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Ekpyrotic/cyclic contracting cosmology proposal

Physical scope

Conditional four-dimensional Einstein-frame canonical-scalar contracting model. It proves the exact exponential-potential branch and linear homogeneous Bianchi-I shear suppression for c^2>6; it asserts no nonsingular bounce, global string realization, or perturbation transfer.

Example

Use natural units $ℏ = c_{l i g h t} = 1$ , and call $g$ Einstein-frame when the gravitational action has the constant coefficient $M_{P}^{2} / 2$ multiplying its Ricci scalar. Let $t, x^{i}$ have mass dimension $- 1$ and $a$ be positive, smooth and dimensionless. The flat FLRW metric is $d s^{2} = - d t^{2} + a (t)^{2} δ_{i j} d x^{i} d x^{j}$ , with $H : = \dot{a} / a$ . For a canonical scalar, meaning kinetic density $- \frac{1}{2} g^{μ ν} \partial_{μ} ϕ \partial_{ν} ϕ$ , set $ρ_{ϕ} : = {\dot{ϕ}}^{2} / 2 + V$ and $P_{ϕ} : = {\dot{ϕ}}^{2} / 2 - V$ , and define $w : = P_{ϕ} / ρ_{ϕ}$ where $ρ_{ϕ} \neq 0$ . For the linear homogeneous Bianchi-I deformation $d s_{B I}^{2} = - d t^{2} + \sum_{i = 1}^{3} a_{i} (t)^{2} (d x^{i})^{2}$ , $a_{i} = a e^{β_{i}}$ , $β_{i} \in C^{2} (I)$ , and $\sum_{i} β_{i} = 0$ , define directional rates $H_{i} : = {\dot{a}}_{i} / a_{i}$ and shear components $σ_{i} : = H_{i} - H$ . With $ϕ$ and $M_{P}$ of mass dimension one, $V_{0}$ of mass dimension four, and $c$ a dimensionless potential slope (not the speed of light), the potential $V = - V_{0} e^{- c ϕ / M_{P}}$ with $c^{2} > 6$ gives a contracting branch with $w > 1$ and suppresses this linear homogeneous shear relative to the scalar density. A proposed bounce and perturbation transfer require additional premises and are not consequences of this calculation.

Use natural units $ℏ = c_{l i g h t} = 1$ . The Einstein-frame FLRW line element is $d s^{2} = - d t^{2} + a (t)^{2} δ_{i j} d x^{i} d x^{j}$ on $I = (- \infty, 0)$ , with signature $(-, +, +, +)$ , $t, x^{i}$ of mass dimension $- 1$ , and dimensionless smooth scale factor $a > 0$ . Let $ϕ \in C^{2} (I, R)$ be a homogeneous canonical scalar, $M_{P} > 0$ a mass, $V_{0} > 0$ of mass dimension four, and $c$ a dimensionless potential slope. Then $V (ϕ) = - V_{0} e^{- c ϕ / M_{P}}$ has mass dimension four, $H = \dot{a} / a$ has mass dimension one, and $w = P_{ϕ} / ρ_{ϕ}$ is dimensionless. Adopt [A1] with $c^{2} > 6$ . The exhibited ekpyrotic branch contracts toward $t = 0$ . For the shear calculation, the Bianchi-I metric is $d s_{B I}^{2} = - d t^{2} + \sum_{i = 1}^{3} a_{i} (t)^{2} (d x^{i})^{2}$ , with $a_{i} = a e^{β_{i}}$ , $\sum_{i} β_{i} = 0$ , and only terms linear in the smooth homogeneous $β_{i}$ retained; shear-energy backreaction is omitted at this order. This is a formal classical Einstein-scalar model: no EFT cutoff or omitted-mode bounds are supplied, and no controlled string compactification or validity at $t = 0$ is asserted. Bounce dynamics, perturbation matching/transfer, and moduli stabilization are additional missing inputs; no bounce or perturbation spectrum is asserted.

Example references

Alternative string-cosmology model data

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 datum states its metric frame, action and fields, background branch, initial and boundary data, transition prescription, perturbation/transfer information, and control regime; an absent bounce or transfer is not inferred.

[A1]

Adopt the four-dimensional Einstein-scalar action $S = \int d^{4} x \sqrt{- g} (\frac{M_{P}^{2}}{2} R - \frac{1}{2} g^{μ ν} \partial_{μ} ϕ \partial_{ν} ϕ - V (ϕ))$ , the tensor equations $M_{P}^{2} G_{μ ν} = T_{μ ν}$ and $□_{g} ϕ - V_{, ϕ} = 0$ , with no other matter or cosmological constant. Here $T_{μ ν} : = \partial_{μ} ϕ \partial_{ν} ϕ - g_{μ ν} (\frac{1}{2} g^{α β} \partial_{α} ϕ \partial_{β} ϕ + V)$ , $G_{μ ν} : = R_{μ ν} - \frac{1}{2} g_{μ ν} g^{α β} R_{α β}$ , and $□_{g} ϕ : = g^{μ ν} (\partial_{μ} \partial_{ν} ϕ - Γ^{α}_{μ ν} \partial_{α} ϕ)$ . Use $Γ^{α}_{μ ν} = \frac{1}{2} g^{α β} (\partial_{μ} g_{β ν} + \partial_{ν} g_{β μ} - \partial_{β} g_{μ ν})$ and $R_{μ ν} = \partial_{α} Γ^{α}_{μ ν} - \partial_{ν} Γ^{α}_{μ α} + Γ^{α}_{α β} Γ^{β}_{μ ν} - Γ^{α}_{ν β} Γ^{β}_{μ α}$ , summing repeated Greek indices from 0 to 3. These are adopted model equations in the displayed action normalization, not a claimed derivation of the gravitational action. For homogeneous $ϕ$ , they give $T^{0}_{0} = - ρ_{ϕ}$ and $T^{i}_{j} = P_{ϕ} δ^{i}_{j}$ . Define $ρ_{ϕ} = {\dot{ϕ}}^{2} / 2 + V$ and $P_{ϕ} = {\dot{ϕ}}^{2} / 2 - V$ . The FLRW equations are $3 M_{P}^{2} H^{2} = ρ_{ϕ}$ , $\dot{H} = - {\dot{ϕ}}^{2} / (2 M_{P}^{2})$ , and $\ddot{ϕ} + 3 H \dot{ϕ} + V_{, ϕ} = 0$ . For the shear calculation use the Bianchi-I metric $d s_{B I}^{2} = - d t^{2} + \sum_{i = 1}^{3} a_{i} (t)^{2} (d x^{i})^{2}$ with $a_{i} = a e^{β_{i}}$ , $a = (a_{1} a_{2} a_{3})^{1 / 3}$ , $β_{i} \in C^{2} (I)$ , $\sum_{i} β_{i} = 0$ , and no anisotropic stress. Keep only linear order in $β_{i}$ ; the shear-energy backreaction is quadratic.

Verification

Given: [A1], $V = - V_{0} e^{- c ϕ / M_{P}}$ , $V_{0} > 0$ , $c^{2} > 6$ , and $t < 0$ .

1.1givenA1algebra

Put $p = 2 / c^{2}$ , so $0 < p < 1 / 3$ , and choose $a_{0} > 0$ and $A > 0$ by $A^{2} = V_{0} / [p (1 - 3 p) M_{P}^{2}]$ . The dimensions are $[A] =$ mass and $[A τ] = 1$ , so $a_{0}$ and $A τ$ are dimensionless. For $τ = - t > 0$ , set $a (t) = a_{0} (A τ)^{p}$ and $ϕ (t) = (2 M_{P} / c) \log (A τ)$ . Then $H = p / t < 0$ , $\dot{ϕ} = 2 M_{P} / (c t)$ , $K = {\dot{ϕ}}^{2} / 2 = p M_{P}^{2} / t^{2}$ , and $V = - p (1 - 3 p) M_{P}^{2} / t^{2}$ .

1.2givenA1algebra

For the Bianchi-I metric, the only nonzero connection coefficients from [A1] are $Γ^{0}_{i i} = a_{i}^{2} H_{i}$ and $Γ^{i}_{0 i} = Γ^{i}_{i 0} = H_{i}$ (no sum over $i$ ). In the Ricci formula, the derivative term is $\partial_{t} (a_{i}^{2} H_{i}) = a_{i}^{2} ({\dot{H}}_{i} + 2 H_{i}^{2})$ , the connection trace product is $a_{i}^{2} H_{i} \sum_{k} H_{k}$ , and the last two nonzero products total $- 2 a_{i}^{2} H_{i}^{2}$ ; spatial derivatives vanish. Thus $R^{i}_{i} = {\dot{H}}_{i} + H_{i} \sum_{k} H_{k}$ and $G^{i}_{i} - G^{j}_{j} = {\dot{H}}_{i} - {\dot{H}}_{j} + 3 H (H_{i} - H_{j})$ . The homogeneous stress in [A1] has equal spatial diagonal entries, so this difference vanishes. Write $σ_{i} = H_{i} - H = {\dot{β}}_{i}$ , with $\sum_{i} σ_{i} = 0$ . The pairwise equations make all ${\dot{σ}}_{i} + 3 H σ_{i}$ equal; their sum is zero, so each is zero. Consequently $a^{3} σ_{i} = C_{i}$ is constant on the interval and $σ^{2} : = \frac{1}{2} \sum_{i} σ_{i}^{2} = \frac{1}{2} \sum_{i} C_{i}^{2} a^{- 6}$ . These identities hold for the exact diagonal homogeneous equations; in the stated perturbation calculation retain $σ_{i}$ to first order and its squared amplitude to second order, while evaluating $a$ and $ρ_{ϕ}$ on the unperturbed branch.

2.1A1step 1.1algebra

The expressions in step 1.1 give $K + V = 3 p^{2} M_{P}^{2} / t^{2} = 3 M_{P}^{2} H^{2}$ . Also $\ddot{ϕ} = - 2 M_{P} / (c t^{2})$ and $V_{, ϕ} = - (c / M_{P}) V$ ; substituting $p = 2 / c^{2}$ gives $\ddot{ϕ} + 3 H \dot{ϕ} + V_{, ϕ} = 0$ . Thus the prescribed contracting branch solves the adopted flat-FLRW equations.

3.1A1step 2.1algebra

Its pressure is $P_{ϕ} = K - V$ , so $w = P_{ϕ} / (K + V) = (2 p - 3 p^{2}) / (3 p^{2}) = 2 / (3 p) - 1 = c^{2} / 3 - 1 > 1$ . Since $a \to 0$ as $t \to 0^{-}$ , the background scalar density $ρ_{ϕ} = 3 M_{P}^{2} H^{2}$ scales as $a^{- 2 / p} = a^{- c^{2}}$ .

4.1F1step 3.1step 1.2algebra

Comparing this linearized homogeneous shear with the FLRW background density from step 3.1 gives $σ^{2} / ρ_{ϕ} \propto a^{c^{2} - 6} \to 0$ during contraction because $c^{2} > 6$ . This proves suppression of the stated shear mode only; it does not prove nonlinear stability to every anisotropy or a controlled bounce.

5.1F1A1step 2.1algebra∎

On any smooth canonical-scalar FLRW segment, [A1] gives $\dot{H} = - {\dot{ϕ}}^{2} / (2 M_{P}^{2}) \leq 0$ . Therefore $H$ cannot pass from $H < 0$ to $H > 0$ on such a segment. A bounce requires an additional premise or a different regime, consistent with the source's statement that the ekpyrotic bounce is a separate unresolved construction.

Depends on

Alternative string-cosmology model data

Dependency tree · 2 levels

Sources

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