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Unless we are very “lucky” and the ended star happen to be sitting exactly on the source of the event. this can be attributed to the fact that all the blending ight comes from the lensing object.
Unless we are very “lucky” and the blended star happen to be sitting exactly on the source of the event, this can be attributed to the fact that all the blending light comes from the lensing object.
Hence. the lenses must xe luminous.
Hence, the lenses must be luminous.
Their locations on the CMD. calculated from he blended microlensing mocel. indicate they belong to the LMC star locus.
Their locations on the CMD, calculated from the blended microlensing model, indicate they belong to the LMC star locus.
This clearly hints towards the selt-Iensing nature of the detected events.
This clearly hints towards the self-lensing nature of the detected events.
The two events found. manually are also cillicult to attribute to ALACΠΟ lensing.
The two events found manually are also difficult to attribute to MACHO lensing.
OGLE-LAIC-05 is likely to be caused by a very. red lens. probably located in the Galaxy disk.
OGLE-LMC-05 is likely to be caused by a very red lens, probably located in the Galaxy disk.
OGLE-LAIC-06 is generally very puzzling. with its light curve resembling a binary lens/source event and its very red colour. suggesting rather a variable star than a microlensing event.
OGLE-LMC-06 is generally very puzzling, with its light curve resembling a binary lens/source event and its very red colour, suggesting rather a variable star than a microlensing event.
It means we have no strong candidates for microlensing events caused. by lenses from the halo.
It means we have no strong candidates for microlensing events caused by lenses from the halo.
We can. however. »ut an upper limit on the NLACTIO presence in. the alo. similarly as done in Tisserandetal.(2007)... who detected no events in their bright star sample.
We can, however, put an upper limit on the MACHO presence in the halo, similarly as done in \cite{TisserandEROSLMC}, who detected no events in their bright star sample.
In our case. iowever. we have two events we associated with origin other than NLACTIO lensing.
In our case, however, we have two events we associated with origin other than MACHO lensing.
This imply we can not apply straightforward zero-detection Poisson statistics. but should ollow the suggestion of Moniez.(2010)... also applied in "aper HE and treat our SL candidate events as an expected xickeround.
This imply we can not apply straightforward zero-detection Poisson statistics, but should follow the suggestion of \cite{Moniez2010}, also applied in Paper II and treat our SL candidate events as an expected background.
This is an obvious estimate. as more cetailed studies involving LMC modelling are necessary in order to obtain exact amount of expected: self-Iensing events over he entire OGLELL LMC sky coverage.
This is an obvious estimate, as more detailed studies involving LMC modelling are necessary in order to obtain exact amount of expected self-lensing events over the entire OGLE–III LMC sky coverage.
Such analysis is Xanned to be performed in a way similar to the OGLEIL LMC study in CalehiNovatietal.(20090).
Such analysis is planned to be performed in a way similar to the OGLE–II LMC study in \cite{CalchiNovati2009}.
Based on a mean detection ellicieney over all fields we were able to estimate the number of expected. events due to NACIHIOs considering model 78 of Alcockctal. (2000).
Based on a mean detection efficiency over all fields we were able to estimate the number of expected events due to MACHOs considering model “S” of \cite{AlcockMACHOLMC}.
. This number was calculated. for a wide range of ALACLIO masses [rom LO7 to 107.AZ. and was translated toa fraction of halo mass using zero-statistics of FeldmanCousins (1998).
This number was calculated for a wide range of MACHO masses from $10^{-8}$ to $10^2~\msun$ and was translated to a fraction of halo mass using zero-statistics of \cite{FeldmanLowStatistics}.
. Ht is shown in Fig. 1H4..
It is shown in Fig. \ref{fig:taulimit}.
For masses around AL=OAM. we expected. Nox,=69 events for a halo full of NLACTIOs. it translates to an upper limit of f.«7 per cent at 95 per cent confidence and. f.«6 per cent at 90 per cent.
For masses around $M=0.4\msun$ we expected $N_{\rm exp}=69$ events for a halo full of MACHOs, it translates to an upper limit of $f<7$ per cent at 95 per cent confidence and $f<6$ per cent at 90 per cent.
The limit reaches its minimum at around AJ=0.1AL. with f«4 per cent and is less rigid on masses higher than O.4AL. reaching around 20 per cent at Al=10M. and more al higher masses.
The limit reaches its minimum at around $M=0.1~\msun$ with $f<4$ per cent and is less rigid on masses higher than $0.4\msun$ reaching around 20 per cent at $M=10\msun$ and more at higher masses.
Our result is in agreement not only with previous LAIC microlensing findings (Visserancetal.2007)... but also with studies of the microlensing elfects. of compact objects in distant. galaxies observed in the Llensecl quasars (e.g. Aleciavillaetal. 2009)). which ruled out NLACTIOs in mass range between 0.1 and LO AZ..
Our result is in agreement not only with previous LMC microlensing findings \citep{TisserandEROSLMC}, but also with studies of the microlensing effects of compact objects in distant galaxies observed in the lensed quasars (e.g. \citealt{Mediavilla2009}) ), which ruled out MACHOs in mass range between 0.1 and 10 $\msun$.
For the mass window 10|30AZ. there is still no reasonable constraint.
For the mass window $10-30~\msun$ there is still no reasonable constraint.
There is actually some hint for heavy mass (around 1047.) compact objects in the halo - Dongetal.(2007). studied the OGLE-2005-SALC-001 microlensing event and concluded: it was caused by a binary black hole most. likely located. in. the halo.
There is actually some hint for heavy mass (around $\msun$ ) compact objects in the halo - \citet{Dong2007} studied the OGLE-2005-SMC-001 microlensing event and concluded it was caused by a binary black hole most likely located in the halo.
However. it still remains a mystery why we don't see such events towards the LMC. therefore it is too early to conclude on dark matter compact objects existing in that mass window.
However, it still remains a mystery why we don't see such events towards the LMC, therefore it is too early to conclude on dark matter compact objects existing in that mass window.
OGLETLL SMC data. including that unique event. will be presented. and. studied. in the forthcoming paper (Wvrzvkowski ct al.
OGLE–III SMC data, including that unique event, will be presented and studied in the forthcoming paper (Wyrzykowski et al.
in prep.).
in prep.).
1n this study we analysed. almost ὃ vears of observations of the Large Magellanic Cloud by OGLEILL
In this study we analysed almost 8 years of observations of the Large Magellanic Cloud by OGLE–III.
The data set with its volume. coverage and quality supersedes all previous determinations of the microlensing optical depth. including he one based on the OGLELl data (Paper D.
The data set with its volume, coverage and quality supersedes all previous determinations of the microlensing optical depth, including the one based on the OGLE–II data (Paper I).
We detected wo sound candidates for microlensing events and. further wo possible candidates.
We detected two sound candidates for microlensing events and further two possible candidates.
Neither of them. however. is likely o be caused by dark matter compact lenses from the halo of our Galaxy.
Neither of them, however, is likely to be caused by dark matter compact lenses from the halo of our Galaxy.
The two best candidates can be explained as an expected signal from the selt-Iensing within the LMC.
The two best candidates can be explained as an expected signal from the self-lensing within the LMC.
Of he remaining two. one is either a binary event or a some kind of chromatic outburst. whereas the other is a candidate or galactic clisk lens.
Of the remaining two, one is either a binary event or a some kind of chromatic outburst, whereas the other is a candidate for galactic disk lens.
Such null. detection. for NLACTHIO. lensing led to estimating the upper limit on their contribution to the mass of the Halo of the Galaxy.
Such null detection for MACHO lensing led to estimating the upper limit on their contribution to the mass of the Halo of the Galaxy.
Phe upper limit set at a level of 6-7 per cent at AZ=OAAL. leaves very little room for dark matter compact objects.
The upper limit set at a level of 6-7 per cent at $M=0.4\msun$ leaves very little room for dark matter compact objects.
Still. at the moment we can not exclude more heavy dark matter lenses. like black holes.
Still, at the moment we can not exclude more heavy dark matter lenses, like black holes.
Our survey puts a 20 per cent halo mass fraction limit on compact objects with masses of Ad=10M. and actually πο limit on higher masses.
Our survey puts a 20 per cent halo mass fraction limit on compact objects with masses of $M=10\msun$ and actually no limit on higher masses.
This heavy mass end window should be now explored with more attention.
This heavy mass end window should be now explored with more attention.
Asa side product of our analysis we also discovered that event NLACTIO-LMC-7. reported by the ALACILO group and used in their final optical depth determination. exhibited couple of additional brightening episodes in the ΟΚΤΙΟ data. a feature which excludes it as a gonuine microlensing event.
As a side product of our analysis we also discovered that event MACHO-LMC-7, reported by the MACHO group and used in their final optical depth determination, exhibited couple of additional brightening episodes in the OGLE-III data, a feature which excludes it as a genuine microlensing event.
With the OGLE project continuing now in its fourth phase we hope the sensitivity to extremely long events will improve significantly within next vears.
With the OGLE project continuing now in its fourth phase we hope the sensitivity to extremely long events will improve significantly within next years.
It should result. in the increase in the statistics of potential black-hole lenses or allow us to rule out heavy. dark matter compact objects as well and close that topic delinitively.
It should result in the increase in the statistics of potential black-hole lenses or allow us to rule out heavy dark matter compact objects as well and close that topic definitively.
We would like to thank for their help at various stages of this work to Drs Nicholas Rattenbury. Vasily Belokuroy and Patrick “Visseranc.
We would like to thank for their help at various stages of this work to Drs Nicholas Rattenbury, Vasily Belokurov and Patrick Tisserand.
We also thank the anonymous referee. for their invaluable comments. and remarks.
We also thank the anonymous referee for their invaluable comments and remarks.
“Phis work was partially supported. by EC PRT grant. PERCO+L- to LW. JS acknowledges support through the Polish ALNISW grant. no.
This work was partially supported by EC FR7 grant PERG04-GA-2008-234784 to W. JS acknowledges support through the Polish MNiSW grant no.
N20300832/0709 ancl Space Exploration Research Fund of The Ohio State University.
N20300832/0709 and Space Exploration Research Fund of The Ohio State University.
For Arp 2, Terzan 7 and Terzan 8, the number counts follow a Gaussian distribution with no deviation larger than the Poisson noise (see Figure 3)).
For Arp 2, Terzan 7 and Terzan 8, the number counts follow a Gaussian distribution with no deviation larger than the Poisson noise (see Figure \ref{f:magshift}) ).
This indicates either that (1) the clusters are in or near the mean distance of the Sgr stream; (2) that the Sgr background
This indicates either that (1) the clusters are in or near the mean distance of the Sgr stream; (2) that the Sgr background
As an example of the potential importance of including the effects. of both photo-ionization and heavy elements. we considered the so-called warm-hot intergalactic medium (ΑΔΗΔΕΟ which is thought to contain a large fraction. of the barvons at redshifts 2«1.
As an example of the potential importance of including the effects of both photo-ionization and heavy elements, we considered the so-called warm-hot intergalactic medium (WHIM), which is thought to contain a large fraction of the baryons at redshifts $z < 1$.
We demonstrated that the overdensities for which gas at typical WIILM. metallicities (Z~104 Z.)and temperatures (£7710"10* IX) can cool within a Llubble time. can shift bx an order of magnitude depending on whether photo-ionization ancl metal-line cooling are taken into account.
We demonstrated that the overdensities for which gas at typical WHIM metallicities $Z\sim 10^{-1}~Z_\odot$ ) and temperatures $T\sim 10^5 - 10^7\,\K$ ) can cool within a Hubble time, can shift by an order of magnitude depending on whether photo-ionization and metal-line cooling are taken into account.
Hence. photo-ionization of heavy elements may. have important consequences. for predictions ofthe amount of matter contained in this elusive gas phase.
Hence, photo-ionization of heavy elements may have important consequences for predictions of the amount of matter contained in this elusive gas phase.
Because chemical enrichment happens in a number of stages. involving a number of processes with cdilleren timescales. the relative abundances of the heavy. elements varies with redshift and. environment by factors of a Lew.
Because chemical enrichment happens in a number of stages, involving a number of processes with different timescales, the relative abundances of the heavy elements varies with redshift and environment by factors of a few.
llence. computing cooling rates on an element-by-elemen basis rather than scalingIn] all elements by the metallicity. wil change the cooling rates by [actors of a few.
Hence, computing cooling rates on an element-by-element basis rather than scaling all elements by the metallicity, will change the cooling rates by factors of a few.
The cdillerence is therefore typically somewhat smaller than the ellect. of neglecting metals or photo-ionization altogether. but stil uehly significant.
The difference is therefore typically somewhat smaller than the effect of neglecting metals or photo-ionization altogether, but still highly significant.
While it was known that different elements dominate he cooling for dillerent. temperatures in CLE. we shower hat photo-ionization both shifts the peaks cue to individual elements to smaller. temperatures ancl reduces heir amplitude.
While it was known that different elements dominate the cooling for different temperatures in CIE, we showed that photo-ionization both shifts the peaks due to individual elements to smaller temperatures and reduces their amplitude.
Note that. since. photo-ionization over-ionizes the gas. this elfect is similar (but not equivalent to) hat found in non-equilibrium calculations without ionizing radiation (e.g..??)..
Note that since photo-ionization over-ionizes the gas, this effect is similar (but not equivalent to) that found in non-equilibrium calculations without ionizing radiation \cite[e.g.,][]{Sutherland1993, Gnat2007}.
Because the importance of photo-ionization depends on the ionization parameter. the relative contributions of individual elements exposed. to a fixed ionizing radiation field depends also on the gas density.
Because the importance of photo-ionization depends on the ionization parameter, the relative contributions of individual elements exposed to a fixed ionizing radiation field depends also on the gas density.
Would dropping our assumption of ionization equilibrium have a large οσο on the cooling rates?
Would dropping our assumption of ionization equilibrium have a large effect on the cooling rates?
lonizing radiation results in a plasma that is overionized relative to its temperature.
Ionizing radiation results in a plasma that is overionized relative to its temperature.
Is effect is therefore similar to that of non-equilibrium ionization following rapicl cooling (ie. if the cooling time is shorter than the recombination times of the ions dominating the cooling. see citealtIxafatos1973.5hapiro1976)).
Its effect is therefore similar to that of non-equilibrium ionization following rapid cooling (i.e., if the cooling time is shorter than the recombination times of the ions dominating the cooling, see \\citealt{Kafatos1973,Shapiro1976}) ).
We therefore. anticipate that the effect. of non-equilibrium ionization will be much smaller for our cooling rates than for those that assume CLIE.
We therefore anticipate that the effect of non-equilibrium ionization will be much smaller for our cooling rates than for those that assume CIE.
The assumptions that the gas is optically thin. and exposed only to the meta-galactic background: radiation are likely to be more important than the assumption of ionization equilibrium. particularly since. non-equilibrium collisional cooling rates only diller from those assuming CLE by factors of a few or Less (e...2?2)..
The assumptions that the gas is optically thin and exposed only to the meta-galactic background radiation are likely to be more important than the assumption of ionization equilibrium, particularly since non-equilibrium collisional cooling rates only differ from those assuming CIE by factors of a few or less \cite[e.g.,][]{Schmutzeler1993,Sutherland1993,Gnat2007}.
For column densities Ngco10Uem self-shielding becomes important. and only part of the L-ionizing radiation will penetrate the eas cloud. which would particularly alfect the cooling rates for T10"Ix.
For column densities $N_{\rm HI} > 10^{17}~\cm^{-2}$ self-shielding becomes important and only part of the H-ionizing radiation will penetrate the gas cloud, which would particularly affect the cooling rates for $T\la 10^5\,\K$.
At higher temperatures line cooling is dominated by heavier ions. which can only be ionized by higher energy photons and which therefore. remain optically thin up to much higher column densities.
At higher temperatures line cooling is dominated by heavier ions, which can only be ionized by higher energy photons and which therefore remain optically thin up to much higher column densities.
This is because the photo-ionization cross sections of HL and Le drop rapidly with increasing frequency for energies exceeding their ionization potentials.
This is because the photo-ionization cross sections of H and He drop rapidly with increasing frequency for energies exceeding their ionization potentials.
Moreover. for £»107IK hydrogen is collisionally ionized to à high degree and consequently the optical depth for ionizing radiation will be significantly. reduced.
Moreover, for $T\gg 10^4~\K$ hydrogen is collisionally ionized to a high degree and consequently the optical depth for ionizing radiation will be significantly reduced.
lt is. however. far from clear that high column densities would reduce the elfeet of radiation.
It is, however, far from clear that high column densities would reduce the effect of radiation.
For self-shielded clouds the cooling radiation. may itself. be trapped. providing a source of ionizing radiation even in the absence of an external one (e...22)..
For self-shielded clouds the cooling radiation may itself be trapped, providing a source of ionizing radiation even in the absence of an external one \cite[e.g.,][]{Shapiro1976,Gnat2007}.
Moreover. gas clouds with columns that exceed. 107.em are on average expected to. be sulliciently. close to a galaxy. that local sources of ionizing radiation dominate over the background (?7)..
Moreover, gas clouds with columns that exceed $10^{17}~\cm^{-2}$ are on average expected to be sufficiently close to a galaxy that local sources of ionizing radiation dominate over the background \citep{Schaye2006,Miralda2005}.
Ultimately. these issues can only be resolved. if non-equilibrium cooling rates are computed including radiative transfer and if the locations of all relevant sources of ionizing radiation are known.
Ultimately, these issues can only be resolved if non-equilibrium cooling rates are computed including radiative transfer and if the locations of all relevant sources of ionizing radiation are known.
lt will be some time before it is feasible to carry out such a calculation in. say. a cosmological hvdrocdvnamical simulation.
It will be some time before it is feasible to carry out such a calculation in, say, a cosmological hydrodynamical simulation.
In the mean time. we believe that our element-by-element. caleulation of the equilibrium cooling rates for an optically thin gas exposed to the CMD and an evolving UV/X-ray background. provides a marked improvement over earlier treatments.
In the mean time, we believe that our element-by-element calculation of the equilibrium cooling rates for an optically thin gas exposed to the CMB and an evolving UV/X-ray background provides a marked improvement over earlier treatments.
In future publications we will present cosmological. hydrodynamical simulations using these cooling rates.
In future publications we will present cosmological, hydrodynamical simulations using these cooling rates.
We are grateful to the anonymous referee. whose helpful comments greatly improved the manuscript.
We are grateful to the anonymous referee whose helpful comments greatly improved the manuscript.
We are also erateful to Gary Ferland for help with ancl to irent Groves for help withIL.
We are also grateful to Gary Ferland for help with and to Brent Groves for help with.
We would. also like to thank "Tom Abel and. the members of the OWLS collaboration for discussions.
We would also like to thank Tom Abel and the members of the OWLS collaboration for discussions.
In particular. we are grateful to ‘Tom Theuns for showing us the benefits of HIDE.
In particular, we are grateful to Tom Theuns for showing us the benefits of HDF5.
This work was supported by Marie Curie Excellence Grant MIZNT-C'T-
This work was supported by Marie Curie Excellence Grant MEXT-CT-2004-014112.
Although attempts are now being made to include effective energy feedback from supernovae within cosmological simulations (e.g. Kay et al.
Although attempts are now being made to include effective energy feedback from supernovae within cosmological simulations (e.g. Kay et al.
2001). the modelling can only be done at a phenomenological level since the true physical processes occur on much smaller scales than are resolved.
2001), the modelling can only be done at a phenomenological level since the true physical processes occur on much smaller scales than are resolved.
For the purposes of this paper. we adopt a particularly simple model. which ts to take the output from the radiative simulation at z24 and raise the specific thermal energy of all gas particles by 0.1 keV (an equivalent temperature of 1.2106 K). before evolving to z20 as before.
For the purposes of this paper, we adopt a particularly simple model, which is to take the output from the radiative simulation at $z=4$ and raise the specific thermal energy of all gas particles by 0.1 keV (an equivalent temperature of $1.2\times 10^{6}\,$ K), before evolving to $z=0$ as before.
This choice of energy is motivated by observations of an "entropy floor in X-ray groups by Lloyd-Davies. Ponman Cannon (2000): we define specific "entropy as s=KpT/n? and our injection model would give sg,~SOkeVeni? were the gas all at mean density.
This choice of energy is motivated by observations of an `entropy floor' in X-ray groups by Lloyd-Davies, Ponman Cannon (2000); we define specific `entropy' as $s=k_BT/n^{2/3}$ and our injection model would give $s_{\rm floor} \sim 80 \, {\rm keV \, cm^2}$ were the gas all at mean density.
This reduces the amount of cooled material to a more acceptable level (~ 10%)). although a greater amount of heating would have been needed to give the best match to the X-ray scaling relations.
This reduces the amount of cooled material to a more acceptable level $\sim 10$ ), although a greater amount of heating would have been needed to give the best match to the X-ray scaling relations.
These simulations have two advantages over those we studied previously (da Silva et al.
These simulations have two advantages over those we studied previously (da Silva et al.
2000. 2001).
2000, 2001).
Firstly. those simulations did not include radiative cooling. and so contain hot high-density gas where the cooling time is much shorter than a Hubble time.
Firstly, those simulations did not include radiative cooling, and so contain hot high-density gas where the cooling time is much shorter than a Hubble time.
Secondly. the new simulations have improved numerical resolution (the particle masses are a factor of 19 smaller) allowing the SZ effect to be studied at smaller scales.
Secondly, the new simulations have improved numerical resolution (the particle masses are a factor of 19 smaller) allowing the SZ effect to be studied at smaller scales.
We analyze the SZ effect by constructing simulated SZ maps of area one square stacking simulation boxes from low to high redshift (up to redshift z~ 6.5) as described by da Silva et al. (
We analyze the SZ effect by constructing simulated SZ maps of area one square stacking simulation boxes from low to high redshift (up to redshift $z \sim 6.5$ ) as described by da Silva et al. (
2000).
2000).
The vast majority of the thermal SZ signal is produced well within this redshift interval.
The vast majority of the thermal SZ signal is produced well within this redshift interval.
We repeat this process to produce 30 random map realizations for each run. using the same sequence of random seeds for the different models.
We repeat this process to produce 30 random map realizations for each run, using the same sequence of random seeds for the different models.
The thermal effect is described by the Comptonization parameter y. and the kinetic by the temperature perturbation AT/T.
The thermal effect is described by the Comptonization parameter $y$ , and the kinetic by the temperature perturbation $\Delta T/T$.
The mean y distortion. obtained by averaging v over lines of sight. is an important measure of the global state of the gas in the Universe.
The mean $y$ distortion, obtained by averaging $y$ over lines of sight, is an important measure of the global state of the gas in the Universe.
In our simulations we find (For comparison. the mean mass-weighted temperature at in the three simulations are 0.39. 0.35 and 0.38 keV.) At present. the best upper limit is aj<1.5«107? fromFIRAS (Fixsen et al.
In our simulations we find (For comparison, the mean mass-weighted temperature at $z=0$ in the three simulations are 0.39, 0.35 and 0.38 keV.) At present, the best upper limit is $y_{{\rm mean}} < 1.5 \times 10^{-5}$ from (Fixsen et al.
1996). and all our models are well below this.
1996), and all our models are well below this.
The cooling simulation gives the lowest level of distortion and the pre-heating the highest.
The cooling simulation gives the lowest level of distortion and the pre-heating the highest.
The factor of two difference between these runs shows that the mean v Is quite sensitive to pre-heating.
The factor of two difference between these runs shows that the mean $y$ is quite sensitive to pre-heating.
The mean v distortion in the non-radiative model is slightly smaller than the value found in da Silva et al. (
The mean $y$ distortion in the non-radiative model is slightly smaller than the value found in da Silva et al. (
2000); part of this difference is attributable to a different os. and the remainder lies within the uncertainty expected from changes to the simulation technique.
2000); part of this difference is attributable to a different $\sigma_8$, and the remainder lies within the uncertainty expected from changes to the simulation technique.
To study the relative contribution to the SZ effect from the different gas phases in the simulations. we sorted the particles in each simulation according to their temperature. 7. and overdensity. à. and computed the fraction of SZ signal arising from all particles at a given T and à.
To study the relative contribution to the SZ effect from the different gas phases in the simulations, we sorted the particles in each simulation according to their temperature, $T$ , and overdensity, $\delta$, and computed the fraction of SZ signal arising from all particles at a given $T$ and $\delta$.
By carrying out cuts in the 6-T plane we are able to determine the type of structures contributing to the SZ effect.
By carrying out cuts in the $\delta$ $T$ plane we are able to determine the type of structures contributing to the SZ effect.
In Figure |.. the top panels show the distribution of the mean v as a function of overdensity and temperature.
In Figure \ref{f:ytd}, , the top panels show the distribution of the mean $y$ as a function of overdensity and temperature.
The area under the curves are the γω listed earlier.
The area under the curves are the $y_{{\rm mean}}$ listed earlier.
The bottom panels show the corresponding cumulative mean y fractions. F-VineanCà)Jinean and Fy.=NMineanCT)Jinean+
The bottom panels show the corresponding cumulative mean $y$ fractions, $F_{y}=y_{{\rm mean}}(<\!\delta)/y_{{\rm mean}}$ and $F_{y}=y_{{\rm mean}}(<\!T)/y_{{\rm mean}}$.