Expansion of the universe

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The expansion of the universe is the increase in

Cosmic expansion is a key feature of

In 1912–1914, Vesto M. Slipher discovered that light from remote galaxies was redshifted,^[7][8] a phenomenon later interpreted as galaxies receding from the Earth. In 1922, Alexander Friedmann used the Einstein field equations to provide theoretical evidence that the universe is expanding.^[9]

Swedish astronomer Knut Lundmark was the first person to find observational evidence for expansion, in 1924. According to Ian Steer of the NASA/IPAC Extragalactic Database of Galaxy Distances, "Lundmark's extragalactic distance estimates were far more accurate than Hubble's, consistent with an expansion rate (Hubble constant) that was within 1% of the best measurements today."^[10]

In 1927, Georges Lemaître independently reached a similar conclusion to Friedmann on a theoretical basis, and also presented observational evidence for a linear relationship between distance to galaxies and their recessional velocity.^[11] Edwin Hubble observationally confirmed Lundmark's and Lemaître's findings in 1929.^[12] Assuming the cosmological principle, these findings would imply that all galaxies are moving away from each other.

Astronomer Walter Baade recalculated the size of the known universe in the 1940s, doubling the previous calculation made by Hubble in 1929.^[13][14][15] He announced this finding to considerable astonishment at the 1952 meeting of the International Astronomical Union in Rome. For most of the second half of the 20th century, the value of the Hubble constant was estimated to be between 50 and 90 km⋅s⁻¹⋅Mpc⁻¹.

On 13 January 1994, NASA formally announced a completion of its repairs related to the main mirror of the

, in which objects recede from each observer with velocities proportional to their positions with respect to that observer. That is, recession velocities ${\displaystyle {\vec {v}}}$ scale with (observer-centered) positions ${\displaystyle {\vec {x}}}$ according to

${\displaystyle {\vec {v}}=H{\vec {x}},}$

where the Hubble rate ${\displaystyle H}$ quantifies the rate of expansion. ${\displaystyle H}$ is a function of cosmic time.

Dynamics of cosmic expansion

The expansion of the universe can be understood as a consequence of an initial impulse (possibly due to inflation), which sent the contents of the universe flying apart. The mutual gravitational attraction of the matter and radiation within the universe gradually slows this expansion over time, but expansion nevertheless continues due to momentum left over from the initial impulse. Also, certain exotic relativistic fluids, such as dark energy and inflation, exert gravitational repulsion in the cosmological context, which accelerates the expansion of the universe. A cosmological constant also has this effect.

Mathematically, the expansion of the universe is quantified by the scale factor, ${\displaystyle a}$, which is proportional to the average separation between objects, such as galaxies. The scale factor is a function of time and is conventionally set to be ${\displaystyle a=1}$ at the present time. Because the universe is expanding, ${\displaystyle a}$ 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 13.787 ± 0.020 billion years ago. If the universe continues to expand forever, the scale factor will approach infinity in the future. It is also possible in principle for the universe to stop expanding and begin to contract, which corresponds to the scale factor decreasing in time.

The scale factor ${\displaystyle a}$ is a parameter of the

shows how the contents of the universe influence its expansion rate. Here, ${\displaystyle G}$ is the gravitational constant, ${\displaystyle \rho }$ is the energy density within the universe, ${\displaystyle p}$ is the pressure, ${\displaystyle c}$ is the speed of light, and ${\displaystyle \Lambda }$ is the cosmological constant. A positive energy density leads to deceleration of the expansion, ${\displaystyle {\ddot {a}}<0}$, and a positive pressure further decelerates expansion. On the other hand, sufficiently negative pressure with ${\displaystyle p<-\rho c^{2}/3}$ leads to accelerated expansion, and the cosmological constant also accelerates expansion.

is essentially pressureless, with ${\displaystyle |p|\ll \rho c^{2}}$, while a gas of

) has positive pressure ${\displaystyle p=\rho c^{2}/3}$. Negative-pressure fluids, like dark energy, are not experimentally confirmed, but the existence of dark energy is inferred from astronomical observations.

Distances in the expanding universe

Comoving coordinates

Main article:

Comoving coordinates

In an expanding universe, it is often useful to study the evolution of

FLRW metric

.

Shape of the universe

The universe is a four-dimensional spacetime, but within a universe that obeys the cosmological principle, there is a natural choice of three-dimensional spatial surface. These are the surfaces on which observers who are stationary in comoving coordinates agree on the age of the universe. In a universe governed by special relativity, such surfaces would be hyperboloids, because relativistic time dilation means that rapidly receding distant observers' clocks are slowed, so that spatial surfaces must bend "into the future" over long distances. However, within general relativity, the shape of these comoving synchronous spatial surfaces is affected by gravity. Current observations are consistent with these spatial surfaces being geometrically flat (so that, for example, the angles of a triangle add up to 180 degrees).

Cosmological horizons

An expanding universe typically has a finite age. Light, and other particles, can have propagated only a finite distance. The comoving distance that such particles can have covered over the age of the universe is known as the particle horizon, and the region of the universe that lies within our particle horizon is known as the observable universe.

If the dark energy that is inferred to dominate the universe today is a cosmological constant, then the particle horizon converges to a finite value in the infinite future. This implies that the amount of the universe that we will ever be able to observe is limited. Many systems exist whose light can never reach us, because there is a

Within the study of the evolution of structure within the universe, a natural scale emerges, known as the

Consequences of cosmic expansion

Velocities and redshifts

An object's

More generally, the peculiar

, scales inversely with the scale factor (i.e. ${\displaystyle T\propto a^{-1}}$). The temperature of nonrelativistic matter drops more sharply, scaling as the inverse square of the scale factor (i.e. ${\displaystyle T\propto a^{-2}}$).

Density

The contents of the universe dilute as it expands. The number of particles within a comoving volume remains fixed (on average), while the volume expands. For nonrelativistic matter, this implies that the energy density drops as ${\displaystyle \rho \propto a^{-3}}$, where ${\displaystyle a}$ is the

scale factor

.

For ultrarelativistic particles ("radiation"), the energy density drops more sharply, as ${\displaystyle \rho \propto a^{-4}}$. This is because in addition to the volume dilution of the particle count, the energy of each particle (including the

rest mass energy

) also drops significantly due to the decay of peculiar momenta.

In general, we can consider a perfect fluid with pressure ${\displaystyle p=w\rho }$, where ${\displaystyle \rho }$ is the energy density. The parameter ${\displaystyle w}$ is the equation of state parameter. The energy density of such a fluid drops as

${\displaystyle \rho \propto a^{-3(1+w)}.}$

Nonrelativistic matter has ${\displaystyle w=0}$ while radiation has ${\displaystyle w=1/3}$. For an exotic fluid with negative pressure, like dark energy, the energy density drops more slowly; if ${\displaystyle w=-1}$ it remains constant in time. If ${\displaystyle w<-1}$, corresponding to phantom energy, the energy density grows as the universe expands.

Expansion history

Cosmic inflation

Inflation is a period of accelerated expansion hypothesized to have occurred at a time of around 10⁻³² seconds. It would have been driven by the

During inflation, the cosmic scale factor grew exponentially in time. In order to solve the horizon and flatness problems, inflation must have lasted long enough that the scale factor grew by at least a factor of e⁶⁰ (about 10²⁶).^{[citation needed]}

Radiation epoch

The history of the universe after inflation but before a time of about 1 second is largely unknown.^[20] However, the universe is known to have been dominated by ultrarelativistic Standard Model particles, conventionally called radiation, by the time of neutrino decoupling at about 1 second.^[21] During radiation domination, cosmic expansion decelerated, with the scale factor growing proportionally with the square root of the time.

Matter epoch

Since radiation redshifts as the universe expands, eventually nonrelativistic matter came to dominate the energy density of the universe. This transition happened at a time of about 50 thousand years after the Big Bang. During the matter-dominated epoch, cosmic expansion also decelerated, with the scale factor growing as the 2/3 power of the time (${\displaystyle a\propto t^{2/3}}$). Also, gravitational structure formation is most efficient when nonrelativistic matter dominates, and this epoch is responsible for the formation of

Dark energy

Around 3 billion years ago, at a time of about 11 billion years, dark energy is believed to have begun to dominate the energy density of the universe. This transition came about because dark energy does not dilute as the universe expands, instead maintaining a constant energy density. Similarly to inflation, dark energy drives accelerated expansion, such that the scale factor grows exponentially in time.

Measuring the expansion rate

The most direct way to measure the expansion rate is to independently measure the recession velocities and the distances of distant objects, such as galaxies. The ratio between these quantities gives the Hubble rate, in accordance with Hubble's law. Typically, the distance is measured using a

Supernovae are observable at such great distances that the light travel time therefrom can approach the age of the universe. Consequently, they can be used to measure not only the present-day expansion rate but also the expansion history. In work that was awarded the 2011 Nobel Prize in Physics, supernova observations were used to determine that cosmic expansion is accelerating in the present epoch.^[23]

By assuming a cosmological model, e.g. the

A third option proposed recently is to use information from gravitational wave events (especially those involving the merger of neutron stars, like GW170817), to measure the expansion rate.^[25][26] Such measurements do not yet have the precision to resolve the Hubble tension.

In principle, the cosmic expansion history can also be measured by studying how redshifts, distances, fluxes, angular positions, and angular sizes of astronomical objects change over the course of the time that they are being observed. These effects are too small to have yet been detected. However, changes in redshift or flux could be observed by the Square Kilometre Array or Extremely Large Telescope in the mid-2030s.^[27]

Conceptual considerations and misconceptions

Measuring distances in expanding space

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At cosmological scales, the present universe conforms to Euclidean space, what cosmologists describe as geometrically flat, to within experimental error.^[28]

Consequently, the rules of Euclidean geometry associated with Euclid's fifth postulate hold in the present universe in 3D space. It is, however, possible that the geometry of past 3D space could have been highly curved. The curvature of space is often modeled using a non-zero Riemann curvature tensor in curvature of Riemannian manifolds. Euclidean "geometrically flat" space has a Riemann curvature tensor of zero.

"Geometrically flat" space has three dimensions and is consistent with Euclidean space. However, spacetime has four dimensions; it is not flat according to Einstein's general theory of relativity. Einstein's theory postulates that "matter and energy curve spacetime, and there is enough matter and energy to provide for curvature."^[29]

In part to accommodate such different geometries, the expansion of the universe is inherently general-relativistic. It cannot be modeled with

The images to the right show two views of

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 the 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. The distance traveled is thus inherently ambiguous because of the changing scale of the universe. Nevertheless, there are two distances that appear to be physically meaningful: the distance between 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 defined as the spatial dimension). The former distance is about 4 billion light-years, much smaller than ct, whereas the latter distance (shown by the orange line) is about 28 billion light-years, much larger than ct. In other words, if space were not expanding today, it would take 28 billion years for light to travel between 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. In fact, the light emitted towards Earth was actually moving away from Earth when it was first emitted; the metric distance to Earth increased with cosmological time for the first few billion years of its travel time, also indicating that the expansion of space between Earth and the quasar at the early time was faster than the speed of light. None of this behavior originates from a special property of metric expansion, but rather from local principles of special relativity integrated over a curved surface.

Topology of expanding space

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 3D manifold only; that is, the description involves no structures such as extra dimensions or an exterior universe.^[30]

The ultimate

Despite being extremely dense when very young and during part of its early expansion – far denser than is usually required to form a black hole – the universe did not re-collapse into a black hole. This is because commonly used calculations for gravitational collapse are usually based upon objects of relatively constant size, such as stars, and do not apply to rapidly expanding space such as the Big Bang.^{[citation needed][dubious – discuss]}

Effects of expansion on small scales

The expansion of space is sometimes described as a force that 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.^[32]

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

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 that is proportional (not inversely proportional) to distance. Unlike inertia it actively "pulls" on objects that 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 that 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, 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 around 3 billion years.

Metric expansion and speed of light

At the end of the

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 that 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 conceptually 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 that expands uniformly in all directions. The addition of a second spatial dimension allows for 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 that is inflated from an initial size of zero (representing the Big Bang). A balloon has positive Gaussian curvature, even though 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 within the limits of observation. This analogy is potentially confusing since it could wrongly suggest 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 or will be occupied later.

In the "raisin bread model", one imagines a loaf of raisin bread expanding in an oven. The loaf (space) expands as a whole, but the raisins (gravitationally bound objects) do not expand; they merely move farther away from each other. This analogy has the disadvantage of wrongly implying that the expansion has a center and an edge.

See also

• Comoving and proper distances

References

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 Lyth, David H. Cosmological Inflation and Large-Scale Structure. Cambridge University Press, 2000.
• Lineweaver, Charles H. and Davis, Tamara M. "Misconceptions about the Big Bang", Scientific American, March 2005 (non-free content).
• Mook, Delo E. and Thomas Vargish. Inside Relativity. Princeton University Press, 1991.

External links

Wikimedia Commons has media related to Expansion of the universe.

• Swenson, Jim, Answer to a question about the expanding universe Archived 11 January 2009 at the Wayback Machine
• Felder, Gary, "The Expanding universe".
• WMAP team offers an "Explanation of the universal expansion
  " at an elementary level.
• Hubble Tutorial from the University of Wisconsin Physics Department Archived 9 June 2014 at the Wayback Machine
• 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).