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2014-08-13T13:14:18Z

StarlaDIOA:

In [[mathematics]], the '''Gaussian binomial coefficients''' (also called '''Gaussian coefficients''', '''Gaussian polynomials''', or '''''q''-binomial coefficients''') are [[q-analog|''q''-analog]]s of the [[binomial coefficients]].

==Definition==

The Gaussian binomial coefficients are defined by

:<math>{m \choose r}_q
= \begin{cases}
\frac{(1-q^m)(1-q^{m-1})\cdots(1-q^{m-r+1})} {(1-q)(1-q^2)\cdots(1-q^r)} & r \le m \\
0 & r>m \end{cases}</math>

where ''m'' and ''r'' are non-negative integers. For {{nowrap|''r'' {{=}} 0}} the value is 1 since numerator and denominator are both [[empty product]]s. Although the formula in the first clause appears to involve a [[rational function]], it actually designates a polynomial, because the division is exact in '''Z'''<nowiki>[</nowiki>''q''<nowiki>]</nowiki>. Note that the formula can be applied for {{nowrap|''r'' {{=}} ''m'' + 1}}, and gives 0 due to a factor {{nowrap|1 − ''q''0 {{=}} 0}} in the numerator, in accordance with the second clause (for even larger ''r'' the factor 0 remains present in the numerator, but its further factors would involve negative powers of ''q'', whence explicitly stating the second clause is preferable). All of the factors in numerator and denominator are divisible by {{nowrap|1 − ''q''}}, with as quotient a [[Q-analog#Introductory examples|''q'' number]]:
:<math>[k]_q=\frac{1-q^k}{1-q}=\sum_{0\leq i<k}q^i=1+q+q^2+\cdots+q^{k-1};</math>
dividing out these factors gives the equivalent formula
:<math>{m \choose r}_q=\frac{[m]_q[m-1]_q\cdots[m-r+1]_q}{[1]_q[2]_q\cdots[r]_q}\quad(r\leq m),</math>
which makes evident the fact that substituting {{nowrap|''q'' {{=}} 1}} into <math>\tbinom mr_q</math> gives the ordinary binomial coefficient <math>\tbinom mr.</math> In terms of the [[Q-analog#Introductory examples|''q'' factorial]] <math>[n]_q!=[1]_q[2]_q\cdots[n]_q</math>, the formula can be stated as
:<math>{m \choose r}_q=\frac{[m]_q!}{[r]_q!\,[m-r]_q!}\quad(r\leq m),</math>
a compact form (often given as only definition), which however hides the presence of many common factors in numerator and denominator. This form does make obvious the symmetry <math>\tbinom mr_q=\tbinom m{m-r}_q</math> for {{nowrap|''r'' ≤ ''m''}}.

Instead of these algebraic expressions, one can also give a combinatorial definition of Gaussian binomial coefficients. The ordinary binomial coefficient <math>\tbinom mr</math> counts the {{math|''r''}}-[[combination]]s chosen from an {{math|''m''}}-element set. If one takes those {{math|''m''}} elements to be the different character positions in a word of length {{math|''m''}}, then each {{math|''r''}}-combination corresponds to a word of length {{math|''m''}} using an alphabet of two letters, say {{math|{0,1},}} with {{math|''r''}} copies of the letter 1 (indicating the positions in the chosen combination) and {{math|''m'' − ''r''}} letters 0 (for the remaining positions). To obtain from this model the Gaussian binomial coefficient <math>\tbinom mr_q</math>, it suffices to count each word with a factor {{math|''q''''d''}}, where {{math|''d''}} is the number of "inversions" of the word: the number of pairs of positions for which the leftmost position of the pair holds a letter 1 and the rightmost position holds a letter 0 in the word. It can be shown that the polynomials so defined satisfy the Pascal identities given below, and therefore coincide with the polynomials given by the algebraic definitions. A visual way to view this definition is to associate to each word a path across a rectangular grid with sides of length {{math|''r''}} and {{math|''m'' − ''r''}}, from the bottom left corner to the top right corner, taking a step left for each letter 0 and a step up for each letter 1. Then the number of inversions of the word equals the area of the part of the rectangle that is to the bottom-right of the path.

Unlike the ordinary binomial coefficient, the Gaussian binomial coefficient has finite values for <math>m\rightarrow \infty</math>:

:<math>{\infty \choose r}_q = \lim_{m\rightarrow \infty} {m \choose r}_q = \frac{1}{[r]_q!\,(1-q)^r}</math>

==Examples==

:<math>{0 \choose 0}_q = {1 \choose 0}_q = 1</math>

:<math>{1 \choose 1}_q = \frac{1-q}{1-q}=1</math>

:<math>{2 \choose 1}_q = \frac{1-q^2}{1-q}=1+q</math>

:<math>{3 \choose 1}_q = \frac{1-q^3}{1-q}=1+q+q^2</math>

:<math>{3 \choose 2}_q = \frac{(1-q^3)(1-q^2)}{(1-q)(1-q^2)}=1+q+q^2</math>

:<math>{4 \choose 2}_q = \frac{(1-q^4)(1-q^3)}{(1-q)(1-q^2)}=(1+q^2)(1+q+q^2)=1+q+2q^2+q^3+q^4</math>

==Properties==
Like the ordinary binomial coefficients, the Gaussian binomial coefficients are center-symmetric, i.e., invariant under the reflection <math> r \rightarrow m-r </math>:

:<math>{m \choose r}_q = {m \choose m-r}_q. </math>

In particular,

:<math>{m \choose 0}_q ={m \choose m}_q=1 \, ,</math>

:<math>{m \choose 1}_q ={m \choose m-1}_q=\frac{1-q^m}{1-q}=1+q+ \cdots + q^{m-1} \quad m \ge 1 \, .</math>

The name ''Gaussian binomial coefficient'' stems from the fact{{cn|date=February 2014}} that their evaluation at {{nowrap|''q'' {{=}} 1}} is

:<math>{m \choose r}_1 = {m \choose r}</math>

for all ''m'' and ''r''.

The analogs of [[Pascal's triangle|Pascal identities]] for the Gaussian binomial coefficients are

:<math>{m \choose r}_q = q^r {m-1 \choose r}_q + {m-1 \choose r-1}_q</math>

and

:<math>{m \choose r}_q = {m-1 \choose r}_q + q^{m-r}{m-1 \choose r-1}_q.</math>

There are analogs of the binomial formula, and of Newton's generalized version of it for negative integer exponents, although for the former the Gaussian binomial coefficients themselves do not appear as coefficients:

:<math>\prod_{k=0}^{n-1} (1+q^kt)=\sum_{k=0}^n q^{k(k-1)/2}
{n \choose k}_q t^k </math>

and

:<math>\prod_{k=0}^{n-1} \frac{1}{(1-q^kt)}=\sum_{k=0}^\infty
{n+k-1 \choose k}_q t^k. </math>

which, for <math>n\rightarrow\infty</math> become:

:<math>\prod_{k=0}^{\infty} (1+q^kt)=\sum_{k=0}^\infty \frac{q^{k(k-1)/2}t^k}{[k]_q!\,(1-q)^k} </math>

and

:<math>\prod_{k=0}^\infty \frac{1}{(1-q^kt)}=\sum_{k=0}^\infty
\frac{t^k}{[k]_q!\,(1-q)^k} . </math>

The first Pascal identity allows one to compute the Gaussian binomial coefficients recursively (with respect to ''m'' ) using the initial "boundary" values

:<math>{m \choose m}_q ={m \choose 0}_q=1 </math>

and also incidentally shows that the Gaussian binomial coefficients are indeed polynomials (in ''q''). The second Pascal identity follows from the first using the substitution <math> r \rightarrow m-r </math> and the invariance of the Gaussian binomial coefficients under the reflection <math> r \rightarrow m-r </math>. Both Pascal identities together imply

:<math>{m \choose r}_q = {{1-q^{m}}\over {1-q^{m-r}}} {m-1 \choose r}_q </math>

which leads (when applied iteratively for ''m'', ''m'' − 1, ''m'' − 2,....) to an expression for the Gaussian binomial coefficient as given in the definition above.

==Applications==

Gaussian binomial coefficients occur in the counting of [[symmetric polynomial]]s and in the theory of [[partition (number theory)|partitions]]. The coefficient of ''q''''r'' in

:<math>{n+m \choose m}_q</math>

is the number of partitions of ''r'' with ''m'' or fewer parts each less than or equal to ''n''. Equivalently, it is also the number of partitions of ''r'' with ''n'' or fewer parts each less than or equal to ''m''.

Gaussian binomial coefficients also play an important role in the enumerative theory of [[projective space]]s defined over a finite field. In particular, for every [[finite field]] ''F''''q'' with ''q'' elements, the Gaussian binomial coefficient

:<math>{n \choose k}_q</math>

counts the number ''v''''n'',''k'';''q'' of different ''k''-dimensional vector subspaces of an ''n''-dimensional [[vector space]] over ''F''''q'' (a [[Grassmannian]]). When expanded as a polynomial in ''q'', it yields the well-known decomposition of the Grassmannian into Schubert cells. Furthermore, when ''q'' is 1 (respectively -1), the Gaussian binomial coefficient yields the Euler characteristic of the corresponding complex (respectively real) Grassmannian. For example, the Gaussian binomial coefficient

:<math>{n \choose 1}_q=1+q+q^2+\cdots+q^{n-1}</math>

is the number of different lines in ''F''''q''''n'' (a [[projective space]]).

In the conventions common in applications to [[quantum groups]], a slightly different definition is used; the quantum binomial coefficient there is
:<math>q^{k^2 - n k}{n \choose k}_{q^2}</math>.
This version of the quantum binomial coefficient is symmetric under exchange of <math>q</math> and <math>q^{-1}</math>.

==Triangles==

The Gaussian binomial coefficients can be arranged in a triangle for each ''q'', which is [[Pascal's triangle]] for ''q''=1. 
Read line by line these triangles form the following sequences in the [[On-Line Encyclopedia of Integer Sequences|OEIS]]:
* [[oeis:A022166/table|A022166]] for ''q''= 2
* [[oeis:A022167/table|A022167]] for ''q''= 3
* [[oeis:A022168/table|A022168]] for ''q''= 4
* [[oeis:A022169/table|A022169]] for ''q''= 5
* [[oeis:A022170/table|A022170]] for ''q''= 6
* [[oeis:A022171/table|A022171]] for ''q''= 7
* [[oeis:A022172/table|A022172]] for ''q''= 8
* [[oeis:A022173/table|A022173]] for ''q''= 9
* [[oeis:A022174/table|A022174]] for ''q''= 10

==References==
*Exton, H. (1983), ''q-Hypergeometric Functions and Applications'', New York: Halstead Press, Chichester: Ellis Horwood, 1983, ISBN 0853124914, ISBN 0470274530, ISBN 978-0470274538

* {{cite web
|first1=Eugene
|last1=Mukhin
|url=http://mathcircle.berkeley.edu/BMC3/SymPol.pdf
|title= Symmetric Polynomials and Partitions
}} (undated, 2004 or earlier).
* Ratnadha Kolhatkar, [http://www.math.mcgill.ca/goren/SeminarOnCohomology/GrassmannVarieties%20.pdf Zeta function of Grassmann Varieties] (dated January 26, 2004)
* {{MathWorld| urlname=q-BinomialCoefficient|title=q-Binomial Coefficient}}
* {{cite journal
|first1=Henry
|last1=Gould
|journal=[[Fibonacci Quarterly]]
|year=1969
|title=The bracket function and Fontene-Ward generalized binomial coefficients with application to Fibonomial coefficients
|mr=0242691
|volume=7
|pages=23–40
}}
* {{cite journal
|first1= G. L.
|last1=Alexanderson|author1-link=Gerald L. Alexanderson
|journal=[[Fibonacci Quarterly]]
|year=1974
|volume=12
|title=A Fibonacci analogue of Gaussian binomial coefficients
|mr=0354537
|pages=129–132
}}
* {{cite journal
|first1=George E.
|last1=Andrews
|author1-link=George Andrews (mathematician)
|title=Applications of basic hypergeometric functions
|journal=SIAM Rev.
|year=1974
|volume=16
|number=4
|jstor=2028690
|mr=0352557
|doi=10.1137/1016081
}}
* {{cite journal
|first1=Peter B.
|last1=Borwein
|title=Padé approximants for the q-elementary functions
|journal=Construct. Approx.
|year=1988
|volume=4
|number=1
|pages=391–402
|doi=10.1007/BF02075469
|mr=0956175
}}
* {{cite journal
|first1=John
|last1=Konvalina
|title=Generalized binomial coefficients and the subset-subspace problem
|journal=Adv. Appl. Math.
|year=1998
|doi=10.1006/aama.1998.0598
|volume=21
|pages=228–240
|mr=1634713
}}
* {{cite journal
|first1=A.
|last1=Di Bucchianico
|title=Combinatorics, computer algebra and the Wilcoxon-Mann-Whitney test
|journal=J. Stat. Plann. Inf.
|year=1999
|doi=10.1016/S0378-3758(98)00261-4
|volume=79
|pages=349–364
}}
* {{ cite journal
|first1=John
|last1=Konvalina
|title=A unified interpretation of the Binomial Coefficients, the Stirling numbers, and the Gaussian coefficients
|journal=Am. Math. Monthly
|jstor=2695583
|year=2000
|pages=901–910
|volume=107
|number=10
|mr=1806919
}}
* {{cite journal
|first1=Boris A.
|last=Kupershmidt
|title=q-Newton binomial: from Euler to Gauss
|journal=J. Nonlin. Math. Phys.
|year=2000
|volume=7
|number=2
|pages=244–262
|mr=1763640
|bibcode=2000JNMP....7..244K
|arxiv = math/0004187 |doi = 10.2991/jnmp.2000.7.2.11 }}
* {{ cite journal
|first1=Henry
|last1=Cohn
|journal=Am. Math. Monthly
|year=2004
|title=Projective geometry over F1 and the Gaussian Binomial Coefficients
|volume=111
|number=6
|jstor=4145067
|mr=2076581
|pages=487–495
}}
* {{cite journal
|first1=T.
|last1=Kim
|title=q-Extension of the Euler formula and trigonometric functions
|journal=Russ. J. Math. Phys.
|volume=14
|number=3
|pages=-275–278
|year=2007
|doi=10.1134/S1061920807030041
|mr=2341775
|bibcode = 2007RJMP...14..275K }}
* {{cite journal
|first1=T.
|last1=Kim
|title=q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients
|journal = Russ. J. Math. Phys.
|volume=15
|number=1
|pages=51–57
|doi=10.1134/S1061920808010068
|mr=2390694
|year=2008}}
* {{cite journal
|first1=Roberto B.
|last1=Corcino
|title= On p,q-binomial coefficients
|journal=Integers
|volume=8
|year=2008
|pages=#A29
|mr=2425627
}}
* {{cite web
|first1=Gevorg
|last1=Hmayakyan
|url=http://ghmath.files.wordpress.com/2010/06/mobius.pdf
|title= Recursive Formula Related To The Mobius Function
}} (2009).

[[Category:Q-analogs]]
[[Category:Factorial and binomial topics]]

Main Page

2014-08-13T05:54:18Z

StarlaDIOA:

In mathematics, the '''Bruhat decomposition''' (named after [[François Bruhat]]) G = BWB into cells can be regarded as a general expression of the principle of [[Gauss–Jordan elimination]], which generically writes a matrix as a product of an upper triangular and lower triangular matrices—but with exceptional cases. It is related to the [[Schubert cell]] decomposition of Grassmannians: see [[Weyl group]] for this.

More generally, any group with a [[(B,N) pair]] has a Bruhat decomposition.

==Definitions==
*''G'' is a [[connected space|connected]], [[reductive group|reductive]] [[algebraic group]] over an [[algebraically closed field]].
*''B'' is a [[Borel subgroup]] of ''G''
*''W'' is a [[Weyl group]] of ''G'' corresponding to a maximal torus of ''B''.

The '''Bruhat decomposition''' of ''G'' is the decomposition
:<math>G=BWB =\coprod_{w\in W}BwB</math>
of ''G'' as a disjoint union of [[double coset]]s of ''B'' parameterized by the elements of the Weyl group ''W''. (Note that although ''W'' is not in general a subgroup of ''G'', the coset ''wB'' is still well defined.)

== Examples ==
Let ''G'' be the [[general linear group]] '''GL'''n of invertible <math>n \times n</math> matrices with entries in some algebraically closed field, which is a reductive group. Then the Weyl group ''W'' is isomorphic to the [[symmetric group]] ''Sn'' on ''n'' letters, with [[permutation matrices]] as representatives. In this case, we can take ''B'' to be the subgroup of upper triangular invertible matrices, so Bruhat decomposition says that one can write any invertible matrix ''A'' as a product ''U1PU2'' where ''U1'' and ''U2'' are upper triangular, and ''P'' is a permutation matrix. Writing this as ''P = U1-1AU2-1'', this says that any invertible matrix can be transformed into a permutation matrix via a series of row and column operations, where we are only allowed to add row ''i'' (resp. column ''i'') to row ''j'' (resp. column ''j'') if ''i>j'' (resp. ''i<j''). The row operations correspond to ''U1-1'', and the column operations correspond to ''U2-1''.

The [[special linear group]] '''SL'''n of invertible <math>n \times n</math> matrices with [[determinant]] 1 is a [[semisimple algebraic group|semisimple group]], and hence reductive. In this case, ''W'' is still isomorphic to the symmetric group ''Sn''. However, the determinant of a permutation matrix is the sign of the permutation, so to represent an odd permutation in '''SL'''n, we can take one of the nonzero elements to be -1 instead of 1. Here ''B'' is the subgroup of upper triangular matrices with determinant 1, so the interpretation of Bruhat decomposition in this case is similar to the case of '''GL'''n.

== Geometry ==
The cells in the Bruhat decomposition correspond to the [[Schubert cell]] decomposition of Grassmannians. The dimension of the cells corresponds to the [[length function|length]] of the word ''w'' in the Weyl group. [[Poincaré duality]] constrains the topology of the cell decomposition, and thus the algebra of the Weyl group; for instance, the top dimensional cell is unique (it represents the [[fundamental class]]), and corresponds to the [[longest element of a Coxeter group]].

==Computations==
The number of cells in a given dimension of the Bruhat decomposition are the coefficients of the ''q''-polynomial<ref>[http://math.ucr.edu/home/baez/week186.html This Week's Finds in Mathematical Physics, Week 186]</ref> of the associated [[Dynkin diagram]].

==See also==
* [[Lie group decompositions]]
* [[Birkhoff factorization]], a special case of the Bruhat decomposition for affine groups.

==Notes==
<references/>

==References==
*[[Armand Borel|Borel, Armand]]. Linear Algebraic Groups (2nd ed.). New York: Springer-Verlag. ISBN 0-387-97370-2.
*[[Nicolas Bourbaki|Bourbaki, Nicolas]], ''Lie Groups and Lie Algebras: Chapters 4-6 (Elements of Mathematics)'', ISBN 3-540-42650-7

[[Category:Lie groups]]
[[Category:algebraic groups]]

[[ja:ブリュア分解]]

Main Page

2014-08-13T02:41:40Z

StarlaDIOA:

'''Semiclassical gravity''' is the approximation to the theory of [[quantum gravity]] in which one treats [[Field (physics)|matter fields]] as being quantum and the [[Gravitation|gravitational field]] as being classical.

In semiclassical gravity, matter is represented by quantum matter fields that propagate according to the theory of [[quantum field theory in curved spacetime|quantum fields in curved spacetime]]. The spacetime in which the fields propagate is classical but dynamical. The curvature of the spacetime is given by the ''semiclassical Einstein equations'', which relate the curvature of the spacetime, given by the [[Einstein tensor]] <math>G_{\mu\nu}</math>, to the expectation value of the [[Stress–energy tensor|energy–momentum tensor]] operator, <math>T_{\mu\nu}</math>, of the matter fields:

:<math> G_{\mu\nu} = \frac{ 8 \pi G }{ c^4 } \left\langle \hat T_{\mu\nu} \right\rangle_\psi </math>

where ''G'' is [[Gravitational constant|Newton's constant]] and <math>\psi</math> indicates the quantum state of the matter fields.

==Stress–energy tensor==
There is some ambiguity in regulating the stress–energy tensor, and this depends upon the curvature. This ambiguity can be absorbed into the [[cosmological constant]], [[Newton's constant]], and the [[f(R) gravity|quadratic couplings]]<ref>See Wald (1994) Chapter 4, section 6 "The Stress-Energy Tensor".</ref>
:<math>\int d^dx \,\sqrt{-g} R^2</math> and <math>\int d^dx\, \sqrt{-g} R^{\mu\nu}R_{\mu\nu}</math>.
There's also the other quadratic term
:<math>\int d^dx\, \sqrt{-g} R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}</math>,
but (in 4-dimensions) this term is a linear combination of the other two terms and a surface term. See [[Gauss–Bonnet gravity]] for more details.

Since the theory of quantum gravity is not yet known, it is difficult to say what is the regime of validity of semiclassical gravity. However, one can formally show that semiclassical gravity could be deduced from quantum gravity by considering ''N'' copies of the quantum matter fields, and taking the limit of ''N'' going to infinity while keeping the product ''GN'' constant. At diagrammatic level, semiclassical gravity corresponds to summing all [[Feynman diagram]]s which do not have loops of gravitons (but have an arbitrary number of matter loops). Semiclassical gravity can also be deduced from an axiomatic approach.

==Experimental status==
There are cases where semiclassical gravity breaks down. For instance,<ref>See Page and Geilker; Eppley and Hannah; Albers, Kiefer, and Reginatto.</ref> if ''M'' is a huge mass, then the superposition
:<math>\frac{1}{\sqrt{2}} \left( \left| M \text{ at } A \right\rangle + \left| M \text{ at } B \right\rangle \right)</math>
where ''A'' and ''B'' are widely separated, then the expectation value of the stress–energy tensor is ''M/2'' at ''A'' and ''M/2'' at ''B'', but we would never observe the metric sourced by such a distribution. Instead, we [[decohere]] into a state with the metric sourced at ''A'' and another sourced at ''B'' with a 50% chance each.

==Applications==
The most important applications of semiclassical gravity are to understand the [[Hawking radiation]] of [[black hole]]s and the generation of random gaussian-distributed perturbations in the theory of [[cosmic inflation]], which is thought to occur at the very beginnings of the [[Big Bang|big bang]].

==Notes==
{{Reflist}}

==References==
* Birrell, N. D. and Davies, P. C. W., ''Quantum fields in curved space'', (Cambridge University Press, Cambridge, UK, 1982).
* Don N. Page, and C. D. Geilker, "Indirect Evidence for Quantum Gravity." ''Phys. Rev. Lett.'' '''47''' (1981) 979–982. DOI:[http://dx.doi.org/10.1103/PhysRevLett.47.979 10.1103/PhysRevLett.47.979]
* K. Eppley and E. Hannah, "The necessity of quantizing the gravitational field." ''Found. Phys.'' '''7''' (1977) 51–68. [[Digital object identifier|doi]]:[http://dx.doi.org/10.1007/BF00715241 10.1007/BF00715241]
* Mark Albers, Claus Kiefer, Marcel Reginatto, "Measurement Analysis and Quantum Gravity." ''Phys.Rev.D'' '''78''' 6 (2008) 064051, [http://dx.doi.org/10.1103/PhysRevD.78.064051 DOI:10.1103/PhysRevD.78.064051]. Eprint [http://arxiv.org/abs/0802.1978 arXiv:0802.1978] [gr-qc].
* Robert M. Wald, ''Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics''. University of Chicago Press, 1994.
*[http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=-43587 Semiclassical gravity on arxiv.org]

{{theories of gravitation}}
{{quantum gravity}}

[[Category:Theories of gravitation]]
[[Category:Quantum field theory]]
[[Category:Quantum gravity]]

Main Page

2014-08-12T20:13:35Z

StarlaDIOA:

{{About|scientific estimates of the age of the universe|religious and other non-scientific estimates|Dating creation}}
{{Cosmology}}

In [[physical cosmology]], the '''age of the universe''' is the [[cosmological time|time]] elapsed since the [[Big Bang]]. The [[Planck (spacecraft)#2013 data release|best measurement]] of the age of the universe is {{val|13.798|0.037}} billion years ({{val|13.798|0.037|e=9}} years or {{val|4.354|0.012|e=17}} seconds) within the [[Lambda-CDM model|Lambda-CDM concordance model]].<ref name='planck_overview'>
{{cite arXiv
|author=Planck Collaboration
|year=2013
|title=Planck 2013 results. I. Overview of products and scientific results
|class=astro-ph.CO
|eprint=1303.5062
}}</ref><ref name="arxiv-20121220">
{{cite arXiv
|last=Bennett |first=C.L.
|author2=''et al.''
|year=2013
|title=Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results
|eprint=1212.5225
|class=astro-ph.CO
}}</ref> The [[measurement uncertainty|uncertainty]] of 37 million years has been obtained by the agreement of a number of scientific research projects, such as [[microwave background radiation]] [[measurement]]s by the [[Planck satellite]], the [[Wilkinson Microwave Anisotropy Probe]] and other probes. Measurements of the cosmic background radiation give the cooling time of the [[universe]] since the Big Bang,<ref name="arxiv-20121220" /> and measurements of the [[Red shift#Extragalactic observations|expansion rate]] of the universe can be used to calculate its approximate age by extrapolating backwards in time.

== Explanation ==
The [[Lambda-CDM model|Lambda-CDM concordance model]] describes the evolution of the universe from a very uniform, hot, dense primordial state to its present state over a span of about 13.8 billion years<ref>{{cite web
|date=2 April 2013
|title=Cosmic Detectives
|url=http://www.esa.int/Our_Activities/Space_Science/Cosmic_detectives
|publisher=[[European Space Agency]]
|accessdate=2013-04-15
}}</ref> of [[cosmological time]]. This model is well understood theoretically and strongly supported by recent high-precision astronomical observations such as [[WMAP]]. In contrast, theories of the origin of the primordial state remain very speculative. If one extrapolates the Lambda-CDM model backward from the earliest well-understood state, it quickly (within a small fraction of a second) reaches a [[Gravitational singularity|singularity]] called the "Big Bang singularity." This singularity is not understood as having a physical significance in the usual sense, but it is convenient to quote times measured "since the Big Bang" even though they do not correspond to a physically measurable time. For example, "10−6 seconds after the Big Bang" is a well-defined era in the universe's evolution. If one referred to the same era as "13.8 billion years minus 10−6 seconds ago," the precision of the meaning would be lost because the minuscule latter time interval is swamped by uncertainty in the former.

Though the universe might in theory have a longer history, the [[International Astronomical Union]]<ref>
{{cite news
|last=Chang |first=K.
|date=9 March 2008
|title=Gauging Age of Universe Becomes More Precise
|url=http://www.nytimes.com/2008/03/09/science/space/09cosmos.html?_r=1&oref=slogin
|work=[[The New York Times]]
|accessdate=
}}</ref> presently use "age of the universe" to mean the duration of the Lambda-CDM expansion, or equivalently the elapsed time since the Big Bang in the current [[observable universe]].

==Observational limits==
Since the universe must be at least as old as the oldest thing in it, there are a number of observations which put a lower limit on the age of the universe; these include the temperature of the coolest [[white dwarf]]s, which gradually cool as they age, and the dimmest [[turnoff point]] of [[main sequence]] [[stars]] in clusters (lower-mass stars spend a greater amount of time on the main sequence, so the lowest-mass stars that have evolved off of the main sequence set a minimum age).

==Cosmological parameters==
[[Image:Universe.svg|thumb|400px|The age of the universe can be determined by measuring the [[Hubble constant]] today and extrapolating back in time with 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 Ω''m''. Note that the [[accelerating universe]] has the greatest age, while the [[Big Crunch]] universe has the least age.]]
[[File:Age Universe Planck 2013.png|thumb|400px|The value of the age correction factor, ''F'', is shown as a function of two [[cosmology|cosmological parameters]]: the current fractional matter density Ω''m'' and cosmological constant density Ω''Λ''. The [[Lambda-CDM model|best-fit values]] of these parameters are shown by the box in the upper left; the matter-dominated universe is shown by the star in the lower right.]]

The problem of determining the age of the universe is closely tied to the problem of determining the values of the cosmological parameters. Today this is largely carried out in the context of the [[Lambda CDM model|ΛCDM]] model, where the universe is assumed to contain normal (baryonic) matter, cold [[dark matter]], radiation (including both [[photon]]s and [[neutrino]]s), and a [[cosmological constant]]. The fractional contribution of each to the current energy density of the universe is given by the [[density parameter]]s Ω''m'', Ω''r'', and ΩΛ. The full ΛCDM model is described by a number of other parameters, but for the purpose of computing its age these three, along with the [[Hubble constant|Hubble parameter]] <math>H_0</math>, are the most important.

If one has accurate measurements of these parameters, then the age of the universe can be determined by using the [[Friedmann equations|Friedmann equation]]. This equation relates the rate of change in the [[scale factor (cosmology)|scale factor]] ''a''(''t'') to the matter content of the universe. Turning this relation around, we can calculate the change in time per change in scale factor and thus calculate the total age of the universe by [[Integral|integrating]] this formula. The age ''t''0 is then given by an expression of the form
:<math>t_0 = \frac{1}{H_0} F(\Omega_r,\Omega_m,\Omega_\Lambda,\dots) </math>
where <math>H_0</math> is the [[Hubble's law|Hubble parameter]] and the function ''F'' depends only on the fractional contribution to the universe's energy content that comes from various components. The first observation that one can make from this formula is that it is the Hubble parameter that controls that age of the universe, with a correction arising from the matter and energy content. So a rough estimate of the age of the universe comes from the [[Hubble time]], the inverse of the Hubble parameter. With a value for <math>H_0</math> around {{val|68|u=km/s/Mpc}}, the Hubble time evaluates to <math>1/H_0</math> = {{val|14.4}} billion years.<ref>
{{Cite book
|last=Liddle |first=A. R.
|year=2003
|title=An Introduction to Modern Cosmology
|edition=2nd |page=57
|publisher=[[John Wiley & Sons|Wiley]]
|isbn=0-470-84835-9
}}</ref>

To get a more accurate number, the correction factor ''F'' must be computed. In general this must be done numerically, and the results for a range of cosmological parameter values are shown in the figure. For the [[Lambda CDM model|Planck values]] (Ω''m'', Ω''Λ'') = (0.3086, 0.6914), shown by the box in the upper left corner of the figure, this correction factor is about ''F'' = 0.956. For a flat universe without any cosmological constant, shown by the star in the lower right corner, ''F'' = {{Frac|2|3}} is much smaller and thus the universe is younger for a fixed value of the Hubble parameter. To make this figure, Ω''r'' is held constant (roughly equivalent to holding the [[Cosmic Microwave Background|CMB]] temperature constant) and the curvature density parameter is fixed by the value of the other three.

Apart from the Planck satellite, the Wilkinson Microwave Anisotropy Probe ([[WMAP]]) was instrumental in establishing an accurate age of the universe, though other measurements must be folded in to gain an accurate number. [[CMB]] measurements are very good at constraining the matter content Ω''m''<ref>
{{cite web
|last=Hu |first=W.
|title=Animation: Matter Content Sensitivity. The matter-radiation ratio is raised while keeping all other parameters fixed.
|url=http://background.uchicago.edu/%7Ewhu/physics/anim2.html
|publisher=[[University of Chicago]]
|accessdate=2008-02-23
|archiveurl=http://web.archive.org/web/20080223184613/http://background.uchicago.edu/%7Ewhu/physics/anim2.html
|archivedate=23 February 2008 
|deadurl=no
}}</ref> and curvature parameter Ω''k''.<ref name="anim3">
{{cite web
|last=Hu |first=W.
|title=Animation: Angular diameter distance scaling with curvature and lambda
|url=http://background.uchicago.edu/%7Ewhu/physics/anim3.html
|publisher=[[University of Chicago]]
|accessdate=2008-02-23
|archiveurl=http://web.archive.org/web/20080223184618/http://background.uchicago.edu/%7Ewhu/physics/anim3.html
|archivedate=23 February 2008 
|deadurl=no
}}</ref> It is not as sensitive to ΩΛ directly,<ref name="anim3"/> partly because the cosmological constant becomes important only at low redshift. The most accurate determinations of the Hubble parameter ''H''0 come from [[Type Ia supernova]]e. Combining these measurements leads to the generally accepted value for the age of the universe quoted above.

The cosmological constant makes the universe "older" for fixed values of the other parameters. This is significant, since before the cosmological constant became generally accepted, the Big Bang model had difficulty explaining why [[globular cluster]]s in the Milky Way appeared to be far older than the age of the universe as calculated from the Hubble parameter and a matter-only universe.<ref>
{{cite web
|date=1 July 2011
|title=Globular Star Clusters
|url=http://www.seds.org/messier/glob.html
|publisher=[[SEDS]]
|accessdate=2013-07-19
|archiveurl=http://web.archive.org/web/20080224064318/http://seds.org/messier/glob.html
|archivedate=24 February 2008 
|deadurl=no
}}</ref><ref>
{{cite web
|last=Iskander |first=E.
|date=11 January 2006
|title=Independent age estimates
|url=http://www.astro.ubc.ca/people/scott/bbage.html
|publisher=[[University of British Columbia]]
|accessdate=2008-02-23
|archiveurl=http://web.archive.org/web/20080306024809/http://www.astro.ubc.ca/people/scott/bbage.html
|archivedate=6 March 2008 
|deadurl=no
}}</ref> Introducing the cosmological constant allows the universe to be older than these clusters, as well as explaining other features that the matter-only cosmological model could not.<ref>
{{cite arXiv
|title=Cosmic Concordance
|last1=Ostriker |first1=J. P.
|last2=Steinhardt |first2=P. J.
|year=1995
|class=astro-ph
|eprint=astro-ph/9505066
}}</ref>

==WMAP==
[[NASA]]'s [[Wilkinson Microwave Anisotropy Probe]] (WMAP) project's [[Wilkinson Microwave Anisotropy Probe#Nine-year data release|nine-year data release]] in 2012 estimated the age of the universe to be {{val|13.772|0.059|e=9}} years (13.772 billion years, with an uncertainty of plus or minus 59 million years).<ref name="arxiv-20121220" />

However, this age is based on the assumption that the project's underlying model is correct; other methods of estimating the age of the universe could give different ages. Assuming an extra background of relativistic particles, for example, can enlarge the error bars of the WMAP constraint by one order of magnitude.<ref>
{{cite journal
|last1=de Bernardis |first1=F.
|last2=Melchiorri |first2=A.
|last3=Verde |first3=L.
|last4=Jimenez |first4=R.
|year=2008
|title=The Cosmic Neutrino Background and the Age of the Universe
|journal=[[Journal of Cosmology and Astroparticle Physics]]
|volume=2008 |issue=3 |pages=20
|arxiv=0707.4170
|bibcode=2008JCAP...03..020D
|doi=10.1088/1475-7516/2008/03/020
}}</ref>

This measurement is made by using the location of the first acoustic peak in the [[cosmic microwave background radiation|microwave background]] power spectrum to determine the size of the decoupling surface (size of the universe at the time of recombination). The light travel time to this surface (depending on the geometry used) yields a reliable age for the universe. Assuming the validity of the models used to determine this age, the residual accuracy yields a margin of error near one percent.<ref name="wmap">
{{cite journal
|first=D. N. |last=Spergel
|author2=''et al.''
|year=2003
|title=First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters
|journal=[[The Astrophysical Journal Supplement Series]]
|volume=148 |issue=1 |pages=175–194
|arxiv=astro-ph/0302209
|bibcode=2003ApJS..148..175S
|doi=10.1086/377226
}}</ref>

==Planck==
In 2013, the [[European Space Agency]]'s [[Planck (spacecraft)|Planck spacecraft]] team estimated the age of the universe to be {{val|13.82}} billion years,<ref name="ESA-20130321">
{{cite web
|author=Staff
|title=Planck Reveals An Almost Perfect Universe
|url=http://www.esa.int/Our_Activities/Space_Science/Planck/Planck_reveals_an_almost_perfect_Universe
|publisher=[[European Space Agency]]
|date=21 March 2013
|accessdate=2013-03-21
}}</ref><ref name="NASA-20130321">
{{cite web
|last1=Clavin |first1=W.
|last2=Harrington |first2=J. D.
|title=Planck Mission Brings Universe Into Sharp Focus
|url=http://www.jpl.nasa.gov/news/news.php?release=2013-109&rn=news.xml&rst=3739
|date=21 March 2013
|publisher=[[NASA]]
|accessdate=2013-03-21
}}</ref><ref name="NYT-20130321">
{{cite news
|last=Overbye |first=D.
|date=21 March 2013
|title=An Infant Universe, Born Before We Knew
|url=http://www.nytimes.com/2013/03/22/science/space/planck-satellite-shows-image-of-infant-universe.html
|work=[[New York Times]]
|accessdate=2013-03-21
}}</ref><ref name="NBC-20130321">
{{cite web
|last=Boyle |first=A.
|date=21 March 2013
|title=Planck probe's cosmic 'baby picture' revises universe's vital statistics
|url=http://cosmiclog.nbcnews.com/_news/2013/03/21/17397298-planck-probes-cosmic-baby-picture-revises-universes-vital-statistics
|work=[[NBC News]]
|accessdate=2013-03-21
}}</ref> slightly higher but within the uncertainties of the earlier number derived from the WMAP data. By combining the Planck data with previous missions, the best combined estimate of the age of the universe is [[Planck (spacecraft)#2013 data release|{{val|13.798|0.037|e=9|u=years}} old]].<ref name="planck_overview" />

<center>
{| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-size: 70%; text-align:center;"
|- bgcolor="#B0C4DE" align="center"
|+ [[Lambda-CDM model|Cosmological parameters]] from 2013 Planck results<ref name="planck_overview" /><ref name='planck_cosmological_parameters'>
{{cite arXiv
|author=Planck collaboration
|year=2013
|title=Planck 2013 results. XVI. Cosmological parameters
|class=astro-ph.CO
|eprint=1303.5076
}}</ref>
! Parameter !! Symbol !! Planck Best fit !! Planck 68% limits !! Planck+lensing Best fit !! Planck+lensing 68% limits !! Planck+WP Best fit !! Planck+WP 68% limits !! Planck+WP +HighL Best fit !! Planck+WP +HighL 68% limits !! Planck+lensing +WP+highL Best fit !! Planck+lensing +WP+highL 68% limits !! Planck+[[WMAP|WP]] +highL+[[baryon acoustic oscillations|BAO]] Best fit !! Planck+[[WMAP|WP]] +highL+[[baryon acoustic oscillations|BAO]] 68% limits
|-
| Age of the universe (Ga) || <math>t_0</math> || 13.819 || {{val|13.813|0.058}} || 13.784 || {{val|13.796|0.058}} || 13.8242 || {{val|13.817|0.048}} || 13.8170 || {{val|13.813|0.047}} || 13.7914 || {{val|13.794|0.044}} || 13.7965 || {{val|13.798|0.037}}
|-
| [[Hubble's constant]] ( {{frac|km|Mpc·s}} ) || <math>H_0</math> || 67.11 || {{val|67.4|1.4}} || 68.14 || {{val|67.9|1.5}} || 67.04 || {{val|67.3|1.2}} || 67.15 || {{val|67.3|1.2}} || 67.94 || {{val|67.9|1.0}}|| 67.77 || {{val|67.80|0.77}}
|}
</center>
{{-}}

==Assumption of strong priors==
Calculating the age of the universe is accurate only if the assumptions built into the models being used to estimate it are also accurate. This is referred to as [[strong priors]] and essentially involves stripping the potential errors in other parts of the model to render the accuracy of actual observational data directly into the concluded result. Although this is not a valid procedure in all contexts (as noted in the accompanying caveat: "based on the fact we have assumed the underlying model we used is correct"), the age given is thus accurate to the specified error (since this error represents the error in the instrument used to gather the raw data input into the model).

The age of the universe based on the best fit to [[Planck (spacecraft)#2013 data release|Planck 2013 data]] alone is {{val|13.813|0.058}} billion years (the other estimate of {{val|13.798|0.037}} billion years uses Gaussian [[Prior probability|prior]]s based on earlier estimates from other studies to determine the combined uncertainty). This number represents the first accurate "direct" measurement of the age of the universe (other methods typically involve [[Hubble's law]] and the age of the oldest stars in globular clusters, etc.). It is possible to use different methods for determining the same parameter (in this case – the age of the universe) and arrive at different answers with no overlap in the "errors". To best avoid the problem, it is common to show two sets of uncertainties; one related to the actual measurement and the other related to the systematic errors of the model being used.

An important component to the analysis of data used to determine the age of the universe (e.g. from [[Planck (spacecraft)|Planck]]) therefore is to use a [[Bayesian statistics|Bayesian statistical]] analysis, which normalizes the results based upon the priors (i.e. the model).<ref name="wmap" /> This quantifies any uncertainty in the accuracy of a measurement due to a particular model used.<ref>
{{cite conference
|last=Loredo |first=T. J.
|year=1992
|title=The Promise of Bayesian Inference for Astrophysics
|url=http://www.astro.cornell.edu/staff/loredo/bayes/promise.pdf
|editor1-last=Feigelson |editor1-first=E. D.
|editor2-last=Babu |editor2-first=G. J.
|booktitle=Statistical Challenges in Modern Astronomy
|pages=275–297
|publisher=[[Springer-Verlag]]
|bibcode=1992scma.conf..275L
|doi=10.1007/978-1-4613-9290-3_31
|isbn=978-1-4613-9292-7
}}</ref><ref>
{{cite journal
|last1=Colistete |first1=R.
|last2=Fabris |first2=J. C.
|last3=Concalves |first3=S. V. B.
|year=2005
|title=Bayesian Statistics and Parameter Constraints on the Generalized Chaplygin Gas Model Using SNe ia Data
|journal=[[International Journal of Modern Physics D]]
|volume=14 |issue=5 |pages=775–796
|arxiv=astro-ph/0409245
|bibcode=2005IJMPD..14..775C
|doi=10.1142/S0218271805006729
}}</ref>

==History==
{{Expand section|date=February 2014|1= The estimated age of the universe for certain crucial points in history, e.g.:
* before Einstein
* after Einstein
* before Hubble
* after Hubble
* etc}}
In the 18th century, the concept that the [[age of the Earth]] was millions, if not billions, of years began to appear. However, most scientists throughout the 19th century and into the first decades of the 20th century presumed that the universe itself was [[Steady State theory|Steady State]] and eternal, with maybe stars coming and going but no changes occurring at the largest scale known at the time.

The first scientific theories indicating that the age of the universe might be finite were the studies of [[thermodynamics]], formalized in the mid-19th century. The concept of [[entropy]] dictates that if the universe (or any other closed system) were infinitely old, then everything inside would be at the same temperature, and thus there would be no stars and no life. No scientific explanation for this contradiction was put forth at the time. In 1915 [[Albert Einstein]] published the theory of [[general relativity]].<ref>
{{cite journal
|last=Einstein |first=A.
|year=1915
|title=Zur allgemeinen Relativitätstheorie
|journal=[[Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften]]
|pages=778–786
|language=German
|bibcode=1915SPAW.......778E
}}</ref> Based on Einstein's theory, [[Georges Lemaître]]'s work showed that the universe cannot be static and must be either expanding or contracting. Einstein himself did not believe this result and so he added what he called a [[cosmological constant]] to his equations in an unsuccessful attempt to produce a theory consistent with a steady state universe.

The first direct observational evidence that the universe has a finite age came from the observations of astronomer [[Edwin Hubble]] published in 1929.<ref>
{{cite journal
|last=Hubble |first=E.
|year=1929
|title=A relation between distance and radial velocity among extra-galactic nebulae
|journal=[[Proceedings of the National Academy of Sciences]]
|volume=15 |issue=3 |pages=168–173
|url=http://www.pnas.org/cgi/reprint/15/3/168
|bibcode=1929PNAS...15..168H
|doi=10.1073/pnas.15.3.168
|pmid=16577160
|pmc=522427
}}</ref> Earlier in the 20th century, Hubble and others resolved individual stars within certain [[nebula]]e, thus determining that they were [[Galaxy|galaxies]], similar to, but external to, our [[Milky Way Galaxy]]. In addition, these galaxies were very large and very far away. [[electromagnetic spectrum|Spectra]] taken of these distant galaxies showed a [[red shift]] in their [[spectral lines]] presumably caused by the [[Doppler effect]], thus indicating that these galaxies were moving away from the Earth. In addition, the farther away these galaxies seemed to be, the greater the redshift and thus the faster they seemed to be moving away. This was the first direct evidence that the universe is not static but expanding. The first estimate of the age of the universe came from the calculation of when all of the objects must have started speeding out from the same point. Hubble's initial value for the universe's age was very low, as the galaxies were assumed to be much closer than later observations found them to be.

The first reasonably accurate measurement of the rate of expansion of the universe, a numerical value now known as the [[Hubble constant]], was made in 1958 by astronomer [[Allan Sandage]].<ref>
{{cite journal
|last1=Sandage |first1=A. R.
|title=Current Problems in the Extragalactic Distance Scale
|year=1958
|journal=[[The Astrophysical Journal]]
|volume=127 |issue=3 |pages=513–526
|bibcode=1958ApJ...127..513S
|doi=10.1086/146483
}}</ref> His measured value for the Hubble constant came very close to the value range generally accepted today.

However Sandage, like Einstein, did not believe his own results at the time of discovery. His value for the age of the universe{{Elucidate|date=February 2014|reason=What was his value for the age of the universe?}} was too short to reconcile with the 25-billion-year age estimated at that time for the oldest known [[star]]s. Sandage and other astronomers repeated these measurements numerous times, attempting to reduce the Hubble constant and thus increase the resulting age for the universe. Sandage even proposed new theories of [[cosmogony]] to explain this discrepancy. This issue was finally resolved by improvements in the theoretical models used for estimating the ages of stars. As of 2013, using the latest models for stellar evolution, the estimated age of the [[HD 140283|oldest known star]] is {{val|14.46|0.8}} billion years.<ref name=arxiv>
{{cite journal
|last1=Bond |first1=H. E.
|last2=Nelan |first2=E. P.
|last3=Vandenberg|first3=D. A.
|last4=Schaefer|first4=G. H.
|last5=Harmer |first5=D.
|title=HD 140283: A Star in the Solar Neighborhood that Formed Shortly After the Big Bang
|year=2013
|journal=[[The Astrophysical Journal]]
|volume=765 |pages=L12 |issue=12
|arxiv=1302.3180
|bibcode=2013ApJ...765L..12B
|doi=10.1088/2041-8205/765/1/L12
}}</ref>

The discovery of [[microwave]] [[cosmic background radiation]] announced in 1965<ref name="apj142:419">
{{cite journal
|last=Penzias |first=A. A.
|last2=Wilson |first2=R .W.
|year=1965
|title=A Measurement of Excess Antenna Temperature at 4080 Mc/s
|journal=[[The Astrophysical Journal]]
|volume=142 |pages=419–421
|doi=10.1086/148307
|bibcode=1965ApJ...142..419P
}}</ref> finally brought an effective end to the remaining scientific uncertainty over the expanding universe. The recently launched space probes WMAP, launched in 2001, and Planck, launched in 2009, produced data that determines the Hubble constant and the age of the universe independent of galaxy distances, removing the largest source of error.<ref name="wmap"/>

== See also ==
{{Portal|Astronomy}}
* [[Age crisis]]
* [[Age of the Earth]]
* [[Anthropic principle]]
* [[Cosmology]]
* [[Hubble Deep Field]]
* [[Metric expansion of space]]
* [[Multiverse]]
* [[Observable universe]]
* [[Red Shift#Observations in astronomy|Red shift observations in astronomy]]
* [[Static universe]]
* [[The First Three Minutes: A Modern View of the Origin of the Universe]] an essay written by [[Steven Weinberg]] and published in 1977
* [[Dark Ages Radio Explorer (DARE)]]

==References==
{{Reflist|30em}}

==External links==
*[http://www.astro.ucla.edu/~wright/cosmolog.htm Ned Wright's Cosmology Tutorial]
*{{cite web | url =http://www.astro.ucla.edu/~wright/age.html | first =Edward L. | last =Wright | title =Age of the Universe | date =2 July 2005}}
*Wayne Hu's [http://background.uchicago.edu/~whu/metaanim.html cosmological parameter animations]
*{{cite arXiv|eprint=astro-ph/9505066|author1=Ostriker|author2=Steinhardt|title=Cosmic Concordance|class=astro-ph|year=1995}}
*SEDS page on [http://www.seds.org/messier/glob.html "Globular Star Clusters"]
*Douglas Scott [http://www.astro.ubc.ca/people/scott/bbage.html "Independent Age Estimates"]
*KryssTal [http://www.krysstal.com/scale.html "The Scale of the Universe"] Space and Time scaled for the beginner.
*[http://icosmos.co.uk/ iCosmos: Cosmology Calculator (With Graph Generation )]
*[http://www.aip.org/history/cosmology/ideas/expanding.htm The Expanding Universe] (American Institute of Physics)

{{DEFAULTSORT:Age Of The Universe}}
[[Category:Universe]]
[[Category:Physical cosmology]]
[[Category:Big Bang]]