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Figure Lb shows detailed views of the light curves to demoustrate the | Figure 4b shows detailed views of the light curves to demonstrate the |
Introduct ion becauseecodesics.the parametermisleadiug. is becau sehivpersurfa (Thelehceslike heht-likeLightslicing.of Nery|dack holes.e.ginthe propagation ofwave frouts and ligh-Teque mCY waves. aud in initial value Ingoine or asviuptoticallvoutfeoiug) οσα]“quasi-spherical”at infinity.lightThese coucs.hvpersurfacesdefined asbecome degenerate the 3-spacesandard whosespatial lightsectious whenare errs parameter vandshes:do Ον they thesfamiliar Miukowskispherical Πο cones fer the constaut.mass These appealing features "unctionisWe | We write the Kerr metric in its standard (Boyer-Lindquist) form where and we note the useful identities The equation represents an axisymmetric lightlike hypersurface (ingoing for $\epsilon=1$, outgoing for $\epsilon=-1$ ) if $g^{\alpha \beta}(\partial_\alpha v)(\partial_\beta v)=0$ , i.e., if It is easy to obtain a particular (separable) solution of \ref{pde}) ) depending on two arbitrary constants (a “complete integral”) by adding and subtracting an arbitrary separation constant $a^2\lambda$ on the right -hand side. |
ofthe briefly line the Bover-Lindquistcoordients lates.ofthe (e.g. asviupt | Let us define (Note the useful identity which follows from \ref{ident}) ).) |
otic) «liapeof thesrfaceds given.(1 i0the AlinkowsliΛατ lightThecous we lakefje special clioicefreeM ofcaus | A complete integral of \ref{pde}) ) is then obtained by integrating the exact differential at fixed $\lambda$. |
ticsf£ap oxopriatefor0.ais axvclenij ototicallynoustratedin sphericalSec.6. lightIn conesSees. inTand space. howsev result. thatthese arecoordi« construct8we hieb-frequshow the solutions | When \ref{drb}) ) is integrated, a second, additive integration constant appears which we shall denote as $a^2 f(\lambda)/2$, where $f$ is an arbitrary function. |
the “quasi-spherical” equationjafes rid.0 INxuskal-likedefinedbv. coordinatestheselight conesfor Ίναcan blackbeusedholes.to Sec.9 encv to wave a*ceable sinnuarnzesthe limiting | We next proceed in the usual way to promote this complete integral, depending upon the arbitrary constants $\lambda$ and $f(\lambda)$, to a general solution involving an arbitrary function. |
casesformthatofcanthe metric.provideFinasey Iv. in Sec. 10 approximapreseutions for the results ellipticof uunericalfunctious which iutegratiousappearforiuthe παν quasi-splericalevolution ofthe Axisymmetriclightlike termsof elliptic integrals aswe nowprocee | In \ref{rho}) ), $\rho$ is a function of three independent variables $r$, $\theta$ and $\lambda$ (not counting $m$ and $a$ ), and a more complete expression for its differential is where $F$ is the partial derivative Its explicit form may be taken to be |
casesformthatofcanthe metric.provideFinasey Iv. in Sec. 10 approximapreseutions for the results ellipticof uunericalfunctious which iutegratiousappearforiuthe παν quasi-splericalevolution ofthe Axisymmetriclightlike termsof elliptic integrals aswe nowproceed | In \ref{rho}) ), $\rho$ is a function of three independent variables $r$, $\theta$ and $\lambda$ (not counting $m$ and $a$ ), and a more complete expression for its differential is where $F$ is the partial derivative Its explicit form may be taken to be |
occurrence and the position of the scattering sites along the path of the photon through the wind. | occurrence and the position of the scattering sites along the path of the photon through the wind. |
Indeed by using the transformation equation (Eq.1)) we can determine the random optical depth της at which the photon interacts: Because we stored the run of r,,=τω.) along the photon path. it is easy to invert this relation and find the point s» where r,,=tac. | Indeed by using the transformation equation \ref{transfo}) ) we can determine the random optical depth $\tau_{MC}$ at which the photon interacts: Because we stored the run of $\tau_{tot} = \tau_{tot}(s)$ along the photon path, it is easy to invert this relation and find the point $s$ where $\tau_{tot}=\tau_{MC}$. |
At this location the photon is radiatively absorbed and then instantaneously re-emitted at a frequency and in a direction chosen by assuming a complete redistribution in frequency and direction (CRFD. Lucy 1971. Mihalas et al.1976)). | At this location the photon is radiatively absorbed and then instantaneously re-emitted at a frequency and in a direction chosen by assuming a complete redistribution in frequency and direction (CRFD, Lucy \cite{lu71}, Mihalas et \cite{mi76}) ). |
This photon may then either be absorbed somewhere else in the wind or leave it and be detected by one of the detector (spectrographs. imagers) located around the wind. | This photon may then either be re-absorbed somewhere else in the wind or leave it and be detected by one of the detector (spectrographs, imagers) located around the wind. |
To decrease the simulation time in the case of non-spherically symmetric winds we make an intense use of the advanced concepts of “first forced interaction" (e.g. Cashwell Everett 1959... Witt 1977)) and "peeling off” (e.g. Zadeh et al. 1984.. | To decrease the simulation time in the case of non-spherically symmetric winds we make an intense use of the advanced concepts of “first forced interaction" (e.g. Cashwell Everett \cite{ca59}, , Witt \cite{wi77}) ) and “peeling off" (e.g. Yusef-Zadeh et al. \cite{yu84}, |
Wood Reynolds 1999)) where we follow a photon packet rather than a single photon. | Wood Reynolds \cite{wo99}) ) where we follow a photon packet rather than a single photon. |
We checked the validity of our MCRT code by comparing the line profiles we obtained to line profiles computed with two traditional methods for spherical winds that allow an exact integration. of the transfer equation. | We checked the validity of our MCRT code by comparing the line profiles we obtained to line profiles computed with two traditional methods for spherical winds that allow an exact integration of the transfer equation. |
These benchmarks. are the well-known SEI method of Lamers et al. (1987)) | These benchmarks are the well-known SEI method of Lamers et al. \cite{la87}) ) |
and the comoving frame method of Hamann et al. (1981)). | and the comoving frame method of Hamann et al. \cite{ha81}) ). |
We noted good agreement between the general shape of the computec profiles. regardless of the considered turbulence (AF/Fx56€ on the normalized emission peak flux. as well as a good match between the absorption profiles). | We noted good agreement between the general shape of the computed profiles, regardless of the considered turbulence $\Delta F/F \leq 5 \%$ on the normalized emission peak flux, as well as a good match between the absorption profiles). |
We also tested MCRT Π the case of axi-symmetric winds by comparing the profiles obtained with MCRT to those produced by the SEI methoc adapted byBjorkman et al. (1994)). | We also tested MCRT in the case of axi-symmetric winds by comparing the profiles obtained with MCRT to those produced by the SEI method adapted byBjorkman et al. \cite{bj94}) ). |
Once again we observec good agreement between the line profiles produced by both methods (see Borguet 2009 for details). | Once again we observed good agreement between the line profiles produced by both methods (see Borguet \cite{bo09} for details). |
While the shape of the line in BAL QSOs can be mostly governed by resonance scattering (Scargle et al. 1972)). | While the shape of the line in BAL QSOs can be mostly governed by resonance scattering (Scargle et al. \cite{sc72}) ), |
the presence of Cin] emission constitutes evidence that part of the emission ts due to collisional excitation (Turnshek 1984a.. Turmshek 1988.. Hamann et al. 1993)). | the presence of $]$ emission constitutes evidence that part of the emission is due to collisional excitation (Turnshek \cite{tu84a}, Turnshek \cite{tu88}, , Hamann et al. \cite{ha93}) ). |
To account for this second source of photons. we allow the production directly in the wind of a fraction of photons f=Dueemission/Leoninnm- | To account for this second source of photons, we allow the production directly in the wind of a fraction of photons $f_e=I_{pure~emission}/I_{continuum}$. |
The choice of the location of the emission (7.0...6,) of these photons is made using a random sampling of the corresponding NPDF: where the 70.6.6) is a function. that deseribes the emissivity throughout the wind. | The choice of the location of the emission $(r_e, \theta_e, \phi_e)$ of these photons is made using a random sampling of the corresponding NPDF: where the $\eta(r,\theta,\phi)$ is a function that describes the emissivity throughout the wind. |
Once again. our goal here is not to provide a detailed self-consistent model of the wind so we choose as a first guess an emissivity function of the form where nr.8.à) 15 the density of the ton through the wind and where the second term allows taking the temperature distribution and the ionization fraction into account. | Once again, our goal here is not to provide a detailed self-consistent model of the wind so we choose as a first guess an emissivity function of the form where $n(r, \theta, \phi)$ is the density of the ion through the wind and where the second term allows taking the temperature distribution and the ionization fraction into account. |
In. the following we simply take y=| in order to reduce the number of free parameters. | In the following we simply take $\gamma=1$ in order to reduce the number of free parameters. |
Each photon emitted then makes its way through the wind where it can be scattered and then finally escapes the wind in order to be detected by a distant observer. | Each photon emitted then makes its way through the wind where it can be scattered and then finally escapes the wind in order to be detected by a distant observer. |
As stated in the introduction. it is difficult to give a simple explanation to the observed BAL profiles when using a spherically symmetric expanding wind. | As stated in the introduction, it is difficult to give a simple explanation to the observed BAL profiles when using a spherically symmetric expanding wind. |
The obvious next type of geometry that can then be considered is the axi-symmetric one. | The obvious next type of geometry that can then be considered is the axi-symmetric one. |
The simplest of these models would still consist in a wind originating from the central core. | The simplest of these models would still consist in a wind originating from the central core. |
Such à generic model. with polar and equatorial components like the one presented in details by Bjorkman et al. (1994)) | Such a generic model, with polar and equatorial components like the one presented in details by Bjorkman et al. \cite{bj94}) ) |
produces line profiles remarkably similar to those observed in some BAL QSOs, | produces line profiles remarkably similar to those observed in some BAL QSOs. |
Although simple. this model is versatile enough to produce a variety of line profiles. as observed in BAL QSOs. | Although simple, this model is versatile enough to produce a variety of line profiles, as observed in BAL QSOs. |
It constitutes a first good approximation to the more complex wind from disk models proposed for AGN outflows in which the BALR and the BELR are generally cospatial (Sect. 1)). | It constitutes a first good approximation to the more complex wind from disk models proposed for AGN outflows in which the BALR and the BELR are generally cospatial (Sect. \ref{lintro}) ). |
In our model. we adopt stellar wind laws to describe the kinematies of the winds observed in quasars. | In our model, we adopt stellar wind laws to describe the kinematics of the winds observed in quasars. |
Indeed. quasar winds are also supposed to be driven by radiation (Arav Li 1996.. Murray et al. 1995)) | Indeed, quasar winds are also supposed to be driven by radiation (Arav Li \cite{ar96}, Murray et al. \cite{mu95}) ) |
as suggested by the line-locking in the spectra of some BAL QSOs (e.g. Weymann et al. 199].. | as suggested by the line-locking in the spectra of some BAL QSOs (e.g. Weymann et al. \cite{we91}, |
Korista et al. 1993.. | Korista et al. \cite{ko93}, |
Arav 1996. 1997)). | Arav \cite{ar96,ar97}) ). |
However. there are important. differences between stars and quasars (e.g. Arav 1994)). | However, there are important differences between stars and quasars (e.g. Arav \cite{ar94}) ). |
One of them is the so-called overionization problem caused by the strong UV/X-ray central source in quasars (e.g. Proga et al. 2000)). | One of them is the so-called overionization problem caused by the strong UV/X-ray central source in quasars (e.g. Proga et al. \cite{pr00}) ). |
Several scenarios have been suggested to solve this problem: Murray et al. (1995)). | Several scenarios have been suggested to solve this problem: Murray et al. \cite{mu95}) ), |
Murray Chiang (1997)). Risalitt Elvis (2009)). Punsly (1999)) and Ghosh Punsly (2007)). | Murray Chiang \cite{mu97}) ), Risaliti Elvis \cite{ri09}) ), Punsly \cite{pu99}) ) and Ghosh Punsly \cite{gh07}) ). |
In our study we assume the existence of shielding material between the radiation source and the outflow that prevents the total ionization. of the outflow (see Krolik 1999)). | In our study we assume the existence of shielding material between the radiation source and the outflow that prevents the total ionization of the outflow (see Krolik \cite{kr99}) ). |
Another difference between stellar objects and quasars comes from a significant fraction of the radiation in quasars supposedly being emitted from an accretion disk rather than from a spherically symmetric. photosphere (e.g. Proga et al. 2000)) | Another difference between stellar objects and quasars comes from a significant fraction of the radiation in quasars supposedly being emitted from an accretion disk rather than from a spherically symmetric photosphere (e.g. Proga et al. \cite{pr00}) ). |
Outflows with axial geometries have been studied by several authors. who show that the flow can be launched vertically from the diskand then pushed away by the radiation from the central source (Murrayet al. 1995.. | Outflows with axial geometries have been studied by several authors, who show that the flow can be launched vertically from the diskand then pushed away by the radiation from the central source (Murrayet al. \cite{mu95}, , |
Proga et al. 2000)) | Proga et al. \cite{pr00}) ) |
with the elevation of the wind over the disk is still small when the flow starts to expand radially. | with the elevation of the wind over the disk is still small when the flow starts to expand radially. |
Since the launching | Since the launching |
by Bertoldi Melxec (1992) although this is iu part. due to a selection effect which excludes regions with sinall velocity dispersious. | by Bertoldi McKee (1992) although this is in part, due to a selection effect which excludes regions with small velocity dispersions. |
This may also be due to the spatial variation of pressure throughout the outer Galaxy aud the underestimate of the eravitational parameter for lower mass objects (see 52. 1 and 63.3). | This may also be due to the spatial variation of pressure throughout the outer Galaxy and the underestimate of the gravitational parameter for lower mass objects (see $\S$ 2.4 and $\S$ 3.3). |
The iudex is similar to values derived from oobservatious of targeted cloud complexes. | The index is similar to values derived from observations of targeted cloud complexes. |
The results of Dobashi (1996). Yonekura (1997). and TIocvanmra (1998) show even shallower slopes (0.2-0.3). | The results of Dobashi (1996), Yonekura (1997), and Kawamura (1998) show even shallower slopes (0.2-0.3). |
The preceding sections demonstrate that there are a large number of molecular regions whose internal motions are not bound by self exavity. | The preceding sections demonstrate that there are a large number of molecular regions whose internal motions are not bound by self gravity. |
To remain bound in the observed configuration. these iiotious must be confined by the pressure of the external medi. | To remain bound in the observed configuration, these motions must be confined by the pressure of the external medium. |
To gauge the maguitude of the required external pressures. the full virial theorem is rewritten. where lis the measured minor axis length. DP. is the external pressure. and Nyy, is the mean molecular colin cdensits over the surface of the object. | To gauge the magnitude of the required external pressures, the full virial theorem is rewritten, where is the measured minor axis length, $P_\circ$ is the external pressure, and $N_{H_2}$ is the mean molecular column density over the surface of the object. |
This expression assunes that the clouds are prolate. | This expression assumes that the clouds are prolate. |
Figure 11. shows the quantity. o7/7,,;,, plotted as a function of Nyy, for each ideutified object. | Figure \ref{pext} shows the quantity, $\sigma_v^2/l_{min}$ plotted as a function of $N_{H_2}$ for each identified object. |
Also shown are the variations of OF2fling, With coluun density for different values of the external pressure for bouud objects. | Also shown are the variations of $\sigma_v^2/l_{min}$ with column density for different values of the external pressure for bound objects. |
Self gravitating objects lie along the curve P/k=0. | Self gravitating objects lie along the curve P/k=0. |
The primary cluster of points lie well off this line. | The primary cluster of points lie well off this line. |
The distribution of required pressure is shown in Fiewe 15.. | The distribution of required pressure is shown in Figure \ref{pextdistr}. . |
The mean and median of the distribution are ἐν 101 and. 6700 cnm7C>3A respectively, | The mean and median of the distribution are $\times$ $^4$ and 6700 $^{-3}\;K$ respectively. |
No significant variation of the required pressure with ealactoceutric radius can be determined given the limited dvuamic rauge of ρω aud the small απνο. of ideutified objects at 2,4> 11 kpc. | No significant variation of the required pressure with galactocentric radius can be determined given the limited dynamic range of $R_{gal}$ and the small number of identified objects at $R_{gal} >$ 14 kpc. |
Given the maenitude of measured line widths. the internal pressure arises from the non thermal. turbulent notions of the gas. | Given the magnitude of measured line widths, the internal pressure arises from the non thermal, turbulent motions of the gas. |
The required pressures to bind these motions are larger than the measured thermal oressures of the interstellar medium although thermal pressure fiuctuatious of the required maecnitude are observed within a small fraction of the atomic gas volume (Jeukius. Jura. Lowenstein 1983: Wannier 1999). | The required pressures to bind these motions are larger than the measured thermal pressures of the interstellar medium although thermal pressure fluctuations of the required magnitude are observed within a small fraction of the atomic gas volume (Jenkins, Jura, Lowenstein 1983; Wannier 1999). |
ILoxcever. even in the case of comparable external thermal pressure. the cloud boundary can tot be maintained due the anisotropy of internal. turbulent eas flow. | However, even in the case of comparable external thermal pressure, the cloud boundary can not be maintained due the anisotropy of internal, turbulent gas flow. |
Au initial perturbation of the cloud voundary by a turbulent fluctuation generates an iubalauce of the pressure force perpendicular to the surface which in turn. causes a larger distortion of the boundary (Vishniac 1983). | An initial perturbation of the cloud boundary by a turbulent fluctuation generates an imbalance of the pressure force perpendicular to the surface which in turn, causes a larger distortion of the boundary (Vishniac 1983). |
Tf tlhe molecular clouds are simply Hel density regions resulting from converging eas streams within a larger turbulent flow. as sugeested by Munerical simulations. there is au effective external ram pressure component. | If the molecular clouds are simply high density regions resulting from converging gas streams within a larger turbulent flow, as suggested by numerical simulations, there is an effective external ram pressure component. |
However. this componcut is similarly :anisotropic and therefore. cau not provide the necessary pressure to confine the cutive boundary of 16 cloud (Ballesteros-Paredes- ,1999). | However, this component is similarly anisotropic and therefore, can not provide the necessary pressure to confine the entire boundary of the cloud (Ballesteros-Paredes 1999). |
The: static: maguetic: field⋅ applies⋅ au effective⋅ pressure. DonsD/8zk. to 16 1nolecular gas and may contribute to the pressure support of the cloud. | The static magnetic field applies an effective pressure, $B^2/8{\pi}k$, to the molecular gas and may contribute to the pressure support of the cloud. |
For a 5jigauss field. this pressure is 7200 en7IK aud comparable to the values required to bind the non-soltf-eravitatiug clouds observed iu us study. | For a $\mu$ gauss field, this pressure is 7200 $cm^{-3}\;K$ and comparable to the values required to bind the non-self-gravitating clouds observed in this study. |
Bertoldi MelIxee (1992) propose that the weight of the sclferavitatine cloud complex squeezes the interchump inediuu to provide an effective mean pressure upon a constituent clump. | Bertoldi McKee (1992) propose that the weight of the self-gravitating cloud complex squeezes the interclump medium to provide an effective mean pressure upon a constituent clump. |
The magnitude of ds- pressure. <Pepm. is. equivalent. to the gravitationalHB euergy deusitvB of the cloud complex. GAL?4D/R! where M. aud AR, are the mass and radius of the cloud. complex respectively, | The magnitude of this pressure, $<P_G>$, is equivalent to the gravitational energy density of the cloud complex, $G M_c^2/R_c^4$ where $M_c$ and $R_c$ are the mass and radius of the cloud complex respectively. |
They. demonstrate that the -iaenitude of this pressure is simular to the required pressure to bind the chumps within the four targeted cloud complexes which they analvzed. | They demonstrate that the magnitude of this pressure is similar to the required pressure to bind the clumps within the four targeted cloud complexes which they analyzed. |
Tf the weielt of the molecular couples is a significant component o the equilibriu of clips.then the required pressure should vary with location of the clump within | If the weight of the molecular complex is a significant component to the equilibrium of clumps,then the required pressure should vary with location of the clump within |
Over the past vears. there have been ereat successes ii nieasureineut of CAIB anisotropy bv eround aud satellite observatious (?7777).. | Over the past years, there have been great successes in measurement of CMB anisotropy by ground and satellite observations \citep{WMAP7:basic_result,ACBAR2008,QUaD2,Planck_mission}. |
Siuco the release of he WMAP data (2? there have been reports on various anolatlies (2°Εν, Tha | Since the release of the WMAP data \citep{WMAP3:temperature,WMAP5:basic_result,WMAP7:basic_result}, there have been reports on various anomalies \citep{cold_spot1,cold_spot2,cold_spot_wmap3,cold_spot_origin,Tegmark:Alignment,Multipole_Vector1,Multipole_Vector2,Multipole_Vector3,Axis_Evil,Axis_Evil2,Axis_Evil3,Universe_odd,Park_Genus,Chiang_NG,correlation_Copi1,correlation_Copi2,Hemispherical_asymmetry,power_asymmetry_subdegree,Power_Asymmetry5,odd,odd_origin,lowl_anomalies,odd_bolpol,WMAP7:anomaly,odd_tension}. |
articular. there are reports on the lack of augular correlation at large angles. which are observed in COBE-DM data aud subsequently in WMAP data (2?????).. | In particular, there are reports on the lack of angular correlation at large angles, which are observed in COBE-DMR data and subsequently in WMAP data \citep{correlation_COBE,WMAP1:Cosmology,correlation_Copi1,correlation_Copi2,lowl_anomalies,lowl_bias}. |
Iu order to figure out the cause of the anomaly. we lave investigated non-cosmnological contamuünation and additionally WALAP team's simulated data. | In order to figure out the cause of the anomaly, we have investigated non-cosmological contamination and additionally WMAP team's simulated data. |
However. we iive not found a definite cause. which makes us believe he anomaly is produced by uuknowan svstematics or Way ρα, Indeed. cosmological. | However, we have not found a definite cause, which makes us believe the anomaly is produced by unknown systematics or may be, indeed, cosmological. |
Iu the anugulhw power spectrmu of WAIAP data. here exist anomalous odd-paritv prefercuce at low uultipole« C2????y.. | In the angular power spectrum of WMAP data, there exist anomalous odd-parity preference at low multipoles \citep{Universe_odd,odd,odd_origin,odd_bolpol,odd_tension}. |
Noting the equivalence between »ower spectrum aud correlation. we have investigated the association between the odd-paritv preference and the ack of lavec-anele correlation. | Noting the equivalence between power spectrum and correlation, we have investigated the association between the odd-parity preference and the lack of large-angle correlation. |
Frou. our iuvestigation. we fd that the odd-parity preference at low iultipoles is in fact. : phenomenological origin of the lack of he luge-augle correlation. | From our investigation, we find that the odd-parity preference at low multipoles is, in fact, a phenomenological origin of the lack of the large-angle correlation. |
Even though it still leaves he fundamental question on its origin unanswered. the association between seenüuglv distinct anomalies will iclp the investigation ou the uucderling origin. whetler cosmological or unaccounted svstematies. | Even though it still leaves the fundamental question on its origin unanswered, the association between seemingly distinct anomalies will help the investigation on the underlying origin, whether cosmological or unaccounted systematics. |
The outline of this paper is as follows. | The outline of this paper is as follows. |
Iu Section 2.. we briefiv discussthe statistical properties of CAIL anisotropy. | In Section \ref{CMB}, we briefly discussthe statistical properties of CMB anisotropy. |
Iu Section 3.. we investigate the angular correlation anomalies of WALAP data. and show the lack of correlation at sinall angles in addition to that at large angles. | In Section \ref{wmap}, we investigate the angular correlation anomalies of WMAP data, and show the lack of correlation at small angles in addition to that at large angles. |
In Section lL. we investigate uou-cosmoloeical contamination and WAIAP team’s simulated data. | In Section \ref{systematics}, we investigate non-cosmological contamination and WMAP team's simulated data. |
Tn Section 5.. we show the odd-paritv prefercuce at low multipoles is a phenomenological origin of the lack of the large-augle correlation. | In Section \ref{odd}, we show the odd-parity preference at low multipoles is a phenomenological origin of the lack of the large-angle correlation. |
Ii Section 7... we stuuimarize our investigation. | In Section \ref{discussion}, we summarize our investigation. |
CXMB anisotropy over a whole-sky is) couvenicutly decomposed in terms of spherical harmonics: where ej, and τω) are a decomposition cocfiicicut anda spherical harmonic function. | CMB anisotropy over a whole-sky is conveniently decomposed in terms of spherical harmonics: where $a_{lm}$ and $Y_{lm}(\hat {\mathbf k})$ are a decomposition coefficient and a spherical harmonic function. |
Iu most of inflationary models. decomposition cocficicuts of CALIB anisotropy follow the Gaussian distribution of the following statistical properties: where denotes the average over an ensenible of universes. and C7; denotes CAIB power spectruni. | In most of inflationary models, decomposition coefficients of CMB anisotropy follow the Gaussian distribution of the following statistical properties: where $\langle \ldots \rangle$ denotes the average over an ensemble of universes, and $C_l$ denotes CMB power spectrum. |
Given CMD anisotropy data. we may estimate two poiut aneular correlation: where 0=cosH“(y+m»). | Given CMB anisotropy data, we may estimate two point angular correlation: where $\theta=\cos^{-1}(\hat{\mathbf n}_1\cdot \hat{\mathbf n}_2)$. |
Using Eq. | Using Eq. |
1. aud 2.. we may easilv show that the expectation value of the correlation is given by (23: where ϐ is a separation anesle. Wy is the window function of the observation aud 2 is a Legeudre polvuonuidal | \ref{T_expansion} and \ref{C_lm}, we may easily show that the expectation value of the correlation is given by \citep{structure_formation}: : where $\theta$ is a separation angle, $W_l$ is the window function of the observation and $P_l$ is a Legendre polynomial. |
From Eq. L. | From Eq. \ref{cor}, |
we may easily see the augular correlation C'(0) and power spectrum Cp possess sole equivalence. | we may easily see the angular correlation $C(\theta)$ and power spectrum $C_l$ possess some equivalence. |
Iun Fig. l.. | In Fig. \ref{C_data}, , |
we show the augular correlation ofthe WMAP 7 year data. which are estimated respectively | we show the angular correlation ofthe WMAP 7 year data, which are estimated respectively |
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