Observational consequences of $H_0$ tension in de Sitter Swampland

Realising de Sitter vacua in string theory is challenging. For this reason it has been conjectured that de Sitter vacua inhabit the Swampland of inconsistent low-energy effective theories coupled to gravity. Since de Sitter is an attractor for $\Lambda$CDM, the conjecture calls $\Lambda$CDM into question. Reality appears sympathetic to this idea as local measurements of the Hubble constant $H_0$ are also at odds with $\Lambda$CDM analysis of Planck data. This tension suggests that the de Sitter state is unstable, thereby implying a turning point in the Hubble parameter. We present a model relieving this tension, which predicts a turning at small positive redshift $z_*$ that is dictated by present-day matter density $\omega_m$. This feature is easily identified by homogeneous surveys covering redshifts $z \leq 0.1$. We comment on the implications for the Swampland program.

For late-time cosmology, z < 2, physics is dominated by only two parameters from the six of ΛCDM: the Hubble parameter H and the matter density Ω m .While ΛCDM leads to a Hubble parameter that is monotonically increasing with redshift, H ′ (z) > 0, model-independent fits to direct measurements of H(z) [24] provide evidence of a stationary point, H ′ (z) = 0 [25].This result dovetails with independent observations based on Baryon Acoustic Oscillations (BAO) [26,27] favouring a Hubble parameter that is constant for small z, but becomes monotonically increasing when type Ia supernovae data is included [3].
Recently, questions about ΛCDM have arisen from an unexpected quarter: string theory.In essence, string theory permits an embarrassingly rich landscape of vacua, necessitating the task of telling consistent low-energy effective theories coupled to gravity apart from inconsistent counterparts, which are deemed to inhabit the Swampland [28,29].Born out of longstanding frustrations in realising de Sitter vacua in string theory (see [30]), two Swampland criteria have been proposed [31], which have served as the catalyst for further debate [32][33][34][35][36][37][38][39][40].
The first criteria affirms that theories with large scalar field excursions in reduced Planck units M p = (8πG) −1/2 are inconsistent, while the second states that the gradient of the potential V of a canonically normalised scalar field in a consistent gravity theory must satisfy the bound, where c is a constant of order unity, c ∼ O(1).
This bound is interesting for a number of reasons.First and foremost, it precludes de Sitter vacua where ∇ φ V = 0, thus calling ΛCDM into question.Even relaxing c ≪ 1, this is still true.Secondly, ( 1) is striking as it represents a marriage of the requirements of early inflationary and late-time dark energy cosmology.In the wake of Planck, potentials are required to be flat, i. e. ∇ φ V small, whereas we also require that any cosmological constant, in other words, the potential, at late times also be small.In principle, this requires double fine-tuning, but this can be avoided if the potential and its gradient are related in the fashion above.It has been suggested [32] that quintessence models represent a natural way to satisfy this bound.
The purpose of this letter is to examine (1) in the light of H 0 tension.However, before getting so far, we notice a potential aversion to quintessence models in the data.This may be expected as an inferred bound 1 + w ≥ 0.15c 2 [32] on the equation of state w disagrees with a number of data-driven studies [21][22][23] favouring 1 + w < 0. This disagreement in turn may come as little surprise, as the latter is expected to violate the Null Energy Condition (NEC), a sacred cow for theorists [41][42][43], and in addition has the theoretically unsavory prospect of a "Big Rip" [44].However, the conflict appears to run deeper and taking data at face value, quintessence, the model favoured by (1), is difficult to reconcile with tension in H 0 .
More concretely, in this letter we present a cubic fit of a compilation of Hubble parameter measurements over redshifts up to about two [24].The resulting cubic fit provides evidence for a stationary point H ′ (z) = 0 associated with a large curvature in H(z) about z ≃ 0 ( §2).Translated into a quintessence model, the presence of a stationary point is problematic and difficult to ignore.While quintessence can easily accommodate ΛCDM and a de Sitter attractor asymptotically (z → −1), it is hard to reconcile any stationary point in the Hubble parameter at small z with widely accepted notions of matter.Therefore, if the data signaling tension in H 0 is correct, then quintessence cannot be a valid description.Ultimately, this precludes us from evaluating (1), but it may hold in other models.Even if the conjecture (1) turns out to be false in the end, the suspicion of de Sitter vacua may be well-founded and worth salvaging.
Regardless, the data point to a relatively high value of the Hubble constant H 0 = 74.4± 4.9 km s −1 Mpc −1 , essentially as an analytic continuation of these data to z = 0 consistent with the local value H 0 = 73.48± 1.66 km s −1 Mpc −1 of [2].In the latter part of the letter ( §4), working outside the framework of quintessence models, we present a simple dynamical dark energy model with only two parameters, the same as ΛCDM, which agrees remarkably well with the cubic fit and in turn the data.
Data fits and ΛCDM -In this section we review a modelindependent fit of the Hubble parameter as a function of redshift z [25].Concretely, we define the normalised Hubble parameter h(z) ≡ H(z)/H 0 .Analyticity at z = 0 permits a Taylor expansion, with coefficients expressed in terms of the values (q 0 , Q 0 ) at z = 0 of the deceleration parameter q(z) and its derivative Q(z) [47], A cubic polynomial provides a minimal setting to capture (H 0 , q 0 , Q 0 ) of an analytic cosmological evolution with vastly distinct future (−1 < z < 0) and past (z > 0).
To get oriented, we recall that late-time cosmology of ΛCDM is governed by two parameters: H 0 and ω m , the combined baryonic and cold dark matter (CDM) density today (z = 0).Assuming pressure due to matter is zero, p m = 0, one can solve the continuity equation for matter density ρ m , before substituting back into the remaining Einstein equation to find an analytic solution for h(z) , Note, in deriving this expression, we have made use of the following relation between time and redshift, thus allowing us to switch between time and the more natural parameter relevant for astronomy.
Further expanding in z one can identify h ′ (0) and (q 0 , Q 0 ) for ΛCDM: The canonical estimate ω m ≃ 0.3 gives The task now is to import existing measurements of the Hubble parameter H(z) [24] and to fit our cubic expansion to the data to provide a model-independent reference.The result can be found in Figure 1 and Table 1.
According to Table 1, the data point to an appreciable curvature Q 0 = 2.49 ± 0.55 above Q 0 = 1.00 ± 0.03 of ΛCDM at 2.7σ level of confidence.Thus, extrapolation  of H(z) data to z = 0 by analytic continuation -here by our cubic fit -gives H 0 = 74.4± 4.9 km s −1 Mpc −1 that is greater than H 0 in ΛCDM.Remarkably, the former identifies with H 0 = 73.48± 1.66 km s −1 Mpc −1 obtained from surveys of the Local Universe [2].
Implications for Quintessence -It is hopefully evident from the best-fit plot that there is an apparent stationary point where H ′ (z) = 0.In fact, extrapolating the cubic fit to small z, one finds H ′ (z) < 0, but let us be conservative and focus solely on the stationary point.Assuming H ′ (z) = 0 for a given z, we can study the implications for a quintessence model with action, where V a potential for the quintessence scalar and S m is the action for the matter sector.The ultimate stumbling block we encounter is the second Friedmann equation, It is worth noting that we have introduced the matter equation of state w through p m = wρ m .Now, it is usu-ally assumed for late-time cosmology that w = 0.This reduces the matter continuity equation to ρm + 3Hρ m = 0, which can be easily solved in terms of the scale factor up to a constant, ρ m = a −3 .Substituting this back into (11) we see that when Ḣ = 0 that ρ m = φ = 0. Of course, we could complexify φ, but this is not in the spirit of quintessence.While the latter is easy to impose, vanishing matter density can only be realised asymptotically when the scale factor blows up.Since H ′ (z) = 0 implies Ḣ = 0, this suggests that quintessence is not a valid description of cosmological Hubble data and without adopting unphysical assumptions [48] ultimately precludes us from bringing the data directly to bear on (1).Dynamical dark energy -Even if the Swampland conjecture turns out to be invalid, the idea that we should view de Sitter vacua with disdain has merit.Recalling that de Sitter is an attractor for ΛCDM, this appears to rule it out.Indeed, the small but persistent H 0 tension appears to reflect a feature of late-time cosmology, e.g., a value of Q 0 that is not accessible by ΛCDM according to Table 1.This points to a dynamical dark energy that appears to be increasingly relevant at low z.Extrapolating, it points to a future (−1 < z < 0) distinct from the de Sitter state predicted by ΛCDM.The present H 0 tension may reflect the fact that the de Sitter state is unstable, whereby ΛCDM cannot be true to all orders today (z = 0).
In a bid to find an improved model, we consider a simple dynamical dark energy model that has the attractive feature that it shares the same number of late-time cosmology parameters as ΛCDM and exhibits greater agreement with the data.In what follows, we set Newton's constant and the velocity of light equal to 1, leaving M 2 p = 1/8π.We recall that a key feature of FLRW cosmologies is the absence of asymptotically flat infinity in the face of a cosmological horizon H at the Hubble radius R H = 1/H.This apparent horizon introduces a fundamental frequency ω 0 = √ 1 − qH, where q is defined in (3), which renders the cosmological vacuum dispersive: super-horizon scale fluctuations (ω < ω 0 ) acquire imaginary wave numbers which gives rise to negative pressures p -a hallmark property of dark energy, where Λ = ω 2 0 = (1 − q)H 2 .Using this Λ as an input for the Friedmann equations, and assuming a three-flat FLRW metric, we have: In addition, we have made use of ( 5) and an expression for the scale factor in terms of redshift, a(z) = (1 + z) −1 .Next, we introduce the deceleration parameter and employ (10) to recast (11) as which we take as a definition of the "dark pressure".This has a holographic origin as explained in [45].This leaves us with a single equation to solve.When rewritten in FIG.2: Normalised cosmological evolution according to (13) and ΛCDM (4).While evolution are relatively similar in the past (z > 0), they are vastly distinct in the future (z < 0).
terms of y(z) = log h(z), we find the ODE: Quite remarkably, this ODE permits an exact solution [25], leaving us just two parameters H 0 and ω m to be determined through a fit to the data.Figure 2 illustrates this cosmological evolution alongside that of ΛCDM in the same two parameters.It highlights a common epoch of matter dominated evolution (q = 1/2, total pressure zero) yet distinct behavior towards the present and future.By (12), the associated dark matter satisfies w + 1 = 2(1 + q)/(2 + q) > 0 as long as q > −1, i.e., away from the turning point in (14) at z ≃ 1/3 − ω m .In the presence of a small baryonic mass density ρ b , ρ = ρ M + ρ b , the dark matter density ρ M satisfies w M = p M /ρ M ≃ q/(2 + q) − ρ b /ρ M , permitting w M < −1 on ρ M alone, even when q > −1.
Applying the same analysis as the cubic fit above, we find H 0 = 74.9± 2.6 km s −1 Mpc −1 (15) with ω m = 0.2719 ± 0.028 and Q 0 = 2.75 ± 0.15.This result agrees with both our cubic fit and H 0 = 73.48± 1.66 km s −1 Mpc −1 from surveys of the local Universe [2].The NEC T ab ν a ν b ≥ 0 for all null-vectors ν b imposes the condition ρ + p ≥ 0, where p is the dark pressure, and hence [46] H ′ (z) ≥ 0. ( We see that monotonically increasing H(z) is consistent with the NEC, but when it is decreasing across a turning point near z = 0, NEC is violated, in line with expectations.This corresponds to w < −1, which places it at odds with the bound 1 + w ≥ 0.15c 2 [32].The fact that the NEC is violated tells us that we cannot expect to embed this model in a quintessence model.Indeed, attempts to do so lead to the scalar φ becoming complex before the turning point in the Hubble parameter is reached.This is in line with expectations.
Outlook -Cosmology is currently a data-driven subject.Attempts to better fit data by introducing extra parameters, such as a dynamical equation of state w, lead to models that are deeply challenging for theory.It is in fact telling that the currently favoured model ΛCDM saturates the NEC w = −1, so any dip below this value is a concern.On the flip side, theorists may attempt to make progress starting from EFT and ruling out inconsistent low-energy effective theories coupled to gravity.This is the philosophy of the Swampland program.The current tension in H 0 , assuming it persists, constitutes an affront to quintessence and potentially too the Swampland criteria.That being said, there appears to be a silver lining in that both the Swampland conjecture and existing data point to a future without de Sitter vacua.
and work with quintessence as an effective description in the vicinity of the stationary point.Linearising the equations around this point, we find that consistent fluctuations can only be defined if V ′′ < 0 at the point.This precludes exponential potentials, cf.[32] and under the assumption that V ′′ > 0 at large φ implies the existence of a point in time where V ′ = 0.

3 FIG. 1 :
FIG.1:(Top panels.)Fits to a heterogeneous data set of Hubble parameters H(z) over 0.07 < z < 1.965 by a cubic expansion (2) and ΛCDM (4), the latter by nonlinear model regression.This distinct behavior at z close to zero is apparent in curvature expressed by Q(z) with Q0 > 2.5 from the data versus Q0 1, respectively.By this large Q0, the cubic expansion provides an analytic extrapolation to the relatively high value H0 obtained from surveys of the Local Universe in tension with H0 obtained from ΛCDM.(Bottom panel.)More broadly, this tension of ΛCDM extends over an extended redshift, apparent in the qQ-diagram derived from the same data.