### Expansion of space

The **metric expansion of space** is the increase of the distance between two distant parts of the universe with time. It is an intrinsic expansion whereby *the scale of space itself is changed*. That is, a metric expansion is defined by an increase in distance between parts of the universe even without those parts "moving" anywhere. This is not the same as any usual concept of motion, or any kind of expansion of objects "outward" into other "preexisting" space, or any kind of explosion of matter which is commonly experienced on earth.

Metric expansion is a key feature of Big Bang cosmology and is modeled mathematically with the FLRW metric. This model is valid in the present era only on large scales (roughly the scale of galaxy clusters and above). At smaller scales matter has become bound together under the influence of gravitational attraction and such bound objects clumps do not expand at the metric expansion rate as the universe ages, though they continue to recede from one another. The expansion is a generic property of the universe we inhabit, though the reason we are expanding is explained by most cosmologists as having its origin in the end of the early universe's inflationary period which set matter and energy in the universe on an inertial trajectory consistent with the equivalence principle and Einstein's theory of general relativity (that is, the matter in the universe is separating because it was separating in the past). Additionally, the expansion rate of the universe has been measured to be accelerating; to explain this, physicists postulate a repulsive force of dark energy, which appears in the theoretical models as a cosmological constant. This acceleration of the universe has only recently become measurable. Due to recent measurements, it is now thought that up until about five billion years ago, the universe's expansion rate was actually *decelerating* due to the gravitational attraction of the matter content of the universe. According to the simplest extrapolation of the currently-favored cosmological model (known as "ΛCDM"), however, the dark energy acceleration will dominate on into the future.

While special relativity constrains objects in the universe from moving faster than the speed of light with respect to each other, it places no theoretical constraint on changes to the scale of space itself. It is thus possible for two objects to be stationary or moving at speeds below that of light, and yet to become separated in space by more than the distance light could have travelled, which can suggest the objects travelled faster than light. For example there are stars which may be expanding away from us (or each other) faster than the speed of light, and this is true for any object that is more than approximately 4.5 gigaparsecs away from us. We can still see such objects because the universe in the past was expanding more slowly than it is today, so the ancient light being received from these objects is still able to reach us, though if the expansion continues unabated there will come a time that we will never see the light from such objects being produced *today* (on a so-called "space-like slice of spacetime") because space itself is expanding between Earth and the source faster than their light can reach us.

Because of the changing rate of expansion, it is also possible for a distance to exceed the value calculated by multiplying the speed of light by the age of the universe. These details are a frequent source of confusion among amateurs and even professional physicists.^{[1]}

Due to the non-intuitive nature of the subject and what has been described by some as "careless" choices of wording, certain descriptions of the metric expansion of space and the misconceptions to which such descriptions can lead are an ongoing subject of discussion in the realm of pedagogy and communication of scientific concepts.^{[2]}^{[3]}^{[4]}^{[5]}

## Contents

- 1 Basic concepts and overview
- 2 Understanding the expansion of Universe
- 2.1 How is the expansion of the universe measured and how does the rate of expansion change?
- 2.2 How are distances between two points measured if space is expanding?
- 2.3 What space is the universe expanding into?
- 2.4 Is the expansion of the universe felt on small scales?
- 2.5 Scale factor
- 2.6 Other conceptual models of expansion

- 3 Overview of metrics
- 4 Theoretical basis and first evidence
- 5 Observational evidence
- 6 Notes
- 7 Printed references
- 8 External links

## Basic concepts and overview

### Metric expansion

The metric expansion of space is a key part of science's current understanding of the universe, whereby space itself is described by a metric tensor which changes over time. It explains how the universe expands in the Big Bang model, a feature of our universe supported by all cosmological experiments, astrophysics calculations, and measurements to date.

This kind of expansion is different from all kinds of expansions and explosions commonly seen in nature. What we see normally as "space" and "distance" are not absolutes, but are determined by a metric that can change. In the metric expansion of space, rather than objects in a fixed "space" moving apart into "emptiness", it is space itself which is changing. It is as if without objects themselves moving, space is somehow growing or shrinking between them: if it were possible to place a tape measure between even stationary objects, one would observe the scale of the tape measure changing to show more distance between them.

Because this expansion is caused by changes in the distance-defining metric, and not by objects themselves moving in space, this expansion (and the resultant movement apart of objects) is not restricted by the speed of light upper bound of special relativity. So objects can be moving at sub-light speed yet appear to be moving apart faster than light.

Theory and observations suggest that very early in the history of the universe, there was an "inflationary" phase where this metric changed very rapidly, and that the remaining time-dependence of this metric is what we observe as the so-called Hubble expansion, the moving apart of all gravitationally unbound objects in the universe. The expanding universe is therefore a fundamental feature of the universe we inhabit - a universe fundamentally different from the static universe Albert Einstein first considered when he developed his gravitational theory.

### Measuring distances in expanding spaces

A *metric* defines how a distance can be measured between two points in space. In Euclidean geometry, this distance can be measured by tracking a straight line between two points. However, in non-Euclidean geometry, the notion of a "straight line" or "distance" may not be the same, and the notion of "distance" varies depending upon the actual metric involved. In this sense, a metric is a generalization of the concept of "distance" for other geometries.

In expanding space, proper distances are dynamical quantities which change with time. An easy way to correct for this is to use comoving coordinates which remove this feature and allow for a characterization of the universe as a whole without having to characterize the physics associated with metric expansion. In comoving coordinates, the distances between all objects are fixed and the instantaneous dynamics of matter and light are determined by the normal physics of gravity and electromagnetic radiation. Any time-evolution however must be accounted for by taking into account the Hubble law expansion in the appropriate equations. Cosmological simulations that run through significant fractions of the universe's history therefore must be able to work in physical units which can directly predict observational cosmology.

## Understanding the expansion of Universe

### How is the expansion of the universe measured and how does the rate of expansion change?

In principle, the expansion of the universe can be measured by taking a standard ruler and measuring the distance between two cosmologically distant points, waiting a certain time, and then measuring the distance again. In practice, standard rulers are not straightforward to find on cosmological scales and the time-scales for waiting to see a measurable expansion of the universe today are too long to be observable by even generations of humans. Instead, the theory of relativity predicts and observations show phenomena associated with the expansion of the universe, notably the redshift-distance relationship known as Hubble's Law, functional forms for cosmological distance measurements that differ from what would be expected if space were not expanding, and an observable change in the matter and energy density of the universe seen at different lookback times.

The first measurement of the expansion of space occurred with the creation of the Hubble diagram. Using standard candles with known intrinsic brightness, the expansion of the universe has been measured using redshift to derive Hubble's Constant: H_{0} = 67.15 ± 1.2 (km/s)/Mpc. For every million parsecs of distance from the observer, the rate of expansion increases by about 67 kilometers per second.^{[6]}^{[7]}^{[8]} Since distant objects are observed further back in time, there is a one-to-one correspondence between the distance to a distant galaxy and the amount of time that has passed since the light being observed was emitted from that galaxy. Thus, Hubble's Constant can be thought of as an acceleration which in the local universe is equivalent to approximately $7\; \backslash times\; 10^\{-10\}$ meters per second per second, though this value is on large-scales dependent on how one defines the distance between two points and how one measures the elapsed time. Cosmologists often adopt comoving coordinates which remove the expansion altogether.

The acceleration of objects moving away from each other in an expanding universe is not the sort of acceleration which can be associated with a force as in Newton's Second Law because the expansion is an intrinsic property of the way space and time are measured rather than being due to dynamical interactions. Nevertheless, because the dimensional form of Hubble's Constant can yield an acceleration this has caused some confusion associated with the so-called "accelerating universe" which was first discovered and characterized in the late 1990s. In a universe that is undergoing a constant Hubble expansion, the universal Hubble Constant can be conceptualized as a universal acceleration, but Hubble's Constant is not constant through time since there are dynamical forces acting on the particles in the universe which affect the expansion rate. It was expected that the Hubble Constant would be decreasing as time went on due to the influence of gravitational interactions in the universe, and thus there is an additional observable quantity in the universe called the deceleration parameter which cosmologists expected to be directly related to the matter density of the universe. Surprisingly, the deceleration parameter was measured by two different groups to be less than zero (actually, consistent with -1) which implied that today Hubble's Constant is increasing as time goes on. Since Hubble's Constant can be associated with an acceleration, the change in Hubble's Constant over time can be associated with the time derivative of acceleration, and so some cosmologists have whimsically called the effect associated with the "accelerating universe" the "cosmic jerk".^{[9]} The 2011 Nobel Prize in Physics was given for the discovery of this phenomenon.^{[10]}

### How are distances between two points measured if space is expanding?

At cosmological scales the present universe is geometrically flat, which is to say that the rules of Euclidean geometry associated with Euclid's fifth postulate hold, though in the past spacetime could have been highly curved. In part to accommodate such different geometries, the expansion of the universe is inherently general relativistic; it cannot be modeled with special relativity alone, though such models can be written down, they are at fundamental odds with the observed interaction between matter and spacetime seen in our universe.

The images to the right show two views of spacetime diagrams that show the large-scale geometry of the universe according to the ΛCDM cosmological model. Two of the dimensions of space are omitted, leaving one dimension of space (the dimension that grows as the cone gets larger) and one of time (the dimension that proceeds "up" the cone's surface). The narrow circular end of the diagram corresponds to a cosmological time of 700 million years after the big bang while the wide end is a cosmological time of 18 billion years, where one can see the beginning of the accelerating expansion as a splaying outward of the spacetime, a feature which eventually dominates in this model. The purple grid lines mark off cosmological time at intervals of one billion years from the big bang. The cyan grid lines mark off comoving distance at intervals of one billion light years. Note that the circular curling of the surface is an artifact of the embedding with no physical significance and is done purely to make the illustration viewable; space does not actually curl around on itself. (A similar effect can be seen in the tubular shape of the pseudosphere.)

The brown line on the diagram is the worldline of the Earth (or, at earlier times, of the matter which condensed to form the Earth). The yellow line is the worldline of the most distant known quasar. The red line is the path of a light beam emitted by the quasar about 13 billion years ago and reaching the Earth in the present day. The orange line shows the present-day distance between the quasar and the Earth, about 28 billion light years, which is, notably, a larger distance than the age of the universe multiplied by the speed of light: *ct*.

According to the equivalence principle of general relativity, the rules of special relativity are *locally* valid in small regions of spacetime that are approximately flat. In particular, light always travels locally at the speed *c*; in our diagram, this means, according to the convention of constructing spacetime diagrams, that light beams always make an angle of 45° with the local grid lines. It does not follow, however, that light travels a distance *ct* in a time *t*, as the red worldline illustrates. While it always moves locally at *c*, its time in transit (about 13 billion years) is not related to the distance traveled in any simple way since the universe expands as the light beam traverses space and time. In fact the distance traveled is inherently ambiguous because of the changing scale of the universe. Nevertheless, we can single out two distances which appear to be physically meaningful: the distance between the Earth and the quasar when the light was emitted, and the distance between them in the present era (taking a slice of the cone along the dimension that we've declared to be the spatial dimension). The former distance is about 4 billion light years, much smaller than *ct* because the universe expanded as the light traveled the distance, the light had to "run against the treadmill" and therefore went farther than the initial separation between the Earth and the quasar. The latter distance (shown by the orange line) is about 28 billion light years, much larger than *ct*. If expansion could be instantaneously stopped today, it would take 28 billion years for light to travel between the Earth and the quasar while if the expansion had stopped at the earlier time, it would have taken only 4 billion years.

The light took much longer than 4 billion years to reach us though it was emitted from only 4 billion light years away, and, in fact, the light emitted towards the Earth was actually moving *away* from the Earth when it was first emitted, in the sense that the metric distance to the Earth increased with cosmological time for the first few billion years of its travel time, and also indicating that the expansion of space between the Earth and the quasar at the early time was faster than the speed of light. None of this surprising behavior originates from a special property of metric expansion, but simply from local principles of special relativity integrated over a curved surface.

### What space is the universe expanding into?

Over time, the space that makes up the universe is expanding. The words 'space' and 'universe', sometimes used interchangeably, have distinct meanings in this context. Here 'space' is a mathematical concept that stands for the three-dimensional manifold into which our respective positions are embedded while 'universe' refers to everything that exists including the matter and energy in space, the extra-dimensions that may be wrapped up in various strings, and the time through which various events take place. The expansion of space is in reference to this 3-D manifold only; that is, the description involves no structures such as extra dimensions or an exterior universe.^{[11]}

The ultimate topology of space is something which in principle must be observed as there are no *a priori* constraints on how the space in which we live is connected or whether it wraps around on itself as a compact space. Though certain cosmological models such as Gödel's universe even permit bizarre worldlines which intersect with themselves, ultimately the question as to whether we are in something like a "pac-man universe" where if traveling far enough in one direction would allow one to simply end up back in the same place like going all the way around the surface of a balloon (or a planet like the Earth) is an observational question which is constrained as measurable or non-measurable by the universe's global geometry. At present, observations are consistent with the universe being infinite in extent and simply connected, though we are limited in distinguishing between simple and more complicated proposals by cosmological horizons. The universe could be infinite in extent or it could be finite; but the evidence that leads to the inflationary model of the early universe also implies that the "total universe" is much larger than the observable universe, and so any edges or exotic geometries or topologies would not be directly observable as light has not reached scales on which such aspects of the universe, if they exist, are still allowed. For all intents and purposes, it is safe to assume that the universe is infinite in spatial extent, without edge or strange connectedness.^{[12]}

Regardless of the overall shape of the universe, the question of what the universe is expanding into is one which does not require an answer according to the theories which describe the expansion; the way we define space in our universe in no way requires additional exterior space into which it can expand since an expansion of an infinite expanse can happen without changing the infinite extent of the expanse. All that is certain is that the manifold of space in which we live simply has the property that the distances between objects are getting larger as time goes on. This only implies the simple observational consequences associated with the metric expansion explored below. No "outside" or embedding in hyperspace is required for an expansion to occur. The visualizations often seen of the universe growing as a bubble into nothingness are misleading in that respect. There is no reason to believe there is anything "outside" of the expanding universe into which the universe expands.

Even if the overall spatial extent is infinite and thus the universe can't get any "larger", we still say that space is expanding because, locally, the characteristic distance between objects is increasing. As an infinite space grows, it remains infinite.

### Is the expansion of the universe felt on small scales?

The expansion of space is sometimes described as a force which acts to push objects apart. Though this is an accurate description of the effect of the cosmological constant, it is not an accurate picture of the phenomenon of expansion in general. For much of the universe's history the expansion has been due mainly to inertia. The matter in the very early universe was flying apart for unknown reasons (most likely as a result of cosmic inflation) and has simply continued to do so, though at an ever-decreasing rate due to the attractive effect of gravity.

In addition to slowing the overall expansion, gravity causes local clumping of matter into stars and galaxies. Once objects are formed and bound by gravity, they "drop out" of the expansion and do not subsequently expand under the influence of the cosmological metric, there being no force compelling them to do so.

There is no difference between

- the inertial expansion of the universe and
- the inertial separation of nearby objects in a vacuum;

the former is simply a large-scale extrapolation of the latter.

Once objects are bound by gravity, they no longer recede from each other. Thus, the Andromeda galaxy, which is bound to the Milky Way galaxy, is actually falling *towards* us and is not expanding away. Within our Local Group of galaxies, the gravitational interactions have changed the inertial patterns of objects such that there is no cosmological expansion taking place. Once one goes beyond the local group, the inertial expansion is measurable, though systematic gravitational effects imply that larger and larger parts of space will eventually fall out of the "Hubble Flow" and end up as bound, non-expanding objects up to the scales of superclusters of galaxies. We can predict such future events by knowing the precise way the Hubble Flow is changing as well as the masses of the objects to which we are being gravitationally pulled. Currently, our Local Group is being gravitationally pulled towards either the Shapley Supercluster or the "Great Attractor" with which, if dark energy were not acting, we would eventually merge and no longer see expand away from us after such a time.

An interesting consequence of metric expansion being due to inertial motion is that a uniform local "explosion" of matter into a vacuum can be locally described by the FLRW geometry, the same geometry which describes the expansion of the universe as a whole and was also the basis for the simpler Milne universe which ignores the effects of gravity. In particular, general relativity predicts that light will move at the speed *c* with respect to the local motion of the exploding matter, a phenomenon analogous to frame dragging.

The situation changes somewhat with the introduction of dark energy or a cosmological constant. A cosmological constant due to a vacuum energy density has the effect of adding a repulsive force between objects which is proportional (not inversely proportional) to distance. Unlike inertia it actively "pulls" on objects which have clumped together under the influence of gravity, and even on individual atoms. However, this does not cause the objects to grow steadily or to disintegrate; unless they are very weakly bound, they will simply settle into an equilibrium state which is slightly (undetectably) larger than it would otherwise have been. As the universe expands and the matter in it thins, the gravitational attraction decreases (since it is proportional to the density), while the cosmological repulsion increases; thus the ultimate fate of the ΛCDM universe is a near vacuum expanding at an ever increasing rate under the influence of the cosmological constant. However, the only locally visible effect of the accelerating expansion is the disappearance (by runaway redshift) of distant galaxies; gravitationally bound objects like the Milky Way do not expand and the Andromeda galaxy is moving fast enough towards us that it will still merge with the Milky Way in 3 billion years time, and it is also likely that the merged supergalaxy that forms will eventually fall in and merge with the nearby Virgo Cluster. However, galaxies lying farther away from this will recede away at ever-increasing rates of speed and be redshifted out of our range of visibility.

### Scale factor

At a fundamental level, the expansion of the universe is a property of spatial measurement on the largest measurable scales of our universe. The distances between cosmologically relevant points increases as time passes leading to observable effects outlined below. This feature of the universe can be characterized by a single parameter that is called the scale factor which is a function of time and a single value for all of space at any instant (if the scale factor were a function of space, this would violate the cosmological principle). By convention, the scale factor is set to be unity at the present time and, because the universe is expanding, is smaller in the past and larger in the future. Extrapolating back in time with certain cosmological models will yield a moment when the scale factor was zero, our current understanding of cosmology sets this time at 13.798 ± 0.037 billion years ago. If the universe continues to expand forever, the scale factor will approach infinity in the future. In principle, there is no reason that the expansion of the universe must be monotonic and there are models that exist where at some time in the future the scale factor decreases with an attendant contraction of space rather than an expansion.

### Other conceptual models of expansion

The expansion of space is often illustrated with conceptual models which show only the size of space at a particular time, leaving the dimension of time implicit.

In the "ant on a rubber rope model" one imagines an ant (idealized as pointlike) crawling at a constant speed on a perfectly elastic rope which is constantly stretching. If we stretch the rope in accordance with the ΛCDM scale factor and think of the ant's speed as the speed of light, then this analogy is numerically accurate—the ant's position over time will match the path of the red line on the embedding diagram above.

In the "rubber sheet model" one replaces the rope with a flat two-dimensional rubber sheet which expands uniformly in all directions. The addition of a second spatial dimension raises the possibility of showing local perturbations of the spatial geometry by local curvature in the sheet.

In the "balloon model" the flat sheet is replaced by a spherical balloon which is inflated from an initial size of zero (representing the big bang). A balloon has positive Gaussian curvature while observations suggest that the real universe is spatially flat, but this inconsistency can be eliminated by making the balloon very large so that it is locally flat to within the limits of observation. This analogy is potentially confusing since it wrongly suggests that the big bang took place at the center of the balloon. In fact points off the surface of the balloon have no meaning, even if they were occupied by the balloon at an earlier time.

In the "raisin bread model" one imagines a loaf of raisin bread expanding in the oven. The loaf (space) expands as a whole, but the raisins (gravitationally bound objects) do not expand; they merely grow farther away from each other.

All of these models have the conceptual problem of requiring an outside force acting on the "space" at all times to make it expand. Unlike real cosmological matter, sheets of rubber and loaves of bread are bound together electromagnetically and will not continue to expand on their own after an initial tug.

## Overview of metrics

Metric expansion is not something that most humans are aware of, on a day to day basis. To understand the expansion of the universe, it is helpful to discuss briefly what a metric is, and how metric expansion works.

### Definition of a metric

A **metric** defines how a distance can be measured between two *nearby* points in space, in terms of the coordinates of those points. A coordinate system locates points in a space (of whatever number of dimensions) by assigning unique numbers known as coordinates, to each point. The metric is then a formula which converts coordinates of two points into distances.

### Metric for Earth's surface

For example, consider the measurement of distance between two places on the surface of the Earth. This is a simple, familiar example of spherical geometry. Because the surface of the Earth is two-dimensional, points on the surface of the earth can be specified by two coordinates—for example, the latitude and longitude. Specification of a metric requires that one first specify the coordinates used. In our simple example of the surface of the Earth, we could choose any kind of coordinate system we wish, for example latitude and longitude, or X-Y-Z Cartesian coordinates. Once we have chosen a specific coordinate system, the numerical values of the coordinates of any two points are uniquely determined, and based upon the properties of the space being discussed, the appropriate metric is mathematically established too. On the curved surface of the Earth, we can see this effect in long-haul airline flights where the distance between two points is measured based upon a Great circle, and not along the straight line that passes through the Earth. While there is always an effect due to this curvature, at short distances the effect is so small as to be unnoticeable.

## Theoretical basis and first evidence

### Hubble's law

Technically, the metric expansion of space is a feature of many solutions to the Einstein field equations of general relativity, and distance is measured using the Lorentz interval. This explains observations which indicate that galaxies that are more distant from us are receding faster than galaxies that are closer to us (Hubble's law).

### Cosmological constant and the Friedmann equations

The first general relativistic models predicted that a universe which was dynamical and contained ordinary gravitational matter would contract rather than expand. Einstein's first proposal for a solution to this problem involved adding a cosmological constant into his theories to balance out the contraction, in order to obtain a static universe solution. But in 1922 Alexander Friedmann derived a set of equations known as the Friedmann equations, showing that the universe might expand and presenting the expansion speed in this case.^{[13]} The observations of Edwin Hubble in 1929 suggested that distant galaxies were all apparently moving away from us, so that many scientists came to accept that the universe was expanding.

### Hubble's concern over the large expansion rate

While the metric expansion of space is implied by Hubble's 1929 observations, Hubble was concerned with the observational implications of the precise value he measured:

"… if redshift are not primarily due to velocity shift … the velocity-distance relation is linear, the distribution of the nebula is uniform, there is no evidence of expansion, no trace of curvature, no restriction of the time scale … and we find ourselves in the presence of one of the principle of nature that is still unknown to us today … whereas, if redshifts are velocity shifts which measure the rate of expansion, the expanding models are definitely inconsistent with the observations that have been made … expanding models are a forced interpretation of the observational results"— E. Hubble,

Ap. J., 84, 517, 1936^{[14]}

"[If the redshifts are a Doppler shift] … the observations as they stand lead to the anomaly of a closed universe, curiously small and dense, and, it may be added, suspiciously young. On the other hand, if redshifts are not Doppler effects, these anomalies disappear and the region observed appears as a small, homogeneous, but insignificant portion of a universe extended indefinitely both in space and time."— E. Hubble,

Monthly Notices of the Royal Astronomical Society, 97, 513, 1937^{[15]}

In fact, Hubble's skepticism about the universe being too small, dense, and young was justified, though it turned out to be an observational error rather than an error of interpretation. Later investigations showed that Hubble had confused distant HII regions for Cepheid variables and the Cepheid variables themselves had been inappropriately lumped together with low-luminosity RR Lyrae stars causing calibration errors that led to a value of the Hubble Constant of approximately 500 km/s/Mpc instead of the true value of approximately 70 km/s/Mpc. The higher value meant that an expanding universe would have an age of 2 billion years (younger than the Age of the Earth) and extrapolating the observed number density of galaxies to a rapidly expanding universe implied a mass density that was too high by a similar factor, enough to force the universe into a peculiar closed geometry which also implied an impending Big Crunch that would occur on a similar time-scale. After fixing these errors in the 1950s, the new lower values for the Hubble Constant accorded with the expectations of an older universe and the density parameter was found to be fairly close to a geometrically flat universe.^{[16]}

### Inflation as an explanation for expansion

While the detailed measurements of the precise rate of expansion were being worked out, the notion of the expansion of the universe became consensus. Until the theoretical developments in the 1980s no one had an explanation for why this seemed to be the case, but with the development of models of cosmic inflation, the expansion of the universe became a general feature resulting from vacuum decay. Accordingly, the question "why is the universe expanding?" is now answered by understanding the details of the inflation decay process which occurred in the first 10^{−32} seconds of the existence of our universe. It is suggested that in this time the metric itself changed exponentially, causing space to change from smaller than an atom to around 100 million light years across.

### Measuring distance in a metric space

In expanding space, distance is a dynamic quantity which changes with time. There are several different ways of defining distance in cosmology, known as *distance measures*, but the most common is **comoving distance**.

The metric only defines the distance between nearby points. In order to define the distance between arbitrarily distant points, one must specify both the points and a specific curve connecting them. The distance between the points can then be found by finding the length of this connecting curve. Comoving distance defines this connecting curve to be a curve of constant cosmological time. Operationally, comoving distances cannot be directly measured by a single Earth-bound observer. To determine the distance of distant objects, astronomers generally measure luminosity of standard candles, or the redshift factor 'z' of distant galaxies, and then convert these measurements into distances based on some particular model of space-time, such as the Lambda-CDM model. Unfortunately, there is no evidence for any 'slowing down' of the expansion.

## Observational evidence

Theoretical cosmologists developing models of the universe have drawn upon a small number of reasonable assumptions in their work. These workings have led to models in which the metric expansion of space is a likely feature of the universe. Chief among the underlying principles that result in models including metric expansion as a feature are:

- the Cosmological Principle which demands that the universe looks the same way in all directions (isotropic) and has roughly the same smooth mixture of material (homogeneous).
- the Copernican Principle which demands that no place in the universe is preferred (that is, the universe has no "starting point").

Scientists have tested carefully whether these assumptions are valid and borne out by observation. Observational cosmologists have discovered evidence - very strong in some cases - that supports these assumptions, and as a result, metric expansion of space is considered by cosmologists to be an observed feature on the basis that although we cannot see it directly, scientists have tested the properties of the universe and observation provides compelling confirmation.^{[17]} Sources of this confidence and confirmation include:

- Hubble demonstrated that all galaxies (with the exception of a few very near ones) and distant astronomical objects were moving away from us, as predicted by a universal expansion.
^{[18]}Using the redshift of their electromagnetic spectra to determine the distance and speed of remote objects in space, he showed that the speed of such distant objects is roughly proportional to their distance, a feature of metric expansion. Further studies have since shown the expansion to be extremely isotropic and homogeneous, that is, it does not seem to have a special point as a "center", but appears universal and independent of any fixed central point. - In studies of large-scale structure of the cosmos taken from redshift surveys a so-called "End of Greatness" was discovered at the largest scales of the universe. Until these scales were surveyed, the universe appeared "lumpy" with clumps of galaxy clusters and superclusters and filaments which were anything but isotropic and homogeneous. This lumpiness disappears into a smooth distribution of galaxies at the largest scales.
- The isotropic distribution across the sky of distant gamma-ray bursts and supernovae is another confirmation of the Cosmological Principle.
- The Copernican Principle was not truly tested on a cosmological scale until measurements of the effects of the cosmic microwave background radiation on the dynamics of distant astrophysical systems were made. A group of astronomers at the European Southern Observatory noticed, by measuring the temperature of a distant intergalactic cloud in thermal equilibrium with the cosmic microwave background, that the radiation from the Big Bang was demonstrably warmer at earlier times.
^{[19]}Uniform cooling of the cosmic microwave background over billions of years is strong and direct observational evidence for metric expansion.

Taken together, these phenomena overwhelmingly support models that rely on space expanding through a change in metric. Interestingly, it was not until the discovery in the year 2000 of direct observational evidence for the changing temperature of the cosmic microwave background that more bizarre constructions could be ruled out. Until that time, it was based purely on an assumption that the universe did not behave as one with the Milky Way sitting at the middle of a fixed-metric with a universal explosion of galaxies in all directions (as seen in, for example, an early model proposed by Milne). Yet before this evidence, many rejected the Milne viewpoint based on the mediocrity principle.

The spatial and temporal universality of physical laws was until very recently taken as a fundamental philosophical assumption that is now tested to the observational limits of time and space.

## Notes

## Printed references

- Eddington, Arthur.
*The Expanding Universe: Astronomy's 'Great Debate', 1900-1931*. Press Syndicate of the University of Cambridge, 1933. - Liddle, Andrew R. and David H. Lyth.
*Cosmological Inflation and Large-Scale Structure*. Cambridge University Press, 2000. - Lineweaver, Charles H. and Tamara M. Davis, "Scientific American, March 2005.
- Mook, Delo E. and Thomas Vargish.
*Inside Relativity*. Princeton University Press, 1991.

## External links

- Swenson, Jim Answer to a question about the expanding universe
- Felder, Gary, "The Expanding universe".
- Explanation of the universal expansion" at a very elementary level
- Hubble Tutorial from the University of Wisconsin Physics Department
- Expanding raisin bread from the University of Winnipeg: an illustration, but no explanation
- "Ant on a balloon" analogy to explain the expanding universe at "Ask an Astronomer". (The astronomer who provides this explanation is not specified.)
- Researched Essay: "The Big Bang" - Proof that the Universe is Expanding

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