DDimensional Reductionand (Anti) de Sitter Bounds
Tom Rudelius ∗ Physics Department, University of California, Berkeley CA 94720 USA
Abstract
Dimensional reduction has proven to be a surprisingly powerful tool for delineating theboundary between the string landscape and the swampland. Bounds from the Weak GravityConjecture and the Repulsive Force Conjecture, for instance, are exactly preserved underdimensional reduction. Motivated by its success in these cases, we apply a similar dimen-sional reduction analysis to bounds on the gradient of the scalar field potential V and themass scale m of a tower of light particles in terms of the cosmological constant Λ, whichideally may pin down ambiguous O (1) constants appearing in the de Sitter Conjecture andthe (Anti) de Sitter Distance Conjecture, respectively. We find that this analysis distin-guishes the bounds |∇ V | /V ≥ (cid:112) / ( d − m (cid:46) | Λ | / , and m (cid:46) | Λ | /d in d -dimensionalPlanck units. The first of these bounds precludes accelerated expansion of the universe inEinstein-dilaton gravity and is almost certainly violated in our universe, though it may ap-ply in asymptotic limits of scalar field space. The second bound cannot be satisfied in ouruniverse, though it is saturated in supersymmetric AdS vacua with well-understood upliftsto 10d/11d supergravity. The third bound likely has a limited range of validity in quantumgravity as well, so it may or may not apply to our universe. However, if it does apply, it sug-gests a possible relation between the cosmological constant and the neutrino mass, which (bythe see-saw mechanism) may further provide a relation between the cosmological constantproblem and the hierarchy problem. We also work out the conditions for eternal inflation ingeneral spacetime dimensions, and we comment on the behavior of these conditions underdimensional reduction.January 29, 2021 ∗ e-mail: [email protected] a r X i v : . [ h e p - t h ] J a n ontents The search for universal features of quantum gravity–also known as the swampland program[1, 2]–has seen a resurgence in recent years. Strong evidence has been given in favor ofcertain conjectured properties of quantum gravities (so-called “swampland conjectures”),some longstanding conjectures have been discarded as counterexamples have emerged, andmany seemingly-unrelated aspects of physics and mathematics have been connected throughan ever-growing swampland web.But the swampland program also faces some serious difficulties. First and foremost is theinfamous swampland tradeoff: conjectures that tend to have more evidence in their favor tendto have less interesting applications to cosmology and phenomenology, whereas conjecturesthat have significant implications for observable physics tend to be more speculative. Relatedto this is a lack of precision: many swampland statements are plagued by squiggles ( ∼ , (cid:46) , (cid:38) ) and O (1) constants to be determined later. These squiggles make the conjectures difficultto test, and they suggest that we may not yet understand the underlying physics responsiblefor such conjectures. Without this, it is difficult to say when exactly the conjectures may beexpected to hold, and when they may cease to be valid.The goal of this paper is to sharpen some of these more speculative swampland conjec-tures, primarily through a mechanism that has successfully sharpened swampland conjecturesin the past: dimensional reduction. As we will see in Section 2 below, O (1) factors in theWeak Gravity Conjecture (WGC) bound and the Repulsive Force Conjecture (RFC) boundcan be fixed precisely by demanding that these bounds should be exactly preserved underdimensional reduction from D to d = D − d di-mensions if and only if it is saturated in D dimensions. In these cases, dimensional reductionsucceeds in selecting bounds that are both physically meaningful and seemingly-universal:all known quantum gravities satisfy both the WGC and the RFC, and compelling evidencehas been provided that these conjectures should hold more generally.1f course, this does not prove that any bound that is preserved under dimensional reduc-tion is necessarily a universal constraint on quantum gravity. But the success of dimensionalreduction in the above cases suggests that it may serve as a useful diagnostic for distinguish-ing certain O (1) values in conjectured bounds when previous arguments have left us withambiguity.Relatedly, this paper will not address the ultimate question of whether certain proposedconjectures are universally true in quantum gravity. It seems very likely that many of theseconjectures are true within a limited domain of validity. For instance, the relationshipbetween the masses of a tower of light particles and the cosmological constant proposed inthe AdS Distance Conjecture [3] is likely true in infinite families of quantum gravities, eachmember of which is distinguished by some discrete flux parameter. The lower bound on thegradient of a scalar field potential proposed in the de Sitter Conjecture [4] is likely true inasymptotic regions of scalar field space, and relatedly eternal inflation is likely forbidden insuch regions [5]. The bounds we will obtain in this paper do not necessarily apply to allquantum gravities–in fact, some of them are violated even in our own universe–but they maypoint us to interesting, universal behavior in certain limits of quantum gravity. At the least,they give us a mark to aim for and to explore further.The main results and structure of our paper is as follows: in Section 2, we will review howdimensional reduction distinguishes the WGC and RFC bounds, fixing their O (1) factors.In Section 3, we will apply a similar dimensional reduction analysis to bounds on themasses of light particles in terms of the cosmological constant of the form m (cid:46) | Λ d | α d M − dα d d , (1.1)where Λ d and M d are the cosmological constant and the Planck mass in d dimensions,respectively. In [3], it was conjectured that a tower of light particles should satisfy thisbound in any AdS (and perhaps dS) vacuum. We will see that dimensional reduction picksout α d = 1 / α d = 1 /d as special values of α d , depending on whether or not weinclude the contribution from 1-loop Casimir effects. The former value is saturated in knownsupersymmetric compactifications [3] and was previously singled out in [6], whereas the latteris interesting from the perspective of the standard model, since neutrinos have a mass ofroughly m ν ∼ Λ / . If this mass comes from the see-saw mechanism, m ν ∼ y v / Λ UV , where y is the Yukawa coupling, v is the Higgs vev, and Λ UV is some UV scale, this bound furtherbecomes v (cid:46) Λ / Λ UV /y , (1.2)which bounds the Higgs vev in terms of the cosmological constant and therefore relates thehierarchy problem to the cosmological constant problem. In particular, if Λ UV ∼ M GUT , y ∼ .
1, then (1.2) becomes v (cid:46)
100 GeV, which is not far from the measured value v = 246GeV. (Note that we have swept a number of O (1) factors here under the rug.)In Section 4, we will derive constraints for eternal inflation in d ≥ d = 4 resultof [5]. We will find that eternal stochastic inflation occurs only if the following conditionsare satisfied on the first and second derivatives of the potential: |∇ V | V ( d +2) / < d − π Ω d − (cid:18) d − d − M d − d (cid:19) ( d +2) / , (1.3) V (cid:48)(cid:48) V > − d − d − M d − d , (1.4)where Ω d − = 2 π ( d − / / Γ( d − ). In multifield landscapes, V (cid:48)(cid:48) above should be replaced by asum over the negative eigenvalues of the Hessian of the potential.In Section 5, we will consider bounds of the form |∇ V d | V γ d d ≥ c d M d (1 − γ d ) − ( d − / d . (1.5)Preservation under dimensional reduction distinguishes γ d = 1 as a special value of γ d ,in agreement with the de Sitter Conjecture (dSC) bound of [4] and the Trans-PlanckianCensorship Conjecture (TCC) bound. It further selects c d = (cid:112) / ( d −
2) as a special valueof c d , in which case (1.5) corresponds exactly to the condition required to avoid acceleratedexpansion of the universe. Since this particular value of c d is larger than the coefficient c d = 2 / (cid:112) ( d − d −
2) that appears in the TCC bound, we learn that a theory satisfyingthe TCC bound will still satisfy the TCC bound after dimensional reduction. Likewise, sinceour value γ d = 1 is smaller than the value γ d = ( d + 2) / In this section, we review how the Weak Gravity Conjecture (WGC) [7] bound and theRepulsive Force Conjecture (RFC) [8] bounds are preserved under dimensional reduction.Our starting point is an Einstein-Maxwell-dilaton action for a P -form gauge field A µ ...µ P in D = d + 1 dimensions: S = 12 κ D (cid:90) d D x √− g (cid:18) R D −
12 ( ∇ φ ) (cid:19) − e P ; D (cid:90) d D x √− g e − α P ; D φ F P +1 . (2.1)3ere, F P +1 = d A P is the field strength for P -form gauge field, and F q := 1 q ! F µ ...µ q F µ ...µ q . (2.2)Note that the action (2.1) is rather special and does not describe the low-energy theory of ageneric quantum gravity, but it will suffice for our purposes to restrict our attention to thiscase.We define 1 κ D := M D − D , (2.3)where M D is the reduced Planck mass in D dimensions. With this convention, the WGCbound for a ( P − q and tension T P is given by: e P ; D q M D − D ≥ (cid:20) α P ; D P ( D − P − D − (cid:21) T P := γ − P ; D ( α ) T P . (2.4)The RFC bound in this theory is given by e P ; D q M D − D ≥ ∂ φ T P ) + P ( D − P − D − T P , (2.5)where ∂ φ T P := ∂T P /∂φ is the partial derivative of the brane tension with respect to thedilaton φ , holding the Planck scale fixed. A ( P − P − s = e λ ( x ) d − dˆ s ( x ) + e − λ ( x ) d y , (2.6)where y ∼ y + 2 πR . This ansatz is chosen so that the dimensionally reduced action isin Einstein frame, i.e., there is no kinetic mixing between λ and the d -dimensional metric.For simplicity, we do not include a Kaluza-Klein photon, but we do include a masslessradion λ ( x ), which controls the radius of the circle. Under such a dimensional reduction, theEinstein-Hilbert term reduces as12 κ D (cid:90) d D x √− g R D → κ d (cid:90) d d x (cid:112) − ˆ g R d − (cid:90) d d x (cid:112) − ˆ g G λλ ( ∇ λ ) , (2.7)4here 1 κ d = M d − d = (2 πR ) M D − D , (2.8) G ( d ) λλ = ( d − κ d ( d −
2) = M d − d d − d − . (2.9)The P -form in D dimensions gives both a P -form and a p := P − d dimensions.The former comes from taking all of the legs of the P -form to lie along noncompact directions,while the latter comes from taking one of the legs of the P -form to wrap the compact circle.Similarly, a ( P − P -form descends to both a ( P − p − P -form gauge field in d dimensions. The associated gauge coupling isgiven by e P ; d = 12 πR e P ; D e Pλd − . (2.10)The tension of a ( P − T ( d ) P = e Pλ d − T ( D ) P . (2.11)Upon reduction, the radion λ and dilaton φ each couple exponentially to the Maxwell termin the action, as in (2.1). We may then redefine the dilaton as the linear combination of λ and φ that couples to the Maxwell term. This effectively shifts the coupling α appearing inthe WGC bound to α P ; d = α P ; D + 2 P ( d − d − . (2.12)Plugging this into the WGC bound (2.4) with D → d , we find e P ; d q M d − d ≥ (cid:20) α P ; d P ( d − P − d − (cid:21) T P = (cid:20) α P ; D P ( D − P − D − (cid:21) T P . (2.13)The right-hand side of this bound matches that of (2.4). This shows that the WGC boundhas been exactly preserved by the dimensional reduction process. An analogous result holdsfor the case of decreasing P → P − P = 1 provided weinclude a Kaluza-Klein photon in the dimensional reduction ansatz of (2.6) [9]: in all suchcases, the WGC bound is exactly preserved under reduction.The RFC bound in d dimensions is then given by e P ; d q ≥ πR ( ∂ φ T ( d ) P ) + ( G ( d ) ) λλ ( ∂ λ T ( d ) P ) − P ( d − P − d − (cid:0) T ( d ) P (cid:1) M d − d . (2.14)5sing (2.9) and (2.11), we have( G ( d ) ) λλ ( ∂ λ T ( d ) P ) = 4( d − M d − d ( d − (cid:18) P d − (cid:19) (cid:0) T ( d ) P (cid:1) = P ( d − d − (cid:0) T ( d ) P (cid:1) M d − d . (2.15)This combines with the last term in (2.14) to give (cid:20) P ( d − d −
2) + P ( d − P − d − (cid:21) (cid:0) T ( d ) P (cid:1) M d − d = P ( D − P − D − (cid:0) T ( d ) P (cid:1) M d − d . (2.16)With this, we see that the RFC bound in d dimensions (2.14) is exactly the same as theRFC bound in D dimensions (2.5). This result may be generalized to multiple scalar fields,multiple gauge fields, and general scalar field couplings. It holds also for the case of P reduced to p = P − P = 1 [10]: in all cases, the RFCbound is exactly preserved under dimensional reduction.Within the string landscape, there are no known counterexamples to either the WGC orthe RFC. Many examples have been confirmed to satisfy these conjectures [7, 10–18], and anumber of arguments suggest that they should hold more generally [19–24].The takeaway lesson here is that preservation under dimensional reduction can be a use-ful tool for identifying universal, physically meaningful behavior of quantum gravities. Thevalue of γ P ; D in (2.4) is distinguished in that it dictates the extremality bound for chargedblack branes in the theory of (2.1), and it is also distinguished in that it is preserved underdimensional reduction. Similarly, the bound (2.5) is distinguished by the fact that it dictatesthe self-repulsiveness of a brane at long distances, and simultaneously it is distinguished inthat it is preserved under dimensional reduction. Both of these bounds are further dis-tinguished by the fact that they seem to be satisfied in all quantum gravity theories. Wesee that, at least in these two cases, dimensional reduction picks out universal, physicallymeaningful constraints on quantum gravities. Suppose that a family of AdS or dS quantum gravities contain a tower of light particleswhose masses satisfy m n (cid:46) n | Λ D | α D M − Dα D D , n ∈ Z > . (3.1)Such towers of light particles occur, for instance, in all known AdS vacua of string theory,typically arising as Kaluza-Klein modes of some compactified dimensions. Such towers werediscussed in detail in [3], where it was further conjectured that there exists a universal, O (1)value of α D that is obeyed in all (A)dS vacua of string theory. This conjecture was calledthe (A)dS Distance Conjecture ((A)dSDC).We now apply a dimensional reduction analysis to the bound (3.1). Our starting point6s the D -dimensional action: S = (cid:90) d D x √− g (cid:34) M D − D R D − Λ D − (cid:88) n (cid:0) ( ∇ φ n ) + m n φ n (cid:1)(cid:35) , (3.2)where φ n represents the n th particle in the tower. We have written the action as if theseparticles are scalar fields, but this assumption is not necessary.We now perform a dimensional reduction to d = D − s = e λ ( x ) d − dˆ s ( x ) + e − λ ( x ) d y , (3.3)The resulting action in d dimensions takes effectively the same form as (3.2): S = (cid:90) d d x (cid:112) − ˆ g (cid:34) M d − d (cid:18) R d − d − d −
2) ( ∇ λ ) (cid:19) − Λ d − (cid:88) n (cid:0) ( ∇ φ n ) + ( m ( d ) n ) φ n (cid:1)(cid:35) , (3.4)At a classical level, the parameters are related by M d − d = (2 πR ) M D − D , Λ d = (2 πR )Λ D e λd − , m ( d ) n = m ( D ) n e λ d − . (3.5)There will also be Kaluza-Klein modes, which we have ignored. For simplicity, we assumehere that (cid:104) λ (cid:105) = 0: we can ensure this by shifting λ → λ − (cid:104) λ (cid:105) , if necessary.From this, we see that if the masses m ( D ) n satisfy (3.1) in D dimensions, then the masses m ( d ) n will satisfy m n (cid:46) n | Λ d | α d M − α d d , n ∈ Z > , (3.6)provided | Λ d | α d M − dα d d ≥ | Λ D | α D M − Dα D D = | Λ d | α D M d − d − (1 − Dα D ) d (2 πR ) D − (2 α D − . (3.7)From this, we see that the value α d = 1 / R dependence cancels, and the bound (3.1) is exactly preservedunder dimensional reduction. As pointed out in [3], the value α d = 1 / α d = 1 / α d was also singled out it [6], where it was connected to the dSC, the TCC, and the SwamplandDistance Conjecture (SDC) [2]. Once again, dimensional reduction has distinguished a boundof physical interest.It is worth pointing out that this bound cannot be satisfied in our own universe, however.A tower of particles of this mass would run in loops, correct the graviton propagator, and7ead gravity to become strongly coupled at the “species bound” scale E QG of order E QG ∼ M / (cid:113) N ( E QG ) , N ( E QG ) ∼ E QG M / Λ / . (3.8)This in turn implies E QG ∼ Λ / M / ∼ .
02 GeV , (3.9)which conflicts with the experimental fact that gravity remains weakly coupled at energiesaccessible at the LHC.Our above analysis takes place entirely at a classical level. As shown in (3.5), the cosmo-logical constant Λ d acquires an exponential dependence on the radion λ . If the radion is notstabilized by quantum effects, the dimensionally reduced theory does not have a vacuum.Suppose that we now insist that the theory should in fact have a vacuum, so the radionmust be stabilized. This can be done most simply by including the 1-loop Casimir energycontributions to the radion potential from the light particles. Upon dimensional reductionto d dimensions, the contribution from a particle of mass m takes a very simple form in themassless limit ( mR (cid:28)
1) [25]: V C ( λ ) = ∓ πR ) d Ω d ζ ( d + 1)e d ( d − d − λ , Ω d = 2 π ( d +1) / Γ( d +12 ) . (3.10)Here, ζ ( x ) is the Riemann zeta function, Ω d is the volume of the unit d -sphere, and the +sign is for bosons or fermions with antiperiodic boundary conditions, while the − sign isfor fermions with periodic boundary conditions. The 1-loop Casimir energy for a particlemuch heavier than 1 /R is exponentially suppressed as e − πmR , so for our purposes, it suf-fices to consider contributions only from light particles with m (cid:46) /R . For simplicity, weapproximate all such particles to be massless.The full potential at 1-loop, including both the classical contribution and the 1-loopCasimir energy from light particles, is then given by V ( λ ) = V Λ ( λ ) + V C ( λ ) = (2 πR )Λ D e λd − − (cid:88) n | m n < /R ( − ) F n πR ) d Ω d ζ ( d + 1)e d ( d − d − λ . (3.11)We differentiate with respect to λ and set this to 0 to find a critical point:0 = ∂ λ V ( λ ) = 1 d − πR )Λ D e λd − − d ( d − d − (cid:88) n | m n < /R ( − ) F n πR ) d Ω d ζ ( d + 1)e d ( d − d − λ . (3.12)The exponent of the second term is larger than that of the first term. This means that inorder to find a minimum, the second term must be positive, while the first is negative. ForΛ D <
0, we may therefore obtain a minimum in d dimensions provided that fermions withperiodic boundary conditions dominate the Casimir energy. For Λ D >
0, on the other hand,8e may find a maximum in d dimensions provided that bosons or fermions with antiperiodicboundary conditions dominate the Casimir energy. Creating a de Sitter minimum in thisway would require a delicate interplay between bosons and fermions, and it would requireus to go beyond the massless particle approximation we have employed here.Let us now suppose that we have a tower of particles of equal spin F , with masses m n = nm , n ∈ Z > , (3.13)so that the sum in (3.12) becomes (cid:88) n | m n < /R ( − ) F n = / ( mR ) (cid:88) n =1 ( − ) F = ( − ) F mR . (3.14)In order for the two terms in (3.12) to balance, we must therefore have (ignoring O (1)factors): 1 mR D ∼ | Λ d | . (3.15)We further impose that the bound (3.1) must be saturated both before and after reduction: m ∼ | Λ D | α D M − α D D ∼ | Λ d | α d M − α d d . (3.16)Putting these together, we may eliminate R and m to find a relation between Λ D and Λ d ,which gives the following relation between α D and α d : α D = α d − α d + α d − α d − d + 3 . (3.17)This is a recursive relation between α d and α d +1 , which has a 1-parameter family of solutions.One such solution is the value α d = 1 /
2, which we found above from a strictly classicalanalysis. Another solution is α d = 1 /d : this value is special in that it is the smallest possiblevalue of α d consistent with the recursion relation (3.17) that remains non-negative in thelimit d → ∞ . Note that it is also the unique value of α d for which the Planck scale M d dropsout of (3.1), and in the reduction studied above it leads to m ∼ /R ∼ | Λ d | /d ∼ | Λ D | /D ,so the sum over n in (3.14) runs over an O (1) number of terms.The bound m (cid:46) | Λ d | /d is rather tantalizing, as the current upper bound on the sum of themasses of the three standard model neutrinos is 0 . .
05 eV. The cosmological constant in our universeis measured to be Λ / (cid:39) − M (cid:39) .
002 eV , (3.18)9hich is quite close to the neutrino mass scale of roughly 0 . − . d <
0, whereas our universe has Λ d >
0, for which our simpleanalysis yields maxima of the potential. This could be remedied by moving beyond themassless limit and considering both fermions and bosons (or fermions with antiperiodicboundary conditions). Indeed, balancing the 1-loop Casimir energies of particles in thestandard model leads to a vacuum in three dimensions upon dimensional reduction [25].
We now take a short break from our dimensional reduction analysis to work out the conditionsfor eternal inflation in general spacetime dimension d by solving the Fokker-Planck equation,generalizing the computations of [5] (see also [29–39]). In Section 5 below, we will commentbriefly on the behavior of the conditions for avoiding eternal inflation under dimensionalreduction.We begin from the d -dimensional metric ds = − dt + a ( t ) d(cid:126)x . (4.1)and the action S = (cid:90) d d x √− g (cid:20) M d − d R −
12 ( ∇ φ ) − V ( φ ) (cid:21) . (4.2)For simplicity, we take φ to be canonically normalized, and at 0th order we take it to behomogenous so that its spatial derivatives vanish. The equation of motion for the scalar fieldis thus ¨ φ + ( d − H ˙ φ = − V (cid:48) ( φ ) , (4.3)where H = ˙ a/a . The stress-energy tensor is given by T µν = − g µδ √− g δSδg δν = diag (cid:18) −
12 ˙ φ − V ( φ ) ,
12 ˙ φ − V ( φ ) , ...,
12 ˙ φ − V ( φ ) (cid:19) . (4.4)This is the stress-energy tensor of a perfect fluid of density ρ , pressure p , with ρ = 12 ˙ φ + V ( φ ) , p = 12 ˙ φ − V ( φ ) . (4.5) We are very thankful to Liam McAllister for discussions on the computations in this section. G µν = κ d T µν = 1 M d − d T µν . (4.6)Here, we have G = 12 ( d − d − H = ρM d − d , (4.7)1 a G = ... = 1 a G d − ,d − = − ( d −
2) ¨ aa −
12 ( d − d − H = pM d − d . (4.8)This gives the Friedmann equations: H = 2 ρ ( d − d − M d − d , ¨ aa = − p ( d − M d − d − ( d − H . (4.9)Inflation takes place in the slow-roll regime,˙ φ (cid:28) V ( φ ) , | ¨ φ | (cid:28) H | ˙ φ | , | V (cid:48) ( φ ) | , (4.10)in which case the equation of motion for φ and the first Friedmann equation become( d − H ˙ φ = − V (cid:48) ( φ ) , H = 2 V ( φ )( d − d − M d − d . (4.11)To study eternal inflation, we further need to incorporate backreaction from quantumfluctuations of the scalar field. The scalar 2-point function in dS d takes the form [40] (cid:104) φ (cid:105) = H d − π Ω d − t + . . . , Ω d − = 2 π ( d − / Γ( d − ) . (4.12)where . . . represents terms that are not linear in t . This linear term encodes the effectof Gaussian quantum fluctuations that exit the horizon, decohere, and backreact on theclassical slow-roll equation of motion (4.11). This backreaction takes the form( d − H ˙ φ + V (cid:48) ( φ ) = N ( t ) , (4.13)where N ( t ) is a Gaussian noise term, which induces a random walk of the field φ in thepotential V ( φ ). In other words, in an infinitesimal time δt , φ will vary according to δφ = − d − H V (cid:48) ( φ ) δt + δφ q ( δt ) , δφ q ( δt ) ∼ N (0 , H d − π Ω d − δt ) , (4.14)where the variance of the normal distribution comes from the coefficient of the linear term11n (4.12).If we further approximate H as a constant, which is well-justified in the slow-roll regime,the evolution of the probability distribution P [ φ, t ] of φ as a function of t is then describedby a Fokker-Planck equation [29–31]:˙ P [ φ, t ] = 12 (cid:18) H d − π Ω d − (cid:19) ∂ φ P [ φ, t ] + 1( d − H ∂ φ (cid:16) ( ∂ φ V ( φ )) P [ φ, t ] (cid:17) . (4.15)This equation is difficult to solve for a general potential V ( φ ), but the solution takes a simple,analytic form when the potential is linear or quadratic in φ . In both cases, the solution takesthe form of a Gaussian distribution: P [ φ, t ] = 1 σ ( t ) √ π exp (cid:20) − ( φ − µ ( t )) σ ( t ) (cid:21) . (4.16)For a linear potential V = V − αφ , the parameters µ ( t ), σ ( t ) are given by µ ( t ) = α ( d − H t , σ ( t ) = H d − π Ω d − t . (4.17)For a quadratic hilltop potential V = V − m φ , the parameters are instead given by µ ( t ) = 0 , σ ( t ) = ( d − H d π Ω d − m (cid:20) − (cid:18) m ( d − H t (cid:19)(cid:21) . (4.18)For the linear potential, inflation occurs if φ < φ c , where φ c is some critical value whoseprecise value will be unimportant to us. The probability that φ < φ c at time t is given byPr[ φ > φ c , t ] = (cid:90) φ c −∞ dφ P [ φ, t ] , (4.19)where P [ φ, t ] is the probability density function for a Gaussian distribution, given in (4.16),with mean µ ( t ) and variance σ ( t ) given by (4.17). The result isPr[ φ > φ c , t ] = 12 erfc (cid:20) µ ( t ) − φ c σ ( t ) √ (cid:21) = 12 erfc α ( d − H t − φ c (cid:16) H d − tπ Ω d − (cid:17) / , (4.20)with erfc the error function. For large t , this error function can be approximated to leadingorder as an exponential,Pr[ φ > φ c , t ] ∼ exp − α ( d − H t − φ c (cid:16) H d − tπ Ω d − (cid:17) / ∼ exp (cid:20) − π Ω d − α d − H d +1 t (cid:21) , (4.21)12his means that the probability of inflation occurring at time t for a fixed comoving observerdecays exponentially with time. On the other hand, the volume of the inflating region growsexponential in time as exp(( d − Ht ). Eternal inflation occurs if this exponential growthbeats the exponential decay, i.e., if( d − H > π Ω d − α d − H d +1 . (4.22)Substituting V for H using (4.11) and setting α = ∇ V ( φ ), this becomes |∇ V | V ( d +2) / < d − π Ω d − (cid:18) d − d − M d − d (cid:19) ( d +2) / . (4.23)A similar analysis applies to the quadratic hilltop potential. Now, inflation occurs if | φ | < φ c , and the probability of this occurring at time t is given byPr[ | φ | < φ c , t ] = (cid:90) φ c − φ c dφ P [ φ, t ] , (4.24)The probability density function is a Gaussian with mean and variance given by (4.18). Thisgives Pr[ | φ | < φ c , t ] = erf (cid:20) φ c − µ ( t ) σ ( t ) √ (cid:21) = erf π Ω d − m φ c ( d − H d (cid:16) − (cid:16) m ( d − H t (cid:17)(cid:17) / ∼ exp (cid:20) − m ( d − H t (cid:21) , (4.25)where in the last line we have Taylor-expanded the error function at the origin. As before,eternal inflation occurs if the exponential expansion of the universe dominates the exponentialdecay of (4.25), which is equivalent to the condition( d − H > m ( d − H . (4.26)Substituting V for H using (4.11) and setting m = − V (cid:48)(cid:48) , this becomes V (cid:48)(cid:48) V > − d − d − M d − d . (4.27)As in [5], this analysis may be generalized straightforwardly to theories with multiplescalar fields: as long as the potential V ( φ i ) separates into a sum (cid:80) i V i ( φ i ), where each V i is linear or quadratic and depends only on φ i , the solution to the Fokker-Planck equation13ill separate into a product of Gaussian wavepackets. The main upshot of this is that, at acritical point of a multifield potential, one should replace V (cid:48)(cid:48) in (4.27) with a sum over thenegative eigenvalues of the Hessian.To conclude this section, let us compare the bound (4.27) to the analogous bound in theRefined de Sitter Conjecture (RdSC) [41–43]:min( ∇ i ∇ j V ) V ≤ − c (cid:48) d M d − d , (4.28)where c (cid:48) d is some O (1) constant, and min( ∇ i ∇ j V ) is the minimum eigenvalue of the Hessian.We see here that the RdSC is incompatible with eternal inflation provided c (cid:48) d > d − / ( d − c (cid:48) d seems to be violated in some 4d examples with10d supergravity uplifts [44,45], but these solutions do not lie in the classical regime of stringtheory, so they may be modified by stringy effects [46]. Furthermore, even if the RdSC boundwith this value of c (cid:48) d is violated for the smallest eigenvalue of the Hessian, one might stillviolate the necessary conditions for eternal inflation upon summing over the other negativeeigenvalues of the Hessian. Next, we consider a bound on scalar field potentials of the form |∇ V D | V γ D D ≥ c D M D (1 − γ D ) − ( D − / D , (5.1)for V D >
0, where γ D and c D are O (1) constants. Here, |∇ V D | = G ij ∂ i V D ∂ j V D , and theaction takes the form S = (cid:90) d D x √− g (cid:20) M D − D R D − G ij ( φ ) ∇ φ i ∇ φ j − V D ( φ ) (cid:21) . (5.2)We want to understand how this behaves under dimensional reduction. We take our usualdimensional reduction ansatz: ds = e λ/ ( d − d ˆ s + e − λ dy . (5.3)The resulting action in d dimensions is given by S = (cid:90) d d x (cid:112) − ˆ g (cid:20) M d − d (cid:18) R d − d − d −
2) ( ∇ λ ) (cid:19) − G ( d ) ij ( φ ) ∇ φ i ∇ φ j − V d ( φ ) (cid:21) , (5.4)14here we have M d − d = (2 πR ) M D − D , G ( d ) ij = (2 πR ) G ( D ) ij , V d = V D (2 πR ) e λ/ ( d − . (5.5)There is also a contribution to V d from the Casimir energy, but for large R , this will beparametrically subdominant to the classical term, and we will neglect it here.Thus we have |∇ V d | = ( G ( d ) ) ij ∂ i V d ∂ j V d + 4( d − d − M d − d ( ∂ λ V d ) , = (2 πR )( G ( D ) ) ij ∂ i V D ∂ j V D + 4( d − d −
2) 2 πRM D − D V D , (5.6)= (2 πR ) |∇ V D | + 4( d − d −
2) 2 πRM D − D V D , where we have set (cid:104) λ (cid:105) = 0 after taking the λ derivative. This yields |∇ V d | V γ d d M d − − d (1 − γ d ) d = (2 πR ) d − ( γ d − M d − d − (2 dγ d − d − D |∇ V D | + d − d − M − DD V D V γ d D . (5.7)From this, we see that the value γ D = γ d = 1 is distinguished by dimensional reduction: forthis particular value, the R -dependence cancels, and if we ignore the λ dependence of V d ,the bound (5.1) is exactly preserved under dimensional reduction. With γ D = 1, this boundmatches the “de Sitter Conjecture” bound of [4].Setting γ D = γ d = 1, we may further fix the constant c D by including the λ dependenceof V d . In particular, assuming that the bound (5.1) is exactly preserved under dimensionalreduction, so that V D saturates the bound in d dimensions precisely when V d saturates thebound in d dimensions, we have c d = |∇ V d | V d M d − d = c D V D + d − d − V D V D , (5.8)where in the first equality we have assumed that the bound is saturated d dimensions, andin the second we have used (5.6) and assumed that it is saturated in D dimensions. Thisgives a recursive relation for c D : c d = c D + 4( d − d − , (5.9)which is solved by c d = β + 4 d − , (5.10)15here β is a free parameter. This leads to the bound |∇ V d | V d ≥ (cid:114) β + 4 d − M ( d − / d . (5.11)Our dimensional reduction argument leaves β unfixed, but the value β = 0 is distinguishedfor two reasons: first of all, it is the smallest value of β such that c d remains non-negativein the limit d → ∞ . Secondly, the bound with β = 0, |∇ V d | V d ≥ (cid:114) d − M ( d − / d , (5.12)has physical meaning: it is exactly the condition required for a theory of Einstein gravitycoupled to a dilaton field to avoid accelerated expansion of the universe in d dimensions [4].Observational constraints rule out single-field quintessence models satisfying (5.12) [47].Even if one entertains the possibility of fine-tuned, multi-field quintessence models [48–50], itis very hard to imagine that (5.12) could be satisfied at the maximum of the Higgs potential[51]. However, this bound may yet be satisfied in asymptotic limits of scalar field space, asis evidenced by many examples in string theory. For instance, toroidal compactifications of O (16) × O (16) heterotic string theory to d dimensions satisfy the bound [4]: |∇ V d | V d ≥ min (cid:32) (cid:114) d − d − , √ (cid:112) (10 − d )( d − (cid:33) M ( d − / d . (5.13)This bound implies (5.12), and it implies (5.11) in d ≥ β ≤ / V ( φ ) ∼ exp (cid:18) −√ φM (cid:19) , (5.14)and the LVS potential behaves asymptotically as V ( φ ) ∼ exp (cid:32) − (cid:114) φM (cid:33) . (5.15)Thus, in asymptotic limits of field space, both of these satisfy (5.12), and more generallythe former satisfies (5.11) provided β ≤
4, whereas the latter requires β ≤ /
2. Moreexamples of 4d theories satisfying (5.12) in asymptotic regions of scalar field space can befound in [4, 6, 45, 55, 56].However, some of the string compactifications considered in [55, 56, 6] naively seem to16iolate (5.12). For instance, in Type IIA compactifications in the presence of O4-planes butno D4-brane sources, [55] derived a lower bound on |∇ V | M /V of (cid:112) /
3, which if saturatedwould violate (5.12) while saturating the Transplackian Censorship Conjecture (TCC) bound[54]: | V (cid:48) | V ≥ (cid:112) ( d − d −
2) 1 M ( d − / d . (5.16)However, there is an important subtlety here: the lower bound in this example comes fromconsidering only the volume modulus and the dilaton, so it is plausible that including otherscalar fields could lead to consistency with (5.12) in asymptotic limits of scalar field space.Likewise, the lower bounds on | V (cid:48) | M /V listed in Table 2 of [6] come from considering thederivative V (cid:48) of the potential with respect to the geodesic distance along certain geodesics inscalar field space, not from considering the gradient ∇ V of the potential with respect to all scalar fields in the theory. Once contributions to the gradient ∇ V from additional scalarfields are included, it is plausible that our bound (5.12) may yet be satisfied. It may sound like wishful thinking to hope that contributions to the gradient from addi-tional scalar fields will always come to the rescue, ensuring consistency with (5.12). However,we already know of at least one simple example where this is precisely what happens: in het-erotic string theory compactified to four dimensions, one has | ∂ ρ V | M /V = (cid:112) /
3, where ρ isthe volume modulus of the compactification manifold (see Table 2 of [6]). This saturates theTCC bound (5.16) along a geodesic in the ρ direction and naively violates our bound (5.12).However, the potential also depends asymptotically on the dilaton, | ∂ τ V | M /V = √
2. Thisadditional contribution to the gradient leads to consistency with (5.12).Note, however, that while this example is a useful proof of principle, it does not representthe typical scenario. In general, it is quite difficult to determine the dependence of thepotential on every scalar field in the theory, so falsifying (5.12) in asymptotic regions ofscalar field space is not easy to do. Finally, note that (5.12) is incompatible with the condition (4.23) needed for eternal infla-tion, since γ d = 1 < γ EI d = ( d + 2) /
4. This is rather unsurprising from a physical perspective:(5.12) forbids accelerated expansion of the universe, which is obviously a prerequisite foreternal inflation. Dimensional reduction induces an exponential potential for the radion λ , V d = V D exp( λ/ ( d − The fact that the derivative with respect to geodesic distance V (cid:48) may saturate the TCC bound (5.16) iscrucial for the connection between the TCC and the Swampland Distance Conjecture (SDC) proposed in [6]. We are very thankful to David Andriot for discussions on these points. Conclusions
We have seen that dimensional reduction distinguishes particular O (1) constants appearingin the WGC, RFC, (A)dSDC, and dSC. In the case of the (A)dSDC, one mass scale thatemerges is suggestively close to the neutrino mass scale, which may be related to the Higgsvev via the see-saw mechanism. It might be worthwhile to search for other swampland-related reasons for a connection between the neutrino mass/Higgs vev and the cosmologicalconstant.In the case of the dSC, the bound we obtain precisely forbids accelerated expansion ofthe universe in Einstein-dilaton gravity. This is interesting in that it offers an alternativephysical explanation to Trans-Planckian Censorship [54] as to why the dSC seems to holduniversally in asymptotic limits of scalar field space: perhaps quantum gravity simply forbidsaccelerated expansion in such regions. This possibility merits further investigation.We have also worked out conditions for eternal inflation in d ≥ d = 4. Models of inflation in more than fourdimensions have not received very much attention, due to the obvious fact that such modelsare of little experimental interest. But given the recent, renewed interest in scalar fieldpotentials and de Sitter vacua in quantum gravity, it may be worthwhile to temporarilyignore the question of experimental relevance and instead explore more general theories ofinflation and cosmology in diverse dimensions, as well as their possible embeddings in stringtheory. It would be interesting to study de Sitter critical points of scalar field potentialsin string compactifications to d > Acknowledgements
We thank David Andriot, Ben Heidenreich, Juan Maldacena, Liam McAllister, GeorgesObied, Eran Palti, Matthew Reece, and Cumrun Vafa for useful discussions. We thankDavid Andriot, Ben Heidenreich, Liam McAllister, and Matthew Reece for comments on adraft of this paper. We acknowledge hospitality from the 2019 Simons Summer Workshopat the Simons Center for Geometry and Physics at Stony Brook University, where some ofthese discussions took place. The work of TR was supported by NSF grant PHY1820912,the Simons Foundation, and the Berkeley Center for Theoretical Physics.18 eferences [1] C. Vafa, “The String landscape and the swampland,” arXiv:hep-th/0509212[hep-th] .[2] H. Ooguri and C. Vafa, “On the Geometry of the String Landscape and theSwampland,”
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