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We investigate up to which order quantum effects can be neglected in calculating cosmological correlation functions after horizon exit. As a toy model, we study $\phi^3$ theory on a de Sitter background for a massless minimally coupled scalar field $\phi$. We find that for tree level and one loop contributions in the quantum theory, a good classical approximation can be constructed, but for higher loop corrections this is in general not expected to be possible. The reason is that loop corrections get non-negligible contributions from loop momenta with magnitude up to the Hubble scale H, at which scale classical physics is not expected to be a good approximation to the quantum theory. An explicit calculation of the one loop correction to the two point function, supports the argument that contributions from loop momenta of scale $H$ are not negligible. Generalization of the arguments for the toy model to derivative interactions and the curvature perturbation leads to the conclusion that the leading orders of non-Gaussian effects generated after horizon exit, can be approximated quite well by classical methods. Furthermore we compare with a theorem by Weinberg. We find that growing loop corrections after horizon exit are not excluded, even in single field inflation., Comment: 44 pages, 1 figure; v2: corrected errors, added references, conclusions unchanged; v3: added section in which we compare with stochastic approach; this version matches published version

The electroweak symmetry breaking transition may supply the appropriate out-of-equilibrium conditions for baryogenesis if it is triggered sufficiently fast. This can happen at the end of low-scale inflation, prompting baryogenesis to occur during tachyonic preheating of the Universe, when the potential energy of the inflaton is transfered into Standard Model particles. With the proper amount of CP-violation present, the observed baryon number asymmetry can be reproduced. Within this framework of Cold Electroweak Baryogenesis, we study the dependence of the generated baryon asymmetry on the speed of the quenching transition. We find that there is a separation between ``fast'' and ``slow'' quenches, which can be used to put bounds on the allowed Higgs-inflaton coupling. We also clarify the strong Higgs mass dependence of the asymmetry reported in a companion paper (hep-ph/0604263)., Comment: 18 pages, 20 figures

We present real-time lattice simulations of Cold Electroweak Baryogenesis, in which the baryon asymmetry of the Universe is generated during tachyonic electroweak symmetry breaking at the end of inflation. In the minimal realisation of the model, only three parameters remain undetermined: the strength of CP-violation, the Higgs mass and the speed of the symmetry breaking quench. The dependence of the asymmetry on these parameters is studied., Comment: 4 pages. Presented at International Conference on Strong and Electroweak Matter (SEWM 2006), Upton, New York, 10-13 May 2006

Cold electroweak baryogenesis was proposed as a scenario to bypass generic problems of electroweak baryogenesis within the Standard Model. In this scenario, baryogenesis takes place during an electroweak symmetry breaking transition, which is also responsible for preheating after inflation. In the simplest modelling of the scenario, only two parameters remain undetermined: The Higgs mass and the strength of CP violation. Using full real-time lattice simulations, we compute the dependence of the asymmetry on these parameters., Comment: 14 pages, 13 figures

We investigate the development of winding number and Chern-Simons number in a tachyonic transition in the SU(2) Higgs model, motivated by the scenario of cold electroweak baryogenesis. We find that localized configurations with approximately half-integer winding number, dubbed half-knots, play an important role in this process. When the Chern-Simons number adjusts locally to the winding number, the half-knots can stabilize and acquire half-integer Chern-Simons number as well. We present two examples from numerical simulations: one half-knot that stabilizes early and one that gives rise to a late sphaleron transition. We also study the winding number distribution after the transition and present new results on the development of the Chern-Simons susceptibility., Comment: 36 pages, 38 figures

The process of equilibration in phi^4 theory is investigated for a homogeneous system in 3+1 dimensions and a variety of out-of-equilibrium initial conditions, both in the symmetric and broken phase, by means of the 2PI effective action. Two Phi-derivable approximations including scattering effects are used: the two-loop and the ``basketball'', the latter corresponding to the truncation of the 2PI effective action at O(lambda^2). The approach to equilibrium, as well as the kinetic and chemical equilibration is investigated., Comment: 32 pages, 14 figures, uses axodraw, minor corrections added

We study the process of tachyonic preheating using approximative quantum equations of motion derived from the 2PI effective action. The O(N) scalar (Higgs) field is assumed to experience a fast quench which is represented by an instantaneous flip of the sign of the mass parameter. The equations of motion are solved numerically on the lattice, and the Hartree and 1/N-NLO approximations are compared to the classical approximation. Classical dynamics is expected to be valid, since the occupation numbers can rise to large values during tachyonic preheating. We find that the classical approximation performs excellently at short and intermediate times, even for couplings in the larger region currently allowed for the SM Higgs. This is reassuring, since all previous numerical studies of tachyonic preheating and baryogenesis during tachyonic preheating have used classical dynamics. We also compare different initializations for the classical simulations., Comment: 32 pages, 21 figures. Published version: Some details added, section added, references added, conclusions unchanged

We present arguments that the CKM CP-violation in the standard model may be sufficient for the generation of the baryon asymmetry, if the electroweak transition in the early universe was of the cold, tachyonic, type after electroweak-scale inflation. A model implementing this is described which complies with the CMB constraints and which is falsifiable with the LHC. Numerical simulations of the tachyonic transition with an effective CP-bias indicate that the observed baryon asymmetry can be generated this way., Comment: 10 pages, 4 figures, presented at Strong and ElectroWeak Matter 2004

We study the strength of effective CP violation originating from the CKM matrix in the effective action obtained by integrating out the fermions in the Standard Model. Using results obtained by Salcedo for the effective action in a general chiral gauge model, we find that there are no CKM CP-violating terms to fourth order in a gauge-covariant derivative expansion that is non-perturbative in the Higgs field. The details of the calculation suggest that, at zero temperature, the strength of CP violation is approximately independent of the overall scale of the Yukawa couplings. Thus, order of magnitude estimates based on Jarlskog's invariant could be too small by a factor of about 10^{17}., Comment: 19 pages, no figures

We construct a phenomenological model of electroweak-scale inflation that is in accordance with recent cosmic microwave background observations by WMAP, while setting the stage for a zero-temperature electroweak transition as assumed in recent models of baryogenesis. We find that the scalar spectral index especially poses tight constraints for low-scale inflation models. The inflaton-Higgs coupling leads to substantial mixing of the scalar degrees of freedom. Two types of scalar particles emerge with decay widths similar to that of the Standard Model Higgs particle., Comment: 21 pages, 4 figures. v2: Two references added, matches version to be published in JCAP

We consider a scenario in which the baryon asymmetry was created in the early universe during a cold electroweak transition. The spinodal instability of the Higgs field caused by a rapid change of sign of its effective mass-squared parameter induces tachyonic preheating. We study the development of Chern-Simons number in this transition by numerical lattice simulations of the SU(2)-Higgs model with an added effective CP-violating term. A net asymmetry is produced, and we study its dependence on the size of CP violation and the ratio of Higgs to W mass., Comment: 30 pages, 9 figures

We consider the out-of-equilibrium (quasi-) particle number distributions of the Higgs and W-fields during electroweak tachyonic preheating. We model this process by a fast quench, and perform classical real-time lattice simulations in the SU(2)-Higgs model in three dimensions. We discuss how to define particle numbers and effective energies using two-point functions in Coulomb and unitary gauge, and consider some of the associated problems. After an initial exponential growth in effective particle numbers, the system stabilises, allowing us to extract effective masses, temperatures and chemical potentials for the particles., Comment: 3 pages, Lattice2003(higgs)

We study out-of-equilibrium quasi-particle distributions of the Higgs and W fields during the zero-temperature tachyonic electroweak transition that has been assumed in recent scenarios of baryogenesis. Approximating the process by a fast quench, we perform classical real-time lattice simulations in the SU(2)-Higgs model. The emerging quasi-particle numbers and energies are then used to determine the effective temperatures, chemical potentials and masses of the particles shortly after the transition., Comment: 28 pages including 17 figures

We consider the creation of non-zero Chern-Simons number in a model of the early Universe, where the Higgs field experiences a fast quench at the end of inflation and subsequently rolls down its potential barrier. Neglecting the expansion, we perform numerical lattice simulations in the Abelian Higgs model in 1+1 dimensions with an added phenomenological C and P violating term during this stage of so-called tachyonic preheating. The results suggest that even the sign of the Chern-Simons and thus baryon number is dependent on the ratio of the Higgs to W mass. We also discuss the appropriate choice of vacuum initial conditions for classical simulations., Comment: 24 pages, 17 figures, added references, published version

Results are presented of a numerical study of the distribution of W-bosons generated in a tachyonic electroweak pre-heating transition, Comment: Contribution to Strong and ElectroWeak Matter 2002, 5 pages

The Hartree ensemble approximation is studied in the ``symmetric phase'' of 1+1 dimensional lambda phi^4 theory. In comparison with the ``broken phase'' studied previously, it is shown that the dynamical evolution of observables such as the particle distribution, energy exchange and auto-correlation functions, is substantially slower. Approximate thermalization is found only for relatively large energy densities and couplings., Comment: 17 pages RevTeX, 16 figures, 3 tables, uses amsmath and feynmp. Extended some sections, reordered Sec.IV, added 3 refs, numerical typo corrected, published version

As a toy model for dynamics in nonequilibrium quantum field theory we consider the abelian Higgs model in 1+1 dimensions with fermions. In the approximate dynamical equations, inhomogeneous classical (mean) Bose fields are coupled to quantized fermion fields, which are treated with a mode function expansion. The effective equations of motion imply e.g. Coulomb scattering, due to the inhomogeneous gauge field. The equations are solved numerically. We define time dependent fermion particle numbers with the help of the single-time Wigner function and study particle production starting from inhomogeneous initial conditions. The particle numbers are compared with the Fermi-Dirac distribution parametrized by a time dependent temperature and chemical potential. We find that the fermions approximately thermalize locally in time., Comment: 16 pages + 6 eps figures, some clarifications and two references added, typos corrected; to appear in Phys.Rev.D

The dynamics of the 1+1 abelian Higgs model with fermions is studied in the large N_f approximation, on a real-time lattice. The Bose fields obey effective classical equations of motion which include the fermion back reaction. The dynamics of the quantized fermion field is treated with a mode function expansion. Numerical results are shown for renormalizability, nonequilibrium dynamics and the anomalous charge, and Pauli blocking., Comment: 5 pages, 6 eps figures, talk presented by G. Aarts at Strong and Electroweak Matter '98, Kopenhagen, Denmark, December 2-5, 1998

The 1+1 dimensional abelian Higgs model with fermions is a toy model for the theory of electroweak baryogenesis. We study the dynamics of the model with axially coupled fermions in real-time. The model is defined on a spacetime lattice to preserve gauge invariance and to obtain numerical stability in a simple way. We take into account the phenomenon of lattice fermion doubling. The dynamics is approximated by treating the inhomogeneous Bose fields classically, which is justified in a large N_f approximation. The back reaction on the Bose fields due to fermion field fluctuations is calculated using a mode function expansion. We discuss and present numerical results for the response of fermions to sphaleron transitions, the renormalizability of the effective equations of motion and non-perturbative dynamics in the framework of non-equilibrium quantum field theory. The long-time behaviour of the system is discussed and we speculate about applications to finite density calculations., Comment: 34 pages + 20 eps figures, improved presentation, discussion of the figures and figure captions expanded, references added; to appear in Nucl.Phys.B

We consider the dynamics of the 1+1 dimensional abelian Higgs model with axially coupled fermions, in the large N_f limit, on a lattice in space and real-time. We allow for inhomogeneous classical Bose fields. In order to deal with the lattice doublers, we use Wilson's lattice fermions. The lattice formulation leads to a stable integration algorithm. We demonstrate the practical implementation with numerical results., Comment: 4 pages, 6 eps figures, poster presented at the 5th International Workshop on Thermal Field Theories and Their Applications, Regensburg, Germany, August 10-14, 1998. Short discussion on the large N_f limit added

We discuss an explorative computation of real time autocorrelation functions, in the classical approximation. The results for the `plasmon' frequencies and damping rates appear compatible with the divergencies expected from perturbation theory., Comment: 5 pages, latex, 4 figures; to appear in the proceedings of `Strong and Electroweak Matter '97 (SEWM97)', Eger, Hungary, May 1997

We study time dependent correlation functions in hot quantum and classical field theory for the $\lambda\phi^4$ case. We set up the classical analogue of thermal field theory and make a direct comparison between the quantum and classical diagrams. A restriction to time independent correlation functions gives the connection with conventional dimensional reduction. If the parameters in the classical theory are chosen according to the dimensional reduction matching relations, the classical expressions are cutoff independent and they approximate the quantum expressions, provided that the external momenta and frequencies are small with respect to the temperature., Comment: 30 pages, 10 eps figures

We investigate the renormalizability of the classical $\phi^4$ theory at finite temperature. We calculate the time-dependent two point function to two loop order and show that it can be rendered finite by the counterterms of the classical static theory. As an application the classical plasmon damping rate is found to be $\gamma = \lambda^2 T^2/1536 \pi m$. When we use the high temperature expression for $m$ given by dimensional reduction, the rate is found to agree with the quantum mechanical result., Comment: 11 pages, 4 eps figures, minor typographic corrections and two references added

The rate of $B$-violation in the standard model at finite temperature is closely related to the diffusion rate $\Gamma$ of Chern-Simons number. We compute this rate for $m_H \approx m_W$ in the classical approximation in an effective SU(2)-Higgs model, using Krasnitz's algorithm. The parameters in the effective hamiltonian are determined by comparison with dimensional reduction. In the high temperature phase we find $\Gamma/V (\alpha_W T)^4 \approx 1$, neglecting a finite renormalization. In the low temperature phase near the transition we find the rate to be much larger than might be expected from previous analytic calculations based on the sphaleron., Comment: 23 pages, 8 Postscript figures, uses epsf.sty

New results for the rate are presented using the canonical ensemble in the classical approximation on a spatial lattice. We find that the rate at high temperatures is proportional to $T^2$, and strongly dependent on the lattice spacing $a$. We conclude that a better effective action is needed for the classical approximation., Comment: 3 pages, uuencoded compressed postscript file, contribution to the Conference Proceedings "LATTICE 94", Bielefeld- Germany

Lattice proposals for a nonperturbative formulation of the Standard Model easily lead to a global U(1) symmetry corresponding to exactly conserved fermion number. The absence of an anomaly in the fermion current would then appear to inhibit anomalous processes, such as electroweak baryogenesis in the early universe. One way to circumvent this problem is to formulate the theory such that this U(1) symmetry is explicitly broken. However we argue that in the framework of spectral flow, fermion creation and annihilation still in fact occurs, despite the exact fermion number conservation. The crucial observation is that fermions are excitations relative to the vacuum, at the surface of the Dirac sea. The exact global U(1) symmetry prohibits a state from changing its fermion number during time evolution, however nothing prevents the fermionic ground state from doing so. We illustrate our reasoning with a model in two dimensions which has axial-vector couplings, first using a sharp momentum cutoff, then using the lattice regulator with staggered fermions. The difference in fermion number between the time evolved state and the ground state is indeed in agreement with the anomaly. A study of the vacuum energy shows that the perturbative counterterm needed for restoration of gauge invariance is insufficient in a nonperturbative setting. For reference we also study a closely related model with vector couplings, the Schwinger model, and we examine the emergence of the $\theta$-vacuum structure of both theories., Comment: 31 pages, LaTeX + uuencoded figs file (=5 PS figs). UvA-ITFA 94-17, UCSD/PTH 94-07

As a first step towards constructing chiral models on the lattice with staggered fermions, we study a U(1) model with axial-vector coupling to an external gauge field in two dimensions. In our approach gauge invariance is broken, but it is restored in the classical continuum limit. We find that the continuum divergence relations for the vector and axial-vector currents are reproduced, up to contact terms, which we determine analytically. The current divergence relations are also studied numerically for smooth external gauge fields with topological charge zero. We furthermore investigate the effect of fluctuating gauge transformations and of gauge configurations with non-trivial topological charge., Comment: 17 pages, 5 postscript figures (appended), ITFA 93-13, UCSD/PTH 93-13

We present results for the renormalized quartic self-coupling $\lambda_R$ and the Yukawa coupling $y_R$ in a lattice fermion-Higgs model with two SU(2)$_L$ doublets, mostly for large values of the bare couplings. One-component (`reduced') staggered fermions are used in a numerical simulation with the Hybrid Monte Carlo algorithm. The fermion and Higgs masses and the renormalized scalar field expectation value are computed on $L^3 24$ lattices, where $L$ ranges from $6$ to $16$. In the scaling region these quantities are found to have a $1/L^2$ dependence, which is used to determine their values in the infinite volume limit. We then calculate the $y_R$ and $\lambda_R$ from their tree level definitions in terms of the masses and renormalized scalar field expectation value, extrapolated to infinite volume. The scalar field propagators can be described for momenta up to the cut-off by one fermion loop renormalized perturbation theory and the results for $\lambda_R$ and $y_R$ come out to be close to the tree level unitarity bounds. There are no signs that are in contradiction with the triviality of the Yukawa and quartic self-coupling., Comment: 36 pages + 12 postscript figures (appended), Amsterdam ITFA 92-23, HLRZ Juelich 92-58, UCSD/PTH 92-38