Detecting relic gravitational waves in the CMB: A statistical bias
ABSTRACT Analyzing the imprint of relic gravitational waves (RGWs) on the cosmic microwave background (CMB) power spectra provides
a way to determine the signal of RGWs. In this Letter, we discuss a statistical bias, which could exist in the data analysis
and has the tendency to overlook the RGWs. We also explain why this bias exists, and how to avoid it.
arXiv:1107.4504v1 [astro-ph.CO] 22 Jul 2011
Detecting relic gravitational waves in the CMB: A statistical bias
Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, Copenhagen DK-2100, Denmark
(Dated: July 25, 2011)
Analyzing the imprint of relic gravitational waves (RGWs) on the cosmic microwave background
(CMB) power spectra provides a way to determine the signal of RGWs. In this Letter, we discuss a
statistical bias, which could exist in the data analysis and has the tendency to overlook the RGWs.
We also explain why this bias exists, and how to avoid it.
A stochastic background of relic gravitational waves
was produced in the very early stage of the Universe due
to the superadiabatic amplification of zero point quan-
tum fluctuations of the gravitational field [1, 2]. The
relic gravitational waves have a wide range of spreading
of the spectra, and their detection provides a direct way
to study the physics in the early Universe.
Recently, there have been several experimental efforts
to constrain the amplitude of relic gravitational waves
in different frequencies.Among various direct obser-
vations, LIGO S5 has experimentally obtained so far
the most stringent bound Ωgw(f) ≤ 6.9 × 10−6around
f ∼ 100Hz , which will be much improved by fu-
ture observations, including the third-generation Ein-
stein Telescope . The timing studies on the millisec-
ond pulsars by the PPTA and EPTA teams also re-
ported upper limits Ωgw(f) ? 10−8at f ∼ 1/yr [5, 6].
In addition, there are two bounds on the integration
?Ωgw(f)dlnf ? 1.5 × 10−5, obtained by the big bang
background radiation observation .
In this paper, we shall focus on the detection of relic
gravitational waves by the cosmic microwave background
(CMB) radiation observations.
understood imprints on the anisotropies in temperature
and polarization of CMB [9, 10]. The theoretical analysis
of these imprints along with the data (including T, C,
E, B) from CMB experiments allows one to determine
the RGW background by constraining the parameters:
the tensor-to-scalar ratio r and the spectral index nt.
The current observations of CMB by WMAP satellite
place an interesting bound r ≤ 0.20  by assuming
nt = −r/8, which has been generalized in . These
bounds are equivalent to the constraints on the energy
density Ωgw(f) of relic gravitational waves at the lowest
frequency range f ∼ 10−17Hz.
Detecting the relic gravitational waves remains one of
the most important tasks for the upcoming CMB obser-
vations (see  for reviews). Due to the various large
contaminations, in the near future, we can only expect
to detect a signal of RGWs in a relative low signal-to-
nucleosynthesis observation  and the cosmic microwave
The RGWs leave well
∗Electronic address: email@example.com
noise ratio (S/N). This result would guide the far future
As for the whole data analysis, we expect that, the
maximum value of the parameters in the posterior pos-
sibility density function (pdf) is unbiased for the ‘true’
values of the parameters, which is auto-satisfied when
the S/N is high. However when S/N is low, the maxi-
mum values of the parameters sometimes lead to a biased
guide for the ‘true’ values, which can be generated either
by some systematics or by the statistics, and should be
avoided in any data analysis.
In this Letter, we will point out that, a statistical bias
could exist in the CMB data analysis for the detection
of RGWs. We also explain why the bias does exist, and
suggest the way to avoid it.
II.THE STATISTICAL BIAS
The primordial power spectrum of relic gravitational
waves can be simply described by the following power-law
Pt(k) = At(k0)(k/k0)nt,(1)
where k0 is the pivot wavenumber, which can be arbi-
trarily chosen. At(k0) is the amplitude of RGWs, and nt
is the spectral index. The value of nt is quite close to
zero, predicted by the physical models of the early Uni-
verse. As usual, we can define the tensor-to-scalar ratio
r ≡ At(k0)/As(k0), where As(k0) is the amplitude of
the density perturbations. Obviously, assuming As(k0)
is known as in this Letter, r is just At(k0) normalized by
the constant As(k0).
In order to discuss the statistical bias for the detection
of RGWs in the data analysis, let us simulate the ob-
servable data for the Planck satellite, where we only con-
sider the Planck instrumental noises at the 143GHz fre-
quency channels . We adopt the ‘input’ cosmological
models as Ωbh2= 0.02267, Ωch2= 0.1131, ΩΛ= 0.726,
τreion= 0.084, h = 0.705, As= 2.445×10−9and ns= 1.
The RGWs parameters are adopted as r = ˆ r = 0.05,
nt= ˆ nt= 0. As we have discussed in the previous paper
, this small r is expected to be detected at 2σ for the
assumed noise level.
Based on this input cosmological model, and the as-
sumed noise level, we simulate 500 data samples. For
FIG. 1: The distribution of rML, ntML and zML for the 500
simulated samples. The blue shadow shows the results by
adopting the free parameters r and nt and the flat prior of
them. The red column shows the results by adopting the free
parameters r and z and the flat prior of them.
every sample, we can probe the likelihood function by ap-
plying the Markov Chain Monte Carlo (MCMC) method.
In the data analysis, we assume all the parameters, ex-
cept for r and nt, are all fixed as their input values.
For the parameter nt, one always presumes the relation
nt= ns− 1 or nt= −r/8 in the data analysis .
However, this assumption does depend on the special cos-
mological models. If they are not the truth, but pre-
sumed, the finial conclusion of the data analysis would
deviate from the real physics.
In order to avoid this danger, the natural way is setting
r and ntas free parameters. We choose the flat priors of
them in the range r ∈ [0,1] and nt∈ [−3,3]. We adopt
the best-pivot wavenumber, which is k0= 0.0006Mpc−1
for the input model and the assumed noise level .
The most interesting final result is the maximum value
in the 1-dimensional posterior pdf for the parameters r
and nt. In this paper, we denote them by rMLand ntML.
Of course, their values do depend on the simulated data.
For different data samples, they have different values.
We expect the distribution of these 500 rML and ntML
are around their input values. However, it may be not
the truth in the real analysis. In Fig.1, we plot the distri-
bution of rMLand ntMLwith blue shadows. This figure
shows that, the distribution of ntML is peaked at zero,
the input value. However, the distribution of rMLobvi-
ously approaches to r = 0, and biased the input value
at r = 0.05. This suggests that, if we deal with the
data analysis in this way, the resulting conclusion has
the tendency to deviate from the ‘true’ value of r, and to
overlook the RGWs.
III.UNDERSTANDING THE STATISTICAL
It is important to understand why this statistical bias
does exist. In order to realize it, let us proceed the follow-
ing analytical approximation for the likelihood analysis.
The primordial power spectrum of RGW in (1) can be
Pt(k) = At(k0)(k/k0)nt= As(k0)rexp[ntln(k/k0)],(2)
which can be approximated as
Pt(k) ≃ As(k0)[r + rntln(k/k0)].(3)
In this approximation, we have used |nt| ≪ 1.
The total CMB power spectra CY
include the contributions of density perturbations and
gravitational waves, i.e.
(Y = T,C,E,B)
turbations and gravitational waves, separately. Note that
function of r and nt, can be approximated as 
ℓ,tare the contributions of density per-
ℓ,s= 0. By considering |nt| ≪ 1, the spectra CY
ℓ,t, as a
ℓ,t[r + rntln(ℓ/ℓ0)].(5)
ℓ0= k0×104Mpc . So Pt(k) and CY
combinations of the parameters r and rnt.
Now, let us turn to the likelihood function. The exact
form can be found in the previous works [15, 16, 18].
In the analytical approximation, it can be well approxi-
mated by 
ℓ,t(r = 1,nt= 0), and best-pivot multipole
ℓ,tare all the linear
− 2lnL =
ℓis the observable data, and ˆ σDY
ℓ. The likelihood function (6) can be rewritten as
ℓis standard deviation
− 2lnL =
ℓ− (r + rntbℓ)aY
where we have defined the quantities
which are all independent of the variables r and nt.
Obviously, the value of dY
ℓdepends on the data. For
a larger number of different sample, the average value
ˆ nt) ≃ CY
ℓ(ˆ r + ˆ rˆ ntbℓ).
ℓ? = aY
ℓ(ˆ r + ˆ rˆ ntbℓ), due to the facts of
ℓ,t(r = ˆ r,nt= ˆ nt) and CY
ℓ? = CY
ℓ,t(r = ˆ r,nt=
Since we have adopted the best-pivot multipole ℓ0,
which is defined by requiring ?
?r − rp
where C is a constant, and the other quantities are de-
ℓ)2bℓ = 0,
the likelihood (7) can be rewritten as 
− 2lnL =
The posterior pdf relates to the likelihood by the prior.
Here, let us adopt the flat prior for the parameters r and
nt, the 2-dimensional posterior pdf for the variables is
− 2lnP(r,nt) =
?r − rp
which follows the 1-dimensional posterior pdf for r as
?r − rp
We notice that, when rp ≫ rs, corresponding to
S/N ≫ 1 (see  for details), this pdf can be reduced
?r − rp
This is gaussian function for r, and peaks at r = rpwith
spread rs. From the expression of rp, we know that, the
value of rpdepends on the data DY
However, the average value of rpfor a larger number of
sample is ¯ rp= ˆ r, i.e. rpis an unbiased estimator for ˆ r.
This has been mentioned in the previous paper .
But here, we want to emphasize that, when rpis not
much larger than rs, the peak of the posterior pdf in (13)
is smaller than rp, due to the term 1/r. Especially when
rp< 3rs, the peak of the pdf is very close to zero, which
is never an unbiased estimator for the input value ˆ r. This
explains what we have found in the left panel of Fig.1.
ℓby the quantity dY
IV.AVOIDING THE STATISTICAL BIAS
Now, let us consider the possible way to avoid this
bias in the data analysis. Let us return to the likelihood
function in (9). We find that, if considering r and z ≡ rnt
as two independent parameters, this likelihood is a simple
gaussian function for the uncorrected parameters r and
Now, we adopt the flat prior for the variables r and z,
and the posterior pdf for r and z becomes
− 2lnP(r,z) =
?r − rp
?z − zp
from which follows that the 1-dimensional posterior pdf
for r is
P(r) = exp
?r − rp
+ C′. (16)
This pdf peaks at r = rp, which is an unbiased estimator
for the input value ˆ r. Similarly, we can also find that,
the 1-dimensional posterior pdf for z peaks at z = zp,
which is also an unbiased estimator for ˆ z ≡ ˆ r ˆ nt. So the
statistical bias in data analysis is elegantly avoided.
In order to clearly show this result, we have analyzed
the same 500 samples, by adopting the flat prior on r
and z. In Fig.1, we plot the distribution of the rMLand
zMLwith the solid columns. As expected, we find that,
these rML and zML are all distributed around at their
input values ˆ r = 0.05 and ˆ z = 0, and the bias for the
tensor-to-scalar ratio is naturally avoided. In this figure,
we also plot the distribution of ntML, which also unbiased
distributed around its input value ˆ nt= 0.
It is interesting to compare the difference between the
prior f(r,z) and the general prior f(r,nt). They can be
related by the Jacobi, i.e.
????f(r,z) = rf(r,z).
So, comparing with the
This relation shows that, the flat prior f(r,z) = 1 exactly
corresponds to f(r,nt) = r.
analysis with flat prior f(r,nt) = 1, the new flat prior
f(r,z) induces a larger value of the variable r.
In this Letter, we find a statistical bias in the CMB
data analysis for the detection of RGWs, when the signal-
to-noise ratio is not very high. This could overlook the
signal of RGWs in the CMB data analysis. We explain
why this bias does exist by the analytical approximation
of the likelihood function, and also find this bias can be
elegantly avoided by adopting the orthogonalized param-
eters r and z ≡ rnt, instead of the general parameters r
In the end, we should emphasize that a similar statis-
tical bias might exist for any data analysis , which
should be carefully treated.
Acknowledgement: The author thanks D.Baskaran,
L.P. Grishchuk, P.Coles, H.Chiaka, S.Gupta for helpful
discussions. This work is supported by NSFC grants Nos.
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