# Ultra-high frequency gravitational waves from cosmic strings with friction

S. Mukovnikov<EMAIL_ADDRESS>Centro de Astrofísica da
Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal  Instituto
de Astrofísica e Ciências do Espaço, CAUP, Rua das Estrelas, 4150-762 Porto,
Portugal  Departamento de Física e Astronomia, Faculdade de Ciências,
Universidade do Porto, Rua do Campo Alegre 687, PT4169-007 Porto, Portugal  L.
Sousa<EMAIL_ADDRESS>Centro de Astrofísica da Universidade do Porto,
Rua das Estrelas, 4150-762 Porto, Portugal  Instituto de Astrofísica e
Ciências do Espaço, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal
Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do
Porto, Rua do Campo Alegre 687, PT4169-007 Porto, Portugal

###### Abstract

We include the effect of the frictional force caused by interactions between
cosmic strings and the particles of the background plasma in the computation
of the stochastic gravitational wave background generated by cosmic string
loops. Although our results show that friction leads to a partial suppression
of the emission of gravitational radiation by cosmic string loops, we also
find that loop production is very intense in the early stages of the Kibble
regime. We show that, in many instances, this leads to a prominent signature
of friction in the ultra-high frequency range of the spectrum, in the form of
a secondary peak. The signature of friction is not only sensitive to cosmic
string properties, but also to the initial conditions of the network and its
surroundings. A detection of this signature would then allow us to extract
information about the physics of the early universe that cannot be uncovered
when probing the rest of the stochastic gravitational wave background
spectrum.

##  I Introduction

We are swiftly moving into the era of Gravitational Wave (GW) Astronomy. The
LIGO-Virgo Collaboration has observed, over the past years, numerous compact
object coalescence events [1, 2] and very recently major millisecond pulsar
timing arrays have announced the detection of a Stochastic Gravitational Wave
Background (SGWB) [3, 4, 5, 6] in the nanohertz frequency range. Gravitational
radiation, unlike electromagnetic radiation, travels freely through spacetime
even in the very early stages of the evolution of the universe and may then
allow us to study previously undetected sources. Cosmic strings may be one
such source. The early universe is expected to have undergone a series of
symmetry-breaking phase transitions that may lead to the production of these
linelike topological defects. Cosmic strings are generally expected to survive
until the present time and, since they concentrate a significant amount of
energy and move with relativistic velocities, they may potentially leave
different observational signatures [7, 8]. These remnants of the very early
universe may then allow us to probe particle physics up to very high energies,
even beyond the reach of current and future collider experiments.

Although standard observational probes have failed to detect cosmic strings,
with the onset of GW Astronomy, we have a new promising way to probe them: the
SGWB they generate [9, 10]. The SGWB generated by cosmic string networks is
mainly sourced by the emissions of the closed loops of string that are
continuously produced in string interactions. In the initial stages of the
evolution of the network, when the universe is quite dense, strings are damped
by the frictional force caused by the frequent interactions with particles of
the surrounding plasma [11, 12, 13]. Studies of the cosmic string SGWB [9, 10,
14, 15, 16, 17] often assume that loops created in this friction-dominated era
do not provide a significant contribution to this spectrum and that GW
production only starts once the universe has rarified enough for friction to
become irrelevant for string dynamics. However, although friction should in
fact lead to a decrease of the GW emission of loops, string networks are
expected to be quite dense in the friction era, which means that many loops
should be produced. As a result, the contribution of the loops created in the
friction era may not, in fact, be negligible as is usually assumed. Here, we
study the impact of friction on the GW emission of loops and on the number of
loops produced, with the objective of computing the contribution of the loops
created in the friction era to the SGWB. We show that friction may lead to a
distinct signature in the high frequency range of the spectrum, in the form of
a secondary peak, and show that its properties are dependent on cosmic string
properties and on the properties of the background plasma. The SGWB generated
by cosmic strings may then extend further into the ultra-high frequency range
of the GW spectrum than previously anticipated, allowing us then to look
further into the early stages of the evolution of the universe.

This paper is organized as follows. In Sec. II, we introduce the Nambu-Goto
equations of motion for cosmic strings with friction and outline how these may
be used to derive a model to describe the cosmological evolution of cosmic
string networks. Then, in Sec. III, we provide a brief overview of the
computation of the SGWB genereated by cosmic string networks. In Sec. IV, we
describe the impact of friction on the evolution of a cosmic string network.
In Sec. V, we study the impact of friction on the evolution of cosmic string
loops numerically and derive an analytical approximation to describe the
evolution of the length of loops during this era. Finally in Sec. VI, we
characterise the signature of friction on the SGWB and study its dependence on
the network and background plasma properties. We discuss the results and
conclude in Sec. VII.

Throughout this paper, we will use natural units with $c=\hbar=1$, where $c$
is the speed of light in vacuum and $\hbar$ is the reduced Planck constant.
Moreover, we will use cosmological parameters determined using Planck 2018
data [18], where the values of the density parameters for radiation, matter
and dark energy at the present time are respectively given by $\Omega_{\rm
r}=9.1476\cdot 10^{-5}$, $\Omega_{\rm m}=0.308$,
$\Omega_{\Lambda}=1-\Omega_{\rm r}-\Omega_{\rm m}$ and the Hubble constant is
$H_{0}=2.13\cdot h\cdot 10^{-33}\,\rm eV$, with $h=0.678$.

##  II The evolution of cosmic string networks

The evolution of cosmic string networks is determined mostly by four main
physical processes. Strings have tension that force them to move and, in the
absence of strong damping forces, this accelerates them to relativistic
velocities. Cosmological expansion also has a significant impact throughout
their evolution: it stretches long cosmic strings, rarefies the network
density, and decelerates the strings. Also, as long strings move, they often
collide with other strings or self-intersect. When this happens, the colliding
strings exchange partners and reconnect, which may lead to the formation of
closed loops of string. These loops detach from the network and evolve
independently, thus resulting in energy loss. Finally, since cosmic string
networks are expected to form deep in the early universe, strings inevitably
interact with the particles of the background plasma in the early stages of
their evolution. As a result, strings also experience a frictional force,
caused by these interactions, which plays a key role in the early stages of
their evolution, when the universe was very dense.

###  II.1 Cosmic String Evolution with Friction

In the cosmological context, local cosmic strings may be regarded as
infinitely-thin and featureless objects. They sweep, in their movement, a
$1+1$-dimensional worldsheet in spacetime, represented by

$x^{\mu}=x^{\mu}(\sigma^{0},\sigma^{1})\,,$ (1)

where $\sigma^{0}$ and $\sigma^{1}$ are, respectively, a timelike and a
spacelike parameter of the worldsheet. Cosmic string dynamics may then be
described by the Nambu-Goto action

$S=-\mu\int\sqrt{-\gamma}d^{2}\sigma\,,$ (2)

where $\mu$ is the cosmic string mass per unit length (which, in this case,
coincides with the cosmic string tension), $\gamma$ is the determinant of the
worldsheet metric $\gamma_{ab}=g_{\mu\nu}x^{\mu}_{,a}x^{\nu}_{,b}$, with
$a,b=0,1$, and $g_{\mu\nu}$ is the spacetime metric. This action, however,
does not provide a full description of the evolution of cosmic strings in the
early universe, as it does not take the interactions with particles of the
background plasma into account. The frequent scattering of particles by cosmic
strings leads, in fact, to a frictional force per unit length of the form
[11]:

$\mathbf{F}=-\frac{\mu}{l_{f}}\frac{\mathbf{v}}{\sqrt{1-v^{2}}}\,,$ (3)

where $\mathbf{v}$ is the string velocity. Here, we also introduced the
friction length, $l_{f}$, that describes the characteristic lengthscale for
which friction plays an important role in the dynamics of cosmic strings. For
gauge strings, friction is mainly caused by Aharonov-Bohm scattering [19] and
this lengthscale assumes the form:

$l_{f}=\frac{\mu}{\beta T^{3}}\,,$ (4)

where $T$ is the background temperature and $\beta$ is a parameter that
depends on the number of particle species that interact with the string.
Including the effect of friction, the equation of motion for a cosmic string
is of the form [11]:

${x^{\nu}_{,a}}^{;a}+\Gamma^{\nu}_{\sigma\mu}x^{\sigma}_{,a}x^{\mu,a}=\frac{1}{l_{f}}\left(U^{\nu}-x^{\nu}_{,a}x^{\sigma,a}U_{\sigma}\right)\,,$
(5)

where $U^{\nu}$ is the 4-velocity of the background fluid and
${x^{\nu}_{,a}}^{;a}=\partial_{a}(\sqrt{-\gamma}\gamma^{ab}x^{\nu}_{,b})/\sqrt{-\gamma}$
is the covariant Laplacian.

In a Friedmann-Lemaitre-Robertson-Walker (FLRW) background, the line element
is given by

$ds^{2}=a(\eta)^{2}(d\eta^{2}-d\mathbf{y}\cdot d\mathbf{y})\,,$ (6)

where $a$ is the cosmological scale factor, $d\eta=dt/a$ is the conformal
time, $t$ is the physical time and $\mathbf{y}$ are cartesian coordinates, and
the $4$-velocity of radiation is given by $U^{\nu}=(a^{-1},0,0,0)$. In this
case, it is convenient to choose the temporal-transverse gauge, in which

$\sigma^{0}=\eta\quad\mbox{and}\quad\dot{\mathbf{x}}\cdot{\mathbf{x}}^{\prime}=0\,,$
(7)

where $x^{\mu}=(\eta,\mathbf{x})$ and dots and primes represent, respectively,
derivatives with respect to $\eta$ and $\sigma\equiv\sigma^{1}$. The equations
of motion for a cosmic string take then the form [20, 11]:

$\displaystyle\ddot{\mathbf{x}}+\left(2\frac{\dot{a}}{a}+\frac{a}{l_{f}}\right)\left(1-\dot{\mathbf{x}}^{2}\right)$
$\displaystyle=$
$\displaystyle\epsilon^{-1}\left(\epsilon^{-1}\mathbf{x}^{\prime}\right)^{\prime}\,,$
(8)
$\displaystyle\dot{\epsilon}+\left(2\frac{\dot{a}}{a}+\frac{a}{l_{f}}\right)\dot{\mathbf{x}}^{2}\epsilon$
$\displaystyle=$ $\displaystyle 0\,,$ (9)

where $\epsilon^{2}=\mathbf{x}^{\prime 2}/(1-\dot{\mathbf{x}}^{2})$.

###  II.2 Velocity-dependent One-Scale model

The cosmological evolution of a cosmic string network may be described, on
sufficiently large scales, using the Velocity-dependent One-Scale (VOS) model
[13, 21]. This model provides a quantitative description of the cosmological
evolution of cosmic string networks by following the evolution of two
variables: the Root-Mean-Squared (RMS) velocity $\bar{v}$ of the network,
defined as

$\bar{v}^{2}\equiv\langle\dot{\mathbf{x}}^{2}\rangle=\frac{\int\dot{\mathbf{x}}^{2}\epsilon
d\sigma}{\int\epsilon d\sigma}\,,$ (10)

and its characteristic lengthscale $L$, defined in terms of the energy density
of long strings $\rho$ as

$\rho=\mu/L^{2}\,,$ (11)

where $E=\mu a(\eta)\int\epsilon d\sigma$ is the energy of cosmic strings and
$\rho\propto Ea^{-3}$. For standard strings, without internal degrees of
freedom, $L$ also measures roughly the average distance between long strings.
The evolution equations for $\bar{v}$ and $L$ may then be found by averaging
Eqs. (8) and (9) respectively111Note however that, in their derivation, it is
assumed that
$\langle\dot{\mathbf{x}}^{4}\rangle=\langle\dot{\mathbf{x}}^{2}\rangle^{2}=\bar{v}^{4}$.
and by including a phenomenological term to account for the impact of
collisions and interactions between cosmic strings. These interactions lead,
as previously explained, to the production of cosmic string loops that detach
from the Hubble flow and that are expected to decay and evaporate. This then
results in an energy loss of the form [22]:

$\left.\frac{d\rho}{dt}\right|_{loops}={\tilde{c}}\bar{v}\frac{\rho}{L}\,,$
(12)

where ${\tilde{c}}=0.23\pm 0.04$ [21] is a phenomenological parameter that
quantifies the efficiency of loop chopping. By using Eqs. (8)-(12), one finds
that the evolution of a cosmic string network on cosmological scales may be
described by [13, 21]:

$\displaystyle\frac{d\bar{v}}{dt}$ $\displaystyle=$
$\displaystyle(1-\bar{v}^{2})\left[\frac{k(\bar{v})}{L}-\bar{v}\left(2H+\frac{1}{l_{f}}\right)\right]\,,$
(13) $\displaystyle 2\frac{dL}{dt}$ $\displaystyle=$ $\displaystyle
2HL(1+\bar{v}^{2})+\frac{L\bar{v}^{2}}{l_{f}}+{\tilde{c}}\bar{v}\,,$ (14)

where $k(\bar{v})$ is a phenomenological parameter that accounts, to some
extent, for the effects of small-scale structure on long strings (here, we
will use the ansatz proposed in Ref. [21]), and $H=da/dt/a$ is the Hubble
parameter. These equations enable us to describe quantitatively the
cosmological evolution of cosmic string networks from early to late
cosmological times. We briefly outline this evolution, with particular focus
on the impact of friction in early cosmological times, in Sec. IV.

##  III The Stochastic Gravitational Wave Background generated by cosmic
string networks

After cosmic string loops are created, they detach from the Hubble flow and
start to evolve under the effect of their tension. As a result, they oscillate
periodically with relativistic velocities and they are then generally expected
to decay by emitting GWs222The potential role of the emission of scalar and
gauge radiation is currently still a matter of debate [23, 24, 25, 26].
Abelian-Higgs simulations indicate that this radiation may lead to the fast
decay of the majority of loops [23] and that less than $10\%$ of the loops
produced would decay by emitting GWs [24]. It is argued in [25, 26], however,
that this may be a transient phenomenon.. The superimposition of the GW bursts
generated by the copious amounts of loops that are created throughout the
evolution of the cosmic string network gives rise to a Stochastic
Gravitational Wave Background (SGWB) [27, 9, 10]. This SGWB is generally
characterized by the spectral energy density of GWs,

${\Omega}_{\rm gw}(f)=\frac{1}{\rho_{c}}\frac{d{\rho}_{\rm gw}}{d\log f}\,,$
(15)

where ${\rho}_{\rm gw}$ is the energy density of gravitational radiation,
$\rho_{c}=3H_{0}^{2}/(8\pi G)$ is the critical density of the universe at the
present time $t_{0}$ (for the remainder of this paper, the subscript ‘$0$’ is
used to refer to the value of the corresponding quantity at $t=t_{0}$).

The frequency of the GWs emitted by cosmic string loops is determined by
harmonics of their length $\ell$ at the time of emission $t$. They then arrive
to an observer at the present time $t_{0}$ with a frequency

$f_{j}=\frac{2j}{\ell}\frac{a(t)}{a_{0}}\,,$ (16)

where $j$ is the harmonic mode of emission and the subscript ‘$j$’ is used to
label the contribution of the $j$-th mode of emission to the corresponding
quantity. The spectrum of emission of cosmic string loops follows roughly a
power law [28, 29, 30, 31, 32] of the form

$\frac{dE_{j}}{dt}=\Gamma_{j}G\mu^{2}\,,\quad\mbox{with}\quad\Gamma_{j}=\frac{\Gamma}{\zeta(q)}j^{-q}\,,$
(17)

where $E=\mu\ell$ is the energy of the loops, $\zeta(q)$ is the Riemann Zeta
function and $\Gamma\sim 50$ [33, 28, 34, 35, 30, 32] is the GW emission
efficiency 333Note that these studies only apply to cosmic strings without
internal degrees of freedom. If strings carry currents, for instance, $\Gamma$
should be suppressed and the spectrum of emission may be altered [36].. The
spectral index $q$ depends on the small-scale structure present in the cosmic
string loops. Loops are generally expected to have points that move
instantaneously at the speed of light, known as cusps, that give rise to a
spectrum characterized by $q=4/3$. Moreover, collisions between strings lead
to discontinuities in the string that travel along the string at the speed of
light. These kinks generate a spectrum characterized by $q=5/3$ and, when two
kinks collide, by $q=2$.

The amplitude of the SGWB generated by cosmic string loops is then given by
(see e.g. [8]):

${\Omega}_{\rm gw}(f)=\sum_{j=1}^{+\infty}\Gamma_{j}{\Omega}_{\rm
gw}^{j}(f)\,,$ (18)

where

$\displaystyle{\Omega}_{\rm
gw}^{j}(f)=\frac{16\pi}{3f}\left(\frac{G\mu}{H_{0}}\right)^{2}\int_{t_{i}}^{t_{0}}j\,n\left(\ell_{j}(t),t\right)\left(\frac{a(t)}{a_{0}}\right)^{5}dt$

is the contribution of the $j$-th harmonic mode of emission and
$\ell_{j}=2ja(t)/(fa_{0})$ is the length of loops that radiate, in the $j$-th
harmonic, GWs that have a frequency $f$ at the present time. Moreover, $t_{i}$
is the time in which significant emission of gravitational radiation by cosmic
string loops starts, which is often assumed to be the end of the friction-
dominated era. In this paper, we will study the validity of this assumption
and examine whether there are signatures of friction on the SGWB. We will then
generally assume that $t_{i}$ coincides with the time of creation of the
network $t_{c}$. We will also consider the contribution of the fundamental
mode of emission (and drop the superscript ‘$1$’ that denotes this mode). Note
however that it is straightforward to compute the contribution of any mode of
emission once that of the fundamental mode is known:

${\Omega}_{\rm gw}^{j}(jf)={\Omega}_{\rm gw}^{1}(f)\,,$ (20)

and thus considering $j=1$ is sufficient to fully characterize the SGWB.

The crucial ingredient to compute the SGWB generated by cosmic string loops is
then the loop distribution function, $n(\ell,t)d\ell$, which describes the
number density of string loops with lengths between $\ell$ and $\ell+d\ell$
that exist at the time $t$:

$n(\ell,t)=\int_{t_{i}}^{t}dt_{b}f(\ell_{b},t_{b})\left(\frac{a(t_{b})}{a(t)}\right)^{3}\,,$
(21)

where $\ell_{b}$ is the length of the loop at the time of birth $t_{b}$. Here
$f(\ell,t)d\ell$ is the loop production function, which represents the number
density of loops with lengths between $\ell$ and $\ell+d\ell$ produced per
unit time. The amount of energy that is lost as a result of loop production
may be inferred from the large-scale dynamics of the cosmic string network
using Eq. (12). If one assumes that the length of loops at the moment of
creation is proportional to the characteristic length of the long string
network at that instant of time,

$\ell(t_{b})=\alpha L(t_{b})\,,$ (22)

where $\alpha<1$ is a constant loop-size parameter, we then have that [16, 17,
37]:

$f(\ell,t)=\frac{{\tilde{c}}}{\sqrt{2}\ell}\frac{\bar{v}(t)}{L(t)^{3}}\delta(\ell-\alpha
L)\,,$ (23)

where the factor of $\sqrt{2}$ was introduced to account for the effect of the
peculiar velocities of loops [8]. Note that, although the assumption that all
loops are created with the same length seems rather strong, this form of the
loop production function may be used to construct the loop production function
for any distribution of loop lengths at birth [15, 37]. Moreover, the impact
of the length of loops following a distribution at the moment of creation on
the SGWB may also, to some extent, be described by including a factor
$\mathcal{F}<1$ in Eq. (22) [17, 37]. Such a factor could also account for the
possibility that not all loops decay by emitting GWs. Here, for simplicity, we
take $\mathcal{F}=1$. We then have that the loop distribution function is
given by [16]:

$\displaystyle
n(\ell,t)=\frac{dt_{b}}{dl_{b}}\frac{{\tilde{c}}\bar{v}\left(t_{b}\right)}{\sqrt{2}\alpha
L\left(t_{b}\right)^{4}}\left(\frac{a\left(t_{b}\right)}{a\left(t\right)}\right)^{3}\,.$
(24)

##  IV Impact of friction on the evolution of cosmic string networks

Cosmic string networks are expected to form in the early universe at a
critical temperature $T_{c}$ that determines their mass per unit length:
$\mu\sim T_{c}^{2}$ [38]. The time of string formation is then roughly given
by:

$t_{c}=\frac{1}{\chi(t_{c})}\frac{t_{pl}}{G\mu},\quad\mbox{with}\quad\chi(t)=4\pi\left(\frac{\pi
g_{*}(t)}{45}\right)^{1/2}\,,$ (25)

where $t_{pl}=G^{1/2}$ is the Planck time and $g_{*}(t)$ is the effective
number of massless degrees of freedom (see e.g. [39]). The friction epoch
happens, in general, long before the first change $g_{*}(t)$ predicted by the
Standard Model of Particle Physics (which is assumed throughout our
calculations). For the remainder of this paper, we will then assume that
$\chi(t)=\chi(t_{c})\equiv\chi$ throughout the friction era.

Cosmic string networks, however, are generally expected to survive throughout
cosmic history and, depending on what damping mechanism dominates their
dynamics, they can go through different scaling regimes in their evolution. In
the early universe, interactions between cosmic strings and the particles of
the surrounding plasma should be quite frequent and, as a result, their
dynamics should be dominated by friction. In this case, string movement is
heavily damped and they are expected to move with non-relativistic velocities
and, as can be inferred from Eq. (13), their velocity is roughly given by
$\bar{v}\simeq k_{c}l_{f}/L$, where $k_{c}\equiv k(0)=2\sqrt{2}/\pi$. During
this friction-dominated era, two different scaling regimes may emerge. The
first is the Stretching Regime, during which strings are frozen in comoving
coordinates and stretched by expansion:

$L\propto a\quad\mbox{and}\quad\bar{v}\propto a^{2}\,.$ (26)

Assuming that the friction-dominated epoch occurs deep in the radiation era
(which, as we shall see, should in general be the case) we should have that,
during this regime,

$L=L_{c}\left(\frac{t}{t_{c}}\right)^{1/2},\quad\mbox{and}\quad\bar{v}=\bar{v}_{c}\left(\frac{t}{t_{c}}\right)\,,$
(27)

where $L_{c}$ and $\bar{v}_{c}=k_{c}l_{f}(t_{c})/L_{c}$ are the initial
characteristic lengthscale and the RMS velocity, respectively. Note that, in
general, we should have that $l_{f}(t_{c})<L_{c}<t_{c}$ [13].

During the Stretching regime, the movement of strings is so damped by friction
that there are almost no interactions between strings. However, as the
universe expands and cools, the friction lengthscale $l_{f}$ grows much faster
than the characteristic length $L$ (cf. Eq. (4) and (27)) and interactions
become increasingly relevant. As the Hubble damping and friction terms in Eq.
(14) become comparable (i.e., when $l_{f}/L\sim HL$), the Kibble Regime,
characterized by

$L\propto\left(\frac{l_{f}}{H}\right)^{1/2}\quad\mbox{and}\quad\bar{v}\propto\left(l_{f}H\right)^{1/2}\,,$
(28)

emerges. In this regime, a considerable amount of energy is lost as a result
of interactions, since $HL\sim{\tilde{c}}\bar{v}$, which explains why the
network evolves differently. Loop production is not negligible during the
Kibble regime. As a matter of fact, as we will show, it is quite significant,
since networks are expected to be quite dense during this stage.

More precisely, the Kibble regime should emerge when the following condition
is satisfied

$AHL=\frac{l_{f}}{2L}k_{c}(k_{c}+{\tilde{c}})\,,$ (29)

where $A$ is a constant of order unity that was introduced to obtain a better
fit to the numerical evolution. During the radiation era, we may express the
friction lengthscale as [13]:

$l_{f}=\frac{1}{\theta}\frac{t^{3/2}}{t_{c}^{1/2}}\,,\quad\mbox{with}\quad\theta=\frac{\beta}{\chi\sqrt{G\mu}}\,.$
(30)

We then have that, during the Kibble regime,

$\displaystyle\frac{L}{t_{c}}$ $\displaystyle=$
$\displaystyle\left(\frac{k_{c}(k_{c}+{\tilde{c}})}{\theta
A}\right)^{1/2}\left(\frac{t}{t_{c}}\right)^{5/4}\,,$ (31)
$\displaystyle\bar{v}$ $\displaystyle=$ $\displaystyle
k_{c}\left(\frac{A}{\theta
k_{c}(k_{c}+{\tilde{c}})}\right)^{1/2}\left(\frac{t}{t_{c}}\right)^{1/4}\,.$
(32)

These equations may be confronted with the full numerical evolution during the
Kibble regime — described by Eqs. (13) and (14) — to determine the value of
the constant $A$ in Eq. (29). Our analysis indicates that $A=3/2$ provides an
excellent fit for the relevant range of initial conditions and for different
values of $\beta$.

To estimate the time in which the network enters the Kibble regime, we may
assume that the transition between the Stretching and Kibble regimes happens
suddenly at a time $t_{k}$, corresponding to the time in which the condition
in Eq. (29) is first satisfied. Assuming that the network is in the Stretching
regime characterized by Eq. (27) for $t\leqslant t_{k}$, we have that

$t_{k}=\left(\frac{3\theta}{2k_{c}(k_{c}+{\tilde{c}})}\right)^{\frac{2}{3}}\left(\frac{L_{c}}{t_{c}}\right)^{\frac{4}{3}}t_{c}\,.$
(33)

This expression shows that the time of emergence of the Kibble regime depends
on the initial conditions and thus so does the duration of the Stretching
regime. As a matter of fact, the network only goes through the Stretching
regime if the conditions of existence of the Kibble regime are not met
initially (or, in other words, provided that the initial characteristic length
is not significantly smaller than the horizon). However, the cosmic string
network necessarily goes through the Kibble regime in the early universe,
either following the Stretching regime or right after creation if its initial
density is large enough. This may be seen clearly in Fig. 1, where the
evolution of $L/t$ and $\bar{v}$ are plotted for different initial conditions.
Therein, one may see that, if the initial characteristic length of the network
is smaller, the Kibble regime starts earlier and the duration of the
Stretching regime decreases.

Figure 1: Evolution of a cosmic string network with friction for
$G\mu=10^{-12}$. The top panel displays the characteristic length divided by
the physical time and we plot the RMS velocity on the bottom panel. The solid
lines of different colours correspond to different initial conditions. The
vertical dashed line represents the value of the scale factor at the end of
the friction dominated epoch, $a_{f}$.

The friction-dominated era of cosmic string evolution is, however, necessarily
transient. As the friction lengthscale continues to increase rapidly, it will
eventually become larger than the characteristic length of the network and the
impact of friction on the dynamics of cosmic string networks will become
negligible. Assuming as before that this transition to the frictionless epoch
happens suddenly at a time $t_{f}$ in which the Hubble damping term $2H$
becomes comparable to the friction damping term $l_{f}^{-1}$, we have that:

$t_{f}=\theta^{2}t_{c}\,.$ (34)

Using this one may show that, unless cosmic string tension is very small,
string dynamics should become frictionless before any variation of $\chi(t)$
(according to the Standard Model of Particle Physics) and well before the
radiation-matter transition, which justifies our assumption that friction
happens deep in the radiation era.

After $t_{f}$, when friction becomes negligible to cosmic string dynamics, the
movement of strings is no longer so heavily damped and the network evolves
towards a Linear scaling regime characterized by:

$L=\xi t,\;\;\frac{d\bar{v}}{dt}=0\,,$ (35)

with

$\xi^{2}=\frac{k(k+{\tilde{c}})}{4\nu(1-\nu)},\;\;\bar{v}^{2}=\frac{k}{k+{\tilde{c}}}\frac{1-\nu}{\nu}\,.$
(36)

This regime is an attractor solution of the VOS equations when $a\propto
t^{\nu}$, with $0<\nu<1$, and its existence has been verified in numerical
simulations. In this regime, cosmic strings reach relativistic velocities and,
as a result, interactions between strings and loop production are quite
frequent. As Fig. 1 illustrates, the evolution of the cosmic string network
during the frictionless epoch, for $t>t_{f}$, does not depend on the initial
conditions. Note that the cosmic string network is temporarily ‘‘knocked out’’
of the Linear Scaling regime as a result of the decrease in the effective
number of massless degrees of freedom, as the universe cools. This is the
cause of the small bumps seen, during the radiation-era Linear scaling regime,
in Fig. 1 (these are more evident in the plot for $L$ due to the scale of the
plot, but also show up in the plot for $\bar{v}$ in the form of a small
temporary decrease). Also, later the universe transitions to the matter-
dominated era and the network starts to evolve towards a new Linear Scaling
regime. However, this regime is not reached before the present time due to the
emergence of dark energy. Once this happens, cosmic string movement is heavily
damped by the fast expansion, which leads to a new stretching regime of the
form $L\propto a$ and ${\bar{v}}\to 0$. In this case, the Stretching regime is
sustainable because the accelerated expansion ensures that the network remains
frozen, with non-relativistic velocities.

##  V Impact of friction on the evolution of cosmic string loops

To include the contribution of the loops created during the friction-dominated
regime in the computation of the SGWB, we need to start by studying the
evolution of a cosmic string loop in the presence of friction. Although this
has been studied previously in [40], therein the impact on gravitational wave
emission — which is crucial for our study — was not considered. Here, we
follow a similar approach and consider a circular loop with a coordinate
radius $q$:

$\mathbf{x}=q(\eta)(\sin\phi,\cos\phi,0)\,,$ (37)

and choose the spacelike worldsheet parameter to be $\phi$. In a FLRW
universe, the physical length of the loop $\ell$ — defined such that
$E=\mu\ell$, where $E$ is the energy of the loop — is given by

$\ell=2\pi\gamma|q|a\,,$ (38)

where $v=\dot{q}$ is the velocity of the loop and $\gamma=(1-v^{2})^{-1/2}$.

Cosmic string loops are usually assumed to emit GWs at a roughly constant rate
given by

$\left.\frac{d\ell}{dt}\right|_{\rm gw}=\Gamma G\mu\,,$ (39)

with $\Gamma=50$ (as may be seen from Eq. (17), by summing over all the
harmonic modes of emission). However, more generally, the energy loss caused
by the emission of gravitational radiation, which may be estimated using the
quadrupole formula, may be written in the following form [13]:

$\frac{d\ell}{dt}\bigg{|}_{GW}=-\Gamma^{\prime}G\mu v^{6}\,,$ (40)

where $\Gamma^{\prime}$ is a parameter evaluating the efficiency of GW
emission. The approximation in Eq. (39), in fact, only describes the decrease
of the length of loops caused by the emission of gravitational radiation in an
averaged sense. Cosmic string loops, in the absence of friction and provided
that expansion has a negligible impact, oscillate periodically with an average
squared velocity of $\langle v^{2}\rangle=1/2$. In these oscillations, they
reach ultra-relativistic velocities periodically and they should, in fact,
emit GWs dominantly in the stages of the oscillation in which their velocities
are significant. In general, for these freely oscillating loops, assuming a
constant rate of GW emission provides an adequate description of their
evolution on time-scales much larger than their oscillation period. Here,
since we are including the impact of friction — which damps the movement of
loops and may significantly delay their oscillations [40] —, we use the form
in Eq. (40) to account for the possibility that the rate of GW emission can no
longer be assumed to be constant. As proposed in [13], for consistency, given
the fact that for freely-oscillating loops $\langle v^{6}\rangle=5/16$, we
take $\Gamma^{\prime}=16/5\cdot\Gamma=160$ 444In Ref. [13], the authors take
$\Gamma^{\prime}=8\Gamma$ since, by mistake, they assume that $\langle
v^{6}\rangle=\langle v^{2}\rangle^{3}$..

By introducing Eq. (37) into Eqs. (8) and (9) and using Eqs. (38) and (40),
one finds that the equation of motion for $q$ is given by:

$\ddot{q}=(1-\dot{q}^{2})\left(-2Ha\dot{q}-\frac{a\dot{q}}{l_{f}}-\frac{1}{q}-\frac{1}{2\pi|q|}\sqrt{1-\dot{q}^{2}}\Gamma^{\prime}G\mu\dot{q}^{5}\right)\,.$
(41)

To study the impact of friction on the evolution of cosmic string loops and on
their GW emission, we solved this equation numerically, with initial
conditions of the form

$q_{b}=\frac{\alpha L(t_{b})}{2\pi a_{b}}\quad\mbox{and}\quad
v_{b}=\dot{q}_{b}=0\,,$ (42)

for a wide range of the loop-size parameter $\alpha$ and of the birth time of
loops $t_{b}$. This choice of initial conditions is natural since it is well
known that a free circular loop will oscillate with amplitude $q_{b}$ having
vanishing velocity when the loop radius is at a maximum. This extensive study
was done with the objective of determining which physical processes play an
important role in the evolution of subhorizon loops — the main contributors to
the SGWB — and to aid us in the development of an analytical approximation to
the evolution of $\ell$.

Our results have shown that for very large loops (with lengths larger or
comparable to the horizon), the expansion of the background stretches loops
significantly thereby increasing their energy — and particularly so in the
presence of friction as was also found in [41, 12] — but small enough loops do
not feel its effects even when friction is present. Here, since we are
considering sub-horizon loops, we will then neglect the expansion term in Eq.
(41), as is usually done in the absence of friction. Notice however that the
impact of expansion on the evolution of the friction lengthscale cannot be
neglected, as it is determined by the density of the background plasma which
is strongly dependent on the expansion.

We have also verified numerically that, for sub-horizon loops, even deep in
the friction-dominated regime, we have $\langle v^{2}\rangle\approx 1/2$ as
for free loops evolving in the absence of expansion, friction and GW emission.
One may then conclude that, during the friction era, sub-horizon loops also
emit GWs at a roughly constant rate, with $\Gamma=50$, and, in fact, we have
verified numerically that the average energy lost in the form of gravitational
radiation in a period of oscillation computed using Eq. (40) is well described
by the simpler expression in Eq. (39).

Finally, our results also show that the dynamics of any loop will eventually
become frictionless. This may be explained by the fact that $\ell$ decreases
over time as a result of the energy loss caused by friction and by the
emission of gravitational radiation, while the friction lengthscale increases
(see Eq. (4)). Thus as time goes by, the impact of friction becomes smaller
and smaller until loop dynamics becomes frictionless and, once this stage is
reached, the rest of the energy of the loop will be radiated in the form of
GWs. Notice that the later in the friction era a loop is created, the sooner
this frictionless regime will be reached and then an increasingly larger
fraction of the energy of loops will be converted into GWs as time ellapses.
For instance, for a loop with $G\mu=10^{-12}$ and $\alpha=0.1$ created deep in
the friction era ($t_{b}=10^{-5}\cdot t_{f}$) only about $0.2\,\%$ of its
energy will remain by the time this frictionless stage is reached. However,
closer to the end of the friction epoch (for $t_{b}=10^{-2}\cdot t_{f}$),
without the inclusion of GW emission, the loop still has about $56\,\%$ of its
initial energy at this stage. This shows that, in fact, there is GW emission
throughout the friction regime.

To calculate the number density of loops with a length $\ell$ at a time $t$,
we need to find the time of formation of these loops, which means that we need
to follow their evolution since the time of birth. However, for the physically
relevant values $\alpha$ and $G\mu$, loops lose energy very slowly during most
of the evolution and undergo a very large number of oscillations before
evaporating. As a result, the numerical computation of their evolution is time
costly. It is then convenient to find an analytical approximation to describe
the evolution of cosmic string loops, including the effects of friction and GW
emission, to make the computation of the SGWB more efficient. Neglecting, as
suggested by our analysis, the Hubble damping term in Eq (41) and assuming
that GW emission happens with the constant rate given by Eq. (39), the
evolution of $\ell$ should then be well described by

$\frac{d\ell}{dt}=-\frac{\ell v^{2}}{l_{f}}-\Gamma G\mu\,,$ (43)

as may be found by combining Eqs. (38) and (41) under these assumptions. Let
us start by considering the case of a loop evolving only under the effect of
friction and curvature and drop the GW emission term in Eq. (43). To find an
analytical approximation to describe the evolution of $\ell$ on time scales
much larger than a period of oscillation, we may then average this expression
over one period of oscillation and assume that $\langle v^{2}\rangle=1/2$. We
then find that

$\ell=\ell_{b}exp\left[t_{f}^{1/2}\left(t^{-1/2}-t_{b}^{-1/2}\right)\right]\,.$
(44)

This expression is equivalent to the expression found in [40] and also
supports the observation that the effect of friction will eventually become
negligible. Indeed, when $t\to\infty$, we see that $\ell$ does not vanish. The
loop then does not evaporate completely as a result of friction and at least a
fraction of $\exp\left(-t_{f}^{1/2}/t_{b}^{1/2}\right)$ of the initial length
of the loop should be converted into GWs.

Introducing the energy loss caused by the emission of gravitational radiation
in Eq. (44), we have:

$\ell=\ell_{b}exp\left[t_{b}^{1/2}\left(t^{-1/2}-t_{b}^{-1/2}\right)\right]-\Gamma
G\mu\left(t-t_{b}\right)\,.$ (45)

We have compared this approximation extensively to the full numerical
evolution described by Eq. (41), including the curvature, friction and GW
radiation terms in their complete form. Our results show that this
approximation provides a good description for the evolution of the length of
the loops produced during the Kibble regime — which is enough since the
contribution of the loops created during the Stretching regime to the SGWB is
negligible (as we shall see later). In Fig. 2, we display some examples of the
evolution of the length of the loops throughout their lifetime (computed
numerically) alongside the analytical approximation derived here. We plot
$(\ell_{b}-\ell)/\ell_{b}$ as a function of $(t-t_{b})/(t_{ev}-t_{b})$, where
$t_{ev}$ denotes the moment of evaporation of the loop, since this choice
allows us to reflect all stages of their evolution and to plot several cases
on the same figure. We display loops with $\alpha=10^{-3}$ for four different
values of cosmic string tension so that, as tension decreases, the impact of
friction becomes stronger. Since the period of oscillation of loops is
proportional to their length and, for small values of $G\mu$, they lose energy
very slowly, the numerical study of the later stages of their evolution is
computationally costly. For this reason, for illustration purposes, we opted
to display the evolution of loops with unphysically large tensions too
($G\mu=10^{-3}$ and $G\mu=10^{-6}$) as this allows us to follow the evolution
of loops numerically until $t_{ev}$ in the case of strong GW emission. The
slope of the tangent vector of each curve at the time of evaporation
characterises the fraction of energy which was lost in the form of GWs,
reaching its maximum when GW emission is at its strongest for the largest
$G\mu$. This figure clearly shows that our approximation provides an excellent
description of the evolution of a loop when friction is strong (the beginning
of the evolution for the lowest tension) — which demonstrates the validity of
our assumption that $\langle v^{2}\rangle\approx 1/2$ — and when its effects
are negligible and the evolution is determined by the emission of GWs instead
(high tensions). When the impacts of friction and GW emission becomes
comparable, however, our approximation predicts a slightly faster decay of the
loops, which results in an underestimation of the time of evaporation of the
loop. Note however that this does not have a significant impact on the quality
of the results we will derive since assuming a faster decay leads, in general,
to smaller contributions to the SGWB. These results are then conservative and
may then be seen as a lower bound on the amplitude of the spectrum.

Figure 2: Evolution of loops with $\alpha=10^{-3}$ and $t_{b}=t_{k}$ in the
presence of friction for different values of cosmic string tension $G\mu$.
Here, the solid lines correspond to the evolution predicted by the analytical
approximation in Eq. (45), while the dashed lines represent that computed
numerically by solving Eq. (41).

##  VI The signature of friction on the SGWB

It is usually assumed in the literature that the loops created during the
friction-dominated epoch do not provide a significant contribution to the
SGWB, since their motion would be effectively damped by friction. However,
during this era, the network is very dense, which makes intersections and,
consequently, the creation of loops more probable. Also, as we have seen,
although friction is effective in the initial stages, the evolution of loops
eventually becomes frictionless and, from this point on, their energy is lost
only in the form of gravitational radiation. The combination of these two
effects may result in a significant contribution to the SGWB.

In this section, we fully characterize the signature of friction on the SGWB,
which involves a detailed computation of the loop distribution function during
this era. Notice that the impact of friction on $n(\ell,t)$ is twofold. On the
one hand, friction, by significantly affecting the large-scale evolution of
the cosmic string network — so much so that it can no longer be assumed to
evolve in a linear scaling regime — affects the loop production function.
Here, we use the method introduced in [16] (and summarized in Sec. III) and
solve the VOS Eqs. (13) and (14), coupled with the Friedmann equation, to
fully characterize $f(\ell,t)$ throughout the evolution of the cosmic string
network. Our results show that indeed there is an intensive loop production
during the friction epoch. For instance, for $G\mu=10^{-12}$ the number of
loops produced at the beginning of the Kibble regime is $400$ times larger
than what one would have if the network was in a linear scaling regime. On the
other hand, $n(\ell,t)$ is also affected by the change in the rate of energy
loss of the loops caused by this additional decay mechanism, which may reduce
the lifetime of the loops significantly. Here, we use the analytical
approximation in Eq. (45) to find the Jacobian $dt_{b}/d\ell_{b}$ needed to
fully characterize the loop distribution function in Eq. (24):

$\frac{dt_{b}}{d\ell_{b}}=\left(\alpha\frac{dL}{dt}\bigg{|}_{t_{b}}+\frac{\alpha
L(t_{b})}{2t_{b}^{3/2}}t_{f}^{1/2}+\Gamma G\mu\right)^{-1}$ (46)

and to find the time of birth $t_{b}$ of the loops that contribute to the
given frequency.

###  VI.1 The signature of friction

Figure 3: The Stochastic Gravitational Wave Background generated throughout
the full evolution of a cosmic string network with $G\mu=10^{-10}$,
$\alpha=10^{-9}$, $\beta=1$, $L_{c}=t_{c}$. Here, the solid line represents
the spectra including the contribution of the loops created during the
friction-dominated era of cosmic string network evolution, while the dashed
line corresponds to the spectra one obtains by assuming that significant GW
emission by cosmic string loops starts only at the beginning of the
frictionless regime.

We now have all the ingredients to characterize the signature of friction on
the SGWB. Since the friction-dominated era of cosmic string dynamics happens
right after the formation of the network, and thus in the very early universe,
this will result in a contribution in the ultra-high frequency range. As a
matter of fact, as illustrated by Fig. 3, the inclusion of the GWs emitted by
the loops that are produced during the friction era extends the range of the
spectrum significantly, typically leading to a prominent bump at very high
frequencies that may be regarded as a secondary peak. Since friction is often
assumed to completely suppress the emission of loops, it is generally assumed
that cosmic string networks start contributing to the SGWB in the end of the
friction era (at $t=t_{f}$). As a result, a cut-off is introduced artificially
in the high-frequency range of the spectrum and this secondary peak is
neglected. However, as the results of Sec. V show, the GW emission of loops
during the friction era, although partially reduced, is not completely
suppressed (especially during the end stages of the life of the loops and/or
towards the end of this era). This partial reduction of the GW emission in
many instances does not lead to a complete suppression of the SGWB, since a
large number of loops may be created during the friction era. This is well
illustrated by Fig. 4, where we plot the SGWB sourced during the friction era
with and without the inclusion of the suppression of GW emission caused by
friction (the latter case was also briefly discussed in [42]). Therein one may
see that, although this suppression may indeed lead to a significant reduction
of the amplitude of the friction signature on the SGWB, the increase in the
number of loops may be so significant that this signature may still survive.

Figure 4: The impact of the inclusion of friction in the evolution of cosmic
string loops on the signature of friction. The solid line represents the full
SGWB generated by a cosmic string network during the friction era for
$G\mu=10^{-10}$, $\alpha=10^{-8}$, $\beta=1$, $L_{c}=t_{c}$. The dashed line
corresponds to the SGWB one would obtain by completely neglecting the
contribution of the loops created during the friction-dominated era of cosmic
string network evolution, while the dash-dotted line represents that one would
obtain if the impact of friction on cosmic string loops was not included.
Figure 5: The signature of friction on the SGWB generated by a cosmic string
network with $G\mu=10^{-10}$, $\alpha=10^{-9}$, $\beta=1$, $L_{c}=t_{c}$. The
shaded areas with different colors correspond to the contributions to the
spectrum generated by loops that where produced before different instants of
time in the friction-dominated era. Here, $\Delta t=t_{f}-t_{k}$.

The signature of friction is, as shown in Fig. 5, generated mostly in the
Kibble regime and, in fact, the contribution of the loops created in the
Stretching regime may be considered generally negligible. During the latter,
cosmic strings are so effectively damped that their velocities are extremely
small, which makes collisions and intercommutations unlikely. In the beginning
of the Kibble regime, however, not only the RMS velocity is somewhat larger
but the networks are also denser (in comoving coordinates), resulting in loops
being profusely produced. Because of that, the dominant contribution to the
peak generated by friction is, in fact, sourced in the earlier stages of the
Kibble regime — when the number of loops produced per unit volume is quite
large and friction is still quite strong — and not when friction is at its
weakest. This shows that the number of loops produced may, in some instances,
be the determinant aspect in having a signature of friction and not how strong
the suppression of GW emission is.

###  VI.2 Impact of cosmic string parameters on the signature of friction

As we have seen, a clear signature of friction is expected if the loops
created in the early stages of the Kibble regime, when the network is very
dense and copious amounts of loops are created, lose a considerable amount of
energy in the form of gravitational radiation. This signature should then
depend not only on the size of loops, parameterized by $\alpha$, but also on
the large-scale properties of the cosmic string network during this regime,
which are determined by cosmic string tension, the initial characteristic
length, and the strength of friction (characterized by $\beta$). In this
section, we investigate these dependencies in detail. Since the friction-
dominated era happens early in the evolution of the cosmic string network
(and, in fact, quite early in cosmic history) and the characteristic
lengthscale may be quite small at this stage, some care must be taken to avoid
considering unphysical scenarios. Here, we only consider the potential
emissions of the loops that are created with lengths that are larger than the
Planck length, $l_{pl}$, and assume that GW emission starts only once the
gravitational backreaction scale is well defined, with $\Gamma G\mu L>l_{pl}$.

Figure 6: The SGWB generated by cosmic string networks with $G\mu=10^{-10}$,
$\beta=1$, $L_{c}=t_{c}$ for different values of $\alpha$. The solid lines
represent spectra with friction, while the dashed lines correspond to spectra
without friction. Figure 7: The SGWB generated by cosmic string networks with
$\alpha=10^{-10}$, $\beta=1$, $L_{c}=t_{c}$ for different values of $G\mu$.
The solid lines represent spectra with friction, while the dashed lines
correspond to spectra without friction. Figure 8: The SGWB generated by cosmic
string networks with $G\mu=10^{-14}$, $\alpha=10^{-12}$, $L_{c}=t_{c}$ for
different values of $\beta$. The solid lines represent spectra with friction,
while the dashed lines correspond to spectra without friction. Figure 9: The
SGWB generated by cosmic string networks with $G\mu=10^{-14}$,
$\alpha=10^{-12}$, $\beta=1$ for different values of $L_{c}$. The solid lines
represent spectra with friction, while the dashed line correspond to spectra
without friction.

Let us start by considering the impact of the loop-size parameter $\alpha$ on
the signature of friction. As previously explained, as loops evolve and their
length decreases, they eventually reach a regime in which their dynamics
becomes essentially frictionless. This happens, roughly, once their length
becomes smaller than the friction lengthscale and, therefore, loops with a
smaller $\alpha$ will reach this regime earlier and consequently lose a larger
portion of their energy in the form of GWs. This generally results in a
stronger signature of friction as $\alpha$ decreases, as is clearly shown in
Fig. 6, and this signature is at its strongest in the small-loop regime, with
$\alpha\ll\Gamma G\mu$. In this case, in fact, GW emission leads to the
evaporation of the loops effectively immediately on cosmological time scales
[43] and, as a result, they are barely affected by friction 555Note that, in
the small-loop regime, if $\alpha$ is decreased further, the shape and
amplitude of the spectrum is no longer affected and the spectrum is merely
shifted towards higher frequencies (see e.g. [16, 43]). Recall, however, that
loops cannot be made to be arbitrarily small.. The secondary peak caused by
friction is then only present for small enough loops, but how small these
loops actually need to be also depends, as we shall see, on the other
parameters of the model. In this particular case, the peak starts to form only
when loop size approaches the gravitational backreaction scale. However, for
larger loops, there is also a signature of friction, although more subtle: the
transition between the cut-off of the spectrum and the linear-scaling
radiation era plateau is somewhat slower than in the frictionless case. Note
also that at the beginning of this transition the spectrum without friction
exceeds the spectrum with friction. This is explained by the fact that, around
$t_{f}$, loops still experience a weak frictional force that is not taken into
consideration in the computation of the spectrum without friction, which
causes a slight suppression of their GW emission (recall that $t_{f}$
corresponds to the instant of time in which the Hubble and friction damping
terms become comparable).

The impact of varying cosmic string tension is slightly more complex. As
tension is decreased and strings become lighter, friction damps strings more
efficiently (see e.g. Eq. (4)). Although one would naively expect this to lead
to a progressive suppression of this signature, the network also becomes
significantly denser during the Kibble regime — as may be seen in Eq. (31) —
and the number of loops then increases with decreasing $G\mu$, leading to a
more prominent signature instead. Moreover, GW emission becomes slower when we
decrease $G\mu$, which means that, for a fixed $\alpha$, we are moving further
into the large-loop regime, in which the signature becomes less visible. These
contradictory effects may be seen in Fig. 7, which shows the signature of
friction in the SGWB for different values of $G\mu$ and a fixed $\alpha$.
Although, initially a decrease of tension leads to an increase of the relative
height of the friction peak 666The very pointy peak and sharp cut-off seen in
the spectra with $G\mu=10^{-10}$ and $G\mu=10^{-8}$ are caused by the
conditions imposed to ensure that gravitational backreaction scale and the
length of loops are physical and well defined. We have, however, verified that
the behaviour is qualitatively similar if these conditions are not enforced.,
the friction signature starts to become less prominent as the tension is
decreased further. Moreover, since $t_{k}\sim(G\mu)^{-4/3}$, decreasing of the
tension also causes a shift of the friction peak towards lower frequencies and
the Kibble regime actually lasts longer (since $t_{f}\sim(G\mu)^{-2}$), thus
affecting a broader range of frequencies.

Increasing $\beta$, which depends on the number of particle species that
interact with the strings and thus quantifies the strength of friction,
increases the duration of the friction era. As a result, the friction peak is
broadened and the SGWB spectrum is affected by friction up to lower
frequencies, as Fig. 8 shows. However, an increase in $\beta$ also means that
the network becomes denser during the Kibble regime, leading to an increase of
the height of the friction peak, and that this regime happens at later times,
causing the very slight shift of the peak frequency that may be seen in this
figure.

Finally, as discussed in Sec. IV, if one decreases the initial characteristic
length of the network, the Kibble regime, in which this signature is created,
will start earlier and the network will be much denser in this stage. This
leads, as shown in Fig. 9, to a significant increase of the height of the peak
and to an extension of its range towards even higher frequencies. Note however
that for very small $L_{c}$, $\Gamma G\mu L$ and $\alpha L$ necessarily become
unphysically small at the very early stages of the evolution of the network.

##  VII Discussion

We have shown that friction may leave a distinct signature, in the form of a
secondary peak, in the ultra-high frequency range of the SGWB spectrum
generated by cosmic string loops. This signature has been neglected in the
literature, since it is usually assumed that friction leads to a very strong
suppression of the GW emission of cosmic string loops. We found, however, that
the end stages of the life of loops will necessarily be frictionless and that
their emission is only partially reduced. As a matter of fact, during the
Kibble regime, when the network is very dense and intercommutations are very
frequent, loop production is very intensive and the superposition of their
numerous individual emissions despite being weakened may still generate a
prominent characteristic imprint.

The signature of friction uncovered here may potentially allow us not only to
probe cosmic string tension (and through it the energy scale of the string-
forming phase transition), but also the details of the underlying particle
physics scenario. As we have seen, the height and broadness of the secondary
peak caused by friction depends on the number of particle species of the
surrounding plasma that interact strongly with the strings, quantified through
the parameter $\beta$. This parameter necessarily depends on the nature of the
fields that constitute the string and, therefore, one may expect different
string models to generate distinguishable signatures. This signature is also
sensitive to the initial conditions of the cosmic string network, which may
also allow us to uncover more about the symmetry breaking process that
originated the strings. As a matter of fact, an initial characteristic length
of the order of the cosmological horizon would be expected in slow first order
phase transitions, while smaller $L_{c}$ (up to $L_{c}\sim t_{c}/\theta$ [13])
may result from second-order phase transitions [8, 13]. Note that the
evolution of the cosmic string network in the frictionless regime is
insensitive to these two aspects and, therefore, this signature of friction
offers the possibility of extracting information about the physics of the
early universe that the rest of the SGWB does not allow us to access.

Our results show that the existence of this signature is also strongly
dependent on the size of loops and, in fact, this signature is more prominent
roughly when the length of loops is not much larger than the gravitational
backreaction scale (but this, as we have seen, depends on a number of other
aspects too). Nambu-Goto numerical simulations [14] indicate that 10% of the
energy lost by the network as a result of interactions is in the form of large
loops with $\alpha\sim 0.34$ and that the rest of the energy is lost in the
form of small loops with $\alpha\sim\Gamma G\mu$ and thus the majority of the
loops created is, in fact, quite small. So, although the contribution of the
large loops (that may provide the dominant contribution to the SGWB in the
frictionless regime) would be highly suppressed in the friction era, the
numerous small loops could still give rise to a prominent signature in the
ultra-high frequency range. Note that loop production in the presence of
friction has not yet been studied in detail and there is the possibility that
the production of small loops may be suppressed in these early stages of the
evolution, since friction could potentially erase small-scale structure on
long strings on time scales smaller than a Hubble time [7, 27]. However, since
during the Kibble regime collisions and intercommutations between strings are
quite frequent, new kinks should be copiously produced during this regime.
Friction should also only be effective in removing structure on scales that
are larger than the friction lengthscale. Whether the kink decay caused by
friction is effective in smoothing long strings and in suppressing the
production of small loops is then a subject that warrants further
investigation.

We also noticed that unphysically small scales for the length of loops and for
the gravitational backreaction scale may arise during the friction epoch.
Since this could mean that the classical treatment of cosmic strings might not
be well justified anymore and that quantum effects may have to be considered
(if thermal fluctuations are not large enough to prevent the existence of
strings at these scales), we have introduced cut-offs to address these
situations in our computations. Quantum fluctuations on straight strings in
Minkowski spacetime in the weak coupling limit were studied in [44] and the
results indicate that these fluctuations are generally small enough, when
compared to string thickness, to be neglected. However, if quantum corrections
are indeed necessary, in principle, they may leave a signature in the SGWB in
the ultra-high frequency range (which would appear in the spectra where the
cut-off we have imposed have a visible effect).

A detection of the signature of friction in the ultra-high frequency range of
the SGWB spectrum is currently challenging. This range of the spectrum has,
however, been garnering a lot of attention in the literature and several
observational concepts to probe it have been proposed (see e.g. [45] for a
recent review). Given the location of the signature, experiments based on the
inverse Gertsenshtein effect [46, 47], also known as magnetic conversion, are
of particular interest, as they may, in principle, be designed to detect GWs
with frequencies above $1$ THz. This effect — the conversion of GWs into
photons in the presence of an external magnetic field — may also take place in
current and upcoming axion-search experiments, in which axion-like particles
may convert into photons in the presence of a strong magnetic field. In [48],
the authors use ALPS, OSQAR and CAST data to constrain the amplitude of
isotropic ultra-high frequency GWs in two frequency bands: $(2.7-14)\times
10^{14}$ Hz and $(5-12)\times 10^{18}$ Hz. Besides experiments based on
artificial magnetic fields, there are suggestions to also use astronomical
magnetic fields such as those of pulsars [49], planetary magnetospheres [50]
or galaxy clusters [51] to detect GWs. The former [49] constrain SGWBs around
$10^{13}-10^{27}$ Hz, precisely in the relevant frequency range to probe
friction. Moreover, in [52], the authors proposed a new method based on the
inverse Gertsenshtein effect that resorts to high-energy laser beams instead
of magnetic fields. They find that the best sensitivity is reached when the GW
frequency is twice the frequency of the lasers used and thus, given the wide
range of lasers’ operational frequencies ($10^{13}-10^{19}$ Hz), this method
may allow us to probe GWs in a broad frequency range in the future. So,
although this signature is currently out of reach of GW detectors, the
sensitivity to SGWBs in the ultra-high frequency range of the spectrum is
expected to increase, thus opening the prospect of probing the imprints of
friction directly in the future.

###### Acknowledgements.

The authors thank Pedro Avelino and Andreas Neitzel for many enlightening
discussions. S. M. is supported by FCT - Fundação para a Ciência e a
Tecnologia through the PhD fellowship UI/BD/152220/2021. L. S. is supported by
FCT through contract No. DL 57/2016/CP1364/CT0001. Funding for this work has
also been provided by FCT through the research grants UIDB/04434/2020 and
UIDP/04434/2020 and through the R & D project 2022.03495.PTDC – Uncovering the
nature of cosmic strings.

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