ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-11-04 (gpt-6.1-sol)

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Rolling open-string tachyon as an effective cosmological component

Physical scope

Conditional source-normalized open-string tachyon effective action on a four-dimensional induced string-frame worldvolume. The stress tensor is derived locally; no microscopic brane-action derivation, four-dimensional Einstein-frame conversion, global compactification, inflation, or observation is asserted.

Example

Use natural units $ℏ = c_{l i g h t} = 1$ and the source convention $2 π α^{'} = 1$ . Let $W = I \times R^{3}$ be a four-dimensional unstable D3-brane worldvolume (three spatial dimensions) with coordinates $x^{μ}$ of length, smoothly embedded by $ι$ in a string-frame Lorentzian spacetime $(X, G_{s t r})$ ; its induced metric is $g^{i n d, s} : = ι^{*} G_{s t r}$ . The instability is modeled by a real tachyon field $T \in C^{2} (W)$ with a potential $V (T)$ that has a maximum and vanishes at its asymptotic minimum. The field has units of length, so $\partial_{μ} T$ is dimensionless; in the stated string units its numerical values are dimensionless. The potential density has mass dimension four, with the positive brane-tension normalization absorbed into $V$ . The adopted action is $S_{s} = - \int_{W} d^{4} x V (T) \sqrt{- \det (g_{μ ν}^{i n d, s} + \partial_{μ} T \partial_{ν} T)}$ . Sen's §5 uses the flat Minkowski metric $η_{μ ν}$ ; replacing it by $g^{i n d, s}$ is an additional covariantization assumption, not a result supplied by Sen. For a homogeneous $T = T (t)$ on the flat-FLRW induced metric $d s_{s}^{2} = - d t^{2} + a_{s} (t)^{2} δ_{i j} d x^{i} d x^{j}$ with $∣ \dot{T} ∣ < 1$ , the stress components $ρ_{s} = V (T) / \sqrt{1 - {\dot{T}}^{2}}$ and $p_{s} = - V (T) \sqrt{1 - {\dot{T}}^{2}}$ have mass dimension four. A dust-like asymptotic statement requires $V (T) \to 0$ and the stated solution regime. A four-dimensional Einstein-frame quantity requires a separate dilaton/volume Weyl transformation and field normalization, not asserted here.

For the calculation, take $X$ to be smooth and time-oriented, $ι : W \to X$ a smooth timelike embedding, and $g^{i n d, s} = ι^{*} G_{s t r}$ a smooth Lorentzian metric of signature $(-, +, +, +)$ . The coordinates $(t, x^{1}, x^{2}, x^{3})$ have dimensions of length; $a_{s}$ and the metric components are dimensionless in these coordinates. The adopted worldvolume model is [A1], with $T = T (t)$ ; a Weyl/dilaton/volume transformation to a four-dimensional Einstein-frame action and any corresponding field normalization would be separate input.

Example references

Alternative string-cosmology model data

A. Sen, Tachyon Matter, arXiv:hep-th/0203265

M. Cicoli, J. P. Conlon, A. Maharana, S. Parameswaran, F. Quevedo and I. Zavala, String Cosmology: from the Early Universe to Today, arXiv:2303.04819v2

Facts & Assumptions

[F1]

Alternative string-cosmology model data: A model datum specifies its spacetime dimension and metric frame, action and fields, background branch, transition data, correction-control regime, and observable outputs; unprovided transfer data remain missing.

[A1]

Adopt the open-string tachyon worldvolume action $S_{s} [T, g] = - \int_{W} d^{4} x V (T) \sqrt{- \det (g_{μ ν}^{i n d, s} + \partial_{μ} T \partial_{ν} T)}$ , with the positive overall source normalization absorbed into $V$ . Sen's flat-space action uses $A_{μ ν} = η_{μ ν} + \partial_{μ} T \partial_{ν} T$ ; the induced-metric replacement in this item is an additional adopted covariantization. Here $V (T) \geq 0$ on the branch and $∣ \dot{T} ∣ < 1$ for a homogeneous real-action solution. The action is an effective model input; this item does not derive it from a general brane action or a global string construction.

Verification

Given: [A1], a compactly supported metric variation, and the homogeneous flat-FLRW setup above.

1.1givenA1algebra

The matrix determinant lemma gives $\det (g^{i n d, s} + \partial T \partial T) = \det (g^{i n d, s}) Q$ , where $Q = 1 + g^{i n d, s, μ ν} \partial_{μ} T \partial_{ν} T > 0$ . Thus [A1] has density $L_{s} = - V (T) \sqrt{- g} \sqrt{Q}$ .

2.1A1step 1.1algebra

Varying the reduced density with respect to $g^{μ ν}$ gives $δ S_{s} = \frac{1}{2} \int d^{4} x \sqrt{- g} (g_{μ ν} V \sqrt{Q} - \frac{V}{\sqrt{Q}} \partial_{μ} T \partial_{ν} T) δ g^{μ ν}$ . Therefore $T_{μ ν} = - \frac{2}{\sqrt{- g}} \frac{δ S_{s}}{δ g^{μ ν}} = \frac{V}{\sqrt{Q}} \partial_{μ} T \partial_{ν} T - g_{μ ν} V \sqrt{Q}$ .

3.1A1step 2.1algebra

For $T = T (t)$ , $Q = 1 - {\dot{T}}^{2}$ . Then $T_{00} = V {\dot{T}}^{2} / \sqrt{Q} + V \sqrt{Q} = V / \sqrt{1 - {\dot{T}}^{2}}$ and $T_{i j} = - g_{i j} V \sqrt{Q}$ . Thus $ρ_{s} = V (T) / \sqrt{1 - {\dot{T}}^{2}}$ and $p_{s} = - V (T) \sqrt{1 - {\dot{T}}^{2}}$ .

4.1step 3.1algebra

Their ratio is $w_{s} = p_{s} / ρ_{s} = - (1 - {\dot{T}}^{2})$ whenever $V > 0$ . Hence $w_{s} \to 0$ as $∣ \dot{T} ∣ \to 1$ . This ratio alone does not guarantee finite nonzero density.

5.1F1A1step 4.1algebra∎

If, in addition, the chosen solution has $V (T) \to 0$ and $ρ_{s} \to ρ_{*} > 0$ while $∣ \dot{T} ∣ \to 1$ , then $p_{s} = - ρ_{s} (1 - {\dot{T}}^{2}) \to 0$ and the homogeneous stress approaches a pressureless component of density $ρ_{*}$ . The action alone selects no such solution and proves no Einstein-frame cosmology.

Depends on

Alternative string-cosmology model data