Could dark energy be more negative than a cosmological constant?

Our Universe, as we understood it, underwent a radical change at the end of the 20th century. We had long assumed — consistent with the evidence we had, mind you — that our Universe was taking part in a great cosmic race that began back at the start of the hot Big Bang. On the one hand, the Universe was born rapidly expanding, but on the other hand, the force of gravity worked to slow the expansion down and pull things back together. For most of the 20th century, the big question for cosmology was, “Which impulse will win out in the end: gravitation or expansion?” Then, in 1998, we got our shocking answer: it will expand forever, but that’s because there’s a new type of energy that we didn’t expect, dark energy.

In the time since, we’ve ruled out alternative explanations and measured dark energy’s properties very well, but many questions still remain. In particular, despite all the ways that our knowledge of cosmology has changed in the 21st century, we still don’t know what dark energy is, or what its properties truly are. Could it be even stranger than most of us imagine? That’s the question of Paolo Craviolatti, who asks about dark energy, noting:

“…you do not say anything about the negative pressure [of dark energy]… [where] the various models assume w=-1 without testing for other larger, [more] negative [values of] w. If they did…things would be so different for us all. BUT should the assumption of w=-1 be tested to see if current (conflicting) cosmological models hold water?”

This is a very deep question, and it requires an understanding of a large set of concepts to make sense out of. Let’s dive in and unpack the idea of dark energy, negative pressure, the parameter “w,” and what the consequences of a “more negative w” would truly mean.

The gravitational behavior of the Earth around the Sun is not due to an invisible gravitational pull, but is better described by the Earth falling freely through curved space dominated by the Sun. The shortest distance between two points isn’t a straight line, but rather a geodesic: a curved line that’s defined by the gravitational deformation of spacetime. The notion of “distance” and “time” is unique for every observer, but under Einstein’s description, all frames of reference are equally valid, and the “spacetime interval” remains an invariant quantity.

Our Universe, as far as we understand it and to the best precisions we’ve ever tested it, appears to be governed by Einstein’s general relativity as our overarching framework for gravity. The presence of matter and energy — in all of their various forms — dictates how space expands, curves, and otherwise evolves, while that curved, expanding, evolving space dictates how the matter and energy within it will then move and evolve itself. On “small” cosmic scales, like on the scale of Earth, our Solar System, black holes, a galaxy, or a galaxy cluster, our Universe is full of individual masses that cause specific effects within general relativity. But on large cosmic scales, like the scale of our entire observable Universe (or close to it), our Universe is remarkably uniform.

This is special and important! Within general relativity, there is a very special exact solution for a Universe that’s uniformly filled with:

• matter,
• radiation,
• a cosmological constant,
• or any other uniform “species” of energy that you can imagine,

and that solution goes all the way back to 1922: the Friedmann–Lemaître–Robertson–Walker (FLRW) solution. On the one hand, your Universe will have both an energy density (what we call ρ) and a pressure (what we call p) for each species of matter/radiation/energy present, and both of those will contribute to how the Universe expands (or contracts) over time, as well as how the expansion (or contraction) rate changes over time.

A photo of Ethan Siegel at the American Astronomical Society’s hyperwall in 2017, along with the first Friedmann equation at right. The first Friedmann equation, an exact solution in general relativity, details the Hubble expansion rate squared on the left hand side, which governs the evolution of spacetime. The right side includes all the different forms of matter and energy, along with spatial curvature (in the final term), which determines how the Universe evolves in the future. This has been called the most important equation in all of cosmology and was derived by Friedmann in essentially its modern form back in 1922.

We know that our Universe contains both matter and radiation because we’ve observed them both for so long: far longer than we’ve known about Einstein’s general relativity.

• If your Universe is filled only with matter, it’s going to expand at a specific rate as a function of time (t): it will grow proportionally to ~t^⅔. Matter is completely pressureless, so its pressure, p, is zero (p = 0).
• If your Universe is filled only with radiation, it will also expand at a specific rate as a function of time, but grows differently: proportional to ~t^½. Radiation has a positive pressure that’s related to its density: p = +⅓ρ, and its energy density drops more quickly (owing to its wavelength, which defines its energy, stretching) than the matter density does as the Universe expands.

For both of these cases, as the Universe expands, the energy density (ρ) drops, and it’s that very energy density that determines the expansion rate. Lower the energy density and you lower the expansion rate, and hence, the expansion of the Universe slows down.

This is where the parameter “w” comes in: it tells us the relationship between energy density (ρ) and pressure (p): specifically that p = wρ. For matter, w = 0; for radiation, w = +⅓. Now, before we get into dark energy and a cosmological constant, it’s important to understand another aspect of general relativity: what we know as energy conditions and theorems.

An animated look at how spacetime responds as a mass moves through it helps showcase exactly how, qualitatively, it isn’t merely a sheet of fabric. Instead, all of 3D space itself gets curved by the presence and properties of the matter and energy within the Universe. Space doesn’t “change shape” instantaneously, everywhere, but is rather limited by the speed at which gravity can propagate through it: at the speed of light. The theory of general relativity is relativistically invariant, as are quantum field theories, which means that even though different observers don’t agree on what they measure, all of their measurements are consistent when transformed correctly.

Even though we think of general relativity as a physical theory with explicit relevance to the Universe we inhabit, its foundations are purely mathematical in nature: specifically in the mathematical field of differential geometry. There was a theorem that was proven more than 40 years ago, first by Richard Schoen and Shing-Tung Yau, and then by Ed Witten (in an alternative fashion) of tremendous importance: the positive energy theorem. It tells us that if we have any isolated system — and yes, you can consider the entire Universe as an isolated system — then the total gravitational energy of that system cannot be negative. Furthermore, if the system has any energy-containing objects in it (like matter or radiation), the total gravitational energy of the system can’t be zero either; it must be positive.

Why is this important?

Because if we treat the Universe as a uniform, expanding system that contains a variety of different types of energy within it, including matter and radiation but also allowing for others, then there are certain energy conditions we can define that have big implications for the Universe. Three of the most important ones are:

• The strong energy condition, which states that the contributions of energy density and pressure to the stress-energy tensor must always be positive. This implies that, for any energy species within the Universe, w can be no smaller than -⅓. However, there are no pathological consequences if this is violated.
• The weak energy condition, which states that the energy density of any species can never be negative and that the pressure of any energy species can never dominate the energy density. In other words, w can be no smaller than -1. If this is violated, then you can travel faster than light, create wormholes, and travel back in time; we would violate causality if this is true.
• And the null energy condition is the most restrictive of all, as it allows for no lower limit to negative energy states; you can just keep “pulling energy out of the quantum vacuum” with no lower limit to it at all.

How matter (top), radiation (middle), and dark energy/inflationary energy (bottom) all evolve with time in an expanding Universe. As the Universe expands, the matter density dilutes, but the radiation also becomes cooler as its wavelengths get stretched to longer, less energetic states. Dark energy’s (or inflationary energy’s) density, on the other hand, will truly remain constant if it behaves as is currently thought: as a form of energy intrinsic to space itself. These three components, together, dictate how the Universe expands at all times from the Big Bang until the present day and beyond.

When it comes to our actual Universe, we believe (but aren’t certain) that general relativity should be allowed to violate the strong energy condition, but not the weak energy condition (i.e., w can be less than -⅓, but not less than -1), and definitely not the null energy condition. If you consider things like quantum field theory in curved spacetimes, you can violate the weak energy condition, but probably even then not the null energy condition for the overall physical system. We’re only dealing with general relativity here, so the full expectation is that w can be negative, and can even be more negative than -⅓, but not more negative than -1; otherwise negative energy states, an Alcubierre drive, backward time travel, and all sorts of other odd behaviors that we think are forbidden would suddenly be allowed.

Nevertheless, just because we think we know how the Universe behaves or should behave — doesn’t mean that’s what the Universe actually has in store for us. Just because there are very good reasons to think that w couldn’t be more negative than -1 doesn’t mean that, when we measure:

• how the Universe is expanding,
• how the expansion rate has changed over the entirety of cosmic history,
• and what types of energy that implies the Universe is filled with,

we’re actually going to find that the Universe aligns with our expectations. After all, the hallmark of discovery and scientific advancement is almost always accompanied by a set of observations or experiments that completely defied our expectations going into it.

Various components of and contributors to the Universe’s energy density, and when they might dominate. Note that radiation is dominant over matter for roughly the first 9,000 years, then matter dominates, and finally, a cosmological constant emerges. (The others, like cosmic strings and domain walls, do not appear to exist in appreciable amounts.) However, dark energy may not be a cosmological constant, exactly, but may still vary with time by up to ~4% or so. Future observations will constrain this further.

Now, let’s come to dark energy. Theoretically, there are two very different avenues we can go down to think about what properties the energy inherent to empty space might possess. Sure, naively, you might expect that space wouldn’t have any energy inherent to it at all, but that’s just one possibility.

1. The Universe, after all, is governed by several different fundamental forces, and (at least three of) those forces can be described by quantum field theories: where these fields permeate all of space. Even in the absence of any particles, these fields still fluctuate, and are allowed to have a positive, non-zero amount of energy in every region of space. This is known as a field’s vacuum expectation value, and if the energy density (ρ) of these fields are non-zero, then we also expect them to have a pressure (p) associated with it, governed by the relationship p = –ρ, or equivalently, where w = -1.
2. Also, the Universe is governed by general relativity, where we don’t just have “matter-and-radiation” on one side and “the curvature of spacetime” on the other, but we are free to add in a cosmological constant, sometimes denoted by the Greek letter Λ. The cosmological constant can take on any value, positive or negative, in theory, but it has an energy density (ρ) and a pressure (p) as well, governed by the same relationship: p = –ρ, where again w = -1.

When we first discovered the evidence for an accelerating Universe, it appeared to be very consistent with a cosmological constant (Λ) or with what we call vacuum energy: where w = -1. Even today, even with all the talk about a Hubble tension, evolving dark energy, and the tension between CMB, supernova, and large-scale structure data, w = -1, or a constant form of dark energy, is extremely consistent with any individual data set.

These graphs show the fit for evolving dark energy, in terms of the parameters w_0 and w_a, where a constant cosmological constant for dark energy corresponds to w_a = 0 and w_0 = -1, exactly. Note that the DESI data on its own is consistent with constant dark energy, but that when you combine CMB and supernova (for example, DESY5, as shown in the middle panel) data with it, it favors evolving dark energy instead.

You might wonder what the point is of wondering so much about the nature of dark energy, and why we should care so deeply about the parameter “w” in dark energy’s equation of state. It’s a good thing to wonder, but the answer is one of the most profound things of all: the entire fate of the Universe hinges on it.

Let’s explain. The way we directly measure the expansion of the Universe is simply by:

• examining a distant object,
• determining how far away it is,
• measure how fast it’s receding from us (through its redshift, which allows us to interpret a recession velocity),
• and then, ideally, to measure (or more practically, to infer) how that recession speed changes over time.

Alongside that is another parameter of great importance: what we call the Hubble constant. Technically, it isn’t even a constant, but rather is a parameter that changes over time; what we call the “Hubble constant” is simply the value of the Hubble parameter today, right now, some 13.8 billion years after the initiation of the hot Big Bang. While the Hubble tension reflects the fact that different methods of measuring the expansion rate, today, yield inconsistent values (of ~67 km/s/Mpc for “early relic” methods, like the CMB, and of ~73 km/s/Mpc for “distance ladder” methods, such as type Ia supernovae), the big question for the fate of the Universe isn’t, “What is the value of today’s Hubble constant?” but rather, “How is the Hubble constant going to evolve far into the future?”

Measuring back in time and distance (to the left of “today”) can inform how the Universe will evolve and accelerate/decelerate far into the future. By linking the expansion rate to the matter-and-energy contents of the Universe and measuring the expansion rate, we can come up with an estimate for the amount of time that’s passed since the start of the hot Big Bang. The supernova data in the late 1990s was the first set of data to indicate that we lived in a dark energy-rich Universe, rather than a matter-and-radiation dominated one; the data points, to the left of “today,” clearly drift from the standard “decelerating” scenario that had held sway through most of the 20th century.

The answer to that, believe it or not, is purely dependent on what w is! Here are some key categories.

• If w is greater than 0, including if it evolves to be that way in the future (even if it’s less than zero today), then the Universe’s expansion rate will decrease, distant galaxies will recede evermore slowly, the Hubble constant will approach zero, and the expansion may have a chance of even reversing, leading to a potential Big Crunch scenario for our fate.
• If w is between 0 and -⅓, again including if it winds up that way in the future, then the Universe’s expansion rate will decrease, distant galaxies will recede evermore slowly, the Hubble constant will approach zero, but there will be no chance of the expansion ever reversing or even of reaching 0; it will expand forever, leading to a heat death scenario for our fate.
• If w is between -⅓ and -1, including if it ends up evolving to that final value, then the Universe’s expansion will accelerate. Distant galaxies will recede at ever greater and greater speeds, and the Hubble constant will continue to drop but won’t necessarily approach zero; for w = -1, in fact, it approaches a finite, constant, positive value instead. The Universe will expand forever, and there will be a maximum size/scale on which gravitationally bound structures form.
• And if w is more negative than -1, then the Universe’s expansion accelerates, distant galaxies recede at ever increasing speeds, the Hubble constant rises over time, and eventually the Universe will rip apart, culminating in a Big Rip scenario.

The far distant fates of the Universe offer a number of possibilities, but if dark energy is truly a constant, as the data best indicates, it will continue to follow the red curve, leading to the long-term scenario frequently described on Starts With A Bang: of the eventual heat death of the Universe. If dark energy can strengthen, weaken, or reverse sign over time, however, all bets are off, and alternative possibilities, like a big crunch or a big rip, suddenly abound.

When we first discovered dark energy, we noticed that it was behaving most consistently with a cosmological constant (or vacuum energy), indicating that w = -1. However, early on, it seemed like values like w = -1.5 might be preferred, and that was the impetus for considering the Big Rip, and for looking more closely at scenarios that violated the weak energy condition. Regardless of whether nature “makes sense” to us or not, we have no choice, as scientists, but to look at the Universe directly, to measure it, and to draw our conclusions for how it’s behaving on that data alone. It shouldn’t be colored by our expectations; observation and experiment is our guide, and theory must reckon with whatever it is that nature tells us it’s doing.

For a long time, it looked like w was most consistent with -1, and the errors on that have shrunk tremendously over time. By 2008, it was clear that the error was less than about ±0.3, which was important because most different species of energy see w change in increments of ±⅓; if w was equal to -1 ± 0.3, that ruled out the most expected, straightforward alternatives. Errors then fell further: to ±0.12 as of about 10 years ago and to ±0.07 based on the most current set of data, assuming a constant, non-evolving form of dark energy.

But now, we have data from large-scale structure (through DESI, for instance), from the CMB (through Planck and other small-angular-scale experiments), and from supernova data (through Pantheon+, Union, and DESY) that all conflict when combined together. Perhaps w once was close to -1, but perhaps it’s weakening recently, maybe evolving to be as “big” as just -0.8 or so today.

Looking at the data points from DESI (top row) or from the various supernova collaborations (bottom row), it’s very clear that the data, at this point in time, is not sufficiently good to robustly discriminate between the various options for how dark energy is behaving in the Universe. The fact that the three different supernova samples, DESY, Union, and Pantheon+, give such different answers from one another should be a troubling indication that we haven’t yet uncovered the full story.

What do we do, in this situation, as responsible scientists? The only acceptable solution must be to go out and acquire superior data: data that will actually allow us to determine, with smaller errors and high confidence, what w is today and to measure how w has evolved over cosmic time. If we hope to assemble a picture of the Universe that’s comprehensive and accurate, we need the data from multiple different, independent lines of evidence to all agree or at least be consistent with one another. If they’re not, that will instead point to something potentially revolutionary: like the notion that our theoretical picture of the Universe needs altering, refining, or perhaps even to be thrown out completely.

While many who talk about the Hubble tension and the implications of evolving dark energy in this last context — the notion that we should throw out our modern cosmology and start from scratch — that’s overwhelmingly thought to be the least likely way forward by the community. Put simply, there are too many successes for our concordance cosmology, with constant dark energy where w = -1, to simply get rid of it all. Each individual data set, on its own, poses no tensions with this picture; it is only when independent data sets are combined that any problem at all arises. Furthermore, the problem is dependent on which data set is chosen; if we look at the Pantheon+ data versus the Union data for supernovae (combined with DESI large-scale structure data), for instance, the significance of the tension varies tremendously: from less than 3-sigma for Pantheon+ to more than 4-sigma for Union data.

We’re compelled to consider that dark energy might be something other than a cosmological constant, but so far, we only have hints, not a full-blown crisis.

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