Dark energy from string theory: an introductory review

Abstract

Dark energy, the main constituent in our expanding universe, responsible for its acceleration, is currently observed at unprecedented precision by different experiments. While several cosmological models can fit this latest data, deriving some of them from string theory would provide a valuable theoretical prior, with information on the nature of dark energy. This article reviews the efforts towards such a derivation, namely the options from string theory to get a cosmological constant (a de Sitter solution) or a dynamical dark energy (via a quintessence model). After a brief historical perspective, we first review proven or conjectured constraints in getting dark energy from string theory, in classical or asymptotic regimes. Circumventing such obstructions, by changing regime or ansatz, one can try to construct a de Sitter solution: we present a long list of such attempts, and the difficulties encountered. Among them, we discuss in detail efforts towards classical de Sitter solutions. Then, we review quintessence from string theory, focusing on single-field exponential models. Related topics are discussed, including the coupling to matter, the comparison to observational data, and the absence of a cosmological event horizon.

Figures (9)

• Figure 1: Schematic representation of a typical scalar potential $V(\varphi)$ obtained in constructed examples of 4d effective string theories. It is common that the 4d scalar field value determines the string regime, namely with non-negligible perturbative or non-perturbative contributions ($\varphi< 1$), or without in the classical ($\varphi> 1$) or asymptotic ($\varphi \rightarrow \infty$) regimes. In turn, constructions of de Sitter minima were proposed with perturbative and non-perturbative terms, while candidate de Sitter maxima, strongly unstable, were found classically. All these attempts were then subject to scrutiny, regarding the corrections to these constructions. Asymptotically, the typical expectation for a positive potential is a runaway, forbidding any de Sitter extremum.
• Figure 2: Candidate realistic cosmological solution of single field exponential quintessence, with $\lambda=\sqrt{\frac{8}{3}}$, $k=0$, $\Omega_{n0}$ in \ref{['Onfiducial']} and $w_{\varphi0}=-0.57196843265$. We observe the successive radiation, matter and dark energy domination phases in Figure \ref{['fig:solO']}, in terms of e-folds with $N=0$ today. The evolution of the field $\varphi$ (shifted by today's value) and of its equation of state parameter $w_{\varphi}$ along the solution history are given in Figure \ref{['fig:solp']} and \ref{['fig:solwp']}. The evolution of the quantity $-w_{{\rm eff}} -1/3$ is displayed in Figure \ref{['fig:solacc']}, indicating acceleration when it is positive. This emphasizes the transient acceleration phase today. The asymptotic values can be verified to match those of $P_{\varphi}$: for example, $-w_{{\rm eff}} -1/3 \rightarrow -2/9$.
• Figure 3: Cosmological solution of single field exponential quintessence, with $\lambda=\sqrt{\frac{8}{3}}$, $k=-1$, $\Omega_{k0}=0.0850$, $\Omega_{\varphi0}=0.6000$, $\Omega_{m0}=0.3149$, and $w_{\varphi0}=-0.63998750867$. It should be compared to the case without curvature, $k=0$, of Figure \ref{['fig:solutionquint']}: we see here in Figure \ref{['fig:solkO']} the modest contribution of $\Omega_k$ in the recent universe, and the modified future, that corresponds to $P_{k\varphi}$. The quantity $-w_{{\rm eff}} -1/3$, given as the plain curve in Figure \ref{['fig:solkacc']}, indicating (transient) acceleration, can be compared to the case $k=0$, i.e. the curve of Figure \ref{['fig:solacc']} represented here as dashed. The future of the two curves differ due to the change of attractor.
• Figure 4: Universal evolution of the scalar field (red dot), here at different e-folds, in a candidate realistic solution of a thawing quintessence model. The example here is the solution of the model of Figure \ref{['fig:solutionquint']} with exponential potential $V(\varphi)$. The field is first rolling in a kination regime due to some initial speed (here starting with $P_{{\rm kin}}^-$), it then gets slowed-down and frozen at a point on the slope (here $\varphi-\varphi_0 \approx -0.63$) due to the high Hubble friction during the radiation-matter domination phase. When the friction drops at the end of matter domination, the field "thaws" and rolls-down the potential while dark energy is rising.
• Figure 5: Evolution of the equation of state parameter of dark energy in terms of the scale factor $a$ in Figure \ref{['fig:solwa']} and the redshift $z$ in Figure \ref{['fig:solwz']}, where $1+z=1/a$. Today corresponds to $a=1$ and $z=0$. The green curve, above $-1$, corresponds to $w_{\varphi}$ for the candidate realistic solution of exponential quintessence with $\lambda=\sqrt{8/3}$, $k=0$, displayed in Figure \ref{['fig:solutionquint']}. Its linear fit over the period $0\leq z \leq 4$ corresponds to the red line, with parameters given by $w_0\approx w_a \approx -0.558$. The orange lines correspond to $w_0,\,w_a$ obtained from the observational data sets DESI+CMB+Union3 DESI:2025zgx, namely $w_0= -0.667^{+0.088}_{-0.088},\, w_a = -1.09^{+0.31}_{-0.27}$; the plain line is the central value, the dashed ones correspond to the errors. Those largely go below $-1$, corresponding to what is known as the phantom regime.