# Detecting the gravitational wave memory effect with TianQin

Shuo Sun  Changfu Shi  Corresponding author. Email<EMAIL_ADDRESS>Jian-dong Zhang  Jianwei Mei  MOE Key Laboratory of TianQin Mission, TianQin
Research Center for Gravitational Physics & School of Physics and Astronomy,
Frontiers Science Center for TianQin, Gravitational Wave Research Center of
CNSA, Sun Yat-sen University (Zhuhai Campus), Zhuhai 519082, China.

###### Abstract

The gravitational wave memory effect is a direct prediction of general
relativity. The presence of memory effect in gravitational wave signals not
only provides the chance to test an important aspect of general relativity,
but also represents a potentially non-negligible contribution to the waveform
for certain gravitational wave events. In this paper, we study the prospect of
directly detecting the gravitational wave memory effect with the planned
space-based gravitational wave detector — TianQin. We also determine the
region of parameter space in which the contribution of memory effect is
significant to waveform modeling.

##  I Introduction

The observations of gravitational waves from binary black hole mergers Abbott
_et al._ (2019a, 2021a, 2021b) have opened a new window to observe the
universe, which not only promise a deeper understanding of compact objects in
the universe Abbott  _et al._ (2019b, 2021c, 2021d) but also provide a new way
to test general relativity Abbott  _et al._ (2019c, 2021e, 2021f). Both the
detection of GWs and the testing of general relativity require accurate
waveforms of GWs. Among all the contributions to the GW waveforms, the GW
memory effect is particularly interesting not only in that it can potentially
be directly observed and thus serves as an important test of general
relativity, but also in that it depends on the entire past history of the
gravitational systems Blanchet and Damour (1992); Blanchet (2014).

Under real physical conditions, the spacetime background seen by an observer
near the null infinity before and after a flux of GW radiation is not purely
vacuum, but contains GW sources at a long distance. Due to the loss of energy
and other conserved charges of the GW sources, the radiation of GWs typically
leads to a permanent change to the spacetime, and this has been known as the
linear memory effect since the early 1970s Zel’dovich and Polnarev (1974);
Braginsky and Throne (1987). A second type of memory effect was later
discovered by Christodoulou Christodoulou (1991) and has been called the
nonlinear memory effect, for it involves the contribution of the GW strain at
the quadratic order. The Christodoulou effect does not appear as a change of
the charges of the GW sources but is still related to the permanent change of
the background spacetime. It has been suggested to rename the linear and
nonlinear memory effect as the ordinary and null memory effect, respectively,
to be more clear on what they truly represent Bieri and Garfinkle (2014).

Since the memory effect is a persistent change of the spacetime, it is natural
to relate it to the symmetries and the corresponding charges that characterize
different spacetimes (see Nichols (2018) for some early references). However,
it was only in the past few years that the relations among the various types
of memory effect and the various types of Bondi-Metzner-Sachs (BMS)
transformations Bondi  _et al._ (1962); Sachs (1962); de Boer and Solodukhin
(2003); Barnich and Troessaert (2010a, b); Kapec  _et al._ (2014, 2017); He
_et al._ (2017) were elucidated and established Strominger and Zhiboedov
(2016); Pasterski  _et al._ (2016); Nichols (2018); Compère  _et al._ (2020).
Such understanding has enabled systematic calculations of the memory effect
using the so called BMS flux-balance laws Barnich and Troessaert (2011);
Flanagan and Nichols (2017); Nichols (2017); Compère  _et al._ (2020); Mitman
_et al._ (2020, 2021).

The detection of memory effect has attracted much attention in recent years.
Attempts have been made to calculate the memory effect for cosmic strings
Jenkins and Sakellariadou (2021) and core-collapse supernovae Mukhopadhyay
_et al._ (2021). There have also been calculation of memory effect in
alternative theories of gravitation, such as scalar tensor theory Du and
Nishizawa (2016), Brans-Dicke theory Seraj (2021); Tahura  _et al._ (2021) and
Chern-Simons modified gravity Hou  _et al._ (2022). Using the memory effect to
test alternative theories of gravitation has also been studied, such as for
the Brans-Dicke theory with screening Koyama (2020).

The strategies for detecting the GW memory effect were first proposed by
Braginsky and Thorne in the 1980s Braginsky and Grishchuk (1985); Braginsky
and Throne (1987). Since the first detection of GWs by LIGO in 2015 Abbott
_et al._ (2016), many works about detecting memory effect with current and
future GW detectors have appeared Seto (2009); van Haasteren and Levin (2010);
Pshirkov  _et al._ (2010); Cordes and Jenet (2012); Madison  _et al._ (2014);
Arzoumanian  _et al._ (2015); Lasky  _et al._ (2016); McNeill  _et al._
(2017); Nichols (2017); Divakarla  _et al._ (2020); Boersma  _et al._ (2020);
Khera  _et al._ (2021); Hübner  _et al._ (2020, 2021); Islam  _et al._ (2021);
Zhao  _et al._ (2021). It has been found that the memory effect is difficult
to detect directly with the current ground-based GW detectors, but the future
ground-based detectors such as Cosmic Explorer Abbott  _et al._ (2017) and
Einstein Telescope Punturo  _et al._ (2010) may be able to detect the memory
effect Yang and Martynov (2018). GW sources that are more massive than those
detectable with ground-based detectors can radiate more energy through GW
emission, and thus have better chance at revealing the memory effect. Indeed,
the prospect of detecting memory effect with the planned space-based GW
detector LISA has been studied in Islo  _et al._ (2019), and it has been shown
that LISA can directly detect the displacement memory from the merger of
massive black hole binarys. It is also possible to detect memory effect with
pulsar timing array Seto (2009); van Haasteren and Levin (2010); Pshirkov  _et
al._ (2010); Cordes and Jenet (2012); Madison  _et al._ (2014); Arzoumanian
_et al._ (2015).

TianQin is a space-based GW detector targeting GWs in the frequency band
$10^{-4}$ Hz$\sim 1$ Hz Luo  _et al._ (2016); Mei  _et al._ (2021). Among the
promising scientific discovery potential of TianQin in many different
directions Hu  _et al._ (2017); Fan  _et al._ (2020); Liu  _et al._ (2020,
2022); Huang  _et al._ (2020); Liang  _et al._ ; Shi  _et al._ (2019); Bao
_et al._ (2019); Zi  _et al._ (2021); Zhu  _et al._ (2022a), the detection of
MBHBs is an outstanding one Wang  _et al._ (2019); Feng  _et al._ (2019).
Motivated by the known result on LISA, it is likely that TianQin can also
detect memory effect directly. But the orbit, the constellation size and
motion of TianQin are quite different from those of LISA, and so are the
sensitive frequency band and the response of the detector to GWs. So it is
necessary to study the related problems for TianQin independently, especially
when it comes to the question of when the memory effect will become a must-
consider factor during science and data analysis.

In this paper, we study the prospect of directly detecting the gravitational
wave memory effect with TianQin and also determine the region of parameter
space in which the contribution of memory effect is potentially non-
negligible.

The paper is organized as following. In Sec. II, we recall the basic results
on memory effect and on the calculation of the corresponding waveform
corrections. In Sec. III, we study the potential of detecting memory effect
directly with TianQin. In Sec. IV, we study the parameter space in which the
memory effect could become an adverse factor if not properly taken into
consideration. Finally, we summarize in Sec. V.

##  II memory effect and waveform

In this section, we recall some of the basics of memory effect and the
calculation of the corresponding waveform. The purpose is to have all the
relevant nomenclature clearly defined and to keep the logic in a self-
contained manor. There is nothing particularly new in this section and we
mainly follow the treatment and notation of Flanagan and Nichols (2017);
Mitman  _et al._ (2020).

Physically, there are three different types of memory effect: the displacement
memory Zel’dovich and Polnarev (1974); Braginsky and Throne (1987), which
causes the relative displacement between two test masses to change after the
passage of GWs, the spin memory Pasterski  _et al._ (2016), which causes the
change in the relative time delay of two free-falling test masses that are
initially on anti-orbital trajectories, and the center-of-mass memory effect
Nichols (2018), which causes the change in the relative time delay of two
free-falling test masses that are initially on anti-parallel trajectories. All
three types of memory effect contain the previously mentioned linear
(ordinary) and nonlinear (null) contributions.

The physical consequences of the memory effect described above are easy to
visualize, but they do not constitute the best ways for detection. Due to the
intrinsic weakness of GWs, the permanent change in the displacement and time
delays in various detector dependent setup is also very small and cannot be
directly measured with any currently known technology. The practically more
relevant consequence of memory effect is the shifts and corrections to the GW
waveforms, and it is the knowledge about the waveform correction that offers
the best chance for a detection.

Earlier methods for computing the waveform of memory effect include the post-
Newtonian (PN) approximation and various kinds of postprocessing techniques
based on existing numerical waveforms Favata (2009a, b, 2010); Talbot  _et
al._ (2018); Nichols (2017, 2018). Recently, the first numerical relativity
waveform with memory effect has been obtained by using the BMS flux-balance
law Mitman  _et al._ (2020). We will follow Mitman  _et al._ (2020) to
calculate the memory waveforms in this paper.

Following Flanagan and Nichols (2017); Mitman  _et al._ (2020), the Bondi-
Sachs metric of a generic asymptotically flat spacetime containing GWs can be
written as Bondi  _et al._ (1962); Sachs (1962),

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-e^{2\beta}\Big{(}Udu^{2}+2dudr)$ (1)
$\displaystyle+r^{2}\gamma_{AB}(d\theta^{A}-{\cal U}^{A}du)(d\theta^{B}-{\cal
U}^{B}du)\,,$

where $u=t-r$ is the retarded time, $\theta^{A}\in\\{\theta^{1},\theta^{2}\\}$
are coordinates on the two-sphere, and $U$, $\beta$, ${\cal U}^{A}$ and
$\gamma_{AB}$ are functions of $u$, $r$ and $\theta^{A}$. A special feature of
the Bondi-Sachs metric is

$\displaystyle g_{rr}=g_{rA}=0\,.$ (2)

The radial coordinate $r$ is determined by requiring that the determinant of
$\gamma_{AB}$ is the same of that of the fixed round metric, denoted as
$q_{AB}$, on the unit two-sphere,

$\displaystyle\det(\gamma_{AB})=\det(q_{AB})\,,$ (3)

where, if written in the usual spherical coordinates,

$\displaystyle q_{AB}d\theta^{A}d\theta^{B}=d\theta^{2}+\sin^{2}\theta
d\phi^{2}\,.$ (4)

Note the metric $q_{AB}$ will be used to raise and lower all capital Latin
indices (e.g., $A,B$) throughout this paper.

A general solution to the source-less Einstein equations can be found as

$\displaystyle U$ $\displaystyle=$ $\displaystyle
1-\frac{2m}{r}-\frac{2M}{r^{2}}+{\cal O}(r^{-3})\,,$ $\displaystyle\beta$
$\displaystyle=$
$\displaystyle-\frac{C_{2}}{32r^{2}}-\frac{C_{2}^{2}}{48r^{3}}+{\cal
O}(r^{-4})\,,$ $\displaystyle\gamma_{AB}$ $\displaystyle=$ $\displaystyle
q_{AB}+\frac{C_{AB}}{r}+{\cal O}(r^{-3})\,,$ $\displaystyle{\cal U}^{A}$
$\displaystyle=$
$\displaystyle-\frac{D_{B}C^{AB}}{2r^{2}}+\frac{1}{r^{3}}\Big{[}-\frac{2}{3}N^{A}+\frac{1}{16}D^{A}C_{2}$
(5) $\displaystyle\qquad\qquad\qquad+\frac{1}{2}C^{AC}D^{B}C_{BC}\Big{]}+{\cal
O}(r^{-4})\,,$

where all coefficient functions on the right-hand sides are functions of
$(u,\theta^{A})$ only, $D^{A}$ is the covariant derivative associated with
$q_{AB}$, and

$\displaystyle C_{2}$ $\displaystyle=$ $\displaystyle C_{AB}C^{AB}\,,\quad
q^{AB}C_{AB}=0\,,$ $\displaystyle\dot{m}$ $\displaystyle=$
$\displaystyle-\frac{1}{8}\dot{C}_{AB}\dot{C}^{AB}+\frac{1}{4}D_{A}D_{B}\dot{C}^{AB}\,,$
$\displaystyle\dot{N}_{A}$ $\displaystyle=$ $\displaystyle
D_{A}m+\frac{1}{4}D_{B}D_{A}D_{C}C^{BC}-\frac{1}{4}D_{B}D^{B}D^{C}C_{CA}$
$\displaystyle+\frac{1}{4}D_{B}(\dot{C}^{BC}C_{CA})+\frac{1}{2}D_{B}\dot{C}^{BC}C_{CA}\,,$
$\displaystyle\dot{M}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}D_{A}D^{A}\Big{(}m+\frac{3}{8}C_{BC}\dot{C}^{BC}\Big{)}+\frac{1}{3}D_{A}\dot{N}^{A}$
(6)
$\displaystyle-\frac{1}{4}(D_{A}\dot{C}_{BC})\Big{(}D^{A}C^{BC}-2D^{B}C^{AC}\Big{)}$
$\displaystyle+\frac{1}{8}\dot{C}_{AB}\Big{(}D_{C}D^{C}+\frac{3}{2}\Big{)}C^{AB}\,.$

Here an overdot means a derivative with respect to $u$. The solution solves
Einstein’s equations to the ${\cal O}(r^{-3})$ order, i.e.,

$\displaystyle R_{\mu\nu}dx^{\mu}dx^{\nu}\leq{\cal O}(r^{-4})\,,$ (7)

where $R_{\mu\nu}$ is the Ricci tensor of the Bondi-Sachs metric (1),
$dx^{\mu}\in\\{du,dr,d\theta^{1},d\theta^{2}\\}$, and we take

$\displaystyle du,dr\sim{\cal O}(r^{0})\,,\quad
d\theta^{1},d\theta^{2}\sim{\cal O}(r^{-1})\,.$ (8)

All dynamical properties of the solution are encoded in the two unconstrained
functions of the shear $C_{AB}\,$.

Solutions with all possible values of $C_{AB}$ form the solution space that
preserves the structure of the Bondi-Sachs metric with the boundary conditions
(2), (3) and (5). The solutions are related to each other through the BMS
transformations,

$\displaystyle\xi$ $\displaystyle=$ $\displaystyle
f\partial_{u}+\Big{[}Y^{A}-\frac{1}{r}D^{A}f+\frac{1}{2r^{2}}C^{AB}D_{B}f+{\cal
O}(r^{-3})\Big{]}\partial_{A}$ (9)
$\displaystyle-\Big{[}\frac{1}{2}rD_{A}Y^{A}-\frac{1}{2}D^{A}D_{A}f+\frac{1}{4r}\Big{(}D_{A}C^{AB}\Big{)}D_{B}f$
$\displaystyle\quad+\frac{1}{4r}D_{A}(D_{B}fC^{AB})+{\cal
O}(r^{-2})\Big{]}\partial_{r}\,,$

where

$\displaystyle f=\alpha(\theta^{A})+\frac{1}{2}uD_{B}Y^{B}(\theta^{A})\,.$
(10)

Here $\alpha(\theta^{A})$ is unconstrained and $Y^{A}(\theta^{B})$ must obey
the conformal Killing equation on the unit two-sphere,

$\displaystyle D_{A}Y_{B}+D_{B}Y_{A}=q_{AB}D_{C}Y^{C}\,.$ (11)

Under the action of (9), the shear transforms as

$\displaystyle\delta C_{AB}$ $\displaystyle=$ $\displaystyle
f\dot{C}_{AB}-2D_{A}D_{B}f+q_{AB}D_{C}D^{C}f$ (12)
$\displaystyle+\frac{1}{2}(D_{C}Y^{C})C_{AB}+{\cal L}_{\vec{Y}}C_{AB}\,,$

where ${\cal L}_{\vec{Y}}$ is the Lie derivative with
$\vec{Y}=\\{Y^{1},Y^{2}\\}\,$. Other functions in the solution will transform
accordingly, and we refer to Flanagan and Nichols (2017) for details on the
transformation of the Bondi mass aspect $m$ and the angular-momentum aspect
$N^{A}$.

The BMS transformations (9) are examples of asymptotic symmetries which are
associated with conserved charges that characterize different solutions in the
solution space. In general, only the difference between the charges of a pair
of solutions is defined (see Compère and Fiorucci (2018) for a reader friendly
exposition). In GW-free and matter-free cases, however, it is possible to
write the charges for each solution directly,

$\displaystyle Q$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi}\int
d^{2}\Omega\Big{[}4\alpha m-2u_{0}Y^{A}D_{A}m+2Y^{A}N_{A}$ (13)
$\displaystyle\qquad\qquad-\frac{1}{8}Y^{A}D_{A}C_{2}-\frac{1}{2}Y^{A}C_{AC}D_{B}C^{BC}\Big{]}\,,$

where the integral is over a two-sphere near the null infinity and at the
retarded time $u_{0}\,$.

In the usual spherical coordinates, the standard BMS algebra is generated by

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle
t^{0}-t^{i}n_{i}+\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}\alpha_{\ell
m}Y_{\ell m}\,,$ $\displaystyle Y^{A}$ $\displaystyle=$ $\displaystyle
w^{0i}e^{A}_{i}+w^{ij}e^{A}_{[i}n_{j]}\,,$ (14)

where $t^{\mu}=\\{t^{0},t^{i}\\}$ and $w^{0i}\,,\,w^{ij}$ are free parameters,
$n_{i}=\\{\sin\theta\cos\phi\,,\,\sin\theta\sin\phi\,,\,\cos\theta\\}$, and
$e^{A}_{i}=D^{A}n_{i}\,$. The corresponding charges are

$\displaystyle
Q=-P^{\mu}t_{\mu}+\frac{1}{2}J^{\mu\nu}w_{\mu\nu}+\frac{1}{4\pi}\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}\alpha_{\ell
m}{\cal P}^{\ast}_{\ell m}\,.$ (15)

Here $P^{\mu}$ is the usual four-momentum, $J^{\mu\nu}$ is the usual angular
momentum, and ${\cal P}_{\ell m}$ are the supermomenta associated with the
supertranslations, i.e., the $\ell\geq 2$ terms in $\alpha$. So the standard
BMS algebra is simply the Poincaré algebra extended with the supermomenta.

The BMS algebra can be extended if one allows for singular Killing vectors on
the future null infinity Barnich and Troessaert (2010c, 2011). To do this, it
is more convenient to use the complex stereographic coordinates
($z,{\bar{z}}$) with $z=\cot(\theta/2)e^{i\phi}$ instead of the usual
spherical coordinates. In this case, the conformal Killing equation requires
$Y^{z}=Y^{z}(z)$ and $Y^{\bar{z}}=Y^{\bar{z}}({\bar{z}})$, i.e., both to be
meromorphic functions of their arguments. The corresponding Killing vector can
be expanded with the set of vectors,

$\displaystyle
l_{m}=-z^{m+1}\partial_{z}\,,\quad{\bar{l}}_{m}=-{\bar{z}}^{m+1}\partial_{\bar{z}}\,,$
(16)

where $m\in\mathbb{Z}\,$. Among the vectors, $l_{0}\,,\,l_{\pm
1}\,,\,{\bar{l}}_{0}$ and ${\bar{l}}_{\pm 1}$ have already appeared in Eq.
(14), while the rest are called the superrotations. Although the Killing
vectors are singular for the superrotations, the corresponding charges have
been shown to be finite Flanagan and Nichols (2017). The charges of the
superrotations can be split to the spin parts, corresponding to operations
analogues to the rotations in the Lorentz group, and the center-of-mass parts,
corresponding to operations analogues to the Lorentz boost.

It has been realized that the displacement memory is related to changes in the
supermomenta and the corresponding flux, the spin memory is related to changes
in the spin parts of the super-angular momenta and the corresponding flux, and
the center-of-mass memory effect is related to changes in the center-of-mass
parts of the super-angular momenta and the corresponding flux Strominger and
Zhiboedov (2016); Pasterski  _et al._ (2016); Nichols (2018); Compère  _et
al._ (2020). As we have mentioned earlier, however, that for a true detection
of the memory effect we have to focus on the waveforms. From this perspective,
the displacement memory is characterized by a finite change in the GW strain,
the spin memory is characterized by a change in the time integral of the
magnetic part of the GW strain, while the center-of-mass memory effect is
characterized by a change in the time integral of certain expression that has
the dimension of the GW strain Nichols (2018).

It was only recently that the displacement and spin memorys are successfully
captured in the SXS catalog sxs ; Boyle  _et al._ (2019) by using the Cauthy-
characteristic extraction (CCE), in which one has to evolve a world tube
produced by a Cauchy evolution to asymptotic infinity and then to extract the
GW strain Moxon  _et al._ (2020). In Mitman  _et al._ (2020), the authors use
the BMS flux-balance law to calculate the memory effect from numerical
waveforms known at the null infinity, and they find that the results agree
with those from CCE well.

In the treatment of Mitman  _et al._ (2020), the key quantity to calculate is
the leading order spin-weight -2 GW strain,

$\displaystyle h=\frac{1}{2}\bar{q}^{A}\bar{q}^{B}C_{AB}=\sum_{\ell\geq
2}\sum_{|m|\leq\ell}h_{\ell m}~{}_{-2}Y_{\ell m}(\theta,\phi)\,,$ (17)

where $q^{A}$ is the complex dyads Moxon  _et al._ (2020),
$q^{A}=-\\{1,i\csc\theta\\}\,$. In reverse, one has

$\displaystyle
C_{AB}=\frac{1}{2}\Big{(}q_{A}q_{B}h+\bar{q}_{A}\bar{q}_{B}\bar{h}\Big{)}\,.$
(18)

Splitting the shear into the electric and the magnetic components,

$\displaystyle
C_{AB}=\Big{(}D_{A}D_{B}-\frac{1}{2}q_{AB}D_{C}D^{C}\Big{)}\Phi+\epsilon_{C(A}D_{B)}D^{C}\Psi\,,$
(19)

where $\epsilon_{AB}$ is the Levi-Civita tensor on the two-sphere, one has for
the electric and magnetic memory effect, respectively,

$\displaystyle\Delta J^{(E)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bar{q}^{A}\bar{q}^{B}\Delta
C^{(E)}_{AB}=\frac{1}{2}\bar{\eth}^{2}\Delta\Phi$ (20) $\displaystyle=$
$\displaystyle\frac{1}{2}\bar{\eth}^{2}\mathfrak{D}^{-1}\left[\Delta
m+\frac{1}{4}\int^{u}_{-\infty}|\dot{h}|^{2}du\right]\,,$ $\displaystyle\Delta
J^{(B)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bar{q}^{A}\bar{q}^{B}\Delta
C^{(B)}_{AB}=-\frac{1}{2}i\bar{\eth}^{2}\Delta\Psi$ (21) $\displaystyle=$
$\displaystyle\frac{1}{2}i\bar{\eth}^{2}\mathfrak{D}^{-1}D^{-2}\text{Im}\Bigg{\\{}\bar{\eth}(\partial_{u}\hat{N})$
$\displaystyle+\left.\frac{1}{8}\left[\eth(3h\bar{\eth}\dot{\bar{h}}-3\dot{h}\bar{\eth}\bar{h}+\dot{\bar{h}}\bar{\eth}h-\bar{h}\bar{\eth}\dot{h})\right]\right\\}$

Here $\rm Im$ means the imaginary part of the function, $D^{2}=\bar{\eth}\eth$
is the usual Laplacian on the two-sphere,
$\mathfrak{D}=\frac{1}{8}D^{2}(D^{2}+2)$, and $\eth$ and $\bar{\eth}$ are the
spin-weight operators in the Newman-Penrose convention Ezra and Roger (1962),

$\displaystyle\eth_{s}Y_{\ell m}=+\sqrt{(\ell-s)(\ell+s+1)}_{s+1}Y_{\ell
m}\,,$ $\displaystyle\bar{\eth}_{s}Y_{\ell
m}=-\sqrt{(\ell+s)(\ell-s+1)}_{s-1}Y_{\ell m}\,.$ (22)

For the spin weight 0 spherical harmonics $Y_{\ell m}\,$,

$\displaystyle D^{2}Y_{\ell m}$ $\displaystyle=$
$\displaystyle-\ell(\ell+1)Y_{\ell m}\,,$ $\displaystyle\mathfrak{D}Y_{\ell
m}$ $\displaystyle=$
$\displaystyle\frac{1}{8}(\ell+2)(\ell+1)\ell(\ell-1)Y_{\ell m}\,.$ (23)

The $\ell\leq 1$ modes of $Y_{\ell m}$ are in the kernel of $\mathfrak{D}$, so
$\mathfrak{D}^{-1}$ is defined by projecting out the $\ell\leq 1$ modes Mitman
_et al._ (2020),

$\displaystyle\mathfrak{D}^{-1}Y_{\ell m}=\left\\{\begin{matrix}0&:&\ell\leq
0\,,\cr\left[\frac{1}{8}(\ell+2)(\ell+1)\ell(\ell-1)\right]^{-1}Y_{\ell
m}&:&\ell\geq 2\,.\end{matrix}\right.$ (24)

The first term in the right hand side of Eq. (20) is the ordinary part of the
displacement memory, and the second term is the null part. The Bondi mass
aspect $m$ can be written in terms of Weyl scalar $\Psi_{2}$ and $h$ as

$\displaystyle m=-{\rm Re}\left[\Psi_{2}+\frac{1}{4}\dot{h}\bar{h}\right]\,.$
(25)

Similarly, the first term in the right hand side of Eq. (21) is the ordinary
part of spin memory and the second part is the null part. The Bondi angular
aspect $\hat{N}$ can be written in terms of Weyl scalar $\Psi_{1}$ and $h$ as

$\displaystyle\hat{N}=2\Psi_{1}-u\eth
m-\frac{1}{8}\eth(h\bar{h})-\frac{1}{4}\bar{h}\eth h\,.$ (26)

As discussed in Mitman  _et al._ (2020), $\Delta J^{(E)}$ is dominated by the
displacement memory, $\int\Delta J^{(B)}(u)du$ is the spin memory, while the
explicit formula for computing the center-of-mass memory by using the BMS
flux-balance law is not known yet. The detection of the center-of-mass memory
effect using PN waveform has been studied in Nichols (2018). It has been found
that, for a future ground-based detector such as the Einstein-Telescope, which
can achieve a signal-to-noise ratio (SNR) at the order ${\cal O}(10^{3})$ for
a signal like GW150914, can only detect the ordinary part of the center-of-
mass memory effect with a SNR several orders of magnitude below unit. The
prospect for space-based detector will be similar, so the chance for detection
is negligible for TianQin. Therefore, in this work, we only consider the
displacement and the spin memorys.

We use the IMRPhenomXHM waveform García-Quirós  _et al._ (2020); Colleoni  _et
al._ (2021); Pratten  _et al._ (2020) to calculate the displacement memory
strain by Eq. (20) and the spin memory strain by Eq. (21),

$\displaystyle h_{\rm dis}=\Delta J^{(E)}\,,\quad h_{\rm spin}=\Delta
J^{(B)}\,.$ (27)

The IMRPhenomXHM waveform is an aligned-spin model with the most dominant
higher-order modes of (2,2), (2,1), (3,3), (3,2) and (4,4). Beside the
frequently used frequency domain waveform, it also provides the time domain
waveform, which is more convenient for the calculation of the memory effect.
All these features are important for calculating the memory effect. From Eq.
(24), one can see that higher $\ell$ modes are strongly suppressed. So we will
only consider the dominant mode of the displacement memory and the spin
memory, i.e., $h_{(2,0)}$ and $h_{(3,0)}$, respectively. In Fig. 1, we give an
example of the waveforms generated with IMRPhenomXHM.

Figure 1: Example of waveforms generated with IMRPhenomXHM. Plotted with a
non-spinning MBHB with mass ratio q=1, total mass $M=10^{6}{\rm M}_{\odot}$ at
luminosity distance $D_{\rm L}=2\rm Gpc$. The top panel is the dominant GW
mode $h_{(2,2)}$, the middle panel is the dominant displacement memory mode
$h_{(2,0)}$, and the bottom panel is the dominant spin memory mode
$h_{(3,0)}$. The cycles of the waveform for this plot is about 57.

It’s convenient to use the time-domain waveforms of IMRPhenomXHM to calculate
the displacement and spin memory. But in order to assess the detection
potential of memory effects, we need to convert waveforms to the frequency
domain by Fourier transform. We use Planck-taper window McKechan  _et al._
(2010) to window the time-domain waveform to remove edge effects. In the
calculation, we choose $\epsilon=0.04$ (see Eq. (7) in McKechan et al.
McKechan  _et al._ (2010) for the detail of window function).

The displacement and spin memory are mainly accumulated during the merger
phase of a BBH system since the most energy and angular momentum of the system
are radiated away during the merger. However, the inspiral phase will have a
relatively effective contribution to the spin memory, because the source of
the nonlinear spin memory is the angular momentum flux of the system Mitman
_et al._ (2020); Nichols (2017). Since the memory effect is always subdominant
compared to the full GW strain, one should search for memory effect in GW
signals with the highest possible SNR. So we will focus on MBHBs with masses
in the range $10^{4}~{}{\rm M}_{\odot}\sim 10^{7}~{}{\rm M}_{\odot}$ in this
study. The number of inspiraling cycles in the detector bandwidth
$f\in[f_{min},f_{max}]$ is defined as Maggiore (2007)

$\mathcal{N}=\frac{1}{32\pi^{8/3}}\left(\frac{GM_{c}}{c^{3}}\right)^{-5/3}\left(f_{\min}^{-5/3}-f_{\max}^{-5/3}\right)$
(28)

where $M_{c}$ is the chirp mass of the masses $m_{1}$, $m_{2}$, and is defined
as

$\displaystyle M_{c}=\frac{(m_{1}m_{2})^{3/5}}{(m_{1}+m_{2})^{1/5}}.$ (29)

We examined the SNRs of the memory effects generated by the waveform for about
150 cycles, and we found that the change in SNR was negligible compared to the
waveform for about 50 cycles. There is no need to use a very long waveform
since the vast majority of the SNRs produced by the MBHBs on TianQin can be
obtained within one hour before merger Wang  _et al._ (2019); Feng  _et al._
(2019) and memory effects accumulate mainly during the merging. To improve the
efficiency of the calculation, we use the time-domain waveform of one day of
IMRPhenomXHM for total mass $M=2\times 10^{6}{\rm M}_{\odot}$ generated by the
LALsimulation LIGO Scientific Collaboration (2018). Thus, for symmetric mass
ratios $\eta$ between $0.04$ and $0.25$, we generate waveforms with
approximately 57 to 113 cycles.

##  III Detection potential

In this section, we study the prospect of using TianQin to detect the memory
effect.

The data stream output from a detector can be schematically written as

$\displaystyle d(t)=s(t)+n(t)\,,$ (30)

where $s(t)$ is the GW signal registered by the detector and $n(t)$ is the
noise in the data stream.

The registered signal depends on the detector through the antenna pattern
functions. For space-based GW detectors like TianQin and LISA, they are given
by

$\displaystyle F_{+}(\theta,\phi,\psi)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}}{2}\bigg{[}\frac{1}{2}(1+\cos^{2}\theta)\cos
2\phi\cos 2\psi$ $\displaystyle-\cos\theta\sin 2\phi\sin 2\psi\bigg{]}\,,$
$\displaystyle F_{\times}(\theta,\phi,\psi)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}}{2}\bigg{[}\frac{1}{2}(1+\cos^{2}\theta)\cos
2\phi\sin 2\psi$ (31) $\displaystyle+\cos\theta\sin 2\phi\cos
2\psi\bigg{]}\,,$

where the $\theta$ and $\phi$ are the position of the source in the detector
frame, and $\psi$ is the polarization angle. For any incoming GW signal with
the usual plus and cross modes, $h_{+,\times}(t,\iota,\varphi_{c})$, where
$\iota$ and $\varphi_{c}$ are the inclination angle and the reference phase
(i.e., the phase at the final coalescence), respectively, of the GW source,
the registered signal is given by

$\displaystyle
s(t)=F_{+}(\theta,\phi,\psi)h_{+}(t,\iota,\varphi_{c})+F_{\times}(\theta,\phi,\psi)h_{\times}(t,\iota,\varphi_{c})\,.$
(32)

For a preliminary estimation, we focus on the sky-averaged response of GWs,

$\displaystyle\left\langle
s^{\ast}(t)s(t)\right\rangle=\int_{0}^{+\infty}S_{h}(f)\bar{R}(f)df\,,$ (33)

where $S_{h}(f)$ is the power spectral density (PSD) of the incoming GW signal
and $\bar{R}(f)$ is the sky-averaged response of the detector. In this work,
we use

$\displaystyle S_{h}(f)$ $\displaystyle=$
$\displaystyle\frac{2}{T}\Big{|}\tilde{h}_{+,\times}(f)\Big{|}^{2}\equiv\frac{2}{T}\Big{|}\tilde{h}(f)\Big{|}^{2}\,,$
$\displaystyle\bar{R}(f)$ $\displaystyle=$
$\displaystyle\frac{3}{10}\left[1+\left(\frac{2fL_{0}}{0.41}\right)^{2}\right]^{-1}\,,$
(34)

where $T$ is the time span of the data stream, $\tilde{h}_{+,\times}(f)$ are
the Fourier components of $h_{+,\times}(t)$, and $L_{0}$ is the arm-length of
the detector.

The noise $n(t)$ is usually given in terms of its PSD,

$\displaystyle S_{N}(f)=\frac{2}{T}\Big{|}\tilde{n}(f)\Big{|}^{2}\,,$ (35)

where $\tilde{n}(f)$ is the Fourier component of $n(t)\,$. For a space-based
detector like TianQin and LISA, the noise can be grossly grouped into two
categories: those from the propagation of the laser used to measure the change
of spacetime due to the passage of GWs and those from the irregular motion of
the test masses serving as reflecting endpoints for the lasers. The former is
quantified with the total displacement measurement noise in a single laser
link, denoted by $S_{x}$, and the latter is quantified with the residue
acceleration noise of a test mass in the sensitive direction, denoted by
$S_{a}$. The structure of $S_{N}(f)$ and the constants $L_{0}\,$, $S_{x}\,$
and $S_{a}$ are different for different detectors. For TianQin,
$L_{0}=\sqrt{3}\times 10^{8}{\rm~{}m}\,$, and the following noise model is
used in the process of mission development Luo  _et al._ (2016); Mei  _et al._
(2021),

$\displaystyle S_{N}(f)$ $\displaystyle=$
$\displaystyle\frac{1}{L_{0}^{2}}\Big{[}S_{x}+\frac{4S_{a}}{(2\pi
f)^{4}}\Big{(}1+\frac{10^{-4}{\rm Hz}}{f}\Big{)}\Big{]}\,,$ $\displaystyle
S_{x}^{1/2}$ $\displaystyle=$ $\displaystyle 1\times 10^{-12}{\rm~{}m}/{\rm
Hz}^{1/2}\,,$ $\displaystyle S_{a}^{1/2}$ $\displaystyle=$ $\displaystyle
1\times 10^{-15}{\rm~{}m/s}^{2}{\rm/Hz}^{1/2}\,.$ (36)

The huge number of Galactic compact binaries, most of which are ultra-compact
double white dwarf systems, can generate a foreground confusion noise that may
affect the detection of GWs from other sources. However, it has been shown
that the expected foreground confusion noise for the TianQin is relatively
weak Huang  _et al._ (2020). So we will not consider it in this work.

The SNR is defined as

$\displaystyle\rho=\sqrt{4\int^{f_{max}}_{f_{min}}\frac{|\tilde{h}(f)|^{2}\bar{R}(f)}{S_{N}(f)}df}\,,$
(37)

where $f_{min}$, $f_{max}$ are the minimal and maximal frequency limits of the
chosen PSD. The sky averaged sensitivity, or equivalently the amplitude
spectral density (ASD), of TianQin is define as

$\displaystyle h_{\rm eff}(f)=\sqrt{S_{n}(f)}\,,\quad
S_{n}(f)\equiv\frac{S_{N}(f)}{\bar{R}(f)}\,.$ (38)

The numerical result is plotted in Fig. 2. As a comparison, we also plot the
sensitivity curve of LISA Robson  _et al._ (2019), for which the contribution
of the foreground confusion noise is non-negligible.

Figure 2: The sensitivity curve of TianQin (solid/red), compared to that of
LISA (dashed/black). For LISA, we adopt 4 years of the the parameters of the
analytic fit the Galactic confusion noise.

###  III.1 SNR

A comparison of the strain for the dominant mode $h_{(2,2)}$, the displacement
memory $h_{(2,0)}$, and the spin memory $h_{(3,0)}$ is plotted in Fig. 3,
together with the ASD of TianQin. With the source parameters used in the plot,
the SNRs are: $\rho_{(2,2)}=1112.2\,$, $\rho_{(2,0)}=14.7$ and
$\rho_{(3,0)}=3.8\,$. As we will discuss in the next section, the detection
threshold can be set at $\rho_{th}=3\,$. So with the chosen source, it is
possible to directly detect the displacement memory with high significance,
but the spin memory can only be detected with close to the threshold.

Figure 3: A comparison of the dominant modes, $h_{(2,2)}$ (black/solid),
$h_{(2,0)}$ (blue/dashed) and $h_{(3,0)}$ (green/dashdot), for the usual GW
strain, the displacement memory strain and the spin memory strain,
respectively. Plotted using a non-spinning MBHB with mass ratio $q=1$, total
mass $M=10^{6}~{}{\rm M}_{\odot}\,$, inclination $\iota=\pi/4$ and luminosity
distance $D_{\text{L}}=2~{}\text{Gpc}\,$.

Interestingly, we find that there are several parameters that can strongly
affect the SNR.

The most significant one is the inclination angle $\iota$, as one can see in
Fig. 4. For the displacement memory, the SNR will get its maximum at
$\iota=\pi/2$, and approaches to $0$ as $\iota$ approaches to $0$ and $\pi$.
On the other hand, for the spin memory, the SNR will be $0$ while
$\iota=\pi/2$, and get its maximum near $\iota=3\pi/10$ and $\iota=7\pi/10$.

Figure 4: Dependence of SNR on the inclination angle $\iota$. Note different
values of $\rho_{\rm max}$ are used for the displacement memory and the spin
memory. Plotted using a MBHB with mass ratio $q=1$, total mass
$M=10^{6}~{}{\rm M}_{\odot}\,$ and luminosity distance
$D_{\text{L}}=2~{}\text{Gpc}\,$.

Another important parameter is the spin of the components of the binaries,
which can increase the efficiency of energy loss through GW radiation Reisswig
_et al._ (2009); Lousto  _et al._ (2010). The dependence of SNR on the
effective spin $\chi$ is plotted in Fig. 5. One can see that the SNR of the
displacement memory increases nearly monotonically with $\chi$, while the SNR
of the spin memory does not vary too much with $\chi\,$.

Figure 5: Dependence of SNR on the effective spin $\chi$. Plotted using a MBHB
with mass ratio $q=1$, total mass $M=10^{6}{\rm M}_{\odot}$ and luminosity
distance $D_{L}=2\text{Gpc}$ . The blue horizontal line represents SNR equal
to 3. We choose inclination $\iota=\pi/2$ for displacement memory, and
inclination $\iota=\pi/4$ for spin memory.

It is expected that the SNR will strongly depend on the source distance, for
it is inversely proportional to the magnitude of the GW strain. In Fig. 6, we
plot the horizon distance for detection in terms of the redshift $z$ as a
function of the redshifted total mass $M\,$. For the displacement memory, the
maximum redshifts for SNRs equal to 3, 5, 8 is $z=4.6$, $z=2.98$ and $z=2.03$,
respectively. For the spin memory, the maximum redshift for SNR equal to 3 is
$z=0.48$. Using the detection threshold $\rho_{th}=3\,$, one can see that
TianQin can detect the displacement memory from MBHBs at the redshift
$z\lesssim 4.6$ for GW sources with favorable parameters.

Figure 6: The redshift of detection horizons of SNRs equal to 3, 5, 8 for
displacement memory and SNRs equal to 3 for spin memory. The memory effect are
calculated by the MBHB with mass ration $q=1$ and effective spin $\chi=0.8$.
We choose inclination $\iota=\pi/2$ for displacement memory, and inclination
$\iota=\pi/4$ for spin memory.

In Fig. 7, we also plot the dependence of SNR on the total mass $M$ and the
symmetric mass ratio $\eta$, for different values of $\chi$. The maximal SNRs
for both displacement memory and spin memory appear around $\eta=0.25\,$,
which is consistent with the expectation that systems of equal mass can
radiate the most energy and angular momentum during GW radiation and thus can
incur the largest memory effect. For the displacement memory, it is possible
to have SNRs passing the threshold $\rho_{th}=3\,$ for $\eta$ roughly in the
range $0.06\sim 0.25\,$, while for the spin memory, the allowed range for
$\eta$ is narrowed to the small region of roughly $0.22\sim 0.25\,$. Depending
of the values of the $\chi$, the maximal SNRs for the displacement memory
appear roughly in the range $4\times 10^{5}~{}{\rm M}_{\odot}\sim 8\times
10^{5}~{}{\rm M}_{\odot}\,$, while the maximal SNRs for the spin memory appear
roughly in the range $8\times 10^{5}~{}{\rm M}_{\odot}\sim 1.5\times
10^{6}~{}{\rm M}_{\odot}\,$.

Figure 7: The dependence of SNRs on the symmetric mass ration $\eta$ and the
total mass $M\,$, for three values of the effective spin, $\chi=-0.8,0,0.8$,
and fixed luminosity distance $D_{\text{L}}=2{\rm~{}Gpc}\,$. In calculation,
we choose inclination $\iota=\pi/2$ for displacement memory, and inclination
$\iota=\pi/4$ for spin memory. The upper panel is for the displacement memory
while the bottom panel is for the spin memory. The numbers on the red dash
contours indicate the values of the corresponding SNRs.

###  III.2 Detection number

It is known that TianQin can detect several to dozens of MBHBs per year Wang
_et al._ (2019). So a natural question to ask is how many of the detected
events will contain memory effect that is directly detectable.

There is still a lot of uncertainty in the astrophysical models that determine
the population of MBHBs in the universe. In this work, we use three different
models for merger history of massive black holes which have been used in Wang
_et al._ (2019). These three models are generated using the semi-analytic
model proposed in Barausse (2012) and successively improved in Sesana  _et
al._ (2014); Antonini  _et al._ (2015). One model is referred as ”popIII”
which corresponds to a light seed model Madau and Rees (2001), and the other
two models are referred as ”Q3_d” and ”Q3_nod”, which correspond to two heavy
seed models Bromm and Loeb (2003); Begelman  _et al._ (2006); Lodato and
Natarajan (2006) with and without the time delay between the merger of a MBHB
and that of their host galaxies.

In Wang  _et al._ (2019), 1000 mock catalogs have been generated for each of
the astrophysical models. We search in each of these mock catalogs for events
containing memory effect with SNRs exceeding the threshold, $\rho_{th}=3\,$,
counting the detection number, and then average over the 1000 mock catalogue
for each of the astrophysical models. The result is given in TABLE. 1, where
we also present the result for LISA as a comparison. One can see that, on
average, TianQin is projected to detect $0.5\sim 2.0$ MBHB events containing
the displacement memory with SNRs passing the threshold $\rho_{th}=3\,$. The
corresponding number for MBHB events containing the spin memory is no greater
than $0.2$, indicating that chance for a direct detection of the spin memory
is low.

| IMR | $\mathcal{M}>Threshold$ | $\rho_{dis}>3$ | $\rho_{dis}>5$ | $\rho_{dis}>8$ | $\rho_{spin}>3$ | $\rho_{spin}>5$ | $\rho_{spin}>8$
---|---|---|---|---|---|---|---|---
popIII TianQin | $56.8$ | 0.6 | $0.5$ | 0.3 | 0.1 | $\sim 0$ | $\sim 0$ | $\sim 0$
Q3_d TianQin | 18.1 | 0.7 | 0.6 | 0.3 | 0.2 | $\sim 0$ | $\sim 0$ | $\sim 0$
Q3_nod TianQin | 271.4 | 2.3 | 2.0 | 1.2 | 0.7 | $0.2$ | 0.1 | $\sim 0$
popIII LISA | $148.35$ | 1.9 | $1.6$ | 0.7 | 0.4 | $0.1$ | $\sim 0$ | $\sim 0$
Q3_d LISA | 37.4 | 3.1 | 2.6 | 1.4 | 0.8 | 0.2 | 0.1 | $\sim 0$
Q3_nod LISA | 295.5 | 7.0 | 5.8 | 2.6 | 1.4 | 0.4 | 0.2 | 0.1
Table 1: The expected IMR detection number and both displacement memory and
spin memory detection number on TianQin and LISA detectors for ”popIII”,
”Q3_d” and ”Q3_nod” astrophysical models.

##  IV Relevant parameter space

Due to the important application of MBHB events in fundamental physics Shi
_et al._ (2019); Bao  _et al._ (2019), astrophysics Wang  _et al._ (2019) and
cosmology Zhu  _et al._ (2022b), each detected MBHB event would be extremely
precious and it is extremely important to have accurate waveform models to
precisely measure the source parameters. As we have mentioned before, although
the memory effect is physically interesting on its own right, it can also lead
to systematic errors if not properly taken into consideration in waveform
modeling. In this section, we study when the contribution of memory effect
will become non-negligible. One can intuitively reason that this should be
equivalent to when the memory effect will become detectable. So our key task
here is to determine the relevant parameter space.

The effect of neglecting the memory effect in the waveforms can be quantified
using the mismatch between the waveforms with and without the memory effect.
For two different waveform models $\tilde{h}_{1}(f)$ and $\tilde{h}_{2}(f)$,
the mismatch is defined as

$\displaystyle{\cal
M}=1-\frac{\left\langle\tilde{h}_{1},\tilde{h}_{2}\right\rangle}{\sqrt{\left\langle\tilde{h}_{1},\tilde{h}_{1}\right\rangle\left\langle\tilde{h}_{2},\tilde{h}_{2}\right\rangle}}\,,$
(39)

where

$\displaystyle\left\langle\tilde{h}_{1},\tilde{h}_{2}\right\rangle=4\text{Re}\int^{f_{max}}_{f_{min}}\frac{\tilde{h}_{1}(f)\tilde{h}_{2}^{*}(f)}{S_{n}(f)}df\,.$
(40)

The threshold of the mismatch is determined when the systematic error caused
by the wrong waveform equals the statistical errors caused by the instrumental
noise. In the high SNR regime, the threshold of the mismatch is given by
Chatziioannou  _et al._ (2017); Mangiagli  _et al._ (2019)

$\displaystyle{\cal M}_{th}=\frac{D}{2\rho^{2}}\,,$ (41)

where $D$ is the number of parameters used to determine the waveforms, in our
case, we have $D=8$. One needs ${\cal M}<{\cal M}_{th}$ to ensure that the
parameter estimation is unbiased.

The dependence of the mismatch on the symmetric mass ratio $\eta$ and the
total mass $M$, for several values of the effective spin $\chi$ and the fixed
luminosity distance $D_{\rm L}\,$, is plotted in Fig. 8. One can see the
contours for the mismatch threshold and those for the SNR threshold are very
close to each other. This is because the SNR threshold is in fact inferred
from the mismatch threshold, but has been rounded to integer numbers to make
it easier to use.

Figure 8: The dependence of mismatch on the symmetric mass ration $\eta$ and
the total mass $M\,$, for three values of the effective spin,
$\chi=-0.8,0,0.8$, and fixed luminosity distance
$D_{\text{L}}=2{\rm~{}Gpc}\,$. In calculation, we choose inclination
$\iota=\pi/2$ for displacement memory, and inclination $\iota=\pi/4$ for spin
memory. The upper panel is for the displacement memory while the bottom panel
is for the spin memory. The contours indicated where the mismatch
(black/dashed) and the SNR (red/solid) reach their thresholds.

We also plot the critical contour which represent for the mismatch equals to
its threshold for different redshift values on the symmetric mass ratio $\eta$
and redshifted total mass $M$ in Fig. 9. Then for the sources closer than the
corresponding redshift, the memory effect must be considered in order to avoid
possible systematic errors. One can see that the critical contours shrink with
the increasing of the redshift. As we mentioned before, if we don’t consider
the memory effects in the waveform modelling, systematic errors will be
introduced for the relevant parameters such as $M$ and $\eta$. For the
displacement memory, the influence may exist even for a source at $z=4$, if
its about $3\times 10^{5}{\rm M}_{\odot}$ with almost equal masses. However,
for the spin memory effect, it can only have influence for very small
redshift, which means that it can be neglected safety.

Figure 9: The critical contour which represent for the mismatch equals to its
threshold for different redshift values on the symmetric mass ratio $\eta$ and
the total mass $M$ for different redshifts. For calculation of displacement
memory we choose effective spin $\chi=0.98$ and inclination $\iota=\pi/2$,
while for spin memory, we choose effective spin $\chi=0.98$ and inclination
$\iota=\pi/4$. The upper panel is results of the displacement memory and the
bottom panel is results of the spin memory.

In Fig. 10, we plot the horizon distance in redshift for the detection with a
criterion of mismatch and SNR as the function of redshifted total mass $M$.
The horizon distance to pass the threshold of mismatch are always larger than
that for SNR equals to 3. The results also tell us that when the SNRs of the
memory effect are 3, bias must be introduced in the parameter estimates if we
do not consider the memory effect. Using the criterion of mismatch, for the
displacement memory, the maximum redshift is $z=4.74$, and for the spin
memory, the maximum redshift is $z=0.51$.

Figure 10: The horizon distance in redshift for the detection with a criterion
of mismatch and SNR in terms of reshifted total mass. The calculation is made
by the MBHB with $M=10^{6}~{}{\rm M}_{\odot}$, $q=1$ and $\chi=0.8$. In
calculation, we choose inclination $\iota=\pi/2$ for displacement memory, and
inclination $\iota=\pi/4$ for spin memory. The black lines are the results of
displacement memory and the red lines are the results of spin memory. The
dashed lines use the criterion of SNR equal to 3 for both displacement and
spin memory.

##  V Conclusion

Among all the curious effects, the radiation of GWs also cause permanent
changes in the background spacetime. Such changes are caused not only by the
changes in the BMS charges in spacetime, but also by the energy and angular
momentum fluxes that go to the null infinity. So the detection of memory
effect will serve as an important portal to study the nature of gravity and
spacetimes.

These permanent changes, which are called the displacement, spin and center-
of-mass memory, are related to the various types of BMS transformations and
their extensions. Such connection has enabled the calculation of the
displacement memory strain and the spin memory strain using the BMS flux-
balance laws.

In this paper, we study the prospect of using TianQin, a planned space-based
GW detector, to directly detect the memory effect.

We find that the SNRs depend strongly on many of the source parameters. For
the displacement memory, the maximal SNRs can be achieved with the inclination
angle $\iota=\pi/2$, large effective spin, nearly equal mass and with total
masses in the range $4\times 10^{5}~{}{\rm M}_{\odot}\sim 8\times
10^{5}~{}{\rm M}_{\odot}\,$. For the spin memory, the maximal SNRs can be
achieved with the inclination angle near $\iota=3\pi/10$ and $\iota=7\pi/10$,
nearly equal mass and with total masses in the range $8\times 10^{5}~{}{\rm
M}_{\odot}\sim 1.5\times 10^{6}~{}{\rm M}_{\odot}\,$. With favorable source
parameters, the detection horizon can reach $z\approx 4.6$ for the
displacement memory and $z\approx 0.48$ for the spin memory. By using
currently available astrophysical models for MBHBs, we find that TianQin can
detect about $0.5\sim 2.0$ MBHBs events that will contain detectable
displacement memory throughout its five years of mission lifetime, while the
corresponding number for spin memory is no greater than $0.2$, meaning that
the chance for a direct detection is low.

We also study the question of when the contribution of memory effect will
become non-negligible in the waveform modeling of MBHBs. We find that the
criterion for the mismatch beyond the threshold and for SNR=3 are almost
equivalent. By using (nearly) optimal values for the inclination angle and the
effective spin, we determine the parameter space in which the memory effect
can be safely neglected.

##  VI Acknowledgement

We are grateful to Keefe Mitman, Michael Boyle, Moritz Hübner, Oliver Boersma,
Alberto Mangiagli and Neev Khera for their kind helps. We thank Yiming Hu,
Qingfei Zhang and Han Wang for many useful discussions. We also thank Alberto
Sesana and Enrico Barausse for sharing their simulated catalogue of massive
black holes. We acknowledge the usage of the calculation utilities of LALsuite
LIGO Scientific Collaboration (2018), SXS Boyle (2020), NUMPY van der Walt
_et al._ (2011), and SCIPY Virtanen  _et al._ (2020), and plotting utilities
of MATPLOTLIB Hunter (2007). This work has been supported by Guangdong Major
Project of Basic and Applied Basic Research (Grant No.2019B030302001).

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  *[GWs]: gravitational wave
  *[GW]: gravitational wave
  *[BMS]: Bondi-Metzner-Sachs
  *[MBHBs]: massive black hole binary
  *[PN]: post-Newtonian
  *[CCE]: Cauthy-characteristic extraction
  *[SNR]: signal-to-noise ratio
  *[SNRs]: signal-to-noise ratio
  *[PSD]: power spectral density
  *[ASD]: amplitude spectral density
  *[MBHB]: massive black hole binary