Macroscopic Behaviours of Palatini Modified Gravity Theories [gr-qc] and [gr-qc] Baojiu Li, David F. Mota & Douglas J. Shaw Portsmouth, June 23, 2008

Palatini Modified Gravity Theories Recently considered as an alternative to Λ as the explanation to the accelerated cosmic expansion, but has a much longer history… Recently considered as an alternative to Λ as the explanation to the accelerated cosmic expansion, but has a much longer history… The Ricci scalar in the Einstein-Hilbert action replaced by a function f(R, R ab R ab, R abcd R abcd ). The Ricci scalar in the Einstein-Hilbert action replaced by a function f(R, R ab R ab, R abcd R abcd ). Variation of the action taken wrt the metric g ab and the connection field Γ a bc, which are considered to be independent, respectively. Variation of the action taken wrt the metric g ab and the connection field Γ a bc, which are considered to be independent, respectively. Field equations second order, avoiding stability problems… Field equations second order, avoiding stability problems…

Palatini Modified Gravity Theories Reduce to standard General Relativity when f = R - 2Λ, otherwise leads to different field equations from the latter. Reduce to standard General Relativity when f = R - 2Λ, otherwise leads to different field equations from the latter. In this talk I will concentrate on Palatini f(R) theory, bearing in mind that Palatini f(R ab R ab ) theories share many similarities if is more complicated in formula… In this talk I will concentrate on Palatini f(R) theory, bearing in mind that Palatini f(R ab R ab ) theories share many similarities if is more complicated in formula… However, there is some difference between these two theories in special cases like a Universe dominated by radiation (photons) as we shall see below… However, there is some difference between these two theories in special cases like a Universe dominated by radiation (photons) as we shall see below… Time limited, so only the idea is presented, not any mathematical details Time limited, so only the idea is presented, not any mathematical details

Field Equations Metric that couples to matter Trace Varying action wrt the connection: Varying action wrt the metric: Algebraic

Field Equations It is possible to write the field equations in terms of the barred metric, as could be found in many references. However, those equations involves 2 nd order derivatives of F(R), and is thus complicated. Instead we write all the field equations in terms of the unbarred metric. This is not the metric coupling to matter, but due to the conformal relation between these two metrics the quantities in terms of the barred metric are easy to recover… The Einstein tensor in terms of the unbarred metric f ’ = F

Microscopic Viewpoint Since we start from an action, it is more appropriate to think the above field equations as microscopic ones, i.e., they are only surely correct on small scales. On large scales, however, we need to be careful to use an averaged version of them. Since we start from an action, it is more appropriate to think the above field equations as microscopic ones, i.e., they are only surely correct on small scales. On large scales, however, we need to be careful to use an averaged version of them. On small scales (at classical level), matter can be treated as a collection of small rigid particles which has the density of order nuclear density and are distributed in vacuum. The particles are so small that the majority of the space is vacuum (except in extreme environments like neutron stars…). On small scales (at classical level), matter can be treated as a collection of small rigid particles which has the density of order nuclear density and are distributed in vacuum. The particles are so small that the majority of the space is vacuum (except in extreme environments like neutron stars…).

Two Averaging Approaches: 1 Averaged energy density Volume V Particle size S, density ρ p N particles in V Apply field equations to the averaged configuration

Two Averaging Approaches: 2 In vacuum, from the algebraic field equation Φ is a constant, so that V(Φ) and f’(Φ) are also constants – we retrieve a cosmological constant. Inside particles the local energy density determines Φ again via the algebraic field equation. V(Φ) and f’(Φ) thus take different values from the vacuum, and these values vary inside the particles indeed…

Two Averaging Approaches: 2 In our averaging approach, we apply the field equations to the particles and the vacuum respectively firstly to express any quantity we want to calculate in terms of the local energy density, and then average the quantity itself to obtain its large-scale averaged value. In our averaging approach, we apply the field equations to the particles and the vacuum respectively firstly to express any quantity we want to calculate in terms of the local energy density, and then average the quantity itself to obtain its large-scale averaged value. In old averaging approach, people average the energy density firstly, and then apply the field equations to calculate physical quantities in terms of the averaged energy density. In old averaging approach, people average the energy density firstly, and then apply the field equations to calculate physical quantities in terms of the averaged energy density. In GR due to the linearity of field equations the above two approaches lead to the same result. In general because of the microscopic configuration depicted above, the former approach should be more appropriate… In GR due to the linearity of field equations the above two approaches lead to the same result. In general because of the microscopic configuration depicted above, the former approach should be more appropriate…

Average: How and What? The most direct and simplest approach: volume average The most direct and simplest approach: volume average In most cases, except things like neutron star, the volume occupied by particles is so much smaller than that is occupied by vacuum, while the values of the quantity Q inside and outside the particles are of same order. Thus the averaged value of Q is almost the same as its vacuum value. In most cases, except things like neutron star, the volume occupied by particles is so much smaller than that is occupied by vacuum, while the values of the quantity Q inside and outside the particles are of same order. Thus the averaged value of Q is almost the same as its vacuum value. The large-scale averaged theory should behave as LCDM. The large-scale averaged theory should behave as LCDM.

Average: How and What? If one uses the naïve averaging process, the theory is not so trivial – there people have shown that the model will predict so different CMB and matter power spectra from LCDM that it is almost ruled out by observations. If one uses the naïve averaging process, the theory is not so trivial – there people have shown that the model will predict so different CMB and matter power spectra from LCDM that it is almost ruled out by observations. We now save the model by showing that it is trivial… We now save the model by showing that it is trivial… Note that the Palatini gravity is a different theory from GR, despite of their same behaviour on large scales. Due to the differences in the field equations, the internal structures of a particle are different in these two theories. Note that the Palatini gravity is a different theory from GR, despite of their same behaviour on large scales. Due to the differences in the field equations, the internal structures of a particle are different in these two theories.

Average: How and What? In contrast to what is claimed by some authors, we find that the WEP is not violated; the active gravitational, passive gravitational and inertial masses of a particle are the same, which anyway is the mass we actually measure In contrast to what is claimed by some authors, we find that the WEP is not violated; the active gravitational, passive gravitational and inertial masses of a particle are the same, which anyway is the mass we actually measure The spacetime outside a spherical particle is exactly Einstein-de-Sitter, with the metric determined by that mass. No fifth force between different particles. No way to distinguish between Palatini and GR classically. The spacetime outside a spherical particle is exactly Einstein-de-Sitter, with the metric determined by that mass. No fifth force between different particles. No way to distinguish between Palatini and GR classically. There are ways to distinguish (in fact constrain Palatini gravity) with quantum effects… There are ways to distinguish (in fact constrain Palatini gravity) with quantum effects…

Radiation: there is a difference Photon (EM) field is important at early times. Photon (EM) field is important at early times. Unlike classical particle, that are tiny entities in between of which is vacuum, EM field permeates everywhere in the space. Unlike classical particle, that are tiny entities in between of which is vacuum, EM field permeates everywhere in the space. We can use a naïve averaging process taking into account that the electric and magnetic components of the EM field orient randomly… We can use a naïve averaging process taking into account that the electric and magnetic components of the EM field orient randomly… In principal, nontrivial large-scale behaviours should be found, as we have found for Palatini f(R ab R ab ) theory. For Palatini f(R), however, from the field equations we see that what matters is only the combination ρ-3p (the trace), which is always zero for EM field, so the averaged behaviour is still same as GR. In principal, nontrivial large-scale behaviours should be found, as we have found for Palatini f(R ab R ab ) theory. For Palatini f(R), however, from the field equations we see that what matters is only the combination ρ-3p (the trace), which is always zero for EM field, so the averaged behaviour is still same as GR.

Conclusion The cosmological and astrophysical behaviours of the Palatini modified gravity theories are the same as those of GR, if the system, whose energy density is much smaller than nuclear density, is dominated by particles. The cosmological and astrophysical behaviours of the Palatini modified gravity theories are the same as those of GR, if the system, whose energy density is much smaller than nuclear density, is dominated by particles. The cosmological behaviour of Palatini f(R) theory is the same as that of GR, if the universe is dominated by EM field. This is not true for Palatini f(R ab R ab ) theory. The cosmological behaviour of Palatini f(R) theory is the same as that of GR, if the universe is dominated by EM field. This is not true for Palatini f(R ab R ab ) theory. The averaging issue raised here is relevant to other modified gravity theories like ω = -3/2 Brans-Dicke theory and other modified source gravity theories. The averaging issue raised here is relevant to other modified gravity theories like ω = -3/2 Brans-Dicke theory and other modified source gravity theories. The averaging is not a problem for metric f(R) theories. The averaging is not a problem for metric f(R) theories.