# CMB Temperature Polarization Correlation and Primordial Gravitational Waves: WMAP5

Authors: N. J. Miller; B. G. Keating; A. G. Polnarev
Journal: Advances in Astronomy
 (2009)

---

## Abstract

we continue our study of the CMB temperature polarization (TE) cross-correlation as a source of
information about primordial gravitational waves (PGWs). In a previous paper, we considered two
methods for detecting PGWs using the TE cross-correlation. The first method is the zero multipole
method, where we find the multipole,ℓ0, where the TE cross-correlation power spectrum, CℓTE, first changes sign. The second method Wiener filters the CMB TE data to remove the density
perturbation contribution to the TE power spectrum. We then use statistical tests to determine
if there is a detection of negative residual TE correlation and hence a detection of primordial
gravitational waves, the only source of negative TE correlation at these superhorizon scales. In this
paper, we will apply these tests to the WMAP 5-year data. We find that the TE power spectrum
consistent with r < 2.0 at 95% confidence with no additional assumptions about the PGWs. If we
assume that the PGWs are generated by inflation, then we get r < 1.0 at 95% confidence.


---

## Body

## 1. Introduction

The cosmic microwave background (CMB) has
provided a wealth of information about cosmological phenomena. The CMB
temperature anisotropy has been measured extremely accurately by the WMAP team over the past five years [1]. The parameters describing
the standard ΛCDM model have been
measured to outstanding precision. The current focus of CMB telescopes is to
measure the primordial gravitational wave (PGW) background. Most attention has
been focused on detecting the BB power spectrum of the CMB which can only be
generated by PGWs at large scales. In the previous paper [2], we considered a different
approach and looked at how well the TE power spectrum could constrain PGWs. In
this paper, we will apply the methods formulated in [2] to the publically available
WMAP5 data [3].PGWs (tensor perturbations) generate negative
temperature-polarization (TE) correlation for low multipoles, while primordial
density (scalar) perturbations generate positive TE correlation for low
multipoles (see [2, 4–6]). This signature can be used to detect or constrain
the amount of PGWs. The tests formulated from this signature are useful as an
insurance against false detection of PGWs using the BB power spectrum or as a
way to monitor imperfectly subtracted systematic effects and foregrounds.In this paper, we apply these tests to the WMAP5 data,
to see how well PGWs can be detected or constrained using real data. By
detection of PGWs, we mean the measurement of the parameterr defined as the
ratio of the primordial tensor power spectrum, Pt(k), to the primordial scalar power spectrum, Ps(k), measured at the wave number k0=0.002Mpc−1.The plan of this paper is as follows.
In Section2, we will give a brief description of
the two methods based on the TE cross-correlation which can detect or
constrain PGWs. In Section 3, we will apply these two methods to the WMAP5 data
and discuss the results.


## 2. TE Cross-Correlation and Detection of PGWs

In [2], we looked at two different methods for detecting PGWs
based on the TE cross-correlation. The first method, called the zero multipole
method, is based on the calculation of the lowest ℓ, where CℓTE=0. This value, denoted as ℓ0, will be smaller for larger r. To find ℓ0, the TE cross-correlation is approximated around ℓ0 by a linear
dependence on ℓ. We can use observed data to fit this linear
dependence. Finally, ℓ0 is determined
as the crossing of this fitted line with the ℓ-axis. The
errors on ℓ0 are calculated
by using a Monte Carlo simulation. We generate many mock datasets, using the
best fits Cℓ's and the
errors provided by the WMAP5 data for the TE cross-correlation. We then
calculate ℓ0 for each of
these datasets. The statistical deviation in these ℓ0-values is
considered as the uncertainty in the determination of ℓ0.The second method we look at is what we called in
[2] and will call
hereafter, for brevity, the Wiener filtering method. In actuality, the operator
employed is the square root of the Wiener filter, as described in [7]. It has been used before,
where it was called “power filtration”, in [8]. We initially filter out the
contributions to the TE power spectrum due to density perturbations. We then
use several nonparametrical statistical tests to determine if there is a
residual nonzero TE power spectrum due to PGWs. The statistical tests used are (a) the sign test, (b) a signal-to-noise test, and (c) the Wilcoxon
rank-sum test.(a) The sign
test is applied to the tensor contribution to the TE cross-correlation power
spectrum, denoted asCℓ,tTE (t for tensor). If r=0, then Cℓ,tTE=0 for all ℓ. If the measured data are equally distributed around Cℓ,tTE=0 as one would
expect, then there should be roughly an equal number of positive and negative
measured values. If r>0, then there will be more negative measured values
than positive.(b) The
signal-to-noise test deals with the sum of the signal-to-noise ratios for manyℓ. If, by definition, S/N=∑iCℓi/ΔCℓi, then a negative value of S/N is a signature
of PGWs in the data.(c) The
Wilcoxon rank sum test requires a second mock dataset. For this test, we
combine theCℓ,tTE from both the
WMAP5 data and the mock data into one large dataset, which contains m multipoles.
Then, we rank all multipoles in the large data set from (1) to m. The variables R1 and R2 are defined as
the sums of the ranks for the first original dataset and the second original
dataset, correspondingly. Then, we introduce the variables U and Ui as(1)U=min(U1,U2),Ui=Ri−ni(ni+1)/2,where ni is the number
of multipoles in that dataset (i=1,2). The distribution of U is known if r=0. Its value is used to determine the probability that
the two datasets do not have the same r. For a more detailed discussion of these two methods
for detection of PGWs using the TE cross correlation and the statistical tests
see [2, Sections 3 and 4].


## 3. Discussion and Results

We apply the test described in Section2
to the publically available WMAP5 data. We use both the binned
and unbinned versions of the data. The unbinned version of the data provides
values and errors for every multipole. The binned version of the data provides
fewer measurements, but smaller error bars because binning means the averaging
over ℓ from ℓ to ℓ+Δℓ. The current best constraint on r is r<0.2 at 95% confidence
level. This constraint is provided by WMAP5 in combination with distance
measurements of Type Ia supernovae and imprints of baryon acoustic oscillations
on the spatial distribution of galaxies ([9]). By itself, WMAP5 provides a constraint of r<0.43 at 95% confidence
level [10]. The ΛCDM model of WMAP5
gives ℓ0=52. A plot of the WMAP5 best fits within the framework
of the ΛCDM model, and
their results of CℓTE measurements
are shown in Figure 1.Figure 1
The solid black line
is the TE cross-correlation power spectrum predicted by theΛCDM model with no PGWs, which is the best fit for
all WMAP5 data. This plot also shows WMAP5 results for TE cross-correlation
with error bars (blue crosses).We use several different fitting routines to determine
the value ofℓ0 from the WMAP5
data.(a)
The first
routine is a linear fit. This routine minimizes theχ2 error
criterion. In other words, this is the least squares technique. We minimize the
value S1=∑i(f(ℓi)−Cℓi)2, where f is our linear
fitting function, and ℓi and Cℓi are the data.(b)
The
second routine, which is also a linear fit, is a least absolute deviations
fitting routine. Instead of minimizingS1, we minimize S2=∑i|f(ℓi)−Cℓi|. An advantage to this routine is that it should be
more robust to outlying data compared to the least squares technique. A
disadvantage is that it is unstable. By instability, we mean that small
variations in ℓ can cause
considerable variations in the slope of the fitted line. In contrast, the least
squares technique is stable. The second disadvantage is that the minimum of S2 may correspond
to more than one fitting line.(c)
Our third
fitting routine, in contrast to both previous ones, is a polynomial fit rather
than linear. We use the same least squares fitting routine as our first
routine, except we fit to a quadratic polynomial instead of a linear one.For all our routines, we fit over the rangeℓ=35 to ℓ=70. The TE correlations can be approximated as a line
over this range for our first two routines, and ℓ0 will be within
this range unless r is extremely
large.The results for the calculation ofℓ0 for the two
linear fitting routines are shown in Figure 2, and the result for the
polynomial fitting routine is shown in Figure 3. A table of the calculated ℓ0 and their
standard deviation is shown in Table 1. There is no detection of PGWs using
this method. The measured values are all consistent with ℓ0=52. To go from a limit on ℓ0 to a limit on r, we first calculate the ℓ0 as a function
of r and nt. We then marginalize over nt by integrating
the likelihood function to remove the dependance on nt. Doing this, we get a limit of r<2.0 at 95% confidence. If
we use the assumption that the PGWs are generated by inflation and apply the
inflationary consistency relation, nt=−r/8 [11], we get r<1.0 at 95% confidence.
Neither of these is better than r<0.43 at 95% confidence.Table 1
A table of the
zero multipole,ℓ0, and the uncertainty in ℓ0, Δℓ0, for the different fitting routines and datasets
considered in Section 3
FittingDataℓ0Δℓ0linear,χ2Unbinned46.214.2linear, abs. dev.Unbinned55.514.8quadratic,χ2Unbinned46.811.2linear,χ2Binned52.2613.6linear, abs. dev.Binned51.714.3quadratic,χ2Binned41.310.1These plots show the
results of the calculation ofℓ0 for the two linear fitting routines. The top
row is for the unbinned data, and the bottom row is for the binned data. The
left column is the routine which minimizes the χ2 error criterion. The right column is the least
absolute deviation fitting routine.
(a)(b)(c)(d)These plots show the
results of the calculation ofℓ0 for the polynomial fitting routine. The top
plot is for the unbinned data, and the bottom plot is for the binned data.
(a)(b)Applying the filtering method, we assume that the
power spectrum of scalar perturbations is given by theΛCDM model, which is
the best fit to the WMAP5 data. Then, we apply the statistical tests (a), (b),
and (c) to the difference between the power spectrum of this model and
power spectrum formed from the raw data.The results of the filtering are shown in Figure4.
For the Wilcoxon rank sum test, the plots show the mean value of U if r=0 in the WMAP5
dataset and the mock dataset (red dashed line) and the distribution of U given the WMAP5
dataset and a mock dataset with r=0 (solid black
line). Results using both the unbinned WMAP5 data along with the binned WMAP5
data are shown in Figure 4. For the Wilcoxon rank sum test, the distribution of U is centered
somewhat below the mean value we would get if r=0 in the WMAP5
data. The difference is only 0.34σ for the
unbinned data and 0.6σ for the binned
data, so we cannot reject the hypothesis that r=0. The S/N values are less
than half of a standard deviation away from a value of zero, which they would
be if the TE power spectrum due to density perturbations matched perfectly to
the measured TE power spectrum. We cannot place a limit on r using these
tests because they are only testing hypotheses that r=0. Since these are nonparametric statistical tests,
they can help to answer the question whether r=0 or not. In
other words, all these tests do not help constrain r.The upper left plot
shows the results for the Wilcoxon rank sum test using the unbinned data. The
dashed red line shows the mean value ofU if r=0. The solid black line is the
distribution of U with the WMAP5 dataset and a mock dataset with
WMAP5 error bars and r=0. The upper right plot is the same as
the upper left plot except it uses the binned data. The lower left plot shows
the results for the S/N test using the unbinned data. The dashed red
line is the value of S/N for WMAP5. The solid black line is the
distribution of the S/N value for a mock dataset with r=0 and WMAP5 error bars. The lower right plot is
the same as the lower left plot except it uses the binned data.
(a)(b)(c)(d)


## 4. Conclusion

The large angular scale part of the CMB TE power
spectrum can be used to constrain PGWs. The WMAP5 data was chosen as WMAP5 has
made the best measurements of the TE power spectrum so far at the scales we are
looking at. QUAD has also made high-sensitivity measurements of the TE power
spectrum, but on smaller scales [12] We used two different methods for detecting PGWs.
Neither technique was able to detect PGWs in the released data as expected. We
are able to say that r<2.0 at 95% confidence,
which is nowhere near the current r<0.43 limit at 95% confidence
provided by WMAP5 alone. Using this technique, it will be impossible to improve
this constraint. As shown in [2], a cosmic variance limited experiment would only be
able to set a limit of r<0.3 at 99% confidence or r<0.2 at 95% confidence from
TE data. In other words, a cosmic variance limited experiment could set
constraints using these technique that are similar to current constraints
provided by WMAP using different techniques. This is also similar to the
constraints provided by a cosmic variance limited detection of just the TT
power spectrum [13].
These techniques, however, are helpful, as mentioned above, as insurance against
a false detection of r from such
effects as beam systematics [14, 15]. These effects could be potential pitfalls for
upcoming experiments with higher signal-to-noise than WMAP. In the future, we
will consider the insurance against false detection separately and in more
detail. For this purpose, we supplement TE cross-correlation measurements with
large scale structure data, pulsar timing measurements, baryon acoustic
oscillations, supernovae measurements, and so on.



---

*Source: 309024-2009-03-16.xml*