Timelike self-similar spherically symmetric models with two scalar fields
Bibliographic record
Abstract
We investigate timelike self-similar spherically symmetric cosmological models with two scalar fields with exponential potentials. We determine the governing autonomous system of ordinary differential equations for the model, and by defining bounded variables we are able to apply the theory of dynamical systems. We obtain all of the equilibrium points in the compact dynamical phase space and analyse their local stability. We find a monotonic function, so that we can rule out the existence of periodic orbits or recurrent orbits in the physical phase space and conclude that it is plausible that all orbits asymptote towards equilibrium points, both to the past and to the future. We also identify all equilibrium points on the null surface, through which orbits can be continued into the spacelike self-similar regime. Defining a parameter K , which depends on the two scalar potential parameters, we find that for K < 1, the orbits generically expand away from a massless scalar field model or a static single scalar field model, recollapse and then evolve towards a contracting massless scalar field model or a static single scalar field model. The evolution from a non-static massless scalar field model to a second non-static massless scalar field model is also possible, and in this case the orbits cannot be extended into the spacelike self-similar regime. We note that there are static single scalar field models that are physical attractors, so that there appear to be multiple-scalar field models with exponential potentials for which only a single scalar field dominates at late-times or at early-times. For the parameter range K > 1, all orbits generically expand away from a massless scalar field model, then recollapse and subsequently contract to a massless scalar field model. However, since in this case there always exist equilibrium points on the null boundary, there are orbits that can be continued into the spacelike self-similar regime and subsequently either continue expanding towards an assisted inflationary solution or recollapse to a massless scalar field model which is again on the null boundary.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".