Percolating Cosmic String loops from evaporating primordial black holes

The Pulsar timing data from NANOGrav Collaboration has regenerated interest in the possibility of observing stochastic gravitational wave background arising from cosmic strings. In the standard theory, the cosmic string network forms during spontaneous symmetry breaking (SSB) phase transition in the whole universe via the so called Kibble mechanism. This scenario would not be possible, e.g., in models of low energy inflation, where the reheat temperature is much lower than the energy scale of cosmic strings. We point out a very different possibility, where a network of even high energy scale cosmic strings can form when the temperature of the Universe is much lower. We consider local heating of plasma in the early universe by evaporating primordial black holes (PBHs). It is known that for suitable masses of PBHs, their Hawking radiation may re-heat the surrounding plasma to high temperatures, restoring certain symmetries {\it locally} which are broken at the ambient temperature at that stage. Expansion of the hot plasma cools it so that the {\it locally restored symmetry} is spontaneously broken again. If this SSB supports formation of cosmic strings, then string loops will form in this region around the PBH. Further, resulting temperature gradients lead to pressure gradients such that plasma develops radial flow with the string loops getting stretched as they get dragged by the flow. For a finite density of PBHs of suitable masses, one will get local hot spots, each one contributing to expanding cosmic string loops. For suitable PBH density, the loops from different regions may intersect. Intercommutation of strings can then lead to percolation, leading to the possibility of formation of infinite string network, even when the entire universe never goes through the respective SSB phase transition.

There is a renewed interest in cosmic strings in view of the Pulsar timing array (PTA) data from NANOGrav Collaboration raising the possibility of observing stochastic gravitational wave background arising from cosmic strings [1] (see, also, ref. [2]).With PTA, and upcoming Laser Interferometer Space Antenna (LISA) [3], search for cosmic strings using their gravitational wave signatures has reached new levels of excitement [4].Formation of cosmic strings, and their possible observation, is of great importance as they can provide a direct possible window to the physics of ultra-high energy scales.Apart from the standard spontaneous symmetry breaking related cosmic strings, these also arise in superstring theories [5], making a compelling case for the existence of cosmic strings in the Universe.In the conventional picture, strings are produced via the so called Kibble mechanism [6] when the universe goes through a symmetry breaking phase transition.This string network rapidly evolves and reaches a scaling solution with a given spectrum of string loops and long strings at any given time.This necessarily requires that the entire observable Universe goes through the SSB phase transition.This scenario may not always be possible, for example, in models of inflation which have low reheat temperature, or in models of TeV scale gravity [7].
Here we propose a very different possibility involving evaporating primordial black holes (PBHs).There have been a host of investigations exploring the possibility that small black holes can form in the early universe.(See, ref. [8] for a recent review.Also, see refs.[9] for early discussions.)Black holes evaporate by emitting Hawking radiation, and for sufficiently small black hole masses, they can completely evaporate away during early stages of the universe.There has been extensive discussion in the literature regarding the possibility of formation of a hot plasma surrounding primordial black holes (see [10][11][12][13]).In these references the plasma consists of black hole radiation and particles produced by the interaction of the black hole radiation with itself.However, for PBH embedded in an ambient thermalized plasma in the early universe, it is rather straightforward to see that the plasma near the black hole will be heated up as the black hole radiation propagates through the ambient plasma and loses energy.A detailed investigation of this was carried out in [14] (see, also [15]) utilizing known results about the energy loss of quarks and gluons traversing a region of quark-gluon plasma [16].(We mention here that the results of ref. [16] are certainly applicable for high energy partons, with energies of several 100 GeV, it is not clear whether they can be extrapolated to the case of Hawking radiation with energies almost near the GUT scale.Still, the overall picture of PBH heating the local plasma to very high temperatures may remain valid even for such large Hawking temperatures).It was shown in [14] that this energy loss leads to rapid heating of the plasma near the black hole.Further, resulting temperature gradients lead to large pressure gradients such that plasma develops radial flow.One therefore gets the situation that evaporating primordial black holes in the early universe lead to radially expanding hot plasma regions For suitable masses of PBHs, Hawking radiation of evaporating primordial black holes may re-heat the surrounding plasma to high temperatures, restoring certain symmetries locally which are broken at the ambient temperature of the Universe at that stage.Expansion of the hot plasma cools it so that the locally restored symmetry is spontaneously broken again [14].If this SSB supports formation of cosmic strings, then string loops will form in this region around the PBH.Further, resulting temperature gradients lead to large pressure gradients such that plasma will develop radial flow.String loops can get stretched as they flow with this expanding plasma.For a finite density of PBHs of similar masses, one will get local hot spots, each one contributing to expanding cosmic string loops.For suitable PBH density, the loops from different regions may intersect.If that happens, then intercommutation of strings can lead to percolation, leading to the possibility of formation of infinite string network, even when the entire universe never goes through the respective SSB phase transition.
Possibility of formation of primordial black holes, and various observational constraints on them are well discussed in the literature [8,9].In first order phase transitions primordial black holes can be produced by collapsing regions of false vacuum [17] or due to inhomogeneities formed during bubble wall collisions [18].They could also form by large amplitude density perturbations produced due to fluctuations in the inflaton field [19], or by shrinking cosmic string loops [20].
For primordial black holes produced from large amplitude density fluctuations produced by the inflation, one might ask whether CMBR data puts any constraints.However, as discussed in ref. [21], one should note that CMBR data imposes restrictions on the power spectrum at large scales (of order 1 -10 3 Mpc), whereas, the primordial black holes form due to large amplitude and much smaller scale density fluctuations produced by the inflation.Hence they do not affect CMBR results at all but imposes constraints on the spectral index and consequently puts restrictions on certain models of inflation.The constraints on primordial black hole formation come from other sources like nucleosynthesis, gamma ray background etc. [21].However, all these phenomena essentially impose constraints on primordial black holes with a mass range which is much larger than the ones relevant here.We consider primordial black holes with masses at most few tonnes and such black holes evaporate away much above the QCD scale, without causing any conflict with current observations.Our main aim is to illustrate a completely novel possibility of forming infinite string networks via black holes.Hence, we will simply assume the required masses and number densities of black holes, without discussing any specific mechanism which could give rise to the formation of such black holes.
Consider the case when the plasma heated by the black hole leads to local restoration of a gauge symmetry (with an energy scale and associated phase transition temperature of order η) which allows for the existence of cosmic strings [6].As the plasma flows out radially away from the black hole, its temperature will decrease and fall below T c (= η) at some distance r η where the symmetry will be broken.Near that region, again, string defects will form either via the Kibble mechanism, or due to turbulence.(For internal symmetries, spatial variations of the order parameter could lead to formation of non-trivial windings when turbulent motion of the plasma folds up extended spatial regions, essentially compactifying parts of spatial regions, i.e. plasma).
If the friction forces dominate over the string tension so that strings are effectively frozen locally in the plasma then the string loops will be carried out by the radially expanding plasma.They will then stretch out to large sizes instead of shrinking due to tension.Note that cosmic string loops shrink when energy can be dissipated in other modes, such as gravitational waves, in particle emission, or dissipation in background plasma.For strong friction case (which is likely in the early stages of hot plasma) strings can expand, with the plasma flow increasing the energy of the expanding string loop.
If there is a uniform density of similar primordial black holes, then each black hole will emit string loops which will be stretched to large sizes as they are carried away by convective flow away from each black hole.We now recall a very crucial property of string-string interaction.It is well established that string defects when crossing each other, intercommute.This always happens as long as strings are not crossing with ultra-relativistic velocities [22].Thus intercommutation will almost always occur here as string velocities are not expected to increase beyond the sound velocity.If a string loop emitted by one black hole intersects the string loop emitted by a nearby black hole, this intercommutation will lead to formation of a much larger string loop.Iteration of this process where string loops of different black holes start intercommuting with each other leads to the remarkable possibility that these string loops may percolate and lead to formation of an infinite string network.This possibility is remarkable because such an infinite string network would normally only arise when the entire universe undergoes phase transition.
It is interesting to note here that this picture is quite the opposite of the standard evolution picture of cosmic string network which are formed in an overall SSB phase transition.In that case, large string loops fold on themselves, and intercommutation at the intersecting point leads to the large string loop breaking into smaller loops, which decay away by emitting radiation (gravitational waves, or particles).This is the most crucial aspect of the scaling behavior of the cosmic string network evolution.Completely opposite to this, in our model, string loops first expand (due to expanding plasma).Intercommutation of these expanding smaller loops leads to formation of much larger loops, eventually forming infinite string network.However, This reverse evolution only happens during the period when black hole evaporation fuels expanding plasma shells.After the black holes evaporate away (within one Hubble period of the beginning of this process), one may expect that the final infinite string network will be similar to the conventional string network, and may undergo same evolution, achieving scaling solution eventually.However, there is a subtle point here [23].In the conventional scenario, the strings form due to random variation of the order parameter field (for U(1) case, the phase of the scalar field), in uncorrelated domains leading to non-trivial windings.In contrast, in the present scenario, expanding hot plasma produces string loops (which have zero net winding for regions outside the loop) in the background of a relatively cold Universe which initially has topologically trivial order parameter configuration (ignoring any previously present strings).Thus, on global scales, order parameter field behaves very differently in the two cases.It may then be possible that there is some difference in the nature of the string network in the two cases.We will discuss this issue further in the following, after discussing the details of the model.
Let us discuss the conditions for percolation of strings.A black hole of mass M bh evaporates by emitting Hawking radiation with an associated temperature Here GeV is the Planck mass.We use natural units with = c = 1.The rate of loss of mass by the evaporating black hole is given by Here, α accounts for the scattering of emitted particles by the curvature and depends on T bh .For different values of T bh values of α have been tabulated in [24].For T bh = 1, 200, and 10 15 MeV, the corresponding values of α are 3.6 × 10 −4 , 2.3 × 10 −3 and 4.5 ×10 −3 respectively.For the range of black hole temperatures for our model, we will set α to be 3 × 10 −3 .The lifetime τ bh of the black hole can be obtained by integrating Eq.( 2).We get where M 0 is the initial mass of the black hole.Eq.( 2) implies that very little energy is emitted until time of the order of τ bh which is when most of the energy of the black hole gets emitted.Thus for a black hole formed early in the Universe, it is reasonable to assume that the black hole essentially evaporates only when the age of the Universe is of order τ bh .Let us assume that black holes formed in the early Universe have masses so that they evaporate when the temperature of the Universe is T U .The age of the Universe t U when its temperature is where g * (≃ 100) is the number of relevant degrees of freedom [25].By equating τ bh = t U , we can get the mass M 0 of the black hole such that its evaporation becomes effective when the temperature of the Universe is T U , Black hole with this mass will have the temperature, For example, with T U = 1 GeV we get M 0 = 4 × 10 11 M P l , T bh = 10 6 GeV and τ bh = 5 × 10 17 GeV −1 = 3 × 10 −7 s.The picture then is that these black holes emit particles with energies roughly equal to T bh (= 10 6 GeV for this sample case) into the background plasma which is at a temperature T U (∼ 1 GeV) to start with.These 10 6 GeV particles will scatter with the particles in the background plasma and will heat it up through their energy loss.For the black hole masses considered here only elementary particles will be emitted, such as quarks, gluons, photons, leptons, etc. (We mention here that the mass of a primordial black hole can also increases due to accretion of background plasma particles.However, for relevant ranges of black hole masses here, all the dominant particle species are ultra relativistic, so only the geometric cross-section of black hole is relevant which is not very effective [26].Some slow growth of the mass of the black hole could occur because of this, and the black hole masses we use here should be taken to be the final mass of the black hole when its evaporation becomes effective.) In ref. [14], conditions for the equilibration of the Hawking radiation in the ambient plasma were discussed.Further, estimates of plasma velocity were made in [14] using the Euler equation for a relativistic fluid and it was shown that plasma velocity rapidly becomes relativistic.This is natural to expect because of sharp temperature gradients close to the evaporating black hole.Assuming a steady state situation where the luminosity L(r) is independent of r and equal to −dM bh /dt, we can get the relation between the temperature T of the expanding plasma shell and the radius r of the shell (see, ref. [14] for details).
where γ is the Lorentz factor for the plasma velocity.This expression is valid for distances where bulk plasma flow dominates over diffusion of particles.As shown in [14], this will be applicable for the entire range of distances of relevance to our scenario in the present paper.We will take the plasma velocity to be of order sound velocity v ≃ 1/ √ 3. Though it is important to realize that much larger velocities will be expected during last explosive stages of black hole evaporation, and shocks should develop during those stages.We will not consider shocks to keep calculations simple.(Hence, for estimates, we ignore the Lorentz factor in the above equation.) Eq.( 6) gives the temperature of the black hole when its evaporation becomes significant.Within one Hubble time the entire black hole evaporates away.However, Eqs.( 1), (2) show that as the black hole mass decreases, its temperature, and the rate of energy loss, also increase.Thus it is the last stages of black hole evaporation which will lead to highest temperatures for the plasma in the nearby region.As our interest is in determining how far the string loops can be dragged by expanding hot shells, we need to consider the plasma flow during these last stages of black hole evaporation.Initial mass M 0 of the black hole is determined by the ambient temperature T U when black hole evaporation becomes significant.However, we consider the situation only when its mass has reduced to a value M x , given by M x is determined by the following considerations.Let us assume that the cosmic strings under consideration correspond to a symmetry breaking scale η. (For simplicity we will consider gauge strings for the estimation of string tension, but ignore the length scales associated with gauge fields.)Thus black holes which are relevant to our scenario must have temperatures larger than η.Strings will be produced in expanding shells as they cross the critical distance r η where temperature drops to a value below the critical temperature, which we take to be of order η.For the self consistency of this picture, we must have the thickness ∆r of the plasma shell to be at least of the order of η −1 which is roughly the thickness of the string.With a spherical shell around the black hole, with this thickness, several correlation domains will then form, leading to string formation via the Kibble mechanism [6].Thickness of the shell is determined by the life time of the remaining black hole with mass M x .We thus require (using Eq.( 3)), Initial temperature T x of this remaining black hole is given by Eq.( 1) with M bh = M x (Eq.( 8)).We take this temperature to be effectively at a distance of order of the Schwarzschild radius x .(This is reasonable within an order of magnitude, see discussion in [14].)Using the temperature distance relationship from Eq.( 7), we can get the temperature of the plasma from this remaining (last stage) of the black hole as, We mention here that though Eq.( 7) (and hence Eq.( 10)) was derived in ref. [14] with considerations of bulk plasma and particle diffusion, essentially similar profile is obtained if we take T = T (r x ) effectively at a distance of order of the Schwarzschild radius x along with the condition T 4 r 2 = constant.(Which essentially means conservation of energy-momentum of the expanding shells).We will be considering black hole masses such that T x > η.Temperature will drop to the value η at a distance r η which is obtained by setting T (r η ) = η in the above equation.We get, String loops will be produced at this distance.We then expect these string loops to be dragged (and consequently stretched) by expanding plasma.At certain distance r U , the temperature will drop to the ambient value T U and plasma flow should stop roughly at that distance.String stretching is not possible beyond distance of this order.r U , therefore, sets an upper limit on the distance R stretch up to which strings can be carried out by plasma flow.Using Eqs.( 5), (10), we can get r U by setting T (r U ) = T U .we get, It should be clear that r U sets an upper limit on the distance R stretch up to which strings can be carried out by plasma flow.To determine R stretch we need to find the condition when the string motion is dominated by friction forces.Friction forces on strings arise from the scattering of plasma particles from the string.It has been discussed in the literature that the dominant contribution to this comes from Aharanov-Bohm scattering (of particles with appropriate).For simplicity, we will assume this to be the case.Friction force per unit length on the cosmic string is then given by [27], where v is the string velocity through the plasma, T is the plasma temperature, and β is a numerical parameter related to the number of relevant particle species [27].In specific grand unified theory models [28] it could be of order one, and we will assume that to be the case here.(See, also, ref. [29] for discussions of scattering of particles from strings.)We mention here that stretching of string due to plasma flow has been discussed by Chudnovsky and Vilenkin in ref. [30] where they considered light superconducting cosmic strings getting stretched due to turbulent plasma flow in the galactic disc and even in stellar interiors.One could also consider formation and stretching of such strings around primordial black holes.This will require considerations of magnetic fields in the cosmic plasma etc.We consider a simple situation when a string encircling the black hole is stretched symmetrically by radial flow of the plasma.(Order of magnitudes of our estimates should remain same for other geometries of strings, e.g.string ends may lie on the boundary of the symmetry restored region around the black hole) Such large strings should easily form during string formation on the surface of the sphere with radius r η near the black hole.String tries to collapse due to its tension.For a string loop of radius R, one can estimate the tension force as [30], At the formation stage, near r ≃ r η , the string will be at rest w.r.t local plasma frame.In such a situation v in Eq.( 13) will be zero and there will be no friction force on the string.This will lead to collapse of the string under its tension.As the string loop starts collapsing, it will develop non-zero velocity w.r.t. the local plasma frame, and consequently a non-zero friction force via Eq.( 13).As the plasma flow velocity is taken to be the sound velocity (= 1/ √ 3) w.r.t. the black hole rest frame, we can deduce that the friction force on the string will be large as the string collapses, and will decrease as string collapse decreases.When string is also at rest w.r.t. the black hole, then the friction force will be given by Eq.( 13) with v = 1/ √ 3, and should balance the string tension force (Eq.( 14)).String loop will expand because of plasma drag as long as F f rict (v = 1/ √ 3) > F tension .Using r dependence of the temperature from Eq.( 10), this condition gives the upper limit for R as, As plasma flow stops beyond a distance of order r U , we conclude that plasma flow can stretch string loops to sizes of order R stretch ≃ min(R max , r U ).We note that the largest values of R max , (as well as r U (Eq.( 12)), and r η (Eq.( 11)), correspond to the largest value x max of x obtained by taking equality sign in Eq.( 9), and we will use this.
We should also consider the effect of black hole gravity on the string loops.Gravitational force per unit length on the string due to the black hole of mass M bh can be roughly estimated as where R is the separation of the string segment from the black hole.(We are taking string as a simple gravitating system, as for a string loop.For special geometries, such as for a straighter string the gravitational force will be different, and will be typically smaller).We see that, per unit length, this gravitational force becomes much less than the force f tension ∼ µ R due to string tension when R > GM bh .Thus for any distances much larger than the Schwarzschild radius of the black hole, gravity of the black hole is sub-dominant compared to string tension forces.We are considering the situation when plasma drag forces completely dominate over the string tension forces, leading to stretching of string to large distances (where black hole gravity becomes even more negligible).Thus black hole gravity is not relevant for string dynamics in our case.
If black hole number density is such that inter-black hole separation d bh is smaller than R stretch then loop intersections will be frequent.As string velocity is not expected to be ultra-relativistic here (neglecting shocks), this will lead to intercommutation of strings and hence percolation of strings resulting in an infinite string network.By assuming that energy density of these black holes (with mass M 0 ) contributes only a fraction f of the total energy density of the universe, we get (with g * ≃ 100), Percolation of loops (leading to infinite string network formation) will happen when d bh < R stretch .We can now determine conditions on various parameters which satisfy all the constraints for string percolation.For given values of the fraction f , and string scale η, the condition for percolation (d bh < R stretch ) shows that percolation can happen if the Universe temperature T U lies in the range T min U < T U < T max U , which is determined as follows (using largest value of x from Eq.( 9), as we discussed above).Note that as R stretch ≃ min(R max , r U ), condition for percolation requires that d bh should be smaller than both r U and R max .
If R stretch = r U , then the condition d bh < r U implies If R stretch = R max , then the condition d bh < R max implies In Table 1 we have given several sets of values for various parameters for which string percolation happens.We see that for all these cases, R stretch >> r η , (where R stretch ≃ min(R max , r U )).This means that, after getting produced near r η , the string loops are dragged (via friction forces) and stretched by the plasma flow up to very large distances.There are several points to be noted from numbers given in table I.We have included two cases, represented by boldfaced digits in the column of d bh .For d bh = 520.0GeV −1 and 5.4 × 10 11 GeV −1 , string loops will not percolate.This is because in these cases, though d bh is smaller than R max , but it is larger than r U .As string loops can get stretched only up to a distance R stretch ≃ min(R max , r U )), in these cases, loops emanating from different black holes will not intersect, hence no possibility of percolation.We have also chosen parameters to show cases such that in some case r U > R max , while in other cases R max > r U , (both lengths being larger than d bh allowing for string percolation).
We also see that the percolation of strings is not a rare occurrence, but happens for rather generic set of parameter values.It is important to see that strings with energy scale η = 10 7 GeV can percolate and form an infinite network when the ambient temperature of the Universe is as low as T U = 1 GeV.Similarly, strings with η = 10 5 GeV scale can form with T U = 50MeV.(Though, for such low temperatures, one should change value of g * in Eq.( 4) etc.).Further, even with almost negligible fraction of energy density in black holes, f = 10 −10 , string percolation is achieved.
However, for large values of η, percolation is achieved only during very late stages of black hole evaporation when its mass has become very small (i.e.x becomes very large, with mass of black hole in this late stage being M 0 /x), only few 100 times the Planck mass.Eqs.( 1),(2) describe black hole evaporation process when quantum gravity effects have been neglected.It can then be of concern whether these results can be used for such late stages of black hole evaporation when its mass is not very much larger than the Planck mass.We, therefore, consider smaller values of η also, for which much larger black holes masses can also lead to string percolation.Though, we mention that the numbers given in table I do not represent any optimized set of values of various parameters.These are just taken as sample examples, showing that string percolation in this model can happen for a range of parameter values.For example, small values of M x here are due to the fact that we take maximum allowed value of x from the inequality in Eq.( 9).We do that so that we get largest values of r U and R max (as we mentioned above).However, even with smaller values of x string percolation will be possible, if suitable values of other parameters (e.g.T U ) are chosen.We have also ignored shock formation in the expanding plasma.During last stages of exploding black hole, shocks will be expected and they can significantly add to the possibility of expanding string loops attaining large sizes.A more detailed investigation is needed for all these considerations to better optimize these parameter values.
We mention that we have considered here black holes of similar masses as an example.Clearly, the basic physics of this model will remain applicable for black holes with a range of masses.Though, simplest situation for this model will be to have a black hole mass spectrum which is peaked at some specific value, for example, if the black holes form in a first order phase transition [18].
We now come back to the issue we mentioned above in the beginning relating to possible differences between the final infinite string network in the present model, and the conventional string network formed via the Kibble mechanism.First, we clarify that we have not actually demonstrated that an infinite string network will necessarily form in this scenario.Even if all requirements for overlapping string loops (arising from different black holes) are satisfied, including intercommutation at string intersection points, one still needs to check that intersecting loops do percolate to give rise to infinite string network.Proper answer for this question can be given by a detailed numerical simulation of this scenario.Even if we accept that an infinite string network will form in this scenario, there are reasons to expect that there may be important differences in the nature of string network on large scales [23].In the conventional scenario of Kibble mechanism, the starting point is the existence of uncorrelated domains after an overall phase transition in the Universe, with random variation of the order parameter field (say, the phase of scalar field for U(1) strings) across different domains.This, with the use of geodesic rule, leads to non-trivial windings, and hence the formation of strings.We may think of this conventional scenario as top to bottom model, i.e. "from order parameter field variations to strings" model.A very important feature of this scenario is that non-trivial windings are present at all scales larger than the correlation domains, and in particular they are necessarily present at scales larger than the horizon (Hubble) scale [23].Indeed, precisely this causality argument was the one which led to the monopole problem in the Universe, and which strongly constrains the domain wall models.The situation, in this respect, is very different in the scenario we have discussed in this paper.Here, PBHs evaporate in a relatively cold Universe where the order parameter field has settled to a roughly uniform, topologically trivial, configuration (ignoring any previously present strings).Starting point here is small string loops forming around evaporating black holes, which have topologically trivial winding numbers on scales larger than the loop size.Even during the period when these loops expand, due to plasma expansion, order parameter field remains topologically trivial on larger scales, and this remains true even if string loops from different black holes intersect and percolate.This scenario may thus be taken as bottom to top model, i.e. "from string loops to order parameter field variations" model.Even if an infinite string network forms in this scenario, it is no longer clear if actual distribution of order parameter field on large scales, in particular on Hubble scales, is the same here as in the conventional case.If there are important differences, it will raise important questions whether the nature of the infinite string network in the two cases has any crucial differences.Though, approach to scaling solution does not retain much memory of the initial string network, as it involves robust steps, like folding of large loops to self intersection, and small loops radiating away.Thus one may expect that such a scaling solution may be expected in the present scenario also.Still, it will be very interesting to explore this issue in detail whether the final string networks in the two cases, even in the scaling regime, differ in any qualitative manner.It is possible that the cases of gauge strings and global strings may behave very differently in this respect.One may expect stronger differences in the two scenarios of string formation for the global string case, where order parameter field variation plays a more direct role in the string network evolution.In comparison, for gauge string case, the string tension is the primary relevant factor for late stages, thus making it more likely that the string networks in these two scenarios may have similar late time evolution for the gauge string case.
We conclude by emphasizing that our results present a possible scenario for the formation of an extended (infinite) cosmic string network in the Universe without the entire Universe undergoing a spontaneous symmetry breaking phase transition.Importantly, this happens when the ambient temperature of the Universe is much below the relevant spontaneous symmetry breaking scale.String loops are generated by evaporating primordial black holes leading to local symmetry restoration.Expansion of locally heated plasma cools it, leading to symmetry breaking (again locally), producing cosmic string loops.Expanding plasma stretches these loops, and intersection of loops originating from different regions (heated by different PBHs) can lead to percolation, hence the formation of an extended (infinite) string network without an overall phase transition.Though there appear to be some important differences in the distribution of the order parameter field on large scales in this scenario, as compared to the conventional Kibble mechanism (as discussed above), the robust mechanisms responsible for achieving a scaling solution for the string network may hold out here also leading to the standard scaling solution.It needs to be further explored if, even in the scaling regime, there are any important differences between the string networks in the two cases.
I would like to thank R. Rangarajan, S. Digal, S. Sengupta, R. Ray, B. Layek, and A.P. Mishra for useful discussions and comments.