CMB Temperature Polarization Correlation and Primordial Gravitational Waves
We examine the use of the CMB’s TE cross correlation power spectrum as a complementary test to detect primordial gravitational waves (PGWs). The first method used is based on the determination of the lowest multipole, , where the TE power spectrum, , first changes sign. The second method uses Wiener filtering on the CMB TE data to remove the density perturbations contribution to the TE power spectrum. In principle this leaves only the contribution of PGWs. We examine two toy experiments (one ideal and another more realistic) to see their ability to constrain PGWs using the TE power spectrum alone. We found that an ideal experiment, one limited only by cosmic variance, can detect PGWs with a ratio of tensor to scalar metric perturbation power spectra at confidence level using only the TE correlation. This value is comparable with current constraints obtained by WMAP based on the upper limits to the B-mode amplitude. We demonstrate that to measure PGWs by their contribution to the TE cross correlation power spectrum in a realistic ground based experiment when real instrumental noise is taken into account, the tensor-to-scalar ratio, , should be approximately three times larger.
keywords:cosmic microwave background – polarization – gravitational waves – cosmological parameters
Primordial gravitational waves (PGWs) polarize the cosmic microwave background (CMB) (see for example Basko & Polnarev (1980); Polnarev (1985); Crittenden et al. (1993); Frewin et al. (1994); Coles et al. (1995); Kamionkowski et al. (1997); Seljak (1997); Seljak & Zaldarriaga (1997); Kamionkowski & Kosowsky (1998); Baskaran et al. (2006); Keating et al. (2006)). Current experiments are using the polarization of the CMB to search for this PGW background (Taylor et al. (2004); Bowden et al. (2004); Yoon et al. (2006). This polarization can be used as a direct test of inflation. An alternative probe of the inflationary epoch which does not use the PGW background was studied by (Spergel & Zaldarriaga (1997)). This probe was used in recent analyses by the WMAP team (Peiris et al. (2003)) to provide plausibility for the inflationary paradigm. This paper presents a test similar in spirit to that of Spergel & Zaldarriaga (1997).
CMB polarization can be separated into two independent components: E-mode (grad) polarization and B-mode (curl) polarization. B-mode polarization can only be generated by PGWs (see for example Seljak (1997); Seljak & Zaldarriaga (1997); Kamionkowski & Kosowsky (1998)), therefore most CMB polarization experiments which are searching for evidence of PGWs focus on measuring the BB power spectrum. However the TE cross correlation power spectrum offers another method to detect PGWs (Crittenden et al. (1995)). The TE power spectrum is two orders of magnitude larger than the BB power spectrum and it was suggested that it may therefore be easier to detect gravitational waves in the TE power spectrum (Baskaran et al. (2006); Grishchuk (2007)).
In this paper we first discuss the method of detection of PGWs by measuring the TE power spectrum for low . This method, originally proposed in Baskaran et al. (2006), is based on a measurement of , the multipole where the TE power spectrum first changes sign. Hereafter we will call this method “the zero multipole method”. The TE power spectrum due to density perturbations is positive on large scales, corresponding to , changes sign at , and then oscillates for , while for PGWs the TE power spectrum must be negative for small and then also oscillates for larger . The current best set of cosmological parameters, obtained in Spergel et al. (2007), gives, in the absence of PGWs, . Therefore, the measurement of the difference between the multipole number, , where the TE power spectrum changes sign, and is the way to detect PGWs. We will then consider an alternative method based on Wiener filtering, removing the contribution to the TE power spectrum due to density perturbations. Since the TE power spectrum due to PGWs is megative on large scales a test of negativity of the resulting TE power spectrum is a test of PGWs. In this paper, we present an analysis of both of these methods, based on Monte Carlo simulations.
At the present time, the main priority and the main challenge in CMB polarization observations is the detection of the PGW background via the BB power spectrum. In connection with BB experiments, the methods based on the TE cross correlation can be considered as very useful auxiliary measurements of PGWs because systematic effects in TE measurements are not degenerate with those in BB measurements. For example, T/B leakage or even E/B leakage could swamp a detection of BB, whereas T/E leakage would be small and well controlled (see Shimon et al. (2007)). These BB systematics could falsely indicate a detection of PGWs, but measurements of the TE power spectrum provide insurance against such a spurious detection. Additionally, galactic foreground contamination affects BB and TE in different ways, which enables us to perform powerful cross-checks and subtraction of foregrounds in BB measurements.
Another advantage of TE measurements for experiments which measure a small fraction of the sky, is related to the fact that a significant contaminant to the B modes is caused by E/B mixing. This limits the power spectrum of PGWs that can be detected (Challinor & Chon (2005)). The E-modes are practicly unaffected by E/B mixing so, in contrast to the BB measurements, the TE power spectrum should be nearly the same for both full and partial sky measurements.
The plan of this paper is the following. In Section 2, we introduce the primordial power spectra of scalar (density) and tensor (PGW) perturbations (2.1). Then following Crittenden et al. (1995) and Baskaran et al. (2006), we explain why the sign of the TE power spectra for scalar and tensor perturbations is opposite for large scales (2.2). In Section 3, we describe in more detail the zero multipole method for the detection of PGWs. In Section 4, we describe the method for detection of PGWs based on Wiener filtering along with the statistical tests used and a comparison of the tests. In Section 5, we present results of numerical Monte Carlo simulations for two toy experiments. In the first toy experiment we neglect instrumental noise and the uncertainties are limited only by cosmic variance (5.1). In the second toy experiment, along with cosmic variance, we take into account instrumental noise which is comparable to real noise in current ground experiments (5.2). For comparison, we also present results of simulations for the two satellite experiments, WMAP (5.3) and Planck (5.4). In Section 6, we compare the the signal-to-noise ratio of the TE measurements with those of BB measurements.
2 TE Cross Correlation
The power spectrum of TE correlations is determined by primordial power spectra of scalar and tensor perturbations and time evolution of these perturbations during the epoch of recombination.
2.1 Primordial Power Spectra
The primordial power spectra describing the initial scalar (density) perturbations (denoted by ) and tensor (PGW) perturbations (denoted by ) are (see, for example, Spergel et al. (2007))
where Mpc, this value of is obtained by fitting of CMB data (Smith et al. (2006)). The variables and are the scalar and tensor spectral indices respectively. The variable is the running of the scalar spectral index. In terms of and , the tensor-to-scalar ratio, , is
The location of is determined by the parameters and . In this paper, we do not specify particular cosmological models considering the generation of primordial spectra, and , which means that for our purposes we consider , , and as independent parameters. (This is not true if we use some particular cosmological model. For example, in standard inflation models, the parameters and are related by the consistency relation, (see, for example, Peiris et al. (2003)).) In other words, we consider all parameters , , and as independent except in 5.1 and 5.2, where along with model-independent we give also model-dependent constraints on .
2.2 Opposite Signs of Scalar and Tensor Perturbations to TE Correlation
Taking into account that scalar and tensor perturbations are not correlated, the TE power spectrum is simply a sum of two TE power spectra for scalar and tensor perturbations correspondingly.
First, the physical motivation for the difference in the cross correlation contributions produced by scalar and tensor perturbations for small was demonstrated and physically interpreted for the cross correlation of the Stokes parameters and in Crittenden et al. (1995). For scalar perturbations the Stokes parameter contains only E-modes, hence the TE correlation is identical with the TQ correlation and is positive for small . As was then emphasized in Baskaran et al. (2006), the sign of the TE correlation for tensor perturbations is negative for small . The simple qualitative physical interpretation of the fact that the contributions of the TE correlation are different for scalar and tensor perturbations is the following. For both scalar and tensor perturbations, the temperature fluctuations, , for small (when oscillations of are absent) are proportional to the metric perturbations at the moment of recombination, while the E-mode fluctuations, , are proportional to at the moment of recombination. Hence, the TE correlation is proportional to . Taking into account the growth of scalar perturbations and tensor perturbations decay, one can see that the contributions to the TE correlation for scalar and tensor perturbations are opposite.
To understand this in more detail, following Baskaran et al. (2006), we consider the multipole expansion of the TE cross correlation with coefficients . These coefficients are related to the spherical harmonic expansion coefficients of the temperature anisotropy and polarization by
where the brackets denote averaging over all possible statistical realizations. The statistical properties of the CMB field in general, and the TE cross correlation specifically, follow from the statistical properties of the underlying scalar or tensor metric field. Assuming gaussianity together with statistical isotropy and homogeneity, the TE cross correlation takes the form
where is the contribution from temperature perturbation while is the contribution from E-polarization. The integration over takes into account the contribution from all the possible wavenumbers.
It was shown in Baskaran et al. (2006)
where is the mode function of the metric perturbation, and is the conformal time at recombination. It follows that the TE correlation is approximately
where is a strictly positive function which peaks at . Heuristically, the function projects the -space onto the -space. Therefore the sign of the integral in the RHS of Eq. 7 evaluated at around determines the sign of on large scales.
The adiabatic decrease of the gravitational wave amplitude upon entering the Hubble radius is preceded by the monotonic descrease of the gravitational wave mode function as a function of . Since is decreasing the integral on the RHS of Eq. 7 is negative. The RHS of Eq. 7 is negative for since is decreasing over that range. Therefore, for the correlation must be negative. For larger s, the in Eq. 7 and, hence, the TE cross correlation power spectrum changes sign as a function of .
Thus the TE cross correlation, due to density pertrbations, must be positive at lower (as mentioned above, the TE cross correlation in absence of PGWs changes sign at ). If we were able to separate them we could use this signature for detection of PGWs. However, even without such separation the presence of PGWs manifests itself in the value of , which is the smallest where the total TE correlation power spectrum (scalar plus tensor) changes its sign. Thus, the sign of the TE correlation is a very prominent signature of PGWs. For this reason, in the next section, we investigate the dependance of on , , and .
3 Dependance of on Parameters of PGW Power Spectrum
The method of detecting PGWs which implies a calculation of , where the TE power spectrum first goes to zero, will be called hereafter as the zero multipole method. We take into account uncertainties in determination of s which are unavoidable in any experiment:
(see, for example, Dodelson (2003)). Even in an ideal experiment, when we neglect instrumental noise () and measure the full sky (), we still have uncertainties related with cosmic variance (which arises from the fact that we have only one realization of the sky in CMB measurements) (see, for example, Dodelson (2003)). For a more realistic experiment, we take into account noise and partial sky coverage (see Section 5). Over small multipole bands it is reasonable to approximate the power spectrum as linear. In the range , it seems reasonable to use a linear approximation for . It seems unlikely that in this range any deviations from a linear approximation can be larger than mentioned above uncertianties.
Plots of for different values of are shown in Fig. 1 plotted for . It can be seen that a linear fit to the TE power spectrum do well approximates near .
Thus near , can be approximated as a line with negative slope , where and are positive real numbers. For any set of experimental data, we can find and by applying a least squares fit. The values and corresponding to the best fit obviously can be used for prediction of . This value, , can then be used to constrain the parameter under some assumptions about spectral indices and .
We need to investigate how depends on the cosmological parameters , , , , and the optical depth to reionization, . The value of for a standard CDM cosmology described in Spergel et al. (2007) as a function of and is shown in Fig. 2 and 3. All power spectra were generated with the code CAMB111see http://camb.info on web (Lewis et al. (2000)). If , , while if (the WMAP3 upper limit on the tensor-to-scalar ratio) and , we find that .
From Fig. 2 and 3, one can see that decreases with increase of . This effect is more pronounced for smaller . For example, if , then for , for , and for . The fact that is discrete (the plots are composed of a set of step functions) puts limitations on using this method for determination of . For example, a value of and will most likely correspond to the same and therefore no matter what the sensitivity is this method cannot distinguish between absence of PGWs and PGWs corresponding to such small . For (the Harrison-Zel’dovich scale-free spectrum), corresponds to of . For negative values of , corresponds to smaller . For , for example, requires .
The effect of variations of the scalar spectral index on is opposite: A decrease of with increase of is more pronounced for larger , however we do not need to worry about it because is well constrained by the observations of TT and EE power spectra (see for example Spergel et al. (2007)) along with Ly- measurements (see for example Viel et al. (2004)). Thus everywhere in this paper we use (the value given by WMAP3 Spergel et al. (2007)) with no running of the scalar spectral index, . A change of in the running of the scalar spectral index has no effect in the value of when using Mpc. Even if it did have an effect on the value of , it can be well constrained by the same observations that constrain .
The value of does not depend on , because for fixed , must change by the same factor as leaving unchanged, i.e. any rescaling of the primordial power spectra does not change . The same thing happens when one varies the optical depth to reionization, . The value of is in the range where the TE power spectrum for scalar and tensor perturbations depend on in the same way for instantaneous reionization (they are damped by the factor, , since the relevant scales are within the cosmological horizon at the time of reionization (Dodelson (2003)). Thus any variation of can be considered just as a rescaling of the TE power spectrum which leaves the value of unchanged. It is possible for different reionization histories to cause a change in as shown in Kaplinghat et al. (2003), however we will assume instantaneous reionization for the purpose of this paper.
Thus, even if we cannot separate the contributions of scalar and tensor perturbations to the TE power spectra, PGWs still leave their imprint on the value of . In the next section, we will consider the possibility of such separation with the help of Wiener filtering.
4 Wiener Filtering of the TE Cross Correlation Power Spectrum
Wiener filtering has been used often in the case of CMB data analysis. For example, it was used to combine multi-frequency data in order to remove foregrounds and extract the CMB signal from the observed data (Tegmark & Efstathiou (1996); Bouchet et al. (1999)). Here we examine the use of the Wiener filter to subtract the PGW signal from the total TE correlation signal. This is done because the Wiener filter reduces the contribution of noise in a total signal by comparison with an estimation of the desired noiseless signal (Vaseghi (2006)). In our case, the signal is the one due to PGWs only, and the signal contributed by density perturbations is considered to be “noise”.
The observed signal can be written as
where and refer to the contributions to the power spectrum due to scalar and tensor perturbations respectively. The values and refer to the spherical harmonic coefficients of the temperature and polarization maps. In our application to TE correlation, we consider the Wiener filter, :
The filtered signal, (for and ), is obtained from the measured signal, , as
In this paper, we assume the Wiener filter is perfect, in the sense that it leaves the signal due to PGWs only. We then get, for the filtered multipoles ,
In practice this is not true, because we are trying to determine , which is not known in advance. Nevertheless, the assumption that the Wiener filter is perfect is good as a first approximation and illustrates the detectability of PGWs with the help of TE correlation measurements.
The filtering can reduce the measured signal to the desired signal, but, since we are trying to remove the density perturbations and not the actual noise, we can not reduce the measurement uncertainties. These uncertainties in are then entirely determined by the noise in the original signal.
We have shown that the TE power spectrum due to PGWs is negative on large scales, hence a test determining whether the Wiener filtered power spectrum is negative or not is a probe of PGWs.
There are three different statistical tests we use to see if we can measure a negative TE power spectrum. The first test is a Monte Carlo simulation to determine signal-to-noise ratio, (Section 4.1). The other two tests are standard non-parametric statistical tests: the sign test (Section 4.2) and the Wilcoxon rank sum test (Wilcoxon (1945)) (Section 4.3).
For all of our tests, we calculate a random variable. If the data satisfies the hypothesis that , we can calculate the mean and uncertainty in the variables. If we make one realization of data, the random variable is determined from its distribution. Because we are not using any real observational data, we must run a Monte Carlo simulation to reduce the risk of randomly getting a value for the variable taken from the outlying area of its distribution. To do this, the filtered multipoles, , are randomly chosen from a gaussian distribution with mean and standard deviation where
(see, for example, Dodelson (2003)), the variable refers to the fraction of the sky covered by observations and is the effective power spectrum of the instrumental noise (see Dodelson (2003) for details on how is related to actual instrumental noise).
Our determination of is dependent on . However, for two of our tests we ignore the value of in the calculation of the random variable. We assume that the calculated random variable is gaussian. In order for this to work, the random variable must be calculated from gaussian variables. The errors on the multipoles for the “ideal” toy experiment are large enough so that we can assume the multipoles are taken from a single distribution and not from a distribution that depends on .
4.1 Monte Carlo S/N Test
For this test, the random variable we calculate, , is defined as
The reason why the sum in this equation is taken in the range is because only in this range . In other words, if we include higher multipoles we confront with a danger of a false detection, because the total TE power spectrum is negative for .
The value of is gaussian distributed because it is a sum of many modes of squares of gaussian distributed values, . We approximate each as being gaussian distributed for the purpose of this paper. For each set of parameters we run this simulation one million times to determine the mean, , and standard deviation, . The mean of this distribution is determined by the preassumed value of , while the standard deviation is determined by parameters of the experiment and gives the confidence level of detection. We run such Monte Carlo simulations for different values of to determine in what range of we can detect PGWs. When then using real observational data, we can compare the actual value of with the results of Monte Carlo simulations to infer the likelihood, as function of , which determines the probability that , or that PGWs exist at detectable levels.
4.2 Sign Test
The sign test is a test of compatability of observational data with the hypothesis that . If we do have , then will be equally distributed around zero. Application of this test to the filtered data is very simple. In practice, all observational data are distributed between several bins and the averaging of the signal is produced in each bin separately. Let be the number of such bins. The sign test actually gives the probability that in bins the average is negative and in it is positive, if . This probability, , is given by the binomial distribution
The probability that the hypothesis is wrong is
The value is the probability that we would get positive values given . This is the same as the probability of getting negative values given . Therefore our confidence that is just minus the sum of the probabilities describe above (the probability that the is closer to the mean, , if ). This equation only makes sense if , since that is required for . If , that would imply , which is not physical. We would have to interpret the result as a random realization of , with the most likely result of . Therefore we would not be able to say with any confidence.
Let us consider the following example: we put all measurements of into bins and in three of them the average is positive. In this example, the probability that the hypothesis is wrong is equal to .
One possible drawback of this method is that it does not take into account any measure of the signal-to-noise ratio of individual measurements. As we show in Section 4.4, it is possible to have two completely different sets of data with the same probability of having . This test is also unable to make any prediction as to the value of , only that it differs from zero.
4.3 Wilcoxon Rank Sum Test
This statistical test deals with two sets of data. The first set of data is taken from a real experiment which measures with some unknown . The second set of data is generated by Monte Carlo simulations (see Section 4.1) with . The objective of the Wilcoxon rank sum test is to give the probability that the hypothesis is wrong (Wilcoxon (1945)).
First, we choose some random variable , whose probability distribution is known if . For that, let us combine all data from first set with multipoles and second set with multipoles into one large data set, which obviously contains multipoles. Then, we rank all multipoles in the large data set from to according to their amplitude (rank for the smallest and rank for the largest). Now, the variables and are defined as the sum of the ranks for the first original data set and the second original data set, correspondingly. Finally, the variable , is
If all multipoles of the first data set are larger than all multipoles of the second data set, then and . It is not difficult to show that . If both sets of measurements have no evidence for PGWs, . It is also simple to see that .
It is important to emphasize that the ranks of multipoles are random variables because all multipoles themselves are random variables, hence , , and are random variables. If is large, the distribution of can be approximated as a gaussian with a known mean and standard deviation. In this approximation we have
In some cases, instead of , the variables or are used. The reason is used here is because is symmetric in the data sets. If in both sets of data, then the distributions of and are the same, no matter what and are. The distributions of and would be the same only if . The probability that the first data set corresponds to obtained from the test in which or is used is the same as if is used.
Since this test requires Monte Carlo simulations for the second set of data, we ran this test many times for many different data sets to get an accurate mean value for .
To reject the hypothesis means to detect PGWs. Using the Wilcoxon rank sum test the allowable value of is determined, if instead of comparing with simulated data with , we compare with simulated data with . In order to get a range of allowable values for , we need to run multiple Monte Carlo simulations with multiple values for . This is where the assumption that the are from a random distribution that is independent of is used. This implies that the ranks are random variables. If the errors on the are small enough, then the ranks will be predetermined. Therefore, our assumption about the distribution of will not be true and the test would have to be modified. Fortunately, this is not the case for even an experiment only limited by cosmic variance.
To illustrate how this test works, let us consider the following example. Assume there are multipoles in the first set of data and consider that is the correct value. There are also multipoles in the second set of data (which for sure corresponds to ). All quantities below are expressed in K. The value for the first data set are , , , and . The values for the second data set are , , , and . A ranking of multipoles gives the ordering from lowest to highest, with referring to the first data set and referring to the second data set, as . This results in , , and . Therefore . For , to reject the hypothesis that at confidence level, should be less than one (see, for example, Lehmann (1975)). In this example, since , the first set of data cannot be considered as a detection of PGWs.
4.4 Comparison of Tests
The test is greatly affected by outlying measurements. A measurement of one large negative multipole could falsely imply a detection. Both the sign test and the Wilcoxon rank sum test are not affected by individual outlying measurements. In the sign test, the value of individual measurements is irrelevant, because the test is sensitive only to the sign of individual measurements. The Wilcoxon rank sum test is affected by outliers, but considerably less than the test. If the outlier is larger (or smaller) than every other multipole, its rank does not depend on its particular value.
If we have two completely different sets of data, the main disadvantage of the sign test, as mentioned in Section 4.2, is that it could give the same result, while for the two other tests the chance to obtain the same value of is negligible. For example, one set of data, consisting of small negative multipoles and large positive multipoles, gives the same result as another set of data, consisting of large negative multipoles and small positive multipoles. The test gives two very different values of for these two sets of data. We can also use the Wilcoxon rank sum test to compare these two sets of data. In this case , which corresponds to a confidence level of hypothesis that of less than .
With observational data, the sign test can be applied and does not require any Monte Carlo simulations (which could be considered as an advantage of this test). The test requires Monte Carlo simulations, but only for the distribution of the random variable . The Wilcoxon rank sum test requires large Monte Carlo simulations and combines the data sets generated by these simulations with observational data. In other words, Monte Carlo simulations are absolutely necessary after obtaining observational data, which may be considered a disadvantage of this test. Thus, each of the three tests has advantages and disadvantages, suggesting that the best way to work out observational data is to apply all these three tests.
5 Discussion and Results
Baskaran et al. (2006) used equal amplitudes of scalar and tensor perturbations to sharpen the discussion in their plots. They defined the tensor-to-scalar ratio, , as the ratio of the temperature quadrupoles, and set . Using standard WMAP3 cosmological parameters (Spergel et al. (2007)), the definition of tensor-to-scalar ratio, , used in this paper is approximately twice as large as their definition of . The exact relationship between and will depend on the cosmological parameters used. This means that is equivalent to , which has currently been strongly ruled out by WMAP in combination with previous experiments (Spergel et al. (2007)). We need to see if this method can detect a value of that is currently within the limits. We assume that there is no foreground contamination. In reality foregrounds affect the measured location of (we will consider the effects of foregrounds on elsewhere). For the experiments that do not observe the full sky, correlations between multipoles must be taken into account. The multipoles are binned together of such width that the correlations between the bins are sufficiently small.
Two different toy experiments, along with the two satellite experiments WMAP and Planck, are considered to constrain . The first toy experiment is a full sky experiement. It is idealized in two aspects. The first idealization is that we can take measurements over the full sky while the second idealization is that we assume there is no detector noise. The only uncertainty is due to cosmic variance. Such experiment represents the best limit to which the gravitational waves can be detected by the CMB TE correlation. This toy experiment is close to a space-based experiment with access to the full sky. It is similar to what the Beyond Einstein inflation probe would be able to detect. This toy experiment will be hereafter referred to as the ideal experiment. The second toy experiment is a more realistic one. In this experiment, measurements are on of the sky, the frequency is GHz, and the duration of the experiment is years. The noise in each detector of the polarization sensitive bolometer pairs can be described by their noise equivalent temperature (NET) of . The detectors’ beam profile is assumed to be gaussian and and it is described by the width at half of the maximum sensitivity, abbreviated as FWHM of .
This second toy experiment is similar to current ground-based experiments and the constraints from this experiment represent those that can and will be obtained in the next several years. This will be referred to as the realistic experiment.
The predicted errors for Planck are based on using the GHz, GHz, and GHz channel in the High Frequency Instrument (HFI). The numbers are gotten from the Planck science case, the “bluebook”222http://www.rssd.esa.int/index.php?project=Planck. The WMAP noise was obtained by years of observations of the Q-band, V-band, and W-band detectors.
5.1 Ideal Experiment
A plot of TE for and with error bars for binned in bins of width is shown in Fig. 4. This figure separately shows the contribution to the TE mode of density perturbations, contribution of PGWs with , when the TE power spectrum due to density perturbations is approximately five times larger than the power spectrum due to PGWs at .
The Monte Carlo simulation for the calculation of , with an input model of and , results in the value of with an uncertainty of . A contour plot of the limits on the resulting measurement of is shown in Fig. 5. The white is the allowed region for and that falls within the errors of . The black is the region forbidden with confidence. If , then we measure . If we consider the inflationary consistency relation, (Peiris et al. (2003)), we then get the constraint . The uncertainty is smaller, but not by much. We predict a detection of PGWs by the zero multipole method.
The detectability of using the ideal experiment is shown in Fig. 6. For the effective number of detection is . We make this approximation by determining the detectability for several values of and then approximating a line. For comparison the results are also shown for the zero-multipole method with the realistic experiment and for measurements of the BB power spectrum with the realistic experiment described above. We assume we can make measurements over a range of multipoles for BB measurements.
The Monte Carlo simulation for the Wiener filtering gives an average of measured TE power spectrum multipoles greater than zero out of a total of independent multipoles. If the null hypothesis was true, the sign test would indicate there is a chance of measuring positive multipoles. This is equivalent to a detection. A plot of the distribution of the number of positive multipoles is shown in the upper panel plot of Fig. 7. There is an chance for the observed to give a detection of PGWs.
The test gives a mean value of and standard deviation of . The upper left panel in Fig. 8 shows the distribution of the values for the Monte Carlo simulation with . If we would have a probability of the measured . This negative value signifies that a non-zero tensor-to-scalar ratio produced an anti-correlation. We can assume that the standard deviation would be the same if the mean of was (equivalent to ), because it is equivalent to adding a constant value to every measured value (and hence adding a constant to which would not change the error). Therefore, if , the probability of getting is , and hence we have a chance that . A plot of and as a function of is shown in Fig. 9. As can be seen from the plot, we can predict a value of for any value of . The value of is a relatively constant function of and so our prediction about the distribution of for different value of is a good approximation to the true distribution.
The Wilcoxon rank sum test gives . The variable is the mean value for in the Monte Carlo simulations described earlier. The values and are given in Section 4.3. The distribution of for the Monte Carlo simulations with is shown in Fig. 10. The standard deviation of the distribution of measured is the same as the standard deviation of the distribution of assuming the hypothesis that . The only difference between the distributions is that is shifted by a constant value. Therefore, there is a chance that . There is also a chance that we measure , and are not even able to make a detection of PGWs.
A comparison of the three tests is shown in Fig. 11. This is obtained by simulated with with several values of and then interpolating between them. A detection is obtained for ( test), (sign test), and (Wilcoxon rank sum test), highlighting its intended use as a monitor of a false positive detection for large .
5.2 Realistic Ground Based Experiment
A plot of the error bars for the realistic experiment is shown in Fig. 12 with . Observations on an incomplete sky require the multipoles to be binned in sizes of . This experiment has much larger error bars than the ideal experiment and it is not able to detect low values of with the TE cross-correlation only. Plots of the TE power spectrum due to density perturbations and PGWs are shown in Fig. 12 along with the combined TE power spectrum.
For this experiment, the constraints on measuring are significantly larger than those for the ideal experiment. The uncertainty on is . This corresponds to a limit of with confidence. If we want a limit, then the constraint expands to . If we assume the inflationary consistency relation, then this error on would correspond to a upper limit of about . Fig. 13 shows the region of and allowed with confidence of .
As mentioned earlier, Fig. 6 shows the signal-to-noise ratio of the zero multipole method for the realistic experiment. The measurements of the BB power spectrum are much more sensitive to PGWs and the sensitivity is roughly the same as in the ideal experiment. This shows that the zero multipole method is less sensitive to PGWs than measurements of the BB power spectrum.
The results for the Wiener filtering method are much worse than those for the ideal experiment for . Since this experiment observes a small portion of the sky, the multipoles are correlated and we must bin together to get reasonably uncorrelated measurements. For this experiment, we only have to uncorrelated multipoles, instead of uncorrelated multipoles in the case where the full sky is observed. Getting out of negative multipoles is a probability if there are no PGWs. For the Monte Carlo simulations of the realistic experiment, on average, half of measured multipoles are positive and half are negative. A plot of the distribution of the number of positive multipoles is shown in the upper right panel of Fig. 7. In this case, we cannot distinguish from with any significance.
The test gives an average value of with standard deviation of . For the realistic toy experiment, the distribution of for is shown in the upper right panel of Fig. 8. In order to obtain confidence detection of PGWs, we must use . In this sense the TE test provides monitoring and insurance against false positive detection with , which could arise, for example, if foregrounds or other systematic effects arer improperly removed.
The last statistical test, the Wilcoxon rank sum test, gives . The distribution of for is shown in the upper right panel of Fig. 10. This gives the weakest result in terms of the three tests for the Wiener filtered data. The realistic experiment will not be able to constrain using the TE cross correlation power spectrum. Its limit is closer to at only confidence depending on the test used. For a higher confidence in a detection of PGWs, the value of would need to be much higher. Since the observed distribution of corresponds almost exactly to the simulated distribution of under the assumption that , therefore we have a chance of measuring .
A constraint on using a measurement of for WMAP is almost impossible. Using error bars consistent with WMAP noise, we get for an input of and . The published results of WMAP give limits of so adding this method to the WMAP results would not change constraints significantly. In fact, using the real WMAP data333http://lambda.gsfc.nasa.gov/ we get . With an uncertainty of , the probability of getting a value farther away from is larger than , so we cannot detect primordial gravitational waves in the published WMAP data using the zero multipole method.
The results of the Wiener filtering showed that the WMAP cannot make a detection of gravitational waves using the TE cross correlation power spectrum alone. As with the two toy experiments, the result of the scalar and tensor separation was similar. The Monte Carlo simulation gave on average gave positive multipoles out of a total of uncorrelated multipoles. We would get the same result if the input data had so we cannot detect PGWs with WMAP using only the TE power spectrum. A plot of the distribution of the number of positive multipoles is shown in the lower right panel of Fig. 7. As can be seen, this distribution of for WMAP noise and is simply the distribution for .
For WMAP, the S/N test gives the value of with a standard deviation of . The distribution is shown in the lower right panel of Fig. 8. The distribution is centered around so there is no chance of using this test to detect PGWs in WMAP’s TE power spectrum. The probability of getting a or detection is the same probability that we would randomly get a detection if there are no PGWs.
The rank sum test gives a value of , which is implies no ability to distinguish WMAP’s observed TE data from a data set with no PGWs. A plot of the distribution of for WMAP error bars is shown in the lower right panel of Fig. 10. We reach the same conclusion for WMAP noise as for the realistic experiment. There is only a chance that we can measure and make a detection of
The published WMAP results show an anti-correlation of TE power spectrum at large scales. Unfortunately this is not a detection of PGWs as theorized in Baskaran et al. (2006). The contribution to the TE power spectrum due to PGWs only changes sign once for . If a claimed evidence for gravitational waves is to be believed, then the TE power spectrum would have to change sign three times for . In fact, other than the two anticorrelations at low , the rest of the multipoles, up to , are consistent with . None of the described tests applied to the current WMAP data will give any detection of PGWs.
The uncertainty in is much better for Planck than for the realistic experiment and about twice as large for the ideal experiment. The Monte Carlo simulations resulted in for an input TE power spectrum with and . This results in confidence that , under the assumption that .
The sign test gives on average positive measurements of the TE power spectrum out of a total of uncorrelated multipoles. There is a chance of getting positive multipoles if . A plot of the distribution of the number of positive multipoles for Planck is shown in the lower left panel of Fig. 7. There is a chance that we will measure and hence have a detection of .
The test gives a value of with a standard deviation of . There is only a chance that the test results in a value of larger than zero, if , and a chance getting if . This is close to a probability of detection. The distribution of the variable is shown in lower left panel of Fig. 8.
Again, the rank sum test gives the lowest confidence result with a value of . A plot of the distribution of is shown in the lower left panel of Fig. 10. There is a probability that we will measure and a probability that we measure for Planck.
6 Comparison of Measurements of the TE Power Spectrum with the BB Power Spectrum
As mentioned earlier, it was originally suggested that it might be easier to detect PGWs using the TE power spectrum instead of the BB power spectrum. For both methods, this turned out not to be true. The reason for this is because we are trying to measure the TE power spectrum at the place where the signal is lowest (). In measurements of the BB power spectrum, if we neglect instrumental noise, the signal decreases with a decrease in and so does the cosmic variance limited uncertainty. This is not the case for the TE power spectrum. The uncertainty in the measurement of the TE power spectrum due to PGWs is determined by the total TE, TT, and EE power spectra. When the TE power spectrum goes to zero, the TT and EE power spectrum do not approach zero (in fact, they increase as we approach to ). We therefore have a low signal-to-noise ratio around making it very hard to detect PGWs using the zero multipole method. Below we give simple summarizing arguments why the same is true for the Wiener filtering of the TE power spectrum
If , the signal-to-noise ratio for the BB power spectrum is
If then we will not be able to detect PGWs and a comparison with the TE power spectrum is not worthwhile.
If and , for the TE power spectrum, the signal-to-noise ratio is
where and are
One can see that and are on the order of unity. Therefore, the signal-to-noise ratio is approximated as
In other words if , BB measurements have the obvious advantage in comparison with the Wiener filtering of the TE power spectrum. Indeed if , , while . This is because in BB measurements, applying proper data analysis, we can entirely eliminate contributions of scalar perturbations to CMB polarization signal as well as to the uncertainties. For the perfect Wiener filtering of the TE power spectrum, we can eliminate the contribution of scalar perturbations to the signal only, but cannot eliminate their contribution to the uncertainties.
The measurement of where the TE cross correlation first changes sign can be used to detect or put constraints on PGWs. Such constraints are not as strong as the ones given by measurements of the BB power spectrum, however it is useful to have a supplementary method to detect PGWs. We have shown how well the TE mode can constrain the amount of PGWs from just a measurement of the angular scale where it first changes sign for two different toy experiments and two real satellite experiments. The absolute best limit with which we can measure only gives us less than a detection of the PGW component if . The current confidence limits gives us at confidence level. Current and future experiments are optimized to measure the BB power spectrum if even in the presence of foregrounds, which are not taken into account in this paper. Future satellite experiments should be able to detect which is times better than the sensitivity to than the result of the ideal experiment. If one neglects even cosmic variance, the discreteness of limits the calculation of , and the sensitivity to , to values considerably larger than . The cosmic variance is largest at low and is proportional to the total power spectrum. Since the TE cross correlation has contributions from density perturbations the errors in the measured TE power spectrum make detecting deviations of from difficult, though they also provide insurance against a false detection or imperfect subtraction of instrumental and foreground systematic effects.
The other method described in this paper is one in which we filter out the signal due to density perturbations, leaving only the contribution to the TE power spectrum due to PGWs. We then test the resulting TE power spectrum to see if it is negative. Three different statistical tests were used to see if there was a significant detection of PGWs. The test can give a value for using a comparison with Monte Carlo simulations, while the Wilcoxon rank sum test can only give an allowable range for . The sign test will only tell us if .
Using the Wiener filtering method, we are unable to make as significant of a detection as using the zero multipole method. The best result was for the test which would give a detection of . To detect PGWs on the level of , the tensor-to-scalae ratio should be . The sign test would give detection for and a detection for . The Wilcoxon ranked sum test gives only a detection for and a detection for . Similar results were gotten for the other three experiments tested. Thus in the sense of potential to detect PGWs, the zero multipole method is the best, next best is the test, then the sign test, and the worst is the Wilcoxon ranked sum test.
Baskaran et al. (2006) present illustrative examples in which high is consistent with measured TT, EE, and TE correlations. The value of is so high in these examples that if PGWs with such really existed, current BB experiments would already detect PGWs. All models predict that the TE cross correlation power spectrum change sign only once for . The fact WMAP cannot exclude several multipoles with in between multipoles of means that the TE cross correlation power spectrum either changes sign several times for or there is some instrumental noise which causes some anticorrelation measurements. Using instrumental noise consistent with WMAP, our Monte Carlo simulations give and , which means that there is no evidence of PGWs in the TE correlation power spectrum.
AGP would like to thank the Center for Astrophysics and Space Sciences at UCSD for hosting him while working on parts of this paper and Deepak Baskaran and Leonid Grishchuk for helpful discussions. NJM would like to thank the Astronomy Unit, School of Mathematical Science at Queen Mary, University of London for hosting him while working on parts of this paper. BGK gratefully acknowledges support from NSF PECASE Award AST-0548262. We acknowledge helpful comments on this manuscript by Kim Griest and Manoj Kaplinghat. We acknowledge using CAMB to calculate the power spectra in this work.
- Baskaran et al. (2006) Baskaran D., Grishchuk L. P., Polnarev A. G., 2006, Phys. Rev. D, 74, 083008
- Basko & Polnarev (1980) Basko M. M., Polnarev A. G., 1980, MNRAS, 191, 207
- Bouchet et al. (1999) Bouchet F. R., Prunet S., Sethi S. K., 1999, MNRAS, 302, 663
- Bowden et al. (2004) Bowden M., et al., 2004, MNRAS, 349, 321
- Challinor & Chon (2005) Challinor A., Chon G., 2005, MNRAS, 360, 509
- Coles et al. (1995) Coles P., Frewin R. A., Polnarev A. G., 1995, LNP Vol. 455: Birth of the Universe and Fundamental Physics, 455, 273
- Crittenden et al. (1993) Crittenden R., Davis R. L., Steinhardt P. J., 1993, ApJL, 417, L13+
- Crittenden et al. (1995) Crittenden R. G., Coulson D., Turok N. G., 1995, Phys. Rev. D, 52, 5402
- Dodelson (2003) Dodelson S., 2003, Modern cosmology. Modern cosmology / Scott Dodelson. Amsterdam (Netherlands): Academic Press. ISBN 0-12-219141-2, 2003, XIII + 440 p.
- Frewin et al. (1994) Frewin R. A., Polnarev A. G., Coles P., 1994, MNRAS, 266, L21+
- Grishchuk (2007) Grishchuk L. P., 2007, arXiv:0707.3319 [astro-ph], 707
- Kamionkowski & Kosowsky (1998) Kamionkowski M., Kosowsky A., 1998, Phys. Rev. D, 57, 685
- Kamionkowski et al. (1997) Kamionkowski M., Kosowsky A., Stebbins A., 1997, Physical Review Letters, 78, 2058
- Kaplinghat et al. (2003) Kaplinghat M., Chu M., Haiman Z., Holder G. P., Knox L., Skordis C., 2003, ApJ, 583, 24
- Keating et al. (2006) Keating B. G., Polnarev A. G., Miller N. J., Baskaran D., 2006, International Journal of Modern Physics A, 21, 2459
- Lehmann (1975) Lehmann E. L., 1975, Nonparametric Statistical Methods Based on Ranks. McGraw-Hill
- Lewis et al. (2000) Lewis A., Challinor A., Lasenby A., 2000, ApJ, 538, 473
- Peiris et al. (2003) Peiris H. V., et al., 2003, ApJS, 148, 213
- Polnarev (1985) Polnarev A. G., 1985, SvA, 29, 607
- Seljak (1997) Seljak U., 1997, ApJ, 482, 6
- Seljak & Zaldarriaga (1997) Seljak U., Zaldarriaga M., 1997, Physical Review Letters, 78, 2054
- Shimon et al. (2007) Shimon M., Keating B., Ponthieu N., Hivon E., 2007, arXiv:0709.1513 [astro-ph]
- Smith et al. (2006) Smith T. L., Kamionkowski M., Cooray A., 2006, Phys. Rev. D, 73, 023504
- Spergel et al. (2007) Spergel D. N., et al., 2007, ApJS, 170, 377
- Spergel & Zaldarriaga (1997) Spergel D. N., Zaldarriaga M., 1997, Phys. Rev. Letters, 79, 2180
- Taylor et al. (2004) Taylor A. C., Challinor A., Goldie D., Grainge K., Jones M. E., Lasenby A. N., Withington S., Yassin G., Gear W. K., Piccirillo L., Ade P., Mauskopf P. D., Maffei B., Pisano G., 2004, arXiv:astro-ph/0407148
- Tegmark & Efstathiou (1996) Tegmark M., Efstathiou G., 1996, MNRAS, 281, 1297
- Vaseghi (2006) Vaseghi S. V., 2006, Advanced Digital Signal Processing and Noise Reduction. John Wiley & Sons
- Viel et al. (2004) Viel M., Weller J., Haehnelt M. G., 2004, MNRAS, 355, L23
- Wilcoxon (1945) Wilcoxon F., 1945, Biometrics Bulletin, 1, 80
- Yoon et al. (2006) Yoon K. W., et al., 2006, in Millimeter and Submillimeter Detectors and Instrumentation for Astronomy III. Edited by Zmuidzinas, Jonas; Holland, Wayne S.; Withington, Stafford; Duncan, William D.. Proceedings of the SPIE, Volume 6275, pp. 62751K (2006). Vol. 6275 of Presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference, The Robinson Gravitational Wave Background Telescope (BICEP): a bolometric large angular scale CMB polarimeter