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The power spectral sensitivity over such an | The power spectral sensitivity over such an |
order to self-consistently treat the cosmic magnetic fields. we make use of several cosmological MHD simulations which compute the resulting magnetization of the cosmological structures (e.g. amplitude and structure) following different models for the origin and seeding process of such magnetic fields. | order to self-consistently treat the cosmic magnetic fields, we make use of several cosmological MHD simulations which compute the resulting magnetization of the cosmological structures (e.g. amplitude and structure) following different models for the origin and seeding process of such magnetic fields. |
We also construct magnetic field models with much higher magnetization amplitude in the low density regions to test how the resulting signatures of more extreme models affect our results. | We also construct magnetic field models with much higher magnetization amplitude in the low density regions to test how the resulting signatures of more extreme models affect our results. |
Here we scale up the predicted amplitude of the magnetic field in filaments by several orders of magnitude to test if such strong magnetic fields in low density regions significantly effect the expected correlation signal. | Here we scale up the predicted amplitude of the magnetic field in filaments by several orders of magnitude to test if such strong magnetic fields in low density regions significantly effect the expected correlation signal. |
By introducing GF and adding noise to the signal on top of the underlying cosmological signal. we can study how the shape and amplitude of the cross-correlation function would be modified when considering actual observations. | By introducing GF and adding noise to the signal on top of the underlying cosmological signal, we can study how the shape and amplitude of the cross-correlation function would be modified when considering actual observations. |
To avoid further complications we ignore the cosmological evolution of magnetic fields. which. in principle. would be consistently treated within our cosmological MHD simulations. | To avoid further complications we ignore the cosmological evolution of magnetic fields, which, in principle, would be consistently treated within our cosmological MHD simulations. |
Hence. we neglect the evolution of the cosmic magnetic field seen in the simulation as a result of the structure formation process. and assume the present day magnetization of the simulated universe to be present up to the redshift of the sources. | Hence, we neglect the evolution of the cosmic magnetic field seen in the simulation as a result of the structure formation process, and assume the present day magnetization of the simulated universe to be present up to the redshift of the sources. |
The paper is organized as follows. | The paper is organized as follows. |
In Section ?? we describe the cosmological MHD simulations used and how we compute the synthetic RM catalogs. | In Section \ref{sec:simul} we describe the cosmological MHD simulations used and how we compute the synthetic RM catalogs. |
In Section 3. we discuss the cross-correlation estimators used. the estimation of the intrinsic uncertainties due to the limited number of lines of sight probing the magnetization of the cosmological structures. the different signals expected for the various magnetization of the universe. as well as the uncertainties induced by the redshift distribution ofthe sources. | In Section \ref{sec:cross} we discuss the cross-correlation estimators used, the estimation of the intrinsic uncertainties due to the limited number of lines of sight probing the magnetization of the cosmological structures, the different signals expected for the various magnetization of the universe, as well as the uncertainties induced by the redshift distribution of the sources. |
In Section + we show how the shape and amplitude of the signal is affected by the recipe normally used to remove the foreground signal. due to observational noise and to the Galaxy itself. | In Section \ref{sec:obs} we show how the shape and amplitude of the signal is affected by the recipe normally used to remove the foreground signal, due to observational noise and to the Galaxy itself. |
In Section 5. we summarize the combination of all the effects. and present the resulting observable signal of the different magnetic field models. | In Section \ref{sec:res} we summarize the combination of all the effects, and present the resulting observable signal of the different magnetic field models. |
Finally. our conclusions are given in Section 6.. | Finally, our conclusions are given in Section \ref{sec:conc}. |
We used results from one of the constrained. cosmological MHD simulations presented in ?. and ?.. | We used results from one of the constrained, cosmological MHD simulations presented in \citet{2005JCAP...01..009D} and \citet{2009MNRAS.392.1008D}. |
In both simulations. the initial conditions for a constrained realization of the local Universe were the same as used in ?.. | In both simulations, the initial conditions for a constrained realization of the local Universe were the same as used in \citet{2002MNRAS.333..739M}. |
The initial conditions were obtained based on the the/RAS |.2-Jy galaxy survey (see?.formoredetails). | The initial conditions were obtained based on the the 1.2-Jy galaxy survey \citep[see][ for more details]{2005JCAP...01..009D}. |
Its density field was smoothed on a scale of 7Mpc. evolved back in time to >=50 using the Zeldovich approximation. and used as an Gaussian constraint (2). for an otherwise random realization of à ACDM cosmology (i0.3. AQ.T. h 0.7). | Its density field was smoothed on a scale of $7\, \mathrm{Mpc}$, evolved back in time to $z=50$ using the Zeldovich approximation, and used as an Gaussian constraint \citep{1991ApJ...380L...5H} for an otherwise random realization of a $\Lambda$ CDM cosmology $\Omega_M=0.3$, $\Lambda=0.7$, $h=0.7$ ). |
The observations constrain a volume of z115Mpc centered on the Milky Way. | The observations constrain a volume of $\approx 115 \, \mathrm{Mpc}$ centered on the Milky Way. |
In the evolved density field. many locally observed galaxy clusters can be identified by position and mass. | In the evolved density field, many locally observed galaxy clusters can be identified by position and mass. |
The original initial conditions were extended to include gas by splitting dark matter particles into gas and dark matter. obtaining particles of masses 6.0.107.M. and 4.4510"AL. respectively. | The original initial conditions were extended to include gas by splitting dark matter particles into gas and dark matter, obtaining particles of masses $6.9 \times 10^8\; {\rm
M}_\odot$ and $4.4 \times 10^9\; {\rm M}_\odot$ respectively. |
The gravitational softening length was set to LOkpe. | The gravitational softening length was set to $10\,\mathrm{kpc}$. |
The magnetic field was followed by our MHD simulations through the turbulent amplification. driven by the structure formation process. | The magnetic field was followed by our MHD simulations through the turbulent amplification driven by the structure formation process. |
For the magnetic seed fields. the first simulation (labeled ATHD) followed a cosmological seed field (see Fig. 1») | For the magnetic seed fields, the first simulation (labeled ) followed a cosmological seed field (see Fig. \ref{fig:bmodels}) ), |
while in the second (labeled Gal) we used a semi-analytic model for galactic winds. | while in the second (labeled ) we used a semi-analytic model for galactic winds. |
In particular. we considered the result of theDipole simulation from ?.. | In particular, we considered the result of the simulation from \citet{2009MNRAS.392.1008D}. |
In both simulations. the resulting magnetic field at 2=0 reproduce the observed Rotation Measure in galaxy clusters very well. | In both simulations, the resulting magnetic field at $z=0$ reproduce the observed Rotation Measure in galaxy clusters very well. |
A visual impression for the magnetic field within the two different simulations and their corresponding galaxy distribution is shown in Fig. 2.. | A visual impression for the magnetic field within the two different simulations and their corresponding galaxy distribution is shown in Fig. \ref{fig:simul}. |
As is clearly visible in Fig. . | As is clearly visible in Fig. \ref{fig:simul}, |
such cosmological simulations usually predict relatively low magnetic fields in low density regions. | such cosmological simulations usually predict relatively low magnetic fields in low density regions. |
To explore more extreme models. we scaled up the magnetic field of the simulation by a factor with a being 1/3 D. 1/2 2) and 2/3 3). | To explore more extreme models, we scaled up the magnetic field of the simulation by a factor with $\alpha$ being 1/3 ), 1/2 ) and 2/3 ). |
Here Pocale denotes density scale for fixing the magnetic field. which we choose to be 10! times the mean cosmic baryon density. | Here ${\rho_{\rm scale}}$ denotes density scale for fixing the magnetic field, which we choose to be $10^4$ times the mean cosmic baryon density. |
The resulting behaviour of the mean magnetic field as a function of baryon density for the original runs. as well as for the scaled-up models. are shown in Fig. |.. | The resulting behaviour of the mean magnetic field as a function of baryon density for the original runs, as well as for the scaled-up models, are shown in Fig. \ref{fig:bmodels}. |
Note that the lines shown reflect the mean value of the magnetic field at the corresponding overdensity. while the dispersion of its amplitude can span several orders of magnitude in each density bin (see Dolag et al. | Note that the lines shown reflect the mean value of the magnetic field at the corresponding overdensity, while the dispersion of its amplitude can span several orders of magnitude in each density bin (see Dolag et al. |
2005). | 2005). |
We want to stress that such scaled-up magnetic fields are artificial models. as the primordial field needed to generate them would be well above current cosmological constraints (e.g. from CMB). | We want to stress that such scaled-up magnetic fields are artificial models, as the primordial field needed to generate them would be well above current cosmological constraints (e.g. from CMB). |
Such strong seed fields would lead to an overprediction of the magnetic field amplitude in galaxy clusters by the simulations and it is quite unclear which physical process could be responsible to avoid this. | Such strong seed fields would lead to an overprediction of the magnetic field amplitude in galaxy clusters by the simulations and it is quite unclear which physical process could be responsible to avoid this. |
We also remark that the sealed-up models lead to slightly lower central values for the magnetic field inside of galaxy clusters. | We also remark that the scaled-up models lead to slightly lower central values for the magnetic field inside of galaxy clusters. |
This | This |
may be written where w is the fluxl velocity. uis the vorticity. aud the Bernoulli finction B is Both the fIukl speed aud deusity approach coustaut. values far (rom the mass. | may be written where $\bu$ is the fluid velocity, is the vorticity, and the Bernoulli function $B$ is Both the fluid speed and density approach constant values far from the mass. |
Hence. B is a spatial constant throughout the flow. aud Since wis a poloidal vector. the vorticity w is toroidal. | Hence, $B$ is a spatial constant throughout the flow, and Since $\bu$ is a poloidal vector, the vorticity $\bomega$ is toroidal. |
The last equation then implies that0.. as Claimed. | The last equation then implies that, as claimed. |
We will not need to invoke the irrotational character of the flow until Section 5.. when we explicitly evaluate the dyvuamical friction force. | We will not need to invoke the irrotational character of the flow until Section \ref{sec:friction}, when we explicitly evaluate the dynamical friction force. |
Throughout our analysis. it will be more convenient to employ. not the vector fluid. velocity u(r0). but the scalar stream [uuctiou er). | Throughout our analysis, it will be more convenient to employ, not the vector fluid velocity $\bu (r,\theta)$, but the scalar stream function $\psi (r,\theta)$. |
We may recover the individual velocity components [rom the stream function through the staucdarcd relations where is the mass density. | We may recover the individual velocity components from the stream function through the standard relations where is the mass density. |
The velocity. as given by equations (1)) and (5)). automatically obeysmass continuity: Far [rom the mass. as the density approaches po. the velocity has ouly a z-component. which is V. | The velocity, as given by equations \ref{eqn:ur}) ) and \ref{eqn:ut}) ), automatically obeysmass continuity: Far from the mass, as the density approaches $\rho_0$, the velocity has only a $z$ -component, which is $V$. |
Equivalently. we have in this limit audsin0. | Equivalently, we have in this limit and. |
. IH follows that the Car-feld limit of the stream function is For a more complete analysis of the flow in this region. we take equation (7)) to represent the leading term of a perturbation expansion. | It follows that the far-field limit of the stream function is For a more complete analysis of the flow in this region, we take equation \ref{eqn:sfunc1}) ) to represent the leading term of a perturbation expansion. |
Introducing the sonic radius ο... we first rewrite equation (7)) as | Introducing the sonic radius , we first rewrite equation \ref{eqn:sfunc1}) ) as |
We use and test two relevant models: the ACDAMr model. that is the standard ACDAL model αιemeuted by the tensortoscalar ratio r. and the ACDAIVT model. that is the AC'DAIr model in which the doudewell inflaton potential (see Eq. (1)) | We use and test two relevant models: the $\Lambda$ $r$ model, that is the standard $\Lambda$ CDM model augmented by the tensor–to–scalar ratio $ r $, and the $\Lambda$ $r$ T model, that is the $\Lambda$ $r$ model in which the double–well inflaton potential (see Eq. \ref{binon}) ) |
im the next section) is 1uposed. | in the next section) is imposed. |
Namely 04 aud are coustrained by the analytic relation r=rr.) to lav on the theoretical banana-shaxb curve (the upper border of the banana-shaped region Fig. 1)). | Namely, $ n_s $ and $ r $ are constrained by the analytic relation $ r = r(n_s) $ to lay on the theoretical banana-shaped curve (the upper border of the banana-shaped region Fig. \ref{banana}) ). |
The ioveltv in the MCMC analysis of the CMB data with theACDAIVT inodoel is in the fact tha we nmupose the analytical expressions for po, and r derived from the inflaton potential as a hard coustraint (Destrietal.2008a). | The novelty in the MCMC analysis of the CMB data with the$\Lambda$ $r$ T model is in the fact that we impose the analytical expressions for $
n_s $ and $ r $ derived from the inflaton potential as a hard constraint \citep{mcmc1}. |
. We take oth models. CDM and ACDAIVT. as fica(lal models in our Moute Carlo Markov Chains (AICAIC) simulations to produce the corresponding skies (1iock data). | We take both models, $\Lambda$ $r$ and $\Lambda$ $r$ T, as fiducial models in our Monte Carlo Markov Chains (MCMC) simulations to produce the corresponding skies (mock data). |
Iu the ACDAIy inodel the independent cosmologica paralcters are Qih?.οςΠτι0.τιA.na andr. while all other independent parameters are assumed to vanish. QO,=0. or have the standard values. uw=Ll. | In the $\Lambda$ $r$ model the independent cosmological parameters are $\Omega_b \, h^2, \; \Omega_c \, h^2, \; \theta , \; \tau, \; A_s ,
\; n_s $ and $ r $, while all other independent parameters are assumed to vanish, $ \Omega_\nu=0 $ , or have the standard values, $ w=-1 $. |
The aforemetioned AC'DAI-T inodel includes the same parameters but with ος and or not being independent. but related by the curve kr—rly) as widely diseussed in Sect. 2.. | The aforementioned $\Lambda$ $r$ T model includes the same parameters but with $ n_s $ and $ r $ not being independent, but related by the curve $r=r(n_s)$ as widely discussed in Sect. \ref{LGtheory}. . |
We xoduce oue skv (mock. data) for the anisotropy CAIB multipoles CIL.CER and. CPP from the Αλ [η iiocel aud from the ACDAIYT model. with the parainecters in Table 2.. C | We produce one sky (mock data) for the anisotropy CMB multipoles $ C_l^{TT}, \; C_l^{TE}, \;
C_l^{EE} $ and $ C_l^{BB} $ from the $\Lambda$ $r$ model and from the $\Lambda$ $r$ T model, with the parameters in Table \ref{tab2}. . |
H.We describe the detailed xocedure m Sect. 5.. | We describe the detailed procedure in Sect. \ref{mock}. |
We run Monte Carlo Markov Chains frou this sky and obtain the mareinalized. likelihood distributionsfor the cosuxlogical parameters (QuAT.Q.B7.0r.Oa. Ave o the Universe. i4Ην.Ay.ne andr) in the two test models ACDM and ACDAIFT . We study the indedeut AC DALY pariuneters with the mock data produced from ACDAM (first ic»v of Table 2)) aud the independent parameters of both ACDALr aud ACDAU-T with the mocks «ata produced from AC DAV T (secoud row in Table 2)). | We run Monte Carlo Markov Chains from this sky and obtain the marginalized likelihood distributionsfor the cosmological parameters $ \Omega_b \, h^2 , \;
\Omega_c \, h^2 ,\; \theta \; \tau, \; \Omega_{\Lambda} $, Age of the Universe, $ z_{re}, \; H_0, \; A_s, \; n_s $ and $ r $ ) in the two test models $\Lambda$ $r$ and $\Lambda$ $r$ T. We study the independent $\Lambda$ $r$ parameters with the mock data produced from $\Lambda$ CDM (first row of Table \ref{tab2}) ) and the independent parameters of both $\Lambda$ $r$ and $\Lambda$ $r$ T with the mock data produced from $\Lambda$ $r$ T (second row in Table \ref{tab2}) ). |
The fiducial valies. re=000127 and ος=0.9611 correspoud to the best fit to the CAIB-LSS data with the ACDMLUT iiecl using the doublewell inflaton potential expressed by Eq. (1)). | The fiducial values, $ r = 0.0427 $ and $ n_s = 0.9614 $ correspond to the best fit to the CMB-LSS data with the $\Lambda$ $r$T model using the double–well inflaton potential expressed by Eq. \ref{binon}) ). |
Namely. these are the best fit values to 7 aud ος within the Cinshtve-Laida effective theory approach. | Namely, these are the best fit values to $ r $ and $ n_s $ within the Ginsburg-Landau effective theory approach. |
No usine +1ο Cansbure-Laixau approac1. lower bounds for k& are not obtained aud the best fit value for r can be much simaler +an =0.01 (INiunevetal.2Ws:Poiris&Easther2008). | Not using the Ginsburg-Landau approach, lower bounds for $ r $ are not obtained and the best fit value for $ r $ can be much smaller than $ r = 0.04 $ \citep{kinney08,PeirisEasther}. |
. We consider two choices for the € likelihood. one without the B uodes al one with the Bu jocles and take iuto account the white roise seusitivitv ofPlanck (LEI aud WEFT) iu the 70. |OO and 113 GIIz chanuels 2006).. | We consider two choices for the $C_\ell-$ likelihood, one without the $ B $ modes and one with the $B$ modes and take into account the white noise sensitivity of (LFI and HFI) in the 70, 100 and 143 GHz channels \citep{PlanckBlueBook}. . |
We also consider a eunmlative channel whose q? is the sun of he von softhe three channels above. | We also consider a cumulative channel whose $ \chi^2 $ is the sum of the $ \chi^2 $ 's of the three channels above. |
When using different chaunels in the MCAIC analysis. we use differcut noise realizatious Wwule keeping the same sikv. that is the sale realization of the Ciaussiau process that eeucrated the primordial fluctuations. | When using different channels in the MCMC analysis, we use different noise realizations while keeping the same sky, that is the same realization of the Gaussian process that generated the primordial fluctuations. |
Iu our MCMC analysis we always take standard flat priors for the cosmological parameters. | In our MCMC analysis we always take standard flat priors for the cosmological parameters. |
Iu. particuar we assume the flat priors θ<r<0.2 in the ACDAMLI model aud 0x+«S/60. where 8/60~0.133 is the theoretical upper limit for r in the ACDAIYT model. | In particular we assume the flat priors $ 0 \leq r < 0.2 $ in the $\Lambda$ $r$ model and $ 0 \leq r< 8/60$, where $8/60 \simeq 0.133$ is the theoretical upper limit for $ r $ in the $\Lambda$ $r$ T model. |
We performed t1ο MCMC sniuimlatious using the publicly available Cosimo (Lewis&Bridle2002) interfaced to the Doltziuaun code (sce Lewisοἳal.(2000) and references therein).ICOur fiuciugs without includiue the svstematie effects are summarized in Fies. 3- | We performed the MCMC simulations using the publicly available CosmoMC \citep{mcmc} interfaced to the Boltzmann code (see \citet{2000ApJ...538..473L} and references therein).Our findings without including the systematic effects are summarized in Figs. \ref{r0r04}- |
6 where the mareinaized likelihood distributions of the cosnological paraimcters are plotted for several different setups. | \ref{ban_y} where the marginalized likelihood distributions of the cosmological parameters are plotted for several different setups. |
In Tables 3. to 5. we list the correspouding relevaut munerical values. | In Tables \ref{tab3} to \ref{tab5} we list the corresponding relevant numerical values. |
Clearly. in the case of the ratio r. due to the specific form of its likelihood distribution. it is nore interesting to exhibit upperaud lower bounds rather than mean values aud stamard deviations as dm Tables3 aid [.. | Clearly, in the case of the ratio $ r $ , due to the specific form of its likelihood distribution, it is more interesting to exhibit upperand lower bounds rather than mean values and standard deviations as in Tables\ref{tab3} and \ref{tab4}. . |
We report the uw»per bounds and. when preseut. the lower bounds in Tables 5 and 6.. | We report the upper bounds and, when present, the lower bounds in Tables \ref{tab5} and \ref{tab6}. . |
Our conclusious without iucποιο the svstematiceffects are: lu Sect. | Our conclusions without including the systematiceffects are: In Sect. |
7? we inchde m the forecaststhe svsclnatic effects discussed i1i Sects | \ref{fortoy} we include in the forecaststhe systematic effects discussed in Sects. |
2? and L. | \ref{sensit} and \ref{toym}. |
Our couclusious includiue the xvstemiatic effects and foreground residuals are: | Our conclusions including the systematic effects and foreground residuals are: |
kknot. is 850 pe across. | knot, is 850 pc across. |
Holes of this size are often detected in Ims and are thought to be the result of star formation activity. | Holes of this size are often detected in Ims and are thought to be the result of star formation activity. |
The largest hole in DDO 43 resembles the one formed by star formation that surrounds the OD association NGC 206 in M31. | The largest hole in DDO 43 resembles the one formed by star formation that surrounds the OB association NGC 206 in M31. |
However. there are no voung star associations in DDO 43's holes: the stellar colors are slightly redder than those in the rest of the galaxy and they do not appear to be expanding. so thev must be relatively old. | However, there are no young star associations in DDO 43's holes; the stellar colors are slightly redder than those in the rest of the galaxy and they do not appear to be expanding, so they must be relatively old. |
The knots in the high density ridge form a broken ring interspersed with the holes. | The knots in the high density ridge form a broken ring interspersed with the holes. |
The structure of knots and holes we see today is most likely the result of (he overlapping of expanding sshells caused by the stellar winds and supernovae explosions rom earlier star. formation episodes. | The structure of knots and holes we see today is most likely the result of the overlapping of expanding shells caused by the stellar winds and supernovae explosions from earlier star formation episodes. |
Most of the current star formation is taking place in the ridge. possibly induced bv the sweeping up of gas as the holes were formed. | Most of the current star formation is taking place in the ridge, possibly induced by the sweeping up of gas as the holes were formed. |
DDO 43's current star formation rate is normal. falling in the middle range for Ins. | DDO 43's current star formation rate is normal, falling in the middle range for Ims. |
There are several moclerate-size rregions scattered around the galaxy. but there is on average less star formation activity in the inner 0.75 kpe than expected based on a comparison of the azimuthallv-averaged aad: V-band surface densities. | There are several moderate-size regions scattered around the galaxy, but there is on average less star formation activity in the inner 0.75 kpc than expected based on a comparison of the azimuthally-averaged and V-band surface densities. |
Most of the rregions are located in areas of high ssurlace densitv. and many of those are located near local peaks in theILL. as is seen in other Ims. | Most of the regions are located in areas of high surface density, and many of those are located near local peaks in the, as is seen in other Ims. |
A few of the rregions are associated with lower ssirface density areas however. including the hholes. | A few of the regions are associated with lower surface density areas however, including the holes. |
lxinematicallv. (here is nothing strikingly unusual about. DDO 43. | Kinematically, there is nothing strikingly unusual about DDO 43. |
It is undergoing the primarily solic-body rotation seen in many dwarls. with an S-distorGon visible in the velocity lied. possibly indicating the presence of a warp in the gas disk. | It is undergoing the primarily solid-body rotation seen in many dwarfs, with an S-distortion visible in the velocity field, possibly indicating the presence of a warp in the gas disk. |
The rotation curve levels off and possibly (urns over. | The rotation curve levels off and possibly turns over. |
The maximum rotation velocity is 25I... which is higher than the original estimate. | The maximum rotation velocity is 25, which is higher than the original estimate. |
It was the low rotation velocity [or its luminosity Chat cased this galaxy to be classified as a candidate “fossil” tidal dwarf. | It was the low rotation velocity for its luminosity that caused this galaxy to be classified as a candidate “fossil” tidal dwarf. |
Dwarl galaxies formed. from tidal interactions are predicted to contain less dark matter than primordially formed divas. so should have lower total masses and subsequently. lower rotation speeds. | Dwarf galaxies formed from tidal interactions are predicted to contain less dark matter than primordially formed dwarfs, so should have lower total masses and subsequently, lower rotation speeds. |
Llowever. the revised velocity estimate puts it back in the normal area of the Tull-Fisher relation for Dus. so (here is no reason to think that DDO 43 formed from an interaction. | However, the revised velocity estimate puts it back in the normal area of the Tully-Fisher relation for Ims, so there is no reason to think that DDO 43 formed from an interaction. |
The dispersion velocities in the iindicate an average of ~ 10ον, which is normal. | The dispersion velocities in the indicate an average of $\sim$ 10, which is normal. |
There is one area of higher dispersion: {his is associated with the large hole. | There is one area of higher dispersion; this is associated with the large hole. |
The sspectra through the hole indicate that the gas motions are complex. wilh most of the spectra | The spectra through the hole indicate that the gas motions are complex, with most of the spectra |
In section ?? we review the estimator of the (binned) angular bispectrum ancl fixe and develop a parametrisation of the bispectrum to display and. visualise it efficiently. | In section \ref{sect:3pNGe} we review the estimator of the (binned) angular bispectrum and $f_\mathrm{NL}$ and develop a parametrisation of the bispectrum to display and visualise it efficiently. |
In section 3.. we develop a prescription to infer the bispectrum from the power spectrum. for clustered: sources. ancl for dillerent populations. | In section \ref{sect:prescription}, we develop a prescription to infer the bispectrum from the power spectrum for clustered sources and for different populations. |
In section ??. we use publicly available full-sky simulations of radio and infrared sources to compute and characterise their bispectrum at CMD frequencies and we compare them to the predictions from the prescription. | In section \ref{sect:resultsehgal} we use publicly available full-sky simulations of radio and infrared sources to compute and characterise their bispectrum at CMB frequencies and we compare them to the predictions from the prescription. |
We examine theconfiguration dependence of the. point-source bispectra ancl study the bias they induce on the estimation of the primordial local non-Ciaussianity in section 77.. | We examine theconfiguration dependence of the point-source bispectra and study the bias they induce on the estimation of the primordial local non-Gaussianity in section \ref{sect:ngcsqce}. |
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