String-Inspired Running Vacuum—The “Vacuumon”—And the Swampland Criteria
Abstract
:1. Introduction: Embedding Effective Field Theory Models into Quantum Gravity and the Problems of Cosmology
2. The Swampland Conjectures
- (1)
- The Field-theory-space Distance Conjecture (DfsC) [36]If an effective low-energy field theory (EFT) of some string theory contains a scalar field $\varphi $, which changes by a “distance” (in field space) $\Delta \varphi $, from some initial value, then for the EFT to be valid one must have$$\kappa \phantom{\rule{0.166667em}{0ex}}|\Delta \varphi |\lesssim {c}_{1}\phantom{\rule{3.33333pt}{0ex}},$$
- (2)
- The First Swampland Conjecture (FSC) [37]This conjecture provides restrictions on the scalar field(s) self interactions within the framework of a EFT stemming from some microscopic string theory. In particular, for the EFT to be valid, the gradient of the scalar potential V in field space must satisfy:$$\frac{|\nabla V|}{V}\gtrsim {c}_{2}\phantom{\rule{0.166667em}{0ex}}\kappa >0$$The FSC (2) rules out slow-roll inflation (implying that the standard slow-roll parameter $\u03f5={\left(\sqrt{2}\phantom{\rule{0.166667em}{0ex}}\kappa \right)}^{-1}\phantom{\rule{0.166667em}{0ex}}\left(\right|{V}^{\prime}|/V)$ is of order one), and with it several phenomenologically successful inflationary models.
- (3)
- This conjecture states that, if the scalar potential possesses a local maximum, then near that local maximum the minimum eigevalue of the theory-space Hessian $\mathrm{min}\left({\nabla}_{i}\phantom{\rule{0.166667em}{0ex}}{\nabla}_{j}V\right)$, with ${\nabla}_{i}$ denoting gradient of the potential $V\left({\varphi}_{j}\right)$ with respect to the (scalar) field ${\varphi}_{i}$, should satisfy the following constraint:$$\frac{\mathrm{min}\left({\nabla}_{i}\phantom{\rule{0.166667em}{0ex}}{\nabla}_{j}V\right)}{V}\phantom{\rule{0.166667em}{0ex}}\le \phantom{\rule{0.166667em}{0ex}}-{c}_{3}\phantom{\rule{0.166667em}{0ex}}{\kappa}^{2}\phantom{\rule{0.166667em}{0ex}}<0$$We should stress at this point that in models for which the SSC applies but not the FSC, the entropy-bound-based derivation of FSC [38] still holds, but for a range of field values outside the regime for which the local maximum of the potential occurs. The critical value/range on the magnitude of the fields for the entropy-bound implementation of FSC depends on the details of the underlying microscopic model, as do the parameters $\gamma ,b$ appearing in (13).
- (4)
- Warm Inflation modified First Swampland Conjecture (WImFSC) [40]A slight modification of the SSC has been discussed in [40], for the case of warm inflation, that is, inflationary models for which there is an interaction of the inflaton field $\varphi $ with, say, the radiation fields, of energy density ${\rho}_{r}$, via:$$\begin{array}{cc}& {\dot{\rho}}_{r}+4H\phantom{\rule{0.166667em}{0ex}}{\rho}_{r}=\mathcal{Y}\phantom{\rule{0.166667em}{0ex}}{\dot{\varphi}}^{2}\\ & \ddot{\varphi}+(3H+\mathcal{Y})\phantom{\rule{0.166667em}{0ex}}\dot{\varphi}+{V}^{\prime}=0,\end{array}$$$$\frac{|{V}^{\prime}|}{V}\phantom{\rule{0.166667em}{0ex}}>\phantom{\rule{0.166667em}{0ex}}\frac{2\phantom{\rule{0.166667em}{0ex}}\gamma \phantom{\rule{0.166667em}{0ex}}b}{3-b}\phantom{\rule{0.166667em}{0ex}}{\left|1-\frac{1}{3-\delta}\phantom{\rule{0.166667em}{0ex}}\frac{1-\frac{{\kappa}^{2}\phantom{\rule{0.166667em}{0ex}}\dot{s}}{6\pi \dot{H}}}{1+\frac{{\kappa}^{2}\phantom{\rule{0.166667em}{0ex}}s}{6\pi H}}\right|}^{-1}\phantom{\rule{0.166667em}{0ex}}\kappa ,$$The aforementioned swampland conditions FSC, SSC and WImFSC, are all incompatible with slow-roll inflation, in the sense that in string theory de Sitter vacua seem to be excluded.
- (5)
- Non-Critical String Modification of the FSC [45]At this point we remark that in non-critical (supercritical) string cosmology models, where the target time is identified with the (world-sheet zero mode of the) (time-like) Liouville mode [46,47], the swampland criteria are severely relaxed, as a result of drastic modifications in the relevant Friedmann equation [45], arising from the non-criticality of the string.$$\frac{|\frac{d}{d\mathsf{\Phi}}V(\mathsf{\Phi},\dots )|}{V(\mathsf{\Phi},\dots )}\phantom{\rule{0.166667em}{0ex}}>\phantom{\rule{0.166667em}{0ex}}\mathcal{O}\left({e}^{-\left|\mathrm{constant}\right|\phantom{\rule{0.166667em}{0ex}}}{}^{\mathsf{\Phi}}\right)\phantom{\rule{0.166667em}{0ex}}>\phantom{\rule{0.166667em}{0ex}}0\phantom{\rule{0.166667em}{0ex}},$$
3. The Weak Gravity Conjectures
4. The Running Vacuum Model (RVM): A Brief Review of Its Most Important Features
4.1. RVM Cosmic Evolution
4.2. Embedding RVM in String Theory
4.3. The “Vacuumon” Representation of the Running Vacuum Model: Early Eras
- (i)
- Rhe generic RVM, assuming relativistic matter,$$\begin{array}{cc}\hfill U\left(\varphi \right)& ={U}_{0}\phantom{\rule{0.277778em}{0ex}}\frac{2+{\mathrm{cosh}}^{2}\left(\kappa \varphi \right)}{{\mathrm{cosh}}^{4}\left(\kappa \varphi \right)}\phantom{\rule{0.277778em}{0ex}},\hfill \end{array}$$$$\begin{array}{cc}\hfill {U}_{0}& =\frac{{H}_{I}^{2}}{\alpha {\kappa}^{2}}\phantom{\rule{0.277778em}{0ex}},\hfill \\ \hfill \kappa \phantom{\rule{0.166667em}{0ex}}\varphi \left(a\right)& ={\mathrm{sinh}}^{-1}\left(\sqrt{D}{a}^{2}\right)=ln\left(\sqrt{D}\phantom{\rule{0.166667em}{0ex}}{a}^{2}+\sqrt{D\phantom{\rule{0.166667em}{0ex}}{a}^{4}+1}\right).\hfill \end{array}$$
- (ii)
- The specific string-inspired RVM [1,2], with “stiff” gravitational axion b “matter” at early epochs, (33).$$\begin{array}{cc}\hfill U\left(\varphi \right)& ={\tilde{U}}_{0}\phantom{\rule{0.277778em}{0ex}}\frac{{\textstyle \frac{2}{3}}+{\mathrm{cosh}}^{2}\left(\kappa \varphi \right)}{{\mathrm{cosh}}^{4}\left(\kappa \varphi \right)}\phantom{\rule{0.277778em}{0ex}},\hfill \end{array}$$$$\begin{array}{cc}\hfill {\tilde{U}}_{0}& =\frac{9{H}_{I}^{2}}{\alpha {\kappa}^{2}}\phantom{\rule{0.277778em}{0ex}},\hfill \\ \hfill \kappa \phantom{\rule{0.166667em}{0ex}}\varphi \left(a\right)& =\sqrt{\frac{2}{3}}\phantom{\rule{0.166667em}{0ex}}{\mathrm{sinh}}^{-1}\left(\sqrt{{D}_{\mathrm{string}}}{a}^{3}\right)=\sqrt{\frac{2}{3}}\phantom{\rule{0.166667em}{0ex}}ln\left(\sqrt{{D}_{\mathrm{string}}}\phantom{\rule{0.166667em}{0ex}}{a}^{3}+\sqrt{{D}_{\mathrm{string}}\phantom{\rule{0.166667em}{0ex}}{a}^{6}+1}\right).\hfill \end{array}$$
5. String-Inspired RVM and the Swampland
- (i)
- On plotting the quantity ${\kappa}^{-2}\phantom{\rule{0.166667em}{0ex}}{V}^{\u2033}/V$ versus the field $\varphi >0$, for positive values, as required by the vacuumon representation [34], we observe from Figure 2 (upper panels) that the SSC, (3), is satisfied (${\kappa}^{-2}\phantom{\rule{0.166667em}{0ex}}{V}^{\u2033}/V\lesssim -1$) for small vacuumon field values $0<\kappa \phantom{\rule{0.166667em}{0ex}}\varphi \lesssim 0.4$ in both cases (38) and (40). This range defines the bulk of the de Sitter (inflationary) phase for which $D\phantom{\rule{0.166667em}{0ex}}{a}^{4}\ll 1$ or ${D}_{\mathrm{string}}\phantom{\rule{0.166667em}{0ex}}{a}^{6}\ll 1$, in the generic and string-inspired RVM cases, respectively. We interpret this result as implying that in this range the vacuumon EFT are consistent with an UV completion from string theory.
- (ii)
- As the early-era vacuumon field continues to grow, exceeding a “critical range” range $\kappa \varphi >0.4$ (see discussion in Section 1), towers of string states become light and affect the EFT (the “critical range”). This phase characterises the exit from inflation phase of the RVM, in which most of its entropy is generated due to such states. For $\kappa \phantom{\rule{0.166667em}{0ex}}\varphi \phantom{\rule{0.166667em}{0ex}}\gg \phantom{\rule{0.166667em}{0ex}}1$ we observe (Figure 2, lower panels) that the quantities $|{V}^{\prime}|/V$ approximate a constant value, $-2$ for the generic RVM with radiation-type matter, and $-2.45$ for the string-inspired RVM with “stiff” stringy matter.These values coincide with the value of ${V}^{\prime}/V$ in the regime$$D\phantom{\rule{0.166667em}{0ex}}{a}^{4}\gg 1,\phantom{\rule{2.em}{0ex}}{D}_{\mathrm{string}}\phantom{\rule{0.166667em}{0ex}}{a}^{6}\gg 1$$Indeed, in this range, the entropy of the RVM, which is mainly due to the tower of string states that are becoming light (cf. (8)), saturates the Bousso upper bound (10) and becomes the Bekenstein–Gibbons–Hawking entropy of the RVM [52,53,54,55,56,57,58]. This is a standard property of the RVM [73], and we used it in [1,2] in order to connect smoothly the exit from inflation, corresponding to an almost constant (slowly varying) background axion field b, $\dot{b}\phantom{\rule{0.166667em}{0ex}}\simeq \phantom{\rule{0.166667em}{0ex}}\mathrm{constant}$, with a temperature dependent configuration for this field $b\left(T\right)$ during the radiation era, which leads to leptogenesis [91,92].Let us now demonstrate this. Consider the saturated from below (12) and saturate also the Bousso bound for the entropy (10), assuming that the tower of string states that descend from the UV are mainly point like, which implies $\delta \simeq 0$. We have then:$$\frac{|{V}^{\prime}|}{V}\simeq {(ln\left[{N}^{\gamma}\right])}^{\prime}=-2\phantom{\rule{0.166667em}{0ex}}\frac{{H}^{\prime}}{H},\phantom{\rule{1.em}{0ex}}\kappa \phantom{\rule{0.166667em}{0ex}}\varphi \gg 1.$$We now use the set of equations (23) together with (38), or (34) together with (40), depending on whether we consider generic RVM with relativistic matter [70,71] or the string-inspired RVM with stiff stringy b-axion matter of [1,2], respectively. We consider the range (43), appropriate for the exit-from-inflation phase. From (44), we then obtain$$\frac{|{V}^{\prime}|}{V}\simeq {(ln\left[{N}^{\gamma}\right])}^{\prime}=-2\phantom{\rule{0.166667em}{0ex}}\frac{{H}^{\prime}}{H}$$$$\frac{|{V}^{\prime}|}{V}\simeq 4\phantom{\rule{0.166667em}{0ex}}\frac{1}{a}\phantom{\rule{0.166667em}{0ex}}\frac{da}{d\varphi}\simeq 2\phantom{\rule{0.166667em}{0ex}}\kappa \phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}D\phantom{\rule{0.166667em}{0ex}}{a}^{4}\gg 1,$$$$\frac{|{V}^{\prime}|}{V}\simeq 6\phantom{\rule{0.166667em}{0ex}}\frac{1}{a}\phantom{\rule{0.166667em}{0ex}}\frac{da}{d\varphi}\simeq \sqrt{6}\phantom{\rule{0.166667em}{0ex}}\kappa \phantom{\rule{0.166667em}{0ex}}\simeq 2.45\phantom{\rule{0.166667em}{0ex}}\kappa \phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}{D}_{\mathrm{string}}\phantom{\rule{0.166667em}{0ex}}{a}^{4}\gg 1$$The reader can readily verify from the lower panels of Figure 2 that these are indeed the asymptotic limit of the function $\frac{|{V}^{\prime}|}{V}$ for large $\kappa \phantom{\rule{0.166667em}{0ex}}\varphi $.
6. Conclusions, Vacuumon and the Weak Gravity Conjecture
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Mavromatos, N.E.; Solà Peracaula, J.; Basilakos, S. String-Inspired Running Vacuum—The “Vacuumon”—And the Swampland Criteria. Universe 2020, 6, 218. https://doi.org/10.3390/universe6110218
Mavromatos NE, Solà Peracaula J, Basilakos S. String-Inspired Running Vacuum—The “Vacuumon”—And the Swampland Criteria. Universe. 2020; 6(11):218. https://doi.org/10.3390/universe6110218
Chicago/Turabian StyleMavromatos, Nick E., Joan Solà Peracaula, and Spyros Basilakos. 2020. "String-Inspired Running Vacuum—The “Vacuumon”—And the Swampland Criteria" Universe 6, no. 11: 218. https://doi.org/10.3390/universe6110218