# The dominating mode of two competing massive modes of quadratic gravity

Avijit Chowdhury<EMAIL_ADDRESS>Semin Xavier
<EMAIL_ADDRESS>S. Shankaranarayanan<EMAIL_ADDRESS>Department
of Physics, Indian Institute of Technology Bombay, Mumbai 400076, India

###### Abstract

Over the last two decades, motivations for modified gravity have emerged from
both theoretical and observational levels. $f(R)$ and Chern-Simons gravity
have received more attention as they are the simplest generalization. However,
$f(R)$ and Chern-Simons gravity contain only an additional scalar (spin-0)
degree of freedom and, as a result, do not include other modes of modified
theories of gravity. In contrast, quadratic gravity (also referred to as
Stelle gravity) is the most general second-order modification to 4-D general
relativity and contains a massive spin-2 mode that is not present in $f(R)$
and Chern-Simons gravity. Using two different physical settings — the
gravitational wave energy-flux measured by the detectors and the backreaction
of the emitted gravitational radiation on the spacetime of the remnant black
hole — we demonstrate that _massive spin-2 mode carries more energy than the
spin-0 mode_. Our analysis shows that the effects are pronounced for
intermediate-mass black holes, which are prime targets for LISA.

## Introduction

In quantum gravity, it is impossible to localize an event with a precision
smaller than the Planck length ($\ell_{\rm Pl}$) Ng and Van Dam (1994); Garay
(1995); Adler and Santiago (1999); Calmet  _et al._ (2004). From the hoop
conjecture and the uncertainty principle, we can deduce the existence of a
Planck-sized ball Thorne . Thus, the operational significance of the concept
of space-time points is lost DeWitt (1964, 1981); Kuo and Ford (1993);
Padmanabhan (1997). Most approaches to quantum gravity incorporate the Planck
length by considering extended structures, rather than point particles, as
fundamental blocks Hossenfelder (2013); Amelino-Camelia (2013). The
Generalized Uncertainty Principle (GUP) is a phenomenological approach that
introduces modifications to the Heisenberg uncertainty principle in the
ultraviolet regime and studies its consequences Kempf  _et al._ (1995); Pedram
(2010). Kempf et al. considered the following simplest modification to the
canonical commutation relation Kempf  _et al._ (1995):

$[\hat{x},\hat{p}]=i\hbar\left(1+\beta p^{2}\right)\quad\mbox{leading
to}\quad\Delta_{x}\geq\hbar\sqrt{\beta}\,.$

$\beta$ is a parameter characterizing the GUP whose value needs to be fixed by
observations. Since the above GUP is non-relativistic, it is impossible to
compute GUP corrections in relativistic field theories. Recently, Todorinov et
al. Todorinov  _et al._ (2019) extended the above GUP to a generic class of
covariant GUPs:

$\begin{split}[x^{\mu},p^{\nu}]=i\hbar\eta^{\mu\nu}(1-\gamma
p^{\sigma}p_{\sigma})-2i\hbar\gamma
p^{\mu}p^{\nu},~{}\gamma=\gamma_{0}\ell_{\text{Pl}}^{2}\end{split}$ (1)

and studied the phenomenological features of such GUPs for scalar, spinor, and
vector fields Bosso  _et al._ (2020a, b). $\gamma_{0}$ is a positive numerical
constant to be fixed by observations. There exists a correspondence between
the GUP-modified dynamics of a massless spin-2 field and quadratic gravity
(QG) with suitably constrained parametersNenmeli  _et al._ (2021),.
Specifically, the authors showed that the 4-D gravity action Stelle (1977,
1978)

$S_{\rm QG}=\int{d^{4}x\sqrt{-g}\left[\frac{1}{2\kappa^{2}}R-\alpha
R_{\mu\nu}R^{\mu\nu}+\beta R^{2}\right]},$ (2)

with $\kappa^{2}=\frac{8\pi G}{c^{4}}$ is a classical manifestation of the
imposition of a momentum cutoff at the quantum gravity level when
$\alpha=2\beta=\gamma/\kappa^{2}$. QG or Stelle gravity — unlike $f(R)$ — has
extra massive spin-2 and spin-0 modes Stelle (1978). Intriguingly, for this
class of Stelle theories, the masses of the spin-0 and spin-2 modes _coincide_
$(1/\sqrt{2\gamma})$. The issue of unitarity in these theories is not
completely settled yet Salam and Strathdee (1978); Fradkin and Tseytlin
(1982); Antoniadis and Tomboulis (1986); Johnston (1988) the massive spin-2
mode may be of ghost nature. See, for instance, the fakeon approach for the
same Anselmi (2017); Anselmi  _et al._ (2020). Given that the Stelle gravity
is the most general QG in 4-D, we ask the following questions: For identical
masses, what is the role of the massive spin-2 mode compared to the spin-0
mode? What are the leading order corrections of QG compared to $f(R)$ gravity?
This work addresses these questions by evaluating the corrections to the
radiation in QG.

Over the last two decades, motivations for modified gravity have emerged from
both theoretical and observational levels Alexander and Yunes (2009); De
Felice and Tsujikawa (2010); Will (2014a); Nojiri  _et al._ (2017);
Shankaranarayanan and Johnson (2022). Since $f(R)$ and Chern-Simons gravity
are the simplest generalization, they have received more attention. However,
$f(R)$ and Chern-Simons gravity contain _only the additional scalar degree of
freedom_ and hence do not contain other modes of modified theories of gravity
Odintsov  _et al._ (2022). We explicitly show that the massive spin-2 mode
carries _more energy than_ the spin-0 mode using two different physical
settings: the energy flux of the gravitational waves, measured by an
asymptotically placed gravitational wave detector, and the backreaction of the
emitted gravitational radiation on the spacetime of the remnant black hole.
Thus, we show that the leading order gravitational radiation correction in QG
is linear in the coupling constant compared to $f(R)$, where the corrections
are quadratic in flat space-time Berry and Gair (2011).

To show this explicitly, we start with the GUP-inspired version of QG action
(2), where Nenmeli  _et al._ (2021)

$\alpha={\gamma}/{\kappa^{2}};\quad\beta={\gamma}/{2\kappa^{2}}\,.$

Varying action (2) with respect to the metric leads to:

$\begin{split}{\mathcal{G}}_{\mu\nu}\equiv&G_{\mu\nu}-2\gamma\Box
G_{\mu\nu}-4\gamma R^{\rho\sigma}R_{\mu\rho\nu\sigma}+2\gamma
RR_{\mu\nu}+\gamma
g_{\mu\nu}\left[R^{\rho\sigma}R_{\rho\sigma}-\frac{R^{2}}{2}\right]=0\,.\end{split}$
(3)

Linearizing the field equation (3) about the Minkowski space-time
($\eta_{\mu\nu}$), we get,

$g_{\mu\nu}=\eta_{\mu\nu}+\epsilon
h_{\mu\nu}~{},~{}g^{\mu\nu}=\eta^{\mu\nu}-\epsilon
h^{\mu\nu}~{}\mbox{and}~{}h=\eta^{\mu\nu}h_{\mu\nu},$ (4)

where $h_{\mu\nu}$ is the metric perturbation, and $\epsilon$ is a book-
keeping parameter. This leads to the following linearized equations:

$\mathcal{\delta G}_{\mu\nu}\equiv\left(1-2\gamma\bar{\Box}\right)\left(\delta
R_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}\delta R\right)=0~{},$ (5)

where $\bar{\Box}\equiv\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}$ and

$\displaystyle\delta R_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\partial_{\mu}\partial_{\rho}h^{\rho}_{\nu}+\partial_{\nu}\partial_{\rho}h^{\rho}_{\mu}-\partial_{\mu}\partial_{\nu}h-\bar{\Box}h_{\mu\nu}\right),$
(6) $\displaystyle\delta R$ $\displaystyle=$ $\displaystyle\eta^{\mu\nu}\delta
R_{\mu\nu}=\left(\partial_{\mu}\partial_{\rho}h^{\rho\mu}-\bar{\Box}h\right)\,.$
(7)

As expected in higher-derivative gravity theories, Eq. (5) contains fourth-
order derivatives whose trace is given by:

$\bar{\Box}\delta R-\delta R/{2\gamma}=0.$ (8)

To separate the contribution of the different spin-components from linearized
field equation (5), we use the following ansatz for the metric perturbations
(See Appendix A for a detailed calculation in Ricci-flat spacetime):

$h_{\mu\nu}=\left[\psi_{\mu\nu}-\frac{\eta_{\mu\nu}}{2}\psi\right]+\left[C_{1}+\frac{C_{2}}{4}\right]R^{(1)}\eta_{\mu\nu}-C_{2}\hat{R}^{(1)}_{\mu\nu}~{},$
(9)

where $\psi=\eta^{\alpha\beta}\psi_{\alpha\beta}$ and $C_{1},C_{2}$ are
arbitrary constants, and
$\hat{R}^{(1)}_{\mu\nu}=R^{(1)}_{\mu\nu}-\frac{1}{4}\eta^{\mu\nu}R^{(1)}$ is
the traceless part of $R^{(1)}_{\mu\nu}$. To our knowledge, this is the first
time an explicit separation of metric fluctuations is used in this context,
and we would like to emphasize the following points: First, in general
relativity (GR), the constants $C_{1}$ and $C_{2}$ vanish because the
linearized field equations only contain massless spin-2 (graviton) mode. Thus,
in GR, $\psi_{\mu\nu}$ reduces to the familiar trace-reversed metric
perturbations. Second, in the case of Starobinsky model, the field equations
contain additional contribution from the massive spin-0 mode $(R^{(1)})$ only,
hence $C_{1}=-6\gamma$ and $C_{2}=0$ Berry and Gair (2011); Bhattacharyya and
Shankaranarayanan (2017). Third, the existence of the massive spin-2 ghost
degrees of freedom may suggest a potential pathology due to Ostrogradsky
instability Stelle (1977), in the present context it is possible to avoid such
a pathology by treating the massive and massless spin-2 modes as a single
structureAlves  _et al._ (2022). Lastly, like in GR, we use the de-Donder
gauge on the residual massless graviton mode ($\psi_{\mu\nu}$) to restrict the
gauge freedom:

$\partial^{\mu}\psi_{\mu\nu}=0\,.$ (10)

Substituting Eq. (9) in Eqs. (6,7) and using the gauge conditions (10), we
get:

$2\,\mathcal{\delta G}_{\mu\nu}=-\bar{\Box}\psi_{\mu\nu}~{}.$ (11)

where we set $C_{1}=-2\gamma$ and $C_{2}=4\gamma$. Taking cognizance of Eqs.
(5, 11) and (8), we get the propagation equation for the graviton, the massive
spin-2 and spin-0 modes as,

$\displaystyle\bar{\Box}\psi_{\mu\nu}=0~{},$ (12)
$\displaystyle\bar{\Box}\hat{R}^{(1)}_{\mu\nu}-\hat{R}^{(1)}_{\mu\nu}/{2\gamma}=0~{},$
(13) $\displaystyle\bar{\Box}R^{(1)}-R^{(1)}/{2\gamma}=0~{}$ (14)

(See Appendix B for a detailed derivation of the propagation equations in a
Ricci-flat background). Thus, the theory (2) is described by a massless
graviton ($\psi_{\mu\nu}$); a massive spin-2 ($\hat{R}^{(1)}_{\mu\nu}$) and a
massive spin-0 (${R}^{(1)}$) Stelle (1977, 1978); Nenmeli  _et al._ (2021);
Tachinami  _et al._ (2021) (The massive spin-2 mode can be further decomposed
into two tensor (helicity-2) modes, two vector (helicity-1) modes, and scalar
(helicity-0) mode, whereas the massless spin-2 graviton gives rise two tensor
(helicity-2) modes and the massive spin-0 particle has one scalar (helicity-0)
mode. The different helicity modes can, in principle, be mapped to the six
metric degrees of freedom denoting the six polarization modes of the
gravitational waves Tachinami  _et al._ (2021); Eardley  _et al._ (1973); Will
(2014b); Lee  _et al._ (1974)). The positivity of the parameter $\gamma$
ensures that the massive modes are non-tachyonic. Repeating the analysis for
the Ricci-flat background by using the following ansatz:

$h_{\mu\nu}=\left(\psi_{\mu\nu}-\frac{1}{2}\psi\bar{g}_{\mu\nu}\right)-\bar{g}_{\mu\nu}\gamma
R^{(1)}-4\gamma\hat{R}^{(1)}_{\mu\nu}~{},$ (15)

the equations of motion for the massless (12) and massive spin-2 modes (13) in
the transverse traceless gauge
$\left(\bar{\nabla}^{\mu}\psi_{\mu\nu}=0,~{}\bar{g}^{\mu\nu}\psi_{\mu\nu}=0\right)$
are:

$\displaystyle\bar{\Box}\psi_{\mu\nu}+2\bar{R}_{\mu\alpha\nu\beta}\psi^{\alpha\beta}=0~{},$
(16)
$\displaystyle\bar{\Box}\hat{R}^{(1)}_{\mu\nu}+2\bar{R}_{\mu\alpha\nu\beta}\hat{R}^{(1)\alpha\beta}-\hat{R}^{(1)}_{\mu\nu}/{(2\gamma)}=0,$
(17)

where $\bar{g}_{\mu\nu}$ is the background metric,
$\bar{R}_{\mu\alpha\nu\beta}$ is the background Riemann tensor,
$\bar{\Box}\equiv\bar{\nabla}_{\sigma}\bar{\nabla}^{\sigma}$ with
$\bar{\nabla}^{\sigma}$ being the covariant derivative due to the background
spacetime and the _traceless_ tensor $\hat{R}^{(1)}_{\mu\nu}$ is

$\hat{R}^{(1)}_{\mu\nu}=R^{(1)}_{\mu\nu}-\bar{g}^{\mu\nu}R^{(1)}/4~{}.$ (18)

The propagation of the massive spin-0 mode is still governed by Eq. (14), with
the above-defined D’Alembertian operator. The mass degeneracy between the two
massive (spin-0 and spin-2) modes demonstrates that they are not completely
independent and are related by linearized Bianchi identities:

$4\,\bar{\nabla}_{\mu}\hat{R}^{(1)\mu\nu}-\
\bar{g}^{\mu\nu}\bar{\nabla}_{\mu}R^{(1)}=0\,.$ (19)

Having separated the metric fluctuations into massless and massive modes in
Ricci flat background, we now evaluate the energy and momentum carried by the
gravitational waves in degenerate-Stelle gravity. To go about that, we expand
the field equation to second-order in $\epsilon$:

$\mathcal{G}_{\mu\nu}=\bar{\mathcal{G}}_{\mu\nu}+\epsilon\mathcal{\delta
G}_{\mu\nu}+\epsilon^{2}\delta^{2}\mathcal{G}_{\mu\nu}=0,$ (20)

where $\bar{\mathcal{G}}_{\mu\nu}$ represents the background quantity,
$\delta\mathcal{G}_{\mu\nu}$ are linear in perturbations ($h_{\mu\nu}$) and
$\delta^{2}\mathcal{G}_{\mu\nu}$ are quadratic in $h_{\mu\nu}$. Through
second-order perturbations, Isaacson established a procedure to obtain an
effective stress-energy tensor for gravitational radiation Isaacson (1968a,
b). Specifically, the effective stress-energy tensor of the emitted
gravitational waves is obtained by averaging over a length-scale $l$ such that
$\lambdabar\ll l\ll\Lambda$, where $\lambdabar$ is the wavelength of the
fluctuations and $\Lambda$ is the system size. The short wavelength components
will be averaged out, yielding a gauge-invariant measure of the effective
gravitational wave (GW) stress-energy tensor Brill and Hartle (1964):

$t_{\mu\nu}^{\rm
GW}=\frac{1}{\kappa^{2}}\left<\bar{\mathcal{G}}_{\mu\nu}\right>=-\frac{1}{\kappa^{2}}\left<\delta^{2}\mathcal{G}_{\mu\nu}\right>.$
(21)

In the Ricci-flat background, we get

$\begin{split}t_{\mu\nu}^{\rm
GW}=&\frac{1}{\kappa^{2}}\Bigl{[}\left<\frac{1}{4}\bar{\nabla}_{\mu}\psi^{\rho\sigma}\bar{\nabla}_{\nu}\psi_{\rho\sigma}-\gamma\left(\bar{\nabla}_{\mu}\psi^{\rho\sigma}\bar{\nabla}_{\nu}\hat{R}^{(1)}_{\rho\sigma}+\bar{\nabla}_{\nu}\psi^{\rho\sigma}\bar{\nabla}_{\mu}\hat{R}^{(1)}_{\rho\sigma}\right)\right.\\\
&\left.-\gamma^{2}\left(4\bar{\nabla}_{\mu}\hat{R}^{(1)\rho\sigma}\bar{\nabla}_{\nu}\hat{R}^{(1)}_{\rho\sigma}-\bar{\nabla}_{\mu}R^{(1)}\bar{\nabla}_{\nu}R^{(1)}\right)\right>+\left<\mathcal{A}_{\mu\nu}+\gamma\mathcal{B}_{\mu\nu}+\gamma^{2}\mathcal{C}_{\mu\nu}+\gamma^{3}\mathcal{D}_{\mu\nu}\right>\Bigr{]},\end{split}$
(22)

where
$\mathcal{A}_{\mu\nu},\mathcal{B}_{\mu\nu},\mathcal{C}_{\mu\nu},\mathcal{D}_{\mu\nu}$
are related to the background Riemann tensor (and are explicitly given in
Appendix C). This is the first key result of this work, regarding which we
would highlight the following points: First, in the Minkowski limit (as in the
case of GW detectors),
$\mathcal{A}_{\mu\nu},\mathcal{B}_{\mu\nu},\mathcal{C}_{\mu\nu},\mathcal{D}_{\mu\nu}$
vanish and $t_{\mu\nu}^{\rm GW}$ is proportional to partial derivatives of
$\psi^{\rho\sigma}$, $\hat{R}^{(1)}_{\rho\sigma}$ and $R^{(1)}$. Second, the
first term within the triangular bracket gives the dominant contribution — the
contribution of the graviton mode as in GR Isaacson (1968a). However, the
_crucial difference is the dominant contribution_ of the massive spin-2 mode.
The massive spin-2 mode contribution is proportional to $\gamma$, whereas the
massive spin-0 mode contribution is proportional to $\gamma^{2}$. Thus, the
above expression explicitly shows that the massive spin-2 mode carries more
energy than the massive spin-0 mode. Third, while we have used Isaacson’s
approach to obtain the stress-tensor, other approaches also give similar
resultsSaffer  _et al._ (2018). Fourth, the leading order contribution of the
massive spin-2 mode is opposite to that of the graviton mode. Lastly, the
corrections by QG to GR are much larger than $f(R)$ gravity Berry and Gair
(2011); Bhattacharyya and Shankaranarayanan (2018). Consequently, our study
demonstrates that the $f(R)$ theories _overlook crucial information_
concerning the massive spin-2 mode. In what follows, we use the GW stress-
energy tensor (22) to examine the effect of the massive spin-2 mode under two
distinct physical settings. First, we investigate the energy flux of GWs as
measured by the _GW detectors at asymptotic infinity_. We then estimate the
change in the spherically symmetric metric caused by the backreaction of the
_emitted GWs near the horizon._

## Gravitational wave energy flux:

The energy of the gravitational wave within a spatial volume $V$ is Misner
_et al._ (1973); Maggiore (2007)

$E_{V}=\int_{V}d^{3}x~{}t^{00}_{\rm GW}.$ (23)

Using the stress-tensor conservation equation ($\partial_{\mu}t^{\mu\nu}_{\rm
GW}=0$) for the far-away observer, the power carried by the gravitational
waves is Saffer  _et al._ (2018); Maggiore (2007)

$\frac{dE}{d(ct)}=-\oint_{S}t^{0i}_{\rm GW}\,n_{i}dA$ (24)

where $S$ is the surface enclosing the volume $V$ and $n_{i}$ is the unit
outward normal to $S$. The negative sign signifies that an outward propagating
gravitational wave carries away energy from the source. Thus, plugging the
Minkowski-limit of Eq. (22) in Eq. (24) and considering $S$ to be a spherical
surface at a large distance from the source, the gravitational wave energy
flux passing through the detector is:

$\begin{split}\frac{dE}{dtdA}=\frac{c}{\kappa^{2}}\left<\frac{1}{4}\partial^{0}\psi^{\rho\sigma}\partial_{r}\psi_{\rho\sigma}\right.\left.-\gamma\left[\partial^{0}\psi^{\rho\sigma}\partial_{r}\hat{R}^{(1)}_{\rho\sigma}+\partial^{0}\hat{R}^{(1)}_{\rho\sigma}\partial_{r}\psi^{\rho\sigma}\right]\right.\left.-\gamma^{2}\left[4\partial^{0}\hat{R}^{(1)\rho\sigma}{\partial_{r}}\hat{R}^{(1)\rho\sigma}-\partial^{0}R^{(1)}{\partial_{r}}R^{(1)}\right]\right>\end{split}$
(25)

To make the calculations transparent, we assume that all the three (graviton,
massive spin-2, and spin-0) modes of the following form:

${\begin{split}&\psi_{\mu\nu}=\zeta_{\mu\nu}e^{ik_{\rho}x^{\rho}}+\zeta^{*}_{\mu\nu}e^{-ik_{\rho}x^{\rho}};~{}R^{(1)}=\phi
e^{iq_{\rho}x^{\rho}}+\phi^{*}e^{-iq_{\rho}x^{\rho}};\\\
&\hat{R}^{(1)}_{\mu\nu}=\theta_{\mu\nu}e^{iw_{\rho}x^{\rho}}+\theta^{*}_{\mu\nu}e^{-iw_{\rho}x^{\rho}}\end{split}}$
(26)

where $\zeta_{\mu\nu},\theta_{\mu\nu}$ and $\phi$ depend on $r$ and $t$,

$|\mathbf{k}|=\omega/c~{}~{};~{}~{}|\mathbf{w(q)}|=\sqrt{\omega^{2}/c^{2}-1/2\gamma}$
(27)

with $k^{0}=w^{0}=q^{0}=\omega/c$, $\mathbf{k}=\left(k^{i}\right)$,
$\mathbf{w}=(w^{i})$, and $\mathbf{q}=(q^{i})$. $\omega>c/\sqrt{2\gamma}$ and
$q>c/\sqrt{2\gamma}$ corresponds to oscillatory solution for
$\hat{R}^{(1)}_{\mu\nu}$ and $R^{(1)}$, respectively. A wave propagating
radially outward $(\Phi_{\mu\nu})$ at large distances from the source can be
represented to fall-off radially Maggiore (2007); Poisson and Will (2014):

$\Phi_{\mu\nu}=\frac{1}{r}\chi_{\mu\nu}(t_{r});$ (28)

where $\chi_{\mu\nu}(t_{r})$ is an arbitrary function of the retarded time
$t_{r}=t-r/v$, and
$v=c/\sqrt{1-\frac{1}{2\gamma}\left(\frac{c}{\omega}\right)^{2}}$ is the speed
of the massive modes. Thus,

$\begin{split}\frac{\partial}{\partial
r}\chi_{\mu\nu}(t-r/v)=-\frac{1}{v}\frac{\partial}{\partial
t}\chi_{\mu\nu}(t-r/v)\\\ \Longrightarrow\frac{\partial}{\partial
r}\zeta_{\mu\nu}=-\frac{c}{v}\partial_{0}\zeta_{\mu\nu}+O\left(\frac{1}{r^{2}}\right)\end{split}$
(29)

Substituting Eq. (26) in Eq. (25) and using Eq. (29) in the resultant
expression leads to the energy flux on the GW detectors:

$\begin{split}\frac{dE}{dtdA}=\frac{c^{3}}{8\pi
G}\left<\frac{1}{4}\dot{\psi}^{\rho\sigma}\dot{\psi}_{\rho\sigma}-\gamma\left(1+\frac{c}{v}\right)\dot{\psi}^{\rho\sigma}\dot{\hat{R}}^{(1)}_{\rho\sigma}\right.\left.-\gamma^{2}\frac{c}{v}\left(4\dot{\hat{R}}^{(1)\rho\sigma}\dot{\hat{R}}^{(1)}_{\rho\sigma}-\left(\dot{{R}}^{(1)}\right)^{2}\right)\right>~{},\end{split}$
(30)

where dot denotes derivative w.r.t $t$. Here again, we note that the dominant
contribution comes from the graviton mode with leading order corrections
$(\gamma)$ from the massive spin-2 mode; the massive spin-0 mode contributes
only in the second order. Hence, as expected, the measured energy energy-flux
in the case of QG is lower than that predicted by GR. In other words, the
massive spin-2 mode carries more energy than the spin-0 mode. Since, this
analysis is for the Minkowski background, $\gamma{\cal B}_{\mu\nu}$ (in Eq.
22) is zero. However, in the case of curved geometry, $\gamma{\cal
B}_{\mu\nu}$ contribution might be significantly larger than the linear order
term in the above expression.

In the case of GW detectors, the average $<\ldots>$ is purely a temporal
average Maggiore (2007), and the total energy flowing through the unit area of
the detector is:

$\begin{split}\frac{dE}{dA}=\frac{c^{3}}{8\pi
G}\int_{-\infty}^{\infty}dt\left[\frac{1}{4}\dot{\psi}^{\rho\sigma}\dot{\psi}_{\rho\sigma}-\gamma\left(1+\frac{c}{v}\right)\dot{\psi}^{\rho\sigma}\dot{\hat{R}}^{(1)}_{\rho\sigma}\right.\left.-\gamma^{2}\frac{c}{v}\left(4\dot{\hat{R}}^{(1)\rho\sigma}\dot{\hat{R}}^{(1)}_{\rho\sigma}-\left(\dot{{R}}^{(1)}\right)^{2}\right)\right]~{}.\end{split}$
(31)

Note that the above analysis is strictly valid for $\omega>c/\sqrt{2\gamma}$
and $q>c/\sqrt{2\gamma}$ corresponding to oscillatory solutions for the two
massive modes. In the second physical setting, we will relax this condition
and obtain the contribution of the massive spin-2 mode.

## Backreaction of the emitted gravitational waves

To study the backreaction of the emitted GWs on the background black hole
space-time, we assume that the background space-time is spherically symmetric
Ricci flat and is an exact solution of the QG action (2). Though Schwarzschild
solution in Stelle gravity is known to suffer from $\ell=0$ mode instability
(Gregory-Laflamme instability) when the spin-0 mode is non-propagating Myung
(2013); Myung and Park (2018); Brito  _et al._ (2013); Lü  _et al._ (2017), in
the present work, we consider the spin-0 mode to be dynamical and concentrate
on the $\ell=2$ modes. The backreaction on the background space-time is
obtained by evaluating the GW stress-energy tensor ($t_{\mu\nu}^{\rm GW}$)
w.r.t the Schwarzschild metric. Having obtained this, we then solve the
effective Einstein’s equations:

${G_{\mu}^{\nu}}^{\rm mod}=\kappa^{2}~{}{t_{\mu}^{\nu}}^{\rm GW}\,,$ (32)

where the Einstein tensor ${G_{\mu}^{\nu}}^{\rm mod}$ is evaluated for
spherically symmetric space-time in dimensionless, Eddington-Finkelstein (EF)
coordinates Misner  _et al._ (1973):

$ds^{2}=-e^{2\nu}dV^{2}+2e^{\nu+\lambda}dVd\rho+\rho^{2}d\Omega^{2},$ (33)

$\nu\equiv\nu\left(V,\rho\right)$ and
$\lambda\equiv\lambda\left(V,\rho\right)$ encode the corrections from the
emitted GWs, and $d\Omega^{2}$ is the metric on unit 2-sphere. Note that $V$
and $\rho$ are dimensionless like $\theta$ and $\phi$. Regarding Eqs. (32,
33), the following points are in order: First, the ansatz (15) assumes that
all the metric components are dimensionless. Hence, we have rescaled all the
coordinates to be dimensionless. Second, $V=constant$ hypersurfaces represent
the ingoing null geodesics. As mentioned, the background metric is assumed to
be Schwarzschild black hole; hence, $e^{2\nu}=e^{-2\lambda}=1-2M_{0}/\rho$
where $M_{0}$ is the dimensionless mass parameter (setting $c=G=1$). For
$M(V)$, the above metric gives the Vaidya line element. Third, the ingoing EF
coordinates are smooth across the horizon for ingoing null geodesics and are
suitable for analyzing the gravitational waveform close to the horizon Philipp
and Perlick (2015a, b) as well as the shift in the horizon radius due to the
backreaction. Finally, since $t_{\mu\nu}^{\rm GW}$ contains the contributions
of graviton and massive modes, the LHS of Eq. (32) only contains the Einstein
gravity. Note that the radiation from a remnant black hole decreases its
energy content, inducing a change in the metric.

Since the dominant contributions to the GW stress-energy tensor (22) come
_only from the spin-2 mode_ , to evaluate their effects on the metric, we use
the following ansatz:

$\psi^{\mu\nu}=o^{\mu\nu}\psi(V,\rho)Y^{m}_{\ell}(\theta,\phi);~{}~{}\hat{R}^{(1)\mu\nu}=\iota^{\mu\nu}P(V,\rho)Y^{m}_{\ell}(\theta,\phi),$
(34)

where $o^{\mu\nu}$, $\iota^{\mu\nu}$ are constant, traceless (polarization)
tensors, $\psi(V,\rho)$, $P(V,\rho)$ are scalar functions and
$Y^{m}_{l}(\theta,\phi)$ are spherical harmonics. The above assumptions
essentially replaces the spin-2 modes $\psi^{\mu\nu},~{}\hat{R}^{(1)\mu\nu}$
by scalar functions, where we ignored the non-linear transformation among the
components of the individual spin-2 modes. To obtain the leading order
corrections, we concentrate on the backreaction effect due to the $\ell=2$ and
$m=0$ mode of the gravitational waveform and obtain the average contribution
over the entire solid angle. Note that the $(\ell=2,m=0)$ mode contributes to
the non-linear memory of the GWs, which is otherwise difficult to observe in
ground-based GW detectors Favata (2010); Mehta  _et al._ (2017) and nontrivial
to extract in numerical relativity simulations. We can trivially extend the
analysis to other modes.

We assume the modified black hole to be described by the generalized
spherically symmetric metric ansatz proposed by Johannsen and Psaltis (JP)
Johannsen and Psaltis (2011); Rezzolla and Zhidenko (2014). In the
(dimensionless) EF coordinates (33), we have:

$\begin{split}e^{2\nu}=f(\rho)\left[1-\frac{2\tilde{M}}{\rho}\right];\quad
e^{2\lambda}=\frac{f(\rho)}{1-2\tilde{M}/\rho};\quad
f(\rho)=\sum_{n=0}^{\infty}\tilde{\epsilon}_{n}\left(\frac{\tilde{M}}{\rho}\right)^{n}.\end{split}$
(35)

where $\tilde{\epsilon}_{0}=1$ and the first few coefficients of the expansion
can be constrained from the PPN-like parameters Johannsen and Psaltis (2011).
In the limit of $\tilde{\epsilon}_{n}=0,~{}(n>0)$, JP metric reduces to the
Schwarzschild metric. The event horizon of the corresponding black hole is at
$\rho=2\tilde{M}$ and the (dimensionless) ADM mass is $M_{\rm
ADM}=\tilde{M}(1-\tilde{\epsilon}_{1}/2)$ Rezzolla and Zhidenko (2014). As
mentioned earlier, the remnant black hole decreases its energy content,
inducing a change in the metric mass-function from the initial, dimensionless
Schwarzschild value $(M_{0})$ Abramo and Finelli (2001); Kimura  _et al._
(2021):

$\tilde{M}=M_{0}+\Delta_{M}(V)$ (36)

Assuming the mode functions to be _regular and slowly varying_ in $\rho$ close
to horizon $(\Delta=\rho-2M_{0}<<1)$ Philipp and Perlick (2015a, b);
Bhattacharyya  _et al._ (2020), and expanding both sides of Eq. (32) for the
$\rho-V$ component, we get:

$\sum_{n=0}^{\infty}\frac{2^{n-1}}{\tilde{\epsilon}_{n}}~{}~{}\frac{\Delta_{M}^{\prime}(V)}{M_{0}^{2}}\approx\frac{M_{0}^{4}\psi^{\prime
2}}{21\pi}\sum_{\begin{subarray}{c}(k,l)=\\\
\\{(0,0),(2,3)\\}\end{subarray}}C^{kl}\Biggl{[}1-\frac{\iota^{kl}\gamma
P^{\prime}}{o^{kl}\psi^{\prime}}\Biggr{]}$ (37)

where $\\{C^{00},C^{23}\\}=\\{29(o^{00})^{2},20(o^{23})^{2}\\}$ and ′ denotes
derivative with respect to $V$. The expression is evaluated to the leading
order in $\Delta$ and $M_{0}$. Integrating from some initial $V=V_{0}$, when
$\Delta_{M}(V=V_{0})=0$ to some final $V=V_{1}$, we get,

$\Delta_{M}(V_{1})=A\int_{V=V_{0}}^{V1}dV\sum_{\begin{subarray}{c}(k,l)=\\\
\\{(0,0),(2,3)\\}\end{subarray}}C^{kl}\Biggl{[}1-\frac{\iota^{kl}\gamma
P^{\prime}}{o^{kl}\psi^{\prime}}\Biggr{]},$ (38)

where

$A=\frac{M_{0}^{6}\psi^{\prime
2}}{21\pi}\sum_{n=0}^{\infty}\frac{\tilde{\epsilon}_{n}}{2^{n-1}}$ (39)

This is the third key result regarding which we would like to stress the
following: First, the massive and massless spin-2 modes contribute oppositely
to the change in the mass function, hence the shift in the horizon radius.
Second, since $\Delta_{M}$ is proportional to the sixth power of $M_{0}$, it
implies that the larger the mass of the perturbed black hole, the larger the
corrections to the change in the mass. This assumes particular importance for
intermediate-mass black holes which are prime targets for LISA. Rezzolla and
Zhidenko Rezzolla and Zhidenko (2014) proposed an improved parametric metric
in which, in the near-horizon limit, the Taylor expansion is replaced by an
expansion in continued fractions (CF). The Rezzolla-Zhidenko metric has better
convergence properties and can effectively reproduce any known solution even
in scenarios where the JP parametrization fails. The CF expansion coefficients
in the Rezzolla-Zhidenko metric can be expressed in terms of the JP
parameters; hence it is possible to extend our results to the Rezzolla-
Zhidenko metric Rezzolla and Zhidenko (2014).

## Summary and Discussions

In this work, we examined gravitational radiation in QG. We explicitly
decomposed the GWs in Stelle gravity into massless and massive spin-2 and
spin-0 modes. We demonstrated that the dominant contribution to the GW stress-
energy tensor comes from the graviton mode, with leading order corrections
coming from the massive spin-2 mode, which is absent in $f(R)$ gravity
theories. We can ascertain this because the massive spin-2 and spin-0 have the
same mass in the case of GUP-inspired Stelle gravity. In contrast, the effects
of the massive spin-2 mode are the inverse of those of the GR mode. In the
context of GW detectors, this results in a decreased energy flux measurement.
This result is consistent with a recent finding that this Stelle gravity model
reduces the amplitude of primordial gravitational waves produced by
Starobinsky inflation Das  _et al._ (2022).

We also provided an estimate of the backreaction effect of the GW emission on
the background spacetime, where we once again observe that the massive spin-2
mode decreases the rate of mass-change and the rate of shift in the horizon
radius. Our results indicate that intermediate-mass black holes (prime targets
for LISA) might be good candidates to test these aspects of modified gravity
theories. Focusing on the $\ell=2,m=0$ modes, our analysis suggests that the
backreaction effect may play a crucial role in the study of nonlinear memory
of GWs in modified gravity theories. These are currently under investigation.

The analysis in this work looks at the interesting possible signatures of
strong gravity corrections in future GWs experiments. While the Starobinsky
model and Stelle gravity are the low-energy quantum gravity action, these are
not exhaustive. For instance, we have not included $R\ln\left(\Box R\right)$
and $R_{\mu\nu}\ln\left(\Box R^{\mu\nu}\right)$ Capozziello and Bajardi
(2022). It may be interesting to investigate the potential ring-down
signatures of these terms as these terms may exceed the Stelle gravity
contributions in low momenta. This is currently under investigation.

The decrease in measured energy flux and the decreased rate of horizon shift
and mass change due to the massive spin-2 mode indicate the existence of
quasi-bound states of the massive spin-2 modes surrounding the black hole
Brito (2016). In the context of rotating black hole geometries, where this may
lead to the formation of superradiantly induced spin-2 boson clouds, the
question assumes greater significance. A detailed analysis of the massive
spin-2 dynamics can resolve this question. However, such an analysis is beyond
the scope of this paper so that we will leave it for future work.

## Acknowledgements

The authors thank Susmita Jana, S. Mahesh Chandran, and T. Parvez for the
discussions. S.X. is financially supported by the MHRD fellowship at IIT
Bombay. This work is supported by SERB-MATRICS grant.

##  Appendix A Quadratic gravity as GR “minus” Massive Gravity

As first pointed out by Stelle Stelle (1978), a generic quadratic theory of
gravity is equivalent to GR “minus” massive gravity at a linear level in flat
space. In this section, we will demonstrate an extension of the analysis to a
Ricci-flat spacetime and prove the consistency of the decomposition presented
in Eqs. (15) and (9). The generic quadratic gravity action in four dimensions
is represented in Eq. (2) as,

$\\!\\!\\!\\!S_{\rm
QG}=\\!\\!\frac{1}{2\kappa^{2}}\int\\!\\!{d^{4}x\sqrt{-g}\left[R-2\kappa^{2}\alpha
R_{\mu\nu}R^{\mu\nu}+2\kappa^{2}\beta R^{2}\right]}$ (40)

Perturbing the action in (40) about the background metric ($\bar{g}_{\mu\nu}$)
to second order in $h_{\mu\nu}=g_{\mu\nu}-\bar{g}_{\mu\nu}$, we get,

$\begin{split}S^{(2)}_{QG}[h_{\mu\nu}]=\frac{1}{2\kappa^{2}}\int
d^{4}x\sqrt{-\bar{g}}\Bigl{[}-h^{\mu\nu}\delta
R_{\mu\nu}[h_{\mu\nu}]-2\alpha\kappa^{2}\delta R_{\mu\nu}[h_{\mu\nu}]\delta
R^{\mu\nu}[h_{\mu\nu}]+2\beta\kappa^{2}(\delta
R[h_{\mu\nu}])^{2}+\frac{h}{2}\bar{g}^{\mu\nu}\delta
R_{\mu\nu}\Bigr{]}\end{split}$ (41)

where, $h=\bar{g}^{\mu\nu}h_{\mu\nu}$. We now proceed by treating $\delta
R_{\mu\nu}$ as an independent variable. However, this introduces extra degrees
of freedom. In order to restrict the extra degrees of freedom, we replace
$\delta R_{\mu\nu}$ in the above expression with an auxiliary variable
$A_{\mu\nu}$, thus leading to:

$\begin{split}S^{(2)}_{QG}[-h_{\mu\nu},A_{\mu\nu},\lambda_{\mu\nu}]=\frac{1}{2\kappa^{2}}\int
d^{4}x\sqrt{-\bar{g}}\Bigl{[}-h^{\mu\nu}A_{\mu\nu}-2\alpha\kappa^{2}A_{\mu\nu}A^{\mu\nu}+2\beta\kappa^{2}A^{2}+\frac{h}{2}\bar{g}^{\mu\nu}A_{\mu\nu}+\lambda^{\mu\nu}(A_{\mu\nu}-\delta
R_{\mu\nu}[h_{\mu\nu}])\Bigr{]}\end{split}$ (42)

where $A=\bar{g}^{\mu\nu}A_{\mu\nu}$ and $\lambda_{\mu\nu}$ is the lagrange
multiplier. Varying the above action with respect to $A_{\mu\nu}$, we get:

$\lambda_{\mu\nu}=4\alpha\kappa^{2}A_{\mu\nu}-4\beta\kappa^{2}\bar{g}_{\mu\nu}A+h_{\mu\nu}-\frac{1}{2}h\bar{g}_{\mu\nu}~{}.$
(43)

Rewriting the above expression in terms of $\lambda_{\mu\nu}$ and $h_{\mu\nu}$
and substituting the resultant in Eq. (42), we get:

$\begin{split}S^{(2)}_{QG}[\Phi_{\mu\nu},\tilde{\Phi}_{\mu\nu}]=\\!\\!\frac{1}{2\kappa^{2}}\int\\!\\!d^{4}x\sqrt{-\bar{g}}\Bigl{[}-\Phi^{\mu\nu}\delta
R_{\mu\nu}\bigl{[}\Phi_{\mu\nu}\bigr{]}+\tilde{\phi}^{\mu\nu}\delta
R_{\mu\nu}\bigl{[}\tilde{\phi}_{\mu\nu}\bigr{]}+\frac{m^{2}}{2}\Bigl{(}\tilde{\Phi}_{\mu\nu}\tilde{\Phi}^{\mu\nu}-(1-\epsilon)\tilde{\Phi}^{2}\Bigr{)}\Bigr{]},\end{split}$
(44)

where

$\displaystyle\lambda_{\mu\nu}=\Phi_{\mu\nu}-\tilde{\Phi}_{\mu\nu};$ (45)
$\displaystyle\tilde{h}_{\mu\nu}=h_{\mu\nu}-\frac{1}{2}h\bar{g}_{\mu\nu}=\Phi_{\mu\nu}+\tilde{\Phi}_{\mu\nu}$
(46) $\displaystyle\epsilon=\frac{1+\beta}{(\alpha-4\beta)};\quad
m^{2}=\frac{1}{(\alpha\kappa^{2})}$ (47)

In the case of GUP-inspired Quadratic gravity, Eq. (43) reduces to:

$h_{\mu\nu}=\psi_{\mu\nu}-\frac{1}{2}\bar{g}_{\mu\nu}\psi-4\gamma\hat{A}_{\mu\nu}-\gamma
A\bar{g}_{\mu\nu}$ (48)

where, $\hat{A}_{\mu\nu}=A_{\mu\nu}-\frac{1}{4}\bar{g}_{\mu\nu}A$ is the
traceless part of $A_{\mu\nu}$. Mapping $\lambda_{\mu\nu}\to\psi_{\mu\nu}$,
$\hat{A}_{\mu\nu}\to\hat{R}^{(1)}_{\mu\nu}$ and $A\to R^{(1)}$, leads to Eqs.
(15,9). The above analysis establishes the metric decomposition into massless
spin-2, massive spin-0, and massive spin-2 modes. Combining the results of
Appendix (B), this shows the equations of motion (16, 17, 14) leads to the
correct equations of motion.

It is possible to redo the above analysis for an arbitrary Einsteinian
manifold, interms of the Einstein tensor instead of the Ricci tensor Tachinami
_et al._ (2021). Following Ref. Tachinami  _et al._ (2021), we can represent
the quadratic gravity action (2) as:

$\\!\\!\\!\\!S_{\rm
QG}=\\!\\!\frac{1}{2\kappa^{2}}\int\\!\\!{d^{4}x\sqrt{-g}\left[-G-2\kappa^{2}\alpha
G_{\mu\nu}G^{\mu\nu}+2\kappa^{2}\beta G^{2}\right]}\,.$ (49)

Perturbing the action (49) about the background metric ($\bar{g}_{\mu\nu}$) to
second order in $h_{\mu\nu}=g_{\mu\nu}-\bar{g}_{\mu\nu}$, we get,

$\begin{split}S^{(2)}_{QG}[h_{\mu\nu}]=\frac{1}{2\kappa^{2}}\int
d^{4}x\sqrt{-\bar{g}}\Bigl{[}h^{\mu\nu}\delta
G_{\mu\nu}[h_{\mu\nu}]-2\alpha\kappa^{2}\delta G_{\mu\nu}[h_{\mu\nu}]\delta
G^{\mu\nu}[h_{\mu\nu}]+2\beta\kappa^{2}(\delta
G[h_{\mu\nu}])^{2}-\frac{h}{2}\bar{g}^{\mu\nu}\delta
G_{\mu\nu}\Bigr{]}\end{split}$ (50)

where, $h=\bar{g}^{\mu\nu}h_{\mu\nu}$. Replacing $\delta G_{\mu\nu}$ in the
above expression by an auxiliary variable $A_{\mu\nu}$ and imposing a
constraint, we get,

$\begin{split}S^{(2)}_{QG}[h_{\mu\nu},A_{\mu\nu},\lambda_{\mu\nu}]=\frac{1}{2\kappa^{2}}\int
d^{4}x\sqrt{-\bar{g}}\Bigl{[}-h^{\mu\nu}A_{\mu\nu}-2\alpha\kappa^{2}A_{\mu\nu}A^{\mu\nu}+2\beta\kappa^{2}A^{2}-\frac{h}{2}\bar{g}^{\mu\nu}A_{\mu\nu}+\lambda^{\mu\nu}(A_{\mu\nu}-\delta
G_{\mu\nu}[h_{\mu\nu}])\Bigr{]}\end{split}$ (51)

where $A=\bar{g}^{\mu\nu}A_{\mu\nu}$ and $\lambda_{\mu\nu}$ is the lagrange
multiplier. Varying the above action with respect to $A_{\mu\nu}$, we get

$\lambda_{\mu\nu}=4\alpha\kappa^{2}A_{\mu\nu}-4\beta\kappa^{2}\bar{g}_{\mu\nu}A-h_{\mu\nu}+\frac{1}{2}h\bar{g}_{\mu\nu}~{}.$
(52)

Substituting $A_{\mu\nu}$ in terms of $\lambda_{\mu\nu}$ and $h_{\mu\nu}$ from
Eq. (52) in Eq. (51) and using

$\tilde{h}_{\mu\nu}=h_{\mu\nu}-\frac{1}{2}h\bar{g}_{\mu\nu}=\Phi_{\mu\nu}+\tilde{\Phi}_{\mu\nu}~{};\quad\lambda_{\mu\nu}=\tilde{\Phi}_{\mu\nu}-{\Phi}_{\mu\nu},$
(53)

we get,

$\begin{split}S^{(2)}_{QG}[\Phi_{\mu\nu},\tilde{\Phi}_{\mu\nu}]=\frac{1}{2\kappa^{2}}\int\\!\\!d^{4}x\sqrt{-\bar{g}}\Bigl{[}\Phi^{\mu\nu}\delta
G_{\mu\nu}\bigl{[}\Phi_{\mu\nu}\bigr{]}-\tilde{\Phi}^{\mu\nu}\delta
G_{\mu\nu}\bigl{[}\tilde{\Phi}_{\mu\nu}\bigr{]}+\frac{m^{2}}{2}\Bigl{(}\tilde{\phi}_{\mu\nu}\tilde{\Phi}^{\mu\nu}-(1-\epsilon)\tilde{\Phi}^{2}\Bigr{)}\Bigr{]},\end{split}$
(54)

where, $\epsilon=1+\beta/(\alpha-4\beta)$, $m^{2}=1/(\alpha\kappa^{2})$. For
the GUP-inspired Stelle gravity, we once again recover the ansatz (15) with
the following mapping $\psi_{\mu\nu}\to-\lambda_{\mu\nu}$ ,
$\hat{R}^{(1)}_{\mu\nu}\to-\hat{A}_{\mu\nu}$ and $R^{(1)}\to-A$.

##  Appendix B Reduced equations of motion in Ricci-flat space-times

The field equation derived from the action (2) with arbitrary values of the
parameters $\alpha$ and $\beta$ is given by

$\begin{split}&R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+4\beta R_{\mu\nu}R-4\alpha
R^{\rho\sigma}R_{\mu\rho\nu\sigma}+2(\alpha-2\beta)\nabla_{\mu}\nabla_{\nu}R\\\
&-2\alpha\Box R_{\mu\nu}+g_{\mu\nu}\left(\alpha
R_{\rho\sigma}R^{\rho\sigma}+\beta R^{2}+(\alpha-4\beta)\Box
R\right)=0~{}.\end{split}$ (55)

Linearizing the above field equation in a Ricci-flat background one gets

$\begin{split}\delta R_{\mu\nu}-\frac{1}{2}\bar{g}_{\mu\nu}\delta
R-4\alpha\bar{R}_{\mu\rho\nu\sigma}\delta R^{\rho\sigma}\\\
+2(\alpha-2\beta)\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}\delta
R-2\alpha\bar{\Box}\delta R_{\mu\nu}\\\
-\bar{g}_{\mu\nu}\left(\alpha-4\beta\right)\bar{\Box}\delta R=0\,.\end{split}$
(56)

Equation (56) can be written in terms of the metric perturbations by expanding
$\delta R_{\mu\nu}$ and $\delta R$ in terms of $h_{\mu\nu}$ (see Eqs. (6,7)
and Ref. Misner  _et al._ (1973)). We choose the following ansatz for the
metric perturbations in Ricci-flat spacetime:

$\\!\\!\\!h_{\mu\nu}=\left[\psi_{\mu\nu}-\frac{\bar{g}_{\mu\nu}}{2}\psi\right]+\left[C_{1}+\frac{C_{2}}{4}\right]R^{(1)}\bar{g}_{\mu\nu}-C_{2}\hat{R}^{(1)}_{\mu\nu}~{},$
(57)

where $C_{1},C_{2}$ are constants to be determined and $\psi_{\mu\nu}$
satisfies the transverse-traceless condition
$\left(\bar{\nabla}^{\mu}\psi_{\mu\nu}=0,~{}\bar{g}^{\mu\nu}\psi_{\mu\nu}=0\right)$.
We demand that $R^{(1)}$ and $\hat{R}^{(1)}_{\mu\nu}$ to be respectively equal
to the perturbed Ricci scalar $\delta R$ and the _traceless part_ of the
perturbed Ricci tensor ($\delta R_{\mu\nu}-\frac{1}{4}\bar{g}_{\mu\nu}\delta
R$). This will ensure that the linearized Bianchi identity (19) is satisfied
at all times and in all spaces. Hence, equating $\delta R$ with $R^{(1)}$ and
comparing it with the trace of linearized field equation (56) we get,

$C_{2}=-3C_{1}+4\left(\alpha-3\beta\right)~{}.$ (58)

Thus, we get the propagation equation for the massive spin-0 mode as,

$\bar{\Box}R^{(1)}-R^{(1)}/{4(3\beta-\alpha)}=0~{}.$ (59)

Similarly, equating
$R^{(1)}_{\mu\nu}=\hat{R}^{(1)}_{\mu\nu}+\frac{1}{4}\bar{g}_{\mu\nu}R^{(1)}$
with $\delta R_{\mu\nu}$ and using Eq. (58) we get,

$\begin{split}&{R}^{(1)}_{\mu\nu}+\frac{1}{2}\Bigl{[}\Bigl{(}\bar{\Box}\psi_{\mu\nu}+2\bar{R}_{\mu\rho\nu\sigma}\psi^{\rho\sigma}\Bigr{)}-\left(C_{1}-4\alpha+12\beta\right)\Bigl{(}\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}R^{(1)}-\frac{\bar{g}_{\mu\nu}R^{(1)}}{8(\alpha-3\beta)}\Bigr{)}+\\\
&\left(3C_{1}-4\alpha+12\beta\right)\Bigl{(}\bar{\Box}{R}^{(1)}_{\mu\nu}+2\bar{R}_{\mu\rho\nu\sigma}{R}^{(1)\rho\sigma}\Bigr{)}\Bigr{]}=0\end{split}$
(60)

Comparing with Eq. (56) and using Eq. (8) we get,

$\displaystyle\bar{\Box}\psi_{\mu\nu}+2\bar{R}_{\mu\alpha\nu\beta}\psi^{\alpha\beta}=0~{},$
(61) $\displaystyle C_{1}=-4\beta\mbox{ and }\alpha=2\beta~{}.$ (62)

Equation (61) gives the dynamics of the massless spin-2 graviton mode, as
$\psi_{\mu\nu}$ satisfies the traceless-transverse condition. Equation (62),
on the other hand, implies that the decomposition scheme in Eq. (57) is
specific to the GUP-inspired Stelle gravity $(\alpha=2\beta=\gamma)$, in which
case $C_{1}=-4\beta=-2\gamma$ and $C_{2}=4\gamma$. Thus, substituting $C_{1}$
and $C_{2}$ in Eq. (60) and taking cognizance of Eqs. (59), (61), (19) and
(18), we get the propagation equation of the massive spin-2 mode as given in
Eq. (17)

$\bar{\Box}\hat{R}^{(1)}_{\mu\nu}+2\bar{R}_{\mu\alpha\nu\beta}\hat{R}^{(1)\alpha\beta}-\hat{R}^{(1)}_{\mu\nu}/{(2\gamma)}=0\,.$
(63)

In the case of $\alpha=2\beta=\gamma$, Eq. (59) reduces to Eq. (14). It is
important to note that in the massless limit $\gamma\rightarrow 0$, Eqs. (14)
and (17) suggest that $\hat{R}^{(1)}_{\mu\nu}$ and $R^{(1)}$ vanishes
identically and the theory is described by only one massless spin-2 graviton
$\psi_{\mu\nu}$.

##  Appendix C Coefficients in the effective GW stress-energy tensor

Using xAct Mathematica packages,
$\mathcal{A}_{\mu\nu},\mathcal{B}_{\mu\nu},\mathcal{C}_{\mu\nu}$, and
$\mathcal{D}_{\mu\nu}$ in Eq. (22) are obtained as:

$\displaystyle\mathcal{A}_{\mu\nu}$
$\displaystyle=-\bar{R}_{\mu\lambda\alpha\rho}\psi_{\nu}^{\alpha}\psi^{\lambda\rho}-\frac{1}{4}\bar{g}_{\mu\nu}\bar{R}_{\alpha\rho\lambda\sigma\
}\psi^{\alpha\lambda}\psi^{\rho\sigma}$ (64)
$\displaystyle\mathcal{B}_{\mu\nu}$
$\displaystyle=2\bar{R}_{\nu}{}^{\sigma}_{\alpha}{}^{\eta}\bar{R}_{\lambda\sigma\rho\eta}\psi_{\mu}^{\alpha}\psi^{\lambda\rho}-4\bar{R}_{\mu\alpha\lambda}{}^{\eta}\bar{R}_{\nu\rho\sigma\eta}\psi^{\alpha\lambda}\psi^{\rho\sigma}-2\bar{R}_{\mu\alpha\nu}{}^{\eta}\bar{R}_{\lambda\rho\sigma\eta}\psi^{\alpha\lambda}\psi^{\rho\sigma}-\bar{R}_{\mu\rho\nu\sigma}\bar{\nabla}^{\rho}\psi^{\alpha\lambda}\bar{\nabla}^{\sigma}\psi_{\alpha\lambda}$
$\displaystyle+4\bar{R}_{\mu}{}^{\eta}{}_{\nu\alpha}\bar{R}_{\lambda\rho\sigma\eta}\psi^{\alpha\lambda}\psi^{\rho\sigma}-\bar{g}_{\mu\nu}\bar{R}_{\alpha}{}^{\eta}{}_{\lambda}{}^{h}\bar{R}_{\rho\eta\sigma
h}\psi^{\alpha\lambda}\psi^{\rho\sigma}+2\bar{R}_{\alpha\rho\lambda\sigma}\bar{\nabla}_{\mu}\psi^{\alpha\lambda}\bar{\nabla}_{\nu}\psi^{\rho\sigma}-2\bar{R}_{\alpha\rho\lambda\sigma}\bar{\nabla}_{\mu}\psi^{\rho\sigma}\bar{\nabla}^{\lambda}\psi_{\nu}{}^{\alpha}$
$\displaystyle+2\bar{R}_{\alpha\lambda\rho\sigma}\bar{\nabla}_{\nu}\psi_{\mu}{}^{\alpha}\bar{\nabla}^{\sigma}\psi^{\lambda\rho}-2\bar{R}_{\alpha\lambda\rho\sigma}\bar{\nabla}^{\alpha}\psi_{\mu\nu}\bar{\nabla}^{\sigma}\psi^{\lambda\rho}$
(65) $\displaystyle\mathcal{C}_{\mu\nu}$
$\displaystyle=8\bar{R}_{\nu\lambda\alpha\rho}\hat{R}^{(1)}{}_{\mu}{}^{\alpha}\hat{R}^{(1)}{}^{\lambda\rho}+2\bar{R}_{\mu\lambda\nu\rho}(4\hat{R}^{(1)}{}_{\alpha}{}^{\rho}\hat{R}^{(1)}{}^{\alpha\lambda}-7R^{(1)}\hat{R}^{(1)}{}^{\lambda\rho})-8\bar{R}_{\alpha\lambda\rho\sigma}\bar{\nabla}_{\nu}\hat{R}^{(1)}{}_{\mu}{}^{\alpha}\bar{\nabla}^{\sigma}\psi^{\lambda\rho}$
$\displaystyle+4\bar{g}_{\mu\nu}R_{\alpha\rho\lambda\sigma}\hat{R}^{(1)}{}^{\alpha\lambda}\hat{R}^{(1)}{}^{\rho\sigma}-16\bar{R}_{\alpha\rho\lambda\sigma}\bar{\nabla}_{\mu}\psi^{\rho\sigma}\bar{\nabla}_{\nu}\hat{R}^{(1)}{}^{\alpha\lambda}+2\bar{R}_{\nu\rho\lambda\sigma}\bar{\nabla}_{\mu}\psi^{\rho\sigma}\bar{\nabla}^{\lambda}R^{(1)}$
$\displaystyle+8\bar{R}_{\alpha\rho\lambda\sigma}\bar{\nabla}_{\mu}\psi^{\rho\sigma}\bar{\nabla}^{\lambda}\hat{R}^{(1)}{}_{\nu}{}^{\alpha}+2\bar{R}_{\nu\lambda\rho\sigma}\bar{\nabla}^{\lambda}R^{(1)}\bar{\nabla}^{\sigma}\psi_{\mu}{}^{\rho}+2\bar{R}_{\nu\sigma\lambda\rho}\bar{\nabla}^{\lambda}R^{(1)}\bar{\nabla}^{\sigma}\psi_{\mu}{}^{\rho}$
$\displaystyle+4\bar{R}_{\mu\rho\nu\sigma}\bar{\nabla}^{\rho}\hat{R}^{(1)}{}^{\alpha\lambda}\bar{\nabla}^{\sigma}\psi_{\alpha\lambda}+4\bar{R}_{\mu\sigma\nu\rho}\bar{\nabla}^{\rho}\hat{R}^{(1)}{}^{\alpha\lambda}\bar{\nabla}^{\sigma}\psi_{\alpha\lambda}-10\bar{R}_{\mu\lambda\rho\sigma}\bar{\nabla}_{\nu}R^{(1)}\bar{\nabla}^{\sigma}\psi^{\lambda\rho}$
$\displaystyle+8\bar{R}_{\alpha\lambda\rho\sigma}\bar{\nabla}^{\alpha}\hat{R}^{(1)}{}_{\mu\nu}\bar{\nabla}^{\sigma}\psi^{\lambda\rho}+2\bar{g}_{\mu\nu}\bar{R}_{\lambda\rho\sigma\eta}\bar{\nabla}^{\lambda}R^{(1)}\bar{\nabla}^{\eta}\psi^{\rho\sigma}$
(66) $\displaystyle\mathcal{D}_{\mu\nu}$
$\displaystyle=4\Big{(}8\bar{R}_{\nu}{}^{\sigma}{}_{\alpha}{}^{\eta}\bar{R}_{\lambda\sigma\rho\eta}\hat{R}^{(1)}{}_{\mu}{}^{\alpha}\hat{R}^{(1)}{}^{\lambda\rho}-16\bar{R}_{\mu\alpha\lambda}{}^{\eta}\bar{R}_{\nu\rho\sigma\eta}\hat{R}^{(1)}{}^{\alpha\lambda}\hat{R}^{(1)}{}^{\rho\sigma}+2\bar{R}_{\nu\sigma\lambda\rho}\bar{\nabla}^{\lambda}R^{(1)}\bar{\nabla}^{\sigma}\hat{R}^{(1)}{}_{\mu}{}^{\rho}$
$\displaystyle-8\bar{R}_{\mu\alpha\nu}{}^{\eta}\bar{R}_{\lambda\rho\sigma\eta}\hat{R}^{(1)}{}^{\alpha\lambda}\hat{R}^{(1)}{}^{\rho\sigma}+16\bar{R}_{\mu}{}^{\eta}{}_{\nu\alpha}\bar{R}_{\lambda\rho\sigma\eta}\hat{R}^{(1)}{}^{\alpha\lambda}\hat{R}^{(1)}{}^{\rho\sigma}$
$\displaystyle-4\bar{g}_{\mu\nu}\bar{R}_{\alpha}{}^{\eta}{}_{\lambda}{}^{h}\bar{R}_{\rho\eta\sigma
h}\hat{R}^{(1)}{}^{\alpha\lambda}\hat{R}^{(1)}{}^{\rho\sigma}+8\bar{R}_{\alpha\rho\lambda\sigma}\bar{\nabla}_{\mu}\hat{R}^{(1)}{}^{\alpha\lambda}\bar{\nabla}_{\nu}\hat{R}^{(1)}{}^{\rho\sigma}$
(67)
$\displaystyle+\bar{R}_{\mu\lambda\nu\sigma}\bar{\nabla}^{\lambda}R^{(1)}\bar{\nabla}^{\sigma}R^{(1)}+2\bar{R}_{\nu\lambda\rho\sigma}\bar{\nabla}^{\lambda}R^{(1)}\bar{\nabla}^{\sigma}\hat{R}^{(1)}{}_{\mu}{}^{\rho}-4\bar{R}_{\mu\rho\nu\sigma}\bar{\nabla}^{\rho}\hat{R}^{(1)}{}^{\alpha\lambda}\bar{\nabla}^{\sigma}\hat{R}^{(1)}{}_{\alpha\lambda}\Big{)}$

As mentioned earlier, we assume $\psi(V,\rho)$ and $P(V,\rho)$ to be slowly
varying close to the event horizon $(\rho\sim 2M_{0})$. The leading order
solid angle-averaged value (of the $\rho-V$ component) of the effective GW
stress-energy tensor (up to leading order corrections in $\gamma$) used in Eq.
(37) in the Schwarzschild background is given by

$\displaystyle\\!\\!\\!t_{V}^{\rho\rm GW}\approx$
$\displaystyle\frac{M_{0}^{4}\psi^{\prime}(V,\rho)}{21\pi}\left\\{8[21o^{00}\iota^{22}+20o^{23}\iota^{23}+8o^{33}\iota^{33}]\gamma
P^{\prime}(V,\rho)+[21(o^{22})^{2}+20(o^{23})^{2}+8(o^{33})^{2}]\psi^{\prime}(V,\rho)\right\\}$
$\displaystyle+\frac{1}{21\pi}M_{0}^{2}\psi^{\prime}(V,\rho)\left[-4(21o^{12}\iota^{02}+10o^{13}\iota^{03}+21o^{02}\iota^{12}+10o^{03}\iota^{13})\gamma
P^{\prime}(V,\rho)+\right.$
$\displaystyle\left.\left(21o^{02}o^{12}+10o^{03}o^{13}+\gamma\left(21(o^{22})^{2}-20(o^{23})^{2}+40o^{22}\iota^{33}+8(o^{33})^{2}\right)\right)\psi^{\prime}(V,\rho)\right]$
$\displaystyle+\frac{1}{21\pi}\left(21o^{12}\iota^{02}+10o^{13}\iota^{03}-21o^{02}\iota^{12}-10o^{03}\iota^{13}\right)M_{0}\gamma\left[\psi(V,\rho)\mathrm{P}^{\prime}(V,\rho)-\mathrm{P}(V,\rho)\psi^{\prime}(V,\rho)\right]$
$\displaystyle+\frac{1}{672M_{0}^{2}\pi}\left[84(o^{11}\iota^{00}+o^{00}\iota^{11})\gamma
P(V,\rho)\psi(V,\rho)+\left(-21o^{00}o^{11}+2\left(-42o^{02}o^{12}+21(o^{12})^{2}\right.\right.\right.$
$\displaystyle\left.\left.\left.-20o^{03}o^{13}+10(o^{13})^{2}-63o^{11}o^{22}+38o^{11}o^{33}\right)\gamma\right)(\psi(V,\rho))^{2}+84(o^{01})^{2}\gamma(\psi^{\prime}(V,\rho))^{2}\right]$
$\displaystyle-\frac{1}{32M_{0}^{3}\pi}\left(2(o^{01})^{2}+o^{00}o^{11}-2o^{01}o^{11}\right)\gamma\psi(V,\rho)\psi^{\prime}(V,\rho)+\frac{1}{64M_{0}^{4}\pi}(\psi(V,\rho))^{2}\gamma\left[-2(o^{01})^{2}+o^{00}o^{11}\right]$
$\displaystyle+\frac{\gamma}{168\pi}\left[42(-o^{11}\iota^{00}+o^{00}\iota^{11})P(V,\rho)\psi^{\prime}(V,\rho)+\psi(V,\rho)\left(42(o^{11}\iota^{00}-o^{00}\iota^{11})P^{\prime}(V,\rho)+\left(21o^{02}o^{12}\right.\right.\right.$
$\displaystyle\left.\left.\left.-21(o^{12})^{2}+10o^{03}o^{13}-10(o^{13})^{2}-42o^{11}o^{22}-20o^{11}o^{33}\right)\psi^{\prime}(V,\rho)\right)\right]$
$\displaystyle+\frac{1}{336\pi}\left[4(21o^{12}\iota^{02}+10o^{13}\iota^{03}+21o^{02}\iota^{12}+10o^{03}\iota^{13})\gamma
P(V,\rho)\psi(V,\rho)\right.$
$\displaystyle\left.-\left(21o^{02}o^{12}+42(o^{12})^{2}+10o^{03}o^{13}+20(o^{13})^{2}-42o^{11}o^{22}-20o^{11}\iota^{33}\right)\psi(V,\rho)^{2}\right.$
$\displaystyle\left.+2\psi^{\prime}(V,\rho)\left\\{-84(o^{11}\iota^{00}+2o^{01}\iota^{01}+o^{00}\iota^{11})\gamma
P^{\prime}(V,\rho)+\left(21(o^{01})^{2}\right.\right.\right.$
$\displaystyle\left.\left.\left.+21o^{00}o^{11}+252o^{02}o^{12}\gamma+120o^{03}o^{13}\gamma-84o^{01}o^{22}\gamma-40o^{01}\iota^{33}\gamma\right)\psi^{\prime}(V,\rho)\right\\}\right]$
(68)

where ′ denotes derivative wrt $V$.

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# Essential dimensions of isogenies

Ziquan Zhuang  Department of Mathematics, Johns Hopkins University, Baltimore,
MD 21218, USA<EMAIL_ADDRESS>Dedicated to James McKernan on the occasion of
his sixtieth birthday.

###### Abstract.

We give a formula for the essential dimensions of isogenies $A\to A^{\prime}$
of complex abelian varieties when the degree of the isogeny is coprime to
$(\dim A)!$. As a corollary, we show that the $p$-multiplication map (where
$p$ is a prime) on an abelian variety is incompressible when $p>\dim A$ or
when $\dim A\leq 4$. This confirms a conjecture of Brosnan for most primes.

##  1\. Introduction

The essential dimension $\operatorname{ed}(Y/X)$ (or sometimes denoted
$\operatorname{ed}(f)$) of a generically finite morphism $f\colon Y\to X$ is
the smallest integer $m$ such that, over some dense open subset of $X$, the
map $f$ arises as the pullback of a map of varieties of dimension $m$. This
notion was introduced in [BR-ess-dim] in relation to an algebraic form of
Hilbert’s 13th problem and has been studied in a variety of context. See e.g.
[Rei-ess-dim-survey, Mer-ess-dim-survey] for an overview of some previous
works on this subject.

The purpose of this note is to describe a birational geometric approach to
this invariant. The following is our main result.

######  Theorem 1.

Let $\alpha\colon A\to A^{\prime}$ be an isogeny of complex abelian varieties
whose degree is coprime to $(\dim A)!$. Then the essential dimension of
$\alpha$ equals the minimum of

$\dim A-\dim B+\operatorname{rank}(\ker(\alpha)\cap B)$ 111The rank of a
finite abelian group $G$ is the smallest number of elements needed to generate
$G$.

as $B$ varies among the abelian subvarieties of $A$.

As a direct application, it implies that the $p$-multiplication map $A\to A$
(where $p$ is a prime) is incompressible (i.e. its essential dimension equals
$\dim A$) when $p>\dim A$ (Corollary 11). This confirms the incompressibility
part of a conjecture of Brosnan (see e.g. [FS-incompressibility]*Conjecture
6.1) for most primes. In dimension at most $4$, we are also able to prove
incompressibility for all primes $p$. When the abelian variety $A$ has good
unramified reduction at the prime $p>\dim A$, Brosnan’s conjecture follows
from [FKW-ess-dim-prismatic] via a prismatic cohomological argument. The work
[FS-incompressibility] also establishes a few other special cases of Brosnan’s
conjecture. We should note that [FKW-ess-dim-prismatic, FS-incompressibility]
also prove $p$-incompressibility222This means the map stays incompressible
after base change along any finite morphism whose degree is prime to $p$. in
the cases they consider. The argument we give here is quite different from
theirs.

The proof of our main theorem is based on the following elementary fact: if
the isogeny $\alpha$ is birationally obtained from $Y\to Y^{\prime}$ via base
change, then the Galois group $G\cong\ker(\alpha)$ acts faithfully on (some
birational modification of) $Y$ which is a rational image of the abelian
variety $A$. But rational images of abelian varieties are somewhat restricted.
In particular, we show that the maximal rationally connected quotient
(Definition 3) of $Y$ is again birational to an abelian variety (Theorem 4).
Equivalently, the Albanese morphism of $Y$ has rationally connected general
fibers. We then conclude the proof by using known results on the birational
automorphism groups of rationally connected varieties (e.g. its rank is at
most the dimension of the variety)333It is well known that the essential
dimension $\operatorname{ed}(G)$ of a finite group $G$ is related to Cremona
groups containing $G$. See e.g. [Ser-cremona]*Théorème 3.6 which determines
$\operatorname{ed}(A_{6})$ using the classification of finite subgroups of the
plane Cremona group..

### Acknowledgement

The author is partially supported by the NSF Grants DMS-2240926, DMS-2234736,
a Clay research fellowship, as well as a Sloan fellowship. He would like to
thank Patrick Brosnan, Mark Kisin, János Kollár, Shizhang Li and Jesse Wolfson
for many helpful discussions and comments.

##  2\. Proof

Before we present the proof, we recall a few definitions that will appear
several times in the argument.

######  Definition 2.

An algebraic fibre space is a surjective projective morphism $f\colon X\to Y$
between normal varieties with connected fibers.

######  Definition 3.

A variety $X$ is called uniruled (resp. rationally connected) if through any
point (resp. any pair of points) on $X$ their is a rational curve. See [Kol-
rational-curve] for the general theory of such varieties. For us the most
important point is the following: any variety $X$ has a maximal rationally
connected (MRC) fibration $X\dashrightarrow Y$ whose fibers are rationally
connected while the base is not uniruled ([Kol-rational-curve]*Theorems 5.4
and 5.5 and [GHS-RC-fibration-section]*Corollary 1.4). This MRC fibration is
functorial and unique up to birational modifications. The base of the MRC
fibration is called the MRC quotient of $X$.

As we indicated in the introduction, the key step to the proof of Theorem 1 is
the following description of the rational images of abelian varieties.

######  Theorem 4.

Let $X\to Y$ be an algebraic fiber space where $X$ is birational to an abelian
variety and $Y$ is not uniruled. Then $Y$ is also birational to an abelian
variety.

We should remark that this is essentially a consequence of some well-known
conjectures in birational geometry. More precisely, if we denote the general
fiber of $X\to Y$ by $F$, then the Iitaka conjecture (which is open in general
but known in the case we consider) says that
$\kappa(X)\geq\kappa(F)+\kappa(Y)$. Here $\kappa(X)$ is the Kodaira dimension
of $X$ which takes values in $\\{-\infty\\}\cup\mathbb{N}$. On the other hand,
the non-vanishing conjecture predicts that if $Y$ is not uniruled then
$\kappa(Y)\geq 0$. Combining these two, and since $\kappa(X)=0$, we see that
$\kappa(F)=\kappa(Y)=0$. Although the non-vanishing conjecture is wide open in
general, one can still extract the equality $\kappa(F)=\kappa(Y)=0$ from the
standard approach to the Iitaka conjecture, see e.g. [KL-quotient-of-
CY]*Theorem 14 (for the reader’s convenience, we also provide some argument
below in the special case we consider). By a classical result of Ueno on the
Kodaira dimensions of subvarieties of abelian varieties, it then follows that
the fibers $F$ are abelian subvarieties of $X$ and therefore $Y$ is birational
to the quotient abelian variety $X/F$.

Turning to more details, we need the following statements which we extract
from the previous works on the Iitaka conjecture.

######  Lemma 5.

Let $f\colon X\to B$ be an algebraic fiber space between smooth projective
varieties whose general fibers $F$ are of general type. Then there exists a
divisor $D$ on $X$ such that $\operatorname{codim}f(D)\geq 2$ and

$\kappa(X,K_{X/B}+D)\geq\dim F+\operatorname{Var}(f).$

Here $\kappa(X,L)$ denotes the Iitaka dimension of the line bundle $L$ on $X$
([Laz-positivity-1]*Definition 2.1.3), and $\operatorname{Var}(f)$ denotes the
variation of the family $f$ (a precise definition can be found on [Vie-
Iitaka]*pp. 329, but for us it is enough to know that it’s value is $\geq 0$).

###### Proof.

This follows from [Vie-Iitaka] and [Kol-Iitaka-general-type]. More
specifically, by [Kol-Iitaka-general-type]*Theorem, the assumption of [Vie-
Iitaka]*Theorem II is always satisfied when the general fibers of the
algebraic fiber space are of general type. Then the lemma becomes a special
case of [Vie-Iitaka]*Lemma 7.4 (with $\mathcal{M}$ in _loc. cit._ being the
trivial line bundle). ∎

######  Lemma 6.

Let $f\colon X\to B$ be an algebraic fiber space between smooth projective
varieties with positive dimensional general fibers $F$ of general type. Assume
that $X$ is birational to a normal variety $Y$ with $K_{Y}\sim_{\mathbb{Q}}0$.
Then $B$ is uniruled.

###### Proof.

By [MM-unirule-criterion]*Theorem 1, it suffices to show that there exists a
covering family of curves $C\subseteq B$ such that $(K_{B}\cdot C)<0$.

We make some reductions before we prove this. By blowing up $X$, we may assume
that the birational map $g\colon X\dashrightarrow Y$ is a morphism. One can
find a birational morphism $B_{1}\to B$ such that the main component of
$X\times_{B}B_{1}$ is equidimensional over $B_{1}$ (for example, let
$U\subseteq B$ be an open subset over which $f$ is flat, and take $B_{1}$ as a
log resolution of the closure of $U$ in an appropriate Hilbert scheme that
parameterizes the general fibers of $f$). Replacing $B$ by $B_{1}$ and $X$ by
a log resolution of the main component of $X\times_{B}B_{1}$, we can assume
that for any closed subset $Z\subseteq B$ of codimension at least $2$, the
locus $g(f^{-1}(Z))$ in $Y$ also has codimension at least two (c.f. [Vie-
Iitaka]*Lemma 7.3).

By Lemma 5, there exists a divisor $D$ on $X$ such that
$\operatorname{codim}f(D)\geq 2$ and

$\kappa(X,K_{X/B}+D)\geq\dim F+\operatorname{Var}(f)\geq\dim F>0.$

From the previous reduction step, we see that $\operatorname{codim}g(D)\geq 2$
as well. The positivity of the Iitaka dimension implies that for any $x\in X$,
there exists some effective $\mathbb{Q}$-divisor
$\Delta\sim_{\mathbb{Q}}K_{X/B}+D$ on $X$ such that
$x\in\operatorname{Supp}(\Delta)$.

Let $C_{0}\subseteq X$ be the preimage of a general complete intersection
curve in $Y$. Then $C_{0}$ is disjoint from the exceptional locus of $g$ as
well as the support of $D$, since their images in $Y$ have codimension at
least two. It follows that $(K_{X}\cdot C_{0})=(K_{Y}\cdot g(C_{0}))=0$ and
similarly $(D\cdot C_{0})=0$. Hence if $C=f(C_{0})$, then we get

$(K_{X/B}+D\cdot C_{0})=-(f^{*}K_{B}\cdot C_{0})=-(K_{B}\cdot C)$

where the second equality is by the projection formula.

By Bertini’s theorem, we can make $C_{0}$ pass through any two chosen general
points $x,x^{\prime}\in X$. In particular, we see that $(K_{X/B}+D\cdot
C_{0})>0$ since for the above choice of $\Delta$, we can make
$x^{\prime}\not\in\operatorname{Supp}(\Delta)$ and thus
$C_{0}\not\subseteq\operatorname{Supp}(\Delta)$, while $x\in
C_{0}\cap\operatorname{Supp}(\Delta)$. This proves that $(K_{B}\cdot C)<0$ for
some covering family of curves and therefore $B$ is uniruled. ∎

###### Proof of Theorem 4.

Let $F$ be a general fiber of the rational map $A\dashrightarrow Y$. By [Ueno-
book]*Theorem 10.9, there exists an abelian subvariety $B$ of $A$ such that
$F$ is the preimage of a subvariety of general type in $A/B$. The abelian
subvariety $B$ varies continuously over an open subset of $Y$, hence it stays
constant and is independent of $F$. It follows that the rational map
$A\dashrightarrow Y$ factors through $A/B$. Since $Y$ is not uniruled and the
general fiber of the induced map $g\colon A/B\dashrightarrow Y$ is now of
general type, we deduce from Lemma 6 that $g$ is birational. Therefore, $Y$ is
birational to the abelian variety $A/B$. ∎

Another ingredient in the proof of Theorem 1 is the following result on the
birational automorphism group of rationally connected varieties.

######  Lemma 7 ([Xu-p-group-in-Cremona]*Theorem 2.2).

Let $X$ be a rationally connected variety of dimension $n$ and let
$G\subseteq\operatorname{Bir}(X)$ be a finite $p$-subgroup for some $p>n+1$.
Then $G$ is abelian and the rank of $G$ is at most $n$.

We also record some elementary properties of essential dimensions.

######  Lemma 8.

Let $f_{i}\colon X_{i}\to Y$ $(i=1,2)$ be finite morphisms and let $f\colon
X=X_{1}\times_{Y}X_{2}\to Y$ be the fiber product. Then
$\operatorname{ed}(f)\leq\operatorname{ed}(f_{1})+\operatorname{ed}(f_{2})$.

###### Proof.

Clear from the definition. ∎

######  Lemma 9.

Let $f\colon X\to Y$ be a finite abelian cover with Galois group $G$. Then
$\operatorname{ed}(f)\leq\operatorname{rank}(G)$.

###### Proof.

This follows from [BR-ess-dim]*Theorem 6.1. For the reader’s convenience, we
also include a direct proof. Note that we may write $f$ as the fiber product
of $\operatorname{rank}(G)$ cyclic covers. Since any cyclic cover has
essential dimension one, the statement follows from Lemma 8. ∎

We can now give the proof of Theorem 1.

###### Proof of Theorem 1.

We first prove that

$\operatorname{ed}(\alpha)\leq\dim A-\dim
B+\operatorname{rank}(\ker(\alpha)\cap B)$

for any abelian subvariety $B$ of $A$. Let $L=\ker(\alpha)\cap B$ and let
$K=\ker(\alpha)/L$. Then the isogeny $A\to A^{\prime}$ factors through
$A_{1}=A/L$. We can find another isogeny $A_{2}\to A^{\prime}$ with kernel
rank $r:=\operatorname{rank}(L)$ such that the fiber product
$A_{1}\times_{A^{\prime}}A_{2}$ factors through $A$ (it suffices to lift
$\pi_{1}(A_{1})\to L$ to some $\pi_{1}(A^{\prime})\to L^{\prime}$ with
$\operatorname{rank}(L^{\prime})=r$). By Lemma 9, we have
$\operatorname{ed}(A_{2}/A^{\prime})\leq r$, hence by By Lemma 8 we only need
to show that

$\operatorname{ed}(A_{1}/A^{\prime})\leq\dim A-\dim B.$

To see this, let $A_{0}:=A/B$ and let $A^{\prime}_{0}:=A_{0}/K\cong
A^{\prime}/\alpha(B)$.

$\textstyle{A_{1}=A/L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{0}=A/B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime}_{0}}$

Then the natural map $A_{1}\to A^{\prime}\times_{A^{\prime}_{0}}A_{0}$ is an
isomorphism: in fact, its kernel is the same as the kernel of
$K\cong\ker(A_{1}\to A^{\prime})\to A_{0}$, but the latter map is injective by
construction. It follows that $\operatorname{ed}(A_{1}/A^{\prime})\leq\dim
A_{0}=\dim A-\dim B$ and this gives the desired inequality.

To finish the proof, it remains to show that

$\operatorname{ed}(\alpha)\geq\dim A-\dim
B+\operatorname{rank}(\ker(\alpha)\cap B)$

for some abelian subvariety $B$ of $A$. By the definition of essential
dimension, there exists a Cartesian diagram

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y^{\prime}}$

where the horizontal maps are finite and $\dim Y=\dim
Y^{\prime}=\operatorname{ed}(\alpha)$. Replacing $Y^{\prime}$ by its
normalization in $A^{\prime}$ and $Y$ by the corresponding fiber product, we
may assume that the rational maps $A^{\prime}\dashrightarrow Y^{\prime}$ and
$A\dashrightarrow Y$ have connected fibers. In particular, the function field
$\mathbb{C}(Y)$ of $Y$ is algebraically closed in $\mathbb{C}(A)$. From this
we see that the Galois action of $G:=\ker(\alpha)$ on $\mathbb{C}(A)$ descends
to $\mathbb{C}(Y)$: in fact, any element in the Galois conjugate of
$\mathbb{C}(Y)$ is finite over $\mathbb{C}(Y)$, hence is contained in
$\mathbb{C}(Y)$ since $\mathbb{C}(Y)$ is algebraically closed in
$\mathbb{C}(A)$. It follows that $G$ acts birationally and faithfully on $Y$.

Let $Y\dashrightarrow Z$ be the MRC fibration and let $F$ be its general
fiber. The functoriality of the MRC fibration implies that $G$ also acts
birationally on the MRC quotient $Z$ and the MRC fibration is $G$-equivariant.
By Lemma 4, up to birational modifications we may assume that $Z$ is an
abelian variety. The rational map $A\dashrightarrow Z$ is then a morphism of
abelian varieties. Let $B\subseteq A$ be its kernel which is an abelian
subvariety (it is connected by construction). The induced (translation) action
of $L=G\cap B$ on $Z$ is trivial, hence $L$ must act faithfully on $F$. By the
assumption that $\deg(\alpha)$ is coprime to $(\dim A)!$, we see that there
exists some prime $p>\dim A$ such that the $p$-rank of $L$ computes
$\operatorname{rank}(L)$. Without loss of generality we may assume that
$\operatorname{ed}(\alpha)<\dim A$ (otherwise there is nothing to prove). In
particular, $\dim F\leq\dim Y=\operatorname{ed}(\alpha)\leq\dim A-1<p-1$. It
then follows from Lemma 7 that $\dim F\geq\operatorname{rank}(L)$ and
therefore

$\operatorname{ed}(\alpha)=\dim Y=\dim A-\dim B+\dim F\geq\dim A-\dim
B+\operatorname{rank}(L)$

as desired. ∎

######  Corollary 10.

Let $\alpha\colon A\to A^{\prime}$ be an isogeny of complex simple abelian
varieties whose degree is prime to $(\dim A)!$. Then

$\operatorname{ed}(\alpha)=\min\\{\dim
A,\operatorname{rank}(\ker(\alpha))\\}.$

###### Proof.

Since $A$ is simple, the only abelian subvarieties are the trivial ones, hence
the statement is clear from Theorem 1. ∎

######  Corollary 11.

Let $A$ be an abelian variety of dimension $n$ and let $p$ be a prime. Assume
that $p>n$ or $n\leq 4$. Then the multiplication by $p$ map $A\to A$ is
incompressible.

###### Proof.

When $p>n$, the result follows directly from Theorem 1 since for any abelian
subvariety $B$ of $A$, we have $B\cap A[p]=B[p]$ (as usual, $A[p]$ denote the
$p$-torsion points of $A$) and its rank, which equals $2\dim B$, is $\geq\dim
B$. If $p\leq n\leq 4$, then from the same argument we see that it suffices to
show that any abelian $p$-group acting on a rationally connected variety of
dimension $m\leq 3$ has rank at most $2m$. For $m=1$ this can be seen from the
list of finite abelian subgroups in $PGL_{2}(k)$. The $m=2$ case follows from
[Bea-p-group-Cr2], while the $m=3$ case is obtained by combining [Pro-p-group-
Cr3]*Theorem 1.2 and [Pro-2-group-Cr3]*Theorem 1.2. ∎

We conclude with some questions and remarks. From the proof of Corollary 11,
we see that the multiplication by $p$ map on any abelian variety is
incompressible as long as the following question has a positive answer:

######  Question 12.

Let $X$ be a rationally connected variety and let $G$ be an abelian
$p$-subgroup of the birational automorphism group of $X$. Is it always true
that $\operatorname{rank}(G)\leq 2\dim X$?

In [FKW-ess-dim-prismatic]*Proposition 2.3.10, the authors also prove a more
general result about the essential dimensions of mod $p$ homology covers. It
would be interesting to find a geometric version of their result. A natural
geometric replacement of the mod $p$ homology cover is the étale cover pulled
back from the $p$-multiplication map on the Albanese variety, and one may
guess (as a geometric analog of _loc. cit._) that the essential dimension of
this cover is at least the Albanese dimension of the variety. However, the
following example shows that this is not always true.

######  Example 13.

Let $A$ be an abelian variety of dimension $n$ and let $C$ be a smooth
projective curve with a faithful $G:=(\mathbb{Z}/p)^{2n}$-action such that
$C/G\cong\mathbb{P}^{1}$. By identifying $G$ with $A[p]$ (which acts on $A$
via translation), we get a diagonal $G$-action on $A\times C$ without fixed
point. Let $X:=(A\times C)/G$ and consider the (étale) cover $f\colon A\times
C\to X$. We have a Cartesian diagram

$\textstyle{A\times
C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A/G}$

where the vertical arrows are induced by the natural projections and the
bottom row can be identified with the $p$-multiplication map on $A\cong A/G$.
We claim that the map $X\to A^{\prime}:=A/G$ is the Albanese morphism of $X$.
For this it suffices to show that the induced map
$H_{1}(X,\mathbb{Z})_{\operatorname{tf}}\to H_{1}(A^{\prime},\mathbb{Z})$
between the torsion free part of the integral homology is an isomorphism.

Observe that the projection $X\to A^{\prime}$ is a smooth fibration with
(connected) fiber $C$, hence the induced map
$\pi_{1}(X)\to\pi_{1}(A^{\prime})$ is surjective by the homotopy exact
sequence. This implies that the map $H_{1}(X,\mathbb{Z})\to
H_{1}(A^{\prime},\mathbb{Z})$ is surjective as well. It remains to show that
$b_{1}(X)=b_{1}(A^{\prime})=2n$. By Hodge theory, this is also equivalent to
$h^{1,0}(X)=n$, or $\dim H^{0}(\Omega^{1}_{A\times C})^{G}=n$. This can now be
checked with a direct calculation, noting that by Künneth formula
$H^{0}(\Omega^{1}_{A\times C})\cong H^{0}(\Omega^{1}_{A})\oplus
H^{0}(\Omega^{1}_{C})$, the $G$-action is trivial on the first factor, while
$H^{0}(\Omega^{1}_{C})^{G}=H^{0}(\Omega_{C/G}^{1})=H^{0}(\Omega_{\mathbb{P}^{1}}^{1})=0$.
Thus we see that $X\to A^{\prime}$ is the Albanese morphism and the Albanese
dimension of $X$ is equal to $n=\dim A^{\prime}$. The cover $f\colon A\times
C\to X$ is therefore the pullback of the $p$-multiplication map on the
Albanese variety by the previous diagram. But we have another Cartesian
diagram

$\textstyle{A\times
C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C/G}$

similar to the one we have before. It gives $\operatorname{ed}(f)\leq\dim C=1$
and in particular the essential dimension $\operatorname{ed}(f)$ can be
smaller than the Albanese dimension (as long as $n>1$).

Recall that a surjective morphism $f\colon X\to X$ of normal varieties is
called a polarized (resp. int-amplified) endomorphism if there exists some
ample line bundle $L$ on $X$ such that $f^{*}L=qL$ some integer $q>1$ (resp.
$f^{*}L-L$ is ample). Clearly, every polarized endomorphism is int-amplified.
The $p$-multiplication maps on abelian varieties and the $p$-th power maps on
toric varieties444For any normal toric variety $X$, the $p$-th power map on
the open torus extends to a finite morphism of the toric variety $X$. are
special examples of polarized endomorphisms, but there are many other examples
such as finite self-maps of $\mathbb{P}^{n}$ (or any variety of Picard number
$1$). In positive characteristics there are also the Frobenius maps. We have
already seen that $p$-multiplications on abelian varieties are incompressible
for large $p$. The $p$-th power maps on toric varieties are even
$p$-incompressible for all $p$ by e.g. [RY-ess-dim-fix-point-method]*Theorem
1.2 or [BF-incompressibility]*Proposition 10. Motivated by these results, it
seems natural to ask:

######  Question 14.

Is every polarized (resp. int-amplified) endomorphism incompressible?

One limitation of our current approach is that it does not give much
information about essential dimensions of morphisms between varieties of
general type. It would be interesting to extend our geometric arguments to
this more general setting. Such an extension may also be useful in addressing
the $p$-incompressibility part of Brosnan’s conjecture.

## References