Towards Extra-Terrestrial Particle Detectors:
Superradiant Higgs Fields on Black Hole Spacetimes
by
Luke Edward Sellers
An honors thesis presented for the partial fullfillment of the degree of Bachelor of Science
Department of Physics University of California, Santa Barbara Santa Barbara, CA June 2021 This honors thesis is hereby approved by,
Professor Sathya Professor Douglas Eardley Guruswamy
Signature Signature
Date Date
1 Acknowledgements
My education was starkly interrupted due to medical complications, which kept me out of school for an entire year. During this time, I was unable to use my hands and arms properly, so my submission of this thesis marks a full recovery that seemed doubtful for quite some time. I would therefore like to take this opportunity to, in no particular order, thank those who made it possible for me to return to physics under these exceptional circumstances. First, I would like to thank my academic advisor and long-time professor, Sathya Guruswamy. Her instruction was deeply influential on the way that I think about physics, and her guidance and unwavering support throughout my return to school was indispensable. Second, I would like to thank my first research advisor, Professor Carlos Garcia- Cervera. Our work together saw me rapidly progress in the field of computational physics, the tools of which were pivotal for the completion of this study. Third, I would like to thank Noah Lebovic and Toolchest for collaborating with me and brainstorming ways that parallel cloud computing can increase the efficiency of computational physics. The projects that we worked on allowed me to collect numerical results at a rate that would otherwise not be possible on my local machine, thereby vastly increasing the efficiency of the numerical aspects of this study. Fourth, I would like to thank my research and thesis advisor, Professor Douglas Eardley. His knowledge and guidance were instrumental both for zeroing in on a thesis project and for accumulating the knowledge required for completing this study. I very much enjoyed our weekly meetings, and I appreciate all of the time
2 that he dedicated to this project. Fifth, I would like to thank the Shirley Ryan Ability Lab, where I completed a month long pain management program designed to get me back to school. The knowledge that I acquired during their tutelage has been indispensable in returning to my normal activities. Finally, I would like to thank Sara Sterphone for her continued support while I was away from school, and furthermore for assisting me with my return. Sara always offered a warm presence that made me feel like a priority even while I was away. When I returned to UCSB, Sara continued to check in on me to make sure that I was handling my symptoms. She was a wonderful person and she will be sorely missed.
3 Abstract
In this report, we complete a numerical and analytical study of the evolution of the Abelian Higgs model on black hole spacetimes with an emphasis on conditions that would lead the onset of a superradiant instability. We study the Higgs field in particular to assess the plausibility of probing electro-weak physics that has thus far eluded experimental detection at the LHC. Our study involves the full non-linear evolution of a classical Higgs field for two different systems. The first is the Klein-Gordon equation in Minkowski spacetime with a tunable damping/anti-damping term to emulate a superradiant source. We analyze this system in order to assess whether or not superradiance persists in a system experiencing exponential growth when an inertial, nonlinear potential is present. We then move on to the evolution of the Higgs field on a Kerr background in Boyer-Lindquist and φ˜ coordinates. Our study of this system informs us whether the nonlinearity of the Higgs potential will eclipse the macroscopic process through which the linear field would otherwise acquire an exponential growth rate. We fur- thermore comment on a plausible classical symmetry-breaking mechanism in which fields with a zero vacuum expectation value acquire a radially-dependent expectation value within the ergoregion of the Kerr spacetime. Finally, we apply the results of our classical evolutions to the quantum me- chanical picture via the classical field approximation and draw conclusions on the ensuing particle physics of the Higgs-Kerr system. We then compare these results to the collisions at the LHC and ultimately conclude whether these Higgs-Kerr systems provide a viable alternative to the LHC for encountering electro-weak physics.
4 Contents
1 Motivation: The State of Physics at the LHC 8 1.1 Introduction ...... 8 1.2 The Abelian Higgs Mechanism ...... 8 1.3 The Hierarchy Problem ...... 9 1.3.1 Basic Overview ...... 9 1.3.2 Solutions and Detection Status ...... 11 1.4 Contents: Looking Elsewhere ...... 12
2 Superradiance 14 2.1 Introduction ...... 14 2.2 Superradiance and Dissipation ...... 14 2.2.1 Rotational Superradiance ...... 15 2.3 Black Hole Superradiance ...... 16 2.3.1 Kerr Metric Review ...... 17 2.3.2 Penrose Process ...... 18 2.3.3 Kerr Superradiant Condition ...... 19 2.4 Pertinent Literature Results ...... 20 2.4.1 Black Hole Perturbation Theory ...... 20 2.4.2 Black Hole Instabilities ...... 21
3 Higgs Evolution: Superradiant Model in Minkowski Space 26 3.1 Introduction ...... 26 3.2 Equations of Motion ...... 27
5 3.3 Numerical Methods ...... 28 3.3.1 Finite Differences ...... 29 3.3.2 Code Tests ...... 31 3.4 Examining Superradiance ...... 35 3.4.1 Checking Γ(bt) ...... 35 3.4.2 Testing the Higgs ...... 37 3.5 Conclusion ...... 38
4 Higgs Evolution: Kerr Spacetime 40 4.1 Introduction ...... 40 4.2 Boyer-Lindquist Evolution ...... 42 4.2.1 Equations of Motion ...... 43 4.2.2 Boundary Conditions ...... 46 4.2.3 Numerical Methods ...... 50 4.2.4 Energy Conservation ...... 52 4.2.5 Spontaneous Symmetry Breaking Mechanism ...... 56 4.2.6 Growth in the Massive Linear Regime ...... 59 4.2.7 Higgs Evolution ...... 60 4.3 φ˜ Coordinates ...... 61 4.3.1 Comments on (2 + 1)D Evolutions ...... 62 4.3.2 Evolution in (3 + 1)D ...... 63 4.4 Conclusions ...... 64
5 Discussion: Observational Signatures 65 5.1 Introduction ...... 65 5.1.1 Assessing Black Hole Candidates ...... 65 5.2 Particle Physics Signatures ...... 68 5.2.1 Classical Field Approximation ...... 69 5.2.2 Higgsenova ...... 70 5.3 Conclusion: Comparison with the LHC ...... 70
6 6 Conclusion 72
7 Chapter 1
Motivation: The State of Physics at the LHC
1.1 Introduction
In 2012, the detection of a ∼ 125 GeV Higgs candidate at the LHC provided ex- perimental verification for the final piece of the Standard Model (SM) of particle physics [1]. The Higgs boson is of particular interest due to its role as the catalyst for Spontaneous Symmetry Breaking (SSB). In the context of particle physics, SSB is the mechanism through which particles acquire a mass. This is the reason, for example, why the W ± and Z bosons are observed to have a non-zero mass, but the photon, which does not interact with the Higgs, does not. This mass asymmetry would seem arbitrary for a fundamental theory of physics. However, at energy scales above the SSB threshold (∼ 1 TeV), the weak-nuclear bosons and the photon are each massless, thereby assuming a more unified orientation in the theory.
1.2 The Abelian Higgs Mechanism
Though not the full non-abelian Glashow-Weinberg-Salam theory of weak interac- tions (for details, see section 20.2 of [2]), we can see how a spin-1 gauge boson acquires a mass through the Abelian Higgs Model (AHM). The Lagrangian for the
8 AHM is [2]
1 L = − F F µν + D φDµφ∗ − V (φ) (1.1) 4 µν µ where Dµ = ∂µ + ieAµ is the usual covariant derivative for U(1) gauge invariance, and λ V (φ) = |φ|4 + µ2|φ|2 (1.2) 2
2 p 2 2 has the minimum V = 0 at φ = 0 if µ ≥ 0 or |φ| = µ /λ = φ0 if µ < 0. For
µ the former case, there is no AµA term, so Aµ is a massless particle in the theory. For the latter case, φ is said to take on a Vacuum Expectation Value (VEV). If we expand the degrees of freedom of φ ∈ C about this VEV like
1 φ → φ0 + √ (φ1 + iφ2) (1.3) 2 then the covariant derivative term takes the form
1 1 √ D φDµφ∗ → ∂ φ ∂µφ∗ + ∂ φ ∂µφ∗ + 2eφ A ∂µφ + e2φ2A Aµ + ... (1.4) µ 2 µ 1 1 2 µ 2 2 0 µ 2 0 µ
1 µ 2 2 and we see that Aµ now has a mass term 2 mAA Aµ with mA = 2e φ0. Since Fµν
= ∂µAν − ∂νAµ only depends on derivatives of Aµ and V (φ) depends only on φ, this new mass term will not be canceled by other contributions. We have therefore recovered a massive Aµ particle by breaking the symmetry of V (φ). For the sake of simplicity, we will limit our considerations of the Higgs field to the AHM for the entirety of this thesis.
1.3 The Hierarchy Problem
1.3.1 Basic Overview
Although the Higgs mechanism provides a more unified way of viewing the members of the SM, it comes at the cost of introducing what is known as the Hierarchy
9 Problem. The Hierarchy problem is related to the way that masses are treated in a quantum field theory. In most theories of physics, there is no distinction between the mass pa- rameters that appear in the equations of motion and the observable mass. However, in quantum field theory, the mass parameter that appears in the Lagrangian
1 L = m φ2 (1.5) m 2 0 is known as the ’bare mass’ and need not represent the physical mass m of the φ particle. The reason for this is that quantum field theories are famously plagued by divergent contributions to observable quantities. One such example is the one-loop contribution to φ4 theory [3]:
Z ∞ d4p Σ1 ∼ 2 2 (1.6) 0 p − m + i
R ∞ 2 ∞ 4 3 which diverges like 0 p · dp = p |0 since d p → p dΩ3 in spherical coordinates. A common way to address this divergence is to suppose that the theory is only valid up to a particular energy scale Λ and apply an ultra-violet cutoff to the integral R ∞ R Λ bound → . The choice of Λ clearly affects the value of Σ1, and therefore the contribution to the desired amplitude. In order for observables to be invariant of the cutoff choice, the bare mass receives an adjustment. This process in which coupling parameters of the Lagrangian are adjusted to account for the choice of energy scale is known as Renormalization. In the case of the Higgs boson, the most significant one-loop corrections to the mass are [4] 1 3 9 m2 = m2 + g2 + g2 + 3λ2 − 12λ2 Λ2 (1.7) h 0 16π2 4 1 4 2 h t where g1, g2, λh, and λt are the hypercharge, SU(2), Higgs quartic, and top Yukawa couplings respectively. We can see that the corrections scale quadratically with the
10 energy cutoff Λ. In contrast, the corrections for fermions scale logarithmically with Λ. This discrepancy in scaling behavior causes the Higgs corrections to grow much
19 more rapidly than fermion corrections as Λ → Mpl = 10 GeV. Furthermore, as
Λ approaches Mpl, the Higgs corrections become comparable to the observed mass itself, which requires careful adjustments to the bare mass for large Λ. This process is known as ’fine-tuning’, and it violates what we would expect from perturbation theory, namely that corrections should be sufficiently smaller than the quantity that they are correcting. This discrepancy makes the Higgs and other scalar particles unnatural components of a quantum field theory.
1.3.2 Solutions and Detection Status
Supersymmetry (SUSY)
The Hierarchy problem has motivated a bulk of theoretical work over the past few decades. In particular, it has prompted physicists to postulate several high energy theories for which the SM is an effective, low-energy theory in order to quell the quadratic divergence. One of the most popular of these theories is Supersymmetry (SUSY). The idea behind SUSY is that loop contributions from bosons and fermions can have different signs due to their respective commutation and anti-commutation relations. Therefore, if each boson and fermion had a respective fermion and boson superpartner, then the quadratic Higgs corrections can be canceled. The detection of these superpartners has been one of the focal points for particle physicists at the LHC since its inception. Among the most promising candidates were the gluino and the top squark [5]. Unfortunately, neither of these, nor any superpartner, have been observed at the LHC or elsewhere; with lower mass bounds established at 2.03 and 1.55 TeV for the gluino and top squark by The ATLAS Collaboration [6]. Although superpartners have avoided detection within the energy threshold at the LHC, there still remains evidence in favor of supersymmetry. This includes the fact that the Higgs mass ∼ 125 GeV falls within the range of the Minimal
11 Supersymmetric Standard Model (MSSM), and also that the SM gauge couplings
16 meet at a point under SUSY renormalization MGUT ∼ 2×10 GeV [5, 7]. Therefore, the continued search for SUSY at colliders and elsewhere is well motivated.
Extra-Dimensions
Another way to quell the quadratic divergence as Λ → Mpl would be to reduce the value of Mpl. This is the idea behind extra-dimensional (ED) theories, namely
19 that the calculated value of Mpl ∼ 10 GeV is a (3 + 1)D effective value for the true (d + 1)D value that is on the order of the electro-weak scale ∼ 1TeV. In these theories, gravity propagates in more than 3 spatial dimensions, which limits its flux contribution to our effective (3 + 1)D universe, resulting in its perceived weakness.
d Thus, the ED planck mass Mpl is much smaller than Mpl, provided that the extra- dimensions are relatively small compared to typical length scales. (for details, see
M M d [8]). This reduces the bound on loop corrections R pl → R pl and softens the quadratic divergence. The status for detection of ED theories is similar to that of SUSY in that no significant discoveries have been made with the exception of lower mass bounds established for Md, the energy-scale at which the extra-dimensions would be acces- sible. In particular, experiments at the LHC have place bounds Md > 4.8 − 7.7 TeV [9], and cosmological considerations have established similar constraints. It seems that in the case of both SUSY and ED theories, larger energy scales are required for observational progress.
1.4 Contents: Looking Elsewhere
The LHC has provided a uniquely high experimental energy-scale of up to 13.6 TeV as of Run II for observing high energy physics. It has therefore deservedly served as the first line of attack in the search for physics beyond the standard model. How- ever, with the state of experimental verification of these theories well-documented, and the monetary and infrastructural demands required to raise the energy scale,
12 it seems prudent to explore alternative mechanisms for observing these phenomena. In this thesis, we will explore the possibility of electro-weak scale physics in astro- physical and/or primordial scenarios by studying the Abelian Higgs Model coupled to superradiant black bole spacetimes. In Chapter 2, we will introduce the concept of superradiance and how it applies to black holes, and also detail the literature results that have guided our study of the Abelian Higgs field. In Chapter 3, we will present a Minkowski space superradiant model that we studied in order to asses whether superradiance persists in the presence of an inertial, nonlinear potential. In Chapter 4, we move on to the evolution of the Higgs field on a Kerr background in Boyer-Lindquist and φ˜ coordinates. Our study of this system informs us whether the nonlinearity of the Higgs potential will eclipse the macroscopic process through which the linear field would otherwise acquire an exponential growth rate. We fur- thermore comment on a plausible classical symmetry-breaking mechanism in which fields with a zero vacuum expectation value acquire a radially-dependent expecta- tion value within the ergoregion of the Kerr spacetime. In Chapter 5, we examine what particle processes might be afforded by this superradiant effect and ultimately conclude whether these systems might one day serve as a viable alternative to the LHC for observing electro-weak physics. In Chapter 6, we summarize our findings and discuss ways upon which our study can be elaborated and improved.
13 Chapter 2
Superradiance
2.1 Introduction
As we mentioned in the previous chapter, one way to make progress for the observa- tion of beyond standard model (BYSM) physics is to consider systems with energy scales > 13.6 TeV. One of the most abundant sources of energy in the universe are stars and/or black holes, with astronomical observations documenting supermas-
9 9 sive black holes up to ∼ 10 M , which comes out to 5 × 10 TeV. Therefore, a mechanism to extract and consolidate this energy may provide an avenue towards BYSM physics. One such mechanism that shows promise is Superradiance, which has already inspired studies for the observation of axion dark matter [10, 11] and violations of the Weak Cosmic Censorship Conjecture [12].
2.2 Superradiance and Dissipation
Superradiance is the process through which the energy emitted from a dissipative system is amplified. This occurs when the dissipative object travels faster than the energy that it radiates. For example, the superradiance of sound waves, or a sonic- boom, occurs when an object travels faster than the speed of sound cs. The object emits sound waves as it travels, but since the object travels at speed v > cs, the successively emitted sound waves at the front end collide with each other, causing
14 them to amplify in the case of constructive interference. In addition to sound waves, superradiance is possible for all types of radiation, including electromagnetic waves. This may be surprising to learn, since a radiating object cannot travel faster than light. One way to circumvent this fact is by consid- ering systems with a dielectric, wherein the speed of light is modified c → c/n with n > 1. This can be seen in the case of Cherenkov Radiation, where the radiation emitted from underwater nuclear reactors is blue-shifted, giving the reactors their characteristic blue glow [13].
2.2.1 Rotational Superradiance
The superradiance of light is furthermore possible in the absence of a dielectric provided that the dissipative object is rotating. This is well illustrated by considering cases with cylindrical symmetry worked out by Zel’dovich [14]. Superradiance in this case follows from the fact that orbitals with azimuthal number m rotate with angular velocity ω/m. This can be seen for example if ψ ∼ eimφ with φ = ωt, then ψ has angular period 2π/m. Therefore, if a dissipative cylinder has angular velocity
Ωcyl > ω/m, or the more common expression,
ω < mΩcyl (2.1) then the system is superradiant. Eq (2.1) is know as the superradiant condition for the system. We can see that the condition (2.1) is universal for rotating systems with axial symmetry from Bekenstein’s thermodynamic reasoning [15], for which he showed that the 2nd Law of Thermodynamics requires that
(ω − mΩ)am > 0 (2.2)
where am(ω) is the system’s absorption ratio. We then see that w < mΩ → am < 0, so the system expels energy.
15 Rotational superradiance can furthermore be interpreted as an anti-friction effect by considering the wave equation with a friction term [13]
∂Φ Φ + α = 0 (2.3) ∂t for which Φ behaves like Φ ∼ e−i(ωt−mφ) for a non-trivial dispersion relation. Then shifting to coordinates φ → φ0 − Ωt, ω shifts like ω0 = ω − mΩ, and the friction coefficient αω0 = α(ω−mΩ) shifts to an anti-friction term in the superradiant regime ω < mΩ, causing Φ to grow. Thus, rotational superradiance can be interpreted as an anti-friction effect. This model will serve as the basis for our superradiant Higgs model in Minkowski space that we will study in Chapter 3. In addition to the amplification of a system’s own dissipated energy, superradi- ant systems provide the opportunity for superradiant scattering, or scattering with reflection coefficient R > 1. This opportunity results from the fact that the system does not discriminate whether the radiation is from its own emission or from an ex- ternal source, so superradience occurs in both cases, provided that the superradiant condition is met.
2.3 Black Hole Superradiance
Shifting our focus towards black holes, in 1971 Roger Penrose demonstrated that the energy from rotating (Kerr) black holes can be extracted by a particle collision process known as the penrose process [16]. In this section, we will show that this process is consistent with our working description of superradiance. We will then generalize the result to fields and recover the analog to (2.1) by invoking the ther- modynamics of black holes. But first, we will need to introduce the machinery of the Kerr metric.
16 2.3.1 Kerr Metric Review
In 1963, Kerr showed that there exists a stationary, axis-symmetric solution to Einstein’s equations for a rotating black hole, now known as the Kerr metric, which in Boyer-Lindquist coordinates has the form
2Mr 2Mar sin2 θ ds2 = − 1 − dt2 − (dtdφ + dφdt) ρ2 ρ2 (2.4) ρ2 sin2 θ + dr2 + ρ2dθ2 + r2 + a22 − a2∆ sin2 θ dφ2 ∆ ρ2 where we are following the notation from Carroll [17] with G and c set to 1. Then a = J/M is the fraction of rotational energy with respect to the total energy of the black hole, and
∆ = r2 − 2Mr + a2 (2.5)
ρ2 = r2 + a2 cos2 θ (2.6)
We know that we have event horizons for grr = 0, since this indicates that hyper- surfaces of constant r are null (see Carroll p. 241 for details). In the case of the √ 2 2 Kerr metric, this occurs when ∆ = 0, or at r± = M ± M − a .
Since each metric element gµν is independent of t and φ, we also know that there are two conserved quantities in the t and φ direction for test particles:
2Mr dt 2Mar dφ E = −K pµ = 1 − + sin2 θ (2.7) µ ρ2 dτ ρ2 dτ
2Mar dt (r2 + a2)2 − a2 sin2 θ dφ L = R pµ = − sin2 θ + sin2 θ (2.8) µ ρ2 dτ ρ2 dτ
µ µ where K = ∂t and R = ∂φ are the killing vectors in the t and φ direction and E and L are the energy and angular momentum per unit mass of the test particle.
17 2.3.2 Penrose Process
The key to the Penrose process is then that the killing vector Kµ becomes spacelike for ρ2 < 2Mr in a domain called the ergoregion, which begins at
2 2 2 1/2 rerg = M + (M − a cos θ) (2.9)
In this region, stationary observers do not follow timelike worldlines and are therefore
µ unphysical, forcing them to move. Kµ spacelike gives KµK > 0, then since the four-
µ µ momentum of a massive particle is timelike, pµp < 0, we can have E = −Kµp < 0 [17]. The Penrose process then involves a 4-momentum conserving process in which an inbound object with E(0) > 0 decays into two particles within the ergoregion with E(1) < 0 and E(2) > 0, where particle (1) falls into the black holes and particle (2) escapes to infinity. By 4-momentum conservation, E(1) +E(2) = E(0), so E(2) > E(0), and the system has gained energy by interacting with the black hole. Carroll [17] shows that the condition that particle (2) retains a physical (timelike) trajectory requires that E(2) L(2) < (2.10) ΩH
In the limiting case where E(2) → 0, we see that the superradiant condition becomes
L2 < 0, or from (2.7):
2Mar dt m(r2 + a2)2 − a2 sin2 θ dφ − sin2 θ + sin2 θ < 0 (2.11) ρ2 dτ ρ2 dτ
Then ignoring the poles where sin θ = 0, we have ρ2, sin2 θ 6= 0, so canceling these factors and multiplying by dt/dτ gives,
dφ −2Mar + (r2 + a2)2 − ∆a2 sin2 < 0 (2.12) dt
Then examining this equation at the event horizon r+, we have ∆ = 0, which further
18 2 2 implies that r+ + a = 2Mr+, so we have
dφ −2Mar + (2Mr )2 < 0 (2.13) + + dt or dφ a < = ΩH (2.14) dt 2Mr+ where ΩH is the angular velocity of the black holes event horizon r+, which can be seen by taking −gtφ/gφφ at r+. We therefore recover our notion that superradiance occurs when the radiating object moves faster than the emitted or scattered radiation that it interacts with.
2.3.3 Kerr Superradiant Condition
We have shown that the Penrose process is consistent with our notion of superra- dience for test particles, but what would be more useful for the our purposes is to verify this for fields. Luckily, this result has been worked out by Bekenstein [18] and elaborated upon in [13]. To recover the analog to (2.1) for a Kerr black hole, we first introduce the thermodynamic relation for such a black hole,
κ δM = δA + Ω δJ (2.15) 8π H H where κ is the surface gravity of the black hole. It is then shown in Appendix C of [13] that the ratio of angular momentum flux to energy flux of a wave with time and azimuthal numbers ω and m is δJ/δM = m/ω for similar reasons as (2.1). Then substituting this result into (2.13), we get
ωκ δA δM = H (2.16) 8π ω − mΩH
The second law of thermodynamics for black holes requires that, in analogy to en- tropy, the surface area of the event horizon is non-decreasing, or δAH ≥ 0. Therefore,
19 to extract energy from the black hole δM < 0, we require
ω < mΩH (2.17) in analogy to (2.1). This is the superradiant condition for Kerr black holes, and this will be integral to our analysis of superradiant Higgs fields. We could have similarly arrived at this result by examining the curved-space wave equation at the event horizon and imposing physical boundary condtions, and this is in fact the approach that we will take in Chapter 4.
2.4 Pertinent Literature Results
2.4.1 Black Hole Perturbation Theory
The perturbations of black hole systems is a well-documented field of study extend- ing from the 1960s until today. The field began as an attempt to verify the stability of black hole solutions to Einstein’s equations, thereby assessing their applicability to astrophysical systems spanning large time scales. The first contribution by Vishvesh- wara asserted the stability for the Schwarzschild metric [19], specifically that with the addition of an external field, the metric-field constituent system evolves into an- other Schwarzschild metric, although with perhaps a different mass. Therefore, the Schwarzschild metric is a reasonable model to be applied in astrophysical scenarios, despite the inevitable bombardment from radiation, matter, etc. Press and Teukolsky generalized the Schwarzschild stability result to Kerr black holes for the case of scalar perturbations for the most likely candidates, namely the first few angular modes up to ` = 4 [20], which was later extended by Detweiler and Ipser [21]. The reader may be bothered by these results, since we showed quite generally in subsection 2.3.3 that Kerr black holes suffer from superradiant instabil- ities satisfying (2.17). This is still of course true, modes satisfying (2.17) rebound with reflection coefficient R > 1. As it turns out, the stability of the Kerr metric
20 to massless perturbations rests on the fact that there is no confinement mechanism to create a bound state for the field. The system is therefore superradiant, but not unstable. As we will see next, it is possible to cause the system to become unstable by installing a mirror, thereby creating a ’Black Hole Bomb’.
2.4.2 Black Hole Instabilities
Black Hole Bomb
In order to retain the superradiant waves satisfying (2.17), one can in theory install a mirror at a finite distance from the event horizon of the Kerr black hole. The mirror then asserts a reflecting boundary condition Ψ = 0 on the waves, causing them to rebound back and forth, continuously amplifying each time that they hit the event horizon [22]. Assuming that the mirror is frail to some degree, eventually the radiation inside the system will reach a point where the mirror breaks, expelling the radiation outward just like a bomb. Although this ’black hole bomb’ as de- scribed does not constitute the most physically plausible system, possible ’mirrors’ as provided by nature have been conjectured, one of them being the potential of a massive field.
Massive Mirrors
Zouros and Eardley [23] and Detweiler [24] conducted studies that demonstrated that the Kerr metric is unstable to massive scalar fields in the superradiant regime. The study [23] was within the regime Mµ >> 1, where M is the mass of the black hole and µ is the mass of the scalar field, and they found that the fastest rate of exponential growth was given by Γ = (10−7/M) exp −1.84Mµ. Similarly, Detweiler examined the Mµ << 1 regime, and found a fastest growth rate of Γ = (a/24M)(Mµ)8µ. The confinement mechanism that causes the instability to occur is best illustrated by examining the radial equation for a massive scalar field in the
21 Figure 2.1: Effective potential for a massive scalar field on a Kerr background . Figure taken from [23]. We see that the potential approaches µ2 as r∗ → ∞, providing the confinement mechanism that allows superradiant waves to accumulate. transformed tortoise coordinate dr∗ = (r2 + a2)∆−1dr from [23]
d2u + ω2 − V (ω) u = 0 (2.18) dr∗2 where u = (r2 + a2)1/2R where R is the seperated radial function of Ψ and
∆µ2 1 V (ω) = + O (2.19) r2 + a2 r is the effective potential which can be seen in Fig (2.1). From (2.19), we can see that the mass of the field provides a rising barrier that approaches µ2 as r∗ and r → ∞, since ∆ → r2 in this limit. Furthermore, since V (r → ∞) = µ2 > 0, we can have bound states with E > 0. The extreme regimes of Mµ >> 1 and Mµ << 1 were chosen largely because they lend themselves to analytic techniques. Much later in [25] and several other studies since, the growth rates were computed numerically in the regime where Mµ ∼ 1. The results from [25] showed that the growth rate was maximal for Mµ ∼ 0.42 with Γ = 1.5 × 10−7(GM/c3)−1.
22 Gravitational Atoms
The massive superradiant states will accumulate in region III of Fig (2.1), since waves in region I are either absorbed or extract outgoing energy from the black hole. If we integrate the tortoise transformation dr∗/dr, we get
2M 2 − a2 r − r r∗ = M log ∆ + √ log + (2.20) 2 M 2 − a2 r − r−
∗ ∗ We then see that r → −∞ as r → r+. Then for bound states at finite r , the location of the bulk of the field is therefore in a large r limit hri/r+ >> 1, where hri is the average radius calculated using |Φ|2 as a probability density. In this regime, the Kerr potential looks like (using the Boyer-Lindquist radius)
−M L2 − a2 (E2 − 1) (L − aE)2 V = + − M (2.21) eff r 2r2 r3 and is identical to the result for Schwarzschild taking a → 0. Then in the limit of large r, this potential behaves like Newtonian gravity with V ∼ 1/r. Furthermore, the superradiant condition gives
mJ ω < mΩH = √ (2.22) 2M(M − M 2 − J 2) where we are working in SI units instead of M ∼ 1 units as in (2.14). Then taking m ∼ 1 and for rapidly rotating black holes, J/M ∼ 1, and we get
1 ωM < (2.23) 2
In other words, ω is non-relativistic to a good approximation. The argument here is identical to the corrections ∼ (v/c)2n to relativistic kinetic energy. For v/c = 0.5, the leading correction is ∼ 0.52 = 0.25, which get successfully smaller and smaller. Now, since the field Φ satisfies the Klein-Gordon equation and is approximately non-relativistic, it will also satisfy the Schrodinger equation to a good approxima-
23 tion. Furthermore, since it is also in a potential V ∼ 1/r, its energy eigenstates will be analogous to those of the Hydrogen atom, with spectrum
α2 ω ≈ µ 1 − (2.24) n 2n2 where α = Mµ is the analogous fine structure constant. The only difference is that Φ is a classical field, and is not to be interpreted as a probability density for a quantum mechanical particle.
Bosenovas
Now that were are equipped with the hydrogen-like spectrum of superradiant states, we are prepared to discuss the phenomena of the bosenova. A bosenova is the bosonic cloud analog to a supernova of stars. A supernova occurs when a star runs out of nuclear fuel to resist its own gravitational atttraction, causing it to collapse. Similarly, a bosenova occurs when the growth provided by superradiance is eclipsed by an attractive self-interaction. A bosenova involves a field with a quartic term |Ψ|4 in the potential, which leads to a cubic term |Ψ|2Ψ in the equation of motion when taking ∂V/∂Ψ∗. This term provides an inertial force similar to a mass term m2Ψ, although the influence of the cubic term grows as |Ψ| increases. In terms of superradaince, we saw in the previous subsection that a massive term is not strong enough to eclipse the growth propelled by superradiance. However, a cubic term does turn out to be strong enough for at least part of the equation’s parameter space [11, 26, 27]. The bosenova commences once the potential energy with the respect to the self- interaction becomes comparable to the gravitational potential provided by superra- diance. In the case of axions, this occurs when [26]
α |Ψ|2 ∼ 2 (2.25) r 8fa
where fa is the inverse coupling parameter and the LHS comes from the hydrogen-
24 like spectrum we saw previously. Once the self-interaction energy becomes large enough, the field collapses, just like a supernova. Once the field again reaches a small enough |Ψ|, the superradiant growth again takes over, causing a periodic growth and collapse until all of the energy is extracted from the black hole. The case of the Higgs is slightly different from pure quartic potentials in that it reaches its equilibrium value at a nonzero VEV. Then when considering bosenovas, it seems that the Higgs will periodically grow and collapse about it’s expectation value for similar parameters to the case of the symmetric bosenova. For the purposes of this study, it will be important to examine whether this bosenova effect will admit the sufficient electro-weak signatures for which we are looking.
25 Chapter 3
Higgs Evolution: Superradiant Model in Minkowski Space
3.1 Introduction
We will begin our study of the Higgs field with a superradiant model in flat Minkowski space. With this model, we will be able to observe the interaction between super- radiant growth and the non-linear Higgs potential, thereby providing a model for superradiant Higgs systems and furthermore guiding our study of the Kerr spacetime in Chapter 4. In Section 2.3, we saw that the onset of a superradiant instability can be in- terpreted as an anti-friction effect by examing the wave equation with a friction term ∂Ψ Ψ + α = 0 (3.1) ∂t
For our Minkowski model, we use the covariant anolog to (3.1) in a flat (2 + 1)D spacetime ∂V (Ψ) Ψ + bµ∂ Ψ − = 0 (3.2) µ ∂Ψ∗ where we use the Abelian Higgs potential
λ V (Ψ) = |Ψ|2 − v2 |Ψ|2 (3.3) 2
26 and bµ is a (2+1)-vector to be tuned during our study.
3.2 Equations of Motion
We will proceed in a standard set of polar coordinates (t, r, φ) in which bµ = bt, br, bφ . We can see immediately that for bt > 0, eq (3.2) receives an anti-friction contribution as in eq (3.1) with α > 0. We might therefore suppose a superradiant condition of bt > 0, but this will turn out to be not quite correct. In these coordinates, we decompose Ψ into a semi-seperable solution Ψ → ψ(r, t)eimφ. The equation of motion (3.2) then becomes
∂2ψ ∂ψ ∂2ψ 1 ∂ψ m2 − + bt + + + br − ψ + imbφψ ∂t2 ∂t ∂r2 r ∂r r2 (3.4) λ − |ψ|2 − v2 ψ = 0 2
We can now split (3.4) into real and imaginary parts by parsing out ψ → R(r, t) + iI(r, t), which gives us
∂2R ∂R ∂2R 1 ∂R m2 − + bt + + + br − R − mbφI ∂t2 ∂t ∂r2 r ∂r r2 (3.5) λ − R2 + I2 − v2 R = 0 2 and
∂2I ∂I ∂2I 1 ∂I m2 − + bt + + + br − I + mbφR ∂t2 ∂t ∂r2 r ∂r r2 (3.6) λ − R2 + I2 − v2 I = 0 2
We can get a sense for how these equations will behave by classifying each of the terms. Eq (3.4) has the form of a wave equation with modifying terms ∼ ∂ψ/∂t, ∂ψ/∂r, |ψ|2ψ, and ψ. We saw in the previous chapter when discussing bosenovas that linear mass and cubic self-interacting terms provide an inertial contribution to the evolution of ψ. The reason for this is that if the coefficients of ∂2ψ/∂t2 and ψ have the same sign, then these two terms behave together like a simple harmonic
27 oscillator y00 + y = 0. The resulting behavior for a cubic term is similar, although more reactionary for large y, just like a spring with modified Hooke’s Law F = −ky3. However, if the coefficients of the ∂2ψ/∂t2 and the ψ term are different, they behave together like y00 − y = 0, which will generally contain a growing solution y ∼ eΓt. When this occurs, the ψ terms become ’anti-mass’ terms in the sense that they then precipitate motion instead of providing inertia against it. In this way, we see that we can have additional growth contributions to the anti-friction coupling bt for λv2/2 > m2/r2. Similarly for mbφ 6= 0, we get a growth contribution to R for mbφ < 0 and for I for mbφ > 0, either of which will cause |ψ| to grow. The |ψ|2ψ term is necessarily inertial since the Higgs coupling satisfies λ ≥ 0 necessarily. The ∂ψ/∂r term will have a similar relationship with the ∂2ψ/∂r2 term as the case for time derivatives, although the length of our radial domain is (typically) much shorter than our time domain, so this effect will not be as significant. The br∂ψ/∂r term will however impose a bias for the field to congregate towards smaller
r r for b > 0, just as a radial derivative in a 1st order wave equation ∂tψ+∂rψ = 0 has solutions that travel towards r = −∞. However, by imposing reflecting boundary conditions ψ = 0 at the left and right handed boundaries, the radial derivatives will turn out not to significantly affect the growth rate of the field ψ. We will show this quantitatively in Section 3.4.
3.3 Numerical Methods
Looking at eq (3.4), we can see that we recover a separable equation for the case where |ψ|2 = 1. However, since (3.4) is non-linear, it does not admit an eigenvalue spectrum in either space or time that would yield solutions of the form e−i(ωt−kr). We therefore have no reason to believe that solutions to (3.4) will be separable, since in general |ψ|2 6= 1. Then, as is usually the case for non-separable equations, it seems that a numerical approach will be the most fruitful method for analyzing solutions without enforcing limiting cases.
28 3.3.1 Finite Differences
In order to numerically solve the coupled equations (3.5) and (3.6), we use a finite difference scheme in which we evaluate both equations on a discrete grid with con- stant spacings ∆r and ∆t. Using i and j as out temporal and spatial indices, we have t = i∆t for i = 0, ..., Nt − 1 and r = j∆r for j = 0, .., Nr − 1. Evaluated on this grid, (3.5) and (3.6) become
2 i i 2 i i 2 i ∂ R t ∂R ∂ R 1 r ∂R m − 2 + b + 2 + + b − 2 R ∂t j ∂t j ∂r j r ∂r j r j (3.7) i φ i λ 2 2 2 − mb I j − R + I − v R = 0 2 j and
2 i i 2 i i 2 i ∂ I t ∂I ∂ I 1 r ∂I m − 2 + b + 2 + + b − 2 I ∂t j ∂t j ∂r j r ∂r j r j (3.8) i φ i λ 2 2 2 + mb R j − R + I − v I = 0 2 j
i where each term is evaluated at (r, t) = (j∆r, i∆t). For the terms ∼ ψ, ψj = ψ(j∆r, i∆t). To evaluate the derivative terms, we need a discrete approximation for the derivative operators. The simplest of these approximations for the case of a first
i derivative is the forward difference operator (∂rψ)j ≈ (ψ(r0 + ∆r, t) − ψ(r0, t))/∆r were r0 = n∆r and t = i∆t. This is just the secant line approximation to the slope of the tangent line at r0. The forward difference operator can furthermore be derived from taylor expanding ψ about r0
i ∂ψ 2 ψ(r + ∆r, t) = ψ(r0, t) + ∆r + O (∆r) (3.9) ∂r j
Then by dividing by ∆r, we find that
∂ψ i ψ(r + ∆r, t) − ψ(r , t) = 0 0 + O(∆r) (3.10) ∂r j ∆r
29 is a first order accurate approximation in ∆r. We can further recover a 2nd order
first derivative operator by taylor expanding ψ about r0 + ∆r and r0 − ∆r
i 2 2 i ∂ψ (∆r) ∂ ψ 3 ψ(r0, t) = ψ(r0 + ∆r, t) − ∆r + 2 − O (∆r) (3.11) ∂r j 2! ∂r j
i 2 2 i ∂ψ (∆r) ∂ ψ 3 ψ(r0, t) = ψ(r0 − ∆r, t) + ∆r + 2 + O (∆r) (3.12) ∂r j 2! ∂r j
Then by taking (3.11) − (3.10) and dividing by ∆r, we get that
∂ψ i ψ(r + ∆r, t) − ψ(r − ∆r, t) = 0 0 + O (∆r)2 (3.13) ∂r j 2∆r and we see that this centered difference approximation is second order accurate in ∆r. Similarly, we can find a second order accurate approximation for the second
2 2 order derivative operator ∂r by taking (3.10) + (3.11) and dividing by (∆r)
2 i ∂ ψ ψ(r0 + ∆r, t) − 2ψ(r0, t) + ψ(r0 − ∆r, t) 2 2 = 2 + O (∆r) (3.14) ∂r j (∆r)
We then proceed with a second order accurate, central difference scheme in both
∆r and ∆t with Lt = 50 and Lr = 10. Then by plugging in (3.13), (3.14), and their temporal analogs into eqs (3.7) and (3.8), we get our discrete equations of motion
! ! Ri+1 − 2Ri + Ri−1 Ri+1 − Ri−1 Ri − 2Ri + Ri − j j j + bt i j + j+1 j j−1 (∆t)2 2∆t (∆r)2
i i 2 1 Rj+1 − Rj−1 m (3.15) + + br − Ri + mbφIi r 2∆r r2 j j
λ − (Ri )2 + (Ii)2 − v2 Ri = 0 2 j j j
30 Figure 3.1: Example of a 5 point stencil where the field at the next time step i + 1 depends on the four field values at (i, j), (i − 1, j), (i, j + 1), (i, j − 1). Figure taken from p. 222 of [28]. and
! ! Ii+1 − 2Ii + Ii−1 Ii+1 − Ii−1 Ii − 2Ii + Ii − j j j + bt j j + j+1 j j−1 (∆t)2 2∆t (∆r)2
i i 2 1 Ij+1 − Ij−1 m (3.16) + + br − Ii − mbφRi r 2∆r r2 j j
λ − (Ri )2 + (Ii)2 − v2 Ii = 0 2 j j j
i+1 i+1 Then looking at eqs (3.14) and (3.15), we see that both Rj and Ij depend on
i i−1 i i i i−1 i i the set of values Σ = Rj,Rj ,Rj+1,Rj−1,Ij,Ij ,Ij+1,Ij−1 . This is typical for central difference schemes, where the value at the next time step depends on a stencil of 4 points (see Fig 3.1), or in our case 2 × 4 = 8 points since we have
0 1 0 1 two coupled equations. Therefore, given initial conditions I = Rj ,Rj ,Ij ,Ij for
i i i i all j ∈ [0, Nr − 1] and boundary conditions B = R0,RNr−1,I0,INr−1 for all i+1 i+1 i ∈ [0, Nt − 1], we can iterate through each time step and solve for Rj and Ij algebraically.
3.3.2 Code Tests
Now that we have our numerical method in place, it is of course important to verify whether or not it is working probably. We will proceed by testing both the
31 convergence of the numerical method, as well as by testing the solutions themselves by setting the physical parameters such that the solution should be a plane wave ψ ∼ e−i(ωt−kr).
Convergence
As we stated in the previous section, our finite difference operators are second order accurate in ∆t and ∆r. What we mean by this is that if our continuum equation (3.4) has the solution ψ and our discrete equations of motion (3.15, 3.16) have solution
i i i 2 2 ψj, then we expect ψj to differ from ψ by |ψ − ψj| = O ((∆t) , (∆r) ). Then if we have an analytical solution ψ, we can assess the accuracy of our code by varying ∆t
i and ∆r and verifying whether the error |ψ − ψj| follows a quadratic. In the absence of an analytical solution, we can still assess the order of accuracy of our code using the method of convergence factors (see Appendix C of [28] for details). If our scheme is indeed second order accurate, then if we run the code for time step spacings 4∆t, 2∆t, and ∆t and recover solutions ψ4(r, t), ψ2(r, t), and
ψ1(r, t), then we can define the convergence factor C at a particular radius r0 [28]
|ψ4(r0, t) − ψ2(r0, t)| C(r0, t) = (3.17) |ψ2(r0, t) − ψ1(r0, t)|
Then for a second order accurate scheme, we expect
O(16∆t2) − O(4∆t2) 12 C ∼ ∼ = 4 (3.18) O(4∆t2) − O(∆t2) 3 since the coefficients to the error contribution should cancel in the ratio C (this is not necessarily the case for equations with time-dependent coefficients, but the coefficients in eq (3.4) vary only in space). Then computing C at r0 = 6 for Lt = 10,
Lr = 10, Nt = 12001, and Nr = 40, we find that C ≈ 4 as expected up to some noise (see Fig 3.2). We therefore conclude that our code is convergent to 2nd order in ∆t. It would be instructive to understand what exactly is causing the blips in the plot of C(t), so we leave this for future study.
32 2 Figure 3.2: Left: Cross-section of |ψ| taken at r0 = 6 for m = 1, v = 1, λ = 1, µ and b = h1, 0, −1i. Right: The convergence factor C computed for Lt = 10 and Nt = 4001, 8001, and 12001. We see that C ∼ 4 as expected up to some noise.
Plane Wave Test
In regard to testing the code by comparing it to a known solution, we of course don’t know the solutions to eq (3.4), but we can feel free to plug in any form of ψ = ψs we like into (3.4) and observe the result. If ψs is not a solution of (3.4), then applying the LHS operator to ψs will result in a nonzero source term on the RHS. We can then subtract off this source term from the original equation of motion, and we will be left with an equation whose solution is ψs.
−i(ωt−kr) A typical form of ψs we might choose is a plane wave solution ψs = e . Plugging this into (3.4) for ω, k ∈ R, we are left with
1 m2 λ ω2 − iωbt − k2 + ik + br − + imbφ − 1 − v2 ψ =? 0 (3.19) r r2 2 s
Then to get the LHS to equal 0, we can both subtract appropriate source terms and set the parameters ω, k, and bµ appropriately. To avoid radially dependent parameters, we subtract off the radially dependent terms ∼ 1/r and 1/r2 as source terms. Then by taking real and imaginary parts of (3.18), we are left with two conditions on our set of parameters
λ ω2 − k2 − 1 − v2 = 0 (3.20) 2
ωbt − kbr − mbφ = 0 (3.21)
33 Figure 3.3: Cross-sections of real and imaginary parts R(t) and I(t) taken at r0 = 15 for m = 1, v2 = 1, bt = −10, bφ = 1, and ω = k = −1/10. The result is a plane wave with magnitude |ψs| ∼ 1 as desired. A more quantitative assessment of this result can be seen in Fig (3.3).
2 µ Figure 3.4: Cross-section of |ψ| taken at r0 = 15 for m = 1, v = 1, b = h−10, 0, 1i, and ω = k = −1/10. The result is a plane wave with magnitude ∼ 1, as desired. We can see the error in ψ begin to accumulate as t increases. The cumulative error at t = 100 is ∼ 10−3
q r φ t 2 λ 2 We then proceed with the test by setting ω = (kb +mb )/b and k = + ω − 2 (1 − v ) over a grid with Lt = 100, Lr = 30, Nt = 8000, and Nr = 100. For our results, we record radial cross-sections of the real and imaginary parts R and I and the magnitude |ψ| at r0 = 15 ( see Fig (3.2), (3.3)). The results of the plane wave test are as desired, namely that |ψ| ∼ 1 with error ∼ 10−3 at t = 100. If the linear behavior of the error accumulation continues as in Fig (3.4), then we can expect an
−3 additional ∆ = 10 for every interval of Lt = 100, which is plenty accurate for our purposes.
34 3.4 Examining Superradiance
Now that we know that our code is working properly, we can continue onward and examine the prospect of superradiance for our model. A natural place to start would be the numerical growth rate found for a linear scalar field about a Kerr black hole with Mµ = 0.42 and Γ = 1.5 × 10−7(GM/c3)−1 [25]. This Γ should constitute an upper bound for the growth rate that we might expect in the nonlinear regime because, recalling the confining potential from eq (2.19) for massive linear scalars,
∆µ2 1 V = + O (3.22) r2 + a2 r we see that switching from the linear to the nonlinear regime is tantamount to modifying µ2 → λ (|ψ|2 − v2) for large r. Then for systems where Mµ ∼ 1, we expect similar confining behavior for Mp|ψ|2 − v2 ∼ 1. As |ψ| continues to grow, we expect the growth rate to decrease towards the Mµ >> 1 limit computed by [23] that we saw in Chapter 2.
Unfortunately, simulations for growth rates of this size have e-folding times τe ∼ 107M, the storage for which places computational demands that exceed the resources for our study. We might be tempted to circumvent this issue by setting M ∼ 10−5 or so, but we would hit time step instabilities without scaling Nt → Nt/M as well, making the simulation just as long. We therefore limit our study to growth rates Γ ∼ 10−2 and extrapolate our findings to smaller Γ.
3.4.1 Checking Γ(bt)
Before we begin testing the onset of a superradiant instability, we should check how accurately the exponential growth of our model is given by Ψ ∼ eΓt. In general, for a damped oscillator y00 − bty + ω2y = 0, the growth rate Γ is given by Γ = bt/2 ± pbt2/4 − ω2. We then see that in the two limits bt >> ω (over-anti-damped) and 2ω/bt ≥ 1 (under-anti-damped), we should have Γ = bt and Γ = bt/2, respectively. Then checking the contributions to the ψ term from eq (3.4), we see that we can
35 t 2 −2 Figure 3.5: Cross-sections of |ψ| taken at r0 = 6 for (b , v ) = (10 , 0)(left) and (10−2, −1) (right). We find that |ψ| ∼ eΓt with Γ = 0.0112 (left) and Γ = 0.00447 as expected. Furthermore, we see a greater contribution to the oscillating behavior of ψ in the under-anti-damped case (right) than the over-anti-damped case (left), as expected. source ω2 with −v2 and test these two limits for (bt, v2) = (10−2, 0) and (10−2, −1). To stop other terms from interfering, we set bφ = 0, m = 0, and add a parameter q = 0 that switches off the non-linear term q|ψ2|ψ.
The results of our test can be seen in Fig (3.5). We run the test over Lt = 300 and Lr = 10 with Nt = 60001 and Nr = 40. For our initial data, we use a gaussian
2 2 −(r−p0) /2σ wavepacket ψ0 = Ae with A = 1, p0 = Lr/2, and σ = Lr/4. We then extract the growth rates by iterating in time through a cross-section of |ψ| at r0 = 6 and recording the local maximum for |ψ|i > |ψ|i−1 and |ψ|i > |ψ|i+1. We then take the first and last recorded local maximum |ψ|1 and |ψ|2 and compute
1 |ψ| Γ = ln 2 (3.23) T |ψ|1
where T is the length in time between |ψ|1 and |ψ|2 (it is therefore important to record the time at which the local maximum occur as well). Our results match what we expect: For (bt, v2) = (10−2, 0) we find that Γ = 0.0112, and for (10−2, −1) we find that Γ = 0.00447. So we see that we can recreate exponentially growing solutions with our model. In Chapter 4, we will see that our equation of motion for ψ in the Kerr geometry will
t have a first order derivative in time term with complex coefficient term ∼ imb ∂tψ. We might then be tempted to see whether this term will cause growth similar to
36 2 µ Figure 3.6: Cross-section of |ψ| taken at r0 = 6 for m = 1, v = −1, and b = hi10−2, 0, 0i where bt is now complex. As we can see, ψ does not grow exponentially in this case.
t the b ∂tψ and imψ terms from the model considered here. However, if we make the switch bt → imbt, we find that ψ does not grow exponentially (see Fig (3.6)). For the case of the Kerr metric, this stems from the fact that such a term is derived from the Lagrangian of the system, and is therefore associated with a particular conserved Energy. This is not the case for the model under consideration here, which is what provides the growth in the system.
3.4.2 Testing the Higgs
We see now that our system is well-modeled by the anti-damped harmonic oscillator in the linear regime q = 0. In this regime, the envelope of the field magnitude |ψ| rises like eΓt with |ψ| itself oscillating intermittently. The turning points for these
1 2 2 oscillations occur once the condition E(t) = V (|ψ|) = 2 m |ψ| is reached. Then in the nonlinear regime, we might expect similar behavior with turning points occurring
λ 2 2 2 when E(t) = 2 (|ψ| − v ) |ψ| . In other words, we should expect superradiance to persist in the nonlinear regime. We would then ideally like to test this hypothesis for the maximal growth rate Γ = 1.5 × 10−7(GM/c3)−1 from [25]. However, as we stated earlier, simulations of this size are beyond the scope of our study. Luckily, the previous conjecture is qualitatively indistinguishable for any Γ, so if it applies to Γ = 10−2, it should apply for Γ = 10−7 as well.
37 Figure 3.7: Cross-sections of |ψ| taken at r0 = 6 in the non-linear regime q = 1 and λ = 1 for v2 = −1 and A = 10−7 (left), A = 10−5 (right) and A = 10 (bottom). The results show that superradiance persists in the nonlinear regime of the Higgs potential.
We therefore test the prospect of Higgs superradiance for the model considered here with the same test from Section 3.4.1 with the nonlinear term turned on with q = 1. We modify the simulation length for the nonlinear case with Lt = 1000 and Nt = 200001 to compensate for the smaller growth rates. The results are seen in Fig (3.6). Our findings support what we expect, namely that superradiant growth persists. Quantitatively, for initial data with small amplitudes A = 10−5 and A = 10−7, we find that Γ = 0.49319 and Γ = 0.49320 with v2 = −1, thereby approaching the large ω2 limit Γ = bt/2 from the linear case. This is what we expect, since the nonlinearity is a very small perturbation in the small amplitude regime. For larger amplitudes A = 10, we find Γ = 0.0036, thereby decreasing with the increased stiffness of the nonlinear term.
3.5 Conclusion
The numerical results from our model show that superradiance persists in the non- linear regime of the Higgs potential if it is initially present in the linear regime. That is to say, the nonlinear term in the Higgs potential does not eclipse the on- set of superradiance as it occurs in the model considered here. More specifically,
38 the rising amplitude of Higgs oscillations indicates that energy enters the system according to the turning points of these oscillations E(t) = V (|ψ(t)|). In the nonlin- ear regime, we saw that the increased stiffness of the Higgs potential caused these turning points to occur for lower values of E(t), thereby reducing the growth rate of the field. However, this stiffness effect was not strong enough to eclipse the onset of superradiance.
39 Chapter 4
Higgs Evolution: Kerr Spacetime
4.1 Introduction
In Chapter 3, we saw that, for our model, if a field experiences superradiant growth |ψ| ∼ eΓt in the linear regime of the Higgs potential, then the switch to the nonlinear regime will not eclipse the onset of superradiance. This was because in our model, the bt∂ψt term caused growing oscillations like that of an anti-damped harmonic oscillator, so adding a stiffer potential only changed the turning points of these oscillations. As we saw in Chapter 2, the onset of superradiant growth |ψ| ∼ eΓt in the Kerr geometry is a global effect that requires a sufficient confinement mechanism to trap the superradiant waves. It is therefore entirely possible that, in the time that it would take for these waves to accumulate and cause exponential growth, that the evolution mediated by the Higgs potential could impose some eclipsing effect, similar to how one may not be able to snap a bundle of twigs, but one could certainly snap each individual twig before they are accumulated. Another point worth considering with respect to the onset of superradiance in the Higgs-Kerr system is that the superradiant process is contingent upon the presence of waves in the superradiant regime
ω < mΩH (4.1)
40 Then for linear wave equations, an eigen-mode solution with frequency ω retains the same frequency ω throughout its evolution. For evolutions in the Kerr geome- try, this means that superradiant waves in the linear regime remain superradiant. However, for nonlinear wave equations, this is not the case. For example, for the one-dimensional anharmonically perturbed oscillator,
y00 + ω2y + y3 = 0 (4.2)
if y0 = cos ωt is the simple harmonic solution, then the anharmonic solution to first order in is, 3t 1 y(t) = ( + 1) cos ωt − sin ωt + cos 3ωt (4.3) 8 32ω2 and we see that the nonlinear term causes a mixing of frequencies. Therefore, the nonlinearity of the Higgs potential will have an additional mechanism to affect the onset of superradiance by either knocking waves into or out of the superradiant regime. It is therefore quite necessary to extend our considerations from Chapter 3 to the physical system born out of the Kerr geometry. In this chapter, we will complete a non-linear evolution of the Abelian Higgs field on a Kerr background in Boyer-Lindquist coordinates, which have thus far been largely avoided in the literature. To simplify the numerical complexity of the problem, we limit our considerations to the (2 + 1)D spacetime confined to the equatorial plane θ = π/2. As discussed in Chapter 2, the principle physics of this √ 2 2 2 system is provided by the presence of an ergoregion rerg = M + M − a cos θ √ 2 2 outside of the outer event horizon r+ = M + M − a . For θ = π/2, we have rerg = 2M and r+ < 2M, so setting θ = π/2 alleviates computational costs while retaining the principle physics of the system. We will test the accuracy our nonlinear solver with the convergence test from Chapter 3, by testing the energy conservation law for the system, and finally by recreating results from the literature involving linear scalar fields. We will then cross-examine our conclusions from Chapter 3 by evolving the Abelian Higgs model
41 under different parameter settings in the Kerr geometry. In Chapter 5, we proceed to assess the plausibility for the onset of electro-weak physics within the Higgs-Kerr system based on the ensuing field evolutions.
4.2 Boyer-Lindquist Evolution
In this section, we evolve the Higgs field in both the linear and non-linear regime on a Kerr background in Boyer-Lindquist coordinates. In the linear regime, for both black hole bomb and massive potential confinement mechanisms, our simulations find good agreement with [29], which is the only other time domain study in Boyer- Lindquist coordinates to our knowledge. Field evolutions in these coordinates have otherwise been avoided in the literature due to numerical instabilities within the ergoregion. Yoshida and Kodama [30] cite the reason for these instabilities as being due to the spacelike lines of constant radius. However, we find compelling evidence in our simulations that the propagation of error is due rather to the (∆−a2)(−m2)ψ term in the equation of motion that switches sign in the ergoregion, causing trun- cation errors to grow like y00 − y = 0. Evidence in support of this claim includes the fact that our code satisfies conservation laws within the ergoregion for m = 0, but not for m 6= 0. Furthermore, for the black hole bomb system in the nonlinear regime, the nonlinear term quells the growth of such error, resulting in a modified expectation value for ψ that matches that which would be expected by generalizing
2 2 2 2 2 the expectation value parameter v → v (r) = λ (∆ − a )(−m ). In this way, the Higgs-Kerr system combined with this numerical growth effect constitutes a mech- anism through which the Higgs field undergoes spontaneous symmetry breaking. If this numerical growth contribution can be emulated by a physical energy source, then this system provides an avenue for studying symmetry breaking in the labora- tory using analog superradiant systems, such as the Zel’dovich cylinder discussed in Chapter 2. In regard to assessing the prospect of Higgs superradiance, we find that our simulations are convergent and stable for m ≤ 1, although the energy conservation
42 law of the system is not satisfied for m = 1 due to growth contributions by the
2 2 4 (∆ − a )(−m )ψ term. This results in an e-folding time of τe ∼ 10 M instead of
7 the expected τe ∼ 10 M from [25]. This result is in agreement with [29] where m = 1 is used as well. In this way, the Higgs-Kerr system in Boyer-Lindquist coordinates as presented here can be thought of as the physical system with an energy source provided by the numerical growth contributions of the (∆−a2)(−m2)ψ term. This study therefore provides an upper-bound to the assessment of the onset of superradiance, namely that if superradiance is not present with the extra numerical energy, then it should not be expected in the error-free system. This conjecture should of course be examined in a coordinate system where error propagation is minimized, which we begin to address in Section 4.3.
4.2.1 Equations of Motion
Lagrangian Formulation
Our equation of motion in Chapter 3 (3.1) did not conserve energy in either a local or global sense and was therefore not derivable from a Lagrangian. This was of course by design, since we wanted a term in eq (3.1) to emulate superradiant growth. This lack of a Lagrangian formulation is analogous to the case of frictional systems in classical mechanics. Furthermore, the superradiant growth in our Minkowski
µ model was enforced at each point by the anti-friction term ∼ b ∂µΨ, making the onset of superradiance a local effect. In the case of evolving Ψ on the Kerr metric, the superradiant growth will be provided by scattering waves in the superradiant regime off of the event horizon of the black hole, thereby extracting its rotational energy. A superradiant instability then sets in if there is a sufficient confinement mechanism that allows the growing superradiant waves to accumulate. In this way, the superradiance of the Higgs field in the Kerr geometry is a manifestly global effect. The change in energy of the system will occur only at the boundaries of the system, and energy will be locally conserved (in the absence of numerical error). Therefore,
43 we can implement a Lagrangian formulation of the Higgs-Kerr system
1 L = − gµν (∇ Ψ) (∇ Ψ∗) − V (|Ψ|) (4.4) 2 µ ν
ν ν ν λ where ∇µV = ∂µV + ΓµλV is the covariant derivative for the Kerr geometry, and we recall from eq (2.4) that the metric gµν is
2Mr 2Mar sin2 θ ds2 = − 1 − dt2 − (dtdφ + dφdt) ρ2 ρ2 (4.5) ρ2 sin2 θ + dr2 + ρ2dθ2 + r2 + a22 − a2∆ sin2 θ dφ2 ∆ ρ2 from which we can derive the inverse metric gµν
∂ 2 −1 (r2 + sin θ2a2)2 ∂ 2 1 2Mra ∂2 ∂2 = − a2 sin θ2 − + ∂s ρ2 ∆ ∂t ρ2 ∆ ∂φ∂t ∂t∂φ
∆ ∂2 ∆ − a2 sin θ2 ∂2 + + ρ2 ∂r2 ρ2∆ sin θ2 ∂φ2 (4.6) where we have made the switch to (2 + 1)D by dropping the ∂2/∂θ2 term. We are allowed to do this because in Boyer-Lindquist coordinates, the inverse metric is found by flipping the grr and gθθ terms and then inverting the 2 × 2 metric in t and φ. In general, one needs to compute the (2 + 1)D metric and find the inverse from there, as opposed to setting ∂/∂θ → 0. This then gives us the matter contribution to the action of the system
Z √ 1 S = d4x −g − gµν (∇ Ψ) (∇ Ψ∗) − V (|Ψ|) (4.7) Ψ 2 µ ν
44 Klein-Gordon Equation
We can now find our equation of motion for Ψ on the fixed Kerr background by varying the action Sφ. The result will of course be the Euler-Lagrange equation
∂L ∂L ∗ − ∂µ ∗ = 0 (4.8) ∂Ψ ∂ (∂µΨ )
Then since Ψ is a scalar field, this will just come out to be the Klein-Gordon equation
∂V gµν∇ ∇ Ψ − = 0 (4.9) µ ν ∂Ψ∗
We can then avoid dealing with the Christoffel symbols in ∇µ by switching to the covariant formulation of eq (4.9) [31]
1 √ ∂V √ ∂ gµν −g∂ Ψ − = 0 (4.10) −g µ ν ∂Ψ∗
Then by plugging in the values for gµν, we get the following after multiplying by ρ2
(r2 + sin θ2a2)2 4Mar ρ2 ∆ − − a2 sin θ2 ∂ Ψ − ∂ Ψ + ∂ ∂ Ψ ∆ tt ∆ tφ r r r r (4.11) a2 sin θ2 λ + 1 − ∂ Ψ − ρ2 |Ψ|2 − v2 Ψ = 0 ∆ φφ 2
The discrepancy in this equation from eq (1) of [29] is due to the fact that for our √ (2 + 1)D evolution, −g = r as opposed to ρ2 sin θ in (3 + 1)D. We now make the same semi-separable anzats for Ψ as in Chapter 3 Ψ → ψ(r, t)eimφ. Then plugging this in along with confining eq (4.9) to the equatorial plane θ = π/2 and multiplying by ∆, we arrive at
∆ − (r2 + a2)2 − ∆a2 ∂ ψ − 4Mar(im)∂ ψ + r∆∂ ∂ ψ tt t r r r (4.12) ∆λ + ∆ − a2 (−m2)ψ − ρ2 |ψ|2 − v2 ψ = 0 2 which is our equation of motion for Ψ in Boyer-Lindquist coordinates. There is one more modification that is common in the literature that we will make use of,
45 namely transforming our radial coordinate to the tortoise coordinate r → r∗. The advantages of using such a coordinate are two-fold. First, the transformation maps
∗ the outer event horizon from r = r+ to r = −∞, thereby mandating that our radial domain falls outside of the black hole. Second, the transformation yields a smaller dr∗ closer to the event horizon, thereby acting as its own adaptive-mesh-refinement mechanism. Then using the tortoise transformation from [32] dr∗/dr = r2/∆, we arrive at,
2 2 2 2 4 − (r + a ) − a ∆ ∂ttψ − 4Mar(im)∂tψ + r∆∂r∗ ψ + r ∂r∗r∗ ψ (4.13) λ + ∆ − a2 (−m2)ψ − ∆ ρ2 |ψ|2 − v2 ψ = 0 2 which is the equation of motion that we will use throughout this study.
4.2.2 Boundary Conditions
In Chapter 3, the boundary conditions that we chose were largely inconsequential with respect to the onset of superradiance, prompting us to employ simple reflecting boundary conditions ψ = 0. For the case of the Kerr geometry, getting the bound- ary conditions correct is paramount in order to properly model the superradiant behavior, since the left-handed boundary at the outer event horizon is where energy is flowing into the system.
Event Horizon r∗ → −∞
Bardeen, Press, and Teukolsky first proposed the proper boundary condition for the case of linear massless scalar fields [32]. In this case, the Klein-Gordon equation is completely separable, and the radial equation looks like
d2ψ ∆ ∂ψ + + W (r)ψ = 0 (4.14) dr∗2 r3 ∂r∗ with (r2 + a2)Ω − a2 ∆ W (r) = m2 − λ − a2ω2 (4.15) r2 r4 ml
46 where ω = mΩ, m, and l are the energy, azimuthal, and orbital separation constants, √ respectively. Discrepancy from eq (4.12) of [32] is due to the fact that −g = r in (2 + 1)D, and we also don’t factor out an r from the field ψ. Then at the event horizon r∗ = −∞, the second term vanishes as ∆ = 0, and we are left with the
±ik+r 2 radial solution ψ ∼ e where k+ = W (r+). Recovering the time dependent phase factor, we then have ψ ∼ e−iωte±ik+r. The physically consistent boundary condition at the event horizon is then an in- going boundary condition, which agrees with the intuition that black holes do not allow objects to escape from their event horizon. However, the choice of reference frame when imposing this condition in the case of a Kerr black hole is subtle. The argument from [32] is that the frame should be not with respect to the Boyer- Lindquist frame, but rather with respect to in-going timelike observers. In this frame, the observers are co-rotating with the black hole at the event hoizon with
−i(ω−mΩH )t angular velocity ΩH , so their time dependence is seen as e , making the overall time and radial dependence ψ ∼ e−i(w−mΩH )te±ikr. Then for an in-going condition, we need the sign of the t and r exponent to be the same so that ψ ∼ f(r + vt). In the superradiant regime 0 < ω < mΩH , the proper choice of sign is e+ikr, so in the Boyer-Lindquist frame ψ ∼ e−iωteikr, the wave is observed to be out-going. In this way, we recover the superradiant condition by imposing physically consistent boundary conditions at the event horizon. In order to generalize this result to our non-linear wave equation, we can ’undo’ the separation in r and t made by the anzats for the time dependence ψ ∼ e−iωt =
−imΩt e by substituting mΩ with the operator i∂t in eq (4.13). Upon this switch, we find that, at the event horizon,
2 2 2 i(r+ + a )∂t − am WH ≡ W (r+) = 2 (4.16) r+
2 2 where W (r+) is now an operator. Then since ∆(r+) = r+ − 2Mr+ + a = 0, we can
2 2 substitute r+ + a = 2Mr+. Substituting this and using eq (2.14) ΩH = a/2Mr+,
47 we find that 2 2M 2 WH = − [∂t + imΩH ] (4.17) r+
Then returning to eq (4.12) and restoring the time dependence of ψ → ψ(r, t), we find the wave equation for ψ at the boundary
2 2 ∂ ψ 2M 2 ∗2 − [∂t + imΩH ] = 0 (4.18) ∂r r+
We then might like to make this look more explicitly like a wave equation by defining v+ ≡ r+/2M, which gives
2 2 ∂ ψ 1 2 ∗2 − [∂t + imΩH ] = 0 (4.19) ∂r v+ which can furthermore be factored into two first order derivative operators
−1 −1 (∂r∗ + v+ [∂t + imΩH ])(∂r∗ − v+ [∂t + imΩH ])ψ = 0. (4.20)
Then the equation that yields an in-going wave with respect to the co-rotating observer is
−1 (∂r∗ − v+ [∂t + imΩH ])ψ = 0 (4.21) which we use as our left-handed boundary condition. Eq (4.19) is analogous to the
−1 equation (∂r − v ∂t)ψ = 0 that yields a wave traveling towards r → −∞ in flat space, although in our case we have a complex source term. For the purposes of our numerical analysis, we cannot extend the tortoise coor-
∗ ∗ dinate r → −∞. We therefore enforce eq (4.19) at a minimum rmin. To examine what size of radial grid is required for a reasonably accurate approximation, we can look at the integrated tortoise coordinate transformation,
2M 2 − a2 r − r r∗ = r + M ln ∆ + √ ln + (4.22) 2 M 2 − a2 r − r− a plot of which can be seen in Fig (4.1). From this figure, we can see quite clearly
48 Figure 4.1: Tortoise transformation eq (4.20) plotted for M = 1, a = 0.99, and −3 r ∈ [r+ + , Nr] with = 10 and Nr = 10.
∗ that if our tortoise coordinate r lies on the quasi-vertical line at r ≈ r+, then further decreasing r∗ will get us negligibly closer to the event horizon. We therefore can expect accurate event horizon behavior when employing the boundary condition
∗ (4.19) for rmin ≤ −20M.
Spatial Infinity: r∗ → ∞
In terms of a right handed boundary, our physical system extends all the way out to r = ∞. We therefore might think that we do not have a right handed boundary. However, as we mentioned before, our numerical grid cannot extend indefinitely. We
∗ will therfore need to choose some rmax to act as a proxy for spatial infinity. In order to model the behavior at infinity, we can impose an outgoing boundary condition at
∗ rmax to mimic the effect that waves heading out towards infinity do not return to the bulk of the system. Since the Kerr spacetime is asymptotically flat, ψ will satisfy the flat space wave
∗ equation for large rmax to a good approximation. Then to impose an outgoing boundary condition, we use the flat outgoing wave equation
∂ψ ∂ψ + = 0 (4.23) ∂r∗ ∂t
∗ ∗ at the right handed boundary rmax. This equation is exact in the limit rmax → ∞ and for massless particles, since the phase velocity vph = 1. However, even in the case of massive particles, this boundary condition can strongly suppress the effect
49 that reflected waves have on the internal dynamics of the system. To see this, consider a particle that satisfies the flat wave equation with phase velocity v0. Now consider its behavior under a luminal wave equation with phase velocity c 6= v0 with c − v0 = ∆v > 0. We can express the luminal wave equation in terms of v0 and ∆v like ∂ψ 1 ∂ψ ∗ + = 0 (4.24) ∂r v0 + ∆v ∂t and then express the luminal velocity as a power series in ∆v/v0
! ∂ψ 1 ∆v ∆v 2 ∆v 3 ∂ψ ∗ + 1 − + − O = 0 (4.25) ∂r v0 v0 v0 v0 ∂t
The first term in the taylor expansion then vanishes by ψ’s own wave equation. We are then left with a superposition of ingoing and outgoing waves with powers of
∆v/v0 for phase velocities. The solutions that we would be concerned with are the in-going solutions that occur at even powers of ∆v/v0. For ∆v << v0, the phase velocities of these solutions are very small . Then when these solutions are incident on say, a delta potential barrier V ∼ αδ(r − r0), then their transmission coefficient
2 will be suppressed like T ∼ 1/(1+1/vph) (see p. 75 of [33]). We can therefore expect similar suppression for these waves when they reach the potential barrier seen in Fig (2.1).
In the case that ∆v << v0, is not satisfied, we can simply tune the phase velocity c of the boundary condition until it is satisfied. For our simulations, we find that the resulting reflection coefficient for the condition (4.19) is R ≈ 1/16 (see Fig 4.2).
4.2.3 Numerical Methods
Now that we have our wave equation and boundary conditions setup, we are prepared to begin analyzing eq (4.11) by similar numerical methods as in Chapter 3. We begin by parsing out ψ into real and imaginary parts ψ(r, t) → R(r, t) + iI(r, t). Eq (4.11)
50 Figure 4.2: Evolution of |ψ| for the non-superradiant case ω0 = 1.0 and mΩH = 0.4998. Cross-section is taken at r0 = 0M. We see that the waves colliding with the event horizon after being reflected from the right hand boundary are roughly 1/16 the size of the wave that is initially incident on the event horizon. The two large waves correspond to the incident initial wave and and wave that is reflected back out towards infinity. The wave evidently loses a bit of energy at the event horizon, as expected for the non-superradiant case. then gives us the two coupled equations
2 2 2 2 4 − (r + a ) − a ∆ ∂ttR + 4Marm∂tI + r∂r∗ R + r ∂r∗r∗ R (4.26) λ + ∆ − a2 (−m2)R − ∆ ρ2 R2 + I2 − v2 R = 0 2 and
2 2 2 2 4 − (r + a ) − a ∆ ∂ttI − 4Marm∂tR + r∂r∗ I + r ∂r∗r∗ I (4.27) λ + ∆ − a2 (−m2)I − ∆ ρ2 R2 + I2 − v2 I = 0 2
Then using the same notation and the same centered difference operators as in Chapter 3, we arrive at our discrete equations of motion
! Ri+1 − 2Ri + Ri−1 Ii+1 − Ii−1 − (r2 + a2)2 − a2∆ j j + 4Marm ∆t2 2∆t
i i i i i Rj+1 − Rj−1 Rj+1 − 2Rj + Rj−1 (4.28) + r + r4 + ∆ − a2 (−m2)Ri 2∆r∗ ∆r∗2 j
λ − ∆ ρ2 (Ri )2 + (Ii)2 − v2 Ri = 0 2 j j j
51 and
! Ii+1 − 2Ii + Ii−1 Ri+1 − Ri−1 − (r2 + a2)2 − a2∆ j j − 4Marm ∆t2 2∆t
i i i i i Ij+1 − Ij−1 Ij+1 − 2Ij + Ij−1 (4.29) + r + r4 + ∆ − a2 (−m2)Ii 2∆r∗ ∆r∗2 j
λ − ∆ ρ2 (Ri )2 + (Ii)2 − v2 Ii = 0 2 j j j
We then time evolve ψ on a finite grid with parameters r∗ ∈ [−50M, 950M] and
Nr = 4000. Our temporal parameters will depend on the length of the particu- lar simulation, but in terms of the grid spacing ∆t, we find empirically that our simulations are stable with m = 1 and a Courant number C = ∆t/∆r ≤ 3/8.
Convergence Test
In order to ensure that our code is convergent with the expected 2nd order accuracy, we employ the same convergence test as in Chapter 3. We find similar results for the Higgs-Kerr system, namely that the convergence factor C(t) is centered about C = 4 with some short-lived variation. Again, it would be instructive to examine what is causing these variations, which we leave for future study. Eq (4.19) is significantly less forgiving than eq (3.4) with regards to tuning the parameters in order to test a particular solution. We therefore reserve the testing of the numerical solutions themselves to the testing of the energy conservation law of the system.
4.2.4 Energy Conservation
Since the equation of motion for our system was dervied from the action eq (4.7), we will have an energy conservation law
µν ∂µT = 0 (4.30)
52 Figure 4.3: Plot of the convergence factor C(t) for r0 = 20M. We see that C(t) ∼ 4 with the exception of short-lived variations, as expected for a scheme that is second order accurate in time. for the stress-energy tensor
−2 δS T =√ µν −g δgµν (4.31) 1 =∇ ψ∇ ψ∗ − g gρσ∇ ψ∇ ψ∗ − g V (|ψ|) µ ν 2 µν ρ σ µν
In the Kerr geometry, this is a fairly cumbersome object to calculate, particularly in the upper indices. Alternatively, for wave equations of the form,
−A(r)ψtt + iB(r)ψt + D(r)ψr∗r∗ + C(r)ψr∗ − U(|ψ|)ψ = 0 (4.32) the conserved quantity can be derived from the anzats
1 ∗ ∗ ρ = F (r)[A(r)ψ ψ + D(r)ψ ∗ ψ ∗ + V (|ψ|)] (4.33) num 2 t t r r
where F and V are to be determined. Taking the time derivative of ρnum, we find that
∂ρnum 1 ∗ ∗ ∂V ∗ ∗ ∗ ∂V = F Aψ ψ + Dψ ∗ ψ ∗ + ψ + Aψ ψ + Dψ ∗ ψ ∗ + ψ ∂t 2 t tt r r t ∂ψ∗ t t tt r r t ∂ψ t (4.34)
53 Then to invoke the equation of motion eq (4.32), we add a total derivative term and subtract off the appropriate piece of the product rule,
∂ρnum 1 ∗ ∗ ∂V ∗ ∗ ∗ ∂V = F Aψ ψ + Dψ ∗ ψ ∗ + ψ + Aψ ψ + Dψ ∗ ψ ∗ + ψ ∂t 2 t tt r r t ∂ψ∗ t t tt r r t ∂ψ t (4.35) dF dD ∗ ∗ 1 ∂ ∗ ∗ − D + F (ψ ∗ ψ + ψ ∗ ψ ) + [FD (ψ ∗ ψ + ψ ∗ ψ )] dr dr r t r t 2 ∂r∗ r t r t
∗ and then add and subtract two copies of iBψt ψt to get
∂ρ 1 ∗ 1 dF dD ∂V = F ψ Aψ − iBψ − Dψ ∗ ∗ − D + F ψ ∗ + + c.c ∂t 2 t tt t r r F dr dr r ∂ψ∗
1 ∂ ∗ ∗ + [FD (ψ ∗ ψ + ψ ∗ ψ )] 2 ∂r∗ r t r t (4.36) where c.c is the complex conjugate of the first term. We then find that the first two terms vanish by eq (4.32) provided that U = ∂V/∂ψ∗ and that the ODE
1 dF dD D + F = C(r) (4.37) F dr∗ dr∗ is satisfied. Then for D = r4 and C = r∆, we find that,
1 F = (4.38) r3 so our numerical conservation law is
∂ρnum 1 ∂ ∗ ∗ = [r (ψ ∗ ψ + ψ ∗ ψ )] (4.39) ∂t 2 ∂r∗ r t r t and we see that the rate of change of energy
Z Z dE ∂ρ 2 1 ∂ ∗ ∗ ∗ = d x = 2π (ψ ∗ ψ + ψ ∗ ψ ) dr (4.40) dt ∂t 2 ∂r∗ r t r t
54 can be entirely accounted for by the boundary term
∗ dE ∗ ∗ rmax = π [(ψr∗ ψt + ψr∗ ψt )] |r∗ (4.41) dt min
It is important to note that this ρnum quantity need not represent the physical energy density of the system. For now it is just a particular conserved quantity that is useful for checking our numerical code. For the physical conserved energy of the system, we need to use eq (4.31). Luckily, this quantity was computed in [34] in
∗0 2 2 µν (3 + 1)D with the tortoise coordinate dr /dr = (r + a )/∆. Since gµν and g
θθ in Boyer-Lindquist coordinates are the same apart from dropping the gθθ and g terms, we can find the density and flux at the boundary using dr∗/dr∗0 = r2/(r2 +a2) and multiplying by a factor r/ρ2 to substitute in the (2 + 1)D Jacobian,
1 ∗ ∆ tt ∗ φφ 2 ∗ 2 2 2 ρ = rψ ∗ ψ ∗ + −g ψ ψ + g (m )ψψ + |ψ| − v |ψ| (4.42) sys 2 r r r t t and
∂ρsys 1 ∂ ∗ ∗ = [r (ψ ∗ ψ + ψ ∗ ψ )] (4.43) ∂t 2 ∂r∗ r t r t
But these are exactly the quantities derived for ρnum, so ρnum is indeed the physical energy density of the system. We then proceed to check our code by integrating E(t) = R ρ(r, t)d2x and com- puting dE/dt with a backward difference method dE/dt ≈ (E[i] − E[i − 1])/dt and comparing it to the flux at the boundary eq (4.41). We test four cases in the massless linear regime, namely m = 0 and m = 1 in two radial domains: one r∗ ∈ [10M, 70M] with no overlap with the ergoregion and one r∗ ∈ [−50M, 10M] with significant overlap with the ergoregion. For our initial data, we use an ingoing
2 2 ∗ −(r−p0) /2σ −iω(r −p0) gaussian wavepacket in the superradiant regime ψ0 = Ae e with
ω = 0.5mΩH /M, A = 1, p0 = Lr/2, and σ = Lr/4. We find that for m = 0, dE/dt as calculated by the flux at the boundary and by integrating the interior density over the radial domain are in good agreement. This would seem to contradict the conclusion made by [30] that instabilities in Boyer-Lindquist evolutions are caused
55 dE ∗ Figure 4.4: dt plotted for m = 0 outside of the ergoregion with rmin = 10M and ∗ rmax = 70M. (Left) Plot is with respect to flux at the boundary eq (4.41) and (right) plot is with respect to integrating the density (4.33) over the spatial domain and using a backward difference method for the time derivative. The plots are in good agreement, as expected.
dE ∗ Figure 4.5: dt plotted for m = 0 inside the ergoregion with rmin = −50M and ∗ rmax = 10M. (Left) Plot is with respect to flux at the boundary eq (4.41) and (right) plot is with respect to integrating the density (4.33) over the spatial domain and using a backward difference method for the time derivative. The plots are in good agreement, although with a discrepancy in the amplitude of oscillations. We dE ascribe this discrepancy to the backward difference method used in calculating dt for the bulk density. by the ergoregion itself. For m = 1, we find that the two methods are in decent agreement outside of the ergoregion, but the two quantities start to diverge within the ergoregion. These results give credence to the conclusion that we mentioned at the start of Section 4.2, namely that instabilities are supplied by the change in sign of the (∆ − a2)(−m2)ψ term at the ergoregion boundary. We will substantiate this claim further in Section 4.2.5.
4.2.5 Spontaneous Symmetry Breaking Mechanism
Our test in the previous subsection showed that evolutions in Boyer-Lindquist co- ordinates are accurate in the ergoregion for m = 0. For m = 1, we saw that the
56 dE ∗ Figure 4.6: dt plotted for m = 1 outside of the ergoregion with rmin = 10M and ∗ rmax = 70M. (Left) Plot is with respect to flux at the boundary eq (4.41) and (right) plot is with respect to integrating the density (4.40) over the spatial domain and using a backward difference method for the time derivative. The plots are in decent agreement. We again ascribe the discrepancy to the backward difference method in dE calculating dt for the bulk density.
dE ∗ Figure 4.7: dt plotted for m = 1 inside the ergoregion with rmin = −50M and ∗ rmax = 10M. (Left) Plot is with respect to flux at the boundary eq (4.41) and (right) plot is with respect to integrating the density (4.40) over the spatial domain and using a backward difference method for the time derivative. We see that in this case, the two methods diverge due to a growing mode present in the interior density. The fact that the tail end of the boundary flux begins to resemble the growing interior modes is likely caused by the growing modes coming into contact with the boundary.
57 energy in the interior of the system began to grow faster than the energy entering the system from the boundary, thereby violating the conservation law (4.41). As it turns out, we can investigate this discrepancy further by examining the black hole bomb system, or by setting reflecting boundary conditions ψ = 0 at the
λ 2 right-hand boundary. In the nonlinear regime, the term 2 |ψ| ψ has the ability to quell the growth of truncation errors by acquiring a radially dependent expectation value. To see this, we can manually insert the (∆ − a2)(−m2)ψ term into the
2 ∂V potential term ∆ρ ∂ψ∗ like
∂V λ∆ρ2 2 (a2 − ∆) m2 ∆ρ2 → |ψ|2 − v2 + ψ (4.44) ∂ψ∗ 2 λ ∆ρ2
Then we see that for a2 −∆ > 0, the (∆−a2)(−m2)ψ term will raise the expectation value of ψ. In particular, we see that we can have a non-zero VEV even for v2 = 0. In this way, the growing truncation errors cause ψ to acquire an expectation value in the ergoregion. For v2 = 0, this expectation value is
s 2 (a2 − ∆) m2 vev(r) = (4.45) λ ∆ρ2
We show this numerically in Fig (4.8). For this test, we run our code as a black hole bomb system in the non-linear regime for λ = 1 and v2 = 0 and with boundary
∗ ∗ rmin = −50M and rmax = 10M. We then plot the quantity (4.45) and a slice of constant r∗ of |ψ|. The two plots are in near perfect agreement apart from the
∗ ∗ interference of the left-handed boundary condition as r → rmin. This result is interesting for two reasons. The first is that it provides convincing evidence that the growth in the ergoregion in Boyer-Lindquist evolutions is due to the (∆−a2)(−m2)ψ term. The second is that if the growth of truncation errors can be emulated by a physical energy source, then this system constitutes a mechanism of symmetry breaking that can be studied in the laboratory with use of superradiant analog systems, such as the Zel’dovich cylinder discussed in Chapter 2 [14].
58 Figure 4.8: Black Hole Bomb system in the nonlinear regime with λ = 1 and v2 = 0. (Top left) cross-section of constant r∗ = −30M shows that the field begins to reach an equilibrium at about t = 150M. (Top right) cross-section of constant t = 150M shows the shape of this equilibrium, which is a good match to the expected vev(r) as seen in its own plot (bottom).
4.2.6 Growth in the Massive Linear Regime
To test the growth of massive fields that provide their own confinement mechanism, we set our right-hand boundary condition to the outgoing condition (4.23). Then as we did in Chapter 3, we add a parameter q = 0 to turn off the nonlinear term and set λ = 1 and v2 = −0.4/M, which is approximately the value Mµ ∼ 0.42 for which the maximal growth rate was found in [25]. We then use the same initial data as in subsection 4.2.4 with m = 1 and ω = 0.25/M. The results for the massive evolution can be seen in Fig (4.69). We can see an e-folding time of ∼ 104M, similar to that which was recorded in [29]. Quantitatively, using the same method as for eq (3.23), we find the growth rate to be Γ = 0.00011M, which gives an e-folding time of τe = 9090.9M. This e-folding time is off of the result in [29] by a factor of 5 or so, and the cause of this is likely that our code has more truncation error than the Lax-Wendroff method used in [29].
59 Figure 4.9: Evolution of the massive field in the linear regime with outgoing condi- tions at the right-hand boundary. Field parameters are λ = 1, q = 0, v2 = −0.4, m = 1, and ω = 0.25/M. The field grows with a growth rate of Γ = 0.00011. (Left) The simulation is run up to t = 20000M. (Right) The same simulation is run up to t = 40000M.
4.2.7 Higgs Evolution
Now we assess the onset of superradiance for the Higgs field in the Boyer-Lindquist system. We then proceed by turning the non-linearity back on with q = 1 and setting λ = 10−2, m = 1, ω = 0.25/M, and v2 = 1. The results of this simulation can be seen in Fig (4.10). Our results show that the Higgs field does not grow without bound, but rather reaches an equilibrium state that oscillates about its expectation value |ψ| = |v| = 1. From our results, we see that the evolution is split up into two sets of behavior: The first is the dynamical period for which the initial data is inbound towards the event horizon (t = 1000M), and the second is when the majority of the field is oscillating about |v| with little activity at the event horizon (t = 10000M). It then seems that Higgs superradiance does not persist in the Kerr geometry, at least in the Boyer-Lindquist system with truncation error. This could be due to several reasons, some which could be that the non-linearity inhibits waves from remaining within the superradiant regime. A follow up study that would be inter- esting would be to set the coupling λ within the perturbative regime and to use perturbation theory similar to eq (4.3) to pinpoint exactly how and why this might occur. A similar process might occur as mediated by the growing numerical error as well. In order to test this, we would need to use a new coordinate system, since error is absent for m = 0, but superradiance is also prohibited for this value of m.
60 Figure 4.10: Black Hole Bomb system in the nonlinear regime with λ = 1 and v2 = 0. (Top left) cross-section of constant r∗ = −30M shows that the field begins to reach an equilibrium at about t = 150M. (Top right) cross-section of constant t = 150M shows the shape of this equilibrium, which is a good match to the expected vev(r) as seen in its own plot (bottom).
4.3 φ˜ Coordinates
So we see that for our Higgs-Kerr system with truncation error, superradiance of the Higgs field does not persist as it did in Chapter 3. During our analysis, we saw that, for a more accurate study of the Higgs-Kerr system, we might like a coordinate system whose Klein-Gordon equation does not have an ’anti-mass term’ like the (∆ − a2)(−m2)ψ term within the ergoregion for Boyer-Lindquist coordinates. One such coordinate system is that which uses the modified azimuthal coordinate
a dφ˜ = dφ + dr (4.46) ∆
61 used by [35]. In this coordinate system, the inverse metric, which we find by sub-
µν stituting eq (4.46) into gµν and computing g using Sage Manifolds, is
∂ 2 −1 (r2 + sin θ2a2)2 ∂ 2 1 2Mra ∂2 ∂2 = − a2 sin θ2 − + ∂s ρ2 ∆ ∂t ρ2 ∆ ∂φ∂t ∂t∂φ
∆ ∂2 a ∂2 ∂2 1 ∂2 1 ∂2 + + + + + ρ2 ∂r2 ρ2 ∂φ∂r ∂r∂φ ρ2 ∂θ2 ρ2 sin θ2 ∂φ2 (4.47) for which we can find the equation of motion in (3 + 1)D using eq (4.10)
2 2 2 2 4 − (r + a ) − a ∆ ∂ttψ − 4Mar(im)∂tψ + 2r∆∂r∗ ψ + r ∂r∗r∗ ψ 2 2 ∆(−m ) λ 2 2 2 + 2a(im)r ∂ ∗ ψ + ψ + ∆∂ ψ + cot θ∆∂ ψ − ∆ ρ |ψ| − v ψ = 0 r sin θ2 θθ θ 2 (4.48) and we see that the (−m2)ψ term no longer changes sign inside the ergoregion— or anywhere for that matter. Its contribution to the evolution of ψ and truncation errors is always inertial.
4.3.1 Comments on (2 + 1)D Evolutions
We then might like to proceed as we did in Boyer-Lindquist coordinates, namely by simplifying the computational demands of our problem by looking at the system in (2 + 1)D. Doing so in Boyer-Lindquist was not too problematic — the only modi- √ fication was that, when using the (2 + 1)D Jacobian −g = r, the terms resulting
grr from the ∂r r ψ term in eq (4.10) were slightly more algebraically complicated. grν In general, we see that we get extra product rule terms whenever ∂r r ) 6= 0. In the case of φ˜ coordinates, we get an additional term from the grφ component that looks like −a∆ −a∆(im) ψ = ∂ ψ = ψ (4.49) term r φ r
This term is quite similar to the imbφ term that caused ψ to grow in Chapter 3, regardless of the sign of m. Then using this coordinate system in (2 + 1)D, we
62 Figure 4.11: Plots using the φ˜ coordinates in (2 + 1)D for the massive linear and nonlinear cases with outgoing boundary conditions. For the linear case, we find the growth rate to be Γ = 9.367 × 10−5. encounter the same problem as we did when using Boyer-Lindquist coordinates, namely that the conservation law eq (4.41) is satisfied only for m = 0. However, we find that evolutions in these coordinates are stable for m up to m = 7, and perhaps for larger m as well. This is a substantial improvement to the case of Boyer-Lindquist coordinates, where the evolutions are dominated by the (∆ − a2)(−m2)ψ modes for m ≥ 2. Plots for the massive linear and nonlinear case with outgoing conditions can be seen in Fig (4.11) for m = 3. From the plots, we find that for the massive linear case with outgoing conditions that the field takes on a growth rate Γ = 9.367 × 10−5, which is about the same as that which we found in Boyer-Lindquist coordinates, although for a higher value of m. This demonstrates that the term (4.49) is less sensitive than the (∆ − a2)(−m2)ψ term in terms of error propagation, which we should expect since (4.49) is only linear in m. For the nonlinear case, we find a similar result as before, namely that the Higgs field experiences a growth phase and then heads towards equilibrium at its expectation value v2 = 1.
4.3.2 Evolution in (3 + 1)D
Although the φ˜ coordinates in (3+1)D avoid obvious growth contributing terms, we still find that the conservation law (4.34) is not quite satisfied in these coordinates. However, the results that we find make good sense with respect to what we have seen previously, namely that the nonlinear Higgs field experiences a growth phase,
63 Figure 4.12: Plots using the φ˜ coordinates in (3 + 1)D for the nonlinear cases with outgoing boundary conditions. We see that the Higgs field reaches an equilibrium in t ∼ 800M. and then reaches an equilibrium about its vev (see Fig (4.12)). In (3 + 1)D, this occurs in t = 800M, as opposed to t = 3000M in the (2 + 1)D case. We reason that this is because the (3 + 1)D code does not have explicit growth terms in ∼ m or ∼ m2, so the Higgs field reaches equilibrium more quickly.
4.4 Conclusions
Our results from this section seem to indicate that unbounded superradiant growth of the Higgs field is unlikely to persist in the Kerr spacetime, since it does not for our simulations which have ’extra energy’ due to truncation error. Furthermore, our simulations using the φ˜ coordinates indicate that the Higgs field reaches an equilibrium more quickly for coordinate systems with less truncation energy. This conclusion is not however definitive, since it is entirely possible that the growing truncation errors and the non linearity interact with the physical waves in such a way as to inhibit the superradiant process– either by knocking waves out of the superradiant regime or by some other mechanism. These results should therefore be checked with a numerical implementation where the conservation law (4.41) is satisfied.
64 Chapter 5
Discussion: Observational Signatures
5.1 Introduction
In this chapter, we briefly discuss possible particle physics signatures that might be afforded by the Higgs evolutions examined in Chapters 3 and 4. We namely assess two cases, the first from Chapter 3 where the Higgs field grows without bound, and the second from Chapter 4 where the Higgs field oscillates about its expectation value. But first, we detail what size of black hole we can expect to play a role in these superradiant Higgs systems.
5.1.1 Assessing Black Hole Candidates
We expect the largest contribution to superradiant growth to occur for Mµ ∼ 1 in natural units, at least at the beginning of the growth phase where |ψ| is small (see p. 35). In SI units, since the LHS has dimension [M]2, we need to compare
2 38 2 −16 2 the quantity Mµ to Mpl ∼ 10 (GeV) = 10 (kg) . For a Higgs particle, we
−25 have mH ≈ 125 GeV = 2.23 × 10 kg. Therefore, Higgs-Kerr systems involving black holes of mass ∼ 109 kg will yield the largest growth contributions to the Higgs field. For stellar-mass black holes resulting from gravitational collapse, this is very
65 30 small. A typical stellar-mass black hole is at least one solar mass M = 2 × 10 kg, if not thousands or even millions of solar masses. Despite this fact, one might still wonder in principle if, at the very low end of the spectrum, whether there could be stellar-mass black holes around the 109 kg mark. However, the Tolman- Oppenheimer-Volkoff limit details that the end state of the gravitational collapse of stars under about one solar mass will not be a black hole, so it is unlikely that stellar-mass black holes will provide a reasonable superradiant growth rate. It is however still the case that rotational superradiance does not require an event horizon— the only requirement is the presence of an ergoregion. Ergoregions are still present around the metrics of rotating stars, so one might also wonder whether a star could instead supply the growth to the Higgs field. Unfortunately, the smallest observed red dwarfs are around 0.075M , which is still 3 orders of magnitude off from the optimal mass. Furthermore, stars of this size are generally non-relativistic, and therefore will not have the ergoregions necessary for superradiance. Therefore, Higgs superradiance as mediated by astrophysical stars is also unlikely. We then might like to circumvent this ∼ 109 kg limit by acknowledging that even for Mµ >> 1, we still have a growth rate according to Γ ∼ (10−7/M) exp −1.84Mµ [23]. However, for stellar mass black holes, this gives an e-folding time in SI units
37 of about τe ∼ 10 years, which is not within the dynamic range of the universe ∼ 1010 years. By contrast, for stellar-mass black holes coupled to scalar fields such that Mµ ∼ 0.4, the e-folding time is τe = 16.5 seconds. This would however require a scalar field with a much smaller mass than the Higgs particle, such as the axion [10, 11, 26, 36]. For this reason, superradiant axion systems are a popular candidate for sourcing, at least partially, the Cold Dark Matter (CDM) in our universe that we expect from gravitational observations.
Primordial Black Holes
So it seems that Higgs-Kerr systems are unlikely to be sourced by stellar-mass black holes, since they fall well above the ∼ 109 kg mass threshold required for a reasonable
66 growth rate. The door is however quite open for primordial black holes (PMBHs) to mediate the process. PMBHs are a type of black hole that are believed to have formed in the early universe due to fluctuations in the energy density. During the inflationary phase of the early universe, the universe expands in a manner that is well modelled by the Friedman-Robert-Walker metric [37]
2 2 2 i j ds = −dt + a(t) δijdx dx (5.1)
Then as the scale factor a(t) increases with time, the volume of the 3-geometry of the universe expands. Such inflationary spacetimes result from a sufficient vacuum energy density ρvac. In some of the more standard inflationary models, this vacuum energy density is typically modeled by a homogeneous scalar field V (φ) called the inflaton field. Since φ is a scalar field, in the quantum mechanical picture, the potential V (φ) is subject to quantum corrections, which roughly translate to the loop diagrams that the V term in the system’s Lagrangian would contribute to amplitudes in quantum field theory (for details, see [38]). These fluctuations in V (φ) result in density perturbations in the stress-energy tensor Tµν of the spacetime (5.1). If these fluctuations are sufficiently large and satisfy the Strong Energy Condition (loosely speaking, they are attractive), then these perturbations can cause the local spacetime to collapse in opposition to the inflating spacetime, forming PMBHs. Since PMBHs do not form by a typical gravitational collapse, they circumvent the
TOV condition that would require MBH > M . We can therefore have primordial
9 black holes well below the solar mass scale, including those such that MBH ∼ 10 kg, making them an excellent candidate for sourcing the Higgs-Kerr system.
67 Hawking Evaporation
When considering PMBHs as superradiant candidates, we also need to take into account their Hawking Evaporation time,
3 MBH 66 τhawk ∼ × 10 years (5.2) M after which the black hole will have completely evaporated [39]. For superradiant
9 candidates with MBH ∼ 10 kg, the Hawking evaporation time is τhawk = 1000 years, so these PMBHs would have evaporated by now. However, at some point during their evaporation, more massive PMBHs will hit the 109 kg mark. We then expect our Higgs-Kerr system to be possible towards the ends of the lifetimes of evaporating PMBHs. Since Kerr black holes are necessarily rotating, it should be mentioned that PMBHs that are formed by the fluctuation process mentioned previously are irrota- tional for the most part, since they form from circumstances that are homogeneous and isotropic. However, the PMBHs can acquire rotational energy via collisions and accretion.
5.2 Particle Physics Signatures
Now that we are aware of what types of black holes might source the Higgs-Kerr system, we are prepared to discuss the particle physics that would ensue in these systems and to what degree they might serve as an alternative to terrestrial particle detectors. Thus far, all of our analysis has been for classical fields. To discuss the particle physics of superradiant Higgs systems, we should direct some of our attention towards the quantum mechanical picture.
68 5.2.1 Classical Field Approximation
Upon being introduced to quantum mechanics, undergraduates are told that one of the necessary steps to transition from classical mechanics to the quantum picture is to generalize the observables of a system to operators x → xˆ acting upon a Hilbert space of wave functions ψ ∈ H. In a quantum field theory, we similarly generalize the field to a field operator ψ → ψˆ. We then typically express ψˆ in momentum space as a linear combination of creation and annihilation operators
Z ˆ 4 n −ikx † ikxo ψ = d k aˆke +a ˆke (5.3)
In this way, the annihilation operatora ˆk is analagous to the fourier transform a(k) of the scalar field ψ = R a(k)e−ikxd4k. Furthermore, in this representation, the Hilbert space is the Fock Space |k1, ..., kni ∈ F, which is a set of states of particles with definite momentum ki. Similar to the harmonic oscillator in NRQM, we can define ˆ R † 3 a number operator N = aˆkaˆkd k in some frame whose eigenvectors are states with ˆ definite particle number N |k1, ..., kni = n |k1, ..., kni. To gain insight into the quantum mechanical picture from our classical results, we can undergo the reverse generalizationa ˆk → a(k) [36, 40]. Upon this switch, the number operator is now N ∼ R |a(k)|2d3k. Then by Parseval’s theorem, this is just
Z N ∼ |ψ(x)|2d3x (5.4)
Therefore, in the classical field approximation, we see that unbounded growth in the classical field ψ corresponds to the unbounded creation of Higgs particles within our Higgs-Kerr system. This has direct and interesting consequences to the particle physics that would ensue in the system. Specifically, in a system with a plethora of Higgs bosons, we would expect plentiful production of its most favored decay channels, which are Z and W ± bosons. Once sufficient amounts of the electro-weak bosons are produced, we would then expect to start to see their most populated decay channels, namely hadrons and leptons. The radiation of such products would be an
69 indication that a superradiant Higgs-Kerr system is present and hosting interesting electro-weak physics.
5.2.2 Higgsenova
In Chapter 4, our results indicated that the Higgs field may oscillate about its ex- pectation value in the Kerr geometry, a process we call a ’Higgsenova’. The ensuing particle physics of this system will still operate according to eq (5.4), although with a substantially lower boson density. To compare these two cases quantitatively, consider in both cases that the Higgs field ψ = const, but that ψ ∼ 1000 for the un- bounded case and ψ ∼ 1 for the Higgsenova. Then for a grid of radial width 1000M
2 6 in (2+1)D, the respective occupation numbers come out to Nunb = 1000 M = 10 M and Nhnv = 1000M. This discrepancy is further exacerbated in the (3 + 1) picture. It should be noted that, for lower occupation numbers, the classical field approx- imation becomes less accurate. A more careful analysis would be required to assess the extent to which eq (5.4) applies for low occupation numbers. We reserve this analysis for future studies
5.3 Conclusion: Comparison with the LHC
To compare these results to the LHC, for a single collision within an estimated volume of 1000 m3 and assuming 2 particles per collision, this translates to |ψ|2 ∼ 2 × 10−3. Then in terms of Higgs populations, both the unbounded growth and the Higgsenova systems are capable of matching the quantities at the LHC. In terms of the pure energy scale, both results are capable of matching the LHC as well. However, it should be noted that we would need |ψ| in the Higgs-Kerr system to far exceed the particle density at the LHC, since in Chapter 2 we saw that superradiant populations are non-relativistic. This point requires closer consideration, which we leave for future studies. This should still be feasible at least in the case of unbounded growth, since nearly all of the black holes energy 109 kg ∼ 1033 TeV will be extracted.
70 Furthermore, in consideration of Extra-dimensions where Mpl ∼ 1 TeV, if black hole production occurs at the LHC, then we might expect the Higgs-Kerr system to be present towards the end of the lifetime of these black holes as they evaporate.
71 Chapter 6
Conclusion
The results from our study of superradiant Higgs systems are two-fold. First, for systems for which the exponential growth is explicitly emulated in the Klein-Gordon equation, superradiance of the Higgs field persists in the nonlinear regime. Second, the Higgs field in the Kerr geometry reaches an equilibrium characterized by oscil- lations about its VEV for our numerical implementations in Boyer-Lindquist and φ˜ coordinates. Both of these results are capable of matching the Higgs populations at the LHC, though unbounded growth is more likely to clear the energy threshold for BYSM physics. This study points towards a couple of directions of interest for elaboration. The obvious choice is to confirm these results for a numerical implementation for which the conservation law (4.41) is satisfied to a higher degree of accuracy. Another avenue would be to better understand the mathematics and the mechanism through which the nonlinear potential may eclipse the superradiant effect, whether it is by quelling individual modes before they are accumulated or by knocking waves out of the superradiant regime. A better understanding of this process may point to a way to circumvent the eclipse of superradiance, whether it is by a clever choice of parameters, or by making more explicit changes to the setup of the Higgs-Kerr system. As an auxiliary result, we demonstrated a scenario in which a |ψ|4 theory with a zero VEV acquires a radially-dependent expectation value in the ergoregion caused
72 by growth mediated by the (∆−a2)(−m2)ψ term. It would be interesting to explore whether this growth can be recreated by a physical source with the intent to study this symmetry breaking for superradiant analog systems in the laboratory.
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