DESY 19-206

BSM with Cosmic Strings: Heavy, up to EeV mass, Unstable Particles

Yann Gouttenoirea, Géraldine Servanta,b, Peera Simakachorna,b

a DESY, Notkestraße 85, D-22607 Hamburg, Germany
b II. Institute of Theoretical Physics, Universität Hamburg, 22761 Hamburg,
Germany

Abstract

> Unstable heavy particles well above the TeV scale are unaccessible
> experimentally. So far, Big-Bang Nucleosynthesis (BBN) provides the
> strongest limits on their mass and lifetime, the latter being shorter than
> 0.1 second. We show how these constraints could be potentially tremendously
> improved by the next generation of Gravitational-Wave (GW) interferometers,
> extending to lifetimes as short as $10^{-16}$ second. The key point is that
> these particles may have dominated the energy density of the universe and
> have triggered a period of matter domination at early times, until their
> decay before BBN. The resulting modified cosmological history compared to
> the usually-assumed single radiation era would imprint observable signatures
> in stochastic gravitational-wave backgrounds of primordial origin. In
> particular, we show how the detection of the GW spectrum produced by long-
> lasting sources such as cosmic strings would provide a unique probe of
> particle physics parameters. When applied to specific particle production
> mechanisms in the early universe, these GW spectra could be used to derive
> new constraints on many UV extensions of the Standard Model. We illustrate
> this on a few examples, such as supersymmetric models where the mass scale
> of scalar moduli and gravitino can be constrained up to $10^{10}$ GeV.
> Further bounds can be obtained on the reheating temperature of models with
> only-gravitationally-interacting particles as well as on the kinetic mixing
> of heavy dark photons at the level of $10^{-18}$.

###### Contents

  1. 1 Introduction
  2. 2 Gravitational waves from cosmic strings
    1. 2.1 The micro-physics assumptions
    2. 2.2 The gravitational-wave spectrum
    3. 2.3 The Velocity-dependent One-Scale model
    4. 2.4 The reach of GW interferometers
  3. 3 The imprints of an early era of matter domination
    1. 3.1 Modified spectral index
    2. 3.2 How to detect a matter era with a GW interferometer
    3. 3.3 Model-independent constraints on particle physics parameters
  4. 4 Application to benchmark models
    1. 4.1 Oscillating scalar moduli
    2. 4.2 Scalar particles produced gravitationally
    3. 4.3 Scalar particles produced through the Higgs portal
    4. 4.4 Heavy dark photons
  5. 5 Summary

##  1 Introduction

The existence of very massive particles $X$, with mass $m_{X}\gg$ TeV, is a
generic prediction of many well-motivated extensions of the Standard Model
(SM) of particle physics, such as Grand Unified Theories, extra-dimensional
models inspired by String Theory or supersymmetric constructions. If such
particles are stable and still present in our universe today, they can
contribute to the dark matter, in which case a variety of detection strategies
has been explored depending on their mass range and the nature of their
interactions. On the other hand, unstable particles beyond the Standard Model
(BSM) are very difficult to probe experimentally. The best chances are through
their effects on cosmological observables. The strongest limits come from Big-
Bang Nucleosynthesis (BBN), since any heavy relic which decays after BBN would
ruin the predicted abundances of light elements. From BBN, one obtains general
model-independent bounds in the plane $(\tau_{X},\,m_{X}Y_{X})$ where
$\tau_{X}$ is their lifetime, $m_{X}$ their mass and $m_{X}Y_{X}$ is their
would-be contribution to the total energy density of the universe today if
they had not decayed [1, 2, 3, 4]. We can therefore infer indirect information
on their couplings through the constraints on their lifetime and the
efficiency of their production mechanism in the early universe. In the present
work, we show how a large new region unexplored so far in the
$(\tau_{X},m_{X}Y_{X})$ plane can be probed using future gravitational-wave
observatories.

Our starting assumption is that these particles can temporarily dominate the
energy density of the universe, and therefore induce a period of matter
domination within the radiation era after post-inflationary reheating. This
leads to a modified expansion of the universe compared to the usually assumed
single radiation era. Interestingly, such modified cosmological history can be
probed if during this period, there is an active source of gravitational
waves, in which case the resulting GW spectrum would imprint any modification
of the equation of state of the universe. Particularly well-motivated are the
long-lasting GW production from cosmic string networks.

Cosmic strings (CS) are topological defects produced when a $U(1)$ symmetry
gets spontaneously broken in the early universe [5, 6, 7]. CS behave as
dynamical classical objects, moving at relativistic speed. They can also be
described by fundamental or composite objects in string theory [8, 9, 10, 11,
12, 13, 14, 15]. CS networks offer promising prospects for the detection of GW
of cosmological origin. They are a generic prediction of many Standard Model
extensions, such as models of Grand Unification [16, 17, 18], or the seesaw
mechanism for neutrino masses when $U_{B-L}(1)$ is broken spontaneously [19].

The string network is characterized by its correlation length $L$. While
strings are stretched by cosmic expansion, they form loops. One would naively
expect $L$ to evolve linearly with the scale factor $a$ due to the Hubble
expansion, such that $L\propto t^{1/2}$ in radiation domination and $L\propto
t^{2/3}$ is matter domination. However, a remarkable feature of CS is that,
after a transient evolution, the system reaches its scaling regime where the
energy loss of long strings into loop formation, is exactly such that $L$
scales linearly with the Hubble horizon $t$ [20, 21, 22, 23, 24]. During this
regime, the string network is only characterized by the string tension $\mu$,
roughly given by the square of the phase transition temperature $T_{p}$

$G\mu\simeq 10^{-15}\left(\frac{T_{p}}{10^{11}~{}\text{GeV}}\right)^{2},$ (1)

and the long string density, $\rho_{\infty}=\mu/L^{2}$, redshifts as radiation
in radiation domination and as matter in matter domination.

It has since long been conjectured that the oscillations of the CS loops may
be the dominant source of the Stochastic Gravitational Waves Background
(SGWB), principally because they constitute a long-standing source starting GW
emission after the network formation and still radiating today [25, 26, 27,
28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41]. The interesting
aspect of the SGWB from CS is its flat frequency spectrum, assuming standard
cosmology, spanning many orders of magnitude in frequency. Hence, the
capability of the next generation of GW interferometers, LISA [42], Einstein
Telescope [43, 44], Cosmic Explorer [45], BBO and DECIGO [46] to detect the
SGWB from CS, opens a unique observational window, on any new physics likely
to change the thermal history of the universe, and imprint features in the GW
spectrum. In [47], we analysed in detail how much information on the early
universe equation of state could be extracted from the observation of a SGWB
generated by CS, adding up on recent works [48, 49, 50, 51].

In this study, we assume the presence of a heavy, cold, particle $X$
dominating the energy density of the universe at the temperature $T_{\rm dom}$
and decaying at the temperature $T_{\rm dec}$. Currently, the success of BBN
in a standard radiation dominated universe provides the strongest constraint
on such scenario, $T_{\rm dec}\gtrsim 1$ MeV. The key point of our study is
that the observation of a flat GW spectrum from CS would extend by far the BBN
constraints on heavy relics. All assumptions relevant for our conclusions are
discussed in detail in our companion article [47], where we refine and extend
the work of [48, 49] beyond the scaling regime during the change of cosmology.
We also take into account the recent discussion on the effect of particle
production on the GW emission [52]. We only recap briefly the main points here
and refer the reader to [47] for more details.

In Sec. 2, we review the main assumptions about the microscopic description of
CS. In Sec. 3, we compute the GW spectrum in the presence of an early matter
era. We then derive the improvement by many orders of magnitude of the current
model-independent BBN constraints on the abundance and lifetime of a particle,
c.f. Fig. 3, that can be inferred from the detection of GW produced by CS. In
Sec. 4, we provide unprecedented exclusion bounds on four particle physics
models leading to an early matter domination era: oscillating scalar moduli in
supersymmetric theories, secluded scalar sectors which are only
gravitationally produced, scalars produced through the Higgs portal, and
massive dark photons. At the very end, we also study the scenario where the
dark photon mass and the cosmic string network are generated by the
spontaneous breaking of the same $U(1)$ symmetry.

##  2 Gravitational waves from cosmic strings

###  2.1 The micro-physics assumptions

A detailed discussion of the assumptions made for the prediction of SGWB from
cosmic strings is provided in our companion article [47]. We summarize them
here.

  * •

We consider CS from the breaking of a gauge symmetry.

  * •

We assume first that CS can be described by Nambu-Goto strings which are
1-dimensional classical object, characterized solely by their tension $\mu$.
The main decay channel of loops are through gravitational waves and their
sizes shrink with a rate $\Gamma G\mu$ with $\Gamma\simeq 50$ [53].

  * •

As a correction to the Nambu Goto approximation, we include the possibility of
producing massive particles due to the presence of string segments with
curvature comparable to their thickness, cusps and kinks.

  * •

We consider a monochromatic loop distribution, where loops are produced with a
length being a fraction $\alpha=0.1$ of the Hubble size. Hence, we neglect the
contributions to the SGWB from the loops produced at smaller scales. In
particular, we neglect the contributions from the loops produced at the
gravitational back reaction scale, the scale below which small-structures are
smoothened.

  * •

GW production at time $\tilde{t}$ is dominated by the emission from small
loops of size $\sim\Gamma G\mu\,\tilde{t}$, which have been produced at the
horizon size, i.e. much earlier, at time $t_{i}\sim\Gamma
G\mu\,\tilde{t}/\alpha$.

  * •

We compute the loop-formation efficiency $C_{\rm eff}$ by solving the
Velocity-dependent One-Scale equations [54, 55, 56, 57, 58]. Hence, we account
for the extra time needed by the long-string network to respond to the change
of cosmology, inducing a shift by one or two orders of magnitude of the
turning-point frequency characterizing the non-standard matter era.

###  2.2 The gravitational-wave spectrum

For our study, we follow the assumptions listed above and the expression given
in [49] for the computation of today GW spectrum generated from CS (for a
derivation and discussion of this formula, see [47])

$\Omega_{\rm{GW}}(f)\equiv\frac{f}{\rho_{c}}\left|\frac{d\rho_{\rm{GW}}}{df}\right|=\sum_{k}{\Omega^{(k)}_{\rm{GW}}(f)},$
(2)

with

$\Omega^{(k)}_{\rm{GW}}(f)=\frac{1}{\rho_{c}}\cdot\frac{2k}{f}\cdot\frac{(0.1)\,\Gamma^{(k)}G\mu^{2}}{\alpha(\alpha+\Gamma
G\mu)}\int^{t_{0}}_{t_{F}}d\tilde{t}\,\frac{C_{\rm{eff}}(t_{i})}{t_{i}^{4}}\left[\frac{a(\tilde{t})}{a(t_{0})}\right]^{5}\left[\frac{a(t_{i})}{a(\tilde{t})}\right]^{3}\Theta(t_{i}-t_{F})\Theta(t_{i}-\frac{l_{*}}{\alpha}),$
(3)

where we integrate over all emission times $\tilde{t}$ starting from the
network formation time $t_{F}$ up to today $t_{0}$ and where the sum is
performed over the mode frequencies of the loop. We assume that a loop created
at $t_{i}$, with a length at a fraction $\alpha$ of the Hubble horizon,
shrinks by emission of GW with a rate $\Gamma G\mu$

$l(t)=\alpha t_{i}-\Gamma G\mu(t-t_{i}),$ (4)

where the gravitational loop-emission efficiency is independent of the lenght
of the loop, $\Gamma\simeq 50$ [53]. The 1-loop power spectrum
$\Gamma^{(k)}\,G\mu^{2}$ assumes cusp domination

$\Gamma^{(k)}=\frac{\Gamma\,k^{-4/3}}{\sum_{p=1}^{\infty}p^{-4/3}}\simeq\frac{\Gamma\,k^{-4/3}}{3.60}.$
(5)

From Eq. (4), the time $t_{i}$, when the loops, sourcing GW emitted at time
$\tilde{t}$ and detected with frequency today $f$, have been created can be
expressed as

$t_{i}(f,\,\tilde{t})=\frac{1}{\alpha+\Gamma
G\mu}\left[\frac{2k}{f}\frac{a(\tilde{t})}{a(t_{0})}+\Gamma
G\mu\,\tilde{t}\right].$ (6)

As discussed in [52, 47], loops with length smaller than $l_{*}$ decay
dominantly into massive particles

$\displaystyle l_{*}=\beta_{m}\frac{\mu^{-1/2}}{(\Gamma G\mu)^{m}},$ (7)

where $m=1$ or $2$ for kink-dominated or cusp-dominated loops, respectively,
and $\beta_{m}\sim\mathcal{O}(1)$.

Figure 1:  SGWB generated by the gravitational decay of cosmic strings
compared to the reach of different GW interferometers. We show the impact of a
long (red) or a short (blue) intermediate matter era, starting at the
temperature $r\,T_{\Delta}$ and ending at $T_{\Delta}=10$ MeV or
$T_{\Delta}=10$ TeV. Black lines show the results obtained assuming standard
cosmological evolution. The dashed-lines assume that the scaling regime
switches on instantaneously during the change of cosmology whereas the solid
lines incorporate the transient behavior, solution of the VOS equations, as
discussed in [47]. Limitations due to particle production assuming that the
small-scale structures are dominated by cusps are shown with dotted lines
[47]. The dotted vertical lines indicate the relation in Eq. (10) between the
temperature $T_{\Delta}$ and the frequency $f_{\Delta}$ of the turning point,
where the matter-era-tilted spectrum meets the radiation-era-flat spectrum.

###  2.3 The Velocity-dependent One-Scale model

Numerical simulations realized by Blanco-Pillado et al. [59] have shown that a
fraction $\mathcal{F}\simeq 10\%$ of the loops are produced with a length
equal to $\alpha\simeq 10\%$ of the horizon size and with a Lorentz boost
factor $\gamma\simeq\sqrt{2}$. The remaining $90\%$ of the energy loss by long
strings goes into highly boosted smaller loops whose contribution to the GW
spectrum is sub-dominant. Under those assumptions, the Velocity-dependent One-
Scale (VOS) model [56, 57, 55, 58] predicts the loop-formation rate to be

$\frac{dn}{dt_{i}}=\mathcal{F}\frac{C_{\rm eff}(t_{i})}{\alpha\,t_{i}^{4}}.$
(8)

Concretely, the VOS model describes the evolution of Nambu-Goto long strings
through two macroscopic quantities, their mean velocity $\bar{v}$ and
correlation length $\xi\equiv L/t$, from which one can compute the loop-
formation efficiency

$C_{\rm eff}\equiv\frac{\tilde{c}\,\bar{v}}{\sqrt{2}\xi^{3}},$ (9)

where $\tilde{c}$ is the loop-chopping efficiency, $\tilde{c}=0.23$ based on
Nambu-Goto numerical simulations [55]. In [47], we study the implication of
the recent refinement of the VOS model [60], which includes particle
production and is based on abelian-Higgs field theory numerical simulations.
Out of the competing dynamics of Hubble stretching and energy loss through
loop chopping, the network reaches an attractor solution, the scaling regime,
in which both $\bar{v}$ and $\xi$ are constant. In our companion paper [47],
we show that the long-string network deviates from the scaling regime during a
change of cosmology. As an illustration, Fig. 1 compares the spectrum computed
by assuming that the scaling regime is reached instantaneously during the
change of cosmology with the full solution of the VOS equations. It shows that
the turning point frequency which is a signature of the change of cosmology
from matter to radiation, is over-estimated by more than one order of
magnitude in the scaling approximation. Ref. [47] refines the study of [49] by
going beyond the scaling regime as well as considering particle production and
additional non-standard cosmological histories.

###  2.4 The reach of GW interferometers

GW spectra from CS for two values of $G\mu=10^{-15}$, $10^{-11}$ are plotted
in Fig. 1, together with the power-law sensitivity curves of NANOGrav [61],
EPTA [62], SKA [63], LIGO [64], DECIGO, BBO [46], LISA [42], Einstein
Telescope [43, 44] and Cosmic Explorer [45]. Only EPTA, NANOGrav and LIGO
O1/O2 (which are not visible on the plots) are current constraints, the other
ones being projected sensitivities of future projects. The power-law
integrated sensitivity curves are computed in [47] with a signal-to-noise
ratio SNR=$10$ and an observation time of $10~{}$years. At lower string
tension $G\mu$, the GW power emission per loop is smaller, hence the amplitude
is suppressed. Also, for lower values of $G\mu$, loops decay more slowly and
GW are emitted later, which implies a lower red-shift factor and a global
shift of the spectrum towards higher frequency.

The strongest constraints come from pulsar timing array EPTA, $G\mu\lesssim
8\times 10^{-10}$ [65], and NANOGrav, $G\mu\lesssim 5.3\times 10^{-11}$ [61],
therefore we limit ourselves to $G\mu<10^{-11}$. Our analysis is based on the
assumption that the astrophysical foreground can be subtracted. The GW
spectrum generated by the astrophysical foreground increases with frequency as
$f^{2/3}$ [66], differently from the GW spectrum generated by CS during
radiation (flat) or during matter ($f^{-1}$).

##  3 The imprints of an early era of matter domination

###  3.1 Modified spectral index

The part of the spectrum coming from loops produced and emitting during
radiation is flat since there is an exact cancellation between the red-tilted
red-shift factor and the blue-tilted loop number density. However, in the case
of a matter era, a mismatch induces a slope $f^{-1}$. The impact of a non-
standard matter era is shown in Fig. 1. The frequency detected today
$f_{\Delta}$ of the turning point between the end of the matter domination and
the beginning of the radiation-domination can be related to the temperature of
the universe $T_{\Delta}$ when the change of cosmology occurs

$f_{\Delta}=(2\times
10^{-3}~{}\textrm{Hz})\left(\frac{T_{\Delta}}{\textrm{GeV}}\right)\left(\frac{0.1\times
50\times 10^{-11}}{\alpha\Gamma
G_{\mu}}\right)^{1/2}\left(\frac{g_{*}(T_{\Delta})}{g_{*}(T_{0})}\right)^{1/4}.$
(10)

The GW measured with frequency $f_{\Delta}$ have been emitted by loops
produced during the change of cosmology at $T_{\Delta}$. An extensive
discussion of this frequency-temperature relation as provided in [47]. The
above formula entirely relies on the assumptions that the back-reaction scale
is $\Gamma G\mu$ as claimed by Blanco-Pillado et al. [67, 68, 69] and not much
lower as claimed by Ringeval et al. [59, 53] .

Figure 2:  Left: SGWB for $G\mu=10^{-11}$ assuming that a heavy cold particle
dominates the energy density of the universe at the temperature $T_{\rm dom}$
and decays at the temperature $T_{\Delta}=T_{\rm dec}$. Right: Considering the
particular case of the Einstein Telescope, we illustrate how the constraints
on the abundance and lifetime of a heavy relic depend on the choice of the
prescription, Rx 1 or Rx 2 defined in Sec. 3.2.

###  3.2 How to detect a matter era with a GW interferometer

For a first qualitative analysis, we start with two simple prescriptions for
detecting a matter era from the measurement of a SGWB from CS by a GW
interferometer.

  * •

Rx 1 (turning-point prescription): The turning point, namely the frequency at
which the spectral index of the GW spectrum changes, corresponding to the
transition from the matter to the radiation era, defined in Eq. (10), must be
inside the interferometer window, as shown for instance in Fig. 1.

  * •

Rx 2 (spectra-index prescription): The measured spectral index must be smaller
than $-0.2$, namely $\beta<-0.2$ where $\Omega_{\rm GW}h^{2}\propto
f^{\beta}$.

In Fig. 2, we compare the above two prescriptions. The prescription Rx 1 is
more conservative but enough to measure the lifetime of the particle. In our
study, we use the prescription Rx 1 and, in Fig. 4, we show how to extend the
constraints with Rx 2.

We note here that the presence of the turning point and the changed spectral
index at high frequencies would be similar in the case of a long intermediate
inflation era instead of an intermediate matter era. Disentangling the two
effects deserves further studies. Interestingly, high-frequency burst signals
due to cusp formation could be a way-out [47]. In the analysis of this paper,
we interpret the suppression of the GW spectrum as due to an intermediate
matter era.

Figure 3:  Constraints on the lifetime $\tau_{X}$ and would-be abundance
$m_{X}Y_{X}$ of a heavy unstable particle inducing an early-matter era,
assuming the observation of a SGWB from CS by a GW interferometer, c.f. Sec.
3.3. We compare the new prospects with the current limits inferred from BBN
[2, 3, 4]. We assume the detectability of the turning point in the GW spectrum
at the frequency $f_{\Delta}$, induced by the decay of the particle at $T_{\rm
dec}=T_{\Delta}$, c.f. turning-point description Rx 1 in Sec. 3.2. Limitation
due to particle production in the cusp-domination case [47] are shown in
purple.

###  3.3 Model-independent constraints on particle physics parameters

A matter-dominated era may result from an oscillating scalar field [70], such
as a moduli field, or a relativistic plasma with a non-vanishing tensor bulk
viscosity [71], or simply a massive particle dominating the energy density of
the universe. A matter-dominated era may be motivated by the possibillity to
enhance structure growth at small scales, since density perturbations start to
grow linearly earlier [72, 73], hence boosting the dark matter indirect
detection signals [74, 75], or the possibility to enhance the primordial black
holes production [76, 77, 78].

We suppose an early-matter era is caused by the energy density of a cold
particle $X$, meaning that $X$ is non-relativistic and is decoupled chemically
and kinetically from the visible sector. The energy density $m_{X}n_{X}$ of
$X$ dominates over the energy density of the SM radiation, with entropy
$s_{\rm SM}$, at the temperature $T_{\rm dom}$

$T_{\rm dom}=\frac{4}{3}m_{X}Y_{X},\qquad Y_{X}\equiv n_{X}/s_{\rm SM}.$ (11)

Then, the cold relic decays when its lifetime $\tau_{X}$ is equal to the age
of the universe, corresponding to the temperature

$T_{\rm dec}=1~{}\text{GeV}\left(\frac{80}{g_{\rm
SM}}\right)^{1/4}\left(\frac{2.7\times
10^{-7}~{}\text{s}}{\tau_{X}}\right)^{1/2}.$ (12)

Note that the above relation between $T_{\rm dec}$ and $\tau_{X}$ only assumes
that the decay is followed by a radiation dominated era and is independent of
the previous thermal history of the universe. $T_{\rm dec}$ is sometimes
referred, mistakenly though [79], as the reheating temperature following the
decay. We propose to use the third generation of GW interferometers to
constrain cold relics responsible for early-matter domination. The constraints
we will derive rely on the following assumptions:

  * 1)

A SGWB from CS with tension $G\mu$ is measured by a GW interferometer $i$.

  * 2)

The cold particle is abundant enough to lead to a matter-dominated era before
it decays

$T_{\rm dom}>T_{\rm dec}\ ,$ (13)

where $T_{\rm dom}$ and $T_{\rm dec}$ satisfy Eq. (11) and Eq. (12).

  * 3)

The prescription Rx 1 of Sec. 3.2 is used, i.e. the turning point in the GW
spectrum is in the observation window of the detector and

$\Omega_{GW}(f_{\Delta}(T_{\rm dec},\,G\mu),\,G\mu)h^{2}\,>\,\Omega_{\rm
sens}^{(i)}h^{2},$ (14)

where $\Omega_{GW}(f,\,G\mu)h^{2}$ is the predicted scale-invariant GW
spectrum from Eq. (2), and $\Omega_{\rm sens}^{(i)}h^{2}$ is the power-law
sensitivity curve of the detector $i$.

Fig. 3 shows these new constraints in comparison with the current
complementary constraints from BBN, usually represented in the plane
$(\tau_{X},\,m_{X}Y_{X})$ [2, 3, 4]. We can translate the sensitivity of each
interferometer to probe the particle lifetime into typical mass windows,
assuming some decay width. This is illustrated in Fig. 4 with a Planck-
suppressed decay width $\Gamma_{X}\propto m_{X}^{3}/M_{\rm pl}^{2}$.

Figure 4: Reach of future GW interferometers on the mass of a heavy particle
decaying through a Planck suppressed operator, $\Gamma_{X}\propto
m_{X}^{3}/M_{\rm pl}^{2}$, supposing that it is sufficiently produced to
induce a matter era before the decay. We compare the turning-point
prescription (Rx 1) and the spectral-index prescription (Rx 2) discussed in
Sec. 3.2. In Sec. 4, we study three different production mechanisms of such
particle with Planck suppressed decay width: scalar oscillating moduli
produced after supersymmetry breaking in Sec. 4.1, scalar particle
gravitationally produced at the end of inflation in Sec. 4.2, or scalar
particle produced via thermal freeze-in assuming a Higgs-mixing in Sec. 4.3.

##  4 Application to benchmark models

###  4.1 Oscillating scalar moduli

String theory vacua feature moduli fields which characterize the size and
shape of the compactification manifold. From a 4D effective field theory
perspective, they are fields with flat potential e.g. axions or dilatons.
After supersymmetry (SUSY) breaking, one expects moduli fields to acquire a
mass of the order of the gravitino mass scale for the lightest [80], e.g.
$m_{3/2}\sim$ TeV for low-scale SUSY. As soon as the Hubble rate satisfies
$H\lesssim m_{\phi}$, the scalar field starts coherent oscillations and its
energy density redshifts as matter. We assume that the onset of oscillations
occurs during radiation domination, at the temperature $T_{\rm osc}$

$\frac{\pi^{2}g_{*}T_{\rm osc}^{4}}{90\,M_{\rm pl}}\equiv m_{\phi}^{2}$ (15)

where we fix the number of relativistic degrees of freedom to the fiducial
value $g_{*}=106.75$. Then, the moduli starts dominating the energy density of
universe at the temperature, c.f. Eq. (11)

$T_{\rm dom}=\frac{4}{3}\frac{\rho_{\phi}^{\rm osc}}{s_{\rm
osc}}\equiv\frac{\tfrac{1}{2}m_{\phi}^{2}\phi_{0}^{2}}{\tfrac{2\pi^{2}}{45}g_{*}T_{\rm
osc}^{3}},$ (16)

where $\phi_{0}$ is the vacuum expectation value of the moduli field when it
starts to oscillate. For concreteness, we consider moduli fields which
interact with the visible sector via Planck-suppressed operators and hence
have decay widths of order

$\Gamma_{\phi}\simeq\frac{c}{8\pi}\frac{m_{\phi}^{3}}{M_{\rm pl}^{2}},$ (17)

where $M_{\rm pl}\simeq 2.4\times 10^{-18}$ GeV and $c$ is a model-dependent
factor which we suppose to be in the range $10^{-2}\lesssim c\lesssim 10^{2}$.
For TeV-scale moduli mass, the moduli lifetime is long,
$\Gamma_{\phi}^{-1}\sim 10^{5}$ s, and the decay occurs much after BBN.
Imposing that the energy density of the moduli decay products, $\rho_{\phi}$,
is smaller than a fraction $10^{-14}$ of the total entropy density of the
universe in order to preserve the predictions of BBN, c.f. Fig. 3 and [2, 3,
4], one constrains the vacuum expectation value of the moduli field, just
after it starts oscillating, to be, c.f. Fig. 5

$\text{BBN is preserved for TeV-scale moduli:}\qquad\phi_{0}\lesssim
10^{-12}~{}M_{\rm pl}.$ (18)

A large moduli VEV is expected from the dependence of the moduli potential on
the inflaton VEV [81, 82, 83, 84], except if the scalar moduli field lies at a
point of enhanced symmetry where the induced minimum at late times coincides
with the minimum at earlier times [85]. However, even in the case where the
moduli VEV after SUSY breaking remains small, one expects moduli to be
copiously produced both through thermal [86, 87, 88] and gravitational
production, c.f. Sec. 4.2 and [89, 90], hence violating the bound in Eq. (18).

Figure 5: Constraints on a moduli field oscillating in the early universe
around its minima with initial amplitude $\phi_{0}$ and mass $m_{\phi}$, from
the non-observation of its signature in the GW spectrum from CS with tension
$G\mu=10^{-11}$. In stringy UV completions, the lightest moduli field is
related to the gravitino scale $m_{X}\sim m_{3/2}$ [80]. The BBN constraints
are taken from [2, 3, 4]. We use the turning-point prescription (Rx 1)
discussed in Sec. 3.2. Constraints are tightened when considering the
spectral-index prescription (Rx 2), as illustrated in Fig. 4.

The scalar moduli problem is similar to the gravitino problem [91], both are
copiously produced relics, with weak-scale mass and Planck-suppressed decay
rate, spoiling the BBN predictions. An important difference is that during
inflation, the energy density is diluted in the fermionic case but is frozen
in the scalar case as long as $H\gg m_{\phi}$. Hence, as opposed to the
gravitino case, only a low-scale (weak-scale) inflation can exponentially
dilute the scalar field and solve the moduli problem [92]. Other proposed
solutions are to increase the moduli mass up to $m_{\phi}\sim
O(10^{3})\,m_{3/2}$, c.f. Fig. 5 and [93], or to form substructures (modular
stars) which enhances the decay [94, 95], or to produce gauge fields from the
tachyonic instability [96].

A pragmatic approach to solve the moduli problem is to break SUSY at a much
higher scale, at the expense of large fine-tuning, like in so-called High-
scale SUSY [97] or Split SUSY [98]. A larger SUSY breaking scale improves
gauge coupling unification [99], is compatible with the Higgs at $125$ GeV
[100] and is free from the main difficulties encountered by low-scale SUSY
such as large flavor and CP violation [101]. In Split and High-scale SUSY, a
$125$ GeV Higgs is compatible with a SUSY breaking scale as high as $10^{8}$
GeV and $10^{12}$ GeV respectively [100]. Moduli fields with masses of the
same order would then induce an early matter era which could lead to
detectable features in the GW background.

As shown in Fig. 5, the observation of a SGWB from CS with one of the next
generation GW interferometers would provide constraints on moduli masses up to
$\lesssim 10^{10}$ GeV, well above the current $\lesssim 100$ TeV currently
probed by BBN. Hence, if detected, GW from CS would be a promising tool to
probe superstring theories.

In addition to being naturally motivated in SUSY constructions, moduli fields
have interesting cosmological consequences: Afflect-Dine baryogenesis [102,
83], non-thermal production of Wino-DM [103, 84, 104], formation of oscillons
or Q-balls [95], the required entropy injection to allow thermal DM much
heavier than the standard unitarity bound $\sim 100$ TeV [105] or to revive
Grand-Unified-Theory-scale QCD axion DM [106], see Ref. [107] for a review on
the moduli problem and its cosmological implications.

###  4.2 Scalar particles produced gravitationally

In the previous subsection, Sec. 4.1, we considered a model of
gravitationally-only interacting particle whose abundance is given by the
misalignement mechanism after SUSY breaking. Instead, we now consider the
possibility to produce such a particle, gravitationnally only, at the end of
inflation. In the next subsection, Sec. 4.3, we also consider the possibility
of a thermal production via freeze-in through a Higgs mixing in the case of
the conformal scalar $\xi=1/6$ where the gravitational production is too small
to lead to an early matter domination.

A massive particle can be gravitationally produced at the end of the inflation
due to the non-adiabatic change of its curvature-induced mass from deep de
Sitter to deep radiation-domination [108, 109, 110, 111, 112], hence possibly
leading to heavy dark matter WIMPzillas [113, 114, 115, 116, 117]. Our
interest here is not to explain DM but to predict a non-standard matter era in
the early universe. If coupled non-conformally to gravity and if the condition
of non-adiabaticity, $m_{X}\lesssim H_{\rm inf}$, is satisfied, then the
particle $\chi$ will be produced abundantly, potentially leading to an early
matter domination. The lagrangian is

$\mathcal{S}=\int d^{4}x\sqrt{-g}\left(\frac{1}{2}\left(M_{{\rm
P}}^{2}-\xi\chi^{2}\right)R-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi-\frac{1}{2}m_{\chi}^{2}\chi^{2}\right],$
(19)

with $\xi$ the non-minimal coupling to gravity. We consider the cases $\xi=0$
(minimal coupling) and $\xi=1/6$ (conformal coupling). We suppose that the
scalar $\chi$ decays gravitationally through Planck suppressed operators

$\Gamma_{\chi}\simeq\frac{1}{8\pi}\frac{m_{\chi}^{3}}{M_{\rm P}^{2}}.$ (20)

A too light scalar would spoil the BBN prediction. The comoving number
density, $Y_{\chi}\equiv n_{\chi}/s$, of a minimal scalar, $\xi=0$, after
gravitational production is [116]

$Y_{\chi}^{\xi=0}\simeq\dfrac{H_{\rm reh}^{2}H_{\rm
inf}}{s}\left\\{\begin{array}[]{ll}96\;\dfrac{H_{\rm inf}}{m_{\chi}}\hskip
78.24507pt\dfrac{m_{\chi}}{H_{\rm inf}}<1,\vspace{0.25 cm}\\\
0.76\;\dfrac{H_{\rm inf}}{m_{\chi}}e^{-2m_{\chi}/H_{\rm
inf}}\qquad\dfrac{m_{\chi}}{H_{\rm inf}}>1,\end{array}\right.$ (21)

Figure 6:  Constraints on a purely gravitationally produced ($\kappa=0$) non-
conformal scalar ($\xi=0$) decaying via a Planck suppressed operator, c.f.
Sec. 4.2, assuming the observation of a SGWB from CS with tension
$G\mu=10^{-11}$ by third-generation GW detectors. We have fixed the inflation
scale $H_{\rm inf}=10^{13}$ GeV. We use the turning-point prescription (Rx 1)
discussed in Sec. 3.2. Constraints are tightened when considering the
spectral-index prescription (Rx 2), as shown in Fig. 4.

Figure 7:  Constraints on a scalar conformally coupled to gravity, $\xi=1/6$,
thermally produced via a mixing with the Higgs $\kappa=10^{-6},10^{-4},1$,
c.f. Sec. 4.3, from the measurement of a SGWB from CS. We have also included
the gravitational production, c.f. 4.2, even though it is negligible. We have
fixed the inflation scale $H_{\rm inf}=10^{13}$ GeV. We use the turning-point
prescription (Rx 1) discussed in Sec. 3.2. Constraints are tightened when
considering the spectral-index prescription (Rx 2), as shown in Fig. 4.

where $H_{\rm inf}$ and $H_{\rm reh}$ are the Hubble factors at the end of
inflation and reheating. We have checked that particle production caused by
the oscillations of the inflaton during preheating, potentially relevant when
$H_{\rm inf}\lesssim m_{\chi}\lesssim m_{\phi}$ where $m_{\phi}$ is the
inflaton mass [118, 119, 117, 120, 121], is not strong enough to ignite an
early matter domination before BBN.

For our study, we fix the inflation scale $H_{\rm inf}=10^{13}$ GeV close to
its upper bound value $6\times 10^{13}$ GeV from the non-detection of the
fundamental B-mode polarization patterns in the CMB [122, 123]. If produced
with a sufficient amount, the scalar field can lead to an early-matter
domination era. In Fig. 6, we show the GW constraints on the $\chi$ scalar
particle, which is non-conformally coupled to gravity. We see that for
reheating temperature larger than $10^{7}$ GeV, the scalar field is
sufficiently produced by gravitational effects at the end of inflation to
dominate the energy density of the universe before BBN starts, and be detected
in GW experiments. Masses as large as $10^{10}$ GeV can be probed.

The gravitational production of particles conformally coupled to gravity, i.e.
scalars with $\xi=1/6$, transverse vectors or fermions, is less efficient than
for a minimal scalar, $\xi=0$ [124, 125, 126, 116, 120]. Here we give the
abundance computed in [116] for a conformal scalar111Here ‘conformal scalar’
means ‘conformally coupled to gravity’ with $\xi=1/6$. In any case, the
conformal symmetry is broken via the scalar mass term.

$Y_{\chi}^{\xi=1/6}\simeq\dfrac{H_{\rm reh}^{2}H_{\rm
inf}}{s}\left\\{\begin{array}[]{ll}0.0010\;\dfrac{m_{\chi}}{H_{\rm inf}}\hskip
71.13188pt\dfrac{m_{\chi}}{H_{\rm inf}}<1,\vspace{0.25 cm}\\\
0.0040\;\dfrac{H_{\rm inf}}{m_{\chi}}e^{-2m_{\chi}/H_{\rm
inf}}\qquad\dfrac{m_{\chi}}{H_{\rm inf}}>1,\end{array}\right.$ (22)

We check that the gravitational production of such particles, conformally-
coupled to gravity, is not strong enough to lead to a matter-domination era
before BBN starts, if the reheating temperature following inflation is below
$10^{13}~{}$GeV. Hence, in the next section we consider another production
mechanism by introducing a mixing with the standard model Higgs.

###  4.3 Scalar particles produced through the Higgs portal

The gravitational production of a scalar conformally-coupled to gravity
($\xi=1/6$), is too small to lead to an early-matter domination. On the other
hand, a mixing with the Higgs $H$

$\mathcal{L}\supset\frac{\kappa}{2}\chi^{2}|H|^{2},$ (23)

can lead to a large abundance via thermal freeze-in [116]

$Y_{\chi}\simeq\dfrac{H_{\rm reh}^{2}H_{\rm
inf}}{s}\left\\{\begin{array}[]{ll}\dfrac{105|\kappa|^{2}}{64\pi^{4}}\dfrac{T_{\rm
max}^{12}}{H_{\rm inf}^{4}\,m_{\chi}^{8}}e^{-2m_{\chi}/T_{\rm
max}}f_{0}(m_{\chi}/T_{\rm max})\hskip 39.83368ptT_{\rm reh}\ll
m_{\chi},\vspace{0.25 cm}\\\ \dfrac{3|\kappa|^{2}}{2048\pi^{3}}\dfrac{T_{\rm
max}^{12}}{T_{\rm reh}^{7}\,H_{\rm inf}^{4}\,m_{\chi}}\hskip
142.26378ptm_{\chi}\ll T_{\rm reh},\end{array}\right.$ (24)

where $T_{\rm max}=T_{\rm reh}\,(H_{\rm inf}/H_{\rm reh})^{1/4}$ and $T_{\rm
reh}$ are respectively the maximal temperature after inflation and the
reheating temperature, and $f_{0}(x)\equiv
1+2x+2x^{2}+4x^{3}/3+2x^{4}/3+4x^{5}/15+4x^{6}/45+8x^{7}/315+2x^{8}/315$. In
Fig. 7, we show that a reheating temperature as low as $10^{3}$ GeV (for
$\kappa=1$) can induce a matter-domination era before BBN and leave an imprint
in the would-be SGWB from CS detectable by GW interferometers. For $\kappa=1$,
masses as large as $10^{10}$ GeV can be probed.

###  4.4 Heavy dark photons

#### The $U(1)_{D}$ dark photon:

We consider a $U(1)_{{\rm D}}$ gauge boson, $V_{\mu}$, the dark photon, of
mass $m_{{V}}$, kinematically coupled to the $U(1)_{{\rm Y}}$ gauge boson of
the SM [127, 128]

$\mathcal{L}\supset-\frac{\epsilon}{2c_{w}}F_{{\rm Y}\mu\nu}F_{{\rm
D}}^{\mu\nu},$ (25)

where $c_{w}$ is the cosine of the weak angle and $\epsilon$ is the dark-SM
coupling constant. The decay width into SM, $\Gamma_{V}$, is computed in
[105]. We here report the expression for $m_{{V}}\gtrsim 2m_{Z}$

$\Gamma_{V}\simeq\left(3\times
10^{-8}~{}\textrm{s}\right)^{-1}\,\left(\frac{\epsilon}{10^{-9}}\right)^{2}\left(\frac{m_{{V}}}{1~{}\text{TeV}}\right).$
(26)

The dark photon leads to an early-matter-dominated era if it has a large
energy density $m_{{V}}\,Y_{V}\gtrsim 10$ GeV and a long lifetime
$\tau_{{V}}\sim 10^{-8}$ s, c.f. Fig. 3 at $G\mu=10^{-11}$. Supposing that the
dark photon abundance is close to thermal, $Y_{{\rm V}}\sim 0.02$, c.f. Eq.
(28), this implies $\epsilon\lesssim 10^{-9}$. At such a low $\epsilon$, the
dark sector and the SM sector may have never been at thermal equilibrium (c.f.
[129] or footnote 8 in [130]) and may have their own distinct temperature. We
assume that the dark sector and the SM have a different temperature by
introducing the dark-to-SM temperature ratio [130]

$\tilde{r}\equiv\frac{\tilde{T}_{D}}{\tilde{T_{SM}}},$ (27)

where quantities with a $\sim$ on top are evaluated at some high temperature
$\tilde{T}$. Thus, the dark photon abundance before its decay is given by

$Y_{{\rm V}}=\frac{n_{{\rm V}}}{s_{{\rm
SM}}}=\frac{45\zeta(3)}{2\pi^{4}}\frac{\tilde{g}_{{D}}}{\tilde{g}_{{\rm
SM}}}\tilde{r}^{3}\simeq
0.0169\left(\frac{\tilde{g}_{{D}}}{6.5}\right)\tilde{r}^{3},$ (28)

where $\tilde{g}_{{D}}$ and $\tilde{g}_{{\rm SM}}$ are the relativistic number
of degrees of freedom in the dark sector and the SM at temperature
$\tilde{T}$. Plugging Eq. (28) into Eq. (11) implies a simple relation between
the temperature at which the dark photon dominates the universe $T_{\rm dom}$
and its mass $m_{{V}}$. We choose to be agnostic about the mechanism setting
the abundances in the dark sector and we enclose all possibilities by
introducing a dark-to-SM temperature ratio $\tilde{r}$222 Production of the
dark photon in the early universe has been studied in the literature. For a
small kinetic mixing $\epsilon$, the abundance of the dark sector can be set
non-thermally either by freeze-in [131, 132, 133, 129], or by a separate
reheating mechanism. In the latter case, the temperature asymmetry in Eq. (27)
results from an asymmetric reheating [134, 135, 136, 137]. For moderate
kinetic mixing $\epsilon\gtrsim 10^{-6}\sqrt{M_{{\rm DM}}/\text{TeV}}$ [129],
the dark sector may have been at thermal equilibrium with the SM, but
asymmetric temperatures can result from asymmetric changes in relativisitic
degrees of freedom [130]. On the other hand, a possibility for thermally
equilibrating the $U(1)_{D}$ sector and the SM in the case of a small kinetic
mixing $\epsilon$ would be to introduce a dark Higgs $\phi$, mixing with the
SM Higgs, which once at thermal equilibrium with SM, decays into dark photons.

As shown in the left panel of Fig. 8, low kinetic mixing $\epsilon$, large
mass $m_{{V}}$ or large dark-to-SM temperature ratio $\tilde{r}$ lead to an
early-matter-dominated era, triggered when $T_{\rm dom}\gtrsim T_{\rm dec}$.
The non-detection with a future GW interferometer, of the imprint left by such
a matter era, in the GW spectrum from CS, would exclude the existence of the
dark photon for given values of the kinetic mixing, the dark photon mass and
the dark-to-SM temperature ratio $(\epsilon,\,m_{{V}},\,\tilde{r})$. We show
the GW-from-CS constraints on the dark photon in the right panel of Fig. 8,
together with existing constraints coming from supernova SN1987 [138, 139] and
beam-dump experiments [130]. Other constraints on lighter dark photons do not
appear on the plot and are summarized in the reviews [140, 141, 142]. We also
include the BBN constraint which imposes the dark photon to decay before
$\tau_{{V}}\lesssim 0.1$ s [2, 3, 4] or later if the energy density fraction
carried by the dark photon is smaller than $\sim 10\%$ [130]. Note, that only
the BBN and the GW-from-CS constraints depend on the dark-to-SM temperature
ratio $\tilde{r}$ which fixes the abundance of the dark photon in the early
universe.

Figure 8:  Left: Constant dark photon lifetime $\tau_{V}$ contours. For a
given dark-to-SM temperature ratio $\tilde{r}\equiv T_{{D}}/T_{{\rm SM}}$, a
non-standard early matter domination is induced below the corresponding orange
line where the dark photon dominates the universe before it decays. Right:
Expected constraints on the dark photon mass $m_{{V}}$ and kinetic mixing
$\epsilon$, assuming the measurement of a GW spectrum from CS with tension
$G\mu=10^{-11}$ by future GW interferometers. We use the turning-point
prescription (Rx 1) discussed in Sec. 3.2.

Figure 9:  Additional constraints when the dark photon is embedded in a DM
model as the mediator of $U(1)_{D}$-charged DM (see text). We compare the
expected GW constraints from cosmic strings with the existing constraints on
the $U(1)_{\rm D}$ DM model: Supernovae bounds from [138] and [139], direct
detection bounds from [130] and the indirect detection + CMB constraints are a
rough estimate from [105]. Beam dump constraints are also taken from [130].
The unitarity bound on the DM mass $M_{{\rm DM}}$ [143] can also be applied on
the mediator mass because of the kinematic condition $m_{{V}}<M_{{\rm DM}}$.
The unitarity bound gets relaxed at small $\epsilon$ because of the larger
entropy injection following the dark photon decay [105].

We can appreciate the complementarity between the well-established supernova,
beam dump, BBN constraints, and the expected constraints assuming the
detection of a SGWB from CS by the GW interferometers. Indeed, whereas
supernova and beam dump do not really constrain above $m_{{V}}\gtrsim 0.1$
GeV, the detection of a SGWB from CS with a string tension $G\mu\simeq
10^{-11}$ would exclude dark photon masses up to the maximal reheating
temperature $m_{{V}}\sim 10^{16}$ GeV allowed by the maximal inflation scale
$H_{\rm inf}\lesssim 6\times 10^{13}$ GeV [122, 123], and kinetic mixing as
low as $\epsilon\sim 10^{-18}$.

#### The dark photon as a dark matter mediator:

An interesting motivation for the dark photon is that it can play the role of
a dark matter mediator. We can suppose that the dark sector also contains a
Dirac fermion $\chi_{{\rm D}}$ charged under $U(1)_{{\rm D}}$, playing the
role of DM [144, 145, 146, 147, 148, 149, 150, 151, 152, 130, 105]

$L\supset\quad\bar{\chi}_{{\rm D}}i\not{D}\chi_{{\rm D}}-M_{{\rm
DM}}\bar{\chi}_{{\rm D}}\chi_{{\rm D}},$ (29)

where $D_{\mu}=\partial_{\mu}+ig_{{\rm D}}V_{{\rm D}\mu}$ is the covariant
derivative with $g_{{\rm D}}$ the $U(1)_{{\rm D}}$ gauge coupling constant. We
suppose that the DM freezes-out by annihilating into pairs of dark photons, we
impose $m_{{V}}<M_{{\rm DM}}$. We assume the dark photon to be non-
relativistic when it decays but relativistic when it is produced, therefore,
we set $\tilde{g}_{{D}}=3+\frac{7}{8}\cdot 4=6.5$ in Eq. (28).

The unitarity bound on the DM mass $M_{{\rm DM}}$ can be applied to the dark
photon mass $m_{{V}}$ upon assuming $m_{{V}}<M_{{\rm DM}}$. In the standard
paradigm, the unitarity bound on s-wave annihilating dirac fermion DM is
$M_{{\rm DM}}\lesssim 140$ TeV [143, 153]. However, if long-lived and heavy,
the decay of the mediator can, by injecting entropy, dilute the DM abundance
and relax the unitarity bound to [105]

$M_{{\rm DM}}\lesssim 140~{}\text{TeV}~{}\sqrt{D},$ (30)

where $D$ is the dilution factor $D\simeq{T_{\rm dom}}/{T_{\rm dec}}$, $T_{\rm
dom}$ and $T_{\rm dec}$ are as defined in Eq. (11) and Eq. (12).

In Fig. 9, we add the contraints on the dark photon when the later plays the
role of the mediator of DM. They come from direct detection [130], CMB [105],
indirect detection, using neutrino, gamma-rays, positrons-electrons and anti-
protons [105], as well as from unitarity [105]. They are complemented by the
GW-from-CS constraints. For $\epsilon\lesssim 10^{-10}$, all the traditional
indirect detection constraints evaporate and the unitarity bound is pushed to
larger masses due to the entropy dilution following the dark photon decay such
that the model is then currently only constrained by BBN. It is remarkable
that GW interferometers could probe this unconstrained region where
$\epsilon<10^{-10}$ and $m_{V}>1$ GeV. In Fig. 8 and 9, we use the turning-
point prescription (Rx 1) discussed in Sec. 3.2. Constraints are stronger when
considering the spectral-index prescription (Rx 2), as shown in Fig. 4 or in
Fig. 2.

Figure 10:  Scenario where the dark photon mass $m_{{V}}$ and the cosmic
string network are generated by the spontaneous breaking of the same $U(1)$
symmetry, such that $m_{{V}}$ is related to the string tension $\mu$. Pale
colors: Constraints on the dark photon parameter space assuming the mere
detection of the GW spectrum from CS by NANOGrav, LISA, ET, CE, DECIGO and
BBO. Opaque colors: Constraints assuming the detection of the turning point in
the GW spectrum induced by the transition from matter to radiation when the
heavy dark photon decays. When combining Eq. (10) and Eq. (12), this last
detection allows to measure the dark photon lifetime. The constraints
described in the following part of this caption are independent of the GW
emission. Pale red: The non-observation of the fundamental tensor B-modes in
the CMB imposes the stringest upper bound on the energy density scale of
inflation [123], $V_{\rm inf}\lesssim 1.6\times 10^{16}~{}\rm GeV$. This
provides an upper-bound on the reheating temperature, which also must satisfy
$\left<\phi_{{\rm D}}\right>\lesssim T_{\rm reh}$ in order for the string
network to be formed. Thus, we impose the CS formation to occur after the end
of inflation with the following criteria: $\left<\phi_{{\rm D}}\right>\lesssim
V_{\rm inf}$. Pale purple: Constraints from the non-observation of line-like
temperature anisotropies in the CMB, e.g. [154], $G\mu\lesssim 2\times
10^{-7}$. Pale sky blue: In order to prevent DM overclosure, we must assume
the $U(1)_{D}$ charged states to be heavier than the reheating temperature
such that they are never produced. A possibility which is constrained by the
Weak Gravity Conjecture, c.f. main text, thus we impose $\left<\phi_{{\rm
D}}\right>\lesssim g_{{\rm D}}M_{\rm pl}$. The last inequality implicitly
assumes $\left<\phi_{{\rm D}}\right>\lesssim T_{\rm reh}$. Note however that
such a charged state could be unstable, e.g. if it is a dark Higgs, in which
case the WGC constraint is relaxed.

#### Scenario where the cosmic string network and the dark photon mass have
the same origin:

As a last remark, we comment on the case where the spontaneous breaking of the
$U(1)_{D}$ symmetry would be responsible for the formation of the cosmic
string network, so that the dark photon mass is no longer a free parameter but
is related to the string tension $\mu$, through the Abelian-Higgs relations
[6]

$\displaystyle\mu=2\pi<\phi_{{\rm D}}>,$ (31) $\displaystyle
m_{{V}}^{2}=2g_{{\rm D}}^{2}<\phi_{{\rm D}}>^{2},$ (32)

where $\phi_{{\rm D}}$ is the scalar field whose vacuum expectation value
$<\phi_{{\rm D}}>$ breaks the $U(1)_{D}$ symmetry spontaneously.

In this case, we find that most of the relevant parameter space is ruled out
due to overabundance of dark matter.333The cross-section of a pair of
$U(1)_{D}$ fermions annihilating into dark photons is given by $\sigma
v\simeq\pi\alpha_{{\rm D}}^{2}/m_{\psi}^{2}$ with $\alpha_{{\rm D}}=g_{{\rm
D}}^{2}/4\pi$. It is way too weak to prevent universe overclosure, except if
we tune the Yukawa coupling of the fermion, $\lambda$, defined by
$m_{\psi}=\lambda\left<\phi\right>/\sqrt{2}$, to very small values. The only
viable solution would be to assume that the states which are charged under
$U(1)_{D}$ and stable under decay, are heavier than the reheating temperature
such that they are never produced. The Weak Gravity Conjecture (WGC) requires
the existence of a charged state with mass smaller than [155]

$m_{\rm X}\lesssim g_{{\rm D}}M_{\rm pl}.$ (33)

Hence, $g_{{\rm D}}M_{\rm pl}$ sets the maximal reheating temperature, above
which charged states responsible for universe overclosure might be produced.
Therefore, we should exclude the parameter space where the temperature of the
$U(1)_{D}$ spontaneous breaking, taken as $\sim\left<\phi_{{\rm D}}\right>$,
is heavier than $g_{{\rm D}}M_{\rm pl}$, c.f. pale sky blue region in right
plot of Fig. 10. Note that the WGC does not specify if the suggested charged
state is stable under decay or not. For instance, it would be stable and
overclose the universe if it is a $U(1)_{D}$ fermion but not if it is a
$U(1)_{D}$ Higgs, which can still decay into a dark photon pair when
$m_{\phi_{{\rm D}}}\gtrsim 2m_{{V}}$. Hence, the WGC constraint in our
parameter space has to be taken with a grain of salt.

Assuming a natural gauge coupling value, $g_{{\rm D}}=10^{-1}$, we find that
dark photons heavier than $\gtrsim 100~{}\rm PeV$ would be accompanied with a
$U(1)_{D}$ cosmic string network producing an observable GW spectrum, see left
plot of Fig. 10. In the case where $g_{{\rm D}}=10^{-4}$, we could probe dark
photon masses down to $\gtrsim 100$ TeV, see right plot of Fig. 10.

On the same plot, we superpose the constraints, shown with pale colors, coming
from the simple observation of the GW spectrum with future experiments (except
NANOGrav which is already operating), and the constraints, shown with opaque
colors, coming from the detection of the turning point where the spectral
index of the GW spectrum changes due to the decay of the dark photon which was
dominating the energy density of the universe. The former detection would
allow to measure the dark photon mass whereas the latter detection would allow
to access its lifetime.

##  5 Summary

If future GW observatories have the sensitivity to detect stochastic GW
backgrounds of primordial origin and to measure precise features in this
spectrum, they can reveal very unique information about very high scale
physics. Particularly relevant sources of GW are cosmic strings. Cosmic
strings are almost ubiquitous in many Grand-Unified Theories. As they keep
emitting throughout the whole cosmological history of the universe, the
resulting GW spectrum covers a wide range of frequencies and can be detected
either by space-based or ground-based observatories. An early era of matter
domination due to new heavy particles generates a clear signature in the GW
spectrum of cosmic strings.

In this study, we assume the existence of an early matter era due to the
presence of a cold particle $X$ temporarily dominating the energy density of
the universe and decaying before the onset of BBN. We compute its impact on
the GW spectrum of CS beyond the scaling regime. We show that detecting such a
feature and interpreting it in terms of a new heavy relic can lead to
unparalleled constraints in the $(\tau_{X},\,m_{X}Y_{X})$ (lifetime, yield)
plane. In Fig. 3, we provide model-independent constraints which extend the
usual BBN constraints on the lifetime $\tau_{X}$ by $15$ orders of magnitude
for $G\mu=10^{-11}$, as we are able to constrain early matter dominated era
ending when the temperature of the universe is between $50$ TeV and $1$ MeV.

Next, we show that this new search strategy is likely to provide unprecedented
constraints on particle physics models. We illustrate this on minimal models
of massive particles. In the first class, the heavy particle has only
gravitational interactions and decays though a Planck-suppressed coupling,
c.f. Fig. 4. In the second class, the heavy relic is a new $U(1)$ gauge boson
that decays to the Standard Model via kinetic mixing to $U(1)_{Y}$
hypercharge. We point out that supersymmetric theories could be probed, well
above the reach of present and future colliders, up to a gravitino mass scale
of $10^{10}$ GeV, due to the presence of oscillating scalar moduli fields
produced after dynamical supersymmetry breaking, c.f. Fig. 5. Secondly, we
study a simple model of massive scalar particle interacting only
gravitationally with the Standard Model and which therefore has no chance to
be observed in collider or direct/indirect detection experiments. If non-
conformally coupled to gravity, it can be abundantly produced by gravitational
effects at the end of inflation, hence leading to a matter era, which
stochastic gravitational-wave backgrounds from cosmic strings can uniquely
probe, c.f. Fig. 6. Finally, we study a model of dark photon kinematically
coupled to the Standard Model hypercharge, possibly embedded in the $U(1)_{D}$
dark-photon-mediated dark matter model. The constraints we obtain from GW on
$U(1)_{D}$ dark matter falls in the large mass/small kinetic mixing ballpark
which is otherwise unreachable by any current and probably future
direct/indirect detection and CMB constraints, c.f. Fig. 9. At last, we
consider the possibility that the dark photon mass and the cosmic string
network are generated by the spontaneous breaking of the same $U(1)$ symmetry
and show that we can use future GW interferometers to probe dark photon masses
above $100~{}\rm PeV$, or even down to the $\rm TeV$ scale if we tune the
gauge coupling to small values, see Fig. 10.

These are only a few minimal examples of particle physics models generating
early matter eras. There are many other well-motivated models which would
deserve consideration in this respect. We will present the corresponding
constraints on axion-like-particles and primordial black holes in a separate
study.

## Acknowledgements

We thank Quentin Bonnefoy, Roberto Contino, Valerie Domcke, Yohei Ema, Filippo
Sala and Jérôme Vandecasteele for useful comments and discussions. This work
is supported by the Deutsche Forschungsgemeinschaft under Germany’s Excellence
Strategy - EXC 2121 “Quantum Universe” - 390833306. The work of Y.G. is partly
supported by a PIER Seed Project funding ‘Dark Matter at 10 TeV and beyond, a
new goal for cosmic-ray experiments’ (Project ID PIF-2017-72). P.S.
acknowledges his master-degree scholarship from the Development and Promotion
of Science and Technology Talents project (DPST), Thailand.

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