Tensor Bounds on the Hidden Universe
PPrepared for submission to JHEP
Tensor Bounds on the Hidden Universe
Adri´an del Rio, a Ruth Durrer b and Subodh P. Patil c a Departamento de Fisica Teorica, IFIC. Centro Mixto Universidad de Valencia-CSIC.Facultad de Fisica, Universidad de Valencia, Burjassot-46 100, Valencia, Spain. b Dept. of Theoretical Physics, University of Geneva,24 Quai Ansermet, CH-1211 Geneva-4, Switzerland c Niels Bohr International Academy and Discovery Center,Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DK 2100, Denmark
E-mail: [email protected], [email protected], [email protected]
Abstract:
During single clock inflation, hidden fields (i.e. fields coupled to the inflaton only gravita-tionally) in their adiabatic vacua can ordinarily only affect observables through virtual effects. Afterrenormalizing background quantities (fixed by observations at some pivot scale), all that remains arelogarithmic runnings in correlation functions that are both Planck and slow roll suppressed. In thispaper we show how a large number of hidden fields can partially compensate this suppression andgenerate a potentially observable running in the tensor two point function, consistently inferable cour-tesy of a large N resummation. We detour to address certain subtleties regarding loop correctionsduring inflation, extending the analysis of [1]. Our main result is that one can extract bounds onthe hidden field content of the universe from bounds on violations of the consistency relation betweenthe tensor spectral index and the tensor to scalar ratio, were primordial tensors ever detected. Suchbounds are more competitive than the naive bound inferred from requiring inflation to occur below thestrong coupling scale of gravity if deviations from the consistency relation can be bounded to withinthe sub-percent level. We discuss how one can meaningfully constrain the parameter space of variousphenomenological scenarios and constructions that address naturalness with a large number of species(such as ‘N-naturalness’) with CMB observations up to cosmic variance limits, and possibly future21cm and gravitational wave observations. a r X i v : . [ g r- q c ] D ec ontents N
104 Discussion 11
A On the (cid:15) dependence of the ζ vertices 15B One loop correction to (cid:104) γγ (cid:105) B.1 Dimensional Regularization on a dS background 20B.2 Dimensional Regularization on a quasi dS background 22B.2.1 Slow-roll corrected loop integral 24B.2.2 Slow-roll corrected counterterms 25
C The strong coupling bound for gravity 27 – 1 – ontents1 Introduction
Observations strongly indicate that the Universe underwent an early phase of primordial inflation.Such an inflationary phase not only solves the horizon and flatness problems [2, 3], it also natu-rally produces a nearly scale invariant spectrum of density fluctuations [4] consistent with what hasbeen observed in the cosmic microwave background (CMB). These fluctuations originated as quantumvacuum fluctuations that were forced out of the horizon by the quasi-exponential expansion of theUniverse and subsequently squeezed, resulting in their phase coherence. The inflationary backgroundalso amplifies vacuum fluctuations of the transverse traceless part of the metric, leading to the gen-eration of primordial gravitational waves [5] as well as fluctuations of all other fields present in thequantum vacuum whether they couple directly to the inflaton or not.In this paper we consider the effects of fields that one would ordinarily be tempted to ignore duringinflation: hidden fields, defined as fields that couple only to gravity and have no direct couplings tothe inflaton. In their adiabatic vcauum, such fields would only serve to renormalize backgroundquantities and induce unobservably small (i.e. Planck and slow roll suppressed) logarithmic runningsin cosmological correlation functions. However, in large enough numbers, their effects can add upto an observable running of the spectral index of the two point function of the tensor perturbation,consistently inferable via a ”large N ” expansion that allows us to resum a restricted class of diagrams.The running induced for correlation functions of the curvature perturbation on the other hand remainsfeeble, since the relative suppression of the interaction vertices by factors of (cid:15) is too great to be overcomeby large N and still consistent with being below the strong coupling scale of gravity.One can thus use this observation to convert bounds on the violation of the tensor to scalarconsistency relation to a bound on the possible number of hidden fields present in the universe withmasses below the scale of inflation, were primordial tensors ever to be observed . For simplicity, wefocus on hidden scalars, although our argument generalizes straightforwardly to particles of other spin[9]. We find that any bound from above (to some confidence level) on deviations from the tensor toscalar consistency relation n t + r ∗ (cid:46) ξ (1.1)for some positive ξ , translates into a bound on the number of hidden species as N (cid:46) . × ξr ∗ ∆ − ζ (1.2)Where ∆ ζ ≈ . × − [10] is the amplitude of the spectrum of the curvature perturbation at thepivot scale where we determine the tensor to scalar ratio r ∗ , with n t being the tilt of the tensorspectrum. If we presume the most optimistic case that r ∗ ∼ .
06 then the best we can hope to bound N through CMB measurements is by N (cid:46) . × r ∗ ξ ∼ × ξ (1.3) Whose effects therefore would simply be absorbed into physical measurements of quantities such as (cid:15) := − ˙ H/H (e.g.through the detection of primordial tensors) and its derivatives or the ratio H /M , all of which denote renormalizedquantities. Although fields with masses much greater than the Hubble scale during inflation also contribute to the runningof the tensor spectrum, their effects are very suppressed at long wavelengths and so will not contribute to the boundsderived here. Fields with masses m ∼ H (cf. [6, 7]) do not affect the running of two point functions, although they canimprint on higher order (cross-)correlation functions with additional interactions not considered here [8]. – 2 –e note that this bound is only interesting if it is stronger than the bound coming from the requirementthat we stay below the scale at which gravity becomes strongly coupled [11, 12] (cf. eq (4.2), reviewedin appendix C): N (cid:46) π M H = 32∆ − ζ r ∗ ≈ . × r ∗ ∼ . (1.4)In order to infer a stronger bound from (1.2) than from consistency imposed by being below the strongcoupling scale (1.4), we would need to bound ξ one order of magnitude better than we the accuracywith which we measure r ∗ . As we shall elaborate upon further, cosmic variance limits us to boundson ξ no better than the percent level (were r ∼ .
06) from CMB observations alone, allowing onlymarginally to bound the parameter space of a variety of models that attempt to address the hierarchyproblem with a large number of sectors [13, 14]. However, as we discuss further, observations ofthe stochastic background at very different comoving scales through future 21cm and space basedgravitational wave interferometer observations could allow us to entertain significant improvementsupon these constraints.We begin this paper with an outline of our calculation with details deferred to the appendix. Itbehoves us to elaborate upon various subtleties encountered in the calculation of loop corrections tocosmological correlation functions relevant to this calculation [1, 15]. In particular, we extend theanalysis of Senatore and Zaldarriaga [1] which pointed out that dimensional regularization had onlybeen partially implemented in previous calculations (e.g. [16] and subsequent studies), where it wasfound that loop corrections induced a running of the form log( k/µ ) in the two point function of thecurvature perturbation, with µ some arbitrary renormalization scale. Including previously neglectedcorrections to the mode functions and to the integration measure in D = 3 + δ spatial dimensions , itwas found that loops instead induce a correction of the form log( H/µ ) [1].At first glance this appears to preclude any running of the loop correction, which cannot be thecase in general as quantum corrections typically induce scale dependence unless we are at a fixedpoint of the theory, e.g. in the dS (de Sitter) limit where an exact dilatation (i.e. scale) invarianceis realized – implicitly assumed in [1]. Since corrections to the correlation functions are being forgedas modes exit the horizon during single clock inflation, it must be the case that what appears insidethe log is in fact H k – the Hubble scale at the time the k -mode exits the horizon. We demonstratethis explicitly in appendix B, where we show how additional slow roll corrections to the mode func-tions and the integration measures within the loop integrals indeed result in a correction of the formlog( H k /µ ). Upon fixing the renormalization conditions at some (pivot) scale µ = H ∗ , one reintroducesa running as one moves away from this scale, but now of the form log( H k /H ∗ ) → − (cid:15) log( k/k ∗ ). Thiscontribution to the running is far too feeble to ever be observed for the curvature perturbation , butdoes have a potentially observable effect on the tilt of the tensor spectrum. In Section 3 we derivethe modifications of the tensor and scalar spectral indices due to the presence of hidden fields and inSection 4, we discuss possible observational bounds on N and generalizations of our results. Notation:
In what follows, we shall consider a spatially flat FRLW universe with line element inCartesian coordinates ds = a ( τ ) (cid:2) − dτ + δ ij dx i dx j (cid:3) = g µν dx µ dx ν , (1.5) A conclusion independently arrived at by working in a mass dependent regularization scheme (a hard cutoff inphysical momenta). In section 4, we discuss the possible implications of the running of the two point function of the curvature pertur-bation for whether or not a given model of inflation is eternal according to criteria derived in [17]. – 3 –here τ denotes conformal time and physical time is given by dt = adτ . Derivatives w.r.t. τ aredenoted by a prime and those w.r.t. t by an overdot. The physical Hubble parameter is H = ˙ a/a . We consider an inflationary Universe with an inflaton φ taken to be the only field with an evolvingbackground (hence energy density) and N additional hidden scalar fields χ n with a flat target space,minimally coupled to gravity and taken to be in their respective adiabatic vacuum states. We onlyconsider hidden fields with masses m (cid:28) H , which can therefore be treated as effectively masslessbut are quantum mechanically excited by the background expansion during inflation. By assumptionthe χ n have no non-gravitational interactions. The action is then given by S = M (cid:90) d x √− gR [ g ] − (cid:90) d x √− g (cid:34) ∂ µ φ∂ µ φ + 2 V ( φ ) + N (cid:88) n =1 ∂ µ χ n ∂ µ χ n (cid:35) , (2.1)where M pl = (8 πG ) − / is the reduced Planck mass . We presume the background to be quaside-Sitter, such that (cid:15) := ˙ φ H M = − ˙ HH (cid:28) , (2.2)so that H = V ( φ ) / (3 M ) ∼ const, and for completeness we define higher order slow roll parameters (cid:15) i as (cid:15) ≡ (cid:15) , (cid:15) i +1 = ˙ (cid:15) i H(cid:15) i , i ≥ . (2.3)In order to discuss perturbations around this background, we first ADM decompose the metric as ds = − N dt + h ij ( dx i + N i dt )( dx j + N j dt ) , (2.4)and work in comoving gauge, defined to be the foliation in which we have gauged away the inflatonfluctuations. In this gauge, the only dynamical degrees of freedom are contained in the 3-metric h ij which has now acquired, or ‘eaten’ a scalar polarization that was the inflaton fluctuation [19] φ ( t, x ) = φ ( t ) , (2.5) h ij ( t, x ) = a ( t ) e ζ ( t,x ) ˆ h ij , ˆ h ij = exp [ γ ij ] , (2.6)where γ ii = ∂ i γ ij = 0 is (transverse traceless) graviton, and ζ is the comoving curvature perturbation.The quasi dS background then results in a nearly scale invariant spectrum of curvature perturbations[20–22] P ζ ( k ) = H ∗ π (cid:15)M (cid:18) kk ∗ (cid:19) n s − , n s − − (cid:15) − (cid:15) . (2.7) Equivalently, thermally excited with the identification T dS = H π in units where k B = (cid:126) = c = 1. This so far bare quantity also gets renormalized via diagrams involving external graviton legs with loops of massivefields. However, in the massless limit, the contributions of each species to the divergent and finite parts of M and thecosmological constant vanishes [18] whilst still lowering the strong coupling scale (cf. appendix C). We note that comoving gauge is defined by the vanishing of δT i . This is still satisfied in the presence of an arbitrarynumber of hidden fields since their contributions to δT i go as ˙ χ∂ i χ which vanishes identically since by assumption the χ fields have no classically evolving background. Note that this statement persists at the quantum level as well, since (cid:104) ˙ χ∂ i χ (cid:105) = 0 by isotropy of the Bunch-Davies vacuum state. – 4 –n addition helicity 2 tensor perturbations of the metric are amplified from their initial quantumvacuum state leading to a power spectrum for primordial gravitational waves given by P γ ( k ) = 2 H ∗ π M (cid:18) kk ∗ (cid:19) n t , n t = − (cid:15) . (2.8)The ratio of these two quantities r = P γ ( k ∗ ) P ζ ( k ∗ ) = 16 (cid:15) (2.9)defines the tensor to scalar ratio. Note that its value, as well as n t and n s depend on the pivot scale, k ∗ , and H ∗ is defined as the value of the Hubble parameter at the time the mode k ∗ exits the horizon.The single field scalar tensor consistency relation is simply the identity r + 8 n t = 0. At present, notensor perturbations have been identified in the observed CMB anisotropies and an upper limit of r (cid:46) .
06 has been derived for k ∗ = 0 .
002 Mpc − [23, 24]. We remind the reader that there are higherorder corrections to the tilt of the scalar and tensor spectra that come from the background dynamicsalone, which we will return to later. We are interested in additional corrections to these from virtualeffects due to the presence of the hidden fiends χ n . Perturbing the action (2.1) in comoving gauge (2.5) results in the quadratic action S ,ζ = M (cid:90) d x a (cid:15) (cid:20) ˙ ζ − a ( ∂ζ ) (cid:21) (2.10) S ,χ = 12 (cid:90) d x a (cid:20) ˙ χ n ˙ χ n − a ∂ i χ n ∂ i χ n (cid:21) (2.11) S ,γ = M (cid:90) d x a (cid:20) ˙ γ ij ˙ γ ij − a ∂ k γ ij ∂ k γ ij (cid:21) (2.12)and the cubic interaction vertices S ,ζχ = (cid:90) d x a (cid:15) (cid:20) ζ (cid:18) ˙ χ n ˙ χ n + 1 a ∂ i χ n ∂ i χ n (cid:19) − ˙ χ n ∂ i χ n ∂ i ∂ − ˙ ζ (cid:21) (2.13)and S ,γχ = 12 (cid:90) d x a [ γ ij ∂ i χ n ∂ j χ n ] = 12 (cid:90) d x aγ ij Π χij . (2.14)Where Π χij is the anisotropic stress of the χ fields and the sum over n is implicit. The form of (2.13) –in particular its (cid:15) suppression – is not immediately obvious from naively expanding the original action(2.1) having solved for the lapse and shift constraints, which results in an expression that is nominallyunsuppressed in (cid:15) (A.11). However as shown in appendix A, similar to what occurs for the cubic andhigher order self interactions for ζ [25], enough integrations by parts show that the ζχχ cubic (andthe ζγχχ quartic) interactions are suppressed by an overall factor of (cid:15) . Similarly, interactions that arehigher order in ζ will be sequentially suppressed by additional powers of (cid:15) , consistent with its natureas an order parameter parameterizing the breaking of time translational invariance by slow roll [26].We are interested in calculating the finite time correlation functions of the curvature perturbations k π (cid:104) ζ k ( τ ) ζ q ( τ ) (cid:105) := (2 π ) δ ( k + q ) P ζ ( k ) , (2.15)– 5 – igure 1 : The S-matrix contour (left) comparedto the Schwinger-Keldysh contour (right). Figure 2 : One loop corrections to (cid:104) ζζ (cid:105) . Solidlines denote the curvature perturbation propaga-tor, dashed lines denote the χ -propagator.and tensor perturbations γ rij k π (cid:104) γ rij, k ( τ ) γ rij, q ( τ ) (cid:105) := (2 π ) δ ( k + q ) P γ ( k ) , (2.16)where we have summed over the two independent polarizations. Both of the above are of the form (cid:104)O ( τ ) (cid:105) where the angled brackets denote expectation values with a given initial density matrix (whichwe take to correspond to the Bunch-Davies vacuum), unitarily evolved forward in the interactionpicture with the Dyson operator U ( τ, −∞ ) = T exp (cid:18) − i (cid:90) τ −∞ H I ( τ (cid:48) ) dτ (cid:48) (cid:19) , (2.17)where T denotes time ordering and where H I is the interaction Hamiltonian (equal to minus theinteraction Lagrangian given in eqs. (2.13) and (2.14) respectively for the interactions in question [16].Reading right to left, one evidently evolves the Bunch-Davies vacuum from the initial time −∞ to τ ,inserts the corresponding free-field operator O ( τ ) at time τ and then evolves back to −∞ : (cid:104)O ( τ ) (cid:105) = (cid:104) in | (cid:20) T exp (cid:18) − i (cid:90) τ −∞ H I ( τ (cid:48) ) dτ (cid:48) (cid:19)(cid:21) † O ( τ ) (cid:20) T exp (cid:18) − i (cid:90) τ −∞ H I ( τ (cid:48) ) dτ (cid:48) (cid:19)(cid:21) | in (cid:105) (2.18)The above can be shown to be formally equivalent to the expression [16] (cid:104)O ( τ ) (cid:105) = ∞ (cid:88) n =0 i n (cid:90) τ −∞ dτ n (cid:90) τ n −∞ dτ n − ... (cid:90) τ −∞ dτ (cid:104) [ H I ( τ ) , [ H I ( τ ) , ... [ H I ( τ n ) , O ( τ )] ... ]] (cid:105) (2.19)provided one is mindful of how one selects the correct initial interacting vacuum [15] – an importantpoint that we will return to shortly.Although useful for practical purposes, such an expectation value does not lend itself to the usualdiagrammatic expansion one uses when dealing with S-matrix elements. In order to implement thisone can equivalently consider the expression (2.18) as the product of an arbitrary operator O ( τ ) with– 6 –he unitary evolution operator: (cid:104)O ( τ ) (cid:105) = (cid:104) in | T C (cid:20) exp (cid:18) − i (cid:73) H I ( τ (cid:48) ) dτ (cid:48) (cid:19) O ( τ ) (cid:21) | in (cid:105) (2.20)with the contour going from −∞ → τ and back again (cf. Fig. 1), and with T C denoting contourordering with fields living on the reverse contour treated as independent fields for intermediate ma-nipulations, only being set equal to the original fields at the end of the calculation. Due to its formalsimilarity with an S-matrix element, the former does indeed lend itself to a diagrammatic expansionwhich we will not make explicit use of in the following, but we nevertheless find useful for reasoningdiagrammatically.Suppressing the difference between the fields that live on the future and past directed contours(as a result of which there are typically many cancellations as one sums up relevant diagrams) asshorthand, one can nevertheless intuit the parametric and external momentum dependences of thevarious graphs that one can write down. For example, at one loop, one has two possible contributionsto the correction to the two point correlation function of the curvature perturbation as indicated inFig. 2. However only the diagram involving two cubic vertices results in a dependence on the externalmomenta and hence contributions to the running of the spectral index, which is the object of ourinterest.At two loops, we notice that the double sunset graphs (involving two independent loops of hiddenfields) dominate when N (cid:29) /(cid:15) relative to all other contributions (Fig. 3) . This structure persistsat each loop order and permits the resummation of a restricted subset of diagrams (consisting onlyof the sunset diagrams) in the large N limit, allowing us to consistently infer the running even in theevent that it could compete with the running induced from the background dynamics alone.It is here that we lose interest in the corrections to the running of the curvature perturbation, sinceit will turn out that no amount of enhancement by factors of N can overcome the slow-roll suppressionof the corrections, consistent with the strong coupling bound (1.4). This is in part because of the (cid:15) suppression of the interaction vertices (A.18) and (A.21), but also since (as we shall see shortly)the corrections must be of the form log H k /µ , as opposed to the log k/µ which eventually introducesadditional slow-roll suppression. Tensor perturbations on the other hand, have interactions that areunsuppressed by (cid:15) and will have potentially observable consequences, which we turn to presently. For the rest of this paper, we shall be interested in the operator expectation value (2.16), which isshorthand for (cid:104) γ sij, k ( τ ) γ s (cid:48) ij, k (cid:48) ( τ ) (cid:105) = (cid:104) (cid:16) T e − i (cid:82) τ −∞ dτ (cid:48) H I ( τ (cid:48) ) (cid:17) † γ ,sij, k ( τ ) γ ,s (cid:48) ij, k (cid:48) ( τ ) (cid:16) T e − i (cid:82) τ −∞ dτ (cid:48) H I ( τ (cid:48) ) (cid:17) (cid:105) (2.21)Nominally the second order correction to (2.21) is equivalent to the following expression [16] There are also contributions from cubic interactions involving ζ alone, but these will be suppressed by two extrapowers of (cid:15) [25, 27]. Although vanishing for massless fields, the quartic ‘seagull’ interactions contributes to wavefunction renormalizationfor any small but finite mass, accounted for in practice by fixing the (fully renormalized) expressions H ∗ /M and (cid:15) ∗ via the amplitude of the power spectrum and the tensor to scalar ratio at some pivot scale k ∗ . There are an additional two loop diagrams corresponding to a single sunset graph with a tadpole insertion to aninternal χ propagator, but this is accounted for by wavefunction renormalization of the χ fields and considering diagramswith internal lines taken to be renormalized propagators when summing graphs. – 7 – igure 3 : Two loop corrections to (cid:104) ζζ (cid:105) . Wavylines denote the graviton propagator. The doublesunset graphs dominate when N (cid:29) /(cid:15) . Figure 4 : Two loop corrections to (cid:104) γγ (cid:105) , wherehere we only require N (cid:29) (cid:104) γ sij, k ( τ ) γ s (cid:48) ij, k (cid:48) ( τ ) (cid:105) (2) = − (cid:90) τ −∞ dτ (cid:90) τ −∞ dτ (cid:104) [ H I ( τ ) , [ H I ( τ ) , γ ,sij, k ( τ ) γ ,s (cid:48) ij, k (cid:48) ( τ )]] (cid:105) (2.22)However, we have to be careful, since we shall be deforming the contour to imaginary time in the pastin order to pick out the correct interacting vacuum. Therefore, we really need to be calculating (cid:104) γ sij, k ( τ ) γ s (cid:48) ij, k (cid:48) ( τ ) (cid:105) = (cid:104) (cid:16) T e − i (cid:82) τ −∞ (1+ iε ) dτ (cid:48) H I ( τ (cid:48) ) (cid:17) † γ ,sij, k ( τ ) γ ,s (cid:48) ij, k (cid:48) ( τ ) (cid:16) T e − i (cid:82) τ −∞ (1+ iε (cid:48) ) dτ (cid:48) H I ( τ (cid:48) ) (cid:17) (cid:105) (2.23)with ε, ε (cid:48) independent. This means that the symmetry in the domains of integration that allow one toexpress a time ordered product of integrals in terms of an integral over a simplex is broken wheneverwe have one operator from the time ordered product and another from the anti-time ordered product –this is a requisite for the expression (2.19) to equal (2.22), as first pointed out in [15]. Not accountingfor this will result in missing contributions to the loop integral in addition to spurious divergences.Mindful of the latter, we go through the details of the calculation in appendix B, considering additionalsubtleties arising from dimensional regularization on a quasi dS background. The intermediate resultis the one loop correction P γ ( k ) = 2 H ∗ π M (cid:34) N π H ∗ M
35 log ( H k /H ∗ ) (cid:35) , (2.24)where for the moment, we suppress slow roll corrections to the external mode functions that gener-ate the usual tilt of the tensor power spectrum. We note that the log( k/µ ) dependence previouslycalculated in the literature (e.g. [15, 16]) is merely the first of multiple logarithmic corrections tothe tree level result. An additive correction of the form log( − H ∗ τ k ) also arises from correctionsto the mode functions proportional to δ in 3 + δ spatial dimensions [1], which goes over into alog( k/µ ) + (1 + (cid:15) ) log( − H ∗ τ k ) = log( H k /µ ) dependence once one accounts for additional slow rollcorrections (B.64) . Upon fixing renormalized quantities at some pivot scale H ∗ , the dependence in(2.24) results.However, once one realizes that H k itself runs as inflation progresses, one finds that (cf. (B.74)and (B.75)) – log H k H ∗ = − (cid:15) ∗ log kk ∗ + O ( (cid:15) ∗ ) . (2.25) See the discussion in section 3.2 of [1], which we extend to quasi dS backgrounds in appendix B. – 8 –ence, the intermediate result for the one loop correction becomes: P γ ( k ) = 2 H ∗ π M (cid:34) − (cid:15) ∗ N π H ∗ M
35 log ( k/k ∗ ) (cid:35) . (2.26)That is, the net result of incorporating terms previously neglected in implementing dimensional regu-larization, whereby log( k/k ∗ ) would have appeared in the intermediate expression (2.24) instead oflog( H k /H ∗ ), is to still induce a log k running, but of the opposite sign and with extra (cid:15) ∗ suppression.Additional corrections to the tensor two point function from the background dynamics suppressed in(2.26) results in the final expression P γ = ∆ γ (cid:18) kk ∗ (cid:19) − (cid:15) ∗ + O ( (cid:15) ) (cid:34) − (cid:15) ∗ N π H ∗ M
35 log ( k/k ∗ ) + O ( (cid:15) ) (cid:35) . (2.27)For completeness, we note that as illustrated for two loops in Fig. 4, in the limit N (cid:29)
1, diagramsconsisting of n independent insertions of hidden loops dominate at the n th loop order and can inprinciple be resummed, allowing us to consistently infer the running if it is of the same order as thatinduced from the background alone. However, when doing this, one must be sure to have taken intoaccount dependence on the slow roll parameters to all orders. Formally: P γ = ∆ γ (cid:16) kk ∗ (cid:17) n t ( (cid:15) ∗ , ˙ (cid:15) ∗ ,... ) ∼(cid:13)∼ (2.28)where n t ( (cid:15) ∗ , ˙ (cid:15) ∗ , ... ) denotes the spectral tilt to all orders in the Hubble hierarchy (cf. (3.4) to secondorder), and where ∼(cid:13)∼ = (cid:15) ∗ N π H ∗ M
35 log kk ∗ + ..., (2.29)where the ellipses denote corrections to the running from non-trivial momentum dependent correctionsto the mode functions, suppressed by extra factors of slow roll parameters that one can in principlecalculate to the desired order (cf. appendix A), although in practice we shall only be interested in theleading order correction to the above.In the limit N (cid:29) /(cid:15) ∗ one can perform a similar resummation for the corrections to the powerspectrum of the curvature perturbation, so that again formally P ζ = ∆ ζ (cid:16) kk ∗ (cid:17) − n s ( (cid:15) ∗ , ˙ (cid:15) ∗ ,... ) −(cid:13)− (2.30)where −(cid:13)− = − c (cid:15) ∗ N π H ∗ M log kk ∗ + ... (2.31)and where n s ( (cid:15) ∗ , ˙ (cid:15) ∗ , ... ) is the spectral tilt for the curvature perturbation. The ellipses again denotecalculable corrections to the running from slow roll corrections the mode functions, and where the We note that the equivalent expression in [15] with a log k/µ correction has a different numerical coefficient andopposite sign to that which would have appeared in (2.24), this error has been acknowledged to us [28]. – 9 –recise numerical coefficient c (calculated to be c = 4 /
15 in [16] ) in the above is unimportant to us other than it positive, so that (2.31) has an overall negative sign. N Recalling the Hubble hierarchy of slow roll parameters (cid:15) ∗ = (cid:15) := − ˙ H ∗ H ∗ , (cid:15) i +1 := ˙ (cid:15) i H ∗ (cid:15) i , (3.1)one finds by taking appropriate logarithmic derivatives of the loop corrections to the scalar and tensorpower spectra, additional corrections to the tilt and the running (to second order in the Hubblehierarchy [29–31]) n s − − (cid:15) ∗ − (cid:15) − (cid:15) ∗ − (2 C + 3) (cid:15) ∗ (cid:15) − C(cid:15) (cid:15) + c (cid:15) ∗ λ (3.2) dn s d log k = − (cid:15) ∗ (cid:15) − (cid:15) (cid:15) (3.3) n t = − (cid:15) ∗ + (cid:15) ∗ λ − (cid:15) ∗ − C + 1) (cid:15) ∗ (cid:15) (3.4) dn t d log k = − (cid:15) ∗ (cid:15) + λ (cid:15) ∗ (cid:15) − (cid:15) ∗ λ (3.5)where C = log 2 − γ E , γ E being the Euler-Mascheroni constant and where we define λ := 35 N π H ∗ M = 3160 N r ∗ ∆ ζ . (3.6)As reviewed in appendix C, we observe that λ is necessarily bounded from above by λ (cid:46)
35 (3.7)since
N H / (16 π M ) (cid:46) (cid:15) ∗ λ ).On the other hand, the spectral tilt for the tensor modes can receive corrections that are morecleanly observable: n t = − (cid:15) ∗ + (cid:15) ∗ λ. (3.8)Using the fact that the relation r ∗ = P γ ( k ∗ ) P ζ ( k ∗ ) = 16 (cid:15) ∗ (3.9)remains unchanged, we see that the consistency relation is modified since we now have n t = − r ∗ (cid:18) − λ (cid:19) (3.10) We note that there is a spurious normalization factor of 1 / (2 π ) in eq 72 relative to eq 71 of [16] given the conventionstherein. Correcting for this results in a loop suppression factor of ∼ / (16 π ), consistent with our findings in appendixB. The (cid:15) ∗ suppression of the ζ vertices can never be compensated by large N consistent with the strong coupling boundand will always result in loop corrections that are subleading to slow roll corrections from the background. – 10 –hich we rewrite as 3 . × r ∗ (cid:16) n t + r ∗ (cid:17) ≈ N, (3.11)where we have evaluated the numerical prefactor using the observed value ∆ ζ ≈ . × − . Therefore,if ( n t + r ∗ /
8) can be bounded from above by some positive number, i.e. if we can ever conclude tosome threshold of confidence that n t + r ∗ (cid:46) ξ (3.12)for some positive ξ , then one can bound N (cid:46) ξr ∗ . × (3.13)If we presume the most optimistic case that r ∗ ∼ .
06 then the best we can hope to bound N is by N (cid:46) × ξ (3.14)Note that the strong coupling bound requires that N (cid:46) π M H = 32∆ − ζ r ∗ ≈ . × r ∗ (3.15)so that in order to infer a stronger bound from (3.13) than from consistency imposed by being belowthe strong coupling scale (3.15), we need to bound ξ by an order of magnitude more accurately thanthe measured r ∗ . For CMB observations, this is on the threshold conceivable within cosmic variancelimits – at r ∗ ∼ O (10 − ), the best one can hope to bound ξ is approximately O (10 − ) [32], whichis not much more constraining that the naive strong coupling bound. Whether future space basedgravitational wave interferometry or ultimate 21 cm observations can improve upon this sensitivity isa possibility we contemplate in the following section. For the purposes of the following, we frame the discussion in terms of an observational challengefor bounding ξ in the context of (3.13) as a null test. We will abuse our privileges as theorists tocontemplate the possibility that one could bound ξ past CMB cosmic variance limits to the level of10 − or beyond. That this may be plausible with a combination of future space based interferom-etry [33] and ground based arrays [34] can be appreciated from the fact that the tilt for the tensorspectrum is no longer as negative as λ approaches its upper bound (3.7), so that at comoving scales k ∼ k ∗ ∼ Mpc − (corresponding to peak interferometer sensitivities in the mHz range) thepower will be enhanced by about 20% relative to the standard case. One might conceive improvedprospects for constraining deviations from the consistency relation from combining observations sensi-tive enough to detect the stochastic primordial background [35] at widely separated scales, with CMBobservations and space-based interferometry sensitive to modes 14 orders of magnitude apart , withSKA like surveys interpolating between them with sensitivity at the nHz frequencies ( k ∼ k ∗ ∼ Mpc − ). Ultimate 21 cm observations also offer the possibility to measure primordial gravitational One might be concerned that higher order corrections might need to be incorporated in order to extrapolate therunning over such a large range of scales. However, such corrections only become important when considering scalessuch that log k/k ∗ ∼ λ(cid:15) ∗ (cid:38) (cid:15) ∗ = r ∗ (cid:38) – 11 –ave background through large scale structure fossils, allowing for an in principle sensitivity to r down to the ∼ − level [36], however, the question of whether foregrounds can be understood tothe required level is far from settled at the present moment. For now, we merely state the obviouscorollary that follows from (3.13) that (for r ∗ ∼ . N (cid:46) ξ · ∼ − (4.1)for ξ ranging from 10 − (cid:46) ξ (cid:46) − where the latter corresponds to CMB cosmic variance bounds, andthe former corresponds to us rather speculatively entertaining bounds that could be obtained by othermeans – combinations of next generation space and ground based gravitational wave observations orultimate 21cm observations. The idea of invoking a large number of hidden sectors to address the electroweak hierarchy problemwas considered in [11–13] – the observation being that a large number of species can be used to makethe scale of quantum gravity Λ QG parametrically lower than the Planck mass (cf. appendix C) Λ QG ∼ πM pl √ N . (4.2)Far from being an ad hoc construction, [12, 13] argue that such a large number of hidden sector arisenaturally as Kaluza-Klein copies of the standard model in scenarios with extra dimensions, althoughtheir origin needn’t be extra dimensional in general (see [37] for an interesting speculation that theseextra species could constitute dark matter). As discussed above, any observation of primordial tensorsin the context of single field inflation immediately implies that in order for inflation to have occurredbelow the strong coupling scale, one must necessarily live in a universe with less than N (cid:46) π M H ≈ r ∗ hidden fields (3.15), with tests of deviations from the tensor to scalar consistency relation tobetter than the percent level allowing us more constraining power than the strong coupling bound.More recently, the authors of [14] proposed an alternative solution to the hierarchy problem thatnecessarily invokes inflation, initially dubbed ‘ N -naturalness’. The idea is that we live in a universewith N copies of the standard model each hidden from the other, with all coupled to a reheating field(the reheaton), not necessarily the inflaton. The mass of the Higgs fields in any of the N copies ofthe standard model is drawn from a uniform distribution that interpolates between − Λ ≤ m H ≤ Λ .Given that reheating will preferentially produce particles in the lightest sector (with masses set bythe Higgs expectation value), one dynamically explains why the universe that emerges from inflationwill have a naturally small Higgs mass. A significant parameter space of interest lies within the range N ∼ − , for which tests of the tensor to scalar consistency relation at the per-mille level orbetter (4.1) could significantly constrain. In general, hidden sectors in string or BSM constructions possess a spectrum that is not restricted toscalar fields. An obvious question therefore is how our results generalize when including higher spins.Leaving aside the precise nature of the running of the two point function (the primary concern of thisinvestigation), one can immediately infer the relative importance of the contributions from particles This bound is often stated as Λ QG ∼ M pl √ N where M pl is the reduced Planck mass and an order unity pre-factor isunderstood. As reviewed in the appendix, repeating the various arguments presented in [11, 12] suggests that the boundis at least (4.2). – 12 –f different spins by consulting the one loop effective action obtained from integrating them out overa fixed background [38]. For a particle of a given spin, the effective action is given by (C.3) (seeappendix C for a discussion of the interpretation of the quantity below) L eff = M R + 12880 π (cid:2) a s R µν R µν + b s R (cid:3) + ... (4.3)where the coefficients a s , b s depend on the spin of the particle integrated out, and where we have usedthe Gauss-Bonnet relations in 4D to eliminate redundant operators. The relative contributions of theterms from which we have extracted our tree level result and our loop contributions can be determinedfrom the ratio of the two contributions in (4.3). From the table in Appendix C and eq. (C.3), we canread off the coefficients a s and b s for different spins to obtain for maximally symmetric backgrounds(where R µν = g µν R/ L − loop L tree = 9 / π RM ; spin 0 , (4.4)= 3 / π RM ; spin 1 / , = − π RM ; spin 1 . From this, we can conclude that in a universe with a spectrum of particles consisting of N φ scalars, N ψ Dirac fermions and N V U (1) gauge fields, the actual quantity one is bounding with (3.13) is therelevant spin weighted sum indicated in (4.4). We leave the explicit computation of this index and therunning induced by higher spin fields for a future study [9].We note in passing that on a dS background, R = 12 H , so that for N scalar fields we have2 × L − loop L tree = 2 N · / π RM = N π H M (4.5)where the factor two is to count the two independent polarizations that contribute to the tensor powerspectrum. This is exactly the relative ratio of the loop contribution to the tree level result calculatedin (2.27), providing a non-trivial check on our results. Although not the primary focus of this investigation, having to come to terms with the precise natureof the slow roll corrections to the loop integrals (and correctly implementing dimensional regularizationon a quasi dS spacetime) has potential implications for eternal inflation . Recalling the discussionof [1], who discovered by using a mass dependent regularization scheme (a hard cutoff in physicalmomentum) that logarithmic corrections to the the two point function of the curvature perturbationof the form log H ∗ /µ resulted. This was in contradiction with the log k/µ form of the loop correctionderived elsewhere in the literature when applying dimensional regularization. Senatore and Zaldarriagareasoned that the former could not be the final answer – taking the result [16] at face value, P ζ = H ∗ π M (cid:15) ∗ (cid:34) − (cid:15) ∗ N π H ∗ M log kµ (cid:35) (4.6) Recently, the authors of [40] have proposed a conjecture motivated from string ‘swampland’ considerations [41] thatsuggest obstacles for accomplishing inflation at all within string theory. Insofar as our study takes a viable inflatingbackground for granted, the presence of additional hidden fields with no potential terms is no more problematic thanassuming an inflationary background in the first place. – 13 –ne finds upon Fourier transforming back to position space, that the variance of the inflaton fluctuation δφ = − ˙ φ H ∗ ζ is given by (cid:104) δφ (cid:105) − loop = − (cid:15) ∗ N π H ∗ M (cid:90) Λ a ( t ) d k H ∗ k log k ∼ − (cid:15) ∗ N π H ∗ M H ∗ (log a ) ∼ − (cid:15) ∗ N π H ∗ M H ∗ t (4.7)implying that the fluctuations of the inflaton field decay monotonically over time, implying that nomodel of inflation is eternal were the form of the correction (4.7) to be trusted , which clearly cannotbe the case.The resolution pointed out by [1] was that the log k/µ corrections found previously were merelythe first of several logarithmic contributions that had to be supplemented with corrections to thedimensionally deformed mode functions and integration measures that went as log( − H ∗ τ k ), where τ k is the time of crossing of the comoving k -mode. Adding up all such corrections resulted in a dependenceof the form log k/µ + log( − H ∗ τ k ) = log H ∗ /µ, (4.8)in agreement with the results obtained with a hard cutoff, which suggest that the correlation functionsdo not run at this order. However as we have shown, this is not the final story either, since the onlypossibility for which correlation functions of an interacting theory remain independent of scale is if weare at a fixed point of the theory, where a scale symmetry is realized. This is indeed the case in thestrict dS limit, where H is constant and one has an exact dilatation invariance. Therefore, movingaway from the strict dS limit must reintroduce a running calculated in appendix B (to next to leadingorder in slow roll) with the result (B.64):log k/µ + (1 + (cid:15) ) log( − H ∗ τ k ) = log H k /µ, (4.9)Upon fixing renormalization conditions at a particular pivot scale µ = H ∗ , this implies that correlationfunctions will run as (B.74) log H k H ∗ = − (cid:15) log kk ∗ (4.10)Therefore, repeating the calculation for the corrections to the curvature two point function, one findsthat a log k running is reintroduced, but of the opposite sign and with additional slow roll suppression.Retracing the argument leading to (4.7) seems to imply that our results imply that all models ofinflation in the presence of hidden fields are eternal. However this is too naive, as we have to factorin corrections from the background as well. If the sign of the net log correction log H k /H ∗ generatedfrom background corrections and cubic self interactions of the curvature perturbation alone were alwayspositive, one would then be able to conclude that indeed, all models of inflation were eternal , as arguedto be the case in [39]. However, it is very likely that loop corrections do not always compete withclassical logarithmic corrections from the background evolution, and the precise conclusion one arrivesat depends on the given background model. This is an important issue which deserves a thoroughseparate investigation. A more thorough treatment is provided in [17] where it is shown that the reheating volume diverges above the criticalinflaton velocity ˙ φ /H > / (2 π ). Here, φ cl is to be understood as the field around which one implements backgroundfield quantization – i.e. the field that minimizes the effective action . For the purposes of the present discussion, wecontent ourselves with observing the growth or decay of the variance, which after repeating the steps of [17] can beshown to result in crossing this critical velocity or not. – 14 – cknowledgements We thank Peter Adshead, Cliff Burgess, Martin Sloth, Filippo Vernizzi and Matias Zaldarriaga forvaluable and informative discussions over the course of this investigation. We acknowledge supportfrom the Swiss National Science Foundation. SP is supported by funds from Danmarks Grund-forskningsfond under grant no. 1041811001. A. d. R. is supported by the Spanish Ph.D fellowshipFPU13/04948 and research stay Grant EST15/00296, and thanks the Theoretical Cosmology Groupof the University of Geneva for hospitality during the initial stages of this collaboration.
A On the (cid:15) dependence of the ζ vertices We consider the action for the zero mode of the (canonically normalized) inflaton plus N hidden scalars. S = M (cid:90) d x √− gR [ g ] − (cid:90) d x √− g [ ∂ µ φ∂ µ φ + 2 V ( φ )] (A.1) − n max (cid:88) n =1 (cid:90) d x √− g ∂ µ χ n ∂ µ χ n By assumption, the χ fields have no classically evolving background, and so appear in the action toleading order as quadratic in perturbations. We ADM decompose the metric ds = − N dt + h ij ( dx i + N i dt )( dx j + N j dt ) , (A.2)and work in comoving gauge φ ( t, x ) = φ ( t ) , (A.3) h ij ( t, x ) = a ( t ) e ζ ( t,x ) δ ij . (A.4)This gauge is defined by the foliation where the inflaton is the clock (no other field has a background).Writing N = 1 + α (A.5) N i = ∂ i θ + N iT , with ∂ i N iT ≡ α , θ and N iT all first order quantities, we find the solutions (we only need to calculate to firstorder for the constraints to obtain the action to cubic order [25]) α = ˙ ζH (A.6) ∂ θ = − ∂ ζa H + (cid:15) ˙ ζ (A.7)where ∂ = ∂ i ∂ i contains no factors of the scale factor, and where (cid:15) is defined as: (cid:15) := ˙ φ H M (A.8)The relevant quadratic and cubic terms are (summation over n implicit) S ,ζ = M (cid:90) d x a (cid:15) (cid:20) ˙ ζ − a ( ∂ζ ) (cid:21) (A.9)– 15 – ,χ = 12 (cid:90) d x a (cid:20) ˙ χ n ˙ χ n − a ∂ i χ n ∂ i χ n (cid:21) (A.10) S ,ζχ = 12 (cid:90) d x (cid:40) a ˙ χ n ˙ χ n (cid:32) ζ − ˙ ζH (cid:33) − a ˙ χ n ∂ i θ∂ i χ n − a (cid:32) ζ + ˙ ζH (cid:33) a ∂ i χ n ∂ i χ n (cid:41) (A.11)We do not write the cubic action for ζ since it all we shall need from it is the fact that it is suppressedby (cid:15) to leading order after enough integrations by parts [25]. A similar thing happens for (A.11) –although it may appear that the cubic interactions between ζ and the χ a might be of order (cid:15) , theseinteractions in fact of order (cid:15) . This is readily seen by realizing that this contribution to the action isnothing other than the variation of the quadratic action for the hidden fields to first order in metricperturbations. That is, if L χ = − ∂ µ χ n ∂ µ χ n , then the cubic interaction action for the hidden fieldsis given merely by the first order variation S ,ζχ = δ g µν (cid:90) √− g L χ = 12 (cid:90) √− g T µνχ δ g µν , (A.12)where T µνχ is given by T µνχ = (cid:20) − g µν ∂ λ χ n ∂ λ χ n + ∂ µ χ n ∂ ν χ n (cid:21) (A.13)From (A.2), (A.6) and (A.7) we see that the first order metric variations can be read off as δ g µν = (cid:32) − ˙ ζH a ∂ i θa ∂ i θ a δ ij ζ (cid:33) (A.14)One can explicitly verify that the trace of the product of the above with (A.13) reproduces (A.11).We observe that one can write (A.14) as δ g µν = ∇ µ β ν + ∇ ν β µ + ∆ µν , (A.15)where β = − ζH , β i ≡ µν := (cid:15) (cid:18) ζ a ∂ i ∂ − ˙ ζa ∂ i ∂ − ˙ ζ (cid:19) (A.17)Clearly only the second term in (A.15) gives a non-vanishing contribution. Therefore the relevantcubic interactions are given by S ,ζχ = (cid:90) d x a (cid:15) (cid:20) ζ (cid:18) ˙ χ n ˙ χ n + 1 a ∂ i χ n ∂ i χ n (cid:19) − ˙ χ n ∂ i χ n ∂ i ∂ − ˙ ζ (cid:21) (A.18)where we take note of the advertised (cid:15) suppression of the cubic interaction vertices. The relevant cubicinteraction term for the tensor perturbations can be read off straightforwardly as S ,γχ = 12 (cid:90) d x a [ γ ij ∂ i χ n ∂ j χ n ] . (A.19)– 16 –or completeness, we note that the (cid:15) suppression accompanying factors of ζ extends to mixed verticesas well. For instance, consider the quartic ζγχχ interaction obtained from expanding (A.1) and solvingfor the lapse and shift constraints S ,γχζ = 12 (cid:90) d x a (cid:32) ˙ ζH + ζ (cid:33) [ γ ij ∂ i χ n ∂ j χ n ] . (A.20)A straightforward integration by parts of the first term in the round parenthesis brings the above tothe form S ,γχζ = − (cid:90) d x a (cid:15) ζ [ γ ij ∂ i χ n ∂ j χ n ] + ... (A.21)with the slow roll suppression now manifesting, and where the ellipses denote terms proportionalto ∂ t [ γ ij ∂ i χ n ∂ j χ n ] which can be eliminated via a suitable field redefinition using the backgroundequations of motion similar to the ones considered in [25]. B One loop correction to (cid:104) γγ (cid:105) Mindful of the cautions articulated in [15], we are interested in expanding (2.23) to second order: (cid:104) γ sij, k ( τ ) γ s (cid:48) ij, k (cid:48) ( τ ) (cid:105) = (cid:104) (cid:16) T e − i (cid:82) τ −∞ (1+ iε ) dτ (cid:48) H I ( τ (cid:48) ) (cid:17) † γ ,sij, k ( τ ) γ ,s (cid:48) ij, k (cid:48) ( τ ) (cid:16) T e − i (cid:82) τ −∞ (1+ iε (cid:48) ) dτ (cid:48) H I ( τ (cid:48) ) (cid:17) (cid:105) , (B.1)where we have switched to conformal time, with the interaction Hamiltonian given by H I = − (cid:88) r (cid:90) d x a (cid:104) γ ,rij ∂ i χ n ∂ j χ n (cid:105) , (B.2)where a sum over n is understood. Consistent with the normalizations (2.15) and (2.16), the free fieldsadmit the expansion χ a ( x , τ ) = (cid:90) d k (2 π ) e i k · x (cid:104) b a k χ k ( τ ) + b a †− k χ ∗ k ( τ ) (cid:105) (B.3) γ ,rij ( x , τ ) = (cid:90) d k (2 π ) e i k · x (cid:104) (cid:15) rij ( k ) a r k γ k ( τ ) + (cid:15) r ∗ ij ( − k ) a r †− k γ ∗ k ( τ ) (cid:105) (B.4)with creation and annihilation operators normalized as (cid:2) a s k , a r † q (cid:3) = (2 π ) δ sr δ ( k − q ) ; (cid:2) b a k , b b † q (cid:3) = (2 π ) δ ab δ ( k − q ) (B.5)and where the polarization tensors are normalized as (cid:15) rij ( k ) (cid:15) ∗ sij ( k ) = 4 δ rs ; with (cid:15) r ∗ ij ( k ) = (cid:15) rij ( − k ) (B.6)with mode functions χ k and γ k canonically normalized such that in the dS limit, they’re given by χ k ( τ ) = iH √ k (1 + ikτ ) e − ikτ (B.7) γ k ( τ ) = iH √ k M pl (1 + ikτ ) e − ikτ . (B.8)We note that there is one term with two interaction operators from the time ordered product in (B.1)plus one term with two operators from the anti-time ordered product, plus one term with interaction– 17 –perators from both the time ordered and the anti-time ordered products. The latter can be writtenas term II below, whereas the first two are complex conjugates of each other, and because of the factthat both contain the same lower limits, can be written as twice the integral over a triangle, which wegroup together to form term I: (cid:104) γ sij, k ( τ ) γ s (cid:48) ij, k (cid:48) ( τ ) (cid:105) (2) = − (cid:60) (cid:90) τ −∞ + dτ (cid:90) τ −∞ + dτ (cid:104) γ ,sij, k ( τ ) γ ,s (cid:48) ij, k (cid:48) ( τ ) H I ( τ ) H I ( τ ) (cid:105) (I)+ (cid:90) τ −∞ + dτ (cid:90) τ −∞ − dτ (cid:104) H I ( τ ) γ ,sij, k ( τ ) γ ,s (cid:48) ij, k (cid:48) ( τ ) H I ( τ ) (cid:105) (II) (B.9)where ∞ ± := ∞ (1 ± iε ). We note that (II) has different lower limits but identical upper limits, andwill turn out to be an absolute value. We insert into the above the expansion (and dropping the zerosuperscripts to denote free field operators) H I ( τ ) = − (cid:88) r (cid:90) d k d p (2 π ) a ( τ ) γ rlm, k ( τ ) χ a p ( τ ) χ a − k − p ( τ ) p l ( k m + p m ) (B.10)and similarly for H I ( τ ), where χ a k ( τ ) = b a k χ k ( τ ) + b a †− k χ ∗ k ( τ ) , (B.11)with creation and annihilation operators normalized as per (B.5), χ a k the mode functions (which dependonly on the magnitude of k ) and where γ rlm, k ( τ ) = (cid:15) rlm ( k ) a r k γ k ( τ ) + (cid:15) r ∗ lm ( − k ) a r †− k γ ∗ k ( τ ) . (B.12)In the basis where the graviton propagates along the z-direction, the polarization tensor correspondingto the normalization (B.6) is given by (cid:15) + lm ( k ) = i i − , (B.13)where the index r = ± . After Wick contracting, and utilizing the relations (B.6) and (B.13) todo the contractions with the remaining momenta, we find (working around a dS background with a = − ( Hτ ) − ) that term (II) is given by(II) = 2 N δ ss (cid:48) H δ ( k + k (cid:48) ) (cid:90) τ −∞ + dτ τ (cid:90) τ −∞ − dτ τ (cid:90) d q q sin θ (B.14) × (cid:40) χ q ( τ ) χ ∗ q ( τ ) χ q − k ( τ ) χ ∗ q + k (cid:48) ( τ ) γ k ( τ ) γ ∗ k ( τ ) γ k (cid:48) ( τ ) γ ∗ k (cid:48) ( τ )+ χ q ( τ ) χ ∗ q ( τ ) χ q − k (cid:48) ( τ ) χ ∗ q + k ( τ ) γ k (cid:48) ( τ ) γ ∗ k (cid:48) ( τ ) γ k ( τ ) γ ∗ k ( τ ) (cid:41) . This expression can be taken on shell as far as the wavefunctions are concerned, and using the explicitforms (B.7) and (B.8), we end up with the expression(II) = N H M δ ss (cid:48) δ ( k + k (cid:48) )(1 + k τ ) (cid:90) d q (cid:90) d ¯ q δ (¯ q + q + k ) q sin θk ¯ q × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) τ −∞ + dτ τ (1 + iqτ )(1 + i ¯ qτ )(1 + ikτ ) e − iτ ( q +¯ q + k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (B.15)– 18 –hich results in, after taking the τ → N H M δ ss (cid:48) δ ( k + k (cid:48) ) (cid:90) d q (cid:90) d ¯ q δ (¯ q + q + k ) q sin θk ¯ q (cid:20) τ + Γ − α + k (cid:21) (B.16)where Γ := ( k + q ¯ q )( q + ¯ q ) + k ( q + ¯ q + 4 q ¯ q )( k + q + ¯ q ) (B.17)and where we shall also define for future convenience α := kq ¯ q ( k + q + ¯ q ) (B.18)The apparently divergent term in the τ → H I and performing the relevantWick contractions and traces over polarization indices, one ends up with(I) = − N H M δ ss (cid:48) δ ( k + k (cid:48) ) (cid:90) d q (cid:90) d ¯ q δ (¯ q + q + k ) q sin θk ¯ q ×(cid:60) (cid:40) (1 − ikτ ) e ikτ (cid:90) τ −∞ + dτ τ (1 − iqτ )(1 − i ¯ qτ )(1 + ikτ ) e iτ ( q +¯ q − k ) (cid:90) τ −∞ + dτ τ (1 + iqτ )(1 + i ¯ qτ )(1 + ikτ ) e − iτ ( q +¯ q + k ) (cid:41) (B.19)Performing the τ integral results in the intermediate expression(I) = − N H M δ ss (cid:48) δ ( k + k (cid:48) ) (cid:90) d q (cid:90) d ¯ q δ (¯ q + q + k ) q sin θk ¯ q (B.20) ×(cid:60) (cid:40) (1 − ikτ ) e ikτ (cid:90) τ −∞ + dτ τ (1 − iqτ )(1 − i ¯ qτ )(1 + ikτ ) e − iτ k (cid:18) − τ + ατ − i Γ (cid:19) (cid:41) Performing the τ integral and taking the τ → − N H M δ ss (cid:48) δ ( k + k (cid:48) ) (cid:90) d q (cid:90) d ¯ q δ (¯ q + q + k ) q sin θk ¯ q (B.21) (cid:40) τ − k (cid:0) k − kα ( q + ¯ q ) + 2 q ¯ qα + 8 k ( q + ¯ q − Γ) + 3 kq ¯ q Γ + 2 k ( q ¯ q + α − ( q + ¯ q )Γ) (cid:1) (cid:41) We note that when one evaluates this integral, there are nominal terms that go like log k and log τ ,however these multiply a term whose coefficients cancel out once imposing the relations (B.17) and(B.18). Adding (I) and (II), the divergent term in 1 /τ cancels, leaving us with(I) + (II) = N H M δ ss (cid:48) k δ ( k + k (cid:48) ) (cid:90) d q (cid:90) d ¯ q δ (¯ q + q + k ) q sin θ ¯ q (B.22) (cid:40) k − α + q ¯ qαk + q (¯ q − Γ) + 4 k ( q + ¯ q − Γ) − ¯ q Γ + Γ − qα + q ( α − ¯ q Γ))2 k (cid:41) – 19 –e now make use of the identity (cid:90) d q d ¯ q δ ( q + ¯ q + k ) f ( q, ¯ q, k ) = 2 πk (cid:90) ∞ dq q (cid:90) q + k | q − k | d ¯ q ¯ q f ( q, ¯ q, k ) (B.23)which can be seen by first doing the ¯ q integral with the delta function, leaving us with 2 π (cid:82) d (cos θ q ) q dq ,and using the relation ¯ q = q + k + 2 qk cos θ q (B.24)to express d cos θ q = ¯ qd ¯ q/ ( qk ) to arrive at the above. One can also use (B.24) to writesin θ = (cid:0) q k − (¯ q − q − k ) (cid:1) q k (B.25)Therefore in (B.23), the integrand is f ( q, ¯ q, k ) = (cid:0) q k − (¯ q − q − k ) (cid:1) q k q ¯ q × (cid:40) k − α + q ¯ qαk + q (¯ q − Γ) + 4 k ( q + ¯ q − Γ) − ¯ q Γ + Γ − qα + q ( α − ¯ q Γ))2 k (cid:41) (B.26)At this stage, we need to evaluate the integral (B.23) via dimensional regularization. However thereare various subtleties one must keep track in doing this correctly (elaborated upon in [1]) which weaddress in what follows. B.1 Dimensional Regularization on a dS background
In order to dimensionally regularize the loop integral (B.23), we realize on dimensional grounds that (cid:90) d q d ¯ q δ ( q + ¯ q + k ) f ( q, ¯ q, k ) = k δ F ( δ ) (B.27)where F ( δ ) is a dimensionless constant which admits an expansion in powers of δF ( δ ) = F δ + F + ... (B.28)so that in the limit δ →
0, we find (cid:90) d q d ¯ q δ ( q + ¯ q + k ) f ( q, ¯ q, k ) = k ( F log k + Λ) (B.29)with Λ a divergent constant to be subtracted with appropriate counter-terms. Hence, multiplying theintegral by k and taking the fifth derivative w.r.t. k implies that the right hand side is given by 24 F /k and the left hand side is given by evaluating (B.23) using (B.26) to result in 12 π · / (5 k ), so that F = 12 π . (B.30)From this we conclude the contribution k π (cid:104) γ sij, k ( τ ) γ s (cid:48) ij, k (cid:48) ( τ ) (cid:105) (2) ⊃ N π H M δ ss (cid:48) δ ( k + k (cid:48) ) (cid:18) π k/µ ) + Λ (cid:19) . (B.31)– 20 –owever, as pointed out in [1], we are not done yet. As elaborate upon further in the next subsection,there are additional contributions coming from corrections to the dS mode functions themselves in D = 3 + δ dimensions: χ k ( τ ) = − √ π e iπδ/ H δ/ µ δ/ ( − kτ ) (3+ δ ) / k (3+ δ ) / H (1)(3+ δ ) / ( − kτ ) , (B.32) γ k ( τ ) = − √ π e iπδ/ H δ/ M pl µ δ/ ( − kτ ) (3+ δ ) / k (3+ δ ) / H (1)(3+ δ ) / ( − kτ ) . (B.33)Expanding in δ results in (for example) χ k ( τ ) = iH (cid:112) µ δ k (1 + ikτ ) e − ikτ (cid:20) δ − Hτ ) + δ u ( − kτ ) + ... (cid:21) (B.34)where u ( − kτ ) is given by [26] u ( − kτ ) = (cid:18) [Ci( − kτ ) + i Si( − kτ )] H (2)3 / ( − kτ ) − iπJ / ( − kτ ) − kτ H (1)1 / ( − kτ ) (cid:19) /H (1)3 / ( − kτ ) + iπ . (B.35)We focus on the δ log( − Hτ ) contribution first. Note that there are six contributions from an inversescale factor that is implicit in the canonical normalization of (B.32) and (B.33) for each mode functionentering the loop integral, to be multiplied by two factors of the scale factor from the integrationmeasures for τ and τ . As argued by Senatore and Zaldarriaga [1], the subsequent time integrals havethe form (cid:90) −∞ dτ (cid:90) τ −∞ dτ τ τ e iτ ( q +¯ q − k ) e − iτ ( q +¯ q + k ) [ ... ] , (B.36)which are dominated by contributions around τ ∼ τ ∼ /k – the time of horizon crossing of the k -mode – so that the net effect after multiplying the integrands in (B.15) and (B.19) with factorsof log( − Hτ , ) and performing the time integrals will be an identical momentum integration as in(B.22), but now with a single multiplicative factor of log( c (cid:48) H k τ k ) where c (cid:48) is some order one constant.Therefore the net effect of this correction will be to correct the r.h.s. of (B.27) with the additionalterm lim δ → δ × log( − cHτ k ) k δ F ( δ ) (B.37)so that (B.29) becomes (cid:90) d q d ¯ q δ ( q + ¯ q + k ) f ( q, ¯ q, k ) = k ( F log ( k/µ ) + F log( − Hτ k ) + Λ (cid:48) ) (B.38)= k ( F log( H/µ ) + Λ (cid:48) )where we absorb log c (cid:48) into Λ, and where we have used the horizon crossing relation − τ k = 1 /k .Now we consider the term proportional to u ( − kτ ) in (B.34), realizing that this term will be of theform δ × k ( k/µ ) δ (cid:101) F ( δ ), where (cid:101) F is dimensionless and is the result of the momentum integrations onehas to do after factoring in u ( − kτ ) corrections (B.35). In order for this to contribute a logarithmicrunning, (cid:101) F is required to have a double pole in δ , which can be shown not to be the case [1], so weare left with the expression k π (cid:104) γ sij, k ( τ ) γ s (cid:48) ij, k (cid:48) ( τ ) (cid:105) (2) ⊃ N π H M δ ss (cid:48) δ ( k + k (cid:48) ) (cid:18) π H/µ ) + Λ (cid:48) (cid:19) (B.39)– 21 –t this stage it might appear as if the correlation function does not run whatsoever. This cannot bethe case generically, as quantum corrections typically induce running unless we are at a fixed point ofthe theory e.g. in the dS limit where an exact dilatation (i.e. scale) invariance is realized. Since thearguments of [1] make it clear that corrections to the correlation functions are being forged as modesexit the horizon, it must be the case that what appears above is in fact H k – the Hubble scale at thetime the k -mode exits the horizon. We demonstrate this explicitly in the following by repeating theargument of [1] including slow roll corrections to the mode functions in the loop integral. B.2 Dimensional Regularization on a quasi dS background
We now retrace the steps above, but carefully considering slow roll corrections to the mode functions inthe loop integral away from the dS limit, erring on the side of detail. To this end, we work in the limitof constant but non-zero (cid:15) := − ˙ H/H , since we are only interested in the leading order corrections inslow roll. We note that the defining equation for (cid:15) can be solved explicitly in cosmological time – H ( t ) = 1 (cid:15)t + H − ∗ , (B.40)which we can integrate once more to obtain an exact expression for the scale factor a ( t ) = a (1 + (cid:15)H ∗ t ) /(cid:15) . (B.41)In the above, H ∗ is some pivot scale which we specify shortly. Note that in the limit (cid:15) →
0, the aboveis none other than the limiting expression for the usual exponential scale factorlim (cid:15) → a ( t ) = a e H ∗ t . (B.42)Switching to conformal time, (B.41) becomes a ( τ ) = 1( − H ∗ τ ) − (cid:15) (B.43)where the normalization is taken to be such that the scale factor is a = − / ( H ∗ τ ) in the dS limit.Everything so far is exact in the constant (cid:15) limit, but we are primarily interested in the limit (cid:15) (cid:28) a ( τ ) = 1( − H ∗ τ ) (cid:15) (B.44)Where from now on, we understand that we work to leading order in (cid:15) . We now note that in orderto arrive at the mode fucntions in D = 3 + δ spatial dimensions, we need to consider the action for ageneric member of the χ fields in conformal time – S = µ δ (cid:90) d δ x a δ [ χ (cid:48) χ (cid:48) − ∂ i χ∂ i χ ] (B.45)making the field redefinition v χ := z χ χ , where z χ = a δ/ µ δ/ (B.46)results in the canonically normalized (Mukhanov-Sasaki) action S = 12 (cid:90) d δ x (cid:20) v (cid:48) − ( ∂v ) + z (cid:48)(cid:48) z v (cid:21) . (B.47)– 22 –imilarly, the field redefinition with corresponding z γ = a δ/ µ δ/ M pl (B.48)will bring the action for the individual graviton polarizations into the form (B.47) . The equationsof motion for the Fourier modes that result from (B.47) will be of the form v (cid:48)(cid:48) k + (cid:18) k − ν − / τ (cid:19) v k = 0 (B.49)with the identification z (cid:48)(cid:48) z = ν − / τ = λ (1 + λ ) τ (B.50)where using (B.44), we find that for both (B.46) and (B.48) λ := (1 + (cid:15) ) (1 + δ/ . (B.51)From (B.50) we see that ν = λ + 1 /
2, which we write as ν = 3 + ¯ δ δ = δ (1 + (cid:15) ) + 2 (cid:15). (B.53)Given that the mode functions that solve (B.49) corresponding to the Bunch-Davies vacuum are givenby v k = √ π e i ( ν + ) π √− τ H (1) ν ( − kτ ) , (B.54)one readily obtains the relevant mode functions on a 3 + δ dimensional quasi de Sitter background χ k = v k /z χ and γ k = v k /z γ to be χ k ( τ ) = − √ π e iπ ¯ δ/ H δ/ ∗ µ δ/ ( − kτ ) (3+¯ δ ) / k (3+¯ δ ) / H (1)(3+¯ δ ) / ( − kτ ) , (B.55) γ k ( τ ) = − √ π e iπ ¯ δ/ H δ/ ∗ M pl µ δ/ ( − kτ ) (3+¯ δ ) / k (3+¯ δ ) / H (1)(3+¯ δ ) / ( − kτ ) . (B.56)which are identical to (B.32) and (B.33) with the replacement δ → ¯ δ everywhere except in the powerof µ δ in the denominator, which book keeps the mass dimension of the Fourier component fields. Asbefore, we expand in powers of ¯ δ to obtain χ k ( τ ) = iH ∗ (cid:112) µ δ k (1 + ikτ ) e − ikτ (cid:20) δ − H ∗ τ ) + ¯ δ ( − H ∗ τ ) + ¯ δ u ( − kτ ) + ¯ δ v ( − kτ ) + ... (cid:21) (B.57)with a similar expansion for γ k ( τ ). The function u ( − kτ ) is given by (B.35), now with ¯ δ given by (B.53),and v ( − kτ ) is a term that arises from the second derivative of the Hankel function with respect toits argument, whose explicit form will not be necessary in what follows. The reason we need to goto second order in ¯ δ can be anticipated from the fact that even though we neglect terms of order δ and (cid:15) , we still have ¯ δ = 4 (cid:15)δ , which can contribute leading slow roll corrections to finite terms whenmultiplying δ poles. Recalling that the 1 / – 23 – .2.1 Slow-roll corrected loop integral The net result of inserting the slow roll corrected mode functions and scale factor in the dimensionallyregularized integral (B.27) to second order in ¯ δ , is to effect the corrections (cid:90) d q d ¯ q δ ( q + ¯ q + k ) f ( q, ¯ q, k ) (B.58)= k (cid:18) kµ (cid:19) δ (cid:34)(cid:18) F δ + F (cid:19) (cid:18) δ log( − H ∗ τ k ) + ¯ δ ( − H ∗ τ k ) (cid:19) + 3¯ δ (cid:18) ¯ F δ + ¯ F (cid:19)(cid:35) The single log term is the result of multiplicative corrections of the form ¯ δ log[ − H ∗ τ k ] from each ofthe six momentum independent corrections (cf. footnote 21) to the mode functions that run throughthe loops in (B.57), compensated by the single log contributions from the dimensionally regularizedmeasures of each contributing interaction Hamiltonian: a δ ) = ( − H ∗ τ ) − (cid:15) )(2+ δ ) = 1( − H ∗ τ k ) (cid:0) − δ log( − H ∗ τ k ) + 2¯ δ log ( − H ∗ τ k ) (cid:1) (B.59)Similarly, one can collect all double log terms from the mode functions and the measure to result in thecontribution proportional to ¯ δ = 2 (cid:15)δ above . As argued in the previous subsection, all momentumindependent corrections serve to multiply the loop integral resulting from inserting the uncorrectedmode functions, but with log factors resulting from the fact that these integrals are dominated bycontributions at horizon crossing. The term proportional to 3¯ δ in the above is the leading ordercontribution from the loop integrations incorporating the momentum dependent corrections to themode functions, given by u ( − kτ ) and v ( − kτ ) in (B.57), which we will not need to calculate explicitlyin what follows. In spite of appearances, the last term of (B.58) does indeed factor in momentumdependent corrections to the wavefunctions of order ¯ δ = 4 (cid:15)δ , whose effects can simply be absorbedinto the definition of ¯ F and ¯ F .One immediately sees that slow roll corrections to the loop integrals result in additional δ polesthat go as (cid:15)/δ , which must correspond to slow roll corrections to the counter terms. We will now showthat this is explicitly the case by calculating these counterterms. The final outcome will be that thefactor in (B.39) becomes log( H ∗ /µ ) → log( H k /µ ) (B.60)where H k is the Hubble factor at horizon crossing of the comoving scale k . Let us now show thisexplicitly. We first observe that expanding (B.58), collecting terms, and recalling that ¯ δ = δ (1 + (cid:15) ) + 2 (cid:15) and ¯ δ = 4 (cid:15)δ , the net result of dimensionally regularizing the loop integral is the correction k π (cid:104) γ sij, k ( τ ) γ s (cid:48) ij, k (cid:48) ( τ ) (cid:105) (2) = N π H ∗ M δ ss (cid:48) δ ( k + k (cid:48) ) (cid:40) F (cid:20) log kµ + (1 + (cid:15) ) log H ∗ k (cid:21) + F δ (cid:20) (cid:15) log H ∗ k (cid:21) + F (cid:20) (cid:15) log H ∗ k (cid:21) + 2 (cid:15)F log H ∗ k + 2 (cid:15)F log kµ log H ∗ k + 6 (cid:15) ¯ F log kµ + 6 (cid:15) ¯ F δ + 3(1 + (cid:15) ) ¯ F + 6 (cid:15) ¯ F (cid:41) (B.61)where the highlighted term in the above is only contribution that will survive after we subtractthe divergences and imposing the renormalization conditions at some finite scale. As is standard in This comes from collecting all terms quadratic in ¯ δ log ( − H ∗ τ k ) from the product of six factors of the term in thesquare brackets of (B.57) and the logarithmic contributions from the measures (B.59). – 24 –ffective field theory, one is entitled to pick the renormalization scale to minimize the logarithms at anyparticular scale of interest (typically representing some mass threshold), which in the present contextis naturally given by µ = H ∗ . Doing so results in the cancellation of the double log contributions, sothat k π (cid:104) γ sij, k ( τ ) γ s (cid:48) ij, k (cid:48) ( τ ) (cid:105) (2) = N π H ∗ M δ ss (cid:48) δ ( k + k (cid:48) ) (cid:40) F (cid:20) log kµ + (1 + (cid:15) ) log H ∗ k (cid:21) + F δ (cid:20) (cid:15) log H ∗ k (cid:21) + F (cid:20) (cid:15) log H ∗ k (cid:21) + 6 (cid:15) ¯ F log kµ + 6 (cid:15) ¯ F δ + 3(1 + (cid:15) ) ¯ F + 6 (cid:15) ¯ F (cid:41) (B.62)We first focus on the highlighted term to show that indeedlog kµ + (1 + (cid:15) ) log H ∗ k = log H k µ (B.63)From (B.44) we see that H = ˙ a/a = a (cid:48) /a = H ∗ (1 + (cid:15) )( − H ∗ τ ) (cid:15) . Furthermore, the horizon crossingcondition k = aH implies that the k -mode exits the Hubble radius at τ k = − (1+ (cid:15) ) k , so that H k := H ( τ k ) = H ∗ (1 + (cid:15) ) (cid:18) H ∗ (1 + (cid:15) ) k (cid:19) (cid:15) = k (cid:18) H ∗ (1 + (cid:15) ) k (cid:19) (cid:15) = k (cid:18) H ∗ k (cid:19) (cid:15) , (B.64)where the difference between the second and third terms is higher order in (cid:15) .Hence H k /k = ( H ∗ /k ) (cid:15) and the final equality in (B.63) immediately follows. It now remains to show that all the un-highlightedterms in (B.62) will be subtracted by the relevant slow-roll corrections to the counterterms. B.2.2 Slow-roll corrected counterterms
We are calculating in an effective theory with a cut-off scale Λ. As discussed in the following section ofthe appendix, since we are considering only the spin two perturbations generated from the Einstein-Hilbert term in the presence of N scalar fields, this cut-off is given byΛ ∼ πM pl √ N (B.65)Where we absorb any order unity coefficients of the precise cutoff in the coefficients of the countertermswe are about to calculate. From the structure of the divergences, we see that these counterterms willtake the form of all possible dimension six operators consistent with the symmetries of the problem.Hence H c.t. = µ δ N (cid:90) d δ x a δ (cid:2) c ∂ k γ (cid:48) ij ∂ k γ (cid:48) ij + c ∂ l ∂ k γ ij ∂ l ∂ k γ ij + c γ (cid:48)(cid:48) ij γ (cid:48)(cid:48) ij (cid:3) (B.66)where we note that the pre-factor of M has been canceled in dividing by Λ . We also note a non-standard feature of the effective theory of inflation – although the coefficients of the dimension sixoperators are constants (i.e. Wilson co-efficients), one could in principle add counterterms where thecoefficients are functions of time (i.e. Wilson functions ), and appear at dimension six as operators ofthe form ∆ H c.t. = µ δ N (cid:90) d δ x a δ (cid:2) c ( aH ) γ (cid:48) ij γ (cid:48) ij + c ( aH ) ∂ k γ ij ∂ k γ ij (cid:3) (B.67)with c , c dimensionless. That these must appear can be seen from the fact that field redefinitionsusing the unperturbed equations of motion will necessarily generate such terms. Although they will– 25 –ot be necessary for the renormalization we are about to perform, loop corrections to more complicatedprocesses will necessitate them, particularly if we are interested in calculating corrections to correlationfunctions at finite times (and not τ → (cid:104) γ sij, k ( τ ) γ s (cid:48) ij, k (cid:48) ( τ ) (cid:105) c.t. = i (cid:90) τ −∞ − (cid:104) [ H c.t. ( τ (cid:48) ) , γ ,sij, k ( τ ) γ ,s (cid:48) ij, k (cid:48) ( τ )] (cid:105) dτ (cid:48) (B.68)= 2 (cid:61) (cid:90) τ −∞ − (cid:104) γ ,sij, k ( τ ) γ ,s (cid:48) ij, k (cid:48) ( τ ) H c.t. ( τ (cid:48) ) (cid:105) dτ (cid:48) = 64 N (2 π ) µ δ δ ss (cid:48) δ ( k + k (cid:48) ) γ k ( τ ) × (cid:61) (cid:90) τ −∞ − dτ (cid:48) a δ (cid:8) c k γ (cid:48)∗ k ( τ (cid:48) ) + c k γ ∗ k ( τ (cid:48) ) + c γ (cid:48)(cid:48)∗ k ( τ (cid:48) ) (cid:9) It is straightforward to calculate the above with the first order corrections to the dimensionally de-formed, slow roll corrected mode functions (B.57) using (B.35), including slow roll corrections to thedimensionally deformed measure a δ . The resulting integrals can be performed analytically in the τ → . The counterterm contributions to the regularized dimensionless power spectrum aregiven by k π (cid:104) γ sij, k ( τ ) γ s (cid:48) ij, k (cid:48) ( τ ) (cid:105) c.t. = 16 πN δ ss (cid:48) δ ( k + k (cid:48) ) H ∗ M (cid:40) (cid:15) ( − c + c + 5 c ) − c (1 + (cid:15) ) δ + ( c − c − c ) (cid:20) (cid:15) (cid:18) log H ∗ k − γ E − log 2 + 2 (cid:19)(cid:21)(cid:41) (B.69)We note that the terms proportional to 2 (cid:15) in the second line above are simply the first order slow rollcorrection to the square of the wavefunction modulus at long wavelengths. Furthermore, in the com-bination 1 + 2 (cid:15) log( H ∗ /k ), we see precisely the slow roll correction to the counterterm needed to cancelthe single δ pole encountered in the dS case. However, other δ poles now appear in (B.62), in additionto to finite terms that must be fixed by renormalizing at a particular scale. It is straightforward tocheck that the choices 128 π c = 1 δ (cid:20)
12 (2 − (cid:15) ) ¯ F + (cid:15) ¯ F (cid:21) (B.70)128 π ( − c + 5 c + c ) = F + 6 (cid:15) ¯ F δ + F − F (B.71)128 π ( c − c − c ) = (cid:18) F − F δ − F (cid:19) [2 − γ E − log 2] (B.72)cancel all unhighlighted terms in (B.62), thus fixing the renormalization condition at µ = H ∗ , whichwe take to be the pivot scale at which we fix the tensor to scalar ratio (i.e. the loop correction isnormalized to vanish there). Hence k π (cid:104) γ sij, k ( τ ) γ s (cid:48) ij, k (cid:48) ( τ ) (cid:105) (2) = N π H ∗ M δ ss (cid:48) δ ( k + k (cid:48) ) F log H k H ∗ (B.73) We note that we didn’t need to dimensionally regularize the mode functions that arose from the external legs inthe loop integral (B.22) since these will be compensated by identical terms from the external legs in the counterterms[1]. The net result is equivalent to only dimensionally regularizing terms that depend on loop momenta. Similarly, wedon’t need to consider slow roll corrections to the external wavefunctions in the above since these will simply add a tiltto the overall spectral index multiplying both the loop integrals and the counterterm contributions. – 26 –ith F = 12 π/ k/k ∗ runningto the loop correction, but with extra (cid:15) suppression and with the opposite sign. This follows fromthe fact that log H k H ∗ = − (cid:15) ∗ log kk ∗ + O ( (cid:15) ∗ ), which we see as follows. Consider the expression (B.44) aswell as H = a (cid:48) /a where prime denote derivatives w.r.t. τ . This gives H = (1 + (cid:15) ) H ∗ ( − H ∗ τ ) (cid:15) . With − τ k = k , recalling that we have identified H ∗ as the pivot scale at which we measure the tensor toscalar ratio, and (cid:15) = (cid:15) ∗ , we obtain log H k H ∗ = − (cid:15) ∗ log kk ∗ + O ( (cid:15) ∗ ) . (B.74)so that k π (cid:104) γ sij, k ( τ ) γ s (cid:48) ij, k (cid:48) ( τ ) (cid:105) (2) = − (cid:15) ∗ N π H ∗ M δ ss (cid:48) δ ( k + k (cid:48) ) F log kk ∗ . (B.75)Comparison with (2.16) after summing over polarizations implies the loop correction to the tensorpower spectrum ∆ P γ, − loop = − (cid:15) ∗ H ∗ π M N π H ∗ M
35 log kk ∗ . (B.76) C The strong coupling bound for gravity
We first present an effective field theory derivation of the strong coupling bound for gravity (asdiscussed in [11, 12]), namely that in the presence of N species, one can only treat gravity semi-classically for scales less than Λ (cid:46) πM pl √ N . (C.1)This is easily seen from the fact that were we only interested in calculating n-point correlation functionsof gravitons, then the relevant quantities can be reproduced from the effective action where the matterfields have been integrated out [18, 38] S = S EH + ∆ W + ... (C.2)where S EH is the usual Einstein-Hilbert action, and where the leading quadratic curvature correctionshave the form (cid:52) W = 12880 π (cid:90) √− g (cid:34) ( b + 2 a ) R µν R µν + (cid:16) c − b + 2 a (cid:17) R (cid:35) (C.3)where the coefficients a, b, c depend on the spin (and in the case of scalars, the non-minimal couplingparameter ξ ) of the field integrated out [38]:spin a b c0 1 1 90( ξ − / − / −
11 01 −
13 62 02 212 0 717/4– 27 –ote that in the above, we have eliminated redundant operators by using the Gauss-Bonnet identityin 4D. For N species of various spins, the coefficients that sit in front of the curvature squared termswill be a weighted sum of the above. From this, one can see that the contribution from the quadraticterms in the effective action become comparable to those from the Einstein-Hilbert term at momentumtransfers approaching p ∼ κM /N , or for background curvatures approaching R ∼ κ M N (C.4)where κ is a spin weighted sum, and N is the number of species we have integrated out. For N mini-mally coupled scalar fields, we find from (C.3) and the table above that on any maximally symmetricbackground, κ = 40 · π . On a dS background, R = 12 H so that the implication of the strongcoupling bound is that H (cid:46)
103 16 π M N , (C.5)or that the effective theory is only reliable for momentum transfers up to the scale Λ (cid:46) πM pl / √ N ,where we have neglected an order unity spin dependent coefficient.Another argument presented in [12] concerns black hole evaporation, where it was argued thatblack holes of the size Λ − also have a lifetime of Λ − , suggesting that this is the scale at whichquantum gravity becomes relevant. Consider the evaporation rate for a black hole of mass M [42]: dMdt = − εσT H A (C.6)where σ is the Stefan-Boltzmann constant, T H is the Hawking temperature of the black hole and ε is the grey body emmissivity factor that is proportional to the number of species the black hole canradiate: ε ∝ N. (C.7)Denoting the constant of proportionality c ε so that ε = c ε N , we then have dMdt = − N c ε σ M πM (C.8)where we have used the fact that the Schwarzschild radius of a black hole of mass M is given by R s = M/ (4 πM ) and T H = M /M in reduced Planck units, so that (C.8) can be expressed as τ BH = 4 πN (cid:90) M in M dMc ε σM = 4 πN M c ε σM , (C.9)where M in is the initial size of the black hole. Now consider a black hole of initial radius R s = M in πM := Λ − , (C.10)where Λ QG is understood to be defined by the above. Setting the lifetime to equal the size of the blackhole allows us to determine the scale at which quantum gravity must become relevant. Setting (C.9)equal to (C.10), and using the latter to express M in in terms of Λ QG , we see that this is whenΛ QG ∼ λ πM pl √ N (C.11)with λ = 4 π/ (3 σc ε ). Given that σ = π k B / (60 (cid:126) c ) ∼ . c ε will necessarily less than unity for a grey body, we see that λ > eferences [1] L. Senatore and M. Zaldarriaga, “On Loops in Inflation,” JHEP , 008 (2010) [arXiv:0912.2734[hep-th]].[2] A. H. Guth, “The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems,”Phys. Rev. D (1981) 347.[3] A. D. Linde, “A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness,Homogeneity, Isotropy and Primordial Monopole Problems,” Phys. Lett. , 389 (1982).[4] V. F. Mukhanov and G. V. Chibisov, “Quantum Fluctuations and a Nonsingular Universe,” JETP Lett. (1981) 532 [Pisma Zh. Eksp. Teor. Fiz. (1981) 549].[5] A. A. Starobinsky, “Spectrum of relict gravitational radiation and the early state of the universe,”JETP Lett. (1979) 682 [Pisma Zh. Eksp. Teor. Fiz. (1979) 719].[6] X. Chen and Y. Wang, “Large non-Gaussianities with Intermediate Shapes from Quasi-Single FieldInflation,” Phys. Rev. D , 063511 (2010) [arXiv:0909.0496 [astro-ph.CO]].[7] X. Chen and Y. Wang, “Quasi-Single Field Inflation and Non-Gaussianities,” JCAP , 027 (2010)[arXiv:0911.3380 [hep-th]].[8] R. Saito and T. Kubota, “Heavy Particle Signatures in Cosmological Correlation Functions with TensorModes,” arXiv:1804.06974 [hep-th].[9] A. del Rio, R. Durrer, S. P. Patil, in preparation .[10] N. Aghanim et al. [Planck Collaboration], “Planck 2015 results. XI. CMB power spectra, likelihoods,and robustness of parameters,” Astron. Astrophys. , A11 (2016) [arXiv:1507.02704 [astro-ph.CO]].[11] G. Dvali, “Black Holes and Large N Species Solution to the Hierarchy Problem,” Fortsch. Phys. , 528(2010) doi:10.1002/prop.201000009 [arXiv:0706.2050 [hep-th]].[12] G. Dvali and M. Redi, “Black Hole Bound on the Number of Species and Quantum Gravity at LHC,”Phys. Rev. D , 045027 (2008) [arXiv:0710.4344 [hep-th]].[13] G. Dvali and M. Redi, “Phenomenology of 10 Dark Sectors,” Phys. Rev. D , 055001 (2009)[arXiv:0905.1709 [hep-ph]].[14] N. Arkani-Hamed, T. Cohen, R. T. D’Agnolo, A. Hook, H. D. Kim and D. Pinner, “Solving theHierarchy Problem at Reheating with a Large Number of Degrees of Freedom,” Phys. Rev. Lett. ,no. 25, 251801 (2016) [arXiv:1607.06821 [hep-ph]].[15] P. Adshead, R. Easther and E. A. Lim, “The ’in-in’ Formalism and Cosmological Perturbations,” Phys.Rev. D , 083521 (2009) [arXiv:0904.4207v3 [hep-th]].[16] S. Weinberg, Phys. Rev. D (2005) 043514.[17] P. Creminelli, S. Dubovsky, A. Nicolis, L. Senatore and M. Zaldarriaga, “The Phase Transition toSlow-roll Eternal Inflation,” JHEP , 036 (2008) [arXiv:0802.1067 [hep-th]].[18] N. D. Birrell and P. C. W. Davies, “Quantum Fields in Curved Space,”[19] C. Cheung, P. Creminelli, A. L. Fitzpatrick, J. Kaplan and L. Senatore, JHEP , 014 (2008)[arXiv:0709.0293 [hep-th]].[20] V.F. Mukhanov, Physical Foundations of Cosmology , Cambridge University Press (2005).[21] R. Durrer,
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