Jeans analysis 	of self-gravitating systems 	

in f(R)- gravity	

PRD 85, 044022 (2012)	

Mariafelicia De Laurentis	

In collaboration with	

S. Capozziello, I. De Martino, 	

M. Fromisano, S.D. Odintsov,	

KMI-IEEC Joint International Workshop,	

Nagoya, August 03, (2013).	

Outlines	Dust- dominated self-gravitating systems	

The Newtonian limit of f(R)-gravity	

Jeans criterion for gravitational instability in f(R)-gravity	

The Jeans mass limit in f(R)-gravity	

Discussion and conclusions	

Next steps	Dust- dominated self-gravitating systems	

The collapse of self-gravitational collisionless systems 	

can be dealt with the introduction of coupled collisionlessBoltzmann and Poisson equations	threedimensional
vectors in thespatial manifold	

A self-gravitating system at equilibrium is described by a time-independent
distribution function f0 (x, v) and a potential Φ0 (x) that are solutions of above
equations	

J. Binney and S. Tremaine, Galactic Dynamics (Princeton University Press, Princeton, NJ, 1994).	

Dust- dominated self-gravitating systems	

Considering a small perturbation to this equilibrium:	

where ε<< 1 and	by substituting in Boltzmann and Poisson equations and by	linearizing, one obtains:	

Dust- dominated self-gravitating systems	

Since the equilibrium state is assumed to be homogeneous	and time-independent, one can set f0 (x,v,t ) = f(v), and theso-called Jeans “swindle” to set Φ0 = 0 	

In Fourier components	

By combining these equations, we obtain the dispersion relation	

Dust- dominated self-gravitating systems	

In the case of stellar systems, by assuming a Maxwellian
distribution function for f0 we have	

By setting ω = 0, the limit for instability is obtained:	

by which it is possible to deﬁne the Jeans mass (MJ) as the mass originally contained
within a sphere of diameter λJ:	

where	is the Jeans length	….and then we can write	Dust- dominated self-gravitating systems	

In order to evaluate the integral in the dispersion relation, wehave to study the singularity at ω = k vx. To this end, it isuseful to write the dispersion relation as	deﬁning	

Whereand 	We set also
andin the unstable modes 	because we are interested

These modes appear when the imaginary part of ω is greater than zero and in thiscase the integral in the dispersion relation can be resolved just with previous	prescriptions.	

Dust- dominated self-gravitating systems	

In order to study unstable models we replace thefollowing identities	

into the dispersion relation obtaining:	

This is the standard dispersion relation describing the criterion to collapse
for inﬁnite homogeneous ﬂuid and stellar systems	

The Newtonian limit of f(R) - gravity	

Field equations in f(R)-gravity give rise to the modiﬁed	

Poisson equations. We know that	

Also we well konwn that 	Ψ is the further gravitational potential related to the	metric component g (2)ii 	

…and then the ﬁeld equations assume this form	

S. Capozziello, M. De Laurentis Phys. Rep. 509, 167-321 (2011)	

S. Capozziello, M. De Laurentis Ann. Phys. 524, 545 (2012)	  Jeans criterion for gravitational instability in f(R)-gravity	Let us assume the standard collisionless Boltzmann equation:	

where, according to the Newtonian theory, only the potentia Φ ispresent	Considering the f(R) Poisson equations, also the potential Ψ
has to be considered so we obtain the coupled equations	

we have replaced f’’(0) with the greek letter α	Jeans criterion for gravitational instability in f(R)-gravity	

As in standard case, we consider small perturbation to
the equilibrium and linearize the equations and inFourier space so they became	Jeans criterion for gravitational instability in f(R)-gravity	

Combining the above equations we obtain a relation between Φ1and Ψ1	

And then the dispersion relation is		  As in standard case, one can write	

By eliminating the higher-order terms (imposing α = 0),	

we obtain again the standard dispersion	Jeans criterion for gravitational instability in f(R)-gravity	

In order to compute the integral in the dispersion relation , we	consider the same approach used in the classical case, and	ﬁnally we obtain:	

Whereand 	To compare the modiﬁed and calssical dispersion relation we to 	

normalize the equation to the classical Jeans length by ﬁxing	

the parameter of f(R)- gravity, that is	This parameterization is correct because the dimension (an inverse ofsquared length) allows us to parametrize as in standard case	Jeans criterion for gravitational instability in f(R)-gravity	

Finally we write and plot this relation	

The bold line
indicates the plotof the modiﬁed
dispersion	

relation.	

The thin line
indicates the plotof the standard
dispersion	equation	

The Jeans mass limit 	

in f(R)-gravity	

A numerical estimation of the f(R) instability length in terms of the standard
Newtonian one can be achieved	

By solving numerically the above equation with the condition ω = 0, we obtain that thecollapse occurs for	

However we can estimate also analytically the limit for the instability	

In order to evaluate the Jeans mass limit in f(R)- gravity, we set ω = 0	The additional condition α < 0 discriminates the class of viable f(R) models: in such a case
we obtain stable cosmological solution and positively deﬁned massive states	

The Jeans mass limit 	

in f(R)-gravity	This α<0 condition selects the physically viable models allowing to solve the above
equation for real values of k. 	

In particular, the above numerical solution can be recast as	

The relation to the Newtonian value of the Jeans instability is	Now, we can deﬁne the new Jeans mass as	

which is proportional to the standard Newtonian value	We will confront this speciﬁc solutions with some observed structures.	

The MJ – T relation	

One can deal with the star formation problem in two ways:	

we can take into account the formation of individual stars and	

we can discuss the formation of the whole star system starting frominterstellar clouds	

To answer these problems it is very important to study then 	interstellar medium (ISM) and its properties	

The ISM physical conditions in the galaxies change in a very wide	range, from hot X-ray emitting plasma to cold molecular gas, so it is very
complicated to classify the ISM by its properties	The MJ – T relation	

However, we can distinguish, in the ﬁrst approximation, between	

Diﬀuse hydrogen clouds. The most powerful tool to	measure the properties of these clouds is the 21 cm	

line emission of HI. They are cold clouds so the	temperature is in the range 10 ÷ 50 K, and their	

extension is up to 50 ÷ 100 kpc from galactic center	

Diﬀuse molecular clouds are generally self-gravitating,	

magnetized, turbulent ﬂuids systems, observed in sub-mm.
The most of the molecular gas is H2, and the rest is CO.
Here, the conditions are very similar to the HI clouds but
in this case, the cloud can be more massive. They have,typically, masses in the range 3 ÷ 100 M¤, temperature in
15 ÷ 50 K and particle density in (5 ÷ 50)×	108 m-3.	

The MJ – T relation	

Giant molecular clouds are very large complexes of particles (dust and gas),in which the range of the masses is typically 105 ÷ 106M¤ but they are verycold. 	

The temperature is ≈15 K, and the number of particles is (1 ÷ 3)× 108 m-3 .
However, there exist also small molecular clouds with masses M< 104 M¤ .
They are the best sites for star formation, despite the mechanism offormation does not recover the star formation rate that would be 250M¤ yr-1 	

The MJ – T relation	

HII regions. They are ISM regions with temperatures in the range 103 ÷
104 K, emitting primarily in the radio and IR regions. At low frequencies,observations are associated to free-free electron transition (thermalBremsstrahlung). Their densities range from over a million particles per cm3	in the ultracompact H II regions to only a few particles per cm3 in thelargest and most extended regions. This implies total masses between 102
and 105 M¤	Bok globules are dark clouds of dense
cosmic dust and gas in which starformation sometimes takes place. Bokglobules are found within H II regions,and typically have a mass of about 2 to50 M¤ contained within a region ofabout a light year.	

The MJ – T relation	

Using very general conditions, we want to show the diﬀerence in the Jeans mass
value between standard and f(R)- gravity.	

Let us take into account	

in which ρ0 is the ISM density and σ is the velocity dispersion of particles dueto the temperature	

These two quantities are deﬁned asand 	

where nH is the number of particles measured in m-3, is the mean molecularweight, kB is the Boltzmann constant and mH is the proton mass	

By using these relations, we are able to compute the Jeans mass for interstellar
clouds and to plot its behavior against the temperature	The MJ – T relation	

Any astrophysical system reported in Table is associated to a particular	(MJ – T)-region.	

Diﬀerences between the two theories for any self-gravitating system 	are clear	  The MJ – T relation	Dashed-line indicates the Newtonian Jeans mass behavior with respect tothe temperature.	

Continue-line indicates the same for f(R)-gravity Jeans mass.	

The MJ – T relation	

By referring to the catalog ofmolecular clouds in RomanDuval et al., Astrophys. J.
723, 492 (2010), we have	

calculated the Jeans mass in
the Newtonian and f(R)cases.	

In all cases we note a
substantial	

diﬀerence between theclassical and f.R) value.	

Discussion and Conclusions	Here we have analyzed the Jeans instability mechanism, adopted for starformation, considering the Newtonian approximation of f(R) gravity	

The related Boltzmann-Vlasov system leads to modiﬁed Poisson equations
depending on the f(R) model	

In particular, it is possible to get a new dispersion relation whereinstability criterion results modiﬁed	

The leading parameter is α, i.e. the second derivative of the speciﬁc f(R)model. Standard Newtonian Jeans instability is immediately recovered
for α=0 corresponding to the Hilbert-Einstein Lagrangian of GR.	

A new condition for the gravitational instability is derived, showing
unstable modes with faster growth rates.	

Discussion and Conclusions	Finally we can observe the instability decrease in f(R)- gravity: suchdecrease is related to a larger Jeans length and then to a lower Jeans mass	

We have also compared the behavior with the temperature of the Jeans	mass for various types of interstellar molecular clouds	

In our model the limit (in unit of mass) to start the collapse of aninterstellar cloud is lower than the classical one advantaging the structureformation.	

Real solutions for the Jean mass can be achieved only for α < 0 andthis result is in agreement with cosmology	In particular, the condition α<0 is essentials to have a well formulated	

and well-posed Cauchy problem in f(R)- gravity	

Finally, it is worth noticing that the Newtonian value is an upper limit
for the Jean mass coinciding with f(R.) = R	Discussion and Conclusions	

It is important to stress that we fully recover the standard collapse
mechanisms but we could also describe proto-stellar systems that escapethe standard collapse model	On the other hand, this is the ﬁrst step to study star formation in alternative
theories of gravity	Next Steps	

From an observational point of view, reliable constraints can be achieved
from a careful analysis of the proto-stellar phase taking into accountmagnetic ﬁelds, turbulence and collisions	Addressing stellar systems by this approach could be an extremely importantto test observationally f(R) gravity	

Moreover, the approach developed in this work admits direct generalizations
for other modiﬁed gravities, like nonlocal gravity, modiﬁed Gauss-Bonnet
theory, string inspired gravity, etc.	

Developing further this approach gives, in general, the possibility toconfront the observable dynamics of astrophysical objects (like stars) withpredictions of alternative gravities.