# Super-accelerating bouncing cosmology in asymptotically free non-local gravity

- 505 Downloads
- 27 Citations

## Abstract

Recently, evidence has been collected that a class of gravitational theories with certain non-local operators is renormalizable. We consider one such model which, at the linear perturbative level, reproduces the effective non-local action for the light modes of bosonic closed string-field theory. Using the property of asymptotic freedom in the ultraviolet and fixing the classical behavior of the scale factor at late times, an algorithm is proposed to find general homogeneous cosmological solutions valid both at early and late times. Imposing a power-law classical limit, these solutions (including anisotropic ones) display a bounce, instead of a big-bang singularity, and super-accelerate near the bounce even in the absence of an inflaton or phantom field.

### Keywords

Beltrami Operator Asymptotic Freedom Wick Rotation Critical Energy Density Minkowski Background## 1 Introduction

Asymptotic freedom is an attribute of field theories such that interactions are negligible in the ultraviolet (UV), where the theory possesses a trivial fixed point. This property has been used, directly or implicitly, to construct field theories of gravity where the Laplace–Beltrami operator \(\Box \) is replaced by one or more non-local operators \(f(\Box )\) in the kinetic terms of the action. On a Minkowski background, if the Fourier transform \(\tilde{f}(k^2)\rightarrow \infty \) for large momenta \(k\), these terms dominate in the UV over the interactions, which can be ignored. Asymptotic freedom then, if realized, ensures that the correct UV behavior of the theory be encoded in the free propagator. A class of these models is of particular relevance inasmuch as the kinetic operator is of the exponential form \(\Box \text {e}^\Box \) or \(\text {e}^\Box \) and is inspired, respectively, by open string-field theory (OSFT; see [1, 2, 3, 4] for reviews) and the \(p\)-adic string [5, 6, 7, 8, 9, 10]. In the simplest classical cosmological applications, gravity is local and the only non-local content is a scalar field [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], sometimes identified with the tachyon of bosonic OSFT or of super-symmetric OSFT on an unstable brane.

Non-local gravity sectors and their cosmology have been proposed in [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57], following various criteria including avoidance of ghosts, improved renormalizability, and the possibility to construct non-perturbative solutions. The gathered results (also in theories without non-local operators at the tree level [58]) point towards a resolution of gravitational singularities thanks to asymptotic freedom. These approaches do not stem from closed string-field theory, i.e., the SFT sector containing the perturbative graviton mode [59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73]. Yet, many of them are inspired by it in the sense that the kinetic functions are exponential (or somewhat more general) operators as in effective closed SFT (see also [74, 75, 76, 77, 78] for other types of non-locality). In this context, one bypasses the technical difficulties in getting effective non-local actions directly from SFT [62, 73] and concentrates on phenomenological but more manageable models.

Cosmological dynamical solutions were obtained either directly, by solving the non-local equations of motion, or indirectly, by solving an *Ansatz* for the Ricci curvature (or via the diffusion equation [44]) respecting the equations of motion. Here we pursue a different but no less economic alternative, using asymptotic freedom as a key ingredient.

^{1}(iii) Next, we write down an effective Friedmann equation of the form

## 2 The model

### 2.1 Effective action from closed string-field theory

### 2.2 Non-local gravity model

^{2}

From the point of view of string-field theory, the action (8) only contains massless fields, but in general the whole tower of string massive particle modes should be taken into account. However, for the cosmology-related purposes of the present paper the massless sector is more than enough, since gravity is included in it and, on the other hand, cosmological matter can be modeled by a perfect fluid as a first approximation, even if at the fundamental level it is constituted by fields with non-local dynamics. In this regard, notice that the form of the kinetic operators in (8) is expected to hold also for massive modes [93] (consult that reference for a longer discussion). One simple example of this is given by the string tachyon, which is a massive field although with negative squared mass; as one can see from Eq. (6), its kinetic operator is exactly of the same form as for the massless fields. Another motivation to drop massive states from the discussion is that a perturbative truncation of the non-local operators would correspond to a small Regge slope. By keeping these operators intact we imply no special requirement on the size of squared momenta \(k^2\) with respect to \(\alpha '\). In particular, one is entitled to explore configurations such that \(\alpha 'k^2\gg 1\). This is the region in parameter space where all the masses of the tower, which are multiples of the fundamental mass \(1/\sqrt{\alpha '}\), can be ignored with respect to kinetic terms \(\propto k^2\gamma (-k^2)\). Therefore, a truncation of the massive tower is in principle compatible with keeping the non-local operators untruncated.

In this paper, we study asymptotic profiles which are approximate cosmological solutions of the gravitational system (8a). The classical action (8a) is a “non-polynomial” or “semi-polynomial” extension of quadratic Stelle theory [90, 95]. All the non-polynomiality is incorporated in the form factor \(\gamma (\Box )\). The entire function \(V\) has no poles in the whole complex plane, which preserves unitarity, and it has at least logarithmic behavior in the UV regime to give super-renormalizability at the quantum level.

Here we only consider corrections to the classical solutions coming from the bare two-point function of the graviton field. The reason is that this class of theories is asymptotically free and the leading asymptotic behavior of the dressed propagator is dominated by its bare part. In fact, according to power counting arguments [37, 47], the self-energy insertions, which are constant or at most logarithmic, do not contribute to it.

## 3 Cosmology

### 3.1 General solution in asymptotically free gravity

*Ansatz*(10) is not to be understood in the sense of cosmological perturbation theory, where \(\kappa h\ll 1\) is a perturbation small everywhere and at any time. Equation (10) is simply a splitting of the FRW metric, not the perturbation of Minkowski background with a generic fluctuation. Therefore, we stress that the present method has nothing to do with cosmological perturbation theory.

^{3}

This procedure is similar to the one employed in [96, 97, 98], but with an important difference. There, in order to go beyond the classical theory, one introduces one-loop quantum corrections to the graviton propagator. In our case, however, we already have modifications at the classical level and, therefore, we use only the bare propagator. This can be justified by noting, as mentioned above, that one-loop corrections to the propagator in this class of non-local theories are UV sub-dominant with respect to the tree-level contribution. Thus, a general conclusion is that any asymptotically free theory of gravity with a two-point function of the form (15), with sufficiently strong damping factor \(V\), will admit an asymptotic UV solution of the form (11) solely found at the tree level in perturbation theory.

### 3.2 General solution with power-law regime

*real*parameter) and it correctly reaches the classical power-law solution of the Lorentzian theory at late times. Therefore, the profile \(a(t)\) is unaffected by the analytic continuation. In Sect. 3.3, we will recover the same solution with an independent method.

We conclude this section with three comments. First, we have not checked the stability of the profile (22). In the case of general non-polynomial theories, it is not obvious whether non-locality may trigger potentially dangerous instabilities. This is not so in the present case, where the non-local operator is the exponential of the d’Alembertian. There is reiterated evidence in the literature that such a strongly damping form factor actually improves any stability-related problem (exponential non-locality is under much greater control than other non-local models; see, e.g., [84, 103]). Moreover, stability of the solution can be checked by looking at the behavior at early and late times separately (the complete perturbation would then be given by joining the two asymptotic behaviors). At late times, however, our solution reduces to standard power-law cosmology with standard dynamical equations, whose properties are well known. Therefore, one would need to verify stability only in the asymptotic-freedom regime. In this limit, however, the dynamical equations are linear in \(a^2\) and the perturbation analysis becomes trivial. Therefore, the linear homogeneous perturbation \(\delta (a^2)\approx 2 a(t) \delta a(t)\) of the solution \(a(t)\) obeys the same linearized equation of the background (with unperturbed differential operators), \(\delta a\propto a\), and the perturbed background \(a+\delta a\propto a\) is only a physically irrelevant re-normalization of the scale factor. As a side remark, there should be no instability issues generated by non-locality even at the level of inhomogeneities. This is expected on the grounds that non-local theories with entire functions, such as exponential operators, do not introduce ghost, tachyon or Laplacian instabilities. We do not expect these arguments to be altered by a full calculation, which could be performed only with the knowledge of the full dynamics. This goes beyond the goals of the present paper.

Finally, we also checked that, in models where a Wick rotation is not necessary, the bounce picture persists. Let us recall that a Wick rotation was required because, in momentum space, the form factor (9) is not convergent when integrating in \(k^0\), \(V(k^2)=\exp (\varLambda ^2k^2)=\exp [\varLambda ^2(k_0^2-|\varvec{k}|^2)]\). On the other hand, using even powers of the Laplace–Beltrami operator renders the form factor convergent without transforming to imaginary time, and the bouncing-accelerating scenario still holds. For instance, in Krasnikov’s model with the operator \(V(\Box )=\text {e}^{-\Box ^2/\varLambda ^4}\), the solution with power-law asymptotic limit is a superposition of generalized hypergeometric functions \( _qF_s\), bounded from below (respectively, above) for \(p>0\) (\(<0\)) by a symmetric bounce at some \(a_*\ne 0\).

### 3.3 Alternative derivation of the solution

The solution (22) can also be found via the diffusion-equation method [82], which has proven to be a powerful tool both to address the Cauchy problem in exponential-type non-local systems [103] and to find non-perturbative tachyon solutions in string theory at the level of target actions [25, 82, 104, 105]. While this method works well for matter fields on Minkowski background, the diffusion equation becomes non-linear when applied to the metric itself, since its solution appears also in the Laplace–Beltrami operator. For this reason, a particular gravitational action was constructed which allowed one to circumvent this problem [44]. Here, on the other hand, the action (8) gives rise to non-local, non-linear equations of motions for which the diffusion approach seems unsuitable. Fortunately, asymptotic freedom guarantees a window of applicability of the method in the limit where all interactions, including those of gravity with itself, are negligible.

^{4}Summarizing the procedure in a nutshell, one expands the initial condition \(a^2(t,0)\) as an integral superposition of the eigenfunctions \(\text {e}^{\pm \text {i}E t}\) of the exponential operator \(V^{-1}(\Box )\). Applying \(V^{-1}(\Box )\) to the initial condition and performing the integration, one obtains a linear combination \(C_1 a^2_1(t,r)+C_2a^2_2(t,r)\) of the two solutions to the second-order equation (30), where

### 3.4 Effective dynamics

## 4 Conclusions

In this paper, we have proposed a non-local model of gravity with improved UV behavior. We have focused only on its cosmology, and used the property of asymptotic freedom to find approximate solutions valid both at early and late times. In general, these solutions possess a bounce and avoid the big-bang singularity, and have an early era of acceleration with a natural exit in the absence of inflaton fields. Specifically, the universe is characterized by a super-acceleration regime at the bounce, with effective barotropic index \(w<-1\).

Since we have not solved the full equations of motion exactly, we have bypassed the problem of getting the dynamics of a theory with all sectors (both gravity and matter) being non-local. Instead, we have matched the solution (22) with an effective dynamics encoded in the modified Friedmann equation (1). The good agreement between this solution and Eq. (1) at all scales suggests that the problem is not unsolvable. The diffusion-equation method, applied in Sect. 3.3 to find an alternative derivation of the profile (22), might be a useful tool in this respect. Intriguingly, Eq. (30) acts as a “beta function,” determining the running of the metric with the cut-off length scale \(1/\varLambda \), which plays the role of diffusion time \(\sim \sqrt{r}\). The possibility to study the dynamics of this class of non-local theories via the diffusion method is a direct consequence of their renormalization properties [82].

It is remarkable that the bouncing dynamics of the present model can be reproduced semi-quantitatively by an effective equation with only one free parameter. The value of the critical energy density \(\rho _*\) is also plausible. From Eq. (34) and setting the cut-off to its natural value \(\varLambda =m_\mathrm{Pl}\), we get \(\rho _*\approx 0.02\,\rho _\mathrm{Pl}\) for \(p=1/2\) and \(\rho _*\approx 0.22\,\rho _\mathrm{Pl}\) for \(p=3\). Models such that \(\rho \le \rho _*\le \rho _\mathrm{Pl}\) must have \(p\le p_\mathrm{max}\approx 12.67\). For \(p> p_\mathrm{max}\), the critical energy density exceeds \(\rho _\mathrm{Pl}\), and the energy density of the universe can become trans-Planckian near and at the bounce. Therefore, scenarios with too-large \(p\) are not well represented by Eq. (1). This is not an issue in our model, since we have early-universe acceleration by default also for small values of \(p\).

The relative error between the approximate solution \(H^2\) and the effective energy density \(\kappa ^2\rho _\mathrm{eff}/3\) could be reduced by a more refined *Ansatz* for the effective Friedmann equation. Further study of this method may turn out to shed some light on the exact dynamics, which should be developed in parallel starting from the actual equations of motion. For instance, once derived the actual equations of motion of the theory one could plug the solution \(a(t)\) found here, and check whether a reasonable matter sector is recovered. In particular, from our modified Kasner solution it should be possible to check whether some form of BKL chaos survives in the full anisotropic dynamics.

This should not only clarify whether the bounce picture is robust in our theory, but also to which class of singularity-free cosmologies the model belongs to. As an exact dynamical equation, expression (1) with \(\beta =1\) appears also in braneworlds with a timelike extra direction [106] and in purely homogeneous loop quantum cosmology (for reviews consult, e.g., [107, 108]) only in the parameter choice for the so-called “improved” mini-superspace dynamics [109, 110] (for other parametrizations, Eq. (1) with \(\beta =1\) no longer holds [111]). Although there is no relation between our framework and these high-energy cosmological models, they all share the same type of bounce where the right-hand side of the Friedmann equation receives a negative higher-order correction in the energy density. On the other hand, there is a different class of models where the correction is of the “dark radiation” form \(-1/a^4\), which is responsible for the bounce at \(H=0\). Such is the case for the Randall–Sundrum braneworld with a spacelike extra dimension [112], Hořava–Lifshitz gravity without detailed balance [113, 114], and cosmologies with fermionic condensates [115, 116]. As far as we pushed the analysis in this paper, the present model apparently lies in the first category, with the added bonus that we have an alternative mechanism of inflation of purely geometric origin.

## Footnotes

- 1.
- 2.
This action differs from the theory of [44], which is of scalar-tensor type. There, the curvature invariants appear only in the exponential non-local operator in order to allow for non-perturbative solutions via a method based on the diffusion equation.

- 3.
Throughout the paper, and unless stated otherwise, we will use the term “perturbation theory” exclusively in the sense of field theory, not of cosmology.

- 4.
The reader can track down the steps in [103] from Eqs. (45) to (81), with the following mapping from the symbols used there to those adopted here: \(\varPsi =\varPsi _3\rightarrow a^2\), \(\phi _0\rightarrow t_\mathrm{i}^{-2p}\), \(\theta /4\rightarrow p\), \(\nu \rightarrow 1/2\), \(H_0\rightarrow 0\), \(\alpha \rightarrow -1\). In [103], the opposite convention \(\eta =\mathrm{diag}(-,+,\cdots ,+)\) is used for the spacetime signature.

## Notes

### Acknowledgments

The authors thank G. Mena Marugán for comments on the manuscript and are grateful to J. Moffat for early discussions on related topics. G.C. and L.M. acknowledge the i-Link cooperation programme of CSIC (project ID i-Link0484) for partial sponsorship. The work of G.C. is under a Ramón y Cajal tenure-track contract; he also thanks Fudan University for the kind hospitality during the completion of this article. The work of P.N. has been supported by the German Research Foundation (DFG) grant NI 1282/2-1, and partially by the Helmholtz International Center for FAIR within the framework of the LOEWE program (Landesoffensive zur Entwicklung Wissenschaftlich-Ökonomischer Exzellenz) launched by the State of Hesse and by the European COST action MP0905 “Black Holes in a Violent Universe”.

### References

- 1.K. Ohmori, A review on tachyon condensation in open string field theories, hep-th/0102085
- 2.A. Sen, Tachyon dynamics in open string theory. Int. J. Mod. Phys. A
**20**, 5513 (2005). hep-th/0410103 - 3.E. Fuchs, M. Kroyter, Analytical solutions of open string field theory. Phys. Rep.
**502**, 89 (2011). arXiv:0807.4722 - 4.Y. Okawa, Analytic methods in open string field theory. Prog. Theor. Phys.
**128**, 1001 (2012)CrossRefADSGoogle Scholar - 5.P.G.O. Freund, M. Olson, Nonarchimedean strings. Phys. Lett. B
**199**, 186 (1987). doi: 10.1016/0370-2693(87)91356-6 MathSciNetCrossRefADSGoogle Scholar - 6.P.G.O. Freund, E. Witten, Adelic string amplitudes. Phys. Lett. B
**199**, 191 (1987). doi: 10.1016/0370-2693(87)91357-8 MathSciNetCrossRefADSGoogle Scholar - 7.L. Brekke, P.G.O. Freund, M. Olson, E. Witten, Nonarchimedean string dynamics. Nucl. Phys. B
**302**, 365 (1988). doi: 10.1016/0550-3213(88)90207-6 MathSciNetCrossRefADSGoogle Scholar - 8.V.S. Vladimirov, Ya.I. Volovich, On the nonlinear dynamical equation in the \(p\)-adic string theory. Theor. Math. Phys.
**138**, 297 (2004). doi: 10.1023/B:TAMP.0000018447.02723.29. math-ph/0306018 - 9.V. Vladimirov, Nonlinear equations for \(p\)-adic open, closed, and open-closed strings. Theor. Math. Phys.
**149**, 1604 (2006). doi: 10.1007/s11232-006-0144-z. arXiv:0705.4600 - 10.T. Biswas, J.A.R. Cembranos, J.I. Kapusta, Thermal duality and Hagedorn transition from \(p\)-adic strings. Phys. Rev. Lett.
**104**, 021601 (2010). doi: 10.1103/PhysRevLett.104.021601. arXiv:0910.2274 - 11.I. Ya, Aref’eva, Nonlocal string tachyon as a model for cosmological dark energy. AIP Conf. Proc.
**826**, 301 (2006). doi: 10.1063/1.2193132. astro-ph/0410443 - 12.I.Ya. Aref’eva, L.V. Joukovskaya, Time lumps in nonlocal stringy models and cosmological applications, JHEP
**0510**, 087 (2005). doi: 10.1088/1126-6708/2005/10/087. hep-th/0504200 - 13.I. Ya, Aref’eva, A.S. Koshelev, S. Yu. Vernov, Stringy dark energy model with cold dark matter. Phys. Lett. B
**628**, 1 (2005). doi: 10.1016/j.physletb.2005.09.017. astro-ph/0505605 - 14.G. Calcagni, Cosmological tachyon from cubic string field theory. JHEP
**0605**, 012 (2006). doi: 10.1088/1126-6708/2006/05/012. hep-th/0512259 - 15.I.Ya. Aref’eva, A.S. Koshelev, Cosmic acceleration and crossing of \(w=-1\) barrier from cubic superstring field theory, JHEP
**0702**, 041 (2007). doi: 10.1088/1126-6708/2007/02/041. hep-th/0605085 - 16.I. Ya, Aref’eva, I.V. Volovich, On the null energy condition and cosmology. Theor. Math. Phys.
**155**, 503 (2008). doi: 10.1007/s11232-008-0041-8. hep-th/0612098 - 17.N. Barnaby, T. Biswas, J.M. Cline, \(p\)-adic inflation. JHEP
**0704**, 056 (2007). doi: 10.1088/1126-6708/2007/04/056. hep-th/0612230 - 18.A.S. Koshelev, Non-local SFT tachyon and cosmology. JHEP
**0704**, 029 (2007). doi: 10.1088/1126-6708/2007/04/029. hep-th/0701103 - 19.I.Ya. Aref’eva, L.V. Joukovskaya, S. Yu. Vernov, Bouncing and accelerating solutions in nonlocal stringy models, JHEP
**0707**, 087 (2007). doi: 10.1088/1126-6708/2007/07/087. hep-th/0701184 - 20.I.Ya. Aref’eva, I.V. Volovich, Quantization of the Riemann zeta-function and cosmology, Int. J. Geom. Methods Mod. Phys.
**4**, 881 (2007). doi: 10.1142/S021988780700234X. hep-th/0701284 - 21.J.E. Lidsey, Stretching the inflaton potential with kinetic energy. Phys. Rev. D
**76**, 043511 (2007). doi: 10.1103/PhysRevD.76.043511. hep-th/0703007 - 22.N. Barnaby, J.M. Cline, Large nongaussianity from nonlocal inflation. JCAP
**0707**, 017 (2007). doi: 10.1088/1475-7516/2007/07/017. arXiv:0704.3426 - 23.G. Calcagni, M. Montobbio, G. Nardelli, Route to nonlocal cosmology. Phys. Rev. D
**76**, 126001 (2007). doi: 10.1103/PhysRevD.76.126001. arXiv:0705.3043 - 24.L.V. Joukovskaya, Dynamics in nonlocal cosmological models derived from string field theory. Phys. Rev. D
**76**, 105007 (2007). doi: 10.1103/PhysRevD.76.105007. arXiv:0707.1545 - 25.G. Calcagni, G. Nardelli, Nonlocal instantons and solitons in string models. Phys. Lett. B
**669**, 102 (2008). doi: 10.1016/j.physletb.2008.09.016. arXiv:0802.4395 - 26.L. Joukovskaya, Rolling solution for tachyon condensation in open string field theory, arXiv:0803.3484
- 27.I.Ya. Aref’eva, A.S. Koshelev, Cosmological signature of tachyon condensation. JHEP
**0809**, 068 (2008). arXiv:0804.3570 - 28.L. Joukovskaya, Dynamics with infinitely many time derivatives in Friedmann–Robertson–Walker background and rolling tachyons. JHEP
**0902**, 045 (2009). doi: 10.1088/1126-6708/2009/02/045. arXiv:0807.2065 - 29.N. Barnaby, N. Kamran, Dynamics with infinitely many derivatives: variable coefficient equations. JHEP
**0812**, 022 (2008). doi: 10.1088/1126-6708/2008/12/022. arXiv:0809.4513 - 30.N.J. Nunes, D.J. Mulryne, Non-linear non-local cosmology. AIP Conf. Proc.
**1115**, 329 (2009). doi: 10.1063/1.3131521. arXiv:0810.5471 - 31.A.S. Koshelev, S. Yu. Vernov, Cosmological perturbations in SFT inspired non-local scalar field models. Eur. Phys. J. C
**72**, 2198 (2012). doi: 10.1140/epjc/s10052-012-2198-4. arXiv:0903.5176 - 32.G. Calcagni, G. Nardelli, Cosmological rolling solutions of nonlocal theories. Int. J. Mod. Phys. D
**19**, 329 (2010). doi: 10.1142/S0218271810016440. arXiv:0904.4245 - 33.S. Yu. Vernov, Localization of non-local cosmological models with quadratic potentials in the case of double roots. Class. Quantum Grav.
**27**, 035006 (2010). doi: 10.1088/0264-9381/27/3/035006. arXiv:0907.0468 - 34.S. Yu. Vernov, Localization of the SFT inspired nonlocal linear models and exact solutions. Phys. Part. Nucl. Lett.
**8**, 310 (2011). doi: 10.1134/S1547477111030228. arXiv:1005.0372 - 35.A.S. Koshelev, S. Yu. Vernov, Analysis of scalar perturbations in cosmological models with a non-local scalar field. Class. Quantum Grav.
**28**, 085019 (2011). doi: 10.1088/0264-9381/28/8/085019. arXiv:1009.0746 - 36.N.V. Krasnikov, Nonlocal gauge theories. Theor. Math. Phys.
**73**, 1184 (1987). doi: 10.1007/BF01017588. http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tmf&paperid=5624&option_lang=eng [Teor. Mat. Fiz.**73**, 235 (1987)] - 37.E.T. Tomboulis, Superrenormalizable gauge and gravitational theories, hep-th/9702146
- 38.N. Arkani-Hamed, S. Dimopoulos, G. Dvali, G. Gabadadze, Nonlocal modification of gravity and the cosmological constant problem, hep-th/0209227
- 39.A.O. Barvinsky, Nonlocal action for long distance modifications of gravity theory. Phys. Lett. B
**572**, 109 (2003). doi: 10.1016/j.physletb.2003.08.055. hep-th/0304229 - 40.A.O. Barvinsky, On covariant long-distance modifications of Einstein theory and strong coupling problem. Phys. Rev. D
**71**, 084007 (2005). doi: 10.1103/PhysRevD.71.084007. hep-th/0501093 - 41.H.W. Hamber, R.M. Williams, Nonlocal effective gravitational field equations and the running of Newton’s \(G\). Phys. Rev. D
**72**, 044026 (2005). doi: 10.1103/PhysRevD.72.044026. hep-th/0507017 - 42.T. Biswas, A. Mazumdar, W. Siegel, Bouncing universes in string-inspired gravity. JCAP
**0603**, 009 (2006). doi: 10.1088/1475-7516/2006/03/009. hep-th/0508194 - 43.J. Khoury, Fading gravity and self-inflation. Phys. Rev. D
**76**, 123513 (2007). doi: 10.1103/PhysRevD.76.123513. hep-th/0612052 - 44.G. Calcagni, G. Nardelli, Nonlocal gravity and the diffusion equation. Phys. Rev. D
**82**, 123518 (2010). doi: 10.1103/PhysRevD.82.123518. arXiv:1004.5144 - 45.T. Biswas, T. Koivisto, A. Mazumdar, Towards a resolution of the cosmological singularity in non-local higher derivative theories of gravity. JCAP
**1011**, 008 (2010). doi: 10.1088/1475-7516/2010/11/008. arXiv:1005.0590 - 46.A.O. Barvinsky, Dark energy and dark matter from nonlocal ghost-free gravity theory. Phys. Lett. B
**710**, 12 (2012). doi: 10.1016/j.physletb.2012.02.075. arXiv:1107.1463 - 47.L. Modesto, Super-renormalizable quantum gravity. Phys. Rev. D
**86**, 044005 (2012). doi: 10.1103/PhysRevD.86.044005. arXiv:1107.2403 - 48.T. Biswas, E. Gerwick, T. Koivisto, A. Mazumdar, Towards singularity and ghost free theories of gravity. Phys. Rev. Lett.
**108**, 031101 (2012). doi: 10.1103/PhysRevLett.108.031101. arXiv:1110.5249 - 49.A.S. Koshelev, Modified non-local gravity, Rom. J. Phys.
**57**, 894 (2012). arXiv:1112.6410. http://www.nipne.ro/rjp/2012_57_5-6.html - 50.L. Modesto, Super-renormalizable higher-derivative quantum gravity, arXiv:1202.0008
- 51.A.S. Koshelev, SYu. Vernov, On bouncing solutions in non-local gravity. Phys. Part. Nucl.
**43**, 666 (2012). doi: 10.1134/S106377961205019X. arXiv:1202.1289 - 52.S. Alexander, A. Marcianò, L. Modesto, The hidden quantum groups symmetry of super-renormalizable gravity. Phys. Rev. D
**85**, 124030 (2012). doi: 10.1103/PhysRevD.85.124030. arXiv:1202.1824 - 53.L. Modesto, Super-renormalizable multidimensional quantum gravity: theory and applications. Astron. Rev.
**8.2**, 4 (2013). arXiv:1202.3151 - 54.L. Modesto, Towards a finite quantum supergravity, arXiv:1206.2648
- 55.T. Biswas, A.S. Koshelev, A. Mazumdar, S. Yu. Vernov, Stable bounce and inflation in non-local higher derivative cosmology. JCAP
**1208**, 024 (2012). doi: 10.1088/1475-7516/2012/08/024. arXiv:1206.6374 - 56.F. Briscese, A. Marcianò, L. Modesto, E.N. Saridakis, Inflation in (super-)renormalizable gravity. Phys. Rev. D
**87**, 083507 (2013). doi: 10.1103/PhysRevD.87.083507. arXiv:1212.3611 - 57.A.S. Koshelev, Stable analytic bounce in non-local Einstein–Gauss–Bonnet cosmology. Class. Quantum Grav.
**30**, 155001 (2013). doi: 10.1088/0264-9381/30/15/155001. arXiv:1302.2140 - 58.B. Hasslacher, E. Mottola, Asymptotically free quantum gravity and black holes. Phys. Lett. B
**99**, 221 (1981). doi: 10.1016/0370-2693(81)91112-6 MathSciNetCrossRefADSGoogle Scholar - 59.M. Saadi, B. Zwiebach, Closed string field theory from polyhedra. Ann. Phys.
**192**, 213 (1989). doi: 10.1016/0003-4916(89)90126-7 - 60.T. Kugo, H. Kunitomo, K. Suehiro, Nonpolynomial closed string field theory. Phys. Lett. B
**226**, 48 (1989). doi: 10.1016/0370-2693(89)90287-6 MathSciNetCrossRefADSGoogle Scholar - 61.T. Kugo, K. Suehiro, Nonpolynomial closed string field theory: action and its gauge invariance. Nucl. Phys. B
**337**, 434 (1990). doi: 10.1016/0550-3213(90)90277-K MathSciNetCrossRefADSGoogle Scholar - 62.V.A. Kostelecký, S. Samuel, Collective physics in the closed bosonic string. Phys. Rev. D
**42**, 1289 (1990). doi: 10.1103/PhysRevD.42.1289 CrossRefADSGoogle Scholar - 63.B. Zwiebach, Closed string field theory: quantum action and the B-V master equation. Nucl. Phys. B
**390**, 33 (1993). doi: 10.1016/0550-3213(93)90388-6. hep-th/9206084 - 64.A. Sen, B. Zwiebach, A proof of local background independence of classical closed string field theory. Nucl. Phys. B
**414**, 649 (1994). doi: 10.1016/0550-3213(94)90258-5. hep-th/9307088 - 65.A. Sen, B. Zwiebach, Quantum background independence of closed string field theory. Nucl. Phys. B
**423**, 580 (1994). doi: 10.1016/0550-3213(94)90145-7. hep-th/9311009 - 66.A. Sen, B. Zwiebach, A note on gauge transformations in Batalin–Vilkovisky theory. Phys. Lett. B
**320**, 29 (1994). doi: 10.1016/0370-2693(94)90819-2. hep-th/9309027 - 67.Y. Okawa, B. Zwiebach, Twisted tachyon condensation in closed string field theory. JHEP
**0403**, 056 (2004). doi: 10.1088/1126-6708/2004/03/056. hep-th/0403051 - 68.H. Yang, B. Zwiebach, Dilaton deformations in closed string field theory. JHEP
**0505**, 032 (2005). doi: 10.1088/1126-6708/2005/05/032. hep-th/0502161 - 69.H. Yang, B. Zwiebach, A closed string tachyon vacuum? JHEP
**0509**, 054 (2005). doi: 10.1088/1126-6708/2005/09/054. hep-th/0506077 - 70.Y. Michishita, Field redefinitions, \(T\)-duality and solutions in closed string field theories. JHEP
**0609**, 001 (2006). doi: 10.1088/1126-6708/2006/09/001. hep-th/0602251 - 71.N. Moeller, Closed bosonic string field theory at quintic order: five-tachyon contact term and dilaton theorem. JHEP
**0703**, 043 (2007). doi: 10.1088/1126-6708/2007/03/043. hep-th/0609209 - 72.N. Moeller, Closed bosonic string field theory at quintic order. II: marginal deformations and effective potential. JHEP
**0709**, 118 (2007). doi: 10.1088/1126-6708/2007/09/118. arXiv:0705.2102 - 73.N. Moeller, A tachyon lump in closed string field theory. JHEP
**0809**, 056 (2008). doi: 10.1088/1126-6708/2008/09/056. arXiv:0804.0697 - 74.S. Nojiri, S.D. Odintsov, Modified non-local-\(F(R)\) gravity as the key for the inflation and dark energy. Phys. Lett. B
**659**, 821 (2008). doi: 10.1016/j.physletb.2007.12.001. arXiv:0708.0924 - 75.S. Capozziello, E. Elizalde, S. Nojiri, S.D. Odintsov, Accelerating cosmologies from non-local higher-derivative gravity. Phys. Lett. B
**671**, 193 (2009). doi: 10.1016/j.physletb.2008.11.060. arXiv:0809.1535 - 76.S.I. Nojiri, S.D. Odintsov, M. Sasaki, Y.-L. Zhang, Screening of cosmological constant in non-local gravity. Phys. Lett. B
**696**, 278 (2011). doi: 10.1016/j.physletb.2010.12.035. arXiv:1010.5375 - 77.Y.-L. Zhang, M. Sasaki, Screening of cosmological constant in non-local cosmology. Int. J. Mod. Phys. D
**21**, 1250006 (2012). doi: 10.1142/S021827181250006X. arXiv:1108.2112 - 78.E. Elizalde, E.O. Pozdeeva, S. Yu. Vernov, Y.-L. Zhang, Cosmological solutions of a nonlocal model with a perfect fluid. JCAP
**1307**, 034 (2013). doi: 10.1088/1475-7516/2013/07/034. arXiv:1302.4330 - 79.S. Alexander, R. Brandenberger, J. Magueijo, Noncommutative inflation. Phys. Rev. D
**67**, 081301 (2003). doi: 10.1103/PhysRevD.67.081301. hep-th/0108190 - 80.M. Rinaldi, A new approach to non-commutative inflation. Class. Quantum Grav.
**28**, 105022 (2011). doi: 10.1088/0264-9381/28/10/105022. arXiv:0908.1949 - 81.T. Biswas, A. Mazumdar, Super-inflation, non-singular bounce, and low multipoles. Class. Quantum Grav.
**31**, 025019 (2014). doi: 10.1088/0264-9381/31/2/025019. arXiv:1304.3648 - 82.G. Calcagni, G. Nardelli, String theory as a diffusing system. JHEP
**1002**, 093 (2010). doi: 10.1007/JHEP02(2010)093. arXiv:0910.2160 - 83.G. Calcagni, L. Modesto, Nonlocality in string theory. J. Phys. A
**47**, 355402 (2014). doi: 10.1088/1751-8113/47/35/355402 - 84.N. Barnaby, N. Kamran, Dynamics with infinitely many derivatives: the initial value problem. JHEP
**0802**, 008 (2008). arXiv:0709.3968 - 85.G.V. Efimov,
*Nonlocal Interactions of Quantized Fields (in Russian)*(Nauka, Moscow, 1977)Google Scholar - 86.A. Smailagic, E. Spallucci, Lorentz invariance, unitarity in UV-finite of QFT on noncommutative spacetime. J. Phys. A
**37**, 7169 (2004). doi: 10.1088/0305-4470/37/28/008. hep-th/0406174 - 87.E. Spallucci, A. Smailagic, P. Nicolini, Trace anomaly in quantum spacetime manifold. Phys. Rev. D
**73**, 084004 (2006). doi: 10.1103/PhysRevD.73.084004. hep-th/0604094 - 88.P. Nicolini, M. Rinaldi, A minimal length versus the Unruh effect. Phys. Lett. B
**695**, 303 (2011). doi: 10.1016/j.physletb.2010.10.051. arXiv:0910.2860 - 89.M. Kober, P. Nicolini, Minimal scales from an extended Hilbert space. Class. Quantum Grav.
**27**, 245024 (2010). doi: 10.1088/0264-9381/27/24/245024. arXiv:1005.3293 - 90.M. Asorey, J.L. López, I.L. Shapiro, Some remarks on high derivative quantum gravity. Int. J. Mod. Phys. A
**12**, 5711 (1997). doi: 10.1142/S0217751X97002991. hep-th/9610006 - 91.L. Modesto, J.W. Moffat, P. Nicolini, Black holes in an ultraviolet complete quantum gravity. Phys. Lett. B
**695**, 397 (2011). doi: 10.1016/j.physletb.2010.11.046. arXiv:1010.0680 - 92.L. Modesto, Finite quantum gravity, arXiv:1305.6741
- 93.G. Calcagni, L. Modesto, Proposal for field M-theory, arXiv:1404.2137
- 94.M.J. Duff, D.J. Toms, Kaluza–Klein–Kounterterms, in
*Unification of Fundamental Particle Interactions II*, ed. by J. Ellis, S. Ferrara (Springer, Amsterdam, 1983). doi: 10.1007/978-1-4615-9299-0_3 - 95.K.S. Stelle, Renormalization of higher-derivative quantum gravity. Phys. Rev. D
**16**, 953 (1977). doi: 10.1103/PhysRevD.16.953 MathSciNetCrossRefADSGoogle Scholar - 96.M.J. Duff, Quantum corrections to the Schwarzschild solution. Phys. Rev. D
**9**, 1837 (1974). doi: 10.1103/PhysRevD.9.1837 CrossRefADSGoogle Scholar - 97.B. Broda, One-loop quantum gravity repulsion in the early Universe. Phys. Rev. Lett.
**106**, 101303 (2011). doi: 10.1103/PhysRevLett.106.101303. arXiv:6257 - 98.B. Broda, Quantum gravity stability of isotropy in homogeneous cosmology. Phys. Lett. B
**704**, 655 (2011). doi: 10.1016/j.physletb.2011.09.087. arXiv:1107.3468 - 99.C. Bambi, D. Malafarina, L. Modesto, Terminating black holes in quantum gravity. Eur. Phys. J. C
**74**, 2767 (2014). doi: 10.1140/epjc/s10052-014-2767-9. arXiv:1306.1668 - 100.A. Accioly, A. Azeredo, H. Mukai, Propagator, tree-level unitarity and effective nonrelativistic potential for higher-derivative gravity theories in \(D\) dimensions. J. Math. Phys.
**43**, 473 (2002). doi: 10.1063/1.1415743 MathSciNetCrossRefMATHADSGoogle Scholar - 101.P. Van Nieuwenhuizen, On ghost-free tensor Lagrangians and linearized gravitation. Nucl. Phys. B
**60**, 478 (1973). doi: 10.1016/0550-3213(73)90194-6 CrossRefADSGoogle Scholar - 102.M.D. Pollock, On super-exponential inflation in a higher-dimensional theory of gravity with higher-derivative terms. Nucl. Phys. B
**309**, 513 (1988). doi: 10.1016/0550-3213(88)90456-7 (erratum ibid. B**374**, 469 (1992). doi: 10.1016/0550-3213(92)90363-G) - 103.G. Calcagni, M. Montobbio, G. Nardelli, Localization of nonlocal theories. Phys. Lett. B
**662**, 285 (2008). arXiv:0712.2237 - 104.G. Calcagni, G. Nardelli, Tachyon solutions in boundary and cubic string field theory. Phys. Rev. D
**78**, 126010 (2008). doi: 10.1103/PhysRevD.78.126010. arXiv:0708.0366 - 105.G. Calcagni, G. Nardelli, Kinks of open superstring field theory. Nucl. Phys. B
**823**, 234 (2009). doi: 10.1016/j.nuclphysb.2009.08.004. arXiv:0904.3744 - 106.Y. Shtanov, V. Sahni, Bouncing brane worlds. Phys. Lett. B
**557**, 1 (2003). doi: 10.1016/S0370-2693(03)00179-5. gr-qc/0208047 - 107.A. Ashtekar, P. Singh, Loop quantum cosmology: a status report. Class. Quantum Grav.
**28**, 213001 (2011). doi: 10.1088/0264-9381/28/21/213001. arXiv:1108.0893 - 108.K. Banerjee, G. Calcagni, M. Martín-Benito, Introduction to loop quantum cosmology. SIGMA
**8**, 016 (2012). doi: 10.3842/SIGMA.2012.016. arXiv:1109.6801 - 109.P. Singh, Loop cosmological dynamics and dualities with Randall–Sundrum braneworlds. Phys. Rev D
**73**, 063508 (2006). doi: 10.1103/PhysRevD.73.063508. gr-qc/0603043 - 110.A. Ashtekar, T. Pawłowski, P. Singh, Quantum nature of the big bang: improved dynamics. Phys. Rev. D
**74**, 084003 (2006). doi: 10.1103/PhysRevD.74.084003. gr-qc/0607039 - 111.G. Calcagni, G.M. Hossain, Loop quantum cosmology and tensor perturbations in the early universe. Adv. Sci. Lett.
**2**, 184 (2009). doi: 10.1166/asl.2009.1025. arXiv:0810.4330 - 112.P. Binétruy, C. Deffayet, U. Ellwanger, D. Langlois, Brane cosmological evolution in a bulk with cosmological constant. Phys. Lett. B
**477**, 285 (2000). doi: 10.1016/S0370-2693(00)00204-5. hep-th/9910219 - 113.G. Calcagni, Cosmology of the Lifshitz universe. JHEP
**0909**, 112 (2009). doi: 10.1088/1126-6708/2009/09/112. arXiv:0904.0829 - 114.E. Kiritsis, G. Kofinas, Hořava–Lifshitz cosmology. Nucl. Phys. B
**821**, 467 (2009). doi: 10.1016/j.nuclphysb.2009.05.005. arXiv:0904.1334 - 115.S. Alexander, T. Biswas, Cosmological BCS mechanism and the big bang singularity. Phys. Rev. D
**80**, 023501 (2009). doi: 10.1103/PhysRevD.80.023501. arXiv:0807.4468 - 116.S. Alexander, T. Biswas, G. Calcagni, Cosmological Bardeen–Cooper–Schrieffer condensate as dark energy. Phys. Rev. D
**81**, 043511 (2010). doi: 10.1103/PhysRevD.81.043511. arXiv:0906.5161 (erratum ibid. D**81**, 069902 (2010). doi: 10.1103/PhysRevD.81.069902)

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Funded by SCOAP^{3} / License Version CC BY 4.0.