Testing gravity theories using tensor perturbations
TTesting gravity theories using tensor perturbations
Weikang Lin ∗ and Mustapha Ishak † Department of Physics, The University of Texas at Dallas, Richardson, Texas 75083, USA (Dated: October 12, 2018)Primordial gravitational waves constitute a promising probe of the very early Universe and thelaws of gravity. We study in this work changes to tensor-mode perturbations that can arise invarious proposed modified gravity theories. These include additional friction effects, nonstandarddispersion relations involving a massive graviton, a modified speed, and a small-scale modification.We introduce a physically motivated parametrization of these effects and use current available data toobtain exclusion regions in the parameter spaces. Taking into account the foreground subtraction,we then perform a forecast analysis focusing on the tensor-mode modified-gravity parameters asconstrained by future experiments COrE, Stage-IV and PIXIE. For a fiducial value of the tensor-to-scalar ratio r = 0 .
01, we find that an additional friction of 3 . ∼ .
5% compared to GR will bedetected at 3- σ by these experiments, while a decrease in friction will be more difficult to detect.The speed of gravitational waves needs to be by 5 ∼
15% different from the speed of light fordetection. We find that the minimum detectable graviton mass is about 7 . ∼ . × − eV ,which is of the same order of magnitude as the graviton mass that allows massive gravity theories toproduce late-time cosmic acceleration. Finally, we study the tensor-mode perturbations in modifiedgravity during inflation using our parametrization. We find that, in addition to being related to r , the tensor spectral index would be related to the friction parameter ν by n T = − ν − r/ r , the future experiments considered here will be able to distinguishthis modified-gravity consistency relation from the standard inflation consistency relation, and thuscan be used as a further test of modified gravity. In summary, tensor-mode perturbations andcosmic-microwave-background B-mode polarization provide a complementary avenue to test gravitytheories. PACS numbers: 95.36.+x,98.80.Es,98.62.Sb
I. INTRODUCTION
Current problems in cosmology such as cosmic accel-eration, or older motivations such as finding unified the-ories of physics have led to searches and proposals oftheories of gravity beyond General Relativity (GR). As-sociated with these proposals are efforts to test GR usingcosmological probes. See, for example [1–7] for reviews ontesting modifications to gravity at cosmological scales. Indoing so, instead of building frameworks to test individ-ual modified gravity models, a common and reasonableapproach is to parametrize and test departures from gen-eral relativity predictions. This approach is well justifiedin view of the success of the relativistic Λ cold dark mat-ter (ΛCDM) standard model when compared to obser-vations so that any deviation from GR should be small.It can be viewed as simply testing GR with no referenceto any modified gravity models. Any difference in themodel parameters from their standard values in GR canpoint us to the right direction of modification to GR. Onecould also argue that an efficient parametrization shouldmeet some minimum criteria. First, it should obviouslyreduce to GR in some limit or given point. Second, itshould assemble the behaviors of more than one theoryof modified gravity. Third, the parametrization should ∗ [email protected] † [email protected] be minimum so that the possibly captured deviation isnot merely due to the increased degrees of freedom to fitthe data. And finally the parametrization should allowus to easily assign physical meanings to the parameters.There has been a considerable amount of work to sys-tematically parametrize scalar-mode-perturbation devia-tions from GR in the literature, and we refer readers tosome reviews on the topic [1–8] and publicly availablecodes to perform such tests [9, 10]. On the other hand,the tensor-mode parametrization for modified gravity hasnot been systematically nor extensively studied, althoughseveral non-GR behaviors in the tensor sector have beenindividually investigated [11–15]. It is worth mentioningthat methods of parametrization come also with somelimitations [16, 17], nevertheless they can be informativein some cases.In this paper, we aim to provide a systematic studyof tensor-mode modified-gravity (MG) parameters in-cluding current bounds on the parameters and futureconstraints. In Sec. II, we discuss a general form ofthe modified tensor-mode propagation equation includ-ing different physical effects. In Sec. III, we investigatethe tensor-mode perturbations during inflation for twoof our parametrization schemes. In Sec. IV, we illus-trate the effects of our MG parameters on the cosmic-microwave-background (CMB) B-mode polarization. InSec. V we use the available BKP [18] and Planck 2015[19] data to put bounds on the parameter spaces. In Sec.VI, we analyze and provide a forecast of constraints on a r X i v : . [ a s t r o - ph . C O ] D ec our tensor-mode MG parameters from some future ex-periments. Finally, we summarize in Sec. VII. II. TENSOR MODES IN MODIFIED GRAVITYAND THEIR PARAMETRIZATION
Scalar-, vector- and tensor-mode perturbations withrespect to rotation symmetry can be treated separately[20, 21]. The line element only with tensor-mode pertur-bations reads, ds = − dt + a ( t )( δ ij + D ij ( x , t )) dx i dx j , (1)where D ij is the traceless (i.e., D ii = 0) and transverse(or divergenceless, i.e., ∂ i D ij = 0) part of the perturbedmetric, t is the cosmic time (or the comoving time), and a ( t ) is the scale factor. When working in Fourier space,the propagation equation for a mode with a comovingwave number k and with either helicity ( λ = ±
2) takesthe following form,¨ h k + 3 ˙ aa ˙ h k + k a h k = 16 πG Π Tk , (2)where ˙ h ≡ dhdt , and Π Tk is the tensor part (i.e., tracelessand divergenceless) of the perturbed energy-stress ten-sor in Fourier space. Since the above equation does notdepend on the helicity λ , we have dropped it from thesubscript, but we still keep the subscript k to remind usthat the amplitude is a function of the wavenmuber. Wecan see from Eq. (2) that the dynamics of the tensor-mode amplitude for each mode behaves like a dampingharmonic oscillator with a source. The second term 3 ˙ aa ˙ h k represents the damping effect (or the friction) caused bythe cosmic expansion. The third term k a h k means thatthe frequency of a free wave ω T is the same as its physicalwave number ka , which consequently means that gravita-tional waves propagate at the speed of light. The termon the right-hand side represents the source that comesfrom the tensor part of the stress-energy anisotropy. InGR, the effects from the source on the dynamics of thetensor-mode perturbations are small [20, chapter 6.6],and we assume this is also true in MG. So we ignore thesource term and assume the major modification to thetensor-mode perturbations is from the change to the freepropagation equation, i.e., the left-hand side of Eq. (2).Here a test particle is assumed to follow a geodesic as inGR and there will be no modification to the Boltzmannequations.Relativistic theories of gravity other than GR can ( i )change the damping rate of gravitational waves (i.e., theterm with ˙ h in the propagation equation), ( ii ) modify thedispersion relation (i.e., rather than k /a in the thirdterm, it can be a generic function of k/a ; see for examplethe Hoˇrava-Lifshitz gravity [22] and the Einstein-æthertheory [23]), and ( iii ) add an additional source term onthe right-hand side even in the situation of a perfect fluid(see, for example, in the generalized single scalar field theory [24, 25], and a recent extension to the Horndeskitheories [26–28]). Ignoring the source term as we assumeit gives small effects, we suggest in this paper the follow-ing practical form of the modified propagation equationfor tensor-mode perturbations,¨ h k + 3 ˙ gg ˙ h k + ω T h k = 0 , (3)where g is a model-dependent function of time via somebackground variables and is k independent in the linearregime, and ω T depends on time and the physical wavenumber k/a . Similar modified equations are found in theliterature [11, 12, 14, 15]. In particular, in some previ-ous papers the coefficient in the ˙ h term has been modi-fied to (3 + α M ) H instead of 3 H , which corresponds to g = a αM with a constant α M in Eq. (3). For the dis-persion relation, a modified speed and a graviton masshave also been considered in the literature. But here weintroduce and use a specific form [Eq. (3)] based on amore generic friction term and modified dispersion rela-tion. A different parametrization scheme is consideredin Ref. [29], in which the friction term and the sourceterm are modified in a way that they are both time andwaven-umber dependent. This is different from our con-sideration: 1. We argue that the friction term is onlytime dependent via some background variables. 2. Weneglect changes to the source term since we assume thatthe effect due to those changes is small in MG. 3. Weconsider a more general dispersion relation.Our proposed form of the friction term has more an-alytical advantages, because it can represent the generalfriction term for a wide range of MG theories. For ex-ample, in f ( R ) theories (with R being the Ricci scalar), g = √ f R × a , where f R = df ( R ) dR and equals 1 in GR.In the Horndeski models, we can combine Eq. (5) andEq. (6) in Ref. [15] and manipulate to get g = ω / × a .In tensor-vector-scalar theory, we can modify Eq. (163)in Ref. [30] and get g = bγ . For all MG theories, thefunction g depends only on time but not on the wavenumber.Our consideration of the modified dispersion relationcan in principle cover more generic cases, and is not lim-ited to a constant modified speed c T or a graviton mass µ . The proposed form of the dispersion relation in Ref.[11] reads, ω T = c T k a + µ , (4)which can be manipulated and written as, ω T k /a − c T −
1) + a k µ . (5)Here we can see clearly from Eq. (4) or Eq. (5) that thedifference from a standard dispersion (i.e., ω T k /a − c T (cid:54) = 1 or by a nonzeromass µ (cid:54) = 0. Note that the squared phase speed of grav-itational waves is actually ω T ( k/a ) , which is different fromthe squared speed c T . In this work, we parametrize thedispersion relation from a different approach. Our start-ing point of the dispersion-relation parametrization is totreat the right-hand side of Eq. (5) as a whole and smallpiece. But we will see that, under a few assumptions,our parametrized dispersion relation corresponds to threephysical cases: a modified speed, a graviton mass, and (inaddition) an ultraviolet (high- k/a or small-scale) modifi-cation.There are already some constraints on the dispersionrelation in the literature. First, the consideration of grav-itational Cherenkov radiation puts a strong lower limiton the phase speed of gravitational waves, which is veryclose to the speed of light [31]. The idea is that, if thephase speed is slower than the speed of light, there mustbe some energetic particles moving faster than the phasespeed of gravitational waves which leads to gravitationalCherenkov radiation. Such gravitational Cherenkov radi-ation should in principle slow down these energetic par-ticles. But the observed energetic particles can have aspeed close to the speed of light, and do not appear tohave been slowed down by this process. Or, such particlescan only have traveled for a short distance, which con-tradicts the assumption that they are from the Galacticcenter or other further sources. In other words, if the ideaof gravitational Cherenkov radiation is correct, a sublu-minal phase speed of gravitational waves is not allowed.Second, for the graviton mass, Ref. [13] estimated anupper limit from the CMB observations for a nonvanish-ing tensor-to-scalar ratio. This bound of graviton massis stronger than those set by the gravitational-wave de-tectors. For a more comprehensive list of observationalbounds of the graviton mass, we refer readers to Ref.[32]. In this work, however, we will release the aboveconstraints on the dispersion relation. We do so in or-der to give independent constraints on the tensor sectorsolely from a Monte Carlo Markov Chain (MCMC) anal-ysis on the current CMB observations.Now we turn to our parametrization. We firstparametrize the dispersion relation. Instead of startingwith modifying the speed and adding a graviton mass, weparametrize the dispersion relation from a mathematicalpoint of view. We assume that the dispersion relationdepends only on the physical wave number k/a , but notexplicitly on time. A general modified dispersion rela-tion that only depends on the physical wave number k/a takes the following form: ω T k /a − ε ( k/a ) , (6)where ε ( k/a ) is an arbitrary function of k/a which van-ishes in GR. In the last step, we have denoted everythingon the right-hand side of (5) as ε ( k/a ). This arrangementis motivated by the fact that the deviation from GR issmall in the scalar sector, and so we assume the deviation is also small in the tensor sector. A positive/negative ε corresponds to a superluminal/subluminal phase speed.To parametrize the k/a dependence of the dispersion re-lation, we model it such that the deviation either hap-pens in the large-scale or the small-scale limit but un-changed on the other limit, or the deviation is k/a inde-pendent. And the dispersion relation should be isotropic,so it should be an even function of k/a . Under the aboveassumptions, the following proposals can capture the de-viation up to the lowest order, (and there are examplesof theories corresponding to each of the following cases,) ε ( k/a ) = ε h (cid:16) k/aK (cid:17) , small scales, ε , k/a independent,( ε l ) n (cid:16) µ k/a (cid:17) , large scales. (7)In the above, ε , ε h and ε l are tensor-mode MG param-eters. The subscripts h and l stand for high- and low-physical wave numbers representively. K and µ arenormalization constants. They are inserted to make ε h and ε l dimensionless and within a practical range (i.e., ofunity). For consistency of the units, k in camb is mea-sured in Mpc − , so K and µ is also in Mpc − . Thereare examples of modified gravity theories that have adispersion relation in each of the three forms in Eq. (7).The first case is a ultraviolet deviation. For example inthe Hoˇrava-Lifshitz theory, the dispersion relation devi-ates from the standard one at small scales [22], whichfalls into the first case to the leading order. More ex-plicitly, in Ref. [22], K ε h = g ζ to the leading order atmoderately small scales. The second case correspondsto a constant nonstandard speed of gravitational waves,which can be found in the Einstein-æther theory [11, 23].For the third case, an example of deviation happeningat large scales is when a graviton mass is added to thepropagation equation, ω T = k a + µ , which can be writ-ten as ω T k /a − µ k /a . And we can identify ( ε l ) n asthe ratio µ /µ in the last case. Then our modified dis-persion relation is divided into three separate cases, eachof which has one parameter, namely ε , ε l and ε h . Thethree parameters characterizing the modified dispersionrelation vanish in GR.For the first case, we find K = 100 Mpc − suitable.Roughly speaking, K / √ ε h is the physical wave numberonset of the small-scale deviation. In the last case we use( ε l ) n instead of simply ε l , and we set n = 4. That is be-cause the current constraint on the graviton mass is veryweak (to be explored in Sec. V), and it can span fourorders of magnitude. Using ( ε l ) roughly make differ-ent orders of magnitude of ε l at the same footing whenusing CosmoMC . If further data can provide strongerconstraints, we can set n to be a smaller value, for ex-ample n = 1. A value of µ = 1 Mpc − corresponds toa graviton mass of ∼ × − M p in the Planck units,or ∼ × − eV . In Ref. [13], they used 3000 H (theexpansion rate at recombination), which is roughly 0 . TABLE I. Table of the tensor-mode MG parameters and their corresponding physical meanings or typical examples. In thiswork, we consider the four MG parameters separately. Each MG parameter corresponds to a one-parameter modification. Allparameters vanish in GR. The physical ranges will be discussed in Sec. IV.Parameters Scales of deviation Physical Meaning or example Physical ranges GR values ν All scales Modulating the friction > − ε h Small scales High ka deviation, like in Ref. [22] ≥ ε All scales Gives a modified speed > − ε l Large scales Gives a finite graviton mass ≥ Mpc − and this suggests µ = 1 Mpc − is suitable. Anyother choices of K and µ can be absorbed into the con-stants ε h and ε l .The necessity of the case separation in eq (7) needsto be justified. We concede that separating the disper-sion relation into cases increases the complexity of theanalysis. It might not be useful if we only have datacorresponding a narrow range of k/a , because we wouldnot be able to determine any dependence on k/a fromthe data. And such case separation does not representa more general situation where the deviation can occurat both small and large scales. However, the above sep-aration clearly describes different physics of the possibledeviations, making it possible to quickly link the mod-ified parameters and the reason for their nonvanishingvalues. Also for a practical reason, the constraints onthe tensor sector are very weak, so it is unrealistic toconsider the three deviations simultaneously. One mightwant to replace the three cases with a power index, suchas ( k/a ) n . Then the positive, zero and negative valuesof n can generalize the above three cases. But a continu-ous n lacks physical meaning and can lead to confusion.Therefore, we choose to separate the dispersion relationinto three cases.For the friction term, we simply assume g = a ν for a constant ν , which is equivalent to the work inRef. [11, 12] as explained earlier in this section. A posi-tive/negative ν means the friction is larger/smaller thanthe one in GR, and consequently the gravitational wavesare more/less damped.In summary, the MG parameters ν , ε , ε l and ε h characterize the modified gravitational-wave-propagationequation in four different cases, and they all vanish inGR. When considered separately (as in this work), thefour MG parameters correspond to four one-parametermodifications. The tensor-mode MG parameters and thecorresponding physical meanings are summarized in Ta-ble I. III. TENSOR-MODE PERTURBATIONSDURING INFLATION WITH CONSTANTFRICTION AND SPEED
Our parametrization of the friction term has more an-alytical advantages. One example is the study of tensor- mode perturbations during inflation. For the case withonly a constant friction parameter ν , Eq. (3) in confor-mal time dτ = dt/a reads, h (cid:48)(cid:48) k + 2 ˜ g (cid:48) ˜ g h (cid:48) k + k h k = 0 , (8)where ˜ g = a (1+ ˜ ν ) for a constant ˜ ν and (cid:48) stands forderivative with respect to the conformal time. Note that,the constant ˜ ν in Eq. (3) is different from the one inEq. (8). But they are simply related to each other, and˜ ν = ν . When we let W = ˜ g × h k , Eq. (8) takes thecanonical form, W (cid:48)(cid:48) + ( k − ˜ g (cid:48)(cid:48) ˜ g ) W = 0 . (9)At the early time of inflation when perturbations wereinside the horizon, Eq. (9) and W = ˜ g × h k suggest thatthe solution is normalized such that, h k ( t ) → √ πG (2 π ) / √ k ˜ g exp( − ik (cid:90) dτ ) . (10)The difference from GR is that we have ˜ g in the de-nominator instead of the scale factor a . We assume theUniverse was in the ground state so that Eq. (10) willserve as an asymptotic initial condition of h k . To get h k outside the horizon (by the end of inflation), we need toknow the expansion background. Here we first assumethe background is exactly exponentially expanding withrespect to the cosmic time t (i.e., de Sitter background).We make this assumption at first in order to isolate theMG effects from the slow-roll inflation. Under this as-sumption, we have a = − Hτ , where H is the constantexpansion rate during inflation. And Eq. (8) becomes, h (cid:48)(cid:48) k − ν ) τ h (cid:48) k + k h k = 0 . (11)If we let x = − kτ and h k = x + ˜ ν y , the above equationbecomes, x d ydx + x dydx + [ x − ( 32 + ˜ ν ) ] y = 0 , (12)which is a Bessel differential equation of order ν = + ˜ ν (and this is the reason we use the notation ν ). The gen-eral solution of (12) is a linear combination of Hankelfunctions of the first and second kinds H (1) ν and H (2) ν .Matching the solution deep inside the horizon [Eq. (10)],we eliminate the H (2) ν component since H (1) ν ( − kτ ) al-ready goes as ∼ exp( − ikτ ). And by taking the outsidehorizon limit − kτ → ∞ , we obtain the tensor-mode spec-trum, | h k | = G (2 H ) ν ) (cid:2) Γ( + ˜ ν ) (cid:3) π · k ν . (13)where G is the Newtonian constant. The result in GR in ade Sitter background is recovered for ˜ ν = 0. Since | h k | is proportional to k − − ν , we can identify the tensorspectral index as, n T = − ν = − ν . (14)So if a ∝ e Ht during inflation, n T and ν should be re-lated by (14).For the case of slow-roll inflation, the background isnot exactly de Sitter and H is not a constant. One ofthe slow-roll parameters (cid:15) (not one of our modified grav-ity parameters) measures the first derivative of H withrespect to time, (cid:15) = − ˙ H/H . (15)In this case, the scale factor a no longer goes as a = − Hτ .Instead it is replaced by aH = − − (cid:15) ) τ , which is obtainedby integrating Eq. (15). As a result, Eq. (11) becomes, h (cid:48)(cid:48) k − ν )(1 − (cid:15) ) τ h (cid:48) k + k h k = 0 . (16)For a small (cid:15) , we have − (cid:15) (cid:39) (cid:15) , and Eq. (16) can beapproximately written as, h (cid:48)(cid:48) k − ν + (cid:15) ) τ h (cid:48) k + k h k = 0 . (17)Note that ˜ ν in (11) is now replaced by ˜ ν + (cid:15) in (17).Consequently, we only need to replace ˜ ν by ˜ ν + (cid:15) in thefinal result, i.e., in Eq. (13). In particular, the tensorspectrum index n T is related to both the MG frictionparameter ν = ˜ ν and the slow-roll parameter (cid:15) by, n T = − ν − (cid:15) . (18)In contrast, the ordinary slow-roll inflation in GR gives n T = − (cid:15) [20]. We can see from (18) that the MG frictionparameter ν and the slow-roll parameter (cid:15) have degen-erate roles in the tensor spectral index n T . This meansthe value of n T can not tell us whether the backgroundis exactly de Sitter with an MG friction parameter ν , orslowly changing with a small slow-roll parameter (cid:15) . Theslow-roll inflation consistency relation, n T = − r/ , (19)is expected to change if the friction parameter ν isnonzero. More explicitly, if we assume the result of the scalar sector is unchanged, the tensor-to-scalar ratio r isstill related to the slow-roll parameter (cid:15) by, r = 16 (cid:15) . (20)Note that we have used the fact that the tensor-modeamplitude is not affected by ν to the leading order. Thenthe inflation consistency relation is now modified in MGand becomes, n T = − ν − r/ . (21)We call Eq. (21) the modified-gravity inflation consis-tency relation (MG consistency relation).Verifying the inflation consistency relation is one ofthe important tasks for future CMB experiments. How-ever the near-future experiments have limited capabilityof doing so [33–35]. The presence of ν in the MG consis-tency relation (21) makes the situation even worse. Forexample, if future experiments falsify the standard con-sistency relation n T = − r/
8, it does not necessarily meanthe slow-roll inflation is wrong: it can be that general rel-ativity needs to be modified so that the friction term ischanged.It will be difficult for the near-future CMB experimentsto disentangle the standard and the MG consistency re-lations. However, in some extreme cases, the two con-sistency relations are very different, and this will helpus to tell which consistency relation is possibly correct.We explain as follows. The current upper bound of thetensor-to-scalar ratio r is around 0 . ν is much larger than r , we can ignore the term − r/ n T (cid:39) − ν in modifiedgravity. In contrast, the standard consistency relationstill gives n T = − r/
8. In this case, the MG consis-tency relation expects n T to be much larger than whatis expected in GR. In the future, if we see n T (cid:39) − ν with ν (cid:29) r , then we can say the MG consistency re-lation is possibly right (or the slow-roll inflation theoryhas some troubles). In Sec. VI 3, we explore how fu-ture experiments can distinguish the standard and theMG consistency relations. For the forecast in Sec. VI 3,we set for our fiducial model r = 0 .
01 and ν = 0 . − r/ n T = − ν − r/ (cid:39) − ν = − . n T = − r/ − . n T are thenvery different according to the two consistency relations.For this fiducial model, future experiments will then beable to verify the MG consistency relation and rule outthe standard consistency relation. We refer readers toSec. VI 3 for some details.It is possible to test the MG consistency relation, Eq.(21), with future CMB experiments, because ν affectsthe CMB B-mode power spectrum. We will explore theseeffects in Sec. IV 1. If we are able to obtain the values of ν , r and n T from observations, we can then test whetherEq. (21) is satisfied. However, we note that it is possibleto do so with CMB data only if ν is constant through-out the history of the Universe, or at least from inflationto recombination. Only in this case, it will be the sameMG friction parameter ν in Eq. (21) that also affectsthe CMB B-mode power spectrum. The value of ν in-ferred from CMB data is actually the one after inflation(let us call it ν ,cmb ), while the ν in the MG consis-tency relation Eq. (21) is the one during inflation (letus call it ν ,inf ). If ν ,cmb (cid:54) = ν ,inf , it will be incorrectto test the MG consistency relation n T = − ν ,inf − r/ ν ,cmb . For example, if ν ,inf = 0 but ν ,cmb (cid:54) = 0, the standard consistency rela-tion is correct but we will see a nonzero ν ,cmb from futureCMB experiments. Another example is if ν ,inf (cid:54) = 0 but ν ,cmb = 0, the MG consistency relation is correct butwe will not see any extra friction effects from CMB data.Fortunately, even if ν changes its value after inflation,we can still test the standard inflation consistency rela-tion in GR. Indeed, a nonzero ν ,inf during inflation stillbreaks the relation between n T and r in Eq. (19). Ifthe standard consistency relation is not satisfied by fu-ture CMB experiments, one can draw a conclusion thateither GR needs to be modified or the slow-roll inflationtheory is inconsistent. In this work, we will assume, forsimplicity, that ν is constant.We will close the section with a brief discussion of pos-sible generalizations of the result of Eq. (13). For exam-ple, the result can be generalized to include a constantmodified speed parameter ε in addition to a constantfriction parameter ν . In this case, equation (13) can beeasily generalized to | h k | = G (2 H ) ν ) (cid:2) Γ( + ˜ ν ) (cid:3) π · (cid:0)(cid:112) (1 + ε ) × k (cid:1) ν . (22)In other words, we have replaced k in Eq. (13) with (cid:112) (1 + ε ) × k to obtain Eq. (22). But this does notchange the dependence of | h k | on k , which means thetensor spectral index n T does not depend on a constantmodified speed of gravitational waves. So the consis-tency relation will not be changed due a modified con-stant speed of gravitational waves. Additionally, sincethe wave-propagation equation (8) is a differential equa-tion in time, mathematically the result (22) can be gener-alized to cover cases where ν and ε are functions of thecomoving wave number k . The only difference for suchgeneral cases will be that ν and ε in Eq. (22) become k dependent. But such generalization is not physicallymeaningful because the function g in the friction term(and hence ν ) is k independent, and the dispersion re-lation usually depends on the physical wave number k/a instead of the comoving wave number k . IV. EFFECTS OF TENSOR MODE MODIFIEDGRAVITY PARAMETERS
After investigating the primordial fluctuation duringinflation (only for the cases of constant ν and ε ), thenext step is to see how the MG parameters change theevolution of tensor-mode perturbations at later times,and use observational data to put constraints on our MGparameters. In order to do so, we used a modified ver-sion of camb [36] and CosmoMC [37]. In addition tothe changes to the scalar sector in
ISiTGR , we add mod-ifications of the wave-propagation equation in the tensorsector. For the scalar modes, we refer the modificationsof these to packages
ISiTGR [10, 38]. We add to the topof these modifications the tensor modes.We already mentioned in Sec. II some of the con-straints on the dispersion relation in the literature. Inparticular, a subluminal phase speed of gravitationalwaves is almost forbidden by consideration of gravita-tional Cherenkov radiation. But, in this work we willnot use those as prior bounds but rather aim to ob-tain independent and complementary constraints. Wewill constrain our MG parameters solely from the cur-rent CMB observations. Our results should thus serve asindependent constraints on the dispersion relation. How-ever, some physical ranges need to be imposed on the MGparameters for the stability of the solutions of the per-turbation equations:1. ν > −
1. If not, the friction term in Eq. (3) hasan enhancing instead of suppressing effect.2. ε > −
1. If ε < − ω T = (1 + ε ) × k a is neg-ative and tensor modes will all be unstable. Wealso exclude the situation ε = − ε = − h k = constant is a solution ofEq. (3). Then tensor modes will not contribute toCMB temperature anisotropy or polarization spec-tra, and the tensor-to-scalar ratio r can be arbi-trarily large. Our allowed range of ε means thatwe are also considering subluminal phase speeds ofgravitational waves (i.e., for − < ε < ε l ) n ≥
0. If not, the squared graviton mass µ = ( ε l ) n × µ is negative. Tensor modes becometachyonic, and ω T will be negative for large-scalemodes with k /a < | µ | . The evolution of thesemodes will then grow exponentially and become un-stable.4. ε h ≥
0. If not, ω T will be negative for small-scalemodes with k /a > | ε h | × K .Those physical ranges of MG parameters are also listedin Table I.
100 200 30020.000.010.020.030.040.05 l ( l + ) C l BB / [ K ] l CDM, tensor CDM, all M =-1 (or =-1/3), tensor M =-1 (or =-1/3), all M =1 (or =1/3), tensor M =1 (or =1/3), all r = 0.2 FIG. 1. Reproducing Fig. 1 from Ref. [12]. Within thefigure,“tenso” refers to the B-mode due to tensor modes only,and “all” includes the lensing in the scalar mode. Notice thatwe set r = 0 .
1. Analyzing the effects of modified friction andnonstandard speed
In this subsection, we explore the effects of the MGparameters ν and ε on the CMB B-mode polarizationpower spectrum. We vary each one of them individually,and set the other MG parameters to their GR value. Toverify our modification in camb , in Fig. 1 and Fig. 2 wereproduced two figures from Refs. [12] and [14].Figure 1 shows the effects due to different values of ν ,corresponding to different strengths of friction. In Fig.1 we have used α M to denote the friction term insteadof ν , in order to be consistent with Ref. [12]. For therest of this paper, we use our notation ν . Again, forconstant ν and α M , they are only different by a factorof , and ν = α M . We refer readers to Ref. [12] formore a detailed analysis of the friction term. For a briefdiscussion, we can see that a larger ν (or α M ) meansa larger damping effect, and generally leads to a smallertensor-mode amplitude. But we need to keep in mindthat, a smaller tensor-mode amplitude does not neces-sarily mean a smaller B-mode polarization induced bytensor-mode perturbations, since it is the time derivativeof the amplitude that is important, see Chap. 7 in Ref.[20]. However, it turns out in this case that a larger ν (or α M ) simply leads to a smaller B-mode, as shown inFig. 1.Figure 2 shows the effects due to different values of ε ,corresponding to different speeds of gravitational waves.We do not restrict our parameter ε to be non-negative,which means we do not use the constraint set by theconsideration of gravitational Cherenkov radiation, in order to derive complementary results as we explainedat the beginning of Sec. IV. A detailed analysis of anonstandard speed was given in Ref. [14], in which thespeed was parametrized as c T . Their parametrizationis the same as our 1 + ε parametrization. The majoreffect of a different ε is a horizontal shift of the peaksin the B-mode power spectrum. The reason for suchpeak shifting can be understood as follows. Roughlyspeaking, for a nonzero ε , solutions of Eq. (3) arechanged so that h k → h (cid:48) k = h √ ε k . For the same k , the frequency (in time) ω T = k/a is now replaced by ω T = √ ε × k/a . Consequently, for the same fre-quency ω T , the corresponding comoving wave number isnow k/ √ ε instead of k . If the original peak is at amultiple of (cid:96) , it will be shifted to (cid:96) √ ε . For example,the B-mode recombination peak in GR is around (cid:96) ∼ ε = 1 . .
5, this peak will be shifted to (cid:96) ∼ ∼
140 respectively, as shown in Fig. 2. Another ef-fect from a nonstandard speed involves the amplitude ofthe reionization peak. We can see in Fig. 2 that a smallerspeed leads to a smaller amplitude of this peak, in ad-dition to a horizontal shift. This is because a smallerspeed makes all modes reenter the horizon later, so thatthe largest-scale modes remain constant for a longer timeand do not contribute to the B-mode production (recallagain that the important part is the time derivative ofthe tensor-mode amplitude). Such a contribution is im-portant for the reionization peak, and so a smaller speedleads to a smaller peak. Vice versa, a larger speed makesthe largest-scale modes reenter the horizon, and oscillateearlier and participate in the B-mode production.
2. Effects of large-scale deviation
The large-scale (low- k/a ) deviation represents a con-stant graviton mass. Again, the squared mass µ needsto be non-negative to avoid small-scale tachyonic insta-bility. If µ is negative, roughly speaking the solution willgrow exponentially for the modes with k /a + µ < µ < ∼ − eV , for a nonvanishing tensor-to-scalar ratio.Here we reproduce some of their numerical results andshow them in Fig. 3. A similar upper bound of thegraviton mass will be obtained in Sec. V 2, where, in-stead of estimating, we will use a MCMC analysis andget constraints from the current available data. In Fig. 3,since the effects are not monotonic with ε l , we show themin two panels. In fact, the effects have an oscillating de-pendence on ε l , as we will explain in the next paragraph.We only show the effects on the B-mode polarization, be-cause the temperature and E-mode are dominated by thescalar modes.Depending on the time ordering of recombination, thehorizon reentering (when k/a ∼ H ), and the transitionfrom being relativistic to nonrelativistic (when k/a ∼ µ ),
10 100 10005 5010 -3 -2 -1
10 100 10005 50 10 -1 l ( l + ) C BB l / [ K ] l, c T2 =1.5, c T2 =1, c T2 =0.5 r = 0.2 , c T2 =1.5, c T2 =1, c T2 =0.5 l ( l + ) C TT l / [ K ] l tesnorall FIG. 2. Reproducing Fig. 1 from Ref. [14]. We also set r = 0 . there are different effects on the evolutions of differentperturbation modes. We can qualitatively see that asfollows. With a finite graviton mass, there is a dis-tinct feature from GR for the perturbation evolutions:all perturbation modes will eventually become nonrela-tivistic (i.e., k/a < µ , or the momentum of a gravitonis smaller than its mass). Since the physical wave num-bers decrease with time, perturbation modes always startout being relativistic (i.e, k/a > µ ), and later transi-tion to nonrelativistic (i.e, k/a < µ ). And once theybecome nonrelativistic, they remain so. The time for therelativistic-to-nonrelativistic transition is roughly deter-mined by the condition k/a ∼ µ , which depends on k .Different modes have different transition times. Con-sider only the polarizations produced near recombina-tion: for the modes whose relativistic-to-nonrelativistictransitions happen after recombination (true for small-scale modes), their evolutions before recombination willbe almost the same as in GR. Therefore, their contri-butions to the CMB temperature and polarization willbe nearly unchanged. For the modes whose transitionshappen before recombination, the situation is differentand interesting effects take place, but the analysis willbe more involved. Detailed discussions were provided inRef. [13], in which perturbation modes were divided intothree classes: class I consists of modes that are relativis-tic at recombination; class II consists of modes that arenonrelativistic as they enter the horizon; and class IIIconsists of modes that are relativistic when they reenterthe horizon and become nonrelativistic during recombi-nation. Depending on whether the graviton mass is largeror smaller than the Hubble rate at recombination, thethird class may or may not exist.Now we discuss whether the largest-scale modes (smallwave number compared to µ and H ) are well behaved for a finite µ . The discussion here will also explain theoscillatory dependence of the large-scale effects. Considerthe largest-scale modes with k/a negligible compared to µ and H . In this simple situation, Eq. (3) becomes,¨ h k + 2 t ˙ h k + µ h k = 0 . (23)Solutions to Eq. (23) are the spherical Bessel functionsof order 0. The asymptotically constant initial conditiongives, h k ( t ) ∝ j ( µt ) = sin( µt ) µt , (24)where j ( x ) is the spherical Bessel function of the firstkind of order 0. It means that with a finite µ , the largest-scale-mode evolutions do not depend on k , and they startto oscillate earlier than they would in GR. So the largest-scale modes are well behaved. If the graviton mass islarge enough (more explicitly, larger than the Hubble rateat recombination, i.e., µ > H recom ), they oscillate beforerecombination, and consequently contribute to the CMBtemperature anisotropy and polarization spectra. In con-trast, in GR, the largest-scale modes remain constantand do not contribute. Since the tensor-mode amplitudehas an oscillating dependence on the graviton mass (andhence on ε l ) as shown in Eq. (24), the largest-scale-modecontribution to the B-mode polarization in MG also hasan oscillating dependence on ε l . As shown in the leftpanel of Fig. 3, for small ε l , the low- (cid:96) spectrum of theB-mode polarization decreases with ε l . But in the rightpanel, for larger ε l , it increases with ε l . A more detailedanalysis and similar numerical results were given in Ref.[13], where they showed two more panels, and the B-mode spectrum decreases and increases again with evenlarger graviton masses. -4 -3 -2 -1 -4 -3 -2 -1 Tensor mode only l ( l + ) C BB l / [ K ] l l = 0 (GR) l = 0.01 l = 0.1 l = 1 l ( l + ) C BB l / [ K ] l l l l l Tensor mode only
FIG. 3. The effects of the large-scale deviation on the tensor-induced B-mode polarization. Both panels have the samehorizontal and vertical scales. In the left panel, for a small ε l , a larger ε l leads to a smaller large-scale B-mode polarization.In the right panel,the opposite effects take place. For a large ε l , a larger ε l leads to greater a large-scale B-mode polarization.These results are consistent with those in Ref. [13], where we can see that the amplitude of the tensor-induced B-mode has anoscillating dependence on the graviton mass µ . See the text for a discussion.
200 400 600 800 100010 -6 -5 -4 -3 -2 l ( l + ) C BB l / [ K ] l h =0 h =0.001 h =0.005 r=0.1 FIG. 4. Effects of small-scale (high- k/a ) deviation on the B-mode power spectrum. Here we only show the tensor-inducedB-mode polarization. The spectrum at small scales (low (cid:96) )is not affected as expected. A larger ε h makes the small-scale modes reenter the horizon earlier, resulting in a smallertensor-mode amplitude and consequently a smaller B-modepolarization. This effect is hard to observe since the domi-nating B-mode polarization at small scales is from the lensedE-mode.
3. Effects of small-scale deviation
In this subsection we investigate the effects of thesmall-scale (high- k/a ) parameter ε h on the B-mode po-larization. Figure 4 shows the results of the B-mode po-larization power spectrum for different values of ε h . Herewe set r = 0 .
1. Recall that we restrict ε h to be non-negative because a negative ε h can lead to small-scaleinstability. This small-scale instability can be seen fromEq. (4) and Eq. (7), and when ε h (cid:16) k/aK (cid:17) < − ω T becomes negative. If one wants toallow a negative ε h , it is necessary to introduce a cutoffor include a positive higher-order term. We will not dothese, because, first, the cutoff is totally arbitrary andthe results are not converging for higher and higher cut-offs. A higher cutoff only leads to a higher amplitude.Second, to include a positive higher-order term requiresanother parameter specifying the physical wave numberfrom which the higher-order term becomes significant.Doing so requires more complicated considerations, suchas analyzing the competition of the second-order termand the higher-order term. So for simplicity we keep thenumber of parameters to be a minimum, but we are stillbe able to catch some (if not most) of the physics ofmodified gravity at small scales.As Fig. 4 shows, the tensor-induced B-mode polar-ization power spectrum can be significantly suppressedat small scales (large (cid:96) ) while keeping it unaffected atlarge scales (small (cid:96) ), as expected. The effects of small-scale deviation can be understood as follows. A nonzero0 -1 -0.5 0 0.5 1000.050.10.150.2 r -1 -0.5 0 0.5 1 1.5 2 2.5000.050.10.150.2 r FIG. 5. The 1- σ (green) and 2- σ (blue+green) confidence levels of marginalized constraints in the r vs ν (left panel) and the r vs ε (right panel) parameter spaces. Equivalently, we can say the white parameter region is disfavored at the 95% confidencelevel. ε h changes the time of horizon reentering. For a cer-tain mode with comoving wave number k , a larger ε h leads to earlier horizon reentering, resulting in a smallertensor-mode amplitude. So the tensor-induced B-modeis expected to be smaller.This small-scale deviation is difficult to observe, be-cause it hardly changes the total B-mode power spec-trum at small scales, where the contribution from lensingis dominating. A larger ε h only makes the tensor-modecontribution less significant in the high- (cid:96) spectrum. Con-sequently, the dominating B-mode from lensing at smallscales makes it very difficult to set a constraint on the pa-rameter ε h . So we will not do the corresponding MonteCarlo analysis for ε h and leave it for future data. Fortu-nately, with the near-future CMB experiments we will beable to see such small-scale effects, if ε h is large enoughso that small-scale deviation begins with a large-enough-scale onset. We will estimate the constraint on ε h withthe Fisher matrix formalism in Sec. VI. V. CONSTRAINTS ON TENSOR MODEMODIFIED GRAVITY PARAMETERS
Tensor-mode perturbations, if present, can smooth outthe temperature-anisotropy power spectrum and gener-ate E-mode and B-mode polarization patterns in theCMB. Therefore, both CMB temperature and polariza-tion maps can be used to constrain the parameters re-lated to tensor-mode perturbations. In the followingsubsections, we study the constraints on the four MGparameters individually. For example, when we are con-straining ν , we fix ε , ε h and ε l to their GR values. We do that for a practical reason since current data givesvery weak constraints on the tensor-mode MG parame-ters. It is computationally expensive to constrain the MGparameters simultaneously. In the MCMC analysis, wealso fix the six standard cosmological parameters to thevalues of the Planck 2015 best fit [19], and constrain thetensor-to-scalar ratio r with one of the tensor-mode MGparameters at a time using the joint data of Planck andBICEP2 [18] and the Planck 2015 low- (cid:96) polarization data[19]. In this section, we use the standard inflation con-sistency relation on the value of n T , namely, n T = − r/ n T since otherwise the parameter space would betoo large and give no useful information.For current data, the tensor-induced B-mode polariza-tion has not been detected yet so we will provide onlysome bounds on the MG parameters. Due to the weakconstraining power of current data, we will also not at-tempt any joint constraints on the four MG parameters.We also do not constrain ε h because the observed high- (cid:96) B-mode polarization is dominated by the lensed E-mode,so current data only give a large and meaningless allowedregion in the r vs ε h parameter space. Instead, we willforecast the constraint on ε h in Sec. VI for some futureexperiments.
1. Updating the constraints on friction andconstant speed using the new BKP data
We first update the constraints on the friction and thespeed by using the data from the Planck-BICEP2 jointanalysis (BKP) [18] and the Planck 2015 low- (cid:96) polariza-1tion data [19]. To validate our modification to camb , wereproduced the marginalized likelihood distributions inthe α M vs r and r vs c T parameter spaces in Ref. [12]using the old BICEP2 data [39], and we got the sameresults.The left panel in Fig. 5 shows the marginalized con-straints in the r vs ν parameter space using the BKPand the Planck 2015 low- (cid:96) polarization data. The blackcurves are iso-likelihood contours, within which the inte-grated probabilities are 68% and 95% respectively. Con-sequently, the green and the blue+green regions respec-tively correspond to the 1- σ (68%) and 2- σ (95%) con-fidence levels (C.L.). There is a probability of 68% forthe true values of r and ν to be located within the greenregion, and 95% within the blue+green region. In otherwords, at the 95% C.L., the white parameter space isruled out. (Note that the blue-only region is ruled outat the 68% C.L., but allowed at the 95% C.L.). We cansee from the left panel of Fig. 5 that the degeneratedirection goes roughly as r − . ν = constant, consis-tent with that in Ref. [12]. The tensor-to-scalar ratio r is consistently zero. We cut out the large ν parameterspace, because a larger ν only leads to a larger allowedtensor-to-scalar ratio r .Using the same data, in the right panel of Fig. 5 weshow the constraints in the r vs ε parameter space. Thegreen and blue regions have the same meanings as thosein the left panel of Fig. 5. Since we have not observedthe tensor-induced B-mode polarization, we should notexpect the peak position of the B-mode power spectrumto constrain the speed of gravitational waves as in Ref.[14]. Instead we see in the right panel of Fig. 5 that asmaller ε (and hence a smaller speed) allows a largertensor-to-scalar ratio. As ε approaches −
1, at the 1- σ C.L., we have an upper limit of r ∼ .
75 shown by thegreen region in the right panel of Fig. 5. As mentionedin Sec. IV, a smaller speed means a later horizon reenter-ing. An extreme case is a vanishing speed ( ε = − r can be arbitrarily large. This is alsowhy we excluded the parameter value ε = − r as ε approaches − ε does not seem to affect the constraint on r very much.This is because, besides making the tensor-mode ampli-tudes vary with time, horizon reentering also makes themsmaller. A larger ε then has both an enhancing effect(due to the time-varying tensor-mode amplitudes) anda suppressing effect (due to smaller amplitudes) on theCMB B-mode polarization. r FIG. 6. Constraints in the r vs ε l parameter space. Theplateau from ε l = 0 to ∼ . ε l makeslittle difference on the constraint of r , which is similar to themassless case. Unless r is very small, the sharp drop of theallowed value of r after ε l ∼ . µ upper ∼ . × − eV , for most allowedvalues of r .
2. Constraints on large-scale deviation
Using the same data, we obtained the constraints in the r vs ε l parameter space as shown in Fig. 6. The conver-sion between ε l and the graviton mass µ [for n = 4 in Eq.(7)] is µ = ε l × . × − M p = ε l × . × − eV .We can see that the constraint of r is insensitive to theparameter ε l for ε l < ∼ .
5, which means a graviton masssmaller than ∼ − eV should have no observationaleffect on the CMB for the current level of sensitivity.The constraint of r in this range of ε l is roughly thesame as the case in GR. Both the 1- σ and 2- σ contourshave relatively sharp turns at ε l ∼ .
5. A larger ε l leadsto significant drops of the allowed value of r for bothcontours. This location ( ε l ∼ .
5) of the sharp turnsroughly corresponds to an upper bound of the gravitonmass µ upper ∼ . × − eV unless r is very small. Thisupper bound is roughly of the same order of magnitude athe estimation in Ref. [13]. Note that, if massive gravityis responsible for the late-time cosmic acceleration, thegraviton mass should be of the order of the Hubble con-stant H (in natural units) [13, 40], which is ∼ − eV and is about 3 ∼ r ) obtained in thiswork.There is an allowed parameter-space “tail” for ε l > ∼ . ε l which has been cutoff in Fig. 6. This “tail” is present because, as r ap-proaches 0, the amplitude of tensor-mode perturbations2 TABLE II. Specifications of the COrE mission obtained from Ref. [33]. f sky = 0 .
7. Here, ν denotes the central frequency ofeach band, (not our friction parameter). ν/ (GHz) 45 75 105 135 165 195 225 255 285 315 375 435 555 675 795∆ ν/ (GHz) 15 15 15 15 15 15 15 15 15 15 15 15 195 195 195 θ fwhm / (arcmin) 23 . . . . . . . . . . . . . . . .
61 4 .
09 3 .
50 2 .
90 2 .
38 1 .
84 1 .
42 2 .
43 2 .
94 5 .
62 7 .
01 7 .
12 3 .
39 3 .
52 3 . µ K · arcmin)TABLE III. Specifications of Stage-IV obtained and calcu-lated from Ref. [35]. f sky = 0 . ν/ (GHz) 40 90 150 220 280∆ ν/ (GHz) 30% fractional bandpass θ fwhm / (arcmin) 11 . . . . . µ K · arcmin) 2 . . .
86 1 . . f sky = 0 . ν/ GHz ∆ ν (GHz) θ fwhm (arcmin) Pol. RJ ( µ K · arcmin)15 : 7665 15 96 The sensitivities of the 511frequency channels are pro-vided by Ref. [41]. approaches 0 as well. Then there would be no tensor-induced effects on the CMB (temperature or polariza-tion), and ε l (and the graviton mass) can be arbitrarilylarge. VI. FORECAST OF CONSTRAINTS ONTENSOR MODE MODIFIED GRAVITYPARAMETERS
In this section, we use the Fisher matrix formalism toforecast the constraints on the tensor-mode MG param-eters that could be obtained by the COrE mission [33],CMB Stage-IV [42] and PIXIE [34]. Tables II, III andIV list the specifications of these three near-future ex-periments. To do the forecast correctly, we need to takeinto account the diffuse foreground components. Follow-ing the method described in Refs. [43], we calculate thedegraded-noise power spectrum N post(cid:96) after a componentseparation. To calculate the foreground residuals, we usethe framework described in Ref. [35, 44]. We include inthe analysis the synchrotron and dust as the dominantdiffuse foregrounds. So the number of signal components n comp is three including CMB. We denote CMB as the 0component, the synchrotron as 1 and the dust as 2.
1. Formalism of CMB forecast and foregroundresiduals estimation
With the likelihood provided in Ref. [33], the Fishermatrix reads, F ij = − (cid:28) ∂ (ln L ) ∂θ i ∂θ j (cid:29) = f sky (cid:88) (cid:96) (2 (cid:96) + 1) T r (cid:20) R − (cid:96) ∂ C (cid:96) ∂θ j R − (cid:96) ∂ C (cid:96) ∂θ j (cid:21) , (25)where θ is the parameter vector of a model, R (cid:96) is thesummation of the theoretical power spectra and the totalnoise-like power spectra R (cid:96) = C (cid:96) + N cmb(cid:96) , where, C (cid:96) = C T T(cid:96) C T E(cid:96) C T E(cid:96) C EE(cid:96)
00 0 C BB(cid:96) , and N cmb(cid:96) = N T T(cid:96) N EE(cid:96)
00 0 N BB(cid:96) . (26)For the B-mode polarization, the theoretical power spec-trum is the summation of the contributions from tensormodes and lensing. We do not consider delensing.Since we are considering foreground subtraction, wetake the summation of the degraded (or post-component-separation) noise N post(cid:96) and the foreground residuals C fg,res(cid:96) as the total noise-like power spectrum [33, 35].For the B-mode, N BB(cid:96) = N post(cid:96) + C fg,res(cid:96) . (27)The degraded-noise power spectrum is obtained by, N post(cid:96) = (cid:16) ( A T N − (cid:96) A ) − (cid:17) cmb,cmb , (28)where N (cid:96) is the instrumental-noise power spectra beforecomponent separation, which is assumed to be a n chan × n chan diagonal matrix for each multiple (cid:96) . The diagonalelement of N (cid:96) is given by, (cid:0) N (cid:96) (cid:1) νν = (∆Ω σ v ) exp (cid:32) − (cid:96) ( (cid:96) + 1) θ fwhm ( ν )8 ln 2 (cid:33) , (29)3where the index ν (not our friction parameter) denotesthe central frequency of a channel, and there are n chan channels. For example, for the COrE mission, there are n chan = 15 frequency channels as shown in the firstrow in Table II. The full-width-at-half-maximum angle θ fwhm ( ν ) and the quantity ∆Ω σ v (inverse of the weight)can be obtained from the third and the forth rows in Ta-ble II. The n chan × n comp mixing metric A in Eq. (28) iscalculated as, A νi = (cid:90) dν (cid:48) δ ν ( ν (cid:48) ) A rawi ( ν (cid:48) ) , (30)where the index i can be cmb , sync or dust , denoting thesignal components. Different components can be sep-arated because they have different emission laws. Dif-ferent emission laws are expressed as different antenna-temperature functions A rawi ( ν (cid:48) ) of frequency ν (cid:48) . In Eq.(30) δ ν ( ν (cid:48) ) is a normalized band-pass-filter function foreach channel. Take the COrE specification for example:the central frequency ν and the frequency width ∆ ν of δ ν ( ν (cid:48) ) are given by the first and second rows in Table II.For CMB, the antenna temperature reads, A rawcmb ( ν ) = ( ν/T cmb ) exp( ν/T cmb )[exp( ν/T cmb ) − . (31)We have set h = k B = 1. The temperature of the CMB T cmb is 2 .
73 K, corresponding to 56 . A rawsync ( ν ) ∝ (cid:18) νν ref,s (cid:19) β s , (32)where the reference frequency ν ref,s will be set to 30GHz to be consistent with that for the Planck 2015 syn-chrotron polarization map [45]. If it is only the CMBcomponent that concerns us, the proportional coefficientin Eq. (32) is irrelevant. Since any other proportionalcoefficient can be absorbed into a redefined ν ref,s , thevalue of ν ref,s is actually also irrelevant when we onlycare about the CMB component. The estimated syn-chrotron spectral index β s is − . A rawdust ( ν ) ∝ (cid:18) νν ref,d (cid:19) β d +1 exp (cid:16) ν ref,d T d (cid:17) − (cid:16) νT d (cid:17) − , (33)The dust reference frequency ν ref,d = 353 GHz is chosento be consistent with the one for the Planck 2015 dustpolarization map, but again its value is irrelevant whenwe only care about the CMB component. The dust tem-perature T d is fixed to 19 . β d = 1 .
59. We assume the emission lawsfor synchrotron and dust are spatially independent.We follow the framework described in Refs. [35, 44] tocalculate the foreground residuals. The idea is as follows. Since we do not exactly know what emission laws are fol-lowed by the synchrotron and the dust, the subtractionof those two components from the signal is not ideal. As-suming that the synchrotron and the dust emission lawstake the form of Eq. (32) and Eq. (33), our uncertaintiesare now on the two spectral indices β s and β d ( T d is fixedhere). One first estimates the uncertainties on the spec-tral indices β s and β d , and then infers the propagatederrors in the foreground subtraction. These errors areidentified as the foreground residuals. According to Ref.[44], the uncertainties of the spectral indices are specifiedby the matrix Σ , which is calculated as, (cid:16) Σ − (cid:17) ββ (cid:48) = − T r (cid:110)(cid:2) ∂ A T ∂β N − AC N A T N − ∂ A ∂β (cid:48) − ∂ A T ∂β N − ∂ A ∂β (cid:48) (cid:3) × ˆ F (cid:111) . (34)where C N = ( A T N − A ) − . Note that the n chan × n chan matrix N here (to be distinguished from N (cid:96) ) is thenoise covariance at each pixel , whose diagonal elementis, N νν = (12 × nside )4 π × (cid:0) ∆Ω σ ν (cid:1) . For three known com-ponent template maps (i.e., s cmb , s sync and s dust ), the n comp × n comp matrix ˆ F in Eq. (34) is, (cid:0) ˆ F (cid:1) ij = (cid:88) p s pi s pj , (35)where i, j = cmb , sync or dust , and the superscript p denotes the pixel location.To calculate the matrix Σ , we need to have the syn-chrotron and the dust polarization template maps (i.e., s sync and s dust ), and a mask that specifies nside andwhich pixels are included in the sum in Eq. (35). We donot actually need a template map for the CMB. That isbecause A rawcmb does not depend on β s or β d , and the cor-responding CMB components do not contribute to thesummation when we take the trace in eq (35). In thiswork, we use the second Planck release of componentpolarization maps and the polarization mask, and we de-grade them to nside = 128 resolution. Once the matrix Σ is obtained, the foreground residuals can be computedas, C fg,res(cid:96) = (cid:88) ββ (cid:48) (cid:88) jj (cid:48) Σ ββ (cid:48) κ jj (cid:48) ββ (cid:48) C jj (cid:48) (cid:96) , (36)where κ jj (cid:48) ββ (cid:48) is given by, κ jj (cid:48) ββ (cid:48) = a jβ a j (cid:48) β (cid:48) , (37)and a jβ is, a jβ = (cid:20) C N A T ( N ) − ∂ A ∂β (cid:21) j . (38)The C jj (cid:48) (cid:96) ’s in Eq. (36) are the auto and cross powerspectra of the synchrotron and dust polarization maps.4We refer readers to Refs. [35, 44] for detailed discus-sions of the above framework. In Fig. 7 we show resultsfor the power spectra of the degraded instrumental noise,the (total) foreground residual and the B-mode polariza-tion with our base fiducial model for the three futureexperiments we considered. Different experiment specifi-cations lead to different degraded noises and foregroundresiduals.
2. Performance forecast of constraints ontensor-mode MG parameters
In this subsection, we consider the following question:how significant do the deviations from GR in the tensorsector need to be, so that we can detect them with thenear-future CMB experiments? To answer this question,we do a performance forecast using the Fisher matrixformalism with the specifications of COrE, Stage-IV andPIXIE listed in Tables II, III and IV.In Table V we list the base fiducial model used in ourFisher matrix analysis. In this subsection, we only con-sider the ΛCDM+ r with the standard inflation consis-tency relation as our base model, where ΛCDM standsfor the six standard cosmological parameters. The testof the standard vs the MG consistency relation will bein the next subsection. On top of the base model, weconsider four extended models, namely, ΛCDM+ r + ν ,ΛCDM+ r + ε , ΛCDM+ r + ε l , and ΛCDM+ r + ε h . Whenwe consider the ΛCDM+ r + ν model, for example, wefix the other MG parameters to their GR values. The sixstandard ΛCDM parameters are then marginalized overto give two-dimensional confidence-region plots in the r + ν . We then derive the minimum detectable values ofthe tensor-mode MG parameters for those future experi-ments. In this work, the minimum detectable value x min of an MG parameter x is conservatively defined as theone when the x -direction half width of the 3- σ likelihoodellipse in the marginalized r - x space equals x min itself(or − x min if x is negative). We will repeat and do thesame for the other extended models. These minimum de-tectable values should depend on the base fiducial model,especially on the fiducial value of r . We do not considerthe constraints on MG parameters simultaneously sincethe near-future CMB experiments all have limited con-straining power. Moreover, we want to explore the indi-vidual minimum detectable value for each MG parameterso we can estimate which modification to GR will be mostlikely detectable with these experiments.In Fig. 8 (for friction) and Fig. 9 (for dispersion re-lation) we show the results of the performance forecast.Take the COrE specification for example: we can inferfrom those plots that the minimum detectable values of ν , | ε | , ε l , and ε h are 0 .
035 ( − .
11 for negative ν ), ∼ .
05, 0 .
035 and 0 .
02 respectively. These minimum de-tectable values tell us that the COrE mission can detectdeviations from GR if 1) the additional friction is at least3 .
5% larger than that in GR, 2) or the friction is sup- l -6 -4 -2 l ( l + ) C l / ( K ) COrE
20 100 500 l -6 -4 -2 l ( l + ) C l / ( K ) Stage-IV l -6 -4 -2 l ( l + ) C l / ( K ) PIXIE
FIG. 7. COrE (top), Stage-IV (middle) and PIXIE (bottom):The power spectra of 1) the tensor B-mode polarization with r = 0 .
01 in ΛCDM (solid green), 2) the total B-mode (dashmagenta), 3) the degraded instrumental noise (solid red), 4)the (total) foreground residual (solid blue), 5) the total noise-like error (solid black), and 6) the foreground signals (shownonly on the top of the COrE panel: dotted for synchrotron-auto, dashed for dust-auto and dot-dashed for synchrotron-dust cross spectra). Note the minimal (cid:96) for Stage-IV is just20. And the maximum (cid:96) for PIXIE is 200. pressed and at least 11% less than that in GR, 3) thespeed of gravitational waves is at least by ∼
5% differentfrom the speed of light, 4) gravitons possess a mass ofat least 7 . × − eV , and 5) the small-scale dispersionrelation is modified with a critical scale of 1 . TABLE V. The base fiducial model (ΛCDM + r ) used in the Fisher matrix analysis. We extend it to four MG models (i.e.ΛCDM + r + 1 MG parameter).Base fiducial parameters r n s τ Ω b h Ω c h H A s Values 0 .
01 0 . .
079 0 . . .
27 2 . × − ν r COrE ν ν r Stage-IV ν =0.04 ν r PIXIE ν =0.045 ν -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 r COrE ν =-0.11 ν -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 r Stage-IV ν =-0.3 ν -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 r PIXIE ν =-0.5 FIG. 8. Results of constraints on the friction term for COrE (left), Stage-IV (middle) and PIXIE (right) specifications. Weshow the 1- σ , 2- σ and 3- σ marginalized confidence-region contours in the r - ν space for the ΛCDM+ r + ν model. We set r fid = 0 .
01. All top (bottom) panels are for positive (negative) ν . These figures show the minimum detectable values of ν ,which can be converted to a minimally required percentage difference in the strength of friction. k / √ ε h , which means the dispersion relation at scalessmaller than this will be modified. In particular, theΛCDM+ r + ε l model corresponds to a massive gravitonmodel. With r = 0 .
01 and the standard inflation con-sistency relation, the minimum detectable graviton massis 7 . × − eV for COrE. This is important, because,as we mentioned earlier, if the massive gravity modelsare responsible for the late-time cosmic acceleration, thegraviton mass will be at the order of 10 − eV .The minimum detectable graviton mass depends on thevalue of n we set in Eq. (7). We set n = 4 for conveniencein the MCMC analysis with the current data. We canchoose a different n for future data. Choosing a different n will give us a different value of ε l,min , and consequentlya different minimum detectable graviton mass. This isbecause changing the value of n effectively sets a differentuniform prior. But this change does not give a very dif- ferent result. For example, we later set n = 1 and obtaina minimum detectable graviton mass of 8 . × − eV .We list all the minimum detectable values and theirphysical meanings in Table VI for the three near-futureexperiments. We found that those three near-future ex-periments are optimistic about the constraints of thetensor-mode MG parameters. For r fid = 0 .
01, the ad-ditional friction only needs to be different from that inGR by 3 . ∼ .
5% to allow detection. If the friction issuppressed (negative ν ), it is required to be 11 ∼ ∼ − eV , comparableto the one in the massive gravity theories that give late-time cosmic acceleration.At the end of this subsection, it is worth clarifying why6 ǫ r COrE ǫ =0.05 ǫ -0.05 0 r COrE ǫ =-0.03 ǫ -0.1 -0.05 0 r Stage-IV ǫ =-0.04 ǫ -0.1 -0.05 0 r PIXIE ǫ =-0.05 ǫ l r COrE ǫ l,fid =0.035 ǫ h r COrE ǫ h,fid =0.02 ǫ r Stage-IV ǫ =0.06 ǫ l r Stage-IV ǫ l,fid =0.0375 ǫ h r Stage-IV ǫ h,fid =0.023 ǫ r PIXIE ǫ =0.15 ǫ l r PIXIE ǫ l,fid =0.035 ǫ h r PIXIE ǫ h,fid =0.07 FIG. 9. Results of constraints on the dispersion relation for the COrE (left), Stage-IV (middle) and PIXIE (right) specifications.First two rows: the ΛCDM+ r + ε model. Take COrE for example: a value of | ε ,min | = 0 .
05 means COrE can observe a speedfractional deviation that is 5% different from the speed of light. Third row: the ΛCDM+ r + ε l model. A value of ε l,min = 0 . n = 4) means the minimum detectable mass of the graviton will (at best) be 7 . × − eV . Fourth row: the ΛCDM+ r + ε h model. This is a high- k/a deviation model, ε h,min = 0 .
02 means the dispersion is not changed for a physical wave numbersmaller than k / √ ε h = 700 Mpc − . Similar interpretations apply to the other two experiments. we can constrain ε h in the presence of lensing. It is truethat ε h only changes the tensor-induced B-mode powerspectrum at small scales, where it is generally consideredto be contaminated by the signal from lensing. But ifthe tensor-to-scalar ratio r is not completely negligible,the tensor-mode contributions are important for B-modepolarization at (cid:96) < ∼ ε h leads to a smaller (cid:96) onset of the damping effects on the B-mode power spec-trum; see Fig. 4. The values of the minimum detectable ε h shown in Tables VI, VII and VIII are large comparedto the ones shown in Fig. 4, which are large enough tosuppress the B-mode power spectrum within (cid:96) < ∼ TABLE VI. Results for the COrE specifications of the minimum detectable values of the tensor mode modified gravity param-eters and their physical meaning with r = 0 . r + Minimumdetectable Physical effects associated with a detection at the 3- σ level ν .
035 An enhanced friction that is 3 .
5% (or more) larger than that in GR can be detectednegative ν − .
11 A suppressed friction that is at least 11% smaller than the GR value can be detected | ε | .
04 A speed deviation from the speed of light of ∼
4% or larger can be detected ε l .
035 A graviton mass > . × − eV can be detected ε h .
02 The small-scale dispersion relation needs to be modified with a critical wave number( k/a ) critical < ∼
700 Mpc − (or critical scale > ∼ . r + Minimumdetectable Physical effects associated with a detection at the 3- σ level ν .
04 An enhanced friction that is 4% (or more) larger than that in GR can be detectednegative ν − . | ε | ∼ .
05 A speed deviation from the speed of light of ∼
5% or larger can be detected ε l .
038 A graviton mass > . × − eV can be detected ε h .
023 The small-scale dispersion relation needs to be modified with a critical wave number( k/a ) critical < ∼
660 Mpc − (or critical scale > ∼ . r + Minimumdetectable Physical effects associated with a detection at the 3- σ level ν .
045 An enhanced friction that is 4 .
5% (or more) larger than that in GR can be detectednegative ν − . | ε | .
15 & 0 .
05 A speed deviation from the speed of light that is 15% faster, or 5% slower can be detected ε l .
035 A graviton mass > . × − eV can be detected ε h .
07 The small-scale dispersion relation needs to be modified with a critical wave number( k/a ) critical < ∼
380 Mpc − (or critical scale > ∼ . suppressing effect due to the MG parameter ε h .
3. Testing the standard consistency relation vs theMG consistency relation
Another question is: can we test the standard consis-tency relation (19) vs the MG consistency relation (21)?We find that in some situations we are able to do so,and we show it with the method of performance forecastdescribed in the previous subsection. We assume in thiswork that the friction parameter ν is constant through-out the history of the Universe.We first extend the model from ΛCDM+ r + ν toΛCDM+ r + ν + n T , where n T is the tensor spectral in-dex. We assume the true value of ν is much largerthan r . Here we set r fid = 0 .
01 and ν ,fid = 0 .
2. Thesmall term − r/ n T (cid:39) − ν = − .
6. Onthe other hand, the standard consistency relation gives n T = − r/ − . | n T | can be largefor the MG consistency relation, it must be small for thestandard one (given the fact that r < . r to get a two-dimensional confidence-region plot inthe n T vs ν parameter space. Once we obtain such atwo-dimensional plot, we will be able to see whether theuncertainty is small enough to rule out the standard con-sistency relation.We take the COrE as an example to examine the abovequestion. In the left panel of Fig. 10, the co-center ofthe three ellipses shows the fiducial model in the n T vs ν parameter space, and the three ellipses are the 1- σ , 2- σ and 3- σ marginalized likelihood contours. The “straight8 ν n t -1-0.8-0.6-0.4-0.20 ν =0.2with MG consistencyrelation n t =-3 ν ↑ GR inflation consistency relation, n t =-r/8zoom in → × -3 -4-202 ν -0.05 0 0.05 0.1 0.15 0.2 0.25 n t -0.6-0.4-0.20 ν =0.11with MG consistencyrelation n t =-3 ν ↑ GR inflation consistency relation, n t =-r/8 FIG. 10. Demonstration of how we can distinguish the standard and the MG consistency relations. We assume that the fiducialmodel satisfies the MG consistency relation with ν = 0 . ν = 0 .
11 on the right. Both panels have a fiducialvalue of r = 0 .
01. For the left panel, the MG consistency relation predicts n T (cid:39) − .
6, which is much larger than the onepredicted in GR ( n T = − . n T according to the standard consistency relation n T = − r/
8. That shaped band is so narrow that it looks likea “straight line” in the ν vs n T parameter space. The side box shows the shaped band with a 3- σ uncertainty of r in amore suitable range. We can see that the three iso-likelihood contours do not intersect with the shaped band. Therefore, suchsimulated data favor the MG consistency relation over the standard consistency relation. However, the true value of ν needsto be large enough in order to distinguish the two consistency relations observationally. The right panel shows the minimumvalue of ν that allows us to distinguish the two consistency relations for COrE, which is ν ,min = 0 . line” shows the standard consistency relation n T = − r/ σ uncertainty of r . This “straight line” is actuallya green shaped band. But its offset from 0 and its uncer-tainty are too small compared to the vertical scale of thegraph, so it looks like a straight line. We zoom in andshow this shaped band in a side box in the top-right cor-ner. The ellipses do not intersect with the shaped band,which means the observation is not consistent with thestandard consistency relation at the 3- σ confidence level.In such a case, we can verify the MG consistency relationand rule out the standard one.The next question is: how large does ν need to befor us to experimentally distinguish the two consistencyrelations? If the fiducial value of ν is small, n T will alsobe small even if it follows the MG consistency relation.The ellipses will then move upwards in the r vs ν plane,and intersect with the shaped band. In that case thedata will be consistent with both consistency relations,and we will not be able the tell which one is correct.The minimum value of ν (for COrE) that allows us toobservationally distinguish the two consistency relations(at the 3- σ C.L.) is demonstrated in the right panel ofFig. 10. There we set the fiducial value of ν to 0 . σ likelihood contour marginally intersects with theshaped band. So if ν > .
11, the ellipses will be belowthe shaped band (like the case in the left panel), and if ν < .
11 they intersect. This minimum value of ν isstill very large compared to r , that is, ν ,min = 0 . (cid:29) r = 0 . ν , the discussion will be sim- ilar to that above. But since the negative ν is moredifficult to observe (see Sec. VI 2), | ν | needs to be verylarge for us to distinguish the standard and the MG con-sistency relations.The conclusion of this subsection is that: yes, in somesituations, we can observationally distinguish the stan-dard and the MG consistency relations. The friction pa-rameter | ν | needs to be much larger than the tensor-scalar-ratio r in order for us to experimentally disentan-gle the standard and the MG consistency relations withthe next-generation CMB experiments. VII. SUMMARY
We proposed a general form of the tensor-mode prop-agation equation, which can be applied to a wide rangeof modified gravity theories. Based on this equation, wewrote four physically motivated parametrization schemeswhich include the changes to the friction, the propaga-tion speed, as well as the dispersion relation at large andsmall scales. Some similar modifications have been in-dividually considered in the literature [12–14], but wecombined them in a different approach and extend themto cover more possible cases. We also derived a consis-tency relation for the MG models. We then performedparameter constraints and forecasts.Before investigating the current and future data con-straints, we studied the parametrized tensor-mode per-turbations during inflation and derived a few useful equa-9tions in the modified gravity case. We obtained an MGinflation consistency relation n T = − ν − r/
8. Besidesrelating the tensor spectral index n T to the tensor-to-scalar ratio r as in the standard inflation consistency re-lation, the MG inflation consistency relation also relates n T to the friction parameter ν . If the friction param-eter is constant throughout the history of the Universe(including inflation and the period after it), we can usethe CMB B-mode polarization data to test the standardand the MG consistency relations. If the friction param-eter is finite but changes its value after inflation, thenat least the standard inflation consistency relation canbe falsified due to the additional contribution from ν tothe value of n T .To see the MG effects on the B-mode polarization andto constrain the MG parameters from the current obser-vations, we modify camb to implement our parametriza-tion and apply a Monte Carlo Markov Chain analysisusing CosmoMC . We studied the effects of the four pa-rameters individually on the B-mode polarization powerspectrum. Then using the currently available data fromthe Planck-BICEP2 joint analysis and the Planck-2nd-released low- (cid:96) polarization, we set exclusion regions onthe MG parameters.Then we calculated performance forecasts on con-straining MG parameters for the next-generation CMBexperiments. We used the specifications of the near-future missions COrE, Stage-IV and PIXIE. We per-formed calculations of the corresponding foregroundresiduals and the degraded noise for the analysis. Fora fiducial cosmological model with a tensor-to-scalar ra-tio r = 0 .
01, we determined the 3- σ confidence contoursin the r + each MG parameter spaces. We found that ( i )an additional relative friction of 3 . ∼ .
5% compared toits GR value will be detected at the 3- σ level by theseexperiments (the details are given in our Tables VI, VII,and VIII); ( ii ) a suppressed friction will be harder to con-strain ( −
11 to −
50% is required for a detection); ( iii ) thespeed of gravitational waves with a relative difference of 5 ∼
15% or larger compared to the speed of light will bedetected; ( iv ) the minimum detectable graviton mass isaround 7 . ∼ . × − eV for these experiments: thisis important because this minimum detectable gravitonmass is of order of 10 − eV , which is the same as theone in the massive gravity theories that can produce thelate-time cosmic acceleration; ( v ) for the small-scale de-viation, the dispersion relation needs to be modified witha critical wave number ( k/a ) critical < ∼ ∼
700 Mpc − (or the critical scale needs to be > ∼ . ∼ . n T = − ν − r/
8) from the standard inflation consis-tency relation ( n T = − r/ | ν | needs to be much larger than thetensor-to-scalar ratio r .In summary, we find that the near-future experimentsprobing tensor-induced B-modes such as the COrE mis-sion [33], PRISM mission [46], POLARBEAR2 [47],CMB Stage-IV [42] and PIXIE [34] will open a newpromising window on testing gravity theories at cosmo-logical scales. ACKNOWLEDGMENTS
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