$H_0$ tension and the String Swampland

[Abridged] We realize the Agrawal-Obied-Vafa (AOV) swampland proposal of fading dark matter by the model of Salam-Sezgin and its string realization of Cvetic-Gibbons-Pope. The model describes a compactification of 6-dimensional supergravity with a monopole background on a 2-sphere. In 4 dimensions, there are 2 scalar fields, $X$ and $Y $, and the effective potential in the Einstein frame is an exponential with respect to $Y$ times a quadratic polynomial in the field $e^{-X}$. When making the volume of the 2-sphere large, namely for large values of $Y$, there appears a tower of states, which according to the infinite distance swampland conjecture becomes exponentially massless. If the standard model fields are confined on Neveu-Schwarz 5-branes the 6-dimensional gauge couplings are independent of the string dilaton in the string frame, and upon compactification to 4 dimensions the 4-dimensional gauge couplings depend on $X$ (rather than the dilaton $Y$) which is fixed at the minimum of the potential. This avoids direct couplings of the dilaton to matter suppressing extra forces competing with gravity. We show that this set up has the salient features of the AOV models, and ergo can potentially ameliorate the tension between local distance ladder and cosmic microwave background estimates of the Hubble constant $H_0$. Although the AOV proposal does not fully resolve the tension in $H_0$ measurements, it provides a dynamical dark energy model of cosmology that satisfies the de Sitter swampland conjecture. We comment on a viable solution to overcome the tension between low- and high-redshift observations within the AOV background and discuss the implications for the swampland program.


I. INTRODUCTION
counts [49], S 8 = 0.773 +0.026 −0.020 from DES [50] and S 8 = 0.745±0.039 from KiDS-450 [51] weaklensing-surveys. The physical origin for the increase of σ 8 in early dark energy models is fairly straightforward, because the new dark-energy-like component acts to slightly suppress the growth of perturbations during the period in which it contributes non-negligibly to the cosmic energy density. Henceforth, if we want to preserve the fit to the CMB data we must increase the CDM component to compensate for the suppression in the efficiency of perturbation growth [52].
A second type of interesting models emerges if dark energy and dark matter interact with each other [53][54][55][56][57][58][59][60][61][62]. The identification of the infinite tower of string states (following the swampland distance conjecture) as inhabiting the dark sector automatically provides a string framework for a concomitant coupling of the scalar field to the dark matter [63,64]. Within this framework there is a continually reduction of the dark matter mass as the scalar field rolls in the recent cosmological epoch. Such a reduction of the dark matter mass is actually compensated by a bigger value of dark energy density, which becomes visible in the present accelerating epoch calling for an increase of H 0 . In this paper we present a well motivated realization of the cosmological string framework put forward by Agrawal, Obied, and Vafa (AOV) [63]. A point worth noting at this juncture is that the AOV models do not fully resolve the tension in H 0 measurements, as they can raise the ΛCDM predicted value of the Hubble constant only up to H 0 = 69.06 +0. 66 −0.73 km s −1 Mpc −1 [63]. Indeed, this maximum value of H 0 is characteristic of all models with late dark energy modification of the ΛCDM expansion history. This is because the local distance ladder calibrates SNe far into the Hubble flow and if dark matter fading takes place too recently then it would raise H 0 but without actually changing the part of the Hubble diagram where the tension is inferred. More concretely, by substituting the SH0ES calibration to the Pantheon SNe dataset, the ability of late times dark energy transitions to reduce the Hubble tension drops effectively to H 0 = 69.17 ± 1.09 km s −1 Mpc −1 [65]. However, the AOV proposal provides a novel cosmological set up that improves the fit to data compared to ΛCDM, while satisfying the dS swampland conjecture. Moreover, the smaller content of CDM at late times in AOV models as compared to ΛCDM yield a slight decrease of S 8 , which can help reduce somewhat the tension between the CMB and LSS datasets.
Our starting point is Salam-Sezgin 6-dimensional supergravity (SUGRA) model, where a supersymmetric solution of the form Minkowski 4 × S 2 is known to exist, with a U (1) monopole serving as background in the two-sphere [66]. This model can be lifted to string (and M) theory [67] and is asymptotic at large distances to the near-horizon limit of NS5branes described by the linear dilaton background which is an exact string solution [68]. Moreover, the cosmological content of this supergravity model provides a solution of the field equations that can accommodate both the observed dark energy density and a fraction of CDM [69]. (Time dependence in the moduli fields vitiates invariance under supersymmetry transformations.) The carrier of the acceleration in the present dS epoch is a quintessence field slowly rolling down its exponential potential. Intrinsic to this model is a second modulus, which is automatically stabilized and acts as a source of CDM, with a mass proportional to an exponential function of the quintessence field. The exponential functional form of the mass spectrum characterizes the infinite tower of mass states (inherent to the swampland distance conjecture), which emerges when the quintessence field moves a distance in field space O(1) in Planck units.
In the proposed cosmological framework, the standard model (SM) fields are confined to a probe brane and arise from quantum fluctuations. On the other hand, by computing the quantum fluctuations of the U (1) field associated to the background configuration it is easily seen that the Kalb-Ramond field generates a mass term of horizon size [69]. These "paraphotons" (denoted herein by Υ ) have been redshifting down since the quantum gravity era without being subject to reheating. The presence of any additional relativistic particle species with g degrees of freedom is usually characterized by where N eff quantifies the total relativistic "dark" energy density (including the three lefthanded SM neutrinos) in units of the density of a single Weyl neutrino species [70] and N SM eff = 3.046 [71], and where T dec is the temperature at which particle species decouple from the primordial plasma and the function g * (T dec ) is the number of effective degrees of freedom (defined as the number of independent states with an additional factor of 7/8 for fermions) of the SM particle content at the temperature T dec . Comparing the 106.75 degrees of freedom of the SM with the 10.75 degrees of freedom of the primordial plasma before neutrino decoupling it is straightforward to see that for a massless (real) spin-0 scalar, spin-1 2 (Weyl) fermion, and massive spin-1 vector boson the contributions to N eff asymptote to specific values of ∆N eff = 0.027, 0.047, and 0.080; respectively [72]. 1 Hence, fluctuations in the Kalb-Ramond field do not influence the primordial abundances of the nuclides produced at big-bang nucleosynthesis (BBN) as the Υ 's only count for ∆N eff 0.080 and the 95% CL limit from a combination of current CMB, BAO, and BBN observations is ∆N eff < 0.214 [10]. 2 The layout of the paper is as follows. In Sec. II we briefly describe the geometrical properties of unified dS-Friedmann models when embedded into Salam-Sezgin 6-dimensional supergravity. In Sec. III we interpret numerical results from data analysis that feature estimates for each free parameter in the model. We show that the Salam-Sezgin cosmological set up has the salient features of the generic Agrawal-Obied-Vafa model, and ergo can potentially ameliorate the tension between local distance ladder and cosmic microwave background estimates of H 0 . In Sec. IV we comment on a viable solution to overcome the tension between low-and high-redshift observations within the AOV background. The paper wraps up with discussion and conclusions presented in Sec. V. Before proceeding, we note that other ideas relating cosmological observations to the swampland conjectures have been presented in [79][80][81][82][83][84][85][86][87][88][89][90][91][92][93][94][95].

II. EMBEDDING OF dS-FRIEDMANN MODEL INTO SALAM-SEZGIN SUGRA
Concentrating on the purely bosonic field content of Salam-Sezgin 6-dimensional SUGRA, we can express the bulk action of the system by 1 Asymptote here refers to relativistic species decoupling just before top quark freeze-out. 2 This limit combines the helium measurements of [73,74] with the latest deuterium abundance measurements of [75] using the the PArthENoPE code [76] considering d(p, γ) 3 He reaction rates from [77]. Should one instead use the helium abundance measurement of [78] in place of [73,74], the 95% CL limit on the equivalent neutrino species shifts, N eff = 3.37 ± 0.22, and is in 2.9σ tension with the SM value.
where g 6 = det g M N , R 6 is the Ricci scalar of g M N , σ is a scalar field, , A N is a gauge field, B N P is the Kalb-Ramond field, g is the U (1) coupling constant, κ the gravitational coupling constant, and capital Latin indices run from 0 to 5 [66]. With redefinition of constants G 6 ≡ 2κ 2 and ξ ≡ 4 g 2 , and rescaling of φ ≡ −κσ the action (2) takes the form where the length dimensions of the fields are: Note that by rescaling the 6-dimensional metric as g M N → e −φ g M N , one finds the action at the string frame where φ-dependence enters as an overall exponential factor e −2φ . φ is then identified with the string dilaton, defining the string coupling e φ and having a tree-level potential corresponding to a non-critical string with the parameter ξ determined by the central charge deficit. The latter is induced by the compactification of the four internal dimensions on a manifold with non-vanishing curvature. Its sign implies that the internal curvature is negative, such as the non-compact H (2,2) × S 1 space considered in [67] to compactify from 10 to 6 dimensions. Its compact analytic continuation is S 3 × S 1 , which has an exact (super)conformal field theory description, since S 3 corresponds to an SU (2) k Wess-Zumino-Witten model with curvature fixed by the level k. The total internal 6-dimensional space of our model is then H (2,2) × S 1 × S 2 , with the monopole field on S 2 . The exponential dilaton potential does not allow for static solutions. One solution is the linear dilaton background along a space direction which has an exact string description in terms of a free coordinate with background charge. It corresponds to the near horizon limit of NS5-branes which is holographic dual to a little string theory [96]. In our case of interest, ξ is positive and the solution becomes linear dilaton in the time coordinate with flat metric in the string frame (σ-model) [68]. In the Einstein frame, the scale factor of the metric in FRW coordinates grows linearly with time while the dilaton dependence becomes logarithmic. This exact time dependent 'vacuum' solution is the only asymptotic at large times, even in the presence of matter, as we will see later.
We can now carry out a spontaneous compactification from six to four dimensions, considering the 6-dimensional manifold M of the base spacetime to be a direct product of 4 Minkowski directions (hereafter denoted by M 4 ) and the 2-sphere, R 1,3 × S 2 . The line element on M locally is given by where (t, x) denotes a local coordinate system in M 4 , r c is the compactification radius, and f is the breathing mode of the compact space. We assume that the scalar field φ depends only on the point of M 4 , i.e., φ = φ(t, x). We further assume that the gauge field A M is excited on S 2 and is of the form this is the monopole configuration detailed in [66]. For the purpose of this work, we will set the Kalb-Ramond field to its zero background value, B N P = 0, and since the term A [M F N P ] vanishes on S 2 , we have G M N P = 0. The field strength becomes Taking the variation of the gauge field A M in (3) we obtain the Maxwell equation It is straightforward to verify that the field strength in (6) satisfies (7). Without loss of generality, the Ricci scalar can be written as where , and R[S 2 ] = 2/r 2 c denote respectively the Ricci scalars of the manifolds M, M 4 and S 2 , with Greek indices running from 0 to 3 [101]. To simplify the notation hereafter R 4 and R 2 indicate R[M 4 ] and R[S 2 ], respectively. The determinant of the metric can be written as √ g 6 = e 2f √ g 4 √ g 2 , where g 4 = det g µν and g 2 = r 4 c sin 2 ϑ is the determinant of the metric of S 2 excluding the factor e 2f . We define the gravitational constant in the four dimension as 1 Thus and so, by using the field configuration given in (5) the action in (3) can be recast as where we included the last term that does vanish identically to show what is the 4dimensional coupling of gauge fields that come from 6 dimensions in the Neveu-Schwarz (NS) sector. In the spirit of [42], we now consider a rescaling of the metric of M 4 such that g µν ≡ e 2f g µν and therefore √ĝ 4 = e 4f √ g 4 . The preceding metric transformation brings the model into the Einstein frame, in which the action given in (10) can be rewritten as and we can useĝ 4 in this frame to define a metric which we use to measure distances in the field space. The effective Lagrangian density in 4 dimensions takes the form with where to simplify the notation we have defined: g ≡ĝ 4 and R ≡ R[ĝ 4 ]. Next, we define a new orthogonal basis, , so that the kinetic energy terms in the Lagrangian are both canonical, i.e., where the potentialṼ (X, Y ) ≡ V (f, φ)/G 4 can be re-written (after some elementary algebra) as [102]Ṽ Note that Y corresponds to the 4-dimensional dilaton. The equations of motion for the X and Y fields are and the Einstein field equations are where we have added the matter and radiation stress-energy tensor T µν , which also contributes to the evolution of the Universe. To allow for a dS era we assume that the metric takes the form and that X and Y depend only on the time coordinate, i.e., X = X(t) and Y = Y (t). Before proceeding, we pause to present our notation. Throughout, the subindex zero indicates quantities which are evaluated today. As usual, we normalize the Hubble parameter to its value today introducing an adimensional parameter H 0 = 100 h 0 km s −1 Mpc −1 . Note that the function h(t) in the metric measures the evolution of H, with h(t 0 ) = h 0 . Now, we can rewrite (16) asẌ and the non-zero independent components of (17) arė and where p and ρ are the pressure and energy density contained in T µν . The terms in the square brackets in (15) take the form of a quadratic function of e − √ G 4 X . This function has a global minimum at e − √ G 4 X 0 = R 2 r 4 c /(2 G 6 b 2 ), and so we expand (15) around the minimum, where and Obviously the scalar field X is stabilized around its minimum X 0 . Its physical mass is Y -dependent, and characterizes the mass scale of the tower of string states, which according to the infinite distance conjecture becomes exponentially massless [20,21,28,[37][38][39]. Indeed, as Y runs to large and negative values the 4-dimensional Planck mass grows exponentially as M Pl ∼ e −Y in string units, and thus string excitations become exponentially light in Planck units. Note though that these states cannot play the role of dark matter since part of the string modes carry also SM gauge charges. The X particle on the other hand can play the role of fading dark matter, as we show in the next section.
In the absence of matter and radiation described by the stress tensor T µν , the equations of motion (19)- (21) have no dS or inflationary solution. As we mentioned above, there is an exact string solution with both functions h and Y logarithmic in time describing a linearly expanding universe, which corresponds in the string frame to the well know linear dilaton and flat metric background. This requires the parameter K in eq. (22) to be positive. As we will see later, this solution becomes asymptotic at large times in the presence of matter and radiation. Moreover, there is a period in time of approximate exponential expansion. The dS (vacuum) potential energy density is given by Now, the requirements for preserving a fraction of supersymmetry (SUSY) in spherical compactifications to four dimension imply b 2 ξ = 1, corresponding to winding number n = ±1 for the monopole configuration [66]. From (24) and (26) it follows that the condition for the potential to show a dS rather than an AdS or Minkowski phase is ξb 2 > 1. Therefore, we conclude that a (Y -dependent) dS background can be obtained only through SUSY breaking; see Appendix for details. We finish this section with a comment on possible SM embeddings. In principle, excitations of the electromagnetic field would seemingly induce variation in the electromagnetic fine structure constant, as well as a violation of the equivalence principle through a long range coupling of the dilaton to the electromagnetic component of the stress tensor [97]. A similar variation would be induced in the QCD gauge coupling and thus in the hadron masses. Although a preliminary analysis seems to indicate that such variations may still be compatible with experimental limits because the resulting range of variation of the quintessence field is about 2.5 Planck units (see next section), a very light dilaton would also mediate extra forces at short and larger distances [98] which are excluded in particular by microgravity experiments [99]. A possible way out would be to confine the SM fields on NS5-branes [100]. The 6-dimensional gauge couplings are then independent of the string dilaton in the string frame and thus come with a factor e φ instead of e −φ in the Einstein frame, see eq. (3). It follows that upon compactification to four dimensions, gauge kinetic terms couple to e 2f +φ , see eq. (11), and thus the 4-dimensional gauge couplings depend on the scalar X (instead of the dilaton Y ) which is fixed at the minimum of the potential, and SM couplings do not vary. Moreover, one avoids direct couplings of the dilaton to matter suppressing extra forces competing with gravity.

III. REDUCING THE H 0 TENSION WITH FADING DARK MATTER
We now turn to investigate the cosmological implications of the Salam-Sezgin model, by accommodating recent cosmological observations, while seeking to diminish the tension between low-and high-redshift measurements. To do so, we adopt the best fit value of H 0 = 69.06 +0. 66 −0.73 km s −1 Mpc −1 in the AOV study [63] and analyze the dependence of the quantities relevant to cosmology on the model parameters.
The total energy density of the Universe, ρ = i ρ i , drives the evolution of the Hubble parameter H, where i = {X, Y, X , b, r} accounts for the X and Y fields, for other types of dark matter X , and for the usual SM components of baryonic matter b and radiation r. For a spatially flat Universe, H 2 = ρ/3, where we have adopted reduced Planck units, i.e., M Pl = 1 and G 4 = √ 2. For convenience, herein we consider the evolution in u ≡ − ln(1 + z) rather than t, where z is the redshift parameter. With this in mind, we express the evolution of the matter and radiation components as and where we remind the reader that the subindex zero indicates quantities which are evaluated today, and f (u) parametrizes the u dependent number of radiation degrees of freedom. For the sake of interpolating the various thresholds appearing prior to recombination (among others, QCD and electroweak), we adopt a convenient phenomenological form derived elsewhere f (u) = e −u/15 [103]. To simplify notation we also conveniently define ρ * = ρ b + ρ X . A point worth noting at this juncture is that the leading term in the expansion of the poten-tialṼ around the local minimum X 0 is quadratic, and therefore the coherent X-field energy behaves like non-relativistic dark matter [104]. Thus, the X pressureless dark matter and X add up to the CDM of our model. All in all, the number density of the field X evolves like that of a matter term (i.e., proportional to e −3u ), while its mass evolves with Y according to (25). Therefore, as in the AOV scheme [63], we have Finally, the energy density for Y is found to be Now, making use of the preceding formulae, we can give an explicit expression for the evolution of the Hubble parameter: where ρ s = ρ * + ρ r + V eff stands for the steady-state energy density in moduli space, in the sense that the field Y is not evolving (Y = 0), with V eff ≡ V Y + ρ X . These definitions allow us to rewrite the evolution equation (19) for Y as Next, to simplify the numerical solution to the last equation, we introduce the parameters and Further definition of ρ s ≡ ρ * ,0 ρ s and V eff ≡ ρ * ,0 V eff , which depend only on the parameters introduced in (32), makes explicit the dependence of the solution to (31) on just α, β, and γ. Following [69], we take as initial conditions Y (−30) = 0 and Y (−30) = 0.08, which are in accordance to equipartition arguments [105,106].
In order to understand to which extent this model can represent cosmological data, we introduce the density parameters Ω i = ρ i /3H 2 and the equation of state for the field Y : At this stage, it is worthwhile to note that although the solution for Y only depends on α, β and γ, the cosmological quantities depend on additional parameters. For instance, the use of (30) and (33) requires the introduction of ρ m,0 = ρ * ,0 + ρ X,0 and h 0 as additional parameters. This amounts to a total of five free parameters in this model. For future convenience, they are chosen to be h 0 , Ω m,0 , Ω r,0 , a ≡ A/(3H 2 0 ) and v 0 ≡ V 0 /(3H 2 0 ). These parameters are constrained by five conditions. One is the use of (30) as an internal consistency condition on the total energy density. Four additional constraints will come as an attempt to reproduce experimental data with this model. In particular, we will fix h 0 to an experimental valuẽ h 0 , and subsequently fixing the radiation content of the universe, since this model does not provide any mechanism to modify it, with the additional constraint The total matter content of our model is similarly adjusted to an experimental value and is given by Before we go any further, we clarify that a tilde on top of a given parameter of the model, identifies its direct experimental measurement, and when the measured quantity is a product of two model parameters then we adopt the subindex exp to indicate the experimental measurement. Finally, the equation of state for Y today is fixed to the value of the dark energy equation of state w Y,0 =w Y,0 . In our calculations we takew Y,0 = −0.80 +0.09 −0.11 , as derived from a combination of multiple observational probes in the Dark Energy Survey (DES) supernovae program (including 207 type Ia supernovae light curves, the BAO feature, weak gravitational lensing, and galaxy clustered, but independent of CMB measurements) [107]. This value ofw Y,0 is consistent at the 1σ level with the one derived from a combination of DES data and CMB measurements [108].
Making use of (33) and (32a), we can rewrite the constraint on the equation of state as Making use of (30) at u = 0 together with (34) and (35) we arrive at which can be substituted into (36) to find the constraint Moreover, this result can be substituted back into (37) to find a second constraint: Y 0 2 = 6c + . We have defined the experimentally determined constants The third independent constraint between the still free parameters Ω * ,0 , a, and v 0 can be found as a result of (38) and Ω X,0 = ae Y 0 / √ 2 , and is given by unless a = Ω X,0 = 0. Under this condition, v 0 can be determined from a and Ω X,0 , in which case the full solution comes from the solution to the system It must be noted that both Y 0 and Y 0 2 are functions of Ω X,0 and a through their dependence on the parameters α, β and γ from (32). Solving separately (41a) and (41b) we can obtain two solutions Ω (1) X,0 (a) and Ω (2) X,0 (a) respectively. A common solution exists if there is some a such that Ω (1) X,0 (a) = Ω (2) X,0 (a). In the case that a = Ω X,0 = 0 and Ω * ,0 = Ω m,0 , the remaining parameter v 0 cannot be determined through (40), and its values v (1) 0 and v (2) 0 will come from the solutions to (38) and (41b), respectively, expressing Y 0 and Y 0 as functions of v 0 .
In the following we will consider the matter and radiation parameters as given by the Particle Data Group, Ω b,0 h 2 0 | exp = 0.02226 (23), Ω CDM,0 h 2 0 | exp = 0.1186 (20), and Ω r,0 h 2 0 | exp = 2.473 × 10 −5 (T γ,0 /2.7255) 4 , where T γ,0 is the temperature of the relic photons [109]. The existence of solutions to (41)  For large values of a, it can be seen that Ω (2) X,0 is consistently larger than Ω (1) X,0 , in a wide region of the (h 0 ,w Y,0 ) parameter space. This can be used to study the existence of solutions.
As Ω X,0 and a go to zero simultaneously, they do it as as follows from (40). To ensure consistency with the solutions at a = Ω X,0 = 0, each function Ω (i) X,0 must have a different slope Ω (i) Using this, if v 0 , both curves must cross, guaranteeing the existence of a solution. The limiting condition v 0 , which determines the existence of a solution with a = 0, separates both regions in the (h 0 ,w Y,0 ) parameter space. In Fig. 1 we show this limiting condition together with several solutions for a = 0. The best fit value of [63], h 0 = 0.69, is indicated by a star. We note that models with Ω X,0 /Ω CDM,0 40% are in 3σ tension with current determinations of w Y,0 .
We now take the best fit solution derived in [63] as  [7] together with a compilation [110] of 38 measurements h(z) in the range 0 ≤ z ≤ 2.36 [111][112][113][114][115][116][117][118][119][120]. These 38 h(z) measurements are not completely independent. For example, the 3 measurements taken from [119] are correlated with each other, and the 3 measurements of [120] are correlated too. In addition, in these and other cases, when BAO observations are used to measure h(z), one has to apply a prior on the radius of the sound horizon, r d = ∞ z d c s (z)dz/H(z), evaluated at the drag epoch z d , shortly after recombination, when photons and baryons decouple. This prior value of r d is usually derived using CMB observations. choice justified below. We can see in Fig. 3 how the X-Y coupling depletes dark matter into dark energy, yielding a larger Ω Y,0 = 0.704 for a = 0.178 than for a = 0 where Ω Y,0 = 0.673. This is the so-called "fading dark matter" effect [63], which tends to favor larger values of h 0 when a = 0; namely, h 0 = 0.69 for a = 0.178, and h 0 = 0.66 for a = 0. The dark energy equation of state also shows striking differences. As we can see in the left panel Fig. 4, for −10 u −2. the dark energy equation of state w Y > 0, so that the energy density redshifts faster than that in ΛCDM [63]. For a = 0, however, the dark energy equation of state mimics that of a cosmological constant, w Y = −1, between −10 u −2. This translates into smaller values of w Y,0 for the decoupled system with a = 0, and closer to the ΛCDM prediction of w Λ = −1.
In the left panel of Fig. 7 we show 1, 3 and 5 σ probability contours in the (h 0 , w Y,0 ) parameter space after performing a least squares fit of the model to the Hubble parameter data. The minimum, which corresponds to a = 0, corroborates that quintessence models exacerbate the H 0 tension since the dark energy density decreases in recent times [63]. As can be seen in the right panel of Fig. 7, the best fit value of h 0 in the OAV-study is consistent with determinations of w Y,0 at < 2σ level. We conclude that the set up introduced in Sec. II has the salient cosmological features of the AOV fading dark matter proposal.

IV. HUBBLE HULLABALOO AND D-BRANE STRING COMPACTIFICATIONS
In this section we comment on additional phenomena that would influence the time evolution of the model parameters and may help solving the H 0 problem. It is common knowledge that D-brane string compactifications provide a collection of building block rules that can be used to build up the SM or something very close to it [121][122][123]. Gauge bosons of the brane stacks belong to N = 1 vector multiplets together with the corresponding gauginos. At brane intersections chiral fermions belong to chiral multiplets denoted by their left-handed fermionic components Q, L, U c , D c , E c , N c , where the superscript c stands for the charged conjugate in the familiar notation.  [130]. The rescaled posterior distribution of h 0 for the AOV fit is indicated with the dashed curve [63].The shaded areas indicate the 1σ and 2σ regions as determined by SH0ES [7]. equivalent to the seesaw mechanism to generate the Weinberg term [124] which gives rise to Majorana neutrinos. 3 Neutrino masses could then depend upon the addition of 3 Dirac right-handed neutrinos. If we now adopt the phenomenological structure of D-brane models to describe the matter fields in the visible sector, then the model parameters of the cosmological set-up introduced herein could be (in principle) affected by the right-handed neutrinos, which would contribute to the total radiation energy density. For a decoupling temperature 1 TeV, we have g * (T dec ) 106.75 and via (1) we find that the ν R contribution to the non-SM relativistic energy density, ∆N eff 0.14, is well within the existing 95% CL upper limit. On the other hand, if ν R 's decouple near the QCD phase transition, a D-brane-like description of the matter fields in our cosmological set-up can accommodate the larger value of ∆N eff derived using the helium abundance measurements of [78], while providing interesting predictions for LHC searches [127][128][129].
More concretely, in Fig. 8 we show the normalized posterior distributions of h 0 for different choices of N eff from the 7 parameter fit of [130]. It is evident that the 95% CL limit on ∆N eff from the combination of CMB, BAO, and BBN observations [10] severely constrains a solution of the H 0 problem in terms of additional relativistic degrees of freedom. Consideration of the larger helium abundance measured in [78], with ∆N eff < 0.544 at the 95% CL still precludes a full solution of the H 0 problem in terms of additional light species at the CMB epoch. However, the combined effect produced by fading dark matter and the extra relativistic degrees of freedom at the CMB epoch appears to have the potential to resolve the H 0 tension; see Fig. 8. A comprehensive study of the parameter space is beyond the scope of this paper and will be presented elsewhere. Needless to say, the helium abundance reported in [78] is in tension with Planck observations, so this solution would require a combination of two datasets which are in tension. On the one hand, the addition of extra relativistic degrees of freedom at the CMB epoch can accommodate the local calibration of SNe luminosities well out into the Hubble flow, avoiding the constraints of late times dark energy transitions [65]. On the other hand, we have noted that increasing the amount of radiation in the early universe leads to a higher value of S 8 . Solutions that could mitigate this problem have been presented in [131].
Future experiments, such as CMBPol (which is expected to reach a 2σ precision of ∆N eff = 0.09 [132]) and eventually CMB-S4 (which is expected to reach a 2σ precision of ∆N eff = 0.06 [133]) will be able to probe the contributions from Υ 's and ν R 's, providing additional constraints on the (extended) string cosmological set-up proposed in this section.

V. CONCLUSIONS
We have realized the Agrawal-Obied-Vafa swampland proposal of fading dark matter for relaxing the H 0 tension [63] by the model of Salam-Sezgin [66] and its string realization of Cvetič-Gibbons-Pope [67]. The model is fairly simple, it describes a compactification from six to four dimensions of a 6-dimensional SUGRA with a monopole background on a 2-sphere, allowing for time dependence of the 6-dimensional moduli fields while assuming a 4dimensional metric with a Robertson-Walker form. In terms of linear combinations of the S 2 moduli field and the 6-dimensional dilaton, the 4-dimensional effective potential consists of a pure exponential function of a quintessence field Y which is the 4-dimensional dilaton and the field X which determines the 4-dimensional gauge couplings of NS5-branes. This avoids direct couplings of the dilaton to matter suppressing extra forces competing with gravity. It turns out that X is a source of CDM, with a mass proportional to an exponential function of the quintessence field. The asymptotic behavior of the Hubble parameter, h ≈ ln t, leads to a conformally flat Robertson-Walker metric for large times. The dS (vacuum) potential energy density is characterized by an exponential behavior V Y ∝ e √ 2Y . Asymptotically, this represents the crossover situation with w Y = −1/3, implying expansion at constant velocity with Y varying logarithmically Y ≈ − ln t [68]. The deviation from constant velocity expansion into a brief accelerated phase encompassing the recent past (z 6) makes the model phenomenologically viable.
We have shown that this set up is well equipped to reproduce the salient features of the AOV fading dark matter proposal. For a = 0.178, the model features a tower of light states X originating in the rolling of the Y field. These X particles constitute a portion of the CDM, and the way in which their mass evolve over time demonstrates that the model may help reducing (though not fully eliminate) the H 0 tension.
As a natural outgrowth of this work, we intend to study higher dimensional SUGRAs, which also admit monopole-like solutions [134]. In some cases, however, there are no compactifications to Minkowski vacuum [135]. Of particular interest is the gauged 8-dimensional SUGRA with matter couplings [136] where a solution of the form Minkowski 6 × S 2 is known to exist. In addition, because the Salam-Sezgin model has N = (1, 0) SUSY in 6 dimensions the U (1) coupling is not fixed. In general it may be a combination of e φ and e −φ determined by chiral anomalies [137]. These may offer new possibilities for models of the type discussed in this paper. However, we remind the reader that late time dark energy transitions do not fully resolve the true source of tension between the distance ladder and high redshift observations [65] and therefore some additional assumptions (like those discussed in Sec. IV) must be adopted in order to solve the H 0 problem in the AOV-type string backgrounds.
In summary, the string cosmological framework put forward in this paper calls for new CMB observations and stimulates the investigation of complex theoretical models of the swampland as possible solutions of the H 0 problem.
With this in mind, the non-zero components of the spin connection are found to be ω i0 ı = e h (ḟ +ḣ), ω 05 5 = r cḟ , ω 06 6 = r cḟ sin ϑ, ω 56 6 = cos ϑ, where we adopted the also familiar notation of carets on the curved indices (which are lowered or raised with the spacetime metric g M N ) to distinguish them from the flat indices (that are lowered or raised with the flat Minkowski metric η AB ), so that g Substituting (49) into (45) we obtain Remarkably, the field equations fixed the monopole charged to be ±1, and lead to the condition [66] Γ 56 = ±i .
Substituting the relation g = √ ξ/2 into (58) while imposing (51) we obtain the contraint b 2 ξ = 1. Finally, the variation of δχ implies which setsφ = 0. The constraints from imposing the SUSY background can be summarized as follows: the relation (59) demands φ to be a constant and when this condition is combined with (52) we see that f must also be a constant. Because f is a constant, we can immediately see by inspection of (53), (55), and (57) that is independent of both t and the coordinates of the compact space ϑ and ϕ. Likewise, we rewrite (54) as The temporal dependence of (60) then becomes and so the scale factor for a SUSY background is found to be with κ 1 and κ 2 constants.