Lecture notes on thermodynamics of ideal string gases and its application in cosmology
LLecture notes on thermodynamics of ideal string gasesand its application in cosmology
Lihui Liu
Institute for Theoretical Physics, KULeuvenCelestijnenlaan 200D, 3001 Heverlee [email protected]
Based on lectures given at the Eighth Modave Summer School in Mathematical Physicson 29th and 30th August 2012, Modave, Belgium
Abstract
In these lecture notes I give a pedagogical introduction to the thermodynamics of ideal stringgases. The computation of thermodynamic quantities in the canonical ensemble formalism willbe shown in detail with explicit examples. Attention will be given mainly to the thermodynam-ical consequences of string degrees of freedom, where I will especially address i) the Hagedorntemperature, a critical temperature above which the canonical ensemble description breaks down,which can be the onset point of some instability of the string gas; ii) the phase structure arisingfrom compactification, embodied in the moduli-dependence of the Helmholtz free energy, whichcorrects the tree-level vacuum and can provide mechanism for moduli stabilization. Then I willbriefly explain the implementation of string gas thermodynamics in cosmology, showing a simpleexample which gives rise to a radiation-dominated early universe. Further phenomenological is-sues and open questions will be discussed qualitatively with references indicated, including theHagedorn instability in the resolution of the initial singularity, moduli stabilization, generation ofhierarchy, radiative symmetry breaking and primordial cosmological fluctuations. a r X i v : . [ h e p - t h ] D ec ontents A Canonical partition function in quantum field theory at one-loop 42B Dedekind η -function, Jacobi elliptic functions and SO ( ) characters 46C Unfolding the fundamental domain 47 Introduction and summary
The pursuit for the resolution of the initial cosmological singularity has long been one of themajor driving forces for research in high energy physics. The common belief is that describing theinitial high energy state of the universe is out of the scope of general relativity, and some quantumgravity theory should exist, which can properly describe the spacetime under extreme conditions,and whose low energy limit contains both the general relativity and the standard model.When in the early 1980s physicists became the more and more aware that superstring theorycould probably be that very theory of quantum gravity [1], attempts also started to investigatestring theory effects in early cosmology [2]. The hope was that the Big Bang picture might beradically corrected by the non-local degrees of freedom of strings activated in the high temperatureprimordial universe, which are not available in quantum field theory. This suggested that thestarting point for working out a string resolution of the initial cosmological singularity, could bethe thermodynamics of string gases, and indeed a considerable amount of work has been motivatedsince then on string statistical mechanics [3–6].Being closely related to early cosmology, the high temperature behavior of ideal string gases hasbeen extensively studied. The aspect which received most attention is the presence of a criticaltemperature, referred to as the
Hagedorn temperature [7], above which the canonical partitionfunction becomes ill defined. In some models the Hagedorn temperature is shown to be themaximum temperature of the string gas, while in others it is shown to lead to a phase transition[3, 5], which was later realized to be related to the condensation of some thermal winding modes[8–11]. Therefore clarifying the behavior of string gases near this critical temperature, and workingout its cosmological implications are of particular interest. However this turned out to be a verytough task due to the lack of analytic control. Although considerable progress has been made, nogeneral conclusive result has been obtained so far.In the pioneering work [12], an intuitive proposal was made for resolving the initial cosmologicalsingularity, where the mechanism involved exploits the Hagedorn temperature as the maximumtemperature of the string gas, as well as the T-duality symmetry which implies that the stringgas cannot distinguish large scales from small. A broad range of work have been inspired forelaborating and developing the ideas in [12], and this gave birth to the scenario of string gascosmology (cf. [13] and the references therein). By considering an ideal string gas coupled to aclassical background of dilaton gravity, the string gas cosmology scenario managed to account forthe dimensionality of the universe, the stabilization of moduli, and later it was shown to be able toproduce a nearly scale-invariant spectrum of primordial cosmological perturbation in a bouncinguniverse, leading to an alternative scenario to inflation.2ore recently, there is another line of work developed in [14–24], which attempts to constructa first-principles string theory scenario for cosmology, simply based on string theory subjectedto cosmological principles. It turns out that this approach gives rise to a universe filled withan ideal string gas, whose thermal effects can be naturally taken into account. The importantphenomenologies such as moduli stabilization, supersymmetry breaking, generation of hierarchy,can be addressed on a solid string theory basis, and most interestingly, analytic non-singularcosmological solutions are found for certain models [22–24].These lecture notes are meant to provide an introductory course on ideal string gas thermo-dynamics, and will also cover, to some extent in the end, its cosmological application figured outin [15–24]. To this end it will be convenient to address the string gas thermodynamics with canon-ical ensemble formalism, and thus most technical computations to be carried out will be centeredaround the canonical partition function
Z = Tr e − βH . At this point we need to illustrate the ideabehind the evaluation of Z for ideal string gases, which provides the line of logic that the lecturenotes will follow.For a generic quantum system at finite temperature, we can perform the Euclidean path integralin the second quantized theory to compute its canonical partition function Tr e − βH = ∫ D φ e − S [ φ ] , (1)where we let φ account collectively for all the degrees of freedom involved and S [ φ ] be the Euclideanaction. Finite temperature is implemented by the Euclidean time circle S ( R ) whose perimeteris the inverse temperature: πR = β = T , and along which bosonic (fermonic) fields take periodic(anti-periodic) boundary condition.When dealing with a gas of weakly interacting particles, the path integral can be schematicallyrepresented and approximated as ( Tr e − βH ) particle = exp [ + + + . . . ] ≈ exp [ ] , (2)which points out that an alternative way to evaluate Eq.(1) for this gas is to compute the one-loopamplitude in the first quantized theory. We stress that here this one-loop amplitude shouldbe computed against a thermal background with compact Euclidean time S ( R ) . In the samespirit for an ideal gas of strings, which is a quantum system containing infinite degrees of freedom,we can still stick to the methodology Eq.(1). The difficulty of direct computation, however, is infiguring out the action S [ φ ] for second-quantized string theories, which is the concern of stringfield theory. We will not confront this challenge directly, but will take the alternative path inspiredby the particle gas diagrams (2), whose string theory version is ( Tr e − βH ) string = exp [ + + . . . ] ≈ exp [ ] . (3)3herefore the problem is converted to the computation of the string one-loop (genus-one) vacuum-to-vacuum amplitude against a thermal background with compact Euclidean time S ( R ) .In fact on the practical level the result of is more useful than the partition function itselfsince generally it is ln ( Tr e − βH ) instead of Tr e − βH that is involved directly in the formulae forcomputing thermodynamic quantities such as the energy, the entropy etc. Especially we have bydefinition = ln ( Tr e − βH ) = − βF , where F is the Helmholtz free energy of the ideal string gas.For the sake of clarity and explicitness, we will focus our attention on the models where exactcomputation of is achievable, which are basically weakly coupled string theories living in flatspacetime, with toroidal or orbifold compactifications. We will see for such cases that the genus-one amplitudes against a thermal background can be most conveniently obtained by deformingthose against a Minkowski background, which we will refer to as thermal deformation . Althoughthe restriction that we adopt is somewhat severe, the various possibilities of compactifications stillleave abundant room for crafting models containing rich phenomenology.Nevertheless it should be mentioned that some particular curved spacetime backgrounds doallow string thermodynamics to be exactly formulated, and/or be studied in a holographic setting,for example the thermodynamics of little string theory [25], and strings in the pp-wave background[26] (c.f. also [27] and the references therein). However these topics are beyond the scope of ourlecture notes which are oriented rather to cosmological applications.The lecture notes will be organized as follows:In Sec.2 we will give a pragmatic account of string one-loop vacuum-to-vacuum amplitudesat zero temperature, providing the preliminary background for the thermal one-loop amplitudes.For simplicity, explicit expressions will be presented only for closed string models, including thebosonic string, the type II and heterotic strings. We will however not display all technical steps,which are some standard textbook computation, but will mainly explain the interpretation ofthe results in terms of string spectrum, which is helpful for understanding the physics of thethermalization.In Sec.3 we will switch on finite temperature to the models covered in Sec.2 and compute theresulting thermal one-loop amplitudes, as well as further thermodynamical quantities. Two specificcases, heterotic string gas in 10 dimensions and 9 dimensions, will be investigated respectively inSec.3.2 and Sec.3.3, by which we aim to unravel their stringy specificities which are also sharedby other types of string gases, and which are relevant to the cosmological application. The 10dimensional example has emphasis on the high temperature behavior where we show especiallyhow the Hagedorn temperature emerges. With the 9 dimensional example we show the phasestructure arising from compactification, where the free energy depends nontrivially on the size ofthe internal space. 4n Sec.4 we will carry out further discussions on high temperature behavior where currentunderstandings of the Hagedorn temperature will be gathered and presented. We will emphasizethe following points roughly covered by the three subsections: i) the Hagedorn temperature isgenerically present in different types of ideal string gases; ii) the breaking down of the canonicalensemble formalism can be attributed to some winding modes along the Euclidean time circlebecoming tachyonic, the condensation of which can lead to a phase transition; iii) clarifying thedetails of the Hagedorn phase is not only significant to the early cosmology, but can also bringalong insight into the fundamental degrees of freedom of string theory.Finally Sec.5 implements the explicit results of string gas thermodynamics in Sec.3 in a cos-mological context. We will first set up the scenario where we will see that string thermodynamicscan naturally arise. In the second part we will deal with a simple example, showing how it leadsto a radiation-dominated early universe, which is of course just a tiny step in the long pursuit ofa realistic model. Qualitative discussion on further topics and open questions will be presentedin the end, including the Hagedorn phase of the universe, moduli stabilization, supersymmetrybreaking and hierarchy problem, radiative electroweak symmetry breaking and matter formation,and the primordial cosmological fluctuation.For notational simplicity, we will use Z ( β ) , instead of or , to denote ln ( Tr e − βH ) . Wewill call Z ( β ) the thermal one-loop amplitude . We will also need the one-loop amplitude at zerotemperature, which we will denote by Z , and it will be called the non-thermal one-loop amplitude .Also the string length will be set to be unit: l s = √ α ′ = . In this section we go through a quick survey on non-thermal string one-loop amplitudes, as apreliminary step towards the thermal one-loop amplitudes in the next section. We will mainly focusour attention on the physical idea and especially to the spectral interpretation of the expressions.We will first introduce formal expressions, and then give explicit examples of closed string modelsin the second part.
Here we adopt a heuristic approach to the formulation of non-thermal string one-loop amplitudes,which resorts to the analogy with one-loop amplitudes of particles. Explicit formulae will beshown only for closed strings, while we will be very qualitative with open strings to avoid lengthytechnical constructions. 5 rom particles to strings
In the Appendix A the one-loop amplitude of a generic ideal particle gas has been derived. Herewe need the result of the non-thermal case Eq.(105), which states that for a point particle gasin d -dimensional Minkowski spacetime of regularized volume V d , with quantum states of masses { M s } and helicities { j s } ( s being the label of states), its non-thermal one-loop amplitude is Z = ∑ s V d ( π ) d ∫ ∞ d(cid:96)(cid:96) + d (− ) j s exp (− π M s (cid:96) ) . (4)Here (cid:96) is the Schwinger time parameter, which tells the proper time that a particle spends forfinishing the loop. In order to better establish the analogy between Eq.(4) and its string theorycounterpart, we introduce the notation q ∶= e − π(cid:96) , and rewrite Eq.(4) as Z = ∑ s V d ( π ) d ∫ ∞ d(cid:96)(cid:96) + d (− ) j s q M s . (5)This shows that to obtain the non-thermal one-loop amplitude, we can coin up a power series of q , whose powers are the mass-sqared over 2 with the masses from the mass spectrum, and whosecoefficients are or − for bosonic or fermionic states, and when this is done, we integrate over theSchwinger parameter with weight (cid:96) − − d / to add up the contribution from the loops of all differentsizes.The non-thermal string one-loop amplitudes can loosely be regarded as constructed in thesame spirit: we simply replace in Eq.(5) the point particle “ q M s ” by string “ q M s ”, where now thesum over s runs over the whole string mass tower. There is however important difference betweenclosed strings and open strings. The closed string mass spectrum separates into the holomorphicsector { M L } and the anti-holomorphic sector { M R } , with the physical masses those satisfyingthe level-matching condition M = M ; for the open string mass spectrum the level matchingcondition is automatically satisfied due to the open string boundary conditions, so that there isno need to distinguish the holomorphic masses from anti-holomorphic masses. We now discussthese two cases separately. Closed strings
Assuming the mass spectra of the holomorphic and the anti-holomorphic sectors of closed stringsare { M L } and { M R } , mimicking Eq.(4), we expect the one-loop amplitudes to take the following In the lightcone gauge, M and M are obtained as the eigenvalues of the lightcone Hamiltonians L and ¯ L .The level matching condition M = M is due to the constraint L = ¯ L , which follows from the circle isometry ofthe closed string worldsheet. ∑ L , R V d ( π ) d (− ) j ( L , R ) ∫ − dv exp [− π ( M − M ) v ] ∫ ∞ d(cid:96)(cid:96) + d exp [− π ( M + M ) (cid:96) ] , (6)where the sum over L and R is understood to run over the holomorphic and the anti-holomorphicsector separately; j ( L , R ) is the helicity of the state ( L , R ) in spacetime, for example in type IIstrings j ( L , R ) is integer for NS-NS (NS for Neveu-Schwarz) or RR (R for Ramond) states, half-integer for NS-R or R-NS states. The integration over v imposes the level matching condition,so that the sum over M L and M R only picks up M = M terms, the contribution from physicalstates. The spacetime dimension is left general, denoted by d , in case the model is compactifieddown to lower dimensions.Defining the complex parameter τ = τ + iτ ∶= v + i(cid:96) , usually referred to as the Teichmülerparameter, the above expression is rewritten as ∑ L , R V d ( π ) d ∫ ⊔ dτ dτ τ + d / (− ) j ( L , R ) q M ¯ q M , (7)where q = e iπτ , ¯ q = e − πi ¯ τ , and ⊔ denotes the strip defined by − < τ < and τ > . The aboveexpression is just one step away from the genuine closed string one-loop amplitudes, because theintegral over τ over-counts physically equivalent states. Actually the integration measure τ − dτ dτ as well as the integrand τ − d ∑ q M ¯ q M are invariant under the SL ( , Z ) group generated by τ → τ + and τ → − τ − , which is the residual symmetry after fixing the worldsheet diffeomorphisminvariance. In order to remove this redundancy, we have to replace the integration domain bythe fundamental domain of the SL ( , Z ) group, which is defined, up to an SL ( , Z ) -transform,by − < τ < and τ > √ − τ . Noticeably, the UV divergence is removed because τ → isexcluded from the integration domain. We denote this fundamental domain by F , and hence ageneric non-thermal one-loop amplitude of closed strings takes the form Z = ∑ L , R V d ( π ) d ∫ F dτ dτ τ + d / (− ) j ( L , R ) q M ¯ q M . (8)We have several comments at this point ● The integral domain F can be rigorously derived from path integral calculation (c.f. [28] Chapter7, [29] Chapter 6, or [30] for details). ● If one is provided with the one-loop amplitude without knowing the mass spectrum, which isthe case in the rest of the lecture notes, one can expand the integrand in a double power seriesin q and ¯ q , and read off the masses from the powers, degeneracies from the coefficients and tellwhether a state is bosonic or fermionic from the sign of the coefficients.7 The “Schwinger parameter” τ , which is in analogy to (cid:96) in the particle case Eq.(5), is two dimen-sional. It is because the one-loop diagram, the torus, has two independent deformations whichcannot be cancelled by worldsheet diffeomorphism, i.e. the relative size of the two independentnon-contractable loops, characterized by τ , and the twist when connecting the two ends of thecylinder to form a torus, characterized by τ . It is τ which is the analogue of the Schwinger time (cid:96) in Eq.(4). Open strings
We will not explicitly formulate the open string one-loop amplitudes, since the procedure is tech-nically more involved, while the physical idea remains quite the same as point particles and closedstrings. The complexity comes from the problems that open string alone fails to provide masslessspacetime states of spin higher than , and it is UV divergent at one-loop level. Two measuresare taken to render the model consistent: including a closed string sector, and considering onlyunoriented string worldsheet. This gives rise to the type I string, whose non-thermal one-loopamplitude is the sum of the amplitudes of four different worldsheet topologies of genus one: Z = + + + . (9)We draw attention to the following points: ● The spectrum of spacetime states consists of both closed string states and open string states,and all these states are invariant under the reversal of worldsheet orientation (positive worldsheetparity). ● The torus and the Klein bottle amplitudes are from the closed string sector, wherethe sum of the two projects out the contribution of states of negative worldsheet parity. The samething is true of the open string sector, where we have the annulus and the Möbius stripamplitudes. ● The expression for the torus amplitude is just that of the closed string amplitude that we havedescribed in the previous part; for the other three amplitudes, the expressions are the same asthat of the point particle amplitude Eq.(5), since the geometries of these three types of worldsheetare characterized by only one real parameter which is identified as the Schwinger time parameter.Detailed construction of consistent open string models and the computation of their one-loopamplitudes can be found in [31], or in textbooks [28, 32, 33].8 .2 Some typical examples of closed strings
To concretize the discussions in the previous subsection, here we show explicitly the one-loopamplitudes of some typical simple models, where the technicalities involved can be applied to morecomplicated models. Since the one-loop amplitudes are always an integral over the Teichmülerparameter τ as in Eq.(8), all we need to show here is how to write down the integrand: the powerseries in q and ¯ q . It turns out that these power series can be packed into the Dedekind η -functionand the Jacobi elliptic functions or θ -functions, of which the useful notations and definitions arecollected in Appendix B. Closed bosonic string
The bosonic string is not a consistent model with its tachyonic ground state, but we still chooseit to begin with due to its technical simplicity. The model lives in a 26-dimensional backgroundspace, and has 26 worldsheet bosons. Its one-loop amplitude reads Z b = V ( π ) ∫ F dτ dτ τ η ¯ η . (10)The integrand can be conveniently obtained in the lightcone gauge, where we are left with 24transverse worldsheet bosons, each contributing a factor of η − ¯ η − , accounting exactly for η − ¯ η − .The mass spectrum of spacetime states can be read off from the powers and coefficients in theexpansion of the integrand as a power series of q and ¯ q . More accurately, using the definition(106) we have the expansion η ¯ η = q − ¯ q − + + q ¯ q + . . . , (11)where higher order terms and terms with different powers of q and ¯ q are omitted. This shows thatwe have in the physical spectrum, one tachyon of mass-squared − l − s ; = massless statestransforming in SO ( ) v × SO ( ) v (the subscript v standing for vector representation), containingthe dilaton, the graviton and the anti-symmetric tensor; states of mass l − s , transforming in SO ( ) ◻◻ × SO ( ) ◻◻ ; etc. Type II string
The type II string lives in a 10-dimensional background space, possessing 10 worldsheet bosonsand their superpartner, 10 worldsheet fermions. The one-loop amplitude, in terms of the Dedekind9 -function and Jacobi θ -functions, is Z II = V ( π ) ∫ F dτ dτ τ η ¯ η ∑ a,b = (− ) a + b + ab θ [ ab ] η ∑ ¯ a, ¯ b = (− ) ¯ a + ¯ b + (cid:15) ¯ a ¯ b ¯ θ [ ¯ a ¯ b ] ¯ η , (12)where (cid:15) = or stand for the type IIA or IIB string respectively. We would like to give thefollowing remarks to help understanding this expression: ● The integrand can be obtained, as in the previous example of bosonic string, through thelighcone-gauge fixed model. In the holomorphic part, θ [ ab ] η arises from the 8 transverse holomorphicworldsheet fermions, the extra η from the 8 transverse holomorphic worldsheet bosons; the samerule applies to the anti-holomorphic part. The sum over ( a, b ) and ( ¯ a, ¯ b ) adds up the contributionfrom all combinations of spin structures. ● The indices a and ¯ a , taking independently the values or , indicate the NS sector or the R sectorrespectively in the holomorphic and anti-holomorphic sector; the sum over b and ¯ b implementsthe GSO projection. In the anti-holomorphic sector, changing the value of (cid:15) from to reversesthe GSO projection in the Ramond sector, which flips the gravitino’s chirality, so that the modelpasses from type IIA to type IIB. ● The modular invariance is manifest with Eq.(12): using the properties of η - and θ -functions inthe Appendix C of [33] we can verify the invariance of Eq.(12) under τ → τ + and τ → − τ − .To let the one-loop amplitude display the structure of the type II string spectrum, one rewritesEq.(12) using the SO ( ) -characters. Using the definitions Eqs (108)–(111), Eq.(12) is rewrittenas Z IIA = V ( π ) ∫ F dτ dτ τ V − S η ¯ V − ¯ C ¯ η = V ( π ) ∫ F dτ dτ τ ⎡⎢⎢⎢⎢⎣( V ¯ V η ¯ η + S ¯ C η ¯ η ) − ( V ¯ C η ¯ η + S ¯ V η ¯ η )⎤⎥⎥⎥⎥⎦ , (13) Z IIB = V ( π ) ∫ F dτ dτ τ V − S η ¯ V − ¯ S ¯ η = V ( π ) ∫ F dτ dτ τ ⎡⎢⎢⎢⎢⎣( V ¯ V η ¯ η + S ¯ S η ¯ η ) − ( V ¯ S η ¯ η + S ¯ V η ¯ η )⎤⎥⎥⎥⎥⎦ . (14)In the second line in each of the above two expressions, the integrand is written in the form of[(NS-NS sector) + (RR sector)] − [(NS-R sector)+(R-NS sector)], where the first half contains bosonicstate contributions and the second half fermionic. Numerically Eqs (12), (13) and (14) vanish dueto spacetime supersymmetry (c.f. the end of Appendix B). However these formal expressions10uffice for extracting the spectrum of spacetime states. For example by just staring at Eq.(13),we can tell the following information about the massless spectrum : ◇ the NS-NS sector, giving rise to the V ¯ V -term, has ground states transforming in SO ( ) v × SO ( ) v , consisting of the dilaton, the graviton and the anti-symmetric tensor; ◇ the RR sector, giving rise to the S ¯ C -term, has ground states transforming in SO ( ) s × SO ( ) c (the subscripts s and c standing for the two spinorial representations of opposite chirality), con-taining the 1-, 3- and 5-forms; ◇ the NS-R and the R-NS sectors, which correspond to the V ¯ C -term and the S ¯ V -term, haveground states transforming in SO ( ) v × SO ( ) c and SO ( ) s × SO ( ) v , containing the two gravitiniof opposite chirality.Thus we have seen how to tell the ground states using the formal expressions of one-loopamplitudes, while in order to read off the full physical spectrum, we need to perform the expansionin power series of q and ¯ q , as in Eq.(11), separately to the four terms in the second line of Eqs (13)and (14). The same analysis can be applied to the type IIB one-loop amplitude Eq.(14), whichwe do not repeat here.Finally we point out that for type II strings, a + ¯ a plays the role of spacetime fermion number.That is, the spacetime bosons are associated to a + ¯ a = while spacetime fermions a + ¯ a = . Indeed in the second line of Eqs (13) and (14), the four terms correspond respectivelyto the values , , , of a + ¯ a . Thus j ( L , R ) in the general expression Eq.(8) can be concretelywritten as a + ¯ a . Toroidal compactification
Upon toroidal compactification of closed strings, the compact worldsheet scalars are quantized onthe complex plane as X I L ( z ) = x I L − i √ p I L ln z + ( oscillators ) , X I R ( ¯ z ) = x I R − i √ p I R ln ¯ z + ( oscillators ) , (15)where L and R stand for “left-moving” and “right-moving” corresponding to the holomorphic andanti-holomorphic sectors, I is the index of compact directions, and x I L , R and p I L , R are the center-of-mass positions and momenta. Generically x I L ≠ x I R and p I L ≠ p I R , where the center-of-mass momentatake discrete values lying on the Narain lattice points generated by the basis vectors spanning theinternal torus. For example compactifying the 9-th direction on a circle S ( R ) results in p , R = mR ∓ nR , m, n ∈ Z , (16)11here m and n are respectively the quantum number of the internal momentum and of windingalong the compact circle S ( R ) .More generally, toroidal compactifications down to spacetime dimension d , with compact co-ordinates Eq.(15), induce in the full-dimensional one-loop amplitudes the following modifications: i) the spacetime dimension is replaced by d ; ii) the integrand of the integration over τ acquires a factor of Narain lattice sum Γ = ∑ p L ,p R q p ¯ q p , (17)where p , R ∶= ∑ I p I L , R p I L , R .We still take the example of compactification on a circle S ( R ) in the 9-th direction wherethe internal momenta are given by Eq.(16). Then the bosonic string one-loop amplitude, given byEq.(10) for full spacetime dimension, now becomes Z b / S ( R ) = V ( π ) ∫ F dτ dτ τ / η ¯ η ∑ m,n ∈ Z q ( mR − nR ) ¯ q ( mR + nR ) ; (18)also for type II strings, the compactification on S ( R ) leads to the following one-loop amplitudein 9 spacetime dimensions: Z II / S ( R ) = V ( π ) ∫ F dτ dτ τ / η ¯ η × ∑ a,b = (− ) a + b + ab θ [ ab ] η ∑ ¯ a, ¯ b = (− ) ¯ a + ¯ b + (cid:15) ¯ a ¯ b ¯ θ [ ¯ a ¯ b ] ¯ η ∑ m,n ∈ Z q ( mR − nR ) ¯ q ( mR + nR ) , (19)where (cid:15) is as in Eq.(12).Compactifications generically modify the spectra of spacetime states, where the masses ofstates can be deformed in function of the moduli fields characterizing the compactification. Herewhen we expand the integrand of Eqs (18) and (19) in power series in q and ¯ q , the powers acquire R -dependence, showing that the compactification induces spacetime states with R -dependentmasses. Heterotic string
The heterotic string demands 10-dimensional background space to be consistent. It has 10 world-sheet bosons, whose holomorphic parts are accompanied with superpartners, the 10 holomorphicworldsheet fermions, and it has 16 internal anti-holomorphic worldsheet bosons compactified on12pin(32) / Z or E × E lattice, giving rise to SO ( ) or E × E gauge symmetry in spacetime.The one-loop amplitude takes the following form: Z het = V ( π ) ∫ F dτ dτ τ η ¯ η ∑ a,b = (− ) a + b + ab θ [ ab ] η ¯Γ int ¯ η , where (20) ¯Γ int = ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ ∑ α,β = ¯ θ [ αβ ] , SO ( ) heterotic string , ( ∑ α,β = ¯ θ [ αβ ] ) , E × E heterotic string . In terms of the lightcone-gauge fixed model, η ¯ η arises from the 8 transverse worldsheet bosons, θ [ ab ] η from the 8 transverse holomorphic worldsheet fermions, and ¯Γ int ¯ η from the 16 internal anti-holomorphic worldsheet bosons, where ¯Γ int is the sum of the Spin(32) / Z lattice or the E × E lattice, which is a power series of ¯ q . The modular invariance of Eq.(20) can be easily verified.We can also use the SO ( ) -characters to display the structure of the spectrum. Using Eqs(109)–(111), we have Z het = V ( π ) ∫ F dτ dτ τ V − S η ¯ η ¯Γ int ¯ η = V ( π ) ∫ F dτ dτ τ ( V η ¯Γ int ¯ η − S η ¯Γ int ¯ η ) , (21)where ¯Γ int is still as defined in Eq.(20). In the last step, the integrand is written in the form of(contribution of bosonic states) − (contribution of fermonic states). Here the spacetime bosons andspacetime fermions arise respectively from the NS ( a = ) and R ( a = ) sector of the holomorphicworldsheet fermions. Thus here a plays the role of spacetime fermion number and therefore j ( L , R ) in the general expression in Eq.(8) can be concretely written as a .Due to spacetime supersymmetry, Eq.(21) is numerically , but still we can read off the spec-trum from the formal expression just as for the type II strings under Eq.(14). We will postponethis analysis to Sec.3.2 where Eq.(21) needs to be more carefully investigated. In this section we consider the thermodynamics of ideal string gases. We will work out, throughthermal deformations, the thermal one-loop amplitudes Z ( β ) based on the non-thermal results inthe previous section, and by virtue of Eq.(3), they are the logarithm of the canonical partitionfunction. Then the whole set of canonical ensemble formalism applies and we can derive all otherthermodynamic quantities, for example the energy E = − ∂Z ( β )/ ∂β , the entropy S = βE + Z ( β ) ,the Helmholtz free energy F = − Z ( β )/ β , and the pressure P = − ∂F / ∂V .13n the following we will first specify the rules of thermal deformation of the string one-loopamplitudes, and apply them to the models discussed in the previous section. Then we will inves-tigate in detail two examples of heterotic string gas, where we will put emphasis on the Hagedorntemperature, and the phase structure arising from compactification, which are general features forideal string gases, and are of interest in string cosmology. To figure out how one-loop amplitudes are to be deformed by thermal effects, it is instructive tofirst look into the same situation for point particles, and we thus invoke the calculations presentedin the Appendix A. Comparing the non-thermal one-loop amplitude Eq.(99) and the thermal oneEq.(105), which we restate here for clarity: Z = ∑ s V d ( π ) d ∫ ∞ d(cid:96)(cid:96) + d / (− ) j s exp (− π(cid:96)M s ) , (22) Z ( β ) = ∑ s βV d − ( π ) d ∫ ∞ d(cid:96)(cid:96) + d / ∑ ˜ m (− ) j s ( ˜ m + ) exp (− πR (cid:96) ˜ m − π(cid:96)M s ) , (23)we find that finite temperature is implemented by inserting in the non-thermal one-loop amplitudethe following factor ∑ ˜ m ∈ Z (− ) j s ˜ m exp (− πR (cid:96) ˜ m ) , (24)and replacing the volume of the full spacetime V d by that of the thermal background space βV d − .We qualify this operation as deformation since Eq.(22) is the β → ∞ limit of Eq.(23). For bosonicstates ( j s even) Eq.(24) is merely an instanton sum associated to the compact Euclidean time S ( R ) , where the instanton number ˜ m is the number that the particle loop winds the Euclideantime circle. For fermionic states ( j s odd), the instanton sum in Eq.(24) is weighted by a factor (− ) ˜ m , which accounts for the Fermi-Dirac statistics. We call the weight (− ) j s ˜ m in Eq.(24) thespin-statistics factor .We want to work out the thermal string one-loop amplitudes by a similar deformation, i.e. in-serting into the non-thermal expressions the string version of Eq.(24) which should be a worldsheetinstanton sum weighted by a proper spin-statistics factor . We know that the worldsheet instantonsum arising from the compactification on a circle is (c.f. for example Sec.4.18 of [33]) ∑ m,n ∈ Z e − πR τ ∣ ˜ m − τn ∣ , (25)14here ˜ m and n are the numbers that the world sheet winds the Euclidean time circle. There aretwo winding numbers since the woldsheet is a torus containing two independent non-contractableloops both of which can wind the compact Euclidean time. We still need to find out the spin-statics factor weighting the instanton sum, and it turns out that its explicit form depends on thetype of string theory.Below we will present the results for the specific models that we have discussed in Sec.2.2,some intuitive reasonings and guessworks are inevitably involved. Rigorous derivation can befound in [8] for heterotic strings, the logic of which can be applied to other types of strings.The formalism that we present here is are those in [34] where the expressions were cast into aconcise form. We will present the formal expressions all at once, leaving explicit calculation andinterpretations in later subsections. Bosonic string
To obtain the thermal bosonic string one-loop amplitude, it suffices to simply insert Eq.(25) intoEq.(10), because there are only bosons in the spectrum of spacetime states. Therefore the thermalone-loop amplitude in 26 dimensions is Z b ( β ) = βV ( π ) ∑ ˜ m,n ∫ F dτ dτ τ e − πR τ ∣ ˜ m − τn ∣ η ¯ η . (26)This means that the thermalization of the bosonic string is just a naive toroidal compactificationof the Euclidean time. Indeed when applying Poisson resummation Eq.(95) to the sum over ˜ m above, we obtain exactly the lattice sum (17) with the internal momenta given by Eq.(16), R replaced by R .In case of additional toroidal compactification in spatial dimensions, the thermal one-loopamplitude is still obtained through the steps leading to Eqs (18) and (19) regardless of ther-malization, that is, replacing in Eq.(26) the full dimension 26 by the number of dimension aftercompactification, and inserting the Narain lattice sum corresponding to the compact space. Type II strings
For type II superstrings, as the spectrum consists of spacetime bosons and fermions, the spin-statistics factor is nontrivial. We have already seen in Sec.2.2 that in type II strings it is a + ¯ a as in Eq.(12) that plays the role of j s in Eq.(24). Therefore the spin-statistics factor contains atleast (− ) ( a + ¯ a ) ˜ m . However there should be more than that: since the modular transforms τ → τ + and τ → − τ − mix up a and b in Eqs (12), these two indices need to be treated on equal footing,15nd the same thing is true of ¯ a and ¯ b . It can be shown that the result is modular invariant if wealso include the factor (− ) ( b + ¯ b ) n . Thus the complete instanton sum for type II strings is ∑ ˜ m,n ∈ Z (− ) ( a + ¯ a ) ˜ m +( b + ¯ b ) n e − πR τ ∣ ˜ m − τn ∣ , (27)and the thermal one-loop amplitude is Z II ( β ) = βV ( π ) ∫ F dτ dτ τ ∑ a,b ∑ ¯ a, ¯ b [ ∑ ˜ m,n (− ) ( a + ¯ a ) ˜ m +( b + ¯ b ) n e − πR τ ∣ ˜ m − τn ∣ ]× η ¯ η (− ) a + b + ab θ [ ab ] η (− ) ¯ a + ¯ b + (cid:15) ¯ a ¯ b ¯ θ [ ¯ a ¯ b ] ¯ η , (28)where (cid:15) is as in Eq.(12), and the instanton sum should be inserted inside the sums over spinstructures ( a, b ) and ( ¯ a, ¯ b ) .It is not surprising that the insertion of Eq.(27) preserves modular invariance, since this de-formation is nothing but a Scherk-Schwarz compactification of the Euclidean time, which is afreely acting orbifold compactification and which by construction preserves modular invariance.The modular invariance at finite temperature was studied as early as [35], but the connection withScherk-Schwarz compactification was realized in [34]. Heterotic string
We have stated by the end of Sec.2 that for the heterotic string, the spacetime fermion numberis given by the index a in Eq.(20). Thus by the same consideration as for type II strings, the fullinstanton sum to be inserted into Eq.(20) to account for finite temperature is ∑ ˜ m,n ∈ Z (− ) a ˜ m + bn + ˜ mn e − πR τ ∣ ˜ m − τn ∣ . (29)The thermal one-loop amplitude is Z het ( β ) = βV ( π ) ∫ F dτ dτ τ ∑ a,b [ ∑ ˜ m,n (− ) a ˜ m + bn + ˜ mn e − πR τ ∣ ˜ m − τn ∣ ] η ¯ η (− ) a + b + ab θ [ ab ] η ¯Γ int ¯ η , (30)where again, ¯Γ int is the lattice sum defined in Eq.(20). Here just as for the type II case, theinsertion of Eq.(29) implements a Scherk-Schwarz compactification of the Euclidean time. It was first introduced in field theory [36] as a mechanism of spontaneous supersymmetry breaking and lateron realized in string theory [34, 37]. pen string in short Recall that the non-thermal one-loop amplitude of open string models is the sum of the contri-butions from four different worldsheet topologies, which is mentioned by the end of Sec.2.1 inEq.(9). Thus once the model is thermalized, those four amplitudes (torus, Klein bottle, annulusand Möbius strip) are deformed separately into their thermal version, and they sum up to thetotal thermal one-loop amplitude. The torus amplitude is just the closed string amplitude, andthus its thermal deformation is just as explained in the previous paragraphs. For the three otheramplitudes, since they take the form of one-loop amplitudes of particles as is stated below Eq.(9),their thermal deformation proceeds like that for particles, yielding expressions of the form Eq.(23).From a technical point of view, the thermal deformation of open string one-loop amplitudesamounts to performing the Scherk-Schwarz compactification separately to the non-thermal torus,Klein bottle, annulus and Möbius strip amplitudes, and then summing them up.More about Scherk-Schwarz compactification in open string models can be found in [31]. Adetailed computation of type I thermal one-loop amplitude is presented in the appendix of [20].
To gain a flavor of what all the formulae in the last subsection are about, we will study simple casesof weakly coupled heterotic string thermodynamics where we will focus on the stringy propertiescommonly shared with other types of string gases. Also the technique to be used is of more generalapplicability.As a first example we consider heterotic string in 10 dimensions, whose thermal one-loopamplitude is given by Eq.(30). To perform analytic calculation, we can use the unfolding techniquesummarized in Appendix C, which decomposes the instanton sum of ( ˜ m, n ) into the ( , ) -orbit,integrated over the fundamental domain F , and the ( ˜ m, ) -orbit where ˜ m ≠ , integrated over theupper strip − < τ < , τ > denoted by ⊔ . The decomposition leads to Z het ( β ) = βV ( π ) ∫ F dτ dτ τ ∑ a,b η ¯ η (− ) a + b + ab θ [ ab ] η ¯Γ int ¯ η + βV ( π ) ∫ ⊔ dτ dτ τ ∑ a,b [ ∑ ˜ m ≠ (− ) a ˜ m e − πR τ ˜ m ] η ¯ η (− ) a + b + ab θ [ ab ] η ¯Γ int ¯ η , (31)17r in terms of the SO ( ) characters, Z het ( β ) = βV ( π ) ∫ F dτ dτ τ V − S η ¯Γ int ¯ η + βV ( π ) ∫ ⊔ dτ dτ τ ∑ ˜ m ≠ e − πR τ ˜ m V − (− ) ˜ m S η ¯Γ int ¯ η . (32)The first lines of the above two equations are the ( , ) -orbit, which is just the non-thermal one-loop amplitude (only with V replaced by βV ), and which vanishes numerically by virtue ofthe identity (113), consequence of spacetime supersymmetry. The second lines of the above twoequations, which vanish when temperature is switched off ( β = πR → ∞ ), account for thermaleffects. In the sum of ˜ m in the second lines, we notice that only the terms with odd ˜ m are non-vanishing; terms with even ˜ m vanish again due to the identity (113). Discarding all the vanishingpieces in Eq.(32), we are left with Z het ( β ) = βV ( π ) ∫ ⊔ dτ dτ τ ∑ a,b [∑ ˜ ∈ Z e − πR τ ( + ) ] η ¯ η (− ) b + ab θ [ ab ] η ¯Γ int ¯ η = βV ( π ) ∫ ∞ dτ τ ∫ − dτ ∑ ˜ ∈ Z e − πR τ ( + ) V + S η ¯Γ int ¯ η . (33)To further on, we can power expand the integrand separately in the holomorphic and the anti-holomorphic sector in terms of q and ¯ q (which can be easily done with Mathematica) and thisyields V + S η = ∞ ∑ K = S K q K , where {S K } K ≥ = { , , , , , , , . . . } ; (34) ¯Γ int ¯ η = ∞ ∑ ¯ L =− T ¯ L ¯ q ¯ L , where {T ¯ L } ¯ L ≥− = { , , , , , , . . . } . (35)Here the powers of q and ¯ q are all integers, which is not necessarily the case in general. NowEq.(33) becomes Z het ( β ) = βV ( π ) ∫ ∞ dτ τ ∫ − dτ ∑ ˜ ∈ Z e − πR τ ( + ) ∑ K ≥ L ≥− S K T ¯ L q K ¯ q ¯ L . (36) We take advantage of these expansions to show explicitly how to read off the spectrum of spacetime states,following the recipe given in Sec.2.2. What we can learn from Eqs (34) and (35) is that, in either the bosonicsector or the fermonic sector, the holomorphic mass tower { M L } is given by M = K ( K = , , , . . . ) in theunit of l − s , with degeneracy S K for the level K ; and the anti-holomorphic mass tower { M R } is given by M = L ( ¯ L = − , , , , . . . ), with degeneracy T ¯ L for level ¯ L . The anti-holomorphic sector contains a tachyonic state ¯ L = − ,but it is not physical since physical states need to be level-matched: M = M , and in the holomorphic sector thelevel at K = − does not exist. Thus the tower of physical bosonic or fermionic states has mass spectrum M = K ( K = , , , . . . ), with degeneracy S K T K for level K , e.g. we have S T = × bosonic (or fermionic) states atthe massless level, S T = × bosonic (or fermionic) states of mass l − s , etc. τ is now easy to perform. Taking into account the definition q = e πiτ andthe fact that K and ¯ L are integers, we see that integrating over τ eliminates all terms with K ≠ ¯ L in the sum in Eq.(36). Thus Z het ( β ) = βV ( π ) ∫ ∞ dτ τ ∑ ˜ ∈ Z e − πR τ ( + ) ∑ K ≥ S K T K ( q ¯ q ) K = βV ( π ) ∫ ∞ dτ τ ∑ ˜ ∈ Z ∑ K ≥ S K T K e − πR τ ( + ) − π τ ( K ) =( βV ) β − ∑ K ≥ S K T K G ( β √ K ) . (37)From Eq.(36) to the first line above, we notice that the tachyonic state contribution in the anti-holomorphic sector, the ¯ L = − term in the expansion (35), is discarded. Therefore in the secondline K is non-negative, which ensures the convergence of the τ -integration for each single termin the summation over K and ˜ . Therefore assuming the integration and the summations areinterchangeable, we did the τ -integration and obtained the third line above, where we have definedthe function G d ( x ) = ∑ j ∈ Z ( x π ∣ j + ∣ ) d K d ( x ∣ j + ∣ ) ; here d = . (38)The function K ν ( ⋅ ) is the modified Bessel function of the second kind, which arises by virtue ofEq.(100). Based on the properties of K ν ( ⋅ ) we know that G d ( x ) is a monotonously decreasingfunction for x > and it has the following asymptotic behaviors for d ≥ : G d ( x ) = c d − c d − π x + O( x ) ( for x ≃ ) , G d ( x ) ∼ ( x π ) d − e − x ( for x ≫ ) . (39)where c d = G d ( ) = Γ ( d ) π d ∑ j ∈ Z ∣ j + ∣ d . (40)The function G d ( ⋅ ) can be written in a more familiar form with the standard canonical ensembleformalism. Referring to the formulae (101)–(104) we have, for a given temperature β and mass M , G d ( βM ) = β d − ( π ) d − ∫ d ⃗ p ln ( + e − β √⃗ p + M − e − β √⃗ p + M ) , (41) Had the power expansion in q in the holomorphic sector of Eq.(34) had a tachyonic q − term, which is the caseof bosonic string as shown in Eq.(11), giving rise to a physical tachyonic state, then the sum over K in Eq.(37)would contain a K = − term, which makes the integral over τ diverge at the upper limit τ → ∞ . Generically thepresence of a tachyonic state in the physical spectrum makes the one-loop amplitude ill defied. ⃗ p is the ( d − ) -dimensional spatial momentum. Inserting this into Eq.(37) setting M = √ K ,we obtain exactly the standard canonical ensemble formalism for a supersymmetric ideal gas.We give some comments to close the investigation on this model, while the points that wemake below can be generalized to other models covered in Sec.2.2. ● This model has spacetime supersymmetry, where the supersymmetric boson-fermion pairs havemasses √ K ( K = , , , . . . ). Each boson-fermion pair of mass √ K contribute to the ther-mal one-loop amplitude the amount βV × β − G ( β √ K ) . More generally for an ideal stringgas having spacetime supersymmetry in dimension d , each boson-fermion pair of mass M phy con-tributes βV d − × β − d G d ( βM phy ) . Referring to the asymptotic behaviors (39) we see that lightstates βM phy ≪ contribute substantially, while contributions from heavy states βM phy ≫ aresuppressed by a Boltzmann factor e − βM phy . ● Taking the low temperature limit of Eq.(37) we will see that the heterotic string gas at lowtemperature looks like a particle gas. We set T = β − ≪ , and using the asymptotic behaviorsEq.(39), we obtain Z het ( β ) = βV × β − S T G ( ) + O( e − β ) = V S T c × T + O( e − / T ) . (42)That is, only massless states give non-negligible contribution, while the contribution from massivemodes in the mass tower are exponentially suppressed. From the above result we derive theenergy density: ρ = − V ∂Z het ( β ) ∂β = S T c × T , which is the Boltzmann-Steffan law for radiationwith c the Stefan constant and S T the degeneracy. We also have the free energy density F = − Z het ( β ) βV = − S T c T , and the pressure P = − ∂ T Z het ( β ) ∂V = S T c T . Also we see that thethermodynamic relation ρ = T ∂ T P − P is satisfied. ● The thermal one-loop amplitude Eq.(37) is well defined only when the sum over K converges.However we will see that the convergence is ensured only when the temperature is below a criticalvalue, which is just the Hagedorn temperature mentioned in the introduction Sec.1. Let us seehow this happens. Referring to the results in [5, 38] we have the following asymptotic behavior ofthe degeneracies as K → ∞ : S K ∼ const . × ( √ K ) − / exp ( π × √ × √ K ) ; (43) T K ∼ const . × ( √ K ) − / exp ( π × × √ K ) . (44)Since the mass of states are √ K , the above expressions show that the degeneracies grow expo-nentially with mass. Using also the asymptotic behavior Eq.(39), we see that in Eq.(37) the termsin the sum over K are proportional to K − exp [( π + √ π − β ) √ K ] for large K . Then obviously20he sign of ( π + √ π − β ) in the exponential should be nonpositive in order that the sum over K converge, which amounts to requiring β > π ( + √ ) . This defines the Hagedorn temperature T het H = / β het H , given by: β het H = π ( + √ ) . (45)When the temperature of the gas is above T het H , Eq.(37) becomes pathologic and the canonicalensemble formalism breaks down. Actually the Hagedorn temperature generically exists in idealgases of different types of critical strings, whose value does not depend on the dimensionality ofspacetime [38]. The detailed properties of ideal string gases near the Hagedorn temperature andtheir cosmological consequences will be of interest to investigate, and we will discuss more thisissue in Sec.4. In this part we consider the effect of compactification on string thermodynamics. In particular,we will show that some phase structure can be induced by compactification, rendering the freeenergy moduli-dependent.We illustrate the idea with the simplest model at hand: the compactification of heteroticstring on a circle S ( R ) in the 9-th direction, with R the radius of the circle. Thus the thermalbackground space is S ( R ) × R × S ( R ) , where S ( R ) is the Euclidean time circle. We can nowwrite down the thermal one-loop amplitude, which is obtained by modifying Eq.(30) accordingthe prescription given in Sec.2.2 about toroidal compactification. The expression is Z ( β ) = βV ( π ) ∫ F dτ dτ τ / ∑ a,b [ ∑ ˜ m ,n (− ) a ˜ m + bn + ˜ m n e − πR τ ∣ ˜ m − τn ∣ ]× η ¯ η (− ) a + b + ab θ [ ab ] η ¯Γ int ¯ η × ∑ m ,n q ( m R − n R ) ¯ q ( m R + n R ) . (46)Here the momentum and winding numbers are subscripted with and to distinguish the timedirection from the 9-th spatial direction. There is a clash of notation: V outside the integral isthe volume of the 8 dimensional space, not to be confused with the SO ( ) -character Eq.(109).Performing the unfolding trick to Eq.(46) to the sum over ( ˜ m , n ) as what we did to obtain21q.(32), we have the following decomposition Z ( β ) = βV ( π ) ∫ F dτ dτ τ / ∑ a,b η ¯ η (− ) a + b + ab θ [ ab ] η ¯Γ int ¯ η ∑ m ,n q ( m R − n R ) ¯ q ( m R + n R ) + βV ( π ) ∫ ⊔ dτ dτ τ / ∑ a,b [ ∑ ˜ m ≠ (− ) a ˜ m e − πR τ ˜ m ]× η ¯ η (− ) a + b + ab θ [ ab ] η ¯Γ int ¯ η ∑ m ,n q ( m R − n R ) ¯ q ( m R + n R ) (47)In terms of SO ( ) -characters, Z ( β ) = βV ( π ) ∫ F dτ dτ τ / V − S η ¯ η ¯Γ int ¯ η ∑ m ,n q ( m R − n R ) ¯ q ( m R + n R ) + βV ( π ) ∫ ⊔ dτ dτ τ / [ ∑ ˜ m ≠ e − πR τ ˜ m ] V − (− ) ˜ m S η ¯Γ int ¯ η ∑ m ,n q ( m R − n R ) ¯ q ( m R + n R ) = βV ( π ) ∫ ⊔ dτ dτ τ / [∑ ˜ e − πR τ ( + ) ] V + S η ¯Γ int ¯ η ∑ m ,n q ( m R − n R ) ¯ q ( m R + n R ) . (48)where in proceeding to the last step, we discarded the ( ˜ m , n ) = ( , ) orbit, as well as the termswith even ˜ m in the ( ˜ m , ) -orbit, since they are numerically zero ( V − S = ). Then we plug inthe expansions (34) and (35), and obtain Z ( β ) = βV ( π ) ∫ ⊔ dτ dτ τ / ∑ ˜ e − πR τ ( + ) ∑ K ≥ , ¯ L ≥− m ,n ∈ Z S K T ¯ L q K + ( m R − n R ) ¯ q ¯ L + ( m R + n R ) = βV ( π ) ∫ ⊔ dτ dτ τ / ∑ ˜ e − πR τ ( + ) ∑ K ≥ , ¯ L ≥− m ,n ∈ Z S K T ¯ L e πiτ ( K − ¯ L − m n ) e − πτ ( K + L + m R + n R ) , (49)where in the second line we used the definition q = e πiτ . We notice that in the first exponential thefactor πiτ is multiplied by an integer K − ¯ L − m n . Therefore only terms with K − ¯ L − m n = would survive the integral over τ . Thus we have Z ( β ) = βV ( π ) ∫ ∞ dτ τ / ∑ ˜ e − πR τ ( + ) ∑ K ≥ , ¯ L ≥− m ,n ∈ Z K − ¯ L = m n S K T ¯ L e − πτ ( K + L + m R + n R ) = βV × β − ∑ K ≥ , ¯ L ≥− m ,n ∈ Z K − ¯ L = m n S K T ¯ L G ⎛⎜⎝ β ¿``(cid:192) K + L + m R + n R ⎞⎟⎠ , (50)where G d ( ⋅ ) is as in Eq.(38) or (41). This shows that the physical states have masses-squared M = K + L + m R + n R , subjected to the level-matching condition K − ¯ L = m n , and they22ave R -dependence. To further on analytically, we consider the low temperature approximation β ≫ of Eq.(50), where only light states give non-negligible contribution. There are four groupsof light states: ◇ states with K = ¯ L = m = n = , which are obviously massless for all value of R ; ◇ pure Kaluza-Klein (KK) modes along S ( R ) with K = ¯ L = n = and m ≠ , of masses ∣ m R ∣ , which become light when R → ∞ ; ◇ pure winding modes along S ( R ) with K = ¯ L = m = and n ≠ , of masses ∣ n R ∣ , whichbecome light as R → ; ◇ states with one unit of momentum and winding along S ( R ) : m = n = ± , with K = , ¯ L = − , whose masses are ∣ R − − R ∣ , becoming massless when R = , responsible for the gaugesymmetry enhancement U ( ) → SU ( ) .All other physical states have masses of at least one unit of l − s for all value of R , whosecontribution to Eq.(50) are suppressed at least by e − β . Therefore in the summation in Eq.(50) weonly need to conserve the terms associated to the light states enlisted above, which yields Z ( β ) = βV × β − {S T G ( ) + S T ∑ m ≠ G ( β ∣ m ∣/ R )+ S T ∑ n ≠ G ( β ∣ n ∣ R ) + S T − G ( β ∣ R − − R ∣) } + O( e − β ) . (51)To show the phase structure, we study the density of the Helmholtz free energy F d het = − Z d het ( β )/ βV .Recalling that G (∣ x ∣) is significantly nonzero in the neibourhood of x = and that it decreasesexponentially as ∣ x ∣ increases, therefore the free energy density are well approximated by thefollowing expressions in different regions of R : F d het ≈ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ − T [S T G ( ) + S T − G ( β ∣ R − − R ∣) ] , ∣ R − − R ∣ < R − , phase I ; − T [S T G ( ) + S T ∑ m ≠ G ( β ∣ m ∣/ R ) ] , R > R , phase III ; − T [S T G ( ) + S T ∑ n ≠ G ( β ∣ n ∣ R ) ] , R < R − , phase III’ ; − T S T G ( ) , otherwise , phase II and II’ . (52)Plotting this free energy density against ln R at fixed temperature we obtain Fig.1, where thereason to choose ln R as variable is to display the symmetry under T-duality R → / R . Wecan distinguish different phases where the string gas behaves differently, numbered with Romannumerals both in Fig.1 and in Eq.(52): 23 IIII’ IIIIII’ R ln R F d het − ln R R T S T G ( ) Figure 1:
Schematic plot of the low temperature free energy density Eq.(52) of an ideal heterotic stringgas compactified on the circle S ( R ) . We use ln R in order that the T-duality symmetry is manifest. ◇ In the phase I we see a local minimum at R = , to which R can be attracted, and thisprovides the mechanism for stabilizing the modulus R . ◇ In the phases III and III’ the radius R is attracted to ∞ and respectively, so that thesystem is decompactified. ◇ In the phases II and II’ R can be fixed at any value.The following observations can be drawn from the computations we have gone through so far: ● The free energy density is lowered whenever there are extra light or massless states appearingin the spectrum. In phase I, we have the local minimum induced by the S T − light states ofmass ∣ R − − R ∣ , which carry one unit of momentum and one unit of winding along S ( R ) ; inphase III the slope is induced by the pure KK modes of masses ∣ m ∣/ R ( m ≠ ); in phase III’ theslope is induced by the pure winding modes of masses ∣ n ∣ R ( n ≠ ). Generically for the modelsmentioned in Sec.2.2, the Helmholtz free energy density at one-loop is always lowered when extramassless or light states emerge in the spectrum, and thus at weak coupling, thermal effects attractthe system to nearby available configurations with more massless or light states. ● Whereas in the previous example in Sec.3.2 without compact spatial dimension, stringy effectswere manifest only at high temperature (where we have observed the Hagedorn temperature), hereupon compactification, strictly stringy effects can show up at low temperature, because we canhave light winding modes which are thermalized at low temperature. The stringy effects observedin Fig.1 are: the local minimum in phase I induced by states with both winding and momentum24umbers; the slope in phase III’ induced by pure winding modes. Also the winding modes areessential to implementing the symmetry under T-duality. Similar features related to the windingmodes can be observed in other types of strings mentioned in Sec.2.2.In Sec.5 we will investigate the cosmological implications of this phase structure. More complexphase structure can be produced when more sophisticated compactifications are implemented,producing a larger moduli space.
In this section we pick up the topic of Hagedorn temperature suspended by the end of Sec.3.2. Thediscussions to be presented apply generically to the class of ideal string gases that we consider:those of critical strings living in flat spacetime with toroidal or orbifold compactifications.We will first show that the presence of a Hagedorn temperature is general in the string gasesof our concern. Then we will show with explicit calculation, that some thermal winding modescan become tachyonic above the Hagedorn temperature, and they are responsible for the breakingdown of the canonical ensemble description. We will hence be led to address the aspect that theHagedorn temperature can be a phase transition point. In the last part of this section we willdiscuss the conditions for the onset of the Hagedorn phase transition, the Hagedorn instabilityviewed from microcanonical ensemble, and also the current understanding of the nature of thisphase transition.
Here we explain how the Hagedorn temperature emerges in generic ideal string gases. The thermalone-loop amplitude has the following general form, Z ( β ) ∶= ln ( Tr e − βH ) = Z ( β ) + Z ( β ) , (53)where Z is the non-thermal vacuum bubble contribution, i.e. the Casimir energy, and Z is thepiece accounting for thermal effect which vanishes when β → ∞ . The examples that we studiedfit into Eq.(53): for instance Eq.(32) and the first two lines of Eq.(48), where the first line is Z and the second line Z . The Casimir energy Z ( β ) is just the non-thermal one-loop amplitudeaddressed in Sec.2, which is free of UV divergence and is irrelevant to the Hagedorn temperature.Then we examine the thermal piece Z ( β ) , which can be written as Z ( β ) = V d − ( π ) d − ∫ ∞ dM ∫ d ⃗ p [ − ρ b ( M ) ln ( − e − β √⃗ p + M ) + ρ f ( M ) ln ( + e − β √⃗ p + M ) ] , (54)25here ⃗ p is the ( d − ) -dimensional spatial momentum, M runs over the mass spectrum of onesingle string, and ρ b / f ( M ) is the density of states of one bosonic/fermionic string at mass M . Incase of unbroken spacetime supersymmetry such as the examples in Sec.3.2 and Sec.3.3, we have ρ b ( M ) = ρ f ( M ) . Then by Eq.(41), we recover the expressions with the function G d ( ⋅ ) as in Eqs(37) and (50).According to the results in [38], the densities of states have the following asymptotic behaviorfor large mass M for a closed string in d dimensions at weak coupling: ρ b , f ( M ) ∼ const . × M − d e π (√ ω L +√ ω R ) M , ( for closed strings ) , (55)where ( ω L , ω R ) = ( , ) , ( , ) and ( , ) for bosonic, type II and heterotic strings respectively.On the other hand using Eq.(101) and the asymptotic behavior of the modified Bessel function ofthe second kind, we have ± ∫ d ⃗ p ln ( ± e − β √⃗ p + M ) ∼ const . × ( T M ) d − e − βM , when βM ≫ . (56)Plugging this and Eq.(55) in Eq.(54), we find that the integration over M becomes divergent when β is below the critical value π (√ ω L + √ ω R ) , which is just the inverse Hagedorn temperature β H .The explicit values for different types of string theories are β H ∶= π (√ ω L + √ ω R ) = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ π bosonic string ;2 √ π type II string ; ( + √ ) π heterotic string . (57)In the same way we can also investigate models containing open strings. The type I string hasthe same Hagedorn temperature as the type II string, because i) the closed string sector is thetype II string with states of negative worldsheet parity removed, where the truncation of spectrumdoes not alter the behavior Eqs (54) and (55); ii) in the open string sector Eq.(54) applies as well,and the open string counterpart of Eq.(55) is [38] ρ b , f ( M ) ∼ const . × M − d e β H M , when M → ∞ ( for open strings ) , (58)where the inverse Hagedorn temperature β H is the same as in the closed string sector, β H = √ π . The same logic applies to the open bosonic string where we have Eq.(58) with Hagedorntemperature given by β H = π , identical to that of the closed bosonic string.We have two further points to make: ● The Hagedorn temperature does not depend on compactification, but only on the type of stringtheory. This can be seen from Eqs (55) and (58), where the coefficient of M in the exponential,26hich defines the inverse Hagedorn temperature, does not depend on the spacetime dimension d but only on the type of the string. ● The behavior of an ideal string gas near the Hagedorn temperature can depend not only on thetype of the string, but also on the spacetime dimension. The power of M dressing the exponentialin Eqs (55) and (58) depend on d , and different d will result in different convergence properties ofthe integration over M in Eq.(54) at the point β = β H . More details will be covered in Sec.4.3. For closed string gases in d dimensions, investigating their Hagedorn temperature from the pointof view of the ( d − ) -dimensional effective theory can lead to another level of insight, where the re-duced dimension is the Euclidean time. Basically, the ( d − ) -dimensional effective theory containsstates whose masses depend on temperature, where some winding states along the Euclidean timecircle can become tachyonic when the temperature exceeds the Hagedorn temperature. This causesthe one-loop vacuum-to-vacuum amplitude to diverge literally, implying that near the Hagedorntemperature some instability can set in. We will discuss this aspect in the next subsection whilefor the moment we explain how the thermal tachyons emerge.On the technical level, the starting point is still the thermal one-loop amplitude. However nowconsidering the thermal gas in a ( d − ) -dimensional effective theory, we need to characterize thecompact Euclidean time S ( R ) with a lattice sum, something looking like Eq.(17), instead of aninstanton sum as Eq.(27) or (29) which has been the case in all the previous calculations. Noticingthat the the Euclidean time is not compactified toroidally but à la Scherk-Schwarz, a freely-actingorbifold compactification, therefore the resulting lattice sum is more complicated than that froma toriodal compactification. It turns out that the lattice is shifted differently following differentcombinations of spin structures, that is, the internal momenta { p I L , R } in Eq.(17) become functionsof spin structure indices: the ( a, b ) and ( ¯ a, ¯ b ) as in Eqs (12) and (20) for instance. The detailedconstruction of such lattice sums can be found many literatures (c.f. for example [9, 10, 34, 39]),while a shortcut available for us here is to by performing a Poisson resummation to the instantonsum.To be more explicit, we invoke the simplest model that we have studied, the heterotic string in10 dimensions, whose thermal one-loop amplitude in terms of worldsheet instaton sum is alreadyobtained in Eq.(30). We proceed by Poisson resumming the index ˜ m using Eq.(95), where weneed to put (− ) a ˜ m = e i π a ˜ m and combine this exponential with e − πR τ ∣ ˜ m − τn ∣ . The computation isslightly lengthy but straightforward, and the result is effectively a non-thermal one-loop amplitude27n 9 dimensions: Z het ( β ) = V ( π ) ∫ F dτ dτ τ / { O η Γ int ¯ η ∑ m,n q [ m + R −( n + ) R ] ¯ q [ m + R +( n + ) R ] + . . . } . (59)Here we have only written down the term relevant to the Hagedorn temperature which can developdivergence for certain values of R . The omitted terms give rise to analytic results integrated over F for all real value of R . Also we notice that the lattice involved above is shifted with respectto the toroidal case given by Eq.(17) with Eq.(16), in that m → m + and n → n + . Using thepower expansion (108) together with the definition of the η -function Eq.(106), we have O η = q − / + q / + q / + O( q / ) . (60)Using also Eq.(35), we have the following expansion of the integrand in Eq.(59) O η Γ int ¯ η ∑ m,n q [ m + R −( n + ) R ] ¯ q [ m + R +( n + ) R ] = ( q ¯ q ) ( R + R − ) + ( higher orders ) , (61)where again we exhibited only the term responsible for the definition of Hagedorn temperature,since all other terms are convergent integrated over F for all real value of R . This term comesfrom the leading terms in Eqs (35) and (60), and the two terms in the lattice sum of ( m, n ) = ( , ) and (− , − ) . Therefore it is contributed by the modes corresponding to closed string windingonce and carrying half unit of momentum along the Euclidean time circle, where coefficient 2accounting for the fact that there are two of them: ( m, n ) = ( , ) and (− , − ) . These thermalwinding modes have temperature-dependent mass that we denote by M ∗ , given by the power of q ¯ q in Eq.(61): M ∗ = R + R − = ( πT ) + ( πT ) − . (62)Displaying in Eq.(59) only the terms corresponding to these thermal winding states, we have Z het ( β ) = V ( π ) ∫ F dτ dτ τ / ( q ¯ q ) ( R + R − ) + . . . = V ( π ) ∫ F dτ dτ τ / e − πτ M ∗ + . . . (63)With Eq.(62) we notice that when − /√ < R < + /√ the thermal winding states of ourconcern are tachyonic ( M ∗ < ), rendering the integral over F above divergent so that the canonicalensemble description breaks down. Therefore if we start from very low temperature (large β ), andheat the gas gradually ( β reduces), we will find the critical point where the canonical ensemblebreaks down is β = π ( + √ ) . Unsurprisingly this critical value is exactly the inverse Hagedorntemperature β het H obtained from 10-dimensional point of view Eq.(45).28enerically when we carry out the same analysis to other types of closed strings in d -dimensionsthat we have discussed in Sec.3.1, the resulting one-loop amplitudes in the ( d − ) -dimensionaleffective theory contain systematically a term of the form ( degeneracy ) × V d − ( π ) d − ∫ F dτ dτ τ d + e − πτ M ∗ ( β ) , (64)induced by some thermal winding modes, whose mass-squared M ∗ ( β ) is positive for large β , butbecomes negative as β decreases (or the temperature T increases) till below a critical value. Thiscritical value is just the inverse Hagedorn temperature β H , which matches the results in Eq.(57). In this part we come back to the point about the Hagedorn temperature as an embarkation pointfor the ideal string gas to a new phase. In the following we will first examine the behavior ofideal string gases approaching the Hagedorn temperature, which is important for judging whetherthe phase transition can happen. Then we will comment on the microcanonical approach to theHagedorn temperature and its link with canonical description. Finally we will give some qualitativediscussions about the phase transition itself.
Limiting and non-limiting Hagedorn temperature
The Hagedorn temperature of an ideal string gas can be qualified as non-limiting or limiting according to whether or not the string gas can reach the Hagedorn temperature by absorbing afinite amount of energy [6, 40]. The Hagedorn phase transition cannot happen for limiting casessince the phase transition point cannot be reached in practice.Restricted to the models of our interest, we can verify, using Eqs (54) to (56), that the thermalone-loop amplitude Z ( β ) = ln Tr e − βH at the limit β → β + H : ◇ tends to a finite value for closed string gas in d ≥ dimensions; ◇ tends to infinity for closed string gas in d = dimension and open string gas in all dimensions,which are inferred from the convergence property of the integral over M in Eq.(54). For closedstring gases, this can also be verified from the viewpoint of the ( d − ) -dimensional effective fieldtheory: at β = β H ( M ∗ ( β H ) = ) the infinite integration over τ in Eq.(64) converges for d ≥ and diverges for d = . We also notify that the presence of D-branes in the background can affectthe limiting behavior of a string gas at β → β + H , which can be studied quantitatively (c.f. forexample [41] and the references therein). 29hus we can easily derive the behavior of the total energy E ( β ) = − ∂Z ( β ) ∂β . Taking into accountthe impact of the derivative with respect to β , we find ◇ E ( β ) tends to a finite value when β H for ideal closed string gases in d ≥ ⇒ the Hagedorntemperature is non-limiting, which is a site of phase transition; ◇ E ( β ) diverges when β → β + H for ideal closed string gases in d ≤ and ideal open string gas ⇒ the Hagedorn temperature is limiting, which is the maximum temperature of the gas.In the limiting case, one can never achieve Hagedorn temperature attempting to heat an idealstring gas by pumping energy into it. In fact as the more and more energy is concentrated in thestring gas, some instability other than the Hagedorn instability would already set in and turn thehomogeneous string gas into something else, for example a black hole. Microcanonical ensemble approach
We have seen in the previous discussions, that for the non-limiting cases, the string gases can gobeyond the Hagedorn temperature at a finite cost of energy, but the canonical ensemble fails totrace the string gases across the Hagedorn temperature from below since it becomes ill defined.Therefore it is of interest to try the more fundamental microcanonical ensemble description, whichcan hopefully uncover properties of string gases beyond the Hagedorn temperature.To this end we need to compute Ω ( E ) , the number of microstates of the string gas. For thenon-limiting cases (closed strings in d ≥ dimensions), it is possible to obtain the asymptoticbehavior of Ω ( E ) at large energy density using the technique in [42] for bootstrap models. We canfind that for closed strings with densities of states given by Eq.(55) and with d ≥ , the bootstrapcondition is satisfied: ρ b , f ( E ) ∼ Ω ( E ) for large enough energy density [3, 5]. Thus we have forthe non-limiting cases Ω ( E ) ∼ E − d e β H E ( d ≥ , E → ∞ ). This allows the computation of thetemperature T = ( ∂ ln Ω ( E ) ∂E ) − V and the specific heat C V = ( ∂E∂T ) V , which yields a temperature higherthan the Hagedorn temperature with the specific heat negative. This shows in the microcanonicallanguage that the system is unstable when its temperature exceeds the Hagedorn temperature,and this instability should reasonably be identified the Hagedorn instability.Another way of computing Ω ( E ) is to start from the canonical partition function Tr e − βH = exp [ Z ( β )] and reverse the relation Tr e − βH = ∫ dE Ω ( E ) e − βE which is just a Laplace transform.Actually the thermal one-loop amplitude Z ( β ) can be mathematically defined beyond the Hage-dorn temperature β < β H for closed strings, although literally the integral over M in Eq.(54)evaluates to infinity. This can be seen explicitly in Eq.(64), where the integration can be definedfor M ∗ ( β ) < (where β < β H ) through analytical continuation. In fact, Z ( β ) as a function of β β H is the rightmost branch pointat which Z ( β ) can either be regular or singular [43]. The asymptotic behavior of Z ( β ) approach-ing β H allows us to compute the asymptotic behavior of Ω ( E ) at large energy density, and theresult for string gases with non-limiting Hagedorn temperature is just as in the last paragraph.More details of this approach can be found in [43–45].We still need to explain one puzzle concerning the relation between the canonical and themicrocanonical description: apparently above the Hagedorn temperature the former breaks downwhile the latter can still be valid. Indeed the equivalence between the canonical and the micro-canonical ensemble no longer holds above the Hagedorn temperature, mathematically since thesaddle point approximation, the key step proving this equivalence using Tr e − βH = ∫ dE Ω ( E ) e − βE ,is no longer available for β < β H , due to the factor e β H E in Ω ( E ) . One can also compute in thecanonical ensemble description the energy fluctuation ⟨( E −⟨ E ⟩) ⟩⟨ E ⟩ which shows that it becomes largewith the growing of energy density [5]. Therefore at some point of high energy density and henceof high temperature, the energy fluctuation becomes so large that the canonical ensemble is nolonger suitable for describing an isolated string gas and one need to switch to the microcanonicalensemble. Phase transition at Hagedorn temperature
In the final part of this section we address the issue of Hagedorn phase. The Hagedorn temperaturewas first unraveled in the dual resonance model of hadrons [7], where the hadron density ρ ( m ) atmass m behaves asymptotically as ρ ( m ) ∼ c m − a e b m ( m → ∞) , (65)where a, b, c are positive constants. We notice at once that the asymptotic behaviors of singlestring state densities Eqs (55) and (58) have the same form, so that in the same way as instring theory, Eq.(65) predicts a critical temperature T = / b (the genuine Hagedorn temperature)above which the canonical ensemble formalism breaks down. It was then observed that when a > (in spacetime dimension 4) the hadronic system exhibits instability around the Hagedorntemperature [42]. Later as QCD proved to be the more fundamental theory of strong interaction,it was realized that this instability is the point of phase transition of the hadron system bridgingthe hadronic phase at low energy and the quark-gluon phase at high energy [47].The work on the Hagedorn instability in hadron systems prompt reflection on the fundamen-tal degrees of freedom of string theory. It may turn out that behind the modes in the masstower, obtained from quantizing the Polyakov worldsheet action, there are more fundamental con-stituents. The latter are the true degrees of freedom of strings, which do not manifest themselves31ntil the temperature is high enough, just like the quarks and gluons which are liberated at hightemperature.Some interesting speculations have been made in [8]. Based on the thermal effective action, it isargued that the system should undergo a first-order phase transition somewhat below the Hagedorntemperature; in the high temperature phase the free energy F scales as F ∼ T as if a ( + ) -dimensional quantum field theory, implying that string theories in UV contain far fewer degreesof freedom per unit volume than even quantum field theories. A more detailed analysis is givenin [9], where the exact thermal effective action is worked out using supersymmetry constraints.However the result shows that the phase transition is not of first order, and that the new phase isa non-critical string theory in ( + ) dimensions. Later on string theory dualities were exploitedto examine the role of non-perturbative effects in the Hagedorn phase transition [10].Although the Hagedorn instability of ideal string gases received immediate attention in thevery beginning stage of string thermodynamics, results obtained so far are based on effective fieldtheories, and it is still difficult to achieve a description at full string level. The issue of Hagedornphase transition, and its implementation in early cosmology, remains an open question. In this section we address the cosmological implications of the string thermodynamics in theprevious sections, based on the work in [15–24]. In particular we will show that a string scenariofor cosmology can be built up from first principles, where the string thermodynamics covered inSec.3 can be naturally incorporated. We will then discuss the important phenomenological issuesand some open questions, where most of the discussions will be qualitative.
Due to the nature of string theory as a candidate “theory of everything” which offers the matter,the interactions and the spacetime in a whole package, string cosmology is not about how theEinstein universe would evolve once filled up with strings instead of particles. Rather, it is abouthow cosmological evolution can eventually arise as a solution of string theory among others.On the practical level, a good point to start with is the the effective field theory, from which wecan derive the equations of motion governing the cosmological evolution. In order that the modelbe analytically tractible, we restrict our attention to critical superstring theory and we assume weakcoupling . We can therefore adopt a perturbative approach, searching for cosmological solutions32rder by order. Since we are meant to take into account thermal effects, the effective action shouldbe computed against a thermal background with compact Euclidean time, and then continued backto Lorentzian signature when we derive the equations of motion.The effective supergravity at tree level cannot feel the thermal effects since it is computed ona genus-0 Riemann surface which cannot wrap the Euclidean time circle. As a result the tree leveleffective theory does not lead to cosmological evolution, since generically the effective supergrav-ities give AdS or Minkowski space as vacuum solutions. Therefore higher order corrections needto be investigated .By phenomenological consideration, it is relevant to choose models with Minkowski spaceas the tree level vacuum since the observed cosmological constant is tiny and positive (roughly − M ). Therefore more accurately, the full tree-level background space is ( Tree-level background space ) = R ,d − × M int , (66)where the visible spacetime is of dimension d and the rest − d dimensions are compactified onsome internal space M int . When computing higher order corrections at finite temperature, weneed to use the corresponding thermal background ( Thermal tree-level background space ) = S ( ˆ R ) × T d − ( ˆ R box ) × M int , (67)where S ( ˆ R ) is the Euclidean time circle and T d − ( ˆ R box ) , the spatial part of visible spacetime,is a ( d − ) -dimensional periodical box with the length of each dimension π ˆ R box , introduced toregularize the space volume: ˆ V d − = ( π ˆ R box ) d − . Here we put hat on R , R box and V d − to indicatethat they are measured in string frame. Since we will work both in string frame and in Einsteinframe, the distinction of the two is necessary. Therefore from now on, we use the conventionthat hatted quantities are string frame quantities, and non-hatted quantities are either Einsteinframe quantities or quantities indifferent of reference frame . Since all the calculations of one-loopamplitudes in the previous two sections were carried out in the string frame, when we use thoseresults here, we need to add hat wherever necessary.The metric of background (67) is d ˆ s = ˆ β du + d − ∑ i = ( π ˆ R box ) ( dx i ) + ds M , ( ≤ u, x i < ) . (68)Here ds M is the metric of the internal space; u and x i ( i = , . . . , d − ) are the Euclidean timeand spatial coordinates chosen such that their periods are ; ˆ β = π ˆ R the perimeter of the Certainly, non-perturbative corrections should equally be considered if they contribute substantially. For a quantity, say L , of dimension of length, its string frame value ˆ L and Einstein frame value L are relatedby a dilaton-dependent rescaling ˆ L = e d − φ d L with d the spacetime dimension and φ d the dilaton in d dimensions. M int which allow exact solution of the worldsheet CFT), and hencethe perturbative quantum corrections at one-loop can be precisely computed. Different from thetree level action, the one-loop correction can feel the temperature since the genus-1 worldsheet(torus for closed strings) has non-contractable loops which can wrap the compact Euclidean time.In fact the one-loop correction computed against the background (67) is nothing but the thermalvacuum-to-vacuum amplitude studied in Sec.3.At this point we can write down the effective action computed to one-loop at finite temperature: S = ∫ d d x √− ˆ g [ e − φ d ( ˆ R + ( ∂φ d ) + . . . ) + Z ( ˆ β ) ˆ β ˆ V d − ] , (69)which is already continued to Lorentzian signature. The first term in the bracket with dilatondressing is the tree-level part, where ˆ R is the Ricci scalar of the string frame metric ˆ g , φ d isthe dilaton in d -dimensions which is set to be small e φ d ≪ to ensure weak coupling, and theellipses stand for the kinetic terms of other scalar fields, whose content is model-dependent. Thesecond term in the bracket in Eq.(69) is the one-loop correction, where Z ( ˆ β ) is just the thermalone-loop amplitude computed in Sec.3, and it can depend on scalar fields, for example Eq.(50)has R -dependence.To simplify further discussions, we switch to the Einstein frame by rescaling the metric ˆ g = e d − φ d g . Also we notice that Z ( ˆ β ) ˆ β ˆ V d − = ˆ F is the string frame density of Helmholtz free energy, relatedto the Einstein frame counterpart by F = Z ( ˆ β ) βV d − = e dd − φ d Z ( ˆ β ) ˆ β ˆ V d − = e dd − φ d ˆ F . Therefore the Einsteinframe action is S = ∫ d d x √− g [( R − d − ( ∂φ d ) + . . . ) − F ( β , φ d , . . . )] . (70)Effectively this action describs a universe filled with a fluid of free energy density F , and thiscosmic fluid is just the ideal string gas. We then look for cosmological solution, which, by virtueof the perturbative approach, is meant to be a perturbative correction added to the tree level flatspacetime. Thus we use the ansatz of flat FRW metric for the solution at one-loop order, whichis written down in the Einstein frame as ds = − N ( x ) ( dx ) + d − ∑ i = a ( x ) ( dx i ) , ( ≤ x i < ) . (71)Here x is the coordinate time, x i ( i = , . . . , d − ) are spatial coordinates as in Eq.(68), a ( x ) = π ˆ R box ( x ) e d − φ d is the scale factor, and N ( x ) = ˆ β ( x ) e d − φ d = β ( x ) is the lapse function34dentified here with the Einstein frame inverse temperature. These quantities, as well as thescalar fields, which are static at tree level, can now have nontrivial time dependence due to theinclusion of one-loop thermal and quantum effects. Moreover these time evolutions, which aremeant to divert only perturbatively from the tree-level static solution as is stated above Eq.(71),should be quasi-static. Otherwise if the metric and the fields evolve violently, kinetic correctionsbecome important and should be included in the action, and thus Eq.(70) does not apply anymore.Now we are ready to derive the equations of motion. For simplicity we switch off all scalarfields but the dilaton, while it is straightforward to include more scalar fields. Plugging Eq.(71)into the effective action (70), varying with respect to N , a and φ d , we obtain ( d − )( d − ) H − d − φ d = β ∂ F ∂β + F , (72) ( d − ) [ H + ( d − ) H ] + d − φ d = ad − ∂ F ∂a + F , (73) ¨ φ d + ( d − ) H ˙ φ d + d − ∂ F ∂φ d = , (74)where we assumed the dilaton, as well as other scalar fields if they were included, is spatiallyhomogeneous. Also H = ˙ a / a is the Hubble parameter, and the dot stands for the derivative withrespect to the cosmic time t = t ( x ) defined by dt = N ( x ) dx . We use t in the equations ofmotion in order to absorb N into the derivatives to simplify the equations. If there were morescalar fields, then their equations of motion, which is of a similar form of Eq.(74), should be addedto the above list, and also their kinetic terms should appear in parallel with that of the dilatonin the first two equations (72) and (73). Referring to the standard form of Friedemann equationsthe right-hand-side of Eqs (72) and (73) should be the energy density ρ and minus the pressure − P of the cosmic fluid. Therefore we read off the energy density and the pressure of the stringgas from the cosmological equations of motion: ρ = β ∂ F ∂β + F , P = − ad − ∂ F ∂a − F , (75)which are measured in the Einstein frame. Then using the relation Z = − βV d − F = − βF and V d − = a d − , we rewrite the above expressions as ρ = − ∂Z∂β , P = − ∂F∂V d − , (76)which are exactly the standard canonical ensemble relations. Therefore the cosmological ρ and P in Eq.(75) are really the energy density and pressure in the thermodynamical sense, as long asthe string gas is constantly in equilibrium during the cosmological evolution. This is actually the35ase, guaranteed by the quasi-static evolution of the universe, stated below Eq.(71). There is onelast equation of importance: the continuity equation, which can be derived from Eqs (72)—(74): ˙ ρ + ( d − ) H ( ρ + P ) = ˙ φ d ∂ F ∂φ d . (77)In case where more scalar fields are present, the right hand side should be ∑ Φ ˙Φ ( ∂ F / ∂ Φ ) where Φ runs over all scalar fields. The continuity equation can be integrated, using ˙ F = ˙ β ∂ F ∂β + ˙ φ d ∂ F ∂φ d +( terms with other scalars ) , to give ddt [ β a d − ( ρ + P )] = ⇔ β a d − ( ρ + P ) = const . (78)which is just the conservation of entropy, given that a d − = V d − so that βa d − ( ρ + P ) = β ( E + P V d − ) = S . This equation holds regardless of the content of scalar fields, and can be used in placeof Eq.(73).We recap the essential points in the construction that we have gone through: ● We examine a critical and weakly coupled string theory at finite temperature, with the aim oflooking for solutions corresponding to cosmological evolution. ● The search for cosmological solution at tree level is unsuccessful, so that we go to one-loop level,where the one-loop correction to the effective action is computed against a thermal background. ● We end up with an effective theory describing a universe filled with an ideal string gas in thermalequilibrium, and the effective theory action holds for a universe in quasi-static evolution.
Now we would like to show a concrete example based on the string theory model in Sec.3.3: theheterotic string in 9 dimensions. Certainly it is phenomenologically more relevant to consider4 dimensional models where the extra dimensions compact of order string length. However weconsider here 9 dimensional cosmology for simplicity, while technically reducing from 9 dimensionsto 4 dimensions amounts to enlarging the moduli space and including more fields in the effectiveaction without changing the form of the equations of motion.The thermal background space at tree level is S ( ˆ R ) × T ( ˆ R box ) × S ( R ) . (79)Here R is indifferent of reference frame so we do not put hat on it. The one-loop amplitudeagainst this background is already worked out in Eq.(50), in terms of string frame quantities. The36ffective action is S = ∫ d x √− g [ R − ∂φ ∂φ − ∂ Φ ∂ Φ − F ( β, φ , Φ )] (80)where we have defined Φ ∶= ln R for notational simplicity, since Φ has canonical kinetic term.Varying Eq.(80), or using directly the general equations (72) to (78), we obtain the set of inde-pendent equations of motion H =
27 ˙ φ +
12 ˙Φ + ρ, (81) ¨ φ + H ˙ φ + ∂ F ∂φ = , (82) ¨Φ + H ˙Φ + ∂ F ∂ Φ = , (83) β a ( ρ + P ) = const . , (84)To further narrow down the discussion, we enforce some extra restrictions: i) we assume thebackground scalar fields are constant, or small fluctuations about some constant values; ii) thestring frame temperature is low with respect to the string scale: ˆ R ≫ , which allows us to usedirectly the result (52) for F ; iii) the internal radius R is restricted in the vicinity of 1, so thatwe focus our attention to phase I in Fig.1, corresponding to the first line in Eq.(52).We look for solutions with static scalar background ˙ φ = ˙Φ = which means ∂ φ F = ∂ Φ F = .Restricted to the phase I of Fig.1, we see that this is possible only at the local minimum R = or Φ = . At this point the dilaton φ becomes a flat direction since F does not depend on it. Thuswe have, referring to the first line of Eq.(52), φ = const . such that e φ ≪
1; Φ = (85) F = − (S T + S T − ) G ( ) T , (86)where T = β − is the Einstein frame temperature. From Eq.(86) we derive the energy density andthe pressure P = ρ = −F = (S T + S T − ) G ( ) T , (87)which is the Stefan’s law for radiation in 9 dimensions. Plugging ρ and P in Eqs (81) and (83), weobtain the following cosmological evolution which describes a 9-dimensional radiation-dominateduniverse a ( t ) ∝ T ( t ) ∝ t / . (88)37e do not intend to show the calculation here but this solution is stable against small perturbations[20]. As a result, the scalar field Φ = ln R , massless at tree level, obtains a mass at one-loop levelat Φ = , the local minimum of F . The mass squared is given by the second derivative of Eq.(52)with respect to Φ : M = ∂ F ∂ Φ ∣ Φ = = π S T − G ( ) e φ T , (89)where the factor e φ appears because we work in Einstein frame. This gives an example ofmoduli stabilization by thermal effects, where a remarkable feature, which is generically trueof such moduli stabilization is that the induced scalar mass is temperature-dependent, and istherefore time-dependent due to Eq.(88).Despite the awkward dimensionality 9, the following features of this model persist in otherdimensions higher than or equal 4, as long as the temperature is well below the Hagedorn tem-perature [18, 20]: ● The cosmological solution converges with cosmic time t to an evolution according to a ( t ) ∝ T ( t ) − ∝ t / d , the behavior of a radiation-dominated universe. ● The solution is stable against fluctuations in the scalar fields, showing that it is possible tostabilize moduli. That is, scalar fields massless at tree level can obtain masses at one-loop levelat local minima of the one-loop free energy density. ● The dilaton is frozen at a fixed value (can decrease logarithmically in 4 dimensions but can neverincrease [20]), so that the model remains in the weak coupling regime once it starts out weaklycoupled.
Despite the radiation-dominated early universe that it gives rise to, the example in the last subsec-tion is of little phenomenological interest, obviously because it misses many pieces for completingthe jigsaw puzzle of the real universe. For one thing, the solution (88) is valid for temperaturewell below the Hagedorn temperature, since otherwise we loose analytic control of thermal andquantum corrections, and we cannot use the results in Sec.3. On the other hand no explicit matterformation process is considered. Thus the solution (88) should correspond to a short time spanbetween the exit of the Hagedorn phase and the electoweak symmetry breaking, not counting theincorrect spacetime dimensionality, and the unbroken spacetime supersymmetry... In this verylast part of our lecture notes, we briefly discuss how thermal string cosmology handle these issues.38 agedorn phase and the initial cosmological singularity
Due to the lack of an analytic description at full string level, in our thermal string scenario it isnot yet possible to work out the cosmology in the Hagedorn phase. However there have been somefully solvable models of ideal string gas constructed, in which the Hagedorn temperature does notarise, and which can lead to the resolution of the initial cosmological singularity.The first such model was constructed in [39], based on perturbative type II strings. Switchingon fluxes along the Euclidean time circle and another internal direction in a certain way, one candeform the mass spectrum such that the thermal winding modes responsible for the Hagedorninstability can no longer become tachyonic. Instead they only become massless at a criticaltemperature β c of about the string scale, and β − c is the maximal temperature of the string gas,at which the thermodynamic quantities are finite. Also the model is invariant under the thermalT-duality β → β c / β .Later on, by the same construction a 2 dimensional model free of Hagedorn temperature wasrealized [48] and its cosmology worked out [22]. It was found that if we let the expanding universeevolve backwards, the temperature raises gradually with the quasi-static contraction of the uni-verse until its maximal value β − c . There some thermal winding modes become massless, inducinga gauge symmetry enhancement in the Euclidean time direction U ( ) → SU ( ) . These masslessthermal winding modes source a space-like brane of positive tension, which supplies a suddennegative pressure preventing the universe from further contraction. The universe then embarkson a quasi-static expanding phase, and therefore we obtain a quasi-static bouncing universe. Thetemperature and the dilaton are at their maximum values exactly at the bouncing point, anddrop when the universe evolves forwards or backwards. The behavior of the dilaton shows thatthe model can be maintained in the weakly coupled regime, which, together with the quasi-staticevolution, ensures that the perturbative calculations are valid.As the generalization of [22], quasi-static bouncing universes were achieved in [23] in arbitraryspacetime dimensions via the same mechanism. Then in [24] the authors studied the case wherethe space-like brane exists for an extended time span, where bouncing cosmological solutions werealso found. Moduli stabilization
The example shown in Sec.5.2 has already touched upon this issue. Being the simplest case ofcompactification, the moduli space is two dimensional, coordinatized by the dilaton φ and theinternal radius R . These flat directions are corrected at one-loop level via the free energy density39q.(52). As a result the internal radius is stabilized at the point R = of enhanced gaugesymmetry U ( ) R → SU ( ) R , while the flat direction φ is not lifted.The conception of such mechanism goes back to [49], where it was shown, for non supersym-metric compactifications of heterotic string, that the cosmological constant generated by stringloops reaches extremum at points of enhanced gauge symmetry. This feature was proposed in [50]for use of moduli stabilization where moduli are attracted to the local minima of the modulidependent cosmological constant generated perturbatively beyond tree level. In the context ofthermal string cosmology this mechanism for moduli stabilization is extensively investigated invarious models. The works in [15, 18, 19] have addressed the moduli stabilization by perturbativeeffects at one-loop level, while non-perturbative effects are studied in [20, 21] via string theorydualities.The cosmological moduli problem should also be taken into account. With the cosmologicalevolution, the oscillations of the stabilized moduli in their potential wells store an energy densitywhich dominates over that of the radiation [51]. This leads to a huge entropy production whenthe scalar particles of the moduli fields decay, which wipes out the baryon-antibaryon asymmetry.Also the production of the decay can alter the primordial abundances of light nuclei producedby nucleosynthesis, leading to contradictions with observation [52]. Usually the problem is solvedby imposing a lower bound of about O( ) TeV to the induced scalar mass of the stabilizedmoduli [53], for example in the KKLT senario [54]. In the thermal string cosmology scenariohowever, the induced scalar mass is time dependent, as is inferred from Eq.(89), based on whichone can show that the energy stored in the moduli field oscillation never exceeds that of theradiation [18–20]. As a result, the cosmological moduli problem does not arise.
Supersymmetry breaking and hierarchy problem
In supergravity unlike in supersymmetric field theory, at tree level it is possible to have spontaneoussupersymmetry breaking with the cosmological constant still being zero. Such models are knownas the no-scale supergravity [55], which are of particular interest in phenomenological applications.In superstring theory, a wide class of orbifold compactifications which spontaneously breakspacetime supersymmetry can lead to the no-scale supergravity as the tree-level effective theory(see [14] for a summary and references), among which the Scherk-Schwarz compactifications havebeen intensively exploited in thermal string cosmology. The vanishing tree-level cosmologicalconstant implies that the tree-level vacuum is still given by Eq.(66), so that the technique depictedin Sec.3 can still be applied when we go to the one-loop level. The behavior of the supersymmetrybreaking scale can be precisely traced, since it is one of the moduli fields (the no-scale modulus)40nd thus its equation of motion can be explicitly written down.Quantitative computations were carried out in [16–19], where heterotic string theory is com-pactified on orbifolds down to four dimensions with supersymmetry spontaneously broken à laScherk-Schwarz, of the pattern N = → [16] and N = → [17]. It was found thatthe resulting cosmological behavior is sensitive to the details of the compactification, where themost interesting situation is where the no-scale modulus, or the supersymmetry breaking scale,say M , decreases proportionally with temperature, and where the scale factor evolves as thatof a radiation-dominated universe (referred to as radiation-like universe in the original work): M ( t ) ∝ T ( t ) ∝ a ( t ) − ∝ t − / with t the cosmic time. We thus achieve in such universes thedynamical generation of the hierarchy M ≪ M Planck . Also the dilaton decreases with cosmic time,which maintains the validity of the perturbative calculations.
Radiative electroweak symmetry breaking
The universe as illustrated in the previous paragraph runs into trouble at late time, due to theproportionality between the supersymmetry breaking scale M ( t ) and the temperature T ( t ) . Asthe temperature decreases to almost zero, so does the supersymmetry breaking scale, and thereforethe universe will end up supersymmetric. However there can be mechanism in certain models whichcan break the proportionality bind M ( t ) ∝ T ( t ) and halts the evolution of M ( t ) before it getstoo small. Radiative symmetry breaking by IR effects is the candidate mechanism, which is wellunderstood in the context of supergravity [56]. Such mechanism defines a transmutation scale Q of about the electroweak scale. As the energy of the system, here characterized by temperature,drops below Q , radiative corrections starts to destabilize the scalar potential and triggers theelectroweak phase transition. Hopefully this is accompanied with the stabilization of M , whiletemperature continues to decrease. It will be interesting to work out the details of this mechanism. Primordial cosmological fluctuation
A further direction to explore is to make connection with the observational data of the cosmicmicrowave background in the context of the bouncing cosmology in [22–24]. In light of the matter-bouncing scenario [57], where it is shown that a nearly scale invariant spectrum of cosmologicalfluctuations can be produced, work is in progress studying the propagation of cosmological fluc-tuations across the space-like brane from the contracting phase to the expanding phase [58]. Ifsuccessful, this can eventually provide an alternative scenario to that of the inflationary cosmology.We would like to notify in the end that the same problem has been addressed recently in string41as cosmology [13], and also there have been large amount of work on string cosmology in favorof inflation [59] which is still an active field.
Acknowledgments
I am especially grateful to the organizers of the Modave Summer School VIII for the invitationto give these lectures. I also thank the Modave Summer School attendees for their feedbacks,which are important to the preparation of these lecture notes. Many thanks to Hervé Partouchewho proofread the manuscript and gave helpful comments and suggestions. Part of this work wasdone at the Centre de Physique Théorique, Ecole Polytechnique, supported by the EU contractsPITN GA-2009-237920, ERC-AG-226371 and IRSES-UNIFY, the French ANR 05-BLAN-NT09-573739 contract, the CEFIPRA/IFCPAR 4104-2 project, and PICS contracts France/Cyprus,France/Greece and France/US. The part of this work accomplished at the Institute for TheoreticalPhysics, KULeuven, is supported by a grant of the John Templeton Foundation.
A Canonical partition function in quantum field theory atone-loop
This appendix reviews the standard computation in finite temperature quantum field theory forobtaining the canonical partition function of an ideal gas of particles. The results are used inSec.2 and Sec.3 for motivating the non-thermal and thermal formalisms in string theory.We start with the field theory action describing one particle degree of freedom of mass M ina flat d -dimensional Euclidean space, which is S = ∫ d d x φ (− ◻ E + M ) φ, (90)where ◻ E = ∂ + ⃗∇ is the d -dimensional Laplacian. The canonical partition function Z = Tr e − βH can be evaluated by the path integral ln Z B , F = ln ( ∫ D φ e − S [ φ ] ) = ln [ Det ( − ◻ E + M )] ∓ / = ∓
12 Tr ln (− ◻ E + M ) , (91)where the Euclidean time is compactified on a circle S ( R ) of radius R = β / π , with β the inversetemperature. The subscripts “B” and “F” stand for boson and fermion, corresponding respectivelyto the signs “ − ” and “ + ” in the second and the third step. Here for simplicity we have cheated42n the fermionic case, computing the path integral as if φ were a real grassmanian variable and − ◻ E + M were an antisymmetric matrix. The result is correct which is not accidental since eachsingle fermionic degree of freedom satisfies the Klein-Gordon equation, the equation of motionderived from the action (90).Eq.(91) is nothing but the one-loop amplitude of the field φ . In evaluating Eq.(91), φ takesperiodic boundary condition along S ( R ) for a boson and anti-periodic boundary condition along S ( R ) for a fermion, leading to the following eigenvalues for the Laplacian operator ◻ E : − ( π rR ) − ⃗ p , r ∈ ⎧⎪⎪⎨⎪⎪⎩ Z boson , Z + fermion , (92)where ⃗ p is the ( d − ) -dimensional spatial momentum in the non-compact directions which takescontinuous values. The discrete momentum modes in the Euclidean time direction are the socalled Matsubara modes.We can evaluate Eq.(91) using the integration representation of the logarithm ln A = − ∫ ∞ d(cid:96)(cid:96) ( e − A (cid:96) − e − (cid:96) ) . (93)For one bosonic degree of freedom, we have ln Z B = −
12 Tr ln (− ◻ E + M ) =
12 Tr ∫ ∞ d(cid:96)(cid:96) [ e − (cid:96) (− ◻ E + M ) − e − (cid:96) ]= V D − ( π ) D − ∑ m ∈ Z ∫ d ⃗ p ∫ ∞ d(cid:96)(cid:96) [ e − (cid:96) ( m R +⃗ p + M ) − e − (cid:96) ] . (94)In this expression (cid:96) is referred to as the Schwinger time parameter, whose physical interpretationis the proper time that the particle spends for completing the loop. The second term in thebracket in Eq.(94) has no temperature dependence, which is thus an irrelevant infinity and canbe discarded naively. In fact the Riemann-Zeta regularization of the sum over m also yields zero.Then we integrate over the spatial momentum ⃗ p which is just a ( d − ) -dimensional Gaussianintegration, and perform Poisson resummation over the Matsubara mode index m using ∑ ˜ m e − π ˜ m T A ˜ m + πib T ˜ m = √ det A ∑ m e − π ( m − b ) T A − ( m − b ) , (95) The fact that the logarithm of the partition function is a one-loop amplitude is obscure in the calculationpresented so far in this section. A more detailed analysis showing ln Z = is presented in the Appendix Aof [60]. ln Z B = βV d − ( π ) d / ∑ ˜ m ∫ ∞ d(cid:96)(cid:96) + d / exp (− π R (cid:96) ˜ m − (cid:96)M )= βV d − ( π ) d ∑ ˜ m ∫ ∞ d(cid:96)(cid:96) + d / exp (− πR (cid:96) ˜ m − π(cid:96)M ) , (96)where in the second line we have simply recaled the Schwinger parameter, so that the expressionis more adapted to string theory language. Also with Poisson resummation we trade the the sumover Matsubara modes m in Eq.(94) for the sum over instanton number ˜ m in Eq.(96), wherethe instanton configuration labeled by ˜ m is such that the particle worldline winds around theEuclidean time circle ˜ m times before completing the loop. For this reason, here we call theseinstantons worldline instanton .The case of one fermionic degree of freedom is calculated similarly: ln Z F =
12 Tr ln (− ◻ E + M ) = −
12 Tr ∫ ∞ d(cid:96)(cid:96) [ e − (cid:96) (− ◻ E + M ) − e − (cid:96) ]= − V D − ( π ) D − ∑ m ∫ d ⃗ p ∫ ∞ d(cid:96)(cid:96) { e − (cid:96) [ ( m + / ) R +⃗ p + M ] − e − (cid:96) } , (97)where the Matsubara modes m are shifted by / due to the anti-periodic boundary conditionas is shown in Eq.(92). Now performing Poisson resummation over m , integrating over ⃗ p , andremoving the physically irrelevant infinity, we arrive at ln Z F = βV d − ( π ) d / ∑ ˜ m ∫ ∞ d(cid:96)(cid:96) + d / (− ) ˜ m + exp (− π R (cid:96) ˜ m − (cid:96)M )= − βV d − ( π ) d ∑ ˜ m ∫ ∞ d(cid:96)(cid:96) + d / (− ) ˜ m exp (− πR (cid:96) ˜ m − π(cid:96)M ) . (98)Here different from the bosonic case Eq.(96), the instanton sum is weighted by an interchangingsign (− ) ˜ m , accounting for the statistics of fermions.It will be useful to consider the case where the ideal gas consists of several species of particles.In this case the total partition function should be the product of the partition functions of eachspecies, since the total Hilbert space is a direct product of the component Hilbert spaces. Thereforethe logarithm of the total partition function is merely an addition of the logarithm of component The power of the Schwinger parameter −( + d / ) is characteristic of one-loop amplitudes, different to − d / inthe case of propagators, where the topology of the particle trajectory is a segment (c.f. for example Eq.(2.1.28)in [33]). The extra − in the one-loop amplitude is for removing the redundancy of the rigid global rotation of theloop, which does not exist for a segment with fixed end points. { M s } with helicities { j s } , where s is the index of quantum states, and we have ln Z = ∑ s ∑ ˜ m βV d − ( π ) d ∫ ∞ d(cid:96)(cid:96) + d / (− ) j s ( ˜ m + ) exp (− πR (cid:96) ˜ m − π(cid:96)M s ) . (99)It is instructive to make connection with the standard canonical ensemble formalism. To thisend we make use of the following mathematical formulae ∫ ∞ dt t − ν e − At − B t = ( BA ) ν − K ν − ( √ AB ) , for Re A and Re B positive; (100) ∫ ∞ p n dp e − β √ p + M = n √ π Γ ( n + ) M ( Mβ ) n K n + ( βM ) , (101)where K ν ( ⋅ ) is the modified Bessel function of the second kind. With the above two relations aswell as the Taylor expansion of the logarithmic function, we can actually show that β π ∞ ∑ ˜ m = ∫ ∞ d(cid:96)(cid:96) + d / exp (− πR (cid:96) ˜ m − π(cid:96)M ) = − ∫ d ⃗ p ln ( − e − β √⃗ p + M ) , (102) β π ∞ ∑ ˜ m = ∫ ∞ d(cid:96)(cid:96) + d / (− ) ˜ m + exp (− πR (cid:96) ˜ m − π(cid:96)M ) = ∫ d ⃗ p ln ( + e − β √⃗ p + M ) , (103)where ⃗ p denotes the ( d − ) -dimensional spatial momentum, and M is some mass. We can nowrewrite Eq.(99) as ln Z = ∑ s βV d − ( π ) d ∫ ∞ d(cid:96)(cid:96) + d / (− ) j s e − π(cid:96)M s − ∑ s V d − ( π ) d − ∫ d ⃗ p (− ) j s ln [ − (− ) j s e − β √⃗ p + M s ] . (104)Thus we recover the standard canonical ensemble formalism, where the first line is merely theCasimir energy.Finally we mention another case of our interest, which is the non-thermal limit of Eq.(99).At this limit the Euclidean time decompactifies R → ∞ , and all the instanton configurations areexponentially suppressed in Eq.(99) except for ˜ m = , and also we replace βV d − by V d the totalspacetime volume. Thus we obtain ln Z∣ T = = ∑ s V d ( π ) d ∫ ∞ d(cid:96)(cid:96) + d / (− ) j s exp (− π(cid:96)M s ) . (105)The same result can be obtained putting β → ∞ in Eq.(104), where the second line vanishes.45 Dedekind η -function, Jacobi elliptic functions and SO ( ) characters In this appendix we give a pragmatic summary of the important functions and notations usedin Sec.2.2 in the string one-loop amplitudes. The Appendix C of [33] is recommended for moredetails. Some detailed explanation about how these functions actually arise in the computationof one-loop amplitudes can be found in [29, 31, 61]. The functions that we need areDedekind η -function: η ( τ ) = q / ∞ ∏ n = ( − q n ) , (106)Jacobi elliptic function: θ [ ab ]( τ ) = ∑ k ∈ Z q ( k − a ) e − iπ b ( k − a ) , (107)where q = e πiτ , with τ sitting in the upper-half complex plane. For our purpose we will onlyencounter cases where a, b = , . We will also use the complex conjugates ¯ η and ¯ θ [ ab ] , whichare obtained by replacing in the above definitions q with ¯ q = e − πi ¯ τ .When we want to write down the string one-loop amplitudes, it is convenient to work in thelightcone gauge, where the general rules are: each holomorphic (anti-holomorphic) worldsheet bo-son contributes a factor of η − ( ¯ η − ), and each holomorphic (anti-holomorphic) worldsheet fermioncontributes a factor of √ η − θ [ ab ] ( √ ¯ η − ¯ θ [ ab ] ) where a , b take or independently, depending onthe spin structure of the worldsheet fermions on the torus (cf for example [61]).It is useful to further define SO ( ) characters out of the η - and θ -functions, with which theexpression of one-loop amplitudes can better reveal the structure of the superstring spectra. Hereit is SO ( ) in particular because the lightcone gauge fixing of superstring theories leaves uswith 8 target space dimensions. As a result the spectrum of spacetime states fall into differentrepresentations of SO ( ) .The SO ( ) characters which we use are [31, 62] O = θ [ ] + θ [ ] η = q − [ + q + q + q + O( q )] , (108) V = θ [ ] − θ [ ] η = q [ + q + q + q + O( q )] , (109) S = θ [ ] + θ [ ] η = q [ + q + q + q + O( q )] , (110) C = θ [ ] − θ [ ] η = q [ + q + q + q + O( q )] , (111)which arise from the sectors of the superstring spectrum whose ground states transform in thescalar ( O ), vector ( V ) and two real chiral spinor ( S and C ) representations of SO ( ) . These46ymbols can also appear as χ O , χ V , χ S , χ C in the literature, for example in [33]. We will equally usethe complex conjugates ¯ O , ¯ V , ¯ S and ¯ C , obtained by replacing q with ¯ q in the above definitions.Upon quantization of worldsheet fermions in superstring theories, the Neveu-Schwarz sector statesgive rise to V , while the Ramond sector states give rise to S or C .The following identities will be useful ∑ a,b = (− ) a + b + ab θ [ ab ] η = V − S , ∑ a,b = (− ) a + b θ [ ab ] η = V − C . (112)which are straightforward to verify using the definitions Eqs (108)–(111). They are relevant whenwe study superstring theories which have spacetime supersymmetry. In those cases the one-loop amplitudes would involve the overall factor V − C or V − S , or their complex conjugates.Numerically they vanish, since we have θ [ ] = so that V − S = V − C ∝ θ [ ] − θ [ ] − θ [ ] ,which vanishes because of the identity θ [ ] − θ [ ] − θ [ ] ≡ . (113)This results in a vanishing one-loop amplitude, accounting for the exact cancellation between thespacetime boson contribution and the spacetime fermion contribution.Despite the complexity in the construction of these compact notations, one can just keep inmind that they are power series of q or ¯ q with positive coefficients, and that the computation ofone-loop amplitudes amounts largely to the manipulation of these characters and some left-over η ’s and θ ’s. C Unfolding the fundamental domain
This appendix presents the scheme which we have used in Sec.3 for evaluating one-loop modularintegrals on the fundamental domain of the group SL ( , Z ) : F = {( τ , τ ) ∶ − < τ ≤ , τ >√ − τ } . It is known as the unfolding trick or the orbit method, which can be applied to theintegrations of the form I = ∫ F dτ dτ τ ∑ m,n f ( m,n ) ( τ, ¯ τ , . . . ) . (114)In the integrand the function f ( m,n ) ( τ, ¯ τ , . . . ) satisfies the following condition: setting τ ′ = ( aτ + b )/( cτ + d ) with M = ( a bc d ) ∈ SL ( , Z ) , then we have f ( m,n ) ( τ ′ , ¯ τ ′ , . . . ) = f ( m,n ) M − ( τ, ¯ τ , . . . ) . It isthen straightforward to verify the modular invariance of Eq.(114). The ellipses stand for spectator47ariables. The orbit method, as is demonstrated in [35, 46, 63], asserts that the sum over ( m, n ) can be decomposed into two orbits: I = ∫ F dτ dτ τ f ( , ) ( τ, ¯ τ , . . . ) + ∫ ⊔ dτ dτ τ ∑ m ≠ f ( m, ) ( τ, ¯ τ , . . . ) , (115)where the ( , ) -orbit is still integrated on the fundamental domain and the ( m, ) -orbit ( m ≠ )is integrated on the upper strip ( − < τ ≤ , τ > ) denoted by ⊔ .The orbit method is efficient for a restrained class of situations, and it has the draw backof obscuring the symmetry of the integral. New techniques for evaluating the one-loop modularintegrals are still developing. Recent progress can be found in [64]. References [1] M. B. Green and J. H. Schwarz, “Supersymmetrical Dual String Theory,” Nucl. Phys. B (1981) 502; M. B. Green and J. H. Schwarz, “Supersymmetrical String Theories,” Phys. Lett.B (1982) 444; J. H. Schwarz, “Superstring Theory,” Phys. Rept. (1982) 223.[2] E. Alvarez, “Superstring Cosmology,” Phys. Rev. D (1985) 418 [Erratum-ibid. D (1986)1206].[3] B. Sundborg, “Thermodynamics Of Superstrings At High-energy Densities,” Nucl. Phys. B (1985) 583.[4] S. H. H. Tye, “The Limiting Temperature Universe and Superstring,” Phys. Lett. B (1985) 388.[5] M. J. Bowick and L. C. R. Wijewardhana, “Superstrings at High Temperature,” Phys. Rev.Lett. (1985) 2485.[6] P. Salomonson and B. -S. Skagerstam, “On Superdense Superstring Gases: A Heretic StringModel Approach,” Nucl. Phys. B (1986) 349; E. Alvarez, “Strings At Finite Temperature,”Nucl. Phys. B (1986) 596; E. Alvarez and M. A. R. Osorio, “Superstrings at FiniteTemperature,” Phys. Rev. D (1987) 1175.[7] R. Hagedorn, “Statistical thermodynamics of strong interactions at high-energies,” NuovoCim. Suppl. (1965) 147. 488] J. J. Atick, E. Witten, “The Hagedorn Transition and the Number of Degrees of Freedom ofString Theory,” Nucl. Phys. B (1988) 291.[9] I. Antoniadis and C. Kounnas, “Superstring phase transition at high temperature,” Phys.Lett. B (1991) 369.[10] I. Antoniadis, J. P. Derendinger and C. Kounnas, “Nonperturbative temperature instabilitiesin N=4 strings,” Nucl. Phys. B (1999) 41 [hep-th/9902032].[11] J. L. F. Barbon and E. Rabinovici, “Closed string tachyons and the Hagedorn transition inAdS space,” JHEP (2002) 057 [hep-th/0112173].[12] R. H. Brandenberger and C. Vafa, “Superstrings in the Early Universe,” Nucl. Phys. B (1989) 391.[13] R. H. Brandenberger, “String Gas Cosmology,” arXiv:0808.0746 [hep-th]; R. H. Branden-berger, “String Gas Cosmology: Progress and Problems,” Class. Quant. Grav. (2011)204005 [arXiv:1105.3247 [hep-th]].[14] C. Kounnas and H. Partouche, “Inflationary de Sitter solutions from superstrings,” Nucl.Phys. B (2008) 334 [arXiv:0706.0728 [hep-th]].[15] T. Catelin-Jullien, C. Kounnas, H. Partouche and N. Toumbas, “Induced superstring cos-mologies and moduli stabilization,” Nucl. Phys. B (2009) 290 [arXiv:0901.0259 [hep-th]].[16] F. Bourliot, C. Kounnas and H. Partouche, “Attraction to a radiation-like era in early super-string cosmologies,” Nucl. Phys. B (2009) 227 [arXiv:0902.1892 [hep-th]].[17] J. Estes, C. Kounnas and H. Partouche, “Superstring cosmology for N = → superstringvacua,” Fortsch. Phys. (2011) 861-895. [arXiv:1003.0471 [hep-th]];[18] F. Bourliot, J. Estes, C. Kounnas and H. Partouche, “Cosmological phases of the stringthermal effective potential,” Nucl. Phys. B (2010) 330 [arXiv:0908.1881 [hep-th]].[19] F. Bourliot, J. Estes, C. Kounnas and H. Partouche, “Thermal and quantum induced earlysuperstring cosmology,” arXiv:0910.2814 [hep-th].[20] J. Estes, L. Liu and H. Partouche, “Massless D-strings and moduli stabilization in type Icosmology,” JHEP (2011) 060 [arXiv:1102.5001 [hep-th]].[21] L. Liu and H. Partouche, “Moduli Stabilization in Type II Calabi-Yau Compactifications atfinite Temperature,” JHEP (2012) 079 [arXiv:1111.7307 [hep-th]].4922] I. Florakis, C. Kounnas, H. Partouche and N. Toumbas, “Non-singular string cosmology in a2d Hybrid model,” Nucl. Phys. B (2011) 89 [arXiv:1008.5129 [hep-th]].[23] C. Kounnas, H. Partouche and N. Toumbas, “Thermal duality and non-singular cosmology ind-dimensional superstrings,” Nucl. Phys. B (2012) 280 [arXiv:1106.0946 [hep-th]].[24] C. Kounnas, H. Partouche and N. Toumbas, “S-brane to thermal non-singular string cosmol-ogy,” Class. Quant. Grav. (2012) 095014 [arXiv:1111.5816 [hep-th]].[25] D. Kutasov and D. A. Sahakyan, “Comments on the thermodynamics of little string theory,”JHEP (2001) 021 [hep-th/0012258]. M. Rangamani, “Little string thermodynamics,”JHEP (2001) 042 [hep-th/0104125].[26] L. A. Pando Zayas and D. Vaman, “Strings in RR plane wave background at finite tempera-ture,” Phys. Rev. D (2003) 106006 [hep-th/0208066].[27] T. Harmark, M. Orselli and , “Matching the Hagedorn temperature in AdS/CFT,” Phys. Rev.D (2006) 126009 [hep-th/0608115].[28] J. Polchinski, “String theory. Vol. 1: An introduction to the bosonic string,” Cambridge, UK:Univ. Pr. (1998) 402 p[29] D. Lust, S. Theisen and , “Lectures on string theory,” Lect. Notes Phys. (1989) 1.[30] J. Polchinski, “Evaluation of the One Loop String Path Integral,” Commun. Math. Phys. (1986) 37.[31] C. Angelantonj, A. Sagnotti and , “Open strings,” Phys. Rept. (2002) 1 [Erratum-ibid. (2003) 339] [hep-th/0204089].[32] J. Polchinski, “String theory. Vol. 2: Superstring theory and beyond,” Cambridge, UK: Univ.Pr. (1998) 531 p[33] E. Kiritsis, “String theory in a nutshell,” Princeton University Press, 2007[34] C. Kounnas and B. Rostand, “Coordinate Dependent Compactifications and Discrete Sym-metries,” Nucl. Phys. B (1990) 641.[35] K. H. O’Brien and C. I. Tan, “Modular Invariance of Thermopartition Function and GlobalPhase Structure of Heterotic String,” Phys. Rev. D (1987) 1184.5036] J. Scherk and J. H. Schwarz, “Spontaneous breaking of supersymmetry through dimensionalreduction,” Phys. Lett. B (1979) 60; J. Scherk and J. H. Schwarz, “How to Get Massesfrom Extra Dimensions,” Nucl. Phys. B (1979) 61.[37] R. Rohm, “Spontaneous supersymmetry breaking in supersymmetric string theories,” Nucl.Phys. B (1984) 553; C. Kounnas and M. Porrati, “Spontaneous supersymmetry breakingin string theory,” Nucl. Phys. B (1988) 355;[38] M. Axenides, S. D. Ellis and C. Kounnas, “Universal Behavior Of D-dimensional SuperstringModels,” Phys. Rev. D (1988) 2964. I. Antoniadis, J. R. Ellis and D. V. Nanopoulos,“Universality Of The Mass Spectrum In Closed String Models,” Phys. Lett. B (1987)402.[39] C. Angelantonj, C. Kounnas, H. Partouche and N. Toumbas, “Resolution of Hagedornsingularity in superstrings with gravito-magnetic fluxes,” Nucl. Phys. B (2009) 291[arXiv:0808.1357 [hep-th]].[40] K. R. Dienes, E. Dudas, T. Gherghetta and A. Riotto, “Cosmological phase transitions andradius stabilization in higher dimensions,” Nucl. Phys. B (1999) 387 [hep-ph/9809406].[41] S. A. Abel, J. L. F. Barbon, I. I. Kogan and E. Rabinovici, “String thermodynamics in D-branebackgrounds,” JHEP (1999) 015 [hep-th/9902058].[42] S. C. Frautschi, “Statistical bootstrap model of hadrons,” Phys. Rev. D (1971) 2821;R. D. Carlitz, “Hadronic matter at high density,” Phys. Rev. D (1972) 3221.[43] N. Deo, S. Jain and C. -ITan, “Strings At High-energy Densities And Complex Temperature,”Phys. Lett. B (1989) 125; N. Deo, S. Jain and C. -ITan, “The ideal gas of strings,” HUTP-90-A079.[44] N. Deo, S. Jain and C. -ITan, “String Statistical Mechanics Above Hagedorn Energy Density,”Phys. Rev. D (1989) 2626.[45][45] N. Deo, S. Jain, O. Narayan and C. -ITan, “The Effect of topology on the thermodynamiclimit for a string gas,” Phys. Rev. D (1992) 3641.[46] B. McClain and B. D. B. Roth, “Modular invariance for interacting bosonic strings at finitetemperature,” Commun. Math. Phys. (1987) 539;5147] N. Cabibbo and G. Parisi, “Exponential Hadronic Spectrum and Quark Liberation,” Phys.Lett. B (1975) 67.[48] I. Florakis, C. Kounnas and N. Toumbas, “Marginal Deformations of Vacua with Massiveboson-fermion Degeneracy Symmetry,” Nucl. Phys. B (2010) 273 [arXiv:1002.2427 [hep-th]].[49] P. H. Ginsparg and C. Vafa, “Toroidal Compactification of Nonsupersymmetric HeteroticStrings,” Nucl. Phys. B (1987) 414. V. P. Nair, A. D. Shapere, A. Strominger andF. Wilczek, “Compactification of the Twisted Heterotic String,” Nucl. Phys. B (1987)402.[50] C. Angelantonj, M. Cardella and N. Irges, “An Alternative for Moduli Stabilisation,” Phys.Lett. B (2006) 474 [hep-th/0608022].[51] J. Preskill, M. B. Wise and F. Wilczek, “Cosmology of the Invisible Axion,” Phys. Lett. B (1983) 127;[52] G. D. Coughlan, W. Fischler, E. W. Kolb, S. Raby and G. G. Ross, “Cosmological Problemsfor the Polonyi Potential,” Phys. Lett. B (1983) 59; B. de Carlos, J. A. Casas, F. Quevedoand E. Roulet, “Model independent properties and cosmological implications of the dilatonand moduli sectors of 4-d strings,” Phys. Lett. B (1993) 447 [hep-ph/9308325]; T. Banks,D. B. Kaplan and A. E. Nelson, “Cosmological implications of dynamical supersymmetrybreaking,” Phys. Rev. D (1994) 779 [hep-ph/9308292].[53] J. R. Ellis, D. V. Nanopoulos and M. Quiros, “On the Axion, Dilaton, Polonyi, Gravitino andShadow Matter Problems in Supergravity and Superstring Models,” Phys. Lett. B (1986)176; T. Moroi, M. Yamaguchi and T. Yanagida, “On the solution to the Polonyi problem with0 (10-TeV) gravitino mass in supergravity,” Phys. Lett. B (1995) 105 [hep-ph/9409367];S. Nakamura and M. Yamaguchi, “Gravitino production from heavy moduli decay and cos-mological moduli problem revived,” Phys. Lett. B (2006) 389 [hep-ph/0602081].[54] S. Kachru, R. Kallosh, A. D. Linde and S. P. Trivedi, “De Sitter vacua in string theory,” Phys.Rev. D (2003) 046005 [hep-th/0301240].[55] E. Cremmer, S. Ferrara, C. Kounnas and D. V. Nanopoulos, “Naturally vanishing cosmologicalconstant in N = supergravity,” Phys. Lett. B (1983) 61; J. R. Ellis, C. Kounnasand D. V. Nanopoulos, “No scale supersymmetric GUTs,” Nucl. Phys. B (1984) 373;J. R. Ellis, C. Kounnas and D. V. Nanopoulos, “Phenomenological SU ( , ) supergravity,”52ucl. Phys. B (1984) 406; J. R. Ellis, A. B. Lahanas, D. V. Nanopoulos and K. Tamvakis,“No-scale supersymmetric standard model,” Phys. Lett. B , 429 (1984).[56] L. Alvarez-Gaume, J. Polchinski et M. B. Wise, “Minimal low-energy supergravity,” Nucl.Phys. B (1983) 495; L. E. Ibanez et G. G. Ross, “ SU ( ) L × U ( ) symmetry breakingas a radiative effect of supersymmetry breaking in GUTs,” Phys. Lett. B (1982) 215;L. Alvarez-Gaume, M. Claudson et M. B. Wise, “Low-energy supersymmetry,” Nucl. Phys. B (1982) 96; J. R. Ellis, D. V. Nanopoulos et K. Tamvakis, “Grand unification in simplesupergravity,” Phys. Lett. B (1983) 123; C. Kounnas, A. B. Lahanas, D. V. Nanopouloset M. Quiros, “Supergravity induced radiative SU ( )× U ( ) breaking with light top quark andstable minimum,” Phys. Lett. B (1982) 95; C. Kounnas, A. B. Lahanas, D. V. Nanopou-los et M. Quiros, “Low-energy behavior of realistic locally supersymmetric Grand UnifiedTheories,” Nucl. Phys. B (1984) 438.[57] F. Finelli and R. Brandenberger, “On the generation of a scale invariant spectrum of adia-batic fluctuations in cosmological models with a contracting phase,” Phys. Rev. D (2002)103522 [hep-th/0112249]; R. H. Brandenberger, “The Matter Bounce Alternative to Inflation-ary Cosmology,” arXiv:1206.4196 [astro-ph.CO].[58] C. Kounnas and N. Toumbas, “Aspects of String Cosmology,” arXiv:1305.2809 [hep-th].[59] L. McAllister and E. Silverstein, “String Cosmology: A Review,” Gen. Rel. Grav. (2008)565 [arXiv:0710.2951 [hep-th]].[60] L. Liu, “Cosmologie quantique à température finie en théorie des supercordes,”[61] P. H. Ginsparg, “Applied Conformal Field Theory,” hep-th/9108028.[62] M. Bianchi and A. Sagnotti, “On the systematics of open string theories,” Phys. Lett. B (1990) 517.[63] C. Bachas, C. Fabre, E. Kiritsis, N. A. Obers and P. Vanhove, “Heterotic/type-I duality andD-brane instantons,” Nucl. Phys. B (1998) 33 [arXiv:hep-th/9707126];[64] C. Angelantonj, I. Florakis and B. Pioline, “A new look at one-loop integrals in string the-ory,” Commun. Num. Theor. Phys. (2012) 159 [arXiv:1110.5318 [hep-th]]; C. Angelantonj,I. Florakis and B. Pioline, “One-Loop BPS amplitudes as BPS-state sums,” JHEP1206