We demonstrate a deep connection between cosmology in thermodynamic description of apparent horizon and information theory. In particular we show how the Landauer principle gets satisfied during the entire cosmic evolution of the universe starting from inflation to the late dark energy era. This in turn reveals why the matter fields are not in thermal equilibrium with the apparent horizon during most of the cosmic era of the universe except for the fluids with omega  13 leading to the transitions of the universe from an accelerating to a decelerating era and viceversa.
['General Relativity and Quantum Cosmology']
Landauer principle in cosmology as the link to information theory
{'No Header': '4)', 'Landauer principle in cosmology as the link to information theory': "Sergei D. Odintsov 1 , 2 , Tanmoy Paul 3 , Soumitra SenGupta 4  \n1) ICREA, Passeig Luis Companys, 23, 08010 Barcelona, Spain 2) Institute of Space Sciences (ICE, CSIC) C. Can Magrans s/n, 08193 Barcelona, Spain 3) Department of Physics, Visva-Bharati University, Santiniketan 731235, India School of Physical Sciences, Indian Association for the Cultivation of Science, Kolkata-700032, India  \nWe demonstrate a deep connection between cosmology in thermodynamic description of apparent horizon and information theory. In particular, we show how the Landauer principle gets satisfied during the entire cosmic evolution of the universe starting from inflation to the late dark energy era. This in turn reveals why the matter fields are not in thermal equilibrium with the apparent horizon during most of the cosmic era of the universe, except for the fluids with ω = -1 / 3 leading to the transitions of the universe from an accelerating to a decelerating era and vice-versa.  \nLandauer principle sets the connection between thermodynamics and information theory, which states that a loss of one bit of information from a system to it's surrounding is associated with a dissipation of energy by ∆ Q ≥ k B T ln 2 where k B is the Boltzmann constant and T is the temperature of the surrounding in which the energy is dissipated [1]. The equality sign in the Landauer principle holds for a reversible thermodynamical process where the system remains in thermal equilibrium to the surrounding; while the inequality sign points to irreversibility causes due to non-equilibrium between system and surrounding. Several recent works give strong support to the principle, both theoretically [2-4] and experimentally [5, 6].  \nOn other hand, the discovery of Bekenstein-Hawking entropy associated with an event horizon of a black hole interestingly brings two apparently different arenas of physics, namely gravity and thermodynamics, on an equal footing [7-10]. However, few works have tried to relate the black hole thermodynamics with the Landauer principle [11-15]. Very recently, [16] shows that the Hawking evaporation from a black hole satisfies the reversible case of the Landauer principle, leading to a firm connection between black hole thermodynamics and the information theory.  \nIn the context of cosmology, the homogeneous and isotropic universe acquires an apparent horizon which, in analogy of black hole thermodynamics, may also be associated with an entropy; for the first studies of horizon related cosmology see for instance [17-20] (for recent review of entropic cosmology, see [21]). While a very few works have been reported to connect black hole thermodynamics with the Landauer principle, a similar correlation in the context of cosmological evolution has not been explored critically so far (in this regard, one may see [22] which showed an approach to holographic dark energy from Landauer principle). A recent attempt [23] could not successfully establish the Landauer principle in the context of cosmology. In this work, we like to demonstrate the connection between cosmology and information theory from the perspective of Landauer principle, based on the thermodynamics of apparent horizon. We find that most of the cosmic evolution of  \nthe universe naturally admits the irreversible case of the Landauer principle.  \nThe spatially flat Friedmann-Lemaˆıtre-RobertsonWalker (FLRW) spacetime with the line element  \nds 2 = -dt 2 + a 2 ( t ) [ dr 2 + r 2 ( dθ 2 +sin 2 θdφ 2 )] , (1)  \nadmits an apparent horizon at the radius (with respect to comoving observer),  \nR h = 1 H , (2)  \nwhere a is the scale factor and H = ˙ a/a represents the Hubble parameter of the universe. The cosmic horizon is dynamical in nature, in particular R h increases with time as long as the matter fields inside the horizon obey the null energy condition. Such dynamical nature of R h plays a pivotal role in demonstrating the Landauer principle in cosmology. Another important quantity is the surface gravity on the apparent horizon: κ = 1 2 √ -h ∂ a ( √ -hh ab ∂ b R ) ∣ ∣ R h (with h ab = diag . ( -1 , a 2 ) defines the induced metric along constant θ and constant φ , i.e. h ab is the induced metric along the normal of the apparent horizon) which, due to the spatially flat FLRW metric, is given by  \nκ = -1 R h ( 1 -˙ R h 2 ) . (3)  \nBelow we briefly demonstrate that the apparent horizon may be associated with a thermal behaviour where the surface gravity defines the horizon temperature, namely,  \nT h = | κ | 2 π = 1 2 πR h ( 1 -˙ R h 2 ) . (4)  \nIn order to demonstrate the thermal behaviour of the cosmic horizon, let us consider the FLRW equation (in terms of R h ):  \nH 2 = 8 πG 3 ρ, (5)  \nwhere ρ is the energy density of the matter fields inside the horizon. Eq. (5) has a differential form like,  \n-( 2 R 3 h ) dR h = 8 πG 3 dρ, (6)  \nwhich, owing to the conservation law of the matter fields: ˙ ρ +3 H ( ρ + p ) = 0, takes the following form,  \nT h ( 2 πR h G ) dR h = 4 πR 2 h ( ρ + p ) ( 1 -˙ R h 2 ) dt . (7)  \nThe total internal energy stored in the matter fields inside the apparent horizon is given by E = ρV , with V = 4 π/ (3 H 3 ) symbolizing the volume enclosed by the apparent horizon, and consequently, one has the following identity:  \n4 πR 2 h ( ρ + p ) dt = -dE + ρdV , (8)  \nwhere, once again, we have used the conservation of the matter fields to arrive at the above expression. Thereby Eq. (7) becomes,  \nT h ( 2 πR h G ) dR h = ( -dE + ρdV ) ( 1 -˙ R h 2 ) . (9)  \nEq. (9) leads to two different formulations of thermodynamic law for the apparent horizon based on two different forms of the horizon entropy. In particular, the fist formulation comes as,  \nT h dS (1) h = -dE + 1 2 ( ρ -p ) dV , (10)  \nwith  \nS (1) h = 2 π G ∫ R h dR h = A 4 G , (11)  \nwhile, the second formulation is given by,  \nT h dS (2) h = -dE + ρdV , (12)  \nwith  \nS (2) h = 2 π G ∫ R h dR h ( 1 -˙ R h / 2 ) = A G (1 -3 ω ) . (13)  \nHere A = 4 πR 2 h represents the area of the horizon, and ˙ R h = 3(1 + ω ) / 2 (with ω being the equation of state of the matter fields inside the horizon) due to the FLRW equations. Note that beside the form of horizon entropy, the work density term in the two thermodynamic formulations are different. However it is clear that S (2) h is ill-defined during the radiation dominated era when ω = 1 / 3. This is the reason we will consider the first thermodynamic formulation for the apparent horizon, given by Eq. (10), in the rest of the paper where we express the horizon entropy by S h = A/ (4 G ) without any superfix.  \nThereby the horizon entropy corresponding to the FLRW equations of Einstein gravity is given by one-quarter of the area of the apparent horizon.  \nOwing to S h ∝ A = 4 π/H 2 , the horizon entropy monotonically increases with cosmic time ( t ), in particular, we have  \ndS h > 0 , (14)  \nprovided the matter fields obey the null energy condition, i.e. ω > -1. In the present work, we will not consider any phantom fields, in particular, the EoS of the matter fields satisfies ω > -1.", 'Thermodynamics of matter fields': "We consider the matter fields inside the horizon to be a perfect fluid with a constant equation of state (EoS) parameter ω . The matter fields behaves like an open system as it exhibits a flux through the apparent horizon. Such kind of matter flux exists due to the the difference between the comoving expansion speed of the universe ( v c ) and the speed of the formation of the apparent horizon ( v h ). In particular, v c = HD (at a physical distance D from a comoving observer) and v h = -˙ H/H 2 . Therefore the thermodynamic law of the matter fields inside the horizon can be expressed by,  \nT m dS m = (increase of internal energy) + (work done) + (energy flux through horizon) , (15)  \nwhere T m and S m represent the temperature and entropy of the matter fields respectively. In general, the matter fields' temperature is considered to be different than the horizon temperature - we will explicitly examine this issue at some stage. The internal energy of the matter fields (at instant t ) is given by E ( t ) = ρ ( t ) V ( t ) and thus the increase of internal energy during the cosmic interval dt becomes (by using the conservation law of matter field),  \ndE = -3 H ( ρ + p ) V dt + ρdV . (16)  \nRegarding the second term in Eq. (15), the work done by the matter fields is expressed as,  \ndW = 1 2 T ab h ab dV = 1 2 ( p -ρ ) dV , (17)  \n̸  \nhere the work density is defined by the projection of the energy-momentum tensor of the matter fields along the normal of the apparent horizon, which takes the above form in terms of ρ and p in the spatially flat FLRW spacetime ( h ab being the induced metric along the normal of the horizon). For the third term in Eq. (15), we need to realize that the matter fields exhibit a flux through the horizon due to v c = v h , and the demonstration is shown in Fig. [1] where the concentric spheres (with respect to the comoving observer labeled by 'O') represent as follows  \n- (a) S 1 : the visible universe bounded by the apparent horizon at time t , having radius OS 1 = 1 /H ( t ); (b) S 2 : the visible universe bounded by the apparent horizon at time t + dt , having radius OS 2 = 1 /H ( t + dt ) = 1 H -˙ H H 2 dt (at the leading order in dt ); (c) S 3 : due to the difference between v c and v h , the sphere S 1 moves from S 1 → S 3 due to the comoving expansion of the universe and thus the radius of S 3 turns out to be OS 3 = 1 H + dt (as v c ( t ) = 1 on the apparent horizon). Thereby we cal-  \nFIG. 1: Comparison between the formation of apparent horizon and the comoving expansion of the universe, in order to calculate the matter flux through the horizon.  \n<!-- image -->  \nculate,  \nV c ( t + dt ) -V ( t + dt ) = 4 π 3 { ( OS 3 ) 3 -( OS 2 ) 3 } = -2 π H 2 (1 + 3 ω ) dt , (18)  \ndepicting the gap between the comoving volume and the visible universe at time t + dt (i.e. between the spheres S 2 and S 3 ). It is important to note that for ω > -1 / 3 (i.e. during deceleration era of the universe), the comoving expansion speed is less than the speed of formation of the apparent horizon, which in turn leads to V c ( t + dt ) < V ( t + dt ). On other hand, we get v c > v h during accelerating universe with -1 < ω < -1 / 3, resulting to the reverse scenario (recall that our regime of interest is ω > -1). Thereby Eq. (18) immediately determines the outward flux of the matter fields' effective energy through the horizon as,  \nFlux = ( ρ +3 p ) × [ V c ( t + dt ) -V ( t + dt )] = -2 πρ H 2 (1 + 3 ω ) 2 dt , (19)  \nwhere ( ρ +3 p ) denotes the effective energy of the matter fields. In spirit of the matter flux, the effective energy ( ρ +3 p ) can be realized as the chemical potential ( µ ) of the matter fields, in particular, µ ∝ -( ρ +3 p ) which depicts that the chemical potential is negative for ω > -1 / 3 while µ > 0 for -1 < ω < -1 / 3. This is expected as the universe with ω > -1 / 3 undergoes through a deceleration era during when the matter fields experience  \nan attractive gravitational force, that in turn results to µ < 0; while the matter fields with ω < -1 / 3 result to an accelerating universe and experience a kind of repulsive force that shows up through a positive valued chemical potential. Owing to Eqs. (16), (17) and (19), the thermodynamic law for the matter fields inside the horizon from Eq. (15) turns out to be,  \nT m dS m dt = -πρ H 2 ( 3 + 10 ω +15 ω 2 ) . (20)  \nThis clearly shows that T m ˙ S m < 0 in the range ω > -1 of our interest, i.e. the entropy of the matter fields proves to monotonically decrease with the cosmic time.  \nFrom the previous discussions, particularly from Eq. (14) and Eq. (20), it becomes clear that the entropy of the horizon increases while the matter fields' entropy decreases with the time. Therefore we have,  \nT h dS h dt > 0 and T m dS m dt < 0 , (21)  \nwhich reveal that the heat energy is released by the matter fields and is absorbed by the apparent horizon. This immediately indicates that the flow of heat energy is directed from the matter fields to the apparent horizon during the cosmic expansion of the universe. Such spontaneous direction of heat flow in turn points the following inequality:  \nT m ≥ T h . (22)  \nThis opens two different possibilities: T m = T h or T m > T h . In the next two subsections we will examine which, out of these two possibilities, is allowed during the entire cosmic evolution of the universe started from inflation to the dark energy era.  \nSince the apparent horizon absorbs heat energy during the cosmic time, the increase of the horizon entropy on gaining one bit of information is given by,  \n∆ S h = k B ln 2 , (23)  \nwith the consideration that each bit of information has two degrees of freedom. However one may associate more than two degrees of freedom with each bit of information, say p degrees of freedom, then the increase of the horizon entropy becomes ∆ S h = k B ln p on gaining one bit of information. With such consideration, our main conclusion remains unchanged and thus we will consider p = 2. During the increasement of ∆ S h = k B ln 2, the Hubble parameter changes by an amount (by using S h = π/ ( GH 2 )):  \n∆ H = -( GH 3 2 π ) k B ln 2 . (24)", 'Examination of first possibility T m = T h : Reversible case of Landauer principle': 'Here we are going to examine whether the possibility T m = T h is allowed by the cosmic evolution of the universe. In this case, the matter fields and the horizon  \nshould be in thermal equilibrium, and the heat flow from the matter fields → the horizon is reversible in nature. Therefore the total change of entropy should vanish, i.e.  \n∆ S h +∆ S m = 0 . (25)  \nIf | ∆ Q m | is the amount of heat released by the matter fields (within the interval when the horizon entropy increases by ∆ S h = k B ln 2), then  \n| ∆ Q m | = T m | ∆ S m | , (26)  \nwhich, due to Eq. (25) along with T m = T h and ∆ S h = k B ln 2, takes the following form:  \n| ∆ Q m | = k B T h ln 2 . (27)  \nThis is the reversible statement of Landauer principle. By using Eq. (20), the above expression turns out to be,  \nπρ H 2 ( 3 + 10 ω +15 ω 2 ) ∆ t = k B T h ln 2 , (28)  \nwhere ∆ t is the time interval during when the horizon entropy increases by ∆ S h = k B ln 2. Therefore, owing to Eq. (24), ∆ t can be expressed by,  \n∆ t = ∆ H ˙ H = -( GH 3 2 π ˙ H ) k B ln 2 . (29)  \nDue to the above expression of ∆ t , along with Eq. (4) and H 2 = 8 πρG/ 3, Eq. (28) results to the following condition on ω :  \n3 + 10 ω +15 ω 2 (1 + ω )(1 -3 ω ) = 1 . (30)  \nTherefore the reversible case of Landauer principle demands the above condition to be hold for the matter EoS parameter. However Eq. (30) holds true only for the fluids with ω = -1 / 3, i.e. when the universe expands with no acceleration (or no deceleration). This clearly demonstrates that the reversible case of Landauer principle is allowed at the transitions from an accelerating universe to a decelerating and vice-versa, for instance, the transitions from inflation to standard Big-Bang cosmology (SBBC) and from SBBC to the late dark energy era.', 'Examination of second possibility T m > T h : Irreversible case of Landauer principle': "In this subsection we will examine the second possibility T m > T h . Here the matter fields are not in thermal equilibrium with the horizon and the heat exchange from the matter fields to the horizon should be irreversible in nature where  \n∆ S h +∆ S m > 0 . (31)  \nOnce again, if | ∆ Q m | is the amount of heat released by the matter fields (within the interval when the horizon entropy increases by ∆ S h = k B ln 2), then  \n| ∆ Q m | = T m | ∆ S m | . (32)  \nOwing to T m > T h and ∆ S h = k B ln 2, the above expression gives rise to the following inequality:  \n| ∆ Q m | > k B T h ln 2 ( | ∆ S m | ∆ S h ) , (33)  \nwhich, due to Eq. (31), gets automatically satisfied if,  \n| ∆ Q m | > k B T h ln 2 . (34)  \nThis is the irreversible expression of Landauer principle. By using Eq. (20), the above expression turns out to be,  \nπρ H 2 ( 3 + 10 ω +15 ω 2 ) ∆ t > k B T h ln 2 . (35)  \nWe now can borrow ∆ t from Eq. (29) obtained in the previous subsection; by doing so, Eq. (35) finally leads to the following condition on ω :  \n3 + 10 ω +15 ω 2 (1 + ω )(1 -3 ω ) > 1 . (36)  \nTherefore the irreversible case of Landauer principle demands the above condition to be hold on ω . The interesting point to be noted that the condition (36) is well satisfied by all possible value of -1 < ω < 1 / 3 except ω = -1 / 3. Actually the situation with ω = -1 / 3 satisfies the reversible case of Landauer principle, as we showed in the previous subsection. Thus it turns out that the irreversible scenario of Landauer principle is indeed satisfied during the entire cosmic expansion of the universe started from inflation to the late dark energy era, except for the fluids having ω = -1 / 3 leading to the transitions from inflation to SBBC and from SBBC to dark energy epoch.  \nIn conclusion, we propose a connection between cosmology and information theory from the perspective of Landauer principle based on the thermodynamics of apparent horizon. It turns out that the entire cosmic evolution of the universe with the matter fluids having -1 < ω < 1 / 3 admits the irreversible scenario of Landauer principle, except for the fluids having ω = -1 / 3 leading to the transitions from an accelerating universe to a decelerating one and vice-versa. At such transition points (for instance - the transitions from inflation to SBBC and from SBBC to the late time acceleration), the cosmic evolution allows the reversible case of Landauer principle. Therefore if we trace the cosmic era from inflation to the dark energy era, then the Landauer principle shows as follows - (a) Inflation and dark energy epochs with -1 < ω < -1 / 3 allow irreversible Landauer principle; (b) SBBC with -1 / 3 < ω < 1 / 3 obeys irreversible Landauer principle; (c) Transitions from inflation to SBBC, as well as from SBBC to dark energy, with  \nω = -1 / 3 admits reversible Landauer principle. The irreversible Landauer principle in cosmology is unlike to the black hole case where the Hawking evaporation ad-  \nits the reversible Landauer principle [16]. This points a crucial difference between black hole thermodynamics and the thermodynamics of cosmology.  \n- [1] R. 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