Relations between canonical and non-canonical inflation
DDecember 17, 2012 DESY 12-245
Prepared for submission to JHEP
Relations between canonical and non-canonical inflation
Rhiannon Gwyn, a Markus Rummel b and Alexander Westphal ca AEI Max-Planck-Institut f¨ur Gravitationsphysik, D-14476 Potsdam, Germany b II. Institut f¨ur Theoretische Physik der Universit¨at Hamburg, D-22761 Hamburg, Ger-many c Deutsches Elektronen-Synchrotron DESY, Theory Group, D-22603 Hamburg, Germany
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We look for potential observational degeneracies between canonical andnon-canonical models of inflation of a single field φ . Non-canonical inflationary mod-els are characterized by higher than linear powers of the standard kinetic term X inthe effective Lagrangian p ( X, φ ) and arise for instance in the context of the Dirac-Born-Infeld (DBI) action in string theory. An on-shell transformation is introducedthat transforms non-canonical inflationary theories to theories with a canonical ki-netic term. The 2-point function observables of the original non-canonical theoryand its canonical transform are found to match in the case of DBI inflation. a r X i v : . [ h e p - t h ] F e b ontents c s <
176 Conclusions 19A Accessing the non-canonical regime in DBI 20B Off-shell canonical transformations 24
B.1 DBI + quadratic potential 26B.2 DBI + “inflection point potential” 26
Since the first conclusive detection [1] of the ∆
T /T = O (10 − ) temperature fluc-tuations in the cosmic microwave background radiation (CMB), a new concordancepicture of cosmology has been established. This is supported by vastly increased ob-servational precision in CMB measurements [2–6] as well as measurements of theredshift-distance relation for large samples of distant type IA supernovae [7, 8],baryon acoustic oscillations (BAO) [9], and the Hubble parameter H by the Hub-ble Space Telescope key project [10]. The results paint a Universe very close tobeing spatially flat, where large-scale structure originates from a pattern of coher-ent acoustic oscillations in the early dense plasma which was seeded by an almostscale-invariant power spectrum of super-horizon size curvature perturbations withGaussian distribution. These initial conditions arise as a direct consequence of awide class of models of cosmological inflation driven by the potential energy of ascalar field. An inflationary origin of the observed curvature perturbation spectrum– 1 –redicts in addition the existence of a similarly almost scale-invariant power spec-trum of super-horizon size primordial gravitational waves. The magnitude of this‘tensor mode’ power spectrum, and in turn its detectability, is determined by theenergy scale at which inflation took place.Such almost scale-invariant power spectra of super-horizon size curvature pertur-bations and tensor mode perturbations with Gaussian distribution can be describedat the Gaussian level by just three observational quantities: The overall normal-ization ∆ s of the curvature perturbation power spectrum (known since the COBEmeasurements [1]), its spectral tilt n s , describing the (small) deviations from scaleinvariance expected in most models of inflation, and the fractional power r in tensormodes. n s has been constrained by various combinations of the WMAP satelliteCMB results [2] with type IA SN, BAO and H data. In combination with the re-cently released ca. two-and-a-half full-sky surveys of PLANCK CMB temperaturedata [5, 6], and earlier the 2012 Atacama Cosmology Telescope [3], and South PoleTelescope CMB data [4], this led to an unambiguous > σ detection of a red tilt n s <
1. The tensor mode power fraction r is so far subject to an upper bound, mostrecently improved to r < .
12 (95%) by the PLANCK analysis [5, 6]. A future anal-ysis of data of the PLANCK satellite CMB B-mode polarization results as well asfuture polarized ground-based CMB detectors may substantially sharpen this upperbound in the next few years.Inflationary theory determines these three numbers in terms of the value of thescalar potential V at the time when the largest observable scales exited the infla-tionary horizon (about 60 e-folds before the end of inflation), and its first and secondderivatives V (cid:48) , V (cid:48)(cid:48) with respect to the inflation scalar field φ at that time. This im-plies that there are huge classes of scalar potentials V ( φ ) even for single-field modelswhich yield identical predictions for ∆ s , n s , and r .In any attempt to connect data with theory, potential degeneracies must be takeninto account before any conclusions can be drawn. In this context it is importantto understand the structure of this very large model space, and look for degenera-cies between large classes of inflationary models with respect to the three observablequantities. We will restrict our attention here to single-field models of inflationwhich partition into two large classes: models with a canonically normalized ki-netic term ( ∂ µ φ ) , and so-called non-canonical inflation models with Lagrangian L = p (( ∂ µ φ ) , φ ). Non-canonical inflation has been studied field-theoretically inthe context of k-inflation [11], and within string theory in DBI-inflation [12]. Inboth cases the function p (( ∂ µ φ ) , φ ) can be written as an (infinite) sum over higherpowers of the derivative ( ∂ µ φ ) with potentially field- dependent pre-factors. Theseterms can lead to additional effective friction terms in the equations of motion forthe inflaton. They can slow down the rolling of the scalar field into a regime of vac-– 2 –um energy domination for potentials which would be too steep to do so in presenceof a canonically normalized kinetic term alone. More general studies of such non-canonical models of inflation can be found in [13], while the effective field theory ofinflationary quantum fluctuations in such general settings is discussed in [14]. Non-canonical inflation quite generally leads to appreciable levels of non-Gaussianity ofthe inflationary quantum fluctuations [11, 12], which has been analyzed more gener-ally in [15], and has its full effective field theory treatment in [14].We will look at the question of whether there are degeneracies between canonicaland non-canonical models of inflation with respect to the three observational quan-tities describing their predicted power spectra at the Gaussian level. This questionhas been attacked from the point of view of reconstructing the inflationary actionfrom observables using Monte Carlo simulations in [16]. The method of canonical transformations for transforming noncanonical kineticterms into canonical kinetic terms, even in 0 + 1D, appears to be limited to thecase where the noncanonical theory has a quadratic potential, as we elucidate inAppendix B. Therefore we work here at the level of the action and of the inflationarysolution itself. While formally non-canonical 2-derivative models of the form L = f ( φ )( ∂ µ φ ) − V ( φ ) can always be transformed off-shell by a local field redefinition intoa canonical model with a transformed scalar potential, this question is rather non-trivial in the presence of higher-power kinetic terms. As the inflationary behaviorof a given model is described in terms of a generalized slow-roll attractor solutionin phase space, we will look at possible on-shell transformations of a given non-canonical model on its inflationary attractor into an equivalent canonical slow-rollinflation model. We find the general formalism for performing this matching oftrajectories, which will give the canonical potential V can leading to slow-roll inflationin a canonical theory, with inflationary trajectory X inf ( φ ) matching exactly that inthe given non-canonical model. This matching is quite general.Furthermore, the 2-point observables ∆ s , n s , and r are shown, numerically andanalytically, to match in the case of DBI inflation, over a range of efolds. This de-generacy is nontrivial, and seen for a large range of field values well outside of thecanonical regime of DBI. It could not be resolved with the currently available data atthe 2-point level, requiring a measurement of the ratio of r and n T to distinguish thetwo theories. Note that 3-point function observables, i.e. non-Gaussianities, whilegenerally negligible in single-field canonical inflationary models, can be appreciablein certain special cases. A sum of oscillating terms in the potential can lead to an ap-proximately equilateral-type non-Gaussian signal [21], while coupling of the inflatonto gauge quanta can also give rise to large equilateral-type non-Gaussianity [22, 23]. Earlier work towards reconstructing the inflationary potential was done for a canonical scalarfield in [17], and for a general action with noncanonical kinetic terms in [18–20]. – 3 –
This becomes even more interesting given that the analysis of the recent PLANCKCMB temperature data constrained local-shape non-Gaussianity arising from multi-field inflation models with f loc.NL = 2 . ± . f equil.NL = 42 ±
75 [25]. Hence, a matching of the 2-point function observables canin principle be extended to 3-point function observables by adding additional cou-plings or features to the potential of the canonical theory. We find matching of the2-point function observables to be possible precisely for the case of DBI inflationwhile failing for simple classes of DBI-inspired generalizations. This may point toa special status for DBI inflation as a member of the non-canonical class in thatit can be related to a canonical model of inflation with matching 2-point functionobservables.Our discussion proceeds as follows. In Section 2 we review briefly the relevantaspects of non-canonical inflation, while in Section 3 we discuss the on-shell transfor-mation of a non-canonical model into a canonical one on the inflationary attractorof the non-canonical model. Section 4.1 discusses the relation of the 2-point func-tion observables under the transformation between several classes of non-canonicalinflation with a speed limit inspired by and including DBI inflation and their associ-ated canonical models. Our main example, DBI inflation, we analyze in Section 4.2.Section 5 treats the corrections from typically the reduced speed of sound in non-canonical inflation to the 2-point function observables, and we conclude in Section 6.There are two appendices which contain a short discussion of the accessibility of thenon-canonical regime for DBI inflation (Appendix A), and an analysis of possibleoff-shell transformations between non-canonical and canonical theories using a formof canonical transformations (Appendix B).
We study inflationary dynamics of a single scalar field φ minimally coupled to gravityvia S = (cid:90) d x √ g (cid:20) M p R + p ( X, φ ) (cid:21) , (2.1)with X ≡ − ( ∂ µ φ ) = ˙ φ / ds = − dt + a ( t ) dx .From an effective field theory point of view, we expect the function p ( X, φ ) to Note that such models may be subject to a strong bound on the power spectrum coming fromthe non-detection of primordial black holes [24]. – 4 –ave the form p ( X, φ ) = (cid:88) n ≥ c n ( φ ) X n +1 Λ n − V ( φ ) = Λ S ( X, φ ) − V ( φ ) , (2.2)with some cutoff scale Λ. In this work, we will restrict ourselves to the case where thecoefficients c n are not field dependent, i.e. c n ( φ ) = c n , such that p ( X, φ ) is separable,i.e. p ( X, φ ) = Λ S ( X ) − V ( φ ) . (2.3)A theory is intrinsically non-canonical if the higher order kinetic terms X n with n > H = ˙ aa , (cid:15) = − ˙ HH , η = ˙ (cid:15)H (cid:15) , κ = ˙ c s H c s , c s = (cid:18) X ∂ p/∂X ∂p/∂X (cid:19) − , (2.4)which reduce to H = ˙ aa , (cid:15) = (cid:15) V = 12 (cid:18) V (cid:48) V (cid:19) , η = 4 (cid:15) V − η V , η V = V (cid:48)(cid:48) V , κ = 0 , c s = 1 . (2.5)in the canonical case p ( X, φ ) = X − V ( φ ). The equations of motion can be derivedas the Friedmann equations of a perfect fluid H = 13 M p ρ , ¨ aa = − M p ( ρ + 3 p ) , (2.6)with pressure p = p ( X, φ ) and energy density ρ = 2 X ∂p∂X − p . (2.7)Inflationary solutions p inf (cid:39) − ρ inf to eq. (2.6) can be found as algebraic solutions X inf = X ( A ) to the equation [13] (cid:114) X Λ ∂p∂X = A , (2.8)with the ‘non-canonicalness’ parameter A ( φ ) = V (cid:48) H Λ . (2.9)– 5 –or A (cid:28) p ( X, φ ) (cid:39) X − V ( φ ) while for A (cid:29) X n with n > X/ Λ < R of S ( X ), it was shownthat a truly non-canonical inflationary solution of eq. (2.8) with A (cid:29) • The derivative ∂ X S ( X ) diverges at the radius of convergence R . • The potential is large in units of the cutoff scale, i.e. V (cid:29) Λ such that theenergy density of the potential always dominates that of the kinetic terms .Note that a finite radius of convergence implies a speed limit X < R Λ . Theorieswithout a speed limit with a p ( X, φ ) monotonically increasing in X might lose validityfor X >
Λ as an effective field theory.The scalar power spectrum ∆ s , the tensor power spectrum ∆ t , the scalar spectralindex n s and the tensor spectral index n t can then be calculated via [11, 15]∆ s ( k ) = 18 π H M p c s (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) c s k = aH , ∆ t ( k ) = 2 π H M p (cid:12)(cid:12)(cid:12)(cid:12) k = aH ,n s ( k ) − − (cid:15) − η − κ | c s k = aH ,n t ( k ) = − (cid:15) | k = aH . (2.10)In the canonical case, this reduces to∆ s ( k ) = 18 π H M p (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) k = aH , ∆ t ( k ) = 2 π H M p (cid:12)(cid:12)(cid:12)(cid:12) k = aH ,n s ( k ) − − (cid:15) − η | k = aH ,n t ( k ) = − (cid:15) | k = aH . (2.11) In any theory, canonical or non-canonical with scalar field χ , the inflationary solutioncan be expressed as a function X inf ( χ ). We want to obtain the solution X inf ( φ ) from Note that the effective field theory description is valid as long as
H <
Λ. This generally allowslarge values of the potential in terms of the cutoff scale since H Λ (cid:39) (cid:0) V Λ (cid:1) / M p . – 6 – canonically normalized Lagrangian with scalar field φ and potential V can ( φ ). Inother words we want to find V can ( φ ) such that the slow-roll inflationary solution ofthe action P = X − V can ( φ ), X caninf ( φ ), has the same functional form as the inflationarysolution X inf ( χ ) coming from a general P ( X, χ ). In the following we describe howto construct V can ( φ ).In a canonically normalized theory that allows slow-roll inflation, the equations ofmotion are approximately˙ φ (cid:39) − V (cid:48) can ( φ )3 H ( φ ) , H ( φ ) (cid:39) V can ( φ )3 , (3.1)where (cid:48) denotes the derivative with respect to φ . Using ˙ φ = −√ X we obtain X (cid:39)
16 ( V (cid:48) can ) V can (3.2) √ X dφ = 1 √ V can dV can , (3.3)where the first expression is a slow-roll approximation (see e.g. [13]). At this pointwe replace the approximation with an equal sign, since we are looking for a potentialwhich satisfies the slow-roll conditions. Now, going on-shell X = X inf ( χ ) and hence dφ = dχ we can integrate both sides of eq. (3.3) to solve for V can : (cid:90) φφ (cid:113) X inf ( χ ) dχ = (cid:90) VV can dV can √ V can , ⇒ V can ( φ ) = (cid:32)(cid:112) V can + (cid:90) φφ (cid:114) X inf ( χ ) dχ (cid:33) , (3.4)with V can = V can ( φ ). Eq. (3.4) can be seen as an on-shell transformation of theoriginally possibly non-canonical theory to a canonical theory. It gives us the poten-tial V can , whose dynamics described in eq. (3.1) give exactly the same trajectory inphase space as in the original theory. In other words, given an inflationary trajectoryin a theory with general kinetic term, we have derived the form of the potential in atheory with canonical kinetic term which will give rise to the same inflationary trajec-tory. We assume that the kinetic term is canonical and X = X caninf = X noncaninf = X inf ,and find the corresponding V can . This is not a field transformation, since we simplymatch the inflationary trajectory in two different theories. Hence for any propertiesregarding the inflationary background solution the fields χ and φ are the same whiletheir general dynamics governed respectively by their non-canonical and canonicalLagrangians are different. Note that we are free to choose V (an integration con-stant) to satisfy the slow-roll conditions, since we are explicitly looking for a slow-roll– 7 –olution in a canonical theory with the same inflationary trajectory as that arisingfrom some given non-canonical theory. If the original theory is canonical with potential V ( χ ), the inflationary trajectoryis given by [13] X caninf = ( V (cid:48) ) V = ( V (cid:48) can ) V can , (3.5)such that V can ( φ ) = V ( χ ). In this section, we compare the number of efolds N e , the scalar power spectrum ∆ s ,the tensor power spectrum ∆ t and the scalar spectral index n s of non-canonical andcanonical inflation. We discuss under what conditions these observables will matchfor a non-canonical theory and a canonical theory whose potential is obtained viaeq. (3.4) such that it describes the same dynamics as the non-canonical theory.The natural time measure during inflation is the number of efolds N e that inflationproduces in the time interval [ t i , t f ]. It is defined as N e = (cid:90) t f t i H ( t ) dt = (cid:90) φ end φ Ne H ( φ )˙ φ dφ = (cid:90) φ Ne φ end (cid:18) V ( φ )6 X inf ( φ ) (cid:19) / dφ , (4.1)where in the last equation we have used H (cid:39) V / and ˙ φ = −√ X on theinflationary trajectory in phase space and φ end is the field value when inflation ends.In the case of a canonically normalized Lagrangian, this reduces to N e = (cid:90) φ Ne φ end V ( φ ) V (cid:48) ( φ ) dφ = (cid:90) φ Ne φ end √ (cid:15) dφ . (4.2)The observables are evaluated as functions of the comoving momentum k . Due tothe fact that the sound speed c s is generically different from one, the time of horizoncrossing for scalar modes is different from the time of horizon crossing for tensormodes. In terms of efolds N e , the different times of horizon crossing are determinedvia scalar modes: c s k = aH ⇔ ln k = ( N e − ln c s ) + ln H , tensor modes: k = aH ⇔ ln k = N e + ln H . (4.3) Here we work on-shell, which is to say at the level of the background equation of motion, ratherthan performing an off-shell field transformation at the level of the action. Offshell transformationsbetween canonical and non-canonical theories are discussed in Appendix B, where we show thatcanonical transformations can be used to transform between canonical and noncanonical theoriesin the case that the theory with noncanonical kinetic term has a dominantly quadratic potential.This method thus appears to be somewhat limited. In the following we restrict our analysis to non-canonical theories where the energy density isdominated by the potential, i.e., H (cid:39) V / – 8 –ence, the moment of horizon crossing of the scalar modes is earlier than that of thetensor modes and the correction is logarithmic in c s with ln c s < c s < c s (cid:38) . s and ∆ t while it is significant for n s . Let us examine under which conditions the observables of non-canonical inflationand canonical inflation obtained as a function of N e , as discussed in section 3, willagree. Let us make two assumptions: • The non-canonical theory has a canonical branch where V can (cid:39) V . • The non-canonical theory has a speed limit R such that X inf (cid:39) Λ R for A (cid:29) V can ( φ ) = 32 R Λ ( φ − C ) , (4.4)with a constant C for the canonical potential in the limit for A (cid:29)
1. This implies (cid:15) can = 12 (cid:18) V (cid:48) can V can (cid:19) = 3 R Λ V can ( φ ) . (4.5)It was shown in [13] that the first slow-roll parameter becomes (cid:15) = √ R (cid:15) V A (4.6)for A (cid:29)
1. Using the definition of A , eq. (2.9), and eq. (4.5), the agreement of ∆ s and ∆ t as a function of φ can be phrased as conditions on the potentials and thespeed of sound, i.e. V can (cid:39) V and c s = √ RA for A (cid:29) . (4.7)Note that the first condition in eq. (4.7) is trivially satisfied in the canonical limit A (cid:28)
1. In the non-canonical limit A (cid:29)
1, the derivative V (cid:48) will generically havelarge values while V (cid:48) can has to be small in order to support slow-roll inflation. Thus,at some value A ∗ in the A (cid:29) V and V can will not agree anymore. However,there can be an intermediate regime A ∈ [1 , A ∗ ] with V can (cid:39) V and V (cid:48) can (cid:28) V (cid:48) . Thisintermediate regime can even serve to describe the complete phenomenologically– 9 –nteresting region if c s ( A ∗ ) < .
1, such that only the region
A > A ∗ is excluded dueto non-observation of equilateral non-Gaussianities.The first condition in eq. (4.7) implies an agreement as a function of N e as wellsince according to eq. (4.1)canonical: N e = (cid:90) φ Ne φ end √ (cid:15) can dφ = (cid:90) φ Ne φ end (cid:18) V can ( φ )6 R Λ (cid:19) / dφ , non-canonical: N e = (cid:90) φ Ne φ end (cid:18) V ( φ )6 X inf ( φ ) (cid:19) / dφ = (cid:90) φ Ne φ end (cid:18) V ( φ )6 R Λ (cid:19) / dφ . (4.8)As far as the spectral indices n s and n t are concerned we do not find agreementin the limit A (cid:29) n s − − (cid:15) can + 2 η can = − R Λ V can ,n t = − (cid:15) can = − R Λ V can , non-canonical: n s − − (cid:15) − η − κ = √ RA ( − (cid:15) V + 2 η V ) − κ ,n t = − (cid:15) = − √ RA (cid:15) V , (4.9)using η = √ R/A (4 (cid:15) V − η V ) as was shown in [13]. However, this does not excludean agreement in an intermediate region A (cid:38)
1. Furthermore, the scalar spectralindex n s receives significant corrections from the fact that c s < c s can be fulfilled. First, we note that using eq. (2.8) the speedof sound can be expressed as c s ( A ) = A ∂X inf /∂A X inf . (4.10)Hence, we need to know the functional dependence X inf ( A ) in order to decidewhether the observables ∆ s and ∆ t of the canonical and non-canonical theory agree.For p ( X, φ ) = Λ S ( X ) − V ( φ ) as defined in eq. (2.2) this dependence is determinedby the identity 2 X Λ (cid:32)(cid:88) n ≥ ( n + 1) c n (cid:18) X Λ (cid:19) n (cid:33) = A , (4.11)using the algebraic equation for the inflationary solution, eq. (2.8). To obtain X inf ( A )we have to invert eq. (4.11), which is impossible for most general coefficients c n .However, we will discuss some closed form expressions for p ( X, φ ) in the following.– 10 –onsider the class of non-canonical Lagrangians defined by p ( X, φ ) = Λ (cid:20) − (cid:18) − a X Λ (cid:19) a (cid:21) − V ( φ ) , (4.12)with 0 < a < ∂p/∂X diverges at the radius of convergence R a = a . Thisclass of non-canonical Lagrangians includes the DBI action via the case a = 1 /
2, i.e. p ( X, φ ) = Λ (cid:34) − (cid:18) − X Λ (cid:19) / (cid:35) − V ( φ ) . (4.13)Squaring the equation for the inflationary solution, eq. (2.8) becomes2 X Λ = A (cid:18) − a X Λ (cid:19) − a . (4.14)If 2 − a is not an integer one has to exponentiate with the denominator of 2 − a tosolve for X inf ( A ). In fact the only value of 0 < a < − a is an integeris a = 1 /
2, i.e. the DBI case, with solution X inf = Λ A A . (4.15)For all a (cid:54) = 1 / X inf will be some function of A n with integer n >
2. For instance,for a = 3 / X inf = Λ A (cid:32)(cid:114) A − (cid:33) . (4.16)Note that for X inf ( A n ), c s is also a function of A n since c s = nA n X (cid:48) inf ( A n )2 X inf ( A n ) , (4.17)where (cid:48) denotes the derivative with respect to A n . Hence, the dominating term in c s will be of the order c s ∼ A n (4.18)up to an O (1) coefficient. For the DBI case, we find c s = 11 + A (cid:39) A for A (cid:29) , (4.19)which fulfills the criterion eq. (4.7) on c s for the agreement of the observables ( R =1 / c s ∼ /A n with n > a such that the condition on c s in eq. (4.7) cannot be satisfied. Forexample, for a = 3 / c s = 1 − √ A − (cid:39)
92 1 A for A (cid:29) , (4.20)There are of course plenty of other models apart from those defined in eq. (4.12)that fulfill the conditions of a canonical branch and a speed limit. The questionof whether there could be other examples than DBI where the conditions on thepotential and speed of sound eq. (4.7) for an agreement of ∆ s and ∆ t are fulfilled ishard to answer in full generality. Consider for example the class of functions p ( X, φ ) = X (cid:34) − a (cid:18) X Λ (cid:19) b (cid:35) c . (4.21)For a = 4, b = 4 and c = 1 / X inf ( A ) of the equationsof motion eq. (2.8) with c s (cid:39) √ A = 2 RA for A (cid:29) , (4.22)such that the second condition in eq. (4.7) on the speed of sound is fulfilled. However,this solution suffers from the absence of a canonical limit X inf ∼ A for A < ∂p/∂X >
0. Due to the lack of other workingexamples where the agreement conditions eq. (4.7) are matched, we suspect that thedescription in terms of a canonical theory may be special to the DBI case. We willstudy this case more explicitly in the following section. We note at this point thatthe matching of the background equation of motion does not necessarily mean thatfluctuations around this background in the two different theories should match. Oneshould thus not expect agreement of the inflationary observables in general, even ifthe inflationary trajectory is the same. This makes the agreement in the DBI caseall the more remarkable. We now want to give an example of our general considerations in section 4.1. Weconsider the DBI action together with an inflection point potential: p ( X, φ ) = − f ( φ ) (cid:16)(cid:112) − f ( φ ) X − (cid:17) − V ( φ ) , (4.23)with V ( φ ) = V + λ ( φ − φ ) + β ( φ − φ ) . (4.24) We thank Bret Underwood for discussions on this point. – 12 –e fix the parameters of this theory to be V = 3 . · − , λ = 1 . · − , β = 1 . · − , φ = 0 . , f = 1 . · . (4.25)These are the values that were considered in [13]. In particular, the field-dependentwarp factor has been set to a constant f = Λ − which is justified if the range of fieldvalues that φ travels during inflation is small. The parameters in eq. (4.25) have beenchosen such that for a canonical kinetic term p ( X, φ ) = X − V the amplitude of thescalar fluctuations and the spectral index agree with observations, i.e. ∆ s = 2 . · − and n s = 0 . A . We find that for φ (cid:46) .
025 we are in the canonicalregime A ≤
1, while for φ (cid:38) .
025 we enter the non-canonical regime
A >
1, seeFigure 3. The phase space trajectory (see also Figure 1) for eq. (4.23) is determinedby eq. (4.15). This determines the potential V can ( φ ) that resembles the trajectory Φ (cid:45) (cid:45) (cid:45) (cid:45) X inf (cid:72) Φ (cid:76) Figure 1 . The phase space trajectory X inf ( φ ) for the DBI action eq. (4.23) with thenumerical values given in eq. (4.25). For large φ the trajectory approaches the limit (2 f ) − ,see eq. (4.15). from a canonical kinetic term via eq. (3.4). We perform the integration numericallyand show V can ( φ ) compared to the original inflection point potential V ( φ ) in Figure 2.We see that, as expected, V can agrees with V in the canonical regime while it is flatterthan V in the non-canonical regime. To see that V can actually supports slow-rollinflation we check (cid:15) and η as functions of φ in Figure 3. Comparison of observables
We compare the observables of the canonical and non-canonical theory in Figure 4.The agreement in ∆ s and ∆ t at the level of ∼
1% is up to values φ < . φ where the non-canonical– 13 – .005 0.010 0.015 0.020 Φ (cid:144) M P (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) V (cid:72) Φ (cid:76)(cid:144) M P V can V infl Φ (cid:144) M P (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) V (cid:72) Φ (cid:76)(cid:144) M P V can V infl Figure 2 . Comparison of the inflection point potential V ≡ V infl of eq. (4.24) and thepotential of the canonical theory V can ( φ ) obtained via eq. (3.4) for φ ∈ [0 , . φ ∈ [0 , .
12] (right). regime begins. So as discussed after eq. (4.7) there is indeed an intermediate regimewhere the observables agree even though V can is much flatter than V . Furthermore,since c s < . φ > .
06 the phenomenologically viable region is included in thisintermediate regime. The agreement of n s − N e holds only up to φ ≤ .
05, see Figure 5. However, there are important correctionsto n s − c s in the non-canonical theoryis smaller than one. We will discuss these corrections in detail in section 5.Note that there is an additional upper bound on c s which has to be fulfilled inorder to treat the inflationary quantum fluctuations perturbatively [14, 26, 27]. If thespeed of sound becomes too small the perturbations become strongly coupled andin particular the expressions for the inflationary observables eq. (2.10) are not valid.For DBI this can be expressed as a bound on the ‘non-canonicalness’ parameter [13] A < (cid:18) (cid:15)V (cid:19) / . (4.26)For our numerical example, this implies A < O (100) and hence φ (cid:46) .
2. Note thatthis is exactly the region where we find agreement between the non-canonical andtransformed canonical theory.We can prove the agreement of ∆ s and ∆ t in the whole intermediate region (notethat in section 4.1 this was shown only in the limit A (cid:29) (cid:15) can = 3Λ V can A A , (4.27)– 14 – .001 0.005 0.010 0.050 0.100 0.500 1.000 Φ (cid:144) M P A can . regime Φ (cid:144) M P (cid:45) (cid:45) (cid:45) Η V Ε V can . regime Φ (cid:144) M P (cid:45) (cid:45) (cid:45) Κ DBI Η DBI Ε DBI can . regime Φ (cid:144) M P (cid:45) (cid:45) (cid:45) Η can Ε can can . regime Figure 3 . The ‘non-canonicalness’ parameter A (top left), the parameters (cid:15) V and η V (topright) and the generalized slow-roll parameters (cid:15) , η and κ (bottom left) for the DBI actioneq. (4.23) with the numerical values of the parameters given in eq. (4.25). Also the slowroll parameters (cid:15) can and η can of the canonical theory are shown (bottom right). and the exact expression for (cid:15) that was found in [13] is (cid:15) = 32 A A
11 + V/ Λ − √ A . (4.28)Now the condition c s (cid:15) = (cid:15) can can be rephrased as V Λ + √ A − V can Λ . (4.29)This condition will be fulfilled to very large A for V (cid:39) V can , since V (cid:29) Λ as wedemanded at the beginning of section 4.1. For instance, in the numerical exampledescribed in eq. (4.25) we have V / Λ (cid:39) such that eq. (4.29) would hold up to A (cid:46) assuming the condition V (cid:39) V can is not violated before A reaches this value.The agreement works out as well for the DBI action with a Coulomb type potential V ( φ ) = V − T ( φ + φ ) n , (4.30)– 15 – .001 0.005 0.010 0.050 0.100 0.500 1.000 Φ (cid:144) M P N e canDBI can . regime Φ (cid:144) M P (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:68) s Φ (cid:144) M P n s canDBI can . regime Φ (cid:144) M P (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:68) t Figure 4 . Comparison of the observables ∆ s (top right), n s (bottom left) and ∆ t (bottomright) of the non-canonical DBI and the transformed canonical theory. Since the numberof efolds (top left) of the two theories agrees as a function of φ , the agreement of theobservables as a function of φ can be read as an agreement as a function of N e . instead of an inflection point potential. The non-canonical regime is accessed for φ < φ while the canonical regime is given by φ > φ . Hence, the agreement withthe transformed canonical theory is trivially found for φ < φ and extends to thenon-canonical regime until the condition V (cid:39) V can is violated. Consistency relation
Canonical and non-canonical theories are usually assumed to be distinguishable, notonly because of the possibility of equilateral-type non-gaussianity in the latter, butbecause of the consistency relation relating r = ∆ t / ∆ s the tensor-to-scalar ratio and n t the tensor spectral index. The relation in the noncanonical case has an additionalfactor of c s [11]: r Can = − n t ; r NC = − c s n t . – 16 –ecause of the appearance of c s , a sufficiently precise measurement of the ratio r/n T would therefore resolve the degeneracy we have found. However, we currently haveno bound on n t and only an upper bound on r : r < .
12 [5]. With the currentstate of observational bounds, these models remain indistinguishable at the 2-pointfunction level. c s < As we discussed in eq. (4.3), the observables have to be evaluated as functions of thecomoving momentum k which implies different times of horizon crossing for scalarand tensor modes respectively. Assuming H can (cid:39) H non − can which actually followsfrom the condition V can (cid:39) V , an agreement of tensor observables T as functions ofln k is equivalent to T can ( N e ) = T non − can ( N e ) , (5.1)having used N e = ln k − ln H .For scalar observables S however, we have to take into account that N e − ln c s =ln k − ln H in the non-canonical theory while N e = ln k − ln H in the canonical theory.Hence, we have to check for the equality S can ( N e ) = S non − can ( N e − ln c s ) . (5.2)Since the non-observation of equilateral non-Gaussianities implies | ln c s | (cid:28) N te , it issufficient to expand S non − can to first order in ln c s , i.e. S non − can ( N e − ln c s ) (cid:39) S non − can ( N e ) − S (cid:48) non − can ( N e ) ln c s . (5.3)In the following, we discuss this expansion for the scalar power spectrum ∆ s and thescalar spectral index n s .Using the definition of ∆ s in eq. (2.10), we find ∂ ∆ s ∂N e = ∆ s ∂ ln ∆ s ∂N e = ∆ s (cid:18) ∂ ln H∂N e − ∂ ln (cid:15)∂N e − ∂ ln c s ∂N e (cid:19) , = ∆ s · ( − (cid:15) − η − κ ) = ∆ s · ( n s − , (5.4)having used (cid:15) = − ∂ ln H∂N e , η = ∂ ln (cid:15)∂N e , κ = ∂ ln c s ∂N e . (5.5)This implies ∆ s ( N e − ln c s ) (cid:39) ∆ s ( N e ) [1 − ( n s −
1) ln c s ] . (5.6)Hence the correction that is induced by ln c s is suppressed by the slow-roll parametersand we can approximate ∆ s ( N e − ln c s ) (cid:39) ∆ s ( N e ).– 17 – .1000.0500.020 0.030 0.1500.070 Φ (cid:144) M P ∆ n s (cid:45) correcteduncorrected Figure 5 . The agreement δ n s − of the canonical and non-canonical theory in n s definedin eq. (5.8) as a function of φ with and without c s corrections. The fluctuations in theuncorrected δ n s − for small φ are due to numerical inaccuracies when obtaining V can vianumerical integration. For the spectral index n s the corrections induced by ln c s are significant. Usingthe definition of n s in eq. (2.10) we find to first order in the slow-roll parameters ∂n s ∂N e = − η ∂ ln η∂N e − κ ∂ ln κ∂N e , (5.7)which implies n s ( N e − ln c s ) − (cid:39) − (cid:15) − η − κ + (cid:18) η ∂ ln η∂N e + κ ∂ ln κ∂N e (cid:19) ln c s . (5.8)Note that ∂ ln η/∂N e corresponds to third derivative terms of the potential V , whilethe ∂ ln κ/∂N e term corresponds to second derivative terms of the speed of sound c s .We show in Figure 5 numerically that the agreement in δ n s − ≡ ( n s − can − ( n s − non − can ( n s − can , (5.9)for the DBI example considered in section 4.2 improves if one takes the correctionsdescribed in eq. (5.8) into account. We find that the regime where n s − φ ≤ . φ ≤ .
08. Consequently, the phenomenologically interesting region where c s > . φ ≤ .
06 is included due to the inclusion of this correction.– 18 –
Conclusions
Cosmological inflation generates almost scale-invariant power spectra of super-horizonsize curvature perturbations and tensor mode perturbations with Gaussian distribu-tion. They can be described at the Gaussian level by just three observational quan-tities: The overall normalization ∆ s of the curvature perturbation power spectrum,its spectral tilt n s , describing the (small) deviations from scale invariance expectedin most models of inflation, and the fractional power r in tensor modes.In this context it is important to understand the structure of this very largemodel space, and look for degeneracies between large classes of inflationary modelswith respect to the three observable quantities. We have restricted our attention hereto single-field models of inflation which partition into two large classes: models witha canonically normalized kinetic term ( ∂ µ φ ) , and so-called non-canonical inflationmodels with Lagrangian L = p (( ∂ µ φ ) , φ ).We have explored the degeneracies between canonical and non-canonical models ofinflation with respect to the three observational quantities describing their predictedpower spectra at the Gaussian level. While formally non-canonical 2-derivative mod-els of the form L = f ( φ )( ∂ µ φ ) − V ( φ ) can be always transformed off-shell by a localfield redefinition into a canonical model with a transformed scalar potential, thisquestion is rather non-trivial in the presence of higher-power kinetic terms. We haveelucidated the method of canonical transformations for transforming noncanonicalkinetic terms into canonical kinetic terms which, even in 0+1D, appears to be limitedto the case where the noncanonical theory has a quadratic potential, see AppendixB. As the inflationary behavior of a given model is described in terms of a generalizedslow-roll attractor solution in phase space, we have therefore looked at possible on-shell transformations of a given non-canonical model on its inflationary attractor intoan equivalent canonical slow-roll inflation model. We have constructed such on-shelltransformations in general, so that given a non-canonical lagrangian which supportsinflation, the potential required to reproduce the inflationary trajectory X inf ( φ ) ina canonical theory can be found. Furthermore, we checked for the matching of the2-point function observables ∆ s , n s , and r . We find a full on-shell match for all2-point function quantities precisely for the case of DBI inflation while failing forthe DBI-inspired generalizations. This can be shown analytically and numerically.This may point to a special status of DBI inflation as a member of the non-canonicalclass in that it can be related to a canonical model of inflation with matching 2-pointfunction observables..Lastly, in the light of the much-awaited Planck data on nongaussianity, we wouldlike to point out that given the data we have, there remains a large degree of degen-– 19 –racy between inflationary models, which we have to bear in mind when interpretingthat data. Since it is often claimed that canonical and noncanonical theories can bedistinguished using the data, we feel this added degeneracy serves as a warning thatthis may not be the case, particularly if large NG is not observed. Unless data onnongaussianities improves drastically and reveals a non-negligible single of equilat-eral nongaussianity, or the consistency relation between r and n T can be accuratelymeasured, one may never be able to distinguish between non-canonical inflation andslow-roll inflation in some canonical theory. In fact even with the observation ofNG, this differentiation may not be possible: Note that appreciable non-Gaussianitycan arise in single scalar field theories of inflation with a canonical kinetic term,from features in the potential [21], or from coupling of the inflaton to gauge quanta[22, 23]. It is possible that by adding additional couplings or features to the potentialof the canonical theory one could match observables at the 3-point function level aswell. We have not addressed the question of matching 3-point observables such asnon-Gaussianity here, and leave investigation of this question for future work. Alsothe deeper reason for the agreement of DBI with its canonical transform at the levelof the 2-point function is yet to be understood on a more fundamental level. Thisdegeneracy thus opens many questions for future study. Acknowledgments
We thank Jan Louis, Raquel Ribeiro and especially Bret Underwood for valuable,and enlightening discussions. This work was supported by the Impuls und Vernet-zungsfond of the Helmholtz Association of German Research Centers under grantHZ-NG-603, the German Science Foundation (DFG) within the Collaborative Re-search Center 676 “Particles, Strings and the Early Universe” and the ResearchTraining Group 1670. R.G. is grateful for support by the European Research Coun-cil via the Starting Grant numbered 256994. R.G. was also supported during theinitial stages of this work by an SFB fellowship within the Collaborative ResearchCenter 676 “Particles, Strings and the Early Universe” and would like to thank thetheory groups at DESY and the University of Hamburg for their hospitality at thistime.
A Accessing the non-canonical regime in DBI
Here we comment on constraints on the allowed phase space of brane inflation modelsgoverned by the DBI action eq. (4.23). It was shown in [28] that the presence of non-canonical kinetic terms can ameliorate the initial conditions fine tuning problem,but this effect is only present when the non-canonical terms are relevant in the– 20 –llowed phase space. The non-canonical kinetic terms are relevant when A ≥ | Π | > Λ [28], where Π is the canonical momentum given byΠ = −√ X ∂p ( X, φ ) ∂X . (A.1)Clearly, however, the available phase space is bounded. For an effective field theorydescription to be valid, we require H <
Λ, which implies | Π | < M p Λ for a canonicalkinetic term and | Π | < M p for a non-canonical kinetic term. This leaves a potentiallylarge range of momenta Λ < Π ≤ M p in which the non-canonical kinetic terms arerelevant and the EFT description remains valid. In addition, there can be boundson the range of the inflaton field φ which restrict the allowed phase space.The DBI action [12] describes inflation in the D3/ D f ( φ ) is the warp factor of the throat in the internal space. Inflation proceeds asthe D3 brane moves towards an D φ : µ (cid:28) φ (cid:28) M p . (A.2)A lower limit can be understood physically from the requirement that the branesmust be initially separated by at least a string length so that inflation does not endimmediately. The upper limit reflects the fact that the inflaton range cannot belarger than the size of the compactified space.Critically, the upper bound on the field range is related to the warp factor. Alongwith the lower bound on the field, this implies an upper bound on the warp factor f ( φ ) = λφ and therefore the ‘non-canonicalness’ parameter [13] A DBI ≡ V (cid:48) ( φ ) f / ( φ )3 H ( φ ) . (A.3)This can be understood as follows [35–37]: the 4-dimensional Planck mass scaleswith the warped volume of the compact space, and is therefore an upper bound onthe volume of the warped throat:Vol(X ) π (cid:90) φ UV φ end dφφ f ( φ ) < M p . (A.4)Using λ = Nπ Vol(X ) from the Klebanov-Strassler throat solution [38], where N is the– 21 –mount of 5-form flux associated with the warping, this gives rise to the bounds π λφ UV / < M p , (A.5)(∆ φ/M pl ) < N , (A.6) (cid:112) f < √ M P πφ UV φ , (A.7)where we took Vol(X ) ∼ π as in [36]. We find the field range bound eq. (A.6), orequivalently a bound on the warp factor eq. (A.7). Since φ also has a lower limit,there is immediately an upper bound on f . The lower limit is found to be φ ≈ D [37], where D ∼ T h IR ∼ m s g s h IR . (A.8)This gives us a bound on A DBI : A DBI < √ V (cid:48) √ πφ UV φ √ V , (A.9)where we have set M p = 1.Using the potential [37] V ( φ ) = Ds (1 − CDφ + αφ + βφ − a ∆ φ ∆ ) , (A.10)where s is some number of order 1, and C = π s (cid:28) a ∆ = 0 for a quadraticpotential and ∆ = for an inflection point potential, the bound becomes [37] A < (cid:114) √ Dsπφ UV φ ( βφ + CDφ ) (cid:113) αφ + βφ − a ∆ φ ∆ − CDφ ,A < (cid:114) √ Dsπφ UV φ ( βφ + CDφ ) (cid:113) αφ + βφ − a ∆ φ ∆ − CDφ . (A.11)It is only if A can be larger than 1 within the allowed field range of φ that the non-canonical regime will be accessed. We can test whether this is the case by consideringthree cases. Note that Dφ has upper bound 1. • Case One: Dφ (cid:28)
1. In this case we are firmly in the canonical regime where A (cid:28) A < ∼ (cid:113) Dφ s (2 φ ) φ UV π ,A < ∼ (cid:115) Dφ φφ UV ,A (cid:28) . (A.12)– 22 – Case Two: Dφ ∼
1: In this case A can be very large because the denominatorblows up. However, φ ≈ φ end so by the time this happens inflation is alreadyover. • Case Three: Dφ is intermediate. Let us be more careful about the approachto large A , taking Dφ to be some intermediate value ≤
1. In this case A < ∼ (cid:114) s Cπ D / φ φ UV + (cid:18) (cid:19) / (cid:115) Dsφ βφπφ UV . (A.13)Note that the denominator we have taken to be 1 will generally be less thanone for this intermediate case, thus weakening the bound. Each factor in thesecond term is smaller than 1, so it will only be possible to get a contributionto A of order 1 or greater from the first term. Then A ∼ (cid:114) s Cπ D / φ φ UV ,A ∼ (cid:114) s Cπ φ IR φ φ UV . (A.14)Taking C = π s , s ≈ φ = aφ IR , where a is some positive number greaterthan 1, this gives A ∼ . φ IR φ φ UV ,A ∼ .
02 1 a φ IR φ UV . (A.15)Then we see that there is a relation between how small φ IR can be and how farfrom the end of inflation (parametrized by a ) we can access the non-canonicalregime. For a fixed A ≥
1, decreasing φ IR implies a larger a from which weaccess the non-canonical regime, but the fraction of the field range in the NCregime is reduced. For φ UV = 0 . φ IR = 0 . a ≤ .
53, giving a very smallrange of φ values for which A ≥ φ − φ IR φ UV − φ IR = ( a − φ IR φ UV − φ IR ≤ . . (A.16)Increasing φ IR will increase this fraction, but of course one must have φ IR (cid:28) φ UV (cid:28) φ UV = 0 . φ IR = 0 . a ≤ .
2, so that A ≥ D φ values, consistent withthe conclusions of [35, 37, 39]. – 23 – Off-shell canonical transformations
In this article we have used the attractor equation to make an on-shell transformationfrom a theory with non-canonical kinetic term to a theory with canonical kineticterm, such that the inflationary trajectory is the same. Off-shell transformations,in which the equation of motion is not used, can also be used to transform betweensuch theories, but have a more limited scope.Canonical transformations were used in [18] to transform from theories with canon-ical kinetic terms to theories with non-canonical kinetic terms, in 0 + 1 dimensions.We shall see that this is possible when, in the case that the Lagrangian is sepa-rable, the form of the kinetic and potential terms are exchanged by the canonicaltransformations.A canonical transformation is defined by a generating function F ( φ, ˜ φ ) via p = ∂F∂φ , ˜ p = − ∂F∂ ˜ φ . (B.1)As an example, let L = ˙ φ − V ( φ ), V ( φ ) = kφ / and F ( φ, ˜ φ, t ) = φf ( ˜ φ ). Thenthe transformations can be written p = f ( ˜ φ ) ,φ = − ˜ pf (cid:48) ( ˜ φ ) . (B.2)We have the energy density H = ˜ H and can find ˙˜ φ = ∂ ˜ H∂ ˜ p :˜ H = 12 f ( ˜ φ ) + k (cid:18) − ˜ pf (cid:48) ( ˜ φ ) (cid:19) / , ˙˜ φ = − k f (cid:48) ( ˜ φ ) (cid:18) − ˜ pf (cid:48) ( ˜ φ ) (cid:19) / . (B.3)Invert this to get ˜ p = ( f (cid:48) ) k ˙˜ φ . Then the transformed Lagrangian is found to be:˜ L = ˜ p ˙˜ φ − ˜ H , ˜ L = 3 ( f (cid:48) ) k ˙˜ φ − f , (B.4)which has a non-canonical kinetic term X ∼ ˙˜ φ . We can now ask what the generalconditions are for it to be possible to obtain an action with canonical kinetic termupon performing a canonical transformation.– 24 – eneral conditions for the existence of a dual canonical theory We start with a separable Hamiltonian H ( p, φ ) = K ( p ) + V ( φ ) , (B.5)with kinetic term K ( p ) and potential V ( φ ). To obtain the transformed Hamiltonianone has to invert the second relation in eq. (B.1) to find the dependencies p = p (˜ p, ˜ φ ) , φ = φ (˜ p, ˜ φ ) . (B.6)The transformed Hamiltonian is then given as˜ H (˜ p, ˜ φ ) = K ( p (˜ p, ˜ φ )) + V ( φ (˜ p, ˜ φ )) . (B.7)This is generally not a separable Hamiltonian, i.e. of the form ˜ K (˜ p ) + ˜ V ( ˜ φ ), let alonein canonical form.By Taylor expanding the transformed Hamiltonian around ˜ p = 0:˜ H (˜ p, ˜ φ ) = ˜ H (0 , ˜ φ ) + ∂ ˜ H∂ ˜ p | ˜ p =0 ˜ p ∞ (cid:88) i =1 ,i (cid:54) =2 (cid:15) i ˜ p i , with (cid:15) i = 1 i ! ∂ i ˜ H∂ ˜ p i | ˜ p =0 , (B.8)we see that the transformed theory is approximately canonical with potential ˜ V ( ˜ φ ) =˜ H (0 , ˜ φ ) iff the generating function can be chosen such that ∂ ˜ H∂ ˜ p | ˜ p =0 = 1 and | (cid:15) i | (cid:28) . (B.9) Simplifying approach K ↔ V The simplest way to obtain a separable dual theory is to demand that the transfor-mations eq. (B.6) exchange the role of the kinetic term K ( p ) and the potential V ( φ ).This happens if p (˜ p, ˜ φ ) is only a function of ˜ φ and φ (˜ p, ˜ φ ) is only a function of ˜ p .This requirement determines the form of the generating function F ( φ, ˜ φ ): First, p = p ( ˜ φ ) = ∂F ( φ, ˜ φ ) ∂φ determines F ( φ, ˜ φ ) to be linear in φ , i.e. F ( φ, ˜ φ ) = a ( ˜ φ ) + b ( ˜ φ ) φ .Second, ˜ p = − ∂F∂ ˜ φ = − a (cid:48) ( ˜ φ ) − b (cid:48) ( ˜ φ ) φ will only give a relation φ = φ (˜ p ) independentof ˜ φ if a (cid:48) ( ˜ φ ) and b (cid:48) ( ˜ φ ) do not depend on ˜ φ . Hence, the most general F ( φ, ˜ φ ) thatexchanges the role of K and V can be parametrized as F ( φ, ˜ φ ) = ( kφ + g ) ˜ φ , with k, g ∈ R , k (cid:54) = 0 . (B.10)The transformation is then linear and given by p = k ˜ φ , φ = − g − ˜ pk . (B.11)We will apply this type of generating function in the following sections B.1 and B.2to obtain a canonical theory that is dual to a non-canonical theory.– 25 – .1 DBI + quadratic potential The Hamiltonian is given by H ( p, φ ) = 1 f (cid:16)(cid:112) f p − (cid:17) + V ( φ ) , with V ( φ ) = 12 m φ . (B.12)For the generating function F = mφ ˜ φ , this becomes the canonical theory˜ H (˜ p, ˜ φ ) = 12 ˜ p + ˜ V ( ˜ φ ) , with ˜ V ( ˜ φ ) = 1 f (cid:18)(cid:113) f m ˜ φ − (cid:19) . (B.13)The parameter that indicates if the theory is in the canonical or non-canonicalregime is A = V (cid:48) ( φ ) f / H (cid:39) V (cid:48) ( φ ) f / √ V , (B.14)where in the last step we have used H (cid:39) V /
3, i.e. the energy is dominated by thepotential energy. If A (cid:29) A (cid:28) A DBI = (cid:114) f m . (B.15)Hence, to be in the non-canonical regime we have to demand √ f m (cid:29)
1. In thislimit, we can approximate the potential ˜ V ( ˜ φ ) of the canonical theory eq. (B.13) by˜ V ( ˜ φ ) = m ˜ φ √ f . (B.16) B.2 DBI + “inflection point potential”
We now want to look at the theory H ( p, φ ) = 1 f (cid:16)(cid:112) f p − (cid:17) + V ( φ ) , with V ( φ ) = V + λ ( φ − φ ) + β ( φ − φ ) . (B.17)This potential is suitable for small field inflation since for λ, β (cid:28) V the slow-rollparameters (cid:15) and η are small at φ = φ , without the necessity of φ having to travela trans-Planckian distance as for instance in chaotic inflation.To obtain a potential where the inflaton is rolling down towards a local minimumwe have to choose λ, β >
0. Eq. (B.17) then only describes the dynamics near theinflection point φ = φ .The A -parameter (B.14) for the theory described by eq. (B.17) generically de-mands f (cid:29) .005 0.010 0.015 0.020 Φ (cid:144) M P (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) V (cid:72) Φ (cid:76)(cid:144) M P Figure 6 . Inflection point potential, eq. (B.17) for the parameters V = 3 . · − , λ = 1 . · − , β = 1 . · − and φ = 0 . Using the generating function F ( φ, ˜ φ ) = ( kφ + g ) ˜ φ we can transform the Hamil-tonian eq. (B.17) into the approximately canonical theory˜ H (˜ p, ˜ φ ) = 12 ˜ p + (cid:15) ˜ p + (cid:15) ˜ p + ˜ V ( ˜ φ ) , (B.18)with ˜ V ( ˜ φ ) = 1 f (cid:18)(cid:113) f k ˜ φ − (cid:19) + V − (cid:15) + (cid:15) (cid:15) , (B.19)where we have used g = − kφ + 16 (cid:15) , β = − k (cid:15) , λ = − k(cid:15) + k (cid:15) . (B.20)The first equation in (B.20) follows from the canonical normalization of the ˜ p termin eq. (B.18) while the other two equations are reparametrizations of the potentialparameters λ and β such that the coefficients of ˜ p and ˜ p are small in eq. (B.18), i.e. (cid:15) , (cid:15) (cid:28)
1. At this point, k , V and φ remain free unfixed parameters.We see that for (cid:15) , (cid:15) (cid:28) λ and β to have the same sign since β ∼ − k (cid:15) and λ ∼ k/ (cid:15) . Hence, the original idea of the inflection point inflatonpotential visualized in Figure 6 does not have a dual canonical theory that is relatedvia a generating function F ( φ, ˜ φ ) = ( kφ + g ) ˜ φ .However, we can still go ahead and try to construct a sensible inflationary theoryon both the canonical and non-canonical side respecting the just discussed constraintson λ and β . In the non-canonical theory H ( p, φ ) we choose λ > β < V (cid:48) ( φ ) <
0, i.e. the inflaton is rolling down towardssmaller field values. Furthermore, we demand the local minimum of the potential tobe at φ = 0 and V ( φ ) = 0 which fixes φ and V to be φ = √ − (cid:15) (cid:15) k(cid:15) , V = (1 − (cid:15) (cid:15) ) / (cid:15) . (B.21)– 27 –nserting eqs (B.21) into V ( φ ) the potential simplifies to V ( φ ) = 12 k φ (cid:0) √ − (cid:15) (cid:15) − k(cid:15) φ (cid:1) (cid:39) k φ , (B.22)i.e. an approximately quadratic potential with mass parameter k , see also Figure 7. Φ (cid:144) M P V (cid:72) Φ (cid:76)(cid:144) M P
10 000 20 000 30 000 40 000 50 000 60 000 70 000 Φ (cid:144) M P (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:180) V (cid:72) Φ (cid:76)(cid:144) M P Figure 7 . Modified inflection point potential, eq. (B.22) for the parameters k = 0 . (cid:15) = (cid:15) = 10 − . In other words, our demand that there exists an approximately canonical dualtheory implies that the potential of the non-canonical theory we started from isapproximately quadratic and we are back to our discussion in section B.1.To complete the analogy with section B.1, we look at the potential ˜ V ( ˜ φ ) of theapproximately canonical theory, inserting the expression (B.21) for V :˜ V ( ˜ φ ) = 1 f (cid:18)(cid:113) f k ˜ φ − (cid:19) − (cid:15) + (cid:15) (cid:15) + (1 − (cid:15) (cid:15) ) / (cid:15) = 1 f (cid:18)(cid:113) f k ˜ φ − (cid:19) + O ( (cid:15) , (cid:15) ) . (B.23)In the limit (cid:15) , (cid:15) →
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