Gravitational wave signature of generic disappearance of Z 2 -symmetry breaking domain walls

Breaking of discrete parity at high scale gives rise to Z 2 -domain walls. The metastability of such walls can make them relatively long lived and contradict standard cosmology. We consider two classes of theories with similar underlying feature, the left right symmetric theories and two Higgs doublet models. These theories eﬀectively possess two steps of breaking of Z 2 discrete symmetries. The domains formed at a high energy scale in the ﬁrst step further decompose into subdomains near the electroweak scale in the second step. Then a QCD instanton induced energy diﬀerence can remove the subdomain walls as well as the domain walls successfully. This result holds regardless of whether the domain wall removal is guided by small symmetry breaking terms or purely statistical outcome of a homogeneous vacuum. We then investigate the gravitational waves arising from the collapse of such domain walls and show that the peak frequency of these waves lies in the 10 − 9 –10 − 7 Hz band, depending on annhilation temperatures of 10 − 2 –1 GeV, which is sensitive to pulsar timing based experiments such as SKA and NANOGrav.


I. INTRODUCTION
Several Beyond Standard Model (BSM) theories of particle physics have been proposed over the years to address various issues that the Standard Model (SM) cannot explain.
Examples of these issues include the existence of small non-zero neutrino masses, necessity of more sources of CP violation and more sources of first order phase transition as required for successful baryogenesis, hints of unification of couplings at high energy etc.Many of these models possess a common feature that we will refer to as Z 2 -symmetry breaking in this paper.In Z 2 -symmetry breaking, the models possess an accidental or deliberate Z 2symmetry at high energy which gets broken at lower energies due to spontaneous symmetry breaking.Breaking of Z 2 -symmetry gives rise to energy barriers called domain walls (DW) which contradict the cosmology that we observe today.In this paper we study this issue and propose a mechanism to remove them successfully in the early universe itself for many models, without invoking any new non-renormalisable physics.We then discuss prospects of direct verification of our proposed mechanism in upcoming experiments.
A famous class of models possessing Z 2 -symmetry breaking are the left right symmetric extensions of the Standard Model (LRSM).First proposed by Mohapatra and Senjanović [1], LRSM is a minimal extension of SM based on the gauge group SU(3) c × SU(2) L × SU(2) R × U(1) B−L augmented with a discrete parity or left-right symmetry P which interchanges the gauge bosons and triplet Higgs fields of SU(2) L and SU(2) R besides implementing spacetime parity.The usual SU(2) L Higgs doublet of SM is extended to a SU(2) L × SU(2) R bidoublet.The model naturally accommodates the discrete parity violation of SM as a result of spontaneous symmetry breaking of SU(2) R × U(1) B−L , and also elegantly explains small neutrino masses via the seesaw mechanism.The model adds new heavy Higgs SU(2) L and SU(2) R triplets, henceforth called ∆ L , ∆ R , in order to implement the seesaw mechanism.The spontaneous breaking of Z 2 -discrete symmetry P together with the gauge group SU(2) R × U(1) B−L is achieved by providing a vacuum expectation value (vev) to ∆ R .This is one of the most important examples of models with Z 2 -symmetry breaking.
An interesting variation of LRSM was provided by Chang, Mohapatra and Parida [2] who proposed a 'parity decoupled' version (PLRSM) of the left right symmetric Standard Model.
The gauge symmetry of this model is the same as that of LRSM.However as shown by the authors, in this model it is possible to break the discrete Z 2 -symmetry P at a higher scale without breaking the left right symmetric gauge group SU(3) c ×SU(2) L ×SU(2) R ×U(1) B−L .This is done by introducing an additional scalar Higgs field η which is singlet under the left right symmetric gauge group, but transforms as η → −η under P .In other words, P implements this discrete symmetry as well as space time parity, without affecting the other particles.At very high energies the model is parity invariant.At a slightly lower scale M P , η takes a non-zero vacuum expectation value (vev) which breaks the Z 2 -symmetry P but leaves the gauge symmetry unaffected.The SU(2) R × U(1) B−L gauge symmetry is broken at an even lower scale M R .This decoupling of parity breaking and gauge symmetry breaking scales allows one to have right handed W R and Z R bosons at a few TeV as well as an intermediate scale of partial unification into gauge groups like the Pati-Salam SU(4) c × SU(2) L × SU(2) R at scales of a few hundred TeV [2].Some of these features may show up in upcoming collider experiments.Other partial unification features have low energy implications like neutronantineutron oscillations and neutrinoless double beta decay which may also be detected in the near future with more sensitive experiments.These issues make this model experimentally attractive.
Two Higgs doublet models (2HDM) [3] are studied as examples of models 'just beyond SM' as they provide more possibilities of electroweak baryogenesis by containing more sources of CP violation and more scope of first order phase transitions, which is not the case with the usual SM [4,5].These models have two (or more) SU(2) L Higgs doublets, conventionally called Φ 1 and Φ 2 .In order to prevent disastrous consequences of flavour changing neutral currents, the quark and Higgs sectors in 2HDMs are often required to satisfy a discrete or some other similar symmetry.This scenario is called Type II 2HDM, where the up-type right handed quarks couple only to Φ 1 and the down-type right handed quarks couple only to Φ 2 .It is the most popular 2HDM as it arises as the effective low energy theory for important BSM models like supersymmetry and axion models [3].As we will see later, it can arise also as the effective low energy theory of left right symmetric models like LRSM and PLRSM.This Type II Z 2 -symmetry is broken when a certain linear combination − sin βΦ 1 + e −iθ cos βΦ 2 gets a vev of v 1 in an extended scenario of electroweak symmetry breaking [3].
In the next section, we review the status of several Z 2 -symmetric models where cosmological domain walls can arise.We also recall the mechanism of Preskill et al [6], relying on QCD instanton anomaly effects, that induce a pressure difference across domain walls arising from Z 2 -symmetry breaking in Type II 2HDM.Following this in Sec.III, we perform a detailed analysis of high scale domain wall creation in PLRSM, incorporating the best known bounds on the parameter space of LRSM.In the next section IV we show how the Type II 2HDM arises as an effective low energy theory in the LRSM and PLRSM models.
Section V describes the two types of domain walls formed one after the other due to two stages of Z 2 -symmetry breaking in PLRSM and LRSM.In Sec.VI we show that both types domain walls that are formed can be successfully removed at almost the same time due to an energy bias caused by the Preskill et al. result and random fluctuations in the numbers of certain types of Type II 2HDM subdomains inside an LRSM/PLRSM domain.The removal occurs before Big Bang Nucleosynthesis (BBN) at temperatures of around 2 GeV, and does not require any Z 2 -symmetry breaking non-renormalisable effects.Annihilation of domain walls gives rise to gravitational waves.In Sec.VII we show that the peak frequency of these waves arising in our mechanism lies in the 10 −7 -10 −6 Hz band, corresponding to annhilation temperatures of 1-10 GeV.These frequencies are detectable by existing and proposed pulsar timing array based gravitational wave detectors like NANOGrav, SKA etc. [7,8].The recent NANOGrav results [8] rule out our DW collapse model for higher values of parity breaking scale above 10 7 GeV.Our DW collapse model with parity breaking scales below 10 7 GeV remains consistent with the current NANOGrav results and has a good chance of being seriously tested in future pulsar timing based experiments.We finally conclude with the implications of this study in Section VIII.

II. COSMOLOGY WITH INTERMEDIATE SCALE DOMAIN WALLS
In this paper we study cosmological implications of the breaking of discrete Z 2 -symmetry in BSM models like the ones discussed above.Breaking of Z 2 -symmetry gives rise to a network of domain walls in the early universe separating domains which can be one of two types [9][10][11].For example, breaking of the η → −η discrete parity symmetry in PLRSM model at energy scale M P gives rise to domain walls separating two types of domains.In one type of domain the η Higgs field takes the vev M P and in the other, the vev −M P .We shall call the first type of domain as 'positive-high' and the second type as 'negative-high'.
The term 'high' is there to remind us that we are at a high energy scale of Z 2 -symmetry breaking viz. at M P .This feature of PLRSM forming its own domain wall has not been noticed before, though in their original paper Chang, Mohapatra and Parida discussed how certain symmetry breaking chains descending from SO (10) to SM avoid SO (10) domain wall formation whereas other symmetry breaking chains do not.
In the usual LRSM model, the breaking of SU( 2 The term 'low' is there to remind us that we are at a lower energy scale of Z 2 -symmetry breaking M 2HDM , near the electroweak scale.
In the absence of other factors, these domain walls will be sufficiently long lived so as to conflict with standard cosmology.Even if the Z 2 -symmetry is only approximate, it is still possible in some scenarios to end up with long lived domain walls [12].The usual way to remove these domain walls is to introduce explicit terms softly breaking the discrete Z 2 symmetry to the Lagrangian.For example, Planck scale suppressed non-renormalizable operators were added in [13,14] to the Lagrangian leading to instability to the domain walls.However this introduction of Planck suppressed operators may not work for all gauge symmetric models.
In the Supersymmetric Left-Right symmetric Models (SUSYLR) or ultraviolet completions theoreof, with all Higgs carrying gauge charges, it is possible to introduce Planck scale sup-pressed terms or soft SUSY breaking terms that are well regulated.One can then demand that the new operators ensure sufficient pressure across the domain walls that the latter disappear before Big Bang Nucleosynthesis (BBN).This requirement has been discussed in detail in [15][16][17][18].It has also been shown that domain walls in SUSYLR can be successfully exploited for baryogenesis via leptogenesis while remaining consistent with low energy observables like the electric dipole moment of the electron, with a gravitational wave signature that can be detected in upcoming experiments [19].More recent works have studied the gravitational wave consequences of domain wall collapse [20,21] as well as bubble wall collapse [21][22][23] in LRSM, where the domain wall collapse is initiated by adding tiny explicit terms breaking the discrete left-right symmetry.In a related vein, there has been work [24] studying the gravitational wave consequences of models providing a seesaw mechanism together with thermal leptogenesis.Domain wall formation due to Z 2 -symmetry breaking in 2HDM models was studied recently by Chen et al. [25].The authors showed that tiny explicit CP violating terms in the 2HDM Higgs potential can remove the domain walls successfully before BBN, and studied the gravitational wave and electron EDM consequences of the consequent domain wall collapse.
Preskill et al. [6] have investigated domain wall removal in Type II 2HDM by showing that the Z 2 -symmetry in Type II 2HDM is anomalous.More precisely, they point out that the instanton vertex in quantum chromodynamics (QCD) makes the two types of domains viz.
positive-low and negative-low non-degenerate.In other words one of the types of domains, say positive-low, has a slightly lower energy density than the other type which cannot be seen in the classical perturbative theory but only in the quantum non-perturbative theory.Preskill et al. [6] relate the QCD instanton vertex to the QCD θ angle and provide a temperature dependent expression for the energy difference between the two types of domains.The energy difference falls rapidly with temperature which implies that the domain walls can only be removed after electroweak symmetry breaking, probably at a temperature not far from the QCD scale of 340 MeV.Their analysis is preliminary but shows that domain walls in Type II 2HDM without explicit CP violation need not be cosmologically catastropic.
In this work, instead of adding non-renormalisable operators to the Lagrangian or invoking explicit CP violating terms, we take a different approach to domain wall removal.We leverage the QCD anomaly based mechanism of Preskill et al. [6], and show that it can lead to successful Z 2 -domain wall removal in all the above examples of Z 2 -symmetry breaking, including LRSM and PLRSM which do not obviously involve the Type II 2HDM, when combined with statistical properties of random events.In particular, our method does not require the addition of explicit CP violating terms to the 2HDM Higgs potential, which was the strategy undertaken by Chen et al. [25].
Let us consider the PLRSM model as a running example.At the high scale M P , two types of domains, positive-high and negative-high, are formed.At a lower energy scale M 2HDM when the Type II Z 2 -symmetry of 2HDM breaks, two new types of subdomains, positive-low and negative-low, are created.At this scale a domain of positive-high type will break up further into a bunch of positive-low and negative-low type subdomains.The same will hold for a domain of negative-high type.Percolation theory describes the numbers, sizes and spatial distribution of the low type subdomains within a high type domain.Whether a particular positive-high domain gives rise to more positive-low or negative-low subdomains is a random variable where either outcome occurs with probability very close to 0.5.The theory of random variables tells us that with probability nearly 0.25, a non-trivial majority of positive-high domains will have a non-trivial excess of positive-low subdomains, and a non-trivial majority of negative-high domains will have a non-trivial excess of negative-low subdomains.Non-trivial majority or non-trivial excess here means that the chance excess is around square root of the total number of a suitable class of domains / subdomains.The QCD anomaly argument of Preskill et al. [6] then kicks in and shows that, on average, the positive-high type domains have lower energy density as compared to negative-high type domains.We show that this feature can give rise to enough pressure difference to remove both the high type as well as the low type domain walls almost simultaneously in the radiation era of the universe well before BBN.Both types of walls are annihilated at temperatures around 2 GeV for parity breaking scales ranging from 10 6 to 10 7 GeV.Previous works like [26] have considered probabilistic percolation based arguments to show the decay of domain walls and eventual domination of one type of domain in the early universe.However they assume the presence of a primordial probabilistic bias between the two types of domains.
The present paper differs from them in assuming no bias between the types of domains at the time of their formation.Rather it shows that natural probabilistic fluctuations in the number of domains of each type combined with a QCD anomaly that manifests itself at a much lower temperature suffice to remove all domain walls effectively in the early universe.
The conclusion is that explicit soft discrete symmetry breaking terms are not the only viable way to successfully remove domain walls; pure statistical fluctuations combined with an inevitable QCD anomaly can also remove them.In other words, it is fully possible that the universe we observe is a purely chance survival of one out of the four distinct possibilities.
Our result is agnostic with respect to this provenance.

III. THE PLRSM MODEL
The gauge group of PLRSM is the left-right symmetric group SU(3 Note that Φ = τ 2 Φ * τ 2 transforms in same manner as Φ under the gauge action.Under discrete parity, these Higgs fields transform as follows: The electric charge is given by The charge zero condition forces the vevs of the Higgs fields to be where the placement of complex phases is the customary one obtained by gauge invariance and the real numbers are all positive [27]. We let g L , g R and g B−L denote the couplings of the gauge groups SU(2) L , SU(2) R and U(1) B−L respectively.Their values can ultimately be related to the SM couplings g 2 and g Y of the SU(2) L and U(1) Y gauge groups.At very high energies, the PLRSM model satisfies discrete parity symmetry in addition to gauge symmetry which implies that g L = g R at very high energies.The renormalisable Higgs potential can be written as where the individual terms are the most general ones allowed by gauge and parity symmetry [2,27] viz.
We assume that µ η is much larger than the other masses in the model.So to a very good approximation we can minimise the total potential by first minimising the term V η followed by sequentially minimising the other terms.Minimising V η in Equation 2 leads to the conclusion that at a high scale M P ≈ µη √ 2δ 1 the η field takes the vev η = M P .This gives Using Equations 6, 4, the effective mass terms for ∆ L and ∆ R become Restricting ourselves to only those terms that survive the setting of the bidoublet vevs to zero, we get The triplet Higgs field ∆ R gets a non-zero vev at a lower scale M R breaking down SU(2) R × U(1) B−L to U(1) Y , Φ gets a non-zero vev at an even lower scale breaking down SU(2) L ×U(1) Y to U(1) em .This decouples the parity breaking scale M P from the gauge breaking scale M R .Note that µ 2 ∆ L > 0 as a result of which ∆ L gets a zero or almost zero vev.The hierarchy achieved by the fine tuning allows us to conclude with a high degree of precision . In fact after parity breaking at scale M P , the ∆ L fields end up with higher masses than the ∆ R fields even before gauge symmetry breaking at the lower scale M R .
A consequence of this is that the gauge couplings g L and g R are no longer equal between the energy M P and M R owing to their different runnings under their respective renormalisation group equations.All these features are achieved by implementing the fine tuning condition Notice now that V η has another minimum at η = −M P .This new minimum cannot be gauged away as the gauge has been fixed by the customary electromagnetic charge zero convention of the vevs of ∆ L , ∆ R and Φ.This new minimum leads to a different vev assignment to the other Higgs fields because, using Equations 6, 4, the effective mass terms for ∆ L and ∆ R now become In this scenario, ∆ L gets a non-zero vev at the lower scale M R breaking down SU(2) L × U(1) B−L to U(1) Y , Φ gets a non-zero vev at an even lower scale breaking down SU(2) R ×U(1) Y to U(1) em .This scenario is the mirror image of the standard scenario and can be thought of as the right handed Standard Model.
From a cosmological viewpoint, the early universe breaks down at the high energy scale of M P into a network of domains of two types.In the positive high domain, η = M P .Later on at a lower energy scale M R , the 'right handed triplet Higgs fields' take non-zero vevs and so the positive high domain eventually leads to the familiar left handed SM domain.In the negative high domain, η = −M P .Later on at a lower energy scale M R , the 'left handed triplet Higgs fields' take non-zero vevs and so the negative high domain eventually leads to the right handed SM domain.To summarise, the two sets of vevs are Since at the left right gauge symmetry breaking scale M R the bidoublet has zero vev, it has not been mentioned in the above equation.
The breaking of the discrete parity symmetry and the formation of the two types of high energy domains also leads to a topological domain wall between them.The presence of this wall or energy barrier conflicts with current cosmology.Several earlier works have discussed how a similar wall that occurs in the minimal LRSM or its supersymmetric variant may be made to disappear fast enough so as to be consistent with present day observations.All these works either add Z 2 -symmetry breaking Planck suppressed non-renormalisable operators [14,15] or soft SUSY breaking terms [17] to the Lagrangian in order to remove the domain walls.However the formation of domain wall in the PLRSM has not been noticed before.
Analogous to domain wall removal in the minimal LRSM, it is possible to add Z 2 -symmetry breaking Planck suppressed non-renormalisable operators to the Lagrangian of PLRSM in order to remove its domain walls.However in this paper we shall not do so.Instead, we shall leverage the QCD anomaly based mechanism of Preskill et al. [6] and combine it with statistical arguments in order to ensure successful domain wall removal.The details follow in the subsequent sections.
The best model independent lower bounds on the energy scale M R of left right gauge symmetry breaking are of the order of around 4.7 TeV and come from the ATLAS experiment [28].Since in the PLRSM model the scale M P of discrete parity breaking is assumed to be decoupled and significantly higher than M R , we shall take a lower bound of M P > 10 6 GeV in our work.

IV. TYPE II 2HDM AS AN EFFECTIVE LOW ENERGY THEORY
In this section we shall see how the Type II 2HDM arises as an effective low energy theory in the LRSM [1] and PLRSM [2] models.
In both PLRSM and LRSM, the Yukawa part of the Lagrangian involving the bidoublet Φ, neglecting generation indices, is of the following form where the quarks and leptons are organised into doublets under the action of SU(2) L or SU(2) R as appropriate i.e.
After the breaking of the SU(2) L × SU(2) R gauge symmetry at scale M R , the above LRSM Yukawa terms become 2HDM Yukawa terms.In the LH or positive-high domain, the resulting 2HDM Yukawa terms become under the gauge group SU(2) L × U(1) Y .Ignoring the high mass right handed neutrino ν R , assuming h 3 , h 4 to be proportional to h 1 , h 2 for simplicity, the Yukawa Lagrangian in the LH domain becomes the standard Type II 2HDM Lagrangian for the two SU(2) L Higgs doublets It can be checked that the two doublets above have weak hypercharge Y = 1, and also that they are orthogonal because the constituent fields φ ij are distinct, as expected from a left handed 2HDM.
In the RH or negative-high domain, the resulting 2HDM Yukawa terms become under the gauge group SU(2) R × U(1) Y .Ignoring the high mass left handed neutrino ν L , assuming h 3 , h 4 to be proportional to h 1 , h 2 for simplicity, the Yukawa Lagrangian in the RH domain becomes the standard Type II 2HDM Lagrangian for the two SU(2) R Higgs doublets Again, the two doublets above have weak hypercharge Y = 1 and are orthogonal, as expected from a right handed 2HDM.
Thus, to summarise both PLRSM and LRSM reduce to Type II 2HDM in both LH and RH domains after the SU(2) L × SU(2) R gauge symmetry breaking, provided h 3 , h 4 are proportional to h 1 , h 2 respectively.Without this proportionality condition however, they do not reduce to 2HDM.
Finally, we would like to discuss the issue of flavour changing neutral current (FCNC) effects in LRSM, and how they do not apply to the low energy effective theory of Type II 2HDM that we get as described above when the proportionality condition on the Yukawa couplings holds.FCNC effects occur in PLRSM and LRSM after the LR symmetry is broken only if the two resulting SU(2) L Higgs doublets get vevs of the form Φ 1 = (v 1 , 0), Φ 2 = (0, v 2 e iθ ) i.e. the vevs are of the bidoublet vev form.This can happen if the LR breaking scale is not too distant from the scale of the Higgs doublet vevs.This was discussed in the paper by Zhang et al. [29].However in this paper we are considering the scale of Higgs doublet vevs close to the electroweak scale and well separated from LR breaking scale.That is why in our scenario, after LR symmetry breaking, the effective theory becomes a Type II 2HDM under the proportionality condition.Hence in our model, the two Higgs doublets take vevs of the form Φ 1 = (0, v 1 ), Φ 2 = (0, v 2 e iθ ) and there are no flavour changing effects at tree level.Thus, our conclusions in this section are consistent with the observations in Section 4.2 of Zhang et al..

V. HIERARCHICAL DOMAIN WALL FORMATION IN PLRSM
In most examples of Z 2 -domain walls formed at high enough energy scales, the pressure difference δV between the two types of domains is much smaller than the average energy density V 0 of a domain.For example, in the PLRSM model because of our fine tuning, the largest contribution to the Higgs potential is from V η .Thus the height V 0,hi of the potential barrier occurring at Hence we can accurately describe the surface energy density of the high type of domain wall by . Because of the Z 2 -symmetry at high scale, δV hi (t fo,hi ) = 0 at tree level, where t fo,hi is the time of formation of the high type domain walls.Non-renormalisable effects and quantum corrections are expected to be quite small compared to V 0,hi .Thus, δV hi (t fo,hi ) ≪ V 0,hi .In particular δV hi (t fo,hi ) V 0,hi < 0.795, as a result of which percolation theory ensures that there are giant domains of both positive-high and negative-high types formed at scale M P together with large scale domains walls separating them [30].Moreover, the probabilities p +,hi , p −,hi of forming positive-high and negative-high type domains are very close to 0.5 each, since their ratio must satisfy [30] p +,hi p −,hi ≈ exp − δV hi (t fo,hi ) V 0,hi ≈ 1.
Some time after the discrete parity breaking in PLRSM at temperature T fo,hi = M P resulting in the creation of giant domains of positive-high and negative-high types, the gauge symmetry SU(2) L × SU(2) R breaks in each domain.In the positive-high domian, SU(2) R breaks at an energy scale M R < M P .In the negative-high domian, SU(2) L breaks at energy scale M R .However these are not discrete symmetry breakings, and so no further subdomains after formed inside the domains yet.
Another way to express the two choices above is by postulating that a certain linear combination of Φ 1 and Φ 2 gets two possible vevs: Note that these ansatz are valid even in the absence of any explicit CP violation assumed in the 2HDM potential.
Due to the above spontaneous Z 2 -symmetry breaking in the Type II 2HDM, a high type domain breaks up into a bunch of positive-low and negative-low subdomains.Let t fo,lo be the time of formation of the low type subdomains.The effective theory inside all the subdomains formed within a positive-high domain is the usual left handed SM.The effective theory inside all the subdomains formed within a negative-high domain is the right handed SM.In this work, we assume that M 2HDM ≈ 10M EW i.e. the 2HDM Z 2 -symmetry breaking scale is well separated by around an order of magnitude above the electroweak scale.
Preskill, Trivedi, Wilczek and Wise [6] show that for temperatures T above the QCD scale Λ QCD = 340 MeV [31] the positive-low subdomain, say, has a lower energy density than the negative low subdomain by an amount equal to where m u = 2. 16  The height V 0,lo of the potential barrier between positive-low and negative-low subdomains is around M 4 EW ≈ 10 8 GeV 4 [25].Hence the probabilities p +,lo , p −,lo of forming positive-low and negative-low type subdomains are very close to 0.5 each, since their ratio must satisfy [30] p +,lo p −,lo ≈ exp − δV lo (T fo,lo ) V 0,lo ≈ 1.

Moreover
δV lo (T fo,lo ) V 0,lo < 0.795; so percolation theory ensures that there are giant subdomains of both positive-low and negative-low types inside each high type domain, together with large scale low type walls separating them [30].

VI. TWO STAGE DOMAIN WALL REMOVAL IN PLRSM
Recall from the previous section that the high type walls are formed at time t fo,hi and temperature T fo,hi .The surface tension inside the high type walls is σ hi ≈ M 3 P , where M P is the scale at which parity is broken.We take T fo,hi = M P .Some time later on, the low type walls are formed at time t fo,lo and temperature T fo,lo ≈ 10M EW .The surface tension inside the low type walls is σ lo ≈ M 3 EW ≈ 10 6 GeV 3 [25].
We assume that the high type walls follow the scaling evolution soon after t fo,hi and all the way till t fo,lo and even after till the time of annihilation of high and low type walls.This is because the high type walls are formed from the η scalar field which only interacts with the singlet η Higgs particle, a very massive Higgs.A similar remark would hold for high type walls in the minimal LRSM where they are formed from the triplet ∆ L , ∆ R Higgs particles, which are again massive Higgs.In particular, the high type walls do not interact with the 2HDM and SM particles which proliferate inside both kinds of high type domains at temperatures significantly below T fo,hi .Hence, the radius of curvature of the high type walls at time t fo,lo is given by the scaling equation [30] R hi (t fo,lo This is the Hubble radius at time t fo,lo divided by a wall parameter A = 0.8 ± 0.1.
When low type walls are formed at time t fo,lo , their correlation length ξ lo is given by [33] where M Pl = (8πG) −1/2 = 2.435 × 10 18 GeV is the reduced Planck mass.The above equation holds because at the time of formation, the surface tension in the low type walls has to balance the frictional force that they encounter due to interaction with the particles in the low type subdomains moving around.
We now need to deduce some properties of the low type subdomains formed within a single high type domain.For concreteness, let us concentrate on the positive low subdomains inside a high domain.We can model the formation of the positive low subdomains via percolation theory, by fitting a cubic lattice inside a high domain.The low subdomain correlation length ξ lo is taken to be the spacing between two nearest lattice points.The number of lattice points N L inside a high domain is now given by Since this is occuring in the radiation dominated era of the early universe, the relation between time and temperature is given by [30] T fo,lo = 8.747 × 10 −4 ( g * (T fo,lo ) 10 where g * (T fo,lo ) is the number of relativistic degrees of freedom for the energy density at temperature T fo,lo .Plugging this into the expression for the ratio gives Because two adjacent lattice points are ξ lo apart, the choices of the types of low domains formed at those two points become two independent random variables with probability 0.5 of either choice occuring at a lattice point.In other words, the formation of positive low subdomains can be treated as a site percolation problem on a simple cubic lattice in three dimensions [34] with p = 0.5 being the probability that a particular lattice point is occupied.
Maximal sets of adjacent occupied sites are known as clusters.Percolation theory says that if the occupation probability p is above the so-called percolation threshold p c , then there is exactly one giant, aka infinite, cluster that touches all faces of the bounding cube of the lattice [34].For site percolation on the simple cubic lattice in three dimensions, p c = 0.311 [34].Since in our case 0.5 = p > p c = 0.311, we have exactly one giant subdomain of positive low type inside a high domain.Similarly we have exactly one giant subdomain of negative low type inside a high domain.
However, in addition to the unique giant or infinite cluster, there are many small, aka finite, clusters.Let P ∞ be the probability that an occupied site belongs to the giant cluster.
Then the fraction V f of lattice points belonging to finite clusters is given by According to the scaling laws of percolation theory, this fraction V f is described in terms of the so-called critical exponent β [34]: For the simple cubic lattice in three dimensions, β = 0.4 [34].In fact, it is believed that β depends only on the dimension of the space and not on the particular lattice used to tile the space, a phenomenon called universality.Plugging in our values for p, p c , we get that 0.24 fraction of lattice points belong to small clusters.This means that 0.26 fraction of lattice points belong to the unique giant cluster.The remaining 1 − p = 0.5 fraction of sites is unoccupied as is to be expected.
Percolation theory also tells us that the number of finite clusters N f is governed by another critical exponent α [34]: For the simple cubic lattice in three dimensions, α = −0.5 [34].In fact, it is believed that α depends only on the dimension of the space and not on the particular lattice used to tile the space, thanks to universality.Plugging in our values for p, p c , N L , we get The discussion in the above paragraphs leads us to conclude that the arrangement of low subdomains inside one high domain follows the so-called Swiss cheese phenomenon of percolation theory.There is a unique giant positive-low subdomain plus a unique giant negative-low subdomain inside a high domain.Interspersed within the giant positive-low subdomain are small negative-low subdomains like holes in Swiss cheese.Similarly, interspersed within the giant negative-low subdomain are small positive-low subdomains.In addition, starting from the outer surfaces of the two giant low subdomains and extending all the way to the inner surface of the enclosing high domain is a network of small positive and small negative low type subdomains.The number of positive-low subdomains N ′ f near the inner surface of the enclosing high domain is thus given by For T fo,lo = 10 3 GeV, g * (T fo,lo ) 10 = 10.675[35].Plugging in the values of M Pl , σ lo gives us The pressure difference on a high type wall occurs due to the chance excess of the surface small positive low subdomains over the surface small negative low subdomains.This excess number N ′′ f is given by random fluctuation theory to be From the above discussions, whether the positive-low subdomains or the negative-low subdomains are in excess inside a high type domain, is an event that occurs with probability very close to 0.5.Thus with probability very close to 0.25 it happens that, say, a nontrivial majority of the positive-high domains contain a non-trivial excess of surface small positive-low subdomains and a non-trivial majority of the negative-high domains contain a non-trivial excess of surface small negative-low subdomains.We now study the cosmological consequences of this pure one out of four chance event.
As the temperature T cools below T fo,lo , the energy bias between positive-low and negativelow types of subdomains due to the QCD instanton anomaly as given by Equation 24increases.We assume that the high type walls continue to evolve according to the scaling limit.The low type walls are constrained by frictional forces because they continue to interact with many 2HDM and SM particles, including somewhat heavy ones like the top quark, SM Higgs, W and Z bosons etc.At temperatures below 4 GeV, the friction between the low type domain walls and the particles in the low type domains decreases significantly since the wall is made from 2HDM Higgs fields which can interact with only the light first and second generation quarks and leptons plus the tau lepton that are left now.As we shall see below, the high and low walls will be annihilated at lower temperatures of around 2 GeV.Following Blasi et al. [36], we assume that the radius of curvature of the low walls asymptotically evolves as a power law for some 0 < λ < 1, for time t > t fo,lo .The paper [36] shows via simulations that the asymptotic evolution of walls under mild friction is rather close to the scaling limit.Hence we take the exponent λ in our power law to range from 0.9-0.99 in this work.The consequence of the power law assumption is that the the radius of curvature of the low walls increases slower than scaling limit while their number inside one high domain remains the same.
This happens because the shapes of the low subdomains, and especially the smaller low subdomains, are highly irregular like fractals [34].
The volume pressure difference acting on a high type wall at temperature T , T < T fo,lo thus becomes At a later time t ann,hi > t fo,lo and correspondingly lower temperature T ann,hi < T fo,lo , the high walls experience instability when the volume pressure difference between high domains starts becoming comparable to the surface tension inside the high walls.The equation for this is given by [30]: where C ann is a constant taking values between 2 and 5.We shall take C ann = 2 in this work.
We take the parity breaking scale M P to range from a lower bound of 10 6 GeV, as discussed in Section III, to an upper bound of 10 7 GeV.As mentioned earlier, σ hi = M 3 P , σ lo = 10 6 GeV 3 .As we shall see shortly, both high and low type walls will annihilate in the temperature range 0.5-5 GeV.In this range,   The low type subdomain walls experience instability at a time t ann,lo > t fo,lo corresponding to temperature T ann,lo < T fo,lo when the volume pressure difference between low domains starts becoming comparable to the surface tension inside the low walls.The equation for this is given by [30]: . This implies that [30] t ann,lo = σ lo C ann A δV lo (t ann,lo ) 1/λ = (6.58× 10 −25 s) (C ann A) Since all this is occuring in the radiation dominated era, plugging Equations ( 39), ( 28  = (2.18× 10 8.5 )(4 × 10 −9.5 ) 1/λ .
At the higher friction end of the evolution of the low walls when λ = 0.9, we get The above analysis shows that for a range of values 10 6 -10 7 GeV of the parity breaking scale M P , and for a range of values 0.9-0.995 of the power in the friction dominated evolution of the low walls, both high and low walls experience instability within one order of magnitude of temperature range 1-25 GeV.So both types of walls annihilate at almost the same time resulting finally in a single domain which is our familiar left symmetric SM current universe.
We have thus shown that both high type and low type domain walls can disappear at almost the same time well before the advent of BBN, which occurs at a temperature of 1 MeV [32].Hence, these walls do not leave any discernible signature in the cosmic microwave background radiation (CMBR) and do not contradict the Zel'dovich-Kobzarev-Okun bound [37] of maximum domain wall tension being around few MeV based on observations of the CMBR.

VII. GRAVITATIONAL WAVES FROM COLLAPSE OF DOMAIN WALLS
The almost simultaneous collapse of the two types of domain walls at temperatures within one order of magnitude around T ann = 2 GeV results in primordial gravitational waves.Detection of these primordial waves, distinct from the presently detected astrophysical gravitational waves, would be a breakthrough discovery giving significant clues about the origin and evolution of the very early universe before BBN [38].Several earlier works [39,40] have specifically studied the primordial gravitational waves produced by collapsing domain walls in this regard.
The peak frequency of gravitational waves caused by the collapse of domain walls red shifted to the present time t 0 is given by [30] f peak = 1.1 × 10 −7 Hz g * (T ann ) 10 1/2 g * s (T ann ) 10 where g * s (T ann ) is the number of relativistic degrees of freedom for the entropy density at temperature T ann .The peak energy density spectrum of the waves from the collapse of the high walls at the present time is given by [30] Ω gw h 2 (t 0 ) peak,hi = 7.2 × 10 −44 ǫgw A 2 g * s (T ann ) 10 where ǫgw ≈ 0.7 ± 0.4 and A ≈ 0.8 ± 0.1 for Z 2 -domain walls annhilated in the radiation era.
Similarly, the peak energy density spectrum of the waves from the collapse of the low walls at the present time is given by At annihilation temperatures T ann ranging from 1-10 GeV, both g * (Tann) The peak energy density of the gravitational waves from the collapse of the high walls for parity breaking scales of 10 6 -10 The peak energy density of the gravitational waves from the collapse of the low walls is insignificant in comparison to the high walls because σ lo ≪ σ hi .
These frequencies are about 7 to 8 orders of magnitude below what LIGO-VIRGO [41] can detect today, and what future ground based gravitational wave telescopes like Einstein Telescope [42] are designed to detect.They are also about 6 orders of magnitude below what future space based gravitational wave detectors like DECIGO [43] are designed to detect.However, proposed pulsar timing array based gravitational wave detectors like SKA [7] as well as existing pulsar timing array based detectors like NANOGrav [8] are expressely designed to detect frequencies in the 10

VIII. CONCLUSION AND DISCUSSION
In this work we have addressed the problem of domain wall creation and removal for a class of BSM theories with the common feature of discrete Z 2 -symmetry breaking.We have shown that the seminal PLRSM model of Chang, Mohapatra and Parida [2], which was proposed to show that it is possible to break discrete parity without breaking left right gauge symmetry, is also one such Z 2 -symmetry breaking theory.In particular, PLRSM has a hitherto unnoticed consequence of creating a domain wall at the time of discrete parity breaking, before gauge symmetry breaking.
Without the proportionality condition on the Yukawa couplings, PLRSM and LRSM have a domain wall problem.The walls are of the high type as discussed in the previous sections of this paper.One needs to add Z 2 -symmetry breaking non-renormalisable operators to the Lagrangian as in [14] to remove these walls successfully before they contradict with standard cosmology.
Our work shows that however, when the proportionality constraint on the Yukawa couplings holds, PLRSM and LRSM reduce to Type II 2HDM at lower energies.We then show that there is indeed a mechanism via the formation of low type plus high type domain walls combined with an energy bias due to QCD instanton anomaly and random fluctuations by which all domain walls disappear well before BBN.Complementary to the possibility of primordial gravitational waves arising from domain wall collapse is the possibility of primordial gravitational waves arising from collapse of cosmic strings in the very early universe and the scope of their detecting by existing and planned experiments, see e.g.[44][45][46][47].Many interesting BSM models predict such strings and gravitational waves may offer a novel way of studying them experimentally, see e.g.[48,49].
Proceeding in a very different direction, we observe via the results of Chen et al. [25] that upcoming experiments for measuring the electric dipole moment (EDM) of the electron will constrain some spontaneous CP violating parameters of the underlying Type II 2HDM model involved in the subdomain formation.In particular, the CP violating mixing angle Though our detailed calculations have been carried out for the PLRSM model only, we believe that the main idea of domain wall removal via the formation of Type II 2HDM subdomains at a lower energy scale, followed by the creation of a certain favourable excess of subdomains statistically with probability of around 0.25, and finally followed by the collapse of the subdomain walls due to the QCD anomaly is sufficiently robust.Our work leads one to the tantalising idea that the universe we observe is a purely chance survival of one out of four distinct possibilities.Of course if there were soft explicit Z 2 -symmetry breaking terms in the Lagrangians involved, the domain wall collapse becomes even easier.Our main result is agnostic with respect this provenance.
) L ↔ SU(2) R left right discrete symmetry occurs at an energy scale M R .In one type of domain the triplet Higgs ∆ R takes vev M R leading to our familiar left handed SM vacuum.In the other type of domain ∆ L takes vev M R leading to a right handed SM vacuum where SU(2) L breaks and SU(2) R × U(1) Y controls the electroweak interactions.This is another example of positive-high and negative-high domain where the term 'high' reminds us that the scale of Z 2 -symmetry breaking is at high energy of M R .In Type II 2HDM models, during Z 2 -symmetry breaking − sin βΦ 1 + e −iθ cos βΦ 2 gets a vev of v 1 in one type of domain and a different vev of v ′ 1 in the other type.Henceforth, we shall call the first type of domain as 'positive-low' and the second type as 'negative-low'.
and a Higgs singlet η.All the Higgs fields are singlets under the SU(3) c colour group.Under the left right electroweak gauge group, the Higgs fields transform as follows: At this stage, as argued in the previous section, the effective theory in the positive-high domain is the left handed Type II 2HDM with the gauge group SU(3) C × SU(2) L × U(1) Y , and the effective theory in the negative-high domain is the right handed Type II 2HDM with the gauge group SU(3) C × SU(2) R × U(1) Y .At a still lower temperature T fo,lo = M 2HDM , the Type II 2HDM Z 2 -symmetry breaks resulting in two types of subdomains being formed inside each domain.This happens because there are two possible choices for the vevs of Φ 1 , Φ 2 viz.( Φ 1 , Φ 2 ) = (−v 1 sin β, e iθ v 1 cos β) positive-low type (20) ( Φ 1 , Φ 2 ) = (−v 1 sin β, e −iθ v 1 cos β) negative low-type.

3 = 4 .54 × 10 4 .× 10 3 .
For M P = 10 7 GeV, we get Solving these transcendental equations numerically gives the temperature T ann,hi of the instability of the high walls to range from 23.4 GeV to 10.7 GeV.

5
. At the lower friction end of the evolution of the low walls when λ = 0.99, we get T = 0.79.Solving these transcendental equations numerically gives the temperature T ann,lo of the instability of the low walls to range from 1.09 GeV to 1.77 GeV.

− 9 - 10 − 7
Hz range.Very recently, NANOGrav has published strong evidence for a stochastic gravitational wave background with frequency around 10 −7 Hz (1 yr −1 ) and integrated energy density of 9.3 × 10 −9 based on their 15 year data set[8].Their results are consistent with astrophysical expectations for a signal from a population of supermassive black hole binaries, although more exotic cosmological and astrophysical sources like domain wall collapse cannot be excluded.The NANOGrav results immediately rule out our DW collapse model with a higher parity breaking scale of 10 7 GeV.However, our DW collapse model with a parity breaking scale of 10 6 GeV remains entirely consistent with the NANOGrav results.Further data collection and improvements in pulsar timing array based detectors in the near future can thus serve as a strong experimental test of our model.
Our domain wall removal mechanism starts by showing that a Z 2 -domain formed at a high energy scale of discrete parity breaking decomposes further into many subdomains due to another Z 2 -symmetry breaking present in the Type II 2HDM model with spontaneous CP violation at a much lower scale.A QCD instanton vertex anomaly first identified by[6], on combining with the percolation properties of the distribution of subdomains within a domain statistical and the properties of random fluctuations, creates enough pressure difference to remove the both high and low types of walls almost simultaneously well before BBN.The probability of getting a favourable excess of low type subdomains near the surface of a high type domain due to random fluctations, which is key for the removal of the walls, is very close to 0.25.We have performed detailed calculations of our main idea for the PLRSM model with parity breaking scale ranging from 10 6 -10 7 GeV and friction power law exponent ranging from 0.9-0.99.The peak frequency of gravitational waves resulting from both types of wall collapse ranges from 10 −6 to 10 −7 Hz.This frequency band is sensitive to pulsar timing array based experiments such as SKA and NANOGrav.The recent NANOGrav results rule out our DW collapse model for higher values of parity breaking scale above 10 7 GeV.Our DW collapse model with parity breaking scales below 10 7 GeV remains consistent with the current NANOGrav results and has a good chance of being seriously tested in future pulsar timing based experiments.Our work gives further impetus for gravitational wave astronomy in the 10 −7 -10 −6 Hz band in the near future.

α
c of Chen et al. will be similarly constrained to lie below 10 −2 for reasonable values of the mass splitting between the two neutral heavy Higgs bosons of the Type II 2HDM due to the upper bound on electron EDM obtained by the ACME2 experiment [50].A negative result in the upcoming ACME3 experiment will further constrain α c to lie below 10 −4 .However, unlike the findings of Chen et al., our domain wall removal mechanism does not need any explicit CP violating term in the Type II 2HDM Higgs potential; instead it exploits of the QCD anomaly of Preskill et al.The main insight provided by our work is the increased importance and ubiquitousness of the 2HDM, as well as the demonstration that successful domain wall removal is possible with only spontaneous CP violation in the 2HDM.
[32] m d = 4.67 MeV, m s = 93.4MeV[32]are the masses of the up, down and strange quarks.The energy difference falls rapidly with temperature.