The Nature of Dark Energy

The Nature of Dark Energy

(Originally posted October 12, 2017 on Blogger)

INTRO
Our universe is expanding, and doing so at an accelerating rate as explained in this blog. The stuff inferred to exist that best explains this accelerating expansion is the rarefied, homogeneous existence of an ultra-low-mass energy dubbed dark energy.

However, dark energy remains an unknown type of hypothetical energy, the nature of which is unknown to us. Hypothetical, and unknown, but nevertheless real, physical stuff. All that is known from inference is that it collectively exhibits a strong, constant, repulsive energy (negative pressure) that appears to be driving the accelerating expansion of the universe. The rate of that expansion is estimated by the Hubble "constant".

In a certain framework, or more accurately, within a certain metric (FLRW metric), such a consistently-repulsive energy can be mathematically shown to drive acceleration within an already expanding universe; one that was already expanding out of the Big Bang. This further supports the existence of some form of repulsive energy in the universe. Without it, one would expect the rate of expansion to be slowing down, halted, or even reversed.

Einstein, along with the consensus of the scientific community at the time, believed the universe was static; neither expanding nor contracting. However, for the field equations at the heart of his general theory of relativity to describe a static universe, a mathematical term had to be added. A term meant to counter the effects of gravity such that the universe didn't collapse in on itself. Einstein added that term to his field equations (denoted as Λ), known as the cosmological constant.

The consequence of that term changed dramatically when it was confirmed that the universe was expanding. Though Einstein abandoned it, the constant never really went away. Jump ahead to the 1990s when it was realized the rate of expansion is accelerating, the shock from which left the scientific community scrambling for answers. It turned out, the answer for the simplest candidate of the mysterious "dark" energy assumed to be driving that acceleration was Einstein's cosmological constant (Λ).

But the nature of this dark energy remains, even to this day, unknown. To know its nature is to know the fate of our universe, and perhaps even lead us toward some new comprehensive fundamental physical theory. So what is its nature? Does it remained unchanged through time (fixed), or does it evolve with time (dynamical)? If it is fixed, then the cosmological constant (a fixed term) is indeed a good candidate. However, what if the nature of dark energy isn't fixed?

Recent observations by the HST/GOODS program along with previous supernova data appear to slightly favor a dynamical form of dark energy. This, of course, is quite unlike the fixed nature of the cosmological constant. But to prove (or disprove) this will require improved abilities to observe and collect information from the universe.

Fret not my dear friends! Such instrumentation is on the horizon. 2018 will (hopefully) see the James Webb Space Telescope, and the Dark Energy Spectroscopic Instrument (DESI) survey put into action. These instruments will vastly enhance our observations of the universe, and may reveal the clues and evidence we need to solve this riddle wrapped in an enigma that has been dubbed, dark energy.

This blog will attempt to provide some basic background to the story of dark energy, then do its best to delve into recent work suggesting dark energy may be dynamical. Though the research has only confirmed its findings to a significance of 3.5 sigma, it is nonetheless intriguing. I briefly discussed confirmation to an accuracy of 5-sigma in this blog last year. 3.5 sigma could account for nothing more than a statistical fluke. If any of my readers visit that blog, please scroll down to where it says "The Coveted 5 Sigma".

The well-known bell-shaped curve (normal distribution) where each band represents 1 standard deviation (  σ). Image Source: Wikipedia

The well-known bell-shaped curve (normal distribution) where each band represents 1 standard deviation (σ).
Image Source: Wikipedia

BACKGROUND
Einstein's general theory of relativity (GTR) was a paradigm shift in cosmology, and his field equations are the very heart of that profound theory. His field equations describe the major relationship and behaviors between matter and the geometry of spacetime. They can be written in this form:

field_equations.png

These are Einstein's field equations (plural). They contain tensors, which are multi-component, multi-dimensional quantities. Rμν is the Ricci curvature tensor, gμν is the Einstein (or metric) tensor, and Tμν is the stress–energy tensor. Consider that just the Einstein and stress-energy tensors alone each have ten independent components each of which gives rise to independent field equations describing the shape (curvature) of spacetime, as well as describing the energy, momentum, pressure, (and more) of all the stuff within that spacetime.

What Einstein's field equations describe is a fluid spacetime affected by the presence of matter. As Richard Feynman's professor, John Archibald Wheeler once summarized, "Spacetime tells matter how to move, while matter tells spacetime how to curve."

But Einstein realized his field equations described a universe that, if expanding, would slow over time, come to a halt, then start contracting. Einstein envisioned a static universe; one that neither expands, nor contracts, but to achieve this ideal his equations needed a term to counter the gravitational effects of all the matter in the universe. This term—as seen in the equation above—is Λ (lambda), known as the cosmological constant. Its inclusion in his field equations was meant to define a static, matter-containing universe with closed spatial geometry (envisioned as a sphere).

However, astronomer Willem de Sitter saw things from a different perspective. He realized the addition of the cosmological constant could describe a universe without matter; a vacuum solution to Einstein's field equations. De Sitter proposed an empty 4-dimensional universe of closed spacetime geometry in place of Einstein's 3-dimensional matter-containing universe (de Sitter, 1917).

This de Sitter space didn't sit well with Einstein, but whether or not he ever outright rejected or supported it remains unknown. Regardless, Einstein certainly realized his genius field equations were fast becoming the fundamental framework out of which new ideas of the universe were being built. Ideas that weren't necessarily in agreement with a matter-containing, static universe.

In fact, physicist & mathematician, Aleksandr Friedmann, applied GTR to the entire universe, thereby being able to reduce Einstein's ten field equations to just two relationships. These two relationships are now known as the Friedmann equations.

The first equation describes the average distance between galaxies (scale factor) from which the size of the universe can be inferred. It also describes how this scale factor evolves over time, and even predicts the rate of that expansion if expansion were proven true (Friedmann, 1922).

The first equation expressed the balance between the outward expansion of the universe (kinetic energy) and the capacity of the universe to resist that expansion due to the effects of gravity (potential energy). If this sounds suspiciously similar to the Conservation of Energy, it's because it is.

If the outward kinetic energy of expansion is equal to the inward potential energy of gravity, then the result is zero (0), and we have Einstein's perfectly static universe. The universe could be expanding in this scenario, but it would slow and eventually come to a balanced halt and remain forever static.

However, if the kinetic energy of expansion is greater than the potential energy of 'gravity', then the universe will continue to expand forever; its scale factor goes to infinity. If, however, its kinetic energy is less than its potential energy, then the expansion will slow over time before coming to a halt, after which it would begin contracting to a point whereupon its scale factor reaches zero; dubbed the "Big Crunch" in later years.

Basically, the first Friedmann equation tells us that dark energy produces exponential change in the size of the universe, not necessarily that it is expanding; it could also apply to it shrinking. It describes the constant rate of expansion or contraction. The second equation, as we'll get to later, describes the change of that rate; its acceleration or deceleration.

The question that needed confirmation was whether or not the universe was expanding, contracting, or static. Even before Einstein published GTR and its associated field equations, a relatively unknown astronomer named Vesto Slipher published his findings of galactic redshift in the inaugural publication of the Lowell Observatory Bulletin. Something it seems no one read at the time, because his findings were either ignored or unknown to the wider scientific community.

Slipher noted the spectral lines of distant galaxies (outside our Local Group) were shifted, and he suggested these shifts were due to the stars receding from us. The red shift, was the Doppler Effect applied to light rather than sound. However, he didn't make the connection that this redshifting was due to the expansion of the universe, instead attributing it to the fact these stars were rotating away from us in spiral galaxies.

The Doppler effect applies to   electromagnetic radiation as it travels across expanding space.

The Doppler effect applies to electromagnetic radiation as it travels across expanding space.

Slipher was right about the redshift being the result of stars rotating away from us, and he is credited with the discovery of the rotation of spiral galaxies. But the redshift was also due to the expansion of the universe. It had been speculated in the years after his obscure publication in the Lowell Bulletin in 1912, but it wasn't confirmed until 1929 when Hubble observed galactic red shift in distant galaxies and attributed it to what others had speculated would be due to the expansion of the universe. I stop short of saying Hubble discovered expansion, as it is more accurate to say he confirmed it.

It was Lemaître who first proposed the expansion. Lemaître also derived and estimated what would later be known as Hubble's law and the Hubble "constant" respectively, long before Hubble refined them. I use scare quotes around "constant" because, well, it hasn't been constant since it was first estimated; having been tweaked no less than 18 times since it was first estimated, with no solid estimate in place even to this day. We'll get back to this later.

Hubble's confirmation of Lemaître's expanding universe may have been the final nail in the coffin for Einstein's cosmological constant being the balance necessary for a static universe. Between Friedmann, de Sitter, Lemaître, and Hubble, it was evident that the cosmological constant (Λ) failed to do what Einstein intended it to do.

Einstein eventually abandoned the cosmological constant, along with any notion of a static universe, having written in a postcard to his friend, Hermann Weyl,“Wenn schon keine-quasistatische Welt, dann fort mit dem kosmologischen Glied." “If there is no quasi-static world, then away with the cosmological term.” As mentioned in the intro, Einstein came up with his field equations long before anyone realize or even hypothesized an expanding universe. It was a time when scientific consensus was that the universe was static with no evidence to suggest otherwise.

But the cosmological constant never really went away. Though it failed to maintain a static universe, it did explain some other things. And even more things as cosmologists uncovered more anomalies in more recent decades.

Jump ahead to present day, and we now know that the universe will likely expand forever, with too little matter in it to cause re-collapse. We know that such a relatively low-density universe should be shaped more like some weird negatively-curved hyperbolic hyperplane, yet we've measured it to be almost perfectly flat. And since the 1990s, we've known the expansion rate of the universe is accelerating as evidenced by mapping its past expansion history using distant supernovae.

None of this makes any sense without some unseen field of energy purveying the universe. This of course being dark energy.

Fixed Dark Energy - Unchanged through time
While the cosmological constant fails to define a static universe, it describes this dark energy quite well. The simplest form of dark energy is fixed; a constant cosmological constant if you will. One that represents a purveying non-zero energy of empty space (vacuum energy) such that the more space there is, the more dark energy there is. More space comes with an expanding universe, therefore more dark energy.

Though the amount of dark energy isn't fixed, its energy density is. Matter will continue to be 'thinned out' over time as the universe continues to expand while the energy density of dark energy remains fixed such that any countering effects of gravity become weaker with time (inverse square law). As mentioned above, while the first Friedmann equation describes the exponential change of the size of the universe, it doesn't address why dark energy 'pushes outwards'.

For that we turn to the second Friedmann equation from which its anti-gravitational effects emerge. This equation describes the acceleration of that expansion (it could also explain deceleration of contraction). It considers both the scale factor (a), and the rate of its expansion (ä) or the acceleration of the scale factor.

We'll recall that the scale factor is sort of like the size of the universe, or more accurately the distances between galaxies. But this acceleration depends on the density (ρ). So a larger (ρ) value equates to more matter and therefore a stronger gravitational effect countering the outward push of dark energy, and vice versa. It's more or less Newton's law of gravity for the entire universe at once; G by the way, being Newton's gravitational constant.

The equation also considers pressure (-). This might seem at odds with the fact the energy density of dark matter is constant, because in a homogeneous universe pressure should be uniform throughout and therefore undermine any possible pressure gradients between regions. However, without getting into details, this isn't the case from a purely relativistic point of view having to do with fast, and faster-moving normal-matter particles in different regions of the universe.

High pressure (fast-moving particles) comes from regular matter and energy. The mass of a region of the universe can have higher pressure if its particles are moving faster than the particles in neighboring regions. Positive pressure pulls the universe inwards, negative pressure pushes it outwards. Since the universe has stuff in it, there is always going to be some level of positive pressure. This inward pull shows up in the above equation with the negative sign.

This overall inward pull is what concerned Einstein. All parts of these equations led to a collapsing universe. To counter this, he added (+) the cosmological constant (Λ).

All indications point towards dark energy being real, physical stuff. As such, it follows that its countering outward push comes from its physical properties. Just as normal matter has density and pressure, dark energy must have its own density and pressure as well. 

If we look at dark energy's density parameter mathematically, we'll see that it is positive just as the density of regular matter is positive. Both act to resist the effects of expansion by pulling inward. We'll recall that these are relativistic effects of fast-moving particles (pressure), so as to avoid confusion by the fact positive pressure has an inward effect.

Having positive pressure, dark energy's physical property of density cannot be what is behind the accelerating expansion of the universe. This leaves dark energy's property of pressure as the cause behind that accelerating expansion.

That pressure must be strongly negative; enormously so. Remember, it is not only enough to counter the inward effects of regular matter's density and pressure, as well as dark energy's own density, but to also overcome them all with increasing relative strength.

This is a good point to inject Equation of State (EoS). It is the ratio of the pressure and energy density we've been discussing. This ratio is characterized by the dimensionless number . The ratio is = p/ρ, where p is pressure and ρ (Greek letter rho) is the energy density (as noted in the 2nd Friedmann equation above). For dark energy in the standard model, ≈ -1. EoS -1 (minus unity) is known as the Phantom Divide Line or Cosmological Constant Boundary across which the cosmological constant never evolves. Hence the "constant" in cosmological constant. We'll get back to the standard model and the EoS of dark energy later, but take away from this that -1 is the Phantom Line that fixed dark energy will never evolve (change) across.

Unlike normal matter, the outward pressure of dark energy does not come from its fast-moving particles relative to neighboring slower-moving dark energy particles. Instead, dark energy's outward push comes from the fact the density of dark energy is constant.

For dark energy to maintain its density distribution in an expanding universe, it must gain energy. If it doesn't, then it will dilute over time along with regular matter and weaken with time. To remain constant, the more the universe expands, the more dark energy is gained. The more gained, the more prominent the outward push becomes, particularly as the inward pull from the gravitational effects of normal matter weakens as normal matter continues to dilute across the ever-expanding universe.

Here we may start to get a glimpse of the power of Einstein's cosmological constant as it applies to fixed dark energy. The addition of dark energy out of nothing to maintain a constant density in an expanding universe might seem to violate the Law of Conservation of Energy, but this law applies only to a Newtonian universe of fixed dimensions of space and time, wherein energy cannot be created nor destroyed. In Einstein's relativistic universe, energy can be gained (or lost) from nothing.

But the cosmological constant comes with some presumed hiccups of a philosophical nature; one of note is the fine-tuning problem. For example, it hardly seems natural that it has a fixed energy scale immensely weaker than other scales in physics, and an energy density that hasn't significantly changed as the universe expands (red shifts) over time. We might not exist (nor anything else for that matter) if its 'tuning' were ever so slightly different; a cosmic coincidence of epic proportions.

But such issues could possibly be resolved if dark energy is dynamic.

Dynamical Dark Energy - Changes through time
The Lambda-Cold Dark Matter Model (ΛCDM) of the universe uses the FLRW metric, Friedmann equations, and cosmological equations of state (EoS) to describe the universe since the end of the inflationary epoch, which are—by no coincidence—things we've discussed already.

ΛCDM is considered the standard model for describing the universe in which we live. Λ (Greek letter lambda) in ΛCDM of course represents the cosmological constant which is essentially dark energy in this model as discussed above. The cold dark matter (CDM) very generally has to do with a bottom-up hierarchy of structure within the universe. I briefly explain CDM in this blog on how scientists have inferred dark matter from cosmic shear for those interested.

Though ΛCDM is considered the standard model preferred by cosmologist around the world, the unfortunate reality is that this model comes with some hiccups of its own; Two in particular:

One has to do with discrepancy (tension) between the density of matter favored by 2015 survey, and that favored by the Cosmic Microwave Background (CMB) (Zhao et al., 2017); the latter being the faint red-shifted afterglow of the early universe after recombination.

The second has to do with tension between the latest Planck determination of the Hubble constant obtained from the CMB, and the local measurement from the Hubble Space Telescope (Solà et al. 2017). I discuss the Hubble "constant" in this blog for those interested. The difference is a notable 3.4 standard deviations (Zhao et al., 2017).

In simplest terms, all this suggests that something is askew, and Zhao et al. suggest these tensions may be resolved by a dynamical form of dark energy with rolling scalar fields, meaning fields who scalar (directionless) values change with time rather than being fixed (unchanging). A scalar field gives a value to every point in space; that value could represent density, temperature, or any number of different parameters that would collectively paint a picture of distribution throughout space. Imagine blades of grass each being points in a literal field, each having values (shades of green and yellow for instance) assigned to them. We can collectively look at that and "see" a field of grass with varying color. That's probably a crappy analogy I'll admit, but one I think might clarify the concept for laymen like myself.

A 3-D representation of a single scalar field. The numbers are meaningless insofar as this blog is concerned. This is just a heuristic visual representation. This could represent temperature, pressure, density etc... Scalar fields  look the same no matter how you view them. They can contain energy or pressure... includong negative pressure and are believed to have existed since the earliest moments of the post-inflation universe. I'm assuming of course that inflation happened, which is the consensus. Again, this is a heuristic depiction at best. :/  Image Source: Wikipedia

A 3-D representation of a single scalar field. The numbers are meaningless insofar as this blog is concerned. This is just a heuristic visual representation. This could represent temperature, pressure, density etc... Scalar fields look the same no matter how you view them. They can contain energy or pressure... includong negative pressure and are believed to have existed since the earliest moments of the post-inflation universe. I'm assuming of course that inflation happened, which is the consensus. Again, this is a heuristic depiction at best. :/
Image Source: Wikipedia

Quintessence
Obviously the Phantom Divide cannot be crossed with a fixed field, so what we need is a more flexible potential to have a (rolling) field that tracks an evolving EoS, and the hypothetical scalar field dubbed "quintessence" does just that. As mentioned at the start of this blog, there is ever so slight statistical evidence that favors an evolving EoS... yet to be confirmed.

Quintessence can be attractive or repulsive depending on the ratio of its kinetic energy (outward push) and potential energy (inward pull). We discussed this ratio of kinetic and potential energy above for reasons... ;) Clearly quintessence would be kinetic energy dominant in today's universe given the fact the universe's accelerating expansion, but it is believe this wasn't always the case. This suggests an EoS that changes with time.

A quintessence scalar field would have become kinetically dominant about 10 billion years ago, putting us in the current 'acceleration epoch'. This isn't the first time the universe has experienced an epoch of accelerated expansion. The consensus is, and considerable evidence suggests, that the universe experienced an inflationary epoch during which expansion accelerated (exponentially so) under the influence of an inflaton field that had (on very short time scales) properties similar to the cosmological constant.

Likely inspired by Alan Guth's inflationary theory, quintessence models of dark energy use single minimally-coupled scalar fields to support forms of dynamic dark energy (Alam et al., 2004). However, evolving across the Phantom Divide in single scalar fields turned out to be prohibited (Vikman, 2005) for reasons beyond the scope of this blog. However, according to Zhao et al., crossing the Phantom Divide is allowed in models with multiple scalar fields.

Some special cases of quintessence have been proposed (phantom energy and kinetic quintessence), but their evolution across the Phantom Divide leads to quantum instabilities and violate the weak energy condition (Feng, 2006). Each proposed model of dynamic dark energy has its pros and cons, so a new model that combines the best of quintessence and phantom has been proposed; Quintom.

Quintom Model of Dark Energy
Quintom is a portmanteau of "quintessence" and "phantom". An oscillating quintom field can unify the early inflationary period with the current accelerated expansion of the universe, which would lead to oscillations of the hard-to-pin Hubble constant. This leads to an eternal universe that does not end in a "Big Crunch" nor a "Big Rip" (Feng, 2006), and feeds nicely into what has become scientific consensus, that the universe is very likely eternal.

See this blog post and/or this documentary video (in which I took part) for more on the incredible evidence for an eternal universe.

The evolution of our universe through time.

Image Credit: NASA/WMAP Science Team, modified by Ryan Kaldari

According to the recent paper by Zhao et al., the soon-to-be-completed Dark Energy Spectroscopic Instrument (DESI) should definitively prove or disprove if the nature of dark energy is fixed. This is very exciting, as this instrument, along with the James Webb Space Telescope will be operational and providing data within a few years. And perhaps, once and for all, we'll be able to answer that riddle wrapped in an enigma that is the nature of dark energy.

As always, thank you for your readership.

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Ripples in Spacetime - Part I

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Preserving Our Species Beyond the Solar System - Part II