# Deducing distance of closest neighbor galaxy from expression of correlation and mean galaxy density

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The original post was on physics exchange but I prefer to move it here :

I am trying to estimate the distance of closest galaxy neighbor knowing the expression of number of neighbors $$ext{d}N$$ into a volume $$ext{d}V$$, the mean density $$n_ ext{gal}$$ and the correlation function, i.e with this expression:

$$ext{d}N=n_{ ext{gal}}, ext{d}V,(1+xi(r))$$

with $$xi(r)=igg(dfrac{r}{r_{0}}igg)^{-gamma}quad ext{with}quadgamma,sim,1.77 quad ext{and},r_{0},sim,5, ext{Mpc}$$.

I must precise that expression of $$xi(r)$$ is valid for $$r$$ between $$0.5, ext{and},10, ext{Mpc}$$

I take into my calculation the following value for $$n_{ ext{gal}}=0.0420, ext{h}^{-3}, ext{Mpc}^{-3}$$ : I tried to choose a typical value for this density but it depends on which scale we consider (galaxy cluster, super cluster, very large scales… ), so maybe don't focus on this value.

What it interests me is to have elements to answer to the following question : How could I infer the distance of the closest galaxy from us?

UPDATE 1: Maybe I should take the lower limit of validity for the definition of $$xi(r)$$, i.e $$r=0.5, ext{Mpc}$$ and so:

$$ext{d}N/ ext{d}V=n_{ ext{gal}},(1+xi(r=0.5))$$

So I would have:

$$ext{D}_{ ext{closest}}=igg(n_{ ext{gal}}(1+(0.5/5)^{-1.77})igg)^{-1/3}=0.7353, ext{Mpc}$$

But by taking $$r=0.5, ext{Mpc}$$, I am already located myself like the closest neighbor was at $$r=0.5, ext{Mpc}$$, am I not ?

Is this reasoning correct?

Any help is welcome

## Deducing distance of closest neighbor galaxy from expression of correlation and mean galaxy density - Astronomy

V. G. Fessenkov Astrophysical Institute, Almaty, Kazakhstan

Received June 29, 2012 revised August 4, 2012 accepted August 15, 2012

Keywords: Anisotropy of the Deceleration Parameter Universe Principal Axis

Purpose: The theoretical description of Hubble’s diagrams asymmetry of and calculating the anisotropy of the deceleration parameter phenomenon, that was recently found by R.-G. Cai and Z.-L. Tuo. Method: For doing this the concepts of Universe rotation and its two-component model were attracted. Result: Our result is in good correlation (case of the upper magnitude index) with the value that was got in [1]. Significance: The result of article gives new basing of the Universe rotation axis existence.

The discovering of the accelerating expansion of Universe though observations of distant supernovae [2,3] were stimulated large numbers of articles in which this effect was interprets not only in the framework of general relativity but from other theoretical viewpoints, also. In fact, in [4] it was considered the anisotropic case of the dark energy equation of state in [5] the anisotropy of cosmic acceleration was searched in the framework of generalized teleparallel gravity in [6] for the one variant of 5-dimensional inhomogeneous space-time it was found the anisotropic acceleration in [7] the deceleration anisotropy was considered by usage of baryonic matter Born-Infeld type electrodynamics, and some others attempts [8-11].

Separately to previous variants it’s necessary to mention the fundamental article [12] especially where the asymmetry of Hubble’s diagrams for the North and the South sky hemispheres was searched. (Remind that the Hubble diagram describes in first approximation the linear proportionality between a galaxy’s distance and its redshift. Later on it was found that the velocity at which a distant galaxy is moving from us should be permanently increasing over time, i.e. the cosmic scale factor has a positive second derivative, while deceleration parameter is negative.) This asymmetry, according to authors, cannot be explains by peculiar motion of the observer, but most apparently due to the any bulk flow along the direction ((l, b) = (,)) in the Universe existence that earlier was argued in article [13,14]. Recently R.-G. Cai and Z.-L. Tuo [1] determined more precisely this direction ((l, b) =,)

and found the maximum anisotropy of the deceleration parameter.

Evidently, these results are possible to summaries as follows—our Universe is anisotropic in realty and possesses by any principal space axis. That is why the cosmological deceleration parameter will be anisotropic, also and must be depend on the principal space direction in definite way. These statements require theoretical basing the direction dependence of the cosmological deceleration parameter phenomenon.

2. Basic Cosmological Equations

Our searching we start from the well-known results. The uniform isotropic metric of the space-flat Universe () have the standard form

(1)

Einstein’s equations for the scale factor are

(2)

(3)

(4)

These equations is possible to deduce and from the Newtonian mechanics in the following way. Let’s consider the spherical volume of radius where concentrates any substance with the density and with the Hubble velocity distribution

(5)

In the motionless frame of reference the equation of motion of a probe particle that locates on the surface of this sphere, have the usual form

(6)

Making the well-known Tolman transformation, that allows taking into account the pressure influence on equation of motion, and putting it into (6) we get Equation (2). Next, multiplying left and right sides of (6) by we get Equation (3), that is connected with (6) by the law of energy conservation (4) [15].

In article [16] it was shown that cosmic vacuum produces not only the Universe expansion but its rotation, also. Here the main results of this article are reproducing briefly.

Let’s start from searching the rotational movement of galaxies caused by the antigravitational vacuum force, only. As the model of examining type of galaxy the elliptical galaxy was chosen. For this shape of galaxy its equations of rotational motion are

(7)

In (7) is the first integral of the rotational motion, i.e.. It describes the component of angular velocity with respect to the specific momentum—C. Next, at deducing (7) it was put forward condition that galaxy angular velocity is very small. This allowed neglect the squared angular velocity components and the corresponding angular accelerations. And at last, it was assumed that arbitrary potential in (7) equals to the vacuum potential, where

(8)

Analysis of Equation (7) shown that solution for the precession angle evolving is. Basing on this result it is easy to calculate the angular velocity of the elliptical galaxy around axis. As for this case the following condition takes place, than its module equals

(9)

This expression describes the angular velocity that galaxy acquires due to the vacuum antigravitational force.

Admitting and putting thatwe find. So, its maximal magnitude will be under the condition. Then expression for the vacuum angular velocity simplifies and takes on the form

(10)

This expression interprets as the minimal angular velocity in the Universe that possesses an arbitrary object due to the vacuum presence. Its present numerical value is. Hence, the vacuum creates the identical initial angular velocity for all of cosmic objects, includeing the Universe itself.

At the earliest stages of the Universe evolution, for instance at the baryonic asymmetry epoch when vacuum density was of order, the angular velocity occurs equal. For the very early Universe when vacuum density was—, the Universe angular velocity is. This magnitude practically equals to the result of article [17], which was done in the framework of general relativity theory ().

Henceforth, from these investigations we get the following conclusion—the Universe rotation leads to picking out the principal direction in the space, it may be named as the Universe rotation axis. (Mark, that measurement along this axis only gives the Hubble parameter for the uniform Universe, because in the perpendicular directions the Carioles and centrifugal forces act, also).

4. Basing the Direction Dependence of the Cosmological Deceleration Parameter

For enriching our target, which was formulated in Section 1, put that distance

(11)

where is the distance in uniform space, while— small addition (perturb term) for describing the possible space anisotropy. Putting (11) into the Newtonian Equation (6) we get the equation

(12)

that may be decomposed on two parts, easily: main part

(13)

(14)

Later on these equations will be considered as are independent each other.

Performing the above mentioned Tolman transformation and substituting it into (13) we find equation

(15)

For the case of vacuum (,) the inflationary regime of the Universe expanding follows from (15) immediately—

(16)

It leads to the Hubble expansion law

(17)

and to the corresponding acceleration

(18)

Now consider the Equation (14). Suppose that in this equation, where is the baryonic substance density. The baryonic substance pressure let equals zero, for simplicity. Last requirement means considering the presence of two-component substance—cosmic vacuum and baryonic dust—in the Universe, that are not interact each other in the main approximation.

By introducing the designation, from (14) it follows

(19)

This oscillatory-type equation possesses by two roots

(20)

They lead to the presence of two perturb (with respect to (17)) velocities

(21)

and two corresponding accelerations

(22)

From physical viewpoint expressions (20)-(22) mean that presence of baryonic dust matter creates two spaceopposite fluxes that are propagate on the background of expanding and accelerating “Hubble vacuum flux” along the Universe rotation axis (see division 3). That is why it possible writes down the expressions for total distance, velocity and acceleration of any probe particle (galaxy)

(23)

Thus the cosmological deceleration parameter q with the accuracy no higher than is

(24)

Basing on the definitions of and we introduce the new coefficient. As in unit of the critical density and vacuum density, coefficient, henceforth.

From (24) it is possible find the relative acceleration difference between two baryonic fluxes with respect to the “Hubble vacuum flux”—

(25)

Assuming that for modern epoch we approximately get. Hence, the first term in right side of (25) tends to 1.2, while the second term tends to zero. So,

(26)

Basing on our assumption, that was argued earlier, we may put that it will satisfy if the ratio . This leads to the estimation that is in good correlation (case of the upper magnitude index) with the value in [1].

From observational data it was established the asymmetry of Hubble’s diagrams for the North and the South sky hemispheres [13,14]. Moreover it was estimated the space anisotropy of the deceleration parameter phenomenon, that was done by R.-G. Cai and Z.-L. Tuo. These facts require the adequate theoretical basing, hence.

For doing this the concepts of Universe vacuum rotation and its two independent component model (cosmic vacuum and baryonic dust) were attracted. Our result on the phenomenon of anisotropy the deceleration parameter calculation——is in good correlation

(case of the upper magnitude index) with the value

, which was evaluated in [1].

I would like to express the gratitude to Ministry of Education and Sciences, Republic of Kazakhstan for supports this searching in the framework of budget program 055, subprogram 101 “Grant financing of the scientific researchers”.

Also I thank a reviewer for his thought-out suggestions on the article’s content clarifying.

## Deducing distance of closest neighbor galaxy from expression of correlation and mean galaxy density - Astronomy

The literature on the cosmological tests is enormous compared to what it was just a decade ago, and growing. Our references to this literature are much sparser than in Sec. III, on the principle that no matter how complete the list it will be out of date by the time this review is published. For the same reason, we do not attempt to present the best values of the cosmological parameters based on their joint fit to the full suite of present measurements. The situation will continue to evolve as the measurements improve, and the state of the art is best followed on astro-ph. We do take it to be our assignment to consider what the tests are testing, and to assess the directions the results seem to be leading us. The latter causes us to return many times to two results that seem secure because they are so well checked by independent lines of evidence, as follows.

First, at the present state of the tests, optically selected galaxies are useful mass tracers. By that we mean the assumption that visible galaxies trace mass does not seriously degrade the accuracy of analyses of the observations. This will change as the measurements improve, of course, but the case is good enough now that we suspect the evidence will continue to be that optically selected galaxies are good indicators of where most of the mass is at the present epoch. Second, the mass density in matter is significantly less than the critical Einstein-de Sitter value. The case is compelling because it is supported by so many different lines of evidence (as summarized in Sec. IV.C). Each could be compromised by systematic error, to be sure, but it seems quite unlikely the evidence could be so consistent yet misleading. A judgement of the range of likely values of the mass density is more difficult. Our estimate, based on the measurements we most trust, is

and we would put the central value at M0 0.25. The spread is meant in the sense of two standard deviations: we would be surprised to find M0 is outside this range.

Several other policy decisions should be noted. The first is that we do not comment on tests that have been considered but not yet applied in a substantial campaign of measurements. A widely discussed example is the Alcock and Paczynski (1979) comparison of the apparent depth and width of a system from its angular size and depth in redshift.

In analyses of the tests of models for evolving dark energy density, simplicity recommends the XCDM parametrization with a single constant parameter wX, as is demonstrated by the large number of recent papers on this approach. But the more complete physics recommends the scalar field model with an inverse power-law potential. This includes the response of the spatial distribution of the dark energy to the peculiar gravitational field. Thus our comments on variable dark energy density are more heavily weighted to the scalar field model than is the case in the recent literature.

The gravitational deflection of light appears not only as a tool in cosmological tests, as gravitational lensing, but also as a source of systematic error. The gravitational deflections caused by mass concentrations magnify the image of a galaxy along a line of sight where the mass density is larger than the average, and reduce the solid angle of the image when the mass density along the line of sight is low. The observed energy flux density is proportional to the solid angle (because the surface brightness, erg cm -2 s -1 ster -1 Hz -1 , is conserved at fixed redshift). Selection can be biased either way, by the magnification effect or by obscuration by the dust that tends to accompany mass. 70 When the tests are more precise we will have to correct them for these biases, through models for the mass distribution (as in Premadi et al., 2001), and the measurements of the associated gravitational shear of the shapes of the galaxy images. But the biases seem to be small and will not be discussed here.

And finally, as the cosmological tests improve a satisfactory application will require a joint fit of all of the parameters to all of the relevant measurements and constraints. Until recently it made sense to impose prior conditions, most famously the hope that if the universe is not well described by the Einstein-de Sitter model then surely it is the case either that is negligibly small or else that space curvature may be neglected. We suspect the majority in the community still expect this is true, on the basis of the coincidences argument in Sec. II.B.2, but it will be important to see what comes out of joint fits of both M0 and 0, as well as all the other parameters, as is becoming the current practice. Our test-by-test discussion is useful for sorting out the physics and astronomy, we believe it is not the prototype for the coming generations of precision application of the tests.

Our remarks are ordered by our estimates of the model dependence.

We are in a sea of radiation with spectrum very close to Planck at T = 2.73 K, and isotropic to one part in 10 5 (after correction for a dipole term that usually is interpreted as the result of our motion relative to the rest frame defined by the radiation). 71 The thermal spectrum indicates thermal relaxation, for which the optical depth has to be large on the scale of the Hubble length H0 -1 . We know space now is close to transparent at the wavelengths of this radiation, because radio galaxies are observed at high redshift. Thus the universe has to have expanded from a state quite different from now, when it was hotter, denser, and optically thick. This is strong evidence our universe is evolving.

This interpretation depends on, and checks, conventional local physics with a single metric description of spacetime. Under these assumptions the expansion of the universe preserves the thermal spectrum and cools the temperature as 72

Bahcall and Wolf (1968) point out that one can test this temperature-redshift relation by measurements of the excitation temperatures of fine-structure absorption line systems in gas clouds along quasar lines of sight. The corrections for excitations by collisions and the local radiation field are subtle, however, and perhaps not yet fully sorted out (as discussed by Molaro et al., 2002, and references therein).

The 3 K thermal cosmic background radiation is a centerpiece of modern cosmology, but its existence does not test general relativity.

The best evidence that the expansion and cooling of the universe traces back to high redshift is the success of the standard model for the origin of deuterium and isotopes of helium and lithium, by reactions among radiation, leptons, and atomic nuclei as the universe expands and cools through temperature T

10 10 . The free parameter in the standard model is the present baryon number density. The model assumes the baryons are uniformly distributed at high redshift, so this parameter with the known present radiation temperature fixes the baryon number density as a function of temperature and the temperature as a function of time. The latter follows from the expansion rate Eq. (11), which at the epoch of light element formation may be written as

where the mass density r counts radiation, which is now at T = 2.73 K, the associated neutrinos, and e ± pairs. The curvature and terms are unimportant, unless the dark energy mass density varies quite rapidly.

Independent analyses of the fit to the measured element abundances, corrected for synthesis and destruction in stars, by Burles, Nollett, and Turner (2001), and Cyburt, Fields, and Olive (2001), indicate

both at 95% confidence limits. Other analyses by Coc et al. (2002) and Thuan and Izotov (2002) result in ranges that lie between the two of Eq. (62). The difference in values may be a useful indication of remaining uncertainties it is mostly a consequence of the choice of isotopes used to constrain B0 h 2 . Burles et al. (2001) use the deuterium abundance, Cyburt et al. (2001) favor the helium and lithium measurements, and the other two groups use other combinations of abundances. Equation. (62) is consistent with the summary range, 0.0095 B0 h 2 0.023 at 95% confidence, of Fields and Sarkar (2002).

The baryons observed at low redshift, in stars and gas, amount to (Fukugita, Hogan, and Peebles, 1998)

It is plausible that the difference between Eqs. (62) and (63) is in cool plasma, with temperature T

100 eV, in groups of galaxies. It is difficult to observationally constrain the idea that there is a good deal more cool plasma in the large voids between the concentrations of galaxies. A more indirect but eventually more precise constraint on B0, from the anisotropy of the 3 K thermal cosmic microwave background radiation, is discussed in test (11).

It is easy to imagine complications, such as inhomogeneous entropy per baryon, or in the physics of neutrinos examples may be traced back through Abazajian, Fuller, and Patel (2001) and Giovannini, Keihänen, and Kurki-Suonio (2002). It seems difficult to imagine that a more complicated theory would reproduce the successful predictions of the simple model, but Nature fools us on occasion. Thus before concluding that the theory of the pre-stellar light element abundances is known, apart from the addition of decimal places to the cross sections, it is best to wait and see what advances in the physics of baryogenesis and of neutrinos teach us.

How is general relativity probed? The only part of the computation that depends specifically on this theory is the pressure term in the active gravitational mass density, in the expansion rate equation (8). If we did not have general relativity, a simple Newtonian picture might have led us to write down / a = - 4 G r / 3 instead of Eq. (8). With r

1 / a 4 , as appropriate since most of the mass is fully relativistic at the redshifts of light element production, this would predict the expansion time a / is 2 1/2 times the standard expression (that from Eq. [61]). The larger expansion time would hold the neutron to proton number density ratio close to that at thermal equilibrium, n/p = e -Q/kT , where Q is the difference between the neutron and proton masses, to lower temperature. It would also allow more time for free decay of the neutrons after thermal equilibrium is broken. Both effects decrease the final 4 He abundance. The factor 2 1/2 increase in expansion time would reduce the helium abundance by mass to Y

0.20. This is significantly less than what is observed in objects with the lowest heavy element abundances, and so seems to be ruled out (Steigman, 2002). 73 That is, we have positive evidence for the relativistic expression for the active gravitational mass density at redshift z

The predicted time of expansion from the very early universe to redshift z is

where E(z) is defined in Eq. (11). If = 0 the present age is t0 < H0 -1 . In the Einstein-de Sitter model the present age is t0 = 2 / (3H0). If the dark energy density is significant and evolving, we may write = 0 f (z), where the function of redshift is normalized to f (0) = 1. Then E(z) generalizes to

In the XCDM parametrization with constant wX (Eq. [45]), f (z) = (1 + z) 3(1+wX) . Olson and Jordan (1987) present the earliest discussion we have found of H0 t0 in this picture (before it got the name). In scalar field models, f (z) generally must be evaluated numerically examples are in Peebles and Ratra (1988).

The relativistic correction to the active gravitational mass density (Eq. [8]) is not important at the redshifts at which galaxies can be observed and the ages of their star populations estimated. At moderately high redshift, where the nonrelativistic matter term dominates, Eq. (64) is approximately

That is, the ages of star populations at high redshift are an interesting probe of M0 but they are not very sensitive to space curvature or to a near constant dark energy density. 74

Recent analyses of the ages of old stars 75 indicate the expansion time is in the range

at 95% confidence, with central value t0 13 Gyr. Following Krauss and Chaboyer (2001) these numbers add 0.8 Gyr to the star ages, under the assumption star formation commenced no earlier than z = 6 (Eq. [66]). A naive addition in quadrature to the uncertainty in H0 (Eq. [6]) indicates the dimensionless age parameter is in the range

at 95% confidence, with central value H0 t0 0.89. The uncertainty here is dominated by that in t0. In the spatially-flat CDM model (K0 = 0), Eq. (68) translates to 0.15 M0 0.8, with central value M0 0.4. In the open model with 0 = 0, the constraint is M0 0.6 with the central value M0 0.1. In the inverse power-law scalar field dark energy case (Sec. II.C) with power-law index = 4, the constraint is 0.05 M0 0.8.

We should pause to admire the unification of the theory and measurements of stellar evolution in our galaxy, which yield the estimate of t0, and the measurements of the extragalactic distance scale, which yield H0, in the product in Eq. (68) that agrees with the relativistic cosmology with dimensionless parameters in the range now under discussion. As we indicated in Sec. III, there is a long history of discussion of the expansion time as a constraint on cosmological models. The measurements now are tantalizingly close to a check of consistency with the values of M0 and 0 indicated by other cosmological tests.

An object at redshift z with physical length l perpendicular to the line of sight subtends angle such that

where a0 = a(t0). The angular size distance r(z) is the coordinate position of the object in the first line element in Eq. (15), with the observer placed at the origin. The condition that light moves from source to observer on a radial null geodesic is

where E(z) is defined in Eqs. (11) and (65).

In the Einstein-de Sitter model, the angular-size-redshift relation is

At z << 1, = H0 l / z, consistent with the Hubble redshift-distance relation. At z >> 1 the image is magnified, 76 1 + z.

The relation between the luminosity of a galaxy and the energy flux density received by an observer follows from Liouville's theorem: the observed energy flux i0 per unit time, area, solid angle, and frequency satisfies

with ie the emitted energy flux (surface brightness) at the source and e = 0(1 + z) the bandwidth at the source at redshift z. The redshift factor (1 + z) 4 appears for the same reason as in the 3 K cosmic microwave background radiation energy density. With Eq. (69) to fix the solid angle, Eq. (73) says the observed energy flux per unit area, time, and frequency from a galaxy at redshift z that has luminosity Le per frequency interval measured at the source is

In conventional local physics with a single metric theory the redshift-angular size (Eq. [69]) and redshift-magnitude (Eq. [7]) relations are physically equivalent. 77

The best present measurement of the redshift-magnitude relation uses supernovae of Type Ia. 78 The results are inconsistent with the Einstein-de Sitter model, at enough standard deviations to make it clear that unless there is something quite substantially and unexpectedly wrong with the measurements the Einstein-de Sitter model is ruled out. The data require > 0 at two to three standard deviations, depending on the choice of data and method of analysis (Leibundgut, 2001 Gott et al., 2001). The spatially-flat case with M0 in the range of Eq. (59) is a good fit for constant . The current data do not provide interesting constraints on the models for evolving dark energy density. 79 Perlmutter et al. (http://snap.lbl.gov/) show that a tighter constraint, from supernovae observations to redshift z

2, by the proposed SNAP satellite, is feasible and capable of giving a significant detection of and maybe its evolution. 80

Counts of galaxies -- or of other objects whose number density as a function of redshift may be modeled -- probe the volume element (dV / dz)dz defined by a solid angle in the sky and a redshift interval dz. The volume is fixed by the angular size distance (Eq. [69]), which determines the area subtended by the solid angle, in combination with the redshift-time relation (Eq. [64]), which fixes the radial distance belonging to the redshift interval.

Sandage (1961a) and Brown and Tinsley (1974) showed that with the technology then available galaxy counts are not a very sensitive probe of the cosmological parameters. Loh and Spillar (1986) opened the modern exploration of the galaxy count-redshift relation at redshifts near unity, where the predicted counts are quite different in models with and without a cosmological constant (as illustrated in Figure 13.8 in Peebles, 1993).

The interpretation of galaxy counts requires an understanding of the evolution of galaxy luminosities and the gain and loss of galaxies by merging. Here is an example of the former in a spatially-flat cosmological model with M0 = 0.25. The expansion time from high redshift is t3 = 2.4 Gyr at redshift z = 3 and t0 = 15 Gyr now. Consider a galaxy observed at z = 3. Suppose the bulk of the stars in this galaxy formed at time tf, and the population then aged and faded without significant later star formation. Then if tf << t3 the ratio of the observed luminosity at z = 3 to its present luminosity would be (Tinsley, 1972 Worthey, 1994)

If tf were larger, but still less than t3, this ratio would be larger. If tf were greater than t3 the galaxy would not be seen, absent earlier generations of stars. In a more realistic picture significant star formation may be distributed over a considerable range of redshifts, and the effect on the typical galaxy luminosity at a given redshift accordingly more complicated. Since there are many more galaxies with low luminosities than galaxies with high luminosities, one has to know the luminosity evolution quite well for a meaningful comparison of galaxy counts at high and low redshifts. The present situation is illustrated by the rather different indications from studies by Phillipps et al. (2000) and Totani et al. (2001).

The understanding of galaxy evolution and the interpretation of galaxy counts will be improved by large samples of counts of galaxies as a function of color, apparent magnitude, and redshift. Newman and Davis (2000) point to a promising alternative: count galaxies as a function of the internal velocity dispersion that in spirals correlates with the dispersion in the dark matter halo. That could eliminate the need to understand the evolution of star populations. There is still the issue of evolution of the dark halos by merging and accretion, but that might be reliably modeled by numerical simulations within the CDM picture. Either way, with further work galaxy counts may provide an important test for dark energy and its evolution (Newman and Davis, 2000 Huterer and Turner, 2001 Podariu and Ratra, 2001).

The probability of production of multiple images of a quasar or a radio source by gravitational lensing by a foreground galaxy, or of strongly lensed images of a galaxy by a foreground cluster of galaxies, adds the relativistic expression for the deflection of light to the physics of the homogeneous cosmological model. Fukugita, Futamase, and Kasai (1990) and Turner (1990) point out the value of this test: at small M0 the predicted lensing rate is considerably larger in a flat model with than in an open model with = 0 (as illustrated in Fig 13.12 in Peebles, 1993).

The measurement problem for the analysis of quasar lensing is that quasars that are not lensed are not magnified by lensing, making them harder to find and the correction for completeness of detection harder to establish. Present estimates (Falco, Kochanek, and Muñoz, 1998 Helbig et al., 1999) do not seriously constrain M0 in an open model, and in a flat model (K0 = 0) suggest M0 > 0.36 at 2. This is close to the upper bound in Eq. (59). Earlier indications that the lensing rate in a flat model with constant requires a larger value of M0 than is suggested by galaxy dynamics led Ratra and Quillen (1992) and Waga and Frieman (2000) to investigate the inverse power-law potential dark energy scalar field case. They showed this can significantly lower the predicted lensing rate at K0 = 0 and small M0. The lensing rate still is too uncertain to draw conclusions on this point, but advances in the measurement certainly will be followed with interest.

The main problem in the interpretation of the rate of strong lensing of galaxies by foreground clusters as a cosmological test is the sensitivity of the lensing cross section to the mass distribution within the cluster (Wu and Hammer, 1993) for the present still somewhat uncertain state of the art see Cooray (1999) and references therein.

Estimates of the mean mass density from the relation between the mass distribution and the resulting peculiar velocities, 81 and from the gravitational deflection of light, probe gravity physics and constrain M0. The former is not sensitive to K0, 0, or the dynamics of the dark energy, the latter only through the angular size distances.

We begin with the redshift space of observed galaxy angular positions and redshift distances z/H0 in the radial direction. The redshift z has a contribution from the radial peculiar velocity, which is a probe of the gravitational acceleration produced by the inhomogeneous mass distribution. The two-point correlation function, v, in redshift space is defined by the probability that a randomly chosen galaxy has a neighbor at distance r|| along the line of sight in redshift space and perpendicular distance r ,

where n is the galaxy number density. This is the usual definition of a reduced correlation function. Peculiar velocities make the function anisotropic. On small scales the random relative peculiar velocities of the galaxies broaden v along the line of sight. On large scales the streaming peculiar velocity of convergence to gravitationally growing mass concentrations flattens v along the line of sight. 82

At 10 kpc hr 1 Mpc the measured line-of-sight broadening is prominent, and indicates the one-dimensional relative velocity dispersion is close to independent of r at

300 km s -1 . 83 This is about what would be expected if the mass two-and three-point correlation functions were well approximated by the galaxy correlation functions, the mass clustering on these scales were close to statistical equilibrium, and the density parameter were in the range of Eq. (59).

We have a check from the motions of the galaxies in and around the Local Group of galaxies, where the absolute errors in the measurements of galaxy distances are least. The two largest group members are the Andromeda Nebula (M 31) and our Milky Way galaxy. If they contain most of the mass their relative motion is the classical two-body problem in Newtonian mechanics (with minor corrections for , mass accretion at low redshifts, and the tidal torques from neighboring galaxies). The two galaxies are separated by 800 kpc and approaching at 110 km s -1 . In the minimum mass solution the galaxies have completed just over half an orbit in the cosmological expansion time t0

10 10 yr. By this argument Kahn and Woltjer (1959) find the sum of masses of the two galaxies has to be an order of magnitude larger than what is seen in the luminous parts. An extension to the analysis of the motions and distances of the galaxies within 4 Mpc distance from us, and taking account of the gravitational effects of the galaxies out to 20 Mpc distance, gives masses quite similar to what Kahn and Woltjer found, and consistent with M0 in the range of Eq. (59) (Peebles et al., 2001).

We have another check from weak lensing: the shear distortion of images of distant galaxies by the gravitational deflection by the inhomogeneous mass distribution. 84 If galaxies trace mass these measurements say the matter density parameter measured on scales from about 1 Mpc to 10 Mpc is in the range of Eq. (59). It will be interesting to see whether these measurements can check the factor of two difference between the relativistic gravitational deflection of light and the naive Newtonian deflection angle.

The redshift space correlation function v (Eq. [76]) is measured well enough at hr

10 Mpc to demonstrate the flattening effect, again consistent with M0 in the range of Eq. (59), if galaxies trace mass. Similar numbers follow from galaxies selected as far infrared IRAS sources (Tadros et al., 1999) and from optically selected galaxies (Padilla et al., 2001 Peacock et al., 2001). The same physics, applied to estimates of the mean relative peculiar velocity of galaxies at separations

10 Mpc, yet again indicates a similar density parameter (Juszkiewicz et al., 2000).

Other methods of analysis of the distributions of astronomical objects and peculiar velocities smoothed over scales 10 Mpc give a variety of results for the mass density, some above the range in Eq. (59), 85 others towards the bottom end of the range (Branchini et al., 2001). The measurement of M0 from large-scale streaming velocities thus remains open. But we are impressed by an apparently simple local situation, the peculiar motion of the Local Group toward the Virgo cluster of galaxies. This is the nearest known large mass concentration, at distance

20 Mpc. Burstein (2000) finds that our virgocentric velocity is vv = 220 km s -1 , indicating M0 0.2 (Davis and Peebles, 1983a, Fig. 1). This leads us to conclude that the weight of the evidence from dynamics on scales

10 Mpc favors low M0, in the range of Eq. (59).

None of these measurements is precise. But many have been under discussion for a long time and seem to us to be reliably understood. Weak lensing is new, but the measurements are checked by several independent groups. The result, in our opinion, is a well checked and believable network of evidence that over two decades of well-sampled length scales, 100 kpc to 10 Mpc, the apparent value of M0 is constant to a factor of three or so, in the range 0.15 M0 0.4. The key point for the purpose of this review is that this result is contrary to what might have been expected from biasing, or from a failure of the inverse square law (as will be discussed in test [13]).

Abell (1958) made the first useful catalog of the rich clusters considered here and in the next test. A typical value of the Abell cluster mass within the Abell radius ra = 1.5 h -1 Mpc is 3 × 10 14 h -1 M . The cluster masses are reliably measured (within Newtonian gravity) from consistent results from the velocities of the galaxies, the pressure of the intracluster plasma, and the gravitational deflection of light from background galaxies.

White (1992) and White et al. (1993) point out that rich clusters likely are large enough to contain a close to fair sample of baryons and dark matter, meaning the ratio of baryonic to total mass in a cluster is a good measure of B0 / M0. With B0 from the model for light elements (Eqs. [62]), this gives a measure of the mean mass density. The baryon mass fraction in clusters is still under discussion. 86 We adopt as the most direct and so maybe most reliable approach the measurement of the baryonic gas mass fraction of clusters, fgas, through the Sunyaev-Zel'dovich microwave decrement caused by Thomson-Compton scattering of cosmic microwave background radiation by the intracluster plasma. The Carlstrom et al. (2001) value for fgas gives M0

0.25, 87 in the range of Eq. (59). This test does not directly constrain K0, 0, or the dynamics of the dark energy.

In the CDM model rich clusters of galaxies grow out of the rare peak upward fluctuations in the primeval Gaussian mass distribution. Within this model one can adjust the amplitude of the mass fluctuations to match the abundance of clusters at one epoch. In the Einstein-de Sitter model it is difficult to see how this one free adjustment can account for the abundance of rich clusters now and at redshifts near unity. 88

Most authors now agree that the low density flat CDM model can give a reasonable fit to the cluster abundances as a function of redshift. The constraint on M0 from the present cluster abundance still is under discussion, but generally is found to be close to M0

0.3 if galaxies trace mass. 89 The constraint from the evolution of the cluster number density also is under discussion. 90 The predicted evolution is slower in a lower density universe, and at given M0 the evolution is slower in an open model with = 0 than in a spatially-flat model with (for the reasons discussed in Sec. III.D). Bahcall and Fan (1998) emphasize that we have good evidence for the presence of some massive clusters at z

1, and that this is exceedingly difficult to understand in the CDM model in the Einstein-de Sitter cosmology (when biasing is adjusted to get a reasonable present number density). Low density models with or without can account for the existence of some massive clusters at high redshift. Distinguishing between the predictions of the spatially curved and flat low density cases awaits better measurements.

Elements of the physics of cluster formation in test (9) appear in this test of the early stages in the nonlinear growth of departures from homogeneity. An initially Gaussian mass distribution becomes skew as low density fluctuations start to bottom out and high density fluctuations start to develop into prominent mass peaks. The early signature of this nonlinear evolution is the disconnected three-point mass autocorrelation function, <(, t) (, t) (, t)>, where (, t) = / is the dimensionless mass contrast. If galaxies are useful mass tracers the galaxy three-point function is a good measure of this mass function.

The form for the mass three-point function, for Gaussian initial conditions at high redshift, in lowest nonzero order in perturbation theory, is worked out in Fry (1984), and Fry (1994) makes the point that measurements of the galaxy three-point function test how well galaxies trace mass. 91 There are now two sets of measurements of the galaxy three-point function on scales

10 to 20 Mpc, where the density fluctuations are not far from Gaussian. One uses infrared-selected IRAS galaxies, 92 the other optically-selected galaxies (Verde et al., 2002). The latter is consistent with the perturbative computation of the mass three-point function for Gaussian initial conditions. The former says infrared-selected galaxies are adequate mass tracers apart from the densest regions, which IRAS galaxies avoid. That has a simple interpretation in astrophysics: galaxies in dense regions tend to be swept clear of the gas and dust that make galaxies luminous in the infrared.

This test gives evidence of consistency of three ideas: galaxies are useful mass tracers on scales

10 Mpc, the initial conditions are close to Gaussian, and conventional gravity physics gives an adequate description of this aspect of the growth of structure. It is in principle sensitive to 0, through the suppression of the growth of small departures from homogeneity at low redshift, but the effect is small.

The wonderfully successful CDM prediction of the power spectrum of the angular distribution of the temperature of the 3 K cosmic microwave background radiation has converted many of the remaining skeptics in the cosmology community to the belief that the CDM model likely does capture important elements of reality.

Efstathiou (2002) provides a useful measure of the information in the present measurements 93 : the fit to the CDM model significantly constrains three linear combinations of the free parameters. We shall present three sets of considerations that roughly follow Efstathiou's constraints. We begin with reviews of the standard measure of the temperature anisotropy and of the conditions at redshift z

1000 that are thought to produce the observed anisotropy.

The 3 K cosmic microwave background temperature T(, ) as a function of position in the sky usually is expressed as an expansion in spherical harmonics,

The square of T averaged over the sky is

where |al m | 2 is statistically independent of m. This may be rewritten as

Since l -1 is close to d ln l, Tl 2 is the variance of the temperature per logarithmic interval of l. A measure of the angular scale belonging to the multipole index l is that the minimum distance between zeros of the spherical harmonic Yl m , in longitude or latitude, is = / l, except close to the poles, where Yl m approaches zero. 94

Now let us consider the main elements of the physics that determines the 3 K cosmic microwave background anisotropy. 95 At redshift zdec

1000 the temperature reaches the critical value at which the primeval plasma combines to atomic hydrogen (and slightly earlier to neutral helium). This removes the coupling between baryons and radiation by Thomson scattering, leaving the radiation to propagate nearly freely (apart from residual gravitational perturbations). Ratios of mass densities near the epoch zdec when matter and radiation decouple are worth noting. At redshift zeq = 2.4 × 10 4 M0 h 2 the mass density in matter -- including the baryonic and nonbaryonic components -- is equal to the relativistic mass density in radiation and neutrinos assumed to have low masses. At decoupling the ratio of mass densities is

at the central values of the parameters in Eqs. (6) and (59). The ratio of mass densities in baryons and in thermal cosmic microwave background radiation -- not counting neutrinos -- is

That is, the baryons and radiation decouple just as the expansion rate has become dominated by nonrelativistic matter and the baryons are starting to lower the velocity of sound in the coupled baryon-radiation fluid (presenting us with still more cosmic coincidences).

The acoustic peaks in the spectrum of angular fluctuations of the 3 K cosmic microwave background radiation come from the Fourier modes of the coupled baryon-radiation fluid that have reached maximum or minimum amplitude at decoupling. Since all Fourier components start at zero amplitude at high redshift -- in the growing density perturbation mode -- this condition is

where cs is the velocity of sound in the baryon-radiation fluid. Before decoupling the mass density in radiation is greater than that of the baryons, so the velocity of sound is close to c / 3 1/2 . The proper wavelength at the first acoustic peak thus is

The parameter dependence comes from Eq. (66). The observed angle subtended by peak is set by the angular size distance r computed from zeq to the present (Eq. [71]). If K0 = 0 or 0 = 0 the angular size distance is

If = 0 this expression is analytic at large zeq. The expression for K0 = 0 is a reasonable approximation to the numerical solution. So the angular scale of the peak varies with the matter density parameter as

The key point from these considerations is that the angle defined by the first peak in the fluctuation power spectrum is sensitive to M0 if = 0 (Eq. [85]), but not if K0 = 0 (Eq. [86]). 96 We have ignored the sensitivity of zdec and tdec to M0, but the effect is weak. More detailed computations, which are needed for a precise comparison with the data, show that the CDM model predicts that the first and largest peak of Tl appears at multipole index lpeak 220 M0 -1/2 if = 0, and at lpeak 220 if K0 = 0 and 0.1 M0 1. 97

The measured spectrum 98 peaks at Tl

200, thus requiring small space curvature in the CDM model. This is the first of Efstathiou's constraints. Because of the geometric degeneracy this measurement does not yet seriously constrain M0 if K0 = 0.

The second constraint comes from the spectrum of temperature fluctuations on large scales, l 30, where pressure gradient forces never were very important. Under the scale-invariant initial conditions discussed in Sec. III.C the Einstein-de Sitter model predicts Tl is nearly independent of l on large scales. A spatially-flat model with M0

0.3, predicts Tl decreases slowly with increasing l at small l. 99 The measured spectrum is close to flat at Tl

30 µK, but not well enough constrained for a useful measure of the parameters M0 and 0. 100 Because of the simplicity of the physics on large angular scales, this provides the most direct and so perhaps most reliable normalization of the CDM model power spectrum (that is, the parameter A in Eqs. [40] and [41]).

The third constraint is the baryon mass density. It affects the speed of sound cs (Eq. [82]) in the baryon-radiation fluid prior to decoupling, and the mean free path for the radiation at z

zdec. These in turn affect the predicted sequence of acoustic peaks (see, e.g., Hu and Sugiyama, 1996). The detected peaks are consistent with a value for the baryon density parameter B0 in a range that includes what is derived from the light elements abundances (Eqs. [62]). 101 This impressive check may be much improved by the measurements of Tl in progress.

The measurements of Tl are consistent with a near scale-invariant power spectrum (Eq. [41] with n 1) with negligible contribution from gravity wave or isocurvature fluctuations (Sec. III.C.1). The 3 K cosmic microwave background temperature fluctuations show no departure from a Gaussian random process. 102 This agrees with the picture in test (10) for the nonlinear growth of structure out of Gaussian initial mass density fluctuations.

The interpretation of the cosmic microwave background temperature anisotropy measurements assumes and tests general relativity and the CDM model. One can write down other models for structure formation that put the peak of Tl at about the observed angular scale -- an example is Hu and Peebles (2000) -- but we have seen none so far that seem likely to fit the present measurements of Tl. Delayed recombination of the primeval plasma in an low density = 0 CDM model can shift the peak of Tl to the observed scale. 103 The physics is valid, but the scenario is speculative and arguably quite improbable. On the other hand, we cannot be sure a fix of the challenges to CDM reviewed in Sec. IV.A.2 will not affect our assessments of such issues, and hence of this cosmological test.

If the bulk of the nonrelativistic matter, with density parameter M0

0.25, were baryonic, then under adiabatic initial conditions the most immediate problem would be the strong dissipation of primeval mass density fluctuations on the scale of galaxies by diffusion of radiation through the baryons at redshifts near decoupling. 104 Galaxies could form by fragmentation of the first generation of protocluster "pancakes," as Zel'dovich (1978) proposed, but this picture is seriously challenged by the evidence that the galaxies formed before clusters of galaxies. 105 In a baryonic dark matter model we could accommodate the observations of galaxies already present at z

3 by tilting the primeval mass fluctuation spectrum to favor large fluctuations on small scales, but that would mess up the cosmic microwave background anisotropy. The search for isocurvature initial conditions that might fit both in a baryonic dark matter model has borne no fruit so far (Peebles, 1987).

The most important point of this test is the great difficulty of reconciling the power spectra of matter and radiation without the postulate of nonbaryonic dark matter. The CDM model allows hierarchical growth of structure, from galaxies up, which is what seems to be observed, because the nonbaryonic dark matter interacts with baryons and radiation only by gravity the dark matter distribution is not smoothed by the dissipation of density fluctuations in the baryon-radiation fluid at redshifts z zeq.

As discussed in Sec. III.D, in the CDM model the small scale part of the dark matter power spectrum bends from the primeval scale-invariant form P(k) k to P(k) k -3 , and the characteristic length at the break scales inversely with M0 (Eq. [42]). Evidence of such a break in the galaxy power spectrum Pg(k) has been known for more than a decade 106 it is consistent with a value of M0 in the range of Eq. (59).

The inverse square law for gravity determines the relation between the mass distribution and the gravitationally-driven peculiar velocities that enter estimates of the matter density parameter M0. The peculiar velocities also figure in the evolution of the mass distribution, and hence the relation between the present mass fluctuation spectrum and the spectrum of cosmic microwave background temperature fluctuations imprinted at redshift z

1000. We are starting to see demanding tests of both aspects of the inverse square law.

We have a reasonably well checked set of measurements of the apparent value of M0 on scales ranging from 100 kpc to 10 Mpc (as reviewed under test [7]). Most agree with a constant value of the apparent M0, within a factor of three or so. This is not the precision one would like, but the subject has been under discussion for a long time, and, we believe, is now pretty reliably understood, within the factor of three or so. If galaxies were biased tracers of mass one might have expected to have seen that M0 increases with increasing length scale, as the increasing scale includes the outer parts of extended massive halos. Maybe that is masked by a gravitational force law that decreases more rapidly than the inverse square law at large distance. But the much more straightforward reading is that the slow variation of M0 sampled over two orders of magnitude in length scale agrees with the evidence from tests (7) to (10) that galaxies are useful mass tracers, and that the inverse square law therefore is a useful approximation on these scales.

The toy model in Eq. (57) illustrates how a failure of the inverse square law would affect the evolution of the shape of the mass fluctuation power spectrum P(k, t) as a function of the comoving wavenumber k, in linear perturbation theory. This is tested by the measurements of the mass and cosmic microwave background temperature fluctuation power spectra. The galaxy power spectrum Pg(k) varies with wavenumber at k

0.1h Mpc -1 about as expected under the assumptions that the mass distribution grew by gravity out of adiabatic scale-invariant initial conditions, the mass is dominated by dark matter that does not suffer radiation drag at high redshift, the galaxies are useful tracers of the present mass distribution, the matter density parameter is M0

0.3, and, of course, the evolution is adequately described by conventional physics (Hamilton and Tegmark, 2002, and references therein). If the inverse square law were significantly wrong at k

0.1h Mpc -1 , the near scale-invariant form would have to be an accidental effect of some failure in this rather long list of assumptions. This seems unlikely, but a check certainly is desirable. We have one, from the cosmic microwave background anisotropy measurements. They also are consistent with near scale-invariant initial conditions applied at redshift z

1000. This preliminary check on the effect of the gravitational inverse square law applied on cosmological length scales and back to redshift z

1000 will be improved by better understanding of the effect on Tl of primeval tensor perturbations to spacetime, and of the dynamical response of the dark energy distribution to the large-scale mass distribution.

Another aspect of this check is the comparison of values of the large-scale rms fluctuations in the present distributions of mass and the cosmic microwave background radiation. The latter is largely set at decoupling, after which the former grows by a factor of about 10 3 to the present epoch, in the standard relativistic cosmological model. If space curvature is negligible the growth factor agrees with the observations to about 30%, assuming galaxies trace mass. In a low density universe with = 0 the standard model requires that mass is more smoothly distributed than galaxies, N / N

3M / M, or that the gravitational growth factor since decoupling is a factor of three off the predicted factor

1000 this factor of three is about as large a deviation from unity as is viable. We are not proposing this interpretation of the data, rather we are impressed by the modest size of the allowed adjustment to the inverse square law.

70 This was recognized by Zel'dovich (1964), R. Feynman, in 1964, and S. Refsdal, in 1965. Feynman's comments in a colloquium are noted by Gunn (1967). Peebles attended Refsdal's lecture at the International Conference on General Relativity and Gravitation, London, July 1965 Refsdal (1970) mentions the lecture. Back.

71 The history of the discovery and measurement of this radiation, and its relation to the light element abundances in test (2), is presented in Peebles (1971, pp. 121-9 and 240-1), Wilkinson and Peebles (1990), and Alpher and Herman (2001). The precision spectrum measurements are summarized in Halpern, Gush, and Wishnow (1991) and Fixsen et al. (1996). Back.

72 To see this, recall the normal modes argument used to get Eq. (7). The occupation number in a normal mode with wavelength at temperature T is the Planck form = [e c / kT - 1] -1 . Adiabaticity says is constant. Since the mode wavelength varies as a(t), where a is the expansion factor in Eq. (4), and is close to constant, the mode temperature varies as T 1 / a(t). Since the same temperature scaling applies to each mode, an initially thermal sea of radiation remains thermal in the absence of interactions. We do not know the provenance of this argument it was familiar in Dicke's group when the 3 K cosmic microwave background radiation was discovered. Back.

73 There is a long history of discussions of this probe of the expansion rate at the redshifts of light element production. The reduction of the helium abundance to Y

0.2 if the expansion time is increased by the factor 2 1/2 is seen in Figs. 1 and 2 in Peebles (1966). Dicke (1968) introduced the constraint on evolution of the strength of the gravitational interaction see Uzan (2002) for a recent review. The effect of the number of neutrino families on the expansion rate and hence the helium abundance is noted by Hoyle and Tayler (1964) and Shvartsman (1969). Steigman, Schramm, and Gunn (1977) discuss the importance of this effect as a test of cosmology and of the particle physics measures of the number of neutrino families. Back.

74 The predicted maximum age of star populations in galaxies at redshifts z 1 does still depend on 0 and K0, and there is the advantage that the predicted maximum age is a lot shorter than today. This variant of the expansion time test is discussed by Nolan et al. (2001), Lima and Alcaniz (2001), and references therein. Back.

76 The earliest discussion we know of the magnification effect is by Hoyle (1959). In the coordinate system in Eq. (15), with the observer at the origin, light rays from the object move to the observer along straight radial lines. An image at high redshift is magnified because the light detected by the observer is emitted when the proper distance to the object measured at fixed world time is small. Because the proper distance between the object and source is increasing faster than the speed of light, emitted light directed at the observer is initially moving away from the observer. Back.

77 For a review of measurements of the redshift-magnitude relation (and other cosmological tests) we recommend Sandage (1988). A recent application to the most luminous galaxies in clusters is in Aragón-Salamanca, Baugh, and Kauffmann (1998). The redshift-angular size relation is measured by Daly and Guerra (2001) for radio galaxies, Buchalter et al. (1998) for quasars, and Gurvits, Kellermann, and Frey (1999) for compact radio sources. Constraints on the cosmological parameters from the Gurvits et al. data are discussed by Vishwakarma (2001), Lima and Alcaniz (2002), Chen and Ratra (2003), and references therein, and constraints based on the radio galaxy data are discussed by Daly and Guerra (2001), Podariu et al. (2003), and references therein. Back.

78 These supernovae are characterized by the absence of hydrogen lines in the spectra they are thought to be the result of explosive nuclear burning of white dwarf stars. Pskovskii (1977) and Phillips (1993) pioneered the reduction of the supernovae luminosities to a near universal standard candle. For recent discussions of their use as a cosmological test see Goobar and Perlmutter (1995), Reiss et al. (1998), Perlmutter et al. (1999a), Gott et al. (2001), and Leibundgut (2001). We recommend Leibundgut's (2001) cautionary discussion of astrophysical uncertainties: the unknown nature of the trigger for the nuclear burning, the possibility that the Phillips correction to a fiducial luminosity actually depends on redshift or environment within a galaxy, and possible obscuration by intergalactic dust. There are also issues of physics that may affect this test (and others): the strengths of the gravitational or electromagnetic interactions may vary with time, and photon-axion conversion may reduce the number of photons reaching us. All of this is under active study. Back.

79 Podariu and Ratra (2000) and Waga and Frieman (2000) discuss the redshift-magnitude relation in the inverse power-law scalar field model, and Waga and Frieman (2000) and Ng and Wiltshire (2001) discuss this relation in the massive scalar field model. Back.

80 Podariu, Nugent, and Ratra (2001), Weller and Albrecht (2002), Wang and Lovelace (2001), Gerke and Efstathiou (2002), Eriksson and Amanullah (2002), and references therein, discuss constraints on cosmological parameters from the proposed SNAP mission. Back.

81 Early estimates of the mean mass density, by Hubble (1936, p. 189) and Oort (1958), combine the galaxy number density from galaxy counts with estimates of galaxy masses from the internal motions of gas and stars. Hubble (1936, p. 180) was quite aware that this misses mass between the galaxies, and that the motions of galaxies within clusters suggests there is a lot more intergalactic mass (Zwicky, 1933 Smith, 1936). For a recent review of this subject see Bahcall et. al. (2000). Back.

82 This approach grew out of the statistical method introduced by Geller and Peebles (1973) it is derived in its present form in Peebles (1980b) and first applied to a serious redshift sample in Davis and Peebles (1983b). These references give the theory for the second moment 2 of v in the radial direction -- the mean square relative peculiar velocity -- in the small-scale stable clustering limit. The analysis of the anisotropy of v in the linear perturbation theory of large-scale flows (Eq. [55]) is presented in Kaiser (1987). Back.

83 This measurement requires close attention to clusters that contribute little to the mean mass density but a broad and difficult to measure tail to the distribution of relative velocities. Details may be traced back through Padilla et al. (2001), Peacock et al. (2001), and Landy (2002). Back.

87 This assumes B0 h 2 = 0.014 from Eqs. (62). For the full range of values in Eqs. (6) and (62), 0.1 M0 0.4 at two standard deviations. Back.

89 For recent discussions see Pierpaoli, Scott, and White (2001), Seljak (2001), Viana, Nichol, and Liddle (2002), Ikebe et al. (2002), Bahcall et al. (2002), and references therein. Wang and Steinhardt (1998) consider this test in the context of the XCDM parametrization to our knowledge it has not been studied in the scalar field dark energy case. Back.

92 Two sub-samples of IRAS galaxies are analyzed by Scoccimarro et al. (2001) and by Feldman et al. (2001). Back.

94 A more careful analysis distinguishes averages across the sky from ensemble averages. By historical accident the conventional normalization replaces 2l + 1 with 2(l + 1) in Eq. (79). Kosowsky (2002) reviews the physics of the polarization of the radiation. Back.

95 The physics is worked out in Peebles and Yu (1970) and Peebles (1982). Important analytic considerations are in Sunyaev and Zel'dovich (1970). The relation of the cosmic microwave background anisotropy to the cosmological parameters is explored in many papers examples of the development of ideas include Bond (1988), Bond et al. (1994), Hu and Sugiyama (1996), Ratra et al. (1997, 1999), Zaldarriaga, Spergel, and Seljak (1997), and references therein. Back.

96 This "geometrical degeneracy" is discussed by Efstathiou and Bond (1999). Marriage (2002) presents a closer analysis of the effect. Sugiyama and Gouda (1992), Kamionkowski, Spergel, and Sugiyama (1994b), and Kamionkowski et al. (1994a) are early discussions of the cosmic microwave background anisotropy in an open model. Back.

97 Brax et al. (2000) and Baccigalupi et al. (2000) compute the angular spectrum of the cosmic microwave background anisotropy in the dark energy scalar field model. Doran et al. (2001) discuss the angular scale of the peaks in this case, and Corasaniti and Copeland (2002), Baccigalupi et al. (2002), and references therein, compare model predictions and observations -- it is too early to draw profound conclusions about model viability, and new data are eagerly anticipated. Wasserman (2002) notes that the cosmic microwave background anisotropy data could help discriminate between different dark energy scalar field models whose predictions do not differ significantly at low redshift. Back.

99 The physics was first demonstrated by Sachs and Wolfe (1967) and applied in the modern context by Peebles (1982). The intermediate Sachs-Wolfe effect that applies if the universe is not Einstein-de Sitter is shown in Eq. (93.26) in Peebles (1980). This part of the Sachs-Wolfe effect receives a contribution from the low redshift matter distribution, so cross-correlating the observed large-scale cosmic microwave background anisotropy with the low redshift matter distribution could provide another test of the world model (Boughn and Crittenden, 2001, and references therein). Back.

100 See, e.g., Górski et al. (1998). This ignores the "low" value of the cosmological quadrupole (l = 2) moment, whose value depends on the model used to remove foreground Galactic emission (see, e.g., Kogut et al., 1996). Contamination due to non-cosmic microwave background emission is an issue for some of the anisotropy data sets (see, e.g., de Oliveira-Costa et al., 1998 Hamilton and Ganga, 2001 Mukherjee et al., 2002, and references therein). Other issues that need care in such analyses include accounting for the uncertainty in the calibration of the experiment (see, e.g., Ganga et al., 1997 Bridle et al., 2002), and accounting for the shape of the antenna pattern (see, e.g., Wu et al., 2001a Souradeep and Ratra, 2001 Fosalba, Dore, and Bouchet, 2002). Back.

101 The B0 h 2 values estimated from the cosmic microwave background anisotropy measured by Netterfield et al. (2002), Pryke et al. (2002), and Stompor et al. (2001), are more consistent with the higher, deuterium based, Burles et al. (2001) range in Eqs. (62). Back.

102 Colley, Gott, and Park (1996) present an early discussion of the situation on large angular scales more recent discussions are in Mukherjee, Hobson, and Lasenby (2000), Phillips and Kogut (2001), and Komatsu et al. (2002). Degree and sub-degree angular scale anisotropy data are studied in Park et al. (2001), Wu et al. (2001b), Shandarin et al. (2002), and Polenta et al. (2002). Back.

103 The model in Peebles, Seager, and Hu (2000) assumes stellar ionizing radiation at z

1000 produces recombination Lyman photons. These resonance photons promote photoionization from the n = 2 level of atomic hydrogen. That allows delayed recombination with a rapid transition to neutral atomic hydrogen, as required to get the shape of Tl about right. Back.

104 Early analyses of this effect are in Peebles (1965), and Silk (1967, 1968). Back.

105 For example, our Milky Way galaxy is in the Local Group, which seems to be just forming, because the time for a group member to cross the Local Group is comparable to the Hubble time. The Local Group is on the outskirts of the concentration of galaxies around the Virgo cluster. We and neighboring galaxies are moving away from the cluster, but at about 80 percent of the mean Hubble flow, as if the local mass concentration were slowing the local expansion. That is, our galaxy, which is old, is starting to cluster with other galaxies, in a "bottom up" hierarchical growth of structure, as opposed to the "top down" evolution of the pancake picture. Back.

## 2 Method

To construct our base conditional density estimator we follow the usual random forest construction with a key modification in the loss function, while retaining algorithms with linear complexity. Here we describe our base algorithm the extension to functional covariates is described in Section

At their simplest, random forests are ensembles of regression trees. Each tree is trained on a bootstrapped sample of the data. The training process involves recursively partitioning the feature space through splitting rules taking the form of splitting into the sets < X i ≤ v >and < X i > v >for a particular feature X i and split point v . Once a partition becomes small enough (controlled by a tuning parameter), it becomes a leaf node and is no longer partitioned.

For prediction we use the tree structure to calculate weights for the training data from which we perform a

using “nearby” points. This is analogous to the regression case which would perform a weighted mean.

Borrowing the notation of Breiman [2001] and Meinshausen [2006] , let θ t denote the tree structure for a single tree. Let R ( x , θ t ) denote the region of feature space covered by the leaf node for input x . Then for a new observation x ∗ we use t -th tree to calculate weights for each training point x i as

We then aggregate over trees, setting w i ( x ∗ ) = T − 1 ∑ T t = 1 w i ( x ∗ , θ t ) . The weights are finally used for the weighted kernel density estimate

where K h is a kernel function integrating to one. The bandwidth can be selected using plug-in methods or through tuning based upon a validation set. Up to this point we have the same approach as Meinshausen [2006] .

Our departure from the standard random forest algorithm is the criterion for choosing the splits of the partitioning. In regression contexts, the splitting variable and split point are often chosen to minimize the mean-squared error loss. For CDE, we instead choose splits that minimize a loss specific to CDE [Izbicki and Lee, 2017]

where P ( x ) is the marginal distribution of X .

This loss is the L 2 error for density estimation weighted by the marginal density of the covariates. To conveniently estimate this loss we can expand the square and rewrite the loss as

with C p as a constant which does not depend on ˆ p . The first expectation is with respect to the marginal distribution of X

and the second with respect to the joint distribution of

X and Y . We estimate these expectations by their empirical expectation on observed data.

To provide intuition why switching the loss function is desirable: consider the example in Figure 1 : we have two stationary distributions separated by a transition point at x = 0.5

: the left is a normal distribution centered at zero and the right is a mixture of two normal distributions centered at 1 and -1. The clear split for a tree is the transition point: however, because the generating distribution’s conditional mean is constant for all

x the mean-square-error loss fits to noise and picks a poor split point far from x = 0.5 . The CDE loss on the other hand is minimized at the split point.

Figure 1: (top) Training data with a switch from a unimodal to bimodal response at x = 0.5 (bottom) the normalized losses for each cut-point. MSE overfits to small differences in the bimodal regime while CDE loss is minimized at the true transition point.

While we use kernel density estimates for predictions on new observations, we do not use kernel density estimates when evaluating splits: the calculations in Equation 2 would be expensive for KDE with the term ∫ ˆ p ( y ∣ x ) 2 d y depending on the O ( n 2 ) pairwise distances between all training points.

For fast computations, we instead use orthogonal series to compute density estimates for splitting. Given an orthogonal basis < Φ j ( y ) >such as a cosine basis or wavelet basis, we can express the density as ˆ p ( y ∣ x ) = ∑ j ˆ β j ϕ j ( y ) where ˆ β j = 1 n ∑ n i = 1 ϕ j ( y i ) . This choice is motivated by a convenient formula for the CDE loss associated with an orthogonal series density estimate

The above expression only depends upon the quantities < ˆ β j >that themselves depend only upon linear sums of ϕ j ( y i ) . This makes it computationally efficient to evaluate the CDE loss for each split.

## Help Humanity

"You must be the change you wish to see in the world."
(Mohandas Gandhi)

"When forced to summarize the general theory of relativity in one sentence: Time and space and gravitation have no separate existence from matter. . Physical objects are not in space, but these objects are spatially extended. In this way the concept 'empty space' loses its meaning. . The particle can only appear as a limited region in space in which the field strength or the energy density are particularly high. .
The free, unhampered exchange of ideas and scientific conclusions is necessary for the sound development of science, as it is in all spheres of cultural life. . We must not conceal from ourselves that no improvement in the present depressing situation is possible without a severe struggle for the handful of those who are really determined to do something is minute in comparison with the mass of the lukewarm and the misguided. .
Humanity is going to need a substantially new way of thinking if it is to survive!" (Albert Einstein)

Our world is in great trouble due to human behaviour founded on myths and customs that are causing the destruction of Nature and climate change. We can now deduce the most simple science theory of reality - the wave structure of matter in space. By understanding how we and everything around us are interconnected in Space we can then deduce solutions to the fundamental problems of human knowledge in physics, philosophy, metaphysics, theology, education, health, evolution and ecology, politics and society.

This is the profound new way of thinking that Einstein realised, that we exist as spatially extended structures of the universe - the discrete and separate body an illusion. This simply confirms the intuitions of the ancient philosophers and mystics.

Given the current censorship in physics / philosophy of science journals (based on the standard model of particle physics / big bang cosmology) the internet is the best hope for getting new knowledge known to the world. But that depends on you, the people who care about science and society, realise the importance of truth and reality.

## Segmentation Algorithms

Automated cell identification algorithms systematically applied to every image are the favored approach to the analysis of complex images because (i) they remove the possibility of human bias and (ii) they pick up patterns that go unnoticed by the naked eye. Despite their many advantages, two-photon in vivo images present one important limitation that any automated method must overcome: nonhomogeneous illumination depending on both depth and location. This lack of homogeneity was dealt with through the use of local thresholding methods, which apply an algorithm to the local distribution of pixel intensities to find an appropriate separation between foreground and background. Astrocyte, plaque, and vessel channels were processed independently. All algorithms were implemented through a combination of the Fiji image processing package with custom Python code.

### Astrocyte Identification.

First, subtract the blood vessel channel, and plaque channel if it is an APP/PS1 mouse, from the astrocyte channel to remove fluorescence bleed across channels. Second, apply 3D Gaussian smoothing, sigma = 2.000. Third, apply rolling ball background subtraction on each plane to remove an approximately constant background (30). Fourth, apply the Bernsen autothresholding method to each plane with radius = 30 pixels (31, 32). Fifth, apply 3D Gaussian smoothing, sigma = 5.00. This has the effect of connecting nearby foreground pixels from the previous step (i.e., isolated foreground pixels will have their intensities greatly reduced, whereas clusters of foreground pixels will be essentially unaffected). Finally, autothreshold using the Otsu method (32, 33) and the intensity distribution of the entire stack. Combined with the previous step this has the effect of preserving clusters of foreground and removing isolated pixels. No hard area cut is required with this method.

### Blood Vessel Identification.

First, apply 3D Gaussian smoothing, sigma = 2.000. Second, autothreshold the stack using the Max-Entropy method (32, 34). Finally, find the 100 largest 3D-connected regions, using ref. 35, because a blood vessel should not appear in isolation except at image boundaries.

### Plaque Identification.

First, autothreshold the stack using the intermodes method (32, 36). Second, apply 3D erosion (37) and despeckle (i.e., median filter) each plane. This removes very small objects. Third, apply 3D dilation (37) to reverse effect of erosion from previous step. Finally, remove plaques with radius <4 pixels using the Fiji 3D Object Counter (38).

The results of image processing were carefully inspected to ensure that they matched the visual inspection of the image. Images were discarded if the processing yielded poor results, which were most commonly caused by cross-contamination between channels, obscured regions of the image, and illumination problems.

## 1. Introduction

[2] The hazards posed by particle radiation in space depicted in Figure 1 pose a serious challenge to human and robotic exploration missions to the Moon, Mars and beyond. The hazards include the following.

[3] 1. Galactic cosmic rays (GCRs), which are always present in the near Earth space environment and throughout the solar system, originate from beyond our heliosphere and produce chronic but not acute exposures. GCRs are very difficult to shield against beyond the Earth's protective atmosphere and magnetosphere. Astronauts under typical shielding of a few g/cm 2 of aluminum could reach their career limit of radiation exposure from GCRs in roughly 3 years [ Cucinotta et al., 2001 ]. Current research in this area is focused on understanding the constraints imposed by GCRs and how they vary with mission transit time, shielding type and thickness, and on developing better techniques to shield against GCRs. The intensities of GCRs vary with the solar cycle with the largest intensities occurring near solar minimum when interplanetary field strengths [e.g., Le Roux and Potgieter, 1995 ] are weakest and there are the fewest number of interplanetary disturbances from transient disturbances such as coronal mass ejections [e.g., Owens and Crooker, 2006 Schwadron et al., 2008 ]. GCRs are modulated by the outflowing solar wind and its embedded magnetic field the modulation is therefore weakest when the interplanetary field strength is low [e.g., Potgieter et al., 2001 ] and the associated intensities of GCRs are commensurately high.

[4] 2. Solar energetic particle (SEP) events (which we define to include ions also solar particle events (SPEs)) are also dangerous to astronauts outside of Earth's protective layers (the atmosphere and magnetosphere). Current research in this area focuses on developing the ability to predict when and where SEP events will occur and finding ways to adequately shield against SEP-associated particle radiation.

[5] 3. There are unique radiation environments at each planet and their satellites. We have thoroughly characterized the locations of the radiation belts at Earth, which allows us to reduce the hazard they pose by rapidly transiting them. Human and robotic exploration of other planets and satellites requires that we adequately characterize planetary radiation environments and develop appropriate mitigation strategies and adequate shielding. Shielding is often considered the solution to space radiation hazards. Very high energy radiation (e.g., >100 MeV), however, produces secondary penetrating particles such as neutrons and nuclear fragments in shielding material. Some types of shielding material may actually increase the radiation hazard [ Wilson et al., 1999 ]. The radiation hazard is not sufficiently well characterized to determine if long missions outside of low-Earth orbit can be accomplished with acceptable risk [ Cucinotta et al., 2001 ].

[6] Estimates of radiation hazards may be inaccurate through incomplete characterization in terms of net quantities such as accumulated dose. Time-dependent characterization often changes acute risk estimates [ Cucinotta, 1999 Cucinotta et al., 2000 George et al., 2002 ]. Events with large accumulated doses but relatively low dose rates (<30 rad/h) pose significantly reduced risks. More complete characterization of radiation hazards requires that models take into account time-dependent radiation effects according to organ type, primary and secondary radiation composition, and acute effects (vomiting, sickness and, at high exposures, death) versus chronic effects (such as cancer). Further, to reduce uncertainties in predictions of radiation hazards, radiation exposure models should be tested with direct observations. This requires detailed knowledge of radiation detectors and accurate detector response models [ Nikjoo et al., 2002 ].

[7] We introduce here the Earth-Moon-Mars Radiation Environment Module (EMMREM), which is designed to predict the radiation environment at Earth, the Moon, Mars and throughout the interplanetary medium in the inner heliosphere. Section 2 describes the EMMREM project and the framework of modules associated with it. Section 3 describes EMMREM results applied to two recent SEP events. We highlight, in particular, areas of development required to improve radiation characterization and hazard prediction. Section 4 summarizes the article.

## NGC 1260

1178 elliptical and S0 galaxies, of which 984had no previous measures. This sample contains the largest set ofhomogeneous spectroscopic data for a uniform sample of ellipticalgalaxies in the nearby universe. These galaxies were observed as part ofthe ENEAR project, designed to study the peculiar motions and internalproperties of the local early-type galaxies. Using 523 repeatedobservations of 317 galaxies obtained during different runs, the dataare brought to a common zero point. These multiple observations, takenduring the many runs and different instrumental setups employed for thisproject, are used to derive statistical corrections to the data and arefound to be relatively small, typically

19.5(Brunzendorf & Meusinger 1999). A sample of 19 IRAS galaxies isfound. According to their redshifts, 17 galaxies are likely members ofthe Perseus cluster, two are background galaxies. The sample-averagedFIR excess is higher than expected for normal galaxies. The opticalmorphology of the IRAS galaxies is evaluated on CCD images taken in theB band at a seeing of about 1'', complemented by CCD images taken in theredshifted Hα band or in the R band. Individual descriptions arepresented along with the B band images for all of the IRAS galaxies. Asubstantial fraction of the galaxies in the IRAS sample exhibit signsfor morphological distortion. A correlation between the FIR activity andthe strength of distortion is indicated. On the other hand, there areapproximately as many disturbed/interacting galaxies in the Perseuscluster region without IRAS counterparts as IRAS galaxies. The IRASgalaxies are much less concentrated towards the cluster centre thantypical bright cluster galaxies. For the distorted non-IRAS galaxies,such a trend is less pronounced. These differences may be related torapid stripping of gas as galaxies enter the cluster.

19.5. The estimatedlimit of completeness is B25

18. Two thirds of the galaxiesare published for the first time. The galaxy positions are measured witha mean accuracy of 0farcs5 , the photometric accuracy is of the order of0.1 to 0.2 mag depending on image crowding and galaxy shape.Morphological properties were evaluated from the visual inspections ofboth deep images obtained from the digital co-addition of a large numberof plates and higher-resolution images from single plates taken undergood seeing conditions. The superimposed images unveil faint structuresdown to mu_B

27 mag arcsec(-2) . The catalogue is applied to a studyof statistical properties of the galaxies in A 426: projecteddistribution of morphological types, segregation of morphological types,position of the cluster centre, distribution of galaxy position angles,type-dependent luminosity functions, and total B-luminosity of the thecluster. In agreement with previous studies, we find a relativespiral-deficiency in the central region (r

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