The Cosmological Constant and the Dark Sector

What is the phenomenology of the dark sector? That is the my question. The dark sector refers to dark energy and dark matter, which are two distinct phenomena which seem to have no direct connection other than in name. In this post I am going to talk about the cosmological constant, dark energy, and look at some landmark literature on the subject. I am going to show the origin of the 10¹²⁰ order of magnitude error that results from the quantum field theory prediction and cosmological observation. I am going to outline the physicists theoretical case and the astronomers observational case, and we will see how deceiving the cosmos can be.

Dark energy is a form of energy attributed to the nature of empty space which increases the rate of expansion of the universe; that is if you observe a distant galaxy not only is it moving away from you in time, but the rate at which it recedes from you is accelerating. In the last 30 years or so a wide range of observations have corroborated a model of the universe wherein a majority of energy is attributed to the dark sector. The current consensus is that there is a dark energy component of our universe that represents 2/3 of the entire energy content of the universe that explains the observed cosmic acceleration. This dark energy can lead to other strange phenomena such as repulsive gravity and ultimately a universe that tears itself apart.

The composition of the cosmos.

The classic and simplest explanation for dark energy is the cosmological constant. The cosmological constant was originally introduced by Einstein as a term in his gravitational field equations in order to allow a steady state non-empty universe solution to his equations. The cosmological constant introduces a non-zero vacuum energy into the universe. This vacuum energy acts as a negative pressure (conversely a negative vacuum energy would result in a positive pressure) and this vacuum energy is known as dark energy. The idea of a vacuum containing energy is very much expected by physicists, but the observed value of the vacuum energy is what is surprising as we will see. The cosmological constant represents the particularly simple case of constant vaccum energy and is represented by the Greek character lambda (Λ). A seminal paper (also see Weinberg 1989 or for more recent general reviews see Carroll 2000, Frieman et al. 2008, and Peebels & Ratra 2002) on the topic was published in 1992 by Carrol, Press & Turner. The abstract

The cosmological constant problem is examined in the context of both astronomy and physics. Effects of a nonzero cosmological constant are discussed with reference to expansion dynamics, the age of the universe, distance measures, comoving density of objects, growth of linear perturbations, and gravitational lens probabilities. The observational status of the cosmological constant is reviewed, with attention given to the existence of high-redshift objects, age derivation from globular clusters and cosmic nuclear data, dynamical tests of Ω_Λ, quasar absorption line statistics, gravitational lensing, and astrophysics of distant objects. Finally, possible solutions to the physicist's cosmological constant problem are examined.

Roughly following Carrol et al. (1992) I will explore further the origin of the cosmological consant and the question of why the observed vaccum energy is so small in compairon to the scales of predicted by fundamental physics. We start with the Friedman equation derived from Einstein's field equations. It relates the Hubble parameter, H, to the scale factor, a, and other basic quantities.

Where the dot denotes a time derivative, G is the gravitational constant, ρ_M is the cosmological density of matter, and k is the curvature parameter which can take on values of -1,0, and +1 corresponding to a negative, flat, and hyperbolic universe geometries respectively. The Friedman equation can be viewed in terms of the contribnutions from matter (ρ_M), curvature (k), and vacuum energy (Λ). It is customary to parametrize these quantities in terms of their fractional value at the current epoch, that is today. We denote the current values with a subscript 0. For example the current value of Hubbles Constant is H₀~70 km/s/Mpc.

So in total then we have simplified the problem to the statement that Ω_M+Ω_k+Ω_Λ=1 for consistency with the Friedman equation. The astronomers cosmological constant problem is whether a nonzero Ω_Λ is required to achieve consistency.

The physicists cosmological constant problem begins with the statement that there are virtual vacuum states present in a vacuum due to the Heisenberg uncertainty principle. For example consider a relativistic field as the collection of harmonic oscillators of all possible frequencies, ω. The vacuum has a zero-point energy E₀ (for a scalar field φ of spinless bosons with mass m) which is the sum of contributions over all possible modes of the field, i.e. over all wave vectors k.

We preform the sum by considering the system in a box of side length L and letting L tend to ∞. An appropriate periodic boundary condition implies λ_j=L/n_j for some integer n_j with a wave vector k_j=2π/λ_j. In the range ( k_j, k_j + dk_j) there are dk_j L_j/(2 π) discrete values of k_j such that the sum becomes the integral:

The energy density of the vaccum is simply this ground state energy divided by the volume, L³. In order to properly obtain an answer we must use ω_k²=k²+m²/h² and most importantly we impose a cutoff at a maximum wave vector k_max»m/h (note that I must use h where I mean hbar here). The result is then

The vaccum energy density can be shown to apporach infinity as k_max (the physicist will recall ultraviolet catastrophe) appraoches infinity therefore it is expected that there is some cutoff value near the Planck energy. The logical choice is to choose  k_max=E/h. The resultant prediction for this vaccum zero-point energy density is that ρ_vac ≈ 10⁷⁴ GeV ⁴h^-3 ≈ 10⁹² g/cm³ (the net cosmological constant is more nuanced in that it can be viewed as the sum of a number of disparate contributions including potential energies from scalar fields, zero-point fluctuations, as well as a pure cosmological constant, so only the dominant term has been addressed here), however observational cosmology has constrainted ρ_obs≈ 10^-47 GeV ⁴h^-3 ≈ 10^-29 g/cm³. The difference between the predicted and observed value is 120 orders of magnitude. This discrepancy is devastatingly imcomprensible large and can only be described as an EPIC FAIL. However, it may be misleading to chracterize the discrepancy this way since energy density can be expressed as a mass scale to the fourth power. Writing ρ_λ= M_vac⁴ we find the difference is only 10³⁰.  The theoretical predictions from quantum field theory have been sound in predicting vacuum effects such as the Casimir force so there is no a priori reason to doubt predictions in this cosmological realm. The unsolved problem in physics is why doesn't the vacuum energy produce a very large cosmological constant?

Sean Carrol recently posted on the Cosmic Variance blog a list of 24 Questions for Elementary Physics for the next 100 years. Number six is, what is the phenomenology of the dark sector? Indeed, while current observations have demonstrated the expansion of the universe is now accelerating, there are questions associated with the exact nature that are some of the most challenging problems in physics. The cosmological constant is only one form of dark energy, but what cosmologists really want to know is what is the dark energy equation of state?

Observational support for accelerated expansion is strong from observations of type Ia supernova, standard rulers, the cosmic microwave background, gravitational lensing, etc. (the interested reader is directed to Percival et al 2009 for results using the standard ruler baryonic acoustic oscillations technique and Riess et al 2004 using the type Ia supernova technique), but some methods could be biased by unknown systematics. The baryonic acoustic oscillations technique is very promising; the introduction to Percival et al 2009 explains:

Distinguishing between competing theories will only be achieved with precise measurements of the cosmic expansion history and the growth of structure within it. Among current measurement techniques for the cosmic expansion, Baryon Acoustic Oscillations (BAO) appear to have the lowest level of systematic uncertainty (Albrecht et al. 2006).

The strategies for distinguishing between a cosmological constant and other forms of dark energy all revolve around precision astrophysics measurements. Measuring the dark energy equation of state will provide a check on fundamental physics and general relativity, however it is of note that as Peebles & Ratra 2002 state:

the empirical basis [for dark energy] is not nearly as strong as it is for the standard model for particle physics: in cosmology it is not yet a matter of measuring the parameters in a well-established theory.

The gravity of the statement is that particle physics is right and observational cosmology is still just grasping in the dark. I am not a particle physicist, but to me the standard model does seem like a prediction machine and observational cosmology is just beginning to hold its own. The resolution of the dark sector may come from strange new physics like modified gravity, brane worlds, and so on (for discussions of various solutions see Carroll et al. 2005 for cosmology of generalized modified gravity models, Deffayet 2002 for modified brane worlds, or Ishak et al. 2006 on measuring the cosmological equation of state). So it is possible that the accelerated expansion of the universe is an illusion of our position in the universe, a misunderstanding of fundamental physics, or an unsatisfying tautology. No matter what it is though, everyone agrees that we need more data. That is where I come in. Consider for example, if we live in a non-homogeneous region of the universe then the solutions to Einstein's equations of general relativity would not result in the standard Friedmann-Lamaire-Robertson-Walker metric from whence we obtained the Friedman equation. Hence our entire model would be wrong. Soon much more powerful precision astrophysics experiments (like BOSS) will give us the observational data we need to achieve precision cosmography. I think that statements from Célérier 2009 sum up the fundamental issue

It is commonly stated that we have entered the era of precision cosmology in which a number of important observations have reached a degree of precision, and a level of agreement with theory, that is comparable with many Earth-based physics experiments. One of the consequences is the need to examine at what point our usual, well-worn assumption of homogeneity associated to the use of perturbation theory begins to compromise the accuracy of our models. It is now a widely accepted fact that the effect of the inhomogeneities observed in the Universe cannot be ignored when one wants to construct an accurate cosmological model. Well-established physics can explain several of the observed phenomena without introducing highly speculative elements, like dark matter, dark energy, exponential expansion at densities never attained in any experiment (i.e. inflation), and the like.

In conclusion we can only say the that cosmological constant or dark energy may be an unexpected component of our universe. Currently, the best observational constraints on dark energy come from the cosmic microwave background (CMB) and type Ia supernova measurements and all the data is largely consistent. We must continue to gather data in order to quantify the large scale structure of our universe, its geometry, and its energy budget. Ultimately, in determining the energy budget of the universe, it helps to have a monetary budget here on earth. The NSF Dark Energy Task Force has taken interest in this fundamental question so soon, we may know, what is the phenomenology of the dark sector?


References:


Carroll, Sean M., Press, William H., & Turner, Edwin L. (1992). The cosmological constant ARA&A, 30, 499-542

Marie-Noëlle Célérier (2009). Inhomogeneities in the Universe with exact solutions of General
Relativity Invisible Universe International Conference arXiv: 0911.2597v1

Peebles, P., & Ratra, B. (2003). The cosmological constant and dark energy Reviews of Modern Physics, 75 (2), 559-606 DOI: 10.1103/RevModPhys.75.559

Will J. Percival, Beth A. Reid, Daniel J. Eisenstein, Neta A. Bahcall, Tamas Budavari, Joshua A. Frieman, Masataka Fukugita, James E. Gunn, Zeljko Ivezic, Gillian R. Knapp, Richard G. Kron, Jon Loveday, Robert H. Lupton, Timothy A. McKay, Avery Meiksin, Robert C. Nichol, Adrian C. Pope, David J. Schlegel, Donald P. Schneider, David N. Spergel, Chris Stoughton, Michael A. Strauss, Alexander S. Szalay, Max Tegmark, Michael S. Vogeley, David H. Weinberg, Donald G. York, & Idit Zehavi (2009). Baryon Acoustic Oscillations in the Sloan Digital Sky Survey Data
Release 7 Galaxy Sample MNRAS arXiv: 0907.1660v3

Einstein’s blackboard

Einstein’s blackboard as used in a lecture in Oxford on 16 May 1931:

At that time Einstein’s theories of relativity were being combined with astronomical data to explain the shifts towards the red in the spectra of distant galaxies, which indicated that the universe was expanding. In his lecture Einstein outlined a fairly simple model to explain this apparent expansion. In the first line on the blackboard, D, the measure of expansion in the universe, is defined in terms of the expansion factor P. The expression for the density of matter in the universe, given by ρ in the third line, is derived from the field equations. The last four lines contain numerical data, giving values for density, radius and age of the universe, where ‘L. J’ stands for ‘Licht Jahr’ (light year) and ‘J’ for ‘Jahr’ (year). According to the last line, the age of the universe is about 10, or perhaps 100 billion years (the bracket indicates an alternative figure, not a product of two figures).

From Bye-bye blackboard... from Einstein and others

Spirit, Already Dead

On January 26th, 2274 Mars days into the mission, NASA declared Spirit a 'stationary research station', expected to stay operational for several more months until the dust buildup on its solar panels forces a final shutdown.

If the Spirit rover is just a little robot crawling around on a little red planet somewhere, then why are we so sad that it is dieing? The rover is certainly easy to anthropomorphize, but I think there is more to it. The spectrum of sentient to insentient, is just that a spectrum. When we speak of these concepts we make subtle judgments through the connotation of our words. For example is the robot going to die, or is it going to shutdown? On some level it must be alive, or, at least it was.

The Entropy of the Universe

First, what is entropy? Is it a philosophers dream?

"Acqua Alta - L'amnésie contagieuse de l'amante religieuse." by jef safi

The chaos is not a shapeless state, or a confused and inert mixing, but rather the place of a plastic and dynamic becoming... Philosophy, science and art are "drawing plans" on the chaos, they are the three chords. Philosophy on its plane of immanence of ideas and concepts, science on its plane of consistency of variables and functives, and art on its plane of composition of affects and percepts." -Gilles Deleuze, Qu'est-ce que la philosophie?

That was about as quantitative as dancing about architecture. Let us define entropy as a measurement of disorder in a system. The entropy of a system can be defined as proportional to (the natural log of) the number of microstates corresponding to the observed system macrostate. Entropy is usually defined by big S.

Where Ω is the number of accessible microstates and where k_B = 1.381×10⁻²³ J K⁻¹ (through out the rest of this post k will represent Boltzmann constant and K the unit of Kelvin) is the Boltzmann constant in units of Joules per Kelvin. Entropy has always intrigued me so when I saw this paper on entropy of the entire Universe I was delighted.

A Larger Estimate of the Entropy of the Universe

Authors: Chas Egan, Charles Lineweaver

Abstract: Using recent measurements of the supermassive black hole mass function we find that supermassive black holes are the largest contributor to the entropy of the observable Universe, contributing at least an order of magnitude more entropy than previously estimated. The total entropy of the observable Universe is correspondingly higher, and is S_obs = 3.1×10¹⁰⁴ k. We calculate the entropy of the current cosmic event horizon to be S_CEH = 2.6±0.3×10¹²² k, dwarfing the entropy of its interior, S_{CEH int}= 1.2×10¹⁰³ k. We make the first tentative estimate of the entropy of dark matter within the observable Universe, S_dm = 10^88±1 k. We highlight several caveats pertaining to these estimates and make recommendations for future work.

Entropy is important in the Universe because it is inexorably linked to the arrow of time and understanding why the Universe began in such a low entropy state, namely the big bang, is an open question (considering the anthropic explanation unsatisfactory or at least not fully quantitative). A better quantifiable question that we can answer is the current total value of entropy in our Universe and the constituent contributions from major astrophysical phenomena. The authors explain that the increase of the entropy budget of our Universe is associated with all irreversible process including gravitational clustering, accretion disks, supernovae, stellar fusion, terrestrial weather, chemical, geological and biological processes. The authors have assumed a flat Universe with standard cosmological parameters (Ω_k=0, h=.705, Ω_b = 0.0224, Ω_m= 0.136 and T_cmb=2.725 K) and applied the second law of thermodynamics to the determine the entropy contribution of the most dominate processes in the entropy budget. The exact volume of the Universe is the dominating error in some of the estimates. In fact the entire definition of the volume of the Universe is questionable. In the case of the observable Universe we are considering a volume with a moving boundary in which matter or information may flow in and out, however we may attempt to dispel this concern that our system is not closed because as the authors state, 'the system is effectively isolated because large-scale homogeneity and isotropy imply no net flows of entropy into or out of the comoving volume'. In the appendix of the paper they explain how to calculate the volume of the Universe in a very simple manner if you are familiar with cosmography. The volume of the observable Universe is 3.65×10⁸⁰ m³ or s 43.2×10⁴ glyr³.

Figure 1 from the paper illustrates the particle and cosmic event horizons. At the origin on the x-axis is a vertical dashed line representing our galaxy. In the top panel the x-axis is the comoving distance, x=D/a where a is the cosmic scale factor. In the bottom panel the x-axis is the proper distance D. The region inside the particle horizon is the observable Universe. The comoving volume that corresponds to the observable Universe today is filled grey.

The largest contributor to the entropy budget of our Universe is super massive black holes (BH or SMBH). In the Schwarzchild case:

where , G is the gravitational constant, h (or rather h-bar) is Planck's constant, c is the speed of light, A is area, and M is mass. The largest contributor to the entropy of the Universe besides SMBH is certainly the cosmic microwave background which can be calculated succinctly using the equation of blackbody from Kolb and Turner (1990):

Where Tγ is the photon temperature and g_γ is the number of photon spin states, namely 2. The results for all phenomena in the entropy budget considered are tabulated in the table below:

Table 1 from the paper. The bracketed numbers refer to previous literature references.

It was interesting to hear the discussion on neutrino and graviton entropy. The total entropy of the neutrino background is a little tricky to compute because the neutrino entropy cannot be calculated directly because the temperature of the cosmic neutrino background has not been measured (to young scientists out there, go measure it and report back for a guaranteed Noble Prize). Additionally, infall of neutrinos into nonlinear structure with significant gravitational potentials may alter the neutrino entropy and the authors see this as possible future work. Then there is the relic graviton radiation, and its entropy contribution which is again insignificant compared to SMBH, but it is interesting to note that by reversing the relationship between the current graviton temperature and photon temperature it may enable, 'calculating the number of relativistic degrees of freedom at the Planck time using future measurements of the graviton background temperature'. Dark matter is the final twist. The authors present the first ever tentative estimates by interpreting it as a weakly-interacting superpartner to conclude its contribution is minimal.

Ultimately the entire entropy budget is dominated by black holes and critically the masses of black holes in our Universe. Entropy increases when gravitons are produced. The contribution of super massive black holes to entropy comes from the production of gravitons. Take for example, not a black hole, but another extreme gravitational system of two black holes or neutron stars inspiralling towards each other; gravitational waves are emitted from the system extracting orbital energy and therefore entropy allowing the system to contract. The authors find the BH entropy to be the dominating factor and larger by at least an order of magnitude compared to previous estimates which have different BH initial mass functions (IMF). Indeed the flaw in any work that attempts to make estimates of populations of stars, galaxies, or BHs faces the issue of ambiguous IMFs for the objects. The primary source of uncertainty that I perceive here is the lack of understanding of the IMF. Regardless if anyone was wondering what the entropy of the observable Universe is, and I know I was, it is 3.1×10¹⁰⁴ k .

References:

Chas A. Egan, & Charles H. Lineweaver (2010). A Larger Estimate of the Entropy of the Universe ApJ arXiv: 0909.3983v3

The Limits of Cosmology

Amedeo Balbi on The Limits of Cosmology

When attempting to discuss what a certain discipline can or cannot know, one should keep in mind, as a cautionary tale, the famous case of philosopher Auguste Comte. Writing in the first half of the nineteenth century, he stated that astronomers would never be able to ascertain the chemical composition of celestial objects. However, only a few decades after Comte’s prediction, Kirchhoff founded spectroscopy and managed to identify chemical elements in the atmosphere of the Sun.
    Cosmology is arguably one of mankind’s boldest enterprises. It tries to scientifically understand the origin, evolution and structure of the universe as a whole. In doing so, it has to rely on a certain set of observational data (what we see of the cosmos) whose collection cannot be repeated under different conditions; furthermore, it has to interpret such data according to a set of physical laws whose validity was mostly assessed in laboratories on Earth. Most cosmology is based on extrapolations of known physics to uncertain territories, and on indirect evidence derived from the behaviour of the part of the universe we can observe. We happen to live in the golden age of cosmology—for the first time in the history of mankind we are able to scientifically describe the overall structure of the universe. However, to some extent, it is surprising that we have managed to make some sense of the universe at all. Continued...

This essay is one of the winners from FQXi's contest on What Is Ultimately Possible in Physics?

Perceiving Itself

Through our eyes, the universe is perceiving itself.

Quote from Alan Watts. Art by Viktor Timofeev.

IceCube

The IceCube Neutrino Observatory is currently under construction in Antarctica. It is a pretty nifty project and this animated video by Casey O'hara is well worth watching for an explanation of the IceCube concept. It is light on the science, but it is delight because it explains neutrinos with Legos, breakfast cereal, and otter pops! I will have to post about IceCube again when they catch a neutrino.

The Moon, where the Helium-3 from the Sun is

Moon is a 2009 science fiction film about astronaut Sam Bell who is the solitary worker on the moon. Sam is at the end of a three-year stint on the Moon so the film begins as if it was the denouement of another quieter story. When an accident occurs Sam suddenly meets himself for the first time.

I am adapt at finding flaws in science fiction films, but Moon nails a lot of science as well as could be expected. The most incredulous point about the film for me was the lack of a radio array on the far side of the moon, I mean why else would we go to the moon? There is a very good and scientifically feasible answer for this. The movie begins, as you can see above with the first seven minutes of the film, with a commercial by Lunar Industries:

There was a time when energy was dirty word, when turning on your light was a hard choice. Cities in brown out, food shortages, cars burning fuel to run, but that was the past, where are we now? How did we make the world so much better? Make deserts bloom? Right now we're the largest producer of fusion energy in the world. The energy of the sun trapped in rock harvested by machine from the far side of the moon. Today we deliever enough clean burning helium-3 to supply the energy needs of nearly 70% of the planet. Who would have thought all the energy we ever needed, right above our heads? The power of the moon, the power of our future.

When I saw this at the beginning of the film I was delighted that they had based the story on a kernel of truthful science. The energy source they are gathering from the moon is Helium-3 (³He), but they aren't exactly burning it for fuel as they say. Helium-3 is a light isotope of helium with two protons and one neutron which is suitable as a fusion fuel. I have done some research into the literature to determine just how feasible this ³He mining on the moon is with two specific questions in mind. Why use ³He? Why go to the moon?

An advanced fusion reactor would combine ³He and deuterium (²H) in a fusion reaction to produce a helium-4 nucleus (⁴He) and a high-energy proton. Energy is released as charged particles and that is what powers the world in this science fiction vision of the future. ³He fuel offers some advantages over other types of fusion fuel because it is efficient and because less radioactive byproducts are produced (it is often stated or assumed that fusion power does not create any dangerous materials, but in reality the reactor housing can become radioactively activated and remain so for a number of years. However, the time scale on which it remains active is comparable to human lifetimes and is overall not as dangerous as the byproducts of fission reactors). ³He is very scarce on earth. It is possible to manufacture ³He on earth though the neutron bombardment of certain elemental targets which results in tritium, which then decays to ³He with a half-life of 12 years. This is a complicated, dangerous and inefficient process so other sources would be required to make ³He a viable fusion fuel.

The cosmological abundance of ³He is paltry, but its abundance and chemical evolution is of interest to astronomers because it can be a tracer of various stellar phenomena so it has been studied for many years. The primordial cosmological ratio of ³He to ⁴He is ~1.4 × 10 ^-6, however this abundance can be thousands of times greater in the solar wind. The solar wind is primarily protium traveling at a velocity of ~450 km/s with a flux of ~6 × 10 ¹⁰  ions/(m² s). Of this flux there is ~4% He which has an unusually high ratio of ³He to ⁴He of ~480 atomic parts per million (Heber V. 2003). ³He should be abundant on the moon's surface of regolith where it has been deposited by solar wind over billions of years.  Hence, we could go to the moon and mine it. However, to gather enough fuel to power the earth at current energy consumption rates more than one Space Shuttle load and the processing of 4 million tons of regolith per week, on the lunar surface, would be necessary. Further, to really nail the science here I cite Fa and Jin who state:

The [lunar] inventory of ³He is estimated as 6.50×10⁸ kg, where 3.72×10⁸ kg is for the lunar nearside and 2.78×10⁸ kg is for the lunar far side.

There is a bounty of lunar fuel available, but I wouldn't place my base on 'the far side of the moon' as Lunar Industries states they have done because it would be cold, there is less ³He, and it would be, well, lonely. In conclusion I have found the academic literature validates the idea that Helium from the moon could power terrestrial fusion reactors one day.

Apparently Sam Bell is working alone on the moon to cut costs for the company. In order to mine the necessary amounts of ³He with minimal overhead costs Lunar Industries has chosen a one man job; and based on the size of the lunar regolith harvesters seen in the movie 4 million tons processed per week would not be unfeasible. As the movie continues themes of alienation and societal deception emerge. I have discussed some science that the film never divulges, but in fact, the film never even mentions anything about ³He or the reasons why any of this is going on again. This is a strong point for the film which actually raises deep philosophical questions, which I could dive into, but I don't want to spoil it for anyone. It is a great film and not the craziest science, really:

Online resources:
Mining the Moon from Popular Mechanics
Lunar ³He and Fusion power by J. Santarius
Non-Lunar  ³He Resources by L. Wittenberg
Moon for Sale from the BBC

References:

FA, W., & JIN, Y. (2007). Quantitative estimation of helium-3 spatial distribution in the lunar regolith layer Icarus, 190 (1), 15-23 DOI: 10.1016/j.icarus.2007.03.014

Heber, V., Baur, H., & Wieler, R. (2003). Helium in Lunar Samples Analyzed by High‐Resolution Stepwise Etching: Implications for the Temporal Constancy of Solar Wind Isotopic Composition The Astrophysical Journal, 597 (1), 602-614 DOI: 10.1086/378402