String Gas Cosmology
1.1 The Current Paradigm of Early Universe Cosmology
According to the inflationary universe scenario  (see also [2, 3, 4]), there was a phase of accelerated expansion of space lasting at least 50 Hubble expansion times during the very early universe. This accelerated expansion of space can explain the overall homogeneity of the universe, it can explain its large size and entropy, and it leads to a decrease in the curvature of space. Most importantly, however, it includes a causal mechanism for generating the small amplitude fluctuations which can be mapped out today via the induced temperature fluctuations of the cosmic microwave background (CMB) and which develop into the observed large-scale structure of the universe  (see also [6, 2, 7, 8]). The accelerated expansion of space stretches fixed co-moving scales beyond the Hubble radius. Thus, it is possible to have a causal mechanism which generates the fluctuations on microscopic sub-Hubble scales. The wavelengths of these inhomogeneities are subsequently inflated to cosmological scales which are super-Hubble until the late universe. The generation mechanism is based on the assumption that the fluctuations start out on microscopic scales at the beginning of the period of inflation in a quantum vacuum state. If the expansion of space is almost exponential, an almost scale-invariant spectrum of cosmological perturbations results, and the squeezing which the fluctuations undergo while they evolve on scales larger than the Hubble radius predicts a characteristic oscillatory pattern in the angular power spectrum of the CMB anisotropies , a pattern which has now been confirmed with great accuracy [10, 11] (see e.g  for a comprehensive review of the theory of cosmological fluctuations, and  for an introductory overview).
To establish our notation, we write the metric of a homogeneous, isotropic and spatially flat four-dimensional universe in the form
where is physical time, denote the three co-moving spatial coordinates (points at rest in an expanding space have constant co-moving coordinates), and the scale factor is proportional to the size of space. The expansion rate of the universe is given by
where the overdot represents the derivative with respect to time.
A space-time sketch of inflationary cosmology is shown in Fig. 1. The vertical axis is time. The inflationary phase begins at the time and lasts until the time , the time of “reheating”. At that time, the energy which is driving inflation must change its form into regular matter. The Hubble radius is labelled by and divides scales into those where micro-physics dominates and thus the generation of fluctuations by local physics is possible (sub-Hubble scales) and those where gravity dominates and micro-physical effects are negligible (super-Hubble). As shown in the sketch, during inflation fixed co-moving scales (labelled by in the sketch) are inflated from microscopic to cosmological. Note also that the horizon, the forward light cone, becomes exponentially larger than the Hubble radius during the inflationary phase.
1.2 Challenges for String Cosmology
Working in the context of General Relativity as the theory of space-time, inflationary cosmology requires the presence of a new form of matter with a sufficiently negative pressure (, where denotes the energy density). In order to obtain such an equation of state, in general the presence of scalar field matter must be assumed. In addition, it must be assumed that the scalar field potential energy dominates over the scalar field spatial gradient and kinetic energies for a sufficiently long time period. The Higgs field used for the spontaneous breaking of gauge symmetries in particle physics has a potential which is not flat enough to sustain inflation. Models beyond the Standard Model of particle physics, in particular those based on supersymmetry, typically have many scalar fields. Nevertheless, it has proven to be very difficult to construct viable inflationary models. The problems which arise when trying to embed inflation into the context of effective field theories stemming from superstring theory are detailed in the contribution to this book by Burgess.
If inflationary cosmology is realized in the context of classical General Relativity coupled to scalar field matter, then an initial cosmological singularity is unavoidable . Resolving this initial singularity is one of the challenges for string cosmology.
The energy scale during inflation is set by the observed amplitude of the CMB fluctuations. In simple single field models of inflation, the energy scale is of the order of the scale of Grand Unification, i.e. many orders of magnitude larger than scales for which field theory has been tested experimentally, and rather close to the string and Planck scales, scales where we know that the low energy effective field theory approach will break down. It is therefore a serious concern whether the inflationary scenario is robust towards the inclusion of non-perturbative stringy effects, effects which we know must not only be present but in fact must dominate at energy scales close to the string scale.
The problem for cosmological fluctuations is even more acute: provided that the inflationary phase lasts for more than about 70 Hubble expansion times, then all scales which are currently probed in cosmological observations had a wavelength smaller than the Planck length at the beginning of the inflationary phase. Thus, the modes definitely are effected by trans-Planckian physics during the initial stages of their evolution. The “trans-Planckian problem” for fluctuations [15, 16] is whether the stringy effects which dominate the evolution in the initial stages leave a detectable imprint on the spectrum of fluctuations. To answer this question one must keep in mind that the expansion of space does not wash out specific stringy signatures, but simply red-shifts wavelengths. For string theorists, the above “trans-Planckian problem” is in fact a window of opportunity: if the universe underwent a period of inflation, this period will provide a microscope with which string-scale physics can be probed in current cosmological observations.
Some of the conceptual problems of inflationary cosmology are highlighted in Figure 2, a space-time sketch analogous to that of Figure 1, but with the two zones of ignorance (length scales smaller than the Planck (or string) length and densities higher than the Planck (or string) density) are shown. As the string scale decreases relative to the Planck scale, the horizontal line which indicates the boundary of the super-string density zone of ignorance approaches the constant time line corresponding to the onset of inflation. This implies that the inflationary background dynamics itself might not be robust against stringy corrections in the dynamical equations.
The sketch in Figure 2 also shows the exponential increase of the horizon compared to the Hubble radius during the period of inflation.
The conceptual problems of inflationary cosmology discussed in the previous subsection motivate a search for a new paradigm of early universe cosmology based on string theory. Such a new paradigm may provide the initial conditions for a robust inflationary phase. However, it may also lead to an alternative scenario. In the following, we will explore this second possibility.
In the best possible world, the initial phase of string cosmology will eliminate the cosmological “Big Bang” singularity, it will provide a unified description of space, time and matter, and it will allow a controlled computation of the induced cosmological perturbations. The development of such a consistent framework of string cosmology will, however, have to be based on a consistent understanding of non-perturbative string theory. Such an understanding is at the present time not available.
Given the lack of such an understanding, most approaches to string cosmology are based on treating matter using an effective field theory description motivated by string theory. However, in such approaches key features of string theory which are not present in field theory cannot be seen. The approach to string cosmology discussed below is, in contrast, based on studying effects of new degrees of freedom and new symmetries which are key ingredients to string theory, which will be present in any non-perturbative formulation of string theory.
2 Basics of String Gas Cosmology
2.1 Principles of String Gas Cosmology
In the absence of a non-perturbative formulation of string theory, the approach to string cosmology which we have suggested, string gas cosmology [17, 18, 19] (see also , and [22, 23] for reviews), is to focus on symmetries and degrees of freedom which are new to string theory (compared to point particle theories) and which will be part of any non-perturbative string theory, and to use them to develop a new cosmology. The symmetry we make use of is T-duality, and the new degrees of freedom are the string oscillatory modes and the string winding modes.
String gas cosmology is based on coupling a classical background which includes the graviton and the dilaton fields to a gas of strings (and possibly other basic degrees of freedom of string theory such as “branes”). All dimensions of space are taken to be compact, for reasons which will become clear later. For simplicity, we take all spatial directions to be toroidal and denote the radius of the torus by . Strings have three types of states: momentum modes which represent the center of mass motion of the string, oscillatory modes which represent the fluctuations of the strings, and winding modes counting the number of times a string wraps the torus.
Since the number of string oscillatory states increases exponentially with energy, there is a limiting temperature for a gas of strings in thermal equilibrium, the Hagedorn temperature  . Thus, if we take a box of strings and adiabatically decrease the box size, the temperature will never diverge. This is the first indication that string theory has the potential to resolve the cosmological singularity problem (see also [25, 26] for discussions on how the temperature singularity can be avoided in string cosmology).
The second key feature of string theory upon which string gas cosmology is based is T-duality. To introduce this symmetry, let us discuss the radius dependence of the energy of the basic string states: The energy of an oscillatory mode is independent of , momentum mode energies are quantized in units of , i.e.
and winding mode energies are quantized in units of , i.e.
where both and are integers. Thus, a new symmetry of the spectrum of string states emerges: Under the change
in the radius of the torus (in units of the string length ) the energy spectrum of string states is invariant if winding and momentum quantum numbers are interchanged
The above symmetry is the simplest element of a larger symmetry group, the T-duality symmetry group which in general also mixes fluxes and geometry. The string vertex operators are consistent with this symmetry, and thus T-duality is a symmetry of perturbative string theory. Postulating that T-duality extends to non-perturbative string theory leads  to the need of adding D-branes to the list of fundamental objects in string theory. With this addition, T-duality is expected to be a symmetry of non-perturbative string theory. Specifically, T-duality will take a spectrum of stable Type IIA branes and map it into a corresponding spectrum of stable Type IIB branes with identical masses .
2.2 Dynamics of String Gas Cosmology
That string gas cosmology will lead to a dynamical evolution of the early universe very different from what is obtained in standard and inflationary cosmology can already be seen by combining the basic ingredients from string theory discussed in the previous subsection. As the radius of a box of strings decreases from an initially very large value - maintaining thermal equilibrium - , the temperature first rises as in standard cosmology since the string states which are occupied (the momentum modes) get heavier. However, as the temperature approaches the Hagedorn temperature, the energy begins to flow into the oscillatory modes and the increase in temperature levels off. As the radius decreases below the string scale, the temperature begins to decrease as the energy begins to flow into the winding modes whose energy decreases as decreases (see Figure 3). Thus, as argued in , the temperature singularity of early universe cosmology should be resolved in string gas cosmology.
The equations that govern that background of string gas cosmology are not known. The Einstein equations are not the correct equations since they do not obey the T-duality symmetry of string theory. Many early studies of string gas cosmology were based on using the dilaton gravity equations [18, 31, 32]. However, these equations are not satisfactory, either. Firstly, we expect that string theory correction terms to the low energy effective action of string theory become dominant in the Hagedorn phase. Secondly, the dilaton gravity equations yields a rapidly changing dilaton during the Hagedorn phase (in the string frame). Once the dilaton becomes large, it becomes inconsistent to focus on fundamental string states rather than brane states. In other words, using dilaton gravity as a background for string gas cosmology does not correctly reflect the S-duality symmetry of string theory. Recently, a background for string gas cosmology including a rolling tachyon was proposed  which allows a background in the Hagedorn phase with constant scale factor and constant dilaton. Another study of this problem was given in .
Some conclusions about the time-temperature relation in string gas cosmology can be derived based on thermodynamical considerations alone. One possibility is that starts out much smaller than the self-dual value and increases monotonically. From Figure 3 is then follows that the time-temperature curve will correspond to that of a bouncing cosmology. Alternatively, it is possible that the universe starts out in a meta-stable state near the Hagedorn temperature, the Hagedorn phase, and then smoothly evolves into an expanding phase dominated by radiation like in standard cosmology (Figure 4). Note that we are assuming that not only is the scale factor constant in time, but also the dilaton.
The transition between the quasi-static Hagedorn phase and the radiation phase at the time is a consequence of the annihilation of string winding modes into string loops (see Figure 5). Since this process corresponds to the production of radiation, we denote this time by the same symbol as the time of reheating in inflationary cosmology. As pointed out in , this annihilation process only is possible in at most three large spatial dimensions. This is a simple dimension counting argument: string world sheets have measure zero intersection probability in more than four large space-time dimensions. Hence, string gas cosmology may provide a natural mechanism for explaining why there are exactly three large spatial dimensions. This argument was supported by numerical studies of string evolution in three and four spatial dimensions  (see also ). The flow of energy from winding modes to string loops can be modelled by effective Boltzmann equations  analogous to those used to describe the flow of energy between infinite cosmic strings and cosmic string loops (see e.g. [38, 39, 40] for reviews).
Several comments are in place concerning the above mechanism. First, in the analysis of  it was assumed that the string interaction rates were time-independent. If the dynamics of the Hagedorn phase is modelled by dilaton gravity, the dilaton is rapidly changing during the phase in which the string frame scale factor is static. As discussed in [41, 42], in this case the mechanism which tells us that exactly three spatial dimensions become macroscopic does not work. Another comment concerns the isotropy of the three large dimensions. In contrast to the situation in Standard cosmology, in string gas cosmology the anisotropy decreases in the expanding phase . Thus, there is a natural isotropization mechanism for the three large spatial dimensions.
At late times, the dynamics of string gas cosmology can be described by dilaton gravity or - if the dilaton is fixed - by Einstein gravity. The dilaton gravity action coupled to string gas matter is
where is the determinant of the metric, is the Ricci scalar, is the dilaton, is the reduced gravitational constant in ten dimensions, and denotes the matter action for which we will use the hydrodynamical action of a string gas. The metric appearing in the above action is the metric in the string frame.
In the case of a homogeneous and isotropic background given by (1.1) the three resulting equations (the generalization of the two Friedmann equations plus the equation for the dilaton) in the string frame are  (see also )
where and denote the total energy and pressure, respectively, is the number of spatial dimensions, and we have introduced the logarithm of the scale factor
and the rescaled dilaton
The above equations are consistent with a fixed dilaton in the radiation phase, but not in the Hagedorn phase (see e.g. ). As we run backwards in time, the dilaton runs off towards a singularity which is inconsistent with the ideas of a quasi-static Hagedorn phase. A detailed study of the dynamics of the background space-time in the presence of string gases with both Hagedorn and radiation equations of state was performed in  †)†)†)Corrections to these equations coming from stringy terms were considered in ..
This set of equations (2.2),(2.2),(2.2) can be supplemented with Boltzmann type equations which describe the transfer of energy from the string winding modes to string loops . The equations describe how two winding strings with opposite orientations intersect, producing closed loops with vanishing winding as a final state (see Figure 5). First, we split the energy density in strings into the density in winding strings
where is the string mass per unit length, and is the number of strings per Hubble volume, and into the density in string loops
where denotes the co-moving number density of loops, normalized at a reference time . In terms of these variables, the equations describing the loop production from the interaction of two winding strings are 
If the spatial size is large in the Hagedorn phase, not all winding strings will disappear at the time . In fact, as is well known from the studies of cosmic strings [38, 39, 40], the above transfer equations (2.2),(2.2) lead to the existence of a scaling solution for cosmic superstrings according to which at any given time in the radiation phase for , there will be a distribution of cosmic superstrings characterized by a constant average number of winding strings crossing each Hubble volume. A remnant distribution of cosmic superstrings at all late times is thus one of the testable predictions of string gas cosmology.
3 Moduli Stabilization in String Gas Cosmology
A major challenge in string cosmology is to stabilize all of the string moduli. Specifically, the sizes and shapes of the extra dimensions must be stabilized, and so must the dilaton. In string gas cosmology based on heterotic superstring theory, all of the size and shape moduli are fixed by the basic ingredients of the model, namely the presence of string states with both momentum and winding modes.
The stabilization of the size moduli was considered in [47, 48, 49, 50], that of the shape moduli in [51, 52] (see  for a review). The basic principle is the following: in a string gas containing both momentum and winding modes, the winding modes will prevent expansion since their energies increase with whereas the momentum modes will prevent contraction since their energies scale as . Thus, on energetic grounds, there is a preferred value for the size of the extra dimensions, namely in string units. In heterotic string theory, there are enhanced symmetry states which contain both momentum and winding quantum numbers and which are massless at the self-dual radius. These are the lowest energy states near the self-dual radius and hence dominate the thermodynamic partition function. These states act as radiation from the point of view of our three large dimensions, and are hence phenomenologically acceptable at late times . The role of these states for moduli stabilization was stressed in a more general context in [54, 55, 56].
It turns out that the shape moduli are also stabilized by the presence of the enhanced symmetry states, without requiring any additional inputs. The only modulus which requires additional input for its stabilization is the dilaton (the problem of simultaneously stabilizing both the dilaton and the radion in the context of dilaton gravity coupled to perturbative string theory states was discussed in detail in ).
3.2 Stabilization of Geometrical Moduli
The stabilization of the geometrical moduli at late times can be analyzed in the context of dilaton gravity (the discussion in this subsection is close to the one given in ). We use the following ansatz for an anisotropic metric with scale factor for the three large dimensions and corresponding scale factor for the internal dimensions (considered here to be isotropic):
where are the coordinates of the three large dimensions and the coordinates of the internal dimensions.
The variational equations of motion for , and which follow from the dilaton gravity action are 
where is the energy density and and are the pressure densities in the large and the internal directions, respectively.
Let us now consider a superposition of several string gases, one with momentum number about the three large dimensions, one with momentum number about the six internal dimensions, and a further one with winding number about the internal dimensions. Note that there are no winding modes about the large dimensions (), either because they have already annihilated by the mechanism discussed in the previous section, or they were never present in the initial conditions. In this case, the energy and the total pressures and are given by
where is the string mass per unit length. Below, we will consider a more realistic string gas, a gas made up of string states which have momentum, winding and oscillatory quantum numbers together. The states considered here are massive, and would not be expected to dominate the thermodynamical partition function if there are states which are massless. However, for the purpose of studying radion stabilization in the string frame, the use of the above naive string gas is sufficient.
We are interested in the symmetric case In this case, it follows from (3.8) that the equation of motion for is a damped oscillator equation, with the minimum of the effective potential corresponding to the self-dual radius. The damping is due to the expansion of the three large dimensions (the expansion of the three large dimensions is driven by the pressure from the momentum modes ). Thus, we see that the naive intuition that the competition of winding and momentum modes about the compact directions stabilizes the radion degrees of freedom at the self-dual radius generalizes to this anisotropic setting.
However, in the context of dilaton gravity, the dilaton is rapidly evolving in the Hagedorn phase. Thus, the Einstein frame metric is not static even if the string frame metric is (see e.g. ). The key question is whether the radion remains stabilized if the dilaton is fixed by hand (or by mechanisms discussed below). For a gas of strings made up of massive states such as considered above this is not the case. In Heterotic string theory, there are enhanced symmetry states which are massless at the self-dual radius, hence dominate the thermodynamic partition function, and can stabilize the radion . In the following we will discuss this mechanism.
The equations of motion which arise from coupling string gas matter to the Einstein (as opposed to the dilaton gravity) action lead to - for an anisotropic metric of the form
where the are the internal coordinates - the following equation for the radion
The vector index pairs label perturbative string states. Note that and are momentum and winding number six-vectors, one component for each internal dimension. Also, is the number density of string states with the momentum and winding number vector pair , is the energy of an individual string, and is the determinant of the metric. The source term depends on the quantum numbers of the string gas, and the sum runs over all and . If the number of right-moving oscillator modes is given by , then the source term for fixed and is
where and indicate scalar products relative to the metric of the internal space. To obtain this equation, we have made use of the mass spectrum of string states and of the level matching conditions. In the case of the bosonic superstring, the mass spectrum for fixed and , where is the number of left-moving oscillator states, on a six-dimensional torus whose radii are given by is
and the level matching condition reads
There are modes which are massless at the self-dual radius . One such mode is the graviton with and . The modes of interest to us are modes which contain winding and momentum, namely
, , and ;
, , and ;
, and .
The above discussion was in the context of bosonic string theory. Due to the presence of the bosonic string theory tachyon, the above states are not the lowest energy states for bosonic string theory and hence do not dominate the thermodynamic partition function. In Heterotic string theory, the tachyon is factored out of the spectrum by the GSO  projection, but the states we discussed above survive. In contrast, in Type II string theory, our massless states are also factored out. Thus, in the following we will restrict attention to Heterotic string theory.
In string theories which admit massless states (i.e. states which are massless at the self-dual radius), these states will dominate the initial partition function. The background dynamics will then also be dominated by these states. To understand the effect of these strings, consider the equation of motion (3.10) with the source term (3.11). The first two terms in the source term correspond to an effective potential with a stable minimum at the self-dual radius. However, if the third term in the source does not vanish at the self-dual radius, it will lead to a positive potential which causes the radion to increase. Thus, a condition for the stabilization of at the self-dual radius is that the third term in (3.11) vanishes at the self-dual radius. This is the case if and only if the string state is a massless mode.
The massless modes have other nice features which are explored in detail in . They act as radiation from the point of view of our three large dimensions and hence do not lead to a over-abundance problem. As our three spatial dimensions grow, the potential which confines the radion becomes shallower. However, rather surprisingly, it turns out the the potential remains steep enough to avoid fifth force constraints.
Key to the success in simultaneously avoiding the moduli over-closure problem and evading fifth force constraints is the fact that the stabilization mechanism is an intrinsically stringy one. In the case of a naive effective field theory approach, both the confining force and the over-density in the moduli field scale as , where is the potential energy density of the field . In contrast, in the case of stabilization by means of massless string modes, the energy density in the string modes (from the point of view of our three large dimensions) scales as , whereas the confining force scales as , where is the momentum in the three large dimensions. Thus, for small values of , one simultaneously gets a large confining force (thus satisfying the fifth force constraints) and a small energy density [49, 59].
In the presence of massless string states, the shape moduli also can be stabilized, at least in the simple toroidal backgrounds considered so far . To study this issue, we consider a metric of the form
where the metric of the internal space (here for simplicity considered to be a two-dimensional torus) contains a shape modulus, the angle between the two cycles of the torus:
where corresponds to a rectangular torus. The ratio between the two toroidal radii is a second shape modulus. However, we already know that each radion individually is stabilized at the self-dual radius. Thus, the shape modulus corresponding to the ratio of the toroidal radii is fixed, and the angle is the only shape modulus which has yet to be considered.
Combining the and the Einstein equations, we obtain a harmonic oscillator equation for with as the stable fixed point.
where is a constant whose value depends on the quantum numbers of the string gas. In the case of an expanding three-dimensional space we would have obtained an additional damping term in the above equation of motion. We thus conclude that the shape modulus is dynamically stabilized at a value which maximizes the area to circumference ratio.
3.3 Dilaton Stabilization
The only modulus which is not stabilized with the basic ingredients of string gas cosmology alone is the dilaton. This situation should be compared to the problems which arise in the string theory-motivated approaches to obtaining inflation, where a number of extra ingredients such as fluxes and non-perturbative effects have to be invoked in order to stabilize the Kaehler and complex structure moduli (see e.g.  for a review).
In string gas cosmology, extra inputs are needed to stabilize the dilaton. One possibility is that two-loop effective potential effects can stabilize the dilaton . There have also been attempts to use extra stringy ingredients such as branes [59, 62, 63] or a running tachyon  to stabilize the dilaton.
The most conservative approach to late-time dilaton stabilization in string gas cosmology , however, is to use one of the non-perturbative mechanisms which is already widely used in the literature to fix moduli, namely gaugino condensation .
Gaugino condensation leads to a correction of the superpotential of the theory, from which the actual potential is derived. The change in the superpotential of the four-dimensional theory is
where is the string coupling constant and is a constant. The potential can be derived from the superpotential and the Kaehler potential via the standard formula
where the indices and run over all of the moduli fields, and the Kaehler covariant derivative is given by
Since the superpotential in our case is independent of the volume modulus, the expression for the potential simplifies to
where and now run only over the modulus
and the complex structure moduli (which we, however, do not include here). In the above, is the four-dimensional dilaton given by
is the axion, and is the four dimensional Planck mass.
From the above, we see that the potential (3.19) depends both on the dilaton and on the radion. It is important to verify that adding this potential to the theory can stabilize the dilaton without de-stabilizing the radion (which is fixed by the string gas matter contributions described in the previous subsection). To investigate this issue , we first need to lift the potential (3.19) to ten space-time dimensions. The result, after expanding about the minimum of the potential, is
where we have written the scale factor in the ten dimensional Einstein frame. In the above, is the volume of the internal space, and is the ten dimensional Planck mass which is given in terms of the volume of the internal space and the four dimensional Planck mass by
Also, is a constant which appears in the superpotential (see  for details).
The effects of gaugino condensation on dilaton and radion stabilization can now be analyzed in the following way : we start from the dilaton gravity action to which we add the potential (3.24). To this action we add the action of a gas of strings, as done in (2.2). We work in the ten-dimensional Einstein frame (and thus have to re-scale the radion, the metric and the matter energy-momentum tensor accordingly). From this action we can derive the equations of motion for the dilaton, the radion and the scale factor of our four-dimensional space-time.
For fixed radion, it follows from (3.24) that the potential has a minimum for a specific value of the dilaton. From the considerations of the previous subsection we know that stringy matter selects a preferred value of the radion, the self-dual radius. To demonstrate that the addition of the gaugino potential can stabilize the dilaton without de-stabilizing the radion we expand the equations of motion about the value of the radion corresponding to the self-dual radius and the value of the dilaton for which the potential (3.24) is minimized for the chosen value of the radion. We have shown  that this is a stable fixed point of the dynamical system. Thus, we have shown with the addition of gaugino condensation, in string gas cosmology all of the moduli are fixed.
4 String Gas Cosmology and Structure Formation
At the outset of this section, let us recall the mechanism by which inflationary cosmology leads to the possibility of a causal generation mechanism for cosmological fluctuations which yields an almost scale-invariant spectrum of perturbations. The space-time diagram of inflationary cosmology is sketched in Figure 1. In this figure, the vertical axis represents time, the horizontal axis space (physical as opposed to co-moving coordinates). The period between times and corresponds to the inflationary phase (assumed in the figure to be characterized by almost exponential expansion of space).
During the period of inflation, the Hubble radius
is approximately constant. In contrast, the physical length of a fixed co-moving scale (labelled by in the figure) is expanding exponentially. In this way, in inflationary cosmology scales which have microscopic sub-Hubble wavelengths at the beginning of inflation are red-shifted to become super-Hubble-scale fluctuations at the end of the period of inflation. After inflation, the Hubble radius increases linearly in time, faster than the physical wavelength corresponding to a fixed co-moving scale. Thus, scales re-enter the Hubble radius at late times.
The Hubble radius is crucial for the question of generation of fluctuations for the following reason: If we consider perturbations with wavelengths smaller than the Hubble radius, their evolution is dominated by micro-physics which causes them to oscillate. This is best illustrated by considering the Klein-Gordon equation for a free scalar field in an expanding universe. In Fourier space, the equation is
where is the physical wavenumber. On sub-Hubble scales , the Hubble damping term in the above equation is sub-dominant, and the micro-physics term leads to oscillations of the field. In contrast, on super-Hubble scale , it is the last term on the left-hand side of (4.2) which is negligible, and it then follows that the fluctuations are frozen in.
Thus, if we want to generate primordial cosmological fluctuations by causal physics, the scale of the fluctuations needs to be sub-Hubble ‡)‡)‡)There is, however, a loophole in this argument: the formation of topological defects during a cosmological phase transition can lead to non-random entropy fluctuations on super-Hubble scales which induce cosmological perturbations in the late universe [38, 39, 40].. In inflationary cosmology, it is the accelerated expansion of space which enables the scale of inhomogeneities on current cosmological scales to be sub-Hubble at early times, and thus leads to the possibility of a causal generation mechanism for fluctuations.
Since inflation red-shifts any classical fluctuations which might have been present at the beginning of the inflationary phase, fluctuations in inflationary cosmology are generated by quantum vacuum perturbations. The fluctuations begin in their quantum vacuum state at the onset of inflation. Once the wavelength exceeds the Hubble radius, squeezing of the wave-function of the fluctuations sets in (see [12, 13]). This squeezing plus the de-coherence of the fluctuations due to the interaction between short and long wavelength modes generated by the intrinsic non-linearities in both the gravitational and matter sectors of the theory (see [66, 67, 68] for recent discussions of this aspect and references to previous work) lead to the classicalization of the fluctuations on super-Hubble scales.
Let us now turn to the cosmological background of string gas cosmology represented in Figure 4. This string gas cosmology background yields the space-time diagram sketched in Figure 6. As in Figure 1, the vertical axis is time and the horizontal axis denotes the physical distance. For times , we are in the static Hagedorn phase and the Hubble radius is infinite. For , the Einstein frame Hubble radius is expanding as in standard cosmology. The time is when the string winding modes begin to decay into string loops, and the scale factor starts to increase, leading to the transition to the radiation phase of standard cosmology.
Let us now compare the evolution of the physical wavelength corresponding to a fixed co-moving scale with that of the Einstein frame Hubble radius . The evolution of scales in string gas cosmology is identical to the evolution in standard and in inflationary cosmology for . If we follow the physical wavelength of the co-moving scale which corresponds to the current Hubble radius back to the time , then - taking the Hagedorn temperature to be of the order GeV - we obtain a length of about 1 mm. Compared to the string scale and the Planck scale, this is a scale in the far infrared.
The physical wavelength is constant in the Hagedorn phase since space is static. But, as we enter the Hagedorn phase going back in time, the Hubble radius diverges to infinity. Hence, as in the case of inflationary cosmology, fluctuation modes begin sub-Hubble during the Hagedorn phase, and thus a causal generation mechanism for fluctuations is possible.
However, the physics of the generation mechanism is very different. In the case of inflationary cosmology, fluctuations are assumed to start as quantum vacuum perturbations because classical inhomogeneities are red-shifting. In contrast, in the Hagedorn phase of string gas cosmology there is no red-shifting of classical matter. Hence, it is the fluctuations in the classical matter which dominate. Since classical matter is a string gas, the dominant fluctuations are string thermodynamic fluctuations.
Our proposal for string gas structure formation is the following  (see  for a more detailed description). For a fixed co-moving scale with wavenumber we compute the matter fluctuations while the scale in sub-Hubble (and therefore gravitational effects are sub-dominant). When the scale exits the Hubble radius at time we use the gravitational constraint equations to determine the induced metric fluctuations, which are then propagated to late times using the usual equations of gravitational perturbation theory. Since the scales we are interested in are in the far infrared, we use the Einstein constraint equations for fluctuations.
4.2 String Thermodynamics
The thermodynamics of a gas of strings was worked out some time ago §)§)§)The initial discussions of the thermodynamics of strings were given in [24, 71]. More detailed studies were performed after the first explosion of interest in superstring theory in the early 1980’s . For some studies of string statistical mechanics particularly relevant to string gas cosmology see [73, 74].. We will consider our three spatial dimensions to be compact, admitting stable winding modes. Specifically, we will take space to be a three-dimensional torus. In this case, the string gas specific heat is positive, and string thermodynamics is well-defined, and was discussed in detail in  (see also [76, 77, 78, 79]). What follows is a summary along the lines of .
The starting point for our considerations is the free energy of a string gas in thermal equilibrium
where is the inverse temperature and the canonical partition function is given by
where the sum runs over the states of the string gas, and is the energy of the state. The action of the string gas is given in terms of the free energy via
Note that the free energy depends on the spatial components of the metric because the energy of the individual string states does. The energy-momentum tensor of the string gas is determined by varying the action with respect to the metric:
Consider now the thermal expectation value
As we will see below, the scalar metric fluctuations are determined by the energy density correlation function
We will read off the result from the expression (4.10) evaluated for . The derivative with respect to can be expressed in terms of the derivative with respect to . After a couple of steps of algebra we obtain
is the specific heat, and
is the internal energy. Also, is the volume of the three compact but large spatial dimensions.
The gravitational waves are determined by the off-diagonal spatial components of the correlation function tensor, i.e.
Our aim is to calculate the fluctuations of the energy-momentum tensor on various length scales . For each value of , we will consider string thermodynamics in a box in which all edge lengths are . From (4.10) it is obvious that in order to have non-vanishing off-diagonal spatial correlation functions, we must consisted a torus with its shape moduli turned on. Let us focus on the component of the correlation function. We will consider the spatial part of the metric to be
with . The spatial coordinates run over a fixed interval, e.g. ), The generalization of the spatial part of the metric to three dimensions is obvious. At the end of the computations, we will set .
From the form of (4.10), it follows that all space-space correlation function tensor elements are of the same order of the magnitude, namely
where the string pressure is given by
In the following, we will compute the two correlation functions (4.12) and (4.17) using tools from string statistical mechanics. Specifically, we will be following the discussion in . The starting point is the formula
for the entropy in terms of , the density of states. The density of states of a gas of closed strings on a large three-dimensional torus (with the radii of all internal dimensions at the string scale) was calculated in  (see also ) and is given by
where comes from the contribution to the density of states (when writing the density of states as an inverse Laplace transform of , which involves integration over ) from the closest singularity point to in the complex plane. Note that , and is real. From [75, 80] we have
In the above, is a (constant) number density of order and is the ‘Hagedorn Energy density’ of the order , and
To ensure the validity of Eq. (4.20) we demand that
by assuming , which corresponds to being in a state in which winding modes and oscillatory modes can be excited and we expect important deviations from point particle thermodynamics.
Combining the above results, we find that the entropy of the string gas in the Hagedorn phase is given by
and therefore the temperature will be
In the above, we have dropped a term which is negligible since (see (4.23)). Using this relation, we can express in terms of and via
In addition, we find
Note that to ensure that and , one should demand
The results (4.24) and (4.26) now allow us to compute the correlation functions (4.12) and (4.17). We first compute the energy correlation function (4.12). Making use of (4.27), it follows from (4.13) that
from which we get
Note that the factor in the denominator will turn out to be responsible for giving the spectrum a slight red tilt. It comes from the differentiation with respect to .
Next we evaluate (4.18). From the definition of the pressure it follows that (to linear order in )
where the final partial derivative is to be taken at constant energy. In taking this partial derivative, we insert the expression (4.26) for and must keep careful account of the fact that depends on the radius . In evaluating the resulting terms, we keep only the one which dominates at high energy density. It is the term which comes from differentiating the factor . This differentiation brings down a factor of , which is then substituted by means of (4.27), thus introducing a logarithmic factor in the final result for the pressure. We obtain
which immediately yields
Note that since no temperature derivative is taken, the factor remains in the numerator. This is one of the two facts which will lead to the slight blue tilt of the spectrum of gravitational waves. The second factor contributing to the slight blue tilt is the explicit factor of in the logarithm. Because of (4.28), we are on the large side of the zero of the logarithm. Hence, the greater the value of , the larger the absolute value of the logarithmic factor.
4.3 Spectrum of Cosmological Fluctuations
We write the metric including cosmological perturbations (scalar metric fluctuations) and gravitational waves in the following form:
In the above, we have used conformal time which is related to the physical time via
We have fixed the gauge (i.e. coordinate) freedom for the scalar metric perturbations by adopting the longitudinal gauge in terms of which the metric is diagonal. Furthermore, we have taken matter to be free of anisotropic stress (otherwise there would be two scalar metric degrees of freedom instead of the single function ). The spatial tensor is transverse and traceless and represents the gravitational waves.
Note that in contrast to the case of slow-roll inflation, scalar metric fluctuations and gravitational waves are generated by matter at the same order in cosmological perturbation theory. This could lead to the expectation that the amplitude of gravitational waves in string gas cosmology could be generically larger than in inflationary cosmology. This expectation, however, is not realized  since there is a different mechanism which suppresses the gravitational waves relative to the density perturbations (namely the fact that the gravitational wave amplitude is set by the amplitude of the pressure, and the pressure is suppressed relative to the energy density in the Hagedorn phase).
Assuming that the fluctuations are described by the perturbed Einstein equations (they are not if the dilaton is not fixed [44, 82]), then the spectra of cosmological perturbations and gravitational waves are given by the energy-momentum fluctuations in the following way 
where the pointed brackets indicate expectation values, and
where on the right hand side of (4.37) we mean the average over the correlation functions with , and is the amplitude of the gravitational waves ¶)¶)¶)The gravitational wave tensor can be written as the amplitude multiplied by a constant polarization tensor..
Let us now use (4.36) to determine the spectrum of scalar metric fluctuations. We first calculate the root mean square energy density fluctuations in a sphere of radius . For a system in thermal equilibrium they are given by the specific heat capacity via (see (4.12)
From the previous subsection we know that the specific heat of a gas of closed strings on a torus of radius is (see 4.29)
Hence, the power spectrum of scalar metric fluctuations can be evaluated as follows
where in the first step we have used (4.36) to replace the expectation value of in terms of the correlation function of the energy density, and in the second step we have made the transition to position space
The first conclusion from the result (4.3) is that the spectrum is approximately scale-invariant ( is independent of ). It is the ‘holographic’ scaling which is responsible for the overall scale-invariance of the spectrum of cosmological perturbations. However, there is a small dependence which comes from the fact that in the above equation for a scale the temperature is to be evaluated at the time . Thus, the factor in the denominator is responsible for giving the spectrum a slight dependence on . Since the temperature slightly decreases as time increases at the end of the Hagedorn phase, shorter wavelengths for which occurs later obtain a smaller amplitude. Thus, the spectrum has a slight red tilt.
4.4 Spectrum of Gravitational Waves
As discovered in , the spectrum of gravitational waves is also nearly scale invariant. However, in the expression for the spectrum of gravitational waves the factor appears in the numerator, thus leading to a slight blue tilt in the spectrum. This is a prediction with which the cosmological effects of string gas cosmology can be distinguished from those of inflationary cosmology, where quite generically a slight red tilt for gravitational waves results. The physical reason is that large scales exit the Hubble radius earlier when the pressure and hence also the off-diagonal spatial components of are closer to zero.
Let us analyze this issue in a bit more detail and estimate the dimensionless power spectrum of gravitational waves. First, we make some general comments. In slow-roll inflation, to leading order in perturbation theory matter fluctuations do not couple to tensor modes. This is due to the fact that the matter background field is slowly evolving in time and the leading order gravitational fluctuations are linear in the matter fluctuations. In our case, the background is not evolving (at least at the level of our computations), and hence the dominant metric fluctuations are quadratic in the matter field fluctuations. At this level, matter fluctuations induce both scalar and tensor metric fluctuations. Based on this consideration we might expect that in our string gas cosmology scenario, the ratio of tensor to scalar metric fluctuations will be larger than in simple slow-roll inflationary models. However, since the amplitude of the gravitational waves is proportional to the pressure, and the pressure is suppressed in the Hagedorn phase, the amplitude of the gravitational waves will also be small in string gas cosmology.
The method for calculating the spectrum of gravitational waves is similar to the procedure outlined in the last section for scalar metric fluctuations. For a mode with fixed co-moving wavenumber , we compute the correlation function of the off-diagonal spatial elements of the string gas energy-momentum tensor at the time when the mode exits the Hubble radius and use (4.37) to infer the amplitude of the power spectrum of gravitational waves at that time. We then follow the evolution of the gravitational wave power spectrum on super-Hubble scales for using the equations of general relativistic perturbation theory.
The power spectrum of the tensor modes is given by (4.37):
for . Note that the factor multiplying the momentum space correlation function of gives the position space correlation function, namely the root mean square fluctuation of in a region of radius (the reader who is skeptical about this point is invited to check that the dimensions work out properly). Thus,
which, for temperatures close to the Hagedorn value reduces to
This shows that the spectrum of tensor modes is - to a first approximation, namely neglecting the logarithmic factor and neglecting the -dependence of - scale-invariant. The corrections to scale-invariance will be discussed at the end of this subsection.
On super-Hubble scales, the amplitude of the gravitational waves is constant. The wave oscillations freeze out when the wavelength of the mode crosses the Hubble radius. As in the case of scalar metric fluctuations, the waves are squeezed. Whereas the wave amplitude remains constant, its time derivative decreases. Another way to see this squeezing is to change variables to
in terms of which the action has canonical kinetic term. The action in terms of becomes
from which it immediately follows that on super-Hubble scales . This is the squeezing of gravitational waves .
Since there is no -dependence in the squeezing factor, the scale-invariance of the spectrum at the end of the Hagedorn phase will lead to a scale-invariance of the spectrum at late times.
Note that in the case of string gas cosmology, the squeezing factor does not differ substantially from the squeezing factor for gravitational waves. In the case of inflationary cosmology, and differ greatly during reheating, leading to a much larger squeezing for scalar metric fluctuations, and hence to a suppressed tensor to scalar ratio of fluctuations. In the case of string gas cosmology, there is no difference in squeezing between the scalar and the tensor modes.
Let us return to the discussion of the spectrum of gravitational waves. The result for the power spectrum is given in (4.44), and we mentioned that to a first approximation this corresponds to a scale-invariant spectrum. As realized in , the logarithmic term and the -dependence of both lead to a small blue-tilt of the spectrum. This feature is characteristic of our scenario and cannot be reproduced in inflationary models. In inflationary models, the amplitude of the gravitational waves is set by the Hubble constant . The Hubble constant cannot increase during inflation, and hence no blue tilt of the gravitational wave spectrum is possible.
To study the tilt of the tensor spectrum, we first have to keep in mind that our calculations are only valid in the range (4.28), i.e. to the large side of the zero of the logarithm. Thus, in the range of validity of our analysis, the logarithmic factor contributes an explicit blue tilt of the spectrum. The second source of a blue tilt is the factor multiplying the logarithmic term in (4.44). Since modes with larger values of exit the Hubble radius at slightly later times , when the temperature is slightly lower, the factor will be larger.
A heuristic way of understanding the origin of the slight blue tilt in the spectrum of tensor modes is as follows. The closer we get to the Hagedorn temperature, the more the thermal bath is dominated by long string states, and thus the smaller the pressure will be compared to the pressure of a pure radiation bath. Since the pressure terms (strictly speaking the anisotropic pressure terms) in the energy-momentum tensor are responsible for the tensor modes, we conclude that the smaller the value of the wavenumber (and thus the higher the temperature when the mode exits the Hubble radius, the lower the amplitude of the tensor modes. In contrast, the scalar modes are determined by the energy density, which increases at as decreases, leading to a slight red tilt.
To summarize this section, we have seen that string gas cosmology provides a mechanism alternative to the well-known inflationary one for generating an approximately scale-invariant spectrum of approximately adiabatic density fluctuations. The model predicts a slight red tilt of the spectrum (as is also obtained in many simple inflationary models). However, as a prediction which distinguishes the model from the inflationary universe scenario, it predicts a slight blue tilt of the spectrum of gravitational waves. Thus, a way to rule out the inflationary scenario would be to detect a stochastic background of gravitational waves at both very small wavelengths (using direct detection experiments such as gravitational wave antennas) and on cosmological scales (using signatures in CMB temperature maps) and to infer a blue tilt from those measurements. The current limits on the magnitude of the blue tilt are not very strong  but can be improved.
The scenario has one free parameter (the ratio of the string to the Planck length) and one free function (the k-dependence of the temperature - in principle calculable if the dynamics of the exit from the Hagedorn phase were better known). There are five basic observables: the amplitudes of the scalar and tensor spectra, their tilts, and the amplitude of the jump in the CMB temperature maps produced by long straight cosmic superstrings via the Kaiser-Stebbins  effect. Thus, there are three consistency relations between the observables which allow the scenario to be falsified.
An important point is that the thermal string gas fluctuations evolve for a long time during the radiation phase outside the Hubble radius. Like in inflationary cosmology, this leads to the squeezing of fluctuations which is responsible for the acoustic oscillations in the angular power spectrum of CMB anisotropies (see e.g.  for a more detailed discussion of this point). Note that the situation is completely different from that in topological defect models of structure formation, where the curvature perturbations are constantly seeded from the defect sources at late times, and which hence does not lead to the oscillations in the angular power spectrum of the CMB.
The non-Gaussianities induced by the thermal gas of strings are large on microscopic scale, but Poisson-suppressed on larger scales. The three point correlation function produced by a string gas and the related non-Gaussianity parameter can be calculated  from the same starting point of string thermodynamics described earlier in this section.
Note that the structure formation scenario discussed in this section relies on three key assumptions - firstly the holographic scaling of the specific heat capacity , secondly the applicability of the Einstein equations to describe metric fluctuations on infrared scales, and thirdly the existence of a phase like the Hagedorn phase at the end of which the matter fluctuations seed metric perturbations (it may be a phase only describable using a truly non-perturbative string theory or quantum gravity framework). For attempts to realize our basic structure formation scenario in a different context see e.g .
Dilaton gravity does not provide a satisfactory framework to implement our structure formation scenario [44, 82] and is unsatisfactory as a description of the Hagedorn phase for other reasons mentioned earlier in this chapter. The criticisms of string gas cosmology in [82, 88] are mostly problems which are specific to the attempt to use dilaton gravity as the background for string gas cosmology. There is a specific background model in which all of the key assumptions discussed above are realized, namely the ghost-free higher derivative gravity action of  which yields a non-singular bouncing cosmology. If we add string gas matter to this action and adjust parameters of the model such that the bounce phase is long, then thermal string fluctuations in the bounce phase yield a realization of our scenario .
The string gas scenario is an approach to early universe cosmology based on coupling a gas of strings to a classical background. It includes string degrees of freedom and string symmetries which are hard to implement in an effective field theory approach.
The background of string gas cosmology is non-singular. The temperature never exceeds the limiting Hagedorn temperature. If we start the evolution as a dense gas of strings in a space in which all dimensions are string-scale tori, then there are dynamical arguments according to which only three of the spatial dimensions can become large . Thus, string gas cosmology yields the hope of understanding why - in the context of a theory with more than three spatial dimensions - exactly three are large and visible to us.
If the Hagedorn phase (the phase during which the temperature is close to the Hagedorn temperature and both the scale factor and the dilaton are static) is sufficiently long to establish thermal equilibrium on length scales of about 1 mm, then string gas cosmology can provide an alternative to cosmological inflation for explaining the origin of an almost scale-invariant spectrum of cosmological fluctuations . A distinctive signature of the scenario is the slight blue tilt in the spectrum of gravitational waves which is predicted .
The inflationary universe scenario has successes beyond the fact that it successfully predicted a scale-invariant spectrum of fluctuations - it also explains why, starting from a hot Planck scale space, an extremely low entropy state - one can obtain a universe which is large enough and contains enough entropy to correspond to our observed universe. In addition, it explains the observed spatial flatness. However, if the Hagedorn phase of string gas cosmology is realized as a long bounce phase in a universe which starts out large and cold, then the horizon, flatness, size and entropy problems do not arise.
A serious concern for the current realization of string gas cosmology, however, is the gravitational Jeans instability problem. This problem was first raised in . In the context of dilaton gravity, it can be shown  that gravitational fluctuations do not grow. However, dilaton gravity is not a consistent background for the Hagedorn phase of string gas cosmology. One might hope that since the string states are relativistic, the gravitational Jeans length will be comparable to the Hubble radius, as it is for a gas of regular radiation. However, a recent computation of the speed of sound in string gas cosmology  has shown that in a background space sufficiently large to evolve into our present universe the overall speed of sound is very small. Further work needs to be done on this issue. This is complicated by fact that string thermodynamics is non-extensive (see e.g. ), which leads to problems in using the usual thermodynamic intuition.
Note that the background space does not need to be toroidal. Crucial for string gas cosmology to yield the predictions summarized above is the existence and stability (or quasi-stability) of string winding modes. Certain orbifolds  have also been shown to yield good backgrounds for string gas cosmology. Non-trivial one cycles will ensure the existence and stability of string winding modes.
If the background space does not have have any non-trivial one-cycles, then it might be possible to construct a cosmological scenario based on stable branes rather than strings. The cosmology of brane gases has been considered in . If there are stable winding strings, and if the string coupling constant is small such that the fundamental strings are lighter than branes, then  it is the fundamental strings which will dominate the thermodynamics in the Hagedorn phase and which will be the most important degrees of freedom for cosmology. However, if there are no stable winding strings, then winding branes would become important.
It appears at the present time that Heterotic string theory is most suited for string gas cosmology since this theory admits the enhanced symmetry states which have been shown to yield a very simple way to stabilize the size and shape moduli of the extra spatial dimensions. It will be interesting to study if string gas cosmology can be embedded into particular models of Heterotic string theory which yield reasonable particle phenomenology.
The presentation we gave of string gas cosmology is based on minimal input. In particular, we did not include fluxes since we assume that the net fluxes should cancel for a situation with the most symmetric initial conditions. The role of fluxes in string gas cosmology has been studied in . Whereas the primary application of string gas cosmology will be to the cosmology of the very early universe, it is also interesting to consider applications of string gas cosmology to later time cosmology. The late time dynamics of string and brane gases has been considered in . In particular, in  applications of string gases to the dark energy problem has been considered (see also ). String and brane gases have also been studied as a way to obtain inflation [101, 102, 103] (see also ), or as a way to obtain non-inflationary bulk expansion which may provide a way to solve the size problem in string gas cosmology if one starts with a spatial manifold of string scale in all directions .
I am grateful to all of my present and former collaborators with whom I have had the pleasure of working on string gas cosmology. For comments on the draft of this manuscript I wish to thank Nima Lashkari and Subodh Patil. This work is supported in part by an NSERC Discovery Grant and by funds from the Canada Research Chairs Program.
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