A new method to probe the mass density and the cosmological constant using configuration entropy
aa r X i v : . [ a s t r o - ph . C O ] F e b MNRAS , 1–5 (2018) Preprint 27 February 2019 Compiled using MNRAS L A TEX style file v3.0
A new method to probe the mass density and thecosmological constant using configuration entropy
Biswajit Pandey ⋆ and Biswajit Das † Department of Physics, Visva-Bharati University, Santiniketan, Birbhum, 731235, India
27 February 2019
ABSTRACT
We study the evolution of the configuration entropy for different combinations of Ω m and Ω Λ in the flat Λ CDM universe and find that the cosmological constantplays a decisive role in controlling the dissipation of the configuration entropy. Theconfiguration entropy dissipates at a slower rate in the models with higher value of Ω Λ . We find that the entropy rate decays to reach a minimum and then increaseswith time. The minimum entropy rate occurs at an earlier time for higher value of Ω Λ . We identify a prominent peak in the derivative of the entropy rate whose locationclosely coincides with the scale factor corresponding to the transition from matter to Λ domination. We find that the peak location is insensitive to the initial conditions andonly depends on the values of Ω m and Ω Λ . We propose that measuring the evolutionof the configuration entropy in the Universe and identifying the location of the peakin its second derivative would provide a new and robust method to probe the massdensity and the cosmological constant. Key words: methods: analytical - cosmology: theory - large scale structure of theUniverse.
Understanding the dark matter and dark energy remain themost challenging problems in cosmology. Observations sug-gest that the baryons or the ordinary matter constitutes only ∼ of the Universe. It is believed that ∼ of the Universe ismade up of the gravitating mass out of which ∼ is in theform of a hypothetical unseen matter dubbed as the “darkmatter”. The remaining ∼ of the Universe is accountedby some mysterious hypothetical component dubbed as the“dark energy”. The dark energy is believed to be respon-sible for driving the current accelerated expansion of theUniverse. The existence and abundance of these mysteriouscomponents are determined from various observations.At present, the Λ CDM model where Λ stands for thecosmological constant and CDM stands for the cold darkmatter stands out as the most successful model in explain-ing most of the cosmological observations till date. TheCDM model was initially introduced by Peebles (1982).Davis et al. (1985) carried out the pioneering numericalstudy of the CDM distribution which paved a new era al-lowing comparison of theory with multitude of observations.The current paradigm of structure formation is sup- ⋆ E-mail:[email protected] † E-mail:[email protected] ported by many complementary observations. The factthat the CMBR angular power spectrum peaks at l ∼ suggests a spatially flat Universe where the mean energydensity of the Universe must be close to the criticaldensity (Komatsu et al. 2011; Planck Collaboration et al.2016). Other observations from dynamics of galaxies andclusters (Carlberg et al. 1996), the X-ray observationsof galaxy clusters (Mohr et al. 1999), Sunyaev-Zeldovicheffect (Grego et al. 2000), weak lensing (Benjamin et al.2007; Fu et al. 2008), baryonic acoustic oscillations(BAO) (Eisenstein et al. 2005), correlation functions(Hawkins et al. 2003) and the power spectrum of den-sity fluctuations (Tegmark et al. 2004; Reid et al. 2010;Percival et al. 2010) revealed that the total mass densityparameter including both the baryonic and non-baryoniccomponents must be ∼ . . A flat Universe with Ω m = . leaves us with no other choice but Ω Λ = . which fitsthe bill perfectly. Further, the existence of dark energy isalso supported by independent observations such as TypeIa supernova (Riess et al. 1998; Perlmutter et al. 1999)and BAO (Wang 2006; Eisenstein 2005) with very highconfidence.The information entropy can be an useful tool forcharacterizing the inhomogeneities in the mass distribution(Hosoya et al. 2004; Pandey 2013). Recently, Pandey (2017)propose that the evolution of configuration entropy of the c (cid:13) Pandey, B. and Das, B. mass distribution in the Universe may drive the cosmic ac-celeration. It has been argued that the configuration entropyof the Universe decreases with time due to the amplifica-tion of the density perturbations by the process of gravi-tational instability. The configuration entropy continues todissipate in a matter dominated Universe. The dissipation ofconfiguration entropy due to this transition from smootherto clumpier state demands existence of some efficient en-tropy generation mechanisms to counterbalance this loss.If the other entropy generation mechanisms are not suffi-cient to counter this loss then the Universe must expand insuch a way so as to prevent the further growth of structuresand stop the leakage of information entropy. Interestingly,the dissipation of the configuration entropy comes to a haltin a Λ dominated Universe due to the suppression of thegrowth of structures on large scales. Recently Das & Pandey(2019) used the evolution of the configuration entropy to dis-tinguish various dynamical dark energy parameterizations.The importance and some interesting implications of inho-mogeneities in cosmology has been highlighted earlier inBuchert & Ehlers (1997) and Buchert (2000).In the present work, we assume that the Λ CDM modelwith flat FRW metric to be the correct model of the Uni-verse and the density parameters associated with matter and Λ are to be determined from observations. We propose a newmethod for the determination of the density parameter as-sociated with the mass and the cosmological constant. Themethod is based on the study of the evolution of the con-figuration entropy in the Universe. In future, the presentgeneration surveys like SDSS (York et al. 2000), 2dFGRS(Colles et al. 2001), dark energy survey (Abbott et al. 2018)combined with various other future surveys like DESI, Eu-clid and different future 21 cm experiments like SKA wouldallow us to measure the configuration entropy at differentepochs and study its evolution. The method presented inthis work would provide an alternative route to measure themass density and the cosmological constant in an indepen-dent and unique way and compare their values obtained bythe other methods from various observations. The observations of the cosmic microwave background radi-ation (CMBR) suggest that the Universe was highly uniformin the past. But the matter distribution in the present dayUniverse is highly clumpy due to the structure formationby gravity. Pandey (2017) defines the configuration entropyof the mass distribution following the idea of informationentropy (Shannon 1948) as, S c ( t ) = − Z ρ ( ~ x , t )log ρ ( ~ x , t ) dV . (1)Here S c ( t ) is the configuration entropy of the mass distribu-tion at time t over a sufficiently large comoving volume V .The volume V is divided into a large number of subvolumes dV and the density ρ ( ~ x , t ) is measured inside each volumeelement.The distribution is treated as a fluid on large scales. Thecontinuity equation for the fluid in an expanding universe is given by, ∂ρ∂ t + aa ρ + a ∇ · ( ρ~ v ) = . (2)Here a is the scale factor and ~ v denotes the peculiar velocityof the fluid inside the volume element dV .If we multiply Equation 2 by (1 + log ρ ) and integrateover the entire volume V , then we get the entropy evolutionequation (Pandey 2017) as, dS c ( t ) dt + aa S c ( t ) − a Z ρ (3˙ a + ∇ · ~ v ) dV = . (3)Changing variable from t to a in Equation 3 we get, dS c ( a ) da ˙ a + aa S c ( a ) − F ( a ) = . (4)where F ( a ) is given by, F ( a ) = MH ( a ) + a Z ρ ( ~ x , a ) ∇ · ~ v dV . (5)Here M = R ρ ( ~ x , a ) dV = R ¯ ρ (1 + δ ( ~ x , a )) dV gives the totalmass inside V and δ ( ~ x , a ) = ρ ( ~ x , a ) − ¯ ρ ¯ ρ gives the density contrastin a subvolume dV centred at the comoving co-ordinate ~ x .The ¯ ρ is the mean density of matter inside the comovingvolume V .One can simplify Equation 4 further to get, dS c ( a ) da + a ( S c ( a ) − M ) + ¯ ρ f ( a ) D ( a ) a Z δ ( ~ x ) dV = . (6)where, D ( a ) is the growing mode of density perturbation and f ( a ) = dlnDdlna = aD dDda is the dimensionless linear growth rate.We need to solve Equation 6 to study the evolutionof the configuration entropy for any given cosmologicalmodel. The time-independent quantities in the third termof Equation 6 are set to for the sake of simplicity. We cal-culate D ( a ) and f ( a ) for the cosmological model under con-sideration. We then numerically solve the Equation 6 usingthe th order Runge-Kutta method.The second and third term in Equation 6 together de-cides the evolution of the configuration entropy. The sec-ond term is decided by the initial condition whereas thethird term is governed by the nature of the growth of struc-tures in a particular cosmological model. At the initial stage,the second term solely dictates the evolution because thegrowth factor remains negligible. The third term only comesinto play when the growth of structures becomes significant.So the cosmology dependence of the configuration entropyarises purely from the third term in Equation 6.The Equation 6 can be solved analytically ignoring thethird term which is given by, S c ( a ) S c ( a i ) = MS c ( a i ) + − MS c ( a i ) ! a i a ! . (7)where, a i is the initial scale factor and S c ( a i ) is the initial en-tropy. We choose a i to be − throughout the present anal-ysis. According to this solution, we expect a sudden growthor decay in the configuration entropy near the initial scalefactor a i when S c ( a i ) < M and S c ( a i ) > M respectively. On theother hand, no such transients are expected when S c ( a i ) = M .Since we are only interested in the cosmology dependence ofthe configuration entropy, we shall focus on the solution ofEquation 6 for S c ( a i ) = M in the present work. The solutions MNRAS , 1–5 (2018) ass density and Λ from entropy in the other two cases are similar other than the transientspresent near the initial scale factor. The CMBR observations show that the Universe is highlyisotropic. But the same observations also reveal that thereare small anisotropies of the order of − imprinted in theCMBR temperature maps. These tiny fluctuations are be-lieved to be the precursor of the large scale structures ob-served in the present day Universe. The primordial densityperturbations were amplified by the process of gravitationalinstability for billions of years. The growth of the densityperturbations δ ( ~ x , t ) can be described by the linear pertur-bation theory when δ << . Considering only perturbationsto the matter sector, the linearized equation for the growthof the density perturbation is given by, ∂ δ ( ~ x , t ) ∂ t + H ∂δ ( ~ x , t ) ∂ t − Ω m H a δ ( ~ x , t ) = . (8)Here Ω m and H are the present values of the mass den-sity parameter and the Hubble parameter respectively. Thisequation has two solutions, one which grows and anotherwhich decays away with time. The growing mode solutionamplifies the density perturbations at the same rate at everylocation so that the density perturbation at any location ~ x can be expressed as, δ ( ~ x , t ) = D ( t ) δ ( ~ x ) . Here D ( t ) is the grow-ing mode and δ ( ~ x ) is the initial density perturbation at thelocation ~ x .The growing mode solution of Equation 8 can be ex-pressed (Peebles 1980) as, D ( a ) = Ω m X ( a ) Z a da ′ a ′ X ( a ′ ) , (9)where X ( a ) = H ( a ) H = [ Ω m a − + Ω Λ ] in an Universe with onlymatter and cosmological constant.In a flat Universe, the dimensionless linear growth rate f ( a ) = d ln δ d ln a can be well approximated (Lahav et al. 1991) by, f ( a ) = Ω m ( a ) . +
170 [1 − Ω m ( a )(1 + Ω m ( a ))] , (10)where the matter density history Ω m ( a ) can be written as Ω m ( a ) = Ω m a − X ( a ) .In the present work, we consider different combinationsof Ω m and Ω Λ within the framework of the flat Λ CDMmodel and calculate D ( a ) and f ( a ) in each case. The D ( a ) and f ( a ) for different models are shown in Figure 1. Theseare then used to solve Equation 6 to study the evolution ofthe configuration entropy in each model. We show the evolution of S c ( a ) S c ( a i ) with scale factor in the topleft panel of Figure 2. We see that the configuration entropyinitially decreases with time. The dissipation of the config-uration entropy is driven by the growth of structures. Thedissipation is higher in the models with a larger value of Ω m .This is directly related to a higher growth factor D ( a ) andgrowth rate f ( a ) in the models with a larger Ω m (Figure 1). Clearly, the dissipation is less pronounced in the models withlarger Ω Λ . The structure formation is the outcome of twocompeting effects: one is the tendency of the overdense re-gions to collapse under their self gravity and the other isthe tendency to move apart with the background expan-sion. The cosmological constant Λ contributes to the laterand thus resists the leakage of the configuration entropy inthe Universe by increasing the Hubble drag and suppressingthe structure formation on large scales.We show the rate of change of the configuration entropyin the top right panel of Figure 2. We find that the cosmo-logical constant Λ plays an influential role in controlling thedissipation of the configuration entropy. Models with largervalue of Ω Λ and smaller value of Ω m show a dip in the slopeof the configuration entropy at a smaller value of the scalefactor. For example in the model with Ω Λ = . , initiallythe slope decreases with increasing scale factor reaching theminimum at a ∼ . . The slope then turns upward remainingnegative upto a ∼ . thereafter upcrossing the zero. It thenslowly plateaus towards a stable value. A similar trend isobserved for the other models with different combinationsof Ω m and Ω Λ . The minimum occur at a ∼ . and a ∼ . in the models with Ω Λ = . and Ω Λ = . respectively. Theminimum of the slope indicates the time since Λ becomesproactive in suppressing the dissipation of the configurationentropy. The minimum appears at a smaller value of scalefactor in the higher Ω Λ models simply because the presenceof the Λ term is felt earlier in these models.The bottom middle panel of Figure 2 shows the deriva-tive of the slopes shown in the top right panel of the figure.Initially, the derivative of the slopes are negative for all themodels which imply that the slopes are decreasing with time.But the second derivative keeps on increasing with time andeventually upcrosses zero at some value of the scale factor.This scale factor corresponds to the minimum of the slope.For example the model with Ω Λ = . exhibit the zero up-crossing of the second derivative at a ∼ . where a minimumwas observed in the slope. Similarly, a zero upcrossing can beseen at a ∼ . and a ∼ . in the Ω Λ = . and Ω Λ = . mod-els respectively. The positive value of the second derivativesafter this scale factor suggest that the slopes are increasingwith time. However, the slopes themselves remain negativeafter the occurrence of the minimum. This suggest that theconfiguration entropy still continues to dissipate with timebut with a gradually decreasing rate. The second derivativesof the configuration entropy do not increase monotonicallybut show a prominent peak at a specific value of the scalefactor. The peaks are clearly identified in the models with Ω Λ = . , Ω Λ = . , Ω Λ = . and Ω Λ = . at the scalefactor a = . , a = . , a = . and a = . respectively.Interestingly, the transition from matter to Λ domination inthe respective models are expected to occur at nearly thesame scale factors. It may be noted that the second deriva-tives remain negative throughout the entire range of scalefactor for Ω Λ = . and Ω Λ = . models. The peaks in thesemodels and the rest of the models are expected to occur infuture and hence are not present in the figure. The dissipa-tion rate of the configuration entropy changes at a slowerrate once the Λ domination takes place. This is related tothe fact that the growth of structures are completely shutoff on larger scales once Λ begins to drive the acceleratedexpansion of the Universe. MNRAS , 1–5 (2018)
Pandey, B. and Das, B. D ( a ) scale factor (a)( (cid:1) m0 , (cid:0)(cid:2) ) = (0.1,0.9)( (cid:3) m0 , (cid:4)(cid:5) ) = (0.2,0.8)( (cid:6) m0 , (cid:7)(cid:8) ) = (0.3,0.7)( (cid:9) m0 , (cid:10)(cid:11) ) = (0.4,0.6)( (cid:12) m0 , (cid:13)(cid:14) ) = (0.5,0.5)( (cid:15) m0 , (cid:16)(cid:17) ) = (0.6,0.4)( (cid:18) m0 , (cid:19)(cid:20) ) = (0.7,0.3)( (cid:21) m0 , (cid:22)(cid:23) ) = (0.8,0.2)( (cid:24) m0 , (cid:25)(cid:26) ) = (0.9,0.1) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f ( a ) scale factor (a)( Ω m0 , Ω Λ ) = (0.1,0.9)( Ω m0 , Ω Λ ) = (0.2,0.8)( Ω m0 , Ω Λ ) = (0.3,0.7)( Ω m0 , Ω Λ ) = (0.4,0.6)( Ω m0 , Ω Λ ) = (0.5,0.5)( Ω m0 , Ω Λ ) = (0.6,0.4)( Ω m0 , Ω Λ ) = (0.7,0.3)( Ω m0 , Ω Λ ) = (0.8,0.2)( Ω m0 , Ω Λ ) = (0.9,0.1) Figure 1.
The left panel and the right panel of this figure respectively show D ( a ) and f ( a ) as a function of scale factor for differentcombinations of ( Ω m , Ω Λ ) within the flat Λ CDM cosmology. The value of D ( a ) is normalized to at present for the combination ( Ω m , Ω Λ ) = (0 . , . . The D ( a ) values in the other models are normalized with respect to this model. M = S c (a i ) S c ( a ) / S c ( a i ) scale factor (a)( Ω m0 , Ω Λ ) = (0.1,0.9)( Ω m0 , Ω Λ ) = (0.2,0.8)( Ω m0 , Ω Λ ) = (0.3,0.7)( Ω m0 , Ω Λ ) = (0.4,0.6)( Ω m0 , Ω Λ ) = (0.5,0.5)( Ω m0 , Ω Λ ) = (0.6,0.4)( Ω m0 , Ω Λ ) = (0.7,0.3)( Ω m0 , Ω Λ ) = (0.8,0.2)( Ω m0 , Ω Λ ) = (0.9,0.1) -0.6-0.5-0.4-0.3-0.2-0.1 0 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 dS c ( a ) / d a scale factor (a)( Ω m0 , Ω Λ ) = (0.1,0.9)( Ω m0 , Ω Λ ) = (0.2,0.8)( Ω m0 , Ω Λ ) = (0.3,0.7)( Ω m0 , Ω Λ ) = (0.4,0.6)( Ω m0 , Ω Λ ) = (0.5,0.5)( Ω m0 , Ω Λ ) = (0.6,0.4)( Ω m0 , Ω Λ ) = (0.7,0.3)( Ω m0 , Ω Λ ) = (0.8,0.2)( Ω m0 , Ω Λ ) = (0.9,0.1)-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 d S c ( a ) / d a scale factor (a)( Ω m0 , Ω Λ ) = (0.1,0.9)( Ω m0 , Ω Λ ) = (0.2,0.8)( Ω m0 , Ω Λ ) = (0.3,0.7)( Ω m0 , Ω Λ ) = (0.4,0.6)( Ω m0 , Ω Λ ) = (0.5,0.5) ( Ω m0 , Ω Λ ) = (0.6,0.4)( Ω m0 , Ω Λ ) = (0.7,0.3)( Ω m0 , Ω Λ ) = (0.8,0.2)( Ω m0 , Ω Λ ) = (0.9,0.1) Figure 2.
The top left panel of this figure shows the evolution of the configuration entropy for different combinations of ( Ω m , Ω Λ ) within the flat Λ CDM model. The top right panel and the bottom middle panel respectively show the first and the second derivative ofthe configuration entropy in these models. The arrows in different colours in the bottom middle panel mark the scale factors a t = (cid:16) Ω m Ω Λ (cid:17) corresponding to the transition from matter to Λ domination in the respective models. The transition scale factors closely coincide withthe peak locations in each model. MNRAS , 1–5 (2018) ass density and Λ from entropy We have repeated these analyses for S c ( a i ) > M and S c ( a i ) < M and recovered the peaks at exactly the same loca-tions. This indicates that the peak locations are insensitiveto the initial conditions and depend only on the values of Ω m and Ω Λ .We expect that combining the measurements of the con-figuration entropy at different redshifts from the present andfuture generation surveys would enable us to study the evo-lution of the configuration entropy. This would allow us toidentify the location of the peak in its second derivative andconstrain the value of both the mass density and the cosmo-logical constant in the Universe.We would like to point out here that the present methodrequires us to measure the configuration entropy over a sig-nificantly large volume of the Universe. This is to ensure thatthere are no net mass inflow or outflow across the neighbour-ing volumes. The present studies suggest that the Universe ishomogeneous on large scales (Yadav et al. 2005; Hogg et al.2005; Sarkar et al. 2009; Sarkar & Pandey 2016). So at eachredshift, a measurement of the configuration entropy over aregion extending few hundreds of Mpc would be sufficientfor the present analysis. However, the mutual informationbetween the spatially separated but causally connected re-gions of the Universe may introduce a non-negligible dy-namical entanglement between them (Wiegand & Buchert2010). Any modification of the evolution equation due tothis entanglement need to be investigated further.Furthermore, the baryonic matter constitutes only atiny fraction of the matter budget. The baryonic matterdistribution is expected to be biased with respect to thedark matter distribution. Currently, the proposed methodrequires us to map the distribution of an unbiased tracer ofthe underlying mass distribution at multiple redshifts. Theintroduction of bias may complicate the analysis which wewould like to address in a future work.Finally, we conclude that the analysis presented in thiswork provides an alternative avenue for the determination ofthe mass density and the cosmological constant Λ by study-ing the evolution of the configuration entropy in the Uni-verse. We expect this to find many useful applications inthe study of the mysterious dark matter and the elusive cos-mological constant. The authors thank an anonymous reviewer for valuable com-ments and suggestions. The authors would like to acknowl-edge financial support from the SERB, DST, Government ofIndia through the project EMR/2015/001037. BP would alsolike to acknowledge IUCAA, Pune and CTS, IIT, Kharag-pur for providing support through associateship and visitorsprogramme respectively.
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