Primordial Black Hole Archaeology with Gravitational Waves from Cosmic Strings

Light primordial black holes (PBHs) with masses smaller than $10^9$ g ($10^{-24} M_\odot$) evaporate before the onset of Big-Bang nucleosynthesis, rendering their detection rather challenging. If efficiently produced, they may have dominated the universe energy density. We study how such an early matter-dominated era can be probed successfully using gravitational waves (GW) emitted by local and global cosmic strings. While previous studies showed that a matter era generates a single-step suppression of the GW spectrum, we instead find a"double-step"suppression for local-string GW whose spectral shape provides information on the duration of the matter era. The presence of the two steps in the GW spectrum originates from GW being produced through two events separated in time: loop formation and loop decay, taking place either before or after the matter era. The second step - called the"knee"- is a novel feature which is universal to any early matter-dominated era and is not only specific to PBHs. Detecting GWs from cosmic strings with LISA, ET, or BBO would set constraints on PBHs with masses between $10^6$ and $10^9$ g for local strings with tension $G\mu = 10^{-11}$, and PBHs masses between $10^4$ and $10^9$ g for global strings with symmetry-breaking scale $\eta = 10^{15}~\mathrm{GeV}$. Effects from the spin of PBHs are discussed.

The detection of gravitational waves (GWs) from astrophysical sources by the LIGO-Virgo collaboration in 2015 [92] opened the door to GW astronomy.Upcoming upgrades of LIGO-Virgo [93] and proposed future detectors such as LISA [94], BBO-DECIGO [95], the Einstein Telescope (ET) [96,97], and Cosmic Explorer (CE) [98] will open up a new observation window of the early universe.Unlike photons, primordial GWs that were emitted in the early Universe can propagate freely throughout cosmic history and therefore would constitute ideal messengers of our Universe history [99][100][101].One of the pinnacles for determining the pre-BBN cosmic history of the universe would be the detection of GW sourced by a network of cosmic strings.Cosmic strings are one-dimensional objects produced by the spontaneous breaking of a U (1) symmetry in the early universe [102,103], the pure Yang-Mills theory [104,105], or fundamental objects in superstring theory [106][107][108][109][110]. The crucial peculiarity of cosmic strings is that they are long-standing GW sources [111][112][113][114]. Once the network of cosmic strings is produced, it occupies a constant fraction of the total energy density of the universe, a property known as the scaling regime [115][116][117][118][119][120].An important consequence is that GW emissions occur during most of the universe history.This generates a GW spectrum spanning many orders of magnitude in frequencies.A measurement of the GW spectrum from high to lower frequencies would allow to determine the universe expansion rate from early to later times [16,[121][122][123][124][125][126][127][128].Instead, the use of short-lived GW sources like the ones from first-order phase transitions can only probe the cosmic history within a small window of time around when the source is active [22,[129][130][131][132].Another long-standing GW source that could bring information about the equation of state of the pre-BBN universe is tensor modes sourced during primordial inflation [133][134][135][136][137]. The BICEP-Keck bound on the tensor-to-scalar ratio r ≲ 0.036 [138] constrains the inflation scale to H inf ≲ 3 × 10 13 GeV which prevents the detectability of tensor modes by GW detectors in a foreseeable future [139][140][141][142].However, intermediate cosmological eras can imprint signatures on such GW spectrum [22,23,[143][144][145][146].
In this work, we explore how a period of early matter domination that originates from a large production of PBHs in the early Universe could leave observable imprints in the spectrum of GWs generated by a network of cosmic strings.We do not assume any connection between the sector producing PBHs and the one producing cosmic strings.The paper is organized as follows: in Sec. 2, we review the calculation of the GW spectrum from cosmic strings, both in the local and global cases, as illustrated in Fig. 1.Readers familiar with cosmic strings can jump directly to Sec. 3 where we study the impact of a matter-dominated era.The main smoking-gun GW signatures are summarized in Fig. 2, including the double-step feature and its associated knee, which are studied for the first time in this work.Finally, in Sec. 4, we use those findings to constrain the abundance and mass of primordial black holes formed with a monochromatic distribution, see Fig. 6 and 7 for local and global strings, respectively.We would like to emphasize that the PBHs considered in this work evaporated well before BBN.Hence the constraints are shown as the fraction of energy density β in PBH at formation, instead of today PBH fraction Ω PBH .We discuss these results and conclude in Sec. 5.

GW from Cosmic Strings
A cosmological phase transition associated with the spontaneous breaking of U (1) symmetry leads to the formation of a network of topological defects called cosmic strings (CS).The U (1) symmetry can be either local or global.We refer to Ref. [103] for the original article, Ref. [114] for a textbook, and Refs.[9,101,124,147,148] for reviews of their GW emission.
Cosmic strings are field configurations at the top of the U (1)-breaking Mexican hat potential.The field energy is localized within a core of inverse symmetry-breaking scale size, much smaller than the cosmological horizon.This motivates the Nambu-Goto approximation which describes CS as infinitely thin classical objects with energy per unit length µ, for global strings, ( where η is the vacuum expectation value of the scalar field constituting CS, and n is the winding number taken to be n = 1 since it is the only stable configuration [149].For global strings, the logarithmic divergence arises due to the massless Goldstone mode leading to the presence of long-range gradient energy [114].The CS network forms at the temperature of the U (1)breaking phase transition, where the second equality refers to local strings.

Evolution of the network
In general, one should expect the motion of cosmic strings to be initially frozen due to thermal friction, characterized by the frictional damping length l f = µ/(βT 3 ) where β = O(1) represents the strength of particle-string interactions [150][151][152].At that time the energy density of frozen CS evolves under Hubble expansion as ρ CS ∝ a −2 where a is the scale factor [114], and would quickly dominate the energy density of the universe.Effects from friction become negligible when the damping length l f becomes larger than the cosmic horizon t, i.e., when (2.3) Below that temperature, CS start to move freely under the work of their own tension, they reach velocities of order O(0.1) and start to interact with each other [114].The intersection of straight strings forms loops that decay by radiating particles and GW.Loop formation acts as a loss mechanism for the infinitely long-string network.Out of those two antagonist effects, Hubble expansion and loop formation, which respectively increase and decrease the fractional CS abundance Ω CS ≡ ρ CS /ρ tot , the CS network reaches a stable state called the scaling regime where Ω CS remains constant over time [115][116][117][118][119][120].In the scaling regime, CS redshifts the same way as the background, e.g., ρ CS ∝ a −4 during radiation-domination (RD) and ρ CS ∝ a −3 during matter-domination (MD).One can say that the equation of state of the CS network tracks the equation of state of the expanding background.
The GW signal from CS is dominated by emission from loops [112,114,153].Loops of strings are constantly being produced as long strings self-intersect.Local string loops decay into GW after 0.001/Gµ ≫ 1 Hubble time, while global string loops decay into Goldstone modes in less than one Hubble time, cf.Eq. (2.16).
• Step 1: Loop Formation.We denote α the size of loops in units of the cosmic horizon t i when they form.The rate of loop formation rate can be written [114] where factor 0.1 indicates that 90% of the loop population is small and highly-boosted loops that are red-shifted away and do not contribute substantially to the GW signal [154].
We have introduced the loop formation efficiency factor C eff , which for local strings reaches the asymptotic value C eff ≃ 0.39, 5.4, 29.6 during matter-domination (MD), radiationdomination (RD), and kination eras, respectively [124].For global strings, the long strings lose energy via Goldstone emission on top of loop formation, which logarithmically suppresses the loop production efficiency.At the analytical level, we take C eff ∼ O(1) for all cosmological eras [124,155,156].However, for the plots and analysis of this paper, we solve C eff (t) as a solution of the velocity-dependent one-scale (VOS) equations governing the string network evolution.As studied thoroughly in Ref. [124,157], the VOS evolution captures the inertia of the network during a change in the equation of state of the universe.Instead, approximating C eff using a piecewise constant function across different eras overestimates the value of the turning-point frequency in Eq. (2.17) by more than one order of magnitude [124].
• Step 2: GW Emission from Loops.Numerical simulations [154] have shown that the GW spectrum is dominantly produced by loops with the largest size, corresponding to 10% of the horizon.We account for this result by choosing a monochromatic loop size probability distribution After their formation, t > t i , loops of length l( t) oscillate and radiate a discrete spectrum of GWs with frequencies given by f = 2k/l( t), k ∈ Z + . (2.6) The frequency today is given by f = f a( t)/a 0 .The GW emission power by a loop is independent of its size, which is a remarkable result of the quadrupole formula [124].For each Fourier mode k, it is given by and Γ = 50 for local [158] and global strings [159].The index δ depends on whether high Fourier modes are dominated by cusps (δ = 4/3), kinks (δ = 5/3), and kink-kink collisions (δ = 2) [160].This choice does not impact much the amplitude of the GW spectrum in the deep RD and MD era, but it substantially impacts the slope around the transition between early MD and RD, see Sec. 3.1.In this work, we assume the small-scale structure to be dominated by cusps δ = 4/3.While loops continuously lose energy as they emit either GWs or Goldstone modes, their length l shrinks as where ΓGµ and κ are the shrinking rates due to GW and particle emissions, respectively.

GW spectrum
From right to left, we write in chronological order the different involved processes leading to the final expression for the spectral energy density of GWs from CS, defined as First, loops are formed at a rate dn loop /dt i (note the chain rule dt i /df ) with a distribution of size P loop (α).They redshift like a −3 before radiating GW with power P GW which subsequently dilute as a −4 .We add two Heaviside functions to cut off the GW spectrum above some characteristic frequencies discussed below.Finally, we integrate over all loop sizes α and GW emission times t and sum over all Fourier modes k.
The first Heaviside function Θ(t i − l * /α) eliminates loops smaller than a critical length l * below which massive particle production is the main decay channel [124,162].The second Heaviside function Θ(t i − t osc ) with t osc = Max [t form , t fric ] gets rid of loops which would have formed before the formation of the network, cf.Eq. (2.2), or which form in the friction-dominated epoch, cf.Eq. (2.3).As shown in [124], these high-frequency cut-offs lie at frequencies higher than the windows of current and future GW interferometers.After this short introduction, we simplifies Eq. (2.9) to the handy form, where Ω r h 2 ≃ 4.2 × 10 −5 [163] is the radiation density today and TM is the temperature at time tM of maximum emission defined in Eq. (2.16).Deviations from flatness arise due to a change in the number of relativistic degrees of freedom captured in [124] = 0.39 106.75 g * (T ) In comparison, the GW spectrum from global strings is suppressed by (η/M pl ) 3 due to the shorter loop lifetime and enhanced by a log 3 factor due to the larger string tension 3 We refer to Ref. [124] for a detailed discussion about the differences between GW from local and global strings.The impact of a non-standard cosmological evolution, such as a PBH-dominated universe, is studied in Sec. 4.

Temperature-frequency relation
The dominant emission in GW arises at the end of the loops lifetime when the loop size in Eq. (2.8) is half its initial size l( tM ) = αt i /2 [122,124]

.16)
The global-string loops decay fast after their production, while the local-string loops live much longer.The frequency today f is related to the emitted frequency f by f = f × a( t)/a 0 .The emitted frequency f is related to the loop length l( tM ) = αt i /2 by Eq. (2.6).Assuming RD followed by the standard ΛCDM evolution after loop formation, we obtain the relation between the GW frequency today and the temperature when the loop dominantly sourcing this frequency mode is produced (local strings), (global strings), (2.17) 2 Keeping all the parameters, we get [124] (2.11) 3 The GW spectrum of global strings in radiation-dominated universe reads [124] (2.14) We have multiplied the numerical-fitted factors4 of ≈ 0.03 and ≈ 0.2 for local and global strings, respectively, to account for VOS evolution [124].We now use Eq.(2.17) to compute the frequencies beyond which the GW spectrum is cut off due to network formation, friction, and particle production.Those cut-offs are away from the reach of current and planned GW interferometers.This has to be contrasted with the frequency of the turning point caused by the presence of an early matter-domination era -and in particular, a PBH-dominated era -which can lie within the detectable windows.
Formation cut-off.-NoGW can be produced before the cosmic strings network is formed.Plugging the temperature at which the U (1) symmetry is spontaneously broken in Eq. (2.2) into Eq.(2.17) leads to the high-frequency cut-off for local, 0.47 GHz for global, ( where T 0 denotes the temperature of the Universe today.We recall that α is the string size at formation in a unit of the cosmic horizon and Γ is the GW emission efficiency of a string loop. Friction cut-off.-GWemission starts when effects from friction become negligible and cosmic strings start oscillating at relativistic speed.Plugging Eq. ( 2.3) into Eq.(2.18), we obtain the high-frequency cut-off due to loop motion being frozen at earlier times g * (T fric ) 100 for local, for global. ( Massive particle production cut-offs.-Aspreviously discussed, global strings exhibit suppressed GW emission due to the efficient production of massless Goldstone modes.On the other hand, particles in the gauge sector of local strings are massive, resulting in significant suppression of their production rate [164][165][166].The emission of massive particles occurs when Fourier modes surpass the mass gap, which happens during cusps or kink-kink collisions.It has been found that the power emitted in massive particles only exceeds the power emitted in GWs for loops smaller than the critical length [124,162] 5 where m = 1 or 2 for loops kink-dominated [170] or cusp-dominated [171][172][173], respectively, and β m ∼ O(1).Loops with lengths smaller than l part should be subtracted when computing the SGWB.Plugging Eq. (2.20) into Eqs.(2.6) and (2.17) gives the frequency cut-offs (2.21)
Other signatures from Nambu-Goto strings result from the static gravitational field around the string.This can induce gravitational lensing and temperature anisotropies in the CMB.The resulting constraint Gµ ≲ few × 10 −7 , e.g., [181,182], is however much looser than the one from GW production.However, a recent study has shown that the strong gravitational lensing of the fast radio bursts could probe down to Gµ ∼ 10 −9 with future radio telescopes [183].
Global strings.-Globalcosmic strings efficiently produce massless Goldstone particles that contribute to the number N eff of effective relativistic degrees of freedom.The precise constraint relies on the abundance of Goldstone particles from strings which is still debatable. 6We quote the upper bound η ≲ 3.5 • 10 15 GeV derived in Ref. [156] and refer to Refs.[159,190] for slightly tighter bounds.
The absence of B-mode polarization in the CMB provides another constraint on global strings.Assuming instantaneous reheating and only SM degrees of freedom, the upper limit on the inflationary Hubble parameter H inf ≲ 3 × 10 13 GeV [138] translates to the maximum temperature of the universe T max ≲ 4 × 10 15 GeV.For the string network to form, the string scale η must be smaller than the maximum temperature η ≲ 4 × 10 15 GeV, up to O(1) modeldependent parameters.
For η ≳ 10 15 GeV, GW from global strings extend to f ≲ 10 −14 Hz which could leave signature in CMB polarization experiments, e.g.Ref. [138].Nonetheless, GW in this frequency range is produced after photon decoupling, and the CMB constraint is evaded; see Eq. (2.17) or Fig. 8 of Ref. [156].

GW signatures from an Early Matter-Dominated Era
We consider a period of early matter domination (EMD) inside the usual radiation era.We parametrize the EMD era by two parameters: 1. the temperature T dec of the thermal plasma when the EMD ends, and 2. the duration of the EMD characterized by the number of e-folds where T dom is the radiation temperature when the EMD era starts.During a matter era, the universe expands faster than during a radiation era, inducing a double-step and single-step suppression of the GW spectrum from local and global strings, respectively, see Fig. 1.
In this section, we scrutinize these step signals and find three smoking-gun features -shown in Fig. 2 -which carry direct information about the EMD: In the following subsections, we discuss their origins, derive the value of the frequency at their position, and calculate their detectability in future GW observatories.We emphasize that the present paper is the first to introduce the double-step feature; see Sec. 3.4 for more details.Fig. 2: Left: Three features in the local-string GW spectrum induced by the EMD era.Their origins can be understood, from the (k = 1) spectrum, as different loop populations produced at time t i and emitting GW at time tM before and/or after the EMD era.The GW spectrum from the local string shows a double-step feature.The steep slope of the 1st step is caused by loops formed during the matter era.The steep slope of the 2nd step results from loops decaying during the matter era.The knee is defined as the local maximum of the second step.Right: For global strings, the GW spectrum exhibits a single step due to the loop lifetime being short.

Spectral index
Cosmic-strings GW via a two-step process.-Asmentioned in Sec. 2, the cosmic-string network first produces string loops whose energy density red-shifts as non-relativistic matter, ρ loop ∝ a −3 .Loops dominantly contribute to the GW spectrum at a time tM .It is defined by the time when the string length has shrunk by a factor two l( tM ) = l(t i )/2; see Eq. (2.16).The fraction of energy density in GW today from cosmic-string loops -produced at time t i and emitting GW at time tM -is where i, M , and 0 denote the epochs of loop formation, main GW emission, and today, respectively.These three epochs are related through Eq. (2.16) and f ≃ (4k/(αt i ))(a M /a 0 ), cf.Eq. (2.6).In Eq. (3.2), we have used that ρ loop,M ≃ ρ GW,M (from energy conservation), ρ loop ∝ a −3 , ρ GW ∝ a −4 , and ρ loop,i ∝ ρ tot,i (due to the scaling regime).The GW amplitude today in Eq. (3.2) is sensitive to the cosmic history around the time t i of loop production and the time tM of GW emission.For global strings, the short loop lifetime implies that the two processes are simultaneous, and the GW emission occurs as soon as the loop is formed tM ≃ t i .Fundamental Fourier mode only.-We assume that the universe evolves as ρ tot ∝ a −n and a −m around loop formation at t i and dominant GW emission at tM respectively.The first Fourier mode k = 1 of the GW spectrum in Eq. (2.10) can be expressed as the power-law where B(f ) and β have analytical expressions [22] (see also [101,121,122,124,155,156]), 7local global where f dec is the GW frequency related via Eq.(2.17) to the temperature when the Universe becomes radiation-dominated again.The GW spectrum exhibits different characteristics depending on when the loop formation and emission occur in the history of the universe.For instance, when both events occur during radiation domination with n = m = 4, the GW spectrum is flat in frequencies f 0 .On the other hand, when a loop forms during matter domination with m = 3 and emits during radiation domination with n = 4, the resulting spectrum has a tilt of f −1 .Finally, when loop formation and GW emission happen during matter domination with m = n = 3, the analytical prediction for the spectral tilt is f −2 .This result contradicts numerical simulations, as shown in Fig. 1, where f −1 is observed.The reason for this discrepancy is that the assumption of a single time tM for GW emission is no longer valid in this regime.Effect from higher Fourier modes.-The discussion in the paragraph above only includes the fundamental mode k = 1.Effects from higher Fourier modes have been shown in [124,156,191] to lead to a departure from f −1 , where k max is the maximal excited Fourier mode, which can be quite large cf.Eq. (A.2) in App.A, and δ is the spectral tilt of the GW emission power of a string loop, cf.Eq. (2.7).
In this paper, we assume a cusp-dominated small-scale structure for which δ = 4/3 [160].We obtain the spectral index f −1/3 .For long matter era, we indeed observe a spectral slope −1/3, see Fig. 2 and 4. For the short matter era, a more complex spectral shape emerges, with a distinctive knee.

Low-frequency turning point
The low-frequency (LF) turning point separates GWs emitted by loops formed during the radiation era, characterized by a slope f 0 , from those formed during the matter era, characterized by a slope f β with 0 < β ≤ 1/3.The frequency f dec of the LF turning point in Eq. (2.10) is sensitive to the temperature T dec at which the EMD era ends.Fig. 3 shows the parameter space where future GW experiments can detect the LF turning point for both local and global cosmic strings.The detection criterion for this analysis is SNR ≥ 10, obtained by comparing the GW signal with the power-law integrated sensitivity curves defined in App.D. The red regions in the figure indicate where the particle production and friction cut-offs, given by Eqs.(2. 19) and (2.21), respectively, lies at higher frequencies than f dec .

High-frequency plateau
The plateau at high-frequency results from GW emitted during the radiation era preceding the EMD era at a temperature T M,HF > T dom .T M,HF is the temperature when loop produced at T HF release most of their energy into GW.From Eq. (3.2), GW in the HF plateau is suppressed with respect to GW at the LF turning point by a HF a M,HF where the second step uses ρ MD ∝ a −3 and ρ RD ∝ a −4 G(T ) with G(T ) in Eq. (2.13), and the third step uses a(t) ∝ G 1/4 (T )t 1/2 (from Friedmann's equation) and the loop lifetime in Eq. (2.16).T M,dec is the temperature when loop produced at T dec release most of their energy into GW.The function G(T ) varies from ≃ 0.4 at high temperatures to 1 at low temperatures.
The last bracket reduces to O(1) and 1 for local and global strings respectively.The suppression of the HF plateau encoded in Eq. (3.5) is visible in Fig. 1.In App.C.1, we also show the cosmicstring GW spectrum when the number of degrees of freedom g * and g * s are fixed, i.e. taking G(T ) ≡ 1.For the global string, the HF plateau f 0 has a distinct HF turning point at f = f dec below which the slope turns to f 1/3 .In App.B, we provide an analytic formula for the HF turning point of global strings and argue that it is difficult to detect, e.g., see Fig. 11.For local string, the spectral slope below f < f dec is not exactly −1/3 due to the 'knee' feature, which we now discuss below.

The knee feature
In the case of local strings, the GW spectrum has an additional feature due to loops living longer than the duration of the EMD.In Fig. 2, we show for the first time that the EMD imprints not only one but two steps in the GW spectrum.They originate from the production of GWs occurring in two steps: first, loops are formed at a time t i , and second, they decay at a later time tM .The first step suppression in the GW spectrum is attributed to the formation of loops during the EMD, while the higher-frequency step suppression results from the decay of loops during the EMD.The intermediate region of the GW spectrum is sourced by string loops that formed prior to the onset of the EMD era and emitted most of their energy into GWs after the EMD era ended.Fig. 4 shows the numerical results of the spectral slopes for different Gµ and EMD durations.We call the knee the local maximum of the second step, between the LF turning point and the HF plateau; see the vertical dashed line.The knee feature can also be seen clearly in Figs. 1 and 2. Such features do not show up in the GW spectrum from global strings because the short lifetime of loops merges the two steps into one.
Although the knee feature is the smoking-gun signature of the EMD era, the same underlying physics can also leave an imprint on the local-string GW spectrum, assuming the standard ΛCDM history [158], shown in Fig. 1.The spectral peak -located around f GW ≃ 150 nHz (50 • 10 −11 )/(ΓGµ) [124] -has its UV slope from loops with t i < t eq < tM and its IR cut-off due to the matter-Λ transition.Nonetheless, no knee feature is present for the standard cosmological history due to the lack of a double-step spectrum, unlike the EMD case.
We now analytically estimate the position of the knee feature and calculate its detectability by future GW observatories.The spectral slope reaches its maximum -the tip of the kneewhen the time tM of dominant GW emission occurs at the very end of the EMD tM = t dec ; see Figs. 2 and 5. Using Eq. (3.2) with a M,knee = a dec , we obtain an analytic estimation for the GW amplitude at the knee, where the second step uses ρ MD ∝ a −3 and ρ RD ∝ a −4 G(T ) with G(T ) in Eq. (2.13), and the third step uses a RD (t) ∝ G 1/4 (T )t 1/2 , a MD (t) ∝ t 2/3 , and the loop lifetime in Eq. (2.16).
Applying Eqs.(2.6) and (2.16), we arrive at the knee frequency, (3.7) where we have used f dec (T dec ) in Eq. (2.17) multiplied by a numerical factor of 33.5) 8 We can see that Eq. (3.7) describes well the numerical results in Fig. 1.In App.C.1, we show GW spectrum at fix g * and g * s in order to make effects from the knee more visible.The knee feature described by Eqs.(3.6) and (3.7) is sensitive to both the duration N MD of the EMD era and its end temperature T dec , unlike the low-frequency turning point feature which only depends on the latter.By detecting the knee feature, future GW observatories could reconstruct the full EMD era, as shown in Fig. 5.Even if the knee feature lies outside the detectability region, Fig. 4 indicates that observing a portion of the GW spectrum at frequencies f > f dec would enable determination of the EMD duration N MD .While an analytical expression for the spectral index in Fig. 4 is challenging to derive due to the effects of higher Fourier modes and GW emission beyond the dominant epoch tM , it remains a topic for future studies.
The presence of a second step in the GW spectrum already appeared in previous numerical calculations, e.g., Fig. 12 of Ref. [124] and Fig. 9 of Ref. [22], 9 but its origin and detectability are studied for the first time in the present paper.This knee feature can potentially serve as a distinguishing characteristic of the EMD signature from the effects of particle production cutoffs [162], which also have a f −1/3 slope or supercooling phase transition effects [38] which has peak plus a plateau feature in the GW spectrum.A short intermediate inflation [22,193,194] could also lead to similar behaviors, we leave its study for future work.

Primordial Black Holes Domination Era
After they form in the early Universe, PBHs may be abundant enough to dominate the total energy density before they evaporate through Hawking radiation.Because their mass is relatively constant before Hawking evaporation becomes sizeable, they behave like a component of cold matter in the early Universe.In this section, we explore the possibility that the early matter domination era studied in the previous section, and its effect on the GW spectrum emitted by cosmic strings, are due to the existence of such PBHs.

PBH Formation
PBHs form as a consequence of density fluctuations present in the early universe.When the density in a region exceeds a certain threshold, the gravitational forces become strong enough to overcome the Hubble expansion and pressure [45], leading to gravitational collapse and the formation of a black hole.These fluctuations can arise from various mechanisms, e.g.[46-50, 52-56, 61-63, 65-74, 76, 77, 80-87].Some scenarios produce PBH distributions that peak at a particular mass, while others result in extended distributions related to the Fourier spectrum of the primordial fluctuations and to the equation of state of the Universe at the time they collapsed [26,45,[195][196][197][198].Our study avoids making assumptions about the specific mechanism of overdensity formation and simplifies the analysis by considering a nearly-monochromatic density distribution.
We introduce the parameter γ, which represents the fraction of mass within the Hubble horizon that collapses into a PBH.We can then establish a relationship between the mass M PBH of the PBH and the temperature T f at which it forms where g * (T f ) is the number of relativistic degrees of freedom in the plasma at temperature T f .For PBH collapse occurring from super-horizon fluctuations entering the horizon during RD, we have γ ≃ c 3 s ≃ 0.2 where c s = 1/ √ 3 is the speed of sound in a relativistic plasma.The energy fraction of PBHs at formation is defined as where ρ tot (T f ) denotes the total energy density in the Universe when the temperature of the SM plasma equals T f .

PBH Evaporation
The presence of a Schwarzschild horizon implies that PBHs emit a distribution of particles that can be well approximated by a thermal distribution with temperature [88, 89] The corresponding production rate for particle j is where g j is the number of internal degrees of freedom, s j is its spin, and Γ j (E, M PBH ) is the greybody factor [199,200].As a result of Hawking evaporation, the PBH mass decreases at a rate where M pl ≃ 2.44 × 10 18 GeV and the function ε is a function that encodes the details of the Hawking emission which depends on the particle physics spectrum considered and the Hawking temperature of the black hole (for a thorough description including the PBH spin, see Refs.[201,202]).In this study, we assume the emitted particles to belong to the SM only, 10 , and since we focus on PBHs with masses M PBH ≲ 10 9 g corresponding to Hawking temperatures T PBH ≳ 10 4 GeV, all the SM degrees of freedom can be assumed to be relativistic, giving the constant evaporation rate ε ≃ 4.4 × 10 −3 .Upon integrating Eq. (4.6) over time, one straightforwardly obtains the lifetime of a PBH with mass M PBH at formation: where we have used Eq.(4.2).After they dominate the total energy density, PBHs reheat the Universe when they evaporate.The reheating temperature T dec immediately after evaporation is the temperature of a radiation-dominated universe whose age is equal to the PBH lifetime in Eq. (4.7): The condition T dom > T dec needed to have an early period of PBH domination is equivalent to demanding the energy fraction β to be larger than the critical value

.10)
Gravitational waves from global cosmic strings < l a t e x i t s h a 1 _ b a s e 6 4 = " / T N j U C T 0   Detectable LF turning point The total number of e-folds of the corresponding early matter-domination era starting at temperature T dom and ending at temperature T dec can then be simply obtained as As discussed in Sec. 3, an early matter-domination era would induce a change of slope in the GW spectrum emitted from a pre-existing cosmic string network.In Fig. 6, 7, and 8, we show the detection prospects of the imprint of a period of PBH domination by future interferometers.We consider the PBH-dominated era to be detectable if it leads to a suppression of the GW spectrum larger than 10% as compared to standard cosmology.Using Eq. (3.6), one can translate this condition into a threshold on the number of e-folds of PBH domination to be N MD > 0.14 or T dom /T dec ≳ 1.11.
As shown in Fig. 6, measuring GWs emitted by local-strings with Gµ = 10 −11 can allow probing the existence of PBHs with masses in the range [10 6 , 10 9 ] g at LISA and [5 × 10 3 , 10 9 ] g at ET.The LF turning-point signature (solid lines) in each detector probes the PBH mass towards the small mass range, while the knee feature (dashed lines) is associated with a larger PBH mass.Interestingly, joint efforts between the different collaborations could allow us to accurately pin down the PBH parameters, e.g., for Gµ ≃ 10 −11 , a PBH of mass O(10 8 ) g, and β ∼ 10 −10 , LISA could observe the LF turning point while ET observes the knee.In the unbounded colored region labeled "featureless part", we show the detectability of the GW spectrum outside the LF turning point and the knee.As shown in Fig. 4, the non-trivial spectral slope in those regions also carries information about the duration of the PBH domination era.
The ability to probe the PBH parameter space with global-string GWs is shown in Fig. 7.The top panel provides the observational bounds on the LF turning point and the featureless part, while the bottom panel gives more details on the former.From the top panel of Fig. 7, PBHs with masses [10 4 , 10 9 ] g and [10 2 , 10 6 ] g can be probed by LISA and ET, respectively, with global strings of η = 10 15 GeV.Instead, the bottom panel shows other GW observatories sensitive to LF turning points at frequencies lower than LISA.In contrast to GWs emitted by local strings, whose frequency corresponding to the BBN scale is 10 −6 Hz, the GWs emitted by global strings can have a LF turning point at smaller frequencies; see Eq. (2.17).In this figure, we also show for comparison bounds on PBHs that could be set in the future by CMB-HD, using the fact that massless goldstone bosons may be produced copiously from the evaporation of PBHs and contribute to ∆N eff .As one can see from the figure, in the case of rotating PBHs with spin parameter a ⋆ = 0.99, GW searches and CMB observations may observe complementary smoking-gun signatures of the existence of a PBH-dominated era, whereas, for Schwarzschild PBHs, GW searches reveal to be much more competitive at low PBH masses.
Fig. 8 shows the string-scale (Gµ and η) dependence of the detectability of PBH parameters for large values of β, above which all the shown parameter spaces are guaranteed to have N MD > 0; see Figs. 6 and 7. Note that one loses sensitivity relatively fast in the case of global strings for a decreasing η, as Ω GW ∝ η 4 .Local-string GWs depend on (Gµ) 1/2 .Therefore, even for local strings with a scale as small as Gµ ∼ 10 −18 , PBHs can be searched for efficiently by future experiments.Nonetheless, the particle production cutoff could erode the signature from EMD; see Eq. (2.21).

Effects from PBH spins
In the early Universe, PBHs may have acquired angular momentum from merging events [203], matter accretion [204], evaporation [205], primordial inhomogeneities [204], collapse during matter domination [196,206], or specific mechanisms of PBH formation, such as scalar fragmentation [207,208], collapse of domain walls [209] or cosmic strings [86,87].In that case, the PBHs can be characterized, on top of their mass, by the spin parameter where J is the angular momentum of the black hole.The PBH spin impacts a variety of particlephysics phenomena: e.g., the production of dark matter through evaporation [201,[210][211][212][213][214][215][216][217][218][219][220][221][222][223][224][225][226], the amount of dark radiation produced through evaporation that could contribute to ∆N eff in future CMB measurements [203,211,212,222,[227][228][229], or even the spectrum of GW induced at second order in perturbation theory [229].In this section, we explore the effect of the PBH spin on the PBH domination era and its signature in the cosmic-string GW spectrum.where F(a ⋆ , M PBH ) is a numerical function which range from O(0.4) for a ⋆ ≃ 0.999 to 1 for a ⋆ = 0 and taken from Table 1 of Ref. [229].While the start of PBH domination remains unchanged by spin (fixed by the energy fraction at formation β, γ, M PBH ), the spinning PBHs evaporate at a temperature higher than the Schwarzschild ones, using ∆t PBH = H −1 PBH , The evaporation temperature of maximally-spinning PBHs (a ⋆ = 0.999 → F ≃ 0.4) is ∼ 1.6 times higher than for static PBH, assuming for simplicity g * s (T Kerr evap ) = g * s (T Sch evap ).Concerning the signature in cosmic-string GW, Eq. (2.17), the turning point in the case of the maximallyspinning PBHs sits at a frequency ∼ 1.6 times higher than in the Schwarzschild case.In Fig. 9, our results show that effects from spin turn out to be relatively minor.Furthermore, effects from the spin are degenerate with the PBH mass, and they are equivalent to decrease M PBH by a factor 1.4, see Eq. (4.9).

Conclusion
The cosmological evolution of the universe below a temperature of 5 MeV is well-established and supported by numerous observational probes, including the nuclear abundances predicted by the Big Bang Nucleosynthesis (BBN) epoch, Cosmic Microwave Background (CMB), largescale structure measurements and supernovae observations.This evolution is characterized by successive epochs of radiation domination, matter domination, and dark energy domination.However, to explore the conditions prevailing in the pre-BBN universe with temperatures above 5 MeV, gravitational waves (GW) are one of the limited observables available.Cosmic strings are one candidate for such sources.They are expected in any high-scale theories of particle physics involving local or global U(1) symmetry breaking ranging from neutrino mass generation, leptogenesis, dark matter, flavor, or Grand Unified Theories in Beyond the Standard Model contexts [145].An important particularity of cosmic strings is their scaling behavior.Their energy density scales with the scale factor precisely as the dominant energy density of the universe, i.e., a −4 during radiation and a −3 during matter.Consequently, the spectral slope of the GW energy density is sensitive to the equation of states of the universe at early times.
This study builds upon the research direction "GW archaeology with cosmic strings", initiated in Refs.[16,121,122,124,125].The objective is to use GW produced by cosmic strings to infer the energy content of the pre-BBN universe.The present work focuses on the GW signatures of an early matter-dominated era.The previous studies assumed that the spectral slope of the GW spectrum emitted by local cosmic strings was independent of the duration N MD of the matter era.An important result of the present work is to reveal the contrary.Figs. 1 and 4 show that any measurement of the spectral slope would give information on the duration of the matter era.Until now, only the temperature at the end of the matter era was believed to be measurable.The present study shows that measuring the temperature at which the matter era starts is also possible.As illustrated in Fig. 2, the impact of the matter era leads to two-step features in the GW spectrum from local strings instead of one, as previously thought.This feature arises due to a GW emission occurring in two steps: first, a string loop is formed at time t i , and second, the loop converts its energy into GW at the end of its lifetime at tM ≫ t i .The first step in the GW spectrum is due to the impact of the matter era on the scale factor a(t i ), while the second step is due to the impact of the matter era on the scale factor a( tM ).We analytically compute the position of the second step, which we call the knee, see Eqs. (3.6)- (3.7).
Assuming the standard cosmological history, the cosmic strings with11 a tension Gµ ≲ 10 −11 (local) and an energy scale η ≲ 10 15 GeV (global) would be observed by LISA, ET, and other future-planned experiments.Our analysis provides a way to extract information about the EMD era, by probing any deviation from the standard-cosmology prediction.As a potential example, we show that the existence of PBHs -if they dominate the energy density of the early universe -can be constrained by LISA and ET for PBH masses between [10 6 , 10 9 ] g and [5 × 10 3 , 10 9 ] g, respectively, for the local strings with Gµ = 10 −11 .Similarly in the case of global strings, LISA and ET can probe PBH masses between [10 4 , 10 9 ] g and [10 2 , 10 6 ] g, respectively, if the symmetry breaking scale is at η = 10 15 GeV.The detectability and the ability to constrain PBHs become weaker as Gµ and η decrease because the amplitude of the SGWB is smaller, as shown in Fig. 8. Finally, we considered in Sec.4.4 the possibility that PBHs are Kerr black holes and therefore evaporate at a slightly earlier time than Schwarzschild black holes.
Current studies [229,[232][233][234][235][236][237][238][239] have demonstrated that scalar-induced GW resulting from adiabatic and isocurvature perturbations can produce distinctive resonant peaks and doublepeaks, which can be used to probe the formation and decay time of PBHs in the early universe.These present additional opportunities to test scenarios such as PBH domination and evaporation.The spectral shapes of those GW differ from those found in the context of GW emitted by cosmic strings studied in this work.
Recent pulsar-timing-array data has revealed a common red-noise signal, which may be interpreted as a stochastic GW background in the frequency range 10 −9 Hz ≲ f ≲ 10 −7 Hz [174-177, 230, 240-244].This signal could potentially be attributed to a cosmic-string network with a tension parameter of Gµ ∈ [10 −11 , 10 −10 ] [ 125,178,230].The early matter era, such as the one induced by PBHs domination, cannot distort the GW spectrum at these low frequencies, as it would require modifying the equation of state of the universe after BBN, which is at odd with concordance cosmology.However, if the signal in the pulsar-timing-array data is caused by cosmic strings, then the high-frequency part of the SGWB spectrum will be detected by future gravitational-wave experiments.Observing this high-frequency component would allow for constraining the presence of any early matter-dominated era occurring below T ≲ 10 5 GeV and for probing the presence of PBHs population down to 10 4 g.
There are some worth-mentioning caveats in our analysis.First, we rely on the detection criterion that requires SNR = 10 by comparing our signal to the power-law integrated sensitivity curves.However, this criterion might not be enough when we confront real data.Because the detection claim of a SGWB requires a signal reconstruction over other detectors' noises at an acceptable confidence level, as shown for LISA in [245,246].For a spectrum with several features like the knee, the detectable signal might have to lie above the detectors' noise level in order to read the SGWB spectrum accurately and to extract precise information about the EMD era.Moreover, the optimal SNR depends on the knowledge about the noises, and one would have to perform a global fit [247] on the combined noises and the SGWB signal.The second caveat is that our work assumes cosmic strings and PBHs do not interact during the PBH-dominated era.However, PBHs can affect the cosmic string network through mechanisms such as chopping off long strings, modifying the network-scaling behavior, suppressing loop production, or leading to necklace-like or net-like structures [248][249][250].These processes could result in additional GW contributions [251] beyond our conservative estimates.We leave the study of the string-PBH network for future work.
If the features found in this paper are observed in the GW spectrum from cosmic strings, additional observations will be needed to distinguish between a PBH-dominated era and other forms of early matter domination.It is worth noting that PBHs generate other signals that could be explored in the future.In Fig. 7, we show possibilities for detecting a non-zero component of dark radiation.Other indicators of the presence of evaporating PBHs in the early universe, such as second-order GWs, dark matter searches, baryogenesis, and structure formation, provide various ways to independently verify the existence of an early PBH-domination era (see e.g.Ref. [202] for a recent review).Such detection channels would not only provide valuable independent confirmations of our results but also offer a unique opportunity for synergies between GW searches and CMB, large-scale structure, or even dark-matter search experiments.
The detection of GW has opened up a new avenue for studying the early universe, which is complementary to other methods.Our analysis shows that cosmic strings can serve as excellent standard candles for probing the pre-BBN universe, should they exist International GW detector networks planned for the future could allow us to explore the equation of state of the universe down to 10 −16 s after the Big Bang.breaks down when the string curvature becomes comparable to its thickness.The thickness of the string is estimated to be of the order of the inverse of the U (1)-breaking scalar vacuum expectation value η.Therefore, if the emitted GW frequency f = 2k/l becomes larger than η, the NG approximation is no longer valid.We deduce the highest Fourier mode that is compatible with the NG approximation (see also Ref. [124]) where t is the time at which the loop is formed, L ≃ αt is the size of the loop, H is the Hubble parameter, g * is the effective number of relativistic degrees of freedom, and T ≃ 1.7 M pl H/g 1/4 * is the temperature of the radiation-dominated Universe.Trading the temperature T of loop formation for the frequency f of GW today, we obtain The maximum mode number k max is estimated to be extremely large, highlighting the importance of properly accounting for the contribution of each mode.In Fig. 10, we show effects from higher modes on the GW spectrum from CS, assuming a long EMD era.In this work, we account for those effects by summing over a large number of modes (k max = 10 12 ).Fig. 10: The slopes of cosmic-string GW spectra assuming a long duration (N MD = 30) of EMD era ending at T dec = 10 2 GeV.We can see that the first mode has spectral slope f −1 , while summation leads to f −1/3 (up to the frequency k max f dec ).

B High-frequency turning point for global strings
As discussed in Sec.3.3, the EMD era leads to the global-string GW spectrum with the slope of −1/3.This leads to a high-frequency (HF) turning point (beyond which the HF plateau sits) at The log-dependence term has the 9th power and strongly shifts the HF turning point.Fig. 11 shows that the HF turning point of global-string GW, defined in Eq. (B.1), sits many order-ofmagnitude higher than the LF one, defined in Eq. (2.17).Consequently, the HF turning point may lie beyond the detectable range of forthcoming GW interferometers.Fig. 11: The HF turning-points f dom (solid) of the global cosmic strings depends strongly on the f dec and N MD .The log dependence shifts the turning point higher than the local case (dotted).In the gray region, the HF turning point sits at a frequency higher than 10 kHz and in the ultra-higher frequency regime.This estimates well the numerical results in Fig. 1.
C More details on the knee feature C.1 Effects purely from the early matter era (unrealistic GW spectrum) In Sec. 3, we provide the analytic estimates for three features in the double-step GW spectrum from local strings.The g * -g * s evolution contaminates the effects from the early matter era and complicates the comparison between Fig. 1 and Eqs.(3.5), (3.6), and (3.7).We now consider the effects purely from the early matter era by setting g * (T ) = g * (T 0 ) and g * s (T ) = g * s (T 0 ).We show the GW spectrum and its spectral indices in Fig. 12.The features can be well described by our estimates in Eqs.(3.5), (3.6), and (3.7).Furthermore, it can be inferred from Fig. 12 that the presence of double steps and knee features in the GW spectrum is not caused by the evolution of g * and g * s , but is solely due to the early matter domination epoch.

C.2 Visibility of the knee feature
As discussed in Sec.3.4, if the EMD era lasts too long, the knee feature disappears.This happens if the loop lifetime becomes shorter than the EMD duration, i.e., if where we have used the lifetime of the loops in Eq. (2.16) and using a MD ∼ t 2/3 .

D Sensitivity curves of GW experiments
The sensitivity of a GW detector is Ω sens (f ) ≡ 2π 2 f 3 S n (f )/(3H 2 0 ), where S n (f ) is the noise spectral density derived from the correlation of the detector noise signal n(f ): ⟨n * (f )n(f ′ )⟩ ≡ δ(f − f ′ )S n (f ) [100,252,253].The ability of a GW detector to detect a GW signal with energy density Ω GW (f ) is quantified by the signal-to-noise ratio (SNR), SNR ≡ T The calculation of the SNR can require expensive computations when scanning over the model parameter space.To avoid this, in this paper, we use the power-law integrated sensitivity curve [254].We approximate the GW spectrum as a power-law Ω GW (f ) = Ω β (f /f ref ) β with a given spectral index β and reference frequency f ref .
We calculate the GW amplitude Ω β which gives a certain SNR after an observation time T , We now sample over all possible spectral index β.One defines the envelope of these curves as the power-law integrated sensitivity curve, For a detector with noise sensitivity Ω sens , the observation during time T of a GW signal with amplitude larger than Ω PI (f ; SNR, T ) has a signal-to-noise ratio > SNR.
The power-law integrated sensitivity curves Ω P I (f ), used in this study, are calculated from the noise spectral density in [97] for ET, [98] for CE, [95] for BBO/DECIGO, [255] for AEDGE, [256] for LISA, and [257] for THEIA.We require SNR = 10 with the optimistic observation time of 10 years 12 .For the sensitivity curves of pulsar timing arrays (EPTA, NANOGrav, and SKA), we directly took from [258].The sensitivity curves of LIGO has been taken into account the improvement from the cross-correlation between multiple detectors [254], where we adopt the noise spectral densities for runs O2, O4, and O5, and the overlap function between the two LIGO detectors from [259].We fixed the LIGO curves at SNR = 10 and the observational time of T = 268 days for LIGO O2 and 1 year for LIGO O4 and O5.

N MD = 1 N MD = 2 N MD = 3 N M D = 4 N M D = 5 N M D = 6 N M D = 7 T 2 N M D = 3 = 4 N MD = 5 N MD ≥ 6 ηFig. 1 :
Fig.1:We assume an early matter domination (e.g., PBH domination era) lasting for N MD e-folds of Hubble expansion and ending at temperature T dec .It impacts the GW spectra of local and global cosmic strings through a double-step (left) and a single-step (right) suppression, respectively.Those features can be characterized by three frequencies: the low-frequency turning point, the high-frequency plateau, and the newly-found knee (the second step in local strings GW spectrum), see Sec. 3.

h 2
II. HF plateauGμ = 10 -11 T dec = 1 GeV N MD = 2 10 -10 10 -7 10 -4 10 -Global cosmic strings s u m m e d t o l a r g e k k  1 s p e c tr u m t i b e fo re E M D t i during EMD t i a ft e r E M D I. Low-frequency turning-point II.High-freq.plateau η = 10 15 GeV T dec = 10 GeV N MD = 3

10 Fig. 3 :
Fig. 3: Detectability of the low-frequency turning point, associated with T dec when the EMD era ends; cf.Eq. (2.17).The regions correspond to the detection of SNR ≥ 10.The red regions correspond to the cusp and friction cut-offs in Eqs.(2.19) and (2.21) that erase the LF turning point.
).The regions correspond to the detection of SNR ≥ 10.The red regions correspond to the cusp and friction cut-offs in Eqs.(2.19) and (2.21) that erase the LF turning point.

f 6 NMD = 8 N MD = 10 N MD ≳ 20 ff 8 N M D = 10 N MD ≳ 20 f knee /f dec Fig. 4 :Fig. 5 :
Fig. 4: Numerical result of the spectral slope (i.e., β ≡ d log Ω GW /d log f ) of the local-string GW experiencing different duration N MD of the EMD era.The vertical dashed lines show the position of the knee feature, specified by Eq. (3.7).Only for N MD ≳ 10 (cf.App.C.2), the knee feature submerges below the −1/3-slope tail from the LF turning point.

x 6 6 e 4 8 2
+ c t l l o 6 4 E L h 3 P u 5 d 5 7 g o Q z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / t l + + C w r e J U U m j R m M e y G x A F n A l o a a Y 5 d B M J J A o 4 d I L x 3 c z j y M m m e V b z L y s X 9 e a l 6 k 8 W R h y M 4 h j J 4 c A V V u I M a N I C A g m d 4 h T f n y X l x 3 p 2 P e W v O y W Y O 4 Q + c z x + l V Z H 5 < / l a t e x i t > f (T dec ): low-frequency turning point

10 N M D = 20 Fig. 7 : 8 ↓Fig. 8 :
Fig. 7: Top panel: Detectability of the single-step feature from PBH domination: LF turning point (bounded by solid line) and the featureless part (unbounded).Bottom panel: Detectable LF turning point associated with the end of PBH domination in the GW background from global cosmic strings.This is the solely detectable signature on the global-string GW spectrum as the HF turning point locates at the ultra-high frequency and the spectral index is always −1/3.Dash lines indicate the ∆N eff possibly observable by CMB-HD due to the massless goldstone of the global-string U (1) symmetry produced by the evaporation PBH of spin parameter a * .

2 h 2 st an da rd co sm ol og y a * = 0 . 9 9 a * = 0 Fig. 9 :
Fig.9: Cosmic-string GW spectra experiencing PBH domination era with the monochromatic PBH mass spectrum M PBH = 10 6 g and β = 10 −4 .We vary the PBH spin parameter a * which affects slightly the lifetime of PBH: T dec ≃ T dec,s F −1/2 (a * , M PBH ) for the end of Schwarzschild PBH domination T dec,s and the function F is taken from Ref.[229].
fmax f min df Ω GW (f ) Ω sens (f ) 2 where T ≡ an observation time.(D.1) .10)This formula can be used for both local and global strings after applying Eqs.(2.6) and (2.8) and choosing an appropriated value of κ.The resulting GW spectra assuming the standard ΛCDM Universe are illustrated by the black dashed lines in Fig.1.For loops formed and emitted during radiation-domination epoch, the GW spectrum emitted by local strings is nearly flat with an amplitude of order 2