{"work":{"id":"6fb7d3e0-e1e0-47ca-85c3-33f1c1d498f7","openalex_id":null,"doi":null,"arxiv_id":"gr-qc/9711068","raw_key":null,"title":"Exponential potentials and cosmological scaling solutions","authors":null,"authors_text":"E","year":1997,"venue":"gr-qc","abstract":"We present a phase-plane analysis of cosmologies containing a barotropic fluid with equation of state $p_\\gamma = (\\gamma-1) \\rho_\\gamma$, plus a scalar field $\\phi$ with an exponential potential $V \\propto \\exp(-\\lambda \\kappa \\phi)$ where $\\kappa^2 = 8\\pi G$. In addition to the well-known inflationary solutions for $\\lambda^2 < 2$, there exist scaling solutions when $\\lambda^2 > 3\\gamma$ in which the scalar field energy density tracks that of the barotropic fluid (which for example might be radiation or dust). We show that the scaling solutions are the unique late-time attractors whenever they exist. The fluid-dominated solutions, where $V(\\phi)/\\rho_\\gamma \\to 0$ at late times, are always unstable (except for the cosmological constant case $\\gamma = 0$). The relative energy density of the fluid and scalar field depends on the steepness of the exponential potential, which is constrained by nucleosynthesis to $\\lambda^2 > 20$. We show that standard inflation models are unable to solve this `relic density' problem.","external_url":"https://arxiv.org/abs/gr-qc/9711068","cited_by_count":null,"metadata_source":"pith","metadata_fetched_at":"2026-05-23T23:23:36.977130+00:00","pith_arxiv_id":"gr-qc/9711068","created_at":"2026-05-10T22:10:48.193285+00:00","updated_at":"2026-05-23T23:23:36.977130+00:00","title_quality_ok":true,"display_title":"J., Liddle, A","render_title":"J., Liddle, A"},"hub":{"state":{"work_id":"6fb7d3e0-e1e0-47ca-85c3-33f1c1d498f7","tier":"hub","tier_reason":"10+ Pith inbound or 1,000+ external citations","pith_inbound_count":11,"external_cited_by_count":null,"distinct_field_count":3,"first_pith_cited_at":"2024-07-01T18:41:04+00:00","last_pith_cited_at":"2026-05-12T15:01:18+00:00","author_build_status":"not_needed","summary_status":"needed","contexts_status":"needed","graph_status":"needed","ask_index_status":"not_needed","reader_status":"not_needed","recognition_status":"not_needed","updated_at":"2026-06-04T01:26:50.925890+00:00","tier_text":"hub"},"tier":"hub","role_counts":[{"context_role":"background","n":6}],"polarity_counts":[{"context_polarity":"background","n":6}],"runs":{},"summary":{},"graph":{},"authors":[]}}