolev space $H^{s}$, $0<s<1$:

(1.1) $\left\\{\begin{aligned} &iu_{t}+\Delta_{\mathbb{H}^{3}}u=|u|^{2}u\\\
&u(0,x)=u_{0}(x)\end{aligned}\right.$

where $u_{0}$ is radial and in the Sobolev space $H^{s}$ with $s<1$.
Defocusing NLS with power nonlinearity has been widely studied on Euclidean
space. Numerous mass critical and energy critical results are proved on
Euclidean space ([Dod12a],[Dod16],[Dod19b], [CKS+04],[Vis07],[RV07]). Various
results regarding the equation with rough data are also available on Euclidean
space, including ([Bou98],[CKS+02],[CKS+04], [CKS+07],[Dod12b],[FG07],
[GC06]). The equation is of particular interest in the Hyperbolic space, due
to the fact that the negative curvature of the manifold results in stronger
dispersive estimates and Strichartz estimates. Weighted dispersive and
Strichartz estimates for Schrödinger equation were obtained in [Ban07] and
[Pie08]. $L^{2}$ scattering and $H^{1}$ subcritical scattering results on
$\mathbb{H}^{3}$ were proved in [BCS08], [AP09] and later $H^{1}$ critical
scattering result on $\mathbb{H}^{3}$ was proved in [IPS12]. Also [BD14]
obtained wellposedness and blow-up results for the focusing NLS. Recently, a
rough data scattering result on $\mathbb{H}^{2}$ was proved in [SY20] using
the frequency high-low method. It was shown that the cubic defocusing NLS with
radial rough data scatters if the initial data lies beyond $H^{\frac{3}{4}}$.
This paper aims to extend their result to three dimension.
The global existence of (1.1) is equivalent to proving the boundedness of the
$H^{\frac{1}{2}}$ norm. In the case when $s\geq 1$, this follows from the
energy conservation and mass conservation. The energy $E(u)$ and mass $M(u)$
are defined by

(1.2) $E(u)=\frac{1}{2}\int_{\mathbb{H}^{3}}|\nabla
u|^{2}dx+\frac{1}{4}\int_{\mathbb{H}^{3}}|u|^{4}dx$ (1.3)
$M(u)=\int_{\mathbb{H}^{3}}|u|^{2}dx$

In the case when $s<1$, the energy is not defined. Thus (1.1) lack a conserved
quantity that controls the $H^{\frac{1}{2}}$ norm. To overcome this
difficulty, we split the initial data into two halves and prove an almost
conservation law for the low frequency evolution. This paper follows the same
idea as in [SY20]. Our main result is the following theorem:

######  Theorem 1.1.

The initial value problem (1.1) is globally wellposed and scatters if
$s>\frac{15}{16}$.

Theorem 1.1 will follow directly from Proposition 4.1 by a continuity
argument.

##  2\. Preliminary Results

In this section we present some of the preliminary results we will need later.

###  2.1. Strichartz Estimates

######  Definition 2.1 (Strichart Spaces).

On $\mathbb{H}^{d}$, We say that (p,q) is an admissible pair if $p,q\geq 2$
and $\frac{2}{p}+\frac{d}{q}=\frac{d}{2}$ and $(p,q,d)\neq(2,+\infty,2)$.
Denote p’,q’ to be the dual of p,q. For $\sigma\in\mathbb{R}$, we define the
Strichartz norm $\|\cdot\|_{S^{\sigma}}$ and its dual $\|\cdot\|_{N^{\sigma}}$
to be

$\|f\|_{S^{\sigma}}=\sup\\{\||\nabla|^{\sigma}f\|_{L^{p}_{t}L^{q}_{x}}\ |\
{(p,q)\ is\ admissible}\\}$
$\|F\|_{N^{\sigma}}=\inf\\{\||\nabla|^{\sigma}F\|_{L^{p^{\prime}}_{t}L^{q^{\prime}}_{x}}\
|\ {(p,q)\ is\ admissible}\\}$

######  Theorem 2.2 (Strichartz Estimate on $\mathbb{H}^{d}$ in [IS09]).

Suppose $\frac{2}{p}+\frac{d}{q}\geq\frac{d}{2}$, where (p,q)
$(\tilde{p},\tilde{q})$ are admissable, Then we have the following estimates

$\|e^{it\Delta}f\|_{L^{p}_{t}L^{q}_{x}}\leq_{d,p,q}\|f\|_{L^{2}_{x}}$

$\|\int_{s<t}e^{i(t-s)\Delta}F(s)ds\|_{L^{p}_{t}L^{q}_{x}}\lesssim_{d,p,q,\tilde{p},\tilde{q}}\|F\|_{L^{\tilde{p}^{\prime}}_{t}L^{\tilde{q}^{\prime}}_{x}}$

We will also need the following inhomogeneous Strichartz estimate which
extends the possible choices of admissible pairs.

######  Theorem 2.3 (Inhomogeneous Strichartz Estimate, [Dod19a]).

Suppose $d(\frac{1}{2}-\frac{1}{q})<1$, where $\tilde{p},p<\infty$, and
$d(\frac{1}{2}-\frac{1}{q})=\frac{1}{p}+\frac{1}{\tilde{p}}$. Then we have the
following estimate

(2.1)
$\|\int_{s<t}e^{i(t-s)\Delta}F(s)ds\|_{L^{p}_{t}L^{q}_{x}}\lesssim_{d,p,\tilde{p},q}\|F\|_{L^{\tilde{p}^{\prime}}_{t}L^{q^{\prime}}_{x}}$

######  Remark 2.4.

Theorem 2.3 is proved in [Dod19a] under the Euclidean settings. However, the
same proof is valid on general manifolds.

###  2.2. Local Smoothing Estimates

The Schrodinger evolution has the following local smoothing effect due to its
dispersive nature.

######  Theorem 2.5 (Local Smoothing Estimates in $\mathbb{H}^{3}$, [LLOS18]
[Kai14]).

For any $\delta>0$, We have

(2.2) $\|\langle
x\rangle^{-\frac{1}{2}-\delta}|\nabla|e^{it\Delta}f\|_{L^{2}_{t,x}}\lesssim\||\nabla|^{\frac{1}{2}}f\|_{L^{2}_{x}}$

and

(2.3) $\|\langle
x\rangle^{-\frac{1}{2}-\delta}|\nabla|\int_{s<t}e^{i(t-s)\Delta}F(s)ds\|_{L^{2}_{t,x}}\lesssim\|\langle
x\rangle^{\frac{1}{2}+\delta}F\|_{L^{2}_{t,x}}$

######  Corollary 2.6.

For $0\leq\alpha\leq 1$,
$\frac{1}{r}=\frac{\alpha}{2}+\frac{3(1-\alpha)}{10}$, we have

(2.4) $\|\langle
x\rangle^{-(\frac{1}{2}+\delta)\alpha}|\nabla|e^{it\Delta}f\|_{L^{r}_{t,x}}\lesssim\||\nabla|^{1-\frac{1}{2}\alpha}f\|_{L^{2}_{x}}$

Proof: Interpolate (2.2) with $\|\nabla
e^{it\Delta}f\|_{L^{\frac{10}{3}}_{t,x}}\lesssim\|\nabla f\|_{L^{2}_{x}}$
yields the result.

###  2.3. Heat-Flow Based Frequency Projection

As in [LLOS18], heat flow based freqency projection operators are defined by

$\displaystyle P_{\geq s}f=e^{s\Delta}f$ $\displaystyle P_{<s}f=f-P_{\geq s}f$
$\displaystyle P_{s}f=(-s\Delta)e^{s\Delta}f$

These operators serve as a substitute of the Littlewood Paley frequency
projections in the Euclidean space. Operators of the form $\chi(-\Delta)$,
where $\chi$ is a compactly supported bump function, might seem like a more
straightforward choice. But unfortunately, the latter are not $L^{p}$ bounded
in general. $P_{\geq s}$ can be viewed roughly as frequency localization to
frequency $\leq s^{-\frac{1}{2}}$, whereas $P_{s}$ is the frequency
localization to frequency $\sim s^{-\frac{1}{2}}$, in the sense that they
satisfy the following Bernstein inequalities:

######  Proposition 2.7 (Bernstein Inequality).

$\displaystyle\|P_{<s}f\|_{L^{p}_{x}}\lesssim s^{\frac{1}{2}}\|\nabla
f\|_{L^{p}_{x}}$ $\displaystyle\|\nabla P_{\geq s}f\|_{L^{p}_{x}}\lesssim
s^{-\frac{1}{2}}\|f\|_{L^{p}_{x}}$ $\displaystyle\|\nabla
P_{s}f\|_{L^{p}_{x}}\sim s^{-\frac{1}{2}}\|f\|_{L^{p}_{x}}$

###  2.4. Radial Sobolev Embeddings on $\mathbb{H}^{3}$

We will need the following radial Sobolev inequality.

######  Proposition 2.8 (Corollary 2.22 in [SY20]).

For any $\frac{1}{4}<\alpha<1$ and f radial,

$\|sinh(r)f\|_{L^{\infty}_{x}(\mathbb{H}^{3})}\lesssim\|f\|_{L^{2}_{x}(\mathbb{H}^{3})}^{1-\frac{1}{4\alpha}}\|(-\Delta)^{\alpha}f\|_{L^{2}_{x}(\mathbb{H}^{3})}^{\frac{1}{4\alpha}}$

Proof: The proof in [SY20] can be modified with little effort to hold in
$\mathbb{H}^{3}$, with the radial term $(sinh(r))^{\frac{1}{2}}$ replaced by
$sinh(r)$.

##  3\. Energy Increment Estimate

In this section we treat the solution of (1.1) by splitting it into two
different evolutions and proving an energy bound for the nonlinear one.
Section 3.1 is devoted to local estimates on time interval small enough.
Section 3.2 extends the estimate of Section 3.1 to any interval where solution
exists. Section 3.3 proves a technical result we needed in Section 3.1.

###  3.1. Local Estimates on Small Interval

In this subsection, we study the solution to (1.1) with initial data $u_{0}$
on a small interval I, where the scattering norm of the solution is bounded by
some small constant

(3.1) $\|u\|_{L^{8}_{t}L^{4}_{x}(I)}^{8}\leq\epsilon$

$0<\epsilon=\epsilon(u_{0})\ll 1$ is a small constant to be determined later.
To begin with, we decompose the initial data $u_{0}$ by

(3.2) $u_{0}=P_{<s_{0}}u_{0}+P_{\geq s_{0}}u_{0}$

where $s_{0}=s_{0}(u_{0})$ is to be chosen later and let $\psi$ be the linear
evolution of $P_{<s}u_{0}$. In other words, $\psi(t,x)$ is the solution to

(3.3) $\left\\{\begin{aligned} &i\psi_{t}+\Delta_{\mathbb{H}^{3}}\psi=0\\\
&\psi(0,x)=P_{<s_{0}}u_{0}\end{aligned}\right.$

We define $\phi(t,x)$ and v(t,x) to be the solution to the following nonlinear
solutions respectively:

(3.4) $\left\\{\begin{aligned}
&i\phi_{t}+\Delta_{\mathbb{H}^{3}}\phi=|\phi|^{2}\phi\\\ &\psi(0,x)=P_{\geq
s_{0}}u_{0}\end{aligned}\right.$

and

(3.5) $\left\\{\begin{aligned}
&iv_{t}+\Delta_{\mathbb{H}^{3}}v=|u|^{2}u-|\phi|^{2}\phi\\\
&v(0,x)=0\end{aligned}\right.$

Write $\zeta=\phi+v$.

Under the assumption that

(3.6) $E(\phi)\sim s_{0}^{-(1-s)}$

we have the follwing estimates on the Strichartz norm of $\psi,\phi$ and v.

######  Proposition 3.1.

We have the following estimates

(3.7) $\displaystyle\|\psi\|_{S^{\sigma}(I)}\lesssim
s_{0}^{\frac{1}{2}(s-\sigma)},\ 0\leq\sigma\leq s$
$\displaystyle\|\phi\|_{S^{\sigma}(I)}\lesssim
s_{0}^{-\frac{\sigma}{2}(1-s)},\ 0\leq\sigma\leq 1$

In particular,

(3.8) $\displaystyle\|v\|_{L^{8}_{t}L^{4}_{x}(I)}\lesssim
s_{0}^{\frac{1}{2}(s-\frac{1}{2})}$
$\displaystyle\|\phi\|_{L^{8}_{t}L^{4}_{x}(I)}\lesssim\epsilon$

Proof: The first inequality in (3.7) follows from Strichartz estimate and
Bernstein inequality.
Now we prove (3.8) first. Note that

(3.9)
$iv_{t}+\Delta_{\mathbb{H}^{3}}v=\mathcal{O}(\psi^{3}+v^{3}+\phi^{2}\psi+\phi^{2}v)$

where we abuse notation since the complex conjugate is not significant in our
estimates. We utilize Thm 2.3 with p=8, q=4

(3.10) $\displaystyle\|v\|_{L^{8}_{t}L^{4}_{x}}$
$\displaystyle\lesssim\|\psi\|^{3}_{L^{8}_{t}L^{4}_{x}}+\|v\|^{3}_{L^{8}_{t}L^{4}_{x}}+\|\phi\|^{2}_{L^{8}_{t}L^{4}_{x}}\|\psi\|_{L^{8}_{t}L^{4}_{x}}+\|\phi\|^{2}_{L^{8}_{t}L^{4}_{x}}\|v\|_{L^{8}_{t}L^{4}_{x}}$
$\displaystyle\lesssim
s_{0}^{\frac{3}{2}(s-\frac{1}{2})}+\|v\|^{3}_{L^{8}_{t}L^{4}_{x}}+(\epsilon+\|v\|^{2}_{L^{8}_{t}L^{4}_{x}})^{2}s_{0}^{\frac{1}{2}(s-\frac{1}{2})}+(\epsilon+\|v\|_{L^{8}_{t}L^{4}_{x}})^{2}\|v\|_{L^{8}_{t}L^{4}_{x}}$

We get the first inequality in (3.8) after absorbing the terms with small
coefficient or higher order exponent into the left hand side. The second
inequality in (3.8) follows from the fact that $\phi=u-\psi-v$ and (3.1). With
the smallness of $\|\phi\|_{L^{8}_{t}L^{4}_{x}}$, the second inequality in
(3.7) is a straightforward result of Strichartz inequalities.

######  Corollary 3.2 (Radial inequalities for $\phi$ and v).

$\displaystyle\|sinh(r)\psi\|_{L^{\infty}_{t,x}}\lesssim
s_{0}^{\frac{1}{2}s-\frac{1}{4}}$
$\displaystyle\|sinh(r)\phi\|_{L^{\infty}_{t,x}}\lesssim
s_{0}^{-\frac{1}{4}(1-s)}$

Proof: This is a direct result from proposition 2.8 and the local estimates
above.

######  Proposition 3.3 ($H^{\sigma}$ estimate of v when $0\leq\sigma\leq s$).

(3.11) $\|v\|_{S^{\sigma}(I)}\lesssim s_{0}^{\frac{1}{2}(\sigma
s+s-\sigma-\frac{1}{2})}$

Proof: Differentiate (3.5) with $|\nabla|^{s}$, we get

$\displaystyle\||\nabla|^{\sigma}v\|_{S^{0}}$
$\displaystyle\lesssim\||\nabla|^{\sigma}\psi\|_{S^{0}}\|\psi\|^{2}_{L^{8}_{t}L^{4}_{x}}+\||\nabla|^{\sigma}v\|_{S^{0}}\|v\|^{2}_{L^{8}_{t}L^{4}_{x}}+\||\nabla|^{\sigma}\phi\|_{S^{0}}\|\phi\|_{L^{8}_{t}L^{4}_{x}}\|\psi\|_{L^{8}_{t}L^{4}_{x}}$
$\displaystyle+\||\nabla|^{\sigma}\psi\|_{S^{0}}\|\phi\|^{2}_{L^{8}_{t}L^{4}_{x}}+\|\phi\|^{2}_{L^{8}_{t}L^{4}_{x}}\||\nabla|^{s}v\|_{S^{0}}+\||\nabla|^{\sigma}\phi\|_{S^{0}}\|\phi\|_{L^{8}_{t}L^{4}_{x}}\|v\|_{L^{8}_{t}L^{4}_{x}}$
$\displaystyle\lesssim s_{0}^{\frac{1}{2}(\sigma
s+s-\sigma-\frac{1}{2})}+o(1)\|v\|_{S^{\sigma}}$

######  Proposition 3.4 ($H^{1}$ estimate of v).

(3.12) $\|v\|_{S^{1}(I)}\lesssim s_{0}^{s-\frac{7}{8}}$

Let us postpone the proof of prop 3.2 to later. For now, let us assume
Proposition 3.4. We are ready to study the solution on longer intervals by
cutting the interval into small subintervals, on each of which our local
smallness assumption 3.1 holds.

###  3.2. Conditional Global Energy Estimates

Suppose the solution exists on the interval I and the scattering norm is
bounded by some $M>0$

(3.13) $\|u\|^{8}_{L^{8}_{t}L^{4}_{x}}([0,T])\leq M$

Divide I into subintervals $I_{j}$ such that on each $I_{j}$ (3.1) holds. On
$I_{j-1}$, evolve $\phi$ and v by the respective equations (3.4) and (3.5).
After the evolution on $I_{j-1}$ is done, we dump the data of v into $\phi$
and evolves $\phi$ with the new initial data on $I_{j}$, whereas v starts with
0 initial data again. In other words, for $t\in(b_{j},b_{j+1})$, if we denote
$\phi(b_{j}^{+})=lim_{t\downarrow b_{j}}\phi(t)$ and
$\phi(b_{j}^{-})=lim_{t\uparrow b_{j}}\phi(t)$, $\phi$ and v satisfy

$\left\\{\begin{aligned}
&i\phi_{t}+\Delta_{\mathbb{H}^{3}}\phi=|\phi|^{2}\phi\\\
&\phi(b_{j}^{+},x)=\phi(b_{j}^{-},x)+v(b_{j}^{-},x)\end{aligned}\right.$

and

$\left\\{\begin{aligned}
&iv_{t}+\Delta_{\mathbb{H}^{3}}v=|u|^{2}u-|\phi|^{2}\phi\\\
&v(b_{j}^{+},x)=0\end{aligned}\right.$

We can repeat the evolution by the process described above throughout the
whole interval I. The local estimates, namely Proposition 3.1-3.4, will hold
true on each $I_{j}$ as long as our assumption (3.6) remains valid. In other
words, the energy increment of $\phi$ resulting from the data of v has to be
comparable to $s_{0}^{-(1-s)}$ in order for our local estimates to remain
valid. In fact, under (3.6), We have the following conditional global energy
increment estimate:

######  Proposition 3.5.

Suppose $T>0$ such that $\|u\|^{8}_{L^{8}_{t}L^{4}_{x}}([0,T])\leq M$ where
$M\sim s_{0}^{-\frac{1}{2}s+\frac{3}{8}}$. Then we have

(3.14) $E(\phi(T))\leq
E(\phi(0))+C\frac{M}{\epsilon}s_{0}^{\frac{3}{2}s-\frac{11}{8}}\sim
s_{0}^{-(1-s)}$

Proof: Divide [0,T] into subintervals $I_{j}=[b_{j-1},b_{j}]$ such that
$\|u\|^{8}_{L^{8}_{t}L^{4}_{x}(I_{j})}\leq\epsilon$. The number of such
intervals is at most $\mathcal{O}(\frac{M}{\epsilon})$. During each iteration,
the energy increment of $\phi$ is

(3.15) $\displaystyle E(\phi(b_{j})+v(b_{j}))-E(\phi(b_{j-1}))$
$\displaystyle\leq\frac{1}{2}\int|\nabla\phi(b_{j})+\nabla
v(b_{j})|^{2}dx-\frac{1}{2}\int|\nabla\phi(b_{j})|^{2}dx$
$\displaystyle+\frac{1}{4}\int|\phi(b_{j})+v(b_{j})|^{4}dx-\frac{1}{4}\int|\phi(b_{j})|^{4}dx$
$\displaystyle\leq\int|\nabla\phi(b_{j})||\nabla v(b_{j})|+|\nabla
v(b_{j})|^{2}dx+\int|\phi(b_{j})|^{3}|v(b_{j})|+|\phi(b_{j})||v(b_{j})^{3}|dx$
$\displaystyle=I+II$

$\displaystyle I$
$\displaystyle\leq\|\phi\|_{S^{1}(I)}\|v\|_{S^{1}(I)}+\|v\|_{S^{1}(I)}^{2}$
$\displaystyle\lesssim s_{0}^{-\frac{1}{2}(1-s)}\cdot s_{0}^{s-\frac{7}{8}}$
$\displaystyle\lesssim s_{0}^{\frac{3}{2}s-\frac{11}{8}}$ $\displaystyle II$
$\displaystyle\leq\|\phi\|_{L^{\infty}_{t}L^{4}_{x}(I)}^{3}\|v\|_{L^{\infty}_{t}L^{4}_{x}(I)}+\|v\|_{L^{\infty}_{t}L^{4}_{x}(I)}^{4}$
$\displaystyle\lesssim
E(\phi)^{\frac{3}{4}}\|v\|_{L^{\infty}_{t}H^{\frac{3}{4}}_{x}(I)}+\|v\|_{L^{\infty}_{t}H^{\frac{3}{4}}_{x}(I)}^{4}$
$\displaystyle\lesssim s_{0}^{\frac{7}{2}s-\frac{5}{2}}$

Combining these terms,

(3.16) $\Delta E\lesssim s_{0}^{\frac{3}{2}s-\frac{11}{8}}$

Thus we obtain the local energy increment $\Delta E$ on each interval $I_{j}$.
To justify using the local estimates, we need to ensure that 3.6 holds
throughout I.

$\frac{M}{\epsilon}\sup_{j}\Delta
E(\phi)\lesssim\frac{M}{\epsilon}s_{0}^{\frac{3}{2}s-\frac{11}{8}}\lesssim
s_{0}^{-(1-s)}$ (3.17) $\implies M\sim s_{0}^{-\frac{1}{2}s+\frac{3}{8}}$

This is the restriction on M arises from.

There are at most $\mathcal{O}(\frac{M}{\epsilon})$ such intervals. Thus 3.14
holds under the assumption on M.

###  3.3. Proof of Proposition 3.4

It remains to prove Proposition 3.4. Take derivative of (3.3),

(3.18) $\left\\{\begin{aligned} &i\nabla v_{t}+\Delta\nabla
v=\mathcal{O}(\nabla\psi\cdot\psi^{2}+\nabla v\cdot
v^{2}+\nabla\phi\cdot\phi\psi+\nabla\psi\cdot\phi^{2}+\nabla\phi\cdot\phi
v+\nabla v\cdot\phi^{2})\\\ &\nabla v(0)=0\end{aligned}\right.$

(3.19) $\|v\|_{S^{1}(I)}\lesssim I+II+III+IV$

where

(3.20) $\displaystyle I=\|\nabla\psi\cdot\psi^{2}\|_{N^{0}(I)}$ $\displaystyle
II=\|\nabla v\cdot v^{2}\|_{N^{0}(I)}$ $\displaystyle
III=\|\nabla\phi\cdot\phi\psi\|_{N^{0}(I)}+\|\nabla\psi\cdot\phi^{2}\|_{N^{0}(I)}$
$\displaystyle IV=\|\nabla\phi\cdot\phi v\|_{N^{0}(I)}+\|\nabla
v\cdot\phi^{2}\|_{N^{0}(I)}$

Estimates for I:

(3.21) $\displaystyle I\leq\|\langle
x\rangle^{-(\frac{1}{2}+\delta)\alpha}\nabla\psi\|_{L^{r}_{t,x}}\|\langle
x\rangle^{(\frac{1}{2}+\delta)\alpha}\psi^{2}\|_{L^{r_{1}}_{t,x}}$

We take $\alpha=\frac{1}{2}$,
$\frac{1}{r}=\frac{\alpha}{5}+\frac{3}{10}=\frac{2}{5}$ and
$\frac{1}{r_{1}}=\frac{2}{5}-\frac{\alpha}{5}=\frac{3}{10}$. Now we apply
local smoothing (Corollary 2.4) to estimate the first term in I. Suppose
$s\geq\frac{3}{4}$,

(3.22) $\|\langle
x\rangle^{-\frac{1}{2}(\frac{1}{2}+\delta)}\nabla\psi\|_{L^{r}_{t,x}}\lesssim\||\nabla|^{\frac{3}{4}}\psi_{0}\|_{L^{2}_{x}}\lesssim
s_{0}^{\frac{1}{2}(s-\frac{3}{4})}$

For the second term in I,

(3.23) $\displaystyle\|\langle
x\rangle^{\frac{1}{2}(\frac{1}{2}+\delta)}\psi^{2}\|_{L^{r_{1}}_{t,x}(B_{1})}$
$\displaystyle\lesssim\|\psi\|^{2}_{L^{2r_{1}}_{t,x}}$
$\displaystyle\lesssim\|\psi_{0}\|^{2}_{H^{\frac{3}{4}}}$
$\displaystyle\lesssim s_{0}^{s-\frac{3}{4}}$ $\displaystyle\|\langle
x\rangle^{\frac{1}{2}(\frac{1}{2}+\delta)}\psi^{2}\|_{L^{r_{1}}_{t,x}(B_{1}^{c})}$
$\displaystyle\leq\|sinhr\psi\|_{L^{\infty}_{t,x}}\|\psi\|_{S^{0}}$
$\displaystyle\lesssim s_{0}^{\frac{1}{2}s-\frac{1}{4}}\cdot
s_{0}^{\frac{1}{2}s}$ $\displaystyle\lesssim s_{0}^{s-\frac{1}{4}}$

Thus

(3.24) $I\lesssim s_{0}^{\frac{1}{2}(s-\frac{3}{4})}\cdot
s_{0}^{s-\frac{3}{4}}\leq s_{0}^{\frac{3}{2}(s-\frac{3}{4})}$

Estimates for II:

(3.25) $II\leq\|\nabla v\|_{S^{1}}\|v\|_{L^{8}_{t}L^{4}_{x}}^{2}\lesssim
o(1)\|v\|_{S^{1}}$

which is absorbed into the left hand side.

Estimates for III: We estimate the first term in III in the similar fashion as
in I. Under the asumption that $\frac{1}{2}+\frac{\alpha}{2}\leq s$,

(3.26) $\displaystyle\|\nabla\psi\cdot\phi^{2}\|_{N^{0}}$
$\displaystyle\leq\|\langle
x\rangle^{-\frac{1}{2}(\frac{1}{2}+\delta)}\nabla\psi\|_{L^{r}_{t,x}}\|\langle
x\rangle^{\frac{1}{2}(\frac{1}{2}+\delta)}\phi^{2}\|_{L^{r_{1}}_{t,x}}$
$\displaystyle\lesssim\||\nabla|^{\frac{3}{4}}\psi_{0}\|_{L^{2}_{x}}\left[\|\phi\|^{2}_{L^{2r_{1}}_{t,x}(B_{1})}+\|sinhr\phi\|_{L^{\infty}_{t,x}}\|\phi\|_{L^{r_{1}}_{t,x}(B_{1}^{c})}\right]$
$\displaystyle\lesssim\||\nabla|^{1-\frac{1}{2}\alpha}\psi_{0}\|_{L^{2}_{x}}\left[\|\phi\|_{S^{1}}\phi\|_{L^{8}_{t}L^{4}_{x}}+\|sinhr\phi\|_{L^{\infty}_{t,x}}\|\phi\|_{S^{0}}\right]$
$\displaystyle\lesssim
s_{0}^{\frac{1}{2}(s-\frac{3}{4})}[s_{0}^{-\frac{1}{2}(1-s)}+s_{0}^{-\frac{1}{4}(1-s)}]$
$\displaystyle\lesssim s_{0}^{s-\frac{7}{8}}$

The second term can be estimated by

(3.27) $\displaystyle\|\nabla\phi\cdot\phi\psi\|_{N^{0}}$
$\displaystyle\leq\|\nabla\phi\|_{L^{\infty}_{t}L^{2}_{x}}\|\phi\|_{L^{8}_{t}L^{4}_{x}}\|\psi\|_{S^{\frac{1}{2}}}$
$\displaystyle\lesssim s_{0}^{-\frac{1}{2}(1-s)}\cdot 1\cdot
s_{0}^{\frac{1}{2}(s-1)}$ $\displaystyle\lesssim s_{0}^{s-\frac{3}{4}}$

Estimates for IV: The first term in IV

(3.28) $\displaystyle\|\nabla\phi\cdot\phi v\|_{N^{0}}$
$\displaystyle\lesssim\|\nabla\phi\|_{L^{2}_{t}L^{6}_{x}}\|\phi\|_{L^{8}_{t}L^{4}_{x}}\|v\|_{L^{8}_{t}L^{4}_{x}}$
$\displaystyle\lesssim s_{0}^{-\frac{1}{2}(1-s)}\cdot 1\cdot
s_{0}^{\frac{1}{2}(s-\frac{1}{2})}$ $\displaystyle\lesssim
s_{0}^{s-\frac{3}{4}}$

The second term in IV

(3.29) $\displaystyle\|\nabla v\cdot\phi^{2}\|_{N^{0}}$
$\displaystyle\lesssim\|v\|_{S^{1}}\|\phi\|_{L^{8}_{t}L^{4}_{x}}^{2}$
$\displaystyle\lesssim o(1)\|v\|_{S^{1}}$

which is absorbed into the left hand side. Combining I to IV, we get
$\|v\|_{S^{1}}\lesssim s_{0}^{s-\frac{7}{8}}$

##  4\. Global Wellposedness and Scattering

We will show that there exists $s_{0}$ small enough and corresponding $M\sim
s_{0}^{-\frac{1}{2}s+\frac{3}{8}}$, such that
$\|u\|^{8}_{L^{8}_{t}L^{4}_{x}[0,T]}\leq M$ for any $[0,T]$ where the solution
exists. We will use a bootstrap argument. In fact, we prove the following
result

######  Proposition 4.1.

For $s>\frac{15}{16}$, there exists $s_{0}>0$ small enough and $M\sim
s_{0}^{-\frac{1}{2}s+\frac{3}{8}}$, such that the following implication holds:

$\|u\|^{8}_{L^{8}_{t}L^{4}_{x}[0,T]}\leq
M\implies\|u\|^{8}_{L^{8}_{t}L^{4}_{x}[0,T]}\leq\frac{1}{2}M$

Denote $\zeta=\phi+v$. On [0,T], $\zeta$ satisfies the modified NLS equation

(4.1)
$i\zeta_{t}+\Delta_{\mathbb{H}^{3}}\zeta=|u|^{2}u=|\zeta|^{2}\zeta+\mathcal{N}$

where
$\mathcal{N}=|u|^{2}u-|\zeta|^{2}\zeta=\mathcal{O}(\psi^{3}+\psi\zeta^{2})$.

Due to our choice of M, the result of Proposition 3.3 applies.

(4.2) $\sup_{t\in[0,T]}E[\zeta(t)]\lesssim s_{0}^{-(1-s)}$

We estimate the $L^{8}_{t}L^{4}_{x}$ norm by interpolation

(4.3)
$\|\zeta\|_{L^{8}_{t}L^{4}_{x}}^{8}\leq\|\zeta\|_{L^{\infty}_{t}L^{4}_{x}}^{4}\|\zeta\|_{L^{4}_{t,x}}^{4}$

$\|\zeta\|_{L^{\infty}_{t}L^{4}_{x}}$ is bounded by the energy. To estimate
$\|u\|_{L^{4}_{t,x}}$, we utilize the Morawetz inequality developed in [IS09]

######  Proposition 4.2 (Morawetz Estimate on $\mathbb{H}^{3}$, [IS09]).

If $\zeta$ satisfies the NLS with error term $\mathcal{N}$,

$i\zeta_{t}+\Delta_{\mathbb{H}^{3}}\zeta=|\zeta|^{2}\zeta+\mathcal{N}$

then the solution $\zeta$ satisfies the following Morawetz estimate

(4.4)
$\|\zeta\|_{L^{4}_{t,x}}^{4}\lesssim\|\zeta\|_{L^{\infty}_{t}L^{2}_{x}}\|\zeta\|_{L^{\infty}_{t}H^{1}_{x}}+\|\mathcal{N}\zeta\|_{L^{1}_{t,x}}+\|\mathcal{N}\nabla\zeta\|_{L^{1}_{t,x}}$

We have the following estimates for the latter two terms in (4.4)

######  Proposition 4.3.

  1.   2. (1)

$\|\mathcal{N}\zeta\|_{L^{1}_{t,x}}\leq\epsilon\|\zeta\|_{L^{4}_{t,x}}^{4}+C_{\epsilon}s_{0}^{2s-\frac{1}{2}}$

  3. (2)

$\|\mathcal{N}\nabla\zeta\|_{L^{1}_{t,x}}\leq\epsilon\|\zeta\|_{L^{4}_{t,x}}^{4}+C_{\epsilon}s_{0}^{2s-\frac{3}{2}}$

Proof: To see (1), by Hölder inequality, we have

$\displaystyle\|\mathcal{N}\zeta\|_{L^{1}_{t,x}}$
$\displaystyle\lesssim\|\psi^{3}\zeta\|_{L^{1}_{t,x}}+\|\psi\zeta^{3}\|_{L^{1}_{t,x}}$
$\displaystyle\leq\|\psi\|^{3}_{L^{4}_{t,x}}\|\zeta\|_{L^{4}_{t,x}}+\|\psi\|_{L^{4}_{t,x}}\|\zeta\|_{L^{4}_{t,x}}^{3}$
$\displaystyle\lesssim\epsilon\|\zeta\|_{L^{4}_{t,x}}^{4}+C_{\epsilon}\|\psi\|_{L^{4}_{t,x}}^{4}$

Similarly for (2),

$\displaystyle\|\mathcal{N}\nabla\zeta\|_{L^{1}_{t,x}}$
$\displaystyle\lesssim\|\psi^{3}\nabla\zeta\|_{L^{1}_{t,x}}+\|\psi\zeta^{2}\nabla\zeta\|_{L^{1}_{t,x}}$
$\displaystyle\leq\|\psi\|_{L^{3}_{t}L^{6}_{x}}^{3}\|\nabla\zeta\|_{L^{\infty}_{t}L^{2}_{x}}+\|\psi\|_{L^{2}_{t}L^{\infty}_{x}}\|\zeta\|_{L^{4}_{t,x}}^{2}\|\nabla\zeta\|_{L^{\infty}_{t}L^{2}_{x}}$
$\displaystyle\lesssim\epsilon\|\zeta\|_{L^{4}_{t,x}}^{4}+C_{\epsilon}s_{0}^{2s-\frac{3}{2}}$

This ends the proof of Proposition 4.2.

Resuming the proof of Proposition 4.1, we plug the results of Proposition 4.3
into (4.4)

(4.5) $\|\zeta\|_{L^{4}_{t,x}}^{4}\lesssim
s_{0}^{-\frac{1}{2}(1-s)}+\epsilon\|\zeta\|_{L^{4}_{t,x}}^{4}+C_{\epsilon}s_{0}^{2s-\frac{1}{2}}+\epsilon\|\zeta\|_{L^{4}_{t,x}}^{4}+C_{\epsilon}s_{0}^{2s-\frac{3}{2}}$

After absorbing $\epsilon\|\zeta\|_{L^{4}_{t,x}}^{4}$ into left hand side, we
get

(4.6) $\|\zeta\|_{L^{4}_{t,x}}^{4}\lesssim s_{0}^{-\frac{1}{2}(1-s)}$

which leads to

(4.7) $\|\zeta\|_{L^{8}_{t}L^{4}_{x}}^{8}\lesssim s_{0}^{-\frac{3}{2}(1-s)}$

by interpolation. And we require the right hand side to be bounded by
$\frac{1}{2}M\sim s_{0}^{-\frac{1}{2}s+\frac{3}{8}}$, i.e.

(4.8) $s_{0}^{-2(1-s)}\lesssim s_{0}^{-\frac{1}{2}s+\frac{3}{8}}$

This can be achieved by $s_{0}$ small enough as long as the exponents satisfy
$-\frac{3}{2}(1-s)>-\frac{1}{2}s+\frac{3}{8}$, hence $s>\frac{15}{16}$ is
required.

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