In single-field, slow-roll inflationary models, scalar and tensorial
(Gaussian) perturbations are both characterized by a zero mean and a non-zero
variance. In position space, the corresponding variance of those fields
diverges in the ultraviolet. The requirement of a finite variance in position
space forces its regularization via quantum field renormalization in an
expanding universe. This has an important impact on the predicted scalar and
tensorial power spectra for wavelengths that today are at observable scales. In
particular, we find a non-trivial change in the consistency condition that
relates the tensor-to-scalar ratio "r" to the spectral indices. For instance,
an exact scale-invariant tensorial power spectrum, n_t=0, is now compatible
with a non-zero ratio r= 0.12 +/- 0.06, which is forbidden by the standard
prediction (r=-8n_t). Forthcoming observations of the influence of relic
gravitational waves on the CMB will offer a non-trivial test of the new
predictions.Comment: 4 pages, jpconf.cls, to appear in the Proceedings of Spanish
  Relativity Meeting 2009 (ERE 09), Bilbao (Spain

Birrel N D

Dodelson S

G J Olmo

I Agulló

J Navarro-Salas

L Parker

Lyth D H

Mukhanov V F

Parker L

Weinberg S

Journal of Physics : Conference Series

English

arXiv

5 páginas.-- Trabajo presentado al Spanish Relativity Meeting (ERE 2009).-- El PDF es la versión pre-print (arXiv:1002.3914v1).In single- eld, slow-roll in
ationary models, scalar and tensorial (Gaussian)
perturbations are both characterized by a zero mean and a non-zero variance. In position
space, the corresponding variance of those  fields diverges in the ultraviolet. The requirement
of a  nite variance in position space forces its regularization via quantum fi eld renormalization
in an expanding universe. This has an important impact on the predicted scalar and tensorial
power spectra for wavelengths that today are at observable scales. In particular, we  find a
non-trivial change in the consistency condition that relates the tensor-to-scalar ratio r to the
spectral indices. For instance, an exact scale-invariant tensorial power spectrum, nt = 0, is now
compatible with a non-zero ratio r ~~ 0:12+/- 0:06, which is forbidden by the standard prediction
(r = -8nt). Forthcoming observations of the in
uence of relic gravitational waves on the CMB
will o er a non-trivial test of the new predictions.This work has been partially supported by the spanish grant FIS2008-
06078-C03-02. I.A. and L.P. have been partly supported by NSF grants PHY-0071044 and
PHY-0503366 and by a UWM RGI grant. G.O. thanks MICINN for a JdC contract and the
"José Castillejo" program for funding a stay at the University of Wisconsin-Milwaukee.Peer reviewe

Agulló, Iván

Navarro-Salas, José

Olmo, Gonzalo J.

Parker, Leonard

Digital.CSIC

arXiv:1002.3914v1  [gr-qc]  20 Feb 2010Inflation, Renormalization, and CMB AnisotropiesI. Agullo´1, J. Navarro-Salas2, Gonzalo J. Olmo3,1, and LeonardParker11. Physics Department, University of Wisconsin-Milwaukee, P.O.Box 413, Milwaukee, WI53201 USA2. Departamento de F´ısica Teo´rica and IFIC, Centro Mixto Universidad de Valencia-CSIC.Facultad de F´ısica, Universidad de Valencia, Burjassot-46100, Valencia, Spain.3. Instituto de Estructura de la Materia, CSIC, Serrano 121, 28006 Madrid, SpainE-mail: ivan.agullo@uv.es, jnavarro@ific.uv.es, olmo@iem.cfmac.csic.es,leonard@uwm.eduAbstract. In single-field, slow-roll inflationary models, scalar and tensorial (Gaussian)perturbations are both characterized by a zero mean and a non-zero variance. In positionspace, the corresponding variance of those fields diverges in the ultraviolet. The requirementof a finite variance in position space forces its regularization via quantum field renormalizationin an expanding universe. This has an important impact on the predicted scalar and tensorialpower spectra for wavelengths that today are at observable scales. In particular, we find anon-trivial change in the consistency condition that relates the tensor-to-scalar ratio r to thespectral indices. For instance, an exact scale-invariant tensorial power spectrum, nt = 0, is nowcompatible with a non-zero ratio r ≈ 0.12± 0.06, which is forbidden by the standard prediction(r = −8nt). Forthcoming observations of the influence of relic gravitational waves on the CMBwill offer a non-trivial test of the new predictions.1. IntroductionInflation has become a fundamental piece of the standard cosmological model [1]. Not onlyit addresses and solves the classical problems of the big bang model but it also provides anelegant quantum mechanical mechanism for the origin of primordial perturbations, essentialfor explaining the structures that we see today [2]. In the simplest models of inflation, ascalar field slowly rolling down its potential causes an exponentially large expansion. In thisbackground, quantum vacuum fluctuations of the inflaton field itself and of purely tensorialmodes (gravitational waves) acquire classical properties due to their interaction with theexpanding geometry. As a result, a classical field of scalar (and tensorial) inhomogeneities arises.Cosmological perturbation theory tells us that these primordial perturbations are responsiblefor the existence and richness of the structure that we observe in the temperature anisotropiesof the cosmic microwave background (CMB) and in the large scale distribution of galaxies.A disturbing aspect of the spectra of scalar and tensorial perturbations generated during inflationis that they have a divergent variance. In the case of classical perturbations, divergences of thevariance are usually removed by means of window functions that filter out the wavelengthsthat cause the problem. However, the primordial spectra have a quantum origin and theirdivergences should be removed, on grounds of theoretical consistency, by means of the well-established methods of renormalization in curved spaces[3]. In this work we show that theregularization of the variance of scalar and tensorial perturbations during inflation has a non-trivial impact on the primordial power spectra of those fields [4, 5]. Though the resulting spectraare almost scale free within the slow-roll approximation, the relation between spectral indicesand the slow-roll parameters is significantly changed as compared to the standard derivation.The ratio of tensor to scalar amplitudes is also modified and gives rise to a number of possibilitieswhich are explicitly forbidden according to the standard predictions. In particular, the tensorialspectral index is allowed to take a zero value without implying a zero tensor to scalar ratio, ormay even take positive values.2. Spectrum of fluctuations from inflationLet us first focus on the production of relic gravitational waves by considering fluctuatingtensorial modes hij(~x, t) in an expanding, spatially flat universeds2 = −dt2 + a2(t)(δij + hij)dxidxj . (1)The wave equation obeyed by these modes is given by− a2h¨ij − 3aa˙h˙ij +∇2hij = 0 . (2)The fluctuating fields hij can be decomposed in two independent polarization modesh(p)(~x, t)e(p)ij , where the index p represents the two polarizations + and ×, and e(p)ij are twoindependent constant matrices that obey the conditions eij = eji,∑i eii = 0, and∑i kieij = 0.Expanding the fields h(p)(~x, t) in modes and omitting the polarization label p, (2) yieldsh¨k + 3Hh˙k +k2a2hk = 0 , (3)with k ≡ |~k|. This equation indicates that the relevant part of the tensorial modes is representedby two independent massless scalar fields h(+,×)(~x, t). The solutions that satisfy the asymptoticadiabatic condition [6, 3] for very high k in a slow-roll inflationary background arehk(t) = (−16piGτpi/4(2pi)3a2)1/2H(1)ν (−kτ) , (4)where τ =∫dt/a(t) is the conformal time, the index of the Bessel function is ν = 3/2 + , and =M2P2(V ′V)2is one of the slow-roll parameters. With these solutions it is easy to evaluate thevariance of the gravitational wave fields h(+,×)〈h2〉 =∫∞0k2dk∫dΩ|hk|2 =∫∞0dkk∆2h(k, t) , (5)where we have defined the power spectrum of the field h as ∆2h(k, t) ≡ 4pik3|hk|2. As we advancedin the introduction, the large k behavior of the modes makes this integral divergent〈h2〉 =∫∞0dkk16piGk34pi2a3[ak[1 +(2 + 3)2k2τ2] + ...]. (6)Since the power spectrum is expressed in momentum space, the natural renormalization schemeto apply is the so-called adiabatic subtraction [4], as it renormalizes the theory in momentumspace. Adiabatic renormalization [3, 7] removes the divergences present in the formal expression(5) by subtracting counterterms mode by mode in the integrand of (5)〈h2〉ren =∫∞0dkk∆˜2h(k, t) =∫∞0dkk[4pik3|h~k|2 −16piGk34pi2a3(w−1k + (W−1k )(2))], (7)with wk = k/a(t). The subtraction of the first term (16piGk3/4pi2a3wk) cancels the typical flatspace vacuum fluctuations, which are responsible for the quadratic divergence in the integral (6).The additional term, proportional to (W−1k )(2) = − 1w2k[12w−1/2kd2dt2w−1/2k − 12w−1k(34a˙2a2 +32a¨a)]and which involves a˙2 and a¨, is necessary to properly perform the renormalization in anexpanding universe, since it cancels the logarithmic divergence in (6).According to expression (5), the unrenormalized power spectrum of tensor perturbationsevaluated a few e-folds after the Hubble horizon exit time tk (defined by k/a(tk) = H(tk))is given byPt(k) = 4∆2h(k) = 16pik3|hk|2 = 8M2P(H(tk)2pi)2(8)The k dependence of H2(tk), d lnH(tk)/d ln k = −, leads to Pt(k, tk) = Pt(k0)(kk0)−2, wherek0 is a pivot scale. The tensorial spectral index nt is defined as the exponent in that expression.So nt = −2(tk). Using the renormalized variance (7) to define the renormalized power spectrumand taking into account the time dependence of the counterterms, we findP rent (k, n) = 4∆˜2h(k, n) ≈8M2P(H(tk)2pi)2(tk)(2n − 3/2) , (9)where n represents the number of e-folds after the Hubble exit at which the spectrum is evaluated(we assume n > 1 but n 1). We note that if one evaluates the power spectra at the end of theslow-roll era (where n ∼ 1) the contribution of the counterterms is still significant. However,we find it more natural to evaluate the spectra soon after tk, when the modes have alreadyacquired classical properties. The spectral index corresponding to the renormalized tensorialpower spectrum is given bynrent ≡d lnP rentd ln k= 2(− η) (10)where the slow-roll parameters  and η = M2P (V′′/V ) are evaluated at tk.The discussion just presented regarding the tensorial modes can be paralleled to the case of scalarperturbations. In this case one deals with the gauge invariant scalar R = Ψ+ Hφ˙0δφ, where Ψ isthe curvature perturbation (R(3) = 4∇2Ψ/a2) of the spatial metric gij = a2[(1−2Ψ)δij+2∂ijE].In momentum space, R obeys the equation [1]d2Rkdτ2+2zdzdτdRkdτ+ k2Rk = 0 , (11)where z ≡ aφ˙0/H. In the slow roll approximation z−1dz/dτ = aH(1+2−η), and the solutionsobeying the adiabatic condition (and the de Sitter symmetry for H constant) areRk(t) = (−piτ/4(2pi)3z2)1/2H(1)µ (−τk) , (12)where µ = 3/2 + 3 − η. The resulting unrenormalized and renormalized scalar power spectraand spectral indices are (PR(k) = PR(k0)(kk0)ns−1)PR(k) =12M2P (H(tk)2pi)2, ns − 1 = −6+ 2η (13)P renR (k, n) ≈(3− η)2M2P (H(tk)2pi)2(2n− 3/2) , ns − 1 = −6+ 2η + (122 − 8η + ξ)3− η (14)where ξ is another slow roll parameter: ξ ≡M4P (V ′V ′′′/V 2). This parameter can be reexpressedin terms of , η and the running of the tensorial index n′t ≡ dnt/d ln k as n′t = 8(− η) + 2ξ.3. ConclusionsThe expressions derived above for the power spectra and spectral indices show thatrenormalization has a significant impact on the predictions of slow-roll inflation. Unlike in thestandard derivation, we see that the renormalized tensorial power spectrum is no longer directlyrelated to the Hubble scale during inflation. The amplitude of this spectrum is now modulatedby the slow-roll parameter , which is zero in exact de Sitter inflation. On the other hand, theappearance of the combination 3 − η in the numerator of the renormalized scalar spectrumsuggests that P renR(k, n) ∼ 12M2P(H(tk)2pi)2(up to the numerical factor 2n− 3/2), which indicatesthat H(tk) ∼ 1015GeV. For the exponential potential [1], we find that H ∼ 1.5 × 1015/√4n − 3GeV, roughly up to an order of magnitude larger than the unrenormalized prediction.The most dramatic difference between the standard predictions and the predictions afterrenormalization presented here affects the so-called consistency condition. The tensor to scalarratio in the standard approach gives r = Pt/PR = 16 = −8nt. This implies that if a backgroundof gravitational waves is observed, then its spectral index must be constrained by the ratio of thetensor to scalar amplitudes. If this constraint is not satisfied, then single field slow-roll inflationwould be ruled out. However, the consistency condition after renormalization becomesr = 4(1 − ns − nt) + 4n′tn2t − 2n′t(1− ns −√2n′t + (1− ns)2 − n2t). (15)This expression is much more involved that the standard one and its implications are far reaching.It allows for a zero nt while having a non-zero ratio r. Also, the values of nt are not constrainedto be negative. In addition, according to the standard derivation, the running of nt is fullydetermined by the values of ns and nt, since then one finds n′t = −nt(1 − ns + nt). On thecontrary, the manipulations that lead to (15) indicate that n′t is now an independent quantitythat needs to be measured in order to check the new consistency relation (15). This aspect couldmake more challenging the experimental verification of the consistency condition of (single-field)slow roll inflation.To conclude, the predictions presented in this talk will soon come within the range of ongoingand planned experiments. If single field slow-roll inflation is correct, then observations will(hopefully) tell us if nature have chosen the renormalized spectra as the seeds of perturbationsor not. Quantum field theory in curved spacetimes will thus face a crucial experimental test.Acknowledgements. This work has been partially supported by the spanish grant FIS2008-06078-C03-02. I.A. and L.P. have been partly supported by NSF grants PHY-0071044 andPHY-0503366 and by a UWM RGI grant. G.O. thanks MICINN for a JdC contract and the“Jose´ Castillejo” program for funding a stay at the University of Wisconsin-Milwaukee.References[1] Lyth D.H. and Liddle A.R. The primordial density perturbation, Cambridge University Press, (2009); WeinbergS., Cosmology, Oxford University Press, (2008); Dodelson S., Modern Cosmology, Academic Press, (2003).[2] Mukhanov V.F. and Chibisov G.V., JETP Letters 33, 532(1981). Hawking, S. W., Phys. Lett. B115, 295(1982). Guth A. and Pi, S.-Y., Phys. Rev. Lett. 49, 1110 (1982). Starobinsky, A. A., Phys. Lett. B117, 175(1982). Bardeen, J.M., Steinhardt, P.J. and Turner, M.S., Phys. Rev.D28, 679 (1983).[3] Parker L. and Toms D.J., Quantum field theory in curved spacetime: quantized fields and gravity, CambridgeUniversity Press, (2009).[4] Parker L., Amplitude of perturbations from inflation, hep-th/0702216.[5] Agullo´ I., Navarro-Salas J., Olmo G.J. and Parker L., Phys.Rev. D81,043514(2010); Phys. Rev. Lett. 103,061301 (2009); Gen. Rel. Grav. 41, 2301 (2009); Phys. Rev. Lett. 101, 171301 (2008).[6] Parker L., Phys.Rev.Lett. 21 562 (1968); Phys. Rev. 183, 1057(1969); Parker L., The creation of particles inan expanding universe, Ph.D. thesis, Harvard University (1966).[7] Birrel N.D. and Davies P.C.W., Quantum fields in curved space, Cambridge University Press, (1982).

Journal of Physics Conference Series

Inflation, renormalization, and CMB anisotropies

Crossref

In single-field, slow-roll inflationary models, scalar and tensorial (Gaussian) perturbations are both characterized by a zero mean and a non-zero variance. In position space, the corresponding variance of those fields diverges in the ultraviolet. The requirement of a finite variance in position space forces its regularization via quantum field renormalization in an expanding universe. This has an important impact on the predicted scalar and tensorial power spectra for wavelengths that today are at observable scales. In particular, we find a non-trivial change in the consistency condition that relates the tensor-to-scalar ratio r to the spectral indices. For instance, an exact scale-invariant tensorial power spectrum, nt 0, is now compatible with a non-zero ratio r 0.12 0.06, which is forbidden by the standard prediction (r -8nt). Forthcoming observations of the influence of relic gravitational waves on the CMB will offer a non-trivial test of the new predictions. © 2010 IOP Publishing Ltd

Agulló, I.

Navarro-Salas, J.

Olmo, G. J.

Parker, L.

LSU Scholarly Repository (Louisiana State Univ.)

Louisiana State University LSU Scholarly Repository Faculty Publications Department of Physics & Astronomy 1-1-2010 Inflation, renormalization, and CMB anisotropies I. Agulló University of Wisconsin-Milwaukee J. Navarro-Salas CSIC-UV - Instituto de Física Corpuscular (IFIC) G. J. Olmo CSIC - Instituto de Estructura de la Materia (IEM) L. Parker University of Wisconsin-Milwaukee Follow this and additional works at: https://repository.lsu.edu/physics_astronomy_pubs Recommended Citation Agulló, I., Navarro-Salas, J., Olmo, G., & Parker, L. (2010). Inflation, renormalization, and CMB anisotropies. Journal of Physics: Conference Series, 229 https://doi.org/10.1088/1742-6596/229/1/012058 This Conference Proceeding is brought to you for free and open access by the Department of Physics & Astronomy at LSU Scholarly Repository. It has been accepted for inclusion in Faculty Publications by an authorized administrator of LSU Scholarly Repository. For more information, please contact ir@lsu.edu. Journal of Physics: Conference SeriesOPEN ACCESSInflation, renormalization, and CMB anisotropiesTo cite this article: I Agulló et al 2010 J. Phys.: Conf. Ser. 229 012058 View the article online for updates and enhancements.You may also likeMaxwell group and HS field theorySergey Fedoruk and Jerzy Lukierski-Unitary symmetry constraints on tensorialgroup field theory renormalization groupflowVincent Lahoche and Dine OusmaneSamary-The impact of renormalization on theobservable predictions of inflationJose Navarro-Salas-Recent citationsRenormalization of the inflationaryperturbations revisitedTommi Markkanen-Power spectrum generated during inflationMar Bastero-Gil et al-The LIGO–LIGO cross correlation for thedetection of relic scalar gravitational wavesChristian Corda-This content was downloaded from IP address 130.39.60.163 on 20/10/2021 at 13:36Inflation, renormalization, and CMB anisotropiesI Agulló1, J Navarro-Salas2, G J Olmo3? and L Parker11 Physics Department, University of Wisconsin-Milwaukee, P.O.Box 413, Milwaukee, WI53201 USA2 Departamento de F́ısica Teórica and IFIC, Centro Mixto Universidad de Valencia-CSIC.Facultad de F́ısica, Universidad de Valencia, Burjassot-46100, Valencia, Spain3 Instituto de Estructura de la Materia, CSIC, Serrano 121, 28006 Madrid, SpainE-mail: ivan.agullo@uv.es, jnavarro@ific.uv.es, olmo@iem.cfmac.csic.es,leonard@uwm.eduAbstract. In single-field, slow-roll inflationary models, scalar and tensorial (Gaussian)perturbations are both characterized by a zero mean and a non-zero variance. In positionspace, the corresponding variance of those fields diverges in the ultraviolet. The requirementof a finite variance in position space forces its regularization via quantum field renormalizationin an expanding universe. This has an important impact on the predicted scalar and tensorialpower spectra for wavelengths that today are at observable scales. In particular, we find anon-trivial change in the consistency condition that relates the tensor-to-scalar ratio r to thespectral indices. For instance, an exact scale-invariant tensorial power spectrum, nt = 0, is nowcompatible with a non-zero ratio r ≈ 0.12± 0.06, which is forbidden by the standard prediction(r = −8nt). Forthcoming observations of the influence of relic gravitational waves on the CMBwill offer a non-trivial test of the new predictions.1. IntroductionInflation has become a fundamental piece of the standard cosmological model [1]. Not onlyit addresses and solves the classical problems of the big bang model but it also provides anelegant quantum mechanical mechanism for the origin of primordial perturbations, essentialfor explaining the structures that we see today [2]. In the simplest models of inflation, ascalar field slowly rolling down its potential causes an exponentially large expansion. In thisbackground, quantum vacuum fluctuations of the inflaton field itself and of purely tensorialmodes (gravitational waves) acquire classical properties due to their interaction with theexpanding geometry. As a result, a classical field of scalar (and tensorial) inhomogeneities arises.Cosmological perturbation theory tells us that these primordial perturbations are responsiblefor the existence and richness of the structure that we observe in the temperature anisotropiesof the cosmic microwave background (CMB) and in the large scale distribution of galaxies.A disturbing aspect of the spectra of scalar and tensorial perturbations generated during inflationis that they have a divergent variance. In the case of classical perturbations, divergences of thevariance are usually removed by means of window functions that filter out the wavelengthsthat cause the problem. However, the primordial spectra have a quantum origin and theirdivergences should be removed, on grounds of theoretical consistency, by means of the well-established methods of renormalization in curved spaces[3]. In this work we show that theregularization of the variance of scalar and tensorial perturbations during inflation has a non-Spanish Relativity Meeting (ERE 2009) IOP PublishingJournal of Physics: Conference Series 229 (2010) 012058 doi:10.1088/1742-6596/229/1/012058c© 2010 IOP Publishing Ltd 1trivial impact on the primordial power spectra of those fields [4, 5]. Though the resulting spectraare almost scale free within the slow-roll approximation, the relation between spectral indicesand the slow-roll parameters is significantly changed as compared to the standard derivation.The ratio of tensor to scalar amplitudes is also modified and gives rise to a number of possibilitieswhich are explicitly forbidden according to the standard predictions. In particular, the tensorialspectral index is allowed to take a zero value without implying a zero tensor to scalar ratio, ormay even take positive values.2. Spectrum of fluctuations from inflationLet us first focus on the production of relic gravitational waves by considering fluctuatingtensorial modes hij(~x, t) in a spatially flat FRW universe ds2 = −dt2 + a2(t)(δij + hij)dxidxj .The wave equation obeyed by these modes is given by−a2ḧij − 3aȧḣij +∇2hij = 0 . (1)The fluctuating fields hij can be decomposed in two independent polarization modesh(p)(~x, t)e(p)ij , where the index p represents the two polarizations + and ×, and e(p)ij are twoindependent constant matrices that obey the conditions eij = eji,∑i eii = 0, and∑i kieij = 0.Expanding the fields h(p)(~x, t) in modes and omitting the polarization label p, eq. (1) yieldsḧk + 3Hḣk +k2a2hk = 0 , (2)with k ≡ |~k|. This equation indicates that the relevant part of the tensorial modes isrepresented by two independent massless scalar fields h(+,×)(~x, t). The solutions that satisfythe asymptotic adiabatic condition [6, 3] for very high k in a slow-roll inflationary backgroundare hk(t) = (−16πGτπ/4(2π)3a2)1/2H(1)ν (−kτ), where τ =∫dt/a(t) is the conformal time, theindex of the Bessel function is ν = 3/2 + ε, and ε = M2P2(V ′V)2is one of the slow-roll parameters.With these solutions it is easy to evaluate the variance of the gravitational wave fields h(+,×)〈h2〉 =∫ ∞0k2dk∫dΩ|hk|2 =∫ ∞0dkk∆2h(k, t) , (3)where we have defined the power spectrum of the field h as ∆2h(k, t) ≡ 4πk3|hk|2. As we advancedin the introduction, the large k behavior of the modes makes this integral divergent〈h2〉 =∫ ∞0dkk16πGk34π2a3[ak[1 +(2 + 3ε)2k2τ2] + ...]. (4)Since the power spectrum is expressed in momentum space, the natural renormalization schemeto apply is the so-called adiabatic subtraction [4], as it renormalizes the theory in momentumspace. Adiabatic renormalization [3, 7] removes the divergences present in the formal expression(3) by subtracting counterterms mode by mode in the integrand of (3)〈h2〉ren =∫ ∞0dkk∆̃2h(k, t) =∫ ∞0dkk[4πk3|h~k|2 − 16πGk34π2a3(w−1k + (W−1k )(2))], (5)with wk = k/a(t). The subtraction of the first term (16πGk3/4π2a3wk) cancels the typical flatspace vacuum fluctuations, which are responsible for the quadratic divergence in the integral (4).The additional term, proportional to (W−1k )(2) = − 1w2k[12w−1/2kd2dt2w−1/2k −12w−1k(34ȧ2a2+ 32äa)]Spanish Relativity Meeting (ERE 2009) IOP PublishingJournal of Physics: Conference Series 229 (2010) 012058 doi:10.1088/1742-6596/229/1/0120582and which involves ȧ2 and ä, is necessary to properly perform the renormalization in anexpanding universe, since it cancels the logarithmic divergence in (4).According to expression (3), the unrenormalized power spectrum of tensor perturbationsevaluated a few e-folds after the Hubble horizon exit time tk (defined by k/a(tk) = H(tk))is given byPt(k) = 4∆2h(k) = 16πk3|hk|2 =8M2P(H(tk)2π)2(6)The k dependence of H2(tk), d lnH(tk)/d ln k = −ε, leads to Pt(k, tk) = Pt(k0)(kk0)−2ε, wherek0 is a pivot scale. The tensorial spectral index nt is defined as the exponent in that expression.So nt = −2ε(tk). Using the renormalized variance (5) to define the renormalized power spectrumand taking into account the time dependence of the counterterms, we findP rent (k, n) = 4∆̃2h(k, n) ≈8M2P(H(tk)2π)2ε(tk)(2n− 3/2) , (7)where n represents the number of e-folds after the Hubble exit at which the spectrum is evaluated(we assume n > 1 but nε 1). We note that if one evaluates the power spectra at the end of theslow-roll era (where nε ∼ 1) the contribution of the counterterms is still significant. However, wefind it more natural to evaluate the spectra soon after tk, when the modes have already acquiredclassical properties. The corresponding tensorial spectral index is (here ε and η = M2P (V′′/V )are evaluated at tk)nrent ≡d lnP rentd ln k= 2(ε− η) . (8)The discussion just presented regarding the tensorial modes can be parallelled to the case ofscalar perturbations. In this case one deals with the gauge invariant scalar R = Ψ+ Hφ̇0δφ, whereΨ is the curvature perturbation (R(3) = 4∇2Ψ/a2) of the spatial metric gij = a2[(1 − 2Ψ)δij +2∂ijE]. In momentum space, R obeys the equation [1]d2Rkdτ2+2zdzdτdRkdτ+ k2Rk = 0 , (9)where z ≡ aφ̇0/H. In the slow roll approximation z−1dz/dτ = aH(1 + 2ε − η), and thesolutions obeying the adiabatic condition (and the de Sitter symmetry for H constant) areRk(t) = (−πτ/4(2π)3z2)1/2H(1)µ (−τk), where µ = 3/2 + 3ε − η. The resulting unrenormalizedand renormalized scalar power spectra and spectral indices are (PR(k) = PR(k0)(kk0)ns−1)PR(k) =12M2P ε(H(tk)2π)2, ns − 1 = −6ε+ 2η (10)P renR (k, n) ≈(3ε− η)2M2P ε(H(tk)2π)2(2n− 3/2) , nrens − 1 = −6ε+ 2η +(12ε2 − 8εη + ξ)3ε− η(11)where ξ is another slow roll parameter, ξ ≡M4P (V ′V ′′′/V 2), which can be re-expressed in termsof ε, η and the running of the renormalized tensorial index n′t ≡ dnt/d ln k as n′t = 8ε(ε−η) +2ξ.3. ConclusionsThe expressions derived above for the power spectra and spectral indices show thatrenormalization has a significant impact on the predictions of slow-roll inflation. Unlike in thestandard derivation, we see that the renormalized tensorial power spectrum is no longer directlySpanish Relativity Meeting (ERE 2009) IOP PublishingJournal of Physics: Conference Series 229 (2010) 012058 doi:10.1088/1742-6596/229/1/0120583related to the Hubble scale during inflation. The amplitude of this spectrum is now modulatedby the slow-roll parameter ε, which is zero in exact de Sitter inflation. On the other hand, theappearance of the combination 3ε − η in the numerator of the renormalized scalar spectrumsuggests that P renR (k, n) ∼ 12M2P(H(tk)2π)2(up to the numerical factor 2n− 3/2), which indicatesthat H(tk) ∼ 1015GeV. For the exponential potential [1], we find that H ∼ 1.5× 1015/√4n− 3GeV, roughly up to an order of magnitude larger than the unrenormalized prediction.The most dramatic difference between the standard predictions and the predictions afterrenormalization presented here affects the so-called consistency condition. The tensor to scalarratio in the standard approach gives r = Pt/PR = 16ε = −8nt. This implies that if a backgroundof gravitational waves is observed, then its spectral index must be constrained by the ratio of thetensor to scalar amplitudes. If this constraint is not satisfied, then single field slow-roll inflationwould be ruled out. However, the consistency condition after renormalization becomesr = 4(1− ns − nt) +4n′tn2t − 2n′t(1− ns −√2n′t + (1− ns)2 − n2t), (12)where we have omitted the indices “ren”. This expression is much more involved than thestandard one and its implications are far reaching. It allows for a zero nt while having a non-zero ratio r. Also, the values of nt are not constrained to be negative. In addition, according tothe standard derivation, the running of nt is fully determined by the values of ns and nt, sincethen one finds n′t = −nt(1 − ns + nt). On the contrary, the manipulations that lead to (12)indicate that n′t is now an independent quantity that needs to be measured in order to checkthe new consistency relation (12). This aspect could make more challenging the experimentalverification of the consistency condition of (single-field) slow roll inflation.To conclude, if single field slow-roll inflation is correct, then observations will (hopefully) tell usif nature have chosen the renormalized spectra as the seeds of perturbations or not. Quantumfield theory in curved spacetimes will thus face a crucial experimental test.Acknowledgements. This work has been partially supported by the spanish grant FIS2008-06078-C03-02. I.A. and L.P. have been partly supported by NSF grants PHY-0071044 andPHY-0503366 and by a UWM RGI grant. G.O. thanks MICINN for a JdC contract and the“José Castillejo” program for funding a stay at the University of Wisconsin-Milwaukee.References[1] Lyth D H and Liddle A R 2009 The primordial density perturbation, (Cambridge University Press)Weinberg S 2008 Cosmology (Oxford University Press)Dodelson S 2003 Modern Cosmology (Academic Press)[2] Mukhanov V F and Chibisov G V 1981 JETP Letters 33 532Hawking S W 1982 Phys. Lett. B115 295Guth A and Pi S -Y 1982 Phys. Rev. Lett. 49 1110Starobinsky A A 1982 Phys. Lett. B117 175Bardeen J M, Steinhardt P J and Turner M S 1983 Phys. Rev. D28 679[3] Parker L and Toms D J 2009 Quantum field theory in curved spacetime: quantized fields and gravity (CambridgeUniversity Press)[4] Parker L Amplitude of perturbations from inflation Preprint hep-th/0702216[5] Agulló I, Navarro-Salas J, Olmo G J and Parker L 2009 Phys. Rev. Lett. 103 061301Agulló I, Navarro-Salas J, Olmo G J and Parker L 2009 Gen. Rel. Grav. 41 2301Agulló I, Navarro-Salas J, Olmo G J and Parker L 2008 Phys. Rev. Lett. 101 171301see also Preprint hep-th/0911.0961[6] Parker L 1968 Phys.Rev.Lett. 21 562Parker L 1969 Phys. Rev. 183 1057Parker L 1966 The creation of particles in an expanding universe, Ph.D. thesis, Harvard University[7] Birrel N D and Davies P C W 1982 Quantum fields in curved space (Cambridge University Press)Spanish Relativity Meeting (ERE 2009) IOP PublishingJournal of Physics: Conference Series 229 (2010) 012058 doi:10.1088/1742-6596/229/1/0120584

Inflation, Renormalization, and CMB Anisotropies

Inflation, Renormalization, and CMB Anisotropies

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