New bounds on macroscopic scalar-field topological defects
from non-transient signatures due to environmental dependence
and spatial variations of the fundamental constants

Topological defects are solitonic configurations of fields that acquire stability via a non-trivial field topology and may be produced during a cosmological phase transition associated with the spontaneous breaking of an underlying symmetry [2]. As the simplest example, consider a single real scalar field $\phi$ with the following $\phi^{4}$ potential:

where $\lambda$ is a dimensionless parameter. The potential in (1) is invariant under the discrete $Z_{2}$ symmetry operation $\phi\to-\phi$. The field $\phi$ may settle in either of the two energetically equivalent minima at $\phi=\pm\phi_{0}$, which are separated by a potential barrier of height $\lambda\phi_{0}^{4}/4$, see Fig. 1(a). In this case, the $Z_{2}$ symmetry is spontaneously broken, since the vacuum states associated with these minima do not exhibit the original symmetry present in the potential. If there exist two spatially separated regions of space with topologically distinct vacua, then a domain wall forms between the two vacua, with the following transverse “kink” profile [38]:

where the transverse size of the wall is set by:

The regions on either side of the wall are known as domains, by analogy with magnetic domains in ferromagnetic materials.

Domain-wall transverse profiles akin to (2) also arise for other potentials, such as the Sine-Gordon potential [3]. With more than one scalar field and more complicated potentials, more complex topological wall structures become possible [39]. Topological strings and monopoles require more complicated setups than topological walls. In particular, the minimal model admitting topological strings involves the Abelian U(1) gauge group with an associated complex scalar field $\phi$ and the following $\phi^{4}$-type potential [40, 41]:

while the minimal model admitting topological monopoles involves the non-Abelian SU(2) gauge group with an associated triplet of real scalar fields $\phi^{a}$, where the index $a=1,2,3$, and the following $\phi^{4}$-type potential [42, 43]:

In our present paper, we focus on the simplest model admitting topological defects, namely “infinite” domain walls that stretch across the horizon size and arise from the $\phi^{4}$ potential in (1). There is no loss of generality in focusing on this minimal model, which was also the only model of topological domain walls explicitly presented in the theory paper [6] (see the Supplementary Information therein¹¹1Let us remark that the single-scalar-field configuration described in the main text of [6], based on a prescription of a single real scalar field $\phi$, which takes a non-zero value inside the “defect” object and tends to zero outside of the object, does not describe a topological defect. Indeed, models of topological strings and monopoles require at least one complex scalar field or several real scalar fields, respectively, as well as associated gauge fields. In the case of wall-like objects, the prescription in [6] implies that the vacua on either side of the wall “defect” are topologically equivalent (i.e., the vacua are identical and are not separated by a potential barrier) and hence this type of wall “defect” would lack topological stability.), on which recent clock-based searches for transient signatures of passing domain walls were predicated. Indeed, the effects of back-action of ambient matter on the scalar field(s) comprising a topological soliton (including walls, strings and monopoles) which we study in our present paper are quite generic. Let us consider the following quadratic-in-$\phi$ interactions of the scalar field with the standard-model fields:

where the first term represents the interaction of the scalar field with the electromagnetic field tensor $F$, while the second term represents the interaction of the scalar field with the standard-model fermion fields $f$, with $m_{f}$ being the “standard” mass of the fermion and $\bar{f}=f^{\dagger}\gamma^{0}$ the Dirac adjoint. Note that the choice of signs in (6) is identical to those used in Ref. [6].

When the ambient-matter fields are associated with non-relativistic atoms, we can present Eq. (6) in the following convenient form:

where $\rho_{\gamma}=|\boldsymbol{E}|^{2}/2\approx-F_{\mu\nu}F^{\mu\nu}/4$ is the Coulomb energy density of a non-relativistic nucleus, and $\rho_{e}$ and $\rho_{N}$ are the electron and nucleon energy densities, respectively. The effective potential experienced by the scalar field in the presence of ambient matter, therefore, reads:

In a low-density environment when $2\rho_{X}/(\Lambda^{\prime}_{X})^{2}\ll\lambda\phi_{0}^{2}$, the effective potential in (9) has two minima at $\phi\approx\pm\phi_{0}$ as shown in Fig. 1(a), while in a high-density environment when $2\rho_{X}/(\Lambda^{\prime}_{X})^{2}\gg\lambda\phi_{0}^{2}$, the effective potential has only a single minimum at $\phi=0$ as shown in Fig. 1(b). The inhibition of spontaneous symmetry breaking in a high-density environment is a characteristic feature of the symmetron model, in which the effective potential takes the form (9) and which has been the subject of numerous earlier works [18, 19, 20, 22, 21, 28, 29, 30, 31, 32], but predominantly in a different context where cosmological domain walls are presumed absent.

When the dense region of ordinary matter has a finite size and is surrounded by a large region of low-density ambient matter, a high density in the dense region is insufficient by itself to ensure that the scalar field will attain its minimum near $\phi=0$ inside the dense region. The characteristic length scale, $d^{\prime}$, over which the scalar field appreciably changes inside the dense region of ordinary matter must also be small compared to the size of the dense region. In the case of a spherical dense region of radius $R$, we thus require $d^{\prime}\ll R$. We refer to this case as the “strong screening” regime, since the scalar field amplitude inside the dense region is suppressed compared to its value far away from the dense region. In the limit when $\rho_{X}/(\Lambda^{\prime}_{X})^{2}\gg\lambda\phi_{0}^{2}$, we have $d^{\prime}\ll d$, where:

Otherwise, the scalar field amplitude inside the dense region will only deviate slightly from its value far away from the dense region and the scalar field will not attain a minimum near $\phi=0$ inside the dense region. We refer to this case as the “weak screening” regime, since the scalar field amplitude is practically unaffected by the presence of the dense region. In both the strong screening and weak screening regimes, the scalar field profile is deformed in the vicinity of a dense body, such as Earth, due to the back-action of ambient matter on the scalar field.

To gain an understanding of the domain-wall parameter values for which the scalar field is strongly or weakly screened near the surface of and inside Earth, we recall that the main mass-energy contributions in an electrically neutral atom containing $A$ nucleons and $Z$ electrons are as follows:

The first two terms in (11) correspond to the nucleon and electron mass-energies, respectively. The third term corresponds to the energy associated with the electrostatic repulsion between protons in a spherical nucleus of uniform electric-charge density, with the coefficient $a_{C}\approx 0.7~{}\textrm{MeV}$ scaling proportionally to the electromagnetic fine-structure constant $\alpha$. The final two terms correspond to the electromagnetic energies of the proton and neutron, respectively, with the coefficients $a_{p}\approx+0.63~{}\textrm{MeV}$ and $a_{n}\approx-0.13~{}\textrm{MeV}$ derived from the application of the Cottingham formula [44] to electron-proton scattering [45]. Assuming that the elemental composition of Earth’s interior is a 1:1:1 ratio of ${}^{24}\textrm{Mg}^{16}\textrm{O}$, ${}^{28}\textrm{Si}^{16}\textrm{O}_{2}$ and ${}^{56}\textrm{Fe}$ by number, we find the fractional mass-energy contributions to Earth’s mass-energy content from the electromagnetic, electron-mass and nucleon-mass components as summarised in Table 1. Furthermore, noting that Earth’s radius is $R_{\oplus}\approx 6400~{}\textrm{km}$ and Earth’s average density is $\rho_{\oplus}\approx 5.5~{}\textrm{g/cm}^{3}$, we find the regions of domain-wall parameters for which the scalar field is strongly or weakly screened near the surface of and inside Earth, as shown in Fig. 2 by the light grey and white regions, respectively. For simplicity, in the present work we neglect Earth’s internal structure and density variations therein.

For sufficiently strong interactions of the form (6,7), the scalar field may even become strongly screened inside the main components of an experimental apparatus and thus evade detection altogether. The transition from the weak screening regime to the strong screening regime in this case depends on the details of the apparatus, including the sizes, materials and geometries of the apparatus components, as well as the details of the surrounding shielding and laboratory environment. For the experiments of interest in our present paper, we shall simply model the whole apparatus or satellite as a uniform sphere of radius $R\sim 30~{}\textrm{cm}$ and with a density comparable to Earth’s average density. In this case, the regions of domain-wall parameters for which the scalar field is strongly screened inside the apparatus or satellite are shown in Fig. 2 by the dark grey regions. In order to probe these dark grey regions, measurements should generally be performed using apparata with sufficiently small components and/or systems with sufficiently low densities.

The scalar field may also be strongly screened by Earth’s atmosphere, in which case the scalar field would evade detection by ground-based experiments on Earth. Treating Earth’s atmosphere as a 4:1 ratio of ${}^{14}\textrm{N}_{2}$ and ${}^{16}\textrm{O}_{2}$ by number, with a constant density of $\approx 10^{-3}~{}\textrm{g/cm}^{3}$ and extending out from Earth’s surface to an altitude of $\approx 10~{}\textrm{km}$, we find the regions of domain-wall parameters for which the scalar field is strongly screened by Earth’s atmosphere, as shown in Fig. 2 by the blue region. The density of Earth’s atmosphere decreases rapidly with increasing height. At an altitude of $\approx 10^{3}~{}\textrm{km}$, above which hydrogen and helium become the main components of Earth’s outer atmosphere, the particle density is $\sim 10^{4}~{}\textrm{cm}^{-3}$ and the transition to the strong screening regime by Earth’s outer atmosphere requires even stronger interactions than for the transition to the regime of strong screening of the scalar field inside a satellite. Additionally, for sufficiently strong interactions, spontaneous symmetry breaking may be inhibited altogether within the Solar System and Galaxy by the interplanetary and interstellar media, respectively, in which case $\phi=0$ in all local experiments. The transition to the interplanetary medium occurs at an altitude of $\sim 10^{4}~{}\textrm{km}$, where the particle density (which varies with time) is typically $\sim 10~{}\textrm{cm}^{-3}$. The transition from the interplanetary medium to the interstellar medium occurs at a distance of $\sim 100$ astronomical units from the Sun. Since the average energy density of the interstellar medium in our local galactic neighbourhood is comparable to the typical interplanetary medium energy density near Earth, for simplicity we neglect variations in the densities of the interplanetary and interstellar media within the Solar System and Galaxy (unless explicitly stated otherwise). Furthermore, since we will be restricting our attention to domain-wall transverse size parameters $d$ less than the size of our Galaxy ($\sim 30~{}\textrm{kpc}$ across), we shall also neglect the finite size of our Galaxy and the surrounding intergalactic medium (which has a much lower average energy density than those of the interplanetary and interstellar media). Assuming that the elemental composition of the interplanetary and interstellar media is $75\%$ ¹H and $25\%$ ⁴He by mass, we find the regions of domain-wall parameters for which spontaneous symmetry breaking is inhibited altogether by the interplanetary and interstellar media, as shown in Fig. 2 by the black region.

In addition to the back-action effects of ambient matter on the scalar field discussed above, the interactions in Eq. (6) also cause the scalar field to affect the properties of ordinary matter via apparent variations of the physical “constants”, including changes in the strengths of fundamental interactions and variations in particle masses. Comparing the terms in Eq. (6) with the relevant terms in the standard-model Lagrangian:

where $q_{f}$ is the electric charge carried by the fermion $f$, $J^{\mu}=\bar{f}\gamma^{\mu}f$ is the electromagnetic 4-current and $A^{\mu}$ is the electromagnetic 4-potential, we see that the interactions in (6) effectively alter the electromagnetic fine-structure constant $\alpha$ and fermion masses according to:

Since the scalar field profile is deformed in the vicinity of a dense body, such as Earth, due to the back-action of ordinary matter on the scalar field, and because more generally the scalar field amplitude depends on the local ambient matter density, the apparent values of the fundamental constants will therefore depend on the environment in which measurements are performed and there will also be spatial variations of the fundamental constants in the vicinity of dense bodies. The phenomenology of these environmental dependencies and spatial variations of the fundamental constants, caused by the reciprocal effects of ambient matter and the scalar field on one another, is the central focus of our present work. Although we focus on the interactions (6) in our present work, we note that similar types of reciprocal effects of ambient matter and the scalar field on one another also arise for other types of interactions (see Sec. IV).

A generic mechanism for the production of topological defects is the so-called Kibble mechanism [46]. The Kibble mechanism operates during the cosmological phase transition epoch during which the underlying symmetry becomes spontaneously broken and the scalar field develops a non-zero expectation value. The phase transition proceeds independently in causally disconnected regions of Hubble size $l_{H}\sim 1/H$, where $H$ is the Hubble parameter as a function of time ($H\sim 1/t$ in a matter- or radiation-dominated Universe). Inside any given Hubble region, the scalar field configuration is correlated up to some correlation length $l_{\textrm{cor}}$, which cannot exceed the Hubble size, $l_{\textrm{cor}}\lesssim l_{H}$. Within a particular correlation volume, the system settles in one and the same vacuum. The vacuum states in different correlation volumes are uncorrelated and so a topological defect may form at the boundary between adjacent correlation volumes. On average, $\mathcal{O}(1)$ topological defect is produced per correlation volume $\sim l_{\textrm{cor}}^{3}$, meaning that there should be at least $\mathcal{O}(1)$ topological defects per Hubble volume $\sim l_{H}^{3}$.

The interactions (6,7) provide a natural mechanism to cosmologically produce topological defects at rather late times via the Kibble mechanism, distinct from inflationary production scenarios (see, e.g., [47]). At early times when $2\rho_{X}/(\Lambda^{\prime}_{X})^{2}\gg\lambda\phi_{0}^{2}$, there is no spontaneous symmetry breaking and the scalar field remains at $\phi=0$. Then at the later critical time when $2\rho_{X}/(\Lambda^{\prime}_{X})^{2}=\lambda\phi_{0}^{2}$, the cosmological phase transition can commence and the scalar field can start to roll towards one of the two new minima of the effective potential (9). The energy density of either a non-relativistic or relativistic component decreases with time at least as rapidly as $\rho_{X}\propto t^{-3/2}$, regardless of whether the Universe is matter or radiation dominated, and so the transition from the regime $2\rho_{X}/(\Lambda^{\prime}_{X})^{2}\gg\lambda\phi_{0}^{2}$ to $2\rho_{X}/(\Lambda^{\prime}_{X})^{2}\ll\lambda\phi_{0}^{2}$ occurs within $\mathcal{O}(1)$ Hubble time $t_{H}\sim 1/H$. In other words, the phase transition occurs rather quickly. This phase transition is second order, since the scalar field smoothly develops a non-zero expectation value during the transition, rather than abruptly (first-order phase transition).

The evolution of a scalar field in an expanding Universe is governed by the following time-dependent differential equation:

where $\boldsymbol{\nabla}$ is the spatial gradient operator with respect to proper (non-expanding) spatial coordinates. In order for the scalar field to roll down from $\phi=0$ to one of the two new minima at $\phi=\pm\phi_{0}$ within $\mathcal{O}(1)$ Hubble time, the scalar-field amplitude must change by the amount $|\delta\phi|\sim|\dot{\phi}/H|\sim\phi_{0}$ in the same time interval. This “fast roll” regime requires $|H\dot{\phi}|\ll|\partial V_{\textrm{eff}}/\partial\phi|\sim|\boldsymbol{\nabla}^{2}\phi|$ during the formation of the topological defect. For the effective potential (9), this is equivalent to the requirement $d\ll l_{H}$, which corresponds to the case when the topological defects are well separated and evolve adiabatically. Otherwise, the scalar field remains stuck near $\phi=0$ due to “Hubble friction” and topological defects do not form. In other words, the maximum transverse size of a topological defect produced during a phase transition at time $t_{*}$ is $d_{\textrm{max}}\sim l_{H}(t_{*})$. In Fig. 2, we show the regions of domain-wall parameters in the case of interactions (6) for which domain walls are produced shortly after the BBN (redshift $z\sim 10^{8}$) or CMB ($z\approx 10^{3}$) epochs by the purple and brown lines, respectively. We focus on these two cosmologically interesting epochs, but note that domain walls may also be produced via the same mechanism at different times.

The subsequent evolution of the network of topological defects depends on the dynamics of the defect network. We focus on a network of infinite domain walls arising from the $\phi^{4}$ potential (1). In the limiting case that the initial network of domain walls undergoes little or no dynamical evolution after formation, the total proper area of the walls simply grows as $\propto a^{2}$ with the scale factor $a(t)$ and so the average energy density associated with the walls scales as $\rho_{\textrm{walls}}\propto a^{2}/a^{3}\propto a^{-1}$. Numerical simulations [3, 48, 49, 50], however, show no evidence of such “frustrated” domain-wall networks, but instead reveal that only $\mathcal{O}(1)$ domain walls survive per Hubble volume, with the average energy density associated with the walls scaling approximately as $\rho_{\textrm{walls}}\propto t^{-1}$ (which decreases faster with time than $\rho_{\textrm{walls}}\propto a^{-1}$ in a matter- or radiation-dominated Universe). Numerical simulations with more complicated potentials and more complicated types of hybrid wall networks [51, 52] also find no evidence of frustrated wall networks. The physical explanation for the apparent lack of frustrated domain-wall networks is that the collisions of domain walls lead to wall reconnection and annihilation, thereby “clearing” the volume around the few surviving domain walls and hence decreasing the energy stored in the wall network. We note that, in the presence of interactions (6,7), the average energy density of the wall network may decrease even faster than $\rho_{\textrm{walls}}\propto t^{-1}$, due to the collisions of walls with dense bodies of ordinary matter potentially leading to the loss of wall energy via “pinching off” of parts of the walls and their subsequent radiation (see Sec. III.1).

To determine the relationship between the average energy density associated with a network of domain walls and the wall parameters, we note that the energy density inside a single wall is given by $\rho_{\textrm{inside}}\sim\phi_{0}^{2}/d^{2}$. The energy associated with a network of $N_{\textrm{walls}}$ walls is hence given by $E_{\textrm{walls}}\sim\rho_{\textrm{walls}}l_{H}^{3}\sim N_{\textrm{walls}}\rho_{\textrm{inside}}dl_{H}^{2}$. We thus arrive at the following relation:

The existence of a present-day network of infinite domain walls would give rise to a quadrupolar temperature anisotropy in the CMB via the Sachs-Wolfe effect [53], with an amplitude of $(\delta T/T)_{\textrm{quad}}\sim\Omega_{\textrm{walls},0}=\rho_{\textrm{walls},0}/\rho_{c,0}$ [3], where $\Omega_{\textrm{walls},0}$ is the present-day mass-energy fraction of the Universe due to the wall network and $\rho_{c,0}\sim 10^{-29}~{}\textrm{g/cm}^{3}$ is the present-day critical density of the Universe. From the Planck 2018 temperature power spectrum [54], the size of an extra source of quadrupolar temperature anisotropy is bounded to be $(\delta T/T)_{\textrm{quad}}\lesssim 10^{-5}$, and so we have the following bound:

We stress that the bound (17) is based purely on the gravitational effects of domain walls of cosmological origin. Regarding the number of domain walls at the present epoch, the earliest numerical simulations in Ref. [3] indicated that $N_{\textrm{walls}}$ grows logarithmically in time according to the empirical relation $N_{\textrm{walls}}\approx 1+0.2\ln(t/d)$, which gives $N_{\textrm{walls},0}\sim\mathcal{O}(10)$ for domain walls with the macroscopic transverse size parameters of interest in our present work. The authors of [3] attribute this apparent logarithmic growth in time to the self-avoiding random-walk nature of the wall collisions and subsequent reconnections and annihilations, and suggest that the walls adopt a “stacked” formation over time to avoid collisions with one another. The more recent numerical simulations in Refs. [48, 49, 50], however, do not provide strong evidence for this logarithmic factor and indicate that only $\mathcal{O}(1)$ walls survive to the present epoch. We shall make the minimalistic assumption that $N_{\textrm{walls}}\sim\mathcal{O}(1)$ at all times, which leads to slightly stronger limits in Sec. II.3 than for $N_{\textrm{walls},0}\sim\mathcal{O}(10)$.

We begin by discussing non-transient signatures that are independent of the parameter $\phi_{0}$ appearing in (1,9). First of all, the interactions in (6) allow the emission of $\phi$ particles from hot environments, such as the interiors of stars. In particular, pairs of $\phi$ particles can be produced via the pair annihilation of photons $\gamma\gamma\to\phi\phi$, electron-positron annihilation $e^{-}e^{+}\to\phi\phi$, the Compton-type process $e\gamma\to e\phi\phi$, and the nucleon-bremsstrahlung-type process $NN\to NN\phi\phi$. Constraints on the emission of new particles from the core of supernova 1987A lead to the following bounds on the interactions (6) [19]:

Let us remark that such emission processes require the effective mass of the $\phi$ particles to be less than the core temperature of the supernova $T_{\textrm{core}}\sim 30~{}\textrm{MeV}$, otherwise the $\phi$ particles will not be emitted from the supernova due to the law of conservation of energy. In a very dense environment, $m_{\textrm{eff}}^{2}\approx 2\rho_{X}/(\Lambda^{\prime}_{X})^{2}$, which for a supernova core density of $\rho_{\textrm{core}}\sim 3\times 10^{14}~{}\textrm{g/cm}^{3}$ requires $\Lambda^{\prime}_{N}\gtrsim\textrm{GeV}$. Additionally, the interactions in (6) induce quantum forces mediated by the exchange of a pair of $\phi$ particles between two bodies. These quantum forces are constrained by short-range tests of gravity [55, 56], leading to the following bound on the nucleon interaction in (6) [19, 31]:

with weaker bounds on the electromagnetic and electron interactions.

We shall now demonstrate, however, that there are generally much stronger bounds on the interactions (6) for macroscopic domain walls, produced under the minimal cosmological assumptions outlined in Sec. II.2 above, stemming from different types of non-transient signatures that depend on the parameter $\phi_{0}$ and arise from an environmental dependence and spatial variations of the fundamental constants due to the reciprocal effects of ambient matter and the domain-wall scalar field on one another. The scalar field profile is deformed in the vicinity of a dense body due to the back-action of ambient matter on the scalar field via the interactions (6,7). The geometrical features of the scalar field deformation are determined by the geometry of the dense body. Around a spherical dense body, such as Earth, the scalar field deformation resembles a spherical “bubble-like” defect structure. We derive the scalar field profile around a homogeneous spherical body of radius $R$ in Appendix A. In the weak screening regime when $2\rho_{X}/(\Lambda^{\prime}_{X})^{2}\ll\lambda\phi_{0}^{2}$ or $d^{\prime}\gg R$, where $d^{\prime}$ is given by Eq. (10) in the limiting case when $\rho_{X}/(\Lambda^{\prime}_{X})^{2}\gg\lambda\phi_{0}^{2}$, the scalar field profile outside of the spherical body reads:

On the other hand, in the strong screening regime when $2\rho_{X}/(\Lambda^{\prime}_{X})^{2}\gg\lambda\phi_{0}^{2}$ and $d^{\prime}\ll R$, the scalar field profile outside of the spherical body reads as follows in the limiting cases when $d\ll R$ or $d\gg R$:

where $r_{*}-R\sim\textrm{min}(d,R)$.

The form of the scalar field profile for a spherical bubble-like defect structure surrounding a dense spherical body in the case when the scalar field is strongly screened near the surface of and inside the dense body is shown in Fig. 3(a) for $d\gg R$; for comparison, in Fig. 3(b), we show the form of the scalar field profile of a domain wall in vacuum, as described by Eq. (2) and arising in the same scalar-field model. We note that, unlike the domain walls in (2) that propagate freely in vacuum, the bubble-like defect structures in Eqs. (22) – (25) are permanently affixed to and follow the motion of the dense body. In the case when the scalar field is strongly screened near the surface of and inside the dense body, the scalar field bubble described by Eqs. (24,25) starts to form when the size of the dense proto-body exceeds the critical size $\sim d^{\prime}$ during the formation of the dense body. In arriving at Eqs. (22), (23), (24) and (25), we have assumed that the dense spherical body is located in a domain with vacuum value $+\phi_{0}$ [the overall signs of $\phi_{\textrm{out}}$ in Eqs. (22), (23), (24) and (25) are reversed if the dense body is located in a domain with vacuum value $-\phi_{0}$] and far away from a wall boundary joining two domains. Since we expect the present-day network of infinite domain walls to contain only $\mathcal{O}(1)$ walls (see Sec. II.2), the probability for a dense body such as Earth to be located near a wall boundary at any given time is extremely small. The deformed scalar field profiles around Earth in Eqs. (22), (23), (24) and (25) lead to a number of interesting non-transient signatures via the apparent variations of the fundamental constants in (13,14). We elucidate these signatures in subsections II.3.1, II.3.2 and II.3.3 below. Additionally, the gravitational effects of these bubble-like defect structures surrounding Earth and the Sun lead to another set of bounds, as we discuss in subsection II.3.4 further below. We summarise our results in Fig. 4, in which we assume the relation (16) with $N_{\textrm{walls}}\sim\mathcal{O}(1)$ and the maximum allowable present-day average energy density $\rho_{\textrm{walls},0}\sim 10^{-5}\rho_{c,0}\sim 10^{-34}~{}\textrm{g/cm}^{3}$, see Eq. (17). Unless explicitly stated otherwise, we assume that all the relevant ground-based laboratory apparata are located within $\sim 1~{}\textrm{m}$ of Earth’s surface.

Refs. [12, 13, 14, 15] performed clock- and cavity-based searches for transient variations of the fundamental constants of the form (13,14) induced by the passage of macroscopic domain walls through a network of detectors. These bounds are shown in Fig. 5 as coloured lines (purple = ground-based laboratory optical clocks, brown = space-based microwave clocks within the GPS network). In these searches, it was assumed that the sought domain-wall-induced transient signals are well separated in time, which necessitates that the signal time $\Delta t\sim d/v_{\textrm{TD}}$ is sufficiently short, $\Delta t\ll\mathcal{T}$. The largest average time between encounters of a wall with Earth that can be probed in a statistically meaningful way with such types of searches is limited by the duration of the experiment $T_{\textrm{exp}}$, $\mathcal{T}\lesssim T_{\textrm{exp}}$.

Comparing with Fig. 2, we see that the purple and brown curves in Fig. 5 all lie in regions of domain-wall parameter spaces where the scalar field is strongly screened near the surface of and inside Earth, and in some cases also in regions of parameter spaces where the scalar field is strongly screened inside the apparatus itself (see Fig. 6 for an overlay in the case of the electromagnetic coupling for $\mathcal{T}=1~{}\textrm{day}$). Thus the unwitting assumption made in the earlier searches [12, 13, 14, 15] that domain walls pass in an unperturbed manner through the strongly repulsive potential generated by Earth (and in some cases the apparatus itself) appears dubious. More specifically, there are at least two possible outcomes for a scalar-field domain wall incident on a strongly repulsive potential generated by Earth: (i) part of the domain wall may tunnel through Earth with a diminished amplitude and/or (ii) part of the domain wall may envelop Earth and “pinch off”, forming a bubble-like defect structure around Earth in the process — this defect structure, which is distinct from the bubble-like defect structures discussed in Sec. II.3, may be either stable (if there is insufficient kinetic energy to overcome the potential barrier inside Earth) or metastable (in which case the bubble might undergo a number of oscillations before collapsing and radiating away its energy in the form of $\phi$ particles). The outcome may even depend on the properties of the incident domain wall, such as its initial speed and transverse size, the interaction strength, as well as the size and density of the target body.

The investigation of this problem, as well as the analogous problem concerning the strongly antiscreened regime for the same types of quadratic-in-$\phi$ interactions in (6) but with the signs reversed, warrants numerical simulations, which go beyond the scope of the present work. These calculations are not only of interest for numerous terrestrial experiments [6, 8, 9, 10, 12, 13, 14, 15], but also for measurements involving networks of pulsars [11]. (Note, however, that laser-interferometry-based searches for transient signatures of passing domain walls using existing gravitational-wave detectors such as LIGO and GEO600 [10] are already expected to be able to probe regions of parameter space that lie deep within the weak screening regime where back-action effects are negligible; see, e.g., the dashed red line in Fig. 6.) Additionally, reanalysis of the data in Refs. [12, 13, 14, 15] is warranted for the strongly screened and strongly antiscreened regimes. The interaction of scalar-field topological defects with strongly repulsive or strongly attractive potentials generated by dense bodies may also affect the evolution and propagation of topological defects at very late cosmological times. For example, if parts of domain walls are “pinched off” by dense bodies and subsequently radiate away, then the energy density associated with the domain-wall network would decrease in time at a faster rate. This possibility also warrants further investigation. Additionally, we note that in the case of sufficiently strong interactions, when spontaneous symmetry breaking is inhibited altogether within the Solar System and Galaxy by the interplanetary and interstellar media, there is no longer topological stability to support the free propagation of a domain wall through the Solar System or Galaxy.

Finally, let us briefly remark that if the velocities of domain walls are taken to be semi-relativistic, in accordance with the results of numerical simulations in Refs. [3, 49, 82], instead of the assumption $v_{\textrm{TD}}\sim 300~{}\textrm{km/s}$ made in Refs. [12, 13, 14, 15], then bounds from experiments searching for transient signatures of passing domain walls would be shifted towards larger transverse wall size parameters, since the characteristic wall passage time is $\Delta t\sim d/v_{\textrm{TD}}$. On the other hand, bounds from experiments searching for non-transient signatures of domain walls would not be shifted in such a manner. The sensitivity of experiments searching for either transient or non-transient signatures of domain walls that depend on $\phi_{0}$ would be increased in the case of a larger value of $v_{\textrm{TD}}$, in accordance with Eq. (42). Searches for transient signatures would also benefit from being shifted up towards larger values of $d$ due to the associated increase in $\phi_{0}$ in this case, see Eq. (42).

We now derive bounds on non-transient variations of the fundamental constants, which arise in the same scalar-field domain-wall model, via the torsion-pendulum-based and clock-based approaches discussed in Secs. II.3.1 and II.3.2, respectively. Like the transient searches described in Sec. III.1 above, we assume that the domain walls are well separated, $\Delta t\ll\mathcal{T}$, and that $\Delta t\ll T_{\textrm{exp}}$. In this case, the effects of the domain-wall scalar field are quasi-non-transient on the experimental timescale, since terrestrial detectors are located very far from a domain wall for the vast majority of the time. Importantly, the sign of $\phi^{2}$ remains unchanged on either side of a domain wall, meaning that the sign of these quasi-non-transient effects remains unchanged on either side of a wall, and so there is no cancellation between contributions from different domains. Therefore, if we neglect the brief transients associated with the passage of domain walls through Earth (which is justified when $\Delta t\ll\mathcal{T}$ and $\Delta t\ll T_{\textrm{exp}}$), then it is possible to utilise our findings from Sec. II.3 with minimal modification. In contrast to transient search methods, searches for quasi-non-transient signatures also retain their sensitivity in the regime $\mathcal{T}\gg T_{\textrm{exp}}$, when the average time between encounters of a wall with Earth greatly exceeds the duration of the experiment, since it is not necessary to be near a passing domain wall to experience non-transient signatures.

Using Eqs. (13), (14), (22), (23), (24), (25), (26), (28), (30) and (42), as well as the fractional mass-energy differences of the electromagnetic, electron-mass and nucleon-mass contributions to the relevant test-mass pairs in Table 1, we derive bounds on the quadratic interactions of the domain-wall-model scalar field $\phi$ with the electromagnetic field, electron and nucleons in Eq. (6). We present our bounds as coloured regions in Fig. 5 [red = Tokyo Skytree experiment (clocks), yellow = Galileo satellites and Gravity Probe A mission (clocks), blue = E$\ddot{\textrm{o}}$t-Wash experiments (torsion pendula), green = MICROSCOPE mission (torsion pendulum)]. We see that our newly derived bounds via effects of non-transient variations of the fundamental constants are significantly more stringent than the previously reported bounds from different types of non-transient signatures and transient signatures under the same set of assumptions. In our analysis, we have restricted ourselves to the values of $d$ satisfying the condition $\Delta t\ll T_{\textrm{exp}}$, for which the signal is quasi-non-transient on the experimental timescale. In the case of the currently available Tokyo Skytree and MICROSCOPE mission datasets, for which $T_{\textrm{exp}}\approx 1~{}\textrm{week}$, it is possible to extend the respective analyses to the higher values of $d$ in Fig. 5(d-f) by additionally considering the transient effects associated with the passage of domain walls through Earth. We do not present bounds of the type discussed in Sec. II.3.3, since in the absence of a concrete mechanism leading to the enhancement of the domain-wall energy density inside galaxies by orders of magnitude compared to an average cosmological value, further assumptions about the domain-wall energy density in low-density astrophysical environments are required.