Can Inflation, Dark Matter and Dark Energy Be Replaced by Spacetime Moving Like a Fluid Warren Davison, University of Arizona, Tucson, Arizona 85712, USA Cosmology has settled on several major theories (Inflation, Dark Matter and Dark Energy) to explain some observations that do not match the observed mass forces expected from Einstein’s General Relativity. I believe you can account for these observations with just Einstein’s General Relativity (Friedmann-LemaitreRobertson-Walker metric) and only one additional theory. If Einstein’s Spacetime as we know it had just one more attribute, motion independent of mass, then many of today’s problematic cosmological observations could be explained. Einstein’s General Relativity completely defines the relations between mass and the distortions in Spacetime. The equations are not Spacetime but a description of how Spacetime acts on mass. We do not need to invent a new Spacetime we only need to add another property governed by other equations. Thus if you add the property of movement like a fluid (Navier Stokes Equations) independent of mass you can explain: cosmic microwave background variance, early universe accelerated galaxy formation, angular momentum source for galaxies, large-scale structure of the Universe, high orbital velocities at the periphery of spiral galaxies, galaxy and cluster Einstein rings, accelerating Universe observations and flatness of the Universe. The transformative action of the fluid is the formation of circular eddies. These eddies have a Spacetime distortion. Although the distortion value is relatively low it adds to gravity distortions at great distance. I can apply classic fluid equations to many Cosmological observations with good results. Eddy Theory is a theory that can account for observations, provide equations and has a logical mechanism.
Introduction The concept of Spacetime is widely accepted and provides an elegant way to account for gravity. The problem is that Spacetime is not composed of anything we know, if anything at all. Yet mass moves in curved motion, accelerates and decelerates. Although Spacetime is generally accepted almost everyone thinks of it still as gravity and not Spacetime. Spacetime is a mystery that we have assigned two properties, the distortion from mass (gravity) and the expansion from the Big Bang. Spacetime must continually expand taking mass with it which is the only way to make the Universe we observe. Can Spacetime have other properties? Properties that are independent of mass. An analogy of this would be like having a boat in a bay. Consider the mass property to be like the buoyancy, the tide like expansion, but there can also be a current that moves the boat that is completely independent. If Spacetime had a property that was movement, movement like a fluid then many Cosmological observations would make more sense. First let’s consider a major property necessary for the Universe as we know it which is the continual expansion of Spacetime. Einstein’s General Relativity (FLRW) dictates the Universe be expanding or collapsing from the gravity. There is common analogy of the Universe as a loaf of rising raisin bread where all the raisins move away from each other to explain the
expansion. The Big Bang provided the impetus. Momentum is a stated mechanism. If you were to observe galaxies in all directions at 5 billion years after the Big Bang and again at 10 billion years you find all observed galaxies almost twice as far away and receding at nearly twice the velocity observed before. This increasing velocity is not the momentum I understand. Consider the Big Bang with Fluid Spacetime, it would have a high “pressure” at first so it would expand taking the mass with it, with everything moving away, decreasing the “density”. The expansion is everywhere so when you have more Spacetime between two galaxies the expansion is more so they recede faster. This is exactly like the raisins in the bread and is logical, plausible and exactly like observations.
Fluid Spacetime If Spacetime had fluid properties, then the early hot dense Spacetime would be turbulent with a large range of eddies. Eddies are a movement of Spacetime with a distortion. Spacetime can only have one value at one position so the distortions add just as two gravity distortions add. Astronomical observations cannot distinguish between the distortions, only if they match the observed mass. Eddy Theory is a simple way to account for many problematic cosmological observations that do not match observed mass.
Fluid Spacetime properties must have values for the Navier Stokes Equations, density, pressure, velocity and viscosity with respect to the spatial and time domains. Remember this is Fluid Spacetime not a matter fluid. Pressure and density must be nearly infinite at the Big Bang. Velocity must be defined to comply with the Hubble constant. The viscosity must be extremely low approaching a superfluid. The specific values of the properties are beyond this paper. There are several other fluid attributes that are useful in applying to the observations. As the Fluid Spacetime expands it reduces the pressure and density. The expansion is an asymptotic function with the velocity decreasing with time. Thus the Fluid Spacetime Universe can only be flat. This matches the observations, “The universe is flat with only a 0.4% margin of error” . But what about mass? There is definitely not enough mass to make the Universe flat. The mass would spread with Spacetime but the attraction to other mass would move it within Spacetime. As the Universe expands from the Big Bang it cools and eventually goes laminar with no eddy formation. The eddies that are around today are decaying remnants of a turbulent past. For the ideal eddy (one with a circular uniformly rotating core) the equations for velocity of rotation match the classic fluid equations of a rotating cylinder inside a rotating outer cylinder. Cuvette solved the equations in the late 19th century. The simplest form is a very large stationary outside boundary condition. Cuvette 𝑣𝑣𝑟𝑟 = 𝑎𝑎𝑟𝑟 + 𝑏𝑏/𝑟𝑟 2 2 2 𝑎𝑎 = (Ω2 𝑅𝑅2 − Ω1 𝑅𝑅1 )/(𝑅𝑅2 − 𝑅𝑅12 ) b=(Ω1 − Ω2 )𝑅𝑅22 𝑅𝑅12 /(𝑅𝑅22 − 𝑅𝑅12 )
Cuvette simplified (Ω2 = 0) 𝑣𝑣𝑟𝑟 = Ω1 ∗ 𝑅𝑅12 /(𝑅𝑅22 − 𝑅𝑅12 ) ∗ (𝑅𝑅22 /𝑟𝑟 − 𝑟𝑟)
Is eddy rotation, Ω2 is the outside boundary Ω1 rotation 𝑟𝑟 Is the radius 𝑅𝑅1 Is the radius of the inside of the eddy, 𝑅𝑅2 Is the radius of the outer boundary 𝑣𝑣 Is the tangential orbital velocity The application of these equations only require laminar conditions. Figure 4 shows this equation fitting the data for NGC 6503 almost perfectly. There are more sophisticated equations for finite length Cuvette and Computational Fluid Dynamics solutions for more complex modeling. The curved Spacetime from the eddy produces a distortion that is proportional to the Shear Strain Rate (Velocity Gradient). The shear strain is the velocity of one part of the fluid flowing faster than the adjacent so it is constantly being stretched between the two. It is defined by 𝑑𝑑𝑑𝑑/𝑑𝑑𝑑𝑑 and is literally a Spacetime distortion.
There must be a proportionality constant to relate the magnitude to a mass Spacetime distortion. Thus the orbital velocity is the sum of the Spacetime mass distortion the Spacetime velocity itself plus the Shear Strain Rate Spacetime distortion. Spacetime behaves in a nonintuitive way. Mass makes distortions and these distortions reduce with the inverse square root of the distance. The problem comes when you have more than one masses. Consider two masses, each mass causes a distortion and they add. When a third mass enters it moves through the distortions with the appropriate path. The problem is the distortion by a mass does not affect itself, so the effective distortion for one body movement is only from the other mass, not the sum. The only conclusion is every distortion is independent, as it should be. So is a Spacetime Shear Strain Rate this kind of distortion? Can it be independent? Can they overlap and their movement add together? To match observations, the eddy must act like a fluid but the surrounding distortion may act like gravity distortions do, independently and sum or cancel together. Eddies now are approximately circular and have an angular velocity. The velocity itself is enough to account for observations but the curved Shear Strain Rate Spacetime also act like curved gravity distortions.
The Early Universe Eddies in Spacetime must start forming just after the Big Bang, in a very hot, dense, rapidly expanding and ionized universe before Recombination. Even in this extreme environment the eddies form potential wells to compress the ions and make a variation of 1 in 100,000 in temperature in the Microwave Background. This is the start of an active mechanism in the early Universe starting compression, beginning to coalesce matter and not just a static reflection of an earlier process. During the Dark Age, from about 300,000 to 200 million years, eddies continue forming, dividing and condensing mass. By definition, the Dark Age ends with the formation of stars, at ~200 million years. This is an extremely short time to condense stars. To start the formation of stars and galaxies there must be a very strong initial force to overcome the uniformity of mass, the high temperature and the rapid expansion of Space-time. Galaxies and clusters of galaxies must also accumulate large amounts of angular momentum. You can speed up a rotation by shrinking it, however the angular momentum is conserved and remains the same. To have angular momentum you must start with it. The only way to create it is to make some in the opposite direction simultaneously. As with fluids; eddies have to preserve angular momentum. In the Fluid Spacetime the eddies provide Spacetime distortions. This can provide the strong forces at distances that are required to start to coalesce early stars and galaxies. In fluids the scale factor for the largest eddy to smallest is the Reynolds Number raised to the ¾ power, so for a
specific fluid condition the size range of the eddies is fixed. As with other fluids the early Universe’s properties would result in eddies with in a large range of spatial scales which would populate the largest scales of the universe down to a minimum size. In the early Universe the largest eddies are too large and do not have enough rotational speed and potential well to condense the matter enough for gravity to finish the formation of stars and galaxies. As with other fluids the edges between the large eddies decay by spawning smaller eddies. These smaller eddies rotate faster so they have deeper potential wells. These eddies have the size and power to actively condense the gas to a point where gravity forms stars and galaxies. In short there is a Goldilocks size for eddies where too large and there is not enough time or potential well to condense gas enough for gravity to complete the process, or just right where the forces and distance the gas must move are just right to coalesce. The gas across this Fluid Spacetime universe is spread evenly but the edge of the largest eddies is the only place there are sufficient smaller eddy “seeds” to form galaxies. This mechanism is all one needs to explain the empty regions, walls of galaxies and clusters in the largescale structure of the Universe. The empty regions are not free of gas, just the accretion mechanism to form galaxies. Stars and galaxies are observed at about 200 million years 400 million years respectively, when the universe was ~1.5% to 3% the size it is today . We are used to gravity being much more powerful than Spacetime expansion but in the early universe, this is not true. At that time, it takes an extremely powerful “seed” to stop expansion and collapse hot matter into galaxies. Matter simply moves with Spacetime until there is a force to cause a relative velocity. With eddies in Spacetime the matter would initially swirl with the eddy while starting to collapse. This is not orbiting but simply a rotating motion where mass moves with the Spacetime. The gas that has angular momentum from the outside of the eddy collapses to the point it orbits the galaxy while forming suns. With time the galaxies collide and the eddies combine. The Universe expansion eventually gets to the value where turbulence and eddies cease formation and Spacetime is laminar.
Eddies decay by spawning other eddies until they reach a minimum scale where they decay by viscous energy loss. The eddies that are around today are stable isolated ones that will not split but they can still collide and combine.
Universe Expansion How does the Universe expand? First the Universe is uniform on large scales. Observations show the expansion rate is not changing very fast. Consider the Big Bang. If it expanded at the speed of light, then the rate of expansion would be the speed of light divided by the radius. Thus if you double the size of the Universe you halve the expansion rate. We can look one way and see a galaxy receding at almost the speed of light. When you look the other way you see the same, so we see some galaxies that are going away from each other at close to twice the speed of light. The Universe is expanding faster than the speed of light! This is only part the Universe we can see so the total rate of expansion is much higher. Although nothing can move through Spacetime at faster than the speed of light, it appears Spacetime expands at a modest rate everywhere so the accumulation eventually expands much faster than the speed of light with respect to something far away. We observe a relatively constant expansion rate per distance so the “edge” must be expanding at an increasing velocity with the passage of time, twice the size yields twice the expansion rate. This is an increasing speed function. The limit to the expansion rate is not momentum since the velocity is continually increasing. It is not the speed of light since the expansion is faster than that. Consider Fluid Spacetime, the pressure would reduce with the volume, again much too fast for observations. A limiting function in fluids is the speed of sound. If Fluid Spacetime had a limiting property like the “speed of sound”, then the expansion rate could not exceed it and it would be relatively constant. It would only change when the “speed of sound” changes or the pressure drops below a threshold. This must be a relatively local rate limit and not an outside boundary effect. This “speed of sound” change can be from varying other properties, like (Spacetime) temperature and density. In a fluid, the fluid Figure 1 Cosmic Background Radiation from WMAP satellite Credit: NASA / WMAP Science Team. Note the larger scale features.
will not move faster than the speed of sound no matter how much pressure is applied, and the pressure behind becomes uniform. So you can have a high pressure and high density that can decrease without changing the expansion rate. Thus the Hubble Constant times a distance can be the “speed of sound” in Spacetime. Unfortunately, it is very hard to relate the “speed of sound” to a fluid analogy. In
Figure 2 a blowup of the WMAP figure 1 showing detail that can be matched by fluid simulation
number), a time of commencement and an expansion rate from that period of time. There are enough variables to make it fit absolutely. Note the range of larger structures. The application of Eddy Theory shows the beginning of a process, not the conclusion. The eddies are starting the process of compression from the Spacetime distortions, which will continue to intensify until gravity eventually
Figure 3 A fluid eddy simulation “DNS of Isotropic Decaying Turbulence” (Credit Oliver Desjardins) 
ideal gasses higher temperature increases the speed of sound. In liquids and solids, the speed is slower with higher density. One potential mechanism to decrease the “speed of sound” is to lower the “temperature”. A potential mechanism to increase the “speed of sound” is to lower the “density”. This is one good way to account for the observed expansion velocity change in the Universe. One expansion possibility for the Universe would be the extreme “temperature” at the Big Bang would make the Spacetime expand at a high rate. As Spacetime cools the expansion slows. Eventually the “density” decreases enough to actually increase the” speed of sound” and thus the expansion rate. This mechanism can provide for a slowing and then speeding up of the universe expansion rate. There are other mechanisms that would influence the expansion rate like gravity and eddy expansions, which I will explain later.
becomes dominant. The similarity of Figure 3, which is a numerical simulation of decaying eddies in fluids and Figure 2 which is a blowup of the Microwave Background demonstrate the Navier Stokes Equations have the potential to produce features that are the same shape and density as the observed Cosmic Background Radiation. Polarization of the Cosmic Microwave Background is easy to explain. Eddies cause distortions in Spacetime and these distortions have the same effect as the distortions from gravity. Radiation emanating from gas in a Spacetime distortion has B-mode polarization. The polarization seems to match the thermal variations, which is what you would expect from Eddy Theory. The existence and magnitude of B-mode polarization from eddy Spacetime distortions are indistinguishable from gravity distortions. Polarization from eddy Spacetime distortions are indistinguishable from gravity distortions.
Early Galaxy Formation Galaxy GN-Z11 is estimated at ~4000 light years in diameter described as being formed 400 million years after the big bang. The "Methuselah" Star, HD 130283 formed after a mere 200 million years and is probably a second generation star. These are examples that I have found but almost certainly are not the earliest. N-body simulations show realistic galaxies in many billions, not millions of years. The first problem is that if the accretion of the matter started at 100 million years, the Universe as well as the gas to form a galaxy would quadruple in size and 64 times in volume by the time we see this first galaxy at 400 million years. If the gas cloud is three times the galaxy size, then the cloud is expanding at about 14 km/sec. Arresting the gas expansion and then collapsing it requires a very large Spacetime distortion. I
Cosmic Background The eddies start forming after the Big Bang. The largest of them account for the structure and the smallest for the microwave background and starting the formation of galaxies. Eddies decay by splitting until they reach a minimum scale where they decay by viscous energy loss. The smallest eddies are stable and can be matched to the WMAP data as growing from just after the Big Bang. Figure 3 shows a computational decay of turbulence. It is remarkably similar to the pattern of the microwave background showing that fluid equations can match the pattern of the Cosmic Background Radiation. To match WMAP data you need a disturbance (eddies at a specific Reynolds
ran a very simple calculation using the gravity or eddy functions from NGC 3198 Observations. The eddy function took tens of million years to arrest and collapse the gas to 1/3 of its size. The gravity (from this fully formed galaxy) function took hundreds of millions years. True galaxy formation is a combination of both functions with the outer portion of the cloud dominated by the acceleration from the eddy and the central part forming suns with gravity. The second problem is the temperature of matter starts around 3000⁰ K at 380,000 years and cools to 10⁰ K in 500 million years, so you have to compress hot gas. The third problem is matter in the universe starts out extremely uniform so there is no appreciable gravity to start compression. It takes extremely strong “seeds” very early in the formation of the Universe. Eddy theory can provide those seeds. The form and magnitude of the seeds need to be just right. To week a seed and the Spacetime expansion just continues without forming stars. Too strong and the seed produces only a black hole. Eddy Theory provides the strong forces for early formation of galaxies.
spin and orientation to make a complex form and not this simple equation. Eddy Theory can be very successful at describing the velocity of the extremities of spiral galaxies and can even match some with an ideal eddy shape. Small galaxies appear to have a lot of excess rotation which combine into spiral galaxies which have less which combine into elliptical galaxies have the least, if any. Why does this function decrease when combined.
Angular momentum Source Angular momentum is very hard to account for in early galaxy formation. Conservation of angular momentum dictates that to have angular momentum you must start with it. Most explanations are you have movement in the gas and when it condenses it swirls. Where did the movement come from? The source of angular momentum does not have a generally accepted source. In the early Fluid Spacetime Universe, the gas has very few external forces so when eddies form they “drag” the gas with it. The gas also responds to the Spacetime deformations, so as the Spacetime swirls the gas moves with it while it accelerating toward the center. This is a great theory to provide angular momentum; it has the physical means and a plausible magnitude. In the three dimensional space the eddies will have all orientations and spins. Eddy Theory can explain angular momentum while many theories cannot.
Figure 4 Computed velocity plus Covette fluid Equation (1) (Credit for the data: Kamionkowski 1998, astro-ph/9809214). I have matched more galaxies. This equation is for a singular rotating core and not for complex geometry.
Excessive Rotational Speed in Spiral Galaxies Rotational velocities at the extremities of spiral galaxies far exceed the orbital velocities expected from the observable mass. The idealized Spacetime eddy is a fluid equation with velocities around a rotating core. The distortion of an eddy in Spacetime would produce a relatively flat base surrounded by a constant slope. The orbit would actually be the sum of the effect of the gravity plus the effect from the eddy. Equation (1) shows the contributions to velocity from an ideal eddy. However, eddies are almost certainly not ideal and should have distortions and complex cores. It is relatively easy to produce the shear function 𝑓𝑓(𝑠𝑠,𝑟𝑟) for an observed galaxy by subtracting the observable matter curve from the observed velocity curve. Note the function matches the observed shear function extremely well in NGC 6503 . There are a lot of individual shapes to the rotational velocity curves, but all can be matched by a Shear function, although more complex. Eddies can combine with ones with different diameters, velocities,
Velocity (k/s) 140 120 100 80 60 40 20 0
NGC 6503 Observed Veloity
Gravity+Eddy Computed Velocity 0
With Dark Matter it cannot decrease. Eddy Theory has the exact properties to match these observations with the combining and canceling of eddies when galaxies collide. Einstein Lensing Sometimes the observable mass is not enough to explain the observed gravitation lens. For strong lensing from galaxies, the Einstein Radius is often very similar in size to the maximum eddy function. The magnitude and position of the lensing fits extremely well with Eddy Theory. For weak lensing, the distortions in Spacetime from Eddies add a significant magnitude at distance. For clusters of galaxies this would cause largescale distortions in Spacetime as though there were a lot more mass. The Fluid Spacetime distortions added with gravity distortions which can easily account for the strong lensing and weak lensing observations. Accelerating Universe Based on limited observations the velocity at which the universe expands appears to be increasing. Gravity always provides an initial deceleration. For Eddy Theory to explain an accelerating Universe, there are two potential mechanisms. First assume that Spacetime is expanding at the “speed of sound” and this decreases and increases as explained above. Another is like the classic analogy of the expansion of the Universe is a loaf of raisin bread, and that is perfect to explain an Eddy Theory accelerating Universe. Assume the eddies are the raisins and they do not grow or grow at a slower rate. Then at the start, the dough is a small proportion with the raisins being the larger proportion so the linear expansion of normal Spacetime gives a low total expansion velocity.
The raisins do not grow so as the loaf expands there is more dough to expand. The high content of raisins to the dough in the early expansion actually slows the expansion rate. As the loaf grows the dough is a larger proportion so it expands faster. This yields an acceleration function that decreases at first, then rising and then asymptotically approaches the Spacetime expansion rate.
 Erin Blakemore, SMITHSONIAN.COM. SMARTNEWS https://www.smithsonianmag.com/smart-news/hubblespotted-oldest-galaxy-it-has-ever-seen-180958288/
Early expansion is decelerated by gravity and may be assisted by eddy formation. I do not have enough information to produce the best fit function but it can be an S shape that fits the data well. Eddies may expand with Spacetime, partially expand or not expand. The amount of acceleration has some uncertainty but this function can fit as well, if not better than the Standard Model. There are two major differences in the theories. The Dark Energy theory continues to accelerate the expansion rate and the Eddy Theory reaches a maximum expansion velocity, then decreases. The Dark Energy Theory needs about 14 times the mass energy of all observable matter and energy in the Universe but the Eddy Theory needs orders of magnitude less, in the realm of known energy. Eddy Theory is based on fluid properties so the Universe would have a “pressure”. The “pressure” would decrease with the expansion slowing eventually to zero. Eddy Theory predicts a Universe that is flat, matching observations.
 Oliver Desjardins, DNS of Isotropic Decaying Turbulence, http://web.stanford.edu/group/pitsch/Movies/LESDNS.mpg
Conclusion Why do we need a new theory? The theories of Inflation, Dark Matter and Dark Energy have been studied by so many astronomers for so many decades. The equations in these theories correctly account for the observations, but each has a serious unexplainable problem. Inflation has a leap at hyper light speed, Dark Matter with all the study has only shown it is not anything known to man, and Dark Energy that is an order of magnitude more than all of everything in the Universe. Eddy Theory can provide plausible mechanisms and equations to explain the data. Eddy Theory is a very powerful and dynamic theory with the capacity to change and adapt to new observations. This theory is new and as such will need to be refined as new challenges are identified. Others have tried to apply Navier Stokes Equations but never in this manner or scale. Eddy Theory does not have to modify Einstein’s Equations, it does not need to invent a new Spacetime, it only needs to add independent motion. Adding fluid properties makes so much sense for explaining the observed Universe. Eddy Theory can account for the observations with a single logical mechanism with the properties and equations already identified. We only need to apply the theory to existing data!
 NASA, Measurements from WMAP, https://map.gsfc.nasa.gov/universe/uni_shape.html
 http://www.nicolascretton.ch/Astronomy/images/ Rotation_curve_N6503.jpg  http://www.guidetothecosmos.com/newsletterAccel-Universe.html)
Warren Davison (Affiliation): University of Arizona, Email: [email protected]