Cosmic Scale Factor Research Articles

A COSMOLOGY WITH CONFORMALLY COUPLED SCALAR FIELD

A theory of gravitation with a conformally coupled scalar field is considered in which the gravitational “constant” is associated with the vacuum expectation value of the scalar field. It is found that the universe will remain dominated by classical radiation unless the conformal symmetry is broken. The equations of motion thus derived bear a de Sitter phase solution, which could have an exponential growth of the cosmic scale factor with no vacuum energy. We discuss the cosmological implications of this kind of “inflation”. We also find that in the present theory the smallness of the vacuum energy for most time of the universe is due to the constancy of the gravitational “constant”.

Conformal invariance beyond the leading order in the supersymmetric nonlinear sigma model with dilaton.

We calculate the O(\ensuremath{\alpha}${\mathcal{'}}^{3}$) contributions to the renormalization-group \ensuremath{\beta} functions in the N=1 supersymmetric \ensuremath{\sigma} model with a dilaton. At this order both metric and dilaton \ensuremath{\beta} functions are found to depend nontrivially on the dilaton field and vanish if the dilaton satisfies ${\mathrm{\ensuremath{\nabla}}}_{\mathrm{\ensuremath{\mu}}}$${\mathrm{\ensuremath{\nabla}}}_{\ensuremath{\nu}}$\ensuremath{\varphi}=0. By employing the Curci-Paffuti relation it is shown that such dilaton backgrounds in Ricci-flat spaces ${\mathit{R}}_{\mathrm{\ensuremath{\mu}}\ensuremath{\nu}}$=0 satisfy the conformal invariance conditions up to this order. The particular class of Ricci-flat, compact, and orientable manifolds naturally emerge as appropriate internal-space configurations consistent with local scale invariance. We further explore the cosmological consequences of these dilaton configurations. In a Robertson-Walker four-dimensional background we find all dilatons satisfying ${\mathrm{\ensuremath{\nabla}}}_{\mathrm{\ensuremath{\mu}}}$${\mathrm{\ensuremath{\nabla}}}_{\ensuremath{\nu}}$\ensuremath{\varphi}=0. Except for the constant and the time-dependent dilaton \ensuremath{\varphi}(t)=-2 lnt+\ensuremath{\lambda} whose cosmological implications have been already discussed in the literature, additional solutions are found. These may be of relevance beyond leading order and for nonvanishing background values for the antisymmetric tensor ${\mathit{B}}_{\mathrm{\ensuremath{\mu}}\ensuremath{\nu}}$. For these solutions, also the cosmic scale factor is at most linear in time therefore giving rise to either a static or a linearly expanding (contracting) universe.

Relativistic stochastic Boltzmann equation and fluctuations in general relativity

Using the Klimontovich method (1982) of microscopic phase densities, a relativistic stochastic Boltzmann equation leading to a fluctuating energy-momentum tensor is derived. Via Einstein's equations the metric quantities themselves become random. They fulful relations of the type of fluctuation-dissipation theorems as is demonstrated for fluctuations of the cosmic scale factor of a flat Friedman universe.

Cosmic strings from superstrings

In the low-energy theory of four-dimensional superstrings the spontaneous breakdown of a pseudo-anomalous U(1) can trigger (but not necessarily) the formation of global and/or local cosmic strings. When cosmic strings are formed, supersymmetry is broken inside the string cores with a positive cosmological constant and an exponential growth of the cosmic scale factor. Strings are conformally stretched with the inflationary expansion, never enter the physics horizon and can have no observable (e.g. optical) effects. Off-string regions are in a (radiation-dominated) FRW universe. Most of our observable universe would still be inflating today unless a late (conventional) period of inflation transforms an off-string region into our present universe. The required amount of inflation coincides with the one needed to solve the usual flatness and horizon problems of the standard cosmological model. We consider, as an example, the chaotic inflation associated with the low-energy breaking (∼1 TeV) of supersymmetry.

Large-scale galaxy flow from a non-gravitational impulse

A theory is presented describing linear perturbations of an expanding universe containing multiple, independently perturbed, collisionless, gravitationally coupled constituents. Solutions are found in the limit where one initially unperturbed component dominates the total density. The theory is applied to perturbations generated by a non-gravitational process in one or more of the light components, as would occur in explosive or radiation-pressure-instability theories of galaxy formation. The apparent dynamical density parameter Ωapp, and correlations between density and velocity amplitude for various populations, are evaluated as a function of cosmic scale factor. The simplest two-component radiation-pressure-instability model predicts a very small apparent Ω, which seems to conflict with current estimates, based on clustering and peculiar velocities measured for different populations. The multicomponent treatment avoids this problem, and makes the testable prediction that the more strongly clustered galaxy populations should have larger flow velocities.

Inflationary solutions in quantum cosmology

The authors prove that in the Hartle-Hawking approach to quantum cosmology the existence of an inflationary phase is a general property of minisuperspace models given by a closed Friedmann-Robertson-Walker universe containing a massless scalar field with a lambda phi n self-interaction. The evolution in time of the cosmic scale factor and of the scalar field in the very early universe is derived, together with the constraints to be satisfied in order to solve the horizon and flatness problems.

Nonsingular cosmological models: the massive scalar field case

The nonminimal coupling of a massive self-interacting scalar field with a gravitational field is studied. Spontaneous symmetry breaking occurs in the open universe even when the sign on the mass term is positive. In contrast to grand unified theories, symmetry breakdown is more important for the early universe and it is restored only in the limit of an infinite expansion. Symmetry breakdown is shown to occur in flat and closed universes when the mass term carries a wrong sign. The model has a naturally defined effective gravitational coupling coefficient which is rendered time-dependent due to the novel symmetry breakdown. It changes sign below a critical value of the cosmic scale factor indicating the onset of a repulsive field. The presence of the mass term severely alters the behaviour of ordinary matter and radiation in the early universe. The total energy density becomes negative in a certain domain. These features make possible a nonsingular cosmological model for an open universe. The model is also free from the horizon and the flatness problems.

Decay of massive particles in Robertson-Walker universes with statically bounded expansion laws

The authors study mutually interacting quantum fields in 3-flat Robertson-Walker universes. Using the interaction picture and an in-out approach based on an S matrix, they give an exact calculation of the decay of a massive scalar Phi particle into two massless scalar Psi particles (both conformally coupled to the background) in lowest order of the ( lambda /a( eta )) Phi Psi 2 interaction. The cosmic scale factor a( eta ) is thereby specified (i) as the step-expansion law, and (ii) as the tanh-expansion law which represents the prototype of a smoothed-out step law. These a( eta ) are typical representatives of monotonic expansions with asymptotically static in and out regions. The characteristic modifications of the Minkowskian decay spectrum like the double-resonance structure are discussed. Comparison of amplitudes and decay probabilities for (i) and (ii) demonstrates that smoothing-out the step introduces a gravitational influence of non-vanishing duration, which leads essentially to finite corrections of the step results. Additionally, it can be shown that the massive particles which are gravitationally created out of the background, decay as well and contribute to the number of outgoing massless particles. From the results obtained, the general structure of amplitudes, probabilities, and decay rates holding for all statically bounded monotonic expansion laws can be read off.

Consistency of canonical quantization of gravity and boundary conditions for the wave function of the Universe.

By using the expression given by Halliwell and Hawking, we discuss consistency of the constraint equations of the superspace-quantized gravity coupled to the minimal massive scalar field, in the lowest nontrivial order in the cosmological perturbation. We show that the consistency imposes highly nontrivial conditions on the behavior of the physical state (the wave function of the Universe) at a (the cosmic scale factor) =0.

Quantum-Classical Correspondence in Wave Functions of the Universe

By using the classically solvable minimal massless scalar minisuperspace model we study two exam­ ples of wave functions of the universe in the semiclassical limit. The first one consists of a Lorentzian component and a Euclidean component and admits a clear semiclassical interpretation as a superposition of universes. The second one consists of a Lorentzian (or Euclidean) component and another oscillatory component which corresponds to (neither Lorentzian nor Euclidean) complex classical solutions. This example has some resemblance to Hawking's minimal massive scalar minisuperspace model. We suggest a possible way of recovering the classical interpretation in such a case. In quantum cosmologyl)-4) we describe the universe by a wave function which satisfies the Wheeler-DeWitt equation_ This is a second-order linear partial differential equation_ In the h -> 0 limit some of its solutions admit clear classical interpretation while others do not. In the present paper we study the semiclassical limit of wave functions in a classical­ ly solvable minisuperspace model and discuss the classical interpretation. In § 2 we present the model and its classical solutions. The model is the minimal massless scalar theoryS) coupled to the Einstein gravity with all the spatial degrees. of freedom suppressed, i.e., in the minisuperspace approximation. By using a convenient set of variables we write down the most general complex classical solutions which include real Lorentzian and Euclidean solutions as special cases. (For another classically solva­ ble model, see Ref.· 6).) In § 3 we show the first set of examples for wave functions of the universe. In the semiclassical sense these wave functions correspond to a set of Lorent­ zian and Euclidean solutions which pass a certain fixed point in the minisuperspace (the configuration space), and therefore can be interpreted as a superposition of universes. In § 4 we rewrite the above examples in the path integral representation, which includes slight refinement of the formula in previous works. We also study the conse­ quence of Hartle and Hawking's initial condition 3 ) which states that only the histories starting from the vanishing cosmic scale factor should be summed. In the present model . this condition gives us a wave function which contains no universe. In § 5 we present the second example of wave functions, in which appears a region that is neither Lorentzian nor Euclidean. We use a path integral representation written in terms of a new set of canonical variables. This is transformed to the wave function in th~ original variables by using the generating function of the canonical transformation. The result shows that the wave function in the saddle point approximation is given by (neither Lorentzian nor Euclidean) complex classical solutions in some region. In § 6 we discuss the picture that the universe is created from nothing by the tunneling

Mutually Interacting Quantum Fields in an Expanding Universe: Decay of a Massive Particle

The authors give an analysis of quantum field theory of mutually interacting fields in a 3-flat Robertson-Walker universe with regard to the questions: What are the measurable quantities? What is the related calculation scheme? The classical curved spacetime acts thereby as an external gravitational field, the influence of which is taken into account without any approximation. As a case study a mathematically rigorous calculation of the decay of a massive phi particle into two massless psi particles is given in first order of the ( lambda /a) phi psi 2 mutual interaction where a is the cosmic scale factor. The particles are conformally coupled scalar particles. The interaction picture and the in-out scheme based on an S-matrix approach are used. No renormalisation is needed. Because the external field is capable of producing pairs of massive phi particles from the vacuum, the specification of the outcome of the mutual interaction is based on the registration of massless particles in the out region. A detailed analysis of the corresponding added-up probability is given. It sheds some light on the imperfection of the usual in-out approach when applied to cosmological situations. The results are used to obtain a limit for the local applicability of flat spacetime results as cross section, decay rates etc. in a cosmological situation.

Mutually interacting quantum fields in an expanding universe: decay of a massive particle

The authors give an analysis of quantum field theory of mutually interacting fields in a 3-flat Robertson-Walker universe with regard to the questions: What are the measurable quantities? What is the related calculation scheme? The classical curved spacetime acts thereby as an external gravitational field, the influence of which is taken into account without any approximation. As a case study a mathematically rigorous calculation of the decay of a massive phi particle into two massless psi particles is given in first order of the ( lambda /a) phi psi 2 mutual interaction where a is the cosmic scale factor. The particles are conformally coupled scalar particles. The interaction picture and the in-out scheme based on an S-matrix approach are used. No renormalisation is needed. Because the external field is capable of producing pairs of massive phi particles from the vacuum, the specification of the outcome of the mutual interaction is based on the registration of massless particles in the out region. A detailed analysis of the corresponding added-up probability is given. It sheds some light on the imperfection of the usual in-out approach when applied to cosmological situations. The results are used to obtain a limit for the local applicability of flat spacetime results as cross section, decay rates etc. in a cosmological situation.

Cosmology with decaying particles.

We consider a cosmological model in which an unstable massive relic particle species (denoted by X) has an initial mass density relative to baryons ..beta../sup -1/equivalentrho/sub X//rho/sub B/>>1, and then decays recently (red-shift z< or approx. =1000) into particles which are still relativistic today (denoted by R). We write down and solve the coupled equations for the cosmic scale factor a(t), the energy density in the various components (rho/sub X/,rho/sub R/,rho/sub B/), and the growth of linear density perturbations (deltarho/rho). The solutions form a one-parameter (..beta..) family of solutions; physically ..beta../sup -1/approx. =(..cap omega../sub R//..cap omega../sub NR/)(1+z/sub D/) = (ratio today of energy density of relativistic to nonrelativistic particles) x (1+ red-shift of decay). We discuss the observational implications of such a cosmological model and compare our results to earlier results computed in ''the simultaneous decay approximation.'' In an appendix we briefly consider the case where one of the decay products of the X is massive and becomes nonrelativistic by the present epoch.

A model of spontaneous symmetry breakdown in spatially flat cosmological spacetimes

This paper is an elaboration of a previous short exposition of a theory of spontaneous symmetry breaking in a conformally coupled, massless λø 4 model in a spatially flat Robertson-Walker spacetime. Under the weakened global boundary condition allowing the physical spacetime to be conformal to only a portion of the Minkowski spacetime, the model admits a pair of degenerate vacua in which the ø → − ø symmetry is spontaneously broken. The model is formulated as a euclidean field theory in a space with a positive-definite metric obtained by analytically continuing the conformal time coordinate. An appropriate time-dependent zero energy solution of the euclidean equation of motion representing the field configuration in the asymmetric vacuum is considered and the corresponding quantum trace anomaly 〈 T μ μ 〉 is computed in the one-loop approximation. The nontrivial infrared behavior of the model due to the singular nature of the classical background field forces a modification of the boundary conditions on the propagator. A general form for an “improved|DD one-loop trace anomaly is found by a simple argument based on renormalization group invariance. Via the Einstein equation, the trace anomaly leads to a self-consistent dynamical equation for the cosmic expansion scale factor. Some physical aspects of the back-reaction problem based on a simple power law model of the expansion scale factor are discussed.