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It is worth mentioning that the region R at the PIL is between flare ribbons/kernels at opposite polarities, hence the observed field changes cannot be attributed to flare emissions (Patterson& 2003).. | It is worth mentioning that the region R at the PIL is between flare ribbons/kernels at opposite polarities, hence the observed field changes cannot be attributed to flare emissions \citep{patterson81,qiu03}. |
Detailed investigation of the flare HXR emission in further relation to the coronal field dynamics is out of the scope of the current study and will be presented in a subsequent publication. | Detailed investigation of the flare HXR emission in further relation to the coronal field dynamics is out of the scope of the current study and will be presented in a subsequent publication. |
We have used the unprecedented SDO/HMI vector field observations to analyze the changes of the photospheric magnetic field associated with the first X-class flare in the solar | We have used the unprecedented SDO/HMI vector field observations to analyze the changes of the photospheric magnetic field associated with the first X-class flare in the solar |
emitted by the outer (optically thin) region In order to find an analvlic expression for Fy. we will suppose that both the temperature in (he outer part of the envelope and the grain opacity can be approximated by power-laws. so we wrile and ∖∖↽↥∐↲↕⋅≼↲⋋∣∣↥⊳∖⊽≀↧↴∐≀↧↴↕⋅∣↽≻∐↕⋅≀↧↴↕⋅⋡∖↽≺∢ | emitted by the outer (optically thin) region In order to find an analytic expression for $F_\lambda$, we will suppose that both the temperature in the outer part of the envelope and the grain opacity can be approximated by power-laws, so we write andwhere $\lambda_0$ is an arbitrary constant. |
∪∐⊳∖⊽↥≀↧↴∐↥⋅↴∏∐↲≀↧⊔∖⇁⊳∖⊽⋯∐↕↽≻∐∪∐↓⋟∪↕⋅⊔∐↲∪↕↽≻≀↧↴≺∢∐⋡∖↽↥⊳∖⊽∖↽≼↲↕⋅⋡∖↽↕⋅≼↲≀↧⊔∖⇁∪∐≀↧↴∣↽≻↥≼↲⋅ because by definition most of the fhix emerges at wavelengths larger than the Ravleigh limit (see Figure 1 and discussion in section 2)). | The assumption for the opacity is very reasonable, because by definition most of the flux emerges at wavelengths larger than the Rayleigh limit (see Figure \ref{effsil} and discussion in section \ref{as}) ). |
The assumption for the temperature is justified bv the fact that the outer part of the envelope is optically thin: it can be shown that. if the dust opacily is given by eq. (12)). | The assumption for the temperature is justified by the fact that the outer part of the envelope is optically thin; it can be shown that, if the dust opacity is given by eq. \ref{kappa}) ), |
the radial dependence of the temperature of an optically thin envelope is Tir)xr7?Ὁ (eg. Martin(1978)... eq. | the radial dependence of the temperature of an optically thin envelope is $T(r) \propto r^{-2/(p+4)}$ (e.g. \citet{mar78}, eq. |
7.3). | 7.3). |
Our detailed calculations. shown below. validate this assumption. | Our detailed calculations, shown below, validate this assumption. |
Finally. assuming a dust densityv. p(r). also given by a power-law we calculate 7,24 using eqs. (9)). (12)) | Finally, assuming a dust density $\rho(r)$ , also given by a power-law we calculate $r_{\tau=1}$ using eqs. \ref{tauc}) ), \ref{kappa}) ) |
ancl (13)). which gives where is the envelope optical depth at A= Ag. | and \ref{dens}) ), which gives where is the envelope optical depth at $\lambda = \lambda_0$ . |
substituting eqs. (11)). (12)). (13)). | Substituting eqs. \ref{temp}) ), \ref{kappa}) ), \ref{dens}) ), |
and in eq. (10)) | and in eq. \ref{emis1}) ) |
and letting ἐς— 2€. we obtain thespectral shapeof the radiation emitted by the outer part of the envelope | and letting $r_{\rm e} \rightarrow \infty$ , we obtain thespectral shapeof the radiation emitted by the outer part of the envelope |
Figure 1 [n]0ives a Compilation of the results from prior searches for dust in rieli aud poor galaxy clusters. as wel as groups. | Figure \ref{compil} gives a compilation of the results from prior searches for dust in rich and poor galaxy clusters, as well as groups. |
There is some observational evidence for the existence of dust in the central regions cX clusters. | There is some observational evidence for the existence of dust in the central regions of clusters. |
For example. 1sine IRAS images of 56 clusters. Wiseetal.(1993) [ouud two clusters wit1 Far-IR color excesses thei are probably due to dust. | For example, using IRAS images of 56 clusters, \citet{WOBR93} found two clusters with Far-IR color excesses that are probably due to dust. |
Meanwhile. Hu(1992) derived all aveage value of the differential reddeune of E(B-V)-0219 on the basis of observed emissior line ratios (Lya/Ha) in a sample of 10 Abell cooling flow clusters. | Meanwhile, \citet{H92} derived an average value of the differential reddening of $E(B - V) \sim 0\fm19$ on the basis of observed UV-to-optical emission line ratios $\alpha / $ $\alpha$ ) in a sample of 10 Abell cooling flow clusters. |
Furthermore. east'einents of X-ray spectra have indicated surprisingly high metallicities. Z~0.2—0.5Z.. iu clusters (Bahca|1999:Arneuid&Nushotzky1998:VoitDonahue1995). | Furthermore, measurements of X-ray spectra have indicated surprisingly high metallicities, $Z \sim 0.2 - 0.5\,Z_{\sun}$, in clusters \citep{B99,AM98,VD95}. |
. Yet it should also be noted hat Auuis&Jewitt(1993) [aileca o detect any subimnillimeter dust emission from the central few teis of kpe in 1)| cooling flow clIslers. | Yet it should also be noted that \citet{AJ93} failed to detect any submillimeter dust emission from the central few tens of kpc in 11 cooling flow clusters. |
Evidence for dist in regious ousicle the central o0.5 Mpe is inconclusive. | Evidence for dust in regions outside the central $\;\simgt\; 0.5$ Mpc is inconclusive. |
Past searches [or dust o Lz lpe scales incorporated several techuiiqles. Zwicky(1962).. Ixaracleutsev&Lipovetskii(1969) eVacouleus.deVaucouleus&Corwin (1972). | Past searches for dust on $\simgt$ Mpc scales incorporated several techniques. \citet{Z62}, \citet{KL69}, \citet{dVC72}, |
. aud Bogart&Wagouer(1973) looked [or the exinctiou ol lelt from distant e:ανν clusters v intervening foregrouud custers. | and \citet{BW73} looked for the extinction of light from distant galaxy clusters by intervening foreground clusters. |
The observed depletion iu ἰle LULLiber deusity of ockgrouud clustersled Zwicky(1962) to suggest that there may be as inauy as 0.E maguitudes of extinction i1 the Coma Cluster. | The observed depletion in the number density of background clustersled \citet{Z62} to suggest that there may be as many as 0.4 magnitudes of extinction in the Coma Cluster. |
Bogar&Wagoner(1973) estimate (he€I ο expain the dearth of backgrouud clusters. foreground clusters uimst produce about 0.12 maenitucdes(uc of 'edadenius over radii of few Λpe. | \citet{BW73} estimate that to explain the dearth of background clusters, foreground clusters must produce about 0.12 magnitudes of reddening over radii of few Mpc. |
However. the results ο“these studies could have been affected by LintluaL bias: Distant galaxy. clusters are much harder to detect if they are superimposed ou [oregrotud clusters. so the apparent clepletion of «instant clusters i tlie directions oL nearby clusters could yO€ue to causes other than dust. | However, the results of these studies could have been affected by human bias: Distant galaxy clusters are much harder to detect if they are superimposed on foreground clusters, so the apparent depletion of distant clusters in the directions of nearby clusters could be due to causes other than dust. |
Romani&Maoz(1992) looked at the apparent auti-correlation of htiel-redsift QSOs with vearby Abell chsters. | \citet{RM92} looked at the apparent anti-correlation of high-redshift QSOs with nearby Abell clusters. |
TIev neasured a deficit of ~25% in the quasar ium)er COULs within of ‘ich clusters ae j0tecl tl€at such a dearth of QSOs cau be explainec N Ap=IMLS «of extinetion in jese clusters. | They measured a deficit of $\sim 25 \%$ in the quasar number counts within of rich clusters and noted that such a dearth of QSOs can be explained by $A_B = 0\fm 15$ of extinction in these clusters. |
However. a folow-up work by Maoz(1995)reached :v cliTereit conel5101. | However, a follow-up work by \citet{M95} reached a different conclusion. |
He found 10 difference iu the colors of QSOs found within 1? it oro,jection [roi LAyell clusters to tlose farther in 37 away alk he placed a 90% coulidence upper liiiit on redeulneg o [E(5b—V)=0m)5. | He found no difference in the colors of QSOs found within $1\degr$ in projection from Abell clusters to those farther than $3\degr$ away and he placed a $90 \% $ confidence upper limit on reddening of $E(B-V) = 0\fm05 $. |
foid that blue Tulls-Fisher cdistauces to Virgo €uster spirals correate tightly with lel: color excesS. while uo such correlation was seel iu field spi‘als. | \citet{B90} found that blue Tully-Fisher distances to Virgo cluster spirals correlate tightly with their color excess, while no such correlation was seen in field spirals. |
They suggest that intracluster dust. with reddeing of ~0.12 maeuituces would mase more clistant galaxies within Virgo appear ecer. | They suggest that intracluster dust with reddening of $\sim 0.12$ magnitudes would make more distant galaxies within Virgo appear redder. |
However. Ferguson(1903) analvzed fits to l ecorrelatious between B—V colors and the 'eddenine-iuseusitive Meo iidex (the ratio of the Mg 6 lue [lux to the adjacent continuum flux that straddles the line) for a sample of elliptical galaxies aud found no evidence for reddening in either he Virgo or the Coma clusers. | However, \citet{F93} analyzed fits to the correlations between $B - V$ colors and the reddening-insensitive $_{2}$ index (the ratio of the Mg $b$ line flux to the adjacent continuum flux that straddles the line) for a sample of elliptical galaxies and found no evidence for reddening in either the Virgo or the Coma clusters. |
He placed 90% coufidetce upper limits of E(B—V)=Q"06 and O05 for the Virgo aud Coma clusters. respectively. | He placed $90\% $ confidence upper limits of $E(B - V) = 0\fm06$ and $0\fm05$ for the Virgo and Coma clusters, respectively. |
Dweς.Rephaeji&Mather(1990) were unable o detect auy IR. emission rom dust in the Coma Cluster. cdespie the earlier optical estimates w Zwicky (1962).. | \citet{DRM90} were unable to detect any IR emission from dust in the Coma Cluster, despite the earlier optical estimates by \citet{Z62}. . |
Iu fact. he upper limit placed ou tle l00pan emission was 100 times lower han the theoretical estimaes of Voshiehinutkoy&IxheποΣΙ](198 [).. which were based ou the interactions between dust aid the hot ICM. | In fact, the upper limit placed on the $100 \micron$ emission was $\sim 400$ times lower than the theoretical estimates of \citet{VK84}, , which were based on the interactions between dust and the hot ICM. |
he image position @ as J;—0;τό). where c ds the deflection potential. indices separated by a com, denote ouwtial derivatives with respect to 0;. and sununation over repeated indices is nuaplied. | the image position $\vc\theta$ as $\beta_i=\theta_i-\psi_{,ij}\theta_j$ , where $\psi$ is the deflection potential, indices separated by a comma denote partial derivatives with respect to $\theta_i$, and summation over repeated indices is implied. |
Note that the fox of us equation iuplies that the origi of the lens plane. 00. is inapped onto the origin of the source lane. | Note that the form of this equation implies that the origin of the lens plane, $\vc\theta=0$, is mapped onto the origin of the source plane. |
The surface mass deusitv & and the complex shear at le origin are eiven iu ferus of the deflection poteutial. A=(Oa,|0o2)/2. 5=(oanCo2)/2|10342. beiug η.! aud spin-2 fields. respectively. | The surface mass density $\kappa$ and the complex shear $\gamma$ at the origin are given in terms of the deflection potential, $\kappa=(\psi_{,11}+\psi_{,22})/2$, $\gamma=(\psi_{,11}-\psi_{,22})/2
+{\rm i}\psi_{,12}$, being spin-0 and spin-2 fields, respectively. |
Iu our complex notation. 16 locally linearized leus equation reads. | In our complex notation, the locally linearized lens equation reads. |
. We next generalize this result to a second-order local expansion of the leus equation. which iu Cartesian coordinates reads 3;=0;tpO;cas ;0u/2. | We next generalize this result to a second-order local expansion of the lens equation, which in Cartesian coordinates reads $\beta_i=\theta_i-\psi_{,ij}\theta_j-\psi_{,ijk}\theta_j\theta_k$ /2. |
The third-order derivatives of c are related to the eradieut of κ aud . | The third-order derivatives of $\psi$ are related to the gradient of $\kappa$ and $\gamma$. |
To write these derivatives also in complex form. we define the differential operators | To write these derivatives also in complex form, we define the differential operators;. |
The differential operator Ve turus a spiu-» field iuto a sputo|1) field. whereas V© reduces the spin by one unit. | The differential operator $\nabla_{\rm c}$ turns a $n$ field into a $(n+1)$ field, whereas $\nabla_{\rm c}^*$ reduces the spin by one unit. |
One finds. for example. and we recognize the combinations of third derivatives of c which form the spin-l and spin-3 combinatio defined in (2)). | One finds, for example, ;, and we recognize the combinations of third derivatives of $\psi$ which form the spin-1 and spin-3 combinations defined in \ref{eq:T13def}) ). |
The final relation iu (6)) is the relation between first derivatives of & aud 5 found by [Kaiser (1995). here expressed in compact form. | The final relation in \ref{eq:kapdiff}) ) is the relation between first derivatives of $\kappa$ and $\gamma$ found by Kaiser (1995), here expressed in compact form. |
It expresses f[um fact that the third-order derivatives of the deflectic- potential can be sununuiarized iu the spin-3 field G=Ver aid the spin-l field F=Vox. where we iutroduced. the usual notation for the two flexion quantities. | It expresses the fact that the third-order derivatives of the deflection potential can be summarized in the spin-3 field ${\cal G}\equiv\nabla_{\rm c}\gamma$ and the spin-1 field ${\cal F}\equiv\nabla_{\rm c}^*\gamma$, where we introduced the usual notation for the two flexion quantities. |
T1C SCCOILCL-order lens equation in our complex notation then reads theta E | The second-order lens equation in our complex notation then reads ^* - ^2 ^* - ^*)^2. |
Since this is no longer a linear equation. a source at Jj uav have more than one image. | Since this is no longer a linear equation, a source at $\beta$ may have more than one image. |
In fact. up to four images of a source can be obtained. as can be seen for the special case of 5=0—F aud bv pacing the source at 30. | In fact, up to four images of a source can be obtained, as can be seen for the special case of $\gamma=0={\cal
F}$ and by placing the source at $\beta=0$. |
In his case. if we set G=|Gc5, then one solution is 0=0. aud the other three are 0=LLwyIGef with o=¢. y=|Br3 and y=¢ἐπ. | In this case, if we set ${\cal G}=|{\cal G}|{\rm e}^{3{\rm i}\zeta}$, then one solution is $\theta=0$, and the other three are $\theta=4(1-\kappa)/|{\cal
G}|\,{\rm e}^{{\rm i}\vp}$, with $\vp=\zeta$, $\vp=\zeta+2\pi/3$ and $\vp=\zeta+4\pi/3$. |
Of course. the origin or the occurrence of these solutions lies in the fact that G is a spin-3 quantity. | Of course, the origin for the occurrence of these solutions lies in the fact that ${\cal G}$ is a spin-3 quantity. |
We shall later need the Jacobian deteruiuaut detA of this le1s equation. which is A-(1-«)? | 2j ο.” 00. avhere the first expression is just the first-order Tavlor expansion of the Jacobian around the origin. aud in the second step we made use of the relation 8«OV) f2. | We shall later need the Jacobian determinant $\det \A$ of this lens equation, which is )^2- ^* + ^2) )^2- ^* +, where the first expression is just the first-order Taylor expansion of the Jacobian around the origin, and in the second step we made use of the relation $\vc\theta\cdot\nabla=(\theta\nabla_{\rm
c}^*+\theta^*\nabla_{\rm c})/2$ . |
We point out that (15)) is not the full Jacobian of the lens equation (7)). but oulv its first-order expausion: the full Jacobian contains quadratic terius in 0, | We point out that \ref{eq:detA}) ) is not the full Jacobian of the lens equation \ref{eq:lenseq}) ), but only its first-order expansion; the full Jacobian contains quadratic terms in $\theta$. |
We will return to this important issue further below. | We will return to this important issue further below. |
The observables of a gravitational leus svstcm are Tuchanecc if the surface mass cleusity «ois fransforiued as κ)>WO)An(@)|(1A) (Corenstei1 ot 119858). | The observables of a gravitational lens system are unchanged if the surface mass density $\kappa$ is transformed as $\kappa(\vc\theta)
\to \kappa'(\vc\theta)=\lambda\kappa(\vc\theta) +(1-\lambda)$ (Gorenstein et 1988). |
Iu the case of weal Ieunsiug. the shape of images is Tuchanecc under this trausformatiou (Schneider Seitz 1995). | In the case of weak lensing, the shape of images is unchanged under this transformation (Schneider Seitz 1995). |
Because of this mass-sheet degeneracy. rot the shear is an observable iu weak lensing. but OlLv the reduced. shear 4—(1HR) | Because of this mass-sheet degeneracy, not the shear is an observable in weak lensing, but only the reduced shear $g=\gamma/(1-\kappa)$. |
In fact. si1ος we expect hat the most promising applications of flexion wil conie you situations where p is not mel smaller than units. he distinction between shear aud reduced shear is likely Fo beiore important for flexion than for the usua weals chasing applications. | In fact, since we expect that the most promising applications of flexion will come from situations where $\kappa$ is not much smaller than unity, the distinction between shear and reduced shear is likely to bemore important for flexion than for the usual weak lensing applications. |
Hence. at best we cau expect from uigher-order shape measurements to obtain an estimate or the reduced shear aud its derivatives. | Hence, at best we can expect from higher-order shape measurements to obtain an estimate for the reduced shear and its derivatives. |
For this reason. weshall rewrite the foregoiug expressions in teris of the reduced shear. | For this reason, weshall rewrite the foregoing expressions in terms of the reduced shear. |
coellicient. | coefficient. |
The qualitative features of the integrated spectra upstream and downstream for p€puo, can easily be understood. | The qualitative features of the integrated spectra upstream and downstream for $p<p_{max}$ can easily be understood. |
Let us for instance consider (he case of Bolin diffusion: (he upstream size of the region where particles with momentum p diffuse is ~D(p)/u,xp. | Let us for instance consider the case of Bohm diffusion: the upstream size of the region where particles with momentum $p$ diffuse is $\sim D(p)/u_{1}\propto p$ . |
Therefore Sy(p)xNo(p)pp+ for r=4. | Therefore $S_{1}(p)\propto N_{0}(p)p\propto p^{-1}$ for $r=4$. |
The spatial size of the region where particles reside in (he downstream region is «cοτίοxL/p. therelore Φορ)xINo(p)uaTisuuoXpE [or r=4. | The spatial size of the region where particles reside in the downstream region is $\sim u_{2}\tau_{loss,2}\propto 1/p$, therefore $S_{2}(p)\propto N_{0}(p)u_{2}\tau_{loss,2}\propto p^{-3}$ for $r=4$. |
Again. this (trend is easily seen in the results of our exact caleulations. shown in the left panel of Fig. 6.. | Again, this trend is easily seen in the results of our exact calculations, shown in the left panel of Fig. \ref{fig:spaceint}. |
The solid line shows the spectrum as spatially integrated in the downstream region. while the upstream integrated spectrum is shown as a dashed line. | The solid line shows the spectrum as spatially integrated in the downstream region, while the upstream integrated spectrum is shown as a dashed line. |
The sum of the two is the dash-dotted line. | The sum of the two is the dash-dotted line. |
Since 54(p) is much flatter. at most energies the largest contribution to the spectrum of radiating electrons in an astrophivsical source comes from the downstream region. | Since $S_{1}(p)$ is much flatter, at most energies the largest contribution to the spectrum of radiating electrons in an astrophysical source comes from the downstream region. |
A small contribution appears only at p7p,4,,,. | A small contribution appears only at $p\sim p_{max}$. |
This ellect is even more visible if one assumes thal the magnetic field downstream is obtained from the compression of (he magnetic field in front of the shock surface. by an amount & | This effect is even more visible if one assumes that the magnetic field downstream is obtained from the compression of the magnetic field in front of the shock surface, by an amount $\kappa^{-1}$. |
In the right panel of Fig. | In the right panel of Fig. |
G6 we show the case in which &!= V1. which is the case of compression of an isotropic turbulent field upstream for r=4. | \ref{fig:spaceint} we show the case in which $\kappa^{-1}=\sqrt{11}$ , which is the case of compression of an isotropic turbulent field upstream for $r=4$. |
One can see that in (his case 1 downstream: component produces a more pronounced bump in the total spectrum. as lue to the fact that particles immediately downstream at ppy, lose energv more rapidly ian upstream. | One can see that in this case the downstream component produces a more pronounced bump in the total spectrum, as due to the fact that particles immediately downstream at $p\sim p_{max}$ lose energy more rapidly than upstream. |
While being hardly visible in the spectrum of svnchrotron emission because is is dominated by the downstream region. this feature might be visible in Che spectrum of inverse Compton radiation in (hose cases in which gamma rav emission can clearly be attributed to this process rather (han to a hadronic origin. | While being hardly visible in the spectrum of synchrotron emission because this is dominated by the downstream region, this feature might be visible in the spectrum of inverse Compton radiation in those cases in which gamma ray emission can clearly be attributed to this process rather than to a hadronic origin. |
The simple trends illustrated. above can be easily extended (to any dependence of the diffusion coelficient on momentum. | The simple trends illustrated above can be easily extended to any dependence of the diffusion coefficient on momentum. |
For instance for a constant dilfusion coefficient. one can expect that S4(p)xp7? and Ss(p)xp*. | For instance for a constant diffusion coefficient, one can expect that $S_{1}(p)\propto p^{-2}$ and $S_{2}(p)\propto p^{-3}$. |
The slope of $5(p) is independent of the diffusion coellicient as itis determined by energy losses at all energies. | The slope of $S_{2}(p)$ is independent of the diffusion coefficient as it is determined by energy losses at all energies. |
Of course this simple prediction is (he consequence of (he assumption of stationarity of the problem. | Of course this simple prediction is the consequence of the assumption of stationarity of the problem. |
In real astrophysical sources. stationarity will be reached only at sufficiently high energies. while at lower energies the size of the downstream region is determined by the age of the source and is therefore independent of momentum. | In real astrophysical sources, stationarity will be reached only at sufficiently high energies, while at lower energies the size of the downstream region is determined by the age of the source and is therefore independent of momentum. |
This is the reason for (he appearance of spectral breaks in the spectra of radiation (see for instance Ixardashev (1962))). | This is the reason for the appearance of spectral breaks in the spectra of radiation (see for instance \cite{karda}) ). |
Another note of caution should be issued about the assumption of planar shock adopted throughout the calculations: in astroplivsical sources the ellects of sphericity mightbecome important. especially on the volume integrated quantities. | Another note of caution should be issued about the assumption of planar shock adopted throughout the calculations: in astrophysical sources the effects of sphericity mightbecome important, especially on the volume integrated quantities. |
On the other hand one may argue | On the other hand one may argue |
Alultiplving (56)) ancl (53)) we get the contribution to 3 from the particles in the shell. | Multiplying \ref{eq:numshell}) ) and \ref{eq:A2fit}) ) we get the contribution to $A_2^2$ from the particles in the shell. |
We convert this to GAA. by multiplying by g77E The average of sin?6 over a sphere gives 2/3. | We convert this to $g^{\mu\nu} A_{\mu} A_{\nu}$ by multiplying by $g^{22}$: The average of $\sin^2 \theta$ over a sphere gives $2/3$. |
For the average of V7 we reason as follows: i£ the elfective m of the particles were purely thermal. we could write V?=1/2. | For the average of $V^2$ we reason as follows: if the effective $m^2$ of the particles were purely thermal, we could write $V^2 = 1/2$. |
But the elfective mass actually increases with time as the ratio p trom (54)) increases [rom zero towards one. | But the effective mass actually increases with time as the ratio $p$ from \ref{eq:p_ratio}) ) increases from zero towards one. |
So a better estimate of V7 is 1/(2|p). | So a better estimate of $V^2$ is $1/(2+p)$. |
n= Oat the surface of the bubble. and jj=jy, at the observation point. | $\eta = 0$ at the surface of the bubble, and $\eta = \eta_{\mathrm{p}}$ at the observation point. |
For some time slice at time 5 between these two limits. we ect the contribution of all spherical shells to £A?) by integrating from y—0 to the light cone al X=onsae We now have to sum over all time slices of duration 7. | For some time slice at time $\eta$ between these two limits, we get the contribution of all spherical shells to $\langle {\mathbf A}^2 \rangle_{\mathrm{nt}}$ by integrating from $\chi = 0$ to the light cone at $\chi = \eta_{\mathrm{p}} - \eta$: We now have to sum over all time slices of duration $\tau$. |
The sum can be converted to an integral by multiplving by di: ln section 4. we defined 7=τι(1). | The sum can be converted to an integral by multiplying by ${\mathrm d} \eta/\tau$: In section \ref{sec:source} we defined $\tau = \tau_t / R_{\mathrm{s}} (t)$. |
For τι we will simply use equation 3.24 from 7.. | For $\tau_t$ we will simply use equation 3.24 from \citet{mont2}. |
The factor C in the numerator will be of order (2|p)3/2i the quantity in the bracket in the denominator is of order unity ancl will be omitted. | The factor $U^3$ in the numerator will be of order $(2+p)^{-3/2}$; the quantity in the bracket in the denominator is of order unity and will be omitted. |
We then have. in our notation. m=m-(2|p2(Sqfo A). | We then have, in our notation, $\tau_t = m^2(2+p)^{-3/2}/(8\pi \nu_{\mathrm{s}} q^4 \ln \Lambda)$ . |
For m we will use the effective mass at time £. | For $m^2$ we will use the effective mass at time $t$ . |
The purely thermal value is dE7T. but we should multiplv this bx (11p) to include the non-thermal part also. | The purely thermal value is $q^2 T_{\mathrm{s}}^2$, but we should multiply this by $(1+p)$ to include the non-thermal part also. |
We have £4,=0.24177 We can now get an expression for p by using (60)) in (59)) and dividing by {πι This integral equation is surely not correct. in detail. but it does provide a general picture of the buildup of p from p—0 at jj,—0. | We have $\nu_{\mathrm{s}} = 0.24 T_{\mathrm{s}}^3$, so We can now get an expression for $p$ by using \ref{eq:tau_est}) ) in \ref{exp:A2both}) ) and dividing by $T_{\mathrm{p}}^2$: This integral equation is surely not correct in detail, but it does provide a general picture of the buildup of $p$ from $p=0$ at $\eta_p=0$ . |
We seek tow. the value of 1, for which p=od. | We seek $\eta_{\mathrm{cw}}$, the value of $\eta_p$ for which $p=1$. |
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