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Over such a short time span (compared to (vpical stellar orbital periods of ~10* v αἱ 0.1 pe radii). stellar motions are verv short orbital ares. | Over such a short time span (compared to typical stellar orbital periods of $\sim10^3$ y at 0.1 pc radii), stellar motions are very short orbital arcs. |
Thus. there is no need to include the gravitational effects of individual stars on each other as is done in N-body calculations: we can assume that stars move along orbits determined by the mass enclosedwithin (heir semi-major axes. | Thus, there is no need to include the gravitational effects of individual stars on each other as is done in N-body calculations; we can assume that stars move along orbits determined by the mass enclosedwithin their semi-major axes. |
By avoiding [ull N-body ealeulations. we are ignoring anv collective effects | By avoiding full N-body calculations, we are ignoring any collective effects |
stronger reflection during the higher flux periods). with minimal changes in fix... | stronger reflection during the higher flux periods), with minimal changes in $F_{\rm K\alpha}$. |
We note that our interpretation that variability largely comes from within the innermost stable orbit for a Schwarzschild black hole may be consistent with the scenario for a very active corona (and hence strong hard X-ray emission) within Gry. proposed by Krolik (1999). | We note that our interpretation that variability largely comes from within the innermost stable orbit for a Schwarzschild black hole may be consistent with the scenario for a very active corona (and hence strong hard X-ray emission) within $6
r_g$, proposed by Krolik (1999). |
In this model. magnetic fields within the radius of marginal stability are strong and amplified through shearing of their footpoints. which can enhance variability. | In this model, magnetic fields within the radius of marginal stability are strong and amplified through shearing of their footpoints, which can enhance variability. |
The constancy of the iron line on day-to-day scales suggests that the timescale for variability (.e. the observed periods for which dramatic flux changes are observed) we are naively probing are much larger din the “standard” scenario) than the fluorescing region. | The constancy of the iron line on day-to-day scales suggests that the timescale for variability (i.e. the observed periods for which dramatic flux changes are observed) we are naively probing are much larger (in the `standard' scenario) than the fluorescing region. |
In other words. slow changes would imply on a naive model much arger crossing times and hence large regions for the crossing times of the continuum. | In other words, slow changes would imply on a naive model much larger crossing times and hence large regions for the crossing times of the continuum. |
A light-crossing time of the fluorescing region arger than 50 Ks (assuming an average radius 200.) will ead to an estimate for the black hole mass 107M.. | A light-crossing time of the fluorescing region larger than $\sim$ 50 ks (assuming an average radius $\sim 20 r_s$ ) will lead to an estimate for the black hole mass $\rm \sim 10^8 M_\odot$. |
Reynolds (1999) points out however that the bulge/hole mass relationship of Tagorrian et al. ( | Reynolds (1999) points out however that the bulge/hole mass relationship of Magorrian et al. ( |
1998) implies amuch lower mass estimate for 6-30-15. by an order of magnitude. of about —10M.. | 1998) implies amuch lower mass estimate for $-$ 6-30-15, by an order of magnitude, of about $\rm \sim 10^7 M_{\odot}$. |
In the scenario of the simple model presented above. and given evidence for short timescale variability of the iron line (here and I99) as well as the location of the flare line found in I99. the constancy of the line suggests that the timescale for variability for which we are probing is much than the fluorescing region and reconciles the above mass problem. | In the scenario of the simple model presented above, and given evidence for short timescale variability of the iron line (here and I99) as well as the location of the flare line found in I99, the constancy of the line suggests that the timescale for variability for which we are probing is much than the fluorescing region and reconciles the above mass problem. |
Of further interest is the apparent break in the power spectrum of 6-30-15 seen in both the and data. | Of further interest is the apparent break in the power spectrum of $-$ 6-30-15 seen in both the and data. |
The origin of the break is not vet known (but see e.g. Edelson Nandra 1999: Poutanen Fabian 1999: Kazanas. Hua. Titarchuk 1997: and Cui et al. | The origin of the break is not yet known (but see e.g. Edelson Nandra 1999; Poutanen Fabian 1999; Kazanas, Hua, Titarchuk 1997; and Cui et al. |
1997 for possible explanations). but does provide a useful means to determine the black hole mass. through sealing from similar breaks in the power spectrum of the famous galactic black hole candidate (GBH) Cygnus X-1. and other objects like it. | 1997 for possible explanations), but does provide a useful means to determine the black hole mass, through scaling from similar breaks in the power spectrum of the famous galactic black hole candidate (GBH) Cygnus X-1, and other objects like it. |
The behaviour of the PDS in 6-30-15 is not unlike that of GBHs in the ‘low’ (hard) state (see e.g. Belloni Hasinger 1990: Mivamoto et al. | The behaviour of the PDS in $-$ 6-30-15 is not unlike that of GBHs in the `low' (hard) state (see e.g. Belloni Hasinger 1990; Miyamoto et al. |
1992: van der Klis 1995). | 1992; van der Klis 1995). |
Power law slopes (with form f") of order a— -I to -2 are observed at high frequencies and flatten to —O at lower frequencies. | Power law slopes (with form $f^{\alpha}$ ) of order $\alpha \sim\ $ -1 to -2 are observed at high frequencies and flatten to $\sim 0$ at lower frequencies. |
If we bridge the gap between AGNs and GBHs and assume that similar physics are at play. we can make predictions for the black hole mass in 6-30-15 (using the values for the cutoff frequencies fi.) by a simple scaling relation with Cygnus X-I. | If we bridge the gap between AGNs and GBHs and assume that similar physics are at play, we can make predictions for the black hole mass in $-$ 6-30-15 (using the values for the cutoff frequencies $f_{br}$ ) by a simple scaling relation with Cygnus X-1. |
Belloni Hasinger (1990) report that fi,~ 0.04-0.4 Hz for Cygnus X-1: for 6- we find evidence that fi,~45.10. Az. | Belloni Hasinger (1990) report that $f_{br} \sim$ 0.04-0.4 Hz for Cygnus X-1; for $-$ 6-30-15, we find evidence that $f_{br} \sim 4-5 \times 10^{-6}$ Hz. |
The resulting ratio between the 2 cutoff frequencies is 107 10°. | The resulting ratio between the 2 cutoff frequencies is $\sim
10^4-10^5$ . |
Herrero et al. ( | Herrero et al. ( |
1995) argue that the black hole mass in Cygnus X-I is | 1995) argue that the black hole mass in Cygnus X-1 is |
The model was computed by GLT97 on the basis of a I-D. implicit hydro code. assuming accretion of solar composition matter (Z= 0.02) onto the surface of a 1Me CO white dwarf at a rate of 5x107Me !. | The model was computed by GLT97 on the basis of a 1-D, implicit hydro code, assuming accretion of solar composition matter $Z=0.02$ ) onto the surface of a $1 \, M_{\sun}$ CO white dwarf at a rate of $5 \times 10^{-9}\, M_{\sun}$ $^{-1}$. |
The accumulation of matter 1 degenerate conditions drives a temperature increase in the envelope. resulting in a superadiabatic temperature gradient and eventually convective transport. | The accumulation of matter in degenerate conditions drives a temperature increase in the envelope, resulting in a superadiabatic temperature gradient and eventually convective transport. |
The initial model corresponds to the time when the temperature at the innermost envelope zone is =105 K. At this stage. the mass of the accreted envelope reaches 2x107Me. | The initial model corresponds to the time when the temperature at the innermost envelope zone is $\approx 10^8$ K. At this stage, the mass of the accreted envelope reaches $2 \times 10^{-5}\,
M_{\sun}$. |
This radial profile has been mapped onto a 2-D cartesian grid of 800x800 km and is initially relaxed to guarantee hydrostatic equilibrium. | This radial profile has been mapped onto a 2-D cartesian grid of $\times$ 800 km and is initially relaxed to guarantee hydrostatic equilibrium. |
The initial computational grid comprises 112. radial layers (including the outermost part of the CO core) and 512 lateral layers. | The initial computational grid comprises 112 radial layers (including the outermost part of the CO core) and 512 lateral layers. |
Calculations rely on the adaptive mesh refinement with a minimum resolution 1.6x1.6 km (simulations with a finer resolution will be presented in a forthcoming publication). | Calculations rely on the adaptive mesh refinement with a minimum resolution $\times$ 1.6 km (simulations with a finer resolution will be presented in a forthcoming publication). |
A reduced nuclear reaction network was used to compute the energetics of the explosion: it consists of 13 isotopes (!H. tHe. FRCL ISBN, HUSTST0, and UE -- as in GLT97 and KHT98 — supplemented with '*F to include the important "Op. y)!SP reaction). linked through a net of 18 nuclear processes (mainly. p-captures and -decays). | A reduced nuclear reaction network was used to compute the energetics of the explosion: it consists of 13 isotopes $^1$ H, $^{4}$ He, $^{12,13}$ C, $^{13,14,15}$ N, $^{14,15,16,17}$ O, and $^{17}$ F – as in GLT97 and KHT98 – supplemented with $^{18}$ F to include the important $^{17}$ O(p, $\gamma$ $^{18}$ F reaction), linked through a net of 18 nuclear processes (mainly, p-captures and $\beta^+$ -decays). |
Reaction rates are taken from Angulo et al. ( | Reaction rates are taken from Angulo et al. ( |
1999) and some more recent updates (see José.. Hernanz. Iltadis 2006. José Shore 2008. and references therein). | 1999) and some more recent updates (see José,, Hernanz, Iliadis 2006, José Shore 2008, and references therein). |
Periodic boundary conditions were adopted at both lateral sides. while hydrostatic boundary conditions are fixed at both the bottom (reflecting) and the top The set of boundary conditions are similar to those implemented in GLT97 and KHT98. but note that the outer computational grid adopted in GLT97 is Lagrangian instead of Eulerian (to follow the late expansion stages of the TNR). | Periodic boundary conditions were adopted at both lateral sides, while hydrostatic boundary conditions are fixed at both the bottom (reflecting) and the top The set of boundary conditions are similar to those implemented in GLT97 and KHT98, but note that the outer computational grid adopted in GLT97 is Lagrangian instead of Eulerian (to follow the late expansion stages of the TNR). |
Finally. energy transport 15 included using an effective thermal diffusion coefficient that includes radiative and conductive opacities (Timmes 2000). | Finally, energy transport is included using an effective thermal diffusion coefficient that includes radiative and conductive opacities (Timmes 2000). |
In GLT97. significant numerical noise was present at the onset of their calculations that produced temperature fluctuations of about10-20%. | In GLT97, significant numerical noise was present at the onset of their calculations that produced temperature fluctuations of about. |
.. We introduced (ust at the initial. time-step) a Gaussian temperature perturbation at the core-envelope interface of5%. | We introduced (just at the initial time-step) a Gaussian temperature perturbation at the core-envelope interface of. |
. For comparison. the value in KHT98 wasΤ | For comparison, the value in KHT98 was. |
σο The size of the initial perturbation was 2 km. much smaller than the contemporary depth of the accreted envelope (~800 km). | The size of the initial perturbation was 2 km, much smaller than the contemporary depth of the accreted envelope $\sim800$ km). |
The initial perturbation produces fluctuations that move along the core-envelope interface during the first seconds of the simulations. | The initial perturbation produces fluctuations that move along the core-envelope interface during the first seconds of the simulations. |
These fluctuations. in turn. spawn Kelvin-Helmholtz vortices. which clearly show up about 200 s later (Fig. | These fluctuations, in turn, spawn Kelvin-Helmholtz vortices, which clearly show up about 200 s later (Fig. |
1). and appear to initiate a turbulent cascade. | 1), and appear to initiate a turbulent cascade. |
Filaments and buoyant plumes are fully resolved in these simulations. | Filaments and buoyant plumes are fully resolved in these simulations. |
At this stage. the fluid is characterized by a large Reynolds number. with a characteristic eddy length of 50 km. fluid velocities of 10°— em s' and a dynamic viscosity of 10+ P. These Kelvin-Helmholtz (KH) instabilities transport unburnt CO-rich material from the outmost layers of the white dwarf core and inject it into the envelope. | At this stage, the fluid is characterized by a large Reynolds number, with a characteristic eddy length of 50 km, fluid velocities of $10^5 -
10^6$ cm $^{-1}$, and a dynamic viscosity of $10^4$ P. These Kelvin-Helmholtz (KH) instabilities transport unburnt CO-rich material from the outmost layers of the white dwarf core and inject it into the envelope. |
The characteristic eddy turnover time Is lvo ~ LO 8. As the KH vortices grow in size. more CO-rich material is transferred into the envelope. | The characteristic eddy turnover time is $_{\rm conv}$ $_{\rm
conv}$ $\sim$ 10 s. As the KH vortices grow in size, more CO-rich material is transferred into the envelope. |
Convection becomes more turbulent. | Convection becomes more turbulent. |
The initially small convective eddies merge into huge shells (Fig. | The initially small convective eddies merge into huge shells (Fig. |
2). as seen also in GLT97. | 2), as seen also in GLT97. |
At this stage. the nuclear energy generation rate reaches 10 erg e! sl. while the characteristic burning timescale decreases to ~5 s. The convective filaments continue growing m size and progressively occupy the whole envelope length. | At this stage, the nuclear energy generation rate reaches $10^{15}$ erg $^{-1}$ $^{-1}$, while the characteristic burning timescale decreases to $\sim$ 5 s. The convective filaments continue growing in size and progressively occupy the whole envelope length. |
Although not resolved in these simulations. and in contrast to the 3-D case. the conservation of vorticity in 2-D forces the largest eddies to grow in an inverse vorticity cascade. while energy flows to the viscous scale with a distribution that deviates from the Kolmogorov spectrum (see e.g.. Lesieur. Yaglom. David 2000; Shore 2007). | Although not resolved in these simulations, and in contrast to the 3-D case, the conservation of vorticity in 2-D forces the largest eddies to grow in an inverse vorticity cascade, while energy flows to the viscous scale with a distribution that deviates from the Kolmogorov spectrum (see e.g., Lesieur, Yaglom, David 2000; Shore 2007). |
At this time. the temperature at the envelope base reaches ~2x10? K. at fluid velocities of 10* cm s! (of the order of the escape velocity. characteristic of the dynamic phases of the explosion). and a nuclear energy generation rate of 10! erg g' sl. | At this time, the temperature at the envelope base reaches $\sim2
\times 10^8$ K, at fluid velocities of $10^8$ cm $^{-1}$ (of the order of the escape velocity, characteristic of the dynamic phases of the explosion), and a nuclear energy generation rate of $10^{16}$ erg $^{-1}$ $^{-1}$. |
The convective turnover time is now 5 s, The mean CNO abundance in the envelope has increased to 0.30. a value that agrees well with both the previous simulations by GLT97 and the mean metallicities inferred from observations of the ejecta in non-neon (CO) novae (see José Shore 2008) | The convective turnover time is now $\sim$ 5 s. The mean CNO abundance in the envelope has increased to 0.30, a value that agrees well with both the previous simulations by GLT97 and the mean metallicities inferred from observations of the ejecta in non-neon (CO) novae (see José Shore 2008). |
At this stage. since the outer envelope layers had started to escape the computational (Eulerian) domain. simulations were stopped. | At this stage, since the outer envelope layers had started to escape the computational (Eulerian) domain, simulations were stopped. |
Our 2-D simulations. in agreement with the results reported in GLT97. show that the progress and extension of the TNR throughout the envelope occurs with almost spherical symmetry. even though the structure of the ignition is not. | Our 2-D simulations, in agreement with the results reported in GLT97, show that the progress and extension of the TNR throughout the envelope occurs with almost spherical symmetry, even though the structure of the ignition is not. |
This explains the suecess of I-D models in reproducing the gross observational properties (light curves. velocities of the ejecta. nucleosynthesis) of nova explosions (Starrfield et al. | This explains the success of 1-D models in reproducing the gross observational properties (light curves, velocities of the ejecta, nucleosynthesis) of nova explosions (Starrfield et al. |
1998. 2009; Kovetz Prialnik 1997; Yaron et al. | 1998, 2009; Kovetz Prialnik 1997; Yaron et al. |
2005; José Hernanz 1998). | 2005; José Hernanz 1998). |
We have analyzed the possible self-enrichment of the solar-composition acereted envelope with material from the underlying white dwarf. during nova outbursts in. a multidimensional framework. | We have analyzed the possible self-enrichment of the solar-composition accreted envelope with material from the underlying white dwarf during nova outbursts in a multidimensional framework. |
We have found that a shear flow at the core-envelope interface (which unlike the spherically symmetric case. does not behave like a rigid wall) drives mixing through KH instabilities. | We have found that a shear flow at the core-envelope interface (which unlike the spherically symmetric case, does not behave like a rigid wall) drives mixing through KH instabilities. |
Large convective eddies develop close to the core-envelope interface. of a size comparable to the height of the envelope (similar to the pressure scale height in I-D simulations). mixing CO-rich material from the outermost layers of the underlying white dwarf into the accreted envelope. | Large convective eddies develop close to the core-envelope interface, of a size comparable to the height of the envelope (similar to the pressure scale height in 1-D simulations), mixing CO-rich material from the outermost layers of the underlying white dwarf into the accreted envelope. |
The metallicity enrichment achieved 1n the envelope. Z~0.30. is 1n. agreement with observations of CO nova ejecta. | The metallicity enrichment achieved in the envelope, $Z \sim 0.30$, is in agreement with observations of CO nova ejecta. |
Our 2-D simulations also | Our 2-D simulations also |
jbend. | band. |
hDuteraction of light with small particles has this effect. | Interaction of light with small particles has this effect. |
It also shifts the ου to the observed peak position. | It also shifts the band to the observed peak position. |
Alternatively. the observed baud widths. aud peak positions⋅⋅ are reasonably . ↽↽↽⋅ ⊔⋜↧↑↸⊳↕∐∖≼↧↖∏↑∐⋀∖⋅≻∶≼⊲∪∶↓∶↓↕↸⊳↸∖↴∖↴∪↥⋅↑↕∐∖↥⋅⋯⋜↧∐↖↽↻↥⋅∪↸⊳↸∖↴∖↴↴∖ | Alternatively, the observed band widths and peak positions are reasonably matched with $_2$ :CO=1:1 ices or thermally processed $_2$ --rich ices. |
↴↸∖≼↧∪⋅≻ M tliHY tiredqq wild1οςdeeree ofo IRSO31dthermal dualaabsorptiou baudrequires: Osb "ices b venlictieb a1- theL] rela THN voltsight isdet further discussed linin 51. | Whether the mild degree of thermal processing of $_2$ ices is realistic in the line of sight is further discussed in 4.1. |
SLI absorption features ↕∐∪↥⋅≼∐∖↥⋅↑∪≼∐∖↑↸∖↥⋅∐∐∐↸∖⋜⋯⋜⋯⊳↿∐⋅⋜↧↑↸∖∟≼⊲∪↓⇝≼⊲∪ ⋅⋅⋅ ∐⋅⋜↧⊓∪↕↑↕↴∖↴∐↸∖↸⊳↸∖↴∖↴↴∖↴⋜∐⋅↖↽↑∪↸⊳∪∐⋅↸∖↸⊳↑↑∐↸∖∟≼∪ for↴⋅∙ absorption inue its extended wines. | In order to determine an accurate ratio it is necessary to correct the absorption band for absorption in its extended wings. |
hi11ο. 100Whether kuowledee- of: the intrinsic: shape of:+ocessine the Twof P responsibleB forH the. apparcuti wines. | This requires knowledge of the intrinsic shape of the absorption features responsible for the apparent wings. |
ine oft lusight⋅ is⋅ gained⋅ by comparing: the observed | Insight is gained by comparing the observed |
a shorter infall-peak time. | a shorter infall-peak time. |
The model predictions are shown as the dot lines in Fig. | The model predictions are shown as the dot lines in Fig. |
1 and 2. | 1 and 2. |
It can be seen that the model adopting later t, predicts bluer color, lower metallicity, higher gas fraction and higher specific star formation rate SFR/M.. both in the local Universe and at intermediate redshift, at least up to z~1. | It can be seen that the model adopting later $t_p$ predicts bluer color, lower metallicity, higher gas fraction and higher specific star formation rate $SFR/M_*$ both in the local Universe and at intermediate redshift, at least up to $z \sim 1$. |
This is mainly due to the fact that, in our model, the setting of later infall-peak time corresponds to slower gas infall process and thus slower star formation processes. | This is mainly due to the fact that, in our model, the setting of later infall-peak time corresponds to slower gas infall process and thus slower star formation processes. |
Furthermore, Fig. | Furthermore, Fig. |
1 and 2 show that the separations between the solid and dash lines almost cover the observed range of different data sets, especially the SFR/M, at intermediate redshifts. | 1 and 2 show that the separations between the solid and dash lines almost cover the observed range of different data sets, especially the $SFR/M_*$ at intermediate redshifts. |
In other words, the infall-peak time t, significantly influence the galactic evolution. | In other words, the infall-peak time $t_p$ significantly influence the galactic evolution. |
The impact of outflow (described by bout) on the model predictions is presented by the dot-dash lines in Fig. | The impact of outflow (described by $b_{\rm out}$ ) on the model predictions is presented by the dot-dash lines in Fig. |
1 and 2, where we adopt bout=1.0 and iy= 10Gyr. | 1 and 2, where we adopt $b_{\rm out}=1.0$ and $t_p=10$ Gyr. |
Comparison between solid lines and dot-dash lines shows that the gas outflow process mainly influences the final gas-phase metallicity, since it takes a fraction of metal away from the disk. | Comparison between solid lines and dot-dash lines shows that the gas outflow process mainly influences the final gas-phase metallicity, since it takes a fraction of metal away from the disk. |
Moreover, after including gas outflow process, the model predicts bluer optical-IR (both g—J and g— K) color, while the impact on the optical color is very small, if there is any. | Moreover, after including gas outflow process, the model predicts bluer optical-IR (both $g-J$ and $g-K$ ) color, while the impact on the optical color is very small, if there is any. |
This is mainly due to the fact that the stellar mean age influences the optical color more strongly than the optical-IR color, while the mean stellar metallicity influences the optical-IR color more strongly than the optical color (Chang et al. | This is mainly due to the fact that the stellar mean age influences the optical color more strongly than the optical-IR color, while the mean stellar metallicity influences the optical-IR color more strongly than the optical color (Chang et al. |
2006b). | 2006b). |
The star formation law is another important ingredient in our model. | The star formation law is another important ingredient in our model. |
In this paper, we adopt a Kennicutt star formation law. | In this paper, we adopt a Kennicutt star formation law. |
Although it works well in the region of high surface density, we should be caution to apply it to low density region, such as the outskirts of disk galaxies. | Although it works well in the region of high surface density, we should be caution to apply it to low density region, such as the outskirts of disk galaxies. |
To explore the influence of the adopted star formation law on our model predictions, we adopt a modified star formation law W(r,t)=0.05X2?(r,t) to reduce the star formation efficiency (SFE) and plot the model predictions as dash lines in Fig. | To explore the influence of the adopted star formation law on our model predictions, we adopt a modified star formation law $\Psi(r,t)=0.05\Sigma_{g}^{1.4}(r,t)$ to reduce the star formation efficiency (SFE) and plot the model predictions as dash lines in Fig. |
1 and 2. | 1 and 2. |
The free parameter combination is adopted | The free parameter combination is adopted |
spectrum CC.A). defined by ada ve ο... of which is This can be seen as au exteusion of the usual 2D power spectrum «οOmm:. | spectrum $C(\ell, k)$, defined by a ve estimator of which is This can be seen as an extension of the usual 2D power spectrum $\left< f_{\ell m}f^*_{\ell' m'}\right>=C_l \delta_{\ell \ell'}\delta_{mm'}$. |
Tho latter arises frou the spherical harmonic παιδίον of a 2D field given ou the sphere f(0.0)=M,fonVin(Go). | The latter arises from the spherical harmonic transform of a 2D field given on the sphere $f(\theta,\phi) = \sum_{\ell m}f_{\ell m}Y_{\ell m}(\theta,\phi)$. |
Iu practice. surveys will only cover a finite amount of volume. linitiug the analysis to a sphere of radius HR. | In practice, surveys will only cover a finite amount of volume, limiting the analysis to a sphere of radius $R$. |
These boundary conditions lead to a discrete spectrum evn}. Which is detailed in the appendices. | These boundary conditions lead to a discrete spectrum $\{ k_{\ell n} \}$, which is detailed in the appendices. |
In this paper. we assunied as a boundary condition that f vanishes at r=RR. | In this paper, we assumed as a boundary condition that $f$ vanishes at $r=R$. |
The spherical Fouricr-Bessel decomposition becomes (2?) which is exact if the ranges of £a) andη are infinite. | The spherical Fourier-Bessel decomposition becomes \citep{Erdogdu:2006dv,Fisher:1995}
which is exact if the ranges of $\ell$ $m$ and$n$ are infinite. |
The Fourier-Bessel cocfiicients are denoted by fr,Fr (δρ). aud he, Us the normalisation constant (see appendices for more details). | The Fourier-Bessel coefficients are denoted by $f_{\ell mn} = f_{\ell m}(k_{\ell n})$ , and $\kappa_{\ell n}$ is the normalisation constant (see appendices for more details). |
Tn various applications. though. the coutinnous field f cannot be directly observed. | In various applications, though, the continuous field $f$ cannot be directly observed. |
This is notably the case in COSDAOogv where galaxy surveys give indirect information about the underline matter density field through their spacial positions. | This is notably the case in cosmology where galaxy surveys give indirect information about the underlying matter density field through their spacial positions. |
Note that these tracers are subject to various distortions and non-lneaties. but these are not the prpose of this work. | Note that these tracers are subject to various distortions and non-linearities, but these are not the purpose of this work. |
Iu this work we oulv consider linear or quasi-linear scales (6 «50. k«02hMpe ή], | In this work we only consider linear or quasi-linear scales $\ell< 50$ , $~k<0.2 {\rm hMpc}^{-1}$ ). |
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