Table Of ContentOscillating modulation to B-mode polarization from varying
propagating speed of primordial gravitational waves
Yong Cai1, , Yu-Tong Wang1 , and Yun-Song Piao1,2
∗ † ‡
1 School of Physics, University of Chinese Academy of Sciences, Beijing 100049, China and
5
1
2 State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
0
2
Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China
y
a
M Abstract
8 In low-energy effective string theory and modified gravity theories, the propagating speed cT
1
of primordial gravitational waves may deviate from unity. We find that the step-like variation of
]
O
c during slow-roll inflation may result in an oscillating modulation to the B-mode polarization
T
C
. spectrum, which can hardly be imitated by adjusting other cosmological parameters, and the
h
p
intensityofthemodulationisdeterminedbythedynamicsofc . Thusprovidedthattheforeground
- T
o
r contributionisundercontrol,high-precisionCMBpolarizationobservationswillbeabletoputtight
t
s
a constraint on the variation of c , and so the corresponding theories.
[ T
2
v PACS numbers:
5
4
3
6
0
.
1
0
5
1
:
v
i
X
r
a
∗ Email: [email protected]
† Email: [email protected]
‡ Email: [email protected]
1
I. INTRODUCTION
Inflation, as the paradigm of the early universe, has not only solved a lot of fine-tuning
problems of the big bang theory, but also predicted the primordial scalar and tensor per-
turbations. The primordial tensor perturbations, i.e. primordial gravitational waves (GWs)
[1][2][3], have arisen great attentions after the BICEP2 collaboration’s announcement of the
detection of B-mode signal in the CMB (around l 80) [4], which was interpreted by them
∼
as the imprint of the primordial GWs, though this result is doubtful due to the foregrounds
of polarized dust emissions [5][6], see also [7].
The detection of primordial GWs would verify general relativity (GR) and strengthen our
confidence in inflation and quantum gravity[8], and also put more constrains on inflation
models and modified gravity at the same time. Besides the CMB experiments which mainly
aimed at detecting low frequencies (10 17 10 15 Hz) GWs, many experiments relate to
− −
−
higher frequencies based on other methods, such as pulsar timing array (10 9 10 8 Hz),
− −
−
laser interferometer detectors (10 4 104 Hz), will be carried out in the coming decades.
−
−
However, since the amplitude of the GWs would stay constant after they are stretched
outside the horizon, and decrease with the expansion of the universe after they reenter the
horizon, the primordial GWs with longer wavelength provide the most of opportunities for
the detection [9]. Therefore, the CMB observations, especially the CMB B-mode detections,
are still the most promising experiments to detect the primordial GWs if they actually exist.
Einstein’s GR is the most accepted theory of gravity. However, it might be required
to modify when dealing with the inflation in the primordial universe and the accelerated
expansion of the current universe. During the matter and radiation dominated era, modified
gravity has several effects on the CMB spectra, such as the lensing contribution to B-modes
[10] and the variation of propagating speed c of primordial GWs [11][12], we are especially
T
interested in the latter in this paper, see e.g.[13] for the case with the scalar perturbation. In
GR, graviton is massless and propagates along the null geodesics, so the propagating speed
c of GWs is naturally set to be unity, i.e. the speed of light. But in modified gravity, e.g.,
T
the low-energy effective string theory with high-order corrections [14][15][16][17][18], and
also modified Gauss-Bonnet gravity [19], and generalized Galileon (Horndeski theory [20])
[21][22][23][24], and beyond Horndeski theories [25],[26], and the effective theory of fluids
at next-to-leading order in derivatives (e.g.[27]), c might deviate from unity. Because the
T
2
value of c determines the time of horizon crossing of GWs, during the recombination the
T
change of c can result in a shift of the peak position of the primordial B-modes, see [11][12],
T
for example.
In this paper, we focus on the effect of the variation of c during slow-roll inflation on the
T
CMB B-mode polarization, and show how it offers a distinct way to test the modified gravity
theories. We find that the step-like variation of c may result in an oscillating modulation
T
to the B-mode polarization spectrum, which can hardly be imitated by adjusting other
cosmological parameters. The intensity of the modulation is determined by the ratio of
c before and after the variation, which depends on the dynamics of theoretical models.
T
This oscillating modulation is so rich in feature that it may easily be discriminated from
the variation of other parameters or other features. Thus provided that the foreground
contribution is under control, high-precision CMB polarization observations will be able to
put tight constraint on the variation of c , and so the corresponding theories.
T
II. OSCILLATING SPECTRUM OF PRIMORDIAL GWS
We begin with the action for the GWs mode h , e.g.[17][23]
ij
a2Q
S = dτd3x T h2 c2(~ h )2 , (1)
(2) 8 ′ij − T ∇ ij
Z h i
where h obeys ∂ h = 0 and h = 0, Q is regarded as effective Planck scale M2 (τ), c
ij i ij ii T P,eff T
is the propagating speed of primordial GWs, and the prime is the derivative with respect
to the conformal time τ, dτ = dt/a. During slow-roll inflation, the slow-roll parameter
ǫ = H˙ /H2 1, as well as
− ≪
Q˙ c˙
T T
ǫ = 1, s = 1 (2)
Q
HQ ≪ Hc ≪
T T
are required, e.g., see [28] for a recent study.
Here, we will mainly focus on the effect of varying c , i.e., the condition s 1 might be
T
≪
broke at some point, on primordial GWs spectrum. Noting that the effects of varying sound
speedofprimordialscalarperturbationsonthescalarpowerspectrumhavebeeninvestigated
in [29][30]. We do not get entangled with the details of (1) and the evolution of background,
and will assume that the background is the slow-roll inflation, which is not affected by the
variation of c , and set Q constant and M2 = 1. We will discuss a possibility of such a
T T P
3
case in Appendix A. In addition, there may be a mass term [31][32] in (1), which might also
be time-dependent [33], but we will not involve it.
We can expand h into Fourier series as h (τ,x) = d3k e ikxˆh (τ,k), where
ij ij (2π)3 − · ij
R
hˆij(τ,k) = hλ(τ,k)aλ(k)+h∗λ(τ,−k)a†λ(−k) ǫi(jλ)(k), (3)
λX=+,×h i
where the polarization tensors ǫ(λ)(k) are defined by k ǫ(λ)(k) = 0, ǫ(λ)(k) = 0, which satisfy
ij j ij ii
ǫi(jλ)(k)ǫ∗ij(λ′)(k) = δλλ′, ǫi∗j(λ)(k) = ǫi(jλ)(−k), the commutation relation for the annihilation
and creation operators aλ(k) and a†λ(k′) is [aλ(k),a†λ′(k′)] = δλλ′δ(3)(k − k′). We define
h (τ,k) = u (τ)/z and z = a√Q /2. Thus we get the equation of motion for u as
λ k T T T k
z
u + c2k2 T′′ u = 0. (4)
′k′ T − z k
(cid:18) T(cid:19)
To phenomenologically investigate the effect on primordial GWs spectrum induced by
varying c , we assume that the variation of c can be described by a step-like function
T T
c (τ < τ )
T1 0
c = , (5)
T
c (τ > τ )
T2 0
where τ < 0 is the transition time.
0
We take the background evolution to be the slow-roll inflation. Of course, it is also
provided that the sudden change of c won’t affect the background evolution. Then, we
T
have z /z a /a (2+3ǫ)/τ2, and the equation of motion (4) becomes
T′′ T ≡ ′′ ≈
ν2 1/4
u + c2k2 − u = 0, (6)
′k′ T − τ2 k
(cid:18) (cid:19)
where ν = 9 +3ǫ 3 +ǫ. The solution to Eq.(6) is familiar, we can write it as
4 ≈ 2
q
u = c kτ C H(1)( c kτ)+C H(2)( c kτ) , τ < τ ,
k1 − T1 1,1 ν − T1 1,2 ν − T1 0
u = p c kτ (cid:2)C H(1)( c kτ)+C H(2)( c kτ)(cid:3), τ > τ , (7)
k2 − T2 2,1 ν − T2 2,2 ν − T2 0
p (cid:2) (cid:3)
(1) (2)
whereH andH arethefirstandsecondkindHankelfunctionsofν-thorder, respectively.
ν ν
ThesecoefficientsC arefunctionsofthecomovingwavenumberk, butconstantswithrespect
to conformal time τ. C and C are determined by the initial condition.
1,1 1,2
Here, we set the initial condition as the standard Bunch-Davies(BD) vacuum. Therefore,
when c k a′′, which corresponds to perturbations deep inside the horizon,
T1 ≫ a
1
u e icT1kτ. (8)
k −
∼ √2c k
T1
4
When c k a′′, u in Eq.(7) should approximate to Eq.(8). Allow for the Hankel function
T1 ≫ a k1
Hv(1)(ξ) = π2ξei(ξ−v2π−π4) and Hv(2)(ξ) = π2ξe−i(ξ−v2π−π4) when |ξ| → ∞, we get
q q
√π
C = , C = 0. (9)
1,1 1,2
2√c k
T1
The coefficients C and C are determined by requiring u and u to be continuous at
2,1 2,2 k ′k
τ = τ , i.e. the matching condition. Then we obtain
0
3 1
C = iπ2τ0k2 c H(1) ( c kτ ) H(2) ( c kτ ) H(2)( c kτ )
2,1 16√cT2 T1 −1+ν − T1 0 − 1+ν − T1 0 ν − T2 0
h (cid:16) (cid:17)
+ c H(2) ( c kτ ) H(2) ( c kτ ) H(1)( c kτ ) ,
T2 − −1+ν − T2 0 − 1+ν − T2 0 ν − T1 0
3 1 (cid:16) (cid:17) i
C = iπ2τ0k2 c H(1) ( c kτ )+H(1) ( c kτ ) H(1)( c kτ )
2,2 16√cT2 T1 − −1+ν − T1 0 1+ν − T1 0 ν − T2 0
h (cid:16) (cid:17)
+ c H(1) ( c kτ ) H(1) ( c kτ ) H(1)( c kτ ) . (10)
T2 −1+ν − T2 0 − 1+ν − T2 0 ν − T1 0
(cid:16) (cid:17) i
The spectrum of primordial GWs is defined by P = (k3/2π2) 0 hˆ (τ, k)hˆ (τ,k) 0
T ij ij
h | − | i
with τ 0, which is only a function of comoving wave number k. After neglecting the
→
slow-roll parameter, from Eq.(7) with ν = 3/2, we have
√2
u = C C . (11)
k2 2,1 2,2
| | c kτ√π | − |
T2
−
Therefore, we obtain the power spectrum of primordial GWs as
k3 4k
P = h (τ,k) 2 = Pinf C C 2, (12)
T 2π2 | λ | T Q πc2 | 2,1 − 2,2|
λ=+, T T2
X×
where
Pinf = 2H2 /π2 (13)
T inf
is that of standard slow-roll inflation without modified gravity, i.e. Q = 1 and c = c =
T T1 T2
1, and H is the Hubble parameter during inflation, which sets the scale of inflation.
inf
The effect of varying c is encoded in C and C . We set x = c /c and defined a
T 2,1 2,2 T2 T1
new function
4c k
f(k,k ,x) = T1 C C 2, (14)
0 πx2 | 2,1 − 2,2|
where k = 1/(c xτ ). Then, the GWs spectrum (12) may be rewritten as
0 T1 0
−
f(k,k ,x)
P = Pinf 0 , (15)
T T · c3 Q
T1 T
5
1.6 1.1
1.5
1.0
1.4
Lx 1.3 Lx 0.9
k,0 k,0
Hfk, 1.2 Hfk, 0.8
1.1
0.7
1.0
0.9 0.6
0.01 0.1 1 10 100 0.01 0.1 1 10 100
k(cid:144)k0 k(cid:144)k0
FIG. 1: The function f(k,k ,x), where x = c /c . We have set x = 0.9 on the left panel and
0 T2 T1
x = 1.1 on the right panel, respectively.
where f(k,k ,x) is obtained as
0
2
1 k 1 k k k
f(k,k ,x) = sin2 + cos( ) (1 x2) 0 sin( ) . (16)
0 x2 k x4 k − − k k
(cid:18) 0(cid:19) (cid:20) 0 0 (cid:21)
We plot f(k,k ,x) with respect to k/k in Fig.1, in which we set x = 0.9 on the left panel
0 0
and x = 1.1 on the right panel, respectively. Here, the transition time τ = 1/(c xk )
0 T1 0
−
sets a character scale 1/k . When k k , i.e. the perturbation mode has longer wavelength
0 0
≪
than 1/k , we have f(k,k ,x) 1, and P = Pinf/(c3 Q ) is scale invariant, which is the
0 0 ≈ T T T1 T
result of slow-roll inflation with almost constant c and Q [23][34]. When k k , we have
T T 0
≫
1 1 k
f(k,k ,x) 1+( 1)cos2( ) , (17)
0
≈ x2 x2 − k
(cid:20) 0 (cid:21)
thus f(k,k ,x) oscillates between 1/x2 and 1/x4, and P oscillates correspondingly, just as
0 T
we can see from Fig.1.
In the case of varying sound speed c of primordial scalar perturbations, the sudden
S
change of c may lead to the oscillating modulation to the primordial scalar spectrum, as
S
well as the CMB TT-mode spectrum, just as found in [29]. In addition, the oscillation
in the primordial scalar spectrum can also be attributed to some other effects, such as
inflaton potential with a small oscillation [35][36], a sudden change in inflaton potential or
its derivative, e.g., [37][38][39][40]. Thus the oscillation in the primordial scalar spectrum
may be implemented without modified gravity, as has been mentioned.
However, the oscillation in the primordial GWs spectrum can only be attributed to the
modified gravity. When the gravity is GR, P equals to Pinf, which is given in Eq.(13). The
T T
6
10−2 6000
102
xx==10.1 5000100
BB2πµl(l+1)C / 2 [K]l10−3 TT2πµl(l+1)C / 2 [K]l2340000000001100−−42 10 xx==10.1100 1000
x=1
1000
x=0.1
10−4 0
10 100 1000 10 100 1000
Multipole l Multipole l
10−2 6000102
x=1 5000
x=10 100
2K] 2K] 4000 x=1
BBπµl(l+1)C / 2 [l10−3 TTπµl(l+1)C / 2 [l230000001100−−42 10x=10 100 1000
1000 x=1
x=10
10−4 10 100 1000 0 10 100 1000
Multipole l Multipole l
FIG. 2: Theoretical CMB BB and TT-mode power spectra for our oscillating GWs spectrum
(15)(brown line in the left panel and red solid line in the right panel) and the power-law GWs
spectrum for reference(blue dashed line in the left panel and green dashed line in the right panel).
The insets of the right panels are the TT-mode spectra for our oscillating GWs spectrum (the
yellow solid lines) and the power-law GWs spectrum (the blue dashed lines) for reference. We set
r = 0.05 and k = 1/30000Mpc 1.
0 −
oscillation of Pinf certainly requires H is oscillating, which is impossible, unless the null
T inf
energy condition is violated periodically. Though the particle production may also modify
the GWs spectrum [41][42][43], it only leads to a bump-like contribution, which is entirely
different from the behavior of oscillation.
III. CMB B-MODE POLARIZATION SPECTRUM
The primordial GWs is imprint in CMB as the B-mode polarization. Thus the oscillation
in the primordial GWs spectrum will inevitably affect the B-mode polarization spectrum.
To see such effects, we plot the CMB BB and TT-mode correlations in Fig.2, in which
7
−3
x 10
3
2.5
inflation , x=1 τ =0.09
ri
2] inflation , x=0.1 τ =0.09
K ri
µ 2 inflation , x=10 τ =0.09
π2 [ inflation , x=1, τri=0.07
B / 1.5 inflation , x=1, τri=0.12
B ri
Cl
)
1
+ 1
(l
l
0.5
2 5 8 11 14 17 20
Multipole l
FIG. 3: BB-mode spectra at low multipoles for our oscillating GWs spectrum (15) with different
x(solid lines) and the power-law GWs spectrum with different τ (dashed lines).
ri
Pinf in (15) is parameterized as
T
ninf
k T
Pinf = rAinf . (18)
T k
R
(cid:18) (cid:19)
∗
Here, we assume that the scalar spectrum is hardly affected by the modified gravity, which
inf
will be clarified in Appendix A. Thus the scalar perturbation spectrum is set as P =
R
Ainf (k/k )niRnf−1, in which Ainf is the amplitude of the scalar perturbations. In addition,
R ∗ R
we also assume that after inflation the propagating speed c is unity, so that the spectrum
T
of B-mode polarization is not affected by relevant evolution at late time, or see [11],[12].
In the left panels of Fig.2, we see some obvious enhancements or suppressions to the
reionization bump in the BB-mode spectrum, which depend on the oscillating effect on
corresponding scales. The height of the reionization bump can be estimated roughly as [44],
1
CBB 1 e τri 2CTT , (19)
T,l 2 ≈ 100 − − T,l 2
∼ ∼
(cid:0) (cid:1)
whereCTT standsfortheTT-modespectrumfromtheprimordialGWswithoutthereioniza-
T,l
tion and τ is the optical depth to the beginning of reionization. The periodic enhancements
ri
and suppressions of the reionization bump are a reflection of the oscillations of primordial
GWs spectrum on large scales. In addition, we can also see some obvious oscillations around
the recombination peak at l 80.
∼
8
In the right panels of Fig.2, we see that the TT-mode spectrum is hardly affected by
the oscillating primordial GWs power spectrum, since the contribution of GWs to TT-
mode spectrum is negligible, compared with the scalar perturbations. The case of EE-
mode polarization spectrum is actually also similar. Thus the main effect of varying speed
of primordial GWs is on the B-mode polarization, which makes the B-mode polarization
spectrum show its obvious enhancements or suppressions to the reionization bump and
oscillations around the recombination peak.
In Ref.[45], the authors have pointed out that it is possible to set to one the propagating
speed of GWs by a proper redefinition of the metric. They got the gravitational waves
spectrum in their Eq.(12) “same” as the standard one in the form. However, since they have
made a redefinition of the time coordinate and the scale factor in Eq.(9), the variation of
their H˜ with respect to t˜after redefinition is not the same as the variation of H with respect
to t. Therefore, the oscillating feature induced by a step-like c is encoded in H˜, the result
T
in both frame should be the same.
It might also be a concern whether such a B-mode polarization spectrum can be imitated
by adjusting other cosmological parameters or not. In Eq.(19), the optical depth τ is
ri
relevant with the height of the reionization bump. We show the BB-mode spectrum with
different τ in Fig.3. We see that the change of τ can only alter the overall amplitude
ri ri
of BB-mode spectrum at low multipoles, but hardly create the oscillation. In addition, in
inflationary models with pre-inflation era, the reionization bump could also be suppressed
(e.g.[46],[47],[48],[49]) or enhanced (e.g.[50]). However, similarly, these models also only
alter the overall amplitude of the BB-mode spectrum, without oscillation, at low multipoles.
These results indicate that although the BB-mode spectrum may be modified by other ways,
the oscillating modulation leaded by varying the speed of primordial GWs is difficult to be
imitated. Thus the measure of B-mode polarization spectrum provides an appropriate way
for testing the corresponding gravity physics in the primordial universe.
IV. DISCUSSION
In low-energy effective string theory and modified gravity theories, the propagating speed
c of primordial GWs may deviate from unity. We calculated the spectrum of primordial
T
GWs, assuming that c has a step-like variation and the background of slow-roll inflation
T
9
is not affected by it. We found the spectrum of primordial GWs acquires an oscillating
modulation, which makes the B-mode polarization spectrum show its obvious enhancements
or suppressions to the reionization bump and oscillations around the recombination peak.
The intensity of the modulation is determined by c /c . The frequency of the modulation
T2 T1
is determined by k = 1/(c τ ). Both depend on the dynamics of theoretical models.
0 T2 0
−
The oscillating behavior of the B-mode polarizationcan only be attributed to the effect of
modified gravity, since it canhardly beimitated by adjusting other cosmological parameters.
The oscillating behavior is so rich in feature that it may be easily discriminated from the
variationofotherparametersorotherfeatures, thustheupcomingCMBexperiments, suchas
CMBPol, B-Pol, will beableto put a tight constraint onthe propagatingspeed of primordial
GWs, and so the corresponding theories, provided that the foreground contribution is under
control. Inacertainsense, ourpaperagainhighlightsthesignificance ofB-modepolarization
measures in exploring the fundamental physics of primordial universe.
Here, we only postulate a simple step-like variation of c , which, however, might be far
T
complicated insome modified gravity models, aswell asaccompanied by the variationof Q .
T
The effect could be non-trivial in more general cases, which is under study. But in a certain
sense, a smooth change of c will induce oscillations too, see e.g.[51] for the case of scalar
T
perturbations, so we expect that the case of GWs is similar. When we focus on the B-mode
polarization spectrum, the assumption we adopted is that after inflation the propagating
speed c is unity, which may also be relaxed. Moreover, it may well be possible to produce
T
oscillatory features beyond the standard slow-roll background. The varying c and Q will
T T
also affect the non-Gaussianities of primordial perturbations. The relevant issues are open.
Acknowledgments
This work is supported by NSFC, No. 11222546, and National Basic Research Program
of China, No. 2010CB832804. We acknowledge the use of CAMB.
Appendix A:
In this Appendix, we will argue how to realize the change of c and the changeless of Q
T T
in (1) in low-energy effective string theory and modified gravity theories.
In low-energy effective action of string theory, the simplest extension of the lowest-order
10
