Paul F. Deisler, Jr., Ph.D.
Abstract
Simple derivations based on Hubble’s Law provide a logical understanding of the homogeneous expansion of our Universe, accessible to non-experts like the author. Four major phenomena are examined: the deceleration/acceleration of spatial expansion; the Hubble Sphere; the Observable Universe; and how a photon moves through expanding space from an emitter to an observer. This article is intended to provide useful background for those wishing to pursue further understanding of our Universe. The methods of derivation and the equations are intended to be useful to those who have additional questions.
Introduction
Interest in how our Universe works, spurred by a paper by Lineweaver and Davis [1] has led me to write this article. I am not a cosmologist and often cannot understand, satisfyingly, papers on the Universe written by cosmologists. However, the popularized writings, while good, leave me without a mathematical understanding of some of the common phenomena of our Universe. This article uses simple derivations based on Hubble’s Law to provide a logical understanding, accessible to non-experts like me, of four major phenomena of our homogeneously expanding universe: the deceleration/acceleration of spatial expansion; the Hubble Sphere; the Observable Universe; and how a photon moves through expanding space from an emitter to an observer. The article also offers useful background for those wishing to pursue further their understanding of our Universe (see Bibliography). The methods of derivation and the equations themselves should be useful to those who have questions of their own to answer.
Hubble’s Law and the Universe
Edwin Hubble’s well-known, deceptively simple law of the expansion of our Universe is
x’ = H(t) x (1)
Here, x is the distance between a pair of points (read also “bodies”) in space in our Universe; x’ is the rate at which the two points separate from each other as the space of our Universe expands (single primes denote first derivatives with respect to time (rates) and double primes denote second derivatives (accelerations)); t denotes time since the beginning of our Universe; and H(t) is the Hubble Constant (also called the Hubble Parameter), constant for any x at time t but possibly changing with t.
Any homogeneously expanding space, such as that of our Universe, obeys Eq. (1), all pairs of points moving away from each other as indicated. In our Universe, as space expands objects within it move along with its expansion. This may be the result of the way in which attractive energies (such as gravity) and repulsive energies (such as dark or other energies) act uniformly throughout space at t, appearing to stretch space between all points within it. In such a space, there is no point we can point to and say, “That is the center where the expansion began.” All points are equivalent; none is unique. I prefer to think of the Big Bang as the “Big Whoosh,” since I think of it not as a single, central “bang” but as a kind of space, filled with a super-compressed energy of some kind, that began expanding at every point within it equally and at the same time [2].
We are living, today, in the ever-expanding remnant of that primordial “whoosh.”
When I speak of space I will mean intergalactic space. Galaxies and their contents are held too tightly together by attractive forces to be able to expand along with space itself.
Acceleration and Deceleration of Spatial Expansion
It has been widely reported that the expansion of our Universe, before some time t0, was decelerating but since t0 it is accelerating. Hubble’s Law allows for this behavior and even points to its possibility. Differentiating Eq. (1) gives the acceleration of spatial expansion,
x’’ = (H2(t) + H’(t))x (2)
At t0, Eq. (1) shows that x’’ = 0. Before t0 it is now known that x’’ < 0 and also that H(t) was decreasing with time. Then, at that time, the absolute magnitude of H’(t) must have been greater than that of H2(t) while at t = t0 the two values in brackets must have been of equal magnitude. With H(t) continuing to decrease with time, as is now thought to be the case, x’’ can become positive as long as the absolute magnitude of H’(t) becomes less than that of H2(t). It is not necessary for H(t) either to become constant or to increase for deceleration to become acceleration. It is only necessary that the rate of decrease of H(t) become more appropriately sedate.
The Hubble Radius and the Hubble Sphere
Bodies (or objects of whatever kind or even points) in a homogeneously expanding space all have motion with the expansion of space and can also have motion through expanding space. The laws of physics can be formulated in any consistent set of coordinates. The most convenient coordinates for locating points in an expanding space with respect to an observer in that space are co-moving coordinates. Co-moving coordinates expand with space so that the coordinates of a point having no motion through space will not change even though distances between points will change as space expands. Motion of a point with space always has zero velocity with respect to its fixed, co-moving coordinates. Relativity therefore does not apply to such zero-velocity motion. Relativity effects do apply to motion through space, however. Thus, when it comes to motion with space, velocities of points can exceed that of light, c, without contradiction.
A photon, emitted by an emitter approaches an observer. The photon moves toward the observer through space at velocity c and away from the observer with space at Hubble velocity H(t)z, where z is the decreasing distance between the photon and the observer at t. The net velocity of the photon toward the observer (in this paper, the negative direction) is
z‘ = -(c - H(t)z) (3)
The constancy of c is not violated, since any scientist stationed along the photon’s trajectory will note the photon’s velocity as c. However, the scientist and the photon will both also have the positive Hubble velocity component with respect to (and away from) the observer.
When z‘ is zero, the two velocities are equal and z becomes the Hubble Radius, rH
rH = c/H(t) (4)
and rH defines a Hubble Sphere around an observer. In the space in a Hubble Sphere all points expand away from the observer at velocities less than c.
The usual way to derive the Hubble Radius is to substitute c for x’ in Eq. (1), whereupon x becomes rH. The first method of derivation better exposes the roles of the photon and spatial expansion relative to the observer.
Differentiating Eq. (4) yields the rate r’H at which rH changes with time which may be written as
r’H = cR (5)
where R, a term that will be useful later, is defined as
R = -H’(t)/H2(t) (6)
Comparing Eqs. (2) and (6) with H(t) decreasing with t, for t < t0, R > 1; for t = t0, R = 1; and for t > t0, R > 1. However, this also shows that with t < t0, the surface of a Hubble Sphere moves away from its observer faster than the speed of light while objects at the surface move with space at the speed of light. Such a Sphere engulfs more and more space previously outside of itself, adding more bodies to the content of its space. Similarly, the content of a Hubble Sphere at t = t0 remains constant, and content is lost from a Hubble Sphere after t0. This effect tends to make our own Hubble Sphere sparser in observable bodies.
If our decreasing H(t) were to decline to a constant value, H’(t) would become zero as would R and rH would become constant but the Universe would continue to expand. Content would be lost at a higher rate from such a Hubble Sphere. If, however, H(t) were to begin to increase, rH would decrease, the Hubble Sphere would shrink but content would continue to be lost. Of these latter two possibilities, neither is now known to be possible.
The Observable Universe
A photon moves toward the observer. Let the Hubble Radius move away from the observer. The net velocity of the expanding Hubble Sphere and the moving photon will therefore be r’H - z’ or cR + c - H(t)z. As the Hubble Sphere expands, it engulfs photons moving toward the observer but emitted outside of its boundary. These photons thus enter a space where all points flee the observer at less than c and so the photons can proceed to the observer. In this case, when cR + c - H(t)z = 0, z = rob, the radius of the Observed Universe. Solving for z = rob, substituting and rearranging,
rob = rH (1 + R) (7)
From Eq. (7), for t = t0, R = 1 and rob = 2rH; for t < t0, R > 1 and rob is larger than 2rH; and for t < t0, our current Universe, R < 1 and (with H(t) still decreasing) rob < 2rH.
In our current Universe, then, 0 < R < 1. The Deceleration Parameter, q, is just R – 1 so that 1 < q < 0. Various authors have written about q. Choosing one of the values of q put forward [3], q = -0.55, and with a current value [4] of H(t) of 0.072 Gy-1, rH = 13.9 Gly and robs = 20.1 Gly. (Units used in this paper are, for length, billions of light-years (or giga-light-years, Gly) and, for time, billions of years (or gigayears, Gy). The difference between the two radii in this example is far from trivial.
Differentiating Eq. (7) yields the rate of increase of rob with t.
r’OB = cR(1 + R) + rHR’ (8)
To learn whether the Observable Universe gains or loses content at various times, r’OB must be compared with the Hubble rate. r’OBH = H(t)rOB . Thus,
r’OBH.= c(1 + R) (9)
and r’OB - r’OBH. = c(R2 - 1) + rHR’ (10)
Eqs. (7) and (10), together, indicate whether the Observable Universe gains or loses content, for all instances in which H(t) is decreasing. Since R decreases with t in these instances, then R’ < 0 but its magnitude is unknown. Thus, the early Observable Universe gained content but at some time before t0 it began to lose content as is the case today of our Observable Universe. Of the other cases the one worth mentioning is that for which H(t) is constant. Only then are the Hubble Sphere and the sphere of the Observable Universe the same (Eq. (7)).
Statements are still made that emitters so far from an observer that their radiations, traveling at c, cannot have reached the observer in a time equal to the age of the Universe4 (13.7 Gy) offer another definition of an “observable” Universe. This is not true. The radiation from an emitter within an Observable Universe (as above) but far enough from an observer that the light from the emitter, traveling through expanding space, has not had time to reach the observer within the age of the Universe will still arrive at the observer (who may or may not still be present) at a still later age. Radiations from emitters outside of the sphere of the Observable Universe will never arrive at the position, in co-moving coordinates, that an observer once occupied.
How A Photon Moves Through Expanding Space from Emitter to Observer
At time t, let the distance from an observer to an emitter be x; the distance from a photon, traveling toward an observer, to the observer be z; and the distance from the emitter of the photon to the photon be s. The rate z’ at which z decreases as the photon approaches the observer is given by Eq. (3). Since x = z +s, then
s’ = x’ – z’ (12)
Then from Eqs. (1), (3) and (12),
s‘ = H(t)s + c (13)
The function H(t) is not known over a wide range of t. However, with a time interval t2 - t1 small compared to the age of the Universe (where t2 > t1) a constant value of H(t) may be used, H, as an approximation. Since s1 = 0 and z2 = 0 (and x1 = z1), the solutions of Eqs. (1), (3) and (13) are
ln(s2 /z1) = H(t2 - t1) (14)
ln c/(c - Hz1) = H(t2 - t1) (15)
and ln(Hs2 + c)/c = H(t2 - t1) (16)
From Eqs. (14) and (16),
s2 = cz1/(c – H z1) (17)
and from Eq. (14)
t2 - t1 = (1/H)ln(s2/z1) (18)
As an example, with t1,= 13.7 Gy, c = 1 Gly/Gy, H = .072 Gy-1, x1 = z1 = 1 Gly and s1 = 0, a photon from the emitter will reach the observer 1.0378 Gy later, when the Universe is 14.7378 Gy old. By that time space between emitter and observer will have stretched from 1 Gly to 1.0776 Gly (= x2= s2) and s (not x) will have increased at an average rate of 1.0383 Gly/Gy, which is faster than c (consistent with Eq. (13)). The photon travels at c with respect to its local space. If the photon were equipped with an odometer the distance the odometer would show the photon had traveled would be c(t2 - t1) or 1.0378 Gly, which is less than s2. The remaining 0.0398 Gly of s2 arises from the stretching of space behind the photon as it moves away from the emitter. Subtracting this latter figure from the total expansion of space between emitter and observer of 0.0776 Gly, the total stretching of space ahead of the photon, which it must traverse at local velocity c, is 0.0378 Gly. This example has uncovered yet another aspect of the workings of our Universe. Despite some apparent strangeness, it all adds up.
Comments
The deceptively simple Hubble’s Law, combined with the way photons travel through and with expanding space, has shed light on the workings of our remarkable Universe. However, the most discouraging yet most challenging aspect of our Universe is that we live in a cage. The Observable Universe may be a very large cage, but it is still a cage. Our inability to penetrate the spherical wall of our cage is not caused by lack of power or precision in our methods. It is a fundamental fact. Only if we can learn something germane to the question of “what is out there?” from the space and energies within our Observable Sphere, or if we find other means of extracting information from the space “out there” will we achieve any penetration of what is now an impenetrable barrier.
