{"id":"ab6face6-6a37-4492-abe0-e484c7075f17","arxiv_id":"hep-th/0106179","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Spontaneous symmetry breaking from a tachyonic (negative mass squared) potential completes within one oscillation of the field distribution, because the instability converts potential energy into colliding classical waves almost immediately.","lead":"Lattice simulations show that when a scalar field rolls off the top of a Mexican-hat potential, space breaks into domains of the two vacua within a single oscillation rather than after many. The result matters for how the early universe reheats after inflation and for which approximation schemes can be trusted.","discovery_kind":"extension","skeptic_critique":{"model":"selfhosted/deepseek-v4.1-flash","headline":"The reader's quench-rate concern is real but the paper's own logarithmic-dependence argument (t* ~ (1/2m) ln(C/λ), §II) substantially mitigates it; a gradual-quench run would settle whether it lands.","rationale":"The paper's central claim is supported by three independent lines of evidence: the linear growth estimate (Eq. 5), the PDFs (Fig. 3, 7), and the spatial slices (Fig. 4, 9). The simulations are reproducible (public LATTICEEASY code, per-figure parameters in Appendix A.3). The reader's quench concern is the most natural one, but the paper explicitly argues that t* and l* depend only logarithmically on the seed amplitude, and Fig. 6 checks a displacement larger than the fluctuation level. The remaining gap — quench-rate independence — is a robustness check rather than a refutation. The paper's scope is explicitly zero-temperature, flat-spacetime, which is an acknowledged limitation, not a hidden assumption. The comparison with [3] is argued rather than re-run, but that is a secondary claim. Overall, the central claim holds up; the verdict should remain ACCEPT.","tokens_in":28747,"tokens_out":8915,"duration_ms":158056,"concrete_test":"Re-run the quadratic-model simulation (Fig. 3/4 parameters: N=256^3, L=100, λ=10^-4) with a gradual quench, e.g. m^2(t) = m^2 tanh(t/τ) with τ = 1/m, initializing from the massive vacuum spectrum <|φ_k|^2> = 1/(2√(k^2+m^2)) at t=0. If the PDF still shows the two maxima never returning to φ=0 within one oscillation and the domain size remains within a factor of ~2 of the instantaneous-quench case, the concern does not land; if the field returns to φ=0 or the domain size changes qualitatively, the single-oscillation claim is weakened.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim — that SSB completes within a single oscillation of the field distribution — rests on simulations with an instantaneous quench: at t=0 the negative mass squared is switched on and the field is drawn from the massless vacuum spectrum <|φ_k|^2> = 1/(2k) with random phases and Gaussian amplitudes (Appendix A.1, Eq. A2). The reader correctly identifies this as the weakest assumption. However, the paper explicitly argues (§II, immediately after Eq. 7) that t* depends only logarithmically on the initial fluctuation amplitude, and the domain size l* ~ m^-1 ln^{1/2}(C/λ) is likewise logarithmic. Fig. 6 also checks a homogeneous displacement φ0 = 0.1v and finds completion in two oscillations. So the seed amplitude is not the load-bearing quantity. What is not checked is the shape of the seed spectrum at k < m: a gradual quench (mass ramping over τ ~ 1/m) would populate k < m differently and could in principle change the domain structure and the number of oscillations. The paper's logarithmic argument suggests this is a small effect, but it is not demonstrated. This is the one place where the central claim could in principle be weakened.","agreement_with_reader":"partial"},"referee_report":{"model":"selfhosted/deepseek-v4.1-flash","summary":"This paper presents a combined analytic and lattice study of spontaneous symmetry breaking (SSB) driven by a tachyonic (negative) mass squared at the top of the effective potential. The central claim, stated in the abstract [abstract:s3] and developed in §II–§VI, is that SSB typically completes within a single oscillation of the field distribution, because the tachyonic instability rapidly converts most of V(0) into gradient energy of colliding classical waves and topological defects, so that the long stage of homogeneous oscillations assumed in earlier treatments does not exist. The paper further claims [abstract:s4, abstract:s5] that the dynamics depend qualitatively on the shape of V near its maximum: for V ~ -m^2 phi^2/2 the process is pure tachyonic instability, whereas for -lambda phi^n with n > 2 one encounters a combination of tunneling, tachyonic growth, and bubble-wall collisions. The evidence is a set of 3D lattice simulations (LATTICEEASY, hep-ph/0011159) with all run parameters listed in Appendix A.3, cross-checked against order-of-magnitude analytic estimates such as t_* ~ (1/2m) ln(C/lambda) and n_k ~ 10^2 lambda^{-1}.","tokens_in":57862,"tokens_out":38314,"duration_ms":626441,"significance":"If correct, the result changes the textbook picture of SSB: the homogeneous-oscillation stage that perturbative and Hartree/1/N treatments assume is absent, and the relevant dynamics is the collision of classical waves and defects generated by the instability. The paper is significant for three reasons. First, it ships a documented, reproducible lattice code (LATTICEEASY) and lists every run's N, L, dt, k_cut, and lambda in Appendix A.3, which is a genuine strength and makes the numerical claims checkable. Second, the analytic estimates are independent of the lattice and agree with it at the order-of-magnitude level, so the central claim is not forced by construction. Third, the paper draws explicit, falsifiable consequences for hybrid inflation, new inflation, brane inflation, and the ekpyrotic/pyrotechnic scenario, which is the kind of concrete prediction that makes the work useful beyond its immediate subject.","major_comments":[{"comment":"The simulations assume an instantaneous quench: at t = 0 the negative mass squared is switched on and the field is drawn from the massless vacuum spectrum <|phi_k|^2> = 1/(2k) with random phases and Gaussian amplitudes. The authors acknowledge this in Appendix A.1 ('we were simulating a quench in which the effective squared mass of the fields is presumed to have rapidly become negative'), and the logarithmic argument t_* ~ (1/2m) ln(C/lambda) in §II shows that the seed amplitude enters only logarithmically. However, that argument constrains the amplitude of the seed, not the shape of the k < m spectrum. A gradual quench with a ramp time tau ~ few/m would populate k < m differently and could in principle change the domain structure and the number of oscillations required, which is precisely the quantity the central claim is about. The paper checks independence from the momentum cutoff but not from the quench rate. A single run with a linear ramp of m^2(t) over tau ~ 3/m, or a quantitative argument that the k < m spectral shape also enters only logarithmically, would close this gap. This is the one place where the central claim could in principle be weakened, and it deserves an explicit response rather than only the acknowledgment in Appendix A.1.","section":"Appendix A.1, Eq. (A2); §II","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The paper attributes the disagreement with Boyanovsky et al. [3] to the fact that their initial displacement was chosen two orders of magnitude below the level of the long-wavelength quantum fluctuations, so that their homogeneous component does not track the SSB. This is a claim about another paper's setup, and it is stated without listing the parameters used in [3] or reproducing their initial condition in a lattice run. Since the breakdown of perturbative and Hartree/1/N methods is one of the paper's substantive claims, it would strengthen the case to state [3]'s parameters explicitly and, if feasible, to run the lattice with an initial displacement matching [3] to show that the difference is the initial condition rather than the method. The paper's own Fig. 3 already supports the breakdown claim independently, so this is a strengthening request rather than a correctness objection.","section":"§V, comparison with Ref. [3]","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The abstract says SSB 'typically completes within a single oscillation,' but Fig. 6 shows that for an initial displacement phi_0 = 0.1v the process takes two oscillations. The word 'typically' covers this, but a sentence in the introduction clarifying that 'single oscillation' refers to phi_0 = 0 or phi_0 below the fluctuation level, with two or more oscillations for larger displacements, would prevent misreading.","section":"Abstract vs. Fig. 6","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"Fig. 5 is a 2D simulation, and the text correctly notes that rescattering in 2D differs from 3D. The domain-growth picture drawn from it is therefore qualitative. A sentence stating explicitly which features of Fig. 5 are expected to survive in 3D and which are 2D artifacts would help the reader.","section":"Fig. 5","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The quartic-model runs use the analytic instanton (Eq. 26) as the initial condition rather than waiting for tunneling, because the tunneling time is exponentially long. This is acknowledged in the text, but the expected tunneling rate (Eq. 25) is not compared with the simulation time to justify the choice quantitatively. A short estimate of the tunneling time for the quoted lambda and phi_0 would make the approximation transparent.","section":"§VII, Figs. 10–11","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The definition of n_k in the tachyonic regime depends on an arbitrary choice between omega_k = |k| and omega_k = sqrt(k^2 + |m^2|), and the paper states that the results are qualitatively similar under either choice. Showing one spectrum computed both ways in an appendix figure would substantiate that statement at negligible cost.","section":"Eq. (6) and Appendix A.1","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The stochastic-tunneling estimate uses an undetermined constant C = O(1), and the agreement with the Euclidean action in Eq. (36) depends on C ≈ 2. The text calls the coincidence 'rather impressive,' which is fair, but a sentence on the sensitivity of the exponent to C, and hence on the reliability of the estimate for the cubic model where no instanton exists, would be useful.","section":"§VIII, Eqs. (35)–(37)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The renormalization procedure subtracts lambda v <delta phi^2> phi using <delta phi^2> from the initial quantum fluctuations only, not the dynamical value. The text explains the distinction, but it would help to state how <delta phi^2> is computed on the lattice (e.g., summed over the initial spectrum up to the cutoff) and to confirm that the counterterm is time-independent by construction.","section":"Appendix A.2","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"minor_revision","confidential_remarks":"This is a well-executed and influential paper that extends the authors' own prior work [1] with a much more detailed lattice study and a broader class of potentials. The numerical methodology is documented and reproducible, and the analytic estimates are appropriately order-of-magnitude. My only substantive request is that the instantaneous-quench assumption, which the authors acknowledge in Appendix A.1, be tested or argued more explicitly, since it is the one input that could in principle affect the 'single oscillation' conclusion. The comparison with Ref. [3] in §V is a secondary point that would benefit from explicit parameters. I do not see a correctness problem with the central claim."},"author_rebuttal":null,"desk_editor":{"model":"selfhosted/deepseek-v4.1-flash","letter":"You probably already know the headline from their PRL (hep-ph/0012142), and the paper says as much — it 'extends the previous work [1].' So read it for what's actually new: the n>2 potentials, and the lattice detail behind the quadratic case. The quartic/Coleman-Weinberg section is the interesting part. There is no barrier, tunneling still happens, and the O(3,1) instanton gives a bubble whose interior oscillates and radiates a train of waves rather than a single thin wall. The cubic case is better still: V''(0)=0 kills the tachyonic argument, no instanton exists, and they fall back on Linde's stochastic-tunneling estimate to get a nucleation rate. That estimate reproduces the Euclidean action for -lambda phi^4 to within a factor of two, which is a nice consistency check, and the simulations then show bubble-like growth and collisions from a symmetric start. The lattice work is solid and unusually reproducible for 2001: LATTICEEASY is public, the equations and initial conditions are stated, and Appendix A.3 lists N, L, dt, k_cut and lambda for every figure. The renormalization discussion in A.2 is honest about the induced linear term in the cubic model, which is the kind of thing people usually bury. On soft spots, all minor. The initial condition is an instantaneous quench with a massless vacuum spectrum; they argue t* and l* depend only logarithmically on the seed amplitude, and Fig. 6 checks a homogeneous displacement, but the shape of the spectrum at k<m is never varied. A gradual-quench run would close that, and I'd guess it changes nothing. The O(1) constants (C~10^2, 'a few times' the rms in the cubic case) are hand-chosen, so the quoted agreement between analytics and simulation is partly arranged — they say so themselves ('this estimate is rather crude'). The disagreement with Boyanovsky et al. [3] is argued rather than demonstrated by re-running their setup, though the argument (global <delta phi^2> is the wrong diagnostic when domains form) is correct. The quartic runs start from the analytic instanton rather than waiting for nucleation, which is a fair shortcut given the exponential suppression, but it does mean the tunneling rate itself is not simulated. None of this touches the load-bearing claim. Worth a serious referee, and worth citing.","headline":"The long version of the tachyonic preheating PRL: the quadratic-model headline is a restatement, but the n>2 potentials, the full 3D lattice spectra, and the public code make it the reference to cite.","tokens_in":29653,"tokens_out":968,"would_cite":true,"duration_ms":18327,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":["98.80.Cq","11.30.Qc","11.27.+d","05.10.-a"],"model":"selfhosted/deepseek-v4.1-flash","headline":"A scalar field falling off the top of its potential breaks the symmetry in a single oscillation, because the tachyonic instability turns the vacuum energy into colliding waves and defects before any coherent rolling can set in.","keywords":["tachyonic preheating","spontaneous symmetry breaking","spinodal instability","lattice field simulation","topological defects","cosmological phase transitions","bubble nucleation","parametric resonance"],"falsifier":"A lattice run of the quadratic model in which the negative mass squared is switched on over a time comparable to $m^{-1}$, or in which the initial spectrum is thermal rather than the massless vacuum form, and which still shows the field distribution returning close to $phi = 0$ for several oscillations before settling near $|phi| = v$, would undercut the single-oscillation claim.","tokens_in":30010,"feed_emoji":"⚛️","tokens_out":3661,"duration_ms":66049,"temperature":0.7,"pith_summary":"The paper asks how a scalar field actually leaves the top of its effective potential when a symmetry breaks, and argues that the textbook picture — a homogeneous field oscillating many times about the minimum at $|phi| = v$ before giving up its energy — does not describe what happens. Because the curvature at $phi = 0$ is negative, every mode with wavelength longer than the inverse tachyonic mass grows exponentially, so the field fragments into domains instead of rolling coherently; the authors follow this through three-dimensional lattice simulations of the classical field equations seeded by quantum vacuum fluctuations. They find that most of the potential energy $V(0)$ is converted into gradient energy of colliding classical waves and topological defects within a single oscillation of the field distribution, so the symmetry is effectively broken almost immediately. The claim matters because it fixes the starting conditions for reheating and defect production in cosmological phase transitions, and because it marks a regime where the usual perturbative bookkeeping — Hartree or $1/N$ truncations built on a universe-averaged $langle delta phi^2 rangle$ — gives a demonstrably wrong answer. The authors also show that the mechanism is not universal in detail: for potentials behaving as $-lambda phi^n$ with $n > 2$ the curvature vanishes at the origin, and the instability must be seeded by tunneling, by rare Gaussian peaks, or by both.","feed_headline":"Symmetry breaking finishes in one oscillation, not many","feed_subtitle":"Lattice runs show the unstable vacuum energy becomes colliding waves and defects almost immediately, not after a long roll.","key_machinery":"Two pieces carry the argument. The first is the tachyonic instability band itself: for $V = -lambda phi^n/2$ with the field rolling from rest, energy conservation gives $dot{phi} = sqrt{lambda (phi^n - phi_0^n)}$, and since long-wavelength fluctuations obey the same equation as $dot{phi}$, they grow as $phi_k propto phi^{n/2}$ — so $phi_k phi = text{const}$ for $n = 2$, $phi_k phi propto phi^{1/2}$ for $n = 3$, and $phi_k phi propto phi$ for $n = 4$. This single rule explains why the instability is so much faster than ordinary parametric resonance and why it becomes more violent as $n$ grows. The second is the lattice method: the quantum modes with large occupation numbers are treated as classical waves, evolved under $ddot{phi} - nabla^2 phi + V'(phi) = 0$ on a grid with a staggered leapfrog step, initialised in momentum space with random phases and Gaussian amplitudes $langle |phi_k|^2 rangle = 1/(2k)$. The paper also makes a conceptual point that carries real weight: because a large part of $langle delta phi^2 rangle$ comes from fluctuations longer than the domain scale, averaging over the whole universe is the wrong operation — one must average locally inside each domain, and no amount of accuracy within the Hartree or $1/N$ approximation repairs that. A renormalization step is introduced for the cubic model, where lattice modes induce a spurious linear term $lambda v langle delta phi^2 rangle phi$ that must be subtracted to restore the intended potential.","core_discovery":"The central claim is that spontaneous symmetry breaking driven by a tachyonic mass term completes within a single oscillation of the field distribution, not over a long stage of homogeneous oscillations. For $V = -m^2 phi^2/2 + lambda phi^4/4$ the growth of modes with $k < m$ proceeds until $sqrt{langle delta phi^2 rangle} sim v$, at which point the universe is already divided into domains of size slightly greater than $m^{-1}$ with $phi sim pm v$; the gradient energy of the resulting domain walls is of the same order as the initial potential energy $V(0) = m^4/(4 lambda)$, so the field cannot coherently return to $phi = 0$. Lattice simulations confirm this: the probability distribution of the field never comes back near the origin and settles at two peaks around $pm v$ within one oscillation, with occupation numbers reaching $n_k sim 10^2 lambda^{-1}$ for $k ll m$, far above the thermal value $n_k sim lambda^{-1/4}$ that full thermalization would eventually require. The authors then show that the dynamics depend qualitatively on the shape of the potential near its maximum. In the Coleman-Weinberg and $-lambda phi^4$ cases the curvature vanishes at the origin, so no mode grows from an infinitesimal seed; decay proceeds by barrier-free tunneling described by the $O(4)$-symmetric instanton family $phi(r) = 2sqrt{2/lambda}, rho/(r^2 + rho^2)$, whose action $S_E = 8 pi^2/(3 lambda)$ is independent of $rho$, followed by bubble growth, oscillation of the field inside the bubble, and bubble wall collisions. In the cubic case $-lambda v phi^3/3 + lambda phi^4/4$ there are no instantons at all, yet the state $phi = 0$ is still unstable: rare long-wavelength Gaussian peaks supply a local negative curvature, and the resulting fragmentation looks like bubble formation even though no Euclidean solution describes it.","pith_inferences":["The single-oscillation result is a statement about the shape of the potential at its maximum, so one could recast it as a classification: any potential with $V''(0) < 0$ breaks fast, and the interesting discriminator is not the coupling strength but whether $V''(0)$ is finite, zero, or reached only through fluctuations.","If domain size at formation is set by $m^{-1} ln^{1/2}(C/lambda)$ and grows only by domain wall migration afterwards, the resulting defect network should carry a memory of the coupling constant in its correlation length, which is in principle a distinguishable signature from networks produced by a thermal quench.","The argument against universe-averaged $langle delta phi^2 rangle$ applies wherever an infrared contribution dominates the variance, so it should also constrain how backreaction of long-wavelength inflationary fluctuations on the expansion rate is computed.","The cubic case suggests a broader class of potentials with vanishing curvature at the origin where the decay rate is controlled by the tail of a Gaussian random field rather than by an instanton, which makes the transition rate sensitive to the initial spectrum in a way the quadratic case is not."],"forward_implications":["Reheating in models with concave potentials is dominated at its first stage by tachyonic preheating rather than by parametric resonance of a homogeneous oscillating inflaton, which changes the initial spectrum of produced particles.","Topological defects — domain walls for a real field, strings for a complex field, and their higher-dimensional analogues — are produced as an integral part of symmetry breaking rather than as a small nonperturbative correction, and they absorb a comparable share of the vacuum energy.","Full thermalization lags far behind symmetry breaking: occupation numbers immediately after the transition are many orders of magnitude above the thermal distribution, so a long intermediate stage of quasi-equilibrium with an effective chemical potential must intervene.","In hybrid-inflation-type models with a cubic tachyonic term, the onset of instability is a stochastic process of rare Gaussian peaks growing into bubble-like configurations, so the timing of the transition is set by tail statistics rather than by a homogeneous roll.","In the quartic and Coleman-Weinberg cases the transition proceeds by tunneling without a barrier, and the interior of each bubble behaves like an open universe whose oscillations send out a train of waves of nearly undiminished amplitude rather than a single thin shell."],"supporting_citations":[{"why":"The earlier letter by the same collaboration that first reported symmetry breaking completing within a single oscillation; this paper is the detailed extension of it.","marker":"[1]"},{"why":"Supplies the standard definition of the occupation number $n_k$ used throughout and the parametric-resonance framework the tachyonic growth is compared against.","marker":"[2]"},{"why":"The Hartree and $1/N$ treatment of the same model whose long-lived homogeneous oscillations the present paper argues against.","marker":"[3]"},{"why":"Establishes that large-occupation quantum states can be simulated as classical waves on a lattice, which is the justification for the whole numerical method.","marker":"[10]"},{"why":"The LATTICEEASY code with which every simulation in the paper was run.","marker":"[11]"},{"why":"Source for tunneling without a potential barrier, the mechanism invoked for the quartic and Coleman-Weinberg potentials.","marker":"[17]"},{"why":"The stochastic approach to tunneling used to estimate bubble formation rates in the cubic model, where no instanton exists.","marker":"[19]"},{"why":"The Lamé-equation stability analysis of parametric resonance that the tachyonic band is compared with and shown to dominate.","marker":"[20]"},{"why":"Provides the family of $O(4)$-symmetric instanton solutions with $rho$-independent action used as initial data for the quartic runs.","marker":"[26]"}],"fun_headline_variants":["Tachyonic preheating completes symmetry breaking in one oscillation","Field distribution settles at two peaks within one oscillation","Domain walls form before the field can return to zero","Potential shape sets whether symmetry breaks by rolling or tunneling","Cubic potential breaks symmetry without a Euclidean instanton"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The simulations switch the negative mass squared on instantly at $t = 0$ and seed the field with the massless vacuum spectrum $langle |phi_k|^2 rangle = 1/(2k)$; a gradual quench, a thermal initial spectrum, or an initial displacement comparable to the fluctuation level would change the seed amplitudes, and hence the growth time and the domain size that the single-oscillation conclusion rests on. The authors check independence from the momentum cutoff but not from the quench rate, and in the quartic case they acknowledge starting from an analytic instanton rather than waiting for a bubble to nucleate.","fun_headline_variants_meta":{"raw":{"variants":["Tachyonic preheating completes symmetry breaking in one oscillation","Field distribution settles at two peaks within one oscillation","Domain walls form before the field can return to zero","Potential shape sets whether symmetry breaks by rolling or tunneling","Cubic potential breaks symmetry without a Euclidean instanton"]},"model":"selfhosted/deepseek-v4.1-flash","effort":"low","attempts":1,"cost_usd":0.0,"raw_usage":{"total_tokens":1945,"prompt_tokens":1208,"completion_tokens":737,"prompt_tokens_details":null,"completion_tokens_details":{"reasoning_tokens":673}},"tokens_in":1208,"tokens_out":737,"duration_ms":12893,"image_pages":[],"temperature":1.0,"reasoning_budget":50,"reasoning_tokens":673,"transport_provider":"selfhosted","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-29T17:49:36.219741+00:00","model_set":{"reader":"selfhosted/deepseek-v4.1-flash"},"falsifier":"A lattice run of the quadratic model in which the negative mass squared is switched on over a time comparable to $m^{-1}$, or in which the initial spectrum is thermal rather than the massless vacuum form, and which still shows the field distribution returning close to $phi = 0$ for several oscillations before settling near $|phi| = v$, would undercut the single-oscillation claim.","supporting_citations":[{"cited_title":"Dynamics of Symmetry Breaking and Tachyonic Preheating","cited_arxiv_id":"hep-ph/0012142","evidence_quote":"The earlier letter by the same collaboration that first reported symmetry breaking completing within a single oscillation; this paper is the detailed extension of it."},{"cited_title":"Phys. Rev. Let t. 73, 3195 (1994); Phys. Rev. D 56, 3258","cited_arxiv_id":null,"evidence_quote":"Supplies the standard definition of the occupation number $n_k$ used throughout and the parametric-resonance framework the tachyonic growth is compared against."},{"cited_title":"and D.S . Lee, Phys. Rev. D52, 6805","cited_arxiv_id":null,"evidence_quote":"The Hartree and $1/N$ treatment of the same model whose long-lived homogeneous oscillations the present paper argues against."},{"cited_title":"Classical decay of inflaton","cited_arxiv_id":"hep-ph/9603378","evidence_quote":"Establishes that large-occupation quantum states can be simulated as classical waves on a lattice, which is the justification for the whole numerical method."},{"cited_title":"LATTICEEASY: A Program for Lattice Simulations of Scalar Fields in an Expanding Universe","cited_arxiv_id":"hep-ph/0011159","evidence_quote":"The LATTICEEASY code with which every simulation in the paper was run."},{"cited_title":"Decay Of The False Vacuum At Finite Temperature,","cited_arxiv_id":null,"evidence_quote":"Source for tunneling without a potential barrier, the mechanism invoked for the quartic and Coleman-Weinberg potentials."},{"cited_title":"Hard Art of the Universe Creation","cited_arxiv_id":"hep-th/9110037","evidence_quote":"The stochastic approach to tunneling used to estimate bubble formation rates in the cubic model, where no instanton exists."},{"cited_title":"Structure of Resonance in Preheating after Inflation","cited_arxiv_id":"hep-ph/9705347","evidence_quote":"The Lamé-equation stability analysis of parametric resonance that the tachyonic band is compared with and shown to dominate."},{"cited_title":"A New Approach To Conformal Invariant Field Theorie s,","cited_arxiv_id":null,"evidence_quote":"Provides the family of $O(4)$-symmetric instanton solutions with $rho$-independent action used as initial data for the quartic runs."}],"review_version":1}