Problems with Accelerated Expansion

This is a continuation of the post Accelerated Expansion, which followed on from Extended Consistency Principles. In the latter, I argued that we should not expect to be in a privileged era, which is to say we should not – when looking out on the universe – observe a situation that has never happened before and will never happen again, other than as strictly required for us to exist so that we may make those observations.  But there are what I consider to be major contraventions of these principles.

The age of the universe appears to be the simple inverse of the Hubble parameter, at this time.  If there had been an extremely short period of inflation only, there would be such a small deviation from this situation that inflation would be within the noise, indetectable given the uncertainty inherent in our measurement techniques. If there had been a relatively short period of deceleration in the radiation-dominated era, for about 47,000 years, then the effect of this would also be within the noise and thus indetectable.  It should be noted however that the age of the universe would never have been equal to the simple inverse of the Hubble parameter once inflation had commenced.  (That is to say that although it is likely true that the Hubble parameter may have been related to the age of the universe, that relationship would not have been simple at that time.)

If there was then a period of less extreme deceleration from the end of the radiation-dominated era until about four billion years ago, as described in Decelerated Expansion, then the currently measured values of the Hubble parameter would have been expected to have occurred when the age of the universe was about 9 or so billion years old, or as much as 5 billion years ago (for H=74km/s/Mpc). At no point during the matter-dominated era would the Hubble parameter have been the inverse age of the universe.

Then, as the chronology of the universe is understood, something happened to make the value of the Hubble parameter, right now, extremely close to the inverse of the age of the universe (with the average of the measured values, along with the most recent measure, lying pretty much precisely on the inverse age of the universe.  And, unless the acceleration of the universe stops, right now, the universe will never again have a Hubble parameter that is equal to the inverse of its age (in fact, looking at the value of the Hubble parameter, it would look increasingly younger were we to assume that t=1/H.

There are a few ways, given the assumptions above, that the Hubble parameter could be at its current value:

• the matter-dominated era deceleration was as given (dH/dt=-2H²/3) but ended about 4.6 billion years ago, rather than 4 billion years ago, and thereafter dH/dt=0, meaning that w=-1 and that the Hubble parameter has been about 71km/s/Mpc ever since – and thus only coincidentally is precisely the inverse of the age of the universe.  That would make this a privileged era and there would be no accelerated expansion.

• the matter-dominated era deceleration was as given above and ended about 4 billion years ago, and thereafter there was accelerated expansion, at a higher rate of accelerated expansion than provided by the Planck Collaboration, so that the Hubble parameter (which naturally tends to decrease as the universe ages), would be raised from about 66km/s/Mpc to the current value of about 71km/s/Mpc, rather than the Planck Collaboration’s value of 67.4km/s/Mpc.  As calculated above, that would require an equation of state parameter of about w=-1.13.  This would make this a privileged era and retains accelerated expansion … but requires some fancy footwork by the universe to make things perfect just in time for us to observe it.  Alternatively, we happen to have a Hubble parameter that is just coincidentally within about 5% of the inverse age of the universe.

• the matter-dominated and the dark-energy-dominated eras had neither deceleration nor acceleration and dH/dt=-H² throughout.  This would mean that, at 13.77 billion years of age, the Hubble parameter within our universe would be the inverse of that, about 71km/s/Mpc.  This would mean that we do not live in a privileged era (at least not with respect to the Hubble parameter).

There is the question of the density of the universe.  If the universe is flat and dH/dt=-H², then it follows that the mass (or rather mass-energy) of the universe is increasing with time because M=(c³/2G).t (equivalently E=(c⁵/2G).t)  This means that for every unit of Planck time (t_PL=√(ħG/c⁵)), M changes by half a unit of Planck mass (m_PL=√(ħc/G)), which I’ve mentioned before.  (Planck energy is a derived value, using E_PL=m_PLc²=√(ħc⁵/G)).

There’s a certain elegance to an invariant increase in the mass of the universe, which as described here may be twinned, at a rate of one unit of Planck mass-energy per unit of Planck time.  While I cannot claim that the universe needs to be elegant, if the universe has not increased its mass-energy this way, then we are – again – in a privileged era in which our universe’s mass-energy (our half of it, if you like) is precisely what you’d get if half a unit of Planck mass-energy had been entering every unit of Planck time for the entire age of the universe.

Note that this invariant process would ensure that the universe would remain flat throughout.  For the universe to remain flat during periods of deceleration and acceleration, the rate at which mass-energy enters the universe would be variant.  This is not necessarily a problem, since the precise mechanism by which mass-energy is forced into, channelled into, sucked into or generated in the universe is unclear.  It could just be an artefact of a total zero energy balance as the universe expands, so that more or less mass-energy would appear depending on the expansion rate.  But, unarguably, such a universe would be less elegant.

There is also the mathematics that gets us to dH/dt=-H²(multiplied by some constant) which implies very strongly that H=(constant)/t.  I don’t know how we get away from the problem that, if the value of the constant changes at various times, even if due to conditions that change over time, due to the varying levels of (or domination by) radiation, matter and dark energy, then you have a situation where, in reality, (constant)=f(t).  In which case, the implication that, at t=9.8 billion years, H=66.5km/s/Mpc (implied by Planck Collaboration’s w=-1.03) is highly questionable and would require another level of fancy footwork by the universe.

Then there is dark energy.  According to the standard model, we are in a dark energy dominated phase.  Note that “dark energy” doesn’t necessarily mean that there’s some sort of light-challenged energy, but more that there is a phenomenon that needed some explanation - and the term “dark energy” was assigned to the as yet unknown cause.  Both of the two most common descriptions of dark energy resolve down to the notion that there is a distribution of very low density energy throughout the universe, in the order of 6×10⁻¹⁰ J/m³.  But note that this density is invariant.

Consider a universe in which there is a constant amount of mass-energy, it would look a bit like this:

With dark energy (which can be considered an overlay), it is thought to look more like this:

 

(Note that these images are snipped from a larger image at bigthink.com which credits them to E. Siegel/Beyond the Galaxy.  I’ve not used that larger image because it seems to make an error in the charts that appear to the right of the second of these images.  It could just be a simplification for the purposes of explaining a larger [or different] truth, or what is known as a “lie to children”.)

Now, if the entirety of the standard model is correct, meaning that the entire universe is currently 93 billion light years across (so having a radius of about 46.5 billion light years), and that there is a density of dark energy of 6×10⁻¹⁰J/m³ which makes up about 68% of the total energy, which would be about 8.8×10⁻¹⁰J/m³, then the total mass of the universe would be 3.2×10⁷¹kg, a mass for which the Schwarzschild radius is 550 billion light years.

Note that the Schwarzschild radius that correlates with a density of 8.8×10⁻¹⁰J/m³ is 13.5 billion light years.

Think about that for a moment.  The notion is that there is dark energy (the density of which is invariant), dark matter (the density of which relates to the inverse cube of the scale factor), baryonic matter (the density of which also relates to the inverse cube of the scale factor) and radiation (the density of which, at least during the radiation era, related to the inverse fourth power of the scale factor).  These all combined just perfectly so that, now, when the universe is about 13.7 billion years old, the density is pretty much precisely that of a Schwarzschild black hole of a radius of 13.7 billion light years, making it look like the universe had been expanding at one light year per year, and maintaining a critical density (ρ_c=3H²/8πG where H is and has always been and always will be the inverse of the age of the universe), which it would need to do if it were flat (and, if Sean Carroll is correct, if the universe is ever flat then it always has been and always will be flat).  How incredible is that?  Or should I say, how credible is that?

Penultimately, cosmic acceleration was raised to explain some observations and there exist other potential explanations to account for those observations (also here).  I am not saying that any of those explanations are correct, merely pointing out that the lack of acceleration is not an insurmountable problem in itself.  Its removal would just mean that we would need to look for other, more satisfactory explicatory mechanisms for what has been observed.

And, finally, there is evidence that the methodology that led to the notion of dark energy in the first place was problematic.  Basically, we should expect the universe to homogenous and anisotropic only at sufficiently large scales (because the universe is lumpy at lower scales, such as stellar systems, galaxies, galaxy clusters and so on) – about 200-300 megaparsecs, or one billion light years.  The work that lead to the notion of dark energy assumed homogeneity and anisotropy at 100 megaparsecs, which doesn’t sound too far a stretch, but the analysis showed that the apparent acceleration of the universe disappears once the data is rescaled.  We should not rest on that too much though, because research in the future may reveal that the data is significant at the correct scale and some explicatory mechanism may still be required.

Accelerated Expansion

Observations suggest that, during the dark-energy-dominated era, the expansion of the universe has been accelerating. 

This is acceleration is understood to have happened after two periods of deceleration during the radiation-dominated and the matter-dominated eras (see Decelerated Expansion).

Using the representations from the previous post about inflation, the acceleration could look something like this (somewhat exaggerated, or very much not so, depending on what the time scale is considered to be):

 

Note that the accelerated expansion is not thought to be of the same magnitude as per the inflationary era but, if the equation of state parameter w<-1 (and never rose above w=-1), it would nevertheless lead to a Big Rip eventually (see Equations for Cosmological Expansion).

In the description of the acceleration of expansion, it is stated that:

The most important property of dark energy is that it has negative pressure (repulsive action) which is distributed relatively homogeneously in space.

It is later stated that “Different theories of dark energy suggest different values of w”.  See Equations for Cosmological Expansion for the effects of various values of the equation of state parameter, w.  Note that there is some observational skepticism with regard to dark energy, including an experiment intended to detect associated forces, but which failed to do so (although the associated paper accepted that the expansion of the universe is accelerating).

The value of w established by the Planck Collaboration is consistent with w=-1, but they give the value as w=-1.03±0.03.   (The authors refer to w as the evolution parameter, but this usage may be idiosyncratic.  Note also that w=-1.03 and w=-1.06 are the values used in the charts in Equations for Cosmological Expansion to reflect this Planck Collaboration range.  They also find the universe to be flat.)

With an equation of state parameter of w=-1.03, we have a rate of change of the Hubble parameter of dH/dt=0.045.H² that leads to a Big Rip in about 300 billion years.  That is probably long enough to finish everything I need to get done, but a key question here is how exactly did the Planck collaboration arrive at an equation of state parameter value of w=-1.03?

My suspicion is that it went a bit like this:

At the end of the matter-dominated era about 4 billion years ago, as calculated at Decelerated Expansion, the value of the Hubble parameter would have been in the order of 66.5km/s/Mpc.  The measured value, by the Planck Collaboration, is about 67.4km/s/Mpc.

So, what value of the equation of state parameter do we need to get to today’s Hubble parameter value over a period of about 4 billion years?  First, we need to standardise our units.  That is, express our Hubble parameters in the same units as the inverse of the time delta (so Gy^-1).

So, 66.5km/s/Mpc = 0.0680Gy^-1 and 67.4km/s/Mpc = 0.0689Gy^-1.

Well, what do you know?  The current value of the equation of state parameter is precisely the right value to get the Hubble parameter value that the Planck Collaboration measures today.  It certainly seems like a fudge.  And there are other measurements of the Hubble parameter available, in fact there is a bit of a “crisis” associated with range of Hubble parameter value measurements.

Note that, notionally, at no time since shortly after the inflationary epoch began would the value of the Hubble parameter have been even close to the inverse age of the universe.  I say “notionally” because I am assuming that for the first 10^-36s, w=-1/3 so that dH/dt=-H².  But, conveniently, when we happen to be in the position to measure it, the Hubble parameter is, by some accounts, not just mine, but notably not the Planck Collaboration, exceedingly close to the inverse of the age of the universe, not only within an order of magnitude (which is commonly close enough with things cosmological), but well within the uncertainty of the most recent measurement using a distance ladder of flaring red giant stars and slap in the middle of the current range of authoritative measurements (which, given how disparate they are, may be no more than a coincidence).  See the image below from Scientific American and consider that the Hubble parameter consistent with the inverse age of the universe is 71km/s/Mpc.

If the Planck Collaboration were fudging their value of the equation of state parameter, the question arises, what value would be needed to end up with a Hubble parameter today of 71km/s/Mpc?  That would be about w=-1.13.  And about w=-1.25 to get a value today of 74km/s/Mpc.

In any of those cases, however, the application of the fudge is required to get a value of the Hubble parameter today that is consisted with the inverse age of the universe.  We could avoid this massive temporal coincidence entirely by considering instead that, just maybe, there was no acceleration, nor additional deceleration and that the value of Hubble parameter has always been and will always be given by H=1/t (and that there would be, given a sufficiently large value t, a tiny but “natural” deceleration given only by the inverse square of the age of the universe).

---

It is probably massively arrogant of me to suggest that the expansion of the universe is not accelerating.  Extremely competent astronomers and cosmologists have observed the universe and collated credible evidence that indicates that the expansion of the universe is in fact accelerating.  The equations I have used may be invalid (despite their appearing on Wikipedia without correction – but I accept that errors are made and some are not corrected for many years).  Or I may have used them incorrectly, despite the fact that I arrive at the same value of the equation of state parameter as the Planck Collaboration using their own assumptions.  Or I’m just some random person with access to a computer and few ideas from left field rather than being a credentialed expert in the area.  So, perhaps it behoves me to reiterate and expand on the issues that I have with cosmological acceleration.

---

Rather than make this a massively long post, I’ll do that in the next one.

Decelerated Expansion

The currently understood chronology of the universe has two periods of deceleration after inflation, the radiation-dominated era and the matter-dominated era.

The radiation-dominated era was very short at about 47,000 years, during which the value of the equation of state parameter w=1/3.  Using the equations discussed in Equations for Cosmological Expansion, we find that dH_RE/dt=-2H² and H=1/(2t).

The matter-dominated era extended from 47,000 after the Big Bang to about 4 billion years ago, during which the equation of state parameter w=0.  Using the equations discussed in Equations for Cosmological Expansion again, we find that dH_ME/dt=-3H²/2 and H=2/(3t).

In Inflation, I conclude that having an expansion in the order of 10²⁶ over the 10^-32s available would require a Hubble parameter value of H=~10⁶¹km/s/Mpc.  This is about the same magnitude of Hubble parameter as in the first unit of Planck time.  So, if the deceleration was from that value, it would be as if the clock had been turned back to t=t_PL, so by about 10^-32s.  After that, the expansion of the universe during the radiation-dominated era would be at half pace, H=1/(2t), expanding by half the speed of light, for 47,000 years, reaching about 23,500 light years in radius in addition to the approximately 10^-10 light seconds it was at the end of the inflationary era.  If everything went back to “normal” then (with H=1/t), then the universe would be 23,500 light years smaller than we would have expected, but this is a small fraction of the expected 13.77 billion light years (~0.00017% smaller, or basically indistinguishably smaller).

Then, in the matter-dominated era, the expansion of the universe would be at a rate of 2/3 of the speed of light, for a period of approximately 9.8 billion years, resulting in a radius of about 6.6 billion light years (swamping any delta caused by inflation for 10^-32s or slower expansion during 47,000 years of radiation-dominated era).

The combined effect would look like this, using the same notion as per the Inflation post (approximately):

The H value at the end of that process would be approximately that expected at about 14.7 billion years (assuming a constant H=1/t), or 66.5km/s/Mpc.

To get to where we would expect to be today, assuming a constant H=1/t over 13.77 billion years, it would look like this:

This would imply that a radius at 13.77 billion light years would be receding away from us at about 1.8 times the speed of light.  However, that if objects at (r_H=)13.77 billion light years remove were receding away at 1.8 times the speed of light, that is equivalent to H=128km/s/Mpc, which is not what we measure.  Note however, that this here illustrates a period of faster expansion, not of expansion that accelerates.  I will look at the dark-matter-dominated era with what appears to be accelerated expansion next.

Inflation

This, and the following posts, are expansions on and clarifications of the concerns raised at Avoiding a Contravention of the Extended Consistency Principles.

During the inflationary epoch, between 10^-36s after the Big Bang and approximately 10^-32s later, it is understood that the universe expanded by a factor of at least e⁶⁰=~10²⁶.

In my hypothesis of universal expansion, as described in earlier posts and occasionally referred to as FUGE – Flat Uniform Granular Expansion, the radius of the universe expands by 1 unit of Planck length every unit of Planck time.  A FUGE universe, at 10^-36s after the Big Bang, would have had a radius of 10^-36 light seconds.  In that case, at the end of an inflationary epoch with expansion by a factor of 10²⁶, that universe would have a radius in the order of at least 10^-10 light seconds (which is about 30mm).

There are widely varying estimates for how large the universe was before and/or after inflation.  Some say the size of a grapefruit (after), Alan Guth says a marble (after), some say it was almost infinitesimally small at the beginning (in the order of a Planck length, making the post inflation size in the order of 10^-5 m – diameter or radius, it doesn’t really matter which), and a Forbes article says that the universe was about two AUs across at 10^-12s after the Big Bang (which would imply inflation by a factor of 10¹⁵).

Note that, for the universe to remain flat, this would result in the introduction of a quite substantial amount of mass-energy, equivalent to 10¹⁸ kg (or about 1/10³⁴ of the total mass today) in a period of about 10^-32s, or 10⁵⁰kg/s.  Compare this with the current rate, assuming an age of the universe of 13.77 billion years and H=71km/s/Mpc, of 2.02x10³⁵kg/s.

If, at the end of an inflationary epoch with an expansion by a factor of 10²⁶ in 10^-32s, the universe flipped back to FUGE – that is flat uniform granular expansion, then the universe today would be … pretty much indistinguishable from what we observe.  It would have a radius that is 3cm greater than we would expect in purely FUGE universe, which larger by a factor of about one part in 10²⁷ and the density would be greater by 10^-61kg/m³ (or by one part in 10³⁴), both of which are practically unmeasurable.

So, basically, perhaps inflation happened.  Perhaps it didn’t.  But either way, if the universe otherwise expanded in accordance with the FUGE hypothesis, it could be just as it is observed today.  We could enormously simplify the scenario, by imagining two straight lines with the combined equation r=±ct (t is along the horizontal axis, r is along the vertical axis, c is a conversion factor [and the speed of light]):

This is for values of t between 0 and 20.  Say that we introduce some “inflation” at t=2, then we get:

When we consider a range of t=[0,20], the inflation looks obvious, even with inflation in the order of only 10³.  But when we consider massively greater values of t, we begin to see that the curves are indistinguishable from r=±ct, with t=10⁶:

If we used equivalent scales, it would be inflation of 10²⁶ at t=2 and a final value of t=10⁴³, we get this:

 

This is totally indistinguishable from r=±ct, for sufficiently high values of t (anything above the square root of the order of magnitude of the inflation would be sufficient, assuming that the inflation happens sufficiently early).

---

There are two broad options for what would have happened to the Hubble parameter during an inflationary period.  Either its value snapped to a single value for about 10^-32s, enough to expand the universe out by a factor of 10²⁶ and then relaxed back to some much lower value, or the equation of state parameter flipped below w=-1 for about 10^-32s and then back to above w=-1.

The single value of H required would be enough expand x=10^-36 light seconds into dx=10^-10 light seconds in dt=10^-32 seconds.  Recalling from the previous post that:

We find that H=~10⁵⁸/s=~10⁶¹km/s/Mpc.

If it is a question of the equation of state parameter flipping to below w=-1, then the value needed would be around w=-1.00003 because H is already incredibly high at 10^-36s. But it would have to flip back pretty much instantaneously or it would overshoot.  That is to say, the conditions in the universe that manifest an equation of state parameter of w=~-1.00003 would need to able to change, everywhere in the universe, simultaneously (because you get an inhomogeneity problem otherwise), within less than 10^-33s (or the universe is ripped apart).  Of course, the value of the equation of state parameter could ramp down and ramp back up again to get the amount of inflation that we need, but again you have a simultaneity problem which is precisely what inflation is trying to avoid (see Does Inflation have a Homogeneity Problem?).

---

Note that while the above indicates that, were inflation to have happened, it would be unlikely that we would be able to observe any effect so far as the size or density of the universe goes, there are arguments out there that inflation is not necessary in the first place.  Recall that inflation was introduced as a concept to explain why the cosmic microwave background is so uniform.  There is a paper that argues that gravity is all that is needed to explain homogeneity of the universe and another that argues that an “anti-cosmos” would negate any requirement for inflation.  The latter was referenced at the end of Half a Problem Solved, where I quoted the statement "a CPT-respecting universe naturally expands and fills itself with particles, without the need for a long-theorized period of rapid expansion known as inflation".

---

After the inflationary epoch, it is understood that there were two periods of decelerating expansion, the radiation-dominated era and the mass-dominated era, and then the current, dark-matter-dominated era in which the expansion of the universe is understood to be accelerating.  I’ll take a look at these soon, but first I need to introduce the equations for cosmological expansion and then I need to touch on the inflaverse (as raised in Does Inflation have a Homogeneity Problem?).

Avoiding a Contravention of the Extended Consistency Principles

In the previous post, I concluded that there is an apparent contravention of what I called the extended consistency principles and asked whether that contravention might be resolved?  I think it might.  To explain, I need to beg the indulgence of the reader, to just go with me in the scenario below (at least until I mention Sean Carroll).  There are some issues that might come up, a few of which I am already aware of and have resolved, but I can address them at a later stage.

---

Let us say that there are two conjoined volumes, joined at a null point.  Initially, they are infinitesimally small (ie tending towards zero volume, perhaps even at zero volume) and empty.  Imagine that we can look very, very closely at these volumes at an angle such that they appear to be two areas, purely for illustrative purposes, as below.  (Alternatively, you could imagine a variant of the image illustrating the history of the universe show in the previous post, extending downwards as well as upwards. Just don’t include the exponential curvature.) 

 

Let us say then, that we allow a certain amount of mass-energy into these volumes over the smallest meaningful period of time.  Say that amount is one unit of Planck mass-energy over one unit of Planck time and let us share the mass-energy evenly between the two volumes. 

 

Let us also say that, with every unit of Planck time, the radius of each volume increases by one unit of Planck length.

The equations for Planck mass, Planck time and Planck length are as follows:

This means that after one Planck time, there are (notionally) two tiny volumes of V=4/3.π.l_p³ inside each of which is half a Planck mass (½m_p).  That means there’s a (notional) density of:

After a certain number of Planck times, let us say ꬱ=8.0619x10⁶⁰, the density will be:

Now, these values are:

Planck time – t_P = 5.39x10^-44 s

Gravitational constant – G =6.674x10^-11 m³/kg/s²

Pi – π = 3.14159 (-ish)

Plugging them in, we get 9.47x10^-27 kg/m³.  This is precisely the critical density calculated for of our universe – at this time.  It should be noted that there are other values given, such as by NASA (in 2013), who gave the critical density as 9.9 x 10^-30 g/cm³ but note that they also say that “WMAP determined that the universe is flat, from which it follows that the mean energy density in the universe is equal to the critical density”.  Most of the results that come up, when you search for <<most recent value of the density of the universe>>, are for the critical density, but given that the consensus is that the universe is flat, then the density and the critical density are the same anyway.

Note also that, expressed in energy density, this is 0.85133x10^-9 J/m³ (which is approximately the estimated value for vacuum energy, at 10^-9 J/m³).

The only question of course is: what is 8.0619x10⁶⁰ Planck times in more recognisable units?  It is, of course, 13.77 billion years – the age of our universe as calculated by WMAP.

I just want to summarise here for a moment and make some comments.

In the previous section, I indicated that it would appear that (given the inflationary model of cosmology) we are in a privileged era, because the Hubble parameter is roughly equivalent to the age of the universe.  There are some related facts, so these can’t be thought of as additive coincidences: the universe is (currently) flat, which means that the density is equal to the critical density and the density of the universe is equivalent to the density of a Schwarzschild black hole with a radius of the age of the universe multiplied by the speed of light (which is a consequence of the universe being flat).

However, if the expansion of the universe is currently accelerating (and went through exponential expansion early), then there would not have been any era in the past nor will there be any era in the future when the density would be critical, unless something very strange was happening with the quantity of mass-energy in the universe.  Note that, in the link to Sean Carroll’s article on why we are not inside a black hole, he notes (without support or clarification, unfortunately) “that a spatially flat universe remains spatially flat forever”.  This aligns with the extended consistency principles, but the inflationary cosmological model doesn’t – again unless something odd is happening with mass-energy.

Fortunately, something odd is happening with mass-energy, basically that energy changes when spacetime changes – which is to say that density of the mass-energy of the universe is maintained at the critical density or, as the universe increases in size, the amount of energy increases (at what rate is not explained by Carroll).

Note that this does not mean that the laws of conservation of energy and momentum are broken in such a way that you can get free energy.  What it does mean is that the notion of allowing 1 unit of Planck mass-energy into the universe per unit of Planck time is not a contradiction of existing laws of physics.  Existing laws of physics already imply that mass-energy is being added at rate necessary to ensure that the universe remains flat.

That said, the way that mass-energy would need to enter the universe in order to keep an inflationary/accelerated expansion universe flat (noting that “a spatially flat universe remains spatially flat forever”) would not be consistent.  What we would see is an initial period of exponentially accelerating increase in the mass-energy (well above one unit of Planck mass-energy per unit of Planck time), the rate would then slow down for a while (to less than one unit of Planck mass-energy per unit of Planck time) and then, later, resume a more leisure acceleration (reaching a rate today that is, coincidentally, one unit of Planck mass-energy per unit of Planck time) and, according to predictions, continue to accelerate (to eventually significantly exceed one unit of Planck mass-energy per unit of Planck time).

Perhaps, if the rate of mass-energy entering the universe is entirely governed by rate of expansion (whatever that may be at the time), then I guess that flatness could be maintained.  However, it would not be such that the amount of energy in the universe were proportional to the age of the universe, ever, except for right now.  And the rate at which mass-energy enters the universe would never equal one unit of Planck mass-energy per unit of Planck time, except for right now.  Which puts us back in the situation of being in a privileged era.

There is another issue that I touched on only very lightly.  I pointed out that the critical density, right now, expressed in terms of energy density, is 0.85133x10^-9 J/m³.  In what has been referred to as “the worst theoretical prediction in the history of physics”, quantum field theory calculates a value for the vacuum energy that is 120 orders of magnitude higher than measured (which is in the order of 10^-9 J/m³, and thus in the order of critical density but expressed in terms of energy rather than mass).

Purely out of curiosity at the time, a while ago, I pondered what the energy density would be after the very first unit of Planck time and used my method to get the answer 5.5331x10¹¹² J/m³, which is (approximately) 6.4994x10¹²¹ times higher than currently measured (or within spitting distance of 120 orders of magnitude).  It may come as no surprise to some that this multiplier is the square of 8.0619x10⁶⁰ which the more keen-eyed will remember is the value I assigned to ꬱ above and is the number of units of Planck time in 13.77 billion years.  So again, we look like we might be in a privileged era (since some argue that the vacuum density is invariant and this coincidence would therefore only occur right now).

---

We would not, however, be living in a privileged era – if we were to be living in a two volume universe with smooth consistent expansion, at one unit of Planck length per unit of Planck time, with one unit of Planck energy entering per unit of Planck time (divided equally between the volumes).  If that were the case, then we get the results that we observe without needing to contravene the extended consistency principles.  No matter where or when an observer found herself observing the universe, the universe would be at critical density (which would the density of a Schwarzschild black hole with a radius equal to the age of the universe at the time observed multiplied by the speed of light), a Hubble parameter value that would be the inverse of the age of the universe, the rate at which mass-energy entering the universe would always be one unit of Planck mass-energy per unit of Planck time and, I fully expect, the vacuum energy would be in the ball-park of the critical density (the most significant difference being the units in which they are expressed).

---

The very keen eyed will notice that I use the symbol ꬱ rather than the more easily found æ.  Initially, I took this out of the equation generated in Word, because the rendering looks pretty awful but I have since resurrected it for consistency (which I had lost having inadvertently reverted to using the easy option in later articles).

I prefer ꬱ because it hints at age with the a, along with a reflection of it extending into the other direction, as an echo of the notion of two equal but opposite universe halves, extending in different directions from the same null point (without necessarily being perfect reflections of each other). 

Extended Consistency Principles

There are (at least) three related principles that, in the following posts, I will refer to as “the consistency principles”:

• the Copernican principle,
• the cosmological principle, and
• the principle of relativity.

 

The Copernican principle is the notion that we “are not privileged observers of the universe” and that, therefore, “observations from the Earth are representative of observations from the average position in the universe”.  In other words, we are not the centre of the solar system, nor the centre of the galaxy, nor the centre of the universe.  And we’re not special (although, for some of us, our mothers still love us).

 

The cosmological principle is more technical than the Copernican principle, but basically says the same thing, that “the spatial distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale”.  In other words, what we, as humans, can view from our non-privileged location in the universe … is not special either.  It’s pretty much the same wherever we look.

 

The principle of relativity is the notion that “the equations describing the laws of physics have the same form in all admissible frames of reference”.  Wikipedia says it’s a “requirement” rather than a notion, but this is a requirement in the sense that for science to advance, for any of our testing and collated observations to make sense, the universe cannot be capricious – with different laws (of mathematics and physics and hence all subsequent scientific laws) in different times or locations (or frames).  It’s called a principle of relativity because it’s a necessary postulate for developing relativity, but it really applies to science as a whole.  Unfortunately, there no succinct principle that states that, but we could call it the “the fundamental laws of nature apply throughout the universe” principle.

 

In a sense, the cosmological principle is a consequence of the “the fundamental laws of nature apply throughout the universe” principle since, if we had no expectation that the laws of nature off in the direction of Polaris would be the same as in our solar system or in the direction of, for example, Sigma Octantis or Rasalhague, then we would have no expectation that space (at a sufficiently large scale) would be the same in those directions.

 

There is an extension of the cosmological principle, called “the perfect cosmological principle” – which (as applied) is anything but, since its application infers a steady-state universe.  However, the notion that a principle applies both spatially and temporally could be applied to the Copernican principle without such problem (and it could be modified, one could say “perfected”, to apply to the cosmological principle without implying a steady-state universe).

 

To clarify, the “perfect” cosmological principle is “an extension of the cosmological principle” that “states that the universe is homogeneous and isotropic in space and time”, and that “the universe looks the same everywhere (on the large scale), the same as it always has and always will” (or that “the observable universe is practically the same at any time and any place”) which therefore “underpins Steady State theory”.  To reword the Wikipedia entry slightly, this misguided principle states that the observable universe apparently never changes (at a sufficiently large scale).

 

There is, however, a possible alternative extension to the cosmological principle – that, at all times during its development, the universe is spatially homogenous and isotropic at sufficiently large scales.  This allows the universe to both change (ie develop) and look different at different times (so long as it remains homogenous and isotropic at appropriate scales).  In a sense, it’s not even an extension to the cosmological principle but rather a mere clarification.  The cosmic microwave background (CMB) is predicted to be isotropic by the ΛCDM model and is shown to be isotropic (“to roughly one part in 100,000”).  The implication here is that the universe has been (homogenous and) isotropic since at least the surface of last scattering (when the CMB was generated).  There’s no reason to expect that the universe was not homogenous and isotropic prior to that.

 

An extension to the cosmological principle implies an extension to the Copernican principle, namely that not only are we “not privileged observers of the universe” but also, we do not inhabit a “privileged era” of the universe.  Note that I don’t mean a “privileged era” here as meaning an era that supports life of our type (that is that stars have progressed sufficiently to create the necessary constituents of our bodies, and those constituents have not been destroyed or scattered too thinly).  What I mean is that we should not observe a universe that is in a special condition, beyond that which is necessary to permit our existence as observers.

 

I was toying with calling the extended version of the cosmological principle the “properly perfect cosmological principle”, but I eventually settled on the more obvious “extended cosmological principle” (with the understanding that the “perfect” cosmological principle would become the “overextended cosmological principle”).  Similarly, the notion that we should not be in a privileged era would become the “extended Copernican principle”.

 

There could also be an “extended principle of relativity”, positing that the laws of nature have always and will always be the same everywhere.  I understand that there could be resistance to this notion as the laws of physics are understood to emerge from the state of the universe – but maybe the “extended principle of relativity” could be thought of as applying since the “phase transition” broke the symmetry of a single primordial unified force into a number of distinct forces.  There were no and could be no observers until well after that event, and thus no science being conducted, so it would be an acceptable limitation of the principle’s applicability.

 

Alternatively, we could accept that certain dimensionless constants might change, but the underlying mathematics of the universe would be consistent throughout.  There is a suite of constants that, when expressed in terms of Planck units, all resolve to unity – but that resolution to unity cannot happen with dimensionless constants.  It’s possible that those constants could vary over time.  Similarly, there are different solutions to quantum mechanics, and perhaps some of those solutions dominate in different eras (such as prior to the phase transition mentioned above).

 

In combination, the extended cosmological principle, the extended Copernican principle and the extended principle of relativity would be the “extended consistency principles”.

 

---

 

The question that one must immediately ask, of course, is whether there is any apparent contravention of any of these extended principles.

 

I would say that there is.

 

If standard cosmology is to be believed we live in a privileged era.  Consider this representation of our cosmological history:

 

 

The top-most rim of the bowl represents the universe as it is today, after 13.7 billion years.  (Ignore the placement of apparent galaxies, they are misrepresented in the illustration in an effort to distinguish between the CMB and the universe of today.)

 

The rest of the bowl shows the Big Bang (unfortunately implying that the Big Bang is the inflationary period), followed by the inflationary epoch (a period of approximately 10^-32 seconds, just prior to which the vacuum had a much higher density than now), the photon epoch, the surface of last scattering (which is effectively the CMB), the cosmological dark age (prior to the ignition of the first stars) and, more recently, since about 5 billion years ago, and accelerated expansion – forming the lip of the bowl.

 

The problem is that, today, the Hubble parameter, which is a measure of the expansion of the universe according to Hubble’s law, is currently equivalent to the inverse age of the universe.

 

This would never have been the case is any past era of the universe and, if the rate at which the universe is expanding is accelerating, will never be the case in any future era.  Which makes right now, just when we are around, able to observe the universe, a privileged era.  There is no reason (that I know of) to think that the value of the Hubble parameter makes it possible for us to inhabit the universe.

 

So, there is an apparent contravention of the extended consistency principles.  Can this be resolved?  I think it can, but I will put that in a separate post.