H0 tensions in cosmology and axion pseudocycles in the stringy universe
HH tensions in cosmology and axion pseudocycles in the stringy universe Andrei T. Patrascu ELI-NP, Horia Hulubei National Institute for R&D in Physics and Nuclear Engineering,30 Reactorului St, Bucharest-Magurele, 077125, Romania
The tension between early and late H is revised in the context of axion dark matter arisingnaturally from string theoretical integrations of antisymmetric tensor fields over non-trivial cycles.Certain early universe cycles may appear non-trivial from the perspective of a homology analysisfocused on the early universe, while they may become trivial, when analysed from the perspective ofa homology theory reaching out to lower energies and later times. Such phenomena can introducevariations in the axion potential that would explain the observed H tension. INTRODUCTION
The accumulation of observational data in favour ofa physical tension between low and high redshift deter-minations of the Hubble constant indicates potential newphysics beyond the standard model [1], [2], [3]. There aretwo methods of determining Hubble constant nowadays,one relying on the cosmic microwave background, un-der the assumption of a cold dark-matter universe witha cosmological constant Λ (ΛCDM) and the other re-lying on direct measurements from supernovae. Thesemethods were recently highly refined by employing databoth from the Planck satellite (for the cosmic microwavebackground) and from GAIA, a space telescope systemproviding accurate data for the later method. The Hub-ble constant inferred from CMB has been calculated tobe H = 67 . ± . kms − M pc − while the super-novae observations insist on a value of H = 73 . ± . kms − M pc − with the distinction between the distribu-tions related to the two types of observations being sig-nificant. Basically this leads us to a tension betweenthe ”local” and the ”global” measurements of the Hub-ble constant that can only hardly be considered to re-sult from systematic errors or other data-related biases.It is worth mentioning that while such a discrepancyhas been observed in previous data, the interest of thecosmology community peaked when the latest, far moreaccurate observational results not only re-confirmed theabove mentioned tension, but they also made it sharper.While this observational discrepancy has not been in thefocus of theoretical model-builders, there have been atleast a few attempts to give explanations based on newphysics or modified fundamentals. The reconciliation ofthe theories with observations however is delicate. Neu-trinos [4], axions [5], or other moduli-particles [6] havebeen considered as possible explanations, together withmodified dark energy contributions at the early stagesof cosmological evolution. In all cases it was not pos-sible to fully acknowledge the evolution of the Hubbleconstant as observed presently. In this article I will pro-vide a new theoretical approach, although based on axioncontributions to the cosmological evolution. While axionsolutions have been considered before, no analysis relying on the fundamental origin of axions has been made upto now. Axions indeed are dark matter candidates thatappear as an extension to the standard model of elemen-tary particles and are motivated by the strong CP prob-lem. The strong CP problem refers to the unexpectedabsence of a CP violating term in Quantum Chromo-dynamics (QCD) albeit not forbidden by any usual re-strictions. Given the experimental observations, if sucha CP breaking term were to exist, it would appear tobe unjustifiably small leading to a hierarchy problem.The solution to this problem is given by the so calledPeccei-Quinn mechanism that involves axial degrees offreedom and an additional (Peccei-Quinn) symmetry bro-ken spontaneously, and hence giving rise to the Axion,and further broken explicitly by various instanton-likemechanisms, providing mass to the newly required Ax-ion. As we may notice, while the Axion, and the asso-ciated axial degrees of freedom do solve the strong CPproblem, their existence in the standard model, togetherwith the symmetry they rely upon is in many ways ”ad-hoc”. This changes dramatically however if we considerstring theory and compactified extra-dimensions. Indeed,once extended objects (like strings or branes) are beingconsidered, and once the background manifold is seenas a higher-dimensional manifold with compact extra-dimensions, we may consider various non-trivial topolog-ical cycles emerging, each providing us with axion-likedegrees of freedom. From this perspective, not only isthe axion a common component that should emerge inany description of high energy physics, but it is also, ina sense, unavoidable. Indeed, many non-equivalent cy-cles should give rise to their own specific axions, lead-ing to what is known as axion proliferation, or the ax-iverse. While string theory strongly favours axions, itremains to be seen whether all axions or axion-like par-ticles will have a similar impact on cosmology. As ax-ions appear from the integration of asymmetric tensorfields over non-trivial cycles in string theory, one shouldpay more attention to these elements of string theory. Ithas been shown that early axions (and in general, earlydark matter contributions) would not be able to explainthe tension observed for the Hubble constant, however,it is important to note that the disappearance of cer- a r X i v : . [ phy s i c s . g e n - ph ] A ug tain axion cycles before the recombination period wouldbe able to significantly alleviate the cosmological Hubbleconstant tension. Indeed, this article assumes that thevery early universe already was described by a compact-ified manifold but on this manifold, not all significantcycles were true cycles, some of them being described interms of pseudo-homology groups, giving them a signifi-cant role only at an extremely early stage of the universe.They would dissipate afterwards, becoming basically un-detectable in terms of homology groups at the recombi-nation stage, leaving the true cycles associated to axionswhich we may hope to detect today. Indeed this inter-play between homology and pseudo-homology at an earlystage of the cosmological expansion may offer an expla-nation for the observed Hubble tension without imply-ing exotic/esoteric dark energy contributions at the veryearly cosmological evolution. There are several tools wecan use to describe this observation, pseudo-homologybeing one of them, together with the detectability oftopological features with homology theories with non-trivial coefficients. While these tools are mathematicallyaccurate, they may not be familiar to the reader, henceI will also give an interpretation using a more standardlanguage originating in string theory. I will also give anexample using mostly the language of cosmology. WIMPS AND AXIONS IN COSMOLOGY ANDPARTICLE PHYSICS
Axion like particles are extremely light and weakly cou-pled degrees of freedom which can be considered darkmatter candidates (within certain domains [7]) and arewell motivated both within QCD and string theory. Onone side the QCD axion is an essential component in thesolution of the strong CP problem, on the other side,axion like particles in general emerge naturally from in-tegration over non-trivial cycles in string theory. Thesheer quantity of such cycles leads to the so called ”ax-iverse” which contains an axion like particle for everyenergy decade. From a cosmological perspective a darkmatter candidate must resolve the missing matter issueobserved already in the early observations by Zwicky, andconfirmed to almost perfect accuracy in the first and sec-ond half of the 20th century. While the postulation ofa dark matter particle is obviously required due to cos-mological arguments, there are several particularities re-quired by high energy physics models that add a strongerfoundation to the dark matter claims. On general cos-mological grounds dark matter should have the follow-ing properties: first, they should be non-relativistic can-didates usually represented by massive particles whichare not expected to be faster than the average galac-tic escape velocity, and hence should play a role in ex-plaining the galactic rotation curves observed during thepast decades on a vast number of galaxies. Second, they ought to be non-baryonic candidates, namely candidatescarrying neither electric nor colour charges, and finally,they should be stable enough to ensure their lifetimewould reach out to the current age of the universe andalso to allow us to expect them to continue with a life-time many orders of magnitude greater than the life ofthe universe. Dark matter candidates are produced inthe early universe either through processes taking placeat thermal equilibrium (thermal production) or in pro-cesses taking place away from thermal equilibrium (theso called non-thermal production). The thermal pro-duction usually appears at the freeze-out temperatureand will emerge from relics at thermal equilibrium atthis stage of cosmic evolution, or will appear in scatter-ings and decays of other particles in the original plasma.The non-thermal production involves usually the coher-ent motion of bosons associated to a bosonic field or fromout-of-equilibrium decays of heavier states. Clearly thestandard model of elementary particles cannot accommo-date such dark matter candidates, the only valid alter-native coming from various extensions of the standardmodel both in the direction of heavier and lighter parti-cles. Various observational results excluded most of thecommon standard model candidates, including the mas-sive, compact, and weakly radiating candidates like blackholes, neutron stars, or a potential high density of plan-etary bodies [8], as well as massive neutrinos excludedby calculations involving their relic abundance. The cur-rent dark matter searches focus on the weakly interactingmassive particles, also known as WIMPs which representa broad category of particles usually required by super-symmetry. The gauge hierarchy problem has a simplesupersymmetric solution involving the neutralino. Thestrong-CP problem has another simple solution involv-ing a new Peccei-Quinn symmetry spontaneously (andexplicitly) broken, leading to the axion. The axion is avery well motivated non-thermal relic appearing in SUSYmodels as a supermultiplet containing the axion (a), thespin R -parity odd axino (˜ a ), and the R -parity evenspin-0 saxino ( s ). Their interaction strength is particu-larly weak, the axion, being the fermionic super-partnerof the axion is seen as a WIMP, being on the massiveside of the spectrum, but with an extremely weak inter-action strength. Its mass is strongly model dependent,and they can be either thermal or non-thermal relics. Theaxion on the other side, as an example of a non-thermalrelic has a interaction strength strongly suppressed bythe Peccei-Quinn breaking scale f a ∼ GeV. The in-teraction strength is, as usually, given by ( m W f a ) where m W represents the weak scale. As an additional exam-ple, the gravitino, ˜ G , the SUSY partner of the graviton,is a neutral Majorana fermion with a coupling to ordi-nary particles strongly suppressed by the Planck scalevia ( m W M Planck ) . Particle relics from the early epochs ofthe universe can span a enormous range both in massand in cross section, as they may be generated by verydifferent production mechanisms in the early universe.The WIMP thermal relicts present us with an interest-ing connection between the cold-dark-matter relic densityand the electroweak interaction strength. Because dur-ing the early universe stage the WIMPs are consideredto be in thermal equilibrium at temperature T ≥ m X ,their number density as a function of time is determinedby the Boltzmann equation dn X dt = − Hn X − (cid:104) σ ann v (cid:105) ( n X − n eq ) (1)here H is the Hubble constant which for the radiationdominated universe is given by H = ρ rad M Planck , the equi-librium density is n eq and the term (cid:104) σ ann v (cid:105) representsthe thermally averaged cross section for the WIMP anni-hilation times the relative velocity. In the early universe,the number density for WIMPs follows the equilibriumdensity. As time passes, the temperature reaches a value T fr known as freeze-out point where the expansion ratebecomes larger than the annihilation rate and the Hubbleterm becomes of major importance. After that point, theWIMP’s number density in a co-moving volume becomeseffectively constant. The present day WIMP relic den-sity can be found as a solution of the Boltzmann equationgiven byΩ X h ∼ = s ρ c /h ( 45 π g ∗ ) / x f M P lanck (cid:104) σ ann v (cid:105) (2)where g ∗ is the number of relativistic degrees of freedomat freeze-out, s is the present day entropy density, and x f = T fr /m X the freeze-out temperature scaled to theWIMP mass. Following ref. [10] and introducing thedata from ref. [11] for s , ρ c and M P lanck and using themeasured value for Ω X h ∼ = 0 .
12 we findΩ X h . ∼ = 1 (cid:68) σ ann − cm v/c . (cid:69) (3)The result of this calculation is interpreted in the sensethat a cross section of 1pb and typical WIMP speedsat freeze-out temperature provide the exact present dayrelic density of dark matter. This is why the WIMP darkmatter may be related to new physics which was expectedto appear at or around electroweak level. Another mo-tivation for this was the stabilisation of the Higgs bosonmass, which will not be discussed here. Needless to say,no new dark matter particles around this scale have sofar been detected. We can understand this by think-ing that σ ann ∼ g m X where only the fraction needs befixed, both g and m X being allowed to vary on relativelybroad ranges while still being consistent with the freeze-out mechanism. Our concern in this article however willnot be with the WIMP dark matter candidates, but in-stead with the axions which from a cosmological per-spective can be regarded as a source for bosonic coherent motion (BCM). The BCM involving the axion implies alight boson with a very long lifetime. As there exists oneaxion that solves the strong-CP problem, known as theQCD axion, if this is supposed to make up for the darkmatter, its mass should be smaller than 24 eV to be ableto exist until the current age. The other axions (alsoknown as axion-like particles, short ALP) are very sim-ilar with the QCD axion, arising in a similar way fromstring theory, with the main distinction that their massis not linked to the Peccei-Quinn scale f a . Such axionsare still coupled to the electromagnetic field by means ofa term ( a ALP /f a ) F µν ˜ F µν . When not bound by the re-strictions of the Peccei-Quinn solution of the strong-CPproblem, the axion can couple to the QCD anomaly bya term like L = α s πf a aG aµν ˜ G aµν (4)where the dual gluon field strength is ˜ G aµν = (cid:15) µνρσ G aρσ and α s = g s / π is the strong coupling constant. Suchcoupling can be obtained by integrating the colouredheavy fields below the Peccei-Quinn breaking scale f a but above the electroweak scale v EW . After integratingout all the heavy PQ-charged fields, the axion couplingLagrangian at low energy in terms of the effective cou-plings c i , i = 1 , , L effint = c ∂ µ af a (cid:80) q ¯ qγ µ γ q −− (cid:80) q (¯ q L mq R e ic a/f a + h.c. ) + c π f a aG ˜ G ++ C aWW π f a aW ˜ W + C aY Y π f a aY ˜ Y + L leptons (5)The first term involving the derivative interaction pro-portional to C preserves the U (1) Peccei-Quinn symme-try. The second term proportional to c is related to thephase of the quark mass matrix, and the third term, pro-portional to c is the anomalous coupling. The couplingbetween the axions and the leptons is encoded in theinteraction term L leptons . The axions that are not sup-posed to represent solutions to the strong-CP problem,namely those which are expected to be particularly light,are described by two types of field theoretical models,one is known as the Kim-Shifman-Vainstein-Zakharov(KSVZ) model, and the other is known as the Dine-Fischler-Srednicki-Zhitnitskii (DFSZ) model. In the firstmodel, at the level of field theory, the axion is presentif quarks carry a net PQ charge Γ of the global U (1) P Q symmetry. In general, at the standard model level, thesix quarks are strongly interacting fermions. The elec-troweak scale v EW ∼ = 246 GeV we start taking into ac-count additional, beyond standard model heavy, vecto-rial quarks ( Q i , ¯ Q i ) but these end up being integratedout from the effective Lagrangian written above. In thismodel, the only heavy quarks that may appear beyond v EW is must carry PQ charge and hence, below v EW or below the QCD scale Λ QCD we have c = c = 0and c = 1. The gluon anomaly term given to be pro-portional with c is induced by an effective heavy quarkloop and solves the strong-CP problem. As a byproduct,the axion field appears as a component of the standardmodel singlet scalar field S . The string axions emergingfrom B MN are of this type and are defined by the QCD-anomaly coupling at lower energies. These are like theKSVZ axions. In the second model one does not intro-duce any PQ charge in the heavy quark sector beyondthe standard model. Instead the standard model quarksare assigned a PQ charge with c = c = 0 and c (cid:54) = 0below the electroweak scale v EW . In the same way, theaxion is a part of the standard model singlet scalar field S . Usually string theory gives also rise to componentssimilar to DFSZ axions in addition to the KSVZ axions.The axion has shift symmetry, which is basically just aphase rotation, and the physical observables are invari-ant under this transformation. Below f a the PQ rotationsymmetry is broken into a discrete subgroup which rep-resents the rotation by 2 π . This breaking can be seenthrough the appearance of the c and c terms in theLagrangian. The c term enters as a phase and a shiftby 2 π brings it to the same value, while the c term isthe QCD vacuum angle term, which again, if the vacuumangle is shifted by 2 π comes to the original value. Thesubgroup corresponding to the common intersection ofthe subgroups corresponding to c and c is preserved.The combination c + c is invariant under axion shiftsymmetry and c + c represents the unbroken discretesubgroup of U (1) P Q . This is the domain wall number N DW = | c + c | . STRING THEORY AXIONS
As noted before, axions appear due to integration oftensor fields over non-trivial cycles arising on the com-pactified manfiold of string theory. In QCD, the CP-violating term while being a total derivative and hencebeing trivial from the point of view of classical fieldequations, has a significant quantum impact due to itsnon-trivial topological properties. The topologically non-trivial field configurations can be seen by looking at theterm in the action S θ = θ π (cid:90) d x(cid:15) µνλρ T rG µν G λρ (6)When we shift the parameter θ → θ + 2 π the actionchanges by 2 π and hence leaves the partition functionunchanged. This suggests that the parameter θ repre-sents a periodic parameter with a period equal to 2 π .The introduction of fermions will bring with it the chiralanomaly and the parameter θ will have to include theoverall phase of the quark mass matrix, modifying it as in ¯ θ = θ + arg ( det ( m q )) (7)However, measurements have shown that θ ≤ − . Thesolution to the strong-CP problem implies making the θ parameter a dynamical field a . At a classical level the ac-tion is obviously invariant to any shifts a → a + C . Thismeans that at the classical level, the axion is the Gold-stone boson of a spontaneously broken global symme-try. Quantum perturbative effects preserve this symme-try but non-perturbative, topologically non-trivial QCDfield configurations break it explicitly generating a peri-odic potential for the axion. In the case of the QCD ax-ion, the axion obtains a vacuum expectation value whichadjusts itself to render the resulting ¯ θ small. The ax-ion couples to the gluons, as noted above, but also toother gauge bosons including photons, and to fermionsby means of derivative couplings. It is important to notethat their coupling to photons make the axion detectableat extremely intense laser facilities like the ELI-NP [12].While the justification of the axion resulting from thesolution of the strong-CP problem is clear, one may askmore fundamental questions, namely why should a sym-metry like the Peccei-Quinn even exist and be explic-itly broken by topologically non-trivial QCD fields. Suchangular degrees of freedom are certainly unexpected ina fundamental theory based on standard quantum fieldtheory. Pseudoscalars with axion-like properties are how-ever quite natural in string theory compactifications.They may appear as Kaluza-Klein zero modes of antisym-metric tensor fields. The Neveu-Schwarz 2-form B MN that arises in all string theories, or the Ramond-Ramondforms C , , arising in type IIB string theory as well asthe C , forms arising in IIA string theory are such ex-amples. Higher order antisymmetric tensor fields, uponcompactification, typically give rise to a large number ofKaluza-Klein zero modes which are determined by thetopology of the underlying compact manifold. In partic-ular, considering a single two form B MN or C MN oneobtains a number of massless scalar fields equal to thenumber of homologically non-trivial closed two-cycles inthe underlying manifold. We can look at the Kaluza-Klein expansion for the B MN two-form considering thenon-compact coordinates x and the compact coordinates y B = 12 (cid:88) b i ( x ) ω i ( y ) + ... (8)with ω i being the basis for closed non-exact two forms(cohomologies) dual to the cycles in our manifold, obey-ing the constraint that (cid:90) C i ω j = δ ij (9)Similarly the number of pseudo-scalar zero modes corre-sponding to C is equal to the number of homologicallynon-trivial distinct four-cycles. As it has been noted in[13] the number of cycles in most compactifications isextremely large leading to many axion-like fields beingpredicted in general by string theory. When going tothe four dimensional effective theory the scalar fields re-sulting from the KK reduction are massless and have aflat zero potential resulting from the higher dimensionalgauge invariance of the antisymmetric tensor field ac-tion. This invariance also ensures that no perturbativequantum effect can generate a potential. However, anti-symmetric tensor fields couple by means of Chern-Simonsterms. After KK reduction these terms can couple ax-ion fields to the gauge fields. This has been theoreticallyobserved in type IIB theory with a C axion with a D AXION PSEUDO-CYCLES AND THE H TENSION
As observational evidence for a tension between earlyand late H accumulates, an explanation based on fun-damental physics seems still somehow remote. While theΛ CDM model is successful in describing the large scalestructure of the universe and is well grounded in the pre-cision observations of the cosmic microwave backgroundby Planck, it seems like local observations of supernovaeintroduce a tension with the Hubble rate measured fromthe early cosmic observations, with a statistical signif-icance in several cases already larger than 4 σ . In thisarticle I will provide a model that will increase the num-ber of parameters to be optimised for cosmological databy introducing a new theoretical model based on pseudo-cycles in string theory. The basic idea is as follows: fluc-tuations in the underlying manifold may generate veryearly deformations which may provide pseudo-cycles re-sulting in pseudo-axion fields at a later stage of cosmicevolution. This later stage is still considered early fromthe perspective of cosmological observations. What basi-cally happens is that such deformations, seen exclusivelyat high energies will appear as non-trivial cycles corre-sponding to axion fields which will play a non-trivial rolein the very early universe and will vanish in the successivestages, which will still correspond to the early Λ CDM stages of the cosmological evolution. The pseudo-cyclescorrespond to deformed manifolds which are trivial fromthe perspective of absolute homology theory and hencedo not count in the final number of axion fields in an effec-tive theory. However, at intermediate energies, prior tothe hot-axion cosmology stage (as presented in [9]) theydo have a non-trivial impact as they behave like true ax-ion fields originating from pseudo-cycles perceived as truecycles in the very early stages of cosmic evolution. Thelength of the deformation towards lower energy (latertime) stages represents a new parameter that contributesin the desired way towards the alleviation of the observed H tension. The real axion cycles will become relevantlater on, inducing the effects well known from the hot-axion model, but this time strongly modulated by theearlier disappearance of the ”fake” axions resulting fromearly pseudo-cycles.This gives a mechanism for the ”exotic” early darkenergy presented in [14] however, it is well grounded instring phenomenology and does not rely on a ”myste-rious” early dark energy, but instead on a mechanisminvolving (pseudo) axions which are expected to be de-tected by subsequent radiative emissions. As presentedbefore, axion fields can be regarded as results of non-trivial cycles arising in the process of compactification.However, the extreme environment in the early universeoffers sufficient opportunities for non-trivial topologicaleffects to take place. Among these, we also have a pos-sible local deformation of the underlying manifold thatwould result in more or less extended pseudo-cycles vis-ible around the stages of the early universe. While fun-damentally local and topologically trivial, such deforma-tions can provide non-trivial pseudo-homological effectsclearly distinguishable over a large (albeit local) temporalregion of the early universe. A distinction must be mademanifest. While real axions are defined based on truecycles arising on the compactified directions, the pseu-doaxions arise on deformations of the underlying mani-fold that do not require compactification to begin with.In terms of enumerative geometry, the number of cyclesallowed by a Calabi-Yau manifold is usually large. Thesimplest Calabi Yau manifold, the six-torus, will pro-vide us with (6 × / H tension. This article deals withstring phenomenology as it tries to connect the perturba-tive ten-dimensional effective field theories describing themassless degrees of freedom of string theory at very highenergy (for example type IIA-IIB string theory with D-branes) and the low energy phenomena of the emergingfour-dimensional universe. Indeed, the phenomena dis-cussed are in a sense at an intermediate range betweenthese regions, the H tension observed nowadays in cos-mology being expected to be a result of such intermediatescale phenomena. Moduli stabilisation is a particularlyimportant aspect of string phenomenology, implying thatexpectation values of moduli fields determine many pa-rameters of the low energy effective field theory. More-over, such expectation values also parametrise the shapeand size of the extra dimensions leading to a fascinatingconnection between the string world and the four dimen-sional cosmology. Gauge couplings or Yukawa couplingsarising in the low energy domain are determined by suchmoduli expectation values. Quantum corrections oftenarise in order to fix such expectation values resulting innon-zero masses for their particle excitations. The stan-dard model of elementary particles is expected to exist asa realisation of a stack of spacetime filling branes wrap-ping cycles in the compact dimensions. Gravity on theother side is expected to propagate in the bulk leading toa string scale M s ∼ M P lanck / √ V , of course, with a largecompactification volume V . While real axion fields ap-pear as Kaluza-Klein zero modes of the ten-dimensionalform fields, the pseudo-axions I will consider here areonly visible in the intermediate region, as they arise asmodes over pseudo-cycles which are fundamentally de-pending just on the geometry and pseudo-homology asvisible in the intermediate and high energy (stringy) re-gion. Focusing, for the sake of exemplification, on thetype IIB string theory we consider the Calabi-Yau man-ifolds for the additional spatial dimensions. Fluxes inconformally flat six-dimensional spaces can break N = 4supersymmetry down to N = 3 , , , φ , themetric tensor g MN and the antisymmetric 2-tensor B MN in the NS-NS sector. The massless RR sector contains C , the 2-form potential C MN and the four-form field C MNP Q with the self-dual five-form field strength. Wecan combine the two scalars C and phi into a complexfield τ = C + ie − φ parametrising the SL (2 , R ) /U (1) space. The fermionic superpartners are two Majorana-Weyl gravitinos of the same chirality γ ψ AM = ψ AM andtwo Majorana-Weyl dilatons λ A with opposite chiralitywith respect to gravitinos. The field strength for the NSflux is H = dB and for the RR field strengths, we have F (10) = dC − H ∧ C + me B = ˆ F − H ∧ C (10)where of course ˆ F = dC + me B . The RR fluxes areconstrained by the Hodge duality F (10) n = ( − [ n/ (cid:63) F (10)10 − n (11)The Bianchi identities are dH = 0 , dF (10) − H ∧ F (10) = 0 (12)Sources will clearly alter the potentials leading to noglobally well defined potential. Integrating the fieldstrength over a cycle in the presence of sources does notnecessarily result in a null outcome. This situation im-plies the existence of a non-zero flux. As charges are gen-erally quantised in string theory, the fluxes will also beconstrained by Dirac quantisation prescriptions. Giventhe Bianchi identities, the fluxes will satisfy1(2 π √ α (cid:48) ) p − (cid:90) Σ p ˆ F p ∈ Z (13)for a p -cycle Σ p . The number of 2 and 4 cycles are iden-tical due to hodge duality. The 3-cycles appear as pairs.The axion field arises because of the reduction of a NS-NS two form B on a two-cycle with continuous shiftsymmetry. This appears as a result of some higher di-mensional gauge symmetry of the two-form. Includingbranes and fluxes we also obtain a monodromy whichhas been broadly discussed in [15] and [16].As an example we can consider a D b = α (cid:48) (cid:82) Σ B with the classical shift symmetry b → b + const .The D D S D = 1(2 π ) g s α (cid:48) (cid:90) M× Σ d x (cid:112) − det ( G ab + B ab ) (14)which is the effective field theory associated to string the-ory at lower energies. Integrating over the Σ cycle oneobtains the potential of the axion in the four dimensionaleffective theory V ( b ) = ρ (2 π ) g s α (cid:48) (cid:112) (2 π ) l + b (15)where l represents the size of the two-cycle Σ in stringunits, while ρ is the dimensionless coefficient generatedby the warp factor. Various types of inflation have beenanalysed using the monodromy generated by D-branes.For example in ref. [16] the potential for the inflatonfield has been constructed as a power law with the expo-nent depending on the integral over the internal manifoldand the contributions from a Chern-Simons term. As hasbeen observed in [16] additional flexibility can be gainedby uncertainties in the integration over the internal space.Let us however now consider integration over pseudo-cycles arising in the internal space. These will also giverise to axion fields which however will disappear once thescale of validity of the identification between real homol-ogy and pseudo-homology is reached. There is no muchdifference between the two approaches until the criticalpoint is reached, except for the size of the two cycle (con-sidering the above example) which would be decreasingas a function of time. One must also take into accountthat the proliferation of initial pseudocycles will dependon the fluctuations in the initial manifold.It can be verified as a theorem (for proof see [20]) thatevery pseudocycle f : V → M of dimension k inducesa well defined integral homology class α f ∈ H k ( M ; Z ).Also, any singular cycle α ∈ Z singk ( M ; Z ) gives rise to a k -pseudocycle f : V → M such that α f = α . There-fore integral cycles in singular homology can be repre-sented by pseudo-cycles. This implies that in the re-gion of the early universe, we have a statistical ensembleof indistinguishable cycles and pseudocycles detectableeither by singular homology defined over the early do-main or by pseudohomology capable of detecting suchpseudocycles and associating them to corresponding in-variants. Consider a topological pair ( M, D ) and a do-main S = CP − { } defined by a punctured spherewith a marked point z = ∞ . Take also a cylindricalend near z = 0 given by ( s, t ) → e s + it . Out coordi-nate system ( s, t ) can extend to all of S − z . We canconsider a one-form β which restricts to dt on the cylin-drical end and to zero on a neighbourhood of z . We mayconsider a non-negative, monotone non-increasing cutofffunction ρ ( s ) which is zero for s much larger than zeroand one for s much smaller than zero. In this case wecan write β = ρ ( s ) dt . Considering a fixed pair of com-plex structures J ∈ J c ( M, D ) together with J F we de-note the space of complex structures by J S ( J , J F ) giving J S ∈ C ∞ ( S, J c ( M, D )) in such a way that in a neighbour-hood of z we have J S = J and along the negative stripend we have J S = J F . In the neighbourhood of z we alsofix a distinguished tangent vector pointing in the positivereal direction. For any element α ∈ H ∗ ( ¯ X, ∂ ¯ X ), we canfix a relative pseudocycle representative α c in such a waythat ∂α c ⊂ ∂ ¯ X . Given such a pseudocycle and given anorbit x in χ ( M, H λ,t ) we can choose a surface depen-dent almost complex structure J S ∈ J S ( J , J F ). Givenany orbit x of this type we may define M M ( x ) as thespace of the possible solutions to the map u : S → M satisfying ( du − X H λ,t ⊗ β ) , = 0 (16)satisfying the asymptotic conditionlim s →−∞ u ( (cid:15) ( s, t )) = x (17)Given any x ∈ χ ( X ; H λ,t ) we can consider those u ∈M M ( x ) in such a way that u ( S ) ⊂ X . This modulispace can be called ˆ M M ( x ) and M ( α c , x ) = ˆ M M ( x ) × ev z α c (18)Therefore we can define an extended moduli space de-scribed not only by the parameters that remain valid atlater times, but also by those included in the pseudoho-mological description. This results in an extended spacewhich includes several cycles over which we can integratein a stable way (at least stable at early times). Integra-tion towards the tip of this geometric construction willencounter conic singularities which can be avoided by al-lowing a smooth cut-off. This will allow a smooth, wellbehaved overall spacetime at later times. This techni-cal aspect has been discussed in various references onorbifold compactification. I will not insist on these as-pects here but of course the reader can look up references[17,18].What interests us from a phenomenological point ofview is an additional number of axions arising at highenergies which will have an effect on the cosmologicalparameters. Returning to the four dimensional effectivetheory, we have the general effective action as S = − (cid:90) d x √ g ( ∂ µ φ∂ µ φ + m φ ) (19)where the inflaton field φ becomes a variable in the infla-ton mass, i.e. the inflaton mass varies along the inflatonfield. This all implies the existence of a critical value φ c which plays the role of a scale which triggers a jump inthe potential term of the form m φ = [ m a + ( m b − m a ) θ ( φ − φ c )] φ (20)with m a and m b being the inflaton masses when the in-flaton field is larger and respectively smaller than thecritical value φ c .The discussion about the effects of the axion mon-odromy and its extensions due to D-branes wrappingtorsion cycles have been discussed in ref. [15], [16]. Isuggest here a similar approach with the distinction thatthere will be a certain number of pseudo-cycles gener-ating non-trivial pseudo-homology leading to additionalterms. Following the model for axion monodromy driveninflation by means of torsional cycles presented in [16] weexpand the analysis towards pseudo-cycles to be found inthe early stages. With the usual N D3-branes along the3 + 1 dimensional space-time and the natural warp factor e − A ∼ /N we have the ten dimensional metric ds = e − A ( y i ) dx µ dx µ + e A ( y i ) dy i dy i (21)where the greek indices count the spacetime coordinateswhile the latin ones count the coordinates of the inter-nal manifold. Disregarding certain aspects related to thewarping, in a Calabi-Yau compactification the masslessmodes are in one-to-one correspondence with the ele-ments of the cohomology group of the internal manifold.This observation is again modified by the presence ofD-branes wrapping torsional cycles as presented in [16].Working with the same relevant Σ and Σ cycles as in[16] here we will have to consider the fact that pseudo-cycles will contribute as well. The massless modes arenot capable of detecting D-branes wrapping torsional cy-cles. Designating the unwrapped internal manifold with X and considering the torsion cycles designated above,we consider the Laplacian eigenforms of X in a similarway, namely dγ = pρ , d ˜ ρ = p ˜ γ dη = kω , d ˜ ω = k ˜ η (22)where γ , ˜ ρ , η and ˜ ω are the generators of T or ( H i ( X , Z )) for i = 1 , , , ρ , ˜ γ , ω and ˜ η aretrivial in the de Rham cohomology but non-trivial gen-erators of the H i ( X , Z ) group for i = 2 , , ,
4. We canexpand in terms of this eigenforms and obtain B = bη + ¯ bρ + b ∧ γ + b C = cη + ¯ cρ + c ∧ γ + c C = c ˜ ρ + ¯ c ˜ η + c ∧ ˜ ω + ¯ c ∧ ω ++ c ∧ ρ + ¯ c ∧ η + c ∧ γ + c (23)The distinction with respect to [16] is that here we mustalso include the pseudo-cycles giving rise to a differentform of the potential term. Indeed, we obtain the fourdimensional action S = − √ g ( ∂ µ φ∂ µ φ + m φ ) (24)and we also will have a critical inflaton field occurringaround inflation, φ c . The mass term m φ = [ m a + ( m b − m a ) θ ( φ − φ c )] φ (25)with m a and m b the inflaton masses when inflaton fieldis smaller respectively larger than the critical value, willhave a different structure, involving the statistics of afinite (albeit potentially large) number of pseudo-cyclesmodifying the expected potential term V ( φ ) = 12 m a φ + ( m b − m a ) φ exp ( C H ( φ − φ c ) /M P lanck ))(26)Here the parameter C H is particularly important in de-scribing the phenomenon of pseudo-cycle annihilation. FIG. 1: The potential computed with pseudocyclecorrections (continuous curves) and without (dottedcurve)FIG. 2: Geometric representation of the manifold onwhich pseudo-cycles are being considered, visible atsufficiently high energy and identifiable with real cycles.They can be detected by correspondingpseudo-homologies defined on the region of high energy.They decay as time increases and the energy decreases,i.e. when the distortion of the underlying manifoldbecomes negligibleThis process can be described in the following way. Afterthe cylindrical end extends enough towards lower ener-gies, it will start folding upon itself leading to a decaygeometry leading eventually to some form of cap at lowenough energies. As stated before the associated modulispace will decay whenever the approximation leading toˆ M M ( x ) × ev z α c cannot be sustained anymore. For thesake of simplicity I will consider this process as being gov-erned by a gaussian distribution, modulated by the factthat there is a maximal (albeit large) number of possiblepseudo-cycles supported by the geometry. As the pseudo-cycles disappear, the resulting geometry shifts towardsone that favours accelerated expansion, after a phase ofre-heating which can be seen in Figure 1.The pseudo-homology identifies a non-trivial compo-nent on which non-trivial pseudo-cycles can develop. Itsdefining parameter is given by the amplitude of the de-formation above the flat background, a measure that isdecaying as time advances. The construction has beenrepresented in Figure 2. As the parameter C H measuresthe smoothness of the (pseudo)axion field and the associ-ated potential, it is strongly dependent on the counting ofpseudo-cycles in the early universe. This counting can bedone by means of invariants capable of detecting them,and represents a calculation performed for example in[19]. Given a moduli space associated to the deforma-tion we introduced, the parameter presented previouslyin the construction of the extension of the moduli spacemust be included in the definition of C H . Indeed, oncethe approximation only sees the cylindrical end, the form β restricts to 2 dt and the integration follows the methodof the standard approach. However, the number of seenaxion cycles is substantially increased, being consideredas finite, while large, and the resulting cycles as indis-cernible given the domain of the approximation. Withthis in mind we derive a formula for C H playing also thephenomenological role of a continuation function linkingthe early and the late universe in the form C H ( φ, φ c ) = exp ( − ( φ − φ c ) )(1 + exp ( φ − φ c φ c )) (27)In this way we take into account the proliferation ofpseudo-cycles in the initial phase, expanding the modulispace accordingly and allowing for new axion like parti-cles in the early universe, while taking into account theirdissipation at a later phase. The maximum number ofpseudo-moduli accepted, while large, is finite. Its finitenature will play an important role in the end-stage of thepseudo-cycle proliferation, leading to a re-heating phasecontrolled by φ c and by the length of the cylinder sectionof the distortion. CONCLUSION