String cosmology backgrounds from classical string geometry

Abstract

We introduce a very early universe model based on the thermodynamics of a gas of closed strings in a background that is nonperturbative in α′. Upon considering the fully α′-corrected equations extended to include certain anisotropic cosmological backgrounds, we describe the evolution of the system in three different stages parametrized by the gas’s equation of state. Using standard string thermodynamical arguments, we start with an isotropic ten-dimensional universe inside the string scale and evolve it toward a universe with four large spacetime dimensions and six stabilized internal dimensions in the Einstein frame.

I. Introduction

The ΛCDM model is quite successful. Relying solely on six free parameters, it is able to account for most of the current cosmological data, which has become abundant for the last 30 years. It provides a description of the evolution of the Universe that extends from a fraction of a second to its current age, around 13.8 billion years.

An attachment to the ΛCDM model is the inflationary paradigm for the very early universe. Inflation postulates a phase of accelerated expansion in the early universe that explains why the Universe we live in seems to be so spatially flat, so large, and nearly homogeneous. It also explains how the small fluctuations in the cosmic microwave background are generated and why they are almost scale invariant, and therefore it also explains how structures such as galaxies and galaxy clusters have been formed in our Universe. However, inflation does not explain away all the problems. Both the ΛCDM model and inflation rely on general relativity, which is shown to be unavoidably singular in the very early universe considering these models’ matter content. It is expected that only a fully fledged theory of quantum gravity could yield a nonsingular cosmology, thus explaining what really happens to the spacetime close to diverging curvature regions.

String theory is one of the most promising candidates for a quantum gravity theory. Among its successes, the theory provides a possible framework for unifying all the known interactions of nature. One of the main advantages to consider strings as being fundamental instead of point particles is the fact that the singularity theorems may be avoided. This is easy to understand intuitively, since as the energy scale gets higher, the energy can flow into the additional degrees of freedom present due to the extra dimensionality of the string.

In fact, not only does string theory have new degrees of freedom, it also contains new symmetries. Particularly, on compact manifolds, strings also have winding modes, besides the quantized momentum modes, that correspond to strings wound in closed cycles. Because of the existence of these different types of modes, toroidal compactifications present a new symmetry: T-duality. This symmetry implies that physics in geometries with characteristic radius R is equivalent to physics in geometries with characteristic radius given by the squared string length divided by R, where the string length is the fundamental length of the theory.

Furthermore, it is worth noting that a thermodynamical treatment of a gas of strings also obeys this symmetry, which can be seen from the thermal partition function of a gas of closed strings in a toroidal background. This implies that the temperature as a function of radius remains finite as the torus’s radius runs from zero to the string length while considering the total entropy to be constant. Moreover, if the gas of strings contains a large entropy, then, for a wide range of values of the radius on either side of the string scale, the temperature hovers just below the Hagedorn temperature, the maximal temperature for a gas of closed strings.

Given that thermal effects may be important for realistic cosmological backgrounds, the above considerations gave rise to the string gas cosmology scenario, according to which the spacetime geometry is locally the real line times a nine-torus and the universe emerges from a phase in which matter is made of a gas of strings with temperature close to the Hagedorn temperature, while the T-duality symmetry in the matter sector is unbroken. It was postulated that this phase is quasistatic in the sense that the scale factor in the Einstein frame is nearly constant. Later, it was shown that thermal fluctuations of the string gas lead to a nearly scale-invariant spectrum of cosmological perturbations with a small red tilt for the scalar modes and a slight blue tilt for tensor modes. The fluctuations are Gaussian and have Poisson-suppressed non-Gaussianities on large scales. Hence, string gas cosmology yields an alternative to the cosmological inflationary paradigm for explaining the origin of structure in the universe. (Recently the dynamics of string gas cosmology has been embedded into a more general proposal called the Emergent scenario.)

As studied elsewhere, size moduli of the extra spatial dimensions are naturally stabilized at the string scale by the interplay between momentum and winding modes. Similarly, shape moduli of the extra dimensions can be stabilized by stringy effects. Nonperturbative effects like gaugino condensation can be used to stabilize the dilaton without interfering with the stabilization of the other moduli. This nonperturbative mechanism then leads to supersymmetry breaking at the string scale. The key open issue in string gas cosmology is to justify the assumption that the Einstein frame scale factor is, in fact, nearly constant in the high temperature phase. If we were to use Einstein gravity, we would not obtain an almost constant scale factor in a phase of high string gas energy density.

However, the Einstein equations are clearly not the correct equations to use for the background dynamics since they are inconsistent with the T-duality symmetry of string theory. Pre-big-bang cosmology is an attempt to study early universe cosmology in the context of dilaton gravity where there is a scale factor duality symmetry between solutions. However, the static phase required by string gas cosmology is not a solution of the equations, and even if it were it would not be justified since such equations are not valid anymore for high energy densities.

The dilaton-gravity equations are actually low energy equations for the bosonic sector of the supergravity theory for the background massless fields of superstring theories, once we turn off all the fluxes. In fact, the massless Neveu-Schwarz sector is universal for all ten-dimensional superstring theories and has the same action for closed superstrings. In applications to cosmology, such equations are typically sourced by the energy-momentum tensor of a perfect fluid. For a gas of strings, the energy-momentum tensor has exactly this form with an equation of state that depends on the modes that dominate the gas: for compact directions with size smaller than the string length, winding modes are energetically favorable; for compact directions larger than the string length, momentum modes are.

If we are after solutions with high energy density, such as during the static phase in the Einstein frame of string gas cosmology, we need to correct the bosonic Neveu-Schwarz sector of the supergravity action with higher order operators. These operators are associated with α′ and string-coupling corrections. The former are related to the string length, the square root of α′, which sets the string scale, thus present even at classical level, while the latter are due to string interactions and account for quantum corrections. They correspond to the two-dimensional sigma model and spacetime perturbative expansions, respectively. Having the set of fully corrected equations is one of the most desirable achievements in string theory, as it could be used to answer all sorts of nonperturbative and phenomenological questions.

An interesting point of view is that due to the extensive nature of its fundamental constituents, string theory gives rise to a new kind of geometry at the nonperturbative level, a string quantum geometry. In the limit of vanishing string coupling and vanishing ratio of α′ to the squared radius of spacetime curvature we are back to Einstein theory plus classical fields, while at any given order in both expansions there are corrections to this limit. Note that these limits are not completely independent, since the string coupling is not a free parameter, being fixed by the dilaton’s vacuum expectation value. However, if the equations admit solutions with a small string coupling, then we can neglect the quantum corrections to leading order while keeping all α′ corrections in the nonperturbative regime, which gives rise to the classical string geometry limit, that is, the geometry of the tree-level string theory.

Although it is expected that the final equations are background invariant, significant progress has been made recently for purely time-dependent backgrounds at tree level. This was due to the fact that for such backgrounds there is a noncompact symmetry acting in the field space. Indeed, for a cosmological ansatz in D equal to d plus one dimensions, the scale factor duality is a particular discrete transformation within a global O(d,d) group. Restricted to the lowest order terms, a duality covariant formalism was established, including the energy momentum tensor of a gas of strings, that was shown to transform covariantly under the O(d,d) group. Moreover, it was shown that the O(d,d) symmetry should be present to all orders in α′ and, in fact, it was shown that although the first corrections modify the duality transformation, there are field variables in which they remain unchanged. Assuming that to be the case at any order, all possible corrections were classified for the vacuum case. The formalism was extended to include matter couplings through an O(d,d) invariant matter action, establishing the α′-cosmology framework within which perturbative and nonperturbative solutions were found. Such solutions were later shown to hold even with a nontrivial dilatonic charge, and their stability under homogeneous perturbations was studied.

In the following sections, we propose an early universe cosmological scenario based on these solutions. It starts off with ten dimensions where nine spatial dimensions are smaller than the string length and evolves such that at the end we have four large spacetime dimensions while the other six spatial dimensions remain stabilized around the string length. This is realized after considering a gas of strings sourcing the equations assuming the expected evolution of the equation of state for a gas of strings in the most natural way and then solving the dynamics in three stages. Surprisingly, the α′-corrected equations support a static phase in the Einstein frame as postulated by string gas cosmology, though in ten dimensions.

The outline of the paper is as follows. In Sec. II, we summarize the construction and heuristics of the model. In Sec. III, we discuss technical details, in particular how we can get four-dimensional equations from α′-cosmology after having extended the framework to include a certain class of anisotropic cosmological backgrounds. The quantitative aspects of the model are introduced in Sec. IV, where the dynamics both in the string frame and in the Einstein frame are discussed. Then we conclude in Sec. V.

II. Summary of the Model

In string gas cosmology, the thermodynamics of a gas of strings in a compact space of d spatial dimensions can be separated into three types of equation of state assuming a barotropic perfect fluid, in which the pressure is the equation-of-state parameter times the energy density: a winding equation of state, with parameter minus one over d; a radiation equation of state, with parameter one over d; and a pressureless one, with parameter zero.

Indeed, for a noninteracting isotropic gas of strings on an isotropic toroidal background, winding modes are energetically favorable if the radius of the torus is smaller than the string length, so that the fluid is dominated by these modes and has a winding equation of state. On the other hand, momentum modes dominate when the radius is greater than the string length such that the fluid has a radiation equation of state in this case. Close to the T-duality self-dual radius, both modes contribute with the same magnitude to the pressure, but with opposite signs, giving rise effectively to a dustlike equation of state, since the oscillatory modes which are also excited around the self-dual radius yield a pressureless fluid as well.

Note that in order to calculate how each string state contributes to the energy and pressure of the string gas, the mass spectrum of a single string in a static toroidal spacetime is used. In string gas cosmology, an adiabatic approximation is assumed, such that we can approximate the spectrum in a cosmological spacetime by simply promoting the radius of the torus to be the time dependent scale factor. In the following, we use the results obtained from the adiabatic approximation, in particular the equations of state described above, even though the background is an expanding Friedmann-Lemaître-Robertson-Walker cosmology with Hubble parameter close to the string scale. The justification comes from T-duality, since once it holds to all orders in α′, there should be winding and momentum modes among the states. Thus, even for the full spectrum in the time-dependent background, we expect these modes to dominate the string gas states.

In the string frame, we assume an initial high density string gas phase on a cosmological spacetime that has the topology of the real line times a nine-torus, with all spatial directions compactified on a nine-dimensional torus with radius smaller than the string length. As discussed in previous paragraphs, the string gas starts off with a winding equation of state. Now, given that the energy density is closer to the string scale, the α′-corrected cosmological equations should be the ones to rule the background evolution, which is expected to be nonperturbative in α′. It was shown that there are nonperturbative d-dimensional de Sitter solutions, with constant Hubble parameter in the string frame and constant equation of state, and the dilaton’s evolution completely parametrized by these two constants. Thus, the natural solution for this initial stage is a compactified ten-dimensional de Sitter solution with a winding equation of state, with all directions expanding until their physical radius becomes of the order of the string length. This stage corresponds to a static phase in the Einstein frame.

As the background approaches a characteristic length equal to the string size, the string gas fluid ceases to have a winding equation of state, since oscillatory and momentum modes start to get excited. Thus, the equation of state evolves toward zero as the physical radii get closer to the string scale. This establishes the second stage of the dynamics. There is an important caveat here: as the equation of state approaches zero, the geometry departs from being isotropic in all spatial directions and it divides into two independent isotropic sectors: an internal six-dimensional one, for which the equation of state associated with these dimensions stops evolving as it reaches zero, and an external three-dimensional one, for which the equation of state keeps evolving toward a radiation equation of state. This happens because the winding modes can annihilate completely only in the latter sector, while in the former they remain existing and helping to stabilize the internal equation of state together with the momentum modes. In Sec. III C, we show explicitly that there are solutions with static directions with vanishing pressure and non-evolving equation of state so that the balance between winding and momentum modes at the string scale can potentially stabilize all the internal directions, as previously suggested. This is the end of the dynamics of the internal directions in the string frame, while in the Einstein frame they start to contract.

Since the winding modes completely decay into momentum modes in the external sector, the equation of state continues evolving until becoming a radiation equation of state, which allows the external directions to remain dynamical. This, together with the freezing of the internal sector, is the string gas cosmology mechanism for generating a three-dimensional cosmology from a ten-dimensional one. After the radiation equation of state is settled, we enter the third stage of the model, where we have an anisotropic cosmology with six static spatial internal directions with equation-of-state parameter zero and three evolving external directions with parameter one third. The relevant nonperturbative solution is now locally four-dimensional de Sitter times a six-torus, in the string frame, with a rolling dilaton whose velocity is determined by the evolution of the external directions.

Meanwhile, the dilaton has been evolving so far approximately linearly with time. Although it is possible to choose the dilaton’s initial value such that we reach the third stage in the small string coupling regime, when the internal directions are stabilized and the external ones continue to expand, there is no bound on the dilaton’s time evolution. Thus, we eventually enter in the quantum nonperturbative regime, where the string coupling is of order one. From this moment on, as we would like to have a string theory based model, we cannot fully trust the O(d,d) covariant equations anymore because they do not include string-coupling corrections. Instead, it is known that nonperturbative effects such as D-branes may dominate the theory’s spectrum, giving rise, for instance, to gaugino condensation. Physically the dilaton should acquire a potential that stabilizes it. Thus, in order to potentially make contact with standard big bang cosmology, we seek for perturbative solutions with a constant dilaton. It was shown that this condition completely fixes the solution to be a perturbatively corrected radiation solution of the graviton-dilaton equations that is known to be determined once a constant dilaton is assumed. As time goes by, the perturbative corrections get smaller and the solution approaches the lowest order one with a radiation equation of state.

After the stabilization of the dilaton, there is no difference between the string and Einstein frames. But during the three stages described above, the dilaton time dependence is fixed by the solutions. That is the reason why it is possible to describe how each stage evolves in the Einstein frame. It is important to notice that both frames are equivalent in the sense that any physical observable can be calculated and has the same value regardless of frames. Besides that, describing the Einstein frame evolution is useful when trying to make contact with observations. During the first stage, with a winding equation of state, the Einstein frame scale factor is constant and we have a static ten-dimensional phase as a solution of the nonperturbative equations, in contrast with the four-dimensional static phase postulated by string gas cosmology. In the second and third stages, the dilaton’s evolution is independent of the static internal directions that in the Einstein frame correspond to contracting dimensions, while the external directions first undergo accelerated expansion and then later expand as a radiation dominated universe.

The model is summarized in Fig. 1, where the Hubble radius of the internal and external directions and the equation of state are schematically plotted as a function of time. Quantitative details about the stages can be found in Sec. IV. Finally, it is important to emphasize that prior to α′-cosmology, there were no equations that could describe the dynamics of the model as elucidated above. In particular, the existence of static solutions in the Einstein frame for a winding equation of state and the static solutions for the internal directions in the string frame with a pressureless equation of state, which were essential for the model, are here derived for the first time using the framework discussed here.

III. α′-cosmology: time-dependent backgrounds from classical string geometry

(This section — A, Review of α′-cosmology; B, Anisotropic metric in α′-cosmology; C, Nonisotropic solutions with static internal directions — and Section IV, Emergent cosmological scenario, with its three stages worked through in both the string frame and the Einstein frame, are omitted for length: they consist almost entirely of displayed equations, which do not survive text extraction. The complete text is at the source. The results those sections establish are stated in the summary and the conclusion reproduced here, and the numbered equations referred to in the claims above are found there.)

IV. Emergent cosmological scenario

(Sections omitted for length; the complete text is at the source. Two results from this section are quoted in the conclusion below: Equations 71 and 72 give the Einstein-frame Hubble rate of the external directions during stages two and three, each corresponding to a phase of superexponential accelerated expansion which would reach a finite time singularity if it lasted.)

V. Conclusion and Discussions

In this paper we have built the first very early universe cosmological model based on α′-cosmology and inspired by the string gas cosmology scenario. Our model provides for the first time dynamics for the Einstein frame quasistatic phase advocated by string gas cosmology. Moreover, with reasonable assumptions, we have shown that the nonperturbative equations of α′-cosmology are compatible with the dynamical mechanism of string gas cosmology to generate a four-dimensional cosmology starting from ten dimensions, as required by string theory.

Our dynamical system consists of the equations of α′-cosmology coupled to a matter sector being given by a gas of strings described by a barotropic perfect fluid. From the thermodynamics of the strings, we can model the evolution of the equation of state for both the internal and the external directions. To solve these equations, we break the time evolution into different stages and consider each stage separately.

In our model, all nine spatial dimensions start off with an equal size smaller than the string length, which implies that the dominant modes are winding, with equation-of-state parameter minus one ninth. This corresponds to a de Sitter expansion in the string frame and to a static phase in the Einstein frame. As the dimensions expand in the former, the density of states of winding modes decays as other modes are excited, and the equation of state grows until it becomes that of a pressureless fluid.

As the matter energy drifts from the winding modes to other string excitations and the equation of state parameter approaches zero, the dynamics stops being isotropic since winding modes can completely disappear only in three spatial dimensions. Thus, there result two sectors, each of which we model as isotropic: one with six internal directions and the other with three external ones. The pressure in the former remains around zero due to the interplay of winding and momentum modes, while the equation of state for the latter keeps evolving toward a radiation equation of state; that is, the density of states is dominated by momentum modes. The internal dynamics freezes out completely in the string frame at this point.

Once the equation of state parameter becomes one third for the external directions, the dynamics is divided into two phases: a nonperturbative solution in α′ for when the energy scale is still around the string scale, and a perturbative solution for the low-curvature regime. The transition between these two is given by the stabilization of the dilaton. The first phase is described by a short de Sitter solution in the string frame, which corresponds to superexponential acceleration in the Einstein frame, while the latter converges to a typical radiation dominated solution for both frames.

Needless to say, our model can be further improved, even within the framework of α′-cosmology as well as given that we are at the moment not able to solve the equations for an evolving equation of state parameter. In particular, it would be interesting to explore the phenomenological consequences of our result that there is a short phase of accelerated expansion for the external dimensions in the Einstein frame. Could this be enough to account for the observed spatial flatness of the universe? In a companion work we have shown that the answer is yes, proving that the model can be made compatible with standard cosmology at the background level.

Besides, in order to make contact with the most important cosmological successes of string gas cosmology, namely its prediction of almost scale-invariant power spectra for scalar and tensor perturbations with red and blue tilts, respectively, we would need to consider cosmological perturbations starting in a ten-dimensional isotropic background with a rolling dilaton. The calculations considered in the context of string gas cosmology so far have been made for a constant dilaton and in a four-dimensional space. One reason to believe that the results could be robust is the fact that the earlier results are based mostly on holographic scaling of thermodynamic fluctuations, and this may be robust to the change in the background dynamics.

The reader might have become suspicious about our arguments concerning dilaton stabilization. Indeed, our discussion concerning how this happens remains to be improved in the context of α′-cosmology. In fact, it may be the case that we do not even need to rely on its stabilization by any other mechanism than the sole evolution of the equation of state. The reason for that is that we might be able to make a strong argument purely based on the equations of motion that imply that the Hubble parameter is decaying as the equation of state evolves upward, similarly as derived in Appendix A when the parameter is evolving away from the winding mode dominance. Then, the transition between stages 2 and 3 could end with the Hubble parameter already lower than the string scale, such that we could consider directly the perturbative solution with a rolling dilaton for which its evolution asymptotes to a constant.

Finally, let us comment on the connection of our work to double field theory. The dynamical equations of α′-cosmology as studied in the cited works do not necessarily assume a compact background. Thus, the O(d,d) group explored in those works is present even in the noncompact case. In our model, we have assumed a compact background, so the O(d,d) discussed in the present work is part of the T-duality group. In fact, we can recover it from the generalized coordinate transformations of double field theory. Thus, a possible avenue of exploration is to embed our model into double field theory, or at least to describe its first stage in a T-dual frame, where instead of considering the directions’ size to be smaller than the string scale and expanding, the dimensions are large and contracting.

While the present work was in review, an interesting paper appeared presenting new vacuum solutions including a nontrivial Neveu-Schwarz two-form field. In the model developed here, the energy density of the string gas source cannot be neglected in any phase. Hence, those new solutions cannot be immediately used to improve our model. However, the equations developed there are more general than the ones used to get the solutions of Sec. III, for they include the coupling with the two-form field. It would be of great interest to add matter to that setup and to study whether this would help us improve our model.

Acknowledgments

The research at McGill is supported, in part, by funds from NSERC and from the Canada Research Chair program. R. B. thanks the Pauli Center and the Institutes for Theoretical Physics and of Particle Astrophysics of the ETH for hospitality.

Appendix A: Transition between stage 1 and stage 2

(The appendix shows that as the equation of state parameter increases over time while remaining negative, the Hubble parameter decreases when starting off in a de Sitter phase, by considering linear perturbations of the equations of motion. It is omitted for length; the complete text is at the source.)