We develop a field theoretical approach to the cold interstellar medium ( ISM ) and large structure of the universe .
We show that a non-relativistic self-gravitating gas in thermal equilibrium with variable number of atoms or fragments is exactly equivalent to a field theory of a single scalar field \phi ( { \vec { x } } ) with exponential self-interaction .
We analyze this field theory perturbatively and non-perturbatively through the renormalization group approach .
We show scaling behaviour ( critical ) for a continuous range of the temperature and of the other physical parameters .
We derive in this framework the scaling relation M ( R ) \sim R ^ { d _ { H } } for the mass on a region of size R , and \Delta v \sim R ^ { q } for the velocity dispersion where q = \frac { 1 } { 2 } ( d _ { H } -1 ) .
For the density-density correlations we find a power-law behaviour for large distances \sim| { \vec { r } _ { 1 } } - { \vec { r } _ { 2 } } | ^ { 2 d _ { H } -6 } .
The fractal dimension d _ { H } turns to be related with the critical exponent \nu of the correlation length by d _ { H } = 1 / \nu .
The renormalization group approach for a single component scalar field in three dimensions states that the long-distance critical behaviour may be governed by the ( non-perturbative ) Ising fixed point .
The Ising values of the scaling exponents are \nu = 0.631... , d _ { H } = 1.585... and q = 0.293... .
Mean field theory yields for the scaling exponents \nu = 1 / 2 , d _ { H } = 2 and q = 1 / 2 .
Both the Ising and the mean field values are compatible with the present ISM observational data : 1.4 \leq d _ { H } \leq 2 , 0.3 \leq q \leq 0.6 .
As typical in critical phenomena , the scaling behaviour and critical exponents of the ISM can be obtained without dwelling into the dynamical ( time-dependent ) behaviour .
We develop a field theoretical approach to the galaxy distribution .
We consider a gas of self-gravitating masses on the FRW background , in quasi-thermal equilibrium .
We show that it exhibits scaling behaviour by renormalization group methods .
The galaxy correlations are first computed assuming homogeneity for very large scales and then without assuming homogeneity .
In the first case we find \xi ( r ) \equiv < \rho ( { \vec { r } _ { 0 } } ) \rho ( { \vec { r } _ { 0 } } + { \vec { r } } ) > / < \rho > ^ { 2 } -1 % \sim r ^ { - \gamma } , with \gamma = 2 .
In the second case we find D ( r ) = < \rho ( { \vec { r } _ { 0 } } ) \rho ( { \vec { r } _ { 0 } } + { \vec { r } } ) > \sim r ^ { - \Gamma } with \Gamma = 1 .
While the universe becomes more and more homogeneous at large scales , statistical analysis of galaxy catalogs have revealed a fractal structure at small-scales ( \lambda < 100 h ^ { -1 } Mpc ) , with a fractal dimension D = 1.5 - 2 ( Sylos Labini et al 1996 ) .

We study the thermodynamics of a self-gravitating system with the theory of critical phenomena and finite-size scaling and show that gravity provides a dynamical mechanism to produce this fractal structure .
Only a limited , ( although large ) , range of scales is involved , between a short-distance cut-off below which other physics intervene , and a large-distance cut-off , where the thermodynamic equilibrium is not satisfied .
The galaxy ensemble can be considered at critical conditions , with large density fluctuations developping at any scale .
From the theory of critical phenomena , we derive the two independent critical exponents \nu and \eta and predict the fractal dimension D = 1 / \nu to be either 1.585 or 2 , depending on whether the long-range behaviour is governed by the Ising or the mean field fixed points , respectively .
Both set of values are compatible with present observations .
In addition , we predict the scaling behaviour of the gravitational potential to be r ^ { - \frac { 1 } { 2 } ( 1 + \eta ) } .
That is , r ^ { -0.5 } for mean field or r ^ { -0.519 } for the Ising fixed point .
The theory allows to compute the three and higher density correlators without any assumption or Ansatz .
We find that the connected N -points density scales as r _ { 1 } ^ { N ( D - 3 ) } , when r _ { 1 } > > r _ { i } , 2 \leq i \leq N .
There are no free parameters in this theory .