The universe is accelerating. Do we need a new mass scale?
aa r X i v : . [ a s t r o - ph . C O ] A p r The universe is accelerating. Do we need a new mass scale?
Savvas Nesseris, Federico Piazza,
2, 3 and Shinji Tsujikawa The Niels Bohr International Academy, The Niels Bohr Institute,Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark ∗ Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada Canadian Institute for Theoretical Astrophysics (CITA), Toronto, Canada † Department of Physics, Faculty of Science, Tokyo University of Science,1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan ‡ (Dated: November 19, 2018)We try to address quantitatively the question whether a new mass is needed to fit current su-pernovae data. For this purpose, we consider an infra-red modification of gravity that does notcontain any new mass scale but systematic subleading corrections proportional to the curvature.The modifications are of the same type as the one recently derived by enforcing the “Ultra StrongEquivalence Principle” (USEP) upon a Friedmann-Lemaˆıtre-Robertson-Walker universe in the pres-ence of a scalar field. The distance between two comoving observers is altered by these correctionsand the observations at high redshift affected at any time during the cosmic evolution. While thespecific values of the parameters predicted by USEP are ruled out, there are regions of parameterspace that fit SnIa data very well. This allows an interesting possibility to explain the apparentcosmic acceleration today without introducing either a dark energy component or a new mass scale. I. INTRODUCTION
During the last decade, several observational probes[1–3] have confirmed that our universe is undergoing aphase of accelerated expansion. Beyond the details ofspecific models, one of the most remarkable aspects ofsuch a discovery is the seemingly unavoidable presenceof a new tiny mass scale in the theory that describes ourworld.In the framework of General Relativity (GR), a nega-tive pressure component (“dark energy” [4]) can accountfor the cosmic acceleration. During the cosmologicalexpansion, such a component has to become dominantwhen the average energy density ρ ( t ) drops to about itspresent value ρ ( t is the proper time). Thus, dark en-ergy Lagrangians typically contain a mass parameter ofthe order of M ∼ ρ / ∼ − eV, that triggers the epochwhen dark energy starts to dominate . Models of mas-sive/modified gravity highlight the problem from a dif-ferent perspective. If the graviton is effectively massive,the modified dynamics of gravity at large distances canprovide a mechanism for “self-acceleration” [7] and/or offiltering for the cosmological constant’s zero mode [8]. Inthat case the new mass brought into the theory is themass of the graviton , m g , typically of order the Hubble ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] The mass parameter can be higher e.g. in quintessence mod-els with power-law potentials [5, 6], but at the price of givinga milder equation of state which is now severely challenged byobservations. Interestingly, and as opposed to, e.g., the mass of scalarquintessence fields, such a mass might be protected against –and actually made smaller by – radiative corrections [9]. constant H , i.e. m g ∼ H ∼ ( ρ /M ) / ∼ − eV.In f ( R ) theories of gravity, where the Lagrangian den-sity f is a function of the Ricci scalar R , the late-timeacceleration is realized, again, by introducing a curvaturescale R c of the order of H [10] (see also Refs. [11]). Notethat, although based on different mechanisms, all mod-els follow an analogous pattern : there is a tuned scale“hidden” in the theory which becomes effective, “by co-incidence”, when the appropriate cosmological quantity(the average density ρ ( t ) or the Hubble parameter H ( t ))drops to about its value.But is the acceleration of the universe intrinsically im-plying the presence of a new mass scale? If we allow thepossibility of departures from GR at large distances thereis a logical alternative. High redshift observations havethe unique property of relating objects (e.g., the observerand the supernova) that are placed from each other at arelative distance of the order of the average inverse cur-vature (roughly, the Hubble length ∼ H − ). Therefore,modifying GR in the infrared (IR) at a length scale set bythe curvature – rather than fixed a priori by a parameter– will systematically affect any cosmological observationat high redshift, regardless of when such an observationtakes place and without the need of any external massscale. In other words, we might not need a new mass scalebecause we already have (a dynamical) one, the Hubbleparameter H ( t ); the only “coincidence” that we might beexperiencing is that of observing objects that are placedfrom us as far as the Hubble radius . The few alternatives to this common pattern include the pro-posal that our universe is not homogeneous on large scales [12]and attempts based on possible non-trivial effects of smaller in-homogeneities on the cosmic evolution [13]. The same circumstance does not apply, for instance, to obser-vations within the solar system: typical solar system distances
The point of view sketched above is somewhat com-pelling, it addresses directly the fine tuning and coinci-dence problems, but seems to require a serious revision ofthe current low energy framework for gravity. Any grav-itational operator that becomes effective in the infrared,on dimensional grounds, has to bring in the Lagrangiana mass parameter. Moreover, GR itself is already a ge-ometrical deformation of flat space at distances of theorder of the curvature. What seems to be required is afurther curvature-dependent subleading effect that sys-tematically modifies the geometrical description of GRat large distances.Recently, a proposal along those lines has been madeby one of the present authors. The modification uponthe standard framework is forced by imposing an “Ul-tra Strong” version of the Equivalence Principle (USEP,see Refs. [14, 15] for more details). USEP suggests thatthe usual geometric description of spacetime as a metricRiemannian manifold might hold only approximately, atsmall distances. Such a conjectured “IR-completion” ofgravity, in its full generality, represents a major theoret-ical challenge. However, it can be tentatively exploredwith a Taylor expansion around GR, by applying USEPto a specific GR solution (see Appendix A). For a spec-tator scalar field in a spatially flat Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) universe, the first-order cor-rection to GR is calculable, and few cosmological conse-quences are derivable [15].Consider, as the zeroth-order (GR-) approximation, ahomogeneous, spatially flat FLRW universe with scalefactor a ( t ). It is known that in such a solution the phys-ical distance d ( t ) between a pair of comoving observersgrows proportionally to a ( t ); otherwise stated, the co-moving distance λ ≡ d ( t ) /a ( t ) is a constant. The first-order correction found in Ref. [15] modifies such an ex-pansion law by a subleading distance-dependent contri-bution. As a result, the distance d ( t ) between two comov-ing observers grows as a ( t ) only when small compared tothe Hubble length H − but gets relevant corrections oth-erwise. Thus, the scale factor a ( t ) defines the expansioneverywhere but only in the local limit, and effectivelydetaches from the expansion on the largest scales.The corrected expansion is most easily seen in termsof the above defined comoving distance λ , which is con-stant only in the small distance limit. Its derivative withrespect to observers’ proper time t reads [15]˙ λ = λ ( H a ) · , (1) are always extremely small with respect to, say, the averageWeyl curvature. An order of magnitude estimate indeed gives(Weyl curvature) − / ≈ r ( r/ / , where r is the distancefrom the Sun. In a similar fashion, someone who does not know GR can try toexpand around some point in Riemann normal coordinates andfind, in some specific cases, the first GR corrections to flat space. and is clearly negligible on sub-Hubble scales. For com-pleteness, a basic derivation of Eq. (1) is sketched in Ap-pendix A. The comoving trajectory r ( t ) of a light rayalso receives corrections because the modified global ex-pansion (1) has to be considered on top of the usual con-tribution d r = d t/a . In a matter-dominated universe,where the Hubble parameter at the redshift z = 1 /a − H ( z ) = H (1 + z ) / , we haved( H r )d z = 1(1 + z ) / + ( H r ) , (2)which has to be solved with initial conditions r (0) = 0.The above modification, including the factor of 1 /
4, isforced by requiring that USEP applies for a scalar fieldin a spatially flat FLRW universe [15]. The correctionincreases the luminosity distance d L ( z ) = (1 + z ) r ( z ) , (3)and therefore it effectively goes in the direction of a uni-verse with positive acceleration. However, as the presentanalysis shows as a by-product, the correction given inthe last term of Eq. (2) is too small (of too high order inthe redshift) to explain SnIa data.Equation (2) is an example of a IR-geometrical defor-mation that does not contain any mass scale but only H -dependent subleading terms. In this paper, by study-ing a generalized version of (2), we attempt to addressquantitatively the question whether SnIa data can be ex-plained without introducing any new mass scale. We in-clude terms of lower order in the redshift that are neededto efficiently reproduce SnIa observations. Equation (1)is generalized as follows:˙ λ = A λH + A λ H a + . . . + B λ ( Ha ) · + B λ ( H a ) · + . . . . (4)Note that the above structure of corrections includes (1)as a special case. In practice, the terms in the aboveexpansion rearrange when we calculate the luminositydistance. Therefore, for a matter-dominated universe, aquite general structure of subleading terms is given byd( H r )d z = 1(1 + z ) / F (cid:16) H r (1 + z ) / (cid:17) , (5)where F ( x ) is a generic function with F (0) = 1: F ( x ) = 1 + αx + βx + γx + . . . . (6)By comparison with (4) we have α = − A , β = B / − A , γ = B . Note that Eq. (2) corresponds to α = β = 0and γ = 1 /
4, while in GR all coefficients are set to zero.
II. EFFECTIVE DESCRIPTION
It is possible to obtain analytic solutions of (5) in somerestricted cases (e.g., γ = 0, see Appendix B). However,it is perhaps more useful to study the effective behavior of(5) at low z . In order to make an easy comparison withknown parameterizations of dark energy, we can easilyexpress our first two parameters, α and β , in terms of aneffective density parameter Ω effDE and a constant effectiveequation of state w eff of dark energy, in the presence ofnon-relativistic matter . Such effective parameters [16]are thus defined by r ( z ) = Z z H eff ( x ) d x , (7)where H eff ( z ) ≡ H q (1 − Ω effDE )(1 + z ) + Ω effDE (1 + z ) w eff ) . (8)By expanding Eq. (7) at small redshift we find H r = z − z (cid:0) effDE w eff (cid:1) + z (cid:2) w eff Ω effDE + 3 w Ω DE (3Ω DE − (cid:3) + . . . . (9)On the other hand, the solution of (5) can be expandedas H r = z − z (cid:18) − α (cid:19) + z (cid:0) − α + 4 α + 8 β (cid:1) + . . . . (10)By comparing (9) and (10) we can relate the two setsof parameters: α = − w eff Ω effDE ,β = 38 w eff (cid:2) w eff (cid:0) Ω effDE − (cid:1)(cid:3) Ω effDE . (11)For the ΛCDM model ( w eff = −
1) with Ω effDE = 0 . α = 1 . β = − .
74. Of course the aboveexpansions are valid only for z ≪
1, so it is expectedthat the likelihood analysis including high-redshift datacan give different constraints on the model parameters(as we will see later).The most important correction that leads to a largercomoving distance relative to the Einstein de Sitter uni-verse originates from the α term in Eqs. (5)-(6), i.e., theterm A λH in Eq. (4). Note that, for z ≪
1, the correc-tion γ becomes important only for the terms higher thanorder z in Eq. (10). Hence, we expect that Eq. (2) alonewill not be sufficient to reproduce SnIa data efficiently,at least at low redshift. We can do a similar exercise for an evolving effective equation ofstate w eff ( z ) instead of constant w eff . However the correspondingexpression in this case is much more complicated, so we will notpresent it here. III. THE SnIa DATA ANALYSIS
In this section we shall present a method to place ob-servational constraints on the IR corrections (5)-(6) fromSnIa data. We will use the SnIa dataset of Hicken etal. [17] consisting in total of 397 SnIa out of which 100come from the new CfA3 sample and the rest from Kowal-ski et al. [18]. These observations provide the apparentmagnitude m th ( z ) of the SnIa at peak brightness afterimplementing the correction for galactic extinction, theK-correction and the light curve width-luminosity correc-tion. The resulting apparent magnitude m th ( z ) is relatedto the luminosity distance d L ( z ) = (1 + z ) r ( z ) through m th ( z ) = ¯ M ( M, H ) + 5 log ( d L ( z )) , (12)where ¯ M is the magnitude zero point offset and dependson the absolute magnitude M and on the present Hubbleparameter H as¯ M = M + 5 log (cid:18) H − Mpc (cid:19) + 25 = M − h + 42 . . (13)Here the absolute magnitude M is assumed to be con-stant after the above mentioned corrections have beenimplemented in m th ( z ).The SnIa datapoints are given, after the correctionshave been implemented, in terms of the distance modulus µ obs ( z i ) ≡ m obs ( z i ) − M . (14)The theoretical model parameters are determined byminimizing the quantity χ = N X i =1 [ µ obs ( z i ) − µ th ( z i )] σ µ i , (15)where N = 397, and σ µ i are the errors due to flux uncer-tainties, intrinsic dispersion of SnIa absolute magnitudeand peculiar velocity dispersion. These errors are as-sumed to be Gaussian and uncorrelated. The theoreticaldistance modulus is defined as µ th ( z i ) ≡ m th ( z i ) − M = 5 log ( d L ( z )) + µ , (16)where µ = 42 . − h and µ obs is given by Eq. (14).The steps we have followed for the minimization ofEq. (15) in terms of its parameters are described in detailin Refs. [19–21].We will also use the two information criteria known asAIC (Akaike Information Criterion) and BIC (BayesianInformation Criterion), see Ref. [22] and references therein. The AIC is defined asAIC = − L max + 2 k , (17)where the likelihood is defined as L ∝ e − χ / , the term − L max corresponds to the minimum χ and k is thenumber of parameters of the model. The BIC is definedsimilarly as BIC = − L max + k ln N , (18)where N is the number of datapoints in the set underconsideration. According to these criteria a model withthe smaller AIC/BIC is considered to be the best andspecifically, for the BIC a difference of 2 is consideredas positive evidence, while 6 or more as strong evidencein favor of the model with the smaller value. Similarly,for the AIC a difference in the range 0 − − >
10 the model withthe larger AIC practically irrelevant [22, 23].
IV. RESULTS
We solve Eq. (5)-(6) numerically to find r ( z ) for thematter-dominated model (SCDM). Then the model istested against the SnIa data by using Eqs. (3), (15), and(16). Since the parameter γ does not appear up to thirdorder in the expansion of Eq. (10), we will also considerthe case where γ = 0.In Fig. 1 (left) we present the best fit distance mod-ulus versus the redshift z for the best fit ΛCDM model(dashed line), with the present matter density parameterΩ = 0 . +0 . − . ) and the SCDM model + IR correc-tion of Eq. (10) with all 3 parameters (solid black line)and the 2 parameter case with γ = 0 (dotted line). Forthe two parameter case we find that α = 1 . +0 . − . and β = − . +0 . − . for a χ = 464 .
031 or a χ per degree offreedom ∼ .
17, whereas the best fit ΛCDM has a χ perdegree of freedom ∼ . µ bestfit ( z ) − µ ΛCDM ( z ), for the model with the IR correc-tion of Eq. (5) with all 3 parameters (solid black line) andthe 2 parameter case and γ = 0 (dotted line). In Fig. 2we show the 1 σ and 2 σ contours for the parameters α and β of Eq. (5) with γ = 0. The black dot indicates thebest-fit.We should note that the two parameter model fits verywell the data even if the exact numbers α = 1, β = − / γ = 0 are used. In this case we find that χ = 465 .
462 and a χ per degree of freedom ∼ . w eff ≃ − effDE ≃ .
7, which gives α ≃
1. The value of β ≃ − . β ≃ − . α = β = 0, γ = 1 /
4. In this case, however, the agreement is not very good as the χ per degree offreedom is ∼ .
21 with the difference from the ΛCDMmodel being about 20 σ . This discrepancy can be ex-plained by the fact that the IR correction does not kickin early enough to allow for good compatibility with thedata, while in low redshifts the SCDM behavior of themodel dominates. This latter property comes from thefact that the γ -dependent term appears only at the orderof z in Eq. (10).If we consider all three parameters α , β and γ to befree, then the best fit parameters are α = 0 . β = 1 . γ = − .
51 for a χ = 459 .
424 or a χ per degreeof freedom ∼ .
17. As it can be seen in Fig. 1 (solidblack line), the corresponding luminosity distance showsa more significant departure from ΛCDM at high redshift.Finally, it is interesting to consider the case where wefix the parameters α and β to the exact numbers α = 1, β = − / γ to vary. In this case we expectthat by changing γ we will be able to improve χ and stillbe able to compare with ΛCDM as there will be only onefree parameter in the model. The result is χ = 465 . γ = 0 . δχ ∼
1) than thethree parameter case, but it is slightly better ( δχ ∼ . χ = 465 .
513 for Ω = 0 . V. CONCLUSIONS
We have considered IR modifications of gravity that donot imply the presence of a new mass scale in the theoryand we have studied their compatibility with the SnIa
Table I: Comparison of the one, two and three parametermodels to ΛCDM. Note that the differences for the AIC andBIC are in both cases with regard to the model with the mini-mum value for the corresponding criterion. For the definitionof the AIC and BIC, see Eqs. (17) and (18).Model AIC ∆AIC BIC ∆BICΛCDM 467 .
513 2 .
089 471 .
497 0 . .
082 1 .
658 471 . − . .
031 2 .
607 475 .
999 4 . .
424 0 .
000 477 .
376 5 . Μ H z L - - - Μ H z L - Μ L C D M Figure 1: Left: The distance modulus for the best fit ΛCDM model (dashed line) and the SCDM model + IR correction ofEq. (5) with the 3 parameter case (solid black line) and the 2 parameter case with γ = 0 (dotted line). Right: The residualsrelative to ΛCDM for the model with the IR correction of Eq. (5) with the 3 parameter case (solid black line) and the 2parameter case with γ = 0 (dotted line). The dashed line corresponds to zero. - - - - - Α Β Figure 2: The 1 σ and 2 σ contours in the ( α, β ) plane for the2 parameter model with γ = 0. The black dot ( α = 1 . β = − .
51) corresponds to the best fit. data. Our first result is that the mechanism derived inRef. [15] (see also Appendix A), Eq. (1), is not enough,by itself, to describe the observed amount of acceleration.The absence of free parameters in equation (1) (the modelin [15] has one parameter less than ΛCDM) does notmake up for the very poor fit of the data. However, amore general structure of corrections (4) can lead to asensibly larger luminosity distance than in the Einsteinde Sitter universe. In particular, when γ = 0 in Eq. (5)-(6), we have found that the model fits the data very wellfor the values close to the exact numbers α = 1 and β = − /
2. It is interesting to consider such sharp numericalvalues, not because of abstract numerology, but becausea mechanism analogous to that described in Appendix Avery naturally produces coefficients which are integers orsimple fractions.At present it is not clear if the corrections that onefinds by applying USEP to a field automatically apply to, or are inherited by, other types of fields. The sug-gested luminosity distance may eventually turn out tobe produced by considering other fields than the scalarfield considered in [15] or by means of other theoreticalinsights. We also considered the full three parametersmodel (5)-(6), whose best fit considerably improves the χ and is found to be favored over ΛCDM by the AICbut not by the BIC criterion.It should be noted that the parameters that we arefitting are not coupling constants, and do not appear ina Lagrangian. Rather, they are intended as the termsof a series expansion that approximate the new “IR-completed” theory starting from GR. As mentioned inthe introduction, the proposed deformation is present atany time during the cosmological evolution; it affects anycosmological observation at high redshift, regardless ofwhen such an observation takes place, and therefore ad-dresses the “coincidence problem” in the most direct way.We note that our model requires a rather low value ofthe Hubble constant, H ∼
50 km/s/Mpc, compared tothe constraint from the Hubble Key Project with thedetermination of H = 72 ± H ( z ) ≃ H (1 + z ) / even for z . O (1) due to theabsence of a dark energy component. However, methodsof the determination of H that are largely independentof distance scales of the Large Magellanic Cloud Cepheidtypically give low values of H [26]. For example, Reese et al. [27] showed that Sunyaev-Zel’dovich distances to41 clusters provide the constraint H = 54 +4 − km/s/Mpcin the Einstein de Sitter universe. Thus the informationof H alone is not yet sufficient to rule out our model. It is interesting, for instance, that a different mechanism, basedon a Casimir-like vacuum energy [24], needs a Veneziano ghostin order reproduce a density of the right order of magnitude.
It would be interesting to see the effect of the proposedIR correction on the angular diameter distance to thelast scattering surface and estimate the modification tothe temperature anisotropies in Cosmic Microwave Back-ground (CMB). CMB data will be certainly useful toplace further constraints on our model and to completethe picture at higher redshift.
ACKNOWLEDGEMENTS
F. P. is indebted to Bruce A. Bassett for many valu-able conversations during his visit at Perimeter Insti-tute. We thank Justin Khoury for his suggestions onthe manuscript. S. N. acknowledges support from theNiels Bohr International Academy, the EU FP6 MarieCurie Research & Training Network “UniverseNet” un-der Contract No. MRTN-CT-2006-035863 and the Dan-ish Research Council under FNU Grant No. 272-08-0285.The research of F. P. at Perimeter Institute is supportedin part by the Government of Canada through NSERCand by the Province of Ontario through the Ministry ofResearch & Innovation. S. T. thanks financial supportfor JSPS (Grant No. 30318802).
Appendix A: USEP and the derivation of Eq. (1)
While referring to [15] for details and motivations, itis worth summarizing here the basic steps that lead toEq. (1), based on the “ultra strong equivalence principle”(USEP). The USEP is a statement about the bare energymomentum tensor of the quantized fields on a generalbackground, namely:USEP:
For each matter field or sector suffi-ciently decoupled from all other matter fields,there exists a state, the “vacuum”, for whichthe expectation value of the (bare) energymomentum tensor reads the same as in flatspace, regardless of the configuration of thegravitational field.
Our starting point is a free scalar field with the action S = 12 Z d x √− g (cid:0) ∂φ − m φ (cid:1) , (A1)in a spatially flat FLRW metric:d s = d t − a ( t )d ~x . (A2)The equations of motions of the field and the 00 compo-nent of its energy momentum tensor read, respectively,¨ φ ( t, ~x ) + 3 H ˙ φ ( t, ~x ) − ∂ i a ( t ) φ ( t, ~x ) + m φ ( t, ~x ) = 0 , (A3) T ( t, ~x ) = 12 (cid:20) ˙ φ + 1 a ( ∂ j φ ) + m φ (cid:21) . (A4) In order to apply USEP we now review the standard cal-culation of the energy momentum VEV in such a space-time and compare it to the flat space result.Upon standard quantization in the Heisenberg picturethe field is expanded in creators and annihilators: φ ( t, ~x ) = Z d n h ψ n ( t ) e i~n · ~x A ~n + ψ ∗ n ( t ) e − i~n · ~x A † ~n i , (A5)where ~n is the comoving momentum label in the FLRWspace; ~n is a conserved quantity, related to the properphysical momentum ~p by ~p = ~n/a ( t ). From the canonicalcommutation relations [ φ ( ~x ) , π ( ~x ′ )] = i (2 π ) δ ( ~x − ~x ′ ),the commutation relations among the global Fourier op-erators are easily derivable,[ A ~n , A † ~n ′ ] = δ ( ~n − ~n ′ ) . (A6)It is customary to choose A ~n as the operator that al-ways annihilates the vacuum. The chosen vacuum stateis therefore implicitly characterized by the choice of themode functions ψ n ( t ), that, by (A5) and (A3), satisfy¨ ψ n + 3 H ˙ ψ n + ω n ψ n = 0 , (A7)where ω n = p n /a + m .The solutions of (A7) corresponding to the adiabaticvacuum [28, 29] can be found by a formal WKB expan-sion. After quantization, the energy momentum tensorof the field (A4) becomes an operator whose expectationvalue on the adiabatic vacuum reads [15, 29] h T ( t, ~x ) i = 14 π a Z ∞ n (cid:20) ω n + H a n + O ( n − ) (cid:21) d n . (A8)The above should be compared to the flat space result h T ( t, ~x ) i flat = 14 π a Z ∞ n ω n d n . (A9)It is known that there is a strict connection betweenthe geometric properties of a manifold and the spectrumof the differential operators [30] or the algebra of func-tions [31] therein defined; such abstract characterizationshave occasionally been used for generalizing common ge-ometrical concepts and the description of spacetime it-self [31, 32]. However, so far, attempts in this directionhave always been applied to the UV and intended to mod-ify spacetime at the smallest scales. Here we would liketo modify the IR-spectral properties of the FLRW metric(and therefore its geometry) in order to enforce USEP.The proposed deformation is argued to correspond to abreakdown of the metric Riemannian structure at dis-tances comparable to H − .So the idea is to modify the physics in the infra-red butstrictly maintain the equations and the relations valid lo-cally such as the field equations (A3) and the form of theenergy momentum tensor (A4). We choose a point inFLRW ( ~x = 0) and make a formal Taylor expansion ofwhich GR is the zeroth order. In the spirit of a generalspectral deformation, we conjecture a mismatch betweenthe “metric-manifold” Fourier labels ~n and the physicalmomenta ~k that locally define the infinitesimal transla-tions and the derivatives of the local fields. In otherwords, we now write the field in ~x ≃ φ ( t, ~x ≈
0) = Z d n h ψ k ( t ) e i~k · ~x A ~n + ψ ∗ k ( t ) e − i~k · ~x A † ~n i , (A10)where ~k = ~n (cid:18) − H a n + higher order (cid:19) . (A11)Note that, when derivatives of the field are taken in ~x = 0, a factor of k , instead of n , drops. The form ofabove relation, which is one of the main results of [15],is dictated by the request that the quadratically diver-gent time dependent piece of (A8) disappears. In otherwords, that is the first order correction in order to im-pose USEP upon this particular GR solution. The cor-rected mode equation is in fact obtained by substituting(A10) into (A3), which is assumed to be strictly valid be-cause it applies locally. To the modified mode equation,¨ ψ k ( t ) + 3 H ˙ ψ k ( t ) + ω k ψ k ( t ) = 0, the same WKB expan-sion can be applied and the quadratic divergence in (A8)is reabsorbed just by re-expressing h | T | i in terms ofthe appropriate “flat measure” time-independent Fouriercoordinates n : h T ( t, ~x ≈ i = 14 π a Z ∞ n (cid:20) ω k + H a k + O ( n − ) (cid:21) d n = 14 π a Z ∞ n (cid:2) ω n + O ( n − ) (cid:3) d n . In order to define local quantities away from the originwe exploit the assumption of spatial homogeneity and usethe translation operator T ( ~λ ) = exp (cid:18) − i~λ · Z d n ~k ( ~n ) A † ~n A ~n (cid:19) , (A12)that we obtain by simple exponentiation of the (mod-ified!) momentum operator. The parameter λ is thecomoving distance. The field away from ~x = 0 is thusdefined as φ ( t, ~λ ) ≡ T i ( λ ) φ ( t, T − i ( λ ) and reads φ ( t, ~λ ) = Z d n h ψ k ( t ) e i~k · ~λ A ~n + ψ ∗ k ( t ) e − i~k · ~λ A † ~n i . (A13)From the above expression it is straightforward to cal-culate the modified commutator between the canonicalmomentum π (0) = a ˙ φ (0) and the field at comoving dis-tance ~λ ,[ π (0) , φ ( ~λ )] = − i (cid:18) δ ( ~λ ) + 18 π H a λ (cid:19) . (A14) Note that there is a potential ambiguity in defining thetime derivative of a displaced operator. By deriving(A13) we get π ( ~λ ) = a Z d n h ˙ ψ k ( t ) e i~k · ~λ A ~n + ˙ ψ ∗ k ( t ) e − i~k · ~λ A † ~n i + ia Z d n ( ~k · ~λ )˙ h ψ k e i~k · ~λ A ~n − ψ ∗ k e − i~k · ~λ A † ~n i . The second line in the above equation is there because k is time dependent. However, if we just apply the transla-tion to π (0), instead of deriving φ ( ~λ ), those terms wouldnot be there. Therefore, for consistency, we need tomake them ineffective at the required order of approxi-mation. This can be done by imposing that [ φ (0) , π ( ~λ )] = − [ π (0) , φ ( ~λ )]. Because of the second line the last equa-tion, the commutator between φ (0) and π ( λ ) gives[ φ (0) , π ( ~λ )] = − [ π (0) , φ ( ~λ )] − i a (2 π ) Z d n e − i~k · ~λ | ψ n | ( ~k · ~λ )˙ , (A15)the last term being the spurious contribution. In orderto get rid of it, we have to make the comoving physicaldistance ~λ also time dependent. This effectively meansthat, after an infinitesimal time step d t , we have to re-consider the field translated, from ~x ≈
0, not by the samecomoving distance λ , but by a slightly different amount.At high momenta/small distances, since | ψ k | ∼ /n ,the integral in the second term of (A15) reads Z d n e − i~n · ~λ n (cid:20) ˙ ~λ · ~n (cid:18) − H a n (cid:19) − ~λ · ~n ( H a )˙2 n (cid:21) . We make the ansatz ˙ ~λ = bλ ( H a )˙ ~λ , where b is a num-ber to be determined. We get Z d n e − i~k · ~λ | ψ n | ( ~k · ~λ )˙ ≃ i ( H a )˙ ddα Z d n (cid:18) bλ n − n (cid:19) e − iα ~n · ~λ (cid:12)(cid:12)(cid:12)(cid:12) α =1 . The last integral can be regularized by setting n − → n − ǫ and taking the ǫ → α . The result is null for b = 1 /
4, whichfixes the time dependence of λ :˙ λ = λ ( H a )˙4 . (A16) Appendix B: Exact solutions
Here we present some analytical solutions for the dif-ferential equation (5). If γ = 0 we obtain the followinganalytic solution: H r ( z ) = 4 / √ z − − α + √ δ + 2 √ δ/ [(1 + z ) √ δ/ − , (B1)where δ = (1 + 2 α ) − β .When γ = 0, finding the solution is much more difficultand can only be given in an implicit form. For example,let us consider the correction in Eq. (2), i.e. α = β =0 and γ = 1 /
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