# A simple model describing evolution of genomic GC content with random
perturbations in asexually reproducing organisms

Jon<EMAIL_ADDRESS>
###### Abstract

A genome’s nucleotide composition can usually be summarized with (G)uanine +
(C)ytosine (GC) or (A)denine + (T)hymine (AT) frequencies as GC%=100%-AT%. A
model is presented relating the evolution of genomic GC content over time to
AT$\rightarrow$GC and GC$\rightarrow$AT mutation rates. By employing Itô
calculus it is shown that if mutation rates in asexually reproducing organisms
are subject to random perturbations that can vary over time several
implications follow. For instance, an extra Brownian motion term appears
influencing nucleotide variability; the greater the variability of the random
perturbations on the mutation rates the stronger the impact of the Brownian
motion term. Reducing the influence of the random perturbations, to limit
fitness decreasing and deleterious mutations, will likely imply divesting
resources to genomic repair systems. The stable mutation rates seen in many
organisms could thus be an evolved strategy to reduce the influence of the
Brownian motion term. Furthermore, if change to genomic GC content, _i.e._ the
GC content of variable sites or single nucleotide polymorphisms (SNPs), is
just as likely to increase as to decrease, something that resembles knockout
of repair enzymes and removal of selective pressures seen in evolutionary
laboratory experiments, the species genome will likely decay unless infinite
resources are available. These implications are solely a consequence of
allowing random perturbations affect AT- and GC mutation rates and not
obtainable using standard non-stochastic methodology. Finally, a connection
between the model for genomic GC content evolution and the classical Luria-
Delbrück mutation model is presented in a stochastic setting.

1Division of Infection Control, Department of Methods Development and
Analysis, Norwegian Institute of Public Health

2Centre for Fertility and Health, Norwegian Institute of Public Health

Norwegian Institute of Public Health, P.O. Box 4404, Lovisenberggata 8, 0403
Oslo, Norway.

##  1 Introduction

The hereditary material of living organisms consist of double stranded
deoxyribonucleic acid (DNA) molecules [40]. The building blocks of the DNA
molecule are the nucleotides Adenine (A), Guanine (G), Cytosine (C) and
Thymine (T) [40]. Across each strand Gs pair with Cs and Ts with As only [40].
Within each strand the different nucleotides are stacked in no specific order
connected via a sugar backbone [40]. Virus can have genomes consisting of
single or double stranded DNA or ribonucleic acid (RNA) [14]. RNA is similar
to DNA but is more often single stranded and T is substituted for Uracil (U)
[30]. Genomes consisting of double stranded DNA have approximately as many As
as Ts and Gs as Cs on each strand [10]. These relations were first observed by
Erwin Chargaff [10] and are therefore referred to as one of Chargaff’s parity
laws. Base composition in genomes with double stranded DNA can therefore be
analyzed using either AT- or GC content as GC%=100%-AT%. Base-pairing across
the two DNA strands consists of three hydrogen bonds for G and C and two for A
and T [40]. More energy is in general required to stack and melt G and C
bindings as compared to A and T [11]. Methylation of C, by the addition of a
methyl group, often results in deamination of C transforming it to T [8].
Deamination of methylated Cs is therefore implicated in an AT mutational bias
observed for bacteria and archaea (prokaryotes) [18]. Relaxation of selective
pressures are hypothesized to increase AT content in microbial genomes due to
the failure of repair enzymes to remove methylated cytosines [12]. Indeed,
both modelling and empirical investigations points to an approximately 2:1
relationship of respectively GC$\rightarrow$AT and AT$\rightarrow$GC mutation
rates in prokaryotes [18, 7]. DNA methylation also occurs in organisms with a
cell nucleus (eukaryotes) and it is therefore reasonable to expect an AT
mutational bias for these organisms as well [7]. However, as many larger,
multi-cellular organisms reproduce sexually homologous recombination may
obscure any such mutational AT bias observed for prokaryotes [8]. Homologous
recombination seems instead to increase GC content in eukaryotes as a
consequence of a process referred to as GC-biased gene conversion [38].
Examination of AT- and GC mutation rates in prokaryotes points to an AT bias
also in recombining microbes while AT $\rightarrow$ GC substitutions are more
likely to be retained [6, 7]. It should be noted that AT $\rightarrow$ GC
mutations (and vice versa) are in the present work taken to mean all possible
combinations of A and T to G and C (G and C to A and T).

For larger multi-cellular organisms the GC content is fairly stable between
species although there can be local genomic differences, for instance CpG
islands which are usually not present in prokaryotes [8]. For prokaryotes
within-species genomic GC content is stable while GC content between different
species can vary substantially, from 13.5% GC to 75% GC [9]. This variation in
genomic GC content appears to have both an environmental component as well as
a phylogenetic one [15, 36]. Environmental influence on genomic GC content in
prokaryotes appears to be mediated, at least to some extent, as selective
pressures [20]. In this respect, it is of interest to note that large drops in
genomic GC content is wide-spread in prokaryotes [18] while examples of
substantial increase is practically non-existent as of now [16], although
smaller increases are documented [34, 36, 19, 5].

A previous study [9] modelling the evolution of GC content in microbes living
in a stable symbiotic relationship with an eukaryotic host, usually insects,
suggested that AT $\rightarrow$ GC mutation rates, and vice versa, may
determine the symbiotic species fate early on in the organism’s history
providing mutation rates are approximately constant. Microbial symbionts often
live in low density populations, are unable to perform homologous
recombination and lack many DNA repair genes implying that mutations
accumulate, often in a clock-like manner [29]. If the selective pressures are
not strong enough to purge deleterious and fitness decreasing mutations from
the symbiont’s genome, it will decay [21]. This process is known as Muller’s
ratchet [32]. For clonal, non-recombining symbionts, the evolutionary process
of Muller’s ratchet will always incur, the only question is when [32].

Since microbial symbionts live in a stable environment with their eukaryotic
host it makes sense to model the random perturbations of their AT- and GC
mutation rates as a Gaussian white noise multiplied by a constant, or a fixed
parameter, to be estimated, as has previously been done [9]. For asexually
reproducing organisms in general, subjected to differing selective pressures
over time, it is more natural to model random perturbations of AT/GC mutation
rates as a Gaussian white noise multiplied by a function varying with respect
to time. As such, the purpose of the present work is to expand on the previous
study [9]. That is, to model the evolution of genomic GC content as a
consequence of AT $\rightarrow$ GC, as well as GC $\rightarrow$ AT, mutation
rates subject to random perturbations $c(t)$ over time $t$, as opposed to a
constant $c$, for all non-recombining, Chargaff parity law compliant
organisms, regardless of which kingdom they belong to.

##  2 Methods

###  2.1 The mathematical model

The model presented in this study is an extension of models from previous work
where details on how these formulas were derived can be found [6, 7]. In
particular, the present work is an extension of a model describing the
evolution of genomic GC content as a consequence of AT/GC mutation rates with
random perturbations for microbial symbionts [9]. As there are many
similarities between present- and the previous work on microbial symbionts [9]
only the steps separating these models and details necessary for a complete
comprehension are included.

First, $F_{t}(\omega)$ represents genomic GC content at time $t$ for all
trajectories $\omega\in\Omega$ such that:

$F_{t+\Delta t}(\omega)-F_{t}(\omega)=\alpha F_{t}(\omega)\Delta
t+\beta(1-F_{t}(\omega))\Delta t$ (1)

That is, the change in genomic GC content $F_{t+\Delta
t}(\omega)-F_{t}(\omega)$ during time $\Delta t$ is a fraction multiplied with
genomic GC- and AT content, respectively. Somewhat inaccurately this will be
interpreted as the GC content of single nucleotide polymorphisms
(SNPs/variable sites) in a species during time $\Delta t$. It will be shown
later that there is a natural way of accurately extracting SNP GC/AT content
from eq. (1). If it is assumed that as $\Delta t\rightarrow 0$ eq. (1) can be
written as a differential equation:

$\frac{dF_{t}(\omega)}{dt}=\alpha F_{t}(\omega)+\beta(1-F_{t}(\omega)),$ (2)

GC- and AT mutation rates are described by, $\alpha F_{t}(\omega)$ and
$\beta(1-F_{t}(\omega))$, respectively. The AT$\rightarrow$GC and
GC$\rightarrow$AT mutation rates are subject to random perturbations that can
vary with time multiplied by a function $c(t)$, _i.e._
$\alpha=a+c(t)W_{t}(\omega)$ and $\beta=b+c(t)W_{t}(\omega)$. It is therefore
assumed that $c(t)$ is a measurable function and that $W_{t}(\omega)$ is a
Gaussian white noise [45] with respect to all trajectories $\omega\in\Omega$.
Furthermore, eq. (2) is taken to belong to the probability space
$(\Omega,\mathcal{F}_{t},P)$ while $c(t)$ is an element of the measure space
$(\mathbb{R}^{+},\mathcal{G},dt)$. $\mathcal{F}_{t}$ is the filtration of
$\Omega$ with respect to each time $t\in\mathbb{R^{+}}$ (_i.e._ $[0,\infty)$
of which $\mathcal{G}$ is a Borel algebra and $dt$ the corresponding Lebesgue
measure), and $P$ is a probability (Lebesgue) measure on $\Omega$. The
filtration $\mathcal{F}_{t}$ is interpreted as the evolutionary history of
trajectories $\omega$ up to time $t$. Some further re-arrangements gives:

$\displaystyle\frac{dF_{t}(\omega)}{dt}$
$\displaystyle=(a+c(t)W_{t}(\omega))F_{t}(\omega)+(b+c(t)W_{t}(\omega))(1-F_{t}(\omega))$
$\displaystyle=aF_{t}(\omega)+c(t)W_{t}(\omega)F_{t}(\omega)+$
$\displaystyle+b(1-F_{t}(\omega))+c(t)W_{t}(\omega)(1-F_{t}(\omega))$
$\displaystyle=aF_{t}(\omega)+b(1-F_{t}(\omega))+c(t)W_{t}(\omega).$

The equation:

$\frac{dF_{t}(\omega)}{dt}=aF_{t}(\omega)+b(1-F_{t}(\omega))+c(t)W_{t}(\omega),$
(3)

can be written as a differential form as was explained in the previous study
[9]:

$dF_{t}(\omega)=(aF_{t}(\omega)+b(1-F_{t}(\omega)))dt+c(t)dB_{t}(\omega).$ (4)

If $c(t)=c$, where $c$ is a constant, eq. (4) will coincide with the model for
microbial symbionts [9] and so is an extension containing that model. The
Brownian motion term can additionally be a $c$-scaled Brownian motion which
was previously also shown [9] to be a Brownian motion there termed
$\hat{B_{t}}$. Using the Itô formula [45]:

$dY_{t}(\omega)=\frac{\partial g}{\partial
t}(t,F_{t}(\omega))dt+\frac{\partial g}{\partial
t}(t,F_{t}(\omega))dF_{t}(\omega)+\frac{1}{2}\frac{\partial^{2}g}{\partial
x^{2}}(t,F_{t}(\omega))(dF_{t}(\omega))^{2}$ (5)

eq. (4) can be given an explicit solution through the integrating factor
$g(t,F_{t}(\omega))=Y_{t}(\omega)=e^{(-(a-b)t)}F_{t}(\omega)$:

Because $\frac{\partial^{2}g}{\partial x^{2}}(t,x)=0$ (see [45] for details),
the last term of eq. (5) is equal to zero. As a result,

$\displaystyle dY_{t}(\omega)$ $\displaystyle=\frac{\partial g}{\partial
t}(t,F_{t}(\omega))dt+\frac{\partial g}{\partial
x}(t,F_{t}(\omega))dF_{t}(\omega)$
$\displaystyle=-(a-b)e^{(-(a-b)t)}F_{t}(\omega)dt+e^{(-(a-b)t)}dF_{t}(\omega)$
$\displaystyle=-(a-b)e^{(-(a-b)t)}F_{t}(\omega)dt+$
$\displaystyle+e^{(-(a-b)t)}(((a-b)F_{t}(\omega)+b)dt+c(t)dB_{t})$
$\displaystyle=be^{(-(a-b)t)}dt+c(t)e^{(-(a-b)t)}dB_{t}.$

$F_{t}(\omega)$ can then be give an explicit formula:

$d(e^{(-(a-b)t)}F_{t}(\omega))=be^{(-(a-b)t)}dt+c(t)e^{(-(a-b)t)}dB_{t}$

which, by assuming $s\in[0,t]$, gives

$e^{(-(a-b)t)}F_{t}(\omega)-F_{0}(\omega)=\int_{0}^{t}be^{(-(a-b)s)}ds+\int_{0}^{t}c(s)e^{(-(a-b)s)}dB_{s},$

and therefore the integral

$F_{t}(\omega)=F_{0}(\omega)e^{(a-b)t}+\int_{0}^{t}be^{(a-b)(t-s)}ds+\int_{0}^{t}c(s)e^{(a-b)(t-s)}dB_{s}.$
(6)

An analogous argument to the previous study [9] gives an explicit formula for
the finite variation term $\int_{0}^{t}be^{(a-b)(t-s)}ds$. The quadratic
variation Brownian motion term $\int_{0}^{t}c(s)e^{(a-b)(t-s)}dB_{s}$ however
must be approximated numerically, not least due to the unspecified measurable
function $c(t)$:

$F_{t}(\omega)=-\frac{b}{(a-b)}+(F_{0}(\omega)+\frac{b}{(a-b)})e^{(a-b)t}+\int_{0}^{t}c(s)e^{(a-b)(t-s)}dB_{s}$
(7)

which is subject to the constraints $t\in[0,\infty)$ and $0<F_{t}(\omega)<1$.
The integration constant $c_{0}$ is just included in $F_{0}$. It should be
noted that for $F_{0}=0$,

$\mathbb{E}(F_{t}(\omega))=\frac{b}{(a-b)}(e^{(a-b)t}-1).$ (8)

Since the Brownian motion term vanishes as a consequence of the expectation
operator $\mathbb{E}$ (see p. 30 of [45]), the solution to (7) when $t=x$ is
exactly the model for SNP GC content with respect to core genome GC content
$x$ described in previous work [6, 7]. Moreover, this means that it is not
necessary to calculate the Brownian motion term when estimating the AT- and GC
mutation rate parameters $a$ and $b$.

The variance is given by
$\mathrm{Var}(F_{t}(\omega))=\mathbb{E}((F_{t}(\omega)-\mathbb{E}(F_{t}(\omega))^{2})$,
which can be solved by setting:

$A:=F_{0}(\omega)e^{(a-b)t}+\frac{b}{(a-b)}(e^{(a-b)t}-1)$

and

$B:=\int_{0}^{t}c(s)e^{(a-b)(t-s)}dB_{s}$

which gives:

$\displaystyle\mathrm{Var}(F_{t}(\omega))$
$\displaystyle=\mathbb{E}((F_{t}(\omega)-\mathbb{E}(F_{t}(\omega))^{2}$
$\displaystyle=\mathbb{E}((A+B)^{2}-2(A+B)A+A^{2})$
$\displaystyle=\mathbb{E}(A^{2}+2AB+B^{2}-2A^{2}-2AB+A^{2})$
$\displaystyle=\mathbb{E}(B^{2})=\mathbb{E}((\int_{0}^{t}c(s)e^{(a-b)(t-s)}dB_{s})^{2}).$

Applying the Itô isometry (see pg. 26 [45]):

$\mathbb{E}((\int_{0}^{t}c(s)e^{(a-b)(t-s)}dB_{s})^{2})=\mathbb{E}(\int_{0}^{t}(c(s)e^{(a-b)(t-s)})^{2}ds)$
$=\int_{0}^{t}c(s)^{2}e^{2(a-b)(t-s)}ds.$

The equation cannot be given an analytic representation due to the unspecified
function $c(t)$ but it is clear that the integral
$\int_{0}^{t}c(s)^{2}e^{2(a-b)(t-s)}ds\rightarrow\infty$ as $c(s)$ increases
for $s\rightarrow t$. As was argued in the previous study on microbial
symbionts [9], it will be assumed henceforth that $(a-b)<0$ where $a$ and $b$
are respectively the AT$\rightarrow$GC and GC$\rightarrow$AT mutation rate
parameters that can be estimated (see [6, 7]) or given. The previous study [9]
also showed that $a$ and $b$ could be considered as unspecified measurable
functions. However, as constant mutation rates are not uncommon [27] $a$ and
$b$ will henceforth be regarded as constants, or parameters to be estimated,
to avoid unnecessary complication of interpretation, formulation and
derivation of the model.

###  2.2 The Brownian motion term

The term:

$\int_{0}^{t}c(s)e^{(a-b)(t-s)}dB_{s},$ (9)

depends on the parameters $a$ and $b$ as well as on the duration of the time
period. Since it is assumed that $(a-b)<0$, $c(s)e^{(a-b)(t-s)}\rightarrow
c(s)$ for $s\rightarrow t$. The term can be written as:

$\int_{0}^{t}c(s)e^{(a-b)(t-s)}dB_{s}=\lim_{\Delta s_{i}\rightarrow
0}\sum_{s_{0}}^{s_{N}}c(s_{i})e^{(a-b)(t-s_{i})}(W_{s_{i+1}}(\omega)-W_{s_{i}}(\omega))\Delta
s_{i},$ (10)

where $W_{s}(\omega)$ is scaled white noise with mean $\mu=0$, variance
$\sigma^{2}=1$ (see [6]), $\Delta s_{i}=s_{i+1}-s_{i}$, and
$s_{0}=0,\ldots,s_{i}=t_{i},\ldots,s_{N}=t$. The right hand side term of eq.
(10) can be calculated manually by inserting values for each value $s_{i}$ and
mutation parameters $a$ and $b$. Each Gaussian white noise $W_{s_{i}}(\omega)$
can be sampled from a normal distribution.

###  2.3 The Girsanov Transform

The Girsanov transform implies that eq. (7), if an appropriate transform
exists, can be considered as a Brownian motion. In other words, given the
appropriate transform, genomic GC content can be seen to be just as likely to
increase as to decrease. This is a scenario that can arise when mismatch and
repair enzymes are knocked out and the species is subjected to reduced
selective pressures which is sometimes recreated in laboratory settings such
as the long term evolutionary experiment (LTEE) [12]. The random perturbations
of AT/GC mutation rates are represented here as Gaussian white noise
multiplied by an unspecified deterministic measurable function $c(t)$.
Together with the Gaussian white noise term $W_{t}(\omega)$, the function
$c(t)$ accounts for environmental influences and species-specific traits
perturbing the AT/GC mutation rate variances over time $t$. Since eq. (4) is a
differential equation the Gaussian white noise should turn into Brownian
motion if the SNP GC content is to be modelled as being completely unbiased
with regards to both selective pressures and mismatch and repair enzymes. If
it is assumed that $t\in[0,T]$ for a fixed time $T$, then:

$dF_{t}(\omega)=((a-b)F_{t}(\omega)+b)dt+c(t)dB_{t}(\omega).$ (11)

To see that the Girsanov theorem applies to $F_{t}(\omega)$ recall from
section 2.1 that for $Y_{t}(\omega)=e^{(-(a-b)t)}F_{t}(\omega)$:

$dY_{t}=be^{(-(a-b)t)}dt+c(t)e^{(-(a-b)t)}dB_{t}$ (12)

Since $F_{t}(\omega)$ is a semimartingale, the Girsanov II theorem (pg. 167,
[45]) can be used. Let

$c(t)e^{(-(a-b)t)}u(t,\omega)=be^{(-(a-b)t)}-\alpha(t,\omega)$

and thus:

$u(t,\omega)=c^{-1}(t)(b-\alpha(t,\omega)e^{(a-b)t})$

$\alpha(t,\omega)$ can then be set to zero so that:

$u(t,\omega)=c^{-1}(t)b$

which means that $c(t)$ in eq. (4) is required to have an inverse function
$c^{-1}(t)$. Let

$M_{t}=\exp{\Big{(}-\int_{0}^{t}u(s,\omega)dB_{s}-\frac{1}{2}\int_{0}^{t}u^{2}(s,\omega)ds\Big{)}}$

and set

$dQ(\omega)=M_{t}(\omega)dP(\omega)$

with respect to the filtration $\mathcal{F}_{t}$, assume that the Novikov
condition (see pg. 165 [45]) holds:

$E[\exp{\Big{(}\frac{1}{2}\int_{0}^{t}u^{2}(s,\omega)ds\Big{)}}]<\infty$

so that the Radon-Nikodym derivative $M_{t}$ is a martingale. Then $Q$ is a
probability measure with respect to the filtration $\mathcal{F}_{t}$ and

$\hat{B_{t}}(\omega)=\int_{0}^{t}u(s,\omega)ds+B_{t}(\omega)$

is a Brownian motion with regards to the measure $Q$ and so is $Y_{t}$,
according to the Girsanov theorem:

$dY_{t}(\omega)=c(t)d\hat{B_{t}}(\omega)$

Since $Y_{t}(\omega)=e^{(-(a-b)t)}F_{t}(\omega)$ it is clear that
$F_{t}(\omega)$ is also a Brownian motion with respect to $Q$. Hence, given
the right transform $F_{t}(\omega)$ is just as likely to increase as to
decrease. If the transformed $F_{t}(\omega)$ is considered to represent a
culture in an LTEE-type experiment [22, 31, 12], where mismatch and repair
genes are knocked out and selective pressures are assumed to be at a minimum,
the function $c(t)$ must have an inverse, _i.e._ $c(t)$ must be either
monotonically increasing or decreasing.

##  3 Results and Discussion

###  3.1 Allowing random perturbations to vary as a function of time

By allowing the random perturbation of AT/GC mutation rates to vary according
to a function $c(t)$ a number of differences can be noted as compared to the
previous model [9] where $c(t)=c$. Indeed, while asexually reproducing
organisms still will be subject to the process of Muller’s ratchet [32] this
can no longer be predicted to the same extent previously described for
microbial symbionts [9]. In other words, as the random perturbations of the
AT/GC mutation rates vary according to a function $c(t)$, with respect to time
$t$, it is conceivable that selective pressures could completely alter the
evolutionary trajectory of the organism. One of the surprising results of the
model describing evolution of genomic GC content in microbial symbionts [9]
was that the fate of the symbiont could have been set when it entered it’s
symbiotic relationship possibly millions of years before an extinction event.
Allowing the random perturbations of the AT/GC mutation rates to vary
according to a function with regards to time $t$, as opposed to being fixed,
the genomic GC content of the organism can vary as time progresses. Hence,
selective pressures and genetic drift could radically change the evolutionary
trajectory of the organism.

###  3.2 Comparison of mathematical models

As mentioned above, the present work is based on a previous study [9] where
the aim was to model the evolution of genomic GC content, as a consequence of
AT/GC mutation rates, in microbial symbionts over time $t$, _i.e._ :

$F_{t+\Delta t}(\omega)-F_{t}(\omega)=\alpha F_{t}(\omega)\Delta
t+\beta(1-F_{t}(\omega))\Delta t$

which gave the differential equation:

$\frac{dF_{t}(\omega)}{dt}=\alpha F_{t}(\omega)+\beta(1-F_{t}(\omega))$

when $\Delta t\rightarrow 0$.

In other words, the change in the GC content of SNPs, or variable sites, over
time $t$ is a fraction $\alpha$ multiplied by genomic GC content
$F_{t}(\omega)$ in addition to a fraction $\beta$ multiplied by genomic AT
content ($1-F_{t}(\omega)$, according to Chargaff’s parity rules [10]). Thus,
$\alpha$ and $\beta$ designated the fraction of genomic GC content mutating to
GC (_i.e._ A>C ,A>G ,T>C ,T>G or G>C ,C>G) and to AT (C>A, G>A, C>T, G>T, A>T,
T>A), respectively.

The model also allowed for random perturbations of SNP AT- and GC mutation
rates, _i.e._ $\alpha=a+cW_{t}(\omega)$ and $\beta=b+cW_{t}(\omega)$ where
$W_{t}(\omega)$ is Gaussian white noise with respect to all trajectories
$\omega\in\Omega$. After some rearrangement:

$\frac{dF_{t}(\omega)}{dt}=aF_{t}(\omega)+b(1-F_{t}(\omega))+cW_{t}(\omega)$

Since it was shown [9] that the scaled volatility of Brownian motion is also a
Brownian motion the equation could be solved analytically. It was also shown
that the mutation rates $a$ and $b$ could be described as measurable functions
$a(t)$ and $b(t)$, respectively. However, as $a(t)$ and $b(t)$ are unspecified
analytic solutions would be impossible except for trivial cases of $a(t)$ and
$b(t)$.

The present work is not limited to the genomes of microbial symbionts, which
tend to live in a stable relationship with their hosts, but to all asexually
reproducing organisms regardless of which kingdom they belong to, including
virus. It is still assumed that the genomes of the organisms considered comply
with Chargaff’s parity laws [10]. In particular, it is assumed that % G is
approximately equal to % C and that % A is similar to % T on each strand in
the genomes of the organisms considered. Although Chargaff’s parity laws were
stated for most organisms with double stranded DNA genomes they also apply to
many viruses with single stranded RNA genomes [8]. Indeed, the pathogen
responsible for the currently ongoing Covid-19 pandemic, SARS-CoV-2, has a
single strand, positive sense, RNA genome that obeys Chargaff’s parity rule
with approximately 38% GC [44, 43]. Nevertheless, increasing the generality of
the previous model to the genomes of all asexually reproducing organisms
implies that the assumption of constantly scaled random perturbing AT/GC
mutation rates, which could be justified in a stable host-symbiont
relationship, is no longer tenable. It is now therefore assumed that the
random perturbations to AT/GC mutation rates vary according to a Gaussian
white noise multiplied by a measurable function $c(t)$ with respect to time
$t$ as deduced in section 2.1, eq. (7):

$\frac{dF_{t}(\omega)}{dt}=aF_{t}(\omega)+b(1-F_{t}(\omega))+c(t)W_{t}(\omega)$

If restrictions are imposed on the function $c(t)$ eq. (7) can be applicable
to the Girsanov transform [45]. SNP GC content is then just as likely to
increase as to decrease with respect to the measure resulting from the
Girsanov transform. In other words, it can then be guaranteed that
$F_{t}(\omega)$ is a Brownian motion relative to a measure $Q(\omega)$. SNP GC
content then follows a completely random path $\omega\in\Omega$ according to
the law of Brownian motion. Due to influences from positive- and negative
selection and mismatch and repair systems this is typically not observed
outside laboratories [12].

Figure 1: The figure demonstrates three different (C1-C3) evolutionary
scenarios for the model describing the evolution of genomic GC content
(vertical axis) of a species over time $t$ (horizontal axis) as a consequence
of AT/GC mutation rates. All parameters are the same for all scenarios (_i.e._
$a=-2$, $b=1$, $F_{0}=0.5$ and $T=1$) except for the function $c(t)$ that
determines the influence of the random perturbations of the AT/GC mutation
rates. Panel C1: $c(t)=\sqrt{t}$, Panel C2: $c(t)=T-t$, Panel C3:
$c(t)=e^{{(T-t)^{2}}}$.

###  3.3 The evolutionary dynamics of $c(t)$

AT/GC mutation rates are allowed to have random perturbation varying according
to a function $c(t)$ with respect to time $t$, _i.e._
$\alpha=a+c(t)W_{t}(\omega)$ and $\beta=b+c(t)W_{t}(\omega)$ where
$W_{t}(\omega)$ is a Gaussian white noise with respect to every trajectory
$\omega\in\Omega$. This has implications for the resulting Brownian motion
term eq. (9):

$\int_{0}^{t}c(s)e^{(a-b)(t-s)}dB_{s}$

Indeed, increasing $c(t)$ leads to greater variance in genomic GC content
$F_{t}(\omega)$ as $s\rightarrow t$ while decreasing $c(t)$ reduces it (see
Figure 1). Increased variance of $F_{t}(\omega)$ can be interpreted as
increasing the intrinsic genomic base composition variation. However,
increasing the genomic base composition variance (_i.e._ genomic GC content
variance) may also lead to accumulation of fitness decreasing, and even
deleterious, mutations [2, 5]. Hence, increased variance with regards to
perturbation of AT- and GC mutation rates highlights the trade-off between
more genetic diversity versus an increased chance of accumulating fitness
decreasing and deleterious mutations and subsequent extinction. For decreasing
$c(t)$ approaching $0$ variation in base composition is reduced, something
that is likely to happen in a scenario of increasing purifying selection [1,
5]. Organisms living in large populations [28] or highly adapted to their
environments [5] are therefore more likely to have lower $c(t)$ values than
organisms subjected to more diverse and changing environments. It has
previously been shown that nucleotide diversity in microbes increases with AT
content [5]. That is, GC rich bacteria tend to have a more homogeneous
nucleotide composition while AT rich bacteria have genomes comparably more
heterogeneous in terms of nucleotide composition [4]. The underlying reasons
for this decreasing nucleotide diversity gradient, from AT rich to GC rich
genomes, is not known but years of research has shed some light on the issue.
For instance, GC rich bacteria are mostly found in soil, with excess nitrogen
[1]. Soil bacteria have large genomes and are often capable of metabolising a
wide range of compounds [36]. AT-rich bacteria, on the other hand, are often
symbionts [33] and pathogens [41] with reduced genome sizes. As mutations in
prokaryotes are universally biased towards AT [18] loss of proof reading
enzymes as well as mismatch and repair genes, both hallmarks of relaxed
selection, could be driving the greater genetic diversity found in these
bacteria [37]. It has also been shown that the energetics of stacked A/T and
G/C nucleotides is important in establishing genomic base composition [11].
Between-strand binding of A to T requires only two hydrogen bonds as compared
to the three hydrogen bonds required for G to C bindings [40]. Guanine and
cytosine also require the availability of more nitrogen [39]. Within
eukaryotic genomes, GC content vary considerably more than within prokaryotic
genomes [8]. The genomes of most eukaryotic species have a considerably lower
fraction of gene-coding DNA allowing for greater variation in GC content and
thus a more varied $c(t)$ [8]. Viral genomes often mimic their hosts with
respect to GC content and since their genomes are small, with a high fraction
of coding genes, genomic GC content is typically far more stable than that
observed for eukaryotic genomes and often comparable to prokaryotes [8]. Thus
it would expected that $c(t)$ is lower for both virus and prokaryotes
throughout their genomes as compared to most eukaryotes.

###  3.4 Evolutionary implications

The selective pressures a species is subjected to can, to some extent, be
reflected by the trends of the measurable function $c(t)$. On one hand, $c(t)$
should be as close to zero as possible to avoid excessive hitch-hiking of
fitness decreasing or deleterious mutations [2]. However, it will likely
require several trade-off’s for species’ to reduce the random variation of AT-
and GC mutation rates and thus keep $c(t)$ low; considerable resources must
likely be divested to several genomic processes for $c(t)\rightarrow 0$. On
the other hand, a changing environment may require species’ to adapt rapidly
implying the availability of an increased number of genotypes. A greater
number of genetic variants require that the mutation rates reach a level that
maximizes chance for survival of the species [13]. At the same time, if
mutation rates increase to such an extent that deleterious mutations cannot be
avoided the evolutionary process of Muller’s ratchet will ensue [32].

Whether mutations increase or decrease genomic GC content depends on the
environment and the selective pressures operating on the species’ genomes.
Some environments could favour energetically affordable A/T nucleotides while
others might require the more costly G/C nucleotides [11]. In addition,
phylogeny will also influence the selection of A/T or G/C nucleotides [36] as
a consequence of the mismatch repair system and/or proof reading enzymes [5,
12, 24].

While there are many examples of microbial genomes becoming more AT rich [1]
there are so few examples of genomes becoming more GC rich that it was
recently suggested that it may not happen at all [16]. Some examples have
however been observed in the microbial world although the increase is
miniscule [5, 20, 3]. It is not completely resolved whether the few examples
of microbial genomes becoming more GC rich is tied to recombination [23],
which seems to be the case for recombining eukaryotes, or selection [20, 34,
8].

Muller’s ratchet is, in the model discussed here, linked to the Brownian
motion term and the measurable function $c(t)$ overwhelming the random
perturbations of the AT/GC mutations rates. Hence, restrictions on $c(t)$ will
necessarily influence the species’ trajectory of genomic GC content and thus
it’s life course. It is interesting to note that allowing for random
perturbation of AT/GC mutation rates introduces a term that will unequivocally
lead to greater variability of genomic base composition (see eq. (6)); the
larger the perturbations the greater the impact of the Brownian motion term on
genomic variability. Put differently, genomic base composition can be modified
by varying the random perturbations of the AT/GC mutation rates. It is
therefore not unlikely that this is one of the reasons that mutation rates are
occasionally found to be remarkably stable over a diverse set of organisms
and, in particular, negatively associated with population size [17, 27, 28].

Assuming that eq. (7) is a Brownian motion, in the sense that it is just as
likely that GC content will increase as it will decrease, it is required by
the Girsanov theorem, as seen in section 2.3, that $c(t)$ has an inverse
function meaning that $c(t)$ must either be increasing or decreasing. All
being equal, an increasing $c(t)$ will more likely lead to Muller’s ratchet,
and thus subsequent extinction, due to the accumulation of fitness decreasing
and deleterious mutations. In other words, if it can be argued that $c(t)$
cannot be decreasing (_i.e._ resources are finite), increasing random
perturbations affecting AT- and GC mutation rates will necessarily lead to
decay. Interestingly, this has been demonstrated in a laboratory experiment
and described in a recent study based on the LTEE [12].

###  3.5 Evolution of genomic GC content and the Luria-Delbrück mutation model

By setting $\beta=0$ in

$\frac{dF_{t}(\omega)}{dt}=\alpha F_{t}(\omega)+\beta(1-F_{t}(\omega))$

eq. (3) can be interpreted as a simple model for stochastic population growth
(pg. 65, [45]):

$\frac{dP_{t}(\omega)}{dt}=\pi P_{t}(\omega)$

where $\pi=k+p(t)W_{t}(\omega)$, $k$ is the growth parameter to be estimated
and $p(t)$ is a measurable function with respect to time $t$. By using the Itô
formula, along the lines described by [45] in pg. 66, to solve for
$P_{t}(\omega)$, and multiplying with a parameter $\mu$, taken to be a
parameter designating mutations per unit time, a simple model for calculating
the number of mutations in a study population is obtained. This is a
stochastic version of the model presented in the classical Luria-Delbrück
fluctuation experiment [26], _i.e._

$M_{t}(\omega)=P_{t}(\omega)\mu=P_{0}\mu\exp{((k-\frac{p^{2}(t)}{2})t+p(t)B_{t}(\omega))}$

Hence, in this simple model the number of mutations in the study population is
equal to the population size $P_{t}(\omega)$ multiplied with the mutation rate
$\mu$. It can be seen from the above equations that as $k$ approaches
$\frac{p^{2}(t)}{2}$ a randomly fluctuating population will increasingly
influence mutation rates. In other words, as the size of the population
declines stochastic effects become more dominant [25] resulting not only in
increased genetic variation (see Figure 2) but also the danger of decay due to
accumulation of fitness decreasing and deleterious mutations [28].

$M_{t}(\omega)$ represents the total number of mutations in a population
$P_{t}(\omega)$, typically represented by mutations in one genome. If we
assume that also $M_{t}(\omega)$ complies with Chargaff’s parity rules (see
[6]) we can write:

$M_{t}(\omega)=M_{t}^{AT}(\omega)+M_{t}^{GC}(\omega)$

where $M_{t}^{AT}(\omega)$ and $M_{t}^{GC}(\omega)$ represents the number of
A+T and G+C mutations, respectively. Eq. (3) gives the fraction of GC
mutations at time $t$ and can be interpreted as the GC content of SNPs at that
time. By multiplying eq. (3) with the genome size $g$ the number of GC
mutations is given:

$g\cdot\Big{|}\frac{dF_{t}(\omega)}{dt}=\alpha
F_{t}(\omega)+\beta(1-F_{t}(\omega))$

However, $g\frac{dF_{t}(\omega)}{dt}$ will only be equal to $M_{t}(\omega)$ if
$M_{t}(\omega)=M_{t}^{GC}(\omega)$. If $M_{t}(\omega)=M_{t}^{AT}(\omega)$,
$g\frac{dF_{t}(\omega)}{dt}$ will be negative which is impossible to reconcile
with the fact that $M_{t}(\omega)\geq 0$. But since the right hand side of eq.
(3), multiplied with $g$, can be written as:

$g\cdot\Big{|}\alpha F_{t}(\omega)+\beta(1-F_{t}(\omega))$ (13)

it is clear from eq. (13) that $M_{t}^{GC}(\omega)$ and $M_{t}^{AT}(\omega)$
respectively correspond to $|g\alpha F_{t}(\omega)|$ and
$|g\beta(1-F_{t}(\omega))|$. The triangle inequality with respect to eq. (13)
therefore gives:

$M_{t}(\omega)=M_{t}^{AT}(\omega)+M_{t}^{GC}(\omega)=g\cdot(|\alpha
F_{t}(\omega)|+|\beta(1-F_{t}(\omega))|)$

which is the number of mutations at time $t$.

A model for mutation accumulation can be written as the Lebesgue integral:

$\int_{0}^{t}M_{s}(\omega)ds$

or:

$\int_{0}^{t}g\cdot(|\alpha F_{s}(\omega)|+|\beta(1-F_{s}(\omega))|)ds$

Figure 2: The figure demonstrates three different (P1-P3) scenarios for the
stochastic Luria-Delbrück model describing average number of mutations
$M_{t}(\omega)$ (vertical axis) in a population at time $t$ (horizontal axis).
All parameters are the same for all scenarios ($P_{0}=1$, $\mu=5$, $k=1$ and
$T=1$) except for the function $p(t)$ that determines the influence of random
perturbation on growth rate. Panel P1: $p(t)=\ln{(t)}$, Panel P2: $p(t)=t$,
Panel P3: $p(t)=e^{(T-t^{2})}$.

##  4 Conclusions

The presented work has been concerned with modelling the evolution of genomic
GC content, as a consequence of AT/GC mutation rates, in asexually reproducing
organisms subject to Chargaff’s parity laws [10]. It is an extension of a
previous study modelling genomic GC content in microbial symbionts allowing
for random perturbations of AT- and GC mutation rates by the use of Itô
calculus [45]. In that study [9], it was shown that a symbiont’s life course
could be determined when it entered into a relationship with it’s host. The
present model does not allow for such a conclusion in general as the organisms
modelled can both be diverse and live in very different environments. The fate
of these organisms is thus linked to the function $c(t)$ regulating the
influence of the random perturbations on AT/GC mutation rates that was taken
to be constant for microbial symbionts. Increasing $c(t)$ will increase base
composition diversity but also result in genetic hitch-hiking of fitness
decreasing and deleterious mutations [2]. Keeping $c(t)$ low will likely
reduce genetic variation but require the divestment of increasing resources to
mismatch and repair systems as $c(t)\rightarrow 0$. Interestingly, eq. (7)
implies that increasing the variability of the random perturbations of the AT-
and GC mutation rates impacts genomic GC content through the Brownian motion
term. It is not uncommon to encounter stable mutation rates in different
organisms [27]. Indeed, the presented model (eq. (7)) suggests that this could
be resulting from selection to diminish variability in genomic GC content as
described by the Brownian motion term.

Laboratory based evolutionary experiments [31, 22, 12] often arrange
conditions such that the selective forces subjected to the species’ studied
are as low as possible. Furthermore, recombination related genes as well as
mutation repair enzymes are often knocked out [12] reducing bias considerably
with regards to AT$\rightarrow$GC and GC$\rightarrow$AT mutation rates. The
model presented suggests that if genomic GC content is just as likely to
increase as to decrease $c(t)$ must either be monotonically increasing- or
decreasing due to constraints resulting from the Girsanov transform. Hence, if
not resources are limitless $c(t)$ will increase eventually leading to genomic
disintegration, as described by Muller’s ratchet [32], something that has been
demonstrated experimentally [12].

Finally, it is shown that there exists an intimate relationship with the model
presented here and the classical Luria-Delbrück model for general mutations
[26]. Indeed, disregarding AT mutation rates by setting $\beta=0$ in eq. (3)
gives a model for stochastic population growth $P_{t}(\omega)$ [45] which
multiplied with a mutation rate $\mu$ gives a stochastic Luria-Delbrück model
for the number of mutations $M_{t}(\omega)$. After some re-arrangements, it is
shown in section 3.5 that eq. (3) is related to $M_{t}(\omega)$.

## Acknowledgements

The figures were made with the free statistical software R [35] and the
_ggplot2_ package [42].

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# Echoes of Compact Objects in Scalar-Tensor Theories of Gravity

Christoforos Vlachos, Eleftherios Papantonopoulos
Physics Division, National Technical University of Athens,
15780 Zografou Campus, Athens, Greece
Kyriakos Destounis
Theoretical Astrophysics, IAAT, University of T$\ddot{u}$bingen, 72076
T$\ddot{u}$bingen, Germany<EMAIL_ADDRESS>mail:lpapa<EMAIL_ADDRESS>
###### Abstract

Scalar-tensor theory predicts solutions to the gravitational field equations
which describe compact objects in the presence of a non-minimally coupled
scalar field to the Einstein tensor. These objects are black holes with scalar
hair and wormholes supporting scalar phantom matter. The evolution of test
fields in fixed asymptotically-flat backgrounds of exotic compact objects
leads to the formation of echoes in the ringdown signal, which designate the
existence of trapping regions close to the event horizon. Here, we consider
minimally-coupled test scalar fields propagating on compact object solutions
of the Horndeski action, which possess an effective cosmological constant,
leading to anti-de Sitter asymptotics, and show that echoes can form in the
ringdown waveform due to the entrapment of test fields between the photon
sphere and the effective asymptotic boundary. Although the presence of an
event horizon leads to the usual echoes with decaying amplitude, signifying
modal stability of the scalarized black hole considered, we find that test
scalar fields propagating on a scalarized wormhole solution give rise to
echoes of constant and equal amplitude to that of the initial ringdown,
indicating the existence of normal modes. Finally, we find that, near
extremality, the test field exhibits a concatenation of echoes; the primary
ones are associated with the trapping region between the photon sphere and the
effective anti-de Sitter boundary while the secondary ones are linked to the
existence of a potential well at the throat of the wormhole.

###### Contents

  1. 1 Introduction
  2. 2 Exact compact objects in scalar-tensor theory
  3. 3 Scalar field propagation on fixed gravitational backgrounds
  4. 4 Time-domain integration scheme
  5. 5 Propagation of perturbations on compact objects in scalar-tensor theory
    1. 5.1 Black hole
    2. 5.2 Wormhole
      1. 5.2.1 Extremal wormhole
  6. 6 Conclusions

##  1 Introduction

The direct observation of gravitational waves (GWs) produced during the
relativistic collision of two compact objects, offers exciting new
opportunities for the study of the nature of the colliding bodies. In the near
future, following the recent LIGO detections [1]-[5], GW astronomy will
provide us a new understanding of the gravitational interaction and
astrophysics in extreme-gravity conditions. The recent observations do not yet
probe the detailed structure of spacetime beyond the photon sphere, however
one expects that the strong gravity regime will in the next years come to our
understanding with future GW observations. In particular, the expectation is
to precisely detect the ringdown phase, which is governed by a series of
damped oscillatory modes at early times, named quasinormal modes (QNMs)
[6]-[9], and may potentially contain unexpected anomalies due to new physics
at late times [10].

The expectation is that future GW observations will give us some information
on the nature and physics of the near-horizon region of black holes (BHs) and
if these regions exhibit any unexpected structure. Alternatives to BHs, that
is, objects without event horizons, were recently constructed, known as exotic
compact objects (ECOs) [11]-[14]. The existence of any structure at near-
horizon scales would generate a series of “echoes” of the primary
gravitational wave signal, produced during the ringdown phase [10, 15]. The
LIGO data have already been analyzed on the presence of echoes [16, 17, 18].

It is believed that the ringdown waveform is dominated by the QNMs of the
compact object remnant. Thus, the detection of overtones from the ringdown
signal allows for precision measurements of the characteristic parameters of
compact objects like the mass, charge and angular momentum. Various studies
suggest that the ringdown signal is dominated by the mode excitations of
photons trapped in unstable circular orbits at the photon sphere, namely the
photon sphere (PS) modes [19]-[26]. These QNMs are directly related to the
existence of the PS, and if the compact object is an asymptotically flat BH no
other oscillatory mode is excited. For ECOs, on the other hand, although the
PS excitations still exist at the early stage of the ringdown signal, as in
the case of BHs, they do not belong to the QNM spectrum [10, 15].

Wormholes are ECO solutions of the Einstein equations that connect different
parts of the Universe or two different Universes [27, 28]. Although wormholes
have distinct causal strutures from BHs, they possess PSs and, therefore, can
disguise themselves as BHs in GW data if one only focuses on the early stage
of the ringdown signal. Lorentzian wormholes in General Relativity (GR) were
discussed in [29, 30, 31], where a static spherically-symmetric metric was
introduced and conditions for traversable wormholes were found. Unfortunately,
wormhole solutions of the Einstein equations lead to the violation of null
energy condition (NEC). A matter distribution of exotic or phantom matter
allows the formation of traversable wormhole geometries in GR. There have been
many efforts to build a wormhole with ordinary matter satisfying the NEC [31,
32, 33] in modified gravity theories like Brans-Dicke theory [34],
$f({\mathcal{R}})$ gravity [35], Einstein-Gauss-Bonnet theory [36], Einstein-
Cartan theory and general scalar-tensor theories [37].

A series of contemporary studies [10, 15] suggest that the ringdown signal may
provide a conclusive proof for the formation of an event horizon. Such
expectation is based on the assumption that an ECO would possess a reflective
surface beyond the PS instead of an event horizon. This would lead to the
existence of a trapping region between the surface of the ECO and PS where
perturbations could be confined and manifest themselves as echoes in the late
stage of the gravitational waveform. Then, by considering ringdown waveforms
of ECOs, such as wormholes, it has been claimed [10, 15] that precision
observations of the late-time ringdown signal can distinguish between the
formation of ECOs or BHs.

In this work we will consider the ringdown phase of exact compact object
solutions of scalar-tensor theory, which is part of the Horndeski class of
solutions [38] that give rise to second order field equations in four
dimensions [39, 40, 41] (for a review of this class of Horndeski theories see
[42]). The BH [43]-[48] and wormhole [49] solutions we consider arise from a
gravitational action with a real or phantom scalar field, respectively, non-
minimally coupled to the Einstein tensor. These exact solutions encode the
‘gravitational’ scalar, and its coupling strength, in the metric tensor
components as a primary charge [44, 49], which asymptotically plays the role
of an effective negative cosmological constant. The motivation for considering
these objects is that they have a natural asymptotic reflective boundary, in
the external region of the PS, which may lead to different behaviour compared
with the case of flat spacetimes. The anti-de Sitter (AdS) spacetime is a
necessity for holographic theories which were built by applying the
gauge/gravity duality. The aim of holography is to study strongly coupled
phenomena using dual gravitational systems where the coupling is weak [50].
This duality, which is well founded in string theory, has many interesting
applications and among them is condensed matter physics. In these theories,
AdS BHs play an essential role in the gravity sector in order to achieve the
abundant phase structure of the condensed matter system lying on the conformal
boundary (for a review, see [51]). These holographic theories stimulated the
extended study of AdS BHs, their formation and their stability, with their
QNMs describing the approach to thermal equilibrium in the dual conformal
field theory on the boundary [52].

Wormholes in AdS spacetimes where discussed in [53], in an attempt to yield
some information about the physics of closed Universes. Such discussion is
connected with the physics of inflation, and its connection with vacuum decay.
A unique realization of such ideas is baby-Universe formation by quantum
tunneling which eventually disconnect from the parent spacetime [54].
Recently, these ideas of connecting the physics of wormhole spacetimes to
baby-Universes were revisited in [55], using features associated with a
negative cosmological constant and asymptotically AdS boundaries.

Our main goal is to probe the ringdown of the aforementioned exact compact
object solutions of scalar-tensor theory, by considering the propagation of
linear test scalar perturbations minimally coupled to the metric, with the
hope that these two objects are discernible, even though both possess a PS and
effective AdS asymptotics. We will adopt the methodology used in [56, 57] and
introduce a new minimally-coupled test scalar field in the gravitational
action; the simplest case one can possibly envision. We note that the test
scalar field we utilize is linear and should not be confused with the
gravitational scalar of the theory, which backreacts to the metric to give
rise to the scalarized BH and wormhole solutions we consider. By introducing a
novel minimally-coupled linear test scalar field, we test the response of such
solutions to small fluctuations, which in turn encode the information of the
gravitational scalar that places a natural asymptotic effective boundary,
although a negative cosmological constant is absent from the action. More
complicated non-minimal couplings have been considered in such theories,
though the effect of a coupling between the test and gravitational scalar
leads to a critical coupling constant below which the QNM boundary conditions
are not satisfied [57].

The work is organized as follows. In Section 2 we review the BH and wormhole
solution with the non-minimal derivative coupling in the Horndeski scalar-
tensor theory. In Section 3 we derived the effective potentials for a test
scalar field scattered off from the BH and wormhole. In Section 4 we discuss
the time-domain integration scheme. In Section 5 we study the propagation of
the test scalar field in the background of the BH and the wormhole. Finally,
in Section 6 we discuss our results and possible applications.

##  2 Exact compact objects in scalar-tensor theory

In this section, we briefly review two exact compact object solutions [44, 49]
of the Horndenski Lagrangian with non-minimal kinetic coupling

$S=\int
dx^{4}\sqrt{-g}\left\\{\frac{\mathcal{R}}{8\pi}-\left[\varepsilon\,g_{\mu\nu}+\eta\,G_{\mu\nu}\right]\phi^{,\mu}\phi^{,\nu}\right\\}\leavevmode\nobreak\
.$ (1)

Here, $\mathcal{R}$ is the Ricci scalar, $G_{\mu\nu}$ is the Einstein tensor,
$g_{\mu\nu}$ is the metric tensor with $g=\det g_{\mu\nu}$, $\phi$ is a real
massless scalar field and $\eta$ is a non-minimal coupling constant with
dimensionality length squared. In the case where $\varepsilon=1$, the theory
contains a canonical scalar field with a positive kinetic term, while when
$\varepsilon=-1$ the theory describes a phantom scalar field with a negative
kinetic term.

The BH solution found in [44] is static, spherically symmetric and possesses
an AdS-like boundary. The solution reads

$\displaystyle ds^{2}$
$\displaystyle=-f(r)dt^{2}+g(r)dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})\leavevmode\nobreak\ ,$ (2)

where

$\displaystyle f(r)$
$\displaystyle=\frac{1}{4}\left(3-\frac{8\mu}{r}+\frac{r^{2}}{3l_{\eta}^{2}}+\frac{l_{\eta}}{r}\arctan\frac{r}{l_{\eta}}\right)\,,$
(3) $\displaystyle g(r)$
$\displaystyle=\frac{(r^{2}+2l_{\eta}^{2})^{2}}{(r^{2}+l_{\eta}^{2})^{2}\,4f(r)}\,,$
(4) $\displaystyle\Psi^{2}(r)$
$\displaystyle\equiv\left(\phi^{\prime}(r)\right)^{2}=-\frac{\varepsilon}{8\pi
l^{2}_{\eta}}\,\frac{r^{2}(r^{2}+2l_{\eta}^{2})^{2}}{(r^{2}+l_{\eta}^{2})^{3}\,4f(r)}\,.$
(5)

We would like to stress that this solution describes a BH only in the case
where $\varepsilon=1$ and $\varepsilon\eta<0$ (see [44, 49] for further
details). Here, $\mu$ is an integration constant that plays the role of mass
and $l_{\eta}=\sqrt{|\varepsilon\eta|}$ is a characteristic scale of the non-
minimal coupling. The inverse tangent function restricts the domain of $r$ to
$r\in(0,\infty)$. In the limit $r\rightarrow 0$ the function $f(r)$ yields
Schwarzschild asymptotics, i.e. $f(r)\approx 1-\frac{2\mu}{r}$, while for
$r\rightarrow\infty$ one obtains
$f(r)\approx\frac{3}{4}+\frac{r^{2}}{12l^{2}_{\eta}}$ i.e. AdS-like
asymptotics. Note that in the metric functions (3) and (4) the non-minimal
coupling $l_{\eta}$ is present, which is the strength of the coupling of the
scalar field to curvature. Therefore, the BH solution is dressed with a scalar
field given in Eq. (5).

The wormohole solution found in [49], by following the approach of [44],
reads111Note that the coordinates $(t,\xi,\theta,\phi)$ used here are not the
Schwarzschild coordinates since, $\xi$ is not the curvature radius of a
coordinate sphere $\xi=$const$>0$.

$ds^{2}=-f(\xi)dt^{2}+g(\xi)d\xi^{2}+(\xi^{2}+a^{2})(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})\leavevmode\nobreak\ .$ (6)

where $\varepsilon\eta<0$, $\varepsilon=-1$ and

$\displaystyle g(\xi)$ $\displaystyle=$
$\displaystyle\frac{\xi^{2}(\xi^{2}+a^{2}+2l_{\eta}^{2})^{2}}{(\xi^{2}+a^{2})(\xi^{2}+a^{2}+l_{\eta}^{2})^{2}F(\xi)}\leavevmode\nobreak\
,$ (7) $\displaystyle f(\xi)$ $\displaystyle=$
$\displaystyle\frac{a}{\sqrt{\xi^{2}+a^{2}}}\exp\left[\int_{0}^{\xi}\frac{\xi(\xi^{2}+a^{2}+2l_{\eta}^{2})^{2}}{l_{\eta}^{2}(\xi^{2}+a^{2})(\xi^{2}+a^{2}+l_{\eta}^{2})F(\xi)}d\xi\right]\leavevmode\nobreak\
,$ (8) $\displaystyle\Psi^{2}(\xi)$ $\displaystyle\equiv$
$\displaystyle\left(\phi^{\prime}(r)\right)^{2}=-\frac{\varepsilon}{8\pi
l_{\eta}^{2}}\frac{\xi^{2}(\xi^{2}+a^{2}+2l_{\eta}^{2})^{2}}{(\xi^{2}+a^{2})(\xi^{2}+a^{2}+l_{\eta}^{2})^{3}F(\xi)}\leavevmode\nobreak\
,$ (9)

with

$F(\xi)=3-\frac{8\mu}{\sqrt{\xi^{2}+a^{2}}}+\frac{\xi^{2}+a^{2}}{3l_{\eta}^{2}}+\frac{l_{\eta}}{\sqrt{\xi^{2}+a^{2}}}\arctan\left(\frac{\sqrt{\xi^{2}+a^{2}}}{l_{\eta}}\right).$
(10)

Again, the non-minimal derivative coupling appears in the metric functions and
the phantom scalar field generating the wormhole is given in Eq. (9). The
function $F(\xi)$ has a minimum at $\xi=0$, thus to make it positive definite
one should demand $F(0)>0$. Hence, one can derive the limitation on the upper
value of the parameter mass parameter $\mu$

$2\mu<a\left(\frac{3}{4}+\frac{\alpha^{2}}{12}+\frac{1}{4\alpha}\arctan\alpha\right)\leavevmode\nobreak\
,$ (11)

where $\alpha\equiv a/l_{\eta}$ is a dimensionless parameter which defines the
ratio of the wormhole throat radius $a$ and the scale of the non-minimal
kinetic coupling $l_{\eta}$. Far from the throat, in the limit
$|\xi|\to\infty$, the metric functions $g(\xi)$ and $f(\xi)$ take the
asymptotic form

$g(\xi)=3\frac{l_{\eta}^{2}}{\xi^{2}}+O\left(\frac{1}{\xi^{4}}\right)\leavevmode\nobreak\
,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,f(\xi)=A\frac{\xi^{2}}{l_{\eta}^{2}}+O(\xi^{0})\leavevmode\nobreak\
,$ (12)

where $A$ depends on the parameters $a$, $l_{\eta}$, $\mu$ and can be
calculated only numerically. These asymptotics correspond to AdS space with
constant negative curvature. Close to the throat $\xi=0$ one finds

$g(\xi)=B\frac{\xi^{2}}{l_{\eta}^{2}}+O(\xi^{4})\leavevmode\nobreak\
,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,f(\xi)=1+O(\xi^{2})\leavevmode\nobreak\
,$ (13)

where $B$ depends on $\alpha$ and $\mu$. Moreover, there is a coordinate
singularity at $\xi=0$ where $g(0)=0$.

##  3 Scalar field propagation on fixed gravitational backgrounds

In this study, we will be interested in the response of the compact object
models described above against a linear test scalar field $\Phi$ minimally-
coupled to the metric, but not the gravitational scalar $\phi$ in (1). The
dynamical propagation of a linear massless scalar perturbation $\Phi$ on the
fixed background spacetime of a compact object, described by the metric tensor
$g_{\mu\nu}$, is dominated by Klein-Gordon equation

$\square\Phi=0\Longleftrightarrow\frac{1}{\sqrt{-g}}\partial_{\mu}\left[\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Phi\right]=0\leavevmode\nobreak\
.$ (14)

Due to spherical symmetry we can decompose the scalar field
$\Phi(t,\rho,\theta,\phi)$ into a radial and angular parts, by introducing the
ansatz

$\Phi(t,\rho,\theta,\phi)=\frac{\psi(\rho,t)}{R(\rho)}\,Y_{lm}(\theta,\phi)\leavevmode\nobreak\
,$ (15)

where $Y_{lm}$ are the standard spherical harmonics, $\rho$ is a general
radial-like coordinate and $R(\rho)$222$R(\rho)\equiv R(r)=r$ for the BH and
$R(\rho)\equiv R(\xi)=\sqrt{\xi^{2}+a^{2}}$ for the wormhole. a function of
$\rho$. Equation (14) can, then, be recasted into a Schrodinger-like form

$\displaystyle\left[\frac{\partial^{2}}{\partial
t^{2}}-\frac{\partial^{2}}{\partial\rho_{*}^{2}}+V(\rho)\right]\psi(\rho,t)=0\leavevmode\nobreak\
,$ (16)

with the effective potential given by

$\displaystyle
V(\rho)=f(\rho)\left(\,\frac{\ell(\ell+1)}{R(\rho)^{2}}+\frac{R^{{}^{\prime\prime}}(\rho)}{g(\rho)R(\rho)}+\frac{f^{\prime}(\rho)R^{\prime}(\rho)}{2g(\rho)f(\rho)R(\rho)}-\frac{g^{\prime}(\rho)R^{\prime}(\rho)}{2g^{2}(\rho)R(\rho)}\right)\leavevmode\nobreak\
,$ (17)

where $\ell$ is the angular quantum number and $\rho_{\ast}$ is the usual
tortoise coordinate defined by

$d\rho_{*}=\sqrt{\frac{g(\rho)}{f(\rho)}}\;d\rho\,.$

Equation (16) demonstrates that one is able to reduce the problem of scalar
perturbations around compact objects into a single one-dimensional scattering
problem with an effective potential. Applying this procedure on the BH (3)-(5)
and the wormhole (7)-(9) we find the corresponding effective potentials a test
scalar field “feels” when propagating on these backgrounds.

Figure 1: The effective potential of scalar perturbations with $\ell=1$ for
the black hole (3)-(5) (left) and the wormhole (7)-(9) (right) with throat
radius $a=1$, for three different values of $l_{\eta}$ and $\mu=0.1$.

Fig. 1 shows the effective potentials for various choices of non-minimal
coupling constants. We observe that the BH spacetime has a peak right outside
the event horizon, while asymptotically the effective potential diverges. Such
asymptotic divergence encodes the AdS-like nature of the spacetime. The
increment of $l_{\eta}$ leads to a more distant asymptotic boundary which
could be explained from the fact that the non-minimal coupling has
dimensionality length squared. In a sense, $l_{\eta}$ acts as an inverse
cosmological constant, therefore at the limit $l_{\eta}\rightarrow\infty$ the
spacetime becomes asymptotically flat, which would correspond to a zero
cosmological constant. The wormhole’s effective potential is clearly different
from that of the BH, as seen in Fig. 1. There is a single peak at the throat
$\xi=0$, which corresponds to the PS, while asymptotically the potential
diverges. The effect of $l_{\eta}$ is apparent in this case as well. Although
not proven explicitly here, our numerics indicate that both peaks of $V$ occur
close to the PS, since their amplitude is solely affected by $\ell$ which is
directly associated with the energy of null particles trapped in unstable
circular orbits at the PS.

##  4 Time-domain integration scheme

In this section we briefly demonstrate the numerical scheme of time-domain
integration, first proposed in [58], which yields the temporal response of the
scalar field as it propagates on a fixed background. By defining
$\psi(\rho_{\ast},t)=\psi(i\Delta\rho_{\ast},j\Delta t)=\psi_{i,j}$,
$V(\rho(\rho_{*}))=V(\rho_{\ast},t)=V(i\Delta\rho_{\ast},j\Delta t)=V_{i,j}$,
equation (16) takes the form

$\displaystyle\frac{\psi_{i+1,j}-2\psi_{i,j}+\psi_{i-1,j}}{\Delta\rho^{2}_{\ast}}-\frac{\psi_{i,j+1}-2\psi_{i,j}+\psi_{i,j-1}}{\Delta
t^{2}}-V_{i}\psi_{i,j}=0\,.$ (18)

Then, by using as initial condition a Gaussian wave-packet of the form
$\psi(\rho_{\ast},t)=\exp\left[-\frac{(\rho_{\ast}-c)^{2}}{2\sigma^{2}}\right]$
and $\psi(\rho_{\ast},t<0)=0$, where $c$ and $\sigma$ correspond to the median
and width of the wave-packet, we can derive the time evolution of the scalar
field $\psi$ by

$\displaystyle\psi_{i,j+1}=-\psi_{i,j-1}+\left(\frac{\Delta
t}{\Delta\rho_{\ast}}\right)^{2}\left(\psi_{i+1,j}+\psi_{i-1,j}\right)+\left(2-2\left(\frac{\Delta
t}{\Delta\rho_{\ast}}\right)^{2}-V_{i}\Delta t^{2}\right)\psi_{i,j}\,,$ (19)

where the Von Neumann stability condition requires that $\frac{\Delta
t}{\Delta\rho_{\ast}}<1$. Moreover, the effective potential is positive and
vanishes at the event horizon (but not at the wormhole throat), however, it
diverges as $r\to\infty$ (or $|\xi|\to\infty$). This requires that $\psi$
should vanish at infinity for both compact objects in study, which corresponds
to reflective boundary conditions. To calculate the precise values of the
potential $V_{i}$, we integrate numerically the equation for the tortoise
coordinate and then solve with respect to the corresponding radial coordinate.
Various convergence tests were performed throughout our numerical evolution,
with different integration steps and precision, to reassure the validity of
our ringdown profiles.

##  5 Propagation of perturbations on compact objects in scalar-tensor theory

By applying the numerical procedure outlined above, we calculate the temporal
response of linear massless scalar field perturbations on the BH and wormhole
solutions discussed. In what follows, we assume the mass of both compact
objects to $\mu=0.1$ (if not stated otherwise) and obtain the perturbation
response at a position arbitrarily close to the event horizon for the BH, and
at $\xi=0.01$ for the wormhole.

###  5.1 Black hole

Figure 2: Time evolution of scalar perturbations with $\ell=0$ (top panel)
$\ell=1$ (middle panel) and $\ell=2$ (bottom panel) on the black hole
background (3)-(5) with $\mu=0.1$ and $l_{\eta}=1$.

Fig. 2 displays the evolution of a linear scalar perturbation field on the
background of the BH solution (3)-(5). The most obvious effect we can observe
is the emergence of echoes following the initial quasinormal ringdown. This
pattern becomes more evident for higher $\ell$ due to the fact that more
energy is carried away from the PS when perturbed. For spherically-symmetric
$\ell=0$ perturbations, on the other hand, the echo pattern is not so evident
since the field does not excite the PS significantly, therefore the echoes
fall off rapidly. Our investigation confirms that the decay rate of scalar
perturbations follows an exponential fall-off, as in [52, 59], which is more
evident for the case of $\ell=0$. This behaviour is in contrast to that of
asymptotically flat BH perturbations, where the quasinormal ringing gives way
to a power-law cutoff [58, 60, 61], and its occurrence is related to the
asymptotic nature of timelike infinity in AdS spacetimes which serves as a
reflective boundary. The echoes have significantly smaller amplitudes compared
with the initial ringdown, which is in agreement with the studies in [56, 57,
62] and the dissipative nature of the event horizon, designating modal
stability.

It is worthy to note that further analytical investigations of perturbations
in AdS BHs led to the conclusion that solutions of the Klein-Gordon equation
with fixed angular quantum number $\ell$, indeed decay exponentially [63].
However, an accumulation of all solutions, possessing finite energy, achieves
a logarithmic decay rate, due to the presence of stable trapping [64] (for a
discussion of the wave equation in the interior of AdS BHs see [65, 66]).

Figure 3: Time evolution of scalar perturbations with $\ell=2$ on the black
hole background (3)-(5) with $\mu=0.1$ and $l_{\eta}=4$ (left), $l_{\eta}=5$
(right). The black vertical dashed lines indicate the time one expects the
next echo to arrive, as calculated from (20).

In Fig. 3 the effect of the derivative coupling constant is illustrated. When
$l_{\eta}$ increases the effective AdS boundary moves further away from the
event horizon (see Fig. 1). As a consequence, the scalar wave reflected off
the PS has to travel a larger distance before it reaches the reflective AdS
boundary and return to re-perturb the PS. Thus, the increment of $l_{\eta}$
leads to a delay of the echoes. It is important to note that the echoes appear
in this case, not due to trapping of waves between the PS and the surface of
the compact object, but rather due to the asymptotic nature of infinity. This
effect may have important implications in AdS/CFT correspondence, if an actual
negative cosmological constant is included, where a ringdown in the bulk
corresponds to the approach to thermal equilibrium in the boundary CFT though
a sequence of ringdown signals, such as echoes, does not yet have a proper
boundary interpretation (though see [67, 68]).

To justify our statement we have computed numerically the time interval needed
for light to perform a round trip from the PS to the AdS boundary. For a
metric of the form (2) the characteristic time-scale is given by [10, 69]

$\displaystyle\Delta t=2\int_{PS}^{Boundary}\sqrt{\frac{g(r)}{f(r)}}\;dr\,.$
(20)

As can be seen from Fig. 3, the temporal location of the echoes, as obtained
from the numerical integration, is in good agreement with the values of
$\Delta t$ calculated from Eq. (20) (shown with dashed lines in Fig. 3). This
agreement further supports that the formation of echoes is due on the
secondary perturbations of the PS from the reflected scalar field on the
effective AdS boundary.

###  5.2 Wormhole

In Fig. 4 we demonstrate the behavior of the test scalar field as it
propagates in the wormhole solution (8)-(9). The temporal response exhibits
echoes, as in the BH case above, which follow the initial ringdown due to the
first encounter of the probe field with the PS. In a similar manner, the
$\ell=0$ perturbations do not significantly excite the PS of the wormhole,
thus the echoes are not as oscillatory as the ones obtained for $\ell>0$.
However, we notice that the amplitude of the echoes does not decrease with
time, in contrast to the response in the BH setup.

Through this effect, one then can easily distinguish if the compact object is
a BH or wormhole. The underlying mechanism that leads to such a behavior could
be understood from the fact that instead of an event horizon we have a
wormhole throat, therefore, energy cannot be dissipated. The probe field
travels through the throat and into the second Universe, to be reflected back
from the second effective AdS boundary (see Fig. 1). The small “glitches”
shown in echoes of Fig. 4 (in the linear scale) appear due to the fact that we
measure the response of the test field at $\xi=0.01$, thus the reflected wave
from the AdS boundary of the primary Universe arrives slightly earlier than
the reflected wave from the boundary of the secondary Universe, to superpose
and lead to an echo of equal amplitude to that of the initial outburst.

The effect of the non-minimal coupling $l_{\eta}$ is shown in Fig. 5. Besides
the fact that the initial ringdown and the echoes have the same amplitude, one
can realize that $l_{\eta}$ serves as a scale of the Universe, since for
higher $l_{\eta}$ values, the field has to travel a larger distance from the
throat to the AdS boundary and back. This results to a proportionality between
the coupling $l_{\eta}$ and the echo time. To illustrate this, we have
approximated the time interval between two echoes using the relation

$\displaystyle\Delta
t=2\int_{Throat}^{Boundary}\sqrt{\frac{g(r)}{f(r)}}\;dr\leavevmode\nobreak\ .$
(21)

The agreement between the resulting signal from time integration and the
characteristic time from Eq. (21) (see vertical dashed lines in Fig. 5)
justifies further that the echoes are produced due to the presence of the
effective AdS boundary and not due to the existence of a double barrier
effective potential that usually appears in wormhole solutions (see [10],
[70]-[73]).

Figure 4: Time evolution of scalar perturbations with $\ell=0$ (top panel)
$\ell=1$ (middle panel), $\ell=2$ (bottom panel) on the wormhole background
(8)-(9) with $l_{\eta}=10,\,\mu=0.1$ and $a=1$.

The existence of echoes of equal amplitude to that of the initial ringdown is
an indication that such compact objects may possess normal modes of
oscillation, similar to those found in [74, 75, 76]. In fact, one could
perform a mode decomposition on the test scalar field to calculate these
modes, though the complicated form of the metric components render such
analysis rather challenging. Since the echoes seem to possess a characteristic
timescale (21), it is intriguing to approximate the normal modes as
$\omega\sim 2\pi/\Delta_{t}$. Our numerics indicate that
$\omega\sim\mu/l_{\eta}$, with $\ell$ not playing a dominant role in this
approximation, besides affecting the oscillation rate of each ringdown. A
modal analysis would shed more light to the validity of our approximation and
to the existence of normal modes in such wormholes.

Figure 5: Time evolution of scalar perturbations with $\ell=1$ on the wormhole
background (8)-(9) with $\mu=0.1$, $a=1$ and $l_{\eta}=5$ (left),
$l_{\eta}=10$ (right).

####  5.2.1 Extremal wormhole

So far, the wormhole parameters considered give rise to a single peak at the
effective potential, which leads to two trapping regions separated by a
potential barrier. If we consider a different set of parameters, where the
wormhole mass is nearly extremal $\mu\simeq\mu_{extreme}$ (or exactly
extremal), then the potential peak splits into two potential barriers, for
large angular momenta. In this case, three trapping regions appear, which
depend on the throat radius, near-extremal mass an angular momentum in a
manner which is demonstrated in Fig. 6. The existence of three potential wells
in the wormhole background will lead to echoes which arise from two different
regions: the primary region between the PS and the asymptotic AdS boundary,
and the novel secondary region between the wormhole throat.

Figure 6: Left panel: Effective potential with $l_{\eta}=1\,,\,\ell=10$ and
$\mu=\mu_{extreme}$ for each value of the throat radius $a$. Right panel:
Effective potential with $l_{\eta}=1\,,\,a=1\,,\,\mu=\mu_{extreme}$ for
various angular momenta $\ell$.

Figure 7: Time evolution of scalar perturbations with $\ell=5$ on the wormhole
background with $\mu=\mu_{extreme}\approx 0.37$, $a=1$ and $l_{\eta}=5$ (top
panel), $l_{\eta}=7$ (bottom panel).

In Fig. 7 we observe a new qualitative behaviour of linear scalar
perturbations which is not present in the non-extremal wormhole setups
considered above. Besides the primary echoes arising from the reflection of
the test field at the AdS boundary, a new series of echoes appears in between
them, with smaller amplitude than the primary ones. These secondary echoes are
a product of the new trapping region at the wormhole throat. Although the
first couple of secondary echoes are quite visible in Fig. 7, at sufficiently
late times the superposition of primary and secondary echoes renders them
hardly distinguishable. Nevertheless, the existence of primary and secondary
echoes, associated with different trapping regions, is reported here for the
first time in a wormhole spacetime with trapping regions that arise naturally.

##  6 Conclusions

The GW ringdown, where the final object relaxes to a stable state, contains
key information about the perturbed compact object’s externally observably
quantities. Recently, it has been argued [10, 15] that the late-time ringdown
signal may incorporate signatures for the existence of ECOs, such as
wormholes, or horizon-scale quantum corrections [77, 78, 79] (though see
[80]), in the form of echoes.

Here, we considered minimally-coupled test scalar perturbations on exact BH
and wormhole solutions of scalar-tensor theory, which possess an effective
negative cosmological constant, leading to AdS asymptotics, due to the
presence of a non-minimally coupled ‘gravitational’ scalar to the Einstein
tensor. We find that similar effects arise in the late-time ringdown waveform
for both compact objects under study. After the initial ringdown, the test
field response exhibits echoes, with timescales proportional to the non-
minimal coupling constant. Although the BH considered here does not contain
any quantum corrections at the event horizon, the effective asymptotic AdS
boundary, that the gravitational scalar introduces to the scalarized solution,
forces the partially reflected waves from the PS to mirror off the AdS
boundary and re-perturb the PS to give rise to a damped beating pattern which
is strikingly similar to echoes from quantum corrected compact objects [81].
The existence of echoes in such AdS-like BHs may lead to interesting
interpretations of such a phenomenon on dual holographic descriptions at the
boundary, as AdS/CFT suggests (see [67, 68] for a holographic description of
echoes on a dual CFT at the near-horizon quantum structure).

Interestingly, even though the wormhole studied here possesses a single
barrier effective potential (for non-extremal masses), with its throat located
at the peak [82] (similar to that of Bronnikov-Ellis [83, 84] and Morris-
Thorne [29] wormholes), we still observe echoes in the late-time response of
the test scalar field due to the existence of the effective AdS boundary.
Moreover, if the wormhole is near-extremal or extremal, the response of the
probe scalar field exhibits primary and secondary echoes, associated with the
AdS boundary and throat potential well, respectively. The concatenation of
echoes we observed here are very similar to those found in [85], though our
setup consists of naturally arising trapping regions, solely depending on the
spacetime and perturbation parameters, in contrast to the reflective
boundaries placed by hand in [85].

Contrary to to the BH case, the echoes found in the wormhole background do not
decay with time, but have constant and equal amplitude to that of the initial
ringdown. The constancy of the amplitude of echoes is related to the absence
of dissipation and may be an indication of the existence of normal oscillation
modes, as well as potential instabilities, similar to that found in [86]. The
introduction of gravitational perturbations (such as the ones considered in
[87, 88, 89, 90]) may lead to an unstable wormhole, due to the presence of
phantom matter, which can potentially expand or collapse into a BH [91, 92].

## Acknowledgments

The authors would like to thank Rodrigo Fontana for helpful discussions.

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