Pre-big-bang dilaton-driven cosmology proposal

In string units ${\mathrm{\ell }}_s=\sqrt{{\alpha }^{\mathrm{\prime }}}=1$, use the homogeneous string-frame cosmic-time metric $d{s}_{\mathrm{s}\mathrm{t}\mathrm{r}}^2=-d{t}^2+{\sum }_i{a}_i(t{)}^2(d{x}^i{)}^2$, with dimensionless $t,x^i$ and radii $a_i>0$. Define scale-factor duality by $(a_i,\phi )\mapsto (a_i^{-1},\phi -2{\sum }_i\log a_i)$ and the string coupling by $g_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}=e^{\phi \mathrm{/}2}$. For an isotropic branch let $t_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}={\mathrm{\ell }}_{s}t$ and $H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:={\mathrm{\ell }}_s^{-1}\dot{a}\mathrm{/}a$; acceleration means $d^{2}a\mathrm{/}d{t}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}^2>0$, and deceleration means $d^{2}a\mathrm{/}d{t}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}^2<0$. In cosmic gauge $N=1$, the comoving observer is $u=e_{\widehat{0}}={\mathrm{\ell }}_s^{-1}{\mathrm{\partial }}_t={\mathrm{\partial }}_{t_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}$, with worldlines at fixed torus coordinates, and the spatial orthonormal fields are $e_{\widehat{i}}=({\mathrm{\ell }}_s{a}_i{)}^{-1}{\mathrm{\partial }}_{x^i}$. Thus $g_{\mathrm{s}\mathrm{t}\mathrm{r}}(e_{\widehat{a}},e_{\widehat{b}})=\mathrm{diag}(-1,1,\dots ,1{)}_{ab}$. Use $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z$ for the metric connection and $R_{\widehat{a}\widehat{b}\widehat{c}\widehat{d}}:=g_{\mathrm{s}\mathrm{t}\mathrm{r}}(e_{\widehat{a}},R(e_{\widehat{c}},e_{\widehat{d}})e_{\widehat{b}})$. The specified comoving orthonormal curvature diagnostic is ${\mathcal{K}}_{\mathrm{s}\mathrm{t}\mathrm{r}}:=\mathrm{\mid }d{H}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}d{t}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}+H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}^2\mathrm{\mid }$. The isotropic solution of the adopted tree-level, two-derivative model has an expanding accelerated pre-big-bang branch, but as $t\to 0^{-}$ it reaches $g_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}\to \mathrm{\infty }$ and ${\alpha }^{\mathrm{\prime }}{\mathcal{K}}_{\mathrm{s}\mathrm{t}\mathrm{r}}\to \mathrm{\infty }$; its scale-factor-dual, time-reversed partner is $a(t)=t^{1\mathrm{/}\sqrt{d}}$ for $t>0$, which is decelerating for $d>1$ and coasting for $d=1$. A transition to the post-big-bang hot phase is an additional prescription, and the reduced equations do not establish a nonsingular controlled transition.

Take $d\ge 1$ and a flat torus with dimensionless coordinates $x^i$ of period $2\pi$. Let ${\mathrm{\ell }}_s=\sqrt{{\alpha }^{\mathrm{\prime }}}$ be the string length, ${\alpha }^{\mathrm{\prime }}$ have units of length squared, and use the dimensionless string-unit time $t=t_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}{\mathrm{\ell }}_s$ and dimensionless radii $a_i=R_i\mathrm{/}{\mathrm{\ell }}_s$. On the interval $I=(-\mathrm{\infty },0)$, the functions $a_i,\mathrm{\Phi },N$ are smooth and $N>0$; increasing $t$ fixes the future direction. The string-frame line element is

$d{s}_{\mathrm{s}\mathrm{t}\mathrm{r}}^2={\mathrm{\ell }}_s^2(-N(t{)}^{2}d{t}^2+{\sum }_{i=1}^d{a}_i(t{)}^2(d{x}^i{)}^2),t\in (-\mathrm{\infty },0),N(t)>0.$

Thus $a_i$, ${\lambda }_i:=\log a_i$, the lapse $N$, the dilaton $\phi :=2\mathrm{\Phi }$, and $t$ are dimensionless numerical variables in string units; physical radii are ${\mathrm{\ell }}_s{a}_i$, physical proper time is ${\mathrm{\ell }}_{s}t$ after setting $N=1$, and $g_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}:=e^{\mathrm{\Phi }}=e^{\phi \mathrm{/}2}$ is dimensionless. The metric $g_{\mathrm{s}\mathrm{t}\mathrm{r}}$ is the metric carrying the tree-level dilaton prefactor in the adopted action. Here “tree-level” means no string-loop terms are retained, and “two-derivative” means the reduced action contains no terms with more than two time derivatives. A dot in the reduced action means derivative with respect to dimensionless $t$, and $H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}={\mathrm{\ell }}_s^{-1}\dot{a}\mathrm{/}a$ for the isotropic branch. Write the shifted dilaton as $\bar{\phi }=\phi -{\sum }_{i=1}^d{\lambda }_i$. In this example the adopted homogeneous, tree-level, two-derivative model is the lapse action in [A1], with its overall positive normalization set to one in string units; it is a reduced model input, not a derivation of a globally consistent string compactification. Its isotropic pre-big-bang branch expands from the regime ${\alpha }^{\mathrm{\prime }}{\mathcal{K}}_{\mathrm{s}\mathrm{t}\mathrm{r}}\ll 1$, $g_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}\ll 1$ as $t\to -\mathrm{\infty }$ toward a strong-coupling, high-curvature boundary at $t\to 0^{-}$. The branch is related by scale-factor duality followed by time reversal to an expanding branch for $t>0$, decelerating for $d>1$ and coasting for $d=1$. The transition to the hot post-big-bang phase and its perturbation-transfer rule are recorded as missing; no bounce is inferred.

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