Constraining Planck scale physics with CMB and Reionization Optical Depth
CConstraining Planck scale physics with CMB and Reionization Optical Depth
Brajesh Gupt ∗ Institute for Gravitation and the Cosmos & Physics Department,The Pennsylvania State University, University Park, PA 16802 U.S.A. andDepartment of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, U.S.A.
We present proof of principle for a two way interplay between physics at very early Universe andlate time observations. We find a relation between primordial mechanisms responsible for large scalepower suppression in the primordial power spectrum and the value of reionization optical depth τ .Such mechanisms can affect the estimation of τ . We show that using future measurements of τ , onecan obtain constraints on the pre-inflationary dynamics, providing a new window on the physicsof the very early Universe. Furthermore, the new, re-estimated τ can potentially resolve moderatediscrepancy between high and low- (cid:96) measurements of τ , hence providing empirical support for thepower suppression anomaly and its primordial origin. The ΛCDM model of cosmology explains up to greataccuracy the temperature and polarization spectrum ofthe cosmic microwave background (CMB) measured overthe past three decades. However, the recent precise mea-surements by the WMAP [1] and
Planck [2] missions haverevealed lack of power at large angular scales correspond-ing to (cid:96) <
30 at ∼ σ significance level, also known as thelarge scale power suppression anomaly (PSA) [3]. Whileits origin is still a matter of current investigation, it isenvisaged that PSA could be a relic of pre-inflationarydynamics in the very early Universe [2].In this Letter, we discuss that it is possible to use theplanned observational missions to derive constraints onpotential primordial mechanisms behind PSA. Any newphysics in the early Universe comes with freedom in thechoice initial conditions or physical parameters. We showthat if PSA is indeed primordial in origin, since it affectsEE polarization at low- (cid:96) [4], it can affect the estimation ofThompson scattering optical depth τ of late time reion-ization. This leads to a degeneracy between the value of τ and the aforementioned freedom associated to primor-dial mechanism potentially responsible for PSA. Sincethe power suppression is a low- (cid:96) phenomenon, this de-generacy can be broken via independent measurementsof the optical depth using high- (cid:96) data from future mea-surements. For instance, CMB S4 mission [5] and 21 cmcosmology [6] corresponding to high- (cid:96) physics, are sup-posed to provide independent estimations of τ . Further-more, we find that considering the suppressed power dueto primordial mechanism can alleviate a moderate dis-crepancy that exists in determining mean τ from low- (cid:96) EE polarization in [7, 8] and high- (cid:96) in lensed temperaturedata in [9].One of the most prominent estimations of τ comes fromCMB via the so called “reionization bump” in the E-mode polarization spectrum at (cid:96) <
20, which plays acrucial role in estimating τ [10, 11]. The first constrainton τ from CMB measurements was put by the WMAP 1-year data release to be τ = 0 . ± .
04 using TE-mode po- larization spectrum [10] which was significantly improvedby the 9-year data release to τ = 0 . ± .
014 [12] usingthe EE, TE and TT data at low- (cid:96) . In
Planck τ was estimated using the lensed high- (cid:96) TT spec-trum to be τ = 0 . ± .
016 [9]. In recent
Planck inter-mediate results, τ was re-estimated as τ = 0 . ± . (cid:96) EE data coming from high frequency in-struments [7, 8]. Thus, there is a moderate discrepancyof about ∼ . σ between the mean value of τ from high- (cid:96) TT and low- (cid:96)
EE data by
Planck . We will show that thisdiscrepancy can be alleviated by reestimating τ with thesuppressed scalar power spectrum.For explicit computations we will consider the largescale power suppression due to the quantum gravitationalcorrections of loop quantum cosmology (LQC) proposedin [13] and show that using future measurements, we canobtain constraints on the associated new physics in thepre-inflationary era. For a given inflationary model withan inflaton field in presence of a suitable potential in aFriedmann, Lemaˆıtre, Robertson, Walker (FLRW) space-time, LQC provides a consistent, non-singular extensionof the inflationary scenario all the way up to the Planck-ian curvature scale [20–22]. Let us briefly discuss thesalient features of LQC framework relevant for this pa-per.
Framework:
In the standard inflationary scenariobased on classical GR, the FLRW spacetime is describedby a single spacetime metric g ab ( a, φ ), with a being thescale factor and φ the inflaton field. We will use theStarobinsky potential to drive inflationary dynamics (seee.g. [23] for a detailed analysis). However, the final re-sults of our analysis should hold for other choices of in-flationary potential [13, 23]. There are also other proposals for the power suppression mech-anisms [4, 14–19]. The qualitative results obtained here are ex-pected to hold true for these mechanisms as well. a r X i v : . [ a s t r o - ph . C O ] O c t In LQC, the background spacetime is given by quan-tum Riemannian geometry described by a quantum wave-function Ψ o ( a, φ ) which has support on several g ab ’s. Thequantum wavefunction is obtained by solving the quan-tum Hamiltonian constraint, a difference equation withthe step size given by the minimum non-zero eigenvalue ofthe area operator ∆ o whose value is fixed to ∆ o = 5 . o : k (LQC) a ( t B ) := (cid:115) π ∆ o (cid:96) − ≈ . (cid:96) − , (1)where a ( t B ) is the scale factor at the bounce and k (LQC) is the comoving wavenumber of the characteristic LQCscale today. While the physical wavenumber of the char-acteristic LQC scale ( k (LQC) /a ( t B ))at the bounce is fixedto 3 . (cid:96) Pl in eq. (1), the value of k (LQC) depends onspecific solution to the background equations of motionthrough a ( t B ). There is a one parameter freedom in thechoice of initial conditions that determines a ( t B ) (in theconvention: a ( t ) = 1). Note that this freedom corre-sponds to the freedom in the number of e-folds betweenthe bounce and the onset of slow-roll usually fixed bychoosing the value of inflaton at the bounce φ B . As dis-cussed in [13], physical principles rooted in the simplestquantum geometry can be used to fixed this freedom as a ( t B ) ≈ e − . That, in turn, yields: k (LQC) = 0 . − . (2)Note that eq. (2) represents the value expected from thesimplest quantum geometry in the deep quantum gravity.If this assumptions is dropped, k (LQC) becomes a free pa-rameter and would need to be refined using inputs fromobservations. CMB Polarization spectrum: k (LQC) defines a scaleat which the pre-inflationary effects to the power spec-trum become important. Modes of cosmological pertur-bations with k (cid:29) k (LQC) remain unaffected by LQC cor-rections. However, the infrared modes with k (cid:46) k (LQC) carry an imprint of the quantum gravity era and ar-rive at the slow-roll phase in an excited state [22]. Asshown in [13], with the appropriate choice of initial con-ditions for perturbations, the power spectrum of thesemodes is significantly different from the standard oneand is suppressed at scales corresponding to multipoles (cid:96) <
30. The resulting temperature-temperature spec-trum then fits better with the Planck data than the onecorresponding to the standard, nearly scale invariant pri-mordial power spectrum (PPS). ‘ . . . . . . . ‘ ( ‘ + ) C EE ‘ / π [ µ K ] Standard PPS ( τ = . τ = . ‘ . . . . . . . Standard PPS ( τ = . τ = . FIG. 1. E-mode polarization spectrum for (cid:96) = 2 −
20 for thestandard (dashed blue) and suppressed LQC primordial powerspectra (solid red). In the left panel both curves correspondto τ = 0 . τ has been increased to0 .
072 for the suppressed LQC PPS.
For the analysis in this paper, we will restrict ourselvesto the EE spectrum for (cid:96) = 2 −
20, similarly to the recentanalysis of the reionization history by
Planck [7, 8]. Asdiscussed in [8], this is enough as the high- (cid:96) likelihoods inEE do not contain additional information about reioniza-tion. Fig. 1 shows the EE polarization spectrum for thestandard PPS and the suppressed PPS of LQC, wherethe LQC characteristic scale is fixed as in eq. (2). Theleft panel compares the power spectra for τ = 0 . τ = 0 . τ = 0 . the suppressed power spectrum would predicta larger value of the optical depth . Thus, there is an ap-parent correlation between the values of k (LQC) and τ . Fisher information matrix and error bars:
Toquantify the aforementioned correlation k (LQC) and τ , wecompute the Fisher information matrix [25]. Since the Planck data for low- (cid:96) polarization is not yet available, forthis analysis we will assume that the error bars on C EE (cid:96) atlow- (cid:96) ’s is given by the cosmic variance limit. As discussedin [26], the Fisher matrix then takes the following form: F ij = (cid:96) max (cid:88) (cid:96) =2 (cid:0) ∆ C EE (cid:96) (cid:1) ∂C EE (cid:96) ∂θ i ∂C EE (cid:96) ∂θ j , (3)where ∆ C EE (cid:96) = (cid:114) (cid:96) + 1 C EE (cid:96) , (4)and θ = (cid:0) τ, k (LQC) (cid:1) , while the other cosmological param-eters are fixed at their best fit value given in [9]. Recall .
050 0 .
055 0 .
060 0 .
065 0 .
070 0 . τ k ( L Q C ) [ × − M p c − ] Error Ellipse
FIG. 2. Error ellipse for k (LQC) and τ . The inner and outerellipses show 68 and 95% error contours respectively. Themean value is taken to be k (LQC) = 0 . − and thecorresponding value of optical depth τ = 0 .
062 determinedusing the suppressed LQC PPS. that effects of k (LQC) are limited to very large angularscales and τ is determined from the E-mode reionizationbump which also occurs at (cid:96) <
20. Therefore, we keep (cid:96) max in eq. (3) large enough (at least 50) to include theaffected multipoles in our analysis.Elements of the covariance matrix between k (LQC) and τ are then obtained by inverting the Fisher matrix: C ij = (cid:0) F − (cid:1) ij . Fig. 2 shows the error ellipse corresponding to C ij . The inner and outer contours correspond to the68% and 95% confidence levels respectively. As expectedfrom previous discussions, there is a strong degeneracybetween k (LQC) and τ . Note that C ij only captures theinformation about the error bars and correlation betweenthe two parameters. The mean values of τ and k (LQC) at which the errors ellipse is centered are given by thebest fit values which we have obtained by proceeding asfollows. Implications for future observations and Con-straints on LQC:
As evident from Fig. 2, there is strongdegeneracy between k (LQC) and τ measured from the low- (cid:96) polarization data. In order to break this degeneracy wewould need an independent estimation of either k (LQC) or τ . The LQC scale k (LQC) is a parameter of the underly-ing theory. On the other hand, τ can be measured usinghigh- (cid:96) TT data as well as by upcoming experiments suchas CMB S4 [5] and 21 cm cosmology [6] missions indepen-dent from
Planck measurements.
The measured value of τ from these experiments will break the degeneracy with k (LQC) and put observational constraints on k (LQC) .Recall that the value of k (LQC) determines a ( t B ), i.e.the initial conditions of the background geometry at .
045 0 .
050 0 .
055 0 .
060 0 .
065 0 .
070 0 . τ . . . . . . P o s t e r i o r Standard PPSSuppressed PPS
FIG. 3. 1-dimensional posteriors of τ obtained with the “sim-ulated” data for the standard PPS (dashed blue curve) andthe suppressed LQC PPS (solid red curve) with k (LQC) =0 . − , which is chosen to represent the mean of theerror ellipse in Fig. 2. the bounce. Thus, we can learn about the propertiesof quantum geometry using future observational data.Moreover, as discussed before, since the suppressed powerdetermines a higher value of τ , it can resolve the moder-ate discrepancy in the estimation of τ using the low- (cid:96) po-larization and high- (cid:96) temperature data from Planck mis-sion.
Re-estimating Optical Depth:
Note that, usingthe low- (cid:96)
TT data in [7], τ was determined to be 0 . τ with the low- (cid:96) EE data, thenew value of τ might increase enough and come closerto 0 . (cid:96) TT data [9]—hence resolving an apparent discrepancy between estima-tion of τ from low- (cid:96) EE and high- (cid:96)
TT data. Let us findout if this expectation is borne out in our analysis.To obtain the best fit values of τ for the suppressedPPS we perform the maximum likelihood analysis withlow- (cid:96) C EE (cid:96) by varying τ , keeping k (LQC) = 0 . − (eq. (2)), obtained from consideration of simplest quan-tum geometry in the deep Planck regime [13], while fix-ing other cosmological parameters at their best fit valuesgiven in [9].In order to perform this analysis, we need the EE spec-trum measured from experiments at low- (cid:96) which, how-ever, has not been made available publicly yet. Giventhe lack of real data, we will work with a “simulated” EEdata at low- (cid:96) constructed in the following manner. Wefix τ = 0 .
055 (i.e. the mean value of τ obtained in [7]) andcompute C EE (cid:96) assuming the standard almost scale invari- This assumes that the state for quantum perturbations are fixedusing a quantum generalization of the Weyl curvature hypothesisas discussed in [13, 27] ant power spectrum. We consider this to be the centralvalues of C EE (cid:96) with the errors bars given by the cosmicvariance at low- (cid:96) . In this sense, our “simulated” datarepresents the best ever possible CMB measurements atlow multipoles. Fig. 3 shows the corresponding one dimensional poste-rior distribution of τ for the standard PPS (dashed bluecurve) and the suppressed LQC PPS (solid red curve).The estimated τ with 95% error bars are: τ = 0 . ± .
008 (Standard PPS) τ = 0 . ± .
008 (Suppressed PPS) . (5)Note that the width of the posterior distribution issharper as compared to that obtained in [7], becausehere we have considered the “simulated” data with cos-mic variance error bars which are significantly smaller.It is evident that the peak of the posterior has shiftedto a higher value of τ when the suppressed power spec-trum is used. Moreover, the re-estimated value is closerto the one obtained from high- (cid:96) TT data: τ = 0 . τ purely from low- (cid:96) EE polarization spectrum (estimatedto be 0 .
055 with the standard PPS) and high- (cid:96)
TT spec-trum.
This indicates further empirical support for thepossibility that the PSA could have originated from phys-ical processes in the very early Universe.
In this Letter we have shown that future observationaldata, in particualr giving independent measurement of τ can be used to determine the scale at which PSA isobserved in the TT spectrum, which in turn can con-strain the associated pre-inflationary physics. Here wehave only presented a proof of principle that there is po-tentially a new window on pre-inflationary physics via asymbiotic interplay between observational data and fun-damental. While we performed a case study by consider-ing the pre-inflationary dynamics of loop quantum cos-mology, the overall results of the analysis can be extendedto other primordial mechanisms which introduces a char-acteristic scale for suppression of power at large angularscales. Due to the lack of availability of observationaldata for polarization at low- (cid:96) we used simulated data Cosmic variance is the theoretical lower limit on the error barsat low- (cid:96) and no observational can beat it. However, see [28]for potential way of getting around cosmic variance via carefulmeasurements of quadrapole of the galaxy clusters and CMBsecondaries. Of course, in the real data due to instrumental noise and sys-tematics the real error bars would be larger which we will revisitwhen the low- (cid:96) polarization data from
Planck is released. Weperformed an estimation of the effect of higher error bars on thedegeneracy found here by digitizing figure 33 of [7]. We foundthat the degeneracy is not affected by larger error bars. assuming the error bars on C EE (cid:96) at low- (cid:96) to be givenby the cosmic variance. The actual experimental datafrom Planck expected to be released in upcoming fewmonths, will have higher error bars which is expected toonly increase the width of the error ellipse while keepingthe degeneracy intact. We will revisit this analysis whenmore data from
Planck , CMB S4 and 21 cm cosmology isavailable, which hopefully will provide new observationalinsights on the physics of deep Planck regime in the veryearly Universe.
Acknowledgements:
We would like to thank Nis-hant Agarwal, Abhay Ashtekar, Donghui Jeong and Su-vodip Mukherjee for comments, suggestions and disuc-ssions and Ivan Agullo, B´eatrice Bonga, Anne SylvieDeutsch, Anuradha Gupta, Charles Lawrence, and TarunSouradeep for helpful discussions. This work was sup-ported by NSF grant PHY-1505411 and the Eberly re-search funds of Penn State, and in part by Grant No.NSF-PHY-1603630, funds of the Hearne Institute forTheoretical Physics and CCT-LSU. This work used theExtreme Science and Engineering Discovery Environ-ment (XSEDE), which is supported by National ScienceFoundation grant number ACI-1053575. This work wassupported
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