Cosmic Microwave Background Bispectrum Research Articles

CMB bispectrum, trispectrum, non-Gaussianity, and the Cramer-Rao bound

Minimum-variance estimators for the parameter ${f}_{\mathrm{nl}}$ that quantifies local-model non-Gaussianity can be constructed from the cosmic microwave background (CMB) bispectrum (three-point function) and also from the trispectrum (four-point function). Some have suggested that a comparison between the estimates for the values of ${f}_{\mathrm{nl}}$ from the bispectrum and trispectrum allow a consistency test for the model. But others argue that the saturation of the Cramer-Rao bound---which gives a lower limit to the variance of an estimator---by the bispectrum estimator implies that no further information on ${f}_{\mathrm{nl}}$ can be obtained from the trispectrum. Here, we elaborate the nature of the correlation between the bispectrum and trispectrum estimators for ${f}_{\mathrm{nl}}$. We show that the two estimators become statistically independent in the limit of large number of CMB pixels, and thus that the trispectrum estimator does indeed provide additional information on ${f}_{\mathrm{nl}}$ beyond that obtained from the bispectrum. We explain how this conclusion is consistent with the Cramer-Rao bound. Our discussion of the Cramer-Rao bound may be of interest to those doing Fisher-matrix parameter-estimation forecasts or data analysis in other areas of physics as well.

Odd-parity cosmic microwave background bispectrum

Measurement of the cosmic microwave background (CMB) bispectrum, or three-point correlation function, has now become one of the principle efforts in early-Universe cosmology. Here we show that there is a odd-parity component of the CMB bispectrum that has been hitherto unexplored. We argue that odd-parity temperature-polarization bispectra can arise, in principle, through weak lensing of the CMB by chiral gravitational waves or through cosmological birefringence, although the signals will be small even in the best-case scenarios. Measurement of these bispectra requires only modest modifications to the usual data-analysis algorithms. They may be useful as a consistency test in searches for the usual bispectrum and to search for surprises in the data.

Publisher’s Note: Cosmic microwave background bispectrum of vector modes induced from primordial magnetic fields [Phys. Rev. D82, 121302(R) (2010)

We calculate the cosmic microwave background bispectrum of vector modes induced from primordial magnetic fields. We take into account the full angular dependence of the bispectrum and discuss the amplitude and also the shape of the bispectrum in $\ell$ space. In the squeezed limit as $\ell_1 = \ell_2 \gg \ell_3$, we estimate a typical values of the normalized reduced bispectrum as $\ell_1(\ell_1 + 1)\ell_3(\ell_3+1)|b_{\ell_1\ell_2\ell_3}| \sim 2 \times 10^{-19}$, for the strength of the primordial magnetic field smoothed on $1 {\rm Mpc}$ scale $B_{1 \rm Mpc} = 4.7 \rm nG$ assuming a nearly scale-invariant spectrum of magnetic fields. We find that a constraint on the magnetic field strength will be roughly estimated as $B_{1 \rm Mpc} < 10 {\rm nG}$ if PLANCK places a limit on the nonlinearity parameter of local-type configuration as $|f^{\rm local}_{\rm NL}| < 5$.

Cosmic microwave background bispectrum of vector modes induced from primordial magnetic fields

We calculate the cosmic microwave background bispectrum of vector modes induced from primordial magnetic fields. We take into account the full angular dependence of the bispectrum and discuss the amplitude and also the shape of the bispectrum in $\ensuremath{\ell}$ space. In the squeezed limit as ${\ensuremath{\ell}}_{1}={\ensuremath{\ell}}_{2}\ensuremath{\gg}{\ensuremath{\ell}}_{3}$, we estimate a typical values of the normalized reduced bispectrum as ${\ensuremath{\ell}}_{1}({\ensuremath{\ell}}_{1}+1){\ensuremath{\ell}}_{3}({\ensuremath{\ell}}_{3}+1)|{b}_{{\ensuremath{\ell}}_{1}{\ensuremath{\ell}}_{2}{\ensuremath{\ell}}_{3}}|\ensuremath{\sim}2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}19}$, for the strength of the primordial magnetic field smoothed on 1 Mpc scale ${B}_{1\text{ }\mathrm{Mpc}}=4.7\text{ }\text{ }\mathrm{nG}$ assuming a nearly scale-invariant spectrum of magnetic fields. We find that a constraint on the magnetic field strength will be roughly estimated as ${B}_{1\text{ }\mathrm{Mpc}}l10\text{ }\text{ }\mathrm{nG}$, if PLANCK places a limit on the nonlinearity parameter of local-type configuration as $|{f}_{\mathrm{NL}}^{\mathrm{local}}|l5$.

Primordial magnetic field limits from cosmic microwave background bispectrum of magnetic passive scalar modes

Primordial magnetic fields lead to non-Gaussian signals in the cosmic microwave background (CMB) even at the lowest order, as magnetic stresses and the temperature anisotropy they induce depend quadratically on the magnetic field. In contrast, CMB non-Gaussianity due to inflationary scalar perturbations arises only as a higher order effect. Apart from a compensated scalar mode, stochastic primordial magnetic fields also produce scalar anisotropic stress that remains uncompensated till neutrino decoupling. This gives rise to an adiabaticlike scalar perturbation mode that evolves passively thereafter (called the passive mode). We compute the CMB reduced bispectrum (${B}_{{L}_{1}{L}_{2}{L}_{3}}$) induced by this passive mode, sourced via the Sachs-Wolfe effect, on large angular scales. For any configuration of bispectrum, taking a partial sum over mode-coupling terms, we find a typical value of ${l}_{1}({l}_{1}+1){l}_{3}({l}_{3}+1){b}_{{l}_{1}{l}_{2}{l}_{3}}\ensuremath{\sim}6\ensuremath{-}9\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}16}$, for a magnetic field of ${B}_{0}\ensuremath{\sim}3\text{ }\text{ }\mathrm{nG}$, assuming a nearly scale-invariant magnetic spectrum. We also evaluate, in full, the bispectrum for the squeezed collinear configuration over all angular mode-coupling terms and find ${l}_{1}({l}_{1}+1){l}_{3}({l}_{3}+1){b}_{{l}_{1}{l}_{2}{l}_{3}}\ensuremath{\approx}\ensuremath{-}1.4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}16}$. These values are more than $\ensuremath{\sim}{10}^{6}$ times larger than the previously calculated magnetic compensated scalar mode CMB bispectrum. Observational limits on the bispectrum from WMAP7 data allow us to set upper limits of ${B}_{0}\ensuremath{\sim}2\text{ }\text{ }\mathrm{nG}$ on the present value of the cosmic magnetic field of primordial origin. This is over 10 times more stringent than earlier limits on ${B}_{0}$ based on the compensated mode bispectrum.

O-V-S-Z AND FRIENDS: NON-GAUSSIANITY FROM INHOMOGENEOUS REIONIZATION

We calculate the cosmic microwave background (CMB) bispectrum due to inhomogeneous reionization. We calculate all the terms that can contribute to the bispectrum that are products of first-order terms on all scales in conformal Newtonian gauge. We also correctly account for the de-correlation between the matter density and initial conditions using perturbation theory up to third order. We find that the bispectrum is of local type as expected. For a reasonable model of reionization, in which the universe is completely ionized by redshift zri ∼ 8 with optical depth to the last scattering surface τ0 = 0.087, the signal-to-noise ratio (S/N) for detection of the CMB temperature bispectrum is S/N ∼ 0.1 and confusion in the estimation of primordial non-Gaussianity is fNL ∼ −0.1. For an extreme model with zri ∼ 12.5 and τ0 = 0.14, we get S/N ∼ 0.5 and fNL ∼ −0.2.

Non-Gaussianity and the CMB bispectrum: Confusion between primordial and lensing-Rees-Sciama contribution?

We revisit the predictions for the expected cosmic microwave background bispectrum signal from the cross-correlation of the primary-lensing-Rees-Sciama signal; we point out that it can be a significant contaminant to the bispectrum signal from primordial non-Gaussianity of the local type. This non-Gaussianity, usually parametrized by the non-Gaussian parameter ${f}_{NL}$, arises, for example, in multifield inflation. In particular both signals are frequency-independent, and are maximized for nearly squeezed configurations. While their detailed scale-dependence and harmonic imprints are different for generic bispectrum shapes, we show that, if not included in the modeling, the primary-lensing-Rees-Sciama contribution yields an effective ${f}_{NL}$ of 10 when using a bispectrum estimator optimized for local non-Gaussianity. Considering that expected $1\mathrm{\text{\ensuremath{-}}}\ensuremath{\sigma}$ errors on ${f}_{NL}$ are $&lt;10$ from forthcoming experiments, we conclude that the contribution from this signal must be included in future constraints on ${f}_{NL}$ from the cosmic microwave background bispectrum.

CMB temperature bispectrum induced by cosmic strings

The cosmic microwave background (CMB) bispectrum of the temperature anisotropies induced by a network of cosmic strings is derived for small angular scales, under the assumption that the principal cause of temperature fluctuations is the Gott-Kaiser-Stebbins effect. We provide analytical expressions for all isosceles triangle configurations in Fourier space. Their overall amplitude is amplified as the inverse cube of the angle and diverges for flat triangles. The isosceles configurations generically lead to a negative bispectrum with a power-law decay ${\ensuremath{\ell}}^{\ensuremath{-}6}$ for large multipole $\ensuremath{\ell}$. However, collapsed triangles are found to be associated with a positive bispectrum whereas the squeezed triangles still exhibit negative values. We then compare our analytical estimates to a direct computation of the bispectrum from a set of 300 statistically independent temperature maps obtained from Nambu-Goto cosmic string simulations in a Friedmann-Lema\^{\i}tre-Robertson-Walker universe. We find good agreement for the overall amplitude, the power-law behavior, and the angle dependency of the various triangle configurations. At $\ensuremath{\ell}\ensuremath{\sim}500$ the cosmic string Gott-Kaiser-Stebbins effect contributes approximately the same equilateral CMB bispectrum amplitude as an inflationary model with $|{f}_{\mathrm{NL}}^{\mathrm{loc}}|\ensuremath{\simeq}{10}^{3}$, if the strings contribute about 10% of the temperature power spectrum at $\ensuremath{\ell}=10$. Current bounds on ${f}_{\mathrm{NL}}$ are not derived using cosmic string bispectrum templates, and so our ${f}_{\mathrm{NL}}$ estimate cannot be used to derive bounds on strings. However it does suggest that string bispectrum templates should be included in the search of CMB non-Gaussianities.

Cosmic Microwave Background Bispectrum from Primordial Magnetic Fields on Large Angular Scales

Primordial magnetic fields lead to non-Gaussian signals in the cosmic microwave background (CMB) even at the lowest order, as magnetic stresses and the temperature anisotropy they induce depend quadratically on the magnetic field. In contrast, CMB non-Gaussianity due to inflationary scalar perturbations arises only as a higher-order effect. We propose a novel probe of stochastic primordial magnetic fields that exploits the characteristic CMB non-Gaussianity that they induce. We compute the CMB bispectrum (b(l1l2l3)) induced by such fields on large angular scales. We find a typical value of l1(l1 + 1)l3(l3 + 1)b(l1l2l3) approximately 10(-22), for magnetic fields of strength B0 approximately 3 nG and with a nearly scale invariant magnetic spectrum. Observational limits on the bispectrum allow us to set upper limits on B0 approximately 35 nG.

The radiative transfer at second order: a full treatment of the Boltzmann equation with polarization

This paper investigates the full Boltzmann equation up to second order in the cosmological perturbations. Describing the distribution of polarized radiation by a tensor-valued distribution function, we study the gauge dependence of the distribution function and summarize the construction of the gauge-invariant distribution function. The Liouville operator which describes the free streaming of electrons, and the collision term which describes the scattering of photons on free electrons are computed up to second order. Finally, the remaining dependence in the direction of the photon momentum is handled by expanding in projected symmetric trace-free multipoles and also in the more commonly used normal modes components. The results obtained remain to be used for computing numerically the contribution in the cosmic microwave background bi-spectrum which arises from the evolution of second-order perturbations, in order to disentangle the primordial non-Gaussianity from the one generated by the subsequent nonlinear evolution.

Crinkles in the last scattering surface: Non-Gaussianity from inhomogeneous recombination

The perturbations in the electron number density during recombination contributes to the Cosmic Microwave Background bispectrum through second order terms. Perturbations in the electron density can be a factor of ~5 larger than the baryon density fluctuations on large scales as shown in the calculations by Novosyadlyj. This raises the possibility that the contribution to bispectrum arising from perturbations in the optical depth may be non-negligible. We calculate this bispectrum and find it to peak for squeezed triangles and of peak amplitude of the order of primordial non-Gaussianity of local type with fNL of 0.05 ~ -1 depending on the l-modes being considered. This is because the shape of the bispectrum is different from the primordial one although it peaks for squeezed configurations, similar to the local type primordial non-Gaussianity.

Weak lensing of the primary CMB bispectrum

The bispectrum of cosmic microwave background (CMB) anisotropies is a well-known probe of the non-Gaussianity of primordial perturbations. Just as the intervening large-scale structure modifies the CMB angular power spectrum through weak gravitational lensing, the CMB primary bispectrum generated at the last scattering surface is also modified by lensing. We discuss the lensing modification to the CMB bispectrum and show that lensing leads to an overall decrease in the amplitude of the primary bispectrum at multipoles of interest between 100 and 2000 through additional smoothing introduced by lensing. Since weak lensing is not accounted for in current estimators of the primordial non-Gaussianity parameter, the existing measurements of ${f}_{\mathrm{NL}}$ of the local model with WMAP out to ${l}_{\mathrm{max}}\ensuremath{\sim}750$ is biased low by about 6%. For a high resolution experiment such as Planck, the lensing modification to the bispectrum must be properly included when attempting to estimate the primordial non-Gaussianity or the bias will be at the level of 30%. For Planck, weak lensing increases the minimum detectable value for the non-Gaussianity parameter of the local type ${f}_{\mathrm{NL}}$ to 7 from the previous estimate of about 5 without lensing. The minimum detectable value of ${f}_{\mathrm{NL}}$ for a cosmic variance limited experiment is also increased from less than 3 to $\ensuremath{\sim}5$.

Impact of secondary non-Gaussianities on the search for primordial non-Gaussianity with CMB maps

When constraining the primordial non-Gaussianity parameter ${f}_{\mathrm{NL}}$ with cosmic microwave background anisotropy maps, the bias resulting from the covariance between primordial non-Gaussianity and secondary non-Gaussianities to the estimator of ${f}_{\mathrm{NL}}$ is generally assumed to be negligible. We show that this assumption may not hold when attempting to measure the primordial non-Gaussianity out to angular scales below a few tens arcminutes with an experiment like Planck, especially if the primordial non-Gaussianity parameter is around the minimum detectability level with ${f}_{\mathrm{NL}}$ between 5 and 10. In the future, it will be necessary to jointly estimate the combined primordial and secondary contributions to the cosmic microwave background bispectrum and establish ${f}_{\mathrm{NL}}$ by properly accounting for the confusion from secondary non-Gaussianities.

Primordial non-Gaussianity and the CMB bispectrum

We present a new formalism, together with efficient numerical methods, to directly calculate the cosmic microwave background (CMB) bispectrum today from a given primordial bispectrum using the full linear radiation transfer functions. Unlike previous analyses which have assumed simple separable Ans\atze for the bispectrum, this work applies to a primordial bispectrum of almost arbitrary functional form, for which there may have been both horizon-crossing and superhorizon contributions. We employ adaptive methods on a hierarchical triangular grid and we establish their accuracy by direct comparison with an exact analytic solution, valid on large angular scales. We demonstrate that we can calculate the full CMB bispectrum to greater than 1% precision out to multipoles $ll1800$ on reasonable computational time scales. We plot the bispectrum for both the superhorizon (``local'') and horizon-crossing (``equilateral'') asymptotic limits, illustrating its oscillatory nature which is analogous to the CMB power spectrum.

Optimal estimation of non-Gaussianity

We systematically analyze the primordial non-Gaussianity estimator used by the Wilkinson Microwave Anisotropy Probe (WMAP) science team with the basic ideas of estimation theory in order to see if the limited Cosmic Microwave Background (CMB) data is being optimally utilized. The WMAP estimator is based on the implicit assumption that the CMB bispectrum, the harmonic transform of the three-point correlation function, contains all of the primordial non-Gaussianity information in a CMB map. We first demonstrate that the Signal-to-Noise (S/N) of an estimator based on CMB three-point correlation functions is significantly larger than the S/N of any estimator based on higher-order correlation functions; justifying our choice to focus on the three-point correlation function. We then conclude that the estimator based on the three-point correlation function, which was used by WMAP, is optimal, meaning it saturates the Cramer-Rao Inequality when the underlying CMB map is nearly Gaussian. We quantify this restriction by demonstrating that the suboptimal character of our estimator is proportonal to the square of the fiducial non-Gaussianity, which is already constrained to be extremely small, so we can consider the WMAP estimator to be optimal in practice. Our conclusions do not depend on the form of the primordial bispectrum, only on the observationally established weak levels of primordial non-Gaussianity.

Cosmic microwave background constraints on dark energy dynamics: Analysis beyond the power spectrum

We consider the distribution of the non-Gaussian signal induced by weak lensing on the primary total intensity cosmic microwave background (CMB) anisotropies. Our study focuses on the three point statistics exploiting a harmonic analysis based on the CMB bispectrum. By considering the three multipoles as independent variables, we reveal a complex structure of peaks and valleys determined by the reprojection of the primordial acoustic oscillations through the lensing mechanism. We study the dependence of this system on the expansion rate at the epoch in which the weak lensing power injection is relevant, probing the dark energy equation of state at redshift corresponding to the equivalence with matter or higher (${w}_{\ensuremath{\infty}}$). The variation of the latter quantity induces a geometrical feature affecting distances and growth rate of linear perturbations, acting coherently on the whole set of bispectrum coefficients, regardless of the configuration of the three multipoles. We evaluate the impact of the bispectrum observable on the CMB capability of constraining the dark energy dynamics. We perform a maximum likelihood analysis by varying the dark energy abundance, the present equation of state ${w}_{0}$ and ${w}_{\ensuremath{\infty}}$. We show that the projection degeneracy affecting a pure power spectrum analysis in total intensity is broken if the bispectrum is taken into account. For a Planck-like experiment, assuming nominal performance, no foregrounds or systematics, and fixing all the parameters except ${w}_{0}$, ${w}_{\ensuremath{\infty}}$ and the dark energy abundance, a percent and ten percent precision measure of ${w}_{0}$ and ${w}_{\ensuremath{\infty}}$ is achievable from CMB data only. The reason is the enhanced sensitivity of the weak lensing signal to the behavior of the dark energy at high redshifts, which compensates the reduced signal to noise ratio with respect to the primary anisotropies. These results indicate that the detection of the weak lensing signal by the forthcoming CMB probes may be relevant to gain insight into the dark energy dynamics at the onset of cosmic acceleration.

Constraining the dark energy dynamics with the cosmic microwave background bispectrum

We consider the influence of the dark energy dynamics at the onset of cosmic acceleration on the cosmic microwave background (CMB) bispectrum, through the weak lensing effect induced by structure formation. We study the line of sight behavior of the contribution to the bispectrum signal at a given angular multipole l: we show that it is nonzero in a narrow interval centered at a redshift z satisfying the relation $l/r(z)\ensuremath{\simeq}{k}_{\mathrm{NL}}(z),$ where the wave number corresponds to the scale entering the nonlinear phase and r is the cosmological comoving distance. The relevant redshift interval is in the range $0.1\ensuremath{\lesssim}z\ensuremath{\lesssim}2$ for multipoles $1000\ensuremath{\gtrsim}l\ensuremath{\gtrsim}100;$ the signal amplitude, reflecting the perturbation dynamics, is a function of the cosmological expansion rate at those epochs, probing the dark energy equation of state redshift dependence independent of its present value. We provide a working example by considering tracking inverse power law and supergravity quintessence scenarios, having sensibly different redshift dynamics and respecting all the present observational constraints. For scenarios having the same present equation of state, we find that the effect described above induces a projection feature which makes the bispectra shifted by several tens of multipoles, about ten times more than the corresponding effect on the ordinary CMB angular power spectrum.

Contributions of Point Extragalactic Sources to the Cosmic Microwave Background Bispectrum

All the analyses of Cosmic Microwave Background (CMB) temperature maps up--to--date show that CMB anisotropies follow a Gaussian distribution. On the other hand, astrophysical foregrounds which hamper the detection of the CMB angular power spectrum, are not Gaussian distributed on the sky. Therefore, they should give a sizeable contribution to the CMB bispectrum. In fact, the first year data of the Wilkinson Microwave Anisotropy Probe (WMAP) mission have allowed the {\it first} detection of the extragalactic source contribution to the CMB bispectrum at 41 GHz and, at the same time, much tighter limits than before to non--Gaussian primordial fluctuations. In view of the above and for achieving higher precision in current and future CMB measurements of non--Gaussianity, in this paper we discuss a comprehensive assessment of the bispectrum due to either uncorrelated and clustered extragalactic point sources in the whole frequency interval around the CMB intensity peak. Our calculations, based on current cosmological evolution models for sources, show that the reduced angular bispectrum due to point sources, $b_{ps}$, should be detectable in all WMAP and Planck frequency channels. We also find agreement with the results on $b_{ps}$ at 41 GHz coming from the analysis of the first year WMAP data. Moreover, by comparing $b_{ps}$ with the primordial reduced CMB bispectrum, we find that only the peak value of the primordial bispectrum (which appears at $l\simeq 200$) results greater than $b_{ps}$ in a frequency window around the intensity peak of the CMB. The amplitude of this window basically depends on the capability of the source detection algorithms (i.e., on the achievable flux detection limit, $S_{lim}$, for sources).

Measurement of the Cosmic Microwave Background Bispectrum on theCOBEDMR Sky Maps

We measure the angular bispectrum of the cosmic microwave background (CMB) radiation anisotropy from the COBE Differential Microwave Radiometer (DMR) four-year sky maps. The angular bispectrum is the harmonic transform of the three-point correlation function, analogous to the angular power spectrum, the harmonic transform of the two-point correlation function. First, we study statistical properties of the bispectrum and the normalized bispectrum. We find the latter more useful for statistical analysis; the distribution of the normalized bispectrum is very much Gaussian, while the bare bispectrum distribution is highly non-Gaussian. Then, we measure 466 modes of the normalized bispectrum, all independent combinations of three-point configurations up to a maximum multipole of 20, the mode corresponding to the DMR beam size. By measuring 10 times as many modes as the sum of previous work, we test Gaussianity of the DMR maps. We compare the data with the simulated Gaussian realizations, finding no significant detection of the normalized bispectrum on the mode-by-mode basis. We also find that the previously reported detection of the normalized bispectrum is consistent with a statistical fluctuation. By fitting a theoretical prediction to the data for the primary CMB bispectrum, which is motivated by slow-roll inflation, we put a weak constraint on a parameter characterizing non-linearity in inflation. Simultaneously fitting the foreground bispectra estimated from interstellar dust and synchrotron template maps shows that neither dust nor synchrotron emissions significantly contribute to the bispectrum at high Galactic latitude. We conclude that the DMR map is consistent with Gaussianity.

Dark energy and cosmic microwave background bispectrum

We compute the cosmic microwave background bispectrum arising from the cross-correlation of primordial, lensing, and Rees-Sciama signals. The amplitude of the bispectrum signal is sensitive to the matter density parameter ${\ensuremath{\Omega}}_{0}$ and the equation of state of the dark energy, which we parametrize by ${w}_{Q}.$ We conclude that the data set of the Atacama Cosmology Telescope, combined with MAP 2-year data, or the Planck data set alone will allow us to break the degeneracy between ${\ensuremath{\Omega}}_{0}$ and ${w}_{Q}$ that arises from the analysis of the CMB power spectrum. In particular a joint measurement of ${\ensuremath{\Omega}}_{0}$ and ${w}_{Q}$ with 10% and 30% error on the two parameters, respectively, at the 90% confidence level can realistically be achieved.