Detecting relic gravitational waves in the CMB: Optimal parameters and their constraints
aa r X i v : . [ a s t r o - ph . C O ] M a r Detecting relic gravitational waves in the CMB: Optimalparameters and their constraints
W. Zhao
1, 2, 3, ∗ and D. Baskaran
1, 2, † School of Physics and Astronomy, Cardiff University, Cardiff, CF24 3AA, United Kingdom Wales Institute of Mathematical and ComputationalSciences, Swansea, SA2 8PP, United Kingdom Department of Physics, Zhejiang University of Technology,Hangzhou, 310014, People’s Republic of China
Abstract
The prospect of detecting relic gravitational waves (RGWs), through their imprint in the cos-mic microwave background radiation, provides an excellent opportunity to study the very earlyUniverse. In simplest viable theoretical models the RGW background is characterized by two pa-rameters, the tensor-to-scalar ratio r and the tensor spectral index n t . In this paper, we analyzethe potential joint constraints on these two parameters, r and n t , using the data from the up-coming cosmic microwave background radiation experiments. Introducing the notion of the bestpivot multipole ℓ ∗ t , we find that at this pivot multipole the parameters r and n t are uncorrelated,and have the smallest variances. We derive the analytical formulae for the best pivot multipolenumber ℓ ∗ t , and the variances of the parameters r and n t . We verify these analytical calculationsusing numerical simulation methods, and find agreement to within 20%. The analytical resultsprovides a simple way to estimate the detection ability for the relic gravitational waves by thefuture observations of the cosmic microwave background radiation. PACS numbers: 98.70.Vc, 98.80.Cq, 04.30.-w ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION Detection of relic gravitational waves (RGWs) can be arguably considered one of the mostimportant challenges for current and future cosmic microwave background radiation (CMB) ex-periments [1, 2, 3, 4, 5]. The RGWs are produced in the early Universe due to the superadiabaticamplification of zero point quantum fluctuations of the gravitational field [6]. For this reason, theRGWs carry invaluable information about the early history of our Universe that is inaccessible toany other medium (see review [7] for detailed discussion).A whole range of scenarios of the early Universe, including the inflationary models, genericallypredict a RGW background with a power-law primordial power spectra [6, 8, 9, 10, 11, 12, 13].In fact, the existence of RGWs is a consequence of quite general assumptions. Essentially, theirexistence relies only on the validity of general relativity and basic principles of quantum fieldtheory [6]. The RGW backgrounds are conventionally characterized by two parameters, the so-called tensor-to-scalar ratio r and the primordial power spectral index of RGWs n t (explained indetail below).The RGWs leave well understood imprints on the anisotropies in temperature and polarizationof CMB [14, 15, 16, 17, 18, 19]. More specifically, RGWs produce a specific pattern of polarizationin the CMB known as the B -mode polarization [15]. Moreover, RGWs produce a negative cross-correlation between the temperature and polarization known as the T E -correlation [18, 20, 21, 22](see also [23, 24]). The theoretical analysis of these imprints along with the data from CMB exper-iments allows to place constraints on the parameters r and n t describing the RGW background.The current CMB experiments are yet to detect a definite signature of RGWs. It is hoped that,in the near future, with the launch of the Planck satellite [1] together with a host of ground-based[3] and balloon-borne [4] CMB experiments as well as the proposed satellite mission CMBPol [5],we shall be able to detect a definite signature of the RGW background. In light of this prospect, itis important to be able to effectively constrain the parameters r and n t . A number of papers havediscussed the current and potential constraint on the tensor-to-scalar ratio r [25]. However, most ofthese works either ignore the constraint on the spectral index n t , or make simplifying assumptionsabout its value. One of the common simplifying assumptions is the so-called “consistency relation” n t = − r/ ritical discussion of inflationary predictions and data analysis based on these predictions see [29].In order to keep our discussion sufficiently general we shall not use this consistency relation in ouranalysis.The constraints on the parameters r and n t , characterizing the RGW background, will give usa direct glimpse into the physical conditions in the early Universe. In particular, they will allow toplace constraint on the Hubble parameter of the early Universe [30], which in the case of inflationarymodels would correspond to the constraints on the energy scale of inflation [26]. More specifically,the amplitude of the RGW power spectrum at a particular wavelength, characterized by r and n t , determines the Hubble parameter at the time when the particular wavelength left the horizon.Thus, the determination of r and n t would give a direct measurement of the time evolution of theearly Universe, and provide an observational tool to distinguish between the various inflationarytype models. In addition, the spectral index n t has a special character if the RGW backgroundare generated in a primordial Hagedorn phase of string cosmology [31] or inflation in the loopquantum gravity [32], so the determination of n t provides an observational way to test or rule outthese models.In this paper we shall analyze the joint constraints on two parameters r and n t that wouldbe feasible with the analysis of the data from the upcoming CMB experiments. In general, therewill be a non-vanishing correlation between parameters r and n t [5, 33]. As will be explained inthe following sections, the definition of r and n t depend on a reference scale characterized by amultipole number ℓ , which may be chosen arbitrarily. We shall show that with an appropriatechoice of this multipole number, which we shall call the best multipole number ℓ ∗ t (following theterminology of [34]), the parameters r and n t become uncorrelated and have the smallest possiblevariances. We shall derive approximate analytical expressions for the variances and the correlationcoefficients, followed by an analytical calculation of the pivot multipole ℓ ∗ t . Using the Markov ChainMonte Carlo (MCMC) simulation methods, we shall verify our analytical results and evaluate theexpected constraints for realistic CMB experiments.The outline of the paper is as follows. In Section II we shall introduce and explain the notationsfor the power spectra of gravitational waves, density perturbations and various CMB anisotropyfields and briefly explain how they are calculated. Furthermore, in this section we shall explicitlystate the simplifying assumptions that we shall be using throughout the paper, and explain thelimits of their applicability. Following this, in Section III, we shall calculate analytically the xpected variances and the correlation associated with the parameters r and n t . We shall showexistence of the best pivot multipole scale ℓ ∗ t for which the variances of the corresponding r and n t are minimal and the correlation between them vanishes. In Section IV we shall confirm ouranalytical results using numerical calculations. Finally, Section V is dedicated to a brief discussionand conclusions. II. POWER SPECTRA OF COSMOLOGICAL PERTURBATIONS AND CMBFIELDS
The main contribution to the observed temperature and polarization anisotropies of the CMBcomes from two types of the cosmological perturbations, density perturbations (also known as thescalar perturbations) and RGWs (also known as the tensor perturbations) [10, 11, 14, 15]. Theseperturbations are generally characterized by their primordial power spectra. These power spectraare usually assumed to be power-law, which is a generic prediction of a wide range of scenarios ofthe early Universe, including the inflationary models. In general there might be deviations froma power-law, which can be parametrized in terms of the running of the spectral index (see forexample [27]), but we shall not consider this possibility in the current paper. Thus, the powerspectra of the perturbation fields have the form P R ( k ) = A s ( k ) (cid:18) kk (cid:19) n s − , (1) P h ( k ) = A t ( k ) (cid:18) kk (cid:19) n t , (2)for density perturbations and the RGWs respectively. In the above expression k is an arbitrarilychosen pivot wavenumber, n s is the primordial power spectral index for density perturbations,and n t is the primordial power spectral index for RGWs. A s ( k ) and A t ( k ) are normalizationcoefficients determining the absolute value of the primordial power spectra at the pivot wavenumber k . The choices of n s = 1 and n t = 0 correspond to the scale invariant power spectra for densityperturbations and gravitational waves respectively. The quantity P R ( k ) is the primordial powerspectrum of the curvature perturbation R in the comoving gauge, i.e. P R ( k ) = k h|R k | i / π (see[35] for a detailed exposition). The quantity P h ( k ) is the primordial power spectrum of RGWsand gives the mean-square value of the gravitational field perturbations, in a logarithmic interval f the wave-number k , at some initial epoch when the wavelenghts of interest are well outside thehorizon.The relative contribution of density perturbations and gravitational waves is described by theso-called tensor-to-scalar ratio r defined as follows r ( k ) ≡ A t ( k ) A s ( k ) . (3)Note that, in defining the tensor-to-scalar ratio r , we have not used any inflationary formulae whichrelate r with the physical conditions during inflation and the slow-roll parameters (see for example[26]). Thus, our definition depends only on the power spectral amplitudes of density perturba-tions and RGWs, and does not assume a particular generating mechanism for these cosmologicalperturbations.Assuming that the amplitude of density perturbations A s ( k ) is known, taking into account thedefinitions (2) and (3), the power spectrum of the RGW field may be completely characterizedby tensor-to-scalar ratio r and the spectral index n t . The RGW amplitude A t ( k ) = r ( k ) A s ( k )provides us with direct information on the Hubble parameter of the very early universe [30]. Morespecifically, this amplitude is directly related to the value of the Hubble parameter H at a timewhen wavelengths corresponding to the wavenumber k crossed the horizon [6, 8, 9, 30, 36] A / t ( k ) = √ M pl Hπ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k /a = H , where M pl = 1 / √ πG is the reduced Planck mass.It is important to point out that, for spectral indices different from the invariant case (i.e., when n s = 1 and n t = 0), the definition of the tensor-to-scalar ratio depends on the pivot wavenum-ber k . If we adopt different pivot wavenumber k , the tensor-to-scalar ratio at this new pivotwavenumber r ( k ) is related to original ratio r ( k ) through the following relation (which followsfrom the definitions (1), (2) and (3)) r ( k ) = r ( k ) (cid:18) k k (cid:19) n t − n s +1 . (4)Let us now turn our attention to CMB. Density perturbations and gravitational waves producetemperature and polarization anisotropies in the CMB characterized by the four angular powerspectra C T Tℓ , C EEℓ , C BBℓ and C T Eℓ as functions of the multipole number ℓ . Here C T Tℓ is thepower spectrum of the temperature anisotropies, C EEℓ and C BBℓ are the power spectra of the so-called E and B modes of polarization (note that, density perturbation do not generate B -mode of olarization [15]), and C T Eℓ is the power spectrum of the temperature-polarization cross correlation.In what follows, we shall use the short hand notations C Tℓ , C Eℓ , C Bℓ and C Cℓ to denote these spectra.In general, the various power spectra C Yℓ (where Y = T, E, B or C ) can be presented in thefollowing form C Yℓ = C Yℓ,s + C Yℓ,t , (5)where C Yℓ,s is the power spectrum due to the density perturbations (scalar perturbations), and C Yℓ,t is the power spectrum due to RGWs (tensor perturbations).In the case of RGWs, the various CMB power spectra can be presented in the following form[16, 17, 18] C Yℓ,t = (4 π ) R dkk P h ( k ) h ∆ ( T ) Y ℓ ( k ) i , for Y = T, E, B,C
Cℓ,t = (4 π ) R dkk P h ( k ) h ∆ ( T ) T ℓ ( k )∆ ( T ) Eℓ ( k ) i . (6)Similar expressions hold in the case CMB anisotropies due to density perturbations with a singleexception. Density perturbations do not produce the B -mode of polarization [15]. Thus, the CMBpower spectra have the form [16] C Yℓ,s = (4 π ) R dkk P R ( k ) h ∆ ( S ) Y ℓ ( k ) i , for Y = T, E,C
Cℓ,s = (4 π ) R dkk P R ( k ) h ∆ ( S ) T ℓ ( k )∆ ( S ) Eℓ ( k ) i . (7)The transfer functions ∆ ( S,T ) Y ℓ ( k ) (see [16, 17, 18] for details) in the above expressions translate thepower in the metric fluctuations (density perturbations or gravitational waves) into correspondingCMB power spectrum at an angular scale characterized by multipole ℓ . In general, these transferfunctions are peaked at values ℓ ≃ (1 . · Mpc) × k , which is a reflection of the fact that metricfluctuations at a particular linear scale k − lead to CMB anisotropies predominantly at angularscales θ ∼ kD (where D is the distance to the surface of last scattering). In this work, for numericalevaluation of the various CMB power spectra due to density perturbations and gravitational waves,we use the publicly available CAMB code [37].Since we are primarily interested in the parameters of the RGW field, in the analytical andnumerical analysis below we shall work with a fixed cosmological background model. More specif-ically, we shall work in the framework of ΛCDM model, and keep the background cosmologicalparameters fixed at the values determined by a typical model [38] h = 0 . , Ω b h = 0 . , Ω m h = 0 . , Ω k = 0 , τ reion = 0 . . (8) urthermore, for density perturbations, we shall use a model with primordial scalar perturbationpower spectrum characterized by an amplitude and spectral index A s = 2 . × − , n s = 1 . . (9)In light of the above, CMB power spectra produced by RGWs depend on the tensor-to-scalarratio r and the spectral index n t . In general, this dependence is complicated and requires nu-merical calculations. For analytical calculations in Section III, we shall use a simple analyticalapproximation for this dependence (see for example [39]) C Yℓ,t ≃ ˆ C Yℓ,t (cid:16) r ˆ r (cid:17) (cid:18) ℓℓ (cid:19) n t − ˆ n t = ˆ C Yℓ,t (cid:16) r ˆ r (cid:17) exp [( n t − ˆ n t ) ln ( ℓ/ℓ )] . (10)Here ˆ C Yℓ,t = C Yℓ,t ( r = ˆ r, n t = ˆ n t ) are the spectra calculated for values of tensor-to-scalar ratio andthe spectral index fixed at fiducial values ˆ r and ˆ n t , and ℓ is the pivot multipole. The approximation(10) can be further simplified, for values of spectral index n t sufficiently close to the fiducial valueˆ n t (such that ( n t − ˆ n t ) ln ( ℓ/ℓ ) ≪ C Yℓ,t ≃ ˆ C Yℓ,t (cid:16) r ˆ r (cid:17) [1 + ( n t − ˆ n t ) ln ( ℓ/ℓ )] . (11)The pivot multipole ℓ is closely related to the pivot wavenumber k . The approximation (10)can be derived from (2) and (6) under the assumption that the wavenumber k and multipole ℓ arelinearly related, i.e. k/k ∼ ℓ/ℓ . This assumption is justified due to the peaked nature of transferfunctions ∆ ( T ) Y,ℓ ( k ) entering (6). Numerical evaluations show that the pivot multipole is related topivot wavenumber by ℓ ≈ k × Mpc . (12)For illustration, in FIG. 1 we plot the power spectra C Yℓ,t for different value of the spectral index n t . The pivot wavenumber is taken to be k = 0 . − . As expected, in all the panels thespectra with different values of n t converge at ℓ ≃ C Yℓ are theoretical constructions determined by ensemble averages overall possible realizations of the underlying random process. However, in real CMB observations, weonly have access to a single sky, and hence to a single realization. In order to obtain information onthe power spectra from a single realization, it is required to construct estimators of power spectra.
50 475 500 525 550 575 600 625 6500.0050.0100.0150.0200.0250.0300.0350.0400.0450.0500.0550.060 : r = 1, n t = -0.1: r = 1, n t = -0.05: r = 1, n t = 0: r = 1, n t = 0.05: r = 1, n t = 0.1 l ( l + ) C l C / ( K ) Multipole: l
400 425 450 475 500 525 550 575 6000.0040.0050.0060.0070.0080.0090.010 : r = 1, n t = -0.1: r = 1, n t = -0.05: r = 1, n t = 0: r = 1, n t = 0.05: r = 1, n t = 0.1 l ( l + ) C l B / ( K ) Multipole: l
400 425 450 475 500 525 550 575 600 625 6500.0060.0080.0100.0120.0140.0160.018 : r = 1, n t = -0.1: r = 1, n t = -0.05: r = 1, n t = 0: r = 1, n t = 0.05: r = 1, n t = 0.1 l ( l + ) C l E / ( K ) Multipole: l
450 475 500 525 550 575 600 625 6500.40.50.60.70.80.91.01.11.2 : r = 1, n t = -0.1: r = 1, n t = -0.05: r = 1, n t = 0: r = 1, n t = 0.05: r = 1, n t = 0.1 l ( l + ) C l T / ( K ) Multipole: l
FIG. 1: The CMB power spectra due to RGWs for various values of the spectral index n t : C Tℓ,t (left upper panel), C Eℓ,t (right upper panel), C Bℓ,t (left lower panel), C Cℓ,t (right upper panel). Thepivot wavenumber is chosen k = 0 . − (in all the panels), and the power spectra are shownfor multipoles around the value of the corresponding pivot multipole ℓ ∼ D Yℓ to denote the estimators while retaining the notation C Yℓ to denote the power spectrum. It isimportant to keep in mind that the estimators D Yℓ are constructed from observational data, whilethe power spectra C Yℓ are theoretically predicted quantities. The probability density functions(pdfs) for the estimators are described in detail in Appendix A. In what follows, we shall requirethe data from all the power spectral estimators, i.e. D Yℓ for Y = T, E, B and C . Let us denote thisset of estimators (which we shall sometimes refer to as the sample) as { D Yℓ } ≡ { D Yℓ | Y = C, T, E, B ; ℓ = 2 , , · · · , ℓ max } . To simulate an experiment, we shall randomly draw a data set (cid:8) D Yℓ (cid:9) from the pdf (A1). In alculating the pdf (A1), along with parameters given in (8) and (9), we set the value of the RGWparameters as r = ˆ r, n t = ˆ n t . (13)We shall refer to ˆ r and ˆ n t as the parameters of the input model.For analytical evaluations in Section III, we shall work with Gaussian approximation to the exactpdfs (A1) for the estimators (cid:8) D Yℓ (cid:9) . The Gaussian approximation is characterized by correspondingmean values and standard deviations [22] h D Yℓ i = C Yℓ , ( Y = T, E, B, C ) ,σ D Yℓ = q ℓ +1) f sky ( C Yℓ + N Yℓ W − ℓ ) , ( Y = T, E, B ) ,σ D Cℓ = r ( C Cℓ ) +( C Tℓ + N Tℓ W − ℓ )( C Eℓ + N Eℓ W − ℓ )(2 ℓ +1) f sky . (14)Note that, the above expressions for mean values and standard deviations follow from the exactpdfs considered in Appendix A. In the above expression, N Yℓ are the noise power spectra, f sky isthe cut sky factor, and W ℓ is the window function.In the case of the Planck mission [1], considering the channel at 143GHz (which has the lowerforeground level and lowest noise power spectra) the noise power spectra, the cut sky factor andthe window function are given by [1] (see [22, 40] for further explanations) N Tℓ = 1 . × − µ K , N Eℓ = N Bℓ = 5 . × − µ K ,f sky = 0 . , W ℓ = exp h − ℓ ( ℓ +1)2 θ i , (15)where θ FWHM = 7 . ′ is the full width at half maximum of the Gaussian beam.In this paper, along with predictions for Planck, we shall consider an idealized situation withno instrumental noise, full sky coverage and an idealized window function W ℓ = 1. For this case,we shall assume that the only source of noise comes from contribution of cosmic lensing to the B -mode of polarization. In this case the noise spectrum for the B -mode is close to white with avalue N Bℓ ≃ × − µ K [41, 42]. A number of works have discussed methods to subtract thelensing B -mode signal (see for example [41, 43]). In [43], the authors claimed that a reduction inlensing power by a factor of 40 is possible using approximate iterative maximum-likelihood method.For this reason, as a further idealized but feasible scenario, we shall also consider the case withreduced cosmic lensing noise N Bℓ ≃ × − µ K . Thus in the two described examples the noisesare N Bℓ (lensing) = 2 × − µ K , N Bℓ (reduced lensing) = 5 × − µ K ; N Tℓ = N Eℓ = 0; f sky = 1; W ℓ = 1 . (16) ote that, in addition to the instrumental noises and lensing noise, various foregrounds, suchas the the synchrotron and dust, significantly contaminate the CMB signal. However, it is hopedthat, using multifrequency observations together with ingenious foreground subtraction techniques,future experiments would be able approach the ideal limit of expression (16) (see for instant [44]).Before proceeding, let us briefly mention the notational conventions used in this paper. The star superscript denotes the quantities evaluated at the best pivot multipole ℓ ∗ t . The hat superscriptindicates the parameters of the fixed (input) cosmological model, that are used to generate thesimulated observational data. The summation (product) symbols with subscript ℓ or Y indicatesummation (product) for ℓ = 2 , ..., ℓ max and Y = C, T, E, B respectively. In numerical evaluationwe set ℓ max = 1000. III. ANALYTICAL APPROXIMATION
In this section, we shall derive analytical expressions for the estimation of the parameters r and n t , the associated uncertainties ∆ r and ∆ n t and the correlation between these parameters. Wewill show the existence and explain the significance of the best pivot multipole ℓ ∗ t . Introducing thetensor-to-scalar ratio r ∗ defined at the best pivot multipole, we shall show that this parameter canbe determined with the smallest possible uncertainty, and is not correlated with the spectral index n t . Based on this analysis, we shall discuss the signal-to-noise ratio and detection possibilities forvarious CMB experiments. A. Approximation for the likelihood function
In order to estimate the parameters r and n t characterizing the RGW background, we shall usean analysis based on the likelihood function [45, 46]. The likelihood function is just the probabilitydensity function of the observational data considered as a function of the unknown parameters(which are r and n t in our case). Up to a constant, independent of its arguments, the likelihoodfunction is given by L = Y ℓ f ( D Cℓ , D Tℓ , D Eℓ , D Bℓ ) , where the function f ( D Cℓ , D Tℓ , D Eℓ , D Bℓ ) is explained in detail in Appendix A. or analysis in this section, we shall use a Gaussian function to approximate the pdf of theindividual estimator D Yℓ , and ignore any possible correlation between different estimators. In thiscase the approximate likelihood function can be written as (see [40] for details) − L = X ℓ X Y D Yℓ − C Yℓ σ D Yℓ ! . (17)The parameters r and n t enter the above expression through the quantities C Yℓ and σ D Yℓ . In ouranalytical considerations we shall make a further simplification. We shall assume that σ D Yℓ entering(17) is weekly dependent on parameters r and n t and assume σ D Yℓ = ˆ σ D Yℓ (for a justification ofthis assumption see [22, 40]). With this assumption, the likelihood function can be rewritten asfollows − L = X ℓ X Y D Yℓ − C Yℓ ˆ σ D Yℓ ! . (18)In the likelihood analysis, we shall assume that the value of the sought parameter n t is sufficientlyclose to the input value ˆ n t . In this case, inserting (11) into (18), using (5), we can rewrite thelikelihood in the form − L = X ℓ X Y n a Yℓ h(cid:16) r ˆ r (cid:17) (1 + ( n t − ˆ n t ) b ℓ ) i − d Yℓ o , (19)where a Yℓ ≡ ˆ C Yℓ,t ˆ σ D Yℓ , b ℓ ≡ ln (cid:18) ℓℓ (cid:19) , d Yℓ ≡ D Yℓ − C Yℓ,s ˆ σ D Yℓ . (20)Note that, in the above expression, the dependence on the data (on the estimators D Yℓ ) is solelycontained in the term d Yℓ . Furthermore, a Yℓ , b ℓ and d Yℓ are independent of the RGW parameters r and n t . The dependence on r and n t takes a particularly simple form and is contained within thesquare brackets on the right side in (19).In order to proceed, it is convenient to introduce new variables ξ ≡ r/ ˆ r ζ ≡ ( n t − ˆ n t )( r/ ˆ r ) , (21)in place of r and n t . In terms of these variables, the likelihood (19) can be simplified as − L = X ℓ X Y (cid:2) a Yℓ ( ξ + ζb ℓ ) − d Yℓ (cid:3) . (22)Note that, the dependence on the sought for parameters r and n t , in the above expression, iscontained in the variables ξ and ζ . After a straight forward manipulations (22) can be rewritten s − L = ξ ( P ℓ P Y a Y ℓ ) + ζ ( P ℓ P Y ( a Yℓ b ℓ ) ) + 2 ξζ ( P ℓ P Y a Y ℓ b ℓ ) − ξ ( P ℓ P Y a Yℓ d Yℓ ) − ζ ( P ℓ P Y a Yℓ d Yℓ b ℓ ) + P ℓ P Y d Y ℓ . This expression can be rewritten as of − L = ( P ℓ P Y a Y ℓ ) (cid:16) ξ − P ℓ P Y a Yℓ d Yℓ P ℓ P Y a Y ℓ (cid:17) + ( P ℓ P Y ( a Yℓ b ℓ ) ) (cid:16) ζ − P ℓ P Y a Yℓ b ℓ d Yℓ P ℓ P Y ( a Yℓ b ℓ ) (cid:17) +2 ξζ ( P ℓ P Y a Y ℓ b ℓ ) + C, (23)where C is a constant, independent of r and n t . This constant is responsible for the overallnormalization of the likelihood function and will not participate in estimation of parameters. Inthe following subsection we shall use the approximation (23) for estimating the parameters r and n t . B. Posterior pdf and the best pivot multipole ℓ ∗ t
1. Posterior pdf
The constraint on the parameters r and n t , are determined by the posterior probability densityfunction P ( r, n t ). This posterior pdf is related to the likelihood function L by [45, 46] P ( r, n t ) = f ( r, n t ) L , (24)where f ( r, n t ) is the prior probability density function of the parameters r and n t . In this paper,we adopt a flat prior, i.e. f ( r, n t ) = 1 . (25)Thus, in this case, the posterior pdf P ( r, n t ) becomes equal to the likelihood. Using the approxi-mation (23) for the likelihood, we obtain − P ( r, n t ) = ( X ℓ X Y a Y ℓ ) (cid:18) ξ − P ℓ P Y a Yℓ d Yℓ P ℓ P Y a Y ℓ (cid:19) + ( X ℓ X Y ( a Yℓ b ℓ ) ) (cid:18) ζ − P ℓ P Y a Yℓ b ℓ d Yℓ P ℓ P Y ( a Yℓ b ℓ ) (cid:19) +2 ξζ ( X ℓ X Y a Y ℓ b ℓ ) + C. (26)The parameters r and n t enter the above expression through the variables ξ and ζ . For this reasonit is convenient to firstly consider the posterior pdf for variables ξ and ζ . It will be seen that theposterior pdf for these variables will have a particularly simple form, namely a bivariate normalfunction. The posterior pdf P ( ξ, ζ ) is related to P ( r, n t ) in the following manner P ( ξ, ζ ) = (cid:12)(cid:12)(cid:12)(cid:12) ∂ ( r, n t ) ∂ ( ξ, ζ ) (cid:12)(cid:12)(cid:12)(cid:12) P ( r, n t ) = ˆ rξ P ( r, n t ) , (27) here (cid:12)(cid:12)(cid:12) ∂ ( r,n t ) ∂ ( ξ,ζ ) (cid:12)(cid:12)(cid:12) denotes the Jacobian of the transformation between the two sets of variables calcu-lable from (21). For simplicity and clarity, let us firstly consider the constraints on the parameters ξ and ζ . Following this, we shall return to the discussion on r and n t using relation (27).Introducing the notations ξ p ≡ (cid:0)P ℓ P Y a Yℓ d Yℓ (cid:1) / (cid:0)P ℓ P Y a Y ℓ (cid:1) , ξ s ≡ / qP ℓ P Y a Y ℓ ,ζ p ≡ (cid:0)P ℓ P Y a Yℓ d Yℓ b ℓ (cid:1) / (cid:0)P ℓ P Y ( a Yℓ b ℓ ) (cid:1) , ζ s ≡ / qP ℓ P Y ( a Yℓ b ℓ ) . (28)From (26) and (27), we obtain that expression for the posterior pdf of ξ and ζ in the form P ( ξ, ζ ) = ˆ re C ξ exp (cid:20) − ( ξ − ξ p ) ξ s ) (cid:21) exp (cid:20) − ( ζ − ζ p ) ζ s ) (cid:21) exp " − ξζ ( X ℓ X Y a Y ℓ b ℓ ) . (29)
2. Best pivot multipole ℓ ∗ t Let us concentrate on the posterior pdf (29). As can be seen, there is a non-vanishing correlationbetween the parameters ξ and ζ in the case when P ℓ P Y a Y ℓ b ℓ = 0. From the definition (20), itfollows that the terms b ℓ depend on the arbitrarily chosen pivot multipole ℓ (corresponding to thepivot wavenumber k through the relation (12)). For this reason, we can select the pivot multipole ℓ = ℓ ∗ t so as to require X ℓ X Y a Y ℓ b ∗ ℓ = 0 , (30)where b ∗ ℓ ≡ b ℓ | ℓ = ℓ ∗ t . With this choice of pivot multipole ℓ ∗ t , and the corresponding pivot wavenum-ber k ∗ t , the variables ξ ( k ∗ t ) and ζ ( k ∗ t ) will have no correlation. We shall refer to this pivot multipole ℓ ∗ t as the Best Pivot Multipole number. From definitions (20) of a Yℓ and b ℓ , along with expression(14), it follows that the precise numerical value of the best pivot multipole number ℓ ∗ t depends onthe input cosmological model characterized by the (8), (9) and (13), as well as the specifics of theCMB experiment characterized by noise power spectra, cut sky factor and window function. Weshall discuss this dependence in more detail below.Setting the value of the pivot multipole ℓ = ℓ ∗ t , so as to satisfy (30), we arrive at a simplifiedform for the posterior pdf P ( ξ ∗ , ζ ∗ ) = (ˆ re C ) 1 ξ ∗ exp " − ( ξ ∗ − ξ ∗ p ) ξ ∗ s ) exp " − ( ζ ∗ − ζ ∗ p ) ζ ∗ s ) . (31)As a reminder let us point out that, in the above expression, as well as in what follows, we haveused notations r ∗ , ξ ∗ , ζ ∗ and b ∗ ℓ to denote the corresponding quantities calculated for the pivot ultipole chosen at the best pivot multipole value ℓ ∗ t . Note that for the spectral index of RGWswe shall retain the notation n t , since it does not depend on the choice of the pivot multipole.
3. Constraints on parameters ξ ∗ and ζ ∗ Equipped with the posterior pdf (31), let us analyze the uncertainties in determining the pa-rameters ξ ∗ and ζ ∗ . For simplicity of analysis we shall assume ξ ∗ p ≫ ξ ∗ s . (32)In Section III F, we shall show that this constraint corresponds to a condition that the signal-to-noise ratio is large, i.e. S/N ≫
1. Taking into account this condition, the posterior function (31)may be further approximated in the following manner P ( ξ ∗ , ζ ∗ ) ≃ ˆ re C ξ ∗ p exp " − ( ξ ∗ − ξ ∗ p ) ξ ∗ s ) exp " − ( ζ ∗ − ζ ∗ p ) ζ ∗ s ) . (33)Note that, in the above expression, the factor in front of the exponent ˆ re C /ξ ∗ p now becomes aconstant independent of ξ ∗ and ζ ∗ . Thus, the posterior pdf P ( ξ ∗ , ζ ∗ ) becomes a bivariate normal(Gaussian) function for variables ξ ∗ and ζ ∗ . The position of the maximum and the standarddeviation associated with the posterior pdf P ( ξ ∗ , ζ ∗ ) are given by ξ ∗ ML = ξ ∗ p , ∆ ξ ∗ = ξ ∗ s , ζ ∗ ML = ζ ∗ p , ∆ ζ ∗ = ζ ∗ s . (34)In the above expression, subscript “ M L ” stands for “maximum-likelihood”, since the maximumof the posterior pdf coincides with that of the likelihood function due to (24) and (25). Followingthe maximum likelihood parameters estimation procedure, we shall accept the values ξ ∗ ML and ζ ∗ ML as the estimators for the corresponding quantities ξ ∗ and ζ ∗ . It is worth mentioning that, forthe posterior pdfs considered in this work, the maximum-likelihood values coincide with the meanvalues of the corresponding posterior pdfs. It is worth mentioning that, the assumption ξ ∗ p ≫ ξ ∗ s (which was used to derive the pdf (33)) is equivalent to the requirement ξ ∗ ML ≫ ∆ ξ ∗ .Proceeding further, we can calculate the correlation coefficient for variables ξ ∗ and ζ ∗ . Let usfirstly define the covariance in the following mannercov( x, y ) ≡ ( x − x ) ( y − y ) , (35)where the overline indicates averaging over the corresponding posterior pdf. The correlation coef-ficient can now be explicitly calculated to give ρ ( ξ ∗ ,ζ ∗ ) ≡ cov( ξ ∗ , ζ ∗ ) p cov( ξ ∗ , ξ ∗ )cov( ζ ∗ , ζ ∗ ) = 0 , (36) s expected, the correlation between the variables ξ ∗ and ζ ∗ vanishes.Taking into account (28), we find that the mean values ξ ∗ (also ξ ∗ ML ) and ζ ∗ (also ζ ∗ ML ) dependon data { D Yℓ } through quantities d Yℓ . However, in our approximation, the standard deviations ∆ ξ ∗ and ∆ ζ ∗ are independent of the data. They depend on the input cosmological model determinedby (8), (9), (13), along with noises, cut sky factor and the window function characterizing the CMBexperiment. C. Constraints on parameters r ∗ and n t Let us now return to the parameters of our direct interest, namely the tensor-to-scalar ratio(determined at the best pivot wavenumber) r ∗ and RGW primordial spectral index n t . Theseparameters are related to ξ ∗ and ζ ∗ through relations r ∗ = ˆ r ∗ ξ ∗ , n t = ˆ n t + ζ ∗ /ξ ∗ , (37)which follow from (21). Taking into account the fact that the quantity ξ ∗ is peaked at ξ ∗ ML whichis sufficiently close to the input value ˆ ξ ∗ , and ∆ ξ ∗ /ξ ∗ ML ≪
1, we can approximate the quantity ξ ∗ in the expression for n t (see the second formula in (37)) with 1. Thus (37) can be written as r ∗ = ˆ r ∗ ξ ∗ , n t ≃ ˆ n t + ζ ∗ . (38)Using (27), the posterior pdf for r ∗ and n t is related to P ( ξ ∗ , ζ ∗ ) in the following way P ( r ∗ , n t ) = ξ ∗ ˆ r ∗ P ( ξ ∗ , ζ ∗ ) ≃ ξ ∗ p ˆ r ∗ P ( ξ ∗ , ζ ∗ ) . (39)Note that, in the current approximation, the pdf for variables r ∗ and n t has a bivariate normalform.Based on the above pdf (39), we can now evaluate the maximum likelihood estimators, standarddeviations, and the correlation coefficient for the variables r ∗ and n t . For the maximum likelihoodvalues we get r ∗ ML = ˆ r ∗ ξ ∗ ML = ˆ r ∗ P ℓ P Y a Yℓ d Yℓ P ℓ P Y a Y ℓ ,n tML ≃ ˆ n t + ζ ∗ ML = ˆ n t + P ℓ P Y a Yℓ d Yℓ b ∗ ℓ P ℓ P Y ( a Yℓ b ∗ ℓ ) . (40)The standard deviation are given by∆ r ∗ = ˆ r ∗ ∆ ξ ∗ = ˆ r ∗ / qP ℓ P Y a Y ℓ , ∆ n t ≃ ∆ ζ ∗ = 1 / qP ℓ P Y ( a Yℓ b ∗ ℓ ) . (41) inally, it can be shown that the correlation between r ∗ and n t vanishes ρ ( r ∗ ,n t ) ≡ cov( r ∗ , n t ) p cov( r ∗ , r ∗ )cov( n t , n t ) = 0 . (42)It is interesting to point out that these results are consistent with the results in [40]. The con-straints, presented in this paper, on the tensor-to-scalar ratio r are exactly the same as those in[40]. Unlike the analysis in the current paper, [40] works with a single free parameter r and doesnot consider n t as an independent free parameter. D. Constraints on parameter r In Subsection III C, using the posterior probability function P ( r ∗ , n t ), we have investigatedthe tensor-to-scalar ratio r ∗ and the spectral index n t defined at the best pivot wavenumber k ∗ t (corresponding to the best pivot multipole ℓ ∗ t ). We can now proceed to the analysis of the tensor-to-scalar ratio r , defined at an arbitrary pivot wavenumber k , and determine the possible constraintson this parameter.From the Eq.(4), we can express the tensor-to-scalar ratio r , in terms of r ∗ , k ∗ t and the spectralindices n t and n s , in the following formln r = ln r ∗ + n t ln( k /k ∗ t ) + (1 − n s ) ln( k /k ∗ t ) . (43)It can be seen that, for a fixed value of the spectral index n s (see (9)), r depends on the parameters r ∗ and n t . Thus, the properties of r can be determined using the posterior pdf P ( r ∗ , n t ), whichwas analyzed in detail in Section III C. In the case k = k ∗ t it is more illustrative to consider thevariable ln r instead of the variable r . For this reason, when dealing with the maximum likelihoodestimators of tensor-to-scalar ratio defined at pivot scale different from the best-pivot scale, weshall use the corresponding logarithmsln r ML = ln r ∗ ML + n tML ln( k /k ∗ t ) + (1 − n s ) ln( k /k ∗ t ) , (44)where r ∗ ML and n tML are the maximum likelihood estimators expressible in terms of the inputparameters ˆ r ∗ , ˆ n t and the data { D Xℓ } (see (40)). The uncertainty of r can be expressed in termsof the uncertainties ∆ r ∗ and ∆ n t determined in (41), leading to the following expression∆ln r ≃ q (∆ r ∗ / ˆ r ∗ ) + (ln( k /k ∗ t )∆ n t ) , = q ( ξ ∗ s ) + (ln( k /k ∗ t ) ζ ∗ s ) . (45) he quantities ξ ∗ s and ζ ∗ s , entering the above expression, can be expressed through CMB powerspectra due to RGWs C Xℓ,t using (20) and (28).From (45) it follows that ∆ r/r & ∆ r ∗ /r ∗ , with the equality holding for k → k ∗ t . Thus, thesmallest uncertainty on tensor-to-scalar ratio r is achieved for the choice of the pivot scale k = k ∗ t .This justifies the title “best” pivot wavenumber for k ∗ t . For a choice of pivot wavenumber k = k ∗ t the uncertainty in determining r becomes larger due to the uncertainty in determining the spectralindex n t .Although, as was shown in Subsection III C, the quantities r ∗ and n t are uncorrelated, this isnot true for the quantities r and n t in general. In order to describe the correlation between r and n t , it is convenient to introduce the correlation coefficient ρ ( n t , ln r ) ≡ cov( n t , ln r ) p cov( n t , n t )cov(ln r, ln r ) , (46)where the notation cov( · , · ) for the covariance was defined in (35). Using this definition, along with(43) and (42), the terms entering the above expression can be evaluated ascov(ln r, ln r ) = (∆ln r ) , cov( n t , n t ) = (∆ n t ) , cov( n t , ln r ) = cov( n t , ln r ∗ ) + (ln( k /k ∗ t ))cov( n t , n t ) = (ln( k /k ∗ t ))(∆ n t ) . Taking into account (41), the correlation coefficient can be presented in the following form ρ ( n t , ln r ) = s ζ ∗ s (ln( k /k ∗ t )) ζ ∗ s (ln( k /k ∗ t )) + ξ ∗ s . (47)As expected, for choice the k = k ∗ t , i.e. when the pivot wavenumber is chosen at the value ofthe best pivot wavenumber, the correlation between r and n t vanishes. On the other hand, for | ln( k /k ∗ t ) | ≫
1, i.e. for values of the pivot wavenumber significantly different from the best pivotwavenumber, the correlation coefficient approaches unity, implying a strong correlation between r and n t . E. Statistical properties of maximum likelihood estimators
The exact numerical values of the maximum likelihood (
M L ) estimators ξ ∗ ML , ζ ∗ ML , r ∗ ML , n tML and ln r ML discussed in the previous subsections depend on the CMB data { D Yℓ } . Since theset { D Yℓ } is a single realization of an underlying random process characterized by the pdf (A1),the precise values of the maximum likelihood estimators will depend on this realization. For his reason, it is instructive to analyze the distribution of these maximum likelihood estimators invarious realizations of the underlying random process specified by the pdf for estimators of the CMBpower spectrum { D Xℓ } . Heuristically speaking, the mean value of this distribution characterizes thetypical value for the M L estimators that we are likely to observe (for a specific input cosmologicalmodel), while the standard deviation characterizes the typical departure from the mean value.Let us firstly, for simplicity, consider the estimators ξ ∗ ML and ζ ∗ ML . The expectation values forthese estimators can be calculated in the following manner h ξ ∗ ML i = h (cid:0)P ℓ P Y a Yℓ d Yℓ (cid:1) / (cid:0)P ℓ P Y a Y ℓ (cid:1) i = (cid:0)P ℓ P Y a Yℓ h d Yℓ i (cid:1) / (cid:0)P ℓ P Y a Y ℓ (cid:1) = 1 , h ζ ∗ ML i = h (cid:0)P ℓ P Y a Yℓ d Yℓ b ∗ ℓ (cid:1) / (cid:0)P ℓ P Y ( a Yℓ b ∗ ℓ ) (cid:1) i = (cid:0)P ℓ P Y a Yℓ h d Yℓ i b ∗ ℓ (cid:1) / (cid:0)P ℓ P Y ( a Yℓ b ∗ ℓ ) (cid:1) = 0 . (48)The angle brackets h ... i , in the above expression and elsewhere in the text, denote the ensembleaverage over the joint pdf (A1). Furthermore, in this pdf, the input values for the tensor-to-scalarratio and spectral index are chosen as r = ˆ r and n t = ˆ n t respectively. In deriving the aboveexpressions we have firstly used (34) and (28). We have also used the identity h d Xℓ i = a Xℓ whichfollows directly from (5), (14) and (20). Finally, in the bottom line, we have used the definition ofthe best pivot multipole (30). Similarly, the standard deviations can be calculated to yield σ ξ ∗ ML = ξ ∗ s , σ ζ ∗ ML = ζ ∗ s . (49)Proceeding in an identical manner, the expectation values and standard deviations for themaximum likelihood estimators r ∗ ML , n tML and ln r ML are given by h r ∗ ML i = ˆ r ∗ h ξ ∗ ML i = ˆ r ∗ , h n tML i = ˆ n t + h ζ ∗ ML i = ˆ n t , h ln r ML i = ln ˆ r ∗ + (ˆ n t − n s + 1) ln( k /k ∗ t ) = ln ˆ r, (50)and σ r ∗ ML = ˆ r ∗ σ ξ ∗ ML = ˆ r ∗ ξ ∗ s ,σ n tML = σ ζ ∗ ML = ζ ∗ s ,σ ln r ML ≃ p ( ξ ∗ s ) + (ln( k /k ∗ t ) ζ ∗ s ) . (51)As expected, from expression (50) it can be seen that the constructed M L estimators are unbiased.Furthermore, the standard deviation of the estimator σ ln r ML strongly depends on the choice of thepivot multipole k , and is minimal for the choice k = k ∗ t . We shall numerically verify these resultsin the following section. . The dependence of results on cosmological parameters and experimental noises Let us now address the question of detection of RGWs in various CMB experiments. In orderto quantify the ability to detect the signature of RGWs in the CMB data, it is convenient to definethe signal-to-noise ratio as follows [22, 40]
S/N ≡ ˆ r ∗ ∆ r ∗ . (52)Using expression (41) we arrive at an elegant expression for the signal-to-noise ratio S/N = vuutX ℓ X Y ˆ C Yℓ,t ˆ σ D Yℓ ! . (53)Thus, the signal-to-noise ratio contains contributions from individual power spectra and individualmultipoles. These contributions have a clear physical meaning. For a particular power spectrumand a particular multipole, they represent the ratio of the expected signal due to RGWs to theoverall uncertainty.As was mentioned in Section III B, for the analytical estimations, we had assumed ξ ∗ p ≫ ξ ∗ s (see(32)). We can now relate this condition to the requirement on the value of the signal-to-noise ratio S/N . Using Eqs.(34), (40) and (41), we find that ξ ∗ p ξ ∗ s = r ∗ ML ∆ r ∗ ≃ ˆ r ∗ ∆ r ∗ = S/N. (54)Hence, the condition ξ ∗ p ≫ ξ ∗ s corresponds to the requirement S/N ≫
1, i.e. to the requirementthat the RGW signal may be well determined at a high signal-to-noise ratio.In the discussion above we have mentioned that the best pivot multipole ℓ ∗ t , the signal-to-noiseratio S/N and the uncertainty in determination of the RGW spectral index ∆ n t depend on theinput cosmological model and the specifics of the CMB experiment. Let us analyze this dependencein more detail.The input cosmological model is determined by specifying the background cosmological model,along with the parameters determining the density perturbations and gravitational waves. Thebackground cosmological parameters and contribution from density perturbations are fairly wellconstrained by the current observations [38]. The variation of these parameters within the marginallowed by these constraints will not significantly alter our results. For this reason, we shall fix thebackground cosmological model using the values of the typical ΛCDM model (8). We shall also fix −6 −4 −2 ˆ r ∗ ℓ ∗ t −6 −4 −2 ˆ r ∗ S / N −6 −4 −2 −3 −2 −1 ˆ r ∗ ∆ n t FIG. 2: The figures show the value of the best pivot multipole ℓ ∗ t (left panel), signal-to-noise ratio S/N (middle panel) and the uncertainty in the RGW spectral index ∆ n t (right panel) as functionsof the tensor-to-scalar ratio ˆ r ∗ . The solid lines correspond to the Planck instrumental noises (see(15)); the dashed lines correspond to noises from cosmic lensing (see (16)); and the dotted linescorrespond to reduced cosmic lensing noise (see (16)).the contribution of density perturbations at a value (9). Furthermore, numerical calculations showthat the dependence of various parameters on the input value of the spectral index ˆ n t is weak,for this reason in evaluations of this section we shall set ˆ n t = 0. Thus, we shall be interested inthe dependence of the parameters on value of the input tensor-to-scalar ratio ˆ r . FIG. 2 shows thevalues of quantities ℓ ∗ t , S/N and ∆ n t as functions of ˆ r ∗ , calculated using the expressions (30), (53)and (41).As was explained in Section II, the specifics of the CMB experiment are determined by thenoise power spectra, the cut sky factor and window function. In this section we shall consider theparameters ℓ ∗ t , S/N and ∆ n t for the three cases specified in Section II (see (15) and (16)). Thedifferent curves (solid, dashed and dotted) on the three panels in FIG. 2 show the correspondingvalues of quantities ℓ ∗ t , S/N and ∆ n t for these three noise scenarios.The left panel of FIG. 2 shows the best pivot multipole ℓ ∗ t as a function of the input tensor-to-scalar ratio ˆ r ∗ which is defined with respect to the best pivot multipole. It can be seen that, forsmall values of ˆ r ∗ , the best pivot multipole ℓ ∗ t is small. This behaviour can be easily understood.For small values of ˆ r ∗ , the constraints on r ∗ and n t mainly come from B -mode power spectrum B -mode the main contribution to the signal comes from large angular scalescorresponding to ℓ .
10, where the signal is mainly due to cosmic reionization [22, 40]. Thus, forsmall ˆ r ∗ , the constraints on parameters r and n t are most stringently determined at large angularscales corresponding to multipoles ℓ .
10. For this reason, for small values of ˆ r ∗ the best pivotmultipole is small, corresponding to the scale at which the parameters r and n t are most stringentlydetermined. On the other hand, for large values ˆ r ∗ , the best pivot multipole ℓ ∗ t also becomes large.This happens due to two reasons. Firstly, with an increase in value of ˆ r ∗ , the relative contributionof the reionization contribution to the to the S/N decreases, while the relative contribution of themultipoles around ℓ ≈
90 (where the B -mode spectrum is expected to have a maximum) increases(see FIG. 3). Thus, the contribution of RGWs at higher multipoles ( ℓ ∼ r ∗ is large, thecontributions from the C, T, E power spectra become important in constraining r and n t [40]. Forthese power spectra, the main contribution to the signal comes from the multipoles 10 . ℓ . ℓ ∗ t .The middle panel in FIG. 2 shows the signal-to-noise ratio S/N as a function of ˆ r ∗ . As expected,the signal-to-noise ratio rises with the increase of ˆ r ∗ . Setting the threshold value of S/N = 2, wecan determine the detection possibilities for the three considered examples: ˆ r ∗ ≥ .
05 for Plancknoises; ˆ r ∗ ≥ . × − for the case with cosmic lensing; ˆ r ∗ ≥ . × − for the case with thereduced cosmic lensing. These estimations are consistent with previous results [22, 41, 42, 43].Finally, the right panel in FIG. 2 presents the achievable constraints on the spectral index ∆ n t as a function of ˆ r ∗ . As expected, the uncertainty in determining the spectral index drops with theincrease of the input value ˆ r ∗ . For the case of Planck mission, the uncertainty in estimation of n t always remains fairly large. Even for large value ˆ r ∗ = 1 the constraint on the spectral index is∆ n t = 0 .
08 (for comparison, the Planck mission will be able to achieve constraint of ∆ n s = 0 . r ∗ = 0 .
1, the constraint on thespectral index is ∆ n t = 0 .
25, which is too large to constrain inflationary models or to verify theconsistency relation. Potentially, in an idealized situation with reduced cosmic lensing, for ˆ r ∗ = 0 . n t = 0 . r=0.7 r=0.3 Multipole: l : l ( l + 1 ) C lB / 2 ( K ) : l ( l + 1 ) D lB / 2 ( K ) FIG. 3: The comparison of ˆ C Bℓ and ˆ σ D Bℓ (which enter the expression for signal-to-noise ratio S/N (53)), for models with ˆ r ∗ = 0 . r ∗ = 0 . ℓ ( ℓ + 1) ˆ C Bℓ / π ( µ K ), and dotted lines show the ‘noise’-term ℓ ( ℓ +1)ˆ σ D Bℓ / π ( µ K ). The quantity ˆ σ D Bℓ was calculated using the Planck noises (15). IV. COMPARISON WITH NUMERICAL SIMULATIONS
In Section III we have analytically studied the likelihood analysis of the RGW parameters r ∗ , n t and r , as well as introduced the best pivot multipole ℓ ∗ t (corresponding to the best pivot wavenumber k ∗ t ) and explained its significance. We have analytically derived expressions for the uncertaintiesof the RGW parameters and the value of the best pivot multipole, in terms of the CMB powerspectra, experimental uncertainties and the estimators of the CMB power spectra. In this sectionwe shall compare the analytical results of the previous section with numerical simulations. Weshall show that, although we have used a number of approximations, the analytical results are ingood agreement with the exact numerical results based on the analysis of simulated data.This section is separated into two parts. In the first subsection, using a single simulated data set { D Xℓ } , we shall use the Markov Chain Monte Carlo (MCMC) techniques to construct the posterior df for the RGW parameters. We shall calculate the uncertainties and correlations associatedwith the parameters, and compare these values with the analytical predictions in Sections III Cand III D. In the second subsection we shall generate 300 samples of data sets { D Xℓ } . For eachindividual sample, using the posterior pdf P ( r ∗ , n t ) we shall calculate the estimates for the RGWparameters r ∗ ML , n ∗ tML and r ML . Analyzing the distribution of these estimates, we shall evaluatethe mean values and the standard deviation, and compare these with the analytical predictionsfrom Section III E. A. Likelihood analysis of a single simulated data set
In this subsection, from a single simulated data set { D Xℓ } , using the likelihood analysis pro-cedure, we shall derive the constraints on the tensor-to-scalar ratio and the RGWs primordialspectral index.In order to simulate the CMB data, we shall randomly draw a data set { D Xℓ } , from an underlyingpdf (A1) (see Appendix A). This pdf depends on the input cosmological model and characteristicsof the CMB experiment. We shall choose as an input cosmological model, a model with thebackground cosmological parameters given in (8) and the contribution of density perturbation (9).The input parameters for the RGW field will be chosen asˆ r = 0 . , ˆ n t = 0 . . (55)To characterize the properties of the CMB experiment, namely the power spectra of noises, thecut sky factor and the window function, we shall adopt the values specified for the Planck satellitemission (15) [1].In order to simulate and analyze the data, we proceed as follows:1) We generate a single data sample { D Yℓ | Y = C, T, E, B ; ℓ = 2 , , · · · , } , drawn from the pdf(A1).2) Using (30), we calculate the best pivot multipole scale ℓ ∗ t = 21 . k ∗ t = 0 . − ). Note that, the value of ℓ ∗ t does not depend on the concreterealization generated in Step 1.3) Using the MCMC method (see [45] for details), we construct the likelihood function L as afunction of two free parameters r ∗ and n t , with the other cosmological parameters fixed at their“best-fit” values given by (8) and (9). Choosing a uniform prior we build the posterior pdf P ( r ∗ , n t ) .2 0.3 0.4 0.5−0.6−0.4−0.200.20.4 r * n t * −1 −0.5 0 0.500.20.40.60.81 n t −3 −2 −1 0−0.6−0.4−0.200.20.4 ln r n t −4 −2 0 200.20.40.60.81 ln r 68.3% 95.4% 95.4% 68.3% FIG. 4: 2-dimension and 1-dimension posterior constraints for parameters: r ∗ and n t (upperpanels), and for parameters: ln r and n t (lower panels). The blue ′ + ′ in the left panels indicate thevalue of the input model parameters.(which is exactly equal to the likelihood function L (see (24) and (25))).4) Using the posterior pdf P ( r ∗ , n t ), we find the maximum likelihood values ( r ∗ ML , n tML ), and plotthe contours corresponding to 68 .
3% and 95 .
4% confidence interval regions in the ( r ∗ , n t ) planesurrounding these values. We also calculate the 1-dimensional posterior pdfs for variables r ∗ and n t . From P ( r ∗ , n t ), we calculate the uncertainties ∆ r ∗ and ∆ n t . Using the importance sampletechnique (see [45, 46]), we evaluate the correlation coefficient ρ ( r ∗ ,n t ) defined in (42).5) We now choose a different value of the pivot wavenumber k = 0 . − , corresponding tothe value for the pivot multipole ℓ = 500. Using (43), we calculate the tensor-to-scalar ratio forthis pivot wavenumber r as a function of the parameters r ∗ and n t . From the posterior probabilityfunction P ( r ∗ , n t ), using the importance sample technique, we can obtain the uncertainty ∆ ln r nd the correlation coefficient ρ ( n t , ln r ) defined in (46).The results of the simulation and analysis is shown in FIG. 4. The panels on the top show theconstraints in the r ∗ − n t plane (top-left), and the 1-dimensional posterior pdfs for r ∗ (top-middle)and n t (top-right). The constraint on the parameters r ∗ and n t , together with the correlationcoefficient are as follows r ∗ = 0 . +0 . − . , (68 . . L . ); n t = − . +0 . − . , (68 . . L . ); ρ ( r ∗ ,n t ) = − . . (simulation results)(56)For comparison, the analytical formulae (40), (41) and (42) yield the following results for thesequantities r ∗ ML ± ∆ r ∗ = 0 . ± . n tML ± ∆ n t = − . ± . ρ ( r ∗ ,n t ) = 0 . (analytical results)(57)As can be seen, the analytical results (57) are in good agreement with results of simulation (56).The bottom panels in FIG. 4 show the constraints in the ln r − n t plane (bottom-left), andthe 1-dimensional posterior pdfs for ln r (bottom-middle). As expected, the confidence intervalcontours in the ln r − n t indicate a strong correlation between ln r and n t . The correspondingconstraints and correlation coefficient are as followsln r = − . +0 . − . , (68 . . L . ); ρ ( n t , ln r ) = 0 . . (simulation results) (58)The analytical expressions (44), (45) and (47), yield the following results for these quantitiesln r ML ± ∆ ln r = − . ± . ρ ( n t , ln r ) = 0 . . (analytical results) (59)Once again, we find that analytical and the exact results are consistent with each other.Furthermore, we have applied the same simulation and analysis procedure to the case with the“cosmic lensing” noises (see (16)), for input values of tensor-to-scalar ratio ˆ r ∗ = 0 .
1, 0 . . r ∗ , ∆ln r and ∆ n t agree with theanalytical expression to within 20%. Thus the analytical formulae for ∆ r ∗ , ∆ln r and ∆ n t seem tobe accurate. B. Maximum likelihood analysis in numerous data simulations
In this subsection, we shall discuss the distribution of the maximum likelihood estimators forthe RGW parameters r ∗ ML , r ML and n tML in multiple realizations. We shall generate a simulatedCMB data set { D Xℓ } a number of times. For each individual realization we shall calculate the n t M L ln r ML n t M L r* ML FIG. 5: The values of the ML estimators from 300 simulations are shown projected onto the n tML − r ∗ ML plane (left panel), and n tML − ln r ML plane (right panel). The red ′ + ′ indicate thevalue of the input model parameters.estimators r ∗ ML , r ML and n tML . We shall then analyze the distribution of these parameters andcompare these results with analytical calculations.In order to generate and analyze the data, we proceed in the following manner:1) A collection of 300 samples of data sets { D Yℓ | Y = T, E, B, C ; ℓ = 2 , , · · · , } is randomlygenerated from an underlying pdf f ( D Cℓ , D Tℓ , D Eℓ , D Bℓ ), given in (A1). The input cosmologicalmodel and the noise characteristics of the CMB experiment are chosen in the same manner as inSection IV A.2) Using (30), we calculate the best pivot multipole ℓ ∗ t = 21 . k ∗ t = 0 . − ). Note that, the value of ℓ ∗ t does not depend on the concreterealization generated in Step 1.3) For each individual sample, we construct the likelihood function L as a function of variables r ∗ and n t , which is equal to the posterior pdf P ( r ∗ , n t ) (see (24) and (25))). For each individualsample, an automated search (which uses the numerical technique of the simulated annealing [47]) etermines the maximum likelihood estimators r ∗ ML and n tML (at which the posterior pdf P ( r ∗ , n t )reaches a maximum). The calculated values r ∗ ML and n tML are plotted in FIG. 5 (left panel).4) We adopt a different pivot wavenumber k = 0 . − , corresponding to the value for thepivot multipole ℓ = 500. From the set of values ( r ∗ ML , n tML ), we calculate the correspondingvalues of tensor-to-scalar ratio for the new pivot wavenumber r ML using (43). The resulting valuesare illustrated in FIG. 5 (right panel).The mean values and standard deviations for the quantities r ∗ ML and n tML (shown in FIG. 5(left panel)), obtained from the analysis of the simulated data, are h r ∗ ML i ± σ r ∗ ML = 0 . ± . h n tML i ± σ n tML = − . ± . . (simulation results) (60)For comparison, we can calculate the corresponding quantities using the analytical expressionsderived in Section III. Using (50) and (51), we obtain h r ∗ ML i ± σ r ∗ ML = 0 . ± . h n tML i ± σ n tML = 0 ± . . (analytical results) (61)Comparing (61) with (60), we find that the analytical expressions are in good agreement withresults of numerical simulation.In a similar fashion, for the mean values and the standard deviation of quantity r ML (shown inFIG. 5 (right panel)), we obtain h ln r ML i ± σ ln r ML = − . ± . . (simulation result) (62)The analytical expressions (50) and (51) yield the following results h ln r ML i ± σ ln r ML = − . ± . . (analytical result) (63)Comparing (62) with (63), we find a reasonable agreement to within 10%. V. CONCLUSION
In this paper, we have analyzed the potential joint constraints on the two parameters character-izing the RGW background, the tensor-to-scalar ratio r and the tensor primordial spectral index n t , achievable by the upcoming CMB observations. We have shown that, in general, there exists acorrelation between the parameters r and n t . However, when considering the tensor-to-scalar ratio r ∗ defined at the best pivot multipole number ℓ ∗ t , the correlation between r ∗ and n t disappears.Furthermore, the uncertainty ∆ r ∗ has the least possible value. We have derived analytical formulae or calculating ℓ ∗ t , ∆ r ∗ , ∆ n t , ∆ r , and the correlation coefficient between r and n t . Using numericalsimulations of future CMB data we have verified the robustness of our analytical estimations andhave shown that our fairly simple analytical expressions agree with exact numerical evaluations towithin 20%. We have also discussed the dependence of our results on the background cosmologicalmodel, the amplitude of the RGWs, and the characteristics of the CMB experiment. We havestudied the dependence of the signal-to-noise ratio S/N along with the value of the best pivotmultipole ℓ ∗ t and the uncertainty ∆ n t on the amplitude of the RGWs. We show that, although thePlanck satellite will potentially be able to measure the tensor-to-scalar ratio to a level r & .
05 (at2 σ C.L.), the uncertainty in determining the spectral index will remain fairly large ∆ n t & .
25 (for r = 0 . n t = − r/
8. In an idealized scenario, where the noises are limited by reduced cosmic lensingnoise, the precision ∆ n t & .
007 (for r = 0 .
1) is achievable, thus, potentially allowing tight con-straints on possible inflationary scenarios. The analytical results presented here provide a simpleand quick method to investigate the ability of the future CMB observations to detect RGWs.
Acknowledgement:
The authors thank L. P. Grishchuk for helpful discussions and usefulsuggestions. W. Zhao is partly supported by Chinese NSF under grant Nos. 10703005 and 10775119.In this paper, we have used the CAMB code for calculating the various CMB power spectra [37].
APPENDIX A: EXACT PROBABILITY DENSITY FUNCTIONS FOR D Yℓ ANDLIKELIHOOD FUNCTION
In [22] (see also [40, 48, 49]) we have derived the pdfs for the best unbiased estimators D Yℓ ofthe various CMB power spectra C Yℓ . These were derived under the assumption that the primordialperturbations (density perturbations and RGWs) are isotropic and homogeneous Gaussian randomfields, and that the noises associated with the CMB measurements can be assumed Gaussian. Inthis appendix we shall briefly list the main results that have been used in the present paper.The joint pdf for the estimators D Tℓ , D Eℓ , D Bℓ and D Cℓ has the following form f ( D Cℓ , D Tℓ , D Eℓ , D Bℓ ) = f ( D Cℓ , D Tℓ , D Eℓ ) f ( D Bℓ ) , (A1) here the pdf f ( D Bℓ ) has the form of the χ distribution, f ( D Bℓ ) = ( n e W ℓ ) v ( n e − / e − v/ n e / Γ( n e / σ Bℓ ) , (A2)and the joint pdf f ( D Cℓ , D Tℓ , D Eℓ ) is the Wishart distribution f ( D Cℓ , D Tℓ , D Eℓ ) = n − ρ ℓ )( σ Tℓ σ Eℓ ) o n e / n e W ℓ ) ( xy − z ) ( n e − / π / Γ( n e / n e − / × exp n − − ρ ℓ ) (cid:16) x ( σ Tℓ ) + y ( σ Eℓ ) − ρ l zσ Tℓ σ Eℓ (cid:17)o . (A3)In the above expressions (A2) and (A3), C Yℓ are the corresponding CMB power spectra, N Yℓ arethe noise power spectra, and W ℓ is the window function. The quantity n e = (2 ℓ + 1) f sky is theeffective degree of freedom for a particular multipole ℓ in the case of partial sky coverage with thecut sky factor f sky . The quantities v , x , y , z are defined as follows v ≡ n e ( D Bℓ W ℓ + N Bℓ ) / ( C Bℓ W ℓ + N Bℓ ) ,x ≡ n e ( D Tℓ W ℓ + N Tℓ ) , y ≡ n e ( D Eℓ W ℓ + N Eℓ ) , z ≡ n e D Cℓ W ℓ . In (A2), σ Bℓ is the standard deviation for the multipole coefficient a Bℓm . The quantities σ Tℓ , σ Eℓ and ρ ℓ in (A3) are correspondingly the standard deviations and the correlation coefficient for themultipole coefficients a Tℓm and a Eℓm . These are expressible in terms of the CMB and noise powerspectra in the following form σ Tℓ = q C Tℓ W ℓ + N Tℓ , σ Eℓ = q C Eℓ W ℓ + N Eℓ , σ Bℓ = q C Bℓ W ℓ + N Bℓ ,ρ ℓ = C Cℓ / q ( C Tℓ + N Tℓ W − ℓ )( C Eℓ + N Eℓ W − ℓ ) . Finally, the likelihood function L introduced in Section III A is, up to a constant of normalization,the product of the joint pdf f ( D Cℓ , D Tℓ , D Eℓ , D Bℓ ), i.e. L ∝ Y ℓ f ( D Cℓ , D Tℓ , D Eℓ , D Bℓ ) . (A4)[1] Planck Collaboration, The Science Programme of Planck [arXiv:astro-ph/0604069].[2] M. R. Nolta et al ., Astrophys. J. Suppl. Ser. , 296 (2009).[3] B. G. Keating et al. , in Polarimetry in Astronomy, edited by Silvano Fineschi, Proceedingsof the SPIE, (2003); C. Pryke et al. , QUaD Collaboration, Astrophys. J. , 1247(2009); A. C. Taylor, Clover Collaboration, New Astron. Rev. , 993 (2006); CAPMAPCollaboration, Astrophys. J. , 771 (2008); D. Samtleben, arXiv:0806.4334.
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