Metric expansion of space

2007 Schools Wikipedia Selection. Related subjects: Space (Astronomy)

Physical cosmology
  • Age of the universe
  • Big Bang
  • Comoving distance
  • Cosmic microwave background
  • Dark energy
  • Dark matter
  • FLRW metric
  • Friedmann equations
  • Galaxy formation
  • Hubble's law
  • Inflation
  • Large-scale structure
  • Lambda-CDM model
  • Metric expansion of space
  • Nucleosynthesis
  • Observable universe
  • Redshift
  • Shape of the universe
  • Structure formation
  • Timeline of the Big Bang
  • Timeline of cosmology
  • Ultimate fate of the universe
  • Universe
Related topics
  • Astrophysics
  • General relativity
  • Particle physics
  • Quantum gravity

The metric expansion of space is a key part of science's current understanding of the universe, whereby spacetime itself is described by a metric which changes over time in such a way that the spatial dimensions appear to grow or stretch as the universe gets older. 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.

The expansion of space is conceptually different from other kinds of expansions and explosions that are seen in nature. Our understanding of the "fabric of the universe" ( spacetime) requires that what we see normally as "space", "time", 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 the space that contains the objects which is itself changing. It is as if without objects themselves moving, space is somehow "growing" in between them.

Because it is the metric defining distance that is changing rather than objects moving in space, this expansion (and the resultant movement apart of objects) is not restricted by the speed of light upper bound that results from special relativity.

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.


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 an 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.

For example, consider the surface of the Earth. This is a simple, familiar example of a non-Euclidean 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 chose any kind of coordinate system we wish, for example latitude and longitude, or X-Y-Z Cartesian coordinates. We then choose a metric which will be used to calculate the distance between nearby points in terms of the difference in the coordinates between them. In principle we can choose any one of a multitude of metrics, but in practice we choose a metric that seems to fit our purposes best. For example, local mapmaking often ignores curvature, whereas airline flights and distances are based upon a Great Circle. Since the Earth is not flat, the "distance" between two points (such as London and New York may differs depending on how it is to be measured (ie, the metric which is to be utilized). In measuring small distances on Earth, there is in theory always an effect due to curvature, but in practice the the Earth's curvature is so small as to be almost unnoticable for short distances.

The metric of space appears from current observations to be Euclidean, on a large scale. The same cannot be said for the metric of space-time, however. On the surface of the Earth, we specified points by giving two coordinates. Because space-time is four dimensional, we must specify points in space-time by giving four coordinates. The most convenient coordinates to use for cosmology are called comoving coordinates. Because space appears to be Euclidean, on a large scale, one can specify the spatial coordinates in terms of x,y, and z coordinates, though other choices such as spherical coordinates are also commonly used. The fourth required coordinate is time, which is specified in comoving coordinates as cosmological time. The non-Euclidean nature of space-time manifests itself by the fact that the distance between points with constant coordinates grows with time, rather than remaining constant.

Technically, the metric expansion of space is a feature of many solutions to the Einstein field equations of general relativity. In particular, if the cosmological principle is assumed with a time-varying universe the simplest solution allows for the distances in space to change with an evolving scale factor. This theoretical explanation provides a clean explanation of the observed Hubble's law which indicates that galaxies that are more distant from us appear to be receding faster than galaxies that are closer to us. In spaces that expand, the metric changes with time in a way that causes distances to appear larger at later times, so in our Big Bang universe, we observe phenomena associated with metric expansion of space. If we lived in a space that contracted (a Big Crunch universe) we would observe phenomena associated with a metric contraction of space instead.

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. It wasn't until the observations of Edwin Hubble in 1929 confirmed that distant galaxies were all apparently moving away from us that scientists accepted that the universe was expanding. Until the theoretical developments in the 1980s no one had an explanation for why this was 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 grew exponentially, causing space to change from smaller than an atom to around 100 million light years across.

The expansion of the universe proceeds in all directions as determined by the Hubble constant today. However, the Hubble constant can change in the past and in the future dependent on the observed value of density parameters (Ω). Before the discovery of dark energy, it was believed that the universe was matter dominated and so Ω on this graph corresponds to the ratio of the matter density to the critical density (Ωm).
The expansion of the universe proceeds in all directions as determined by the Hubble constant today. However, the Hubble constant can change in the past and in the future dependent on the observed value of density parameters (Ω). Before the discovery of dark energy, it was believed that the universe was matter dominated and so Ω on this graph corresponds to the ratio of the matter density to the critical density (Ωm).

Measuring distances

In expanding space, distance is a dynamical 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.

Observational evidence

It was not until the year 2000 that scientists finally had all the pieces of direct observational evidence necessary to confirm the metric expansion of the universe. However, before this evidence was discovered, theoretical cosmologists considered the metric expansion of space to be a likely feature of the universe based on what they considered to be a small number of reasonable assumptions in modeling the universe. Chief among these were:

  • 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").

To varying degrees, observational cosmologists have discovered evidence supporting these assumptions in addition to direct observations of space expanding. Today, metric expansion of space is considered by cosmologists to be an observed feature on the basis that although we cannot see it directly, the properties of the universe which scientists have tested and which can be observed provide compelling confirmation. Sources of confirmation include:

Edwin Hubble presented the first observational evidence of an expanding universe.
Edwin Hubble presented the first observational evidence of an expanding universe.
  • Edwin Hubble demonstrated that all galaxies and distant astronomical objects were moving away from us ("Hubble's law") as predicted by a universal expansion. Using the redshift of their electromagnetic spectra to determine the distance and speed of remote objects in space, he showed that all objects are moving away from us, and that their speed is proportional to their distance, a feature of metric expansion. Further studies have since shown the expansion to be extremely isotropic and homogenous, that is, it does not seem to have a special point as a "centre", 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 in much the same way a Jackson Pollock painting looks lumpy close-up, but more regular as a whole.
  • 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 in the dynamics of distant astrophysical systems. As reported by a group of astronomers at the European Southern Observatory, the radiation that pervades the universe is demonstrably warmer at earlier times. Uniform cooling of the cosmic microwave background over billions of years is explainable only if the universe is experiencing a metric expansion.

Taken together, the only theory which coherently explains these phenomena relies 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).

Additionally, scientists are confident that the theories which rely on the metric expansion of space are correct because they have passed the rigorous standards of the scientific method. In particular, when physics calculations are performed based upon the current theories (including metric expansion), they appear to give results and predictions which, in general, agree extremely closely with both astrophysical and particle physics observations. 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. This evidence is taken very seriously because the level of detail and the sheer quantity of measurements which the theories predict can be shown to precisely and accurately match visible reality. The level of precision is difficult to quantify, but is on the order of the precision seen in the physical constants that govern the physics of the universe.

Model analogies

Because metric expansion is not seen on the physical scale of humans, the concept may be difficult to grasp. Three analogies, the ant-on-a-balloon analogy, the expanding rubber sheet analogy, and the raisin bread analogy, have been developed to aid in conceptual understanding. Each analogy has its conceptual benefits and drawbacks.

Ant on a balloon model

The ant on a balloon model is a two-dimensional analog for three-dimensional metric expansion. An ant is imagined to be constrained to move on the surface of a huge balloon which to the ant's understanding is the total extent of space (see article on Flatland for more consequences of a two-dimensional constraint). At an early stage of the balloon-universe, the ant measures distances between separate points on the balloon which serves as a standard by which the scale factor can be measured. The balloon is inflated some more, and then the distance between the same points is measured and determined to be larger by a proportional factor. The surface of the balloon still appears flat, and yet all the points have appeared to recede from the ant, indeed every point on the surface of the balloon is proportionally farther from the ant than earlier in the life of the balloon universe. This explains how an expanding universe can result in all points receding from each other simultaneously. No points are seen to get closer together.

In the limit where the ant is tiny and the balloon is enormous, the ant also cannot detect any curvature associated with the geometry of the surface (which is roughly an elliptical geometry for the outside surface of a curving balloon). To the ant, the balloon appears to be a plane extending out in all directions. This mimics the so-called " flatness" seen in our own observable universe which appears even at the largest scale to follow the geometrical laws associated with flat geometry. Like the ant on an enormous balloon, while we may be unable to detect curvature, on larger, unobservable scales there may be residual curvature. The shape of the universe we observe is driven to be flat no matter what starting conditions the universe had by the same cosmic inflation which caused the universe to begin expanding in the first place.

In the analogy, the two dimensions of the balloon do not expand "into" anything since the surface of the balloon admits infinite paths in all directions at all times. There is some possibility for confusion in this analogy since the balloon can be seen by an external observer to be expanding "into" the third dimension (in the radial direction), but this is not a feature of metric expansion, rather it is the result of the arbitrary choice of the balloon which happens to be a manifold embedded in a third dimension. This third dimension is not mathematically necessary for two-dimensional metric expansion to occur, and the ant that is confined to the surface of the balloon has no way of determining whether a third dimension exists or not. It may be useful to visualize a third dimension, but the fact of expansion does not theoretically require such a dimension to exist. This is why the question "what is the universe expanding into?" is poorly phrased. Metric expansion does not have to proceed "into" anything. The universe that we inhabit does expand and distances get larger, but that does not mean that there is a larger space into which it is expanding.

Expanding rubber sheet model

Similar to the ant on a balloon model, the expanding rubber sheet universe (ERSU) is given as a model that represents the expansion by ignoring the third dimension. Instead of relying on a balloon expanding into three dimensions, the ERSU model describes an infinite rubber sheet that is stretched in both directions. Heavy objects placed on the sheet create dips and dents of local curvature in much the same way massive galaxies curve spacetime in the gravitational wells of our universe. These objects all appear to be receding from each other unless they get caught in each other's gravitational wells (a process called virialization). The infinite rubber sheet stays infinite and two dimensional, but distances between points on the sheet steadily increase with the expansion. This model has the advantage over the balloon model of a macroscopically two-dimensional flat geometry which corresponds well to the measured three-dimensional (lack of) curvature in our observable universe.

Raisin bread model

Animation of an expanding raisin bread model. As the bread doubles in size, the distances between raisins also double.
Animation of an expanding raisin bread model. As the bread doubles in size, the distances between raisins also double.

The raisin bread model imagines galaxies as raisins in a raisin bread dough that will "rise" or "expand" when cooked. As the expansion occurs, each of the raisins gets farther from each of the other raisins while the raisins themselves stay the same size. The dough between raisins in this model acts as the space between galaxies while the raisins as "bound objects" are not subject to the expansion. This model is useful for explaining how it is that a standard ruler can be determined for measuring the expansion. In an empty universe, space serves as the only ruler and as rulers expand with space, there would be no way to distinguish between an expanding universe and a static universe. Only in a universe where there are objects which are bound and do not expand so that the rulers are independent of the expansion can the metric expansion be measured.

Like the ant on the balloon model, this model also suffers from the problem that the raisin bread is expanding into the pan. To make the analogy to the universe, it is necessary to imagine raisin bread that has no observable edge. Expansion would still occur, but the question "what is the raisin bread expanding into?" would be meaningless.

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