IPMU16-0076

Prepared for submission to JCAP

Testing chirality of primordialgravitational waves with Planckand future CMB data: no hopefrom angular power spectraMartina Gerbino,a,b Alessandro Gruppuso,c,d Paolo Natoli,eMaresuke Shiraishi,f Alessandro MelchiorrigaThe Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, StockholmUniversity, AlbaNova, SE-106 91 Stockholm, SwedenbThe Nordic Institute for Theoretical Physics (NORDITA), Roslagstullsbacken 23, SE-10691 Stockholm, SwedencINAF, Istituto di Astrofisica Spaziale e Fisica Cosmica di Bologna,via P. Gobetti 101, I-40129 Bologna, ItalydINFN, Sezione di Bologna, Via Irnerio 46, I-40126 Bologna, ItalyeDipartimento di Fisica e Scienze della Terra and INFN, Università degli Studi di Ferrara,Via Saragat 1, I-44100 Ferrara, ItalyfKavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI),UTIAS, The University of Tokyo, Chiba, 277-8583, JapangPhysics Department and INFN, Università di Roma “La Sapienza”, P.le Aldo Moro 2, 00185,Rome, ItalyE-mail: martina.gerbino@fysik.su.se, gruppuso@iasfbo.inaf.it,paolo.natoli@gmail.com, maresuke.shiraishi@ipmu.jp,alessandro.melchiorri@roma1.infn.it

Abstract. We use the 2015 Planck likelihood in combination with the Bicep2/Keck likelihood(BKP and BK14) to constrain the chirality, χ, of primordial gravitational waves in a scale-invariant scenario. In this framework, the parameter χ enters theory always coupled tothe tensor-to-scalar ratio, r, e.g. in combination of the form χ · r. Thus, the capabilityto detect χ critically depends on the value of r. We find that with present data sets χis de facto unconstrained. We also provide forecasts for χ from future CMB experiments,including COrE+, exploring several fiducial values of r. We find that the current limit on ris tight enough to disfavor a neat detection of χ. For example, in the unlikely case in whichr ∼ 0.1(0.05), the maximal chirality case, i.e. χ = ±1, could be detected with a significance of∼ 2.5(1.5)σ at best. We conclude that the two-point statistics at the basis of CMB likelihoodfunctions is currently unable to constrain chirality and may only provide weak limits on χ inthe most optimistic scenarios. Hence, it is crucial to investigate the use of other observables,e.g. provided by higher order statistics, to constrain these kinds of parity violating theorieswith the CMB.

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Contents

1 Introduction 1

2 Chiral gravity model 2

3 Method and datasets 43.1 Datasets 43.2 Monte Carlo analysis 4

4 Results 54.1 Current limits 54.2 Forecasts for a COrE+ like mission 6

4.2.1 Fiducial models with χ = 0 74.2.2 Fiducial models with χ = 1 7

5 Conclusions 9

A Fisher matrix computations 10

1 Introduction

Parity symmetry is one of the essential properties of the gravity and electromagnetic sectorsin the Universe. It is preserved in the description provided by general relativity and standardMaxwell electromagnetism, while its breaking might give indication of the deviation from sucha standard models. Parity violation may occur within inflationary models through circularlypolarized gravitational waves (GWs), which are referred as chiral gravity models, and also atlate-time Universe through a new Chern-Simons like coupling via the so called cosmologicalbirefringence effect [1–4].

Polarized cosmic microwave background (CMB) observations can be used to test paritysymmetry. Usual four CMB power spectra of the temperature and E/B-mode polarizationanisotropies, i.e., TT, TE, EE, and BB, are always nonzero regardless of parity. On theother hand, additional two combinations, TB and EB, are different from zero when parityis violated, otherwise they are null, and hence unbiased observables of parity violation [1].There are constraints on many kinds of chiral gravity models and cosmological birefringencemodels obtained from many kinds of CMB data [4–21], indicating no significant evidence ofparity violation.

In this paper, we test chiral gravity models with the most recent all-sky polarized dataobserved by the Planck satellite. The shapes of the GW spectrum and the resultant TBand EB spectra are strongly model-dependent (e.g., [1, 22–31, 34–42]), while in this paper,as the simplest example, we constrain an almost scale-invariant template, which has beenanalyzed in [7, 44]. This is motivated by the fact that CMB ` modes of the TB and EBdata used in our analysis are limited to the range 2 ≤ ` ≤ 29 and hence it is hard to extractuseful information on the scale dependence. We then consistently restrict ourselves to thescale-independent case even when combining Planck data with Bicep/Keck measurements orwhen perform forecasts for a COrE+ like experiment, as explained below in more detail. The

– 1 –

analysis with the WMAP TB and EB data have led to an unconstrained result due to the lackof sensitivity [7]. On the other hand, according to the Fisher matrix analyses, there could bea region of the parameter space where visibly large TB and EB correlation are produced, ifa Planck-level sensitivity is realized [7, 44]. This motivates the check with the Planck data,although we would like to address the fact that forecasts provided in [7, 44] assume a specificfiducial value for r and rely on the expected Planck-HFI sensitivity in the 143 GHz channel.

In this paper, we perform a Monte Carlo analysis for deriving updated constraints onthe chirality, conveniently parameterized, both employing current CMB data from the Plancksatellite in combination with Bicep/Keck measurements of the B-modes at degree angularscales and providing forecasts for a future satellite mission like COrE+.

The paper is organized as follows: in Section 2 we introduce the considered chiral gravitymodel giving the main equations and defining the additional parameter χ which basicallyprovides the fraction of circularly polarized gravitational wave; in Section 3 we describe thedata set considered to provide the constraints on χ and the other cosmological parameters;furthermore, in the same section we provide forecast for future CMB experiments, as COrE+.The latter is performed using both a Fisher matrix or Markov Chain Monte Carlo (MCMC)approach; conclusions are drawn in Section 5.

2 Chiral gravity model

The detection of non-vanishing chiral GWs would be a powerful evidence of the Chern-Simonsinteractions in the very primordial Universe. For example, if there exists an axion or a pseu-doscalar field φ in the inflationary era and it couples to a gauge field Aµ via an electromagneticChern-Simons interaction f(φ)FF , the U(1) gauge field is helical and sources chiral GWs dueto the inverse decay process A + A → h [29–33, 35, 36, 39]. The similar production can berealized also by the SU(2) gauge field [37, 40, 41]. Another well-known candidate is a grav-itational Chern-simons term g(φ)RR, motivated by the extention or modification of generalrelativity. Provided that g(φ) shows a time dependence, then it is no longer a topologicalterm and can affect the GW production. This term explicitly breaks parity and hence theinduced GW becomes chiral [1, 22–28].

The shape of resultant GW power spectrum is strongly model-dependent. Specific scaledependence can be created e.g., by choosing time dependence of the running coupling, f(φ)or g(φ). In the next section, we do the data analysis with a nearly scale-invariant powerspectrum template, since our available CMB ` modes, which are limited to 2 ≤ ` ≤ 29, aretoo few to extract useful information on the scale dependence. Such a nearly-scale invariantpower spectrum is realized e.g., in the simplest pseudoscalar inflation models where f(φ) ∝ φwith φ identified with the inflaton field [29, 30, 41, 43].

Let us decompose primordial GWs hij = δgTTij /a2, with a denoting the scale factor, into

two helicity states (λ = ±2):

hij(x) =

∫d3k

(2π)3

∑λ=±2

h(λ)k e

(λ)ij (k)eik·x , (2.1)

where we have used the transverse-traceless polarization tensor e(±2)ij satisfying kie(λ)ij (k) =

e(λ)ii (k) = 0, e(λ)∗ij (k) = e

(−λ)ij (k) = e

(λ)ij (−k) and e(λ)ij (k)e

(λ′)ij (k) = 2δλ,−λ′ . Assuming isotropy

– 2 –

and homogeneity of the Universe, the GW power spectrum can be expressed as

⟨h(λ1)k1

h(λ2)k2

⟩= (2π)3

2π2

k31

P(λ1)h (k1)

2δ(3)(k1 + k2)δλ1,λ2 . (2.2)

In this convention, the GW helicity λ = ±2 is exchanged each other under parity transfor-mation, so parity violation in the power spectrum equates to P(+2)

h 6= P(−2)h . Following the

convention in the previous literature [7, 44], we define the tensor-to-scalar ratio r and thechirality parameter χ as

r ≡⟨hij(k1)hij(k2)

⟩〈ζ(k1)ζ(k2)〉

=P(+2)h (k1) + P(−2)

h (k1)

Pζ(k1), (2.3)

χ ≡ P(+2)h (k)− P(−2)

h (k)

P(+2)h (k) + P(−2)

h (k)=P(+2)h (k)− P(−2)

h (k)

rPζ(k)(2.4)

where Pζ(k) is the power spectrum of the curvature perturbation ζ, defined in

〈ζk1ζk2〉 = (2π)32π2

k31Pζ(k1)δ(3)(k1 + k2) . (2.5)

Note that χ takes nonzero values respecting −1 ≤ χ ≤ 1, if parity is violated.The CMB temperature (X = T ) and polarization (X = E,B) anisotropy is expanded

as X(n) =∑

`m aX`mY`m(n), where n is the unit vector from the observer to CMB photons.

The harmonic coefficients induced by primordial GWs are expressed as [7, 45–47]

aT/E`m = 4πi`

∫d3k

(2π)3∆T/E` (k)

(h(+2)k −2Y ∗`m(k) + h

(−2)k 2Y

∗`m(k)

), (2.6)

aB`m = 4πi`∫

d3k

(2π)3∆B` (k)

(h(+2)k −2Y ∗`m(k)− h(−2)k 2Y

∗`m(k)

), (2.7)

where ∆T/E/B` (k) is the tensor-mode radiation transfer function. Using these, one can for-

mulate the CMB power spectra:⟨aX1`1m1

aX2`2m2

⟩= (−1)m1δ`1,`2δm1,−m2C

X1X2`1

, (2.8)

CX1X2` = 4π

∫ ∞0

dk

k∆X1` (k)∆X2

` (k)rPζ(k)

2×

1 : X1X2 = TT,EE,BB, TE

χ : X1X2 = TB,EB,(2.9)

In the following data analysis, we assume P(±2)h (k) = const (since we are interested in

the scale-invariant case as explained above), however, we take into account a slightly red-tiltedshape of the curvature power spectrum to respect observations.

As seen in eq. (2.9), the TB and EB correlation depend on the combination of r and χ, itis thus essentially difficult to measure χ from TB and EB if r is very small. Nevertheless, theFisher matrix forecast assuming a Planck-level sensitivity1 tells that, even if r ' 0.1, χ = 1can be judged with & 1σ accuracy [7, 44]. This motivates our analysis with the Planck data.

1We recall that previous forecasts like e.g. [7, 44] rely on Planck-HFI expected sensitivity.

– 3 –

3 Method and datasets

3.1 Datasets

We derive constraints on the model under investigation from current data and perform fore-casts for future experiments. As our current dataset, we make use of the full set of the Planck2015 likelihood in both temperature and polarization (referred as Planck TT,TE,EE+lowTEB)[48]. Although we expect that the chirality has the main impact on the large-scale regionof the spectra for the nearly scale-invariant case under examination, we also employ small-scale data in order to better constrain the remaining cosmological parameters and possiblybreak degeneracies among them. In particular, we want to reduce the degeneracy betweenthe tensor-to-scalar ratio and the amplitude of scalar perturbations. We use the Planck like-lihood alone or complemented with results from the Bicep/Keck collaboration. We test boththe data coming from the joint Planck/Bicep/Keck analysis (BKP [49]) and the most recentresults from the Bicep/Keck collaboration (BK14 [50]). In both cases, we include the likeli-hood accounting for the B auto-correlation modes at degree angular scales, where we expectthe recombination bump. We follow the default setting of considering the first five bandpow-ers for BKP, roughly corresponding to the multipole range 40 < ` < 180, while BK14 datasetroughly covers the multipole range 40 < ` < 300 in 9 bandpowers. The inclusion of BKPand BK14 data is motivated by the fact that it allows to better constrain the tensor-to-scalarratio r, thus reducing any possible degeneracy between the latter parameter and χ.

As far as forecasts are concerned, we simulate a COrE+ like mission, assuming theexperimental setup described in [51], which corresponds to the dual-band detector upgradeof the baseline COrE+ proposal. We suppose a nine-frequency measurement of the CMBsignal in the frequency range [90-220] GHz over a fraction fsky = 0.70 of the sky, in the best-case scenario of perfect foreground removal. We employ the full set of lensed temperatureand polarization power spectra up to ` = 3000, which is a reasonable range an experimentlike COrE+ may achieve. Details about the generation of mock CMB data can be found in[52, 53]. We report in Table 1 our fiducial models. We decide to test two different cases,namely the absence of chirality signal χ = 0 and maximal chirality signal χ = 1, for differentamplitudes of the primordial tensor modes. As we can see from Table 1, we choose to consideralso models with a tensor-to-scalar ratio which is already excluded with high significance bycurrent data (see e.g. [50] for the most recent results), i.e. r ≥ 0.5. The reason for this choiceis to highlight that the detectability of the chiral models analysed in this work is cruciallyrelated to the amplitude of the primordial tensor signal, as CTB` and CEB` depend on thecombination of r and χ, as described in Eq. (2.9). The remaining cosmological parametersare set to the Planck TT,TE,EE+lowTEB ΛCDM best-fit model2.

3.2 Monte Carlo analysis

We perform a Monte Carlo Markov Chain (MCMC) analysis by using the public code cosmomc[54], complemented with the Boltzmann solver camb, modifying the relevant subroutines forthe inclusion of non-vanishing primordial TB and EB spectra. We consider an 8-dimensionalparameter space representative of our model. The parameter vector is composed by the baryondensity Ωbh

2, the cold dark matter density Ωch2, the angular size of the sound horizon at

decoupling θ, the reionization optical depth τ , the amplitude ln[1010As] and tilt nS of thepower spectrum of primordial scalar perturbations at a pivot scale of k = 0.05 Mpc−1, the

2The full grid of Planck results can be found at http://pla.esac.esa.int/pla/.

– 4 –

fiducial model color code

r = 0.00 , χ = 0 yellow

r = 0.05 , χ = 0 ; 1 black

r = 0.10 , χ = 0 ; 1 red

r = 0.50 , χ = 0 ; 1 blue

r = 1.00 , χ = 0 ; 1 green

Table 1. Fiducial models used for forecasts. The remaining cosmological parameters are set to thePlanck TT,TE,EE+lowTEB ΛCDM+r best-fit model, while nT = 0. For each value of the tensor-to-scalar ratio r, we consider the two cases of χ = 0 (no chirality) and χ = 1 (maximal chirality). Notethat we just consider the case χ = 0 when r = 0. The color code refers to the figures showing theposterior probability of χ in the next section.

tensor-to-scalar ratio r at a pivot scale of k = 0.05 Mpc−1 and the chirality parameter χ.As anticipated in the previous sections, we choose to test chirality within the framework of ascale-invariant primordial tensor spectrum, i.e. with a vanishing tensor spectral index nT = 0.We note that, given the current upper limit on the r, our choice is not dissimilar from havingassumed the standard inflation consistency relation. We leave to future works the possibilityto test chirality in scenarios where consistency relation does not hold. However, we do notexpect significant deviations with respect to the findings reported in this work, given thecurrent limits on nT [55].

When we perform forecasts, we employ an exact likelihood approach for the Monte Carloanalysis (see e.g. [56, 57]).

In Appendix A, we report results from a Fisher matrix approach as a consistency checkof our findings from the Monte Carlo analysis.

4 Results

In this section, we report our results in terms of both current limits from already existingdata and forecasted sensitivity provided several fiducial models.

4.1 Current limits

Figure 1 and Table 2 show our results for the ΛCDM+r+χmodel for Planck TT,TE,EE+lowTEBalone and in combination with data from Bicep/Keck, either BKP or the most recent BK14.As we can see, Planck TT,TE,EE+lowTEB alone provides the weakest constraints on themodel, given the low sensitivity of the large scale data alone to the tensor signal. The inclu-sion of BKP and BK14 data helps constrain better the tensor-to-scalar ratio, yielding r < 0.09and r < 0.07 at 95% CL respectively. In all cases, the chirality χ is left unconstrained. Giventhe definition of χ and the dependence of EB and TB spectra on the combination r · χ, onewould have expected a degeneracy between the two parameters to arise. This is not the casehere and we argue that it is partly due to the fact that the tensor signal is compatible withzero at high significance and that we do not have enough power to constrain χ.

When we include BK data, we are dramatically limiting the parameter space by betterconstraining r. As a result, the inclusion of chirality only produces a slight broadening of the

– 5 –

0.04 0.08 0.12 0.16

τ

0.08

0.16

0.24

0.32

r

2.96

3.04

3.12

3.20

ln(1

010A

s)

−0.5 0.0 0.5

χ

0.04

0.08

0.12

0.16

τ

0.08 0.16 0.24 0.32

r

2.96 3.04 3.12 3.20

ln(1010As)

Planck TT,TE,EE+lowTEB

Planck TT,TE,EE+lowTEB+BKP

Planck TT,TE,EE+lowTEB+BK14

Figure 1. Triangle plot showing the one-dimensional posterior distribution of the main parameterimpacting the large scale CMB signal (top panel in each column) and their two-dimensional probabilitycontours at 68% and 95% CL, for the indicated datasets.

posterior of r with respect to the bounds reported in [49, 50]. The chirality parameter χ isstill unconstrained, due to the fact that the tensor-to-scalar ratio is highly compatible withzero.

Finally, the inclusion of small scale data is crucial for constraining the amplitude ofscalar perturbation, dramatically reducing the degeneracy between ln[1010As]and r.

4.2 Forecasts for a COrE+ like mission

We report here our forecasts for a future COrE+ like mission. We first consider a set of fiducialmodels with χ = 0 and then turn to analyze the same models with χ = 1. Apart from thetensor-to-scalar ratio r, which is set accordingly to Tab.(1), the other cosmological parametersare always chosen to match the Planck TT,TE,EE+lowTEB bestfit for the ΛCDM+r model.

– 6 –

ParameterPlanck TT,TE,EE Planck TT,TE,EE Planck TT,TE,EE

+lowTEB +lowTEB+BKP +lowTEB+BK14

ln[1010As] 3.093+0.063−0.064 3.095+0.064

−0.063 3.103+0.062−0.063

τ 0.079+0.032−0.033 0.080+0.033

−0.032 0.084+0.033−0.032

r < 0.11 < 0.09 < 0.07

χ unc. unc. unc.

Table 2. Constraints on the main parameters impacting the large scale CMB signal for the indicateddatasets. The remaining cosmological parameters are fixed to the Planck TT,TE,EE+lowTEB bestfitfor the ΛCDM+r model. Limits are 95% CL.

Parameter χ = 0, r = 0 χ = 0, r = 0.05 χ = 0, r = 0.1 χ = 0, r = 0.5 χ = 0, r = 1.0

r < 2.4 · 10−4 0.0500+0.0023−0.0023 0.1001± 0.0035 0.500± 0.011 1.000± 0.019

χ unc. unc. 0.00+0.73−0.72 0.00+0.35

−0.35 0.00± 0.26

Table 3. Constraints on the tensor-to-scalar ratio and the chirality parameter for the indicatedfiducial models and a COrE+-like experiment. Limits are 95% CL.

4.2.1 Fiducial models with χ = 0

In Table 3, we report limits on r and χ for different fiducial models, while in the left panelof Fig.2, we show the one-dimensional posterior probability of the parameter χ for the samechoice of models. As already mentioned, the sensitivity on χ depends strongly on the am-plitude of primordial tensor perturbations. As we can see from the left panel of Fig.2, wecan start constraining χ = 0 at 95% CL for those models with r > 0.05: assuming a fiducialr = 0.10, which is the current 95% upper limit on r, we recover χ = 0.00+0.73

−0.72 at 95% CL.For higher values of r, we find an increasing constraining power on χ, with χ = 0.00 ± 0.26at 95% CL for r = 1.0. However, we stress that these cases with large r are already excludedwith high statistical significance by current CMB data.

4.2.2 Fiducial models with χ = 1

In Table 4, we report limits on r and χ for different fiducial models, while in the right panelof Fig.2, we show the one-dimensional posterior probability of the parameter χ for the samechoice of models. Note that for this class of fiducial models, we do not consider the case r = 0for obvious reasons. As we can see from the right panel of Fig.2, we can start excluding χ = 0at 95% CL for those models with r > 0.05: assuming a fiducial r = 0.10, we get χ > 0.235at 95% CL. Even in this case, the bounds on χ becomes tighter for higher fiducial values ofr. However, considering current limits on the tensor-to-scalar ratio and assuming a maximalparity-violating scenario (i.e., χ = 1), we expect that a future COrE+ like mission would beable to exclude χ = 0 at no better than 2.5σ roughly. For example, taking r = 0.05 as ourfiducial model, the lower bound at 95% CL on χ is χ > −0.11, with χ = 0 excluded at nearly1.7σ.

– 7 –

−1.0 −0.5 0.0 0.5 1.0

χ

0.0

0.2

0.4

0.6

0.8

1.0P/

Pm

ax

−1.0 −0.5 0.0 0.5 1.0

χ

0.0

0.2

0.4

0.6

0.8

1.0

P/

Pm

ax

Figure 2. One-dimensional posterior probability for the parameter χ for the forecasted sensitivityof a COrE+-like experiment for different fiducial models. The left panel refers to fiducial choice ofχ = 0, while the right panel to χ = 1. The color code is the following: r = 0 yellow (only in leftpanel), r = 0.05 black, r = 0.1 red, r = 0.5 blue, r = 1.0 green (see also Table 1). The vertical dashedline in the left panel corresponds to the fiducial value for χ = 0.

Parameter χ = 1, r = 0.05 χ = 1, r = 0.1 χ = 1, r = 0.5 χ = 1, r = 1.0

r 0.0500± 0.0023 0.1000± 0.0035 0.500± 0.011 1.000± 0.019

χ > −0.11 > 0.24 > 0.67 > 0.78

Table 4. Constraints on the tensor-to-scalar ratio and the chirality parameter for the indicatedfiducial models and a COrE+-like experiment. Limits are 95% CL.

We would like to discuss the possible implications that the recent results from [58] interms of a lower value of the optical depth τ could have on the analysis reported above.The dependence of non-vanishing TB and EB signal on the optical depth is mainly relatedto the reionization bump, i.e. to the ` < 20 multipole region, where we expect the greatestcontribution to chirality. A lower value of τ would reflect in less power at large scales, resultingin slightly broader constraints on χ. Indeed, we have checked that this is the case by assuminga fiducial value of τ = 0.06 (instead of the fiducial value τ = 0.079 adopted previously) andperforming again forecasts. We find that limits on χ broadens roughly by a factor of ∼ 0.1σ.As an example, for the fiducial model with r = 0.1 and χ = 0 (χ = 1), we get a 95% CL onthe chirality parameter of χ = 0.00 ± 0.78 (χ > 0.14), to be compared with the equivalentbounds in Tab.3 (Tab.4).

Before concluding this section, we remind that cosmological birefringence models predicta non-vanishing TB and EB signal as well, thus introducing some level of degeneracy withchiral gravitational waves. However, as thoroughly discussed in [44], the two effects are almost

– 8 –

orthogonal, with chirality being a pure tensor contribution, thus mostly affecting large scalesand being dumped at smaller scales.

5 Conclusions

We have discussed the sensitivity of current (Planck and Bicep/Keck) and future (COrE+)CMB experiments to the chirality of primordial gravitational waves χ, employing the full setof temperature and polarization power spectra. Our main conclusion is that unfortunatelythe two-point correlation function currently used to build CMB likelihood, can only weaklyconstrain chirality. This is due to fact that χ can be constrained only if the amplitudeof the primordial tensor signal r is detected to be different from zero and high enough toinduce a detectable signature in the parity violating spectra TB and EB, which is not thecase. Current power spectrum datasets are totally insensitive to chirality models, as shownin Fig.(1). The performed forecast for χ with a COrE+ like experiment are given in Fig.(2),assuming fiducial models with no parity violation (χ = 0, left panel) and maximal parityviolation (χ = 1, right panel), respectively. The most stringent constraint on χ assuming afiducial value r = 0.1 (still marginally in agreement with current bounds on the amplitudeof tensor modes) is χ = 0.00+0.73

−0.72 and χ > 0.24 at 95% CL respectively. In other words, inthe best case scenario of perfect foreground removal and high tensor-to-scalar ratio roughlycompatible with current limits, we could be able to constrain chirality models at ∼ 2.5σ atbest. We stress that when we resctrict to the low-` (where chiral gravity models predictmost of the signal) Planck measurements, we employ only LFI data to derive the constraintson χ3. In the ideal cosmic variance limited scenario, with full sky coverage, a Fisher matrixapproach predicts roughly a 5σ detection of χ for r = 0.1, which, according to the Cramér-Raoinequality, can be considered as a lower limit on achievable sensitivity. Such a neat detection(i.e. 5σ) would be possible only in a very optimistic scenario (perfect foreground removal andfsky = 1). On the other hand, from the same analysis, we find that for values of r ≤ 0.008chirality turns out be undetectable even in the best-case cosmic-variance-limited scenario.

In conclusion, our results suggest that the two-point correlation function is not the righttool to constrain a nearly scale-invariant χ even for high-precision future CMB experiments.However, the results should be sensitive to the scale dependence of χ, and TB and EB will beinformative if the GW power spectrum has a nontrivial peak on large scales [39]. Moreover,tests of higher order statistics 4 including clean information on parity violation, such as odd(even) `1 + `2 + `3 of TTT, TTE, TEE, TBB, EEE and EBB (TTB, TEB, EEB and BBB)[61, 64] in principle, enhance the detectability of the chirality of GWs [39, 65–69]. We leavesuch interesting topics for future investigation.

Acknowledgments

We are grateful to Luca Pagano for precious help with Boltzmann code and forecasts and toGiovanni Cabass, Massimiliano Lattanzi and Sabino Matarrese for useful discussions. This pa-per is based on observations obtained with the satellite Planck (http://www.esa.int/Planck),

3We briefly comment about the future possibility to consider HFI data at low `: even if it would beinteresting to investigate it, we do not expect great improvements on the detectability of χ. The reason canbe easily drawn from the analysis for the COrE+ like experiment and relies on the fact that the currentconstraints on r are already limiting the room for r different from zero.

4See [59–63] for studies on parity-violating GW NG.

– 9 –

σχ

r

COrE+ (fsky = 0.7): TBEB

TB+EBnoiseless (fsky = 1): TB

EBTB+EB

10-1

100

10-2 10-1 100

Figure 3. Expected 1σ errors on χ (σχ) as a function of r, computed from the Fisher matrix(A.1). We here analyze the sensitivities in a COrE+-like experiment and an ideal full-sky noiselessexperiment. Solid (dashed) lines correspond to the results for χ = 0 (1).

an ESA science mission with instruments and contributions directly funded by ESA MemberStates, NASA, and Canada. We acknowledge the support by ASI/INAF Agreement 2014-024-R.1 for the Planck LFI Activity of Phase E2. We acknowledge the use of computingfacilities at NERSC (USA), of the HEALPix package [23], and of the Planck Legacy Archive(PLA). MG was partly supported by the grant “Avvio alla ricerca” for young researchers by“Sapienza” university and is supported by the Vetenskapsrådet (Swedish Research Council).MS was supported in part by a Grant-in-Aid for JSPS Research under Grants No. 27-10917,and in part by the World Premier International Research Center Initiative (WPI Initiative),MEXT, Japan.

A Fisher matrix computations

In this appendix, we estimate the sensitivities to the chirality parameter χ in a simple wayand compare them with the results from a Monte Carlo analysis discussed in Sec. 4.

We here consider the measurements of χ using only TB and EB correlations. Assumingthat off-diagonal components of the covariace matrix are negligibly small compared withdiagonal ones, the Fisher matrix for χ is expressed as [44]

F = fsky

`max∑`=2

∂C`

∂χCov−1`

∂C>`∂χ

, (A.1)

– 10 –

where ∂C`/∂χ ≡ (CTB` |χ=1, CEB` |χ=1) and

Cov` =1

2`+ 1

CTT` CBB` + (CTB` )2 CTE` CBB` + CTB` CEB`

CTE` CBB` + CTB` CEB` CEE` CBB` + (CEB` )2

. (A.2)

with C` denoting the sum of the primordial signal C`, additional signal produced via gravita-tional lensing and instrumental noise spectrum. We assume that the noise spectra of TE, TBand EB are zero, and drop negligibly small contributions of lensed TB and EB modes [70].Expected 1σ error on χ is given by σχ = 1/

√F .

Figure 3 describes our numerical results of σχ in a COrE+ like measurement and an idealnoiseless full-sky measurement, showing that the sensitivity to χ gets worse as r becomes smalldue to the decrease of C`(∝ r). Because of this feature, for r . 0.01, χ is undetectable, evenin an ideal noiseless full-sky measurement. As seen in this figure, σχ depends very weakly onχ for small r, since the contributions of CTB` and CEB` to the covariance (A.2) then becomesubdominant. The results for a COrE+-like survey in Fig. 3 are almost consistent with theresults obtained via a full Monte Carlo analysis in Sec. 4, although we here compute theFisher matrix (A.1) by fixing the cosmological parameters other than χ and r and this leadsto a bit better sensitivity. Also, we remind that Fisher matrix results can be considered as alower bound on the variance, according to the Cramér-Rao inequality.

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